MATHEMATICS - kalvisolaionline · 8. Lg
244
g{g<G ye<hkil< uGh<H kQ{<mijl yV hius<osbz< kQ{<mijl yV ohVr<Gx<xl< kQ{<mijl leqkk< ke<jlbx<x osbz< klqp<fim<Mh< himF~z< gpgl< gz<Z~iqs< sijz? ose<je – 600 006. www.kalvisolai.com
Transcript of MATHEMATICS - kalvisolaionline · 8. Lg
g{g<G!
ye<hkil<!uGh<H!
kQ{<mijl!yV!hius<osbz<!
kQ{<mijl!yV!ohVr<Gx<xl<!
kQ{<mijl!leqkk<!ke<jlbx<x!osbz<!
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klqp<fim<Mh<!himF~z<!gpgl<!gz<Z~iqs<!sijz?!ose<je – 600 006.
© klqp<fiM!nvS Lkz<!hkqh<H! ! .!2003 kqVk<kqb!hkqh<H!.!2004 ! lXhkqh<H!!!!!!.!2006
GPk<kjzuI!
LjeuI!si. dkbhi^<gve< g{qk!-j{h<OhvisqiqbI?!lifqzg<!gz<Z~iq (ke<eim<sq),
ose<je – 600 005.
Olzib<uitIgt<! kqV!w/!nIs<See<! kqV!gq. kr<gOuZ!OkIUfqjzg<!g{qk!uqiqUjvbitI? LKfqjzg<!g{qk!uqiqUjvbitI? d/!fi/!nvS!gz<Z~iq? hs<jsbh<he<!gz<Z~iq? ohie<Oeiq − 601 204. ose<je − 600 030.
NsqiqbIgt<! kqV uQ. >vil< kqV g. nxqupge<!hm<mkiiq!NsqiqbI!)g{qkl<*? ht<tq!dkuq!NsqiqbI!)g{qkl<*? h/os/g/g/!nvsqeI!Olz<fqjzh<!ht<tq? nvsqeI!N{<gt<!Olz<fqjzh<!ht<tq? Ogiml<hig<gl<? ose<je − 600 024. dTf<K~IOhm<jm − 606 107. kqVlkq vi. fl<hqg<jg!o\bvi\< kqVlkq! Os. uq\bi LKfqjzh<!hm<mkiiq!Nsqiqjb!)g{qkl<*? LKfqjzh<!hm<mkiiq!Nsqiqjb!)g{qkl<*? Heqk!ne<eit<!oh{<gt<!Olz<fqjzh<!ht<tq? hib<zI!hqti{<m<!oh{<gt<!Olz<fqjzh<!ht<tq? -vibHvl<?! ose<je − 600 013. jgzisHvl<?!kqVs<sq − 620 014.
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ohiVtmg<gl<! hg<gr<gt<! 1. w{<Ljxgt< 1 - 31 1.1 w{<Ljxgt< 1 1.2 olb<ob{<!OfIg<OgiM 18 2. ntjugt< 32 - 48 2.1 hvh<H!lx<Xl<!Sx<xtU 32 2.2 %m<M!dVur<gt< 37 3. sqz!Lg<gqb!GxqbQM 49 - 90
3.1 nxquqbz<!GxqbQM 49 3.2 lmg<jgbqe<!GxqbQM 52 3.3 g{g<!GxqbQM 73 4. -bx<g{qkl< 91 - 121 4.1 hz<ZXh<Hg<!Ogijugt< 92 4.2 -bx<g{qk!Lx<oxiVjlgt< 97 4.3 giv{qh<hMk<kz< 107
4.4 yV!hz<ZXh<Hg<!Ogijujb!lx<oxiV!hz<ZXh<Hg<! Ogijubiz<!uGk<kz< 116!
5. kQIUgt<!gi[l<!Ljxgt< 122 - 134 5.1 nElier<gt<!lx<Xl<!fq'h{r<gt< 123 5.2 g{qk!likqiqgt< 131 6. nxqLjx!ucuqbz< 135 - 168 6.1 siqhiIk<kZg<Giqb!Okx<xr<gt< 135 6.2 kVg<g!iQkqbig!fq'hqg<g!Ou{<cb!Okx<xr<gt< 158 7. hGLjx!ucuqbz< 169 - 186 7.1 giICsqbe<!ns<S!K~vLjx 169 7.2 Ogim<ce<!sib<U 173 7.3 (x1, y1) lx<Xl<!(x2, y2) Ngqb!-V!Ht<tqgTg<G!! ! !!!-jmObBt<t!okijzU 179
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8. Lg<Ogi{uqbz< 187 - 204 8.1 Lg<Ogi{uqbz<!uqgqkr<gt< 189 8.2 Lg<Ogi{uqbz<!Lx<oxiVjlgt< 198
8.3 fqvh<Hg<Ogi{r<gTg<gie!Lg<Ogi{uqbz<!uqgqkr<gt< 202 9. osb<Ljx!ucuqbz< 205 - 216 9.1 yV!Lg<Ogi{k<kqz<!yVHt<tq!upqg<!OgiMgt< 206 9.2 svisiqgtqe<!ucuqbz<!uqtg<gl< 213 10. uquvr<gjtg<!jgbiTkz< 217 - 228 10.1 jlbfqjzh<!Ohig<G!ntjugt< 219 11. ujvhmr<gt< 229 - 240 11.1 OfIg<OgiM!ujvhmr<gt< 229 11.2 OfIg<OgiM!ujvhmr<gtqe<!hbe<hiM 233 lmg<jgh<!hm<cbz< 241 - 242 wkqIlmg<jgh<!hm<cbz< 243 - 244
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!
1. w{<Ljxgt<!
fl<! ne<xim! uip<uqz<! w{<gt<! wr<Gl<! Okie<Xgqe<xe/ “njek<kqZl<! w{<gt<!dt<te” we! hpl<ohVl<! gqOvg<g! g{qk! uz<Zfi<! hqkigv <̂! %xqBt<tii</! w{<gjth<!
hx<xq! fqjxb! nxqf<K! ogi{<Omioleqz<, g{qkk<kqje! fil<! fe<G! gx<xuIgtiOuil</!!giIz<!h<vm<iqs<!gi <̂!wEl<!o\Ileq!fim<jms<!siIf<k!g{qk!uz<Zfi<, “uqR<Rier<gtqe<!nvsq! g{qkl<! lx<Xl<! g{qkr<gtqe<! nvsq! w{<gtqe<! ogit<jg” weg<! %xqBt<tii</!w{<gt<! lqg! dbiqb! h{<Hgjth<! ohx<Xt<te/! nju! uqR<Riek<kqz<! wPl<! hz!
uqeig<gTg<G!uqjm!gi{!dkUgqe<xe/!-f<kqbg<!g{qk!Oljk!-viliE\e<!“w{<gt<!weK!f{<hIgt<” weg<!%xqBt<tii</!o\Ileq!fim<M!GOviofg<gI!wEl<!uz<ZfI “-bz<!w{<gjt!gmUt<! hjmk<kiI; lx<xju!leqkeqe<!kqxjlbiz<!Wx<hm<mju” we!uqbf<K!%xqBt<tiI/! Lf<jkb! uGh<Hgtqz<! -bz<! w{<gt<, LPg<gt<, uqgqkLX! w{<gt<, uqgqkLxi! w{<gt<! lx<Xl<! olb<ob{<gt<! Ngqbux<jxh<! hx<xq! nxqf<K! ogi{<Omil</!w{<[ukx<G! dku! -bz<! w{<gt<! Wx<hm<me/! OkjugTg<Ogx<h! lx<x! w{<gt<!dVuibq<e/!-h<himk<kqz<!w{<gtqe<!h{<Hgjth<!hx<xq!nxqf<K!ogit<Ouil</!
1.1. w{<Ljxgt<! 1.1.1 -bz<!w{<gt< (Natural Numbers)
1, 2, 3,…. we<he!-bz<! w{<gtiGl</! -ju!w{<{qm!dkUl<! w{<gt<! weUl<!njpg<gh<hMl</! -bz<! w{<gtqe<! okiGkqjb! N weg<! Gxqh<Ohil</ N e<! dXh<Hgt<!njek<jkBl<!hm<cbzqm!Lcbiuqm<miZl<, N = {1, 2, 3, …} we!wPKOuil</!-r<G "…"!wEl<!GxqbQM N!e<!lx<x!dXh<Hgt< 1, 2, 3, e<!Ljxjb!yx<xq! hm<cbzqmh<hMgqe<xe!we<hjk!d{i<k<Kgqe<xK/!-r<G!1, 3, 5,… we<he!yx<jxh<hjm!-bz<!w{<gt<!weUl<, 2, 4, 6, … we<he!-vm<jmh<hjm!-bz<! w{<gt<! weUl<! njpg<gh<ohXl</ x − 9 = 0, x −16 = 0, x − 54 = 0 Ohie<x! sle<hiMgTg<Gk<! kQIU! okiGh<H N zqVf<K! ohxzil</!Neiz<? x + 5 = 5, x + 9 = 9 Ngqb!sle<hiMgtqe<!kQIUgt<<!N z<!-z<jz/!Woeeqz<?!-s<sle<hiMgt< N!z<!-z<zik!w{<!0 Nz<!fqjxU!osb<bh<hMgqe<xe/! 1.1.2 LP!w{<gt< (Whole Numbers)
0, 1, 2, … we<he! LP! w{<gtiGl</! LP! w{<gtqe<! okiGkqjb! W weg<!
Gxqh<Ohil</!NgOu!W = {0, 1, 2,…}. -h<OhiK, x + 5 = 5, x + 9 = 9 Ohie<x!sle<gtqe<!kQIU?!w{<!0 NGl<; -K!W z<!dt<tK/!-bz<!w{<gt<!njek<Kl<!LP!w{<gtiGl<; Neiz<, 0 wEl<!LP!w{<!-bz<!w{<!nz<z/!nOk!Ofvk<kqz< x + 25 = 15, x + 12 = 9
1
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we<he W-z<! kQIUgjtg<! ogi{<cvi/! Woeeqz<?!nux<xqe<! kQIUgtie −10 lx<Xl< −3 we<he!LP!w{<gtigi. 1.1.3 LPg<gt< (Integers)
0, 1, −1, 2, −2, … we<hju!LPg<gtiGl</! -ux<xqz< 1, 2, 3, … we<he! lqjg!LPg<gt<;!!−1, −2, −3,… we<he!Gjx!LPg<gtiGl</!LPg<gtqe<!okiGh<H Z!weg< Gxqh<hqmh<hMgqe<xK/!NgOu, Z = {…, −3, −2, −1, 0, 1, 2, 3,…}. njek<K!LP!w{<gTl<!LPg<gtiGl</! Neiz<! Gjx! LPg<gtie −1, −2, −3,… we<he! LP! w{<gtigi/!-h<OhiK x + 25 = 15, x + 12 = 9 wEl<!sle<gtqe<!kQIUgtie!−10 lx<Xl< −3 we<he Z z< dt<tkiz<, ns<sle<hiMgt< Z z<!kQIUgjth<!ohx<Xt<te/!Neiz<?!!
2x + 5 = 12, 3x + 9 = 4
Ngqbe!Z z<!kQIUgjth<!ohx<xqVg<giK/!Woeeqz<?!-ux<xqe<!kQIUgtie 27 lx<Xl<!
35−
Z z<!njlbiK/!! 1.1.4 uqgqkLX!w{<gt< (Rational Numbers)
p, q LPg<gtigUl<, q ≠ 0 weUl<!ogi{<M!qp wEl<!ucuqz<!wPkh<hMl<!w{<gt<!
uqgqkLX! w{<gtiGl</! uqgqkLX! w{<gtqe<! okiGkqjb!Q weg<Gxqh<Ohil<. qp wEl<!
uqgqkLX! w{<{qz<, q lqjg!LP! w{<{igUl<, p lx<Xl< q gTg<G 1-Jk<! kuqIk<K
Ouoxf<k! ohiKg<! giv{qBlqz<jz! weqz<, qpNeK! OfIucU! nz<zK! kqm<mucuqz<!
-Vg<gqe<xK! we<Ohil</! wMk<Kg<gim<mig, 11
7 , 5
2− we<he! OfIucuqZt<t! uqgqkLX!
w{<gtiGl</! lixig, 1512 lx<Xl<!
2224−
we<he! OfIucuqz<! -z<zik! uqgqkLX!
w{<gtiGl</! Neiz<! yu<ouiV! uqgqkLX! w{<[l<! keqk<k! yV! sllie! OfI! ucu!
njlh<jhh<! ohx<xqVg<Gl</! wMk<Kg<gim<mig, 15
12 =5
4 lx<Xl< 22
24
−=
11
12− we<he!
OfIucuk<kqz<!njlgqe<xe/!yu<ouiV!LPUl<!uqgqkLX!w{<!NGl</!wMk<Kg<gim<mig,
−19 we<x!LPju!1
19− we!wPkzil<; -r<G −19, 1 Ngqb! w{<gt<!Z z<!dt<te!
lx<Xl<!1 ≠ 0. !weOu!−19 NeK!uqgqkLX!w{<!NGl</!weOu!njek<K!LPg<gTl<!
2
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uqgqkLX! w{<gtiGl</! Neiz<! LPg<gtx<x! uqgqkLX! w{<gTl<! d{<M/!
wMk<Kg<gim<mig? 5
4 NeK!uqgqkLX!w{<; Neiz<, -K!LP!nz<z/!
1.1.5 N, W, Z lx<Xl< Q okiGh<Hgtqz< +, −, ×, ÷ Ngqbux<xqe<!sqz!h{<Hgt<
%m<mz< + lx<Xl<! ohVg<gz<! × Ngqb! -v{<Ml<! Q z<! hqe<uVl<! h{<Hgjth<!ohx<xqVh<hjk!w{<gTme<!dt<t!flK!nEhuk<jkg<!ogi{<M!nxqbzil</ 1. x, y Ngqbe! uqgqkLX! w{<gt<! weqz<, x + y l<! uqgqkLX! w{<{iGl</!
wMk<Kg<gim<mig, 11!lx<Xl<!3
2− -v{<Ml< Q!z<!dt<te/!nux<xqe<!%Mkzie
11 +3
31
3)2(33
32
111
32
=−+
=⎟⎠⎞
⎜⎝⎛ −+=⎟
⎠⎞
⎜⎝⎛ − .
-KUl< Q z<!dt<tK/!-h<h{<H Q z<!%m<mZg<gie!njmUh<!h{<H!weh<hMl</!! 2. x, y Ngqbe!uqgqkLX!w{<gt<!weqz<, x + y = y + x. wMk<Kg<gim<mig,
33
34
33
)22()12(32
11
4 −=
−+−=
−+
−⎟⎠⎞
⎜⎝⎛⎟
⎠⎞
⎜⎝⎛ lx<Xl<! .
33
34
33
)12()22(
11
4
3
2 −=
−+−=
−+
−⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
weOu ⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛ −
+−
=−
+−
11
4
3
2
3
2
11
4 . -K!%m<mZg<gie!hiqlix<Xh<!h{<H!we!njpg<gh<hMl</
3. x, y, z Ngqbe!uqgqkLX!w{<gt<!weqz<, x + (y + z) = (x + y) + z. wMk<Kg<gim<mig,
72 ,
54 ,
32 −− Ngqbju Q z<!dt<te;
⎥⎦⎤
⎢⎣⎡ −+−
+⎟⎠⎞
⎜⎝⎛=⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −+⎟
⎠⎞
⎜⎝⎛ −+⎟
⎠⎞
⎜⎝⎛
35)10()28(
32
72
54
32 = ⎟
⎠⎞
⎜⎝⎛ −+⎟
⎠⎞
⎜⎝⎛
3538
32 =
105)114(70 −+ =
10544− ,
⎟⎠⎞
⎜⎝⎛ −+⎥⎦
⎤⎢⎣⎡ −+
=⎟⎠⎞
⎜⎝⎛ −+⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −+
72
15)12(10
72
54
32 = ⎟
⎠⎞
⎜⎝⎛ −+⎟
⎠⎞
⎜⎝⎛ −
72
152 =
105)30()14( −+− = .
10544−
∴ .72
54
32
72
54
32 ⎟
⎠⎞
⎜⎝⎛ −+⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −+=⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −+⎟
⎠⎞
⎜⎝⎛ −+⎟
⎠⎞
⎜⎝⎛
-K Q-z<!%m<mZg<gie!OsIh<Hh<!h{<H!we!njpg<gh<hMl</! 4.!!0!NeK!uqgqkLX!w{</!OlZl< 0 + x = x + 0 = x. wz<zi!uqgqkLX!w{<!! x-g<Gl<!-K!ohiVf<Kl</!wMk<Kg<gim<mig,
3
113110
311
10
3110 =
+=+=+ lx<Xl< .
311
3011
10
3110
311
=+
=+=+
uqgqkLX!w{<!0 ju Q z<!%m<mzqe<!Lx<oxiVjl!dXh<H!we<gqOxil</ 5. yu<ouiV! uqgqkLX! w{ ; x g<Gl< −x wEl<! uqgqkLX! w{<{qje
x + (−x) = (−x) + x = 0 we!njlBliX! gi{zil</ −x J x e< Gjx!nz<zK! x-e<
%m<mzqe<!OfIliX!weg<!%XOuil</!!wMk<Kg<gim<mig, 311− we<gqx!uqgqkLX!w{<{qx<G,
3
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311! we<x!uqgqkLX! w{<{qje 0
30
311)11(
311
311
==+−
=⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛ − we!-Vg<GliX! gi{!
LcgqxK/!weOu!3
11 NeK!311− e<!%m<mzqe<!OfIlixiGl</ !
6. x, y Ngqbe!uqgqkLX!w{<gt<!weqz<, x × y uqgqkLX!w{<{iGl</!x × y J xy
we<Ox!wPKOuil</!wMk<Kg<gim<mig, −5, 32 we<he!uqgqkLX!w{<gt</!OlZl<,
(−5) ⎟⎠⎞
⎜⎝⎛
32
= ⎟⎠⎞
⎜⎝⎛⎟⎠⎞
⎜⎝⎛ −
32
15 =
312)5(
××− =
310− , yV!uqgqkLX!w{</!-f<kh<! h{<hqje!Q z<!
ohVg<gZg<gie!njmUh<!h{<H!weg<%XOuil</
7. x, y Ngqbe!uqgqkLX!w{<gt<!weqz<, xy = yx NGl</!wMk<Kg<gim<mig, −3, 75−
we<he!uqgqkLX!w{<gt</ .7
15)3(75 ,
715
75)3( =−⎟
⎠⎞
⎜⎝⎛ −=⎟
⎠⎞
⎜⎝⎛ −− !!
weOu? (−3) ).3(75
75
−⎟⎠⎞
⎜⎝⎛ −=⎟
⎠⎞
⎜⎝⎛ − -f<kh<!h{<hqje!Q z<!ohVg<gZg<gie!hiqlix<Xh<!h{<H!
we!njph<Ohil</! 8. x, y, z we<he uqgqkLX!w{<gt<!weqz< x(yz) = (xy)z.
wMk<Kg<gim<mig, 2, −3, 57− we<he!uqgqkLX!w{<gt</ ,
542
521)2(
57)3()2( =⎟
⎠⎞
⎜⎝⎛=⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −−
[(2) (−3)] .542
57)6(
57
=⎟⎠⎞
⎜⎝⎛ −−=⎟
⎠⎞
⎜⎝⎛ − weOu! [ ] .
57 )3)(2(
57)3()2( ⎟
⎠⎞
⎜⎝⎛ −−=⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −−
-f<kh<!h{<hqje!Q z<!ohVg<gZg<gie!OsIh<Hh<!h{<H!we!njph<Ohil</! 9. w{< 1 NeK! uqgqkLX! w{<{iGl</ 1x = x1 = x we<hK!njek<K! uqgqkLX
x-g<Gl< d{<jlbiGl</!wMk<Kg<gim<mig, (1) .35
351
35
=×
=⎟⎠⎞
⎜⎝⎛
fil<!gueqh<hK?!
(−x) y = [(−1)(x)] y = (−1)xy = −xy; −(−x) = (−1)[(−1)x] =[(−1)(−1)x] = 1x = x. 10. yu<ouiV!H,s<sqblqz<zik!uqgqkLX!w{< x-g<Gl<,
x1 NeK!uqgqkLX!w{<!NGl</!
OlZl<? .111=⎟
⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛ x
xxx wMk<Kg<gim<mig, x =
421− , yV!uqgqkLX!w{</!-r<G, x ≠ 0,
214
214
42111 −
=−
=⎟⎠⎞
⎜⎝⎛ −
=x
, yV!uqgqkLX!w{<!NGl</!OlZl<!
x .18484
214
4211
==⎟⎠⎞
⎜⎝⎛ −⎟⎠⎞
⎜⎝⎛ −=⎟
⎠⎞
⎜⎝⎛
x !w{<
x1 J Q z<! x e<!kjzgQpq!nz<zK!ohVg<gzqe<!gQp<!
OfIliX!we<gqOxil</
4
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11. x, y, z we<he!uqgqkLX!w{<gt<!weqz<?! x(y + z) = xy + xz, (x+y) z = xz+ yz NGl</
wMk<Kg<gim<mig, 5 ,21
,3
2=
−== zyx weqz<,
x (y + z) = ⎥⎦
⎤⎢⎣
⎡+⎟
⎠⎞
⎜⎝⎛ −
⎟⎠⎞
⎜⎝⎛ 5
21
32 = ⎥⎦
⎤⎢⎣⎡ +−⎟⎠⎞
⎜⎝⎛
2101
32 = ,3
29
32
=⎟⎠⎞
⎜⎝⎛⎟⎠⎞
⎜⎝⎛
xy + xz = )5(32
21
32
⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛ −⎟⎠⎞
⎜⎝⎛ = ⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛ −
310
31 = .3
39
310)1(
==+− weOu, x(y+z) = xy + xz.
-u<uiOx, (x + y)z = 521
32
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −+⎟
⎠⎞
⎜⎝⎛ = 5
6)3(4⎥⎦⎤
⎢⎣⎡ −+ = ,
655
61
=⎟⎠⎞
⎜⎝⎛
xz + yz = 5215
32
⎟⎠⎞
⎜⎝⎛ −+⎟
⎠⎞
⎜⎝⎛ = ⎟
⎠⎞
⎜⎝⎛ −+
25
310 = .
65
6)15(20=
−+ NgOu, (x+y) z = xz + yz.
-f<kh<!h{<hqje Q z<!ohVg<gZg<gie!%m<mzqe<!hr<gQm<Mh<!h{<H!we!njph<Ohil</
1, 2, 3, 6, 7, 8 lx<Xl< 11 Nl<!h{<Hgt<!Q z<!dt<t!Gxqh<hqm<m!dXh<Hg<gjts<!siIf<kqVg<guqz<jz/! weOu?! OlOz! osie<e! h{<Hgt< N, W lx<Xl< Z gtqe<!dXh<HgTg<Gl< d{<jlbiGl</
Neiz<!h{<H 4 NeK!0 jus<!siIf<Kt<tK/!0 we<x!w{<!N z<!-z<jzbikziz<, h{<H 4, N z<!d{<jlbz<z/!lixig 0 we<x!w{< W lx<Xl< Z!z<!dt<tkiz<, h{<H 4!NeK W, Z gtqz<!d{<jlbiGl</
h{<H 5, Gjx! w{<gjts<! siIf<Kt<tK/! weOu N lx<Xl< W gtqz<! -K!d{<jlbz<z/!Neiz<?!-h<h{<H Z !z<!ohiVf<Kl</!
h{<H 9 NeK!w{<!1-Js<!siIf<kqVh<hkiZl<, N, W, Z gtqz<, 1 dXh<H!we<hkiZl< N, W, Z gtqz<!h{<H 9!d{<jlbigqxK/
h{<H! 10 NeK, H,s<sqblx<x! w{<gtqe<! kjzgQp<! w{<gjth<! ohiVk<K!
njlukiZl<, N, W, lx<Xl< Z gtqz<! dt<t! H,s<sqblx<x! dXh<Hg<gtqe<! kjzgQpqgt<!nux<xqz<!-Vg<giK!we<hkiZl<!N, W, Z gtqz<!h{<H!10 d{<jlbigiK/!
gpqk<kz< −, wEl<! osbzqjb! %m<mz<! + &zl<! ujvbXg<gzil</ Q z<! x, y!dt<teoueqz<, x − y = x + (−y) NGl</ − osbzq, Q!z<! hiqlix<Xh<! h{<hqje!fqjxU!osb<biK/ wMk<Kg<gim<mig, 4 − 5 = −1, 5 − 4 = 1. weOu?! 4 −5 ≠ 5 − 4.
uGk<kz< ÷ wEl<!osbzqjb!ohVg<gz<!Gxq × &zl<!hqe<uVliX!ujvbXg<gzil</
Q z<!x, y dt<te!lx<Xl<! y ≠ 0 weqz<, x ÷ y = ⎟⎟⎠
⎞⎜⎜⎝
⎛×
yx 1 . osbzq!÷ NeK Q z<!hiqlix<Xh<!
h{<hqje!fqjxU!osb<biK/!wMk<Kg<gim<mig, 4 ÷ 5 =54 , 5 ÷ 4 =
45 . weOu 4 ÷ 5 ≠ 5 ÷ 4.
gpqk<kz<!osbzq − NeK Q z<!OsIh<Hh<!h{<hqje!fqjxU!osb<biK/!wMk<Kg<gim<mig,
− 3, 11, 32!we<x! Q uqZt<t!w{<gjtg<!gVKg/
(−3) − ⎥⎦⎤
⎢⎣⎡ −
−−=⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−
3233)3(
3211 = ,
340
3319
331)3( −
=−−
=−−
5
www.kalvisolai.com
[ ]32)14(
3211)3( −−=−−− = .
344
32)42( −=
−−
weOu, (−3) − [ ] .3211)3(
3211 ⎟
⎠⎞
⎜⎝⎛−−−≠⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−
-u<uiOx,
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛⎟⎠⎞
⎜⎝⎛÷−=⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛÷÷−
23
111)3(
3211)3(
= (−3) ÷ ⎟⎠⎞
⎜⎝⎛
233 = ,
112
332
13
233
13 −
=⎟⎠⎞
⎜⎝⎛⎟⎠⎞
⎜⎝⎛ −=⎟
⎠⎞
⎜⎝⎛÷⎟
⎠⎞
⎜⎝⎛ −
[ ] ⎟⎠⎞
⎜⎝⎛÷⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛÷⎟
⎠⎞
⎜⎝⎛ −=⎟
⎠⎞
⎜⎝⎛÷÷−
32
111
13
3211)3(
= ⎟⎠⎞
⎜⎝⎛÷⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛⎟⎠⎞
⎜⎝⎛ −
32
111
13 = ⎟
⎠⎞
⎜⎝⎛÷⎟
⎠⎞
⎜⎝⎛ −
32
113 = .
229
23
113 −
=⎟⎠⎞
⎜⎝⎛⎟⎠⎞
⎜⎝⎛ −
weOu, (−3) ÷ [ ] ⎟⎠⎞
⎜⎝⎛÷÷−≠⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛÷
3211)3(
3211 .
gpqk<kz<!osbzq − NeK!N z<, njmUh<! h{<hqje!fqjxU!osb<biK/!Woeeqz<, 5, 7 we<x N e<!dXh<Hgjtg<!gVKg/ 5 −7 = −2, N e<!dXh<hz<z/!-u<uiOx!osbzq ÷ NeK!
N z<!njmUh<!h{<hqje!fqjxU!osb<biK/!Woeeqz<, 5 ÷ 7 = 75 , N e<!dXh<hz<z.
N, W, Z lx<Xl< Q gtqe<!h{<Hgjth<!hbe<hMk<kq!yV!sleqe<!kQIU!Gxqh<hqm<m!
w{<gtqe< okiGkqbqz<!dt<tki! we!nxqbzil</! wMk<Kg<gim<mig, sle< 5x −10 = 0 e< kQIU x = 2 NGl<. 2, N e<!dXh<hikziz<, 5x − 10 = 0 we<x!sleqe<!kQIU!N z<!dt<tK/ 5x = 0 we<x!sleqe<!kQIU x = 0!NGl</!w{< 0!NeK N z<!-z<jz/!Neiz<?!w{< 0!NeK W z<!dt<tK/!weOu? 5x = 0 wEl<!sle<!NeK N z< kQIU ogi{<cVg<guqz<jz/!lixig,!nke<!kQIU W z<!dt<tK/ 5x + 10 = 0 we<x!sljeg<!gVKOuil</!-ke<!kQIU x = −2. w{< −2 NeK N lx<Xl< W -v{<cZl<!-z<jz;!Neiz?< −2 NeK Z z<!dt<tK/ weOu, 5x + 10 = 0 we<x!sleqe< kQIU N lx<Xl< W uqz<!-z<jz;!Neiz<, Z z<!dt<tK/
3x + 5 = 0 we<x!lx<oxiV!sljeg<!gVKOuil</!-ke< kQIU x = 35− . w{<
35−!NeK N!z<!
-z<jz, 35− NeK W uqZl<!-z<jz lx<Xl<
35− NeK Z zqZl<!-z<jz/!weOu, N, W
lx<Xl< Z!Ngqb!w{<!okiGkqgtqz<!sle< 3x + 5 = 0 NeK!kQIuqjeh<!ohx<xqVg<guqz<jz/!
Neiz<?!w{< 35− NeK Q uqz<!-Vh<hkiz<, sle< 3x + 5 = 0 NeK Q uqz<!kQIuqjeh<!
ohx<xqVg<gqxK/ -Vh<hqEl< 2 , 3 , π, … we<he! uqgqkLxi! w{<gt<! we!
nxqf<kqVh<hkiz<, x − 2 = 0, x − 3 = 0, x − π = 0 Ohie<x!sle<hiMgt< Q!z<!kQIUgjth<
6
www.kalvisolai.com
ohx<xqVg<giK/! uqgqkLxi! w{<gjth<! hx<xq! nxqf<K! ogit<ukx<G! Le<hig! uqgqkLX!w{<gjt!ksl ucuqz<!wPKl<!Ljxjb!lQ{<Ml<!Nvib<Ouil</!
1.1.6 uqgqkLX!w{<gjt!ksl!ucuk<kqz<!Gxqk<kz<!
fQt<uGk<kz<! Ljxbqz<! uqgqkLX! w{<gjt! kslg<! GxqbQm<cz<! wPKl<!
Ljxbqje!nxqOuil</!wMk<Kg<gim<mig, 3215− lx<Xl<
751!gtqe<!ksl!hqe<e!njlh<Hgt<!
LjxOb!!
3215− = −0.46875. 51 = 7.28571428…
Olx<Gxqh<hqm<Mogit<Ouil</! -kx
wMk<Kg<gim<mig! 32gi{zil</!nkiuK,
weOu 0, 1, 2, …, 100, 101, 102, …ghqe<er<gjtBl<, 0lmr<Ggtigh<!ohxz
6× 10-1 + 2 × 10
weOu? 85 J!0.625
6 e<!Lglkqh<H 106
hkqe<lieh<! Ht<tq!
dt<t 0 e<!lkqh<H
wPKl<OhiK! LPjwPkqmh<!hbe<hMk<k
www.kalvisolai.com
7
t<t! fQt<! uGLjxbqz<! hqe<hx<xqb!<G! LPg<gjt! wu<uiX! Gxqg<gqOxi
4 J 3 F~Xgt<, 2 hk<Kgt<, 4 ye<Xgt<! 324 = 3 × 102 + 2 × 101 + 4 × 100. -u<uiO
2003 = 2 × 103 + 0 × 102 + 0 × 101 +9 wEl<! w{<!-zg<gr<gjtg<! ogi{<M!tqe<! lmr<Ggtig! LcU! osb<K! wP
, 1, 2, …, 9 Jg<! ogi{<M, 10-1, 10il</!wMk<Kg<gim<mig, -2 + 5× 10-3 =
100020600
105
102
106
32
++=++
weg<!Gxqh<Ohil</!-r<G!Ht<tqbqe<!uzK
, -v{<miuK! w{<! 2 e<!Lglkqh<H 100
2
nz<zK! kslh<Ht<tq! we! njph<hI/! k
0 × 100 = 0!NGl</!weOu!kslh<Ht<tq!N
lh<! hGkq, hqe<eh<hGkq! nz<zK! kslh<hMgqxK/
7
uqkqjb! fil<! nxqf<K!l<! we<hjkg<! gi{<Ohil</!
Ngqbeux<xqe<! %Mkzigg<!
x 3 × 100. nux<xqe<!Lglkqh<Hg<gjt, Kgqe<Oxil</! -K! OhizOu!-2, 10-3, … Ngqbeux<xqe<!
85
12581255
10006255
=××
== .
hg<gk<kqz<!Lkz<!w{<{ie
we!nxqbzil</! Ht<tqjb!
slh<Ht<tqbqe<! -mh<Hxk<kqz<!
eK!kslucuk<kqz< 85 J!
h<hGkq! -ux<jxh<! hqiqk<K!
Olx<%xqb!GxqbQm<cz<, 3.025!e<!ohiVt<
3 × 100 + 0 × 10-1 + 2 × 10-2 + 5 × 10-3 = 1000
5100
21003 +++ = 3 +
100025 = 3 +
401 =
40121 .
-h<ohiPK,3215− we<x!hqe<ek<jk!wMk<Kg<ogit<Ouil</!-kqz<?
⎟⎠⎞
⎜⎝⎛=
32150
101
3215 = ⎟
⎠⎞
⎜⎝⎛ +
32224
101 = ⎟
⎠⎞
⎜⎝⎛+
3222
101
104
= ⎟⎠⎞
⎜⎝⎛+
32220
101
104
2 = ⎟⎠⎞
⎜⎝⎛ ++
32286
101
104
2 = ⎟⎠⎞
⎜⎝⎛++
3228
101
106
104
22
= ⎟⎠⎞
⎜⎝⎛++
32280
101
106
104
32 = ⎟⎠⎞
⎜⎝⎛ +++
32248
101
106
104
32
= ⎟⎠⎞
⎜⎝⎛+++
3224
101
108
106
104
332 = ⎟⎠⎞
⎜⎝⎛+++
32240
101
108
106
104
432
= ⎟⎠⎞
⎜⎝⎛ ++++
32167
101
108
106
104
432 = ⎟⎠⎞
⎜⎝⎛++++
3216
101
107
108
106
104
4432
= ⎟⎠⎞
⎜⎝⎛++++
32160
101
107
108
106
104
5432
= 5432 105
107
108
106
104
++++ (osb<Ljx!Lx<Xh<ohXgqxK)
= 0.46875
∴ 3215− = −0.46875.
nMk<K, hqe<uVl<!osb<Ljxjbg<!gVKOuil</
⎟⎠⎞
⎜⎝⎛+=
727
751 (A)
= ⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+
720
1017 = ⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ ++
762
1017 = ⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛++
76
101
1027
= ⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛++
760
101
1027 2 = ⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ +++
748
101
1027 2
= ⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+++
74
101
108
1027 22 = ⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+++
740
101
108
1027 32
= ⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ ++++
755
101
108
1027 32 = ⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛++++
75
101
105
108
1027 332
= ⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛++++
750
101
105
108
1027 432 = ⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ +++++
717
101
105
108
1027 432
= ⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+++++
71
101
107
105
108
1027 4432 = ⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+++++
710
101
107
105
108
1027 5432
8
www.kalvisolai.com
= ⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ ++++++
731
101
107
105
108
1027 5432
= ⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛++++++
73
101
101
107
105
108
1027 55432
= ⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛++++++
730
101
101
107
105
108
1027 65432
= ⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ +++++++
724
101
101
107
105
108
1027 65432
= 7×100+ ⎟⎠⎞
⎜⎝⎛+⎥⎦
⎤⎢⎣⎡ +++++
72
101
104
101
107
105
108
102
665432 (B)
= 7.285714 + ⎟⎠⎞
⎜⎝⎛
72
101
6 . osb<LjxbieK!(A)-zqVf<K!lQ{<Ml<!Nvl<hligqxK/!
3215− e<! ksl! ucul<! wPKl<OhiK! fQt<uGk<kz<! Ljx! Lx<Xh<ohXujkg<!
gi{zil< (lQkq 0 uVgqe<xhcbiz<). weOu!3215− Lx<Xh<ohXl<! ksl! uqiqjuh<!
ohx<Xt<tK/!Neiz<!751 e<!kslucug<!Gxqk<kzqz<?!fQt<uGk<kz<!Ljx!wf<fqjzbqZl<!
Lx<Xh<ohxuqz<jz (lQkq ≠ 0 we! wf<k! fqjzbqZl<! dt<tK). nOk! Ofvk<kqz< (B)!gm<mk<kqz<! ohxh<hMl<! lQkLl< (A) gm<mk<kqZt<t! lQkLl<! ye<Ox! we! nxqg/! ! weOu?!gm<ml< (A), gm<ml< (B) -ux<xqx<gqjmOb! uVgqe<x! -zg<gr<gt< 2, 8, 5, 7, 1, 4 uiqjs!
lixilz<!kqVl<hk<!kqVl<h gqjmg<Gl</!weOu 751 e<!kslucug<!Gxqk<kjz!Lx<Xh<ohxi!
lx<Xl<!lQt<!okimi<!Ljx (Non-terminating and Recurring)!we<Ohil</!-kje
7751
= .285714285714285714… = 285714.7 ,
we! wPKOuil</! -r<G! 285714 e<! lQK! -mh<hm<Mt<t! OgiM! (bar)?!751 e<! fQt<uG!
Ljxbqz<! 285714 kqVl<hk<! kqVl<h! nOk! uiqjsbqz<! -ml<ohXujkg<! Gxqg<Gl</
3215 g<gie! Lx<Xh<ohXl<! ksl! uqiquie 0.46875! J! Lx<Xh<ohxik! lQt<! okimvigUl<!
gVkzil</!Woeeqz<,
0.46875 = 5432 105
107
108
106
104
++++
= .....10
010
010
510
710
810
6104
765432 +++++++ = 046875.0 .
OlZl<!nOk 0.46875 g<G,
9
www.kalvisolai.com
0.46875 = 5432 105
107
108
106
104
++++
= ( )1410
110
710
810
6104
5432 +++++
= ⎟⎠⎞
⎜⎝⎛+++++ 55432 10
110
410
710
810
6104
= ( )1010
110
410
710
810
6104
65432 +++++
= ( )1910
110
410
710
810
6104
65432 ++++++
= ⎟⎠⎞
⎜⎝⎛++++++ 665432 10
110
910
410
710
810
6104
= ( )1010
110
910
410
710
810
6104
765432 ++++++
= ( )1910
110
910
410
710
810
6104
765432 +++++++
= ⎟⎠⎞
⎜⎝⎛+++++++ 7765432 10
110
910
910
410
710
810
6104 = .946874.0
NgOu, yu<ouiV!uqgqkLX!w{<[l<!Lx<Xh<ohXl<!nz<zK!Lx<Xh<ohxik!lQt<!
okimv<! Ljxbqz<! ksl! uqiquig! njlgqxK/! fQt<! uGLjxbqz<! ohxh<hMl<! lQkr<gt<!Gjxbqz<zi! LPg<gtigUl<?! uG! w{<gjt! uqm! GjxuigUl<! -Vh<hkiz<kie<, uqgqkLX! w{<gt<! -k<kjgb! sqxh<Hh<! h{<hqjeh<! ohx<Xt<te/! Lf<jkb! fqjzbqz<!ohxh<hm<m! lQklieK! OuoxiV! fqjzbqz<! lQklig! lXhcBl<<! uVujkg<! gi{zil</!weOu?! =uqz<! dt<t! -zg<gr<gt<! kqVl<hUl<! uv! Nvl<hqg<gqe<xe/! -ke<! lXkjz!d{<jlbi!we!Nvib<Ouil</!Lx<Xh<ohXl<!nz<zK!Lx<Xh<!ohxik!okimI!ujg!ksl!uqiqUgt<!wkjeg<!Gxqg<gqe<xe@!-f<k!uqeiju!wMk<Kg<gim<Mgt<!&zl<!Nvib<Ouil<. (i) ksl!uqiqU 0.45 Jg<!gVKg.
0.45 = 209
10045
100540
105
104
2 ==+
=+ .
(ii) ksl!uqiqU! 45.0 Jg<!gVKg/!x = 45.0 = 0.454545… we<g/!weOu 100x = 45.454545…
∴ 100 x − x = (45.4545…) − (0.4545…) nz<zK! 99x = 45 nz<zK!x = 115
9945
= .
(iii) ksl!uqiqU 934.0− Jg<!gVKg/ x = 934.0 = 0.349999… we<g/!NgOu!100x = 34.9999… ∴ 1000x = 349.9999… ∴ 1000x − 100x = (349.9999…) − (34.9999…)
∴ 900x = 315 nz<zK! x = .207
10035
900315
==
10
www.kalvisolai.com
∴ 934.0 = 207 .
∴ 934.0− = − 20
7207 −
=⎟⎠⎞
⎜⎝⎛ .
weOu, yu<ouiV!Lx<Xh<ohXl<!nz<zK!Lx<Xh<ohxik!lQt<okimI!ujg!ksl!uqiqU,!yV!uqgqkLX w{<{qjeg<! Gxqg<Gl</! nkiuK, yV!Lx<Xh<ohXl<! nz<zK!Lx<Xh<ohxik!lQt<okimI!ujg!ksl!uqiqju!“LPuqe<!gQp<!LP” we<x!ucuk<kqz<!njlg<g!-bZl</! 1.1.7 uqgqkLxi!w{<gt<
Lx<Xh<ohxik! Neiz<! okimI! uiqjsbx<x! ksl! uqiqUgjtg<! gVKOuil</!
wMk<Kg<gim<mig, 0.101001000100001000001… . -u<uqiquqz< 0 lx<Xl< 1 gOt!dt<te/!OlZl< 1 gt<! NeK 1 H,s<sqbl<, 2 H,s<sqbr<gt<, 3 H,s<sqbr<gt<, 4 H,s<sqbr<gt<, … Ngqbeux<xiz< hqiqg<gh<hm<Mt<tjk!nxqg/!-kqz<!kqVl<hk<!kqVl<h!uVl<! okimI!ujg!nMg<Ggt<!-z<jz/!weOu?!-k<kjgb!ksl!uqiqUgt<!uqgqkLX!w{<gjtg<!Gxqg<giK/!nk<kjgb! ksl! uqiqUgt<! uqgqkLxi! w{<gjtg<! Gxqg<Gl<! we<gqOxil</! uqgqkLX!lx<Xl<! uqgqkLxi! w{<gjtOb! olb<ob{<gt<! we<hv</! yV! ksl! uqiquieK!hqe<uVueux<Xt<!WOkEl<!yV!ucuqz<!lm<MOl!-Vg<Gl</!!
(i) Lx<Xh<ohXl<!ucuqz<!-Vg<Gl</ (ii) Lx<Xh<ohxik!Neiz<!kqVl<hk<!kqVl<h!uVl<!okimI!ujg!ksl!uqiqU/!(iii) Lx<Xh<ohxik!lx<Xl<!kqVl<hk<!kqVl<h!uvik!okimI!ujg!ksl!uqiqU/ NgOu!yu<ouiV!ksl!uqiqUl<!yV!olb<!w{<{iGl</!yu<ouiV!olb<!w{<{qx<Gl<!ksl!
uqiqU!d{<M!we!fil<!%xzil</!yV!olb<ob{< x-e<!ksl!uqiquqz<!10n e<!G{gr<gtqz<!Gjxf<khm<sl<! ye<xiuK! lqjg!LP! w{<!Ng!-Vf<kiz<? nkje! lqjg! olb<! w{<!
we<Ohil</! -u<uiOx, olb<ob{< x NeK! lqjgbigOui! nz<zK! h,s<sqbligOui!
-z<zikqVh<hqe<!nK!Gjx!olb<ob{<!weh<hMl</!wMk<Kg<gim<mig,
7.00252525… = ....10
510
210
510
210
01007 65432 +++++++
we<hK!lqjg!olb<ob{</!Neiz<,
− 3.0202202220… = − ⎥⎦⎤
⎢⎣⎡ +++++++ ...
100
102
102
100
102
1003 65432
Gjx!olb<ob{<{iGl</ olb<ob{<gtqe<! okiGkqjb R weg<Gxqh<Ohil</! NgOu! R NeK! uqgqkLX! lx<Xl<!uqgqkLxi! w{<gtqe<! okiGh<hiGl</! -bz<! w{<gt<, LP!w{<gt<, LPg<gt<, uqgqkLX!lx<Xl<! uqgqkLxi! w{<gt<! njek<Kl<! olb<ob{<gtiGl</! wf<kouiV! uqgqkLX!w{<[l< uqgqkLxi! w{<{igiK/! -jkh<OhizOu?! wf<kouiV! uqgqkLxi! w{<[l<!uqgqkLX!w{<!NgiK/ R z<!upg<glie!%m<mz<, gpqk<kz<, ohVg<gz<!lx<Xl<!uGk<kz<!uqkqgt<! biUl<! d{<jlbiGl</! Gxqh<hig, R z<! dt<t! hiqlix<X! lx<Xl<! hr<gQm<Mh<!h{<Hgtiue;!x, y lx<Xl< z WOkEl<!&e<X!olb<ob{<gTg<G?! (i) x + y = y + x, xy = y x; (ii) x(y + z) = xy + xz; (iii) (x + y)z = xz + yz,
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Olx<!%xqBt<t!h{<HgtqzqVf<K!hqe<uVueux<jxg<!gi{zil</!! (i) x (y − z)=x [y + (− z)]=xy + x(− z) = xy − xz. (ii) (x + y)2 = (x + y)(x + y) = (x + y) z, -r<G z = x + y weh<!hqvkqbqm = xz + yz = x (x + y) + y (x + y) = xx + xy + yx +yy = x2 + xy + xy + y2 = x2 + 2xy + y2 . (iii) (x − y)2 = (x − y)(x − y) = (x − y) z, -r<G z = x − y weh<!hqvkqbqm! = xz − yz = x (x − y) − y (x − y) = xx − xy − (yx − yy) = x2 − xy − xy + y2 = x2 − 2xy + y2 . (iv) (x + y) (x − y) = (x + y) z -r<G z = x − y!weh<!hqvkqbqm = xz + yz = x (x − y) + y (x − y) = xx − xy + yx − yy = x2 − y2. wMk<Kg<gim<M 1: 75.0 Nz<!Gxqg<gh<hMl<!uqgqkLX!w{<{qjeg<!gi{<g/!
kQIU: x = 75.0 we<g/ x = 0.757575… ∴ 100x = 75.757575… ∴ 100x − x = (75.757575…) − (0.757575…) = 75.0000…
∴ 99x = 75 nz<zK! x = .3325
9975
=
wMk<Kg<gim<M 2: 0.1 = 9.0 we<hjk!fqbibh<hMk<kUl</!
kQIU: -r<G! 0.1 = 1 = )10(101 = )19(
101
+ = ⎟⎠⎞
⎜⎝⎛+101
109 = ( )10
1001
109+ = ( )19
1001
109
++
= ⎟⎠⎞
⎜⎝⎛++100
1100
9109 = ( )10
10001
1009
109
++ = ( )191000
1100
9109
+++
= ⎟⎠⎞
⎜⎝⎛+++1000
11000
9100
9109 .
OlOz!dt<t!LjxbieK!Lcuqe<xq!ose<Xogi{<OmbqVg<Gl</!NgOu, 0.1 = 0.9999… nz<zK! 0.1 = .9.0
wMk<Kg<gim<M 3: 52.2 + 25.2 J!LPuqe<!gQp<!LP!ucuqz<!gi{<g/ kQIU: x = 52.2 we<g/ x = 2.525252…. ∴ 100x = 252.5252…
∴ 100x − x = 250.000… nz<zK! 99x = 250 nz<zK! x = .99250
y = 25.2 we<g/ y = 2.52222….. ∴ 10y = 25.2222…
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∴ 100y = 252.2222…
∴ 100y − 10y = 227.0000… nz<zK! 90y = 227 nz<zK y = .90227
weOu? 52.2 + 25.2 = 99250 +
90227 =
99024972500 + =
9904997 .
wMk<Kg<gim<M 4: 0.101001000100001… lx<Xl< 0.010110111011110…Ngqb! uqgqkLxi w{<gjtg<!%m<Mg/! uqjm! yV!uqgqkLX! w{<{ig!-Vg<Glieiz<,!nkje!LPuqe<!gQp<!LP!ucuqz<!gi{<g/ kQIU: x = 0.1010010001… lx<Xl< y = 0.0101101110…we<g/ x + y = 0.1010010001… + 0.0101101110…= 0.111111… = 0. 1 . -r<G x + y e<!ksluqiqU!Lx<Xh<ohxik!Neiz<!kqVl<hk<!kqVl<h!uVl<!okimI!ujgjbs<!siv<f<kK/ weOu, x + y yV!uqgqkLX!w{</!a = 0.11111…we<g/!hqe<H!10a = 1.1111…
∴ 10a − a = 1.1111… − 0.1111 = 1.0000…. nz<zK! 9a = 1 nz<zK! a = 91 . ∴ x + y =
91 .
Gxqh<H; Olx<g{<m! wMk<Kg<gim<czqVf<K, -V! uqgqkLxi! w{<gtqe<! %Mkz<! yV!
uqgqkLxi w{<{ig!-Vg<g!Ou{<cb!nusqblqz<jz/!-jkh<!OhizOu, -V!uqgqkLxi!w{<gtqe<! ohVg<gx<hzEl<! yV! uqgqkLxi! w{<{ig! -Vg<gOu{<cb!nusqblqz<jz/ ( 2 yV!uqgqkLxi w{</!Neiz< 2 × 2 = 2, yV!uqgqkLX!w{<).
2 we<x! uqgqkLxi! w{<j{g<! gVKOuil</ 2 -uqe<! ksl! uqiqju! gi[l<!
Ljxjb!fqjeU!ogit<Ouil</!!
weOu 2 = 1.414213562… . filxqukqVl<h!uVl<!okimI!ujg!-z<zilZl
-u<uiOx, 3 , 5 , 7 , … uqgqkLxi!w
www.kalvisolai.com
K?! 2e<! ksluqiqU!Lx<Xh<ohxilZl<, kqVl<hk<!<!dt<tK/!weOu 2 yV!uqgqkLxi!w{<{iGl</!{<gt<!we!fqXuzil</
13
g{qkk<kqz< π lx<Xl< e wEl<! keqs<sqxh<Hlqg<g! -V! uqgqkLxi! w{<gt<!-ml<ohXgqe<xe/! wf<kouiV! um<mk<kqZl<! nke<! Sx<xtU! lx<Xl<! uqm<ml<!Ngqbux<xqx<gqjmOb! dt<t! uqgqkl<! yV! lixqzq! NGl</! -l<lixqzqjb! gqOvg<g!
wPk<kie! π Nz<! Gxqh<Ohil</ π e<! ksl! uqiqU! 3.1415926… we<hkiGl</! -K!
Lx<Xh<ohxik, kqVl<hk<! kqVl<h! uvik! okimvig! dt<tK/ 722 wEl<! uqgqkLX!
w{<{qje!π e<!Okivib!lkqh<higg<!g{g<gqmzqz<!hbe<hMk<kqBt<Otil</!-f<kqbg<!g{qk!Oljk!-viliE\e< π J!dt<tmg<gqb! hz!$k<kqvr<gjt!dVuig<gqBt<tiI/ 1973 l< N{<M!ujvbqz<, π g<gie!ksl!uqiquqz< 1,000,000 !-zg<gr<gt<!-ml<ohXgqx!ujgbqz<< gi{h<hm<Mt<tK/! hz! Ogic! -mk<! kqVk<kk<Kme< π e<! uqiqjug<! gi{<hK! we<hK!kx<OhiK fjmohx<X! uVgqx! gcelie! Neiz<! uqbk<kG! h{qbiGl</ nMk<kkig?
2
23⎟⎠⎞
⎜⎝⎛ ,
3
34⎟⎠⎞
⎜⎝⎛ ,
4
45⎟⎠⎞
⎜⎝⎛ ,
5
56⎟⎠⎞
⎜⎝⎛ , … we<x! Ljxob{<gtqe<! lkqh<Hgjtg<! g{g<gqMjgbqz<!
-jubjek<Kl<! yV! Gxqh<hqm<<m! olb<ob{<{qx<G! lqg! lqg! nVgijlbqz<! njlujk!
nxqbzil</!nf<k!w{<{qje e weg<!Gxqh<Ohil</ e e<!ksluqiqU e = 2.7182818284…. -KUl<! Lx<Xh<ohxik! lx<Xl<! kqVl<hk<! kqVl<h! uvik! okimvig! dt<tK/! -f<k!
uqgqkLxi!w{< e!Jh<!hx<xq!OlZl<!dbI!uGh<Hgtqz<!fQr<gt<!hck<kxqbzil</ 1.1.8 R-z<!uiqjs!dxU
sqz! ohiVm<gjt! njugtqe<! h{<HgTg<Ogx<h! uiqjs! hMk<Kl<OhiK, nju uiqjsh<hMk<kh<hm<mju! weh<hMgqe<xe/! wMk<Kg<gim<mig, li{uIgjt! nuvuI!
dbvk<kqx<Ogx<h, lqgg<! Gjxf<k! dbvk<kqzqVf<K! lqg! nkqghm<s! dbvl<! wEl<!
nch<hjmbqz<!fqx<gs< osb<kiz<, nl<li{uIgt<!!uiqjsbqz<!fqx<hkigg<!%Xgqe<Oxil</ x lx<Xl< y olb<ob{<gt<! we<g/ y − x NeK! lqjg! w{<! weqz<, x!NeK y-J!uqmg<!Gjxf<k!w{< nz<zK!y NeK x J!uqmh<!ohiqb!w{<!weh<hMl</ x!NeK y J!uqmg<!GjxuieK we<hjkg<!Gxqg<g x < y we<x!GxqbQm<jmBl<, y NeK x J!uqmh<!ohiqbK!we<hjkg<!Gxqg<g y > x we<x GxqbQm<jmBl<!hbe<hMk<KOuil</!NgOu, x < y lx<Xl< y > x we<he! y − x yV lqjg! w{<! we<x! yOv! ohiVt<! kVue/ y − x !Gjx!w{<! weqz<, −(y − x) = x − y lqjgbiGl</ weOu? y < x nz<zK! x > y/!wMk<Kg<gim<mig, 19 lx<Xl< 17 we<x!-V!w{<gjt!wMk<Kg<!ogit<Ouil</!19 − 17 = 2 NeK!lqjg!we<hkiz<, 17 < 19 nz<zK 19 > 17 weg<! gqjmg<gqxK/! nMk<K, −19 lx<Xl< −17 gjtg<! gVKOuil</!filxquK?!
(−19) − (−17) = −19 + 17 = −2, yV!Gjx!w{</!weOu, −19 < −17 nz<zK! −17 > −19.
x lx<Xl< y we<he!olb<ob{<gt<!we<g/!hqe<uVl<!&e<xqz<!WOkEl<!ye<X!lm<MOl!d{<jl/ (i) x − y yV!Gjx!w{<; nkiuK, x < y (ii) x − y yV!lqjg!w{<; nkiuK, x > y (iii) x − y = 0; nkiuK, x = y. x < y nz<zK! x = y we<hjk x!NeK y J uqms<!sqxqb!nz<zK!sl!w{<!we<Ohil</!-kje x ≤ y!we!wPKOuil</!-K?! x!NeK! y!J!uqm!
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lqjgbz<z! we<X! Gxqh<hjk! nxqf<K! ogit<g/! -u<uiOx, x > y nz<zK! x = y weqz<, x NeK y J!uqmh<!ohiqb!nz<zK!sl!w{<!we<Ohil</!-kje!x ≥ y!we!wPKOuil</!-K?!x!NeK!y!J!uqm!Gjxf<kkz<z!we<X!Gxqh<hjk!nxqf<K!ogit<g/ x NeK!lqjg!w{<!weqz<? x > 0 NGl<. x NeK!Gjx!nz<z!weqz<? x ≥ 0 NGl</ x Gjx!w{<!weqz<, x < 0! NGl</ x NeK! lqjg! w{<! nz<z! weqz<? x ≤ 0 NGl</! ! NgOu wu<uqV!olb<ob{<gt< x lx<Xl< y gTg<G, hqe<uVl<!&e<xqz<!ye<X!lm<MOl!d{<jlbiGl</!
x < y, x = y, x > y. -kje!olb<ob{<gTg<gie!Lh<hqvqh<H (trichotomy) uqkq!we<Ohil</ x, y lx<Xl< z we<he!x < y lx<Xl< y < z, wEliX!dt<t!WOkEl<!&e<X!olb<ob{<gt<!weqz<, y − x lx<Xl< z − y gt<!lqjg!w{<gtiGl</!weOu, (y − x) + (z − y) lqjg!NGl</!nkiuK, z – x lqjg!NGl</! weOu? x < z. ∴ x < y lx<Xl< y < z weqz<, x < z d{<jlbiGl</! -h<h{<hqje!lix<Xh<!h{<H!we<Ohil</ x < y lx<Xl< y < z!we<hjk x < y < z!weg<!Gxqh<Ohil</!-r<G y !NeK x lx<Xl< z!gTg<G!-jms<!osVgzig!dt<tK!we<Ohil</
olb<ob{< x lx<Xl< 0 Jg<! gVKg/! hqe<uVhjugtqz<! ye<X! lm<MOl!d{<jlbiGl</
x < 0, x = 0, x > 0.
x < 0 weqz<, x2 = x × x = Gjx!w{< × Gjx!w{< = lqjg!w{</ x = 0 weqz<, x2 = x × x = 0 × 0 = 0. x > 0 weqz<, x2 = x × x = lqjg!w{< ×!lqjg!w{< = lqjg!w{</ weOu! wf<kouiV! olb<ob{< x-g<Gl< x2 ≥ 0/! Gxqh<hig! H,s<sqblqz<zi! wu<ouiV!olb<ob{<!x (> 0 nz<zK! < 0)-g<Gl<!x2 > 0. WOkEoliV!olb<ob{<!x Jg<!gVKg/!x ≥ 0 we<xohiPK?!x e<!lm<Mlkqh<H!x we<Xl<;!x < 0 we<xohiPK?!x e<!lm<Mlkqh<H!−x we<Xl<!ujvbXg<gh<hMgqxK/! x e<! lm<Mlkqh<hqje! |x| we<x! GxqbQm<ceiz<! Gxqh<Ohil</! weOu?!!!|x| = x, x ≥ 0 we<xohiPK; |x| = −x, x < 0 we<xohiPK/ filxquK? |x| ≥ 0, |−x| = |x| lx<Xl<! |x|2 = x2. !
wMk<Kg<gim<M 5: π lx<Xl<722 -ux<jx!!uiqjsh<hMk<K/!
kQIU: π = 3.141528…, 722 = 142857.3 = 3.142857142857…
∴ 722 − π = 3.142857… − 3.141528… = 0.0013…
∴ 722 − π NeK!lqjg!w{</
∴ π < 722 .
wMk<Kg<gim<M 6: uqgqkLX! w{<gt< 1.201 lx<Xl< 1.202 -ux<xqx<G! -jmbqz<! WOkEl<!fie<G!w{<gjt!-jms<osVgzigs<!OsIg<gUl</!
15
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kQIU: -r<G 1.202 − 1.201 = 0.001 > 0 weOu 1.201 < 1.202. 1.2011, 1.2012, 1.2013, 1.2014 Ngqb! w{<gjtg<! gVKg/! -jugt<! Lx<Xh<ohXl<! ksl! uqiqU! ucuqZt<tkiz<, njek<Kl<!uqgqkLX!w{<gtiGl</ 1.2011 − 1.201 = 0.0001 > 0. weOu!1.201 < 1.2011. 1.2012 − 1.2011 = 0.0001 > 0, we<hkiz< 1.2011 < 1.2012. 1.2013 − 1.2012 = 0.0001 > 0,!we<hkiz< 1.2012 < 1.2013. 1.2014 − 1.2013 = 0.0001 > 0, we<hkiz< 1.2013 < 1.2014. 1.202 − 1.2014 = 0.0006 > 0, we<hkiz< 1.2014 < 1.202. ∴ 1.201 < 1.2011 < 1.2012 < 1.2013 < 1.2014 < 1.202.
wMk<Kg<gim<M 7: 1.201 lx<Xl< 1.202 gTg<gqjmOb! fie<G! uqgqkLxi! w{<gjt!-jms<osVgzigs<!OsIg<gUl</! kQIU: 1.202 − 1.201 = 0.001 > 0. weOu?!1.201 < 1.202. a = 1.201101001000100001…, b = 1.201202002000200002…, c = 1.201303003000300003…, d = 1.201404004000400004…, wEl<! fie<G! w{<gjtg<! gVKg/! -jubiUl<! Lx<Xh<ohxik, kqVl<hk<! kqVl<h! uVl<!okimI!ujgjbs<!sivik!ouu<Ouxie!ksl!uqiqUgjtg<!ogi{<Mt<tkiz<?!ouu<Ouxie!
uqgqkLxi!w{<gtiGl</!OlZl<, a − 1.201 = 1.2011010010001… − 1.2010000000000… = 0.0001010010001… > 0, b − a =1.2012020020002… − 1.2011010010001… = 0.0001010010001… > 0, c − b =1.2013030030003… − 1.2012020020002… = 0.0001010010001… > 0, d − c=1.2014040040004… − 1.2013030030003….= 0.0001010010001… > 0, 1.202 − d =1.202000000000… − 1.2014040040004… = 0.0005959959995… > 0. ∴ 1.201 < a < b < c < d < 1.202. wMk<Kg<gim<M 8: 001.2 lx<Xl< 003.2 uqgqkLxi! w{<gTg<gqjmOb! WOkEl<! -V!uqgqkLX!w{<gjt!-jms<osVgzigs<!OsIg<gUl</! kQIU: ∴ 001.2 ∴ 003.2 −
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= 1.41456… lx<Xl< 003.2 = 1.41527… 001.2 = 0.0007 (Okiviblig*!> 0.
16
∴ 001.2 < 003.2 . 1.41461 lx<Xl< 1.41462 Ngqb! w{<gjtg<! gVKg/! Lx<Xh<ohXl<! ksl! uqiqjug<!ogi{<Mt<tkiz<,!-jubqv{<Ml<!uqgqkLX!w{<gtiGl</!OlZl<!1.41461 − 001.2 = 1.414610000… − 1.41456…= 0.00004… > 0 ∴ 001.2 < 1.41461. 1.41462 − 1.41461 = 0.00001 > 0 ∴ 1.41461 < 1.41462.
003.2 − 1.41462 = 1.41527… − 1.41462 = 0.0006…> 0 ∴ 1.41462 < 003.2 . ∴ 001.2 < 1.41461 < 1.41462 < .003.2
Olx<%xqBt<t! wMk<Kg<gim<Mg<gtqzqVf<K, a, b! wEl<! wu<uqV! ouu<OuX!
olb<ob{<gt<! a < b we! -Vh<hqe<, r we<x! uqgqkLX! w{<{qje a < r < b wEliX!gi{zil<. a lx<Xl< b ye<Xg<ogie<X! lqg! nVgijlbqz<! -Vh<hqEl<,! OlOz!%xqBt<t!h{<H! d{<jl/! -h<h{<hqje R z< Q uqe<! nmIk<kqh<! h{<H! we<gqOxil</! yV!OfIg<Ogim<cz<! njlf<k -V! ouu<OuX! Ht<tqgTg<gqjmOb! OuoxiV! keqk<kh<! Ht<tq!
-Vg<Gl<<!we<x!d{<jlBme<!R e<!nmIk<kqh<!h{<hqje!yh<hqmzil</!-k<kjgb!yh<hQM?!olb<ob{<gjt!OfIg<OgiM!ye<xqeiz<!Gxqg<g!upq!uGg<gqxK/!!
njek<K! uqgqkLX! w{<gTl<! olb<ob{<gOt! we<hjk! fqjeuqz<! ogit<g/!
Neiz<?! uqgqkLX! nz<zik! olb<ob{<gTl<! d{<M! we<hjk! nxqOuil</!
wMk<Kg<gim<mig, 2 , 3 , π, … Ngqbe! olb<ob{<gt<;! Neiz<! uqgqkLX! w{<gt<!
nz<z/!-h<OhiK, sle<gt< x − 2 = 0, x − 3 = 0, x − π = 0, R z<!kQIjuh<!ohx<Xt<te/!Neiz<? x2 +1 = 0 we<x!sleqx<G R z<!kQIU!-z<jz/!Woeeqz<? x2 = −1 we!wf<kouiV olb<ob{<! x JBl<! ohx! -bziK/! -k<kjgb! sle<hiMgjt! flK! uGh<hqz<! gVkh<!Ohiukqz<jz/!
hbqx<sq 1.1 1. hqe<uVueux<xqx<G!ksl!uqiquqjeg<!gi{<g.
(i) 6421 (ii) −
611 (iii)
207− (iv)
2411
2. hqe<uVl<!uqgqkLX!w{<gjt!!LPuqe<!gQp<!LP!ucuqz<!gi{<g: (i) 0.34 9 (ii) −0. 7 (iii) 0.125 (iv) 0.5 21 (v) 8. 9
3. siqbi!nz<zK!kuxi!weg<!%Xg/!
(i) 72 we<hK!yV!LP!w{<.
(ii) − 11 we<hK!yV!LP/!
17
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(iii) 31 we<hK!yV!uqgqkLX!w{</!
(iv) 61.0 we<hK!yV!uqgqkLX!w{</ (v) yu<ouiV!ksl!uqiqUl<!yV!olb<ob{<!NGl</! (vi) yu<ouiV!olb<ob{<[l<!yV!uqgqkLX!w{<!NGl</ (vii) 0.1212212221… yV!uqgqkLX!w{</
1.2 olb<ob{<!OfIg<OgiM!
Le<!uGh<Hgtqz<!olb<ob{<!OfIg<Ogim<cje!nxqOuil</!olb<ob{<gt<!wu<uiX!OfIg<Ogim<ce<!Ht<tqgtigg<!Gxqg<gh<hMgqe<xe!we<hjk!!lQ{<Ml<!nxqf<K!ogit<Ouil</!
yV!OfIg<Ogim<cz<!WOkEl<!yV!Ht<tqjbg<!gVKg/!-h<Ht<tqjb O weh<!ohbiqMOuil</!w{< 0 J! -h<Ht<tq! Gxqh<hkigg<! ogit<Ouil</! Ht<tq! O-g<G! uzh<Hxlig! w{< 1Jg<!Gxqg<Gl<!Ht<tqbig!OfIg<Ogim<cz< A Ht<tqjbg<!gVKg/!OfIg<Ogim<Mk<K{<M OA NeK!YvzG! fQtLjmbK! we<g/! Ht<tq! A-jbh<! ohiVk<K! fQtl<! OA -Vg<Gl<<! we<hjk!fqjeuqz<!ogit<g/!Ht<tq A,!w{< 1!Jg<!Gxqg<gqxK!we<X!ogi{<m!hqe<H A fqjzbie Ht<tqbiGl<; OAe<! fQtl< 1! nzgiGl</ OA! J! ntUOgizigg<! ogi{<M! yu<ouiV!olb<ob{<{qjeBl<! ng<Ogim<cz<! yV! Ht<tqbigg<! Gxqg<gzil</! -k<kjgb!
OfIg<Ogim<jm! olb<ob{<! OfIg<OgiM! nz<zK! olb<g<OgiM! (Real Number Line)! we!njph<Ohil</!
olb<g<Ogim<cz<! wu<uiX! lqjg! LPg<gt<! Ht<tqgtigg<! Gxqg<gh<hMgqe<xe!
we<hjk!lQ{<Ml<!fqjeU!ogit<Ouil</ O lx<Xl< A LjxOb!w{<gt< 0 lx<Xl< 1 gjtg< Gxqg<gqe<xe/ O–g<G! uzh<hg<gk<kqz< 2 nzGk<! okijzuqZt<t! Ht<tq (OA-uqe<!fQtk<jkh<! Ohiz 2 lmr<G), 3 nzGk<! okijzuqz<!dt<t! Ht<tq, …, -ju!LjxOb!w{<gt<!2, 3, … Ngqbux<jxg<!Gxqg<Gl<!(!hml< 1.1 Jh<!hiIg<gUl<).
hml<!1.1
!
-u<uiOx, O-g<G!-mh<Hxlig − 1, − 2, −3,… w{<gjtg<!Gxqg<Gl<!Ht<tqgjt!LjxOb 1 nzG, 2 nzGgt<<, 3 nzGgt<, … okijzUgtqz<!Gxqg<gzil</!!!
uqgqkLX! w{<gjt! olb<g<Ogim<cz<! Gxqg<Gl<! Ljxjb! kx<OhiK!
okiqf<Kogit<Ouil</! wMk<Kg<gim<cx<gig, uqgqkLX! w{< 32 Jg<! gVKOuil</! uqgqkLX!
w{< 32 e<! hGkqbqZt<t (Denominator of
32 ) 3-Jg<!Gxqg<Gl<! Ht<tq! P we<g/ OP-g<G!
18
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osr<Gk<kig PQ we<El<! Ogim<cje!32 e<! okiGkq 2 (numerator of
32 ) nzG fQtk<kqz<!
ujvg/!OQ-jus<!OsIg<g/ PQ-g<G!-j{big! A-upqs<!osz<Zl<!Ofi<g<Ogim<cje!!ujvg/!-g<OgiM OQ J R we<x!Ht<tqbqz<!
sf<kqg<gqxK! we<g/! AR =32! nzGgt<!
NGl</! Woeeqz<? ∆OAR lx<Xl< ∆OPQ we<he!ucouik<k!Lg<Ogi{r<gt</!weOu,
zKnz<!zKnz< 132 ==
AROAOP
ARPQ
AR =32 .
O-ju! jlbligUl<> AR-g<G! sllie!fQtk<jk! NvligUl<! ogi{<M! yV! um<ml<!ujvg/! -u<um<ml<! olb<g<Ogim<jm O-g<G uzh<Hxk<kqz<!yV!Ht<tqbqz<!oum<Mgqe<xK/!!
hml< 1.2
-h<Ht<tqbieK uqgqkLX!w{< 32 Jg<!Gxqg<gqe<xK! (hml< 1.2!Jh<! hii<g<gUl<*/!nOk!
um<ml< O-g<G!-mh<Hxlig!olb<g<Ogim<jm!oum<Ml<! Ht<tpbieK 32− wEl<!uqgqkLX!
w{<{qjeg<! Gxqg<gqe<xK/! -Ok! Ljxbqz<?! wf<kouiV! uqgqkLX! w{<{qjeBl<!olb<g<Ogim<cz<! Gxqg<gzil</! olb<g<Ogim<cz< P, Q we<he! ouu<OuX, nVgijlh<!Ht<tqgt<! weqz<!nux<xqx<gqjmbqz<, P lx<Xl< Q gjtk<! kuqv! OuoxiV! Ht<tqbqjeBl<!gi{zil</! nkiuK! olb<g<Ogim<cz<! wf<kuqV! Ht<tqgTg<Gl<! -jmObBl<! -jmoutq!-z<jz! we<hjk! nxqgqe<Oxil</! weOu?! olb<g<OgiM! okimIs<sqh<! Ht<tqgtiz<!(continuum of points) NeK/! yu<ouiV! uqgqkLX! w{<{qx<Gl<! olb<g<Ogim<cz<! yV!keqk<k! Ht<tq! d{<M! we<hjk! -Kujvbqz<! hiIk<Okil</! Neiz<! olb<g<Ogim<cZt<t!njek<Kh<Ht<tqgTl<!uqgqkLX!w{<gjt!lm<MOl!Gxqg<Gli!we!uqei!wPh<HOuil</!
-kx<Giqb! uqjm “-z<jz” we<hKkie</! wMk<Kg<gim<mig, 2 we<x! uqgqkLxi!w{<j{g<!gVKOuil</!!!
hml<< 1.3
!!!
!!!!
hg<gl< OA = 1 weg<!ogi{<M!!sKvl< OABC ujvg/!hqkigv <̂!Okx<xh<hc?!!OB2 = OA2 + AB2 = 1+1 = 2.
weOu!OB = .2 O-ju!jlbligUl< OB J!NvligUl<!ogi{<M yV!um<ml<!ujvg!)hml<! 1.3 Jh<! hiIg<gUl<*/! -u<um<ml<, olb<ob{<! Ogim<cje O g<G! -mh<Hxl<! Ht<tq Q uqZl<, O-g<G!uzh<Hxl<!Ht<tq!P-bqZl<!sf<kqg<Gl</ OP = OB = 2 , OQ = OB = 2 !
we<hkiz<!Ht<tqgt< P, Q uqgqkLX!w{<gjtg<!Gxqg<giK/!NgOu!Ht<tq P, 2 wEl<!
uqgqkLxi!w{<{qjeBl<, Ht<tq Q, − 2 wEl<!uqgqkLxi!w{<{qjeBl<!Gxqg<Gl</!!
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wMk<Kg<gim<M 9: 3 , ,5 − 3 ,− 5 Ngqb!uqgqkLxi! w{<gjt!olb<ob{<! Ogim<cz<!Gxqg<gUl</!
kQIU: olb<ob{< 2 g<Giqb Ht<tq! P-J!olb<g<Ogim<cz<!Gxqg<gUl<. 3 lx<Xl< − 3 hqe<uVliX gi{zil<;
hml< 1.4
OP= 2 lx<Xl< PD = 1 wEliX! osu<ugl<!OPDC (hml< 1.4 Jh<! hiIg<gUl<*! ujvg/!hqkigv <̂!Okx<xh<hc, OD2 = OP2 + PD2 = ( 2 )2 + 12 = 3.
weOu OD = 3 . O-ju!jlbligUl<, OD J NvligUl<!ogi{<M!yV!um<ml<!ujvg/!-u<um<ml<! olb<g<Ogim<cje Q lx<Xl< Q′!gtqz<! sf<kqg<gm<Ml</! hmk<kqzqVf<K filxquK?!OQ = OQ′ = OD = 3 . Q lx<Xl< Q′!wEl; Ht<tqgt< O-g<G!uz!lx<Xl<!-mh<Hxr<gtqz<!
LjxOb! njlukiz<, Ht<tq Q! NeK 3JBl<?! Ht<tq Q′ NeK − 3 JBl<!Gxqg<gqe<xe/
5 , − 5 -ux<xqx<G{<mie!
Ht<tqgjtg<! Gxqg<g! Olx<%xqb!dk<kqjbg<! jgbit<Ouil</! olb<ob{<!
Ogim<ce<!lQK!O uqzqVf<K!2 nzGgt<!okijzuq<z<!O uqx<G!uzh<hg<gk<kqz<!R we<x! Ht<tqjbg<! Gxqg<gUl<!!!
(Wx<geOu! uqgqkLX! w{<gTg<G!d{<mie! Ht<tqgjt! olb<ob{<!Ogim<cz<! wu<uiX!gi{<hK!we<hjkg<!
g{<cVg<gqe<Oxil<)/!osu<ugl< ORFC J!!fQtl<!OR = 2, !ngzl<!RF = 1 we<xuiX!uhqkigv <̂!Okx<xh<hc,
OF2 = OR2 + RF2
weOu?!OF = 5 . O J!jlbliUl<, OFnu<um<ml<!olb<ob{<!Ogim<cje O uqx<G!-mh<hg<gk<kqz< S′ we<x!Ht<tqbqZl<!oum<MgS NeK! 5 JBl<?!S′ NeK − 5 JBl<!G
Olx<gi[l<!Nvib<uqzqVf<K, wf<koukeqk<k!yV!Ht<tqBme<!ohiVk<kh<hMgqxKOgim<ce<!lQKt<t!yu<ouiV!Ht<tqBl<!ke
nxqgqe<Oxil</!O uqx<G!uzh<hg<gk<kqz<!duqx<G! -mh<hg<gk<kqZt<t! Ht<tqgt<!
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hml< 1.5
!!!!!
jvg ( hml<!1.5 Jh<!hiIg<gUl<).
= 22 + 12 = 4 + 1 = 5.
J!NvligUl<! ogi{<M! yV! um<ml<! ujvg/!uzh<hg<gk<kqz<!S we<x!Ht<tqbqZl<, O uqx<G!qe<xK/ -r<G!OS = OS′ = OF = 5 we<hkiz<, xqg<gqe<xe/
iV!olb<ob{<[l<, olb<ob{<!Ogim<ce<!lQK!!we!nxqgqe<Oxil</!lXkjzbig?!olb<ob{<!qk<k!yV!olb<ob{<{qjeg<!Gxqg<Gl<!weUl<!
t<t! Ht<tqgt<! lqjg! olb<ob{<gjtBl<?!O Gjx! olb<ob{<gjtBl<! Gxqg<gqe<xe/!
20
olb<ob{<gt<! a lx<Xl<! b -ux<xqx<gie! olb<ob{<! Ogim<ce<! Ht<tqgt<! LjxOb! P lx<Xl<!Q we<g/!a < b weqz<!P NeK!Q e<!-mh<hg<glig!-Vh<hjk!nxqf<K!ogit<tUl</!nkiuK?! Q uieK! P g<G! uzh<hg<gk<kqz<! njlf<k! Ht<tq! we! nxqf<K! ogit<tUl<!)hml<! 1.6 Jh<! hiIg<gUl<*/! OlZl<?! a < x < b we<xuiXt<t! x we<x! olb<ob{<{qx<G!olb<ob{<!!
!
hml< 1.6
!!!!!
Ogim<ce<! lQK! ohiVk<kh<hm<m! Ht<tq! R weqz<?! R NeK! P g<Gl<! Q g<Gl<! -jmh<hm<m!Ht<tqbig!-Vh<hjk!nxqgqe<Oxil</ 1.2.1 uqgqkLxi!w{<gtqz<!g{g<gqmz<
2 , 3 , 5 , … Ohie<x!uqgqkLxi!w{<gt<!wu<uiX!dVuibqe!we<hjk!fqjeU!
%IOuil</! fil< x − 2 = 0 we<x!sle<him<cjek<!kQIU!gi{!uqjpOuioleqz<?! flg<G!!!!
x = 2 !kQIuigg<!gqjmk<kK/!-jkh<OhizOu!xn = r we<x!sle<him<cx<G!olb<!lkqh<H! x kQIuigh<!ohx!uqjpgqOxil<!we<g/!-r<G!r yV!uqgqkLX!w{<?!n yV!lqjg!LPuiGl</
ujg!1: r < 0, n = 2, 4, 6, … weqz<? xn = r we<xuiX!yV!lqjg!olb<ob{<!x gi{!LcbiK/!Woeeqz<?!yV!olb<ob{< x x<G?! x2 = x × x > 0, x4 = x2 × x2 > 0, …, xn > 0; Neiz< r < 0 .
ujg!2: r > 0, n = 2, 4, 6, … weqz<?!xn = r we<xuiX!yV!lqjg!olb<ob{<!x gi{!-bZl</!wMk<Kg<gim<mig? x4 = 625 g<G!x = 5 gi{zil</!
ujg!3: r > 0, n = 1, 3, 5, … weqz<?!xn = r we<xuiX!yV!lqjg!olb<ob{<!x gi{!LcBl</!wMk<Kg<gim<mig? x3 = 64 g<G!x = 4 gi{zil</!!
ujg!4: r < 0, n = 1, 3, 5, … weqz<?!xn = r we<xuiX!yV!lqjg!olb<ob{<!x gi{!LcbiK/!Woeeqz<?!yV!lqjg!olb<ob{<!x g<G!x > 0, x2 = x × x > 0, x3 = x2 × x > 0, … xn > 0; Neiz<!r < 0.
Olx<gi[l<!Nvib<uqzqVf<K?!fil<!nxquK?!n yV!lqjg!LP! (n = 1, 2, 3, …) lx<Xl< r!yV!lqjg!uqgqkLX!w{<!weqz<?! xn = r we<xuiX!yV!lqjg!olb<ob{<! x gi{!-bZl</! -f<fqjzbqz<! ohxh<hMl<! lqjg! olb<ob{<! x J! n r we! wPKgqOxil<;
nkiuK?!xn = r ⇒!x = n r ; OlZl<!x J!r e<!n uK!&zl<!we<gqOxil</!-r<G n r yV!uqgqkLX! w{<{igOui! nz<zK! uqgqkLxi! w{<{igOui! -Vg<Gl</! wMk<Kg<gim<mig?
,4643 = .244 = Gxqh<hig?!fil<!gueqh<hK?!
21
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ujg! 1: r NeK!qp we<x! yV! lqjg! ! uqgqkLX! w{<{qe<! n uK!&zlig!
-Vh<hqe<, nkiuK?!n
qpr ⎟⎟⎠
⎞⎜⎜⎝
⎛= weqz<?! n r =
qp yV!uqgqkLX!w{<{iGl</!
ujg! 2: r NeK! wf<kouiV! lqjg! uqgqkLX! w{<{qe<! n uK! nMg<gig!-z<jzobeqz<?! n r yV! uqgqkLX! w{<{ig! -Vg<g! LcbiK; Woeeqz<?! n r yV!
uqgqkLX! w{<! qp weqz<?!
n
qpr ⎟⎟⎠
⎞⎜⎜⎝
⎛= ?! ! yV! uqgqkLX! w{<o{e! Lv{<himig!
gqjmg<gqxK/! weOu?! yV! lqjg! uqgqkLX! w{<! r NeK! OuX! wf<k! yV! uqgqkLX!w{<{q<e<!n NuK!nMg<gig!-Vg<guqz<jzobeqz<?! n r NeK!yV!uqgqkLX!w{<{ig!
-Vg<giK; nkiuK?! n r yV!uqgqkLxi!w{<!NGl</!!
r! yV! lqjg! uqgqkLX! w{<{ig! -Vf<K?! n r we<x! lqjg! olb<ob{<?! yV!
uqgqkLxi!w{<{ig!-Vh<hqe<?! n r J!yV!LVm<om{<! )surd) nz<zK!yV!uqgqkLxi!&zl<!we<gqOxil</!!!
Gxqh<H: yV!LVm<om{<! we<hK!Gxqh<hqm<m!ucuqz<!njlf<k!uqgqkLxi! w{<{iGl</! r!NeK!wf<kouiV!uqgqkLX!w{<{qe<!n!NuK!nMg<gig!-z<jz!we<xOhiK?! n r yV!
uqgqkLxi!&zliGl</!-r<G!n!we<hK!uqgqkLxi!&zl<! n r e<<!Gxqob{<!we<gqOxil</!n !we<x!GxqbieK!&zg<Gxq (Radical) we<gqOxil</!Gxqob{<!n = 2 weqz<, 2 r J!r e<!
uIg<g&zl<!we<xjpk<K!nkjes<!SVg<glig! r we<Ox!wPKgqe<Oxil</! n = 3 weqz<, 3 r J!r!e<!ge!&zl<!we<gqOxil</!
LVm<om{<gt<!olb<ob{<gt<!we<hkiz<?!nux<Xme<!-bx<g{qk!osbzqgt<!+, −, ×, ÷ -ux<jxs<! osbz<hMk<kzil</! -V!uqgqkLxi!&zr<gjtg<!%m<m! yV!uqgqkLxi!w{<! gqjmh<hqe<?! nf<k! uqgqkLxi! w{<{qjeBl<! yV! LVm<om{<! we<Ox!njpg<gqe<Oxil</!-u<uiOx?!yV!LVm<om{<{qje?!yV!uqgqkLX!w{<{qeiz<!ohVg<g!yV! uqgqkLxi! w{<! gqjmh<hqe<?! nf<k! uqgqkLxi! w{<{qjeBl<?! yV! LVm<om{<!
we<Ohil</! wMk<Kg<gim<<mig?! 3 2 ?! 5 7 ?! 2 ! we<he! LVm<om{<gt</! 23 ,72 53 −+ we<he!uqgqkLxi!w{<gt</!-ux<jxBl<!fil<!LVm<om{<gt<!we<xjpg<gqe<Oxil</!!
a lx<Xl< b we<he! -V! ouu<Ouxie! uqgqkLX! w{<gt<! we<g/! a , b
-v{<MOl!uqgqkLxi!&zr<gt<! )uIg<g!&zr<gt<*!we<g/!a + b , a − b , a + b , a − b Ngqbju! uqgqkLxi! &zr<gtiGl</! -r<G a − b NeK! a + b e<!
Kj{bqb!uqgqkLxi!&zl<!we<gqOxil</!-u<uiOx?! a + b NeK a − b e<!Kj{bqb!&zl<, a + b NeK! a − b e<!Kj{bqb!&zl<?! a − b NeK! a + b e<!Kj{bqb!&zl<!we!nxqgqe<Oxil</
22
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uqgqkLxi! &zk<jkBl<?! nke<! Kj{bqb! uqgqkLxi! &zk<jkBl<! ohVg<gg<!gqjmh<hK!yV!uqgqkLX!w{<{iGl</!wMk<Kg<gim<mig?
( ) ( ) ( ) ( ) 134332322323222 =−=−+−=−+ lx<Xl<
( ) ( ) ( ) ( ) .253515153535322
−=−=−+−=−+ !uqgqkLxi!&zr<gjtg<!jgbit<ukx<G!hqe<uVl<!uqkqgjth<!hbe<hMk<KOuil</
a, b we<he!lqjg!uqgqkLX!w{<gt<<;!m, n we<he!lqjg!LPg<gt<!weqz<?
(i) ( ) .aann =
(ii) ( ) ( ) .nnn abba =
(iii) .nn
n
ba
ba=
(iv) p n r + q n r = ( p + q) n r , -r<G!p lx<Xl< q -v{<MOl!olb<ob{<gt</
(v) mnn m aa .=
(vi) =n a .n m ma (vii) a < b weqz<, nn ba < NGl</!
fil<!gueqh<hK?!uqkqgt< (ii), (iii) lx<Xl< (iv) z<, uqgqkLxi!&zr<gt<!njek<Kl<!
sllie! Gxqob{<! ogi{<mjubiGl</! weOu?! ouu<OuX! Gxqob{<gt<! ogi{<m!uqgqkLxi! &zr<gt<! kvh<hm<M! nux<jxg<! ogi{<M! fie<G! nch<hjms<! osbzqgjt!fqgp<k<k! Ou{<Moleqz<?! nf<k! uqgqkLxi! &zr<gjt! sl! Gxqob{<! djmb!
&zr<gtig,!uqkq!(vi) Jg<<!ogi{<M!lix<xqb!hqxG!osbz<hm!Ou{<Ml</!uqkq!(vi) NeK!uqkq!(i), uqkq!)v*!-ux<xqe<!-j{h<OhbiGl<!we<hjk!hqe<uVliX!nxqbzil<:!!
uqkq!(i) e<!hc?!a = m ma . ∴ n m mn aa = . uqkq (vii) Jg<!ogi{<M?!sllie!Gxqob{<!djmb!uqgqkLxi!&zr<gjt!yh<hqmzil</
Gxqh<H: n a J! na1
weg<!Gxqh<Ohil</ wMk<Kg<gim<M 10: hqe<uVue!uqgqkLxi!&zr<gti?!-z<jzbi!we<hjk!giv{r<gTme<!uqjmbtqg<gUl</
(a) 225121 (b)
2549 (c)
57
(d) 72
144 (e) 3 216
32 (f) 33 164 ×
kQIU:
(a) 225121 =
1511
1511 2
=⎟⎠⎞
⎜⎝⎛ , yV! uqgqkLX! w{</ ∴
225121 , yV! uqgqkLxi!
&zle<X/!
23
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(b) =2549 2
57⎟⎠⎞
⎜⎝⎛ =
57 , yV! uqgqkLX! w{</ ∴
2549 , yV uqgqkLxi!
&zle<X/
(c) 57 yV! uqgqkLxi! &zliGl</! Woeeqz<?
57 NeK! yV! uqgqkLX!
w{<{qeK!uIg<glz<z/
(d) 272
144= = yV!uqgqkLxi!&zliGl</
(e) 3 216
32 = =×××
3 666216 .
322
624
6216
3 3=
×=
×
322 yV!uqgqkLxi!w{<!we<hkiz<,
3 21632 yV!uqgqkLxi!&zliGl</
(f) 44444164164 3 33333 ==××=×=× , yV!uqgqkLX!w{<. ∴ 3 4 × 3 16 yV!uqgqkLxi!&zlz<z/
uqgqkLxi! &zr<gjt?! +, −, ×, ÷ ogi{<M! jgbiTl<ohiPK! gqjmg<Gl<!
uqjtuieK!yV!uqgqkLX!w{<{ig!-Vg<gzil</ wMk<Kg<gim<M 11: hqe<uVue!uqgqkLxi!&zr<gti?!-z<jzbi!we<hjk!giv{r<gTme<!uqjmbtq:
(i) ( ) ( )33 234545 −++ (ii) ( ) ( )3332 +−+ (iii) ( )( )323 243 −+ (iv) 4 432 ÷ 4 2187
kQIU: (i) ( ) ( )33 234545 −++ = (5 + 4) + ( )33 23227 −×
= 9 + ( )333 3 2323 −× = 9 + ( )33 2323 ×−× = 9.
-K!yV!uqgqkLX!w{<?!nkiuK?! 5 + 3 54 , 4 – 3 3 2 Ngqb!-v{<Ml<! uqgqkLxi!&zr<gtig!-Vh<hqEl<!nux<xqe<!%Mkz<!yV!uqgqkLxi!&zlz<z/!weOu!uqgqkLxi!&zr<gt<!%m<mjzh<!ohiVk<K!njmUh<!h{<hqje!fqjxU!osb<buqz<jz/
(ii) (2 + )3 − (3 + )3 = (2 − 3) + ( )33 − = −1 + 0 = −1 , yV!uqgqkLX!w{</ ∴ kvh<hm<m!Ogiju!yV!uqgqkLxi!&zlz<z/!
(iii) ( )( )323 243 −+ = (3 + 4 )2 (3 − )216× = (3 + 4 )2 (3 − 4 )2 = 221632424333 ××−×+×−× = 322122129 −+− = 9 − 32 = −23.
-K!yV!uqgqkLX!w{</!weOu?!kvh<hm<m!OgijubieK!yV!uqgqkLxi!&zle<X/
24
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(iv) 432 = 2 × 2 × 2 × 2 × 27 = 24 × 27. ∴ 44 .272432 = 2187 = 3 × 3 × 3 × 3 × 27 =34 × 27. ∴ 44 .2732187 =
∴ .32
2732722187432
4
444 ==÷
-K!yV!uqgqkLX!w{</!weOu?!kvh<hm<m!OgijubieK!yV!uqgqkLxi!&zlz<z/ wMk<Kg<gim<M 12: hqe<uVueux<Xt<! yu<ouie<jxBl<! yV! keqk<k! uqgqkLxi! &zlig!wPkUl<;!
(i) 8022520 +− (ii) 33 135240 − (iii) 50272 6 + (iv) 33 20001283 −
kQIU: (i) 52545420 =×=×= , 225 = 259× = 259 × = 3 × 5 = 15,
80 = 516× = 54516 =× .
∴ 5415528022520 +−=+− = 6 .155 −
(ii) 33333 52585840 =×=×= lx<Xl<! 33333 53527527135 =×=×= .
∴ .545)62(53252135240 333333 −=−=×−=−
(iii) 333327 22 3 332 36 ==== × lx<Xl< 2525250 =×= .
∴ 50272 6 + = 2532 + .
(iv) 3 128 = 3 3 24 × = 3 24 lx<Xl<! 33 33 2102102000 =×= .
∴ 3 ( ) 3333 2102432000128 −=− = 12 33 2102 − = (12 − 10) 3 2 = 2 3 2 .
wMk<Kg<gim<M 13: hqe<uVueux<Xt<! yu<ouie<jxBl<! yV! keqk<k! uqgqkLxi! &zlig!wPkUl<:
(i) 7 × 3 6 (ii) 3 5 × 4 3 . (iii) 8 3 4 ÷6 4 2 (iv) 3 81 ÷ ( )205 +
kQIU: (i) -r<G! 7 NeK!Gxqob{<!2!djmbK/! 3 6 NeK!Gxqob{<!3!djmbK/ 2, 3 -ux<xqe<! lQ/ohi/l! = 6. weOu? 7 , 3 6 Ngqb! -v{<jmBl<! Gxqob{<! 6 djmb!
uqgqkLxi!&zr<gtig!wPKOuil</!-r<G! 7 = 2 7 = ,3437 62 3 3 =
25
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3 6 = 63 2 2 .366 = weOu, 7 × 3 6 = 666 3634336343 ×=× = 6 12348 .
(ii) 3 5 × 4 3 = 12 625 × 12 27 = 12 27625× = 12 16875 .
(iii) 8 3 4 ÷ 6 4 2 = 4
3
2648 = 1212
12
12
.3234
8256
34
8256
34
=×=×
(iv) 3 81 ÷ ( )545
327205
8120533
×+×
=+
=+ = ===×
6
633
1259
53
5333 6
1259 .
!
wMk<Kg<gim<M 14: 1264 25,10,3 we<heux<jx! nux<xqe<! lkqh<Hgtqe<! WXuiqjsbqz<!njlg<gUl</!kQIU: Lkzqz<?! kvh<hm<m! uqgqkLxi! w{<gTg<gie! ohiKuie! Gxqob{<! gi{<Ohil</!kvh<hm<m! uqgqkLxi!&zr<gtqe<! Gxqob{<gt<! 4, 6, 12. -ux<xqe<! lQ/ohi/l! =12 NGl</!weOu! kvh<hm<m! uqgqkLxi!&zr<gjt!Gxqob{<! 12 dt<t! uqgqkLxi!&zr<gtig!lix<x!Ou{<Ml</!
124 34 273333 =××= 126 26 100101010 =×= . ∴kvh<hm<m!uqgqkLxi!&zr<gt< .25 ,100 ,27 121212 -h<ohiPK!uqkq!(vii) Jh<!hbe<hMk<k?! 121212 .1002725 << nkiuK?! 6412 .10325 << !
sqz!slbr<gtqz<?!uqgqkl<!yx!z<!dt<t!hGkq!y NeK yV!uqgqkLxi!&zlig!
-Vg<gzil</!nh<hch<hm<m!uqgqkr<gtqz<?!yV!kGf<k!Ljxjbg<!ogi{<M!hGkqjb!yV!uqgqkLX!w{<{ig!lix<xzil</!-l<Ljxjb?!hGkqjb!uqgqkLX!w{<{ig<Gl<!Ljx!we<xjpg<gqe<Oxil</!-l<Ljxjb!hqe<uVliX!kVgqe<Oxil<:
(i) hGkq!y NeK! a we<x!njlh<hqz<!-Vf<K?!a yV!uqgqkLX!w{<!weqz<?!okiGkq?!hGkq!-v{<jmBl< a Nz<!ohVg<gUl</!wMk<Kg<gim<mig,
53 = =
××
5553
515 .
(ii) hGkq! y NeK a + b we<x! njlh<hqz<! -Vf<K?! a lx<Xl< b uqgqkLX!w{<gt<!weqz<?!okiGkq?!hGkq!-v{<jmBl< a − b Nz<!ohVg<g!Ou{<Ml</!-h<ohiPK?!
hGkq = (a + b ) (a − b ) = ( ) babbabaa −=−+− 222 , yV! uqgqkLX! w{</ wMk<Kg<gim<mig,
22 )2(32232
)23)(23()23(2
232
−×−×
=−+
−×=
+ = .
7226
29226 −=
−−
(iii) hGkq! y NeK a − b we<x! njlh<hqz<! -Vf<K?! a lx<Xl< b uqgqkLX!w{<gt<!weqz<?!okiGkq?!hGkq!-v{<jmBl<!a + b Nz<!ohVg<g!Ou{<Ml</!-h<ohiPK?!
hGkq!= (a − b ) (a + b ) = ( ) babbabaa −=−+− 222 , yV!uqgqkLX!w{</!
26
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wMk<Kg<gim<mig?!!22
2
)2(3)23(
)23)(23()23()23(
2323
−+
=+−
+×+=
−+
= ( )7
26117
226929
2232322 +
=++
=−
+××+ .
(iv) hGkq y NeK! ba + !we!-Vf<K?!a lx<Xl< b -ju!uqgqkLX!w{<gt<!weqz<?!okiGkq?!hGkq!-v{<jmBl<! ba − Nz<!ohVg<g!Ou{<Ml</!-h<ohiPK?!
hGkq ,)()())(( 22 babababa −=−=−+= yV!uqgqkLX!w{</!wMk<Kg<gim<mig?!
( )( )( )
( )( ) ( )22
53
5325353
53253
2
−
−=
−+−×
=+
= ( ) ( ) ( ) .355312
53253
532−=−−=
−−
=−−
(v) hGkq y NeK! ba − we<x! njlh<hqz<! -Vf<K?! a, b -ju! uqgqkLX!w{<gt<! weqz<?! okiGkq?! hGkq! -v{<jmBl<! ba + Nz<! ohVg<g! Ou{<Ml</!
-h<ohiPK?! hGkq! ,)()())(( 22 babababa −=−=+−= yV! uqgqkLX! w{</!wMk<Kg<gim<mig?
( )( )( )
( )( ) ( )22
2
75
7557575
75575
5
−
+=
+−+×
=−
= ( )35521
75355
+−=−
+ .
wMk<Kg<gim<M 15: 145
1+
e<!hGkqjb!uqgqkLX!w{<{ig!lix<xUl</
kQIU: 145
1+
= ( )( )( ) ( ) ( )22
145
145145145
1451
−
−=
−+−×
= ( )51491
9145
145145
−=−−
=−− .
Gxqh<H: yx we<x!uqgqkk<kqe<! okiGkq! x NeK! a nz<zK! ba + nz<zK! ba −
nz<zK! ba + nz<zK! ba − we<x! njlh<hqz<! dt<tK/! x J! uqgqkLX!w{<{ig! lix<x! Ou{<Moleqz<?! okiGkqbqe<! Kj{bqb! w{<{qeiz<! okiGkq, hGkq!-v{<jmBl<!ohVg<g!Ou{<Ml</
wMk<Kg<gim<M 16: 4
311 − e<!okiGkqjb!uqgqkLX!w{<{ig<Gg/
kQIU: 4
311 − = ( ) ( )( )3114
311311+×
+×− = ( ) ( )( )3114
31122
+×−
= ( )3114311+− = ( )3114
8+
= 311
2+
.
27
www.kalvisolai.com
wMk<Kg<gim<M 17: 321
1+−
= a + b 2 + c 6 weqz<, a + b + c Jg<!g{<Mhqcg<gUl</
kQIU:321
1+−
= ( )( )( )321321
3211−−+−
−−×
= ( )( ) ( )22
321
321
−−
−−
= ( )( ) 32221
321−+−
−−
= ( ) ( )22
22321
22321
×−
−−=
−−−
= 4
6224
23222−−−
=−
−−
= .6412
41
21
4622
+−=+−
∴ 6412
41
2162 +−=++ cba .
∴ .41,
41,
21
=−
== cba
∴ .21
41
41
21
=+−=++ cba
Gxqh<H;! p, q uqgqkLX! w{<gt<? n a yV! uqgqkLxi! &zl<! weqz<?! p+ n aq we<x!
uqgqkLxi!&zk<kqz<, p J!uqgqkLX!w{<!hGkq!we<Xl<?! n aq J!uqgqkLxi!w{<!hGkq!we<Xl<! njpg<gqe<Oxil</! -V! uqgqkLxi! &zr<gt<! sllibqVg<g?! nux<xqe<! uqgqkLX!w{<! hGkqgt<! sllibqVg<g! Ou{<Ml<! lx<Xl<! nux<xqe<! uqgqkLxi! w{<! hGkqgt<!sllibqVg<g!Ou{<Ml</
wMk<Kg<gim<M 18: 31313
1313 yx +=
−+
++− weqz<?!x2 + y2 Jg<!g{<Mhqc/
kQIU: ( ) ( )( ) ( )
( )( ) 13
1323
13
132313131313
1313
22
2
−+−
=−
+−=
−+−−
=+− = 32
2324
−=− .
∴ ( )( )( ) ( ) .32
3432
32
323232
32132
11313
22+=
−+
=−
+=
+−+×
=−
=−+
( ) ( ) 432321313
1313
=++−=−+
++− = 304 + .
28
www.kalvisolai.com
∴ 3043 +=+ yx
∴ x = 4, y =0
∴ x2 + y2 = 16 + 0 = 16.
wMk<Kg<gim<M 19: x = 2323
−+ weqz<?!
xx 1+ Jg<!g{<Mhqcg<gUl</!
kQIU: x =2323
−+ =
( )( )( )( )2323
2323+−++ = ( )
( ) ( )22
2
23
23
−
+
= .231
232323
+=+
=−+
∴ ( )( )( )23
2323
123
11−−
×+
=+
=x
= ( )( ) ( )
( ) 231
2323
23
23
2322 −=
−=
−−
=−
−
∴ ( ) ( ) .3223231=−++=+
xx
wMk<Kg<gim<M 20: a =223
223
−
+ weqz<?!a2 (a− 6)2e<!lkqh<hqjeg<!g{<Mhqc/
kQIU: a = ( )( )
( )( )223
223223223
++
×−+ = ( )
( ) 22389223
243
22322
2
+=−
+=
−
+
∴ a − ( ) .22362236 +−=−+=
∴ a2(a − ( ) ( )222 223223)6 +−+= = ( )( )[ ]2322 322 −+
= ( ) 222
322 ⎥⎦⎤
⎢⎣⎡ − = ( ) .198 2 =−
wMk<Kg<gim<M 21: 2 ≈ 1.414 weqz<?!1212
−+ e<!Okivib!lkqh<hqjeg<!g{<Mhqcg<gUl</!
kQIU: 1212
1212
1212
++
×−+
=−+ = ( )
( ) 22
2
12
12
−
+ = ( ) ( )22
121212
+=−+ .
∴ 1212
−+ = ( ) 1212
2+=+ = 1.414… + 1 = 2.414…
!
29
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hbqx<sq 1.2 1. hqe<uVue!uqgqkLxi!&zr<gti!-z<jzbi!we<hjk!giv{r<gTme<!uqjmbtq/
(i) 3 16 (ii) 42 (iii) 5 729 (iv) 4 2 4 2 (v) 50 2 2. hqe<uVue!yu<ouie<jxBl<!SVg<gucuqz<!wPkUl<;
(i) 25 + 32 (ii) 33 40320 −
(iii) ( ) ( )23343225 +− (iv) ( ) ( )45751220 +−
(v) 553 (vi) 3 ÷ 3 6
(vii) 155 124 ÷ (viii) 34 53 ÷ (ix) 5253 ÷ 3. WXuiqjsbqz<!njlg<gUl<;
(i) ,3 3 5 , 6 11 (ii) ,5 3 7 , 4 9 (iii) 3 2 , 5 , 4 3 4. hqe<uVue yu<ouie<xqe<!hGkqjb!uqgqkLX!w{<{ig!lix<xUl<:
(i) 6
18 (ii) 321
4+
(iii) 1515
−+
(iv) 53
1+
(v) 3232
+− (vi)
22333223
+−
5. hqe<uVue!yu<ouie<xqZl<!x lx<Xl<!y !-ux<jxg<!gi{<g/
(i) 3232
−+ = x + y 3 (ii) 3
354345 yx +=
++
(iii) 55353
5353 yx +=
+−
+−+ (iv) 35
7575
7575 yx +=
−+
−+−
6. a = 3232
−+
weqz<?!a2 (a − 4)2 e<!lkqh<jhg<!g{<Mhqcg<gUl</!
7. a = 1212
−+ weqz<?!a2 + 2
1a
e<!lkqh<jhg<!g{<Mhqcg<gUl</
8. 3 ≈ 1.732 weqz<?! 1313
−+ e<!Okivib!lkqh<jhg<!g{<Mhqcg<gUl</
9. 2 ≈ 1.414 , 3 ≈ 1.732 weqz<? 32
1+
!e<!Okivib!lkqh<jhg<!g{<Mhqcg<gUl</
30
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uqjmgt<!
hbqx<sq 1.1
1. (i) 0.328125 (ii) − 38.1 (iii) − 0.35 (iv) 3458.0
2. (i) 207 (ii)
97
− (iii) 999125 (iv)
495258 (v)
19
3. (i) kuX (ii) siq (iii) siq (iv) siq (v) siq (vi) kuX (vii) kuX!
hbqx<sq 1.2 1. (i) LVm<om{< (ii) LVm<om{<!nz<z!! (iii) LVm<om{<
(iv) LVm<om{< (v) LVm<om{<!nz<z 2. (i) 29 (ii) 3 52 (iii) 6614 + (iv) 154 (v) 6 3125
(vi) 6
43 (vii) 15
316 (viii) 12
62527 (ix) 6 5
3. (i) 6 11 , 3 5 , 3 (ii) 4 9 , 3 7 , 5 (iii) 3 2 , 4 3 , 5 .
4. (i) 63 (ii) ( )132114
− (iii) ( )5321
+
(iv) ( 3521
− ) (v) 562 − (vi) ( )30613191
−
5. (i) x = 7, y = 4 (ii) x = ,5940
599
=y
(iii) x = 7, y = 0 (iv) x = 0, y = 2 6. a2 (a − 4)2 = 1 7. 34 8. 3.732 9. 0.318
31
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2. ntjugt<!
!fl<!ne<xim!uip<uqz<!hz<OuX!fqjzjlgtqz<!fil<!ntjugt<!!osb<gqe<Oxil</!
wMk<Kg<gim<mig? fil<! K{q! jkh<hkx<gig! fQtk<jkBl<? out<jtbch<hkx<gig! Sux<xqe<!hvh<htuqjeBl<? OuzqbqMukx<gig!fqzk<kqe<!Sx<xtuqjeBl<? fqvh<Hukx<gig!hik<kqvk<kqe<!ogit<ttuqjeBl<! ntg<gqe<Oxil</! ntjugjt! nch<hjmbigg<! ogi{<M! fl<!Okjug<Ogx<h! OlZl<! g{g<gQMgt<! osb<gqe<Oxil</! kt! lx<Xl<! ge! dVur<gtqe<!fQtr<gt<?! Ogi{r<gt<? hvh<htUgt< Sx<xtUgt<! lx<Xl<! ge!ntUgt<! hx<xqb!g{qkh<!hqiquqje! ntuqbz<! we<xjpg<gqe<Oxil</! Lf<jkb! uGh<Hgtqz<! fil<?! Lg<Ogi{r<gt<? fix<gvr<gt<! lx<Xl<! um<mr<gt<! Ohie<x! sqz! kt! ucu! dVur<gjth<! hx<xq!gx<xxqf<Kt<Otil<! )wz<zi! ucu! dVur<gTl<! ktk<kqz<! ujvbh<hm<mjubigg<!ogit<Ouil<*/! -f<k!nk<kqbibk<kqz<? -v{<M!nz<zK!nkx<G! Olx<hm<m!Lg<Ogi{r<gt<? fix<gvr<gt<!nz<zK!um<mr<gt<!-ux<jx!ye<xe<Olz<!ye<xig!fqXk<kq!gqjmg<gh<ohXl<!sqz! kt! %m<M! dVur<gjth<! hx<xq! gx<Ohil</! fil<! gVKl<! njek<K! dVur<gTl<!ktk<kqz<!njlf<kju!we<hkiz<?!kt!dVuk<jk!dVul<!we<X!njph<Ohil<. 2.1! hvh<H!lx<Xl<!Sx<xtU!
fil<?!kt!ucuqbz<!dVur<gtqe<!Sx<xtUgt<!lx<Xl<!hvh<Hgtqe<!uib<h<hiMgjt!
lQt<hiIjubqMOuil</!!2.1.1 osu<ugl;:
hvh<H!= l × b!sKv!nzGgt< Sx<xtU = 2 (l + b) nzGgt< d = 22 bl + !nzGgt<
2.1.2! -j{gvl;:
hvh<H = b × h sKv!nzGgt< Sx<xtU = 2(a + b) nzGgt<
2.1.3 ogiMg<gh<hm<m!nch<hg<gl<!lx<Xl<!
!hvh<H = 21 b × h sKv!nzGgt<.
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hml<!2.1
hml< 2.2
dbvk<jkBjmb!Lg<Ogi{l<;
3
hml<!2.3
2
2.1.4! osr<Ogi{!Lg<Ogi{l<;
hvh<H = 21 b × h sKv!nzGgt< hml< 2.4
Sx<xtU = b + h + d!nzGgt< d = 22 hb + !nzGgt<
2.1.5 slhg<g!Lg<Ogi{l<;
dbvl< = h = 23 a!nzGgt<
hml<!2.5 hvh<H =
43 a2!sKv!nzGgt<
Sx<xtU = 3a!nzGgt< 2.1.6! -Vslhg<g!Lg<Ogi{l<:
hml< 2.6 hvh<H!= h 22 ha − !sKv!nzGgt<
Sx<xtU = 2 ( )22 haa −+ !nzGgt< 2.1.7! nslhg<g!Lg<Ogi{l;:
hvh<H = ))()(( csbsass −−− s/n
-r<G 2
cbas ++= nzGgt<! ! hml< 2.7
Sx<xtU = a + b + c nzGgt< 2.1.8 siqugl<;
hvh<H = 21 (a + b) × h sKv!nzGgt< hml< 2.8
2.1.9! fix<gvl<;
!hvh<H!= 21 d × (h1 + h2).
sKv!nzGgt< 2.1.10!sib<sKvl<;
hvh<H!= 21 d1 × d2!s/nzGgt<
Sx<xtU = 2 22
21 dd + = 4a nz
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Ggt
3
hml<!2.9
<
3
hml< 2.10
2.1.11!um<ml<: um<mk<kqe<!hvh<H = πr2!sKv!nzGgt< um<mk<kqe<!Sx<xtU = 2πr!nzGgt<
njvum<mk<kqe<!hvh<H= 21πr2!sKv!nzGgt<
njvum<m!uqz<zqe<!fQtl< = πr!nzGgt<
giz<um<mk<kqe<!hvh<H = 41πr2!sKv!nzGgt<
giz<um<m!uqz<zqe<!fQtl<!= 21 πr!nzGgt<
Gxqh<H: A lx<Xl<!B Ht<tqgjt!-j{g<Gl<!Ofi<g<Ogim<Mk<!weg<!Gxqg<gqe<Oxil</!OlZl< AB!we<Ox AB e<!fQtk<jkBl<! wMk<Kg<gim<M 1: yV! osu<ug! ucu! SuI?! nch<hg<g! fQtlweUl<! ogi{<Mt<tK/! ns<Sux<xqx<G! u{<{l<! H,s!oszuiGoleqz<? olik<k!Sux<xqx<Gl<!u{<{l<!H,s!NGl<!oskQIU: b = 15 ? h = 10!NGl</!!∴ osu<ugh<!hvh<H!= b × h = 15 × 10 = 150 s/lQ. 1 s/lQ!u{<{l<!H,s!oszU = !'/ 16 ∴ 150 s/lQ!u{<{l<!H,s!oszU = 16 × 150 = '. 2400. !wMk<Kg<gim<M! 2: yV!osu<ugk<! kgm<ce<!ntUgt<! 5lQ! ×!hg<g! fQtl<! ogi{<m! sKvk<! kgMgtig! oum<m! Ouw{<{qg<jgjbg<!g{<Mhqc/!kQIU: osu<ugk<!kgm<ce<!hvh<H = 400 × 300 = 12?0000!os/lQ2. yV!sKvk<!kgm<ce<!hvh<H = 4 × 4 = 16 os/lQ2.
∴ sKvk<!kgMgtqe<!w{<{qg<jg = 160000,12 = 7500.
wMk<Kg<gim<M! 3;! 40!os/lQ2! hvh<Hjmb!-j{gvk<kqe<!dbnch<hg<gl<!g{<Mhqc/!kQIU:!hvh<H = b × h. ∴ 40 = b × 15.
∴ b = 38
1540
= .
∴ nch<hg<gl<!= 38 os/lQ.
34
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hml<!2.11!
K{<cje AB !nz<zK!AB Gxqg<gqe<Oxil</!
<! 26lQ! weUl<! dbvl<! 21lQ!sKv! lQm<mVg<G! '/27!!
zjug<!g{<Mhqc/!
hml<!2.12
4lQ/!nk<kgm<cje! 5! os/lQ!{<Ml</! sKvk<! kgMgtqe<!
v
h
hml< 2.13
l<! 15!os/lQ! weqz<?!nke<!
ml< 2.14
wMk<Kg<gim<M! 4: 11 os/lQ? 60 os/lQ! lx<Xl<! 72! os/lQ! hg<g! fQtr<gt<! ogi{<m! yV!Lg<Ogi{k<kqe<!hvh<jhBl<? Sx<xtjuBl<!g{<Mhqc/!
hml<!2.15
kQIU:!hvh<H = ))()(( csbsass −−− . -r<G 2s = a + b + c = 11 + 60 + 61 = 132. ∴ s = 66? s − a = 66 − 11 = 55? s − b = 66 − 60 = 6? s − c = 66 − 61 = 5. ∴ hvh<H = 565566 ××× = 330!s/os/lQ. Sx<xtU = a + b + c = 11 + 60 + 61 =132!os/lQ.
hml< 2.16
wMk<Kg<gim<M 5:! hml<! 2.16! z<! ogiMg<gh<hm<Mt<t! fix<gvl<! ABCD e<! hvh<hqjeg<!g{<Mhqc/!
kQIU:!hvh<H = )(21
21 hhd + = )2010(5021
+××
= 25 × 30 = 750 lQ 2/ wMk<Kg<gim<M 6:!hml<!3/28!z<!ogiMg<gh<hm<Mt<t!siqugl<!ABCD!e<!hvh<H!g{<Mhqc/!!
kQIU: hvh<H!= hba ×+ )(21
= 4)512(21
×+
= 34 sKv!nzGgt</!!!wMk<Kg<gim<M 7: fqzk<jk! slh<hMk<k! sKfqzk<kqe<! -j{! hg<gr<gtqe<! fQtr<gt<!yu<ouie<Xl<! 6lQ! fQtLjmbK! weqz<?! olg{<Mhqc/ kQIU: ABCD!ogiMg<gh<hm<m! siqugl</!-rDA!g<G -j{big!CE!J!ujvg!)hml<!
Lg<Ogi{l</!nke<!dbvl<!h = 22 35 −
= 16 = 4!lsiqugl<!ABCD!bqe<!hvh<H!
= 21 (a + b) × h =
21 (18 + 12) × 4
= 2 × 30= 60 s/lQ/ 1!s/lQ!slh<hMk<k!NGl<!oszU =!'/12. ∴olik<k!fqzk<jk!slh<hMk<k!NGl<!osz
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hml<!2.17
hml<!2.18
v! lQm<mVg<G! '/12! oszuiGl</! siqug! ucu!18lQ! lx<Xl<! 12lQ/! lx<x! -V! hg<gr<gt<!ik<k! fqzk<jk! slh<hMk<k! NGl<! oszjug<!
<G!AB = 18!lQ? CD = 12 lQ? AD = BC = 5 lQ/!!!!2.18!Jh<!hii<g<gUl<*/!∆EBC!YI!-Vslhg<g!
Q/
U = 60 × 12 = '. 720.
35
wMk<Kg<gim<M!8: yV!sib<sKvk<kqe<!Sx<xtU!31!os/lQ/!nke<!yV!&jzuqm<mk<kqe<!fQtl<!9!os/lQ/!nke<!lx<oxiV!&jzuqm<mk<kqe<!fQtk<jkBl<!hvh<jhBl<!g{<Mhqc/!
kQIU: d1? d2 we<he!&jzuqm<mr<gt<!weg<!ogit<g/!
Sx<xtU = 22
212 dd + /Neiz<! Sx<xtU! 31! os/lQ!
weg<!ogiMg<gh<hm<Mt<tK/
∴ 22
212 dd + = 20 os/lQ!nz<zK! = 100.
-r<G!yV!&jzuqm<mk<kqe<!fQtl<!8!os/lQ/!∴!d
22
21 dd +
1 = 8 we<g/!
∴ 64 + d22 = 100 nz<zK!d2
2 = 36. ∴ d2 = 6!os/lQ/
∴ sib<sKvk<kqe<!hvh<H =21 d1 × d2 =
21 × 8 × 6 = 24 os/lQ2.
wMk<Kg<gim<M 9: 264 os/lQ!fQtLt<t!gl<hq!-V!sl!higr<gtigh<!hqiqg<gh<hMgqxK/!yV!higk<jk! um<mligUl<! lx<oxie<jx! slhg<g! Lg<Ogi{ligUl<! ujtk<K!
osb<bh<hMgqxK/!nju!dt<tmg<Gl<!hvh<Hgtqe<!uqgqkk<kqjeg<!g{<Mhqc (π ≈ 722!we<X!
hbe<hMk<K).!
kQIU: um<mk<kqe<!Sx<xtU = 2
264 =132!os/lQ/!
Neiz<!um<mk<kqe<!Sx<xtU = 2πr.
∴ 2 × 722 × r = 132!nz<zK r = 21!os/lQ/
∴!um<mk<kqe<!hvh<H = πr2 = 722 × 21 × 21 = 1386 os/lQ2.
slhg<g!Lg<Ogi{k<kqe<!Sx<xtU = 3a Neiz<!Sx<xtU = 132 os/lQ/!!∴ 3a = 132!nz<zK!a = 44!os/lQ/
∴ slhg<g!Lg<Ogi{k<kqe<!hvh<H = 2
43 a× = 244
43×
= 484 3 os/lQ2
∴ um<ml<! lx<Xl<! slhg<g! Lg<Ogi{k<kqe<! hvh<Hgtqe<!
uqgqkl<!= 1386 : 484 3 = 21 3 : 22 !
hbqx<sq 2.1 1. gQp<g<g{<m!Lg<Ogi{r<gtqe<!hvh<Hgjtg<!gi{<g/
(i) !!nch<hg<gl<!= 18!os/lQ?!dbvl<! = 3 os/lQ/ (ii) Lg<Ogi{k<kqe<!&e<X!hg<gr<gtqe<!fQtr<gt<!31!o(iii) slhg<g!Lg<Ogi{k<kqe<!hg<g!fQtl<!9!os/lQ/!
36
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hml< 2.21
hml<!2.19
hml< 2.20
s/lQ? 48!os/lQ?!63!os/lQ/!
2. ABCD we<x!fix<gvk<kqe<!&jzuqm<ml< AC e<!fQtl; 44 os/lQ!lx<Xl<!B? D!zqVf<K ACg<G! ujvbh<hMl<! Gk<Kbvr<gt<! LjxOb! 31! os/lQ! lx<Xl<! 12! os/lQ! weqz<?!nf<fix<gvk<kqe<!hvh<H!g{<Mhqc.
3. yV! fix<gvk<kqe<! yV! &jzuqm<mk<kqe<! fQtl<! 26! os/lQ! lx<Xl<!!nl<&jzuqm<mk<kqx<G! ujvbh<hMl<! osr<Gk<Kg<OgiMgtqe<! fQtr<gt<! 4! os/lQ!lx<Xl<!6!os/lQ!weqz<?!nf<fix<gvk<kqe<!hvh<H!g{<Mhqc/!
4.! 260lQ! Sx<xtUt<t! yV! sib<sKvk<kqe<! yV! &jzuqm<ml<! 66lQ/! lx<oxiV!&jzuqm<mk<kqe<!fQtl<!gi{<g/!OlZl<!ns<sib<sKvk<kqe<!hvh<H!gi{<g/!
5. yV!-j{gvk<kqe<!hvh<H!723!os/lQ3!lx<Xl<!nke<!dbvl<!29!os/lQ!weqz<?!nke<!nch<hg<gk<jkg<!g{<Mhqc/
6. 8! os/lQ! lx<Xl<! 8 os/lQ! -j{! hg<gr<gtigg<! ogi{<m! siqugk<kqe<! hvh<H! 41!os/lQ2. -j{!hg<gr<gTg<gqjmh<hm<m!K~vl<!g{<Mhqc/!
7. yV! siqugk<kqe<! -j{! hg<gr<gTg<G! -jmh<hm<m! K~vl<! 5! os/lQ! lx<Xl<! yV!-j{h<hg<g! fQtl<! 8! os/lQ/! ns<siqugk<kqe<! hvh<H! 45! os/lQ2 weqz<? lx<oxiV!-j{!hg<gk<kqe<!fQtl<!gi{<g/
8. 22 os/lQ × 14 os/lQ! ntUt<t! yV! osu<ug! fqzk<kqe<! hvh<hqx<<Gs<! sllie!hvh<Hjmb!yV!um<m!fqzk<kqe<!Sx<xtU!gi{<g/
2.2 %m<M dVur<gt<!
yV! fix<gvl< ABCD Jg<! gVKg! (hml<! 2.22 Jh<! hii<g<gUl<*/ BDjb!-j{g<gUl</! kx<ohiPK! fix<gvlieK ∆ABD lx<Xl<! ∆BCD we! -v{<M!
Lg<Ogi{r<gtigh<!hqiqg<gh<hMgqe<xK/!-v{<M!Lg<Ogi{r<gTg<Gl<! BD hg<gl<!!ohiKuieK/!hqe<Oeig<gqh< hii<g<Gl<!OhiK? Lg<Ogi{r<gt< ABD lx<Xl< BCD!J BD Jh<!ohiKh<hg<gligg<! ogi{<M ye<xe<! Olz<! ye<xig! jug<g! fix<gvl<! ABCD ohxh<hMgqe<xK/!weOu?!fix<gvl< ABCD NeK!-V Lg<Ogi{r<gtqe<!Osi<g<jgbiGl<!!!!
hml< 2.22
!!!!!!
nz<zK ABCD yV!%m<M!dVuliGl</!-uyV! osu<ugk<kqe<! -V! hg<gr<gtqz<! -
dVuig<gh<hm<mK!)hml<!2.23 Jh<!hiIg<gUl<*dVur<gjt! ye<xe<! hg<gk<kqz<! ye<jx!Ou{<Moleqz<! Lkz<! dVuk<kqe<! Wkiu
3
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hml< 2.23
<uiOx?!siqugl<!yV!%m<M!dVuliGl<<!nK!V! osr<Ogi{! Lg<Ogi{r<gt<! Osi<k<K!
/!fil<!nxqf<K!ogi{<mK we<eoueqz<?!-V!juk<K! yV! %m<M! dVul<! dVuig<g!okiV! hg<gl<! -v{<miuK! dVuk<kqe<!
7
WkiuokiV!hg<gk<kqx<Gh<!ohiVf<Kukig!-Vg<g!Ou{<Ml</!sqz!%m<M!dVur<gt<? hml< 2.24!Lkz<!hml<!2.35 ujv ogiMg<gh<hm<Mt<te/!
hml< 2.25
hml< 2.24
hml< 2.32
38
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hml< 2.26
hml<! 2.29
hml< 2.27
hml< 2.28
hml< 2.30
hml< 2.33
hml< 2.31
hml< 2.34
ye<xqe<! Olz<! ye<xig! -Vg<Gl<! hg<gr<gjt! H
hml<! 2.24 zqVf<K! hml<! 2.35 ujvBt<t! %wMk<Kg<gim<mig? hml<!2.25 NeK!yV!Lg<Ogi{dVuliGl</!nK!yV!hl<hvk<kqe<!fqjzg<Gk<K!
hml<!2.27 NeK!yV!osu<ugl<!lx<Xl<!Yi<!nj-u<UVul<? Yi<! njvum<mk<kqje! Olz<!
Okix<xltqg<gqe<xK/! hml<! 2.29 NeK! y
giz<um<mr<gjtg<! ogi{<m! %m<M! dVuliGl<.Lg<Ogi{l<? yV! siqugl<! Ngqbux<jxg<! oWUgj{bqjeh<! Ohiz<! okiqgqxK/! %m<M!dVuum<mr<gt<! Ngqbux<xqe<! %m<mig! -Vh<hkNgqbux<jx Lf<jkb!uGh<hqz<!gx<x!$k<kqvr<g
%m<Mk<!kt!dVur<gtie!siqugr<gt<!
fil<!hii<g<gzil</!! 2.2.1 siqugl<
siqugl<! we<hK! fie<G! hg<gr<gjtBj
hg<gr<gt<!-j{bieju!)hml<!2.36 Jh<!hii<g<gDC Bl<!-j{bieju/ AB = a ? CD = b!we<g/ -j{OgiMgTg<G! -jmh<hm<m! K~vl< h we<g/!fil< siqugl<! ABCD J! Lg<Ogi{r<gt<! ABC!lx<Xl<! ACD gtqe<! %m<M! dVuligg<!
ogit<tzil</!Lg<Ogi{l< ABC! z<!nch<hg<gl<!AB ?!Gk<Kbvl< h. NgOu?
∆ABC bqe<!hvh<H = 21 × a × h !
∆ACD bqe<!hvh<H!= 21 × b × h
∴siqugk<kqe<!hvh<H!= 21 × a × h +
21 × b × h =
21
39
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hml< 2.35
t<tqg<! OgiMgtiz<! gim<cbqVg<gqe<Oxil</!!
m<M! dVur<gjt! wtqkig nxqbzil</!
l<? Yi<!njvum<ml<!Ngqbux<xqe<!%m<M!GXg<G!oum<Mk<!Okix<xligk<!okiqgqxK/!
vum<ml<!Ngqbux<xqe<!%m<M!dVuliGl</!Wx<Gl<! osu<ug! se<ez<! Ohiz<!
V! osu<ugl<! lx<Xl<! -V! sllie!
! hml<! 2.33 NeK! yV! osu<ugl<? yV!gi{<m! %m<M! dVuliGl<; -K! YI!r<gtiue? Lg<Ogi{r<gt<? fix<gvr<gt<? iz<? nux<xqe<! Sx<xtUgt<? hvh<Hgt<!jtg<!ogi{<M!g{g<gqmzil</!
lx<<Xl<!hz<Ogi{r<gjth<!hx<xq!-h<OhiK!
mb! yV! kt! dVul</! -kqz<! -V!
Ul<*/ ABCD we<x!siqugk<kqz< AB Bl<!!
(a
glk; 2.36
+ b)h sKv!nzGgt</
2.2.2 hz<Ogi{l< hz<Ogi{l<!we<hK n Ofi<g<Ogim<Mk<!K{<Mgtiz<!ucujlg<gh<hm<m!kt!dVul<!
NGl</!hz!Lg<Ogi{r<gtqe<!%m<M!dVuOl!hz<Ogi{l<!we<hjk!fil<!nxqgqOxil</!hz<Ogi{k<kqe<!hg<gr<gt<!slligUl<? Ogi{r<gt<!slligUl<! -Vh<hqe<? nh<hz<Ogi{l<! Yi<! yPr<G!hz<Ogi{l<! weh<hMl</! NX! hg<gr<gjtg<!ogi{<m! yPr<G! hz<Ogi{l<! Yi<! yPr<G!nXOgi{l<! weh<hMl</! Yi<! yPr<G!nXOgi{k<kqz<! njek<K! hg<gr<gTl<!slligUl<? dm<Ogi{r<gt<! yu<ouie<Xl<!
120°g<Gs<! slligUl<! -Vg<Gl< (hml<! 2.37! Jh<!hii<g<gUl<). -f<k! Gxqh<hqm<m! yPr<G! hz<Ogi{l<!ncg<gc! hbe<hMk<kh<hMukiz<?! nke<! Sx<xtU!
lx<Xl<!hvh<htU!hx<xqg<!gi{<Ohil</ ABCDEF<!!we<hK!Yi<!yPr<G!nXOgi{l<!we<g/!hg<gr<gt< AB?ntUjmbju/!yu<ouiV!hg<gk<kqe<!fQtLl<!a nzGSx<xtuieK a + a + a + a + a + a = 6a nnXOgi{k<kqe<!hvh<H!gi[l<!$k<kqvk<kqje!!uVuqh
CF sf<kqg<Gl<!Ht<tqbqje? O we<g/!Lg<Ogi{r<gt<
slhg<g! Lg<Ogi{r<gt<! NGl</! weOu? yu<ouiV!L
weOu? yPr<G!nXOgi{k<kqe<!hvh<H!= 6 × 2
43 a =
!wMk<Kg<gim<M!10:!hml< 2.38e<!hvh<hqjeg<!g{<Mhqc/!!
kQIU;!hml<!ABCDE NeK!hg<gl< BD bqjeh<!ohiK!fqjzbigg<!ogi{<m ABDE lx<Xl< BCD e<!%m<miGl</ ABDE we<x!siqugk<kqz<!-j{!hg<gr<gtie! AE ? BD gtqe<!fQtr<gt<!LjxOb 10 os/lQ?!16 os/lQ!NGl</!-j{!hg<gr<gTg<G!-jmh<hm<m!K~vl<!9 os/lQ/!!!NgOu? siqugl; ABDE!e<!hvh<H !
.1179)1610(
21)(
21 2Pos/l=×+=×+ hba
BCD yV!Lg<Ogi{l</!-ke<!nch<hGkq BDbqe<!fQtlNgOu!-ke<!hvh<H
6481621
21
=××=× hb
40
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glk; 2.37
BC? CD? DE? EF lx<Xl< FA sllie!gt<!we<g/!yPr<G!nXOgi{k<kqe<!
zGgt</! fil<! -h<OhiK! yPr<G!
<Ohil</!!&jzuqm<mr<gt< AD ? BE ? OAB? OBC? OCD? ODE? OEF? OFA
g<Ogi{k<kqe<! hvh<H 2
43 a NGl</!
2
233 a sKv nzGgt</
hml< 2.38
<!16!os/lQ ? Gk<Kbvl< 8 os.lQ.
.2nrkP
∴!%m<M!dVul< ABCDE!e<!hvh<H!= ABDE !e<!hvh<H + BCD e<!hvh<H!
= 117 + 64 = 181 os/lQ2. wMk<Kg<gim<M 11: yV! fqzl<! nth<huI! yV! fqzk<kqe<! ntUgjth<! hqe<uVliX!Gxqk<Kt<tii/!fqzk<kqe<!hvh<hqjeg<!g{<Mhqc/!
hml<!!2.39 kQIU: A! bqzqVf<K! D ujv! dt<t! fqzlth<huiqe<! Gxqgt< P? Q? R? S! we<g/! hqe<H AP = 5 lQ? AQ = 7 lQ? AR = 12 lQ? AS = 15 lQ? AD = 17 lQ? BP = 10 lQ? FQ = 8 lQ? CR = 8 lQ? ES = 9 lQ. ogiMg<gh<hm<m! fqzlieK!siqugr<gt<!PRCB? FESQ lx<Xl<!Lg<Ogi{r<gt< AQF? APB? DSE? CRD -ux<xqe<!%m<miGl< )hml< 2.40 Jh<!hiIg<gUl<).
siqugl< PRCB!e<! hvh<H! : BP ? CR -j{
hg<gr<gt</!Gk<Kbvl< PR. fil<!nxquK?
hml< 2.40
BP = 10 lQ? CR = 8 lQ?!lx<Xl< PR = AR − AP = 12 − 5 = 7lQ.
NgOu? PRCB e<!hvh<H!= 21 (BP + CR) × PR
= 21 (10 + 8) × 7 = 63 lQ2
siqugl<!QFES e<! hvh<H; ES ? FQ -j{! hg<gr<gt</! Gk<Kbvl<! QS. ! fil<! nxquK! ES = 9 lQ? FQ = 8 lQ ? QS = AS − AQ = 15 − 7 = 8lQ/!NgOu? QFES e<!hvh<H!=!
21 (ES + FQ) × QS =
21 (9 + 8) × 8 = 17 × 4 = 68 lQ2
Lg<Ogi{l<!AQFe<!hvh<H! = 21 × AQ × FQ =
21 × 7 × 8 = 28 lQ2
Lg<Ogi{l< DSE e<!hvh<H!= 21 × DS × ES =
21 ×(AD − AS) × 9
= 21 (17 − 15) × 9 =
21 × 2 × 9 = 9 lQ2.
Lg<Ogi{l<! CRD e<!hvh<H! = 21 × RD × CR =
21 × (AD − AR) × 8
= 4 × (17 − 12)= 4 × 5 = 20 lQ2. Lg<Ogi{l<!APB e<!hvh<H! =
21 × AP × BP =
21 × 5 × 10 = 25lQ2
∴!fqzk<kqe<!hvh<H!= 63 + 68 + 28 + 9 + 20 + 25 = 213!lQ2.
41
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wMk<Kg<gim<M 12: hml<!2.41 z<!gi{<hqg<gh<hm<m!ucuk<kqe<!hvh<H!g{<Mhqcg<gUl<!(π ≈ 722
weh<!hbe<hMk<kUl<). kQIU;!osu<ugl< ABDE?-j{h<Oh!ogiMg<gh<hm
osu<ugl< ABDE e<!hvnjvum<ml< AFE e<!hvh
slhg<g!Lg<Ogi{l<!BC
∴ ljebqe<!hvh<H = 280
wMk<Kg<gim<M 13: hml<!
weh<!hbe<hMk<kUl<*/ kQIU; osu<ugl< ABCD? -j{h<Oh ogiMg<gh<hmosu<ugl<!ABCD!e<!hvnjvum<ml< CDE!e<!hv
giz<!um<mh<!hGkq AFD
giz<!um<mh<!hGkq BCG
∴ ljebqe<!hvh<H = 48
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hml< 2. 41
njvum<ml< AFE lx<Xl<!slhg<g!Lg<Ogi{l< BCD Ngqbux<xqe<!<m!lje!we!fil<!nxqgqOxil</ h<H = 20 × 14 = 280 os/lQ2. <H =
21 π × r2 =
21 ×
722 × 7 × 7 = 77 os/lQ2
D!e<!hvh<H = 43 a2 =
43 × 14 × 14 = 49 3 os/lQ2.
+ 77 + 49 3 = 357 + 49 × 1.732 = 357 + 84.868 = 441.868 os.lQ2
2.42 z<!gi{<hqg<gh<hm<m!ucuk<kqe<!hvh<H!g{<Mhqcg<gUl< (π≈722
glk; 2.42
njvum<ml<!CDE lx<Xl<!giz<!um<mh<!hGkqgt<! AFD? BCG!e<!<m!lje!we!fil<!nxqgqOxil</!h<H = 12 × 4 = 48!os/lQ2. h<H!=
21 π × 6 × 6 =
722 × 3 × 6 =
7396 = 56
74 os/lQ2.
e<!hvh<H!= 41 π × 4 × 4 =
722 × 4 =
788 = 12
74 os.lQ2.
e<!hvh<H = 1274os.lQ2.
+ 56 74 + 12
74 + 12
74 = 129
75 os.lQ2.
42
wMk<Kg<gim<M 14:!hml<!2.43 Nz<!$ph<hm<Mt<t!hvh<H!g{<Mhqcg<gUl</!
kQIU;!-h<hml<?! osu<ugOsi<g<jgbiGl</!osu<ugl< CDFG e<!hvhsiqugl< ABCG!e<!hvh<H
njvum<ml< DEF e<!hvh
∴ ogiMg<gh<hm<m!njlh
sqz! slbr<gtqzdVur<gjtg<! gVk!hvh<Hgjt!%m<Mukx<G!dVur<gtqe<!hvh<Hgjt
um<m!ucu!ujtbk<kqe
yV! um<m! uj
-jmh<hm<m!hGkqbiGl<!um<mh<hvh<hqzqVf<K!dt
∴ njvum<m!ujtbk
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hml< 2.43
l< CDFG? njvum<ml< DEF lx<Xl<! siqugl<!ABCG Ngqbux<xqe<!
<H!= 28 × 13 = 364 os.lQ2. =
21 (36 + 28) × 14 = 64 × 7 = 448 os.lQ2.
<H = 21 ×
722 × 14 × 14 = 22 × 14 = 308 os.lQ2
<hqe<!hvh<H = 364 + 448 + 308 = 1120 os.lQ2.
<! fil<! ohiqb! kt! dVur<gtqzqVf<K! oum<c! fQg<gqb! lQkq!Ou{<cBt<tK/! nux<xqe<! hvh<Hgjtg<! gi{! Olx<g{<muiX!hkqzig!kvh<hm<m!dVuk<kqe<!hvh<hqzqVf<K?!oum<c!wMg<gh<hm<m!g<!gpqk<K!lQkq!dt<t!dVur<gtqe<!hvh<jhg<!gi{zil</!
<!hvh<H!tbl<! we<hK! ohiK! jlbk<jkBjmb! -V! um<mr<gTg<G!
)hml<!2.44 Jh<!hii<g<gUl<*/!um<m!ujtbk<kqe<!hvh<hieK!outq!<!um<mk<kqe<!hvh<jhg<!gpqk<kkx<Gs<!slliGl</!nkiuK?!
ujtb!hvh<H! = π R2 − π r2 = π (R2 − r2).
hml<!2.44
<kqe<!hvh<H!= 21 π (R2 − r2) sKv!nzGgt</
43
wMk<Kg<gim<M!15: hml<!2.45 z<!fqpzqmh<hm<m!hGkqbqe<!hvh<H!g{<Mhqcg<gUl</!-f<k!hGkq!giz<!um<mr<gtiz<!$ph<hm<Mt<tK!(π ≈
722 weg<!ogit<tUl<).
hml< 2.45
kQIU;! ogiMg<gh<hm<m! fqpzqmh<hm<mh<! hGkq?! 28 os/lQ! hg<gLt<t! sKvk<kqz<! fie<G!&jzgtqZl<!14 os.lQ NvLt<t!giz<!um<mh<!hGkqgjt!fQg<gqbkx<Gs<!slliGl</ sKvk<kqe<!hvh<H!= 28 × 28 = 784 os.lQ2.
yV!giz<!um<mk<kqe<!hvh<H = 41 π × 14 × 14 =
41 ×
722 × 14 × 14 = 154 os.lQ2.
∴!Okjubie!hvh<H = 784 − 4(154) = 784 − 616 = 168!sKv!os/lQ/
glk; 2.46
wMk<Kg<gim<M 16: 7lQ ngzLt<t! YMhijk! hml< 2.46 z<! gim<cbuiX!
njlg<gh<hm<Mt<tK/! -ke<! dt<um<ms<! Sx<xtU! 720lQ lx<Xl<! yu<ouiV! OfIg<OgiMh<!hGkqbqe<!fQtl< 140lQ. ujtUh<!hGkqgt<!njvum<m!ucuk<kqz<!dt<te/!YMhijkbqe<!hvh<jhg<!g{<Mhqc (π ≈
722 !weh<!hbe<hMk<kUl<).
!!!!!!kQIU;!dt<!njvum<mk<kqe<!Nvl<! r we<g/!weOu!dt<!Sx<xtU 2 × 140 + 2 × (π × r) nz<zK 280 + 2πr.!Neiz<!-ke<!lkqh<H 720lQ!we!ogiMg<gh<hm<Mt<tK. ∴ 280 + 2πr = 720 nz<zK!2πr = 440 nz<zK!r =
π2440 =
2227440
×× = 70 lQ.
weOu!dt<!njvum<mk<kqe<!Nvl< r = 70lQ. ∴outq!njvum<mk<kqe<!Nvl< R = 70 + 7 = 77lQ. -h<ohiPK!YM!hijkbqe<!hvh<hieK!njvum<m!ujtbr<gtqe<!hvh<H!lx<Xl<!osu<ug!ucu!hijkbqe<!hvh<Hgtqe<!%MkZg<Gs<!slliGl</!Neiz<?!yV!!njvum<m!hijkbqe<!
hvh<H= 21 π (R2 − r2) =
21 ×
722 × (772 − 702) =
711 × 147 × 7 = 1617 sKv lQ/
44
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yV!osu<ug!ucu!hijkbqe<!hvh<H = 140 × 7 = 980 sKv lQ. ∴!hijkbqe<!hvh<H!= 2 × 1617 + 2 × 980 = 3234 + 1960 = 5194 sKv lQ.
wMk<Kg<gim<M 17: 4.2lQ/! hg<g! ntUt<t! yV! sKv! ucu! gm<cmk<kqe<! yV! outqh<Hx!&jzbqz<! yV! hS! gm<mh<hm<Mt<tK/! gbqx<xqe<! fQtl< 4.9 lQ weqz<?! hS! Olbg<%cb!hGkqbqe<!hvh<H!g{<Mhqc!(π ≈
722 !weh<!hbe<hMk<kUl<).
kQIU;!sKvk<kqe<!A!we<x!Ljebqz<!hS!gm<mh<hm<M!dt<tK (hml< 2.47 Jh<!hii<g<gUl<*/!gbqx<xqe<! fQtl< 4.9lQ lx<Xl<! Sux<xqe<! fQtl< 4.2lQ/! hSuieK! Sux<jxg<! gmg<g!
-bzikkiz<?!!nke<!gbqX!sKvk<kqe<!B lx<Xl< E!Ljegjtk<!ki{<c D!lx<Xl< G ujv fQt<gqxK/! weOu?! hSuieK 4.9lQ NvLt<t! Lg<giz<! um<mh<hGkq! lx<Xl<!!!!!!
4.9 − 4.2 = 0.7lQ!NvLt<t!-V!giz<um<mh<!hGkqgt<!Ngqbux<jx!Olb!LcBl</!!∴ hS!Olbg<%cb!hvh<H!!!!
= 43 × π × 4.9 × 4.9 + 2 ×
41 × π × 0.7 × 0.7
= 43 ×
722 × 4.9 × 4.9 +
21 ×
722 × 0.7 × 0.7
= 2
33 × 0.7 × 4.9 + 11 × 0.1 × 0.7
= 56. 595 + 0.77 = 57.365lQ2.
!
hml< 2.48
wMk<Kg<gim<M! 18:!hml<! 2.48!z<! fqpzqmh<hm<m!hGkqbqe
weg<!ogit<tUl<*/ kQIU;!fqpzqmh<hm<m!hGkqbqe<!hvh<H!= (14 os.lQ NvLt<t!
− (7 os.lQ. Nv
= 21 × π × (14)2 −
21 × π × (7)2
= 21 ×
722 × 14 × 14 −
21 ×
722 × 7 × 7 = 11 × 2 × 14 − 11 × 7
!!
45
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hml< 2.47
<! hvh<hqjeg<!g{<Mhqc! (π≈ 722
njvum<mk<kqe<!hvh<H)
Lt<t!njvum<mk<kqe<!hvh<H*!
= 308 − 77 = 231 os.lQ2.
hbqx<sq 2.2 1. fqzlth<huiqe<! Ofim<Mh<! Hk<kgk<kqZt<t! hqe<uVl<! Gxqh<HgtqzqVf<K! dkuqh<hml<!
ujvf<K!nux<xqe<!hvh<Hgjtg<!g{<Mhqc;!
2. yV! uqjtbim<M! jlkiel<! hml<! 2.52 z<!
gim<cbuiX! -V! OfIOgim<Mk<! K{<Mgt<!lx<Xl<! -V! njvum<m! uqz<gtiz<!dVuig<gh<hmOu{<Ml</! yu<ouiV! njv!um<mk<kqe<! Nvl<! 21lQ. yu<ouiV!
Ofi<g<Ogim<Mk<!K{<ce<!fQtl< 85lQ. uqjtbim<Mk<!kqmzqe<!hvh<jh!g{<Mhqc (π ≈
722
3. yV!siqugk<kqe<!-V!-j{!hg<gr<gtqe<!fQtr<glx<x!-V!hg<gr<gt<!yu<ouie<Xl<!10 os.lQ fQtl
4. YI!yPr<G!nXr<Ogi{k<kqe<!hvh<H!150 3 os.l!!!!g{<Mhqcg<gUl</!! 5. hml<!2.53 z<!fqpzqm<mh<!hGkqbqe<!hvh<hqjeg<!g{
!
hml<!2.53
46
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glk; 2.52
(i)
hml<!2.49
(ii)
hml<! 2.50
weg<!o
t<!22 o<!weqz<?
Q2 weqz<?
<Mhqc/!
(iii)
hml<!2.51!
git<tUl</).
s.lQ lx<Xl<!12 os.lQ. nke<!!nke<!hvh<H!gi{<g/!!
!nke<!hg<gk<jkg<!!!!
6. gQp<g<gi[l<!hmr<gtqe<!fqpzqm<mh<!hGkqgtqe<!hvh<Hgjtg<!g{<Mhqcg<gUl</! (i) (ii)
(iii) 7. y
log
8. h 9. 1
ggO
w 10. A
fyf
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hml<!2.54
(iv)
V!um<mk<kqe<!uqm<ml< 54 os.lQ.> nke<!yV!QK C we<x! Ht<tq?! BC = 10 os.lQ we<xgi{<M! yV! um<ml<! ujvbh<hMgqxK/!{<Mhqcg<gUl</ π ≈
722 weg<!ogit<tUl</!
ml<!2.58 z<!fqpzqm<m!hGkqbqe<!hvh<H!lx<Xl
hml<!2.58
4 lQ hg<gLt<t! sKv! ucu! ubzqe<! fim<mh<hm<Mt<te/!yu<ouiV!hSUl<!lx<x!-Vm<mh<hm<Mt<tK/! -h<hSg<gt<?! nux<xqe<! wzlb<gqe<xe/!hSg<gt<!Olbik!fqzh<hGkqbqe
eg<!ogit<tUl</!!
BCD? 36lQ × 24lQ ntUt<t!yV!osu<ug!uie<G! Gkqjvgt< yu<ouie<Xl<! 10lQ. fu<ouiV! GkqjvBl<! nkx<G! wm<cb! ujvqzh<hGkqbqe<!hvh<hqjeg<!g{<Mhqc. π ≈
722 w
47
hml<!2.55
hml<!2.56
u
un
<!
e!<j<!
cQtbe
hml<!2.57
qm<ml< AB OfIOgim<Mk<K{<M AB e<!iX! dt<tK/! AC ! J! uqm<mligg<!u<um<mr<gTg<gqjmh<hm<m! hvh<jhg<!
Sx<xtjug<!g{<Mhqc/!
<G! &jzgtqZl<! fie<G! hSg<gt<!hSg<gjt!sf<kqg<gqe<x!ntuqx<Gg<!zg<Gm<hm<m! fqzh<hGkqbqe<! Hz<jz!hvh<hqjeg<!g{<Mhqcg<gUl</!π ≈
722
u!fqzliGl</!fie<G!&jzgtqZl<!g<! gbqx<xqeiz<! gm<mh<hm<Mt<te/!qz<! Olb<gqxK/! Gkqjvgt<! Olbik!g<!ogit<tUl</!
uqjmgt;
hbqx<sq 2.1 1. (i) 27 os.lQ2 (ii) 480 os.lQ2 (iii) 27.71 os.lQ2
2. 704 os.lQ2 3. 60os.lQ2 4. 112 lQ? 3696 lQ2
5. 34 os.lQ 6. 4 os.lQ 7. 10 os.lQ
8. 62.22os.lQ
hbqx<sq!2.2
1. (i) 27?200 s/lQ (ii) 15?100 s/lQ (iii) 7?525 s/lQ
2. 4?956!s/lQ 3. 147.22 os.lQ2 4. 10 os.lQ
5. 140 lQ2
6. (i) 36.39!s/os.lQ (ii) 25 lQ2 (iii) 37.71 os.lQ2 (iv) 240.28 os.lQ2
7. 770 os.lQ2 8. 354.37 os.lQ2? 94 os.lQ 9. 42 lQ2 10. 549.71lQ2
48
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3. sqz!Lg<gqb!GxqbQM 3.1 nxquqbz<!GxqbQM
uieqbz<?! -bx<hqbz<?! Oukqbqbz<?! dbqiqbz<! lx<Xl<! ohixqbqbz<! Ohie<x!himr<gtqz<!fil<!lqgh<ohiqb!w{<gjtBl<?!lqgs<!sqxqb!w{<gjtBl<!g{<cVg<gqe<Oxil</!!wMk<Kg<gim<mig?!
(i) H,lqbqzqVf<K!gkqvue<!dt<t!okijzU!92,900,000 jlz<gt</ (ii) yV!svisiqbie!osz<zieK!200, 000,000,000,000 &zg<%Xgjt!ogi{<cVg<gqxK/ (iii) nch<hjmk<!Kgt<!ye<xqe<!uip<fitieK 0.000000000251 uqficgt</ (iv) yV!lqe<e[uqe<!uqm<ml<!Slivig 0.000000000004!ose<clQm<mviGl</ -jkh<Ohie<x! w{<gjt! nh<hc! wPKkZl<?! jgbiTkZl<! nu<utU! wtqkz<z/!!-Vh<hqEl<?! nux<jx! nMg<Gg<! Gxq! uqkqgjth<! hbe<hMk<kq! wPkq! jgbit! LcBl</!!
nMg<Gg<Gxq!uqkqgjt!fqjeU!%i<Ouil</!!-bz<!w{<! m, olb<ob{< a -ux<xqx<G!am = a × a ×…. m giv{qgt</!wMk<Kg<gim<mig?!a5 = a × a × a × a × a. -r<G!a we<hK ncliel< we<Xl<? m NeK hc nz<zK nMg<G we<Xl<! njpg<gh<hMgqe<xe. am we<x!GxqbQm<cje!a e<!nMg<G m nz<zK m hc!dbi<k<kqb a!weh<!hcg<gqe<Oxil</ nMg<Gg<!Gxq!uqkqgt<!hqe<uVliX!kvh<hMgqe<xe: (i) am × an = a m+ n (ohVg<gz<!uqkq)
(ii) n
m
aa = a m − n , a ≠ 0, m > n (uGk<kz<!uqkq)
(iii) (am)n = amn (nMg<G!uqkq) (iv) am × bm = (a × b)m (Osi<g<jg!uqkq)
a ≠ 0! -Vg<Gl<! OhiK ma1 J a−m weg<! Gxqh<hqMgqe<Oxil<<<</ a0 = 1! we!
ujvbXg<gqe<Oxil</! nMg<Gg<Gxq! uqkqgjth<! hbe<hMk<kq?! wf<kouiV! lqjg!olb<ob{<{qjeBl< a × 10n!we<x!njlh<hqz<!wPk!LcBl<A!-r<G 1 ≤ a < 10, n yV!LP!NGl</!!wMk<Kg<gim<mig? (i) 7.32 = 7.32 × 100
(ii) 11.2 = 1.12 × 10 =1.12×101
(iii) 226 = 2.26 × 100 = 2.26 × 102
(iv) 6435.7 = 6.4357 × 1000 = 6.4357 × 103
(v) 92900000 = 9.29 × 10000000 = 9.29 × 107
(vi) 0.256 = 1056.2 = 2.56 × 10−1
(vii) 0.00786 = 1000
86.7 = 7.86 × 10−3
(viii) 0.000000537 = 5.37 × 10−7
(ix) 0.0000000279 = 2.79 × 10−8
49
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-eq?!w{<!we<xiz<!lqjg!w{<j{!lm<MOl!Gxqg<Gl</!!yV!w{<{qje!a × 10n!
we!nxquqbz<!GxqbQm<cz<!wPkqeiz<? a e<!ksl!uqiquqz<!uVl<!LPh<hGkqbieK!NeK 1 zqVf<K 9 ujvbqzie!w{<{ig!-Vg<Gl</!OlZl<!n yV!LPuiGl</!!OlZl<!kvh<hm<m!Yi<!w{<{qje!nxquqbz<!GxqbQm<cz<!lix<xq!wPKl<OhiK?!fil<!kvh<hm<m!w{<{qz<!dt<t!
kslh<!Ht<tqbqje!-mKhg<guig<gqz<!r -mr<gt<!fgi<k<kqeiz<!njk!=M!osb<b!10r!Nz<!
ohVg<GgqOxil</!nu<uiOx?!kslh<! Ht<tqjb!uzKhg<guig<gqz<! r -mr<gt<! fgIk<kqeiz<!njk!=M!osb<b!!10−r !Nz<!ohVg<GgqOxil</ wMk<Kg<gim<mig? !lqgh<! ohiqb!nz<zK!lqgnf<k!njlh<hqz<!ohVg<gO
wMk<Kg<gim<M 1: hqe<!uVl (i) 7493 (iv) 0.0056kQi<U: wMk<Kg<gim<M 2: hqe<uVl<! (i) 3.25 × 105 (iv) 4.02 × 10−
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s<! sqxqb! w{<gjt!nxquqbz<!GxqbQm<cz<!njlg<g?!nux<jx!ui!nz<zK!uGg<gOui!wtqjlbig!-Vg<Gl</
<!w{<gjt!nxquqbz<!GxqbQm<cz<!wPKg; (ii) 105001 (iii) 3449099.93
7 (v) 0.0002079 (vi) 0.000001024
w{<gjt!ksl!ucuqz<!wPKg; (ii) 1.86 × 107 (iii) 9.87 × 109
4 (v) 1.423 × 10−6 (vi) 3.25 × 10−9
50
kQi<U: (i) 3.25 × 105 = 210325 × 105 = 325 × 105 − 2 = 325 × 103 = 325000.
(ii) 1.86 × 107 = 210186 × 107 = 186 × 107−2 = 186 × 105 = 18600000.
(iii) 9.87 × 109 = 210987 × 109 = 987 × 109 − 2 = 987 × 107 = 9870000000.
(iv) 4.02 × 10−4 = 210402 × 10−4 = 402 × 10−4− 2 = 402 × 10−6 = 0.000402.
(v) 1.423 × 10−6 = 3101423 × 10−6 = 1423 × 10−6 − 3 = 1423 × 10−9 =0.000001423.
(vi) 3.25 × 10−9 = 210325 × 10−9 = 325 × 10−9 − 2 = 325 × 10−11 =0.00000000325.
wMk<Kg<gim<M 3: g{g<gqmjz! fqgp<k<kq! hqe<uVueux<xqe<! uqjmbqje! nxquqbz<!GxqbQm<cz<!wPKg; (i) (3000000)3 (ii) (4000)5 × (200)3 (iii) (0.00005)4 (iv) (2000)2 ÷ (0.0001)4
kQi<U : (i) 000000 = 3.0 × 106. ∴ (3
(ii) ∴ (4
(iii) ∴ (0 (iv)
∴ (2
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3
000000)3 = (3.0 × 106)3 = (3.0)3 × (106)3
= 3 × 3 × 3 × 106×3
= 27 × 1018 = 2.7 × 10 × 1018 = 2.7 × 1019.
000 = 4.0 × 103, 00 = 2.0 × 102.
4000)5 × (200)3 = (4.0 × 103)5 × (2.0 × 102)3
= (4.0)5 × (103)5 × (2.0)3 × (102)3
= 1024 × 103×5 × 8 × 102×3 = 1024 × 1015 × 8 × 106
= 92 × 1021= 8.192 × 103 × 1021= 8.192 × 1024.
00005 = 5.0 × 10−5
0.
.00005)4 = (5.0 × 10 = 25 10×
000 = 2.0 × 103,
000)2 ÷ (0.0001)4 =
=
81
−5)4 = (5.0)4 × (10−5)4 = 625 × 10−5×4
−20 = 6.25 × 102 × 10−20 = 6.25 × 10−18.
62
0001 = 1.0 × 10−4
2 0.44
23
)1001()1002(
−××
. . = 444
232
)10()01()10()02(
−××
. . = 44
23
101104
×−
×
××
22)16(616
6
10410410
104×=×=
× −−−
.
51
hbqx<sq 3.1
1. hqe<uVl<!w{<gjt!nxquqbz<!GxqbQm<cz<!njlg<gUl<;
(i) 29980000000 (ii) 1300000000 (iii) 1083000000000 (iv) 4300000000 (v) 9463000000000000 (vi) 534900000000000000 (vii) 0.0037 (viii) 0.000107 (ix) 0.00008035 (x) 0.0000013307 (xi) 0.00000000011 (xii) 0.0000000000009
2. hqe<uVl<!w{<gjt!ksl!uqiquqz<!wPkUl<;
(i) 3.25 × 10−6 (ii) 4.02 × 10−5
(iii) 4.132 × 10−4 (iv) 1.432 × 10−3
(v) 3.25 × 106 (vi) 4.02 × 105
(vii) 4.132 × 104 (viii) 1.432 × 103
3. hqe<uVueux<xqe<!lkqh<Hgjt!nxquqbz<!GxqbQm<cz<!g{<Mhqcg<gUl<;
(i) (100)3 × (40)5 (ii) (21000)2 × (0.001)4
(iii) (18000)4 ÷ (30000)2 (iv) (0.002)8 × (0.0001)3 ÷ (0.01)4
(v) (120000) × (0.0005)2 ÷ (400000) 3.2 lmg<jgbqe<!GxqbQM
\ie<! Ofh<hqbi<! we<El<! Nr<gqOzb! g{qk! Oljkbieui<! g{g<gqmjz!wtqjlbigUl<?! uqjvuigUl<! osb<K! Lcg<g! lmg<jgbqe<! GxqbQm<cje!
nxqLgh<hMk<kqeii</!‘lmg<jg’jbg<!Gxqg<Gl<!Nr<gqzs<!osiz< ‘logarithm’!NeK!‘logos’!lx<Xl<! ‘arithmos’! we<x! -V! gqOvg<gs<! osix<gtqzqVf<K! uVuqg<gh<hm<mkiGl</ ‘logos’ we<xiz<! Äg{g<gQMosb<kz<}! lx<Xl<! ‘arithmos’! we<xiz<! ‘w{<’! we<X! ohiVt<hMl</!!
NgOu! ‘logarithm’ we<xiz<! ‘w{<!g{g<gQM!osb<kz<’!NGl</!lmg<jgbqe<!GxqbQm<cje!nxqLgh<hMk<k?!Lkx<g{<{ig!nMg<Gg<!GxqbQm<jmh<hx<xq!nxqOuil</ 3.2.1 nMg<Gg<!GxqbQM
a we<hK! yV! lqjg! w{<! we<g/! ! x yV! LP! weqz<! ax we<x! GxqbQm<cje!
Wx<geOu!nxqLgh<hMk<kqBt<Otil</!!x yV!uqgqkLX!w{</!!nkiuK x = qp , -r<G!p!
yV!LP? q!yV!lqjg!LP!weqz<?!ax NeK!hqe<uVliX!ujvbXg<gh<hMgqxK/!!pq axa ⎟
⎠⎞
⎜⎝⎛= .
wMk<Kg<gim<mig,
( ) 8338
55 = , ( ) 411114
77−
−
= . x we<hK!yV!uqgqkLxi!w{<! we<xohiPK? ax we<hjk!yV!olb<ob{<{qjeg<!
Gxqh<hkig!ujvbXg<gzil</! !Neiz<!-f<k!ujvbjx!osb<ukx<G!flg<G!dbi<g{qk!
52
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gVk<Kg<gt<! Okjuh<hMgqe<xe/! -Vh<hqEl<?! a > 0 we<xuiXt<t! wf<kouiV!
olb<ob{<{qjeg<!ogi{<M!x!wEl<!yu<ouiV!olb<ob{<{qx<Gl<?!ax we<x!keqk<k!yV!olb<ob{<{qje!ujvbXk<K u = ax!we!wPKgqOxil</!-f<fqjzbqz<!u NeK!nMg<G!njlh<hqz<! nz<zK! nMg<Gg<! GxqbQm<cz<! wPkh<hm<Mt<tK! we<Ohil</! -r<G! a J!ncliel<!we<Xl<?!x J!nMg<G!nz<zK!hc!we<Xl<!njph<Ohil</!LPg<gTg<G!wPkqb!nMg<Gg<Gxq! uqkqgjt! njek<K! olb<ob{<! nMg<GgTg<Gh<! ohiVf<Kukigg<!ogit<Ouil</!!-u<uqkqgjt!-r<G!hqe<uVliX!wPKgqe<Oxil</
Olx<%xlmg<jg!Gxqb 3.2.2 lmg<jg
b ≠ 1!wbx NeK!keqkwe! wPk?! -
njpg<gqe<Oxi
a = bx we<hke
a = bx we<xnjlh<hqZl<!n
-K! a = bx!J
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(i) (iv) yxayaxa +=×xa
xa 1=−
(ii) yxaya
xa −= (v) ( )xbaxbxa ×=×
y
(iii) xyaxa =⎟⎠⎞⎜
⎝⎛ (vi) 10 =a
qb! uqkqgjt! nMg<Gg<Gxq! uqkqgt<! we<xjpg<gqe<Oxil</! -h<ohiPK!Qm<cje!ujvbXg<g!Lx<hMOuil</
g<!GxqbQM e<xuiX!b we<hK!yV!lqjg!w{<!we<g/!!x!we<x!yV!olb<ob{<{qx<Gl< <k!yV!olb<ob{<!a Jg<!Gxqg<Gl<!we!Wx<geOu!nxqf<Kt<Otil</!!a = bx!
r<Gt<t! nMg<G! x J b ncliel<! ogi{<m! a! e<! lmg<jg! we<X!l<. !-kje! x = a!we!wPKgqe<Oxil</! !weOu! x = a we<hK!!!blog blog
<!sllie!njlh<hiGl</!!OlZl<?!x = a we<x!lmg<jg!njlh<hieK!!blog
! nMg<Gg<! Gxq! njlh<hqe<! sllie! njlh<H! we<gqOxil</! ! -v{<M!
cliel<!)b)!ye<Ox/ x = a we<x!GxqbQM?!lmg<jgg<!GxqbQM!weh<hMl</!!blog
!Gxqg<gqe<xK/ wMk<Kg<gim<mig? (i) 3 = 729 we<hK 99log 3 = 729!e<!sllie!njlh<H;
(ii) 2log31
8= we<hK 31
8 = 2 e<!sllie!njlh<HA
(iii) −3 = 0.001 we<hK 1010log −3 = 0.001!e<!sllie!njlh<H; (iv) 2 = we<hK 749log7
2 = 49!e<!sllie!njlh<H;
(v) 3log21
9= e<!sllie!njlh<H 39 39 21
== zKnz< ;
(vi) ⎟⎠⎞
⎜⎝⎛=−
81log
23
4 we<hK 814 2
3
=−
e<!sllie!njlh<H/
53
Gxqh<H: lmg<jgg<! GxqbQm<cz<!ncliel<! Gxqh<hqm! Ou{<Ml</ y = log x we<X! wPKuK!ohiVtx<xK;! Woeeqz<! nkx<Gs<! sllie! nMg<Gg<Gxq! njlh<ohPk! ncliel<!okiqf<kiz<kie<! LcBl</! ! -Vh<hqEl<?! sqz! fqjzgtqz<?! ncliek<kqjeg<! Gxqh<hqmilz<!lmg<jggjt! wPKl<OhiK?! nr<G! njek<K! lmg<jggTl<! yOv! ncliek<jkh<!ohiKuigg<!ogi{<Mt<te!we<X!ogit<t!Ou{<Ml</ wMk<Kg<gim<M 4: hqe<uVl<!lmg<jg!njlh<hqje!nMg<Gg<Gxq!njlh<hig!lix<xUl<;
(i) 5 = 25log21 (ii) ⎟
⎠⎞
⎜⎝⎛
41log 2 = −2 (iii)
316log 216 = (iv) ⎟
⎠⎞
⎜⎝⎛
91log3 = −2
kQi<U: ncliel<!-v{<mjlh<HgtqZl<!slliekiGl</!!weOu
(i) 5 = 25log21 NeK! ( ) 525 2
1= g<Gs<!slliekiGl</!
(ii) ⎟⎠⎞
⎜⎝⎛
41log 2 = −2 NeK ( )
412 2 =− g<Gs<!slliekiGl</
(iii) 316log 216 = NeK ( ) 6216 3
1= g<Gs<!slliekiGl</
(iv) ⎟⎠⎞
⎜⎝⎛
91log3 = −2 NeK (3)−2 =
91 g<Gs<!slliekiGl</
wMk<Kg<gim<M 5: hqe<uVl<!nMg<Gg<Gxq!njlh<hqje!lmg<jg!njlh<hig!lix<xUl<;
(i) 2 = 61
64 (ii) 9−3 = 7291 (iii)
41
81 3
2
=⎟⎠⎞
⎜⎝⎛ (iv) 17
71 −=
kQi<U: -v{<mjlh<HgtqZl<!ncliel<!ye<xig!-Vh<hkiz<?
(i) 2 = 61
64 NeK 61 = g<Gs<!slliekiGl</ 2log64
(ii) 9−3 =7291 NeK −3 = ⎟
⎠⎞
⎜⎝⎛
7291log9 g<Gs<!slliekiGl</
(iii) 41
81 3
2
=⎟⎠⎞
⎜⎝⎛
−
NeK ⎟⎠⎞
⎜⎝⎛=−
41log
32
81 g<Gs<!slliekiGl</
(iv) 71 = 7−1 NeK −1 = ⎟
⎠⎞
⎜⎝⎛
71log7 g<Gs<!slliekiGl</
wMk<Kg<gim<M 6: lkqh<hqMg;
(i) (ii) (iii) 729log9 8log4 ⎟⎠⎞
⎜⎝⎛
271log9 (iv) .)243(log 1
3−
kQi<U : (i) x = we<g/!!hqe<H 9729log9x = 729 = 93. ∴ x = 3.
(ii) x = we<g/!!hqe<H 48log4x = 8 = 23. Neiz< 4x = (22)x = 22x.
∴ 22x = 23. ∴ 2x = 3 nz<zK x = 23
54
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(iii) x = ⎟⎠⎞
⎜⎝⎛
271log9 we<g/!hqe<H 9x= 3
3 331
271 −== . Neiz<!9x = (32)x = 32x.
∴ 32x = 3−3, 2x= −3 nz<zK x =23− .
(iv) x = we<g/ hqe<H 313 )243(log − x = (243)−1 =
2431 = 53
1 = 3−5
nz<zK 3x =3−5 nz<zK x = −5. wMk<Kg<gim<M 7: hqe<uVl<!sle<hiMgjtk<!kQi<g<gUl<; (i) (ii) 2log3 −=x 2100log =b (iii) x = 512log
81
⎟⎠⎞
⎜⎝⎛ (iv) x + 2 .09log27 =
kQi<U:
(i) . ∴ 32log3 −=x −2 = x nz<zK! x = 231 =
91 .
(ii) . ∴ b2100log =b2 = 100 = 102. ∴ b = 10.
(iii) x = .512log81
⎟⎠⎞
⎜⎝⎛ ∴ 38512
81
==⎟⎠⎞
⎜⎝⎛
x
nz<zK (8−1)x = 83 nz<zK 8−x = 83.
∴ −x = 3 nz<zK! x = −3.
(iv) x + 2 . ∴ x = −2 nz<zK 09log27 = 9log27 9log2 27=
− x
∴ 9)27( 2 =− x
nz<zK 223 3)3( =− x
nz<zK 223
3)3( =− x
. ∴ 223
=− x nz<zK! x =
34− .
-h<ohiPK! lqjg! w{<gtqe<! lmg<jggtqe<! h{<Hgjtg<! %xq! fq'h{l<!
osb<Ouil</!!2Jk<!kuqv!njek<K!lqjg!w{<gjtg<!gVKgqe<Oxil</ (i) ohVg<gz<!uqkq: a, m, n we<he!lqjg!w{<gt<? a ≠1!weqz<?
)(log mna = nama loglog + .
fq'h{l<: xma =log , !we<g/ yna =log
hqe<H m = ax, n = ay . ∴ m × n = ax × ay nz<zK! mn = ax+y. -f<k!nMg<Gg<Gxq!njlh<hqje!lmg<jg!njlh<hqx<G!lix<xq!wPk?
!!!uig<gqbk<kqz<?! Olx<%xqb!lmg<jgbieK?!nu<uqV!w{ (ii) uGk<kz<!uqkq: m, n, a
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nmyxmn loglog)(log +=+= .
aaauqkq! %XuK?! -V! lqjg! w{<gtqe<! ohVg<gx<hzeqe<!<gtqe<!lmg<jggtqe<!%MkZg<Gs<!slliGl</
we<he!lqjg!w{<gt<!lx<Xl< a ≠ 1!weqz<?
55
namanm
a logloglog −=⎟⎠⎞
⎜⎝⎛ .
fq'h{l<: xma =log , !we<g/!!hqe<H yna =log
m = ax , n = ay.
∴ nm = y
x
aa = ax − y.
-K!nMg<G!njlh<hqz<!dt<tK/!!-jk!lmg<jg!njlh<hqz<!lix<x?
.logloglog namayxnm
a −=−=⎟⎠⎞
⎜⎝⎛
!!
uig<gqbk<kqz<?! uGk<kz<! uqkq!
!we<x!uqk<kqbisnama loglog −
(iii) hc!uqkq: a, m we<he
fq'h{l<: !we<g/!!hxma =log
mn = (ax)n =axn . -K!nMg<G!wPk? (iv) a lqjg!w{<!weqz<?!
fq'h{l<: we<g/!!hqexa =1log
(v) a yV!lqjg!w{<!weqz<?!!
fq'h{l<: we<g/!!hqeaax log=
(vi) nclie!lix<xz<!uqkq: m, n
lo
fq'h{l<: x = pnymp log,log =
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%XukiuK?!nm wEl<! uqgqkk<kqe<! lmg<jgbieK
k<kqx<Gs<!slliGl</
lqjg!w{<gt<? a ≠ 1 lx<Xl< n yV!olb<ob{<!weqz<?
. mannma loglog =
qe<H m = ax!NGl</!!-VHxl<!n nMg<G!wMg<g?
njlh<hqz<! -Vg<gqe<xK/! ! -kje! lmg<jg!njlh<hqz<!
.loglog mannxnma ==
01log =a .
<H ax = 1 = a0. ∴ x = 0. nkiuK!)n.K*!
.01log =a
<H
, p
ng
!
.1log =aa ax = a = a1. ∴ x = 1 n.K
.1log =aa
lqjg
m =
we<g
!w{<gt<?!n ≠ 1!lx<Xl< p ≠ 1!weqz<?
( )pnmp loglog ×⎟⎠⎞⎜
⎝⎛ .
/
56
hqe<H!px = m, ny = p.!-ux<xqz< p J!fQg<g? ( ) . mnmn xyxy == K.n -K!nMg<G!ucuqz<!dt<tK/!!-kje!lmg<jg!njlh<hqx<G!lix<x?
xymn =log n.K ( ) ( ).logloglog pmm npn ×= (vi) kjzgQp<!uqkq: m , n we !
fq'h{l<: !we<gxnm =log
∴ xnm1
= . -K!nMg<G!nj
xmn
1log = nz<zK
(viii) a, b we<he!-V!lqjgfq'h{l<: x = !we<g/ablog
hqvkqbqm! (ix) m, n lx<Xl< a lqjg!w{
fq'h{l<: !we<g/max log=
∴ ax = n n.K aalog
Gxqh<H: njek<K!lmg<jggWoeeqz<?! ncliel< 1 d
Jg<! gVk, -ke<! lkq9log1
Neiz<?!wf<k!olb<ob{<!x!ws<siqg<jg: ohiKuig!hqe<u
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<he 1!fQr<gzie!lqjg!w{<gt<!weqz<?!
logn m =
/!!hqe<H x
nnnxm xx
x⎟⎟⎠
⎞⎜⎜⎝
⎛===
×11
.
lh<hqz<!dt<tK/!!-kje!lmg<jg!njlh<hqx<G!lix<x?
!w{<gt!!hqe<H
<gt<? a ≠
!!hqe<H
nm
=
tqZl<!njmb!h<H! x! wx<Gl<!1Vl<!ku
(1)
(2)
nlogm
1 .
1
<? b ≠ 1!weqz< . aabb =
log
!bx = a!NGl</!!-kqz<!x x<Gh<!hkqzig! Jh<!ablog
nm
mn log
log = .
alog
1!we<g/!! ,loglog nama = weqz< m = n!NGl</
nax log= .
n.K m = n (uqkq (viii)!e<!hc).
cliek<jk!1!we<xqz<zilz<!hii<k<Kg<ogit<gqe<Oxil</!!lmg<jgjbg<! gVkqOeioleqz<?! wMk<Kg<gim<mig?!eqz<! x = nz<zK 19log1
x = 9 weg<! gqjmg<gqxK/!x = 9!weg<!gqjmg<giK/ Xgt<!fqgp!uib<h<H!d{<M/
.log
loglog
na
manm
a =⎟⎠⎞
⎜⎝⎛
.loglog)(log namanma +=+
ab b = .
57
(1) NeK!kuxiGl</! ! Woeeqz<?! -mK! hg<gk<kqz<!dt<tKl<! uzK! hg<gk<kqz<!
dt<tKl<!sllig!LcbiK/!!nkiuK?! .log
logloglog
na
manama ≠−
(2) NeK! kuX/! ! Woeeqz<?! uzK! hg<gk<kqz<! dt<tKl<! -mK! hg<gk<kqz<!dt<tKl<!sllqz<jz/!!nkiuK? ).(log)(log nmamna +≠
wMk<Kg<gim<M 8: SVg<Gg; (i) 729log27log 33 + (ii) 1000
1log8log 55 +
kQi<U: (i) kvh<hm<m!Ogiju!-v{<M!lmg<jggtqe<!%Mkz<!lx<Xl<!-f<k!lmg<jggtqe<!nclier<gt<!sllibqVg<gqe<xe/!!weOu!ohVg<gz<!uqkqjbh<!hbe<hMk<kzil</ )72927(log729log27log
333 ×=+
= )33(log 633 ×
= = 9 × 1 = 9. 3log93log 39
3 ×=
(ii) ⎟⎠⎞
⎜⎝⎛ ×=+
100018log
10001log8log 555
= ⎟⎠⎞
⎜⎝⎛125
1log5
= ( )3535 5log
51log −=⎟
⎠⎞
⎜⎝⎛
= (−3) × 5log = (−3) × 1 = −3. 5
wMk<Kg<gim<M 9: SVg<Gg:
(i) (ii) 14log98log 77 − 4log34log236log21
999 −+
kQi<U: (i) .17log1498log14log98log 7777 ==⎟
⎠⎞
⎜⎝⎛=−
(ii) 39
29
21
9999 4log4log36log4log34log236log21
−+⎟⎟⎠
⎞⎜⎜⎝
⎛=−+
= 64log16log6log 999 −+ = 64log)166(log 99 −× = 64log96log 99 −
= .23log
6496log 99 ⎟
⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛
wMk<Kg<gim<M 10: fq'hqg<g;
(i) (ii) 2log341250log 1010 −= .2log208log3136log
211875log 32555 +−=
kQi<U: (i) uzKhg<gl< 2log34 10−= = 4 − 310 2log
58
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= 4 × 8log10log 1010 −
= 8log10log 104
10 − = 8log10000log 1010 −
= 1250log8
10000log 1010 =⎟⎠⎞
⎜⎝⎛ = -mK!hg<gl</
(ii) uzK!hg<gl< 2log208log3136log
21
3255 +−= = 2log54)8(log)36(log 3231
521
5 ×+−
= 53255 2log42log6log +−
= 426log32log4
26log 5325 +⎟
⎠⎞
⎜⎝⎛=+⎟
⎠⎞
⎜⎝⎛
= 45555 5log3log5log43log +=+
= 625log3log 55 + = ( )6253log5 × = = -mK!hg<gl</!1875log5
wMk<Kg<gim<M 11: fq'hqg<g; 29log8log7log6log5log4log 876543 =××××× . kQi<U: -mK!hg<gl< = ( ) ( ) ( )9log8log7log6log5log4log 876543 ××××× = = 9log7log5log 753 ×× ( )9log7log5log 753 ×× ! = =!uzK!hg<gl</ 2123log23log9log9log5log 3
23353 =×====×
wMk<Kg<gim<M 12: Jk<!kQi<g<gUl</ 3)502(log10 =+xkQi<U: sle<him<cje!nMg<G!njlh<hqz<!wPk? 2x + 50 = 103 = 1000 nz<zK 2x = 1000 − 50 = 950 nz<zK x = 475.
wMk<Kg<gim<M 13: e<!lkqh<jhg<!g{<Mhqc/ 2log381 9−
kQi<U: = 2log381 9− 2log3299−
⎟⎠⎞⎜
⎝⎛
= = 2log69 9− 6262log9 9 −=
−. ( Woeeqz< ) aab b =log
= .641
21
6 =
wMk<Kg<gim<M 14: .0)1(log2log 66 =+− xx Jk<!kQi<g<gUl</!
kQi<U: uGk<kz<! uqkqjbh<! hbe<hMk<k?! kvh<hm<m! sle<him<cje 01
2log6 =⎟⎠⎞
⎜⎝⎛
+xx we!
wPkzil</!!-kje!nMg<G!ucuqz<!wPk?
161
2 0 ==+xx nz<zK 2x = x + 1 nz<zK x = 1.
!
59
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wMk<Kg<gim<M 15: 1)1(log)7(log 33 =−−− xx Jk<!kQi<g<gUl</!
kQi<U: uGk<kz<!uqkqjbh<!hbe<hMk<k?!kvh<hm<m!sle<him<cje 117log3 =⎟
⎠⎞
⎜⎝⎛
−−
xx
we!wPkzil</!!-kje!nMg<G!ucuqz<!wPk?!
3317 1 ==
−−
xx
nz<zK )1(37 xx −=− nz<zK xx 337 −=− nz<zK 2x = −4 nz<zK x = −2.
wMk<Kg<gim<M 16: ( ) 2loglog 32 =x !Jk<!kQi<g<gUl</ kQi<U: we<g/! ! hqe<H!kvh<hm<m! sle<himieK xy 3log= 2log2 =y we<xigqxK/! !-kje!nMg<G! njlh<hqz<! wPk?! nz<zK .422 ==y 4log3 =x -kjeBl<?! nMg<G! ucuqz<!wPk? nz<zK43=x .81 =x ! wMk<Kg<gim<M 17: 3log1log3log2 595 =+× x Jk<!kQi<g<gUl</ kQi<U: kvh<hm<m!sle<him<cje!lix<xq!wPk? 13loglog3log 59
25 −=× x
nz<zK .53log5log3loglog9log 55595 ⎟
⎠⎞
⎜⎝⎛=−=× x
-mK!hg<gk<kqz<!nclie!lix<x!uqkqjbh<!hbe<hMk<k? ⎟⎠⎞
⎜⎝⎛=
53loglog 55 x . ∴ x = .
53
hbqx<sq 3.2.1
1. hqe<uVue siqbi nz<zK kuxi we<X!uqjmbtqg<gUl<: (i) . (ii) 5243log3 = 327log
31 = .
(iii) .4log3
16log43
16log 22 −=⎟⎠⎞
⎜⎝⎛ − (iv) .4log8log)48(log 222 −=−
(v) 1log 1 −=aa
. (vi) .loglog)(log nmnm aaa +=+
2. hqe<uVueux<xqx<Gs<!sllie!lmg<jg!njlh<hqjeg<!gi{<g:
(i) (ii) .04.05 2 =− .481 3
2
=⎟⎠⎞
⎜⎝⎛
−
(iii) .25644 =
(iv) 36= 729. (v) 216136 2
3
=−
. (vi) .001.010 3 =−
3. hqe<uVueux<xqe<!lkqh<H!gi{<g: (i) . (ii) . (iii) 625log5 216log6 9log 3 .
(iv) 31log9 . (v) 81log
31 . (vi) 24log 2 .
60
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(vii) . (viii) . (ix) . 610log2
1085log
2529log2
9−
4. okiqbikjkk<!kQi<g<gUl<: (i) logx 0.001 = −3. (vi) 3log2 =x . (ii) 7log
21 =x . (vii) = 4. c25log5
(iii) . (viii) 2100log −=x 42
log3 =⎟⎠⎞
⎜⎝⎛ N .
(iv) . (ix) 3125log =b 11000
1log10 =⎟⎠⎞
⎜⎝⎛
a
.
(v) 51log2 =⎟⎠⎞
⎜⎝⎛
x. (x) 2 1log9 =N .
5. hqe<uVl<! yu<ouie<xqx<Gl<! kvh<hm<m! lix<Xgtqz<! siqbie! uqjmjbk<! Oki<f<okMk<K!wPKg: (i) 3 weqz< x NeK ,15log =x
(A) 5 (B) 25 (C) 125 (D) 625
(ii) =+ 144log196
1log 1214
(A) 0 (B) 1 (C) 2 (D) 3 (iii) =1024log4 (A) 10 (B) 8 (C) 7 (D) 5 (iv) weqz< x = ,log24log xaa = (A) 0 (B) 1 (C) 2 (D) 3 (v) weqz< x = ,1log2 16 =x (A) 4 (B) 8 (C) 16 (D) 32 (vi) 2log5 =x weqz< x = (A) 5 (B) 25 (C) 125 (D) 625 6. hqe<uVl<!yu<ouie<xqz<!dt<t!Ogijujb!yOv!lmg<jgbigs<!SVg<gq!wPKg!: (i) .9log2log 1010 + (ii) .83log42log3 33 −+ (iii) .6log24log33log25 222 ++− (iv) 7log15log5log32log2 2224 +−+ . (v) . 4log63log22log5 641010 −+ (vi) .325log24log20log5log 10101010 −+−+ 7. zyx aaa === 5log,3log,2log lx<Xl< ,7log ta = weqz<?! hqe<uVl<! yu<ouie<xqe<!
lkqh<hqjeBl<!x, y, z lx<Xl< t!-ux<xiz<!g{<Mhqcg<gUl</ (i) (ii) (iii) (iv) 6log a 4loga 5.1loga 27log a
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(v) 312log a (vi) (vii) 600log a 27
8log a (viii) 15loga
(ix) (x) (xi) (xii) 35log a 12log a 9.4log a 1514log a
8. hqe<uVl<!yu<ouiV!sle<him<cjeBl<!kQi<g<gUl<!: (i) (ii) 2log3log2 55 =x 11log7loglog 333 =+x (iii) 018log16 =+x (iv) 256log16loglog 444 =×x (v) 23log)2(log 44 =++x (vi) 4log)12(log)12(log 333 =−−+ xx (vii) (viii) 2)1(log10log 33 =+− xx 13log)15(log)37(log 222 −=−−+ xx (ix) )43(log)10(log 55 xx +=+ (x) 2)log5(log 35 =x
(xi) 1log21510log 33 +=−+ xx
9. hqe<uVl<!yu<ouie<xqz<!dt<t!sle<him<cje!fq'hq: (i) 5log3135log 33 += (ii) 2log421600log 1010 += (iii) 5log222500log 1010 += (iv) 2log33125log 1010 −= (v) 2log2225log 1010 −= (vi) 10log430027.0log 33 −= (vii) 10log62000256.0log 1616 −= 10. w{<! 2! fQr<gzie! lqjg! w{<gt<! a, b, c weqz<? 1logloglog =×× cba acb we!fq'hqg<gUl</! 3.2.3 ohiK!lmg<jggt<
lmg<jggjt! ujvbXk<kohiPK?! lqjg! w{<gtqe<! lmg<jg! ujvbXg<gh<hm!
Ou{<Moleqz<?! nl<lmg<jgbqe<! ncliel<! w{<! 1! fQr<gzig! lx<x! wf<kouiV! lqjg!w{<{ig! -Vg<gzil<! we! uzqBXk<kqBt<Otil</! ! ncliek<jk! ‘e’ we<x! uqgqkLxi!w{<{ig! Oki<f<okMk<kiz<?! nh<hch<hm<m! lmg<jggt< -bz<! lmg<jggt<! NGl<. ncliek<kqje! w{< 10 Ngg<! ogi{<miz<?! nk<kG! lmg<jggt<! ohiK! lmg<jggt<!weh<hMl</! ! -bz<! lmg<jggjt!\ie<! Ofh<hqbi<! we<hui<! nxqLgh<hMk<kqeii</! ! ohiK!lmg<jggjt! Ofh<hqbvK! f{<hi<! oae<xq! hqiqg<̂ <! we<x! Nr<gqOzb! g{qk! uz<Zfi<!
nxqLgh<hMk<kqeii</! !-bz<!lmg<jg! J? \ie<! Ofh<hqbVg<Gh<!ohVjl!OsIg<<Gl<!
uqklig ln x we<X!SVg<glig!wPKOuil</! ! ln x Jh<!hx<xq!uqiquig!Olz<!uGh<Hgtqz<!hcg<gzil</! ! -h<ohiPK?! ohiK! lmg<jggtqe<! hbe<hiM! hx<xq! nxqf<Kogit<t!
Lx<hMOuil</! ! ohiK! lmg<jg! J! SVg<glig log x! we<X! ncliel< 10!
Gxqh<hqmilz<!wPKOuil</!!weOu? log x = y we<xiz<
xelog
x10log
yx =10log we<X!ohiVt<!ogit<t!
Ou{<Ml<;!-K!x = 10y g<G!slliekiGl</! y = log x!z< x =1000
1weh<!hqvkqbqm?
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y = log 1000
1 = . 310log 310 −=−
-jkh<OhizOu, for x = 100
1 , 101 , 1, 10, 100, 1000, …weg<!ogi{<miz<? nkx<Ogx<xix<Ohiz<
y = −2, −1, 0, 1, 2, 3, ….weh<!ohXgqe<Oxil</!!-ux<jxh<!hm<cbzqm!flg<Gg<!gqjmh<hK!
x 10−3 10−2 10−1 100 101 102 103
x10log −3 −2 −1 0 1 2 3 fil<! gueqh<hK? x NeK! olb<ob{<! Ogim<ce<! lqjg! ns<Sh<! Ohig<gqz<! nkqgiqg<Gl<!OhiK? l<!nkqgiqg<gqe<xK/!!OlZl<!fil<!gueqh<hK?!x > 1!njlf<k!njek<K!!!x!lkqh<HgTg<G! NeK!lqjg!w{<{ig!-Vg<gqe<xK?! 0 < x < 1!we<xqVg<Gl<!!!x!e<!lkqh<HgTg<G e<!lkqh<Hgjtg<!gQp<g{<muiX!nm<muj{h<hMk<kzil<:
x10logx10logx10log
x!e<!uQs<S! x10log e<!-Vh<hqml<! x10log e<!lkqh<H!
10−5 < x < 10−4 −5 < < −4 x10log −5 + 0.a1a2… 10−4 < x < 10−3 −4 < < −3 x10log −4 + 0.b1b2… 10−3 < x < 10−2 −3 < < −2 x10log −3 + 0.c1c2… 10−2 < x < 10−1 −2 < < −1 x10log −2 + 0.d1d2… 10−1 < x < 100 −1 < < 0 x10log −1 + 0.e1e2… 100 < x < 101 0 < < 1 x10log 0 + 0.f1f2… 101 < x < 102 1 < < 2 x10log 1 + 0.g1g2… 102 < x < 103 2 < < 3 x10log 2 + 0.h1h2… 103 < x < 104 3 < < 4 x10log 3 + 0.i1i2… 104 < x < 105 4 < < 5 x10log 4 + 0.j1j2…
Olx<gi[l<!nm<muj{bqzqVf<K!fil<!gueqh<hK?!ohiK!lmg<jg e<!lkqh<H x10log
(yV!LP) + (0.r1r2r3r4 …) we<X! wPk! LcBl</! ! -f<k! njlh<hqz<?! LPh<hGkqjb e<! Ofi<g<%X! weUl<!kslhqe<eh<hGkqjb e<! hkqe<lieg<%X! weUl<! njph<Ohil</! ! ohiKuig!
hkqe<lieg<%xqje!fie<G!hkqe<lier<gtqz<! )kslk<kiek<kqz<*! wPKOuil</! ! e<!hkqe<lieg<<! %xieK 0 g<Gl< 1! g<Gl<! -jmh<hm<m! lqjg! w{<! we<hjk! nxqgqe<Oxil</!!
e<!OfIg<<%xieK x e<!lkqh<jhh<!ohiVk<K!yV!lqjg!LPuigOui!nz<zK!Gjx!LPuigOui!nz<zK!H,s<sqbligOui!-Vg<Gl</!!x < 1!weqz<?! e<!Ofi<g<%xieK!yV! Gjx! LPuig! -Vg<Gl<A! x >10! weqz<?! e<! Ofi<g<%xieK! yV! lqjg!LPuig!-Vg<Gl<A 1 < x < 10!weqz<? e<!Ofi<g<%xieK!H,s<sqblig!-Vg<Gl</
x10logx10log
x10log
x10logx10log
x10logx10log
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wMk<Kg<gim<M 18: hqe<uVl<! w{<gtqe<! ohiK! lmg<jggtqe<! Ofi<g<%Xgjtg<!g{<Mhqcg<gUl</ (i) 2003 (ii) 200.3 (iii) 20.03 (iv) 2.003 (v) 0.2003 (vi) 0.02003 (vii) 0.002003 (viii) 0.0002003 (ix) 0.00002003 (x) 0.000002003 kQi<U: (i) 2003 NeK 103 g<Gl< 104 g<Gl<!-jmbqz<!-Vh<hkiz<? e<!lkqh<hieK 3+0. d
2003log10
1d2d3d4!we<xqVg<Gl</!!weOu! e<!Ofi<g<%X!3; hkqe<lieg<%X 0. d2003log10 1d2d3d4 NGl</!!OuX!upqbig?!2003!J!nxquqbz<!GxqbQm<cz<!wPk?!2003 = 2.003 × 103.
0 = )10003.2(log2003log 31010 ×= 3
1010 10log003.2log + = 10log3003.2log 1010 + = 432110 .03003.2log3 dddd+=+ . 0! e<!OfIg<%X!= 3. 2003log10
[x! wEl<! lqjg!olb<ob{<{qje! a × 10n we<xuiX!nxquqbz<!GxqbQm<cz<! wPkqeiz<?!-r<G!gqjmg<gh<ohXl<!LP!n!NeK e<!Ofi<g<%xiGl<A! e<!lkqh<hieK
e<!hkqe<lieg<!%xiGl</] x10log a10log
x10log(ii) 200.3 = 2.003 × 102
0 200.3 e<!Ofi<g<%X!= 2 ; loglx<Xl<! log 200.3 e<! hkqe<lieg<%X = 2.003 ( log 200.3 e<! hkqe<lieg<%Xl<?!!!!!
log 2003 e<!hkqe<lieg<%Xl<!sll<!we<hjk!nxqg/* 10log
(iii) 20.03 = 2.003 × 101
∴ log 20.03 e<!Ofi<g<%X!=!1; lx<Xl<!log 20.03 e<!hkqe<lieg<!%X!= 2.003. log
(iv) 2.003 = 2.003 × 100
∴ log 2.003 e<!Ofi<g<%X 0; log 2.003 e<!hkqe<lieg<!%X log 2.003. (v) 0.2003 = 2.003 × 10−1
∴ log 0.2003 e<!Ofi<g<%X −1; log 0.2003 e<!hkqe<lieg<!%X log 2.003. (vi) 0.02003 = 2.003 × 10−2
∴ log 0.02003 e<!Ofi<g<%X −2; log 0.02003 e<!hkqe<lieg<!%X!log 2.003. (vii) 0.002003 = 2.003 × 10−3
∴ log 0.002003 e<!Ofi<g<%X −3; log 0.002003 e<!hkqe<lieg<!%X!log 2.003. (viii) 0.0002003 = 2.003 × 10−4
∴ log 0.0002003 e<!Ofi<g<%X −4; log 0.0002003 e<!hkqe<lieg<!%X!log 2.003. (ix) 0.00002003 = 2.03 × 10−5
∴ log 0.00002003 e<!Ofi<g<%X −5; log 0.00002003 e<!hkqe<lieg<!%X log 2.003 (x) 0.000002003 = 2.003 × 10−6
∴ log 0.000002003 e<!Ofi<g<%X −6; log 0.000002003 e<!hkqe<lieg<!%X log 2.003. Olx<gi[l<!wMk<Kg<gim<MgtqzqVf<K?!fil<!nxquK?!
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(i) lqjg! w{< x e<!LP! w{<! hGkq n -zg<gr<gt<! ogi{<m! w{<! weqz<? e<!Ofi<g<%X = n − 1.
x10log
(ii) lqjg!w{< x e<!LP!w{<!hGkq!0!lx<Xl<!x!e<!hkqe<lie!)kslk<kie*!hGkqbqz<!hkqe<lie! Ht<tqg<G! uzh<Hxk<kqz<! dt<t! Lkz<! H,s<sqblx<x! -zg<gk<kqx<G! Le<! n!H,s<sqbr<gt<!ohx<xqVh<hqe<?! e<!Ofi<g<!%X = − (n + 1). x10log (iii) 2003000, 2003, 200.3, …, 0.000002003 -jubjek<Kl<!yOv!hkqe<lieg<!%xqjeh<!ohx<Xt<te/! ! NgOu?! kvh<hm<m! uiqjsbqz<! njlf<K! sllie! lkqh<Hjmb!-zg<gr<gjth<!ohx<x!njek<K!w{<gTg<Gl<!yOv!hkqe<lieg<!%Xkie<!-Vg<Gl</ wMk<Kg<gim<M 19: log 162 = 2.2095 kvh<hce<?!hqe<uVueux<jxg<!g{<Mhqcg<gUl<;! (i) log 1620 (ii) log 16.2 (iii) log 1.62 (iv) log 0.162 (v) log 0.0162 (vi) log 0.00162. kQi<U: log 162 = 2.2095. (i) log 1620 = 3.2095. (ii) log 16.2 = 1.2095. (iii) log 1.62 = 0.2095. (iv) log 0.162 = −1 + 0.2095. (v) log 0.0162 = −2 + 0.2095. (vi) log 0.00162 = −3 + 0.2095. Gxqh<H: −3 +0.2095 J!SVg<glig 3 .2095!we!wPKOuil<. -jkh<!OhizOu −5 + 0.1023 J 5 .1023 we! wPKOuil</! ! g{g<gqMl<ohiPK?! yV! ohiK! lmg<jgg<G! Gjx! w{<!gqjmg<gzil</! ! kGf<k! lqjg! LPju! nf<k! Gjx! w{<[me<! %m<c! gpqg<g?! Gjx!
w{<{igg<! gqjmk<k! lmg<jgbqje! )Gjx! LP* + 0.d1d2d3d4 we<x! njlh<hqz<! wPk!LcBl</!
wMk<Kg<gim<M 20: log 0.25 = 1 .3979 lx<Xl< log 2003 = 3.3016 we<X! kvh<hce<?!hqe<uVueux<jxg<!g{<Mhqcg<gUl</
(i) log 0.025 (ii) log 0.0025 (iii) log ⎟⎠⎞
⎜⎝⎛
200325 (iv) log ⎟
⎠⎞
⎜⎝⎛
2500003.2
kQi<U: log 0.25 = 1 .3979 = −1 + 0.3979. 0 log 0.25 e<! hkqe<lieg<!%X 0.3979 NGl</!-jkh<OhizOu?!log 2003 = 3.3016 = 3 + 0.3016. 0!log 2003 e<!hkqe<lieg<!%X = 0.3016. (i) log 0.025 = 2 .3979 (ii) log 0.0025 = 3 .3979
(iii) log ⎟⎠⎞
⎜⎝⎛
200325 = log 25 − log 2003 3.3016
1.3979
1.9037
2.0000 1.9037
0.0963
= 1.3979 − 3.3016 = − 1.9037. = − 2 + 2 −1.9037 = 2 .0963.
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(iv) log ⎟⎠⎞
⎜⎝⎛
2500003.2 = log 2.003 − log 2500
= 0.3016 − 3.3979 = −4 + 4. 3016 − 3.3979 = 4 .9037.
3.2.4 lmg<jggtqe<!nm<muj{
1.000 Lkz< 9.999 (3 hkqe<lieh<!hGkqBme<) ujvbqzie!lqjg!w{<gtqe<!ohiK!lmg<jggtqe<! lkqh<Hg<gjtg<! g{g<gqm<M! nm<muj{! njlh<hqz<! hm<cbz<! hMk<kq!dt<tei</! ! )-f<F~zqe<! gjmsq! fie<G! hg<gr<gjth<! hii<g<gUl<*/! ! -u<um<muj{jb!
“lmg<jg!nm<muj{”!we<Ohil</!!-f<k!nm<muj{jbh<!hbe<hMk<kq?!wf<kouiV!lqjg!w{<{qx<Gl<!ohiK!lmg<jgbqjeg<!g{g<gqmzil</! !lmg<jgbqe<! hbe<hiMgjth<! hx<xq!nxqf<K! ogit<ukx<G! Le<! lmg<jg! nm<muj{jb! hck<kxqBl<! Ljxbqjeh<!hpg<gh<hMk<kqg<!ogit<Ouil</ wMk<Kg<gim<M 21: log 36.78 Jg<!g{<Mhqc/ kQi<U: 36.78 = 3.678 × 101.
0!Ofi<g<!%X =1. hkqe<lieg<!%xqjeh<!ohXukx<G? 3.678 Jg<!gVKg/!!lmg<jg!nm<muj{bqe<!-mh<hg<g!uqtql<H!fqvzqz< (column) 3.6 e<!-mk<jk!nxqbUl</!!!!!!!!!
Mean differences (svisiq!uqk<kqbisl<) ! 0 1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9 3.6 ! ! ! ! ! ! ! .5647 ! ! ! ! ! ! ! ! ! 10 !
3.6! dt<t! fqjx! (row)! upqbig! hck<Kg<ogi{<Om! ose<xiz< 7! e<! fqvZg<Gg<! gQp<!0.5647g<!gi{<Ohil</!!OlZl<!nOk!fqjxbqz<!okimi<f<K!ose<X!svisiq!uqk<kqbisk<kqz<!8 e<!fqvzqe<!gQp<!w{<!10Jg<!gi{zil</!!-f<k!w{< 10 J 0.5647 dme<!%m<m!0.5657 gqjmg<Gl</! ! -KOu! 3.678!e<! ohiK! lmg<jgbiGl<! )hkqe<lieg<! %xiGl<*/! ! weOu? log 36.78 = 1 + 0.5657 = 1.5657.
wMk<Kg<gim<M 22: log 0.00200316 Jg<!g{<Mhqcg<gUl</ kQi<U: 0.00200316 = 2.00316 × 10−3.
0! Ofi<g<! %X = −3, hkqe<lieg<! %X = log 2.00316. -h<ohiPK! 2.00316 Jg<!gVKg/! ! -kqz<! 5! hkqe<lier<gt<! dt<te/! ! Neiz<! ohiK! lmg<jgbqz<! 3!hkqe<lier<gt<!dt<t!w{<gTg<Gh<!hm<cbz<!-mh<hm<Mt<te/!!weOu!2.00316!≈ 2.003!we! Okivibh<hMk<kqg<ogit<Ouil<! )4uK! hkqe<lie! -mk<kqz<! 5J! uqmg<! Gjxuie! 2!-Vh<hkiz<*/! !-h<ohiPK?!lmg<jg!nm<muj{jbg<!ogi{<M! log 2.003 = 0.3016!we!nxqbzil</!weOu?! log 0.00200316 ≈ −3 + 0.3016 = 3 .3016. wMk<Kg<gim<M 23: log 730.391Jg<!g{<Mhqcg<gUl</ kQi<U: 730.391 = 7.30391 × 102.
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0!Ofi<g<!%X = 2. hkqe<lieg<!%X = log 7.30391. 7.30391 ≈ 7.304 (Woeeqz< 4 uK!hkqe<liel<! 9.! -K! 5 J! uqms<! sqxqbkz<z). weOu! lmg<jgh<! hm<cbzqz<! -Vf<K!!!!!!log 7.304 = 0.8635. 0 log 730.391 ≈ 2 + 0.8635 = 2.8635. 3.2.5 wkqi<lmg<jg!nm<muj{
log x = y weqz< x J!y!e< wkqi<lmg<jg!we<gqOxil</!!nkiuK?! = y!weqz<?
x = 10
x10logy NeK y! e<! wkqi<lmg<jgbiGl</! weOu? y e<! wkqi<lmg<jg = 10y. wkqi<lmg<jg!
ohXuK!lmg<jgh<!ohXukx<G!Ofi<lixiGl</! !wMk<Kg<gim<mig?! log 20 = 1.3010 weqz<!1.3010 e<!wkqi<lmg<jg 20!NGl</
0.0000 zqVf<K 0.9999 (fie<G! -zg<gr<gt<! dt<t! hkqe<lieh<! hGkq) dt<t!w{<gtqe<!wkqi<!lmg<jggTg<gie!nm<muj{!-f<F~zqe<!gjmsq!-v{<Mh<!hg<gr<gtqz<!ogiMg<gh<hm<cVg<gqe<xK/! ! -u<um<muj{jbh<! hbe<hMk<kq! ogiMg<gh<hm<m! wf<kouiV!w{<{qe<!wkqi<!lmg<jgbqjeBl<!ohxzil</ wMk<Kg<gim<M 24: hqe<uVueux<xqe<!wkqIlmg<jgbqjeg<!g{<Mhqcg<gUl</ (i) 1 .2305 (ii) 3 .4629 (iii) 1.8658 (iv) 2.0578 kQIU: !
!( y e<! wkqIlmg<jgjbg<! g{<Mhqcg<g! Lkx<g{<! y e<! hkqe<lieg<! %xqje!lm<Ml<! gVKg/! hqe<H! hkqe<lieg<! %xqe<! Lke<! &e<X! -zg<gr<gTg<G! Wx<x! wkqI<!
lmg<jgbqjeg<!g{<M!nkEme<!svisiq!uqk<kqbisk<kqz<!dt<t!4 uK!-zg<gk<kqx<gie!lkqh<hqjeg<!%m<mg<!gqjmh<hK!Okjubie!wkqIlmg<jgbiGl</) (i) 1 .2305 e<!wkqIlmg<jg!!
= 2305.110 = 10−1 + 0.2305
= 10 −1 × 100.2305
= 700.1101
×
= 0.1700 (ii) 3 .4629 e<!wkqIlmg<jg =10−3 + 0.4629
= 10− 3× 100.4629
= 903.210
13 ×
= 0.002903. (iii) 1.8658 e<!wkqIlmg<jg = 101.8658
wkqv<lmg<jgh<!hm<cbzqzqVf<K0.230 1.698 svisiq!uqk<kqbisl<!5 g<G!Wx<xK!!!!!!! 2
1.700
2.897 6 2.903
7.328 13 7.341
= 101 + 0.8658
= 101 × 100.8658
= 10 × 7.341 = 73.41
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(iv) 2.0578 e<!wkqIlmg<jg = 102.0578
= 102 + 0.0578 1.140 2 1.142
= 102 × 100.0578
= 102 × 1.142 = 114.2
Gxqh<H: Olx<gi[l<!wMk<Kg<gim<czqVf<K?!fil<!nxquK (i) log x e<!OfIg<!%xieK n wEl<!Gjxbz<zi!LP!weqz<?! log x e<!wkqIlmg<jgbqz<!dt<t!hkqe<lieh<!Ht<tqbieK ( n + 1)!uK!-zg<gk<kqx<Gh<!hqe<!-mh<hMgqe<xK/
OfIg<%X! hkqe<lieh<!Ht<tqbqe<!fqjz!0 1 uK!-zg<gk<kqx<G!nMk<K!1 2 uK!-zg<gk<kqx<G!nMk<K 2 3 uK!-zg<gk<kqx<G!nMk<K 3 4 uK!-zg<gk<kqx<G!nMk<K 4 5 uK!-zg<gk<kqx<G!nMk<K
(ii) log x e<!OfIg<%X!− n we<x!Gjx!w{<!weqz<?!log x e<!wkqIlmg<jgbqz<?!hkqe<lieh<!Ht<tqbieK?! hkqe<lieh<! Ht<tqg<G! nMk<Kt<t! n − 1 -mr<gtqz< H,s<sqbLl<?! n uK!-mk<kqz<!H,s<sqblqz<zi!w{<!uVliX!-mh<hMgqe<xK/
OfIg<%X! hkqe<lieh<!Ht<tqbqe<!fqjz!−1 0.d1d2d3d4
−2 0.0d1d2d3d4
−3 0.00d1d2d3d4
−4 0.000d1d2d3d4
−5 0.0000d1d2d3d4
Olx<gi[l<!nm<muj{bqz<, d1. d2d3d4 we<hK!kvh<hm<m!lmg<jgbqz<!dt<t!hkqe<lieg<!%xqeK!wkqIlmg<jgbiGl</ 3.2.6 lmg<jgbqjeh<!hbe<hMk<kqg<!g{g<gqmz<
-e<jxb!dzgqeqz<?! g{g<gqmjz! lqgOugligUl<?! lqgk<! Kz<zqbligUl<! ohx!
lqe<e[!g{g<gqMuie<! )calculator) lx<Xl<! g{qeq! (computer) -jubqv{<Ml<!dt<te/!Neiz<!-s<!siker<gt<!Wx<hMukx<G!Le<eI!leqk!Nx<xjzg<!ogi{<M!g{g<gqm<Omil</!g{g<gqmjz! lqg! Ouglig! leqk! Nx<xjzg<! ogi{<M! g{g<gqm! lmg<jggt<!nxqLgh<hMk<kh<hm<me/! -kx<gig! lmg<jg?! wkqIlmg<jg! Ngqbux<xqx<gie!nm<muj{gt<!kbiiqg<gh<hm<me/!-h<ohiPK?!g{g<gqmzqz<!lmg<jgbqe<!hbjeg<!gim<m!sqz! wMk<Kg<gim<Mgjtg<! gi{! Lbx<sqh<Ohil</! hqe<uVl<! uib<h<hiMgt<!Okjuh<hMgqe<xe/
68
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(i) namamna loglog)(log +=
(ii) namanm
a logloglog −=⎟⎠⎞
⎜⎝⎛
(iii) .loglog mannma =
wMk<Kg<gim<M 25: gi{<g! (i) 27.91 × 5.49 (ii) 0.02871 × 0.00099 × 482.49 kQIU; (i) x = 27.91 × 5.49 we<g/!hqe<H log x = log (27.91 × 5.49)
= log 27.91 + log 5.49 = 1.4458 + 0.7396 = 2.1854
∴ x = 2.1854 e<!wkqIlmg<jg!
= 102.1854= 102 + 0.1854
= 102 ×100.1854
= 102 × 1.532= 153.2 (ii) x = 0.02871 × 0.00099 × 482.49 we<g/!hqe<H!
log x = log 0.02871 + log 0.00099 + log 482.49
1.4456 2 1.4458
1.531 1 1.532
= 6835.29956.44581.2 ++ = 1372.2 .
x = 1372.2 e<!wkqIlmg<jg!=0.01372. wMk<Kg<gim<M 26: lkqh<hqMg;
(i) 23.16
2003 (ii) 09782.03421.0
kQv<U: (i) x = 23.16
2003 we<g/!hqe<H!3.3016 1.2103 2.0913
log x = log 2003 − log 16.23 = 3.3016 − 1.2103 = 2.0913
∴ x = 2.0913 e<!wkqIlmg<jg = 123.4.
(ii) x = 09782.03421.0 we<g/!hqe<H
log x = log 0.3421 − log 0.09782 = 1 .5341 − 2 .9904 = (−1+0.5341) − (−2+0.9904) = 1 + 0.5341 − 0.9904 = 1.5341 − 0.9904 = 0.5437
∴ x = 0.5437 e<!wkqIlmg<jg!= 3.497.
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wMk<Kg<gim<M 27: hqe<uVueux<xqe<!lkqh<Hgjtg<!g{g<gqMg/ (i) (29.76)5 (ii) (0.3749)7
kQIU: (i) x = (29.76)5 we<g/!hqe<H! log x = log (29.76)5 = 5 × log (29.76) = 5 × 1.4737 = 7.3685. ∴ x = 7.3685 e<!wkqIlmg<jg = 23360000. (ii) x = (0.3749)7 we<g/!hqe<H! log x =7× log( 0.3749) = 7 × 1 .5739
= (−1 + 0.5739) × 7 = −7 + 4.0173 = −7 + 4 + 0.0173 = −3 + 0.0173 = 3 .0173
∴ x = 3 .0173 e<!wkqIlmg<jg = 0.001041. wMk<Kg<gim<M 28: 5 2713.0 e<!lkqh<H!gi{<g/!
kQIU: x = 5 2713.0 = (0.2713) 51
we<g/!hqe<H
log x = log (0.2713) 51
.
= 51 × log 0.2713 =
51 × 1 .4335 =
54335.01+− =
54335.45 +−
= −1 + 0.8867 = 1 .8867. ∴ x = 1 .8867!e<!wkqIlmg<jg =0.7702
wMk<Kg<gim<M 29: 46.1828
159.2223.175 × e<!lkqh<hqjeg<!gi{Ul</!
kQIU: x = 46.1828
159.2223.175 × we<g/!hqe<H
log x = log 175.23 + log 22.159 − log 1828.46 1.000 Lkz<! 9.999 (&e<X! hkqlie! hqe<eh<! hGkqBme<) !ujvbqzie! w{<gtqe<! ohiK!lmg<jgh<! hm<cbzieK! kbiI! osb<K! kvh<hm<Mt<tkiz<?! fil<! hqe<uVl<! Okivibr<gjt!osb<b!Ou{<cbqVg<gqxK/
175.23 = 1.7523 × 102
≈ 1.752 × 102. 22.159 = 2.2159 × 101
≈ 2.216 × 101. 1828.46 = 1.82846 × 103
≈ 1.829 × 103. ∴ log x = 2.2435 + 1.3456 − 3.2622
= 3.5891 − 3.2622 = 0.3269. ∴ x = 0.3269!e<!wkqIlmg<jg = 2.122. !
70
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wMk<Kg<gim<M 30: 04623.0)75.42(
928.1)25.76(5
33
×× e<!lkqh<H!g{<Mhqc/!
kQIU: kvh<hm<m!Ogijubqje!x we<g/!hqe<H log x = log (76.25)3 + log 3 928.1 −[log (42.75)5 + log 0.04623]
= 3 log 76.25 + 31 log 1.928−[5 log 42.75 + log 0.04623]
= 3 × 1.8823 + 31 × 0.2851 − [5 × 1.6309 + 2 .6649]
= 5.6469 + 0.0950 − [8.1545 + 2 .6649] = 5.7419 − 6.8194 = − 2 + (7.7419 − 6.8194) = − 2 + 0.9225 = 2 .9925
x = 2 .9925 !e<!wkqIlmg<jg = 0.09828. wMk<Kg<gim<M 31: 4.3562 e<!lkqh<H!gi{<g/!12logkQIU: -r<G! ncliel<! 12 we! dt<tK/! lmg<jg! nm<muj{jbh<! hbe<hMk<k!
lmg<jgbqe<! ncliel<! 10 we! -Vg<g! Ou{<Ml</! ncliek<jk! lix<Xl<! uqkqjbh<!hbe<hMk<k?
12log 4.3562 = 4.3562 × 10 10log 12log
= 4.3562 × 10log12log
1
10
= x=0792.16391.0 we<g/!hqe<H log x = log ⎟
⎠⎞
⎜⎝⎛
0792.16391.0
= 1 .8056 − (0.0331) = −1 + (0.8056 − 0.0331) = −1 + (0.7725) = 1 .7725
∴ x = 1 .7725 e<!wkqIlmg<jg = 0.5923.
hbqx<sq 3.2.2 1. hqe<uVl<!w{<gtqe<!ohiKlmg<jggtqe<!OfIg<%Xgjtg<!gi{<g; (i) 1234 (ii) 27.36 (iii) 3.65 (iv) 0.7851 (v) 0.084 (vi) 0.00532 (vii) 0.00003 (viii) 0.032 × 104
2. hqe<uVliX! ohiK! lmg<jggt<! ohx<Xt<t! w{<gtqe<! LPh<! hGkqbqz<! dt<t!
-zg<gr<gtqe<!w{<{q<g<jgbqjeg<!g{<Mhqcg<gUl<;! (i) 2.345 (ii) 1.456 (iii) 3.4567 (iv) 0.1234 (v) 0.9876 (vi) 3 × 0.982
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3. w{<gtqe<! ohiK! lmg<jggt<! gQOp! kvh<hm<Mt<te/! w{<gtqe<! hkqe<lieh<!
Ht<tqg<G!nh<hiz<!Lkz<! H,s<sqblqz<zik!-zg<gk<kqx<G!Le<uVl<! H,s<sqbr<gtqe<!w{<{qg<jgjbg<!g{<Mhqc/!
(i) 3456.1 (ii) 2 .2345 (iii) 123.3 (iv) 4 .7877 (v) 5 .7245 (vi) 4 .102 4. 32740 e<!ohiK!lmg<jgbqe<!hkqe<lieg<!%X 0.5151 NGl</!hqe<uVl<!w{<gtqe<!
ohiK!lmg<jggjt!wPKg; (i) 32740 (ii) 3274 (iii) 327.4 (iv) 32.74 (v) 3.274 (vi) 0.3274 (vii) 0.0003274 (viii) 0.03274 × 10−5
5. lmg<jg!nm<muj{jbh<!hbe<hMk<kq!hqe<uVl<!w{<gtqe<!ohiK!lmg<jggjtg<!
g{<Mhqcg<gUl</ (i) 8273 (ii) 843250 (iii) 0.001439 (iv) 0.0000324 (v) 0.00468 (vi) 0.2356 6. hqe<uVl<!ohiKlmg<jggtqe<!wkqIlmg<jggjtg<!g{<Mhqcg<gUl</ (i) 2.8903 (ii) 0.4321 (iii) 1 .4583 (iv) 4261.3 (v) 5 .5201 (vi) .0930.3 7. hqe<uVliX!x e<!ohiKlmg<jg!-Vf<kiz<?!x e<!lkqh<jhg<!gi{<g/ (i) 5.3027 (ii) 1.9168 (iii) − 2.0411 (iv) − 3.1773 (v) − 0.3916 (vi) − 4.1083 (vii) 2.12.3 + (viii) 1.24.5 − (ix) 2.1 − 5.4 (x) 23.1 × (xi) 341.3 × (xii) 5 .5 ÷ 3 8. hq<e<uVueux<jx!lkqh<hQM!osb<g/ (i) 25.46 × 80.17 (ii) 37.42 × 816.3 (iii) 1.231 × 0.0084 (iv) 86.3 ÷ 0.0625 (v) (0.0275)3 (vi) (50.49)5
(vii) (525.9)8 (viii) 3 3.452
(ix) 5 08745.0 (x) 0543.0
37.454935.0 ×
(xi) 2
3
)4.132(5.27)23.4( × (xii)
23.394.163.8471.24
××
(xiii) (xiv) 326.1log5 28.63log9
72
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3.3 g{g<!GxqbQM
fuQe!g{qkk<kqe<!sikjeOb!g{g<GxqbQmiGl</!!njek<K!g{qkh<!hqiqUgtqZl<!g{g<GxqbQM!Okie<Xgqe<xK/!!g{qkk<kqje!ncOgit<!ki<g<g!iQkqbqz<!dVuig<g!g{qk!uz<Zfi<gt<! Lbx<sqk<kke<! uibqzig! g{g<GxqbQMgt<! Okie<xqe/! ! o\i<le<! g{qk!
Oljk! \ii<<\<! Oge<mi<! (1845 − 1918) dVuig<gqb! g{r<gtqe<! ogit<jgOb! g{qk!uti<s<sqg<G! jlz<gz<zig! njlf<kK/! ! g{k<kqe<! gVk<KVuqjeBl<?! g{g<ogit<jg!hx<xqb!nch<hjmg<!gVk<Kg<gjtBl<!nxqLgh<hMk<k!-h<ohiPK!Lbz<Ouil</ 3.3.1 yV!g{k<kqe<!gVk<KV
fe<G! ujvbjx! osb<bh<hm<m! dXh<Hgtiz<! Ne! okiGkqjb! g{l< we<Ohil</!!wMk<Kg<gim<mig?! njek<K! -bz<! w{<gtqe<! okiGh<H?! yV! ktk<kqZt<t! njek<K!slhg<g! Lg<Ogi{r<gtqe<! okiGh<H?! kqVk<k{q! nvsqei<! N{<gt<! Olz<! fqjzh<!ht<tqbqe<! njek<K! ye<hkil<! uGh<H! li{ui<gtqe<! okiGh<H?! njek<K!
olb<ob{<gtqe<!okiGh<H?!Nr<gqz!wPk<Kg<gtqZt<t!njek<K vowels.gtqe<!okiGh<H!Ohie<xjugt<! g{r<gTg<Gs<! sqz! wMk<Kg<gim<Mg<gt<! NGl</! ! OlOz! Gxqh<hqm<Mt<t!okiGh<Hg<gtqz<! dt<t! dXh<Hg<gt<?! fl<liz<! wju! we<X! dXkqbqm<Mg<! %x!Lcukiz<kie<!nju!g{r<gt<!Ngqe<xe/!!hqe<uVl<!wMk<Kg<gim<Mgjtg<!gVKOuil<; (i) de<Ejmb!uGh<hqZt<t!dbvlie!li{ui<gtqe<!g{l</ (ii) fQ!hck<k!fz<z!Hk<kgr<gtqe<!g{l</ “dbvl<”?!“fz<z”!Ohie<x!osix<gTg<G!siqbie!ntuqz<!ohiVt<!ogit<t!Lcbikkiz<!Olx<Gxqh<hqm<m! wMk<Kg<gim<Mgt<! fe<G! ujvbXg<gh<hm<mju!nz<z/! ! weOu!nux<xqe<!dXh<Hgjt!fl<liz<!siqbigg<!%x!Lcbuqz<jz/!weOu!nju!g{r<gt<!NgiK/ Gxqh<H: yV!g{k<kqZt<t!dXh<Hgt<!keqk<kjubiGl</
yV! g{k<kqe<! ohiVtqje! yV! keqll<! nz<zK! Yi<! dXh<H! nz<zK! Yi<!
nr<gk<kqei<!we!njph<Ohil</ g{r<gjtg<!Gxqg<g!A nz<zK B Ohie<x!ohiqb!Nr<gqz!wPk<Kg<gjtBl<?! nux<xqe<! dXh<Hgjtg<! Gxqg<g! x,y,a,b… Ohie<x! sqxqb! Nr<gqz!wPk<Kg<gjtBl<!hbe<hMk<KOuil</!!x NeK!g{l<!A z<!Yi<!dXh<H!we<hjkg<!Gxqg<g!x ∈ A!we!wPKOuil</ ‘∈’ we<x!GxqbQM ‘dt<tK ’!weh<!ohiVt<hMl</ keqll< x NeK! Az<!dXh<hz<z!we<hjkg<!Gxqg<g?!x ∉ A!we!wPKOuil</!!wMk<Kg<gim<mig?!A!we<x!g{k<kqe<!keqlr<gt<!1, 3, 4, 5!weqz<?!-kje!1!∈ A, 3 ∈ A, 4 ∈ A, 5 ∈ A. Neiz<!!!!6 ∉ A, −11 ∉ A, 9 ∉ A, Glii< ∉ A.
yV!g{k<kqje!njlg<g?!hqe<uVl<!-VLjxgjth<!hqe<hx<XOuil<: (1) !nm<muj{!nz<zK!uiqjsh<!hm<cbz<!Ljx!(2) !g{g<gm<mjlh<H!nz<zK!uqkqLjx!!
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3.3.2 nm<muj{ nz<zK uiqjsh<!hm<cbz<!Ljxbqz<!g{k<kqjeg<!Gxqk<kz<
njmh<Hgt< { }! -ux<xqx<Gt<! yV! g{k<kqZt<t! njek<K!dXh<Hg<gjtBl<! ?!wEl<! Gxq! ogi{<Mh<! hqiqk<Kh<! hm<cbzqm<M! ng<g{k<kqje! wPKOuil</!!
wMk<Kg<gim<mig?! ‘mathematics’ we<El<! osiz<zqZt<t! njek<K!dbqi<! wPk<Kg<gtqe<!g{l< {a, e, i}!NGl</ -r<G a wEl<!wPk<K!-VLjx!‘mathematics’!we<El<!osiz<zqz<!uf<kiZl<! g{k<kqz<! Gxqh<hqm<omPKl<OhiK! yOvobiV! Ljx! lm<MOl! dXh<hig!wPKOuil</! ! Woeeqz<! g{k<kqz<! dt<t! dXh<Hgt<! keqk<kjugtig! -Vk<kz<!
Ou{<Ml</! ! -u<uiX! dXh<Hgjt! { }! wEl<! njmh<Hg<Gt<! wPKl<! Ljxbqje!nm<muj{! nz<zK! uiqjsh<hm<cbz<! Ljx! we<Ohil</! ! nm<muj{! Ljxbqz<! sqz!
g{r<gjt!hqe<uVliX!wPKgqOxil<: (i) 13 J!uqmg<!Gjxuie!hgi!w{<gtqe<!g{l<!{2, 3, 5, 7, 11}!NGl</ (ii) FOOTBALL we<El<!osiz<zqZt<t!wPk<Kgjtg<!ogi{<m!g{l<!!
{F, O, T, B, A, L} (iii) njek<K! -bz<! w{<gtqe<! g{l< {1, 2, 3,…}. -r<G… we<x! GxqbQM?!okimi<f<K! uVl<! njek<K! dXh<HgjtBl<! ye<xqjeBl<! uqmiK! Gxqh<hkiGl</!!-g<g{k<kqje N!wEl<!sqxh<ohPk<kiz<!Gxqh<Ohil<A!N = {1, 2, 3, …} (iv) njek<K! LP! w{<gt<! 0, 1, 2, 3, … Ngqbeux<jx! dXh<Hg<gjtg<!ogi{<m!g{k<kqje W we<x!sqxh<ohPk<kiz<!Gxqh<Ohil<A!W = {0, 1, 2, 3,…}. (v) njek<K! LPg<gtie 0, 1, −1, 2, −2, 3, −3,… Ngqbeux<jx!dXh<hqei<gtigg<! ogi{<m! g{l<! Z we<x! sqxh<ohPk<kiz<! Gxqg<gh<hMgqe<xKA!Z={0, 1, −1, 2, −2, 3, −3, …}.
wMk<Kg<gim<M 32: 5 e<!lmr<Ggt<?!Neiz< 50!g<Gg<!Gjxuie!njek<K!-bz<!-vm<jmh<!hjm!w{<gjtg<!ogi{<m!g{k<kqje!nm<muj{!Ljxbqz<!wPKg/ kQi<U: 5 Nz<!uGhMl<!-vm<jmh<hjm!-bz<!w{<gt< 10, 20, 30, 40, 50, …!NGl</!
∴ 50g<Gg<!Gjxuie!5Nz<!uGhMl<!njek<K!-vm<jmh<hjm!w{<gtqe<!g{l< ! ={10, 20, 30, 40}.
3.3.3 g{g<!gm<mjlh<H nz<zK!uqkqLjxbqz<!g{k<kqje!njlk<kz<
g{k<kqjeg<! g{g<gm<mjlh<H! we<El<! hqxqokiV! LjxbqZl<! wPkzil</!!-l<Ljxbqz<!g{k<kqje!wPk?!Lkzqz<!nke<!dXh<HgTg<gqjmOb!dt<t!ohiKuie!h{<hqje!nxqb! Ou{<Ml</! !-h<h{<H?!ng<g{k<kqz<!dt<t!dXh<HgTg<G!lik<kqvOl!
d{<jlbig!Ou{<Ml</!!wMk<Kg<gim<cx<G?!{6, 36, 216}!we<x!g{k<kqjeg<!gVKOuil</!!-ke<! dXh<Hgt<! 6, 36, 216! we<he 6e<! nMg<Ggt<! wEl<! h{<hqjeh<! ohx<Xt<te/!!weOu x = 6n, -r<G n = 1, 2, 3 wEl<! fqhf<kje 6, 36, 216!Ngqb!w{<gjt!lm<MOl!kVgqe<xK/!!-f<k!fqhf<kjeg<Gm<hm<M!Ouoxf<k!w{<j{Bl<!ohx!-bziK/!!NgOu?!
g{l< {6, 36, 216} NeK “x = 6n, n = 1, 2, 3”! wEl<! fqhf<kjejb! fqjxU! osb<Bl<!njek<K!dXh<Hgt<!x!Nz<!NeK!we<X!nxqbzil</!!-kje { x | x = 6n, n = 1, 2, 3}!we!wPKOuil</!!-l<Ljxbqz<?!{}!we<x!njmh<Hg<Gxq!“-ux<jxg<!ogi{<Mt<t!g{l<”!we<hjkg<! Gxqg<gh<! hbe<hMgqxK/! ! ‘ | ’ we<El<! (vertical bar) Ofi<Gk<Kg<Ogim<Mg<Gxq!
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“hqe<uVueux<jxk<!kvuz<z” (such that)!we<x!ohiVtqz<!hbe<hMk<kh<hMgqxK/!!ohiKh<!h{<hie “x = 6n, n = 1, 2, 3”. we<hK! g{k<kqje!dVuig<Gl<! uqkqbig! -Vh<hkiz<!-l<Ljx! uqkqLjx! nz<zK! g{g<gm<mjlh<H! Ljx! weh<hMgqe<xK/! ! Ouoxf<k!
dXh<hqx<ge<xq! ogiMg<gh<hm<m!g{l< Ae<!dXh<hqx<G!lm<MOl!h{<H!P ohiVk<kLx<xiz<?!g{l< A J!A = { x | h{<H!P J!x ohx<Xt<tK} we!wPKOuil</!!-kje?!h{<H!P J!ohx<Xt<t!njek<K!dXh<Hgt<!x gtqe<!g{l< A!weh<!hch<Ohil</ wMk<Kg<gim<M 33: hqe<uVueux<jx!uqkqLjxbqz<!g{ligg<!Gxqg<gUl<: (i) 6J!uqmg<!Gjxuie!njek<K!-bz<!w{<gtqe<!g{l</ (ii) Nr<gqz!wPk<Kg<gtqZt<t!njek<K!dbqi<!wPk<Kg<gtqe<!g{l</
(iii) 2, 4, 6, ….!Ngqb!w{<gjtg<!ogi{<m!g{l</ kQi<U: (i) “x ∈ N, x < 6”!we<hK!x!yV!-bz<!w{<!lx<Xl<!6 J!uqmg<!GjxU!we<hjk!uquiqg<Gl<!%x<xiGl</ ∴ { x | x ∈ N, x < 6}!we<hK!Okjubie!g{k<kqjeg<!Gxqh<hkiGl</ (ii) GxqbQM x we<hK!Yi<!Nr<gqz!dbqi<!wPk<kqjeg<!Gxqh<hkigg<!ogi{<miz<? ∴ {x | x yV!Nr<gqz!dbqi<!wPk<K}!we<hOk!Okjubie!g{liGl</!-g<g{k<kqje!!{ x | x = a, e, i, o, u} nz<zK!{a, e, i, o, u} we!wPkzil</!(iii) 2, 4, 6, … we<x!ucuqZt<t!Yi<!w{<j{ “x = 2n, n ∈ N” we<x! %x<xqeiz<!uquiqg<gzil</!∴ { x | x = 2n, n ∈ N}we<hK!Okjubie!g{liGl</ Gxqh<H: dXh<Hgjth<! hm<cbzqmilz<! nux<xqe<! h{<Hgjtg<! ogi{<M! g{g<gm<mjlh<H!Ljxbqz<! Gxqh<hqmh<hMukiz<! g{g<gm<mjlh<H! Ljxbqje! uquiqk<kz<! Ljx! weUl<!njpg<gzil</ wMk<Kg<gim<M 34: g{l< A = {x | x + 5 = 7, x ∈ N}J!nm<muj{!Ljxbqz<!wPKg/ kQi<U: x + 5 = 7 ⇒ x = 7 − 5 = 2. -r<G 2 ∈ N.
w{<!2Jk<!kuqv!Ouoxf<k!w{<[l<!“x + 5 = 7, x ∈ N” we<x!h{<hqje!fqjxU!osb<biK/!
∴ A = {2}.
wMk<Kg<gim<M 35: g{l< A = ⎭⎬⎫
⎩⎨⎧
71 ,
61 ,
51 ,
41 ,
31 ,
21 1, J!g{g<gm<mjlh<H!Ljxbqz<!wPKg/
kQi<U: g{l< A!e<!dXh<Hg<gt<!Lkz<!WP!-bz<!w{<gtqe<!kjzgQpqgtiGl</!!weOu!!
A = ⎭⎬⎫
⎩⎨⎧ ≤∈= 7 1| nn,
nxx Xl<lx<N .
Gxqh<H: ‘x, y’ we<hK ‘x lx<Xl< y’!weg<!Gxqg<Gl</ 3.3.4 LcUX!lx<Xl<!Lcuqzq!g{r<gt<
g{l< A!e<!dXh<Hg<gjt!ye<xe<!hqe<!ye<xig!w{<{qmk<!okimr<gq?!ns<osbz<!LcU!ohXlieiz<?!g{l<!A!J!LcUX!g{l<!we!njpg<gqOxil</
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nu<uiX!w{<[l<!Ljx!LcU!ohxuqz<jzobeqz<!A J Lcuqzq!g{l< we<Ohil</!!g{l<!A e<!dXh<Hg<gtqe<!w{<{qg<jg?!Nkq!w{<!we<X!njpg<gh<hMl</!!-kje!n(A) !weg<!Gxqh<Ohil</!NgOu!n(A) we<hK!g{l< Ae<!dXh<Hg<gtqe<!w{<{qg<jgbiGl</!!A yV!LcUX!g{l<!weqz<!n(A) yV!LP!w{<!NGl</ wMk<Kg<gim<M 36: hqe<uVueux<xqzqVf<K! LcUX! lx<Xl<! Lcuqzq! g{r<gjtg<!Gxqh<hqMg:
(i) A = {x | x ∈W, x < 5}. (ii) {klqpgk<kqZt<t!njek<Kh<!ht<tqgt<}. (iii) {deK!ht<tqbqZt<t!njek<K!ye<hkil<!uGh<H!li{ui<gt<}. (iv) N. (v) W. (vi) Z. (vii) njek<K!hgi!w{<gt<.
kQi<U: (i) x ∈W, x < 5 ⇒ x = 0, 1, 2, 3, 4. ∴ A = {0, 1, 2, 3, 4}. A e<!dXh<Hg<gt<!0,1, 2, 3, 4 gjt!LjxOb!Lkzil<?!-v{<mil<?!&e<xil<?! fie<gil<! lx<Xl<! Jf<kil<! dXh<Hg<gt<! we! w{<{qMl<! osbz<! LcUXgqxK/!!
weOu! n(A) = 5 lx<Xl< A yV!LcUX!g{l<. (ii) klqpgk<kqZt<t!njek<Kh<!ht<tqgjtBl<!yu<ouie<xig!w{<{qm!Nvl<hqk<kiz<?!ns<osbz<! yV! gm<mk<kqz<! LcUXl</! ! weOu {klqpgk<kqZt<t! njek<Kh<! ht<tqgt<} wEl<!g{l<!LcUX!g{liGl<. (iii) deK! ht<tqbqZt<t! ye<hkil<! uGh<H! li{ui<gt<! njeujvBl<! w{<{qMl<!osbz<! LcUXl</! ! weOu {deK! ht<tqbqZt<t! njek<K! ye<hkil<! uGh<H!
li{ui<gt<} g{l<!LcUX!g{liGl<. (iv) N = {1, 2, 3,…}.!-g<g{k<kqe<!dXh<Hg<gjt!yu<ouie<xig!w{<{qm!Nvl<hqk<kiz<?!-s<osbz<Ljx!Lx<Xh<ohxiK/!!weOu N yV!Lcuqzq!g{liGl</ (v) W = {0, 1, 2, 3,…}. W e<!dXh<Hgjt!yu<ouie<xig!Lkz<!dXh<H?!-v{<mil<!dXh<H?////!we!w{<[l<!Ljx!Lx<Xh<ohxiK/!!weOu!W!yV!Lcuqzq!g{liGl</ (vi) Z = {0, 1, −1, 2, −2,…}!e<!dXh<Hg<gtie 0, 1, −1, 2, −2…. J LjxOb!Lkzil<!dXh<H?!-v{<mil<!dXh<H////!we!w{<[l<!Ljx!Lx<Xh<ohxiK/!!NgOu Z l<!Lcuqzq!g{liGl</ (vii) hgi! w{<gt< 2, 3, 5, 7, 11, 13, 17… gjt! w{<[l<! Ljx! Lx<Xh<ohxikkiz<?!
njek<Kh<!hgi!w{<gtqe<!g{l<!Lcuqzq!g{liGl<. 3.3.5 oux<Xg<g{l<!nz<zK!dXh<hqzq!g{l<
dXh<Hgt<!nx<x!yV!g{k<kqje!oux<Xg<g{l<!nz<zK!dXh<hqzq!g{l<! we!njph<Ohil</!!oux<Xg<g{k<jk!Ø!weg<!Gxqh<Ohil</ Ø = { }!lx<Xl< n(Ø) = 0. wMk<Kg<gim<M 37: hqe<uVueux<Xt<!oux<Xg<!g{r<gt<!biju@ (i) {2Nz<!uGhMl<!yx<jxh<!hjm!-bz<!w{<gt<} (ii) {4J!yV!giv{qbigh<!ohx<x!hgi!w{<gt<} (iii) {x | x ∈W, x ∉ N}. (iv) {Ø}.
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kQi<U: (i) 2Nz<!uGhMl<!yx<jxh<!hjm!-bz<!w{<gt<!wKUlqz<jz!we<hkiz<?
{2 Nz<!uGhMl<!yx<jxh<hjm!-bz<!w{<gt<} = Ø NGl</ (ii) Yi<! w{<{qe<! uG! w{<gt<! 1! lx<Xl<! nOk! w{<! we! lm<MOl! -Vh<hqe<?! nf<k!w{<j{h<!hgi!w{<!we<Ohil</!weOu?!4 J!giv{qbigg<!ogi{<m!hgi!w{<!wKlqz<jz/!∴ {4J!yV!giv{qbigh<!ohx<x!hgi!w{<gt<} = Ø!NGl</ (iii) W g<G!dXh<higUl<!Neiz< Ng<G!dXh<hig!-z<zik!yOv!dXh<H!1!NGl</ nkiuK? x ∈W, x ∉ N ⇒ x = 0. ∴ {x | x ∈W, x ∉ N} = {0}. -K!YVXh<H!g{l<!NGl</!!weOu!-K!yV!oux<xx<x!g{l<!NGl</ (iv) {Ø} z<!oux<Xg<g{l<!Ø Yi<!dXh<hiGl</!!weOu {Ø} oux<Xg<g{lz<z/!!nkiuK! {Ø} ≠ Ø. 0!{Ø} YI!YVXh<H!g{liGl</! 3.3.6 slie!g{r<gt<
g{r<gt< A, B we<he! yOv! w{<{qg<jgbqz<! dXh<Hg<gjtg<! ogi{<cVh<hqe<!njugt<! slie! g{r<gt<! weh<hMl</! ! g{l< A NeK! g{l< Bg<G! slie! g{lig!-Vh<hqe<!n(A) = n(B) NGl</!!-kje A ≈ B!we!wPKOuil</!!wMk<Kg<gim<mig?
A = {1, 2, 3}, B = {11, 9, 23}, n(A) = 3, n(B) = 3. weOu A ≈ B. 3.3.7 sl!g{r<gt<
-V! g{r<gt< A, B we<he! nOk! dXh<Hgjth<! ohx<xqVh<hqe<?! nux<jxs<! sl!g{r<gt<! we<gqOxil</! ! -u<uixibqe<! A = B! we! wPKgqe<Oxil</! ! wMk<Kg<gim<mig?!!!!A = {1, 2, 3, 4}, B = {x | x ∈ N, x < 5}weqz< A = B. Woeeqz<? Bz<!dt<t!uqkq!
x ∈ N, x < 5 ↔ x = 1, 2, 3, 4. fil<! nxquK?! A, B! -v{<cZl<! nOk! dXh<Hgt<! -Vh<hOk! A, B! -ju!
sllibqVk<kjz!dXkq!osb<ukiGl</!!weOu!yV!g{k<kqz<!dt<t!dXh<Hgjt!wf<k!uiqjsbqZl<!uqVh<hl<!Ohiz<!wPkzil</!!wMk<Kg<gim<mig?!!
{1, 2, 3, 4} = {4, 3, 1, 2 } = {3, 1, 2, 4} = {2, 4, 3, 1}! Gxqh<H: -V! g{r<gt<! sloleqz<?! nju! yOv! w{<{qg<jgbqzie! dXh<Hgjtg<!
ogi{<Mt<te/!!weOu!nju!slie!g{r<gt</!wMk<Kg<gim<mig?!{1, −1, 2, −2} lx<Xl<!{1, 2, −1, −2} -v{<Ml<!slg<g{r<gtibqVk<kOziM!slie!g{r<gtigUl<!-Vg<gqe<xe/!Neiz<!slie!g{r<gt<!slg{r<gtig!-Vg<gOu{<Ml<!we<hkqz<jz/!wMk<Kg<gim<mig? {1, −1, 2, −2} lx<Xl< {1, 2, 3, 4} we<El<!g{r<gt<!yOv!w{<{qg<jgbqzie!dXh<Hgjtg<!ogi{<Mt<tkiz<! nju! slie! g{r<gt</! ! Neiz<! nju! sl! g{r<gtigiK/!!
Woeeqz<? −1, −2 ∈ {1, −1, 2, −2} lx<Xl<! −1, −2 ∉ {1, 2, 3, 4}. 3.3.8 YVXh<Hg<!g{l< Yi<! dXh<H! lm<MOl! ogi{<m! g{k<kqje! Yi<! YVXh<Hg<! g{l<! we<gqOxil</!!wMk<Kg<gim<mig?! -vm<jmh<! hjm! hgi! w{<gtqe<! g{l<! Yi<! YVXh<Hg<! g{liGl</!!
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Woeeqz<?!hgi!w{<gtqz<!-vm<jmh<hjm!w{<{ig!-Vh<hK!2!lm<MOl!NGl</!!weOu!{njek<K!-vm<jmh<hjm!hgi!w{<gt<}!= {2} !lx<Xl<! n(A) = 1. 3.3.9 njek<Kg<g{l<
yV!g{qk!Nb<uqz<! Okie<Xl<!g{r<gt<!njek<kqe<!dXh<Hgt<!yV!g{k<kqz<!
dXh<Hgtig! -Vg<Gl</! ! -g<g{k<kqje! njek<Kg<! g{l< (universal set)!we<xjpg<gqe<Oxil</! ! njek<Kg<! g{k<kqje! U nz<zK E! we<x! Gxqbiz<!Gxqh<hqMgqe<Oxil</! ! wMk<Kg<gim<mig?! yV! g{qk! Nb<uqz<! gVKl<! g{r<gt< A = {2, 3, 4, 5}, B = {1, 3, 7, 11}!weqz<?!njek<Kg<!g{l<!U uqje U = {1, 2, 3, 4, 5, 7, 11,} nz<zK U = N nz<zK U = W nz<zK U = Z!weg<!ogit<tzil</ 3.3.10 dm<g{l<
A lx<Xl< B we<he! -V! g{r<gt<! we<g/! ! A e<! yu<ouiV! dXh<Hl< B e<!dXh<hieiz<? A bqje! B e<! Yi<! dm<g{l< (subset) we<xjpg<gqe<Oxil</! ! -kjeOb!!!!!B! NeK! A e<! lqjgg<g{l<! (superset)! we<xjpg<gqe<Oxil</! ! -u<U{<jljb A ⊆ B nz<zK B ⊇ A!we<x!Gxqbiz<! wPKgqOxil</! !-r<G ⊆ we<x!Gxq!dm<g{l<!nz<zK!dt<tmr<gqbK!we<hjkg<!Gxqg<gqe<xK/!!-u<uiOx!⊇ we<x!Gxq!lqjgg<g{l<!nz<zK!ogi{<Mt<tK! we<hjkg<! Gxqg<gqxK/! wMk<Kg<gim<mig? A = {1, −1, 2, −2, 3}, B = {1, 2, −1, −2, 3, −3} we<x!g{r<gjtg<!gVKg/!!-r<G!Az<!dt<t!dXh<Hgtie 1, 2, −1, −2, 3 Ngqbju B bqZl<!-Vg<gqe<xe/!!weOu!A!NeK!B e<!dm<g{liGl<A!nkiuK? A ⊆ B. -r<G −3 ∈ B, −3 ∉ A. weOu B NeK! A e<! dm<g{lz<zA! -f<k!d{<jljb! B ⊄ A we<x! Gxqbiz<! Gxqh<hqMOuil</! ! -r<G! ⊄ we<x! GxqbieK!dm<g{lz<z!nz<zK!dt<tmr<gikK!we<X!ohiVt<hMl</ !
Gxqh<H: X !we<x!g{l< Y !we<x!g{k<kqe<!dm<g{ligUl<?!g{l< Y NeK!g{l< X x<G!dm<g{ligUl<! njlf<kiz<?! X l< Y l<! nOk! dXh<Hgjth<! ohx<Xt<te/! ! weOu?!-f<fqjzbqz<!X l<!Y l<!slg<g{r<gtiGl</!!X = Y weqz<?!X!e<!yu<ouiV!dXh<Hl<!Y e<!dXh<hig!-Vg<Gl<;!OlZl<!Y e<!yu<ouiV!dXh<Hl<!X e<!dXh<hig!-Vg<Gl</!X l<? Y l<!g{r<gt<<<A! w{<gt<! nz<z/! ! -Vh<hqEl<! w{<gTg<gqjmOb! hbe<hMk<Kl< ‘=’ Gxqjb!slg<g{r<gjtg<! Gxqg<gh<! hbe<hMk<Kgqe<Oxil</! ! ‘=’! Gxqjb! -Vg{r<gt<! X, Y -ux<xqx<gqjmOb!hbe<hMk<Kl<! OhiK?!fil<!nxquK?!X e<!dXh<Hgt<!njek<Kl<!Y z<!dt<teA!OlZl<!Y e<!dXh<Hgt<!njek<Kl<!X z<!dt<te!we<hOkA!nkiuK?!X l<!Y l<!nOk!dXh<Hgjtg<!ogi{<Mt<te/ !
Gxqh<H: yu<ouiV! g{l<! A l<! A! g<Og! dm<g{lig! -Vg<gqe<xK/! Woeeqz<! A e<!dXh<Hgt<! A! z<! dt<te/! ! oux<Xg<g{l<! Ø NeK! yu<ouiV! g{l<! A! g<Gl<!dm<g{lig! -Vg<gqe<xKA! Woeeqz<! Ø NeK! A! g<G! dm<g{lig!
njlbuqz<jzobeqz<?! oux<Xg<g{l<! Ø z<! A! -z<! -z<zik! Yi<! dXh<H! -Vg<g!Ou{<Ml<A!-K!Lv{<himz<zui@!weOu!A ⊆ A lx<Xl< Ø ⊆ A.
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4/4/22 kG!dm<g{l<!X lx<Xl< Y we<he!-V!g{r<gt<!we<g/!!g{l< X NeK!g{l< Y e<!dm<g{lig!
-Vf<K?! u ∉ X! we<xuiX! Yi<! dXh<H! u ∈ Y! weqz<? X we<x g{lieK! Y! e<! yV!!!!!!kG! g{l<! )proper subset) we<xjpg<gh<hMgqe<xK/! ! -kje X ⊂ Y! we<X!
Gxqh<hqMgqe<Oxil</! ! -f<fqjzbqz<?! -kje! Y ⊃ X! we<Xl<! wPKgqe<Oxil</!!
wMk<Kg<gim<mig? A = {1, 2, 3, 4}, B = {0, 1, 2, 3, 4, 5} we<g/!!A z<!dt<t!njek<K!dXh<HgTl< B z<! dt<teA! nkiuK! A NeK! B e<! dm<g{liGl</! ! Neiz<! A z<!-z<zik 5 we<x!dXh<ohie<X!B z<!dt<tKA!nkiuK!5 ∉ A, 5 ∈ B. weOu!A NeK!!!B e<!kG!dm<g{liGl<A!nkiuK!A ⊂ B . Gxqh<H: A NeK!A g<Og!dm<g{l<!we<X!Wx<geOu!g{<cVf<Okil</!!Neiz<!A NeK!!A g<G!kG!dm<g{lz<z!we<hjk!nxqf<K!ogit<g/!!weOu!A J!A e<!kgi!dm<g{l< (improper subset) we<xjpg<gqe<Oxil</ 3.3.12 nMg<Gg<!g{l<
yV! kvh<hm<m! g{l<! A e<! njek<K! dm<g{r<gtqe<! okiGh<hqje! A e<!nMg<Gg<g{l<!(power set)!we<gqOxil</!!A e<!nMg<Gg<!g{k<kqje! p(A)!we<x!Gxqbiz<!Gxqh<hqMgqe<Oxil</! ! wMk<Kg<gim<mig? A = {a, b} weqz<? A e<! dm<g{r<gt<
. weOu!p(A) = { . -r<G!A z<! 2 !dXh<Hgt<!dt<te; p(A) !z<!4 dXh<Hgt<!dt<te/!weOu!A e<! Nkq!w{<!= 2A!p(A) e<!Nkq!w{< = 4; nkiuK!n(A) = 2; n[p(A)] = 4. -jkh<OhizOu?!A = {a, b, c} weqz<?
}{ },{ },{ {}, a,bba }}{ },{ },{ {}, a,bba
p(A) = { }. },{},{ },{ },{},{ },{ },{ {}, b,cac,ab,ca,bcba -r<G!A e<!Nkq!w{<!n(A) = 3; p(A)!e<!Nkq!w{<!n[p(A)] = 8 = 23. !-kqzqVf<K!fil<!nxquK?!n(A) = m!weqz< n[p(A)] = 2m.
$k<kqvl<;!n[p(A)] = 2n(A).! !!
hbqx<sq 3.3.1 1. hm<cbz<!njlh<hqjeg<!ogi{<M!hqe<uVl<!g{r<gjt!lQ{<Ml<!wPKg: (i) A = { SUNDAY we<x!osiz<zqz<!dt<t!dbqovPk<Kg<gt<}. (ii) B = {N{<ce<!hVugizr<gt<}. (iii) C = { MATHEMATICS!we<x!osiz<zqz<!dt<t!wPk<Kg<gt<}. (iv) D = { TAMILNADU!we<x!osiz<zqz<!dt<t!wPk<Kg<gt<}. 2. hm<cbz<!njlh<hqz<!hqe<uVl<!g{r<gjt!wPKg: (i) P = {x | x we<hK TAMILNADU!we<x!osiz<zqz<!dt<t!wPk<K}. (ii) Q = {x | x yV!LP!w{<!lx<Xl<! −3 ≤ x < 7}. (iii) R = {x | x NeK!ke<!-zg<gr<gtqe<!%Mkz<!9!we<xuiXt<t!Yi<!=iqzg<g!w{<}.
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3. g{g<gm<mjlh<H!njlh<hqz<!hqe<uVl<!g{r<gjt!wPKg: (i) {3, 6, 9, 12} (ii) {5, 25, 125, 625} (iii) {1, 3, 5,…} (iv) {1, 4, 9, 16, 25, 36, 49, 64, 81, 100}. 4. -mKhg<gl<! hm<cbz<! njlh<hqz<! uquiqk<Kt<t! g{l<! yu<ouie<jxBl<! nOk!!uzh<hg<gl<!g{g<gm<mjlh<H!njlh<hqz<!uquiqk<Kt<t!nOk!g{k<Kme<!ohiVk<kUl<. (i) {1, 2, 3, 6} (a) {x | x yV!hgi!w{<!lx<Xl<!nK!6e<!yV!uGh<hie<}. (ii) {2, 3} (b) {x | x NeK10 J!uqmg<!Gjxuie!Yi<!yx<jxh<!hjm!-bz<!!!!!!!!!!!!!!!!!!!!!!!!!!!w{<}. (iii) {2, 4, 6, 8} (c) {x | x yV!lqjg!LPw{<!lx<Xl<!6e<!uGh<hie<}. (iv) {1, 3, 5, 7, 9} (d) {x | x NeK!10J!uqmg<!Gjxuie!Yi<!-vm<jmh<!hjm!-bz<!!!!!!!!!!!!!!!!!!!!!!!!!!w{<}. 5. hqe<uVl<!g{r<gtqe<!Nkq!w{<gjtg<!g{<Mhqcg<gUl<: (i) Nr<gqz!wPk<Kg<gtqz<!dt<t!njek<K!dbqi<!wPk<Kg<gtqe<!g{l</ (ii) 100!J!uqmg<!Gjxuie!njek<K!ui<g<g!w{<gt<!g{l</ (iii) A = { x | x NeK “mathematics”!we<x!osiz<zqz<!dt<t!wPk<K}. (iv) B = {x | x < 0, x ∈ W}. (v) C = {x | − 3 ≤ x < 4, x ∈ Z}. (vi) 10 g<Gl< 20g<Gl<!-jmh<hm<m!hgi!w{<gtqe<!g{l</ 6. hqe<uVl<!LcUX!g{r<gtqe<!uqMhm<m!dXh<Hgjt!wPkUl<: (i) A = {1, 10, 100, ______, _______, 1,00,000}. (ii) B = {2, 5, 8, 11, ______, _______, 20, 23}. 7. hqe<uVl<!Lcuqzig<!g{r<gtqz<!nMk<k!&e<X!dXh<Hgjt!wPkUl<; (i) C = {3, 6, 12, 24, ______, ______, ______, ….}. (ii) D = {−4, −3, −2, −1, ______, ______, _____, …}. 8. hqe<uVl<!g{r<gt<!oux<Xg<!g{r<gti!we<hjkg<!%Xg: (i) A = {3Nz<!uGhMl<!-vm<jmh<hjm!-bz<!w{<gt<}. (ii) B = { x | x ∈ R , x2 + 1 = 0}. (iii) C = {fie<G!hg<gr<gt<!ogi{<m!hz<Ogi{r<gt<}. (iv) D = {Jf<K!hg<gr<gt<!ogi{<m!fix<gvr<gt<}. 9. A = {p, q, r, s}, B = {1, 3, 5, 7}, C = {q, r}, D = {8, 4, 6, 2}, E = {r, q, s, p} F = {2, 6, 4, 8}we<g/!!hqe<uVue!siqbi!nz<zK!kuxi!we!wPKg: (i) A lx<Xl< C slie!g{r<gt</ (ii) A lx<Xl< E sl!g{r<gt</ (iii) F lx<Xl< B slie!g{r<gt</ (iv) A lx<Xl< B sl!g{r<gt</ (v) F lx<Xl< D sl!g{r<gt</ 10. hqe<uVl<!yu<ouiV!g{k<kqe<!nMg<Gg<!g{k<kqje!wPKg: (i) A = {1, 2} (ii) B = { x, y, z} (iii) C = {a, b, c, d}
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11. (i) n(A) = 5 weqz<? n[p(A)]!Jg<!gi{<g/! (ii) n[p(A)] = 128!weqz<? n(A) Jg<!gi{<g/ 3.3.13 g{s<!osbzqgt< -h<ohiPK?!
(i) -V!g{r<gtqe<!Osi<h<H (ii) -V!g{r<gtqe<!oum<M (iii) yV!g{k<kqe<!lqjg!fqvh<H
Ngqbux<jxg<!gx<xxqOuil</! (i) -Vg{r<gtqe<!Osi<h<H
A lx<Xl< B we<he!-Vg{r<gt<!we<g/!!A bqOzi!nz<zK B bqOzi!nz<zK!-v{<cZOli!-Vg<Gl<!njek<K!dXh<HgjtBl<!ogi{<m!g{k<kqje!A lx<Xl<!B e<!Osi<h<H (union) we<gqOxil</!!A, B!-ux<xqe<!Osi<h<H!g{k<kqje!A U B!weg<!Gxqh<hqMgqe<Oxil</!!weOu
A B = { x | x ∈ A nz<zK x ∈ B nz<zK x ∈ A lx<Xl< B}. U
-kje!A B = {x| x ∈ A nz<zK x ∈ B} wes<!SVg<glig!wPKOuil</!!-r<G!Änz<zK}!we<x! osiz<! Ängh<hMk<kq! -j{f<k}! we<x! ohiVtqz<! ogit<th<hMgqe<xKA! nkiuK!!!!!
‘x ∈ A nz<zK x ∈ B’ we<hK ‘x ∈ A nz<zK x ∈ B nz<zK x ∈ A lx<Xl< B’!we<hjkg<!Gxqh<hkiGl</
U
wMk<Kg<gim<M 38: A = {1, 2, 3, 4}, B = {2, 4, 6} weqz<!A U B!Jg<!g{<Mhqc/ kQi<U: A lx<Xl<! B -ux<xqe<!njek<K!dXh<HgjtBl<! hm<cbzqm<M?! lQ{<Ml<! uVkjz!uqzg<g?! 1, 2, 3, 4, 2, 4, 6. weOu?!A U B = {1, 2, 3, 4, 6}. (ii) -V!g{r<gtqe<!oum<M
A, B we<he! -Vg{r<gt<! we<g/ A, B -ju!-v{<cx<Gl<! ohiKuig!njlf<k!dXh<Hgtiz<!njlbh<! ohx<x! g{k<kqje! A lx<Xl<! B -ux<xqe<! oum<M (intersection) we<gqOxil</!!-u<oum<Mg<!g{k<kqje!A I B!we<x!Gxqbiz<!Gxqh<hqMgqe<Oxil</ weOu?
A I B = {x | x ∈ A lx<Xl< x ∈ B}. wMk<Kg<gim<M 39: A = {1, 2, 3}, B = {2, 3, 4} weqz< A B Jg<!g{<Mhqc/ I
kQi<U: A, B!-ux<xqz<!dt<t!njek<K!dXh<Hgt<; 1, 2, 3, 2, 3, 4 A, B!-ux<xqe<!ohiK!dXh<Hgt<;!2, 3 . ∴ A I B = {2, 3}. 3.3.14 -j{h<hge<x!g{r<gt<
A lx<Xl< B -jubqv{<cx<Gl<! ohiKuie! dXh<H! -z<jzobeqz<?! nu<uqV!g{r<gjt!-j{h<hge<x! g{r<gt< (disjoint sets)! we<gqOxil</! !nkiuK?!A B = Ø nz<zK { } weqz<? A Bl< B Bl<! -j{h<hge<x! g{r<gtiGl</! ! wMk<Kg<gim<mig? A = {1, 2, 3, 7}, B = {4, 5, 6}!weqz<? A B = { }. weOu!ABl<!B Bl<!-j{h<hge<x!g{r<gt</
I
I
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3.3.15 uqk<kqbisg<!g{l<
A lx<Xl<< B we<he! -V! g{r<gt<! we<g. B z<! -z<zik! A! e<! dXh<Hgt<!njek<jkBl<! ogi{<m! g{k<kqje!yV!uqk<kqbis! g{l<! (difference set)! we<gqOxil</!!-kje A – B!weg<!Gxqh<hqMgqOxil</!!weOu? A − B = {x | x ∈ A, x ∉ B}. Gxqh<H: B − A = {x | x ∈ B, x ∉ A}. wMk<Kg<gim<M 40: A = {1, 2, 3, 4, 5, 6}, B = {1, 3, 7} weqz<? A − B lx<Xl< B − A!-ux<jxg<!g{<Mhqcg<gUl</ kQi<U: A−B = {2, 4, 5, 6}. B −A = {7}. !Gxqh<H: A − B ≠ B − A. 3.3.16 yV!g{k<kqe<!fqvh<hq
A we<hK!kvh<hm<m!yV!g{l<? U we<hK!njek<Kg<g{l<!we<g/!!A z<!-z<zik! U e<!dXh<Hgjtg<!ogi{<m!g{k<kqje!g{l<!A e<!fqvh<hq! )complement) we<gqOxil</!!-g<g{k<kqje A′ nz<zK Ac nz<zK A !we<x!Gxqbqeiz<!Gxqh<hqMgqe<Oxil</ Gxqh<H: Ac = U − A. wMk<Kg<gim<M 41: U = {1, 2, 3, 4, 5}, A = {3, 4} weqz<? AcJg<!gi{<g/ kQi<U: Ac = {1, 2, 5}.
A: 1, 2, 3, 4, 5, 6 B: 1, 3, 7
U: 1, 2, 3, 4, 5 A : 3, 4
3.3.17 g{g<!ogit<jgbqz<!yV!Lx<oxiVjl
g{g<ogit<jgbqz<?! g{r<gtqe<! Osi<h<Hg<! g{k<kqz<! dt<t! dXh<Hgtqe<!w{<{qg<jgjbg<! g{g<gqMl<! hbEt<t! Lx<oxiVjl! dt<tK/! ! nK! hqe<uVliX!%xh<hMgqe<xK; A, B we<he!-V!g{r<gt<!weqz<? n(A B) ≡ n(A) + n(B) − n(A B). U I
!
wMk<Kg<gim<M 42: A = {1, 3, 4, 5, 6, 7, 8, 9}, B = {1, 2, 3, 5, 7}weqz< n(A), n(B), n(A B) lx<Xl< n(A B) Ngqbux<jxg<!gi{<g/!!OlZl< n(A B) ≡ n(A) + n(B) − n(A B)!we<x!Lx<oxiVjljbs<!siqhii<g<gUl</
U
I U I
kQi<U: fil<!nxquK? A U B = {1, 2, 3, 4, 5, 6, 7, 8, 9} A I B ={1, 3, 5, 7}. n(A) = 8, n(B) = 5, n(A U B) = 9 lx<Xl< n(A I B) = 4. weOu n(A) + n(B) − n(A B) = 8 + 5 − 4 = 9. I
-r<G n(A B) = 9. weOu n(A B) = n(A) + n(B) − n(A B) we<x!Lx<oxiVjl!siqhii<g<gh<hMgqe<xK/
U U I
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hbqx<sq 3.3.2
1. A U B lx<Xl< A I B -ux<jx!hqe<uVl<!g{r<gTg<G!gi{<g: (i) A = {a, e, i, o, u} lx<Xl< B = {a, b}. (ii) A = {1, 3, 5} lx<Xl< B ={1, 2, 3}. (iii) A = {x | x Yi<!-bz<!w{<!lx<Xl< 1 < x ≤ 6} lx<Xl< B = {x | x Yi<!-bz<!w{<!lx<Xl< 6 < x < 10}. (iv) A = {p, q, r} lx<Xl< B = Ø. 2. hqe<uVl<!g{r<gTg<G A − B, A − C lx<Xl< B − A Ngqbux<jxg<!g{<Mhqcg<gUl<: (i) A = {a, b, c, d, e, f, i, o, u}, B = {a, b, c, d} lx<Xl< C = {a, e, i, o, u}. (ii) A = {3, 4, 5}, B = {5, 6, 7, 8} lx<Xl< C = {7, 8, 9}. 3. U = {a, b, c, d, e, f, g, h}, A = {a, c, g} lx<Xl< B = {a, b, c, d, e, f}weqz<?!hqe<uVueux<jxg<!g{<Mhqcg<gUl<; (i) Ac (ii) Bc (iii) (A U B) c
(iv) (AcI B) c (v) Ac BI c (vi) Ac BU c. 3.3.18 oue<hml<
g{k<kqe<! lQK! fqgp<k<kh<hMl<! osbzqgjt! )Osi<h<Hg<! g{l<?! oum<Mg<g{l<?!fqvh<Hg<g{l<!Ngqbux<jx!njlk<kz<*!g{<upqg<!gi{<hkx<G!dkuq!osb<Bl<!ujgbqz<?!
\ie<! oue<! (John Venn)! we<x!Nr<gqOzb! g{qk! uz<Zfi<! g{r<gjt! ujvhmr<gt<!&zl<! njlg<Gl<! upqbqje! nxqLgh<hMk<kqeii</! ! -u<uiX! g{r<gjtg<! Gxqh<hqMl<!
ujvhmr<gjt! oue<hmr<gt<! (Venn diagrams)! we<xjpg<gqe<Oxil</! ! -l<Ljxbqz<!njek<Kg<!g{lieK!yV!osu<ugk<kqeiz<!Gxqg<gh<hMl<<A!nke<!kG!dm<g{r<gt<!ns<!osu<ugk<kqEt<! um<mr<gtiz<! Gxqg<gh<hMl</! ! -h<ohiPK! oue<hmk<kqz<! hz<OuX!g{r<gtqe<!njlh<hqjek<!kVOuil</!
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A B Uhml< 3.2
83
A B I
hml< 3.3
njek<K!g{l<!hml< 3.1
Ac
hml< 3.4
Bc
hml< 3.5
(A U B) c
hml< 3.6
(A I B) c hml< 3.7
A − B hml< 3.8
B − A hml< 3.9
!!!!!wMk<Kg<gim<M 43: gQOp! (hml<! 3.10Jh<! hii<g<gUl<)! kvh<hm<Mt<t! hmk<kqzqVf<K?!hqe<uVueux<jxg<!gi{<g;
(i) A B (ii) A B (iii) (A B)U I U c
kQi<U: -r<G U = {1, 2, 3, 4, 5, 6, 7}.
hml< 3.10
(i) A B = {1, 2, 3, 4, 5, 6}. U
(ii) A B = {2, 5}. I (iii) (A U B) c = {7}.
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!wMk<Kg<gim<M 44: oue<hmk<jkh<!hbe<hMk<kq?! A = {−2, 3, 5, 7} lx<Xl< B = {3, 9, 11} Ngqb! g{r<gjtg<! gim<Mg/! ! OlZl<! n(A B) = n(A) + n(B) − n(A B) we<x!$k<kqvk<jks<!siqhii<g<gUl</
U I
hml<!3.11
kQi<U: A U B = {−2, 3, 5, 7} U {3, 9, 11} = {−2, 3, 5, 7, 9, 11} ∴ n(A B) = 6. (1) U
A I B = {−2, 3, 5, 7} I {3, 9, 11} = {3} ∴ n(A B) = 1. I
fil<!ohx<xqVh<hK n(A) = 4, n(B) = 3. ∴ n(A) + n(B) = 4 + 3 = 7. ∴ n(A) + n(B) − n(A B) = 7 − 1 = 6. (2) I
(1), (2) -ux<xqzqVf<K (A B) = n(A) + n(B) − n(A B). U I Gxqh<H; A lx<Xl< B -u<uqv{<Ml<!-j{h<hge<xqVh<hqe<?!A B = Ø. OlZl< n(A B) = 0. I IweOu n(A B) = n(A) + n(B). U wMk<Kg<gim<M 45 : Yi<!Diqz<?!jkbz<!uGh<hqx<Gs<!osz<Zl<!oh{<gtqe<!w{<{qg<jg!45?!Okim<m!Oujz!uGh<hqx<Gs<!osz<Zl<!oh{<gtqe<!w{<{qg<jg!70/!!-ui<gtqz<!30!Ohi<!-v{<M! uGh<HgTg<Gl<! osz<gqxii<gt<! weqz<?! oue<hmk<kqjeg<! ogi{<M?! hqe<!uVueux<jxg<!g{<Mhqcg<gUl<;
(i) yV!Gxqh<hqm<m!uGh<hqx<G!lm<Ml<!ose<xui<gt<!wk<kje!Ohi<@ (ii) wk<kje! Ohi<! olik<kk<kqz<! -u<uqv{<M! uGh<Hgtqz<! WOkEl<! ye<xqx<giuK!ose<xui<gt<@!
hml< 3.12
kQi<U: A, B we<he!LjxOb!jkbz<!uGh<H?!Okim<m!Oujz! uGh<H! -ux<xqx<Gs<! osz<Zl<! oh{<gtqe<!g{r<gt<!weqz<?! n(A) = 45, n(B) = 70. kvh<hm<Mt<t! uquvk<kqe<hc? n(AI B) = 30. weOu!oue<hmk<kqzqVf<K?!(i) (a) jkbz<!uGh<hqx<G!lm<Ml<!ose<x!oh{<gtqe<!w{<{qg<jg!n(A − B) = 45 − 30 = 15, (i) (b) Okim<m! Oujz! uGh<hqx<G! lm<Ml<! ose<x!oh{<gtqe<!w{<{qg<jg n(B − A) = 70 − 30 = 40 (ii) -u<uqv{<M!uGh<Hgtqz<!WOkEl<!yV!uGh<hqx<giuK!ose<x!oh{<gtqe<!w{<{qg<jg? n(AU B) = 15 + 30 + 40 = 85. -kje!OuXuqkligUl<!ohxzil<;!
n(A B) = n(A) + n(B) − n(A B) = 45 + 70 − 30 = 85. U I
!wMk<Kg<gim<M 46: Yi<!Diqz<!dt<t! 45 uQMgtqz<? 25 uQMgtqz<! okijzg<gim<sqh<! ohm<c!-Vg<gqe<xKA 30 uQMgtqz<! uioeizqh<! ohm<c!-Vg<gqe<xK. wk<kje!uQMgtqz<!-v{<Ml<!-Vg<gqe<xe/ kQi<U: A = {okijzg<gim<sqh<!ohm<cgt<!dt<t!uQMgt<}
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B = {uioeizqh<!ohm<cgt<!dt<t!uQMgt<} A I B = {okijzg<gim<sqh<!ohm<c!lx<Xl<!uioeizqh<!ohm<c!-v{<Ml<!dt<t!uQMgt<}. n(AI B) = x we<g/!!hqe<H!oue<hmk<kqzqVf<K?
hml< 3.13
25 − x + x + 30 − x = 45 n.K! ! 55 − x = 45 n.K! ! !! −x = 45 − 55 n.K! ! !!!−x = −10 n.K x = 10. ∴okijzg<gim<sqh<!ohm<c!lx<Xl<!uioeizqh<!ohm<c!-v{<jmBl<!ogi{<m!uQMgtqe<!w{<{qg<jg = 10. wMk<Kg<gim<M 47: 35 li{ig<gi<gt<! ogi{<m! uGh<hqz<? 28 Ohi<! uvziX! himk<kqz<!Oki<s<sqBx<xei<;! g{g<Gh<! himk<kqz<! 22! Ohi<! Oki<s<sqBx<xei<A! 18! Ohi<! -v{<M!himr<gtqZl<! Oki<s<sqBx<xei</! ! wk<kje! li{ig<gi<gt<! -v{<M! himr<gtqZl<! Okiz<uq!
ohx<xei<@! ! oue<! hmk<jkg<! ogi{<M! (i) uvzix<Xh<! himk<kqz<! lm<Ml<! Oki<s<sq!
ohx<xui<gt<!wk<kje!Ohi< (ii) g{g<Gh<!himk<kqz<!lm<Ml<!Oki<s<sq!ohx<xui<gt<!wk<kje!Ohi<!we<hjkg<!g{<Mhqcg<gUl</ kQi<U: H, M we<he! LjxOb! uvziX?! g{g<G!himr<gtqz<! Oki<s<sq! ohx<x! li{ui<gtqe<!g{r<gt<!weqz<?
hml< 3.14
n(H) = 28, n(M) = 22, n(H M) = 18 we<X!kvh<hm<Mt<tK/! !H M! we<x! g{l<?! uvziX!nz<zK! g{g<G! -ux<xqz<! Oki<s<sqBx<x!li{ui<gtqe<!g{k<kqjeg<!Gxqg<gqxK/!!weOu?!uvziX! nz<zK! g{g<G! -ux<xqz<! ye<xqz<!Oki<s<sqBx<x!li{ui<gtqe<!w{<{qg<jg?
I
U
n(H M) = n(H) + n(M) − n(H M) U I
nkiuK? n(H M) = 28 + 22 − 18 = 32. U
uGh<hqz<! olik<kl<! 35! Ohi<! -Vg<gqe<xei</! ! weOu! -v{<M! himr<gtqZl<!Okiz<uqBx<xui<gtqe<!w{<{qg<jg = 35 − 32 = 3 NGl</!!oue<!hmk<jkg<ogi{<M?!fil<!gi{<hK?
)i*!uvziX!himk<kqz<!lm<Ml<!Oki<s<sqBx<xui<gtqe<!w{<{qg<jg
= n(H ) − n(H M) I
= 28 − 18 = 10 (ii) g{g<Gh<!himk<kqz<!lm<Ml<!Oki<s<sqBx<xui<gtqe<!w{<{qg<jg
= n(M) − n(H M) I
= 22 − 18 = 4.
!
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!hbqx<sq 3.3.3
1. A = {2, 3, 5, 8, 10}, B = {3, 7, 8, 9}, C = {1, 2, 5, 8, 12}!kvh<hce<? (i) n(A B) (ii) n(B C) (iii) n(A − B) (iv) n(C − B) U I
Ngqbux<jxg<!gi{<g/ 2. n(A) = 30, n(B) = 43 lx<Xl< n(A B) = 11 weqz<? n(A B)!J!gi{<g/!I U
3. ye<hkil<!uGh<H!li{ui<gt< A, B!Ngqb!-V!Oki<Ugtqz<!ye<jxObEl<! wPk!
Ou{<Ml</! ! Oki<U! A! uqje! 40 OhVl<?! Oki<U B! bqje 30 li{ui<gTl<?!-v{<jmBl<! 20! OhVl<! wPKgqe<xei</! ! nu<uGh<hqz<! dt<t! li{ig<gi<gtqe<!w{<{qg<jgjbg<!g{<Mhqcg<gUl</
4. yV!GcbqVh<H!hGkqbqz<!dt<t!400 uQMgtqz<!wMg<gh<hm<m!gVk<Kg<!g{qh<hqe<hc?!
250 Ohi<!Nr<gqz!osb<kqk<!kit<gjtBl<, 170 Ohi<!klqp<!osb<kqk<kit<gjtBl<? 65 Ohi<! -v{<cjeBl<! uir<Ggqe<xii<gt</! ! wk<kjeOhi<! wf<kuqklie! osb<kqk<!kijtBl<!uir<Gukqz<jz!we<hjkg<!g{g<gqmUl</
5. Yi<!Diqz<! 200 GMl<hr<gt<!dt<te/! !nr<G A, B we<x! -V! Osih<H! ujggt<!
lg<gtqjmOb!hvuqBt<te/!!160 GMl<hr<gt<!A ujg!Osih<jhBl<? 140 GMl<hr<gt<!B! ujg! Osih<jhBl<! hbe<hMk<Kgqe<xei</! ! njek<K! GMl<hr<gTl<! -u<uqV!ujg! Osih<Hgtqz<! WOkEl<! ye<xqjeh<! hbe<hMk<Khui<gtibqVh<hqe<?! -v{<M!ujg! Osih<HgjtBl<! hbe<hMk<Kl<! GMl<hr<gt<! wk<kje! we<hjkg<!g{<Mhqcg<gUl</
6. 250 Ohi<! gzf<Kogi{<m! yV! uqVf<kqz<? 210 Ohi<! gihqBl<? 50 Ohi<! CBl<?! sqzi<!
-v{<cjeBl<!hVgqei</!!20 Ohi<!gihq!nz<zK!C!-ux<jx!nVf<kuqz<jz!weqz<?!gihq! lx<Xl<! C! -v{<jmBl<! hVgqbui<gt<! wk<kje! Ohi<! we<hjkg<!g{<Mhqcg<gUl</!
7. yV!gz<Z~iq!-kpqz<?!-bx<hqbz<!GP?!g{g<Gg<!GP?!-jubqv{<cZl<!%m<mig!
150 li{ui<gt<!dXh<hqei<gtig!dt<tei</!!g{g<Gg<!GPuqz<!70 li{ui<gt<!dXh<hqei<gtib<! -Vg<gqe<xei<! lx<Xl< 50 li{ui<gt<! -v{<M! GPuqZl<!dXh<hqei<gtig! -Vg<gqe<xei<! weqz<?! -bx<hqbz<! GPuqz<! dXh<hqevib<!-Vh<hui<gtqe<!w{<{qg<jgjbg<!g{<Mhqc/
8. 30 li{uqbi<!dt<t! uGh<hqz<? 20 li{uqbi<! him<Mh<Ohim<cbqz<! hr<Ogx<xei<A! 10
li{uqbi<! him<M?! fim<cbl<! -v{<cZl<! hr<Ogx<xei</! 5 li{uqbi<! -v{<cz<!
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ye<xqZl<!%mh<!hr<Ogx<guqz<jz/!!wk<kjeOhi<!fim<cbk<kqz<!lm<Ml<!hr<Ogx<xei<!we<hjkg<!g{<Mhqcg<gUl</
9. Ogijmgiz! uqMLjxbqe<! OhiK? XII uGh<H! li{ui<gtqz<! 35 Ohi<! g{q{q!
uGh<HgTg<Gs<! osz<gqe<xei<A! 25 Ohi<! FjpUk<! Oki<uqx<gie! hbqx<sq!uGh<HgTg<Gs<! osz<gqe<xei<A! 15 li{ui<gt<! -v{<M! uGh<HgTg<Gl<!osz<gqe<xei</! ! uGh<hqz<! li{ui<gtqe<! olik<k! w{<{qg<jg 50! weqz<?! wf<k!uGh<hqZl<!hr<Ogx<gik!li{ui<gtqe<!w{<{qg<jg!we<eoue<X!g{<Mhqcg<gUl</
uqjmgt<!
hbqx<sq 3.1 1. (i) (ii) 1.3 × 101010998.2 × 9 (iii) 1.083 × 1012 (iv) 4.3 × 109
(v) 9.463 × 1015 (vi) 5.349 × 1017 (vii) 3.7 × 10−3 (viii) 1.07 × 10−4
(ix) 8.035 × 10−5 (x) 1.3307 × 10−6 (xi) 1.1 × 10−10 (xii) 9 × 10−13
2. (i) 0.00000325 (ii) 0.0000402 (iii) 0.0004132 (iv) 0.001432
(v) 3250000 (vi) 402000 (vii) 41320 (viii) 1432 3. (i) 1.024 × 1014 (ii) 4.41 × 10−4 (iii) 1.1664 × 108
(iv) 2.56 × 10−26 (v) 7.5 × 10−8
hbqx<sq 3.2.1
1. (i) siq (ii) kuX (iii) kuX (iv) kuX (v) siq (vi) kuX
2. (i) 204.0log5 −= (ii) 324log
81
−= (iii) 4256log4 =
(iv) 6729log3 = (v) 23216log36
−= (vi) 3001.0log10 −=
3. (i) 4 (ii) 3 (iii) 4 (iv) 21− (v) −4
(vi) 25 (vii) 36 (viii) 64 (ix)
41
4. (i) 10 (ii) 7 (iii) 101 (iv) 5 (v)
321
(vi) 512 (vii) 2 (viii) 162 (ix) 31− (x) 3
5. (i) C (ii) A (iii) D (iv) C (v) A (vi) D
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6. (i) (ii) 18log10 ⎟⎠⎞
⎜⎝⎛
818log3 (iii) 13 (iv) ⎟
⎠⎞
⎜⎝⎛
3350log 2
(v) ⎟⎠⎞
⎜⎝⎛
2572log10 (vi) ⎟
⎠⎞
⎜⎝⎛
485log10
7. (i) x + y (ii) 2x (iii) y − x (iv) 3y (v) t − y (vi) 3x + y + 2 z (vii) 3(x − y) (viii) y + z (ix) z + t (x) 2x + y (xi) 2t − x − z (xii) x − y − z + t
8. (i) 22 (ii) 7
11 (iii) 34− (iv) 16 (v)
310 (vi)
65
(vii) 9 (viii) 9 (ix) 37 (x) 243 (xi)
72−
hbqx<sq 3.2.2 1. (i) 3 (ii) 1 (iii) 0 (iv) −1 (v) −2 (vi) −3 (vii) −5 (viii) 2 2. (i) 3 (ii) 2 (iii) 4 (iv) 1 (v) 1 (vi) 3 3. (i) 0 (ii) 1 (iii) 2 (iv) 3 (v) 4 (vi) 3 4. (i) 4.5151 (ii) 3.5151 (iii) 2.5151 (iv) 1.5151 (v) 0.5151 (vi) .51511 (vii) .51514 (viii) .51517 5. (i) 4.9177 (ii) 5.926 (iii) .1583 (iv). .51055 (v). .67023 (vi) .37221 6. (i) 776.7 (ii) 2.705 (iii) 0.2873 (iv) 0.002668 (v) 0.00003312 (vi) 0.001239 7. (i) 200700 (ii) 82.56 (iii) 0.01099 (iv) 0.0006
(v) 0.4059 (vi) 0.00007789 (vii) 0.00003981 (viii) 0.0005012 (ix) 0.0005012 (x) 0.03981 (xi) 0.00000001698 (xii) 0.03162.
8. (i) 2041 (ii) 30550 (iii) 0.01034 (iv) 1380
(v) 0.00002079 (vi) 3285000 (vii) 5851 × 1018
(viii) 7.670 (ix) 0.6142 (x) 412.3 (xi) 0.02263 (xii) 0.8339 (xiii) 0.1752 (xiv) 1.893.
!!
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hbqx<sq 3.3.1 1. (i) A = {U, A} (ii) B = { Ogijm!gizl<?!GtqI!gizl<?!-tOueqx<!!gizl<?!-jzBkqI!gizl< } (iii) C = { M, A, T, H, I, C, S, E.} (iv) D = { T, A, M, I, L, N, D, U} 2. (i) P = { T, A, M, I, L, N, D, U} (ii) Q = {−3, −2, −1, 0, 1, 2, 3, 4, 5, 6} (iii) R = { 18, 27, 36, 45, 54, 63, 72, 81, 90} 3. (i) A = {x | x = 3n, n = 1, 2, 3, 4} (ii) B = {x | x = 5n, n = 1, 2, 3, 4}
(iii) C = {x | x ∈ N, x NeK!yx<jxh<hjm!w{<<} (iv) D = {x | x ∈ N, x = n2, 0 < n ≤ 10}
4. (i) → (c) ; (ii) → (a); (iii) → (d); (iv) → (b) 5. (i) 5 (ii) 9 (iii) 8 (iv) 0 (v) 7 (vi) 4 6. (i) 1000, 10000 (ii) 14, 17 7. (i) 48, 96, 192 (ii) 0, 1, 2 8. (i) -z<jz (ii) oux<Xg<!g{l< (iii) -z<jz (iv) oux<Xg<!g{l< 9. (i) kuX (ii) siq (iii) siq (iv) kuX (v) siq 10. (i) p(A) = { } { } { } { }{ },2,1,2,1
(ii) p(B) = { } { } { } { } { } { } { } { }{ } ,,, ,, ,, ,, , , , zyxzyzxyxzyx (iii) p(C) = {{a},{b},{c},{d},{a, b}, {a, c}, {a, d}, {b, c},{b, d},{c, d}, {a, b, d}, {b, c, d},{c, d, a}, {a, b, c},{a, b, c, d},{}}
11. (i) 32 (ii) 7 hbqx<sq 3.3.2
1. (i) A B = {a, e, i, o, u, b}, A I B = {a} U
(ii) A B = {1, 2, 3, 5}, A I B = {1, 3} U
(iii) A U B = {2, 3, 4, 5, 6, 7, 8, 9}, A I B = Ø (iv) A B = {p, q, r}, A B = Ø U I
2. (i) A − B = {e, f, i, o, u }, A − C = {b, c, d, f }, B − A = Ø
(ii) A − B = {3, 4}, A − C = {3, 4, 5}, B − A = {6, 7, 8} 3. (i) Ac = {b, d, e, f, h} (ii) Bc = {g, h}
(iii) (A U B) c = {h} (iv) (Ac I B) c= {a, c, g, h} (v) Ac I B c= {h} (vi) AcU B c = {b, d, e, f, g, h }
hbqx<sq 3.3.3
1. (i) 7 (ii) 1 (iii) 3 (iv) 4 2. 62 3. 50 4. 45 5. 100 6. 30 7. 130 8. 5 9. 5
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4. -bx<g{qkl<
-bx<g{qkk<jkg<! Gxqg<Gl<! Nr<gqzs<! osiz<! ‘Algebra’ NeK! ‘al–jabr’ we<x!nvihqbs<! osiz<zqzqVf<K! uVuqg<gh<hm<mkiGl</! nvihqb! olipqbqz<! ‘al’ we<hK! ‘nf<k’ we<Xl<?! ‘jabr’ we<hK ‘djmf<k! hGkqgtqe<! ye<xqj{h<H’ we<Xl<! ohiVt<hMl</!
-s<osiz<zqe<! hbe<him<cje!YI!wMk<Kg<gim<ce<!&zl<! Hiqf<K!ogit<tzil</! x + 5 = 9 we<x!sle<him<cz<!-mK!hg<glieK!x lx<Xl< 5 -u<uqv{<M!hGkqgtqe<!%MkziGl</!fil<!(–5) J!sle<him<ce<!-Vhg<gr<gtqZl<!%m<ceiz<!)OsIk<kiz<*?!flg<G (x + 5) + (–5) = 9 + (–5) nz<zK x + [5 + (–5)] = 9 – 5 nz<zK x + 0 = 4 nz<zK x = 4 weg<! gqjmg<gqxK/! -r<G 9 lx<Xl< −5 wEl<! -v{<M! hGkqgt<! -j{g<gh<hm<M! 4 ohxh<hm<mK/!-u<ujg!g{qkOl!-bx<g{qkl< weh<hMgqe<xK/!-f<kqb!g{qk!Oljkgt<!NIbhm<mI?! hqvl<lGh<kI?! laiuQvI?! >kvI! lx<Xl<! hi <̂gvI II Ohie<xuIgt<!-h<himh<hqiqju! ohVltuqz<! utIk<Kt<teI/! gqOvg<g! g{qk! Oljk! cObihi{<m <̂!we<huI! -bx<g{qkk<kqz<! liohVl<! utIs<sqjb! dVuig<gqBt<tkiz<! -ujv!-bx<g{qkk<kqe<!kf<jk!we<xjpg<gqe<Oxil</
g{qkk<kqe<!-h<himh<hqiquqz<?!a, b, x, y Ohie<x!wPk<Kg<gjt!w{<gjtg<!Gxqg<gh<!hbe<hMk<KOuil</!-g<Gxqgt<?!w{<gt<!-ux<jxg<!ogi{<M!%m<mz<?!gpqk<kz<?!ohVg<gz<?!uGk<kz<?! uIg<g&zl<! wMk<kz<! Ngqb! osbx<hiMgtqeiz<! -bx<g{qkg<! Ogijugt<!weh<hMujkh<!ohXgqe<Oxil</!-bx<g{qkg<!Ogijugt<!sqz!gQOp!ogiMg<gh<hm<Mt<te;!
2x + 3, (3a + b)(2x – y), yxxx 1914,
7191225
+++ .
-V! -bx<g{qkg<! Ogijugjts<! slh<hMk<Kukiz<?! -bx<g{qks<! sle<hiM!gqjmg<gqe<xK/! -bx<g{qks<! sle<hiMgTg<G! sqz! wMk<Kg<gim<Mgt<! gQOp!
ogiMg<gh<hm<Mt<te; 2x + 3 = x + 6, 2532
7 2 +=+− x
xx , 2x + 11 = 0.
-bx<g{qkg<! Ogijubqz<! dt<t! wPk<Kg<gt<! ng<Ogijubqe<! lixqgt<! weh<hMl</!
wMk<Kg<gim<mig? ax + b we<x! Ogijubqz<?! a lx<Xl<! b Gxqh<hqm<m! w{<gt<?! x Gxqh<hqmh<hmuqz<jz!weqz<?!ax + b g<G!x lixqbiGl</!-u<uiOx!2x2 + 3xy + y2 z<! x lx<Xl< y lixqgt</! yV! Ogijubqz<! dt<t! lixqgTg<G! Gxqh<hqm<m! lkqh<Hgt<! =M! osb<b?!ng<OgijubieK! yV! w{<{qjek<! kVgqe<xK/! -f<k! w{<{qje?! ng<Ogijubqe<!
lkqh<H we<gqOxil</!wMk<Kg<gim<mig?!2x2 + y we<hK!YI!-bx<g{qkg<!Ogiju?!x lx<Xl< y -g<Ogijubqe<!lixqgt</ x x<G!2 l<?!y x<G!1 l<!hqvkqbqm?!2x2 + y e<!lkqh<H! 2(2)2 + 1 = 9. x x<G!–1 l<?!y x<G!2 l< hqvkqbqm<miz<?!2x2 + y = 2(–1)2 + 2 = 4. yV!Ogijubqe<!lixqbqe<!sqz! olb<ob{<! lkqh<HgTg<G! ng<OgijubieK! olb<ob{<! lkqh<hqjek<! kvilx<!
Ohigzil</! wMk<Kg<gim<mig?! 3−x we<x! Ogijubqz<! x = 1 weh<hqvkqbqce<!
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we<gqOxil</! -v{<M! hcBjmb! hz<ZXh<Hg<Ogijujb! -Vhch<! hz<ZXh<Hg<Ogiju!we<gqOxil</! &e<X! hcBjmb! hz<ZXh<Hg<Ogijujb! Lh<hch<! hz<ZXh<Hg<Ogiju!we<gqOxil</! fie<G! hcBjmb! hz<ZXh<Hg<Ogijujb! fix<hch<! hz<ZXh<Hg<Ogiju!
we<gqOxil</! f(x) we<X! yV! hz<ZXh<Hg<Ogijujbh<! ohiKuigg<! Gxqk<kiz<?! f(x)e<!hcbqje deg(f(x)) we<x!Gxqbiz<!Gxqh<hqMOuil</!wMk<Kg<gim<mig?!1 – 3x2 + 5x3 – 2x we<x!hz<ZXh<Hg<Ogijujb! f(x) weg<! Gxqh<hqm<miz<?! deg(f(x)) = 3 NGl</! yV!hz<ZXh<Hg<Ogijubqe<! dXh<Hgt<! w{<gt<! we<hkiz<?! njek<K! hz<ZXh<Hg<!OgijugTl<!w{<gjtg<!Gxqg<gqe<xe/!weOu?!fil<!-ux<jxg<!ogi{<M!%m<m?!gpqg<g?!ohVg<g! lx<Xl<! uGg<g! LcBl</! yV! hz<ZXh<Hg<Ogijujb! lx<oxiV! hz<ZXh<Hg<!Ogijubqeiz<! uGh<hjkh<! hx<xq! hqe<eI! hch<Ohil</! -h<ohiPK!hz<ZXh<Hg<Ogijugjtg<!%m<Mujkh<!hx<xqh<!hch<Ohil</ 4.1.1 hz<ZXh<Hg<!Ogijugtqe<!%m<mz<!
yk<k! nMg<Ggtqe<! ogPg<gjtg<! %m<Mukeiz<?! flg<G! -v{<M! hz<ZXh<Hg<!Ogijugtqe<!%Mkz<!gqjmg<gqe<xK/ wMk<Kg<gim<M 1: 2x4 – 3x2 + 5x + 3, 4x + 6x3 – 6x2 – 1 -ux<xqe<!%Mkz<!gi{<g/ kQIU; olb<ob{<gtqe<! OsIh<Hh<! h{<H?! hr<gQm<Mh<! h{<H!Ngqbux<jxh<! hbe<hMk<KOuil</!(2x4 – 3x2 + 5x + 3) + (6x3 – 6x2 + 4x – 1) = 2x4 + 6x3 – 3x2 – 6x2 + 5x + 4x + 3 – 1 = 2x4 + 6x3 – (3+6)x2 + (5+4)x + 2 = 2x4 + 6x3 – 9x2 + 9x + 2. gQOp!ogiMg<gh<hm<m!kqm<ml<!-v{<M!hz<ZXh<Hg<Ogijugjtg<!%m<Mukqz<!dkuqosb<Bl</
2x4 + 0x3 – 3x2 + 5x + 3 0x4 + 6x3 – 6x2 + 4x – 1
2x4 + 6x3– 9x2 + 9x + 2
4.1.2 hz<ZXh<Hg<!Ogijugtqe<!gpqk<kz<!
hz<ZXh<Hg<!Ogijugjtg<!%m<MuK!Ohie<Ox!gpqk<kjzBl<!osb<gqOxil</ !wMk<Kg<gim<M 2: x3 + 5x2 – 4x – 6 zqVf<K!2x3 – 3x2 – 1 Jg<!gpqg<gUl</!kQIU; olb<ob{<gtqe<!OsIh<Hh<!h{<H?!hr<gQm<Mh<!h{<H!Ngqbux<jxh<!hbe<hMk<kqeiz< (x3 + 5x2 – 4x – 6) – (2x3 – 3x2 – 1) = x3 + 5x2 – 4x – 6 – 2x3 + 3x2 + 1 = x3 – 2x3 + 5x2 + 3x2 – 4x – 6 + 1 = (x3 – 2x3) + (5x2 + 3x2) + (–4x) + (–6+1) = –x3 + 8x2 – 4x – 5. gQOp!ogiMg<gh<hMgqe<x!LjxbqZl<!gpqk<kjzs<!osb<bzil</! uiq (1): x3 + 5x2 – 4x – 6. uiq (2): 2x3 – 3x2 – 1. uiq (2) z<!dt<t!hz<ZXh<Hg<!Ogijubqe<!Gxqgjt!lix<xg<gqjmh<hK, uiq (3): –2x3 + 3x2 + 1. uiq (1) , uiq (3) gtqz<!dt<t!hz<ZXh<Hg<!Ogijugjtg<!%m<mg<gqjmh<hK, –x3 + 8x2 – 4x – 5 . !!Olx<g{<m!Ljx!gQp<g<gi[liX!wPkh<hMl</
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x3 + 5x2 – 4x – 6 2x3 – 3x2 – 1 – + + – x3 + 8x2 – 4x – 5 4.1.3 -V!hz<ZXh<Hg<!Ogijugtqe<!ohVg<gz<!
-V!hz<ZXh<Hg<!Ogijugtqe<!ohVg<gz<!nz<zK ohVg<gx<hze<!gi{?!hr<gQm<Mh<!h{<jhBl<?!nMg<Gg<Gxq!uqkqgjtBl<!hbe<hMk<KgqOxil</ wMk<Kg<gim<M 3: x3 – 2x2 – 4 lx<Xl< 2x2 + 3x – 1 -ux<xqe<!ohVg<gx<hze<!gi{<g/ kQIU; (x3 – 2x2 – 4) (2x2 + 3x – 1)
= x3 (2x2 + 3x – 1) + (–2x2) (2x2 + 3x – 1) + (–4) (2x2 + 3x – 1) = (2x5 + 3x4 – x3) + (–4x4 – 6x3 + 2x2) + (–8x2 – 12x + 4) = 2x5 + 3x4 – x3 – 4x4 – 6x3 + 2x2 – 8x2 – 12x + 4 = 2x5 + (3x4 – 4x4) + (–x3 – 6x3) + (2x2 – 8x2) + (–12x) +4 = 2x5 – x4 – 7x3 – 6x2 – 12x + 4.
-V! hz<ZXh<Hg<! Ogijugtqe<! ohVg<gx<hze<! gi[l<OhiK! yV! hz<ZXh<Hg<!
OgijubqEjmb! yu<OuiI! dXh<jhBl<! lx<oxiV! hz<ZXh<Hg<! Ogijubqe<! yu<OuiI!dXh<hiZl<! ohVg<gq! hqe<eI?! nu<uqV! ohVg<gx<hze<gTl<! %m<mh<hMgqe<xe/! gQOp!ogiMg<gh<hm<m!Ljx!Hkqkigg<!gx<Xg<ogit<huIgTg<G!dkuqbig!-Vg<Gl</
x3 – 2x2 – 4 × 2x2 + 3x – 1
x3 (2x2 + 3x – 1) : 2x5 + 3x4 – x3
–2x2 (2x2 + 3x – 1) : – 4x4 – 6x3 + 2x2
–4(2x2 + 3x – 1) : – 8x2 – 12x + 4
2x5 – x4 – 7x3 – 6x2 – 12x + 4
sqz! Oujtgtqz<! hz<ZXh<Hg<! Ogijugtqe<! ohVg<gx<hzeqz<! sqz! Gxqh<hqm<m!
dXh<Hgtqe<!ogPg<gt<! Okjubig!-Vg<Gl</! Ofvk<jkBl<?!-mk<jkBl<! sqg<geh<hMk<k!fil<! hz<ZXh<Hg<! Ogijugjt! LPuKl<! ohVg<gilOzOb! ogPg<gjtg<! gi{zil</!
wMk<Kg<gim<mig?! A, B we<x! -v{<M! hz<ZXh<Hg<! Ogijugtqe<! ohVg<gx<hzeqz<! x3 e<!ogPjuh<! ohx! Ou{<Ml<! weqz<! gQOp! uVgqe<x! upqLjx?! Hkqkigg<! gx<Xg<!ogit<huIgTg<G!dkuqbig!-Vg<Gl</ hc gQOp!ogiMg<gh<hm<mjugtqe<!ohVg<gx<hze<!gi{<g: A bqz<!dt<t!x3 dXh<hqe<!ogP!× B bqz<!dt<t!lixqzq!dXh<H!(keqdXh<H* A bqz<!dt<t x2 dXh<hqe<!ogP × B bqz<!dt<t!x dXh<hqe<!ogP A bqz<!dt<t!x dXh<hqe<!ogP × B bqz<!dt<t x2 dXh<hqe<!ogP A bqz<!dt<t!lixqzq!dXh<H × B bqz<!dt<t!x3 dXh<hqe<!ogP
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hc 2: OlOz!hc!1 z<!dt<t!wz<zih<!ohVg<gx<hze<gjtBl<!%m<Mg/!gqjmg<Gl<!lkqh<H!A, B Ngqb!-V!hz<ZXh<Hg<!Ogijugtqe<!ohVg<gx<hzeqz<!x3 e<!ogPuiGl</ wMk<Kg<gim<M 4: x4, x3, x2 , x Ngqb!dXh<Hgtqe<!ogPju!7x3 – 6x2 – 9x + 8 lx<Xl<! 5x2 – 3x + 5 gtqe<!ohVg<gx<hzeqzqVf<K!LPuKl<!ohVg<gilOzOb!g{<Mhqc/!kQIU; A = 7x3 – 6x2 – 9x + 8 lx<Xl<!B = 5x2 – 3x + 5 we<g/! dXh<H x4 e<!ogPjug<!gi{z<;!A bqz<!dt<t x4 dXh<hqe<!ogP × B bqz<!dt<t!lixqzq!dXh<H = 0 × 5 = 0. A bqz<!dt<t x3 dXh<hqe<!ogP × B bqz<!dt<t x dXh<hqe<!ogP = 7 × – 3 = –21. A bqz<!dt<t x2 dXh<hqe<!ogP × B bqz<!dt<t x2 dXh<hqe<!ogP = – 6 × 5 = –30. A bqz<!dt<t x dXh<hqe<!ogP × B bqz<!dt<t x3 dXh<hqe<!ogP = – 9 × 0 = 0. A bqz<!dt<t!lixqzq dXh<H × B bqz<!dt<t x4 dXh<hqe<!ogP = 8 × 0 = 0. weOu!A × B z<!x4 e<!ogP = 0 + (–21) + (–30) + 0 + 0 = –51. dXh<H x3!e<!ogPjug<!gi{z<; A bqz<!dt<t x3 dXh<hqe<!ogP × B bqz<!dt<t!lixqzq dXh<H = 7 × 5 = 35. A bqz<!dt<t x2 dXh<hqe<!ogP × B bqz<!dt<t x dXh<hqe<!ogP = – 6 × – 3 = 18. A bqz<!dt<t x dXh<hqe<!ogP × B bqz<!dt<t x2 dXh<hqe<!ogP = – 9 × 5 = –45. A bqz<!dt<t!lixqzq!dXh<H × B bqz<!dt<t x3 dXh<hqe<!ogP = 8 × 0 = 0. Njgbiz< A × B z<!x3 e<!ogP!= 35 + 18 + (– 45) + 0 = 8. dXh<H x2!e<!ogPjug<!gi{z<; A bqz<!dt<t x2 dXh<hqe<!ogP × B bqz<!dt<t!lixqzq dXh<H = – 6 × 5 = –30. A bqz<!dt<t x dXh<hqe<!ogP × B bqz<!dt<t x dXh<hqe<!ogP = –9 × –3 = 27. A bqz<!dt<t!lixqzq!dXh<H × B bqz<!dt<t x2 dXh<hqe<!ogP = 8 × 5 = 40. Njgbiz< A × B z<!x2 e<!ogP = (–30) + 27 + 40 = 37. x dXh<hqe<!ogPjug<!gi{z<: A bqz<!dt<t x dXh<hqe<!ogP × B bqz<!dt<t!lixqzq dXh<H = –9 ×5 = –45. A bqz<!dt<t!lixqzq dXh<H × B bqz<!dt<t x dXh<hqe<!ogP = 8 × –3 = –24. Njgbiz< A × B z<!x e<!ogP = (–45) + (–24) = – 69. 4.1.4 hz!lixqgtqz<!njlf<k!hz<ZXh<Hg<!Ogijugt<!
x, y z<!njlf<k!YI!YVXh<Hg<Ogiju axnym weqz<?!nkqz<!a yV!olb<ob{<; x, y we<hju! lixqgt<; n lx<Xl< m we<he! lqjgLPg<gt</! wMk<Kg<gim<mig?! 5x3y2 we<hK!!!!x, y z<!YI!YVXh<Hg<! Ogiju/! x, y z<!njlf<k!LcUX!w{<{qg<jgbqzie!YVXh<Hg<!Ogijugtqe<! %m<mx<! hze<?! x, y z<! njlf<k! yV! hz<ZXh<Hg<! Ogiju! weh<hMl</!wMk<Kg<gim<mig?! 5x2y + 3x + y2, 3x – 8y, 2x2 + 3xy + 2y2 Ngqbju! x, y z<!njlf<k!hz<ZXh<Hg<! Ogijugt</!-OkOhie<X! hz!lixqgtqz<!njlf<k! hz<ZXh<Hg<! Ogijugt<!d{<M/!yV!lixqbqz<!njlf<k! hz<ZXh<Hg<! Ogijugtqz<!%m<mz<?! gpqk<kz<?! ohVg<gz<!osb<kK!Ohie<Ox!hz!lixqgtqz<!njlf<k!hz<ZXh<Hg<!OgijugtqZl<!osb<bzil</ !
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wMk<Kg<gim<M 5: x3y + x2y2 – 3xy3 , x3 – 3x3y + y3 + 4xy3 Ngqb!hz<ZXh<Hg<!Ogijugtqe<!%Mkz<!gi{<g/!kQIU; (x3y + x2y2 – 3xy3) + (x3 – 3x3y + y3 + 4xy3)
= x3y + x2y2 – 3xy3 + x3 – 3x3y + y3 + 4xy3
= (x3y – 3x3y) + (x2y2) + (–3xy3 + 4xy3) + (x3) + (y3) = –2x3y + x2y2 + xy3 + x3 + y3 .
wMk<Kg<gim<M 6: 2x + 3y, x2 – xy + y2 Ngqbux<xqe<!ohVg<gx<hze<!gi{<g/!kQIU;
nz<zK (2x + 3y) (x2 – xy + y2)
= 2x (x2 – xy + y2) + 3y (x2 – xy + y2)
= 2x3 – 2x2y + 2xy2 + 3x2y – 3xy2 + 3y3
=2x3 + (–2x2y + 3x2y) + (2xy2 – 3xy2) + 3y3
= 2x3 + x2y – xy2 + 3y3.
2x + 3y × x2 – xy + y2
2x(x2 – xy + y2) : 2x3 – 2x2y + 2xy2
3y(x2–xy + y2) : 3x2y – 3xy2 + 3y3
2x3 + x2y – xy2 + 3y3
yV!hz<ZXh<Hg<!Ogijujb!lx<oxiV!hz<ZXh<Hg<!Ogijubiz<!uGg<Gl<!osbz<?!hGkq!4.4 z<!Nvibh<hMgqxK/
hbqx<sq 4.1 1. siq!nz<zK!kuX!we!hkqz<ogiM:
(i) 2x + x3 we<hK!yV!hz<ZXh<Hg<!Ogiju/
(ii) 22x + 3x3 + 1 we<hK!YI!-v{<mil<hc!hz<ZXh<Hg<!Ogiju/ (iii) 5 – 2x – 3x2 + x3 z<!x2 e<!ogP 3. (iv) 5xy we<hK!YI!=VXh<Hg<!Ogiju/ (v) 2x + 3y + 5z we<hK!yV!&UXh<Hg<!Ogiju/ 2−4 ujvbqZt<t!yu<ouiV!g{g<gqZl<!%Mkz<!g{<M!kqm<mucuqz<!wPKg/
2. (x3 + 3x – 1) + (2x2 – 4x + 5) 3. (2x4 + x2 + 3x) + (x4 – 3x2 + 7x – 8) 4. (6 –10x + 5x2 + x3) + (2x3 – 3x – 4)
5−7 ujvbqZt<t! yu<ouiV! g{g<Ggtqz<! gpqk<kz<! osbz<! osb<K?! kqm<m! ucuqz<!
wPKg/!
5. (x3 + 5x2 – 10x + 6) – (2x3 – 3x – 4) 6. (x4 – 3x2 + 7x – 8) – (2x4 + x2 + 3x) 7. (3x5 – 5x2 + 4x – 7) – (1– 2x + 3x2 – x3)
8−10 ujvbqz<!dt<t!g{g<Ggtqe<!ohVg<gx<hze<!gi{<g/!njk!kqm<mucuqz<!wPKg/!
8. (2x2 – 6x + 3) (3x2 – 4x + 9) 9. (3x2 – 4x + 5x3 – 7) (2x2
– x + 4) 10. (7 – x – x2 ) (x3 – 5x2 + 3x)
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11 Lkz<! 13 ujvbqZt<t! g{g<Ggtqz<! LPuKl<! ohVg<gilOzOb x3, x2 , x Ngqb!dXh<Hgtqe<!ogPg<gjtg<!gi{<g/
11. (x2 – 4x + 4) (x2 + 2x –3) 12. (3x – 2 – x2) (1 + 3x – x2) 13. (7x3 – 6x2 – 9x – 1) (2x3 – 3x2 – 1)
14 Lkz<!16 ujvbqZt<t!g{g<Ggtqz<!ohVg<gx<hzjek<!kqm<mucuqz<!gi{<g/
14. (ax + by) (cx + dy) 15. (x + y) (2x2 – 3xy – 2y2) 16. (x2 – xy + y2) (x2 + xy + y2) 17. (x3 – px2 + 9x – 1), (2x3 – 3x2 – x + 2) Ngqbjugtqe<!ohVg<gx<hzeqz< x2 e<!ogP!12
weqz<, pbqe<!lkqh<H!gi{</ 18. (x3 – 2x + 5) (a – 3x – x2) e<!ohVg<gx<hzeqz<!x e<!ogPuieK?!!!!!!!!!!!!!!!! (2x2 + x – 1) (x2 – 3x – 2) e<!ohVg<gx<hzeqz<!x2 e<!ogPuqx<Gs<!sll<!weqz<?!a bqe<!
lkqh<H!gi{<g/ 19. (1 – 2x – x2) (2x2 – mx + 3) we<<gqx!ohVg<gx<hzeqz<!x2 e<!ogP?!x e<!ogP!-ux<xqe<!
%Mkz<! 5 weqz<, m e<!lkqh<jhg<!gi{<g/ 4.2. -bx<g{qk!Lx<oxiVjlgt<!
-bx<g{qk! Lx<oxiVjlgt<! weg<%xh<hMl<?! -bx<g{qk! sle<hiMgjth<! hx<xq!-r<G! hch<Ohil</! -bx<g{qk! sle<hiMgjt! Wx<geOu! gx<Xt<Otil</! YI! -bx<g{qk!sle<himieK!ye<X!nz<zK!nkx<G!Olx<hm<m!lixqgjtg<!ogi{<mK/!wMk<Kg<gim<mig 2x + 3 = 6 – x z< x we<hK!lixq/!x g<G!hkqzig!1 Jh<!hqvkqbqm!sle<himieK 5 = 5 we<El<!d{<jl! uig<gqblig! dt<tK/! x g<G! OuX! wf<k! lkqh<Hgt<! ogiMk<kiZl<, wMk<Kg<gim<mig x = 2 weg<! ogi{<miz<! sle<himieK! 7 = 4 we<El<! kuxie!uig<gqbliGl</! lixqg<gigh<! hqvkqbqmh<hm<m! YI! w{<! nf<k! sle<him<jm! d{<jlbie!uig<gqblig<gqeiz<!nf<k!w{<, sle<him<ce<!kQIU nz<zK!&zl< we<X!%xh<hMl</!YI!w{<!yV!sle<him<ce<!kQIU!weqz<?!nK!nf<k!sle<him<jm!fqjxU!osb<Bl</!wMk<Kg<gim<mig?!2x + 3 = 6 – x we<hjk! 1 ! fqjxU! osb<gqxK/! Neiz<! 2 fqjxU! osb<buqz<jz/ x2 –1 = (x + 1)(x – 1) we<El<! sle<hiM! wf<k! w{<{iZl<! fqjxujmgqxK/! -u<uiX!wz<zi!w{<gtiZl<!fqjxU!ohXl<!YI!-bx<g{qk!sle<hiM?!-bx<g{qk!Lx<oxiVjl!weh<hMl</! A = B we<El<! YI! -bx<g{qk! sle<hiM?! -bx<g{qk!Lx<oxiVjlbieiz<?!fil<!-kje A ≡ B we!wPKOuil</!hg<gr<gjt!lix<x!g{qk!Lx<oxiVjljb? B ≡ A we<Xl<!wPkzil</!fil<!-h<ohiPK!sqz!-bx<g{qk!Lx<oxiVjlgjtg<!ogi{IOuil</ 4.2.1 (x + a)(x + b) g<gie!-bx<g{qk!Lx<oxiVjl
w{<gtqe<!hr<gQm<Mh<!h{<jhh<!hbe<hMk<k? (x + a )(x + b ) = x(x + b) + a(x + b) = x2 + xb + ax + ab = x2 + ax + bx + ab= x2 + (a + b)x + ab.
fil<!ohXuK (x + a)(x + b) ≡ x2 + (a + b) x + ab
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Olx<gi[l<! Lx<oxiVjlg<G! yV!
ucuqbz<!uqtg<gl<!kVOuil</!
ABCD we<gqx! osu<ugk<kqe<!hvh<htuieK?! AHFE we<x! sKvk<kqe<!hvh<htU? HBGF, FGCI lx<Xl< EFID Ngqb osu<ugr<gtqe<! hvh<htU! -ux<xqe<!%MkZg<Gs<! sll< (hml< 4.2.1 Jh<!hiIg<gUl<). Njgbiz<!fil<!ohXuK!
hml< 4.2.1
(x + a)(x + b) = x2 + ax + ab + xb
= x2+ (a+b)x + ab.
OlOz! dt<t! Lx<oxiVjljbh<! hbe<hMk<kq! sqz! Lg<gqblie!Lx<oxiVjlgjtg<!ogi{IOuil</
(i) (x – a)(x + b) = [x + (–a)] (x + b) = x2 + [(–a) + b]x + (–a)b
= x2 + (b – a)x – ab. (ii) (x + a)(x – b) = (x + a) [x + (–b)] = x2 + [a + (–b)]x + a(–b) = x2 + (a – b)x – ab. (iii) (x – a)(x – b) = [x + (–a)] [x + (–b)] = x2 + [(–a) + (–b)] x + (–a) (–b)
= x2 – (a + b)x + ab. (iv) (a + b)2 = (a + b)(a + b) = a2 + (b + b)a + b2 = a2 + 2ab + b2. (v) (a – b)2 = (a – b)(a – b) = [a + (–b)] [a + (–b)]
= a2 + [(–b) + (–b)]a + (–b) (–b) = a2 –2ab + b2. (vi) (a + b)(a – b) = a2 + [b + (–b)]a + (b)(–b) = a2 + 0 × a – b2 = a2 – b2.
weOu?!fil<!ohXuK! (x + a)(x + b) ≡ x2 + (a + b)x + ab (x – a)(x + b) ≡ x2 + (b – a)x – ab (x + a)(x – b) ≡ x2 + (a – b)x – ab (x – a)(x – b) ≡ x2 – (a + b)x + ab (a + b)2 ≡ a2 + 2ab + b2
(a – b)2 ≡ a2 – 2ab + b2
(a + b)(a – b) ≡ a2 – b2
(x – a)(x – b) we<hke<!Lx<oxiVjl x2 – (a + b)x + ab. nK!2 Nl<!hc!hz<ZXh<Hg<!
OgijubiGl</!nke<! x e<!ogP?!keq!dXh<H!Ngqbju!LjxOb –(a + b) lx<Xl< ab NGl</! OlOz! ogiMg<gh<hm<Mt<t! Lx<oxiVjlgt<?! (x + a)(x + b) we<hke<!ohVg<gx<hzeqe<! uqiqju! nch<hjmbigg<! ogi{<cVh<hkiz<?! -ju! ohVg<gx<hze<!$k<kqvr<gt<! we<Xl<! %xh<hMl</! -v{<M! =VXh<Hg<! Ogijugtqe<! ohVg<gx<hzjeg<!g{<mxqbUl<!-f<k!$k<kqvr<gt<!hbe<hMl</ !
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wMk<Kg<gim<M 7: hqe<uVueux<xqe<!ohVg<gx<hze<gt<!gi{<g; (a) (x + 3) (x + 5) (b) (p + 9) (p – 2) (c) (z – 7) (z – 5) (d) (x – 8) (x + 2)
kQIU; (a) (x + 3) (x + 5) = x2 + (3 + 5) x + 3 × 5 = x2 + 8x + 15. (b) (p + 9) (p – 2) = p2 + (9 – 2) p – 9 × 2 = p2 + 7p – 18. (c) (z – 7) (z – 5) = z2 – (7 + 5) z + 7 × 5 = z2 – 12z + 35. (d) (x – 8) (x + 2) = x2 + (2 – 8)x – 8 × 2 = x2 – 6x – 16. wMk<Kg<gim<M 8: ohVg<gx<hze<! $k<kqvk<jkh<! hbe<hMk<kq! hqe<uVueux<xqe<! ohVg<gx<!hze<gtqe<!lkqh<Hgt<!gi{<g/ (a) 107 × 103 (b) 56 × 48 kQIU; (a) 107 × 103 = (100 + 7) (100 + 3) = 1002 + (7 + 3) × 100 + 7 × 3
(b) 56 × 48 = (50 + 6) (50 – 2) = 502 + (6 – 2) × 50 – 6 × 2 = 2500 + 200 – 12 =2688.
= 10000 + 10 × 100 + 21 = 10000 + 1000 + 21 = 11021. wMk<Kg<gim<M 9: hqe<uVueux<jx!uqiquig<Gg; (i) (3x + 7y)2 (ii) (11a – 7b)2 (iii) (2p + 5q)(2p – 5q) kQIU; (i) (3x + 7y)2 = (3x)2 + 2(3x)(7y) + (7y)2 =9x2+ 42xy + 49y2. (ii) (11a – 7b)2 = (11a)2 – 2(11a) (7b) + (7b)2 = 121a2 – 154ab + 49b2. (iii) (2p + 5q)(2p – 5q) = (2p)2 – (5q)2 = 4p2 – 25q2. wMk<Kg<gim<M 10: ohVg<gx<hze<! $k<kqvr<gjth<! hbe<hMk<kq! hqe<uVueux<xqe<! lkqh<H!gi{<g/! (i) 1032 (ii) 982 (iii) 104 × 96 kQIU; (i) 1032 = (100 + 3)2 = 1002 + 2(100)(3) + 32 = 10000 + 600 + 9 = 10609. (ii) 982 = (100 – 2)2 = 1002 – 2 (100)(2) + 22 = 10000 – 400 + 4 = 9604. (iii) 104 × 96 = (100 + 4) (100 – 4) = 1002 – 42 = 10000 – 16 = 9984. ohVg<gx<hze<! $k<kqvk<kqeqe<X! OlZl<! sqz! hbEt<t! Lx<oxiVjlgjtg<! ogi{<M!uVOuil</ (i) (a + b)2 + (a – b)2 = (a2 + 2ab + b2) + (a2 – 2ab + b2) = (a2 + a2) + (2ab – 2ab) + (b2 + b2) = 2a2 + 2b2.
∴ 21 [(a + b)2 + (a – b)2] =
21 [2(a2 + b2)] = a2 + b2.
(ii) (a + b)2 – (a – b)2 = (a2 + 2ab + b2) – (a2 – 2ab + b2)
= a2 + 2ab + b2 – a2 + 2ab – b2 = 4ab.
∴ 41 [(a + b)2 – (a – b)2] =
41 [4ab] = ab.
(iii) (a + b)2 – 2ab = (a2 + 2ab + b2) – 2ab = a2 + 2ab + b2 – 2ab = a2 + b2.
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(iv) (a + b)2 – 4ab = (a2 + 2ab + b2) – 4ab = a2 + 2ab + b2 – 4ab
= a2 – 2ab + b2 = (a – b)2.
(v) (a – b)2 + 2ab = (a2 – 2ab + b2) + 2ab = a2 – 2ab + b2 + 2ab = a2 + b2.
(vi) (a – b)2 + 4ab = (a2 – 2ab + b2) + 4ab = a2 – 2ab + b2 + 4ab
= a2 + 2ab + b2 = (a + b)2.
-u<uiX!flg<G!hqe<uVl<!hbEt<t!Lx<oxiVjlgt<!gqjmk<Kt<te;! 2
1 [(a + b)2 + (a – b)2] = a2 + b2
41 [(a + b)2 – (a – b)2] = ab
(a + b)2 – 2ab = a2 + b2
(a + b)2 – 4ab = (a – b)2
(a – b)2 + 2ab = a2 + b2
(a – b)2 + 4ab = (a + b)2
OlOz!ogiMg<gh<hm<Mt<t!Lx<oxiVjlgtqe<!hg<gr<gjt!lix<xq!wPk?!fil<!ohXuK a2 + b2 = 2
1 [(a + b)2 + (a – b)2] ab = 4
1 [(a + b)2 – (a – b)2] a2 + b2 = (a + b)2 – 2ab (a – b)2 = (a + b)2 – 4ab a2 + b2 = (a – b)2 + 2ab (a + b)2 = (a – b)2 + 4ab wMk<Kg<gim<M 11: a + b, a – b Ngqbux<xqe<!lkqh<Hgt<!LjxOb!7 , 4 weqz<? a2 + b2 , ab Ngqbux<xqe<!lkqh<Hgjtg<!gi{</
kQIU: a2 + b2 = 21 [(a + b)2 + (a – b)2] ab =
41 [(a + b)2 – (a – b)2]
= 41 [(7)2 – (4)2]
= 41 (49 – 16) =
433 .
= 21 [(7)2 + (4)2]
= 21 (49 + 16) =
265 .
wMk<Kg<gim<M 12: a + b, ab Ngqbux<xqe<!lkqh<Hgt<!LjxOb!12 , 32 weqz<, a2 + b2 , (a – b)2 e<!lkqh<Hgjtg<!gi{<g/ kQIU: a2 + b2 = (a + b)2 –2ab (a – b)2 = (a + b)2 – 4 ab
= (12)2 – 4(32) =144 – 128 = 16. = (12)2 – 2(32) = 144 – 64 = 80. wMk<Kg<gim<M 13: a – b, ab Ngqbux<xqe<! lkqh<Hgt<!LjxOb! 6, 40 weqz<?! a2 + b2 , (a + b)2 Ngqbux<xqe<!lkqh<Hgjtg<!gi{<g/
(a + b)2 = (a – b)2 + 4ab = 62 + 4(40)= 36 + 160 = 196.
kQIU: a2 + b2 = (a – b)2 + 2ab = 62 + 2(40) = 36 + 80 = 116.
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wMk<Kg<gim<M 14: (x + p)(x + q) = x2 – 5x – 300 weqz<?!p2 + q2 e<!lkqh<H!gi{<g/!kQIU: ohVg<gx<hze<!$k<kqvk<kqe<hc, (x + p) (x + q) = x2 + (p + q)x + pq. yh<hqMjgbqz<!fil<!nxquK? p + q = –5, pq = –300. -h<ohiPK? p2 + q2 = (p + q)2 – 2 pq = (–5)2 –2(–300) = 25 + 600 = 625. fil<!-eq!(a + b + c)2 g<gie!g{qk!Lx<oxiVjljbg<!g{<mxqOuil</! (a + b + c)2 = [(a + b) + c]2 = (a + b)2 + 2(a + b) c + c2
= (a2 + 2ab + b2) + 2ac + 2bc + c2 = a2 + b2 + c2 + 2ab + 2bc + 2ca = a2 + b2 + c2 + 2(ab + bc + ca). Njgbiz<!fil<!ohXl<!Lx<oxiVjl!
(a + b + c)2 ≡ a2 + b2 + c2 + 2(ab + bc + ca) hg<gr<gjt!lix<xq!wPkqeiz<!fil<!ohXuK!
a2 + b2 + c2 + 2(ab + bc + ca) ≡ (a + b + c)2
OlZl<!fil<!gi{<hK! (a + b + c)2 – 2 (ab + bc + ca) = a2 + b2 + c2 + 2(ab + bc + ca) – 2(ab + bc + ca) = a2 + b2 + c2. weOu?!lx<oxiV!g{qk!Lx<oxiVjljbh<!ohXgqOxil</!
(a + b + c)2 – 2(ab + bc + ca) ≡ a2 + b2 + c2
hg<gr<gjt!lix<xq!wPkqeiz<!fil<!ohXuK!
a2 + b2 + c2 ≡ (a + b + c)2 – 2(ab + bc + ca) wMk<Kg<gim<M 15: hqe<uVueux<jx!uqiqUhMk<Kg/! (i) (2x + y + 2z)2 (ii) (x – 2y + z)2 (iii) (2p – 3q – r)2 (iv) (2a + 3b − 2c)2
kQIU: (i) (2x + y + 2z)2 = [(2x) + y + (2z)]2 = (2x)2 + y2 + (2z)2 + 2(2x)y + 2y(2z) + 2(2z)(2x) = 4x2 + y2 + 4z2 + 4xy + 4yz + 8zx. (ii) (x – 2y + z)2 = [x + (–2y) + z]2= x2 + (–2y)2 + z2 + 2x(–2y) +2 (–2y)z + 2zx = x2 + 4y2 + z2 – 4xy – 4yz + 2zx. (iii) (2p – 3q – r)2 = [(2p) + (–3q) + (–r)]2
= (2p)2 + (–3q)2 + (–r)2 + 2(2p) (–3q) + 2(–3q) (–r) + 2(–r)(2p). = 4p2 + 9q2 + r2 – 12pq + 6qr – 4rp. (iv) (2a + 3b – 2c)2 = [(2a) + (3b) + (–2c)]2
= (2a)2 + (3b)2 + (–2c)2 + 2(2a)(3b) + 2(3b)(–2c) + 2(–2c)(2a) = 4a2 + 9b2 + 4c2 + 12ab – 12bc – 8ca. 4.2.2 (x + a) (x + b) (x + c) g<gie!-bx<g{qk!Lx<oxiVjl! (x + a)(x + b)(x + c) = (x + a)[(x + b)(x + c)] = (x + a)[x2 + (b + c)x + bc] = (x + a)(x2 + bx + cx + bc) = x(x2 + bx + cx + bc) + a(x2 + bx + cx + bc) = x3 + bx2 + cx2 + bcx + ax2 + abx + acx + abc = x3 + (a + b + c)x2 + (ab + bc + ca)x + abc. weOu?! (x + a)(x + b)(x + c) ≡ x3 + (a + b + c)x2 + (ab + bc + ca)x + abc
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OlOz!ogiMg<gh<hm<Mt<t!Lx<oxiVjlbqz<! a, b, c -ux<xqx<G! hkqzig! –a, –b, –c Jh<!hqvkqbqm!flg<Gg<!gqjmh<hK (x – a)(x – b)(x – c) = [x + (–a)][x + (–b)][x + (–c)] = x3 + [(–a) + (–b) + (–c)]x2 + [(–a) (–b) + (–b) (–c) + (–c) (–a)]x + (–a) (–b) (–c) = x3 – (a + b + c)x2 + (ab + bc + ca)x – abc. -u<uiX! (x – a)(x – b)(x – c) ≡ x3 – (a + b + c)x2 + (ab + bc + ca)x – abc. -kqz<, x2 e<!ogP!= – (a + b + c), x e<!ogP!= ab + bc + ca, lixqzq!dXh<H= – abc. -jkh<!OhizOu?!(x + a)(x + b)(x + c) z< x2 e<!ogP!= a + b + c, x e<!ogP!= ab + bc + ca, lixqzq!dXh<H= abc.!!wMk<Kg<gim<M 16 : uqiqU!gi{<g/ (i) (x + 4)(x + 3)(x + 5) (ii) (2x + 1)(2x – 3)(2x + 5) (iii) (3 – 2x)(2x + 7)(2x + 1) (iv) (x – a)(x – 2a)(x – 3a) kQIU: (i) (x + 4)(x + 3)(x + 5) = x3 + (4 + 3 + 5)x2 + [4 × 3 + 3 × 5 + 5 × 4]x + 4 × 3 × 5 = x3 + 12x2 + [12 + 15 + 20]x + 60 = x3 + 12x2
+ 47x + 60. (ii) (2x + 1)(2x – 3)(2x + 5) = (2x + 1)[2x + (–3)](2x + 5) = (2x)3 + [1 + (–3) + 5](2x)2 + [1 × (–3) + (–3) ×5 + 5 × 1](2x) + 1 × (–3) × 5 = 8x3 + 3(4x2) + [–3 – 15 + 5](2x) – 15 = 8x3 + 12x2 – 26x – 15. (iii) (3 – 2x)(2x + 7)(2x + 1) = [–(2x – 3)](2x +7 )(2x + 1) = – [2x + (–3)](2x + 7)(2x + 1) = – [(2x)3 + {(–3) + 7 + 1}(2x)2 + {(–3) × 7 + 7×1+ 1×(–3)}(2x) + (–3) × 7 × 1] = – [8x3 + (5)(4x2) + (–21 + 7 − 3)(2x) –21] = – [8x3 + 20x2 – 34x – 21] = 21 + 34x – 20x2 – 8x3. (iv) (x – a)(x – 2a)(x – 3a) = [x+(–a)][x+(–2a)][x+(–3a)] = x3 + {(–a) + (–2a)+( –3a)}x2+{(–a) (–2a) + (–2a) (–3a) + (–3a) (–a)}x + (–a) (–2a) (–3a) = x3 + {–6a}x2 + {2a2 + 6a2 + 3a2}x – 6a3 = x3 – 6ax2 + 11a2x – 6a3. wMk<Kg<gim<M 17 : ohVg<gx<hze<! $k<kqvr<gjth<! hbe<hMk<kq?! x2 dXh<hqe<! ogP?! x dXh<hqe<!ogP?!lixqzq!dXh<H!Ngqbux<jxg<!gi{<g/ (i) (x + 3)(x + 5)(x + 6) (ii) (x – 7)(x + 2)(x + 4) (iii) (x – 5)(x – 2)(x + 4) (iv) (2x – 3 )(2x – 5)(7 – 2x) kQIU: (i) (x + 3)(x + 5)(x + 6) J! (x + a)(x + b)(x +c ) Bme<!yh<hqMjgbqz<?
a = 3, b = 5, c = 6. ∴ x2 e<!ogP!= a + b + c = 3 + 5 + 6 = 14, x e<!ogP!= ab + bc + ca = (3 × 5) + (5 × 6) + (6 × 3) = 15 + 30 + 18 = 63, lixqzq!dXh<H = abc = 3 × 5 × 6 = 90. (ii) (x – 7)(x + 2)(x + 4) = [x + (–7)](x + 2)(x + 4) -jk!(x + a)(x + b)(x + c) Bme<!yh<hqMjgbqz<? a = –7, b = 2, c = 4. ∴ x2 e<!ogP!= a + b + c = (–7) + 2 + 4 = –1.
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x e<!ogP = ab + bc + ca = (–7) × 2 + 2 × 4 + 4 × (–7) = (–14) + 8 + (–28)= (–42) + 8 = –34, lixqzq!dXh<H = abc = (–7) × 2 × 4 = –56. (iii) (x – 5)(x – 2)(x + 4) = [x + (–5)][x + (–2)](x + 4) -jk! (x + a)(x + b)(x + c) Bme<!yh<hqMjgbqz<?!a = –5, b = –2, c = 4. ∴ x2 e<!ogP = a + b + c = (–5) + (–2) + 4 = (–7) + 4 = –3, x e<!ogP = ab + bc + ca = (–5) × (–2) +(–2) × 4 + 4 × (–5)= 10 – 8 – 20 = –18, lixqzq!dXh<H = abc = (–5) (–2)4 = 40. (iv) (2x – 3)(2x – 5)(7 – 2x) we<hK!A!we<gqx!-bx<g{qk!Ogiju!we<g/!-kqz<!!!!!!!!!!!!y=2x weg<!ogi{<miz<, A = (y – 3)(y – 5)(7 – y) = (y – 3)(y – 5) [–(y – 7)] = –[(y – 3)(y – 5 )(y – 7)] = –[y3 + {(–3) + (–5) + (–7)}y2 + {(–3)(–5) + (–5)(–7) + (–7)(–3)}y + (–3)(–5)(–7)] = –[y3 – 15y2 + (15 + 35 + 21)y – 105] = –y3 + 15y2 – 71y + 105 = –(2x)3 + 15(2x)2 – 71(2x) + 105 = –8x3 + 60x2 – 142x + 105 ∴ x2 e<!ogP = 60, x e<!ogP = –142, lixqzq!dXh<H = 105 !lx<oxiV!Ljx;!(2x – 3)(2x – 5)(7 – 2x)
= ⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −−⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −
272
252
232 xxx = ⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −+⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −+⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −+−
27
25
238 xxx
∴ x2 e<!ogP = –8(a + b + c) = ⎥⎦⎤
⎢⎣⎡ −−−−
27
25
238 = ⎟
⎠⎞
⎜⎝⎛−−
2158 = 60,
x e<!ogP = –8( ab + bc + ca) = ⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −⎟⎠⎞
⎜⎝⎛ −+⎟
⎠⎞
⎜⎝⎛ −⎟⎠⎞
⎜⎝⎛ −+⎟
⎠⎞
⎜⎝⎛ −⎟⎠⎞
⎜⎝⎛ −−
23
27
27
25
25
238
= –8 ⎥⎦⎤
⎢⎣⎡ ++
421
435
415 = –8 ⎥⎦
⎤⎢⎣
⎡ ++4
213515 = (–2) (71)= –142,
lixqzq!dXh<H = –8(abc) = ⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −⎟⎠⎞
⎜⎝⎛ −⎟⎠⎞
⎜⎝⎛ −−
27
25
238 = –8 ⎟
⎠⎞
⎜⎝⎛ −
8105
= 105.
wMk<Kg<gim<M 18 : (x + a)(x + b)(x + c) ≡ x3 – 6x2 + 11x – 6 weqz<?!a2 + b2 + c2!e<!lkqh<H!gi{<g/ kQIU; ohVg<gx<hze<!$k<kqvk<kqzqVf<K!fil<!nxquK?! (x + a)(x + b)(x + c) = x3 + (a + b + c)x2 + (ab + bc + ca)x + abc. yh<hqMjgbqz<!flg<Gg<!gqjmh<hK?!a + b + c = –6, ab + bc + ca = 11, abc = –6. ∴ a2 + b2 + c2 = (a + b + c)2 –2 (ab + bc + ca) = (– 6)2 – 2(11) = 36 – 22 = 14. (x + a)(x + b)(x + c) e<!Lx<oxiVjlbqzqVf<K!sqz!Lx<oxiVjlgjtk<!kVuqh<Ohil</! (i) (a + b)3 g<gie!Lx<oxiVjl! (a + b)3 = (a + b)(a + b)(a + b) = a3 + (b + b + b)a2 + (b × b + b × b + b × b)a + b × b × b = a3 + 3a2b + 3ab2 + b3
(a + b)3 ≡ a3 + 3a2b + 3ab2 + b3
∴
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hg<gr<gjt!lix<xqobPk?! (ii) (a – b)3 g<gie!Lx<oxiV (a – b)3 = [a + (–b)]3
= a3 + 3a2(–b)
∴ !
hg<gr<gjt!lix<x?!
(a + b)3 , (a – b)3 we<x!LxkVuqh<Ohil</
(i) (a + b)3–3ab(a +
∴
hg<gr<gjt!lix<x? (ii) (a – b)3+3ab(a –
(a ∴
hg<gr<gjt!lix<x? (iii) a3 + b3 = (a + b)3
= (a + b)[ = (a + b)(
a3 + ∴
hg<gr<gjt!lix<x? (iv) a3 – b3 = (a – b)3 + = (a – b)[( = (a – b)(a ∴ a3 − !
(a −hg<gr<gjt!lix<x?
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a3 + 3a2b + 3ab2 + b3 ≡ (a + b)3
jl
+ 3a(–b)2 + (–b)3= a3 – 3a2b + 3ab2 – b3
(a − b)3 ≡ a3 − 3a2b + 3ab2 − b3
a3 − 3a2b + 3ab2 − b3≡ (a − b)3
<oxiVjlgtqe<!nch<hjmbqz<!hqe<uVl<!Lx<oxiVjlgjtk<!
b) = a3 + 3a2b + 3ab2 + b3 – 3a2b – 3ab2 = a3+b3.
(a + b)3 − 3ab(a + b) ≡ a3 + b3
a3 + b3 ≡ (a + b)3 − 3ab(a + b)
b) = a3 – 3a2b + 3ab2 – b3 + 3a2b – 3ab2 = a3– b3.
− b)3 + 3ab(a − b) ≡ a3 − b3
a3 − b3 ≡ (a − b)3 + 3ab(a− b)
– 3ab(a + b) = (a + b)(a + b)2 – 3ab(a + b) (a + b)2 – 3ab] = (a + b)[(a2 + 2ab + b2) – 3ab] a2 – ab + b2).
b3 ≡ (a + b)(a2 – ab + b2)
(a + b)(a2 – ab + b2) ≡ a3 + b3
3ab(a – b) = (a – b)(a – b)2 + 3ab(a – b) a – b)2 + 3ab] = (a – b)[(a2 – 2ab + b2) + 3ab] 2 + ab + b2).
b3 ≡ (a − b)(a2 + ab + b2)
b)(a2 + ab + b2) ≡ a3 − b3
104
a2 + b2 + c2 − ab − bc − ca g<gie!Lx<oxiVjl! fil<!ohXuK?!
a2 + b2 + c2 − ab − bc − ca = )222222(21 222 cabcabcba −−−++
= [ ])2()2()2(21 222222 acaccbcbbaba +−++−++−
= [ ].)()()(21 222 accbba −+−+−
-u<uiX?!fil<!hqe<uVl<!Lx<oxiVjljbh<!ohXgqOxil<;!
2221 hg<gr<gj (a + b + c
hr (a + b + c = = = -u<uiX! hg<gr<gj wMk<Kg<gkQIU; (i)
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2 2 2
a + b + c −ab − bc − ca ≡ [ ])()()(2accbba −+−+−
t!lix<xqobPk?
2221 2 2 2
[ )()()( ]2accbba −+−+− ≡ a + b + c −ab − bc − ca
) (a2 + b2 + c2 − ab − bc − ca) uqx<gie!Lx<oxiVjl!
<gQm<Mh<!h{<jhh<!hbe<hMk<kq!fil<!njmuK?!
) (a2 + b2 + c2 − ab − bc − ca) a (a2 + b2 + c2 − ab − bc − ca) + b (a2 + b2 + c2 − ab − bc − ca)
+ c (a2 + b2 + c2 − ab − bc − ca)
a3 + ab2 + c2a − a2b − abc − ca2 + a2b + b3 + bc2 − ab2 − b2c − abc +ca2 + b2c + c3 − abc − bc2 − c2a
a3 + b3 + c3 − 3abc.
fil<!ohXuK?!
(a + b + c) (a2 + b2 + c2 − ab − bc − ca) ≡ a3 + b3 + c3 − 3abc
t!lix<x?
a3 + b3 + c3 − 3abc ≡ (a + b + c) (a2 + b2 + c2 − ab − bc − ca).
im<M 19 : hqe<uVueux<jx!uqiqk<okPKg; (i) (3x + 2y)3 (ii) (2x2 – 3y)3
(3x + 2y)3 = (3x)3 + 3(3x)2(2y) + 3(3x)(2y)2 + (2y)3
= 27x3 + 3(9x2)(2y) + 3(3x)(4y2) + 8y3
= 27x3 + 54x2y + 36xy2 + 8y3.
105
(ii) (2x2– 3y)3 = (2x2)3 – 3(2x2)2 (3y) + 3(2x2) (3y)2 – (3y)3
= 8x6 – 3(4x4)(3y) + 3(2x2)(9y2) – 27y3
= 8x6– 36x4y + 54x2y2 – 27y3
!wMk<Kg<gim<M 20 : a + b , ab Ngqbux<xqe<!lkqh<Hg<gt<!LjxOb!4, 1 weqz<? a3 + b3 e<!lkqh<H!gi{<g/ kQIU; a3+ b3 = (a + b)3 – 3ab(a + b) = (4)3 – 3(1)(4) = 64 – 12 = 52. wMk<Kg<gim<M 21: a – b = 4 lx<Xl< ab = 2 weqz<? a3 – b3
e<!lkqh<H!gi{<g/!kQIU; a3 – b3 = (a–b)3 + 3ab(a–b) = (4)3 + 3(2)(4) = 64 + 24 = 88. wMk<Kg<gim<M 22: a + b = 2, a2 + b2 = 8 weqz<? a3 + b3 , a4 + b4 lkqh<H!gi{<g/!kQIU; a2 + 2ab + b2 = (a + b)2
2ab = (a + b)2 – (a2 + b2) = (2)2 – (8) = 4 – 8 = – 4. ∴ ab = 2
1 (2ab) = 21 (–4) = – 2.
a3+ b3 = (a + b)3 – 3ab(a + b) = (2)3 – 3(–2)(2) = 8 – 3 (– 4) = 8 + 12 = 20. lix<Xupq;! a3 + b3 = (a + b)(a2 – ab + b2) = (a + b)(a2 + b2 – ab) = (2) [8 – (–2)] = 2(10) = 20. a4 + b4 = (a2)2 + (b2)2 = [(a2) + (b2)]2 – 2 (a2)(b2) = (a2 + b2)2 – 2a2b2 = (a2 + b2)2 – 2(ab)2
= (8)2 – 2(–2)2 = 64 – 2(4)= 64 – 8 = 56.
hbqx<sq 4.2 1. ohVg<gx<hze<!$k<kqvk<jkh<!hbe<hMk<kq!hqe<uVueux<jxg<!gi{<g;!
(i) (x + 9) (x + 2) (ii) (x + 8) (x – 2) (iii) (t – 2)(t + 6) (iv) (p – 4)(p – 3) (v) 102 × 106 (vi) 59 × 62 (vii) 34 × 36 (viii) 53 × 55
2. ohVg<gx<hze<!$k<kqvk<jkh<!hbe<hMk<kqg<!gi{<g;! (i) (5x + 8y)2 (ii) (3s – 4t)2 (iii) (4p + 7q)(4p – 7q) (iv) (101)2 (v) (98)2 (vi) 101 × 98 3. a + b = 5 , a – b = 4 weqz<? a2 + b2 lx<Xl< ab bqe<!lkqh<Hg<!gi{<g/ 4. a + b = 10 , ab = 20 weqz<?! a2 + b2, (a – b)2 Jg<!gi{<g/ 5. (x + l)(x + m) = x2 + 4x + 2 weqz<? l2 + m2 , (l – m)2 Jg<!gi{<g/! 6. hqe<uVueux<jx!uqiqk<okPKg;!
(i) (3x + y + 2z)2 (ii) (4x – 2y + 3z)2 (iii) (2p + 3q – 2r)2 (iv) (3a – 2b – 2c)2
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7. a + b + c = 11 , ab + bc + ca = 38 weqz<? a2 + b2 + c2 gi{<g/! 8. ohVg<gx<hze<!$k<kqvk<jkh<!hbe<hMk<kqh<!hqe<uVueux<xqe<!uqiqU!gi{<g/!! (i) (x + 2 )(x + 3)(x + 4) (ii) (x + 2)(x + 3)(x – 4) (iii) (x + 2)(x – 3)(x + 4) (iv) (x + 2)(x – 3)(x – 4) (v) (x – 2)(x – 3)(x – 4) 9. ohVg<gx<hze<!$k<kqvk<jkh<! hbe<hMk<kqh<! hqe<uVueux<xqz< x2 dXh<H? x dXh<H!
Ngqbux<xqe<!ogPg<gjtBl<?!lixqzq!dXh<jhBl<!gi{<g; (i) (x + 10)(x – 3)(x + 2) (ii) (2x – 3)(2x + 4)(2x – 1) (iii) (6x + 1)(6x – 5)(7 – 6x)
10. (x + a)(x + b)(x + c) ≡ x3 – 9x2 + 23x – 15 weqz<? a + b + c, cba111
++ , a2 + b2 + c2!
Ngqbux<jxg<!gi{<g/ 11. ohVg<gx<hze<!$k<kqvk<jkh<!hbe<hMk<kq!uqiquig<Gg;
(i) (2x + y2)3 (ii) (2u – 7v)3 (iii) 31⎟⎠⎞
⎜⎝⎛ −
xx (iv) (x2y3 + 2)3
12. 2a – 3b = 2 , ab = 6 weqz<?!8a3 – 27b3 e<!lkqh<H!gi{<g/
13. 31=+
xx !weqz<?! x2 + 2
1x
, x3+ 3
1x
Jg<!gi{<g/
14. x + y = 6 , xy = 8 weqz<?!x2 + y2, x3 + y3!Jg<!gi{<g/ 15. p + q = 6 , p2 + q2 = 32 weqz<?!pq, p3 + q3 , p4 + q4 Jg<!gi{<g/ 4.3. giv{qh<hMk<kz< Lf<jkb!hGkqbqz<!-v{<M!nz<zK!nkx<G!Olx<hm<m!hz<ZXh<Hg<!Ogijugjt!ohVg<gq?! lx<oxiV! hz<ZXh<Hg<! Ogijujb! wu<uiX! ohxzil<! weg<! g{<mxqf<Okil</!-h<ohiPK! yV! hz<ZXh<Hg<! Ogijujb! wu<uiX! -v{<M! nz<zK! nkx<G! Olx<hm<m!hz<ZXh<Hg<! Ogijugtqe<! ohVg<gzig! wPkzil<! weg<! gx<Ohil</! yV! hz<ZXh<Hg<!Ogijujb! -v{<M! nz<zK! nkx<G! Olx<hm<m! wtqb! hz<ZXh<Hg<! Ogijugtqe<!ohVg<gzig! wPKuK!giv{qh<hMk<kz<! weg<! %xh<hMl</! ohVg<gzqz<!dt<t! yu<ouiV!wtqb! hz<ZXh<Hg<! OgijuBl<! ogiMg<gh<hm<m! hz<ZXh<Hg<! Ogijubqe<! yV! giv{q!
weh<hMl</!wMk<Kg<gim<mig? x + 3, x – 3 we<hju x2 – 9 e<!giv{qgtiGl</ x2 – 9 = (x + 3)(x – 3) we<hkiz<?!-r<G!x2 – 9 we<hK!-Vhc!hz<ZXh<Hg<!Ogiju/ x + 3, x – 3 we<hju!yVhc!hz<ZXh<Hg<! Ogijugt</! -u<uiX! giv{qh<hMk<Kkz<! Ogijugjts<! SVg<Gukx<G!dkUgqxK/! giv{qh<hMk<Kkz<! we<El<! -f<ks<! osbz<?! giv{qgtigh<! hGk<kz<! we<Xl<!%xh<hMl</ 4.3.1. giv{qh<hMk<kz<!upqLjx upq 1: (ohiKuieg<!giv{qjbg<!g{<Mhqck<kz<) A we<gqx!YI!-bx<g{qkg<!Ogijubqe<!dXh<Hgtqz< B we<gqx! ohiKg<giv{q!-Vf<kiz<?! fil<!A bqe<! yu<ouiV!dXh<jhBl< B
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biz<!uGg<g!C we<gqx!Ogiju!gqjmg<Gl</!-h<ohiPK!A J!B × C weg<!giv{qh<hMk<kq!wPkzil</! wMk<Kg<gim<M 23 : giv{qh<hMk<Kg; 6x4y3 – 4x2y2 + 10xy3. kQIU; 2xy2 NeK!ohiKg<giv{q!we<hjkg<!gi{<gqOxil</
∴ 6x4y3 – 4x2y2 + 10xy3 = 2xy2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+− 2
3
2
22
2
34
210
24
26
xyxy
xyyx
xyyx = 2xy2(3x3y – 2x + 5y).
upq 2: (dXh<Hgjtg<! GPh<hMk<Kkz<) YI! -bx<g{qkg<! Ogijubqe<! dXh<HgTg<Gh<!ohiKuie! giv{q! -z<jzobeqz<! nf<k!dXh<Hgjtk<! kGf<k!Ljxbqz<! GPh<hMk<kq!yV!ohiKuie!giv{q!kQIlieqg<gh<hMl</! wMk<Kg<gim<M 24 : giv{qh<hMk<Kg; x2 – 2xy – x + 2y. kQIU; Ogijubqe<!dXh<HgTg<Gh<!ohiKuieg<!giv{q!-z<jz/!weqEl<!dXh<Hgt<!gQOp!g{<muiX!GPh<hMk<kh<hmzil<!weg<!gi{<gqOxil</ x2 – 2xy – x + 2y = (x2 – 2xy) – (x – 2y) = x(x – 2y) + (–1) (x – 2y) = (x – 2y) [x + (–1)] = (x – 2y) (x – 1). wMk<Kg<gim<M 25 : giv{qh<hMk<Kg; 6x5y2 + 6x4y3 + 9x2y4 + 9xy5. kQIU;!Lkz<!upq?!2 l<!upq!-v{<jmBl<!hbe<hMk<k?!fil<!gi{<hK?! 6x5y2 + 6x4y3 + 9x2y4 + 9xy5 = 3xy2(2x4 + 2x3y + 3xy2 + 3y3) = 3xy2 [(2x4 + 3xy2) + (2x3y + 3y3)] = 3xy2 [x(2x3 + 3y2) + y(2x3 + 3y2)] = 3xy2 (2x3 + 3y2) (x + y).
hbqx<sq 4.3.1 ohiKuie!yV!giv{qjbg<!g{<Mhqck<K!giv{qgtigh<!hGg<gUl<;! 1. 9m – 3n 2. 4a3 – 8a2 + 16a 3. x5 + 4x 4. 6x5y5 + 3x2y3 + 14xy3 5. 7pq – 21p2q2
ohiKuie! yV! giv{qjbg<! g{<Mhqck<Oki! nz<zK! GPh<hMk<kz<! Ljxjbh<!hbe<hMk<kqObi?!giv{qgtigh<!hGg<gUl<;! 6. mn – 2p – pn + 2m 7. x3 – 2x2 – 2x + 4 8. x3 – x2 – ax + a 9. 2p3 – p2 + 2p – 1 10. 8x3 + 4x2 + 4x + 2 4.3.2. giv{qh<hMk<kz<!$k<kqvr<gjth<!hbe<hMk<kq?!giv{qh<hMk<kz< sqz! Ofvr<gtqz<! yV! Ogijujbg<! giv{qh<hMk<Kl<OhiK! giv{qh<hMk<kz<!$k<kqvr<gjth<! hbe<hMk<KOuil</! -s<$k<kqvr<gt<! ohVg<gz<! $k<kqvr<gtqzqVf<K!ohxh<hm<mjubiGl</!ohVg<gz<!$k<kqvr<gtiue; (i) (X + Y)2 = X 2 + 2XY + Y 2 (ii) (X – Y)2 = X 2 – 2XY + Y 2 (iii) (X + Y)(X – Y) = X 2 – Y 2 (iv) ( X + Y) (X 2 – XY + Y 2) = X 3 + Y 3 (v) (X – Y)(X 2 + XY + Y 2) = X 3 – Y 3 (vi) (X + Y)3 = X 3 + Y 3 + 3X 2Y + 3XY 2 = X 3 + Y 3 + 3X Y (X +Y ) (vii) (X – Y)3 = X 3 – Y 3 – 3X 2Y + 3XY 2 = X 3 – Y 3 – 3X Y (X –Y )
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(viii) (X + Y + Z)2 = X 2 + Y 2 + Z 2 + 2XY + 2YZ + 2ZX (ix) (X + Y + Z) (X 2 + Y 2 +Z 2 – XY – YZ – ZX) = X 3 + Y 3 + Z 3 – 3XYZ OlOz! ogiMg<gh<hm<Mt<t! $k<kqvr<gjt! uzK! hg<gk<kqzqVf<K! -mKhg<glig!hck<Okiole<xiz<! flg<G! gQOp! ogiMg<gh<hm<Mt<t! giv{qh<hMk<kz<! $k<kqvr<gt<!gqjmg<Gl</
(i) X 2 + 2XY + Y 2 = (X +Y)2
(ii) X 2 – 2XY + Y 2 = (X – Y)2
(iii) X 2 – Y 2 = (X + Y) (X – Y) (iv) X 3 + Y 3 = ( X + Y) (X 2 – XY + Y 2) (v) X 3 – Y 3 = (X – Y) (X 2 + XY + Y 2) (vi) X 3 + Y 3 + 3X 2Y+ 3XY 2 = (X + Y)3
(vii) X 3 – Y 3 – 3X 2Y + 3XY 2 = (X – Y)3
(viii) X 2 + Y 2 + Z 2 + 2XY + 2YZ + 2ZX = ( X + Y + Z)2
(ix) X 3+Y 3+Z 3–3XYZ = (X + Y + Z) (X 2 + Y 2 + Z 2 – XY – YZ – ZX) X 2 + 2XY + Y 2 ≡ (X + Y)2 Jh<!hbe<hMk<kq!giv{qh<hMk<kz<! wMk<Kg<gim<M 26: giv{qgtigh<!hGg<gUl<; 4x2 + 12xy + 9y2. kQIU; ogiMg<gh<hm<m!Ogijujbh<!hqe<uVliX!lix<xqbjlg<gzil</ 4x2+ 12xy + 9y2 = (2x)2 + 2(2x)(3y) + (3y)2
X = 2x, Y = 3y we!wMk<Kg<ogi{<miz< uzKhg<gl<!= X 2 + 2XY + Y 2 . weOu?!nK!(X + Y)2 weg<!giv{qh<hMk<kh<hMl</!weOu!fil<!ohXuK! 4x2 + 12xy + 9y2 = (2x + 3y)2. X 2 – 2XY + Y 2 ≡ (X – Y)2 Jh<!hbe<hMk<kq!giv{qh<hMk<kz< wMk<Kg<gim<M 27 : giv{qh<hMk<Kg;! p2 – 18pq + 81q2. kQIU; ogiMg<gh<hm<m!Ogijujb!hqe<uVliX!wPkzil</ p2 – 18pq + 81q2 = p2 – 2(p)(9q) + (9q)2
X = p, Y = 9q we!wMk<Kg<ogi{<miz<!uzKhg<gl<!X 2 – 2XY + Y 2 NGl</!Njgbiz<!nK (X – Y)2 we!giv{qh<hMk<kh<hMl</!weOu?!fil<!ohXuK! p2 – 18pq + 81q2 = (p – 9q)2. X 2 – Y 2 ≡ (X + Y) (X – Y) Jh<!hbe<hMk<kq!giv{qh<hMk<kz< wMk<Kg<gim<M 28: giv{qh<hMk<Kg; 16x4y2 – 25. kQIU; 16x4y2 = (4x2y)2 , 25 = (5)2, we<hkiz< 16x4y2 – 25 = (4x2y)2 – (5)2 = (4x2y + 5) (4x2y – 5). X 3 + Y 3 ≡ (X + Y) (X 2 – XY + Y 2) Jh<!hbe<hMk<kq!giv{qh<hMk<kz< wMk<Kg<gim<M 29: giv{qgtigh<hG; 125a3 + 64b3. kQIU; 125a3 = (5a)3 , 64b3 = (4b)3 we<hkiz<?!fil< X = 5a, Y = 4b we!wMk<Kg<!ogit<Ouil</!hqe<eI!125a3 + 64b3 = (5a)3 + (4b)3 = X 3 + Y 3 = (X + Y) (X 2 – XY + Y 2) = (5a + 4b) [(5a)2 – (5a)(4b) + (4b)2] = (5a + 4b) (25a2 – 20ab + 16b2).
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X 3 – Y 3 ≡ (X – Y) (X 2 + XY + Y 2) Jh<!hbe<hMk<kq!giv{qh<hMk<kz<! wMk<Kg<gim<M 30 : giv{qh<hMk<Kg;! 216p3– 8q3. kQIU; 216p3 = (6p)3, 8q3 = (2q)3 we!wPkzil</!Njgbiz<? X = 6p, Y = 2q we!wMk<Kg<!ogi{<Omilieiz< 216p3 – 8q3 = (6p)3 – (2q)3
X 3 – Y 3 = (X – Y) (X 2 + XY + Y 2) = (6p – 2q) [(6p)2 +(6p)(2q) +(2q)2] = (6p –2q)(36p2+12pq+4q2). X 3 + Y 3 + 3X 2Y + 3XY 2 ≡ (X + Y)3 Jh<!hbe<hMk<kq!giv{qh<hMk<kz< wMk<Kg<gim<M 31: giv{qh<hMk<Kg; 8x3 + y 3 + 12x2y + 6xy2. kQIU; 8x3 + y3 + 12x2y + 6xy2 = (2x)3 + y3 + 3(2x)2y + 3(2x)y2= [(2x) + y]3 = (2x + y)3. X 3 – Y 3 – 3X 2Y + 3XY 2 ≡ (X – Y)3 Jh<!hbe<hMk<kq!giv{qh<hMk<kz< !wMk<Kg<gim<M 32: giv{qh<hMk<Kg; 8x3 – 27y3 – 36x2y + 54xy2. kQIU; 8x3 – 27y3 – 36x2y + 54xy2 = (2x)3 – (3y)3 – 3(2x)2(3y) + 3(2x)(3y)2= (2x – 3y)3. X 2+ Y 2 + Z 2 + 2XY + 2YZ + 2ZX ≡ (X + Y + Z)2 Jh<!hbe<hMk<kq!giv{qh<hMk<kz< wMk<Kg<gim<M 33 : giv{qh<hMk<Kg; x2+ 9y2 – 6xy + 4x – 12y + 4. kQIU; Ogiju x2 + 9y2 + 4 we<x! %Mkjzg<! ogi{<Mt<tK/! -K!&e<X! uIg<gr<gtqe<!%Mkz</!Njgbiz<?!fil<!wPKuK x2 + 9y2 – 6xy + 4x – 12y + 4 = x2 + (3y)2 + 22 + 2x (–3y) + 2x(2) + 2(2)(–3y) = [x + (–3y) + 2]2 = (x – 3y + 2)2. X 3 + Y 3 + Z 3 – 3XYZ ≡ (X + Y + Z) (X 2 + Y 2 + Z 2 – XY – YZ – ZX) Jh<! hbe<hMk<kq!giv{qh<hMk<kz< wMk<Kg<gim<M 34: giv{qh<hMk<Kg; x3 – 8y3 + 27z3 + 18xyz. kQIU; –8y3 = (–2y)3, 27z3 = (3z)3 we<hkiz<!ogiMg<gh<hm<m!Ogijujb!P we<g/!-K!&e<X!g{r<gtqe<!%Mkjzg<!ogi{<Mt<tK/!weOu!fil<!wPKuK! P = (x)3 + (–2y)3 + (3z)3 – 3(x)(–2y)(3z) = [(x) + (–2y) + (3z)] [(x)2 + (–2y)2 + (3z)2 – (x) (–2y) – (–2y)(3z) – (3z)(x)] = (x – 2y + 3z) (x2 + 4y2 + 9z2 + 2xy + 6yz – 3zx). X + Y + Z = 0 !wEl<OhiK!X 3 + Y 3 + Z 3 Jg<!giv{qh<hMk<kz<! X 3 + Y 3 + Z 3 = (X 3 + Y 3 + Z 3 –3XYZ) + 3XYZ = (X + Y + Z) (X 2 + Y 2 + Z 2 – XY – YZ – ZX) + 3XYZ = (0) (X 2 + Y 2 + Z 2 – XY – YZ – ZX) + 3XYZ = 3XYZ. Gxqh<H;! X + Y + Z = 0 weqz<, X + Y = – Z. ∴ (X + Y)3 = (– Z)3 = – Z3 . n-K X 3 + Y 3 + 3XY(X +Y ) = – Z3 . n-K, X 3 + Y 3 + Z 3 + 3XY(– Z) = 0. ∴ X 3 + Y 3 + Z 3 = 3XY Z.
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wMk<Kg<gim<M 35: giv{qh<hMk<Kg; (x – y)3 + (y – z)3 + (z – x)3. kQIU;! X = x – y, Y = y – z, Z = z – x we<g/ hqe<eI X + Y + Z = (x – y) + (y – z) + (z – x) = x – y + y – z + z – x = 0. ∴ X 3 + Y 3 + Z 3 = 3XYZ X, Y, Z !Ngqbux<xqx<G!hqvkqbqm<miz<!(x – y)3 + (y – z)3 + (z –x)3 = 3(x – y) (y – z) (z – x). GP!F[g<gk<jkh<!hbe<hMk<kqg<!giv{qh<hMk<kz<! wMk<Kg<gim<M 36: giv{qh<hMk<Kg; 4x2 + 20xy + 25y2 – 10x – 25y. kQIU; ogiMg<gh<hm<m!Ogiju!4x2 + 20xy + 25y2 – 10x – 25y = (2x)2 + 2(2x)(5y) + (5y)2 – 5(2x) – 5(5y) = [(2x) + (5y)]2 – 5[(2x) + (5y)] = (2x + 5y)2 – 5(2x + 5y) = (2x + 5y) (2x + 5y –5). !wMk<Kg<gim<M 37: giv{qh<hMk<Kg; 4a2 – 4ab + b2 – 2a + b. kQIU; ogiMg<gh<hm<m!hz<ZXh<Hg<Ogiju 4a2 – 4ab + b2 – 2a + b = (2a)2 – 2(2a)(b) + (b)2 – (2a – b) = (2a – b)2 – 1 (2a – b) = (2a – b) (2a – b – 1). wMk<Kg<gim<M 38: giv{qh<hMk<Kg; 81x2 – 18x + 1 – 25y2. kQIU;! dXh<Hgjtg<!GPh<hMk<k! 81x2 – 18x + 1 – 25y2 = [(9x)2 – 2(9x)(1) + (1)2] – (5y)2
= [(9x) – (1)]2 – (5y)2
= (9x – 1) 2 – (5y)2
= [(9x –1) + (5y)][(9x – 1) – (5y)] = (9x + 5y – 1) (9x – 5y –1). wMk<Kg<gim<M 39: giv{qh<hMk<Kg; x4 +1. kQIU;! 2x2 J!%m<cg<!gpqg<g?!fil<!ohXuK x4 + 1 = (x4 + 2x2 + 1) – 2x2
= [(x2)2 + 2(x2)(1) + (1)2] – 2x2
= (x2 + 1)2 – ( 2 x)2
= [(x2 + 1) + ( 2 x)][(x2 + 1) – ( 2 x)] = (x2 + 2 x + 1) (x2 – 2 x + 1). wMk<Kg<gim<M 40: giv{qh<hMk<Kg; x4 + x2y2 + y4. kQIU;! x2y2 Jg<!%m<cg<!gpqg<g?!fil<!ohXuK x4 + x2y2 + y4 = (x4 + 2x2y2 + y4) – x2y2
= [(x2)2 + 2(x2)(y2) + (y2)2] – (xy)2 = (x2 + y2)2 – (xy)2
= [x2 + y2 + (xy)] [x2 + y2 – (xy)] = (x2 + xy + y2) (x2 – xy + y2). wMk<Kg<gim<M 41: giv{qh<hMk<Kg; x4 + 5x2 + 9. kQIU; x2
Jg<!%m<cg<!gpqg<g , fil<!ohXuK
x4 + 5x2 + 9 = (x4 + 6x2 + 9) – x2
= (x2 + 3)2 – x2 = [(x2 + 3) + x][(x2 + 3) – x] = (x2 + x + 3) (x2 – x + 3).
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wMk<Kg<gim<M 42: x8 – x2y6 Jg<!giv{qgtigh<!hqiq. kQIU;! x2 J!yV!ohiKg<giv{qbig!wMg<g? x8 – x2y6 = x2 (x6 – y6) = x2 [(x3)2 – (y3)2] = x2 (x3 + y3) (x3 – y3) = x2[(x + y)(x2 – xy + y2)] [(x – y)(x2 + xy + y2)] = x2 (x + y)(x – y)(x2 + xy + y2) (x2 – xy + y2).
hbqx<sq 4.3.2 hqe<uVue!siqbi!kuxi!we!uqjmbtq/!
1. 2x2 + 2 2 x + 1 = (1 + x2 )2
2. x6 – 4x3 + 4 = (x3 + 1)2
3. a2 – b2 = (a – b)2
4. a3 + b3 = (a + b)(a2 + ab + b2) 5. a3 – b3 = (a – b)(a2 + ab + b2)
giv{qh<hMk<kz<!$k<kqvr<gjth<!hbe<hMk<kq!giv{qgjtg<!g{<Mhqc/! 6. 1 + 6x + 9x2 7. 144x2 – 72x + 9 8. 4a2b2 + 20abcd + 25c2d2 9. x2 + y2 – a2 – b2 + 2xy + 2ab 10. 3 3 x3y3 + 27z3 11. (x + y)3 + 8y3
12. (x2 + 1)3 + (x2 – 1)3 13. x6 – y6 14. (x + y)3 − (x – y)3 15. (p + q)3 + (p – q)3 + 6p(p2 – q2) 16. 27x3 + y3 + 27x2y + 9xy2 17. x3 – 12x2 +48x – 64 18. 8x3 – 27y3 – 36x2y + 54xy2 19. 4x2 + 9y2 + z2 + 12xy + 4xz + 6yz 20. a2 + b2 + 9c2 + 2ab – 6ac – 6bc 21. x3 – y3 + 1 + 3xy 22. 8x3 – 125y3 + 180xy + 216 23. 8x3 – 27y3 + z3 +18xyz 24. 3 3 a3 – 8b3 – 125c3 − 30 3 abc 25. (a – 2b)3 + (2b – 3c)3 + (3c – a)3
26. (x + y – 2z)3 + (y + z – 2x)3 + (z + x – 2y)3
27. (a2 – b2)3 + (b2 – c2) 3 + (c2 – a2)3 28. a3(b – c)3 + b3(c – a)3 + c3(a – b)3
29. x(x + z) – y(y + z) 30. 1 – 2xy – x2 – y2 31. x4 + 4 4.3.3. ax2 + bx + c we<El<!-Vhcg<!Ogijujbg<!giv{qh<hMk<kz<!! ogPg<gt< a, b, c Ngqbju! njek<Kl<! LPg<gt<! we! wMk<Kg<! ogit<Ouil</!OlZl<!a ≠ 0/!ogPg<gt<!a, b, c sqz!gm<Mh<hiMgTg<G!dme<hm<miz<!ax2 + bx + c Jg< giv{qh<hMk<kzil</!-f<kg<!gm<Mh<hiMgjtBl<!giv{qgjtBl<!-eq!gi{<Ohil</ Lkzqz<? a = 1, b Bl<! c Bl<! LPg<gt<! we<El<! wtqb! ujgjb! wMk<Kg<!ogit<Ouil</!fil<!-h<OhiK x2 + bx + c we<hjkg<!giv{qh<hMk<k!Ou{<Ml</!fil<!lixqzq!dXh<higqb!c we<El<!LPju? p + q = b we!-Vg<GliX?!p?!q we<El<!-V!LPg<gtqe<!ohVg<gx<hzeig!wPKOuil</!fl<Ljmb!Lbx<sqbqz<!fil<!oux<xq!ohx<xiz<? x2 + bx + c = x2 + (p + q)x + pq = (x2 + px) + (qx + pq) = x(x + p) + q(x + p) = (x + p) (x + q) -u<uiX?!fil<!uqVl<hqbjks<!sikqk<Kt<Otil</
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uqkq : x2 + bx + c bqe<! lixqzq! dXh<H! c bieK?! p?! q we<El<! -V! LPg<gtqe<!ohVg<gx<hzeigUl<?!nu<uqV!LPg<gtqe<!%Mkzigqb! p + q uieK! x e<! ogPuigqb!!!!b g<Gs<!slligUl<!-Vg<GliX!hqiqg<gh<hm<miz<?!x2 + bx + c = (x + p) (x + q). !wMk<Kg<gim<M 43: giv{qh<hMk<Kg; x2 + 9x + 18. kQIU;! ogiMg<gh<hm<m!Ogijujb!X2 + XY + Y2 wEl<!ucuk<kqz<!wPk!LcbiK/!weOu? X2 + 2XY + Y2 = (X + Y)2 we<hjkh<!hbe<hMk<k!LcbiK/!weOu?!lixqzq!dXh<H!18 Jg<!giv{qh<hMk<k!Lbz<Ouil</! 18 e<!wz<zi!giv{qs<!Osicgt< 18 = 1 × 18 = 18 × 1 = –1 × –18 = –18 × –1 18 = 2 × 9 = 9 × 2 = –2 × –9 = –9 × –2 18 = 3 × 6 = 6 × 3 = –3 × –6 = –6 × –3 -g<giv{qgtqe<!%Mkjz!wPKOuil</! 18 + 1 = 1 + 18 = 19 (–18) + (–1) = (–1) + (–18) = –19 2 + 9 = 9 + 2 = 11 (–2) + (–9) = (–9) + (–2) = –11 3 + 6 = 6 + 3 = 9 (–3) + (–6) = (–6) + (–3) = –9. x e<! ogPjuBl<?! giv{qgtqe<! %MkjzBl<! yh<hqm<Mh<! hiIk<Okiole<xiz<! 3, 6 Ngqb!giv{qgtqe<!%Mkz<!x e<!ogPuig!dt<tK/!weOu?
x2 + 9x + 18 = (x + 3) (x + 6). wMk<Kg<gim<M 44: giv{qh<hMk<Kg; x2 – 15x + 54. kQIU;! lixqzq = 54 lx<Xl< x e<!ogP!= –15. -r<G!54 e<!giv{qgjtBl<?!nux<xqe<!%MkjzBl<!gQOp!hm<cbzqm<Mg<!gim<MOuil</
giv{qgt< %Mkz< {1, 54} 55 {–1, –54} –55 {2, 27} 29 {–2, –27} –29 {3, 18} 21 {–3, –18} –21 {6, 9} 15 {–6, –9} –15
weOu? x2 – 15x + 54 = (x – 6) (x – 9). wMk<Kg<gim<M 45: giv{qh<hMk<Kg; 15 – 2x – x2. kQIU;! kqm<m!ucuqz<!wPk? 15 – 2x – x2 = –x2 – 2x + 15 = (–1) (x2 + 2x – 15). -r<G!fil<! –15 = 5 × –3, 5 + (–3) = 2 weg<!gi{<gqOxil</! weOu? 15 – 2x – x2 = (–1) [(x + 5) {x + (–3)}] = (–1) (x + 5)(x – 3)= (x + 5) ( 3 – x). wMk<Kg<gim<M 46: giv{qh<hMk<Kg; x2 – x – 132. kQIU;!–132 = (–12) × 11, (–12) + 11 = –1 we!fil<!gi{<gqOxil<</ weOu? x2 – x – 132 = [x + (–12)] (x + 11) = (x – 12) (x + 11).
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nMk<kkig?! a ≠ 0, a, b, c LPg<gtigg<! ogi{<m ax2 + bx + c we<gqx! -Vhc!hz<ZXh<Hg<! Ogijubqjeg<! gVKOuil</! p, q Ngqb!-V!LPg<gjt! pq = ac weUl<?!!! p + q = b weUl<!-Vg<GliX!fl<liz<!g{<Mhqcg<g!Lcf<kiz<?
ax2 + bx + c =a1 (a2x2 + abx + ac)
=a1 [a2x2 + a(p + q)x + pq] =
a1 [a2x2 + apx + aqx + pq] =
a1 [ax (ax + p) + q(ax + p)]
= a1 (ax + p) (ax + q)
-u<uiX?!fl<liz<!Ogijujb!giv{qh<hMk<k!LcgqxK/! uqkq : p?!q wEl<!LPg<gjt? p × q = a × c ? p + q = b we<xqVg<GliX!gi{Lcf<kiz<?!
ax2 + bx + c = a1 (ax + p) (ax + q).
wMk<Kg<gim<M 47: giv{qh<hMk<Kg; 2x2+ 7x + 3. kQIU;! -r<G!!! a = x2 e<!ogP!= 2 b = x e<!ogP!= 7 c = lixqzq!= 3 fil<!gi{<hK a × c = 2 × 3 = 6 = 6 × 1, 6 + 1 = 7 = b. Njgbiz< 2x2 + 7x + 3 =
21 (2x + 6) (2x + 1) =(x + 3)(2x + 1).
uqkqbqe<! Lcjuh<! hbe<hMk<Kukx<Gh<! hkqzig! fM! dXh<jhh<! hqiqk<K?!gQpg<gi[liX!GPLjxbqz<!giv{qh<hMk<kzil</!wMk<Kg<gim<mig?! 2x2 + 7x + 3 = 2x2 + (6 + 1)x + 3 = 2x2 + 6x + x + 3 = 2x(x + 3) + (1)(x + 3) = (2x + 1) (x + 3). wMk<Kg<gim<M 48: giv{qh<hMk<Kg; 8a2 + 2a – 3. kQIU;! -r<G? 8 × – 3 = –24 = 6 × – 4, 6 + (– 4) = 2 weg<!gi{<gqOxil</!hqiqk<okPkqbhqe<?!GPh<hMk<kqeiz<!fil<!njmuK?! 8a2 + 2a – 3 = 8a2 + 6a – 4a – 3 = 2a(4a + 3) – (1)(4a + 3) = (4a + 3) (2a – 1). wMk<Kg<gim<M 49: giv{qh<hMk<Kg; 6 +
211 x + x2.
kQIU;! Ogijujb!lix<xqobPk? 6 + 2
11 x + x2 = 21 (2x2 + 11x + 12).
-r<G? 2 × 12 = 24 = 8 × 3, 8 + 3 = 11. ! weOu?!fM!dXh<jhh<!hqiqk<Kl<?!GPh<hMk<kqBl<!wPk?!
6 + 2
11 x + x2 = 21 (2x2 + 8x + 3x + 12) =
21 [2x(x + 4) + 3(x + 4)] =
21 (x + 4) (2x + 3).
ax2 + bx + c z<! a, b, c Ngqbju! LPg<gtig! -z<zilz<?! olb<ob{<gtig!-Vh<hqEl<! %m! fM! dXh<jhh<! hqiqk<okPkq! GPh<hMk<kqg<! giv{qh<hMk<Kl<! Ljxjb!fil<!hbe<hMk<kzil</ !
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wMk<Kg<gim<M 50: giv{qh<hMk<Kg; 241027 2 −− xx . kQIU;! -r<G? 27 × 24− = –56 = (–14) × 4 OlZl<!(–14) + 4 = –10. weOu?!fM!dXh<jhh<!hqiqk<okPkq?!GPLjxbqz<!wPk? 7 2 x2 – 10x – 4 2 = 7 2 x2 – 14x + 4x – 4 2 = 7x ( 2 x – 2) + 2 2 ( 2 x – 2) = ( 2 x – 2) (7x + 2 2 ). wMk<Kg<gim<M 51: giv{qh<hMk<Kg; 3x2 + 5 3 x + 6 kQIU;! -r<G? 3 × 6 = 18 = (3 3 ) (2 3 ) OlZl< 3 3 + 2 3 = 5 3 . weOu?!fM!dXh<jhh<!hqiqk<okPkq?!GP!Ljxbqz<!giv{qh<hMk<k? 3x2 + 5 3 x + 6 = 3x2 + 3 3 x + 2 3 x + 6 = 3x (x + 3 ) + 2 3 (x + 3 ) = (x + 3 ) (3x + 2 3 ). !wMk<Kg<gim<M 52: giv{qh<hMk<Kg; 15x2 + 17xy + 4y2. kQIU;! fil< 15 × 4 = 60 = 12 × 5, 17 = 12 + 5 weg<!gi{<gqOxil</!weOu?!fM!dXh<jhh<!hqiqk<okPkq?!GP!Ljxbqjeh<!hbe<hMk<kqeiz<, 15x2 + 17xy + 4y2 = (15x2 + 12xy) + (5xy + 4y2) = 3x(5x + 4y) + y(5x + 4y)= (5x + 4y) (3x + y). wMk<Kg<gim<M 53: giv{qh<hMk<Kg; 6(a – 1)2b – 5(a –1)b2 – 6b3
kQIU;! x = a – 1 we!wMk<Kg<ogi{<miz<? 6(a – 1)2b – 5(a–1)b2 – 6b3 = 6x2b – 5xb2 – 6b3
= b(6x2 – 5bx –6b2) -r<G? 6 × – 6 = –36 = (–9) × 4 , OlZl< (–9) + 4 = –5 weg<!gi{<gqOxil</ Njgbiz< = b(6x2 – 9bx + 4bx – 6b2) = b[3x(2x – 3b) + 2b(2x – 3b)] = b(2x – 3b) (3x + 2b) = b[2(a – 1) – 3b] [3(a – 1) + 2b] = b[(2a –3b –2) (3a + 2b –3)]. giv{qh<hMk<k!Lcbik!LPg<gjtg<!ogPg<gtigg<!ogi{<m!-Vhc!hz<ZXh<Hg<!OgijugTl<!d{<M/ wMk<Kg<gim<M 54: giv{qh<hMk<Kg; x2 + 3x – 1. kQIU;! ax2 + bx + c Bme<!yh<hqMjgbqz< a = 1, b = 3, c = –1. -r<G ac = 1 × –1 = –1 = –1 × 1. OlZl<!(–1) + 1 = 0 ≠ b. weOu?!x2 + 3x – 1 J LPg<gjtg<!ogPg<gtigg<!ogi{<m!giv{qgtig<g!LcbiK/!
Gxqh<<H;!x2 + 3x – 1= 49
49
2322 −+⎟⎠⎞
⎜⎝⎛+ xx −1!=
413
23 2
−⎟⎠⎞
⎜⎝⎛ +x
= 22
213
23
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟
⎠⎞
⎜⎝⎛ +x = ⎟⎟
⎠
⎞⎜⎜⎝
⎛++
213
23x ⎟⎟
⎠
⎞⎜⎜⎝
⎛−+
213
23x .
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hbqx<sq 4.3.3. gQOp!uVueux<jxg<!giv{qgtigh<!hqiq/! 1. x2 + 7x + 12 2. x2 + 9x + 20 3. d2 + 10d + 21 4. z2 – 7z – 98 5. a2 – a – 72 6. x2 + x – 90 7. p2 – 8p + 15 8. y2 – 13y + 42 9. y2 – 20y + 99 10. t2 – 28t + 195 !gQOp!uVueux<jxg<!giv{qh<hMk<Kg; 11. 2a2 + 13a + 15 12. 4x2 + 8x + 3 13. 4x2 + 12x + 9 14. 6x2 + x – 1 15. 6p2 + 17p + 10 16. 4a2 – 11a – 15 17. 7m2 + 16m – 15 18. 8p2 + 29p – 12 19. 6x2 + 5x – 6 20. 15y2 – 13y – 6 21. 14x2 – x – 3 22. 9a2 – 9a + 2 23. 2a2 – 13a + 18 24. 12x2 – 7x + 1 25. 16x2 – 32x + 7 !gQp<!uVueux<jxg<!giv{qgtigh<!hqiq/! 26. 9x2 + 24xy + 15y2 27. 4x2 – 16xy – 9y2
28. 6c2 + 11cd – 10d2 29. 5x2 – 11xy + 6y2 30. 2a2 – 15ab + 28b2
!gQp<!uVueux<jxg<!giv{qh<hMk<Kg;
31. 103
10132 −− xx 32.
38
3102 +− uu 33.
161
212 +− xx
34. 4784 2 +− xx 35.
61
354 2 +− xx 36. 232 2 ++ xx
37. 36113 2 ++ xx 38. 532055 2 ++ xx 39. 5532 2 ++ xx 40. 21427 2 ++ xx 4.4 yV! hz<ZXh<Hg<! Ogijujb! lx<oxiV! hz<ZXh<Hg<! Ogijubiz<!uGk<kz< f(x) we<El<! hz<ZXh<Hg<! Ogijujb! g(x) we<El<! hz<ZXh<Hg<! Ogijubiz<!uGg<Gl<!osbzieK!f(x) ≡ q(x) g(x) + r(x) we<X!-Vg<GliXl<?!r(x) = 0 nz<zK!r(x) e<!hcbieK! g(x) e<! hcjb! uqm! sqxqbkigUl<! -Vg<GliX? q(x), r(x) we<El<! -V!hz<ZXh<Hg<! Ogijugjtg<!gi{<hkiGl</!-r<G! f(x) J!okiGkq! we<Xl<?! g(x) J!hGkq!we<Xl<?! q(x) J! =U! we<Xl<?! r(x) J lQkq! we<Xl<! njpg<gqe<Oxil</! f(x) J! g(x) Nz<!uGg<Gl<! osbzqz<! uGhMl<! Ogijubigqb! f(x) e<! hc! wh<ohiPKl<?! uGg<Gl<!Ogijubigqb!g(x) e<!hcg<G!nkqgligOui!nz<zK!slligOui!-Vg<gm<Ml</!Woeeqz<!f(x) e<!hc!!g(x) e<!hcjb!uqms<!sqxqbkig!-Vf<kiz<!q(x) = 0 NGl<; OlZl<!r(x) = f(x) we<xiGl</!=U?!lQkq!gi{!-V!Ljxgt<!dt<te/
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Ljx 1 : fQ{<m!uGk<kz<!Ljx: -f<k!Ljxjb!fil<!YI!wMk<Kg<gim<ce<!&zl<!uqtg<GOuil</ wMk<Kg<gim<M 55: 2 + 5x + 3x2 + 2x3 we<hjk 1 + x Nz<!uGg<gUl</ kQIU;! uGhMl<!Ogiju f(x) = 2 + 5x + 3x2 + 2x3
uGg<Gl<!Ogiju g(x) = 1 + x, f(x) e<!hc = 3, g(x) e<!hc = 1/! flK!Gxqg<Ogit<!=U!q(x) JBl<?!lQkq!r(x) JBl<!gi{<hOkbiGl</ gm<ml< 1: f(x), g(x) Ngqbux<jxk<!kqm<m!ucuqz<!wPKOuil</ f(x) = 2x3 + 3x2 + 5x + 2 g(x) = x + 1 gm<ml< 2: uGhMl<!Ogijubqe<!Lkz<!dXh<higqb (2x3) J!uGg<Gl<!Ogijubqe<!Lkz<!
dXh<higqb!(x) Nz<!uGk<kiz<!flg<G!=uqe<!Lkz<!dXh<H! ⎟⎟⎠
⎞⎜⎜⎝
⎛= 2
3
22 xxx gqjmg<Gl</
gm<ml<!3: uGg<Gl<!Ogijubigqb!(x + 1) J!=uqe<!Lkz<!dXh<higqb (2x2) Nz<!ohVg<gq!uVl<! ohVg<gx<hzeigqb 2x3 + 2x2 ! J! uGhMl<! OgijubqzqVf<K! gpqg<gOu{<Ml</!flg<Gg<!gqjmg<Gl<!lQkq! x2 + 5x + 2. lQkqbqe<! hc?!uGg<Gl<! Ogijubqe<! hcjb!uqmh<!ohiqbK/ gm<ml< 4: OlOz! gqjmk<k! x2 + 5x + 2 we<El<! lQkqjb! Hkqb! uGhMl<! Ogijubig!wMk<Kg<! ogi{<M! gm<ml<! 2 Jh<! hbe<hMk<kqeiz<! =uqe<! -v{<mil<! dXh<H!
⎟⎟⎠
⎞⎜⎜⎝
⎛= x
xx2
gqjmg<Gl</
gm<ml< 5: uGg<Gl<! Ogiju! )x + 1) !J!=uqe<!-v{<mil<!dXh<hiz<! (x) ohVg<gq?!uVl<!ohVg<gx<hzje! (x2 + x) J! Hkqb! uGhMl<! Ogijubqeqe<X! gpqg<g! Ou{<Ml</! flg<G!gqjmg<Gl<!lQkq!(4x + 2). lQkqbqe<!nMg<G!uGg<Gl<!Ogijubqe<!hcbig!dt<tK/ gm<ml<! 6: lQkq (4x + 2) J!Hkqb!uGhMl<! Ogijubig!wMk<Kg<! ogi{<M!gm<ml< 2 Jh<!
hbe<hMk<kqeiz<!=uqe<!&e<xil<!dXh<H ⎟⎠⎞
⎜⎝⎛ = 44
xx gqjmg<Gl</
gm<ml< 7: fil<! uGg<Gl<! Ogiju! (x + 1) J?! =uqe<! &e<xiuK!dXh<hiz<! (4) ohVg<gq!uVl<!(4x + 4) Jh<!Hkqb!uGhMl<!Ogijubqeqe<X!gpqg<g!Ou{<Ml</!gqjmg<Gl<!lQkq!–2 . -f<k! lQkqbqe<! hc! H,s<sqbl</! -K!uGg<Gl<! Ogijubqe<! hcjb!uqms<! sqxqbK/! weOu?!fil<!uGg<Gl<!osbjz!fqXk<kquqm<M?!=U?!lQkq!Ngqbux<jx!wMk<K!wPKgqOxil</ ! !! Olx<%xqb!gm<mr<gt<!gQPt<t!ucuk<kqz<!ogiMg<gh<hm<Mt<te/! 2x2 + x + 4
2x3 + 3x2 + 5x + 2 2x3 + 2x2
– – x2 + 5x + 2 x2 + x – – 4x + 2 4x + 4 – – – 2
x + 1 ⎟⎟⎠
⎞⎜⎜⎝
⎛= 2
3
22 xxx
⎟⎟⎠
⎞⎜⎜⎝
⎛= x
xx 2
⎟⎠⎞
⎜⎝⎛ = 44
xx
!
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-u<uixig!flg<Gg<!gqjmg<Gl<!=U!q(x) = 2x2 + x + 4, lQkq r(x) = –2.
fil< nxquK, f(x)=q(x) g(x) + r(x). ∴ 2x3 + 3x2 + 5x + 2 = (2x2 + x + 4) (x + 1) + (–2) Ljx 2 : uGk<Kg<! giv{qh<hMk<kz<! Ljx;! -f<k! Ljxbqz<?! f(x) J! -v{<M! dXh<Hgtqe<!%Mkzigh<! hqiqk<kOz! Gxqg<Ogit<! NGl</! Lkz<! dXh<H! g(x) J! giv{qbigg<!ogi{<cVg<Gl</! -v{<miuK!dXh<H! g(x) J! uqm! nMg<gqz<! Gjxf<k! yV! hz<ZXh<Hg<!OgijubiGl</!-jk!fil<!ohx?!f(x) dme<!Okjubie!dXh<Hgjtg<!%m<c?!gpqh<Ohil</ wMk<Kg<gim<M 56: 9x3 + 3x2 – 5x + 7 J 3x2 + 2x – 1 Nz<!uG/ kQIU; 9x3 + 3x2 – 5x + 7 = [(3x2 + 2x – 1) (3x) – 6x2 + 3x] + 3x2 – 5x + 7 = (3x2 + 2x – 1)(3x) + [–6x2 + 3x2] + [3x – 5x] + 7 = (3x2 + 2x – 1)(3x) + (–3x2) + (–2x) + 7 = (3x2 + 2x – 1) (3x) + [(3x2 + 2x – 1) (–1) + 2x – 1] + (–2x) + 7 = (3x2 + 2x – 1) (3x) + (3x2 + 2x – 1) (–1) + [(2x) + (–2x)] + [(–1) + 7] = (3x2 + 2x – 1)(3x) + (3x2 + 2x – 1) (–1) + 6 = (3x2 + 2x – 1) (3x – 1) + 6 weOu?!=U!= 3x – 1, lQkq!= 6. wMk<Kg<gim<M 57: 4 – 17x – 22x2 – 12x3 – 2x4 J x2 – 3x + 4 Nz<!uG/ kQIU; fil<!kQIju!-V!upqLjxgtqZl<!gi{<Ohil</ fQ{<m!uGk<kz<!Ljx! – 2x2 – 18x – 68 –2x4 – 12x3 – 22x2 – 17x + 4
–2x4 + 6x3 – 8x2
+ – + –18x3 – 14x2 –17x + 4 –18x3 + 54x2 – 72x + – + –68x2 + 55x + 4 –68x2 + 204x – 272 + – + – 149x + 276
x2 – 3x + 4 ⎟⎟⎠
⎞⎜⎜⎝
⎛−=
− 22
4
22 xx
x
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
− xx
x 18182
3
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
− 68682
2
xx
!! !weOu?!=U!= –2x2 – 18x – 68 , lQkq = –149x + 276.!!uGk<Kg<!giv{qh<hMk<kz<!Ljx! 4 – 17x – 22x2 – 12x3 – 2x4 = – 2x4 – 12x3 – 22x2 – 17x + 4
= [(x2 – 3x + 4) (–2x2) – 6x3 + 8x2] – 12x3 – 22x2 – 17x + 4. = (x2 – 3x + 4) (–2x2) – 6x3 + 8x2 – 12x3 – 22x2 – 17x + 4 = (x2 – 3x + 4) (–2x2) – 18x3 – 14x2 – 17x + 4 = (x2 – 3x + 4) (–2x2) + [(x2 – 3x + 4) (–18x) – 54x2 + 72x] – 14x2 –17x + 4 = (x2 – 3x + 4) (–2x2) + (x2 – 3x + 4) (–18x) – 54x2 + 72x – 14x2 –17x + 4
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= (x2 – 3x + 4) (–2x2) + (x2 – 3x + 4) (–18x) – 68x2 +55x+ 4 = (x2 – 3x + 4) (–2x2) + (x2 – 3x + 4) (–18x) + [(x2 – 3x + 4) (–68) – 204x + 272] +55x + 4 = (x2 – 3x + 4) (–2x2) + (x2 – 3x + 4) (–18x) + (x2– 3x + 4) (–68) –204x + 272 + 55x + 4 = (x2 – 3x + 4) (–2x2 – 18x – 68) + (–149x + 276).
weOu?!=U!= − 2x2 – 18x – 68 , lQkq!= –149x + 276. !
hbqx<sq 4.4
1. 4x3 – 3x2 + x – 7 NeK!gQOp!ogiMg<gh<hm<m!Ogijugtiz<!uGhMl<OhiK!=U?!lQkq!gi{<g/! (i) 2x + 1 (ii) x – 4 (iii) 1– x 2. 15 + x4 – 8x2 NeK!gQOp!ogiMg<gh<hm<m!Ogijugtiz<!uGhMl<OhiK!=U?!lQkq!gi{<g/ (i) (x + 1)(x + 2) (ii) (x – 2)2 (iii) x3 + 2x.
!
uqjmgt<!
hbqx<sq 4.1 1. (i) kuX! (ii) kuX (iii) kuX (iv) kuX (v) siq! 2. x3 + 2x2 – x + 4 3. 3x4 – 2x2 + 10x – 8 4. 3x3 + 5x2 – 13x + 2 5. –x3 + 5x2
– 7x + 10 6. –x4 – 4x2 + 4x – 8 7. 3x5 + x3 – 8x2 + 6x – 8 8. 6x4 – 26x3 + 51x2 – 66x + 27 9. 10x5 + x4 + 9x3 + 2x2 – 9x – 28 10. – x5 + 4x4 + 9x3 – 38x2 + 21x
x3 e<!ogP x2 e<!ogP x e<!ogP
11. –2 –7 20 12. –6 10 –3 13. 18 9 9 14. acx2 + bdy2 + (ad + bc)xy 15. 2x3 – x2y – 5xy2 – 2y3
16. x4 + x2y2 + y4 17. p = –9 18. a=27
− 19. m = 12
hbqx<sq 4.2 1. (i) x2 + 11x + 18 (ii) x2 + 6x – 16 (iii) t2 + 4t – 12 (iv) p2 – 7p + 12 (v) 10812 (vi) 3658 (vii) 1224 (viii) 2915 2. (i) 25x2 + 80xy + 64y2 (ii) 9s2 – 24st + 16t2 (iii) 16p2 – 49q2
(iv) 10201 (v) 9604 (vi) 9898
3. 241 ,9/4 4. 60, 20 5. 12, 8
6. (i) 9x2 + y2 + 4z2 + 6xy + 4yz + 12xz (ii) 16x2 + 4y2 + 9z2 – 16xy – 12yz + 24xz (iii) 4p2 + 9q2 + 4r2 + 12pq – 12qr – 8pr (iv) 9a2 + 4b2 + 4c2 – 12ab + 8bc – 12ac
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7. 45 8. (i) x3 + 9x2 + 26x + 24 (ii) x3 + x2 – 14x – 24 (iii) x3 + 3x2 – 10x –24 (iv) x3 – 5x2 − 2x + 24 (v) x3 – 9x2 + 26x – 24 x2 e<!ogP x e<!ogP lixqzq 9. (i) 9 –16 –60 (ii) 0 –26 12 (iii) 396 –138 35
10. –9, 1523
− , 35
11. (i) 8x3 + 12x2y2 + 6xy4 + y6 (ii) 8u3 – 84u2v + 294uv2 – 343v3
(iii) 33 133
xxxx −+− (iv) x6y9 + 6x4y6 + 12x2y3 + 8
12. 304 13. 7, 18 14. 20,72 15. 2, 180,1016
hbqx<sq 4.3.1 1. 3(3m – n) 2. 4a(a2 – 2a + 4) 3. x(x4 + 4) 4. xy3 (6x4y2 + 3x + 14) 5. 7pq (1–3pq) 6. (m – p) (n + 2) 7. (x + 2 ) (x – 2 ) (x – 2) 8. (x + a ) (x – a ) (x – 1) 9. (p2 + 1) (2p – 1) 10. 2(2x + 1) (2x2 + 1)
hbqx<sq 4.3.2 1. siq 2. kuX 3. kuX 4. kuX 5. siq! 6. (1 + 3x)2 7. (12x – 3)2 8. (2ab + 5cd)2 9. (x + y + a − b) (x + y − a + b) 10. )33( zxy + (3x2y2 – xyz33 + 9z2) 11. (x + 3y) (x2 + 3y2) 12. 2x2 (x4 + 3) 13. (x + y) (x − y) (x2 − xy + y2) (x2 + xy + y2) 14. 2y(3x2 + y2) 15. 8p3 16. (3x + y)3
17. (x – 4)3 18. (2x – 3y)3
19. (2x + 3y + z)2 20. (a + b – 3c)2
21. (x – y + 1) (x2 + y2 + 1 + xy + y – x) 22. (2x – 5y + 6) (4x2 + 25y2 + 36 + 10xy + 30y – 12x) 23. (2x – 3y + z) (4x2 + 9y2 + z2 + 6xy + 3yz – 2zx) 24. ( a3 – 2b – 5c) (3a2
+ 4b2 + 25c2 + ab32 – 10bc + ca35 ) 25. 3 (a – 2b) (2b – 3c) (3c – a) 26. 3(x + y – 2z) (y + z – 2x) (z + x – 2y) 27. 3(a + b) (a – b) (b + c) (b − c) (c + a) (c − a) 28. 3abc (a – b) (b – c) (c – a) 29. (x – y) (x + y + z) 30. (1 + x + y) (1 – x – y) 31. (x2 + 2x + 2) (x2 – 2x + 2)
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hbqx<sq 4.3.3 1. (x + 3) (x + 4) 2. (x + 4) (x + 5) 3. (d + 3) (d + 7) 4. (z + 7) (z − 14) 5. (a + 8) (a – 9) 6. (x + 10) (x – 9) 7. (p – 3) (p – 5 ) 8. (y – 6) (y – 7) 9. (y – 9) (y – 11) 10. (t – 13) ( t – 15) 11. (a + 5) (2a + 3) 12. (2x + 1) (2x + 3) 13. (2x + 3) (2x + 3) 14. (2x + 1)(3x − 1) 15. (p + 2) (6p + 5) 16. (a + 1) (4a – 15) 17. (m + 3) (7m – 5) 18. (p + 4) (8p – 3) 19. (2x + 3) (3x – 2) 20. (3y + 1) (5y – 6) 21. (2x – 1) (7x + 3) 22. (3a – 1)(3a – 2) 23. (a – 2) (2a – 9) 24. (3x – 1) (4x – 1) 25. (4x – 1) (4x – 7) 26. (x + y) (9x + 15y) 27. (2x + y) (2x – 9y) 28. (2c + 5d) (3c – 2d) 29. (x – y) (5x – 6y) 30. (a – 4b) (2a – 7b)
31. 101 (2x – 3) (5x +1) 32.
31 (u – 2) (3u – 4) 33.
161 (4x – 1)(4x – 1)
34. 41 (4x – 1) (4x – 7) 35.
61 (4x – 1) (6x – 1) 36. ( x2 + 1) ( x + 2 )
37. ( x3 + 2) (x + 33 ) 38. ( x5 + 3) ( 5x + 5 ) 39. (x + 5 ) (2x + 5 )
40. ( )( )2727 ++ xx
hbqx<sq 4.4 =U lQkq
1. (i) 2x2 –47
25
+x 4
35−
(ii) 4x2 + 13x + 53 205 (iii) –4x2 –x – 2 −5
2. (i) x2 – 3x – 1 9x + 17 (ii) x2 + 4x + 4 –1 (iii) x − 10x2 + 15
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5. kQIUgt<!gi[l<<!Ljxgt<<<<<<<<<<<<! flg<G!yV!g{g<G!ogiMg<gh<hm<M, nkx<Gk<!kQIU!gi{s<!osie<eiz<, Lkzqz<!nkje!LPjlbig! fe<G! hck<K,! Hiqf<K! ogi{<M,! ng<g{g<gqz<! we<e! ogiMg<gh<hm<Mt<tK,!we<e! g{<Mhqcg<gOu{<Ml<! nz<zK! wkje! fq'hqg<g! Ou{<Ml<! we<hkje! nxqf<K!osbz<hMkz<! Ou{<Ml<.! hqxG! ng<g{g<gqx<Gk<! kQIUgi{! wf<k! g{qk!d{<jlgjth<!hbe<hMk<k! Ou{<Ml<,! wf<k Ljxbqz<!nf<k! g{qk d{<jlgjth<! hbe<hMk<kOu{<Ml<!we<hkjek<<! kQIlieqg<g! Ou{<Ml</! g{g<gqx<Gk< kQIU! gi{k<! Okjubie! g{qk!d{<jlgjts<! siquvh<! hbe<hMk<ks<! osbz<! kqxEl<,! hbqx<sqBl<! OkjubiGl</! osbz<!kqxe<!&zligk<<!kQIU!gi{!fqjxb!upqLjxgjtBl<,!dk<kqgjtBl<!nxqbzil<.!sqz!Ofvr<gtqz<!yV!g{g<gqe<!kQIU!hz<OuX!upqgtqz<!nz<zK!hz<OuX!dk<kqgtqz<!gi{!LcBl<.!wMk<Kg<gim<mig!gQOp!dt<t!g{g<gqje!wMk<Kg<ogit<Ouil</
g{g<G: a − b = 4, a + b = 5 weqz<, ba -e<!lkqh<H!gi{<g/!
-g<g{g<gqe<!kQIU!gi{!gQp<g{<m!Ljxgtqz<!flK!w{<O{im<ml<!osbz<hMgqxK/ uqei: !we<e!ogiMg<gh<hm<Mt<tK? uqjm: a − b = 4,! a + b = 5.
uqei: we<e g{<Mhqcg<g!Ou{<Ml<? uqjm: ba -e<!lkqh<H/!
uqei: wh<hcs<!osbz<hMkz<!Ou{<Ml<? uqjm: dk<kq!1:
a − b = 4. (1) (1) + (2) ⇒ 2a = 9 ⇒ a = .29
a + b = 5. (2) (1) − (2) ⇒ −2b = −1 ⇒ b = .21
21=
−−
∴ ba =
29 ×
12 = 9.
dk<kq!2:
ba =
)()()()(
22
babababa
ba
−−+−++
= = .919
4545
==−+
dk<kq 3: a − b = 4 (1) a + b = 5 (2)
ba = x we<g. hqe<eI?!
1x
ba= n.K! a = bx.
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(1) ⇒ bx − b = 4 ⇒ b (x − 1) = 4. (2) ⇒ bx + b = 5 ⇒ b (x + 1) = 5.
∴ =+−
)1()1(
xbxb
54 n.K
54
11=
+−
xx n.K 5x − 5 = 4x + 4 n.K 5x − 4x = 4 + 5 n.K x = 9.
weOu! fl<Ljmb! F{<{xqUk<kqxe<,! osbz<kqxe<,! kqxjl, OlZl<! nEhul<!
Ngqbju! g{g<Ggjtk<! kQIg<g! Wx<hiMgjtBl<?! dk<kqgjtBl<! dVuig<Gukx<Gk<!Okju/! Olx<%xqb! g{g<jgk<! kQIh<hkx<G! &e<X!dk<kqgt<! lm<MOl! -Vh<hkig!LcU!osb<b! Ou{<mil</! yV! g{g<gqe<<! kQIU! gi{?! Hkqb! dk<kqgjt! nEhuk<kqe<!
uibqzigUl<,!nOk! g{g<gqx<G! lQ{<Ml<<! lQ{<Ml<! kQIU! gi[l<! OhiKl<! ohx!LcBl</!
OlZl<!flK!w{<{lieK!sqf<kqg<Gl<!kqxeqz<!ujtf<K!ogiMh<hkib<!-Vg<g!Ou{<Ml<,!lx<x!upqLjxgjt!Wx<Xg<!ogit<t!kqxf<kkib<!-Vg<g!Ou{<Ml<?!oux<xqml<!-z<zilz<!hz<OuX! gVk<Okim<mr<gjth<! ohXukib<! -Vg<g! Ou{<Ml</! yV! g{g<gqe<! kQIuqjeg<!
gi{! YI! dk<kqbiz<! Lcbuqz<jz! weqz<,! lx<oxiV! dk<kqjbh<! hbe<hMk<kqk<! kQIU!gi[kz<!Ou{<Ml</!-u<uiX? yV!g{g<gqe<!kQIuqjeg<!gi{!flg<Gk<!okiqf<k!hz<OuX!dk<kqgt<!kQIf<K!OhiGl<!ujv!hbe<hMk<kqg<!ogi{<Om!-Vg<g!Ou{<Ml</ 5.1 nElier<gt<!lx<Xl<!fq'h{r<gt<< gQp<gi[l<!w{<gtqe<!njlh<jh!Ofig<Gg/
1, 3, 7, 13, 21, …… fil<! -f<k!njlh<hqe<! ohiK!dXh<hqjeg<! gi{!uqjpgqe<Oxil<! we<g/!dXh<Hgjtg<!gQp<g{<muiX!hGk<kxqbzil</
1=1 3=1 + 2 7=1 + 2 + 4 13=1 + 2 + 4 + 6 21=1 + 2 + 4 + 6 + 8
-h<ohiPK, -f<k!njlh<hqz<!nMk<k!w{<!we<e!we<hjk!fl<liz<!g{<Mhqcg<g!LcBl</
1 + 2 + 4 + 6 + 8 + 10 = 21 + 10 = 31. weOu?!6 NuK!dXh<H!31. OlZl<!! 31 = 1 + 2(1 + 2 + 3 + 4 + 5).
7 NuK!dXh<H = 1 + 2 + 4 + 6 + 8 + 10 +12 = 31 + 12 = 43 = 1 + 2(1 + 2 + 3 + 4 + 5 + 6). 8 NuK!dXh<H = 1 + 2 + 4 + 6 + 8 + 10 +12 +14 = 43 + 14 = 57 = 1 + 2(1 + 2 + 3 + 4 + 5 + 6 + 7). 9 NuK!dXh<H = 1 + 2 + 4 + 6 + 8 + 10 +12 +14 + 16 = 57 + 16 = 73
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= 1 + 2(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8). -r<Og!dXh<Hgtqe<!lkqh<H!gqjmg<Gl<!Ljxjb!Ofig<Gl<!ohiPK, njlh<hqe<!ohiK!dXh<H!)nkiuK!n NuK!dXh<H, n = 1, 2, 3, …)
1 + 2[1 + 2 + 3 + … + (n − 1)] we!fil<!%x!LcgqxK/!
Neiz<!-f<kg<!%x<xqz<! 1 + 2 + 3 + … + (n − 1) we<x!%Mkz<!dt<tK. -g<%Mkjz!g{<mxqbilz<! njlh<hqe<! ohiK! ucuk<jkg<! g{<Muqm<Omil<! we<X! osiz<z! LcbiK/!
wMk<Kg<gim<mig?!%Mkz<!1 + 2 + 3 + 4 + 5 + 6 + … + 100 we<hjkg<!gVKOuil<. -K!!!!!1 -zqVf<K! 100 ujv!dt<t!nMk<kMk<k!-bz<!w{<gtqe<!%Mkz<!NGl<</!S we<hK!-g<%Mkz<!weqz<,! S = 1 + 2 + 3 + … + 98 + 99 + 100. fil<!nxquK?! !!!!!!!!!!!!!!!!!!!!!!!! ! 1 + 2 + 3 + … + 98 + 99 + 100 = 100 + 99 + 98 + … + 3 + 2 + 1. nkiuK!!
S = 1 + 2 + 3 + … + 98 + 99 + 100, S = 100 + 99 + 98 + … + 3 + 2 + 1. ∴2S = 101 + 101 + 101 + …+ 101 + 101 + 101.
-kqz<!F~X!101 gt<!-Vh<hkjeg<!gi{!LcgqxK/!
weOu?!2S = 100 × 101 nz<zK! S = .50502
101100=
×
Olx<gi[l<!%Mkz<!gqjmg<gh<!ohx<x!Ljxjbg<!%If<K!Ofig<gqe<, fil<!
1 + 2 + 3 + … + (n − 1) = 2
)1( nn −
we<x! %x<jx! nEliel<! )Conjecture) osb<bzil</! n = 1, 2, 3, … weg<! ogi{<M!-g<%x<jxs<! siqhiIg<gzil</! -r<G! fq'h{l<! ogiMg<gh<hmilz<! yV!%x<xqje!njlk<K!nkjes<!siqhiIg<gqOxil</!fq'hqg<gh<hmikujv!ng<%x<X!YI!nElief<kie< )gi <̂!we<x!
o\i<lieqb! g{qk! Oljk! ke<! 10uK! ubkqz<! 4uK! ! hck<Kg<! ogi{<cVf<kOhiK?!uGh<hisqiqbi<! li{ig<gIgjt! 2! + 2 + 3 +…+ 100 e<! %Mkz<! gi{s<! osie<eOhiK?!dmOe!gi <̂!5050!we<X!uqjmbtqk<kiI/!NsqiqbI!nujv!wu<uiX!dmOe!hkqztqg<g!Lcf<kK!we<X!Ogm<g!gi <̂?!OlOz!fil<!%xqb!Ljxjb?!NsqiqbI!uqbf<K!hivim<MliX!ntqk<kiI*/!flK!njlh<hqe<!ohiK!dXh<H!!
1 + 2 × 2
)1( −nn = 1 + n(n − 1) = n2 – n + 1.
weOu?! 1, 3, 7, 13, …. we<x!njlh<hqe<!ohiK!dXh<H n2 – n + 1 NGl<. Olx<g{<m! Nb<uqz<, fil<! YI! njlh<hqjeg<! %If<kib<f<K, yV! %x<xqje!
d{<mig<gqOeil</! d{<jlbqz<! -bx<g{qkl<,! ucuqbz<! Ohie<x! hqiqUgtqz<! hz!Okx<xr<gt<!w{<gtiZl<!dVur<gtiZl<!njlf<k!njlh<Hgjt!%If<kib<Us<!osb<kke<!uqjtuiz<! g{<mxqf<kju! NGl</! -h<ohiPK! fil<! g{qkk<kqz<! dt<t! ouu<OuX!%x<Xgt<?!ng<%x<Xgjth<!hGk<kxqukx<Gh<!hbe<hMk<kh<hMl<!hz!dk<kqgt<!Ngqbux<jx!nxqf<K!ogit<Ouil</!
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%x<Xgt< weh<hMhju! uzqBXk<kqg<! %xh<hMl<! uig<gqbr<gtiGl</! gQOp! sqz!%x<Xgt<!ogiMg<gh<hm<Mt<te:
(i) “x + 7 = 5, -r<G x ∈ N.” (ii) “sKvk<kqe<!&jzuqm<<mr<gt<!ye<Xg<ogie<X!osr<Gk<kiGl</” (iii) “(a + b) (a − b) = a2 − b2, -r<G a, b ∈ R.” (iv) “ 2 !yV!uqgqkLX!w{<.” (v) “sib<sKvk<kqe<!&jzuqm<<mr<gt<!ye<Xg<ogie<X!osr<Gk<kigiK.”
(vi) “1 + 2 + …. + n = 2
)1( +nn .”
%x<Xgt<! olb<biejubigOui! nz<zK! olb<bx<xejubigOui -Vg<gzil</!
wMk<Kg<gim<mig! OlOz! dt<t! %x<Xgtqz<! (i), (iv) lx<Xl< (v) Ngqbju! olb<bx<x!%x<Xgt<; (ii), (iii) lx<Xl< (vi) Ngqbju!olb<bie!%x<Xgt<. 5.1.1 ujvbjx!we<xiz<!we<e?
sqz! %x<Xgt<! Wx<geOu! dt<t! gVk<KgtqzqVf<K! Hkqb! gVk<Kg<gjt!dVuig<gq! d{<jlbie! LcUgjth<! ohx! upquGg<Gl</! ng<%x<XgOt! ujvbjx!weh<hMl</!Wx<geOu!Wx<Xg<ogit<th<hm<m!sqz ujvbjxgt<!gQOp!ogiMg<gh<hm<Mt<te/
(i) wz<zih<!hg<gr<gTl<!sl!ntU!ogi{<m!Lg<Ogi{l<!slhg<g!Lg<Ogi{l<!!!!!!!!!weh<hMl</!
(ii) ax = b weqz<,!x NeK!ncliel<!a ogi{<m!b -e<!lmg<jg!weh<hMl</ (iii) -v{<M!Ogi{r<gtqe<!%Mkz<!180° weqz<,!nju!lqjg!fqvh<H!Ogi{r<gt<!!!!!!!!weh<hMl</
5.1.2 ncOgit<gt<< sqz! %x<Xgjt! d{<jlobe! LPjlbig! Wx<Xg<ogit<tzil</! -k<kjgb! %x<Xgt<!ncOgit<gt<<<! nz<zK! yh<Hg<ogi{<m! d{<jlgt<! weh<hMl</! ucuqbzqZl<!-bx<g{qkk<kqZl<!dt<t!sqz!yh<Hg<ogi{<m!d{<jlgt<!gQOp!ogiMg<gh<hm<Mt<te;
(i) -v{<M!Ht<tqgt<!upqOb!yOv!yV!OgiMkie<!-Vg<gqxK/ (ii) WOkEl<!-v{<M!olb<ob{<gt<! x, y gTg<G! x + y , xy !NgqbjuBl<!! olb<ob{<gtiGl</ (iii) n yV!-bz<!w{<!weqz<?!n + 1 dl<!-bz<!w{< NGl</ (iv) yV!Ogim<Mk<!K{<mieK!yOv!yV!jlbh<Ht<tqjbk<kie<!ohx<xqVg<Gl</!!!! (v) yV Ogi{lieK!yOv!yV!-Vsloum<cjbk<kie<!ohx<xqVg<Gl<.
5.1.3 ⇒, , ⇔ Ngqb!Gxqgt<!⇒
g{qkl<?!kVg<g! vQkqbig!fqjeU!%If<K!!ohxh<hm<m!yV!himh<hGkqbiGl</!yV!g{g<gqe<! kQIU! nz<zK! g{g<gqe<! upqLjxgt<! nz<zK! g{g<gqe<! fq'h{l<!hch<hcbig!ohxh<hm<m!yV!njlh<hiGl</!upqLjxbqe<!yu<ouiX!hcBl<!nkx<G!Le<H!dt<t!hcbqjes<! siIf<kkig!njlf<kqVg<Gl</! kk<Ku! vQkqbie!upqgjt!Gxqg<g! fil<!
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⇒ we<x! Gxqbqjeh<! hbe<hMk<KgqOxil</! wMk<Kg<gim<ce<! &zlig! ⇒ we<x! Gxqbqe<!hbje!nxqOuil</!hqe<uVl<!%x<Xgt<!P, Q jug<!gVKg/!
P: x = 2, Q: x2 = 4. P J!d{<jlobeg<!ogi{<miz< x = 2 NGl</ ∴ x2 = x × x = 2 × 2 = 4. ∴ Q d{<jlbiGl</
weOu?! P d{<jlobeqz<! Q d{<jlbiGl</! -jkg<! GxqbQm<cz<!!!!!!!!!!“P d{<jl ⇒ Q d{<jl” nz<zK!wtqkig!“P ⇒ Q” we!wPkzil<.
GxqbQM! ⇒ NeK!d{Ik<KgqxK!we<hkjeg<!Gxqg<Gl</
-h<ohiPK!GxqbQM! ⇒ we<hkje!nxqOuil<. lQ{<Ml<!nOk!%x<Xgt<!P, Q -ux<jx!wMk<Kg<ogit<Ouil</!nkiuK
P: x = 2, Q: x2 = 4 Q d{<jlobeqz<!x2 = 4 NGl<. Neiz<!-kx<G!x = 2 l<<!kQIU; x = −2 l<!kQIuiGl</!weOu?!P olb<bie!%x<X!nz<z/!-f<k!d{<jljb!fil<!Q ⇒ P we!wPkzil</!! nMk<kkig “⇔” we<x!GxqbQm<jm!nxqf<K!ogit<Ouil</!
-v{<M!%x<Xgt<!Q1, Q2 !wMk<Kg<ogit<Ouil</! Q1: 2x + 3y = 12 OlZl<! 5x − 6y = 3. Q2: x = 3 OlZl< y = 2.
Q1 d{<jlobeqz< 2x + 3y = 12, (1) 5x − 6y = 3. (2)
∴ (1) × 2 ⇒ 4x + 6y = 24 (3) (2) ⇒ 5x − 6y = 3. (4) ∴ (3) + (4) ⇒ 9x = 27 ⇒ x = 3. x = 3 J!(1) -z<!hqvkqbqm! 6 + 3y = 12 nz<zK 3y = 6 nz<zK! y = 2. nkiuK! x = 3, y = 2. weOu?!Q2 d{<jlbiGl</!weOu? Q1 ⇒ Q2 . Q2 d{<jlobeqz< x = 3, y = 2. ∴ 2x + 3y = 2 × 3 + 3 × 2 = 6 + 6 = 12; OlZl< 5x − 6y = 5 × 3 − 6 × 2 = 15 − 12 = 3. weOu! Q1 d{<jlbigqxK/!weOu Q2 ⇒ Q1. 0 Q1 ⇒ Q2 OlZl<!Q2 ⇒ Q1. -f<k!LcUgt<!Q1 ⇒ Q2 , Q2 ⇒ Q1 Ngqb!-v{<jmBl<!OsIk<K!Q1⇔ Q2 we!wPkzil<. -k<kjgb!$p<fqjzbqz<, Q1 dl<!Q2 dl<! slliekig!-Vg<Gl<. -ke<!ohiVtiuK?!Q1 olb<!weqz<?!Q2 olb</ Q2 olb< weqz<?!Q1 olb</!nkiuK! Q1 olb<big!-Vf<kiz<! lx<Xl<! -Vf<kiz<! lm<MOl!Q2 olb<biGl</ “⇔” we<x!GxqbQM!“-Vf<kiz<!lx<Xl<!-Vf<kiz<!lm<MOl”!)if and only if) we<hkjeg<!Gxqg<Gl<. 5.1.4 Okx<xl<!weh<hMuK!biK?
yV! %x<X?! siqbieK! nz<zK! kuxieK! we! fq'hqg<gh<hmikujv! njk!
nEliel<! )Conjecture) we<xjpg<gqe<Oxil</! yV! nEliel<! )Conjecture) siq! we!fq'hqg<gh<hm<Muqm<miz<! nK! ! Okx<xligquqMgqxK/ kuX! we! fq'hqg<gh<hm<m! nEliel<!olb<bx<x! %x<X! weh<hMl</! weOu! Wx<geOu! olb<! we<X! fq'hqg<gh<hm<Muqm<m! %x<X!Okx<xl<!weh<hMl</!yV!%x<X!yV!Gxqh<hqm<m!ujgbqz<!d{<jlobeqz<!nf<kg<! %x<X!hz<OuX! fqjzgtqz<! siqhiIg<gh<hm<mK! we<X! osiz<gqOxil</! -bx<<g{qkk<kqZl<?!ucuqbzqZl<!fil<!nxqf<Kt<t!sqz!Okx<xr<gt<!hqe<uVliX!ogiMg<gh<hm<Mt<te;
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(i)!!yV!Lg<Ogi{k<kqz<!-v{<M!hg<gr<gt<!sll<!weqz<?!nux<xqx<G!wkqOv!!!!!!!!dt<t!Ogi{r<gt<!sll</!
(ii) yV!Lg<Ogi{k<kqe<!Ogi{!ntUgtqe<!%Mkz<!-v{<M osr<Ogi{liGl</ (iii) -j{gvk<kqe<!&jzuqm<mr<gt<!ye<jxobie<X!-V!slg<!%xqMl</ (iv) yx<jx!w{<{qe<!uIg<gLl<!yx<jx!w{<O{!NGl</ (v) -vm<jm!w{<{qe<!uIg<gLl<!-vm<jm!w{<O{!NGl</ (vi) 2 yV!uqgqkLxi!w{<!NGl<. (vii) (a + b)2=a2 + 2ab + b2. (viii) log a(mn)=log am + log an.
5.1.5 yV!Okx<xk<kqe<!fq'h{l<!we<xiz<!we<e?
yV!Okx<xk<kqe<!fq'h{l<!weh<hMuK!nke<!d{<jljb!fqI{bqg<Gl<!uquikl<!NGl</!wMk<Kg<gim<mig
“yx<jx!w{<{qe<!uIg<gLl<!YI!yx<jx!w{<O{” we<x!%x<xqje!wMk<Kg<ogit<Ouil</!! -f<k! %x<X! yV! Okx<xl<<! weqz<?! -kx<G! kk<KuvQkqbie! fq'h{l<! -Vg<g!Ou{<Ml</!gQp<<g{<m!uquikk<jk!gueqg<gUl<:!
“n YI!yx<jx!w{<!we<g/!hqe<H!n = 2m + 1, -r<G m yV!LP!w{</ -h<ohiPK? n2 = (2m + 1)2 = 4m2 + 4m + 1 = 2(2m2 + 2m)+1. m yV!LP!w{<!we<hkiz<, 2m2 + 2m dl<!yV!LP!w{<{iGl<. weOu! 2(2m2 + 2m) YI!-vm<jm!w{<{iGl<. Nkziz<! 2(2m2+2m)+1 YI yx<jx w{<{iGl<. weOu n2 YI yx<jx w{<.”
fq'h{l<! ntqg<g! hz<OuX! dk<kqgt<! dt<<te/! ohiKuig! nju! hqe<uVliX!
ujgh<hMk<<kh<hm<Mt<te; (i) Ofvc!fq'h{!Ljx. (ii) ljxLg!fq'h{!Ljx!nz<zK!Lv{<him<M<!fq'h{!Ljx. (iii) wkqI!wMk<Kg<gim<Mgtqe<hc!fq'h{l<. (iv) ucuqbz<!fq'h{!dk<kq/ (v) njlh<H!Ljx!fq'h{l</
(i) Ofvc! fq'h{!Ljx: P ⇒ Q we!fq'hqg<g! Ou{<Ml<! we<g. Lkzqz< P d{<jl!weg<ogi{<M! hch<hcbig! uquikqk<K! Q d{<jl weh<! ohXkz<! Ou{<Ml</ P we<hK!olb<obeqz<! Q NeK!olb<!NGl<! we<X!fq'hqg<Gl<!LjxOb!Ofvc!fq'h{!Ljx!weh<hMl</!
wMk<Kg<gim<M 1: ba = 5 weqz<, =
+−
baba
32 weg<!gi{<hqg<gUl<.
kQIU: ba = 5 we<g. hqe<H!a = 5b. ∴ .
32
64
55
==+−
=+−
bb
bbbb
baba
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wMk<Kg<gim<M 2: Ofvc!Ljxbqz<? ba =
dc weqz<, =
+−
baba
dcdc
+− we!fq'hqg<gUl</!
kQIU: dc
ba= we<g/!-r<G u
ba= !weh<!hqvkqbqmUl</ hqe<H a = ub.
dc
ba= we<hkiz<? .u
dc=
∴ c = ud.
∴ =+−
baba .
11
)1()1(
+−
=+−
=+−
uu
bubu
bubbub -jkh<Ohie<Ox!
11
)1()1(
+−
=+−
=+−
=+−
uu
dudu
duddud
dcdc .
∴ dcdc
baba
+−
=+− .
(ii) ljxLg!fq'h{!Ljx: %x<X!P NeK!%x<X Q J!d{Ik<KgqxK!we!fq'hqg<g!Ou{<Ml<!we<X!ogit<Ouil<. -kx<gig!%x<X!P olb<?!%x<X!Q olb<bz<z!weg<!ogit<g/!hqe<H!hch<hcbig!uquikqk<K!Lv{<himie!uqjtuqje!njmOuil<!)nkiuK!wMOgit<!%x<xqx<G!OfIlixie!%x<X!ohXOuil<*/!weOu!wMOgit<!kuxieK!we<X!LcU!osb<K?!
Q!d{<jlbigk<kie<!-Vg<g!Ou{<Ml<!we!dXkq!osb<gqOxil</!%x<X!P olb<!weqz<?!%x<X!Q dl<! olb<! we! fq'hqg<Gl<!-f<k!Ljx!!ljxLg!fq'h{!Ljx!weh<hMl<!)wMk<Kg<ogi{<m!%x<xqx<G!wkqvie!%x<jx!njmkz<*/ !
wMk<Kg<gim<M 3: ljxLg!fq'h{!Ljxbqz<? 2 yV!uqgqkLxi!w{<!we!fq'hq. kQIU: wMOgitig? 2 yV! uqgqkLX w{<! we! OfIlixigg<! ogit<Ouil<. yu<ouiV!
uqgqkLX!w{<j{Bl<!kq<m<mucuqz< qp we!wPkzil<!we!nxqOuil</!hqe<H!
2 = qp , -r<G!p, q lqjg!w{<gt<<A! p g<Gl<, q g<Gl<!1 Jk<!kuqv!Ouoxf<k!giv{qBl<!
-z<jz. -h<ohiPK!
2 = qp ⇒ p = 2 q ⇒ p2 = 2q2 ⇒ p2 YI!-vm<jm!w{<
⇒ p YI!-vm<jm!w{< ⇒ p = 2m, m yV LP!w{<!⇒ 4m2 = 2q2
⇒ q2 = 2m2
⇒ q2 YI!-vm<jm!w{< ⇒ q YI!-vm<jm!w{< ⇒ q = 2n, n yV LP!w{< ⇒ p g<<Gl<!!q g<<Gl< 2 giv{qbiGl< ⇒ Lv{<hiM! )p g<<Gl<! ! q g<<Gl< 1 Jk<!kuqv! Ouoxf<k!giv{qBl<!-z<jz we<hK!wMOgit<*/
weOu 2 !yV!uqgqkLxi!w{<!NGl<.
128
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!
wMk<Kg<gim<M 4: ljxLg! fq'h{! Ljxg<ogi{<M?! 100 hf<Kgt<! 9 ohm<cgtqz<!jug<gh<hm<cVf<kiz<! WkiuK! yV! ohm<cbqz<! 12 nz<zK! nkx<G! OlZl< hf<Kgt<!jug<gh<hm<cVg<Gl<!we!fq'hq. kQIU: 100 hf<Kgjt 9 ohm<cgtqz<!jug<Gl<ohiPK!wf<k!yV!ohm<cbqZl<<!12 hf<KgOti!nkx<G! OlOzi! jug<gh<hmuqz<jz! we! wMk<Kg<! ogit<Ouil</! NgOu! yu<ouiV!
ohm<cbqZl<! nkqghm<slig! 11 hf<Kgt<! jug<gh<hm<cVg<Gl<. weOu! 9 ohm<cgtqz<!jug<gh<hm<m! hf<Kgtqe<!%Mkz<!nkqghm<slig! 9 × 11 = 99 NGl</ -K!ogiMg<gh<hm<m!ogit<jgg<G! Lv{<himiekiGl</! Woeeqz<?! 100 hf<Kgjt 9 ohm<cgtqz<! jug<g!
Ou{<Ml</! weOu! Gjxf<k! hm<sl<! yV! ohm<cbqziuK! 12 hf<KgOti! nkx<G! OlOzi!jug<gh<hm<cVg<Gl<. wMk<Kg<gim<M 5: P we<x!yV!Ht<tq! AB we<x!Ogim<Mk<K{<jm!m : n we<x!uqgqkk<kqz<!dt<hr<gQM!osb<Bl<!weqz<?! P !keqk<kK!we?!ljxLg!fq'h{!Ljxbqz<!fq'hq/ kQIU: P we<x!yV!Ht<tq! AB we<x!Ogim<jm!dt<hr<gQm<cz<! m : n we<x!uqgqkk<kqz<!hqiqh<hkigg<! ogit<Ouil</! ogit<jgg<G! dm<hm<M?! P we<hK! keqk<kK! -z<jz! weg<!ogi{<miz<?! lx<oxiV! Ht<tq!P1?! AB we<x! Ogim<Mk<K{<jm! m : n we<x!uqgqkk<kqz<!dt<hr<gQM!osb<ukiGl< (hml<!5.1 Jh<!hiIg<gUl<). hqxG!
nm
PBAP
= . OlZl<! .1
1
nm
BPAP
=
∴ n AP = m PB OlZl<∴ n AP =m (AB − AP) O∴ n AP =m AB − m AP O∴ (m + n) AP = m AB Ol
∴ AP = ,nm
mAB+
AP1 = mm
∴ AP = AP1
∴ P Bl< P1 l<!yOv!Ht<t-K!ogit<jgg<G!Lv{ (iii) wkqI!wMk<Kg<gimwe<hK!d{<jlbi! we-Vg<GliX!WOkEl<!Yuf<kuIgtiOuil</!nxqb
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hml<!5.1!
! n AP1 = m P1B lZl< n AP1 = m (AB − AP1) lZl< n AP1 = m AB − m AP1
Zl< (m + n) AP1 = m AB
.n
AB+
qbiGl</ <himig!njlgqxK. weOu!P?!yV!keqk<k!Ht<tqbiGl</
<Mgtqe<!hc!fq'hqk<kz<<: P1, P2 -v{<M!%x<Xgt<!we<g/ P1 ⇒ P2 !Nvib! Ou{<Ml<! we<g/! P1 olb<bigUl<! P2 olb<bx<xkigUl<!I!wMk<Kg<gim<M!gi{Lcf<kiz<! P1 ⇒ P2 we<x!LcUg<G!fil<!h<hm<m!wMk<Kg<gim<M YI!wkqI!wMk<Kg<gim<M!weh<hMl</
129
!wMk<Kg<gim<M 6: x we<hK!yV!olb<ob{<!weqz<?!x2 ≥ x we!njlBli? kQIU: P1: x we<hK!yV!olb<ob{<. P2: x2
≥ x.
x = 21 weqz<?!P1 : x we<hK!yV!olb<ob{<!we<x!%x<X!olb<bigqxK/!Neiz<!
x = 21 g<G!x2 =
41 <
21 = x ; nkiuK! P2 olb<bz<z!we<xigqxK/!weOu? P1 ⇒ P2.
-r<G! x = 21 YI wkqI! wMk<Kg<gim<miGl</! “weOu! x we<x! wz<zi! olb<ob{<gTg<Gl<!!!!!
x2 ≥ x” we<x!%x<X!olb<bigiK!we<x!Lcuqje!njmgqOxil</ !
wMk<Kg<gim<M 7: n yV!-bz<!w{<!weqz<?!n2 ≥ 4n we<hK!d{<jlbi@ kQIU: P1: n yV!-bz<!w{</ P2: n2 ≥ 4n. n =1 we<g/!1 YI!-bz<!w{</!weOu?!P1 d{<jlbiGl</!Neiz<!n2 = n × n = 1 × 1 = 1. 4n = 4 × n = 4 × 1 = 4. weOu!n2 < 4n. weOu?!P2 d{<jlbz<z/!!weOu? P1 ⇒ P2 .!!wMk<Kg<gim<M!8: x we<x!wz<zi!olb<ob{<gTg<Gl<! x2 − 1 > 0 we!njlBli? kQIU: P1: x we<hK!yV!olb<ob{<. P2: x2 − 1 > 0.
x = 21 weqz<?! P1 olb<biGl<. Neiz<! x2 − 1=
41 − 1 = −
43 < 0. weOu?!P2 olb<bz<z.
-r<G!x = 21 YI!wkqI!wMk<Kg<gim<M!NGl</! weOu “x we<x!wz<zi!olb<ob{<gTg<Gl<!
x2 − 1 > 0” olb<bx<x!%x<xiGl</! Gxqh<H: Olx<gi{h<hm<m!wMk<Kg<gim<cz<! x -e<!yV!Gxqh<hqm<m!w{<!lkqh<jhg<!ogi{<M!yV!%x<xqe<!olb<bx<x!ke<jljb!fl<liz<!fqI{bqg<g!Lcf<Kt<tK/ (iv) ucuqbz<!fq'h{!Ljx: -bx<g{qkk<kqz<!sqz!g{g<Ggjt!ucuqbz<!Ljxh<hc!kQIU!gi{zil</ wMk<Kg<gim<M 9: (a + b) (a − b) = a2 − b2 we!fq'hq/!kQIU: hmk<kqz<!ABCD yV!osu<ugl</ AB = AI = a, BE = DI = b (hml<!5.2 Jh<!hiIg<gUl<). hqe<H!AD = AI + DI = a + b, AE = AB − BE = a − b. weOu osu<ugl< AEFD -e<!hvh<htU = (a + b) (a − b) Neiz<!osu<ugl<!AEFD -e<!hvh<htU = sKvl< ABGI -e<!hvh<htU − osu<ugl< BEHG -e<!hvh<htU + osu<ugl< DFHI -e<!hvh<htU
130
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hml< 5.2
= a2 − ab + (osu<ugl< CDIG -e<!hvh<htU!− sKvl< CFHG -e<!hvh<htU) = a2 − ab + (ab − b2) = a2 − b2. ∴ (a + b) (a − b) = a2 − b2.
(v) njlh<H!Ljx!fq'h{l<<:
ucuqbzqz<!sqz!g{g<GgTg<Gk<!kQIU!gi{<hkx<G?!kvh<hm<m!dVur<gtqz<!sqz!OgiMgjt!%Mkzig!njlg<gqe<Oxil<</! hz!ucuqbz<!g{g<Ggtqz<!njlk<kz<<!we<hK!lqgs<sqxf<k! dk<kqbiGl</! siqbie! -mr<gtqz<! njlh<Hgt<<! njlbOu{<Ml</! weOu!njlk<kz<<!&zl<!fq'h{l<!gi{<hK?!g{g<gqe<!kQIU!gi{<huvK!Le<%m<cOb!nxqBl<!Nx<xjz!Ou{<cbqVg<gqxK/ !!wMk<Kg<gim<M 10: yV!Lg<Ogi{k<kqz<!-v{<M!hg<gr<gt<!sll<!weqz<?!nux<xqx<G!wkqOv!dt<t!Ogi{r<gt<!sll<!we!fq'hq/ kQIU: ABC yV!Lg<Ogi{l<? AB = AC we<g. ∠B = ∠C we!fq'hqg<g!Ou{<Ml<. D, BC -e<! jlbh<Ht<tq! we<g. AD Js<!!!!OsIg<g! )ujvkz<!Ljx*/ Lg<Ogi{r<gt< ADB, ADC Ngqbux<Xt< BD = DC, AD! ohiK?!!!!!!!!AB = AC (hml<! 5.3 Jh<! hiIg<gUl<*/! weOu!-l<Lg<Ogi{r<gt<<! sIusll</! weOu! yk<k!Ogi{r<gt<!sll</ weOu!∠B = ∠C.!
hbqx<sq 5.1 1. nEliel<!we<xiz<!we<e? 2. Okx<xl<!we<xiz<!we<e? 3. Okx<xk<kqe<!fq'h{l<!we<hK!we<e? 4. We<!siqhiIk<kz<!fq'h{k<kqx<G!Le<evig!-Vg<g!Ou{<M 5. YI!wMk<Kg<gim<Mme<!ucuqbz<!fq'h{!Ljxjb!uqtg<G 6. wkqI!wMk<Kg<gim<M!fq'h{!Ljxjb!wMk<Kg<gim<Mme<!u 7. njlh<H!Ljx!we<xiz<!we<e@! 8. ljxLg!fq'h{!Ljxbq<z<?!x + y ≥ 2 we<xuiX!x, y Wkiuweqz<?!x ≥ 1 nz<zK!y ≥ 1 we!fq'hq/! 5.2 g{qk!likqiqgt<
dzgqz<! fjmLjxs<! sqg<gz<gjth<! Hiqf<K! ogit<u
dVuig<gh<hMgqe<xe/!likqiqgt<!-bx<g{qkl<?!ucuqbz<!Ohieogi{<M!dVuig<gh<hMgqe<xe/!ucuqbz<!gVk<Kgjtg<!ogilikqiqgt<! ! ucuqbz<! likqiqgt<! weh<hMl</! -bx<g{qk!njlg<gh<hm<m! g{qk! likqiqgt<! -bx<g{qk! likqiqgt<! wehiqOsikje!osb<ukx<G!Le<Oh!g{qk!likqiqgt<!hGk<kxqbhwe<hK! lqGf<k! oszU?! Ofvl<! nkqglikz<?! hiKgih<H! N
131
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hml< 5.3
l<@!g/!!qtg<Gg/
K!-v{<M!-bz<!w{<gt<!
kx<gig! g{qk! likqiqgt<!<x!himh<hGkqgtqe<!Kj{!{<M!njlg<gh<hm<<m!g{qk!gVk<Kgjtg<<! ogi{<M!h<hMl</! sqz! slbr<gtqz<!<hMl</!hiqOsikje!osb<uK!gqbux<jxs<<! siIf<kkiGl<</!
fjmLjxbqz<!dt<t!g{g<Ggjth<!hx<xq!hcg<Gl<ohiPK!fqjxb!g{qk!gVk<Kg<gt<!dVuigqBt<te/! -h<ohiPK! sqz! -bx<g{qk! likqiqgjth<hx<xqBl<?! ucuqbz<!likqiqgjth<!hx<xqBl<!nxqf<K!ogit<Ouil</!!!!!!
fil<! Lkzqz<! yV! wMk<Kg<gim<cjeg<! gVKOuil</! 1 Ofim<Mh<Hk<kgk<kqe<! uqjz!!!'!12 weqz<?
10 Ofim<Mh<Hk<kgr<gtqe<<!uqjz!= 12 × 10 = '!120 NGl</!20 Ofim<Mh<Hk<kgr<gtqe<<!uqjz!= 12 × 20 = '!240 NGl</! 5 Ofim<Mh<Hk<kgr<gtqe<<!uqjz!= 12 × 5 = '!!60 NGl</! 3 Ofim<Mh<Hk<kgr<gtqe<<!uqjz!= 12 × 3 = '!!36 NGl</!!
Ofim<Mh<Hk<kgr<gtqe<!w{<{qg<jg (x) 2 3 5 6 8 10
uqjz!(y) 24 36 60 72 96 120
Ofim<Mh<Hk<kgr<gtqe<! w{<{qg<jg! nkqgliGl<ohiPK! uqjz!nkqgliujkBl<?! w{<{qg<jg! GjxBl<ohiPK! uqjz! GjxujkBl<! nxqbzil</! fil<!wMk<Kg<ogi{<m! -v{<M! lixqgTl<! ye<xig! nkqgiqg<gqe<xe! nz<zK! ye<xig!Gjxgqe<xe/!-u<uiX!njlBl<!-V!lixqgt<!Ofvc!uqgqkk<kqz<!-Vh<hkigg<!%XOuil</!! OlOz!%xqb!wMk<Kg<gim<cz<!Ofim<Mh<Hk<kgr<gtqe<!w{<{qg<jgBl<!nux<xqe<!uqjzBl<!OfI!uqgqkk<kqz<!dt<te!we!nxqbzil</!uip<g<jg!$p<fqjzbqz<!dt<t!yV!g{g<jg!-h<ohiPK! wMk<Kg<ogit<Ouil</! yV! Gxqh<hqm<m! Diqz<?! ubK! 10! zqVf<K! 15! g<Gt<!-Vh<huIgtqe<! w{<{qg<jgjb! x ! we<x! lixq! Gxqg<Gl<! weUl<?! y ! we<x! lixq!nuIgTt<! dbIfqjzh<ht<tqg<Gs<! osz<huIgtqe<! w{<{qg<jgjb! Gxqg<Gl<! weUl<!ogit<Ouil</! 1999!-zqVf<K! 2002!uVml<!ujv!gqjmk<k!uquvr<gjt!gQp<g<g{<muiX!nm<muj{h<hMk<kqBt<Otil<;
x 9200 10200 11600 12400 y 4140 4590 5220 5580
-r<G!yu<ouiV!uVmk<kqx<Gl<!uqgqkl<! =xy 0.45 we!njlukig!nxqbzil<. hqe<H!x, y
we<El<!lixqgjt!-j{g<Gl<! sle<himig! y = 0.45x gqjmg<gqxK/!gmf<k!N{<Mgtqe<!uquvr<gjt! Ofig<Gl<ohiPK! uvh<OhiGl<! N{<MgtqZl<<! uqgqkl<! fqjzbig! njlBl<!we!Le<likqiqbig!wMk<Kg<ogit<Ouil</!!!nMk<k!uVmk<kqz<!nf<k!Diqz<!ubK!uvl<H!10! zqVf<K! 15! g<Gt<! -Vh<huIgt<! w{<{qg<jg! 15000! we! -Vf<kiz<! nuIgTt<!dbIfqjzh<ht<tq!osz<OuiI!w{<{qg<jg!6750!we! y = 0.45x sle<him<miz<!nxqbzil</!weOu!y = 0.45x J!ubKuvl<H!10!zqVf<K!15!g<Gt<!-Vh<huIgTg<Gl<?!nuIgTt<!dbIfqjzh<ht<tq! osz<hui<gtqe<! w{<{qg<jgg<Gl<! okimIHhMk<Kl<! yV! g{qk!likqiqbig! ogit<tzil</! -K! yV! -bx<g{qks<! sle<himig! njlukiz<! -l<likqiqjb!-bx<g{qk!likqiq!we<Ohil<</!! -f<k!wMk<Kg<gim<cz<!ubKuvl<H!10!zqVf<K!15!g<Gt<!-Vh<huIgt<!w{<{qg<jg!!!!)x*! g<Gl<, dbIfqjzh<ht<tq! osz<hui<gtqe<! w{<{qg<jg! )y*! g<Gl<! -jmOb! njlf<k!uqgqkl<!yV!lqjg!w{<!lixqzqbiGl</ weOu!-s<$p<fqjzbqz<!y NeK!x !dme<!Ofvc!liXkz<! ogi{<Mt<tK! we<gqOxil</! sle<hiM! y = 0.45x yV! g{qk! Ofvc! liXkz<!!!Le<likqiqbiGl</!ohiKuig!yV!g{qk!Ofvc!liXkz<!Le<likqiq!y = kx we!njlBl</!-r<G!k NeK!lixqzq!NGl</!-K!yV!ucuqbz<!g{g<gig!-z<jz!weqEl<!-kje!yV! hmlig! ujvf<K! nh<hmk<jkg<ogi{<M! g{g<gqe<! kQIU! gi{! Lbx<sqg<gzil</!
132
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wMk<Kg<gim<mig!OlOz!njlf<k!-bx<g{qk!likqiqjb!gQp<g{<muiX!ucuqbz<!g{qk!likqiqbig! njlg<gzil</! y = 0.45x we<x! sle<him<cx<G! ujvhml<! ujvbzil</! -f<k!ujvhml<!yV!OfIg<Ogimig!njlBl<!)hml<!6/5!Jh<!hiIg<gUl<*/!-K!Ofvc!liXkzqe<!ucuqbz<!likqiqbiGl</
!!nMk<kkig?!
sQvie!Ougk<kqz<!os
l{q!Ofvl<!NGl<!
K~vk<jkg<!gmg<g!3
NGl</! -kqzqVf<K!lmr<giGl<! we!nkqgiqg<Gl<ohiPK!
we!nxqbzil</!nk
we<hkiz<! OugLl<!
weOu! -f<k! -v{nxqbzil</! yV! nGjxBl<ohiPK! lkjzgQp<!uqgqkk<kqzK~vk<jkg<! gmg<gogiMg<gh<hm<Mt<te
Olx<g{<m!nm<muj
t × v = 2 × 80 =we<hkje!nxqbzil
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hml< 5.4
hqe<uVl<! wMk<Kg<gim<jmg<! gVKOuil</! yV! -vbqz<! u{<c! yOv!
z<ukigg<!ogit<Ouil</!-vbqz<!u{<c 160 gq/lQ/!K~vk<jkg<!gmg<g!5!
weqz<?!u{<cbqe<!Ougl<!l{qg<G!4
160 = 40 gq/lQ/!NGl</!160 gq/lQ/!
!l{q!Ofvl<!NGl<!weqz<?!u{<cbqe<!Ougl<!l{qg<G!2
160 = 80 gq/lQ/!
fil<! Ougl<! )v*! -V! lmr<gieiz<! gmg<Gl<! Ofvl<! )t*! njv!nxqbzil</! -f<k! uquvk<kqzqVf<K! fil<! -vbqzqe<! Ougl<!yV!Gxqh<hqm<m!K~vk<jk!gmg<g!wMk<Kg<ogit<Tl<! Ofvl<!GjxBl<!
iuK!v nkqglieiz<!t GjxBl/<!OlZl<!t!nkqglieiz<!v GjxBl<!)v*?! OfvLl<! )t*! kjzgQp<! ohiVk<kl<! ohx<xqVh<hjk! nxqbzil</!<M! ntUgTl<! )v lx<Xl<! t) kjzgQp<! uqgqkk<kqz<! -Vh<hkig!tU! nkqgiqg<Gl<ohiPK! lx<oxiV! ntU! GjxuKl<?! yV! ntU!x<oxiV! ntU! nkqgliuKl<! ohx<xqVh<hqe<! -f<k! -V! ntUgTl<!<!njlgqe<xe!we<Ohil</!gQp<g<g{<m!nm<muj{bqz<!yV!Gxqh<hqm<m!! -vbqz<! u{<c! wMk<Kg<ogit<Tl<! OfvLl<?! OugLl<!/!!!!
Ofvl<! (t) l{qbqz<
2 4 5 8
Ougl< (v) gq/lQ/
80 40 32 20
{bqzqVf<K, 160 , t × v = 4 × 40 = 160, t × v = 5 × 32 = 160, t × v = 8 × 20 = 160 </!nkiuK
133
tv = 160 !nz<zK! .160t
v =
v Bl<! t Bl< wkqIuqgqkk<kqz<!dt<te/!-K! Ofvk<jkh<! ohiXk<K! Ougk<jk!Gxqg<Gl<! YI!-bx<g{qk!likqiqbiGl</!
vt = 160 g<G!yV!ujvhml<!ujvf<kiz<!nf<k! ujvhml< yV OfI! OgimigiK/!nK! hmk<kqz<! dt<tK! Ohiz<! yV!
sqxqb!uqz<!Ohiz<!njlBl<! )hml<!5.5!Jh<! hiIg<gUl<*/! -h<hmk<kqz<! t! Js<!
siIf<K! v! -e<! uQPl<! Ohig<gqjeg<!gi{zil</! -u<uiX! Ougk<kqx<Gl<?!Ofvk<kqx<Gl<! okimIH! hMk<Kl<!ujvhml<!yV!ucuqbz<!likqiqbiGl</!
-bx<hqbzqz<!PV = yV!lixqzq? we<Euqgqk! liXkZg<G! lx<XOliI! wMk<Kg<gimnkqglieiz<!gentU!GjxBl<!weUl<?!nweUl<!nxqOuil</! kjzgQp<!uqgqk!lix<xkhvh<htU!A!ogi{<m!osu<ugk<kqz<!fQtk<kqGjxBl<! weUl<?! fQtk<kqe<!ntU!Gjxfnxqbzil</!nkiuK!xy = A !weqz<?!)x = fQt
y = xA nz
fQtl<!Gjxf<kiz<!ngzl<!nkqgiqg<Gl<?!ng
hb 1. g{qk!likqiq!we<xiz<!we<e@!2. g{qk!likqiqbqe<!Okjujbg<!%X/ 3. OfI!uqgqk!lixz<!uqtg<Gg/!4. wkqI!uqgqk!lixz<!uqtg<Gg/!
www.kalvisolai.com
hml< 5.5
l<!sle<him<jm!nxqOuil</!-K!yV!kjzgQp<!<miGl</! lixik! ouh<h! fqjzbqz<! nPk<kl<!Pk<kl<!Gjxf<kiz<!gentU!nkqgliGl<!<kqx<<G!lx<XOliI!wMk<Kg<gim<mig?!Gxqh<hqm<m!e<!ntU!x nkqglieiz<!ngzk<kqe<!ntU!y <kiz<!ngzk<kqe<!ntU!nkqgliGl<! weUl<!l<, y = ngzl<)
<zK x = yA .
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uqkqkVLjxbqz<!hGk<kxqukx<Og!Lke<jl!kvh<hMgqe<xK/!SliI!330!gq/L/!gizk<kqz<!uip<f<k! ucuqbzqe<! kf<jk! we<xjpg<gh<hMl<! gqOvg<g! nxqRI! B,g<tqm<! we<huI?! ke<!gizk<kqb! ucuqbz<! kk<Kur<gjt! keqjlg<%Xgt<! we<X! ohbiqm<M! hkq&e<X! F~z<!okiGh<Hgtig! upr<gqBt<tiI/! ucuqbjzh<! hch<hK! Hkqbkigs<! sqf<kqh<hkx<Gl<! lx<x!himr<gtqz<!kqxjljb!Wx<hMk<Kukx<Gl<!ohiqKl<!dkUukiGl</ 6.1 siqhii<k<kZg<Giqb!Okx<xr<gt<
flK! Lf<jkb! uGh<Hgtqz<?! ujvbXg<g! -bzik! nch<hjmbie! ucuqbz<!osix<gtigqb!Ht<tq?!OgiM?!ktl<?!Ogi{l<!Ohie<xux<jx!nxqf<Kt<Otil</!!OlZl<!fil<!Lg<Ogi{r<gtqe<! ogit<jggt<?! -j{OgiMgt<?! sqz! sqxh<hie! fix<gvr<gt<!Ngqbux<jxBl<!nxqf<Kt<Otil</!!weqEl<?!nux<jxh<hx<xq!OlZl<!nxqf<K!ogit<ukx<G!
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6.1.1 nch<hjmbie!ucuqbz<!osix<gt<
Ht<tq we<hK! outqbqz<! yV! fqjzbqjeg<! Gxqh<hkx<G! hbe<hMukiGl</!!upg<gk<kqz<?!gigqkk<kitqe<!lQOki!nz<zK!gVl<hzjgbqe<!lQOki!yV!Ht<tqbieK?!yV!sqxqb! njmbitlqm<M! Gxqg<gh<hMl</! Neiz<?! nxqLjxbqe<hc! Ht<tqg<G! ntOui!nz<zK! ucuOli! gqjmbiK/! -v{<M! OgiMgt<! ye<jxobie<X! oum<cg<ogit<Tl<!fqjzbqzqVf<K! Ht<tqbqje! nxqf<K! ogit<tzil</! Neiz<?! -r<G! Ht<tqbqje! nxqf<K!ogit<t!Ogim<jmh<!hx<xqb!gVk<K!hbe<hMk<kh<hm<mK/!-v{<M!ktr<gt<!ye<jxobie<X!oum<cg<ogit<Tl<! -mk<kqz<! njlf<k! Ht<tqgtqe<! okiGh<hqje! OgiM! we!nxqbzil</!Neiz<!-r<G!Ogim<cje!nxqf<Kogit<t!ktk<kqjeh<hx<xqb!gVk<K!hbe<hMk<kh<hm<mK/!
135
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ke<! lQKt<t! wf<kuqV! Ht<tqgjtBl<! -j{g<Gl<! OfIg<Ogim<ce<! njek<K!Ht<tqgjtBl<! ke<lQOk! ogi{<Mt<tuixjlf<k?! njek<K! kqjsgtqZl<! Lcuqz<zilz<!osz<zg<%cb! Olx<hvh<Oh! yV! ktl<! we! nxqf<K! ogit<gqOxil</! weOu?! nch<hjm!ucuqbz<!osix<gtie!Ht<tq?!OgiM?!ktl<!-ux<jxh<hx<xq!nEhuk<kqz<!lm<MOl!nxqf<K!ogit<t! LcBle<xq! nux<jxs<! siqbig! ujvbXh<hK! -bzik! ye<xiGl</! Ht<tqgt<?!
Nr<gqzh<! ohiqb! wPk<Kgtie! A, B, C, D Ohie<xux<xiz<! Gxqk<Kg<! gim<mh<hMl</! A, B we<he!yV!Ogim<ce<!lQKt<t!-V!Ht<tqgt<!weqz<?!ng<OgiM!!
↔AB nz<zK!
↔BA!
weg<!Gxqg<gh<hMl</!!-kje!OgiM!AB!weUl<!hcg<gzil</ -Vkjz!nl<Hg<Gxq?!‘↔’ we<hkqz<! dt<t! -V! nl<Hgt<?! OgiM! -V! kqjsgtqZl<! Lcux<X! osz<gqxK!
we<hkjeg<! Gxqg<gqxK/! SVg<glig?! yV! OgimieK! l we<x! keq! wPk<kiZl<!
Gxqg<gh<hMl</! A, B we<hju!yV! Ogim<ce<! lQKt<t!-V! Ht<tqgt<! weqz<?! A, B Ngqb!Ht<tqgjtBl<! Osi<k<K!nh<Ht<tqgTg<gqjmOb!njlBl<!nf<OfIg<Ogim<ce<! higk<kqje?!
A, B we<x! -V! Ht<tqgTg<gqjmOb!njlf<k! OfIg<Ogim<Mk<K{<M! nz<zK! SVg<glig!Ogim<Mk<K{<M!weh<hMl</!-K!GxqbQm<cz<!!
AB weg<!Gxqg<gh<hMl</! AB -e<!fQtk<jk!SVg<glig
AB nz<zK!BA!we!wPkzil</!yV!Ht<tqbqz<!okimr<gq!WOkEl<!yVkqjsbqz<!Lcuqe<xqs<!osz<Zl<!yV!Ogim<ce<!higl<!gkqi<!weh<hMl</!-r<G!okimr<Gl<!Ht<tqbieK?!ng<gkqiqe<!Nvl<hh<Ht<tq!
nz<zK! okimg<gh<Ht<tq! weh<hMl</! A we<hK! yV! gkqiqe<! okimg<gh<Ht<tq?! B! NeK!ng<gkqiqe<!lQKt<t!WOkEl<!yV!Ht<tqobeqz<?!-g<gkqvieK!
→AB !
weg<!Gxqg<gh<hMl</! !-kje?!yVkjz!nl<Hg<GxqBme<!Gxqg<gilz<?! !gkqi< AB weUl<!Gxqg<gzil</ AB bqe<! lQkjlf<Kt<t! yVkjz! nl<Hg<Gxq! gkqi<! AB bqe<! kqjsjbg<!Gxqg<Gl</!!
Gxqh<H; -eq?!↔AB we<hkje?!-Vkjz!nl<Hg<Gxq! ‘↔’!J! AB! bqe<! lQK!Gxqg<gilz<!
SVg<glig!OgiM!AB!weUl<?! AB we<hkje!GxqbQM!‘⎯’ J!AB!bqe<!lQK!Gxqg<gilz<!
SVg<glig! Ogim<Mk<K{<M!AB!weUl<?! gkqI!→AB J!yVkjz!nl<Hg<Gxq! ‘ → ’ J!AB!
bqe<! lQK! Gxqg<gilz<! SVg<glig! gkqI! AB! weUl<! Gxqh<Ohil</! Ht<tqgt<! A, B !-ux<xqx<gqjmh<hm<m!okijzuqje!AB we<Xl<!Gxqh<Ohil</!!
yV!ohiKh<Ht<tq!A!bqz<!Nvl<hqg<Gl<!-V!gkqi<gt< AB, AC Neju?!Ht<tq!A z<!yV! Ogi{k<kqje! njlg<gqe<xe! we<gqOxil</! Ht<tq A NeK Ogi{k<kqe<! Lje!weh<hMl</!OfIg<Ogim<Mk<K{<Mgt<!ABBl< AC Bl<!ng<Ogi{k<kqe<!-V!gkqIgt<!weh<hMl<.!!-g<Ogi{lieK?! nz<zK !weg<!Gxqg<gh<hMl</!BAC∠ CAB∠ BAC∠ nz<zK we!yV!Ogi{lieK!Gxqg<gh<hm<miz<?!Ht<tq!A!NeK!ng<Ogi{k<kqe<!Lje?!AB!lx<Xl<!AC!ng<Ogi{k<kqe<!gkqIgt<!we<hjkk<!okiqf<K!ogit<g/ Abqz<!njlBl<!Ogi{lieK!
CAB∠
136
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yV! gkqiqzqVf<K! okimr<gq! nMk<k! gkqVg<Gs<! osz<ukig! yV! sqxqb! uqz<ziz<!Gxqg<gh<hMl<!(hml<< 6.1 Jh<!hii<g<gUl<). A!we<x!LjebqZt<t!yV!Ogi{k<kqe<!gkqIgt<!okiqf<k! fqjzbqz<! ng<Ogi{k<jk! wtqkig A∠ ! we! wPkzil</! yV! Ogim<Mk<!K{<jmh<Ohie<X! yV! Ogi{k<kqx<Gl<! ntU! d{<M/! A∠ we<x! Ogi{k<kqe<! ntju m∠A weg<! Gxqg<gzil</! Gph<hl<! WKl<! -z<jzobeqz<?! m∠A J! wtqkig! A∠ weg<!Gxqh<Ohil</!Neiz<?!m∠A we<hK!YI!ntU?! A∠ we<hK!yV!GxqbQM!we<hjkh<!Hiqf<K!ogit<tUl</!!
!!
hml< 6.1
!
!!!!!!!!yV!Ogi{k<kqe<!ntU?!ng<Ogi{k<kqe<!gkqIgtqe<!fQt!ntjuh<!ohiVk<kkz<z/!
Ogi{r<gt<?! Ähijg}! weh<hMl<! nzgqeiz<! ntg<gh<hMgqe<xe/ yV! ktk<kqe<! lQkjlf<k!yV!gkqi<!nke<!okimg<gh<!Ht<tqjbg<ogi{<M!yV!LP!Spx<sqjb!Wx<hMk<kqeiz<?!nK!!360 hijg!ntUt<t! Ogi{k<kqje!njlg<gqe<xK! we<gqOxil</ -f<k!ntU! 360º we!wPkh<hMl<! (hml<!6.2 Jh<!hii<g<gUl<). -r<G!Ogi{k<kqe<!gkqIgt<!ye<xqe<!lQK!ye<xig!njlf<Kt<tK/!!lx<x!Ogi{r<gtqe<!ntUgt<!360º jb!nch<hjmbigg<!ogi{<mkiGl</!!
yV! gkqi<?! yV! LP! Spx<sqbqe<<!41 higk<kiz<! njlg<Gl<! Ogi{k<kqe<! ntU!!!!!!!!!
41 (360º) = 90º NGl<;!
61! higk<kiz<!njlg<Gl<! Ogi{k<kqe<!ntU
61 (360º) = 60º
NGl<;! 360
1!higk<kiz<!njlg<Gl<!Ogi{k<kqe<!ntU!
3601 (360º) = 1º NGl</!
hml< 6.3
hml< 6.4
m = 90º weqz<? yV osr<Ogi{l< (hmm e<! ntU 90º g<Gl<! nkqgoleqz<?
BAC∠ BAC∠BAC∠ BAC∠
hii<g<gUl<) weh<hMl<. m e<! ntU 90º g<Gl<!GXr<Ogi{l< (hml< 6.5 Jh<!hii<g<gUl<) weh<hMl</!!
BAC∠
137
www.kalvisolai.com
hml< 6.5
hml< 6.2
l<<!6.3 Jh<!hii<g<gUl<) weh<hMl<. yV! uqiqOgi{l< (hml< 6.4 Jh<!GjxU! weqz<? ! yV BAC∠
m = 180º weqz<? yV!Ofi<g<Ogi{l< (hml< 6.6 Jh<!hii<g<gUl<) weh<hMl</ yV! Ofi<g<Ogi{oleqz<? BC yV! OfIg<Ogim<Mk<! K{<miGl<A A, B, C we<he!!
yVOgimjlf<k! Ht<tqgtiGl<A! nkiuK?! -h<Ht<tqgt<! yVOgim<cz<! njlBl<!Ht<tqgtiGl</!!!
BAC∠ BAC∠BAC∠
!!!
hml< 6.6
!!!!
-V!OgiMgt< BD Bl< CE Bl<!A!we<x!Ogi{r<gt< Bl< BAC∠ DAE∠ Bl<! Gk<okkqi<!
BAE∠ Ngqb!Ogi{r<gt<!Gk<okkqi<!Ogi{r<gtJh<!hii<g<gUl<).
m Ogi{k<kqe<! ntU 180º g<G!
-Vh<hqe<!ng<Ogi{l<!hqe<ujt!Ogi{l< weh<hMBAC∠
!
hml< 6.9
hml< 6.8
!!!!!
!!
-V!Ogi{r<gTg<G!ohiK!ds<sqBl<?!yVgkqVg<G! -V! hg<gr<gtqZl<! Ogi{r<gt<! n
nMk<kMk<k!Ogi{r<gt<!weh<hMl< (hml< 6.9 Jh<!Ngqbux<xqx<G! ohiKLje A NGl<;! ohiK! gknjlf<kqVh<hkiz<?! BAD∠ Bl< Bl<! nM∠BAD Bl<!ohiKuie!jg!AB!ohx<xqVf<kiZl<?!Woeeqz<?! nju! AB we<x! ohiK! hg<gk<kqx<G-Vh<hKkie<!we<hjk!nxqf<K!ogit<Ouil</!-V!
weqz<?! nju! fqvh<Hg<Ogi{r<gt<! (hml< 6.10 Jhm
DAC∠
BAD∠ + m = 90º; weOu? DAC∠ BAD∠ Bl< ∠m BAD∠ = 90º − m ;! yV! Ogi{l<!-g<Ogi{r<gtqz<!ye<xqe<!ntU!okiqf<kiz<?!lx<x
DAC∠
!
138
www.kalvisolai.com
hml< 6.7
Ht<tqbqz<!oum<cg<ogit<ukiz<!njlBl<!
Ogi{r<gt<! weh<hMl</! OlZl<? , CAD∠ig!njlujkBl<!nxqbzil<!)hml< 6.7
nkqgligUl<?! 360º g<Gg<! Gjxuig!l<!(hml< 6.8 Jh<!hii<g<gUl<).
!
hml< 6.10
!ohiKg<!gkqVl<!njlf<K?!ohiKuie!jlBlieiz<?! nu<uqV! Ogi{r<gTl<!
hii<g<gUl<). hml< 6.9 -z< BAD∠ , qI
DAC∠
AD g<G! -V! hg<gr<gtqZl<! -ju!k<kMk<k! Ogi{r<gt<!NGl</ ∠BAC Bl< nju!nMk<kMk<k!Ogi{r<gt<!NgiK;!!! yOv! hg<gk<kqzjlf<k! Ogi{r<gtig!
nMk<kMk<k!Ogi{r<gtqe<!%Mkz<!90º <! hii<g<gUl<) weh<hMl</ hml< 6.10 -z<
DAC Bl<!fqvh<Hg<Ogi{r<gtiGl<;!-r<G lx<xkqe<! fqvh<Hg<Ogi{liGl</! weOu?!kqe<!ntuqje!nxqb!LcBl</
-V!nMk<kMk<k!Ogi{r<gtqe<!%Mkz< 180º weqz<?!nju!lqjgfqvh<Hg<Ogi{r<gt<!weh<hMl<!(hml< 6.11 Jh<!hii<g<gUl<*/! hmk<kqz<?!!!!!!!
m BAD∠ + m = 180CAD∠ o. weOu?! -r<G! filxquK! BAD∠ Bl<! CAD∠ Bl<!lqjgfqvh<Hg<Ogi{r<gtiGl</!!OlZl<?!
m = 180º − m DAC∠ BAD∠ , m BAD∠ = 180º − m . DAC∠
nkiuK?!yV!Ogi{lieK!lx<xkqe<!lqjgfqvh<Hg<!!Ogi{liGl</!-r<G!yV!Ogi{l<!kvh<hce<?!lx<xjkg<!gi{zil</ 6.1.2 <!OgiMgtqe<!lQkie!h{<Hgt<!lx<Xl<!Okx<xr<gt< nxqf<K! ogit<ukx<gig?! fil<?! OfIg<OgiMgtqe<! lQkie! sqz! h{<HgjtBl<?!Okx<xr<gjtBl<! -h<OhiK! gi{<Ohil</! -r<G! ! Okx<xr<gjt! fq'hqg<gilz<! osbz<!uqtg<gk<kqe<!&zl<!siqhiIk<Kg<!ogit<gqOxil</!!osbz<; yV!ktk<kqz< A, B we<x!-V!Ht<tqgjtg<!Gxqg<g/ Ht<tq! A e<! upqbig! hz! Ogosz<ujkg<! gi{<g/! -jkh<!
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hml< 6.12
iMgt<! ujvg/! ! nux<Xt
Ohie<Ox?! Ht<tq! B! e<!lm<Ml<! A! e<! upqOb
gjt! -j{g<Gl<! Of<H!gqjmg<gqxK/!
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g<!ogit<jgbqzqVf<K!gQp
! -V! ouu<OuX! Ht<t
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139
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Ig<OgiM! AB we! nxqg/! -s<!
<kiz<?!nux<xqe<!upqOb!yOv!yV!
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(iii) &e<X!nz<zK!&e<Xg<G!Olx<hm<m!Ht<tqgtqz<!Wkiuokie<X!lx<xuqV!Ht<tqgtiz<!njlBl<!Ogim<cz<!njlbiokeqz<?!nju!yVOgimjlf<k!Ht<tqgtigi/ !
osbz<;!yV!ktk<kqz<!AB, CD we<x!-V!ouu<OuX!OgiMgt<!ujvg/!-V!OgiMgjtBl<!Ou{<Mltuqx<G! fQm<Mg/! -V! OgiMgTg<Gl<! ohiKuig! yV! Ht<tq! -Vh<hjkObi!
nz<zK!-z<zilz<!OhiujkObi!gi{<g (hml< 6.13 Jh<!hii<g<gUl<). Olx<gi[l<!osbz!h{<H 2: -VOuX!!Gxqh<H;!-V!OgiMoum<Ml<OgiMgt<!OgiMgt<! weh<hMweh<hMl</!!osbz<: OgiM!AB!P bqe<!upqOb!osAB g<G!-j{bi !!!!!!!!!-s<osbzqzqVf<K!!h{<H!3: yV!Ogikvh<hm<m!Ogim<cx<G Gxqh<H;! &e<X! nng<OgiMgt<!yVH!
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hml< 6.13
qzqVf<K!gQp<g<g{<m!h{<H!gqjmg<gqxK/!
OgiMgTg<Gl<!ohiKuig!yV!Ht<tqg<G!Olz<!-Vg<giK/
gTg<gqjmOb!yV!ohiKh<Ht<tq!njlBlieiz<!nju!ye<jxobie<X!weh<hMl</! ohiKh<Ht<tq! njlbik! OgiMgt<! oum<cg<ogit<tik!l</! -k<kjgb! ohiKh<Ht<tq! njlbik! OgiMgt<! -j{! OgiMgt<!
ujvg/!!OgiM!AB bqz<!njlbik P we<x!Ht<tqjb!wMk<Kg<ogit<g/!! z<ZliX!OgiMgt<!ujvg/!nk<kjgb!OgiMgtqz<!yOv!yV!OgiMkie<!g!njlujkg<!gi{zil<!(hml< 6.14 Jh<!hii<g<gUl<).
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hml< 6.14
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jlbik!yV!Ht<tqBl<!kvh<hce<?!kvh<hm<m!Ht<tq!upqOb!!yOv!yV!OgiMkie<!njlBl</
<G! Olx<hm<m! OgiMgt<! yVHt<tq! upqOb! osz<Zoleqz<?!Mgt<!weh<hMl</!
140
osbz<;! AB, CD wEl<! -V! ouu<OuX! OgiMgt<! ye<jxobie<X! oum<MliX! ujvf<K?!!oum<Ml<! Ht<tqjb O weg<! Gxqg<gUl<! (hml< 6.15Jh<! hii<g<gUl<*/! , ,
, !Ngqb!Ogi{r<gjt!ntg<gUl</!AOC∠ BOD∠
AOD∠ BOC∠-kqzqVf<K!gQp<g<g{<mux<jx!nxqgqe<Oxil</!!
m = m AOD∠ ,BOC∠m = m AOC∠ .BOD∠
-r<G! , Ngqb!Ogi{r<gt< yVOsic!Gk<okkqi<! Ogi{r<gtiGl</! -u<uiOx?! filxquK!
Bl<! Bl<! yVOsic! Gk<okkqi<!Ogi{r<gtiGl</!!
AOC∠ BOD∠
BOC∠ AOD∠
!-s<osbzqzqVf<K!gQp<g<g{<m!h{<H!gqjmg<gqxK/ h{<H! 4;! -VOgiMgt<! ye<jxobie<X! oum<cg<ogi{Ogi{r<gt<!sll</!!Gxqh<H: m = xº, m = yº we<g/ Olx<g{<mm = xº, m = yº. Neiz< m
AOC∠ AOD∠BOD∠ BOC∠ AOC∠ + m BOC∠
nz<zK xº + yº + xº + yº = 360º nz<zK 2xº + 2yº = 360º nzlqjgfqvh<Hg<Ogi{r<gt<!NGl</!!-kqzqVf<K?!!
m + mAOC∠ BOC∠ = 180º, m∠BOD + m∠m∠AOC + m∠AOD = 180º, m BOC∠ + m ∠
we!nxqbzil</ Osicgt< ∠AOC? ∠BOC Bl<A ∠BOD, ∠∠BOC? ∠BOD Bl<!yu<ouie<Xl<!lqjgfqvh<Hg<Ogi{r<gjt!osbz<: AB, CD!we<x!-j{OgiMgt<!ujvg/!!AB g<G!-ujvg (hml< 6.16Jh<!hii<g<gU PQ NeK?!-j{OgiMgt< AB jbBl< CD jbBl<!oum<Mu-ux<xqe<! yV! GXg<Goum<cbiGl<! we<hjk! nxqg/ohiKh<Ht<tqjb L weUl<? CD, PQ uqx<G!njlBl<! ohi
,PLB∠ ,PLA∠ ,BLM∠ ,ALM∠ ,LMD∠ ,LMC∠ ∠ntf<K!hiIg<gUl</!filxquK?
m = mPLB∠ LMD∠ , m BLM∠ = mm = m mPLA∠ ,LMC∠ ALM∠ = m
!-s<osbzqzqVf<K!hqe<uVl<!h{<H!gqjmg<gqxK/
141
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hml< 6.15
<miz<! d{<miGl<! Gk<okkqi<!
! nch<hjmg<! ogit<jgbqe<hc + m BOD∠ + m = 360º AOD∠<zK xº + yº = 180º. weOu xº, yº
AOD = 180º, BOD = 180º
AOD!Bl<A ∠AOC, ∠AOD Bl<A !njlg<gqe<xe/
j{big!-z<zilz<!OgiM!PQ
l<).hml< 6.16
jk!nxqg/!PQ NeK!AB, CD ! AB, PQ uq<x<G! njlBl<!Kh<Ht<tqjb M weUl<!Gxqg<g/!CMQ , QMD∠ ! Ngqbux<jx!
,DMQ∠ .CMQ∠
h{<H 5: -V!-j{OgiMgTg<G!yV!OgiM!GXg<Goum<cbig!njlf<kiz<?!nr<gjlBl<!wf<kouiV!Osic!yk<k!Ogi{r<gTl<!ntuqz<!sllig!-Vg<Gl</ !Gxqh<H: (i) Ogi{l< ALM∠ l< LMD∠ Bl<! yV! Osic! ye<Xuqm<m! dm<Ogi{r<gjt!njlg<gqe<xe/!Ogi{l< BLM∠ l< Bl<!lx<oxiV!Osic!ye<Xuqm<m!dm<Ogi{r<gt<!NGl</!Olz<!njlf<k!osbzqe<hc?
LMC∠
m BLM∠ = m LMC∠ , m ALM∠ = m LMD∠ !!we!nxqbzil</!-kjek<!Okx<xligg<!%XOuil</! Okx<xl< 1: -V! -j{OgiMgTg<G! yV! OgiM! GXg<Goum<cbig! njlf<kiz<?!nr<gjlBl<!wf<kouiV!Osic!ye<Xuqm<m!dm<Ogi{r<gTl<!ntuqz<!sll</
(ii) Ogi{l< PLA∠ Ul<! Ul<!yVOsic!ye<Xuqm<m!outqg<Ogi{r<gjt!njlg<gqe<xe/! Ogi{l<
DMQ∠BLP∠ Bl<! CMQ∠ Ul<! lx<oxiV! Osic! ye<Xuqm<m!
outqg<Ogi{r<gt<!NGl</!!Olz<!njlf<k!osbzqe<hc?!!m = m mPLA∠ ,DMQ∠ BLP∠ = m CMQ∠
we!nxqgqe<Oxil</!-kjek<!Okx<xligg<!%XOuil</!!Okx<xl< 2: -V! -j{OgiMgTg<G! yV! OgiM! GXg<Goum<cbig! njlf<kiz<?!nr<gjlBl<!wf<kouiV!Osic!ye<Xuqm<m!outqg<Ogi{r<gTl<!ntuqz<!sll</
(iii) Ogi{r<gt< BLM∠ l< LMD∠ Bl<! GXg<Goum<cbqe<! yOvHxl<! njlf<k!yVOsic! dm<Ogi{r<gjt! njlg<gqe<xe/! -jkh<OhizOu! ALM∠ l< Bl<!GXg<Goum<cbqe<! yOvHxl<! njlf<k! lx<oxiVOsic! dm<Ogi{r<gt<! NGl</! Olz<!njlf<k!osbzqe<hc?!
LMC∠
m BLM∠ + m LMD∠ = 180°, m ALM∠ + m LMC∠ = 180° we!nxqbzil</!-f<k!d{<jljbk<!Okx<xligg<!%xzil</ !Okx<xl< 3: -V! -j{OgiMgTg<G! yV! OgiM! GXg<Goum<cbig! njlf<kiz<?! yOv!hg<gk<kqz<!njlf<k!yVOsic!dm<Ogi{r<gt<!lqjgfqvh<Hg<Ogi{r<gtiGl</ !osbz<: yV!OgiM!AB ujvg/!OgiM AB!g<G!-j{big!njlbik!OgiM!PQ ujvg/!!OgiMgt<! AB Bl<! PQ Ul<! sf<kqg<Gl<! Ht<tqjb! L!weg<!Gxqg<gUl</!L!nz<zik!lx<oxiV!Ht<tq!M J!OgiM PQ!e<!lQK!Gxqg<gUl</! ALM∠ J!ntg<gUl</!M -e<! upqOb! ! m ALM∠ = m LMD∠ we!njlBliX! OgiM CD ujvg (hml< 6.17Jh<!hii<g<gUl<). -kqzqVf<K! OgiM!PQ we<hK! OgiMgt<! AB, CD -ux<xqx<G! GXg<Goum<c! weUl<? AB Bl< CDBl<!ye<jxobie<X!sf<kqg<gik!OgiMgt<!weUl<!nxqbLcgqe<xK/!!-s<osbzqzqVf<K!hqe<uVl<!h{<H!gqjmg<gqxK/ h{<H 6: yV! GXg<Goum<c! -V! OgiMgjt! oumye<Xuqm<mOgi{r<gt<!sllig!-Vh<hqe<?!nu<uqV!OgiM!
142
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hml< 6.17
<Ml<ohiPK! Wx<hMk<Kl<! yVOsic!gt<!-j{OgiMgtiGl</!!!
Olx<g{<m!h{<hqzqVf<K!hqe<uVl<!Okx<xr<gjt!fq'hqg<gzil</ Okx<xl< 4: yV!GXg<Goum<c!-V!OgiMgjt!oum<Ml<ohiPK!Wx<hMk<Kl<!yVOsic!yk<k!Ogi{r<gt<!sllig!-Vh<hqe<?!nu<uqV!OgiMgt<!-j{!OgiMgtiGl</ !
Okx<xl< 5: yV!GXg<Goum<c!-V!OgiMgjt!oum<Ml<ohiPK!Wx<hMk<Kl<!Ogi{r<gtqz<?!GXg<Goum<cbqe<! yOvHxl<! njlf<k! yV! Osic! dm<Ogi{r<gt<! lqjgfqvh<Hg<!Ogi{r<gtig!njlf<kiz<?!nu<uqV!OgiMgt<!-j{!OgiMgtiGl</ !
wMk<Kg<gim<M 1: hml< 6.18!-z<!OgiM l1 Ul<!OgiM l2 Ul<!-j{OgiMgtiGl</!!OgiM l3 !NeK OgiMgt< l1, l2 gTg<G GXg<Goum<cbiGl</! 1∠ , 2∠ Ngqb! Ogi{r<gtqe<!ntUgtqe<! uqgqkl< 4 : 5! weqz<? 1∠ , 2∠ , 3∠ , 4∠ , 5∠ , 6∠ , , ! Ngqb!Ogi{r<gtqe<!ntUgjtg<!gi{<g/
7∠ 8∠
kQi<U: m : m = 4 :1∠ 2∠
lqjgfqvh<Hg<Ogi{r<gtqe
∴ 54
× m + m = 182∠ 2∠
(n.K* 9× m = 180 ×2∠∴ m = 180° − m =1∠ 2∠-h<ohiPK?!m = m = 80° ( , 3∠ 1∠ 1∠ ∠m = m = 100°( 4∠ 2∠ 2∠m = m = 80° (5∠ 1∠ 1∠ ,m = m = 100° (6∠ 2∠ 2∠m = m = 80° (7∠ 5∠ 5∠ ,m = m = 100° (8∠ 6∠ 6∠!!
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hml< 6.18
5 ⇒ 54
2m1m
=∠∠ ⇒ m 1∠ =
54
× m 2∠ . Neiz<?! , wEl<!
<!Osic!we<hkiz<?!m
1∠ 2∠
1∠ + m 2∠ = 180°. !
0° (n.K*! 5
2m52m 4 ∠×+∠× = 180°
5 (n.K*! m 2∠ = 100°. 180° − 100° = 80°.
we<he!yV!Osic!Gk<okkqi<!Ogi{r<gt<), 3, we<he!yV!Osic!Gk<okkqi<!Ogi{r<gt<), 4∠
we<he!yV!Osic!yk<k!Ogi{r<gt<), 5∠, we<he!yV!Osic!yk<k!Ogi{r<gt<), 6∠
we<he!yV!Osic!Gk<okkqi<!Ogi{r<gt<), 7∠, we<he!yV!Osic!Gk<okkqi<!Ogi{r<gt<). 8∠
143
wMk<Kg<gim<M 2: hml< 6.19 -z<!AB NeK CD g<G!-j{OgiM!weUl<? AC NeK BD g<G -j{OgiM!weUl<!fq'hq/ !!!!!!! !!kQi<U: OgiM! CD NeK! Ogi
= 50° + 13dm<Ogi{r<gtie! Bl<!BD Bl<! -j{! OgiMgt</! we
CDBACD ∠+∠ mmACD∠
weOu?!m = 130°. nMk<km = m = 130°. n
XCA∠XCA∠ CAB∠
sll</!!weOu?! AB Bl< CD Bl<!- 6.1.3 Lg<Ogi{k<jkh<!hx<xqb!h{
!Lg<Ogi{k<jkh<! hx<xq!
we<hK!&e<X!hg<gr<gtizie!yXY, PQ &e<X!OgiMgt<. OgiMgtHt<tq A -z<!oum<cg<!ogit<gqeDE Bl< XY Bl< Ht<tq! B ogit<gqe<xe/!OgiMgt<!PQ UlC -z<! oum<cg<! ogit<gqe<xe/Ngqb!Ogim<Mk<!K{<Mgtiz<!nLg<Ogi{l<! ABC! weh<hMl</!K{<Mgt< AB, BC, CA Lhg<gr<gt<! weh<hMl</! Ht<tqgLg<Ogi{k<kqe<! ds<sqgt<!Ogi{r<gt< ,BAC∠ ABC∠ , B∠Lg<Ogi{k<kqe<!dm<Ogi{r<gtGxqg<gh<hMl</!!Ogi{l< L
, , we!nxqf<K!ogit<Ouil</
PAB∠DBA∠ ,YBC∠ ,BCQ∠ ECA∠
Le<uGh<hqz<!hck<kux<xqz
(i) yV!Lg<Ogi{k<kqz<!wf<kuqnslhg<g!Lg<Ogi{liGl</ (ii) yV! Lg<Ogi{k<kqz<! Wki-Vslhg<g!Lg<Ogi{liGl</ (iii) yV!Lg<Ogi{k<kqz<!wz<ziLg<Ogi{liGl</
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hml< 6.19
M AC g<Gl<! BD g<Gl< GXg<Goum<cbiGl</! -r<G?!0° = 180°.! weOu?!GXg<Goum<cbqe<! yVHxl<!njlf<k!
Bl<!lqjgfqvh<Hg<Ogi{r<gtiGl</!weOu?!AC Bl<!Ou?! yk<k! Ogi{r<gt<
CDB∠XCA∠ Bl< Bl<! sll</!!
kig?!AB g<Gl< CD g<Gl<?!AC GXg<Goum<cbiGl</ -r<G?! kiuK!ye<Xuqm<m!Ogi{r<gt<
CDB∠
XCA∠ Ul< Bl<!j{OgiMgtiGl<.
CAB∠
<HgTl<!!Okx<xr<gTl<
Wx<geOu! fil<! nxqf<Kt<Otil</! ! Lg<Ogi{l<!V!ucuqbz<!dVuliGl</!!hml< 6.20!-z< DE, < PQ Ul< XY!Bl<!!<xe/!OgiMgt<!-z<! oum<cg<!< DE Bl< Ht<tq! AB, BC, CA jlBl<!ucul<!OfIg<Ogim<Mk<!g<Ogi{k<kqe<!
t< A, B, C weh<hMl</!
CA<!weh<hMl</!-g<Ogg<Ogi{k<kqe<!ou
we<he!LgCAX∠
qVf<K!hqe<uVl<!uV!hg<gr<gTl<!sl
uK! -V! hg<gr
!hg<gr<gTl<!sl
144
hml< 6.20
i{r<gt<!wtqkig A∠ , B∠ , weg<!tqg<Ogi{liGl</!!OlZl<!Ogi{r<gt< <Ogi{l<!ABC!bqe<!outqg<Ogi{r<gt<!
C∠
jvbjxgjt!fqjeU%i<Ouil<; l<!-z<jz!weqz<?!nl<Lg<Ogi{l<!yV!
<gt<! sll<! weqz<?! nl<Lg<Ogi{l<! YI!
l<!weqz<?!nl<Lg<Ogi{l<!yV!slhg<g!
(iv) yV!Lg<Ogi{k<kqz<!&e<X!Ogi{r<gTl<!GXr<Ogi{r<gtibqe<?!nl<Lg<Ogi{l<!!yV!GXr<Ogi{!Lg<Ogi{liGl</ (v) yV!Lg<Ogi{k<kqz< WkiuK!yV!Ogi{l<!uqiqOgi{l<!weqz<?!nl<Lg<Ogi{l<!yV!uqiqOgi{!Lg<Ogi{liGl</ (vi) yV! Lg<Ogi{k<kqz< yV! Ogi{l<! osr<Ogi{libqe< (Ogi{k<kqe<! ntU 90°)? nl<Lg<Ogi{l<!yV!osr<Ogi{!Lg<Ogi{liGl</ !
Lg<Ogi{k<jkh<!hx<xqb!hqe<uVl<!Okx<xr<gjtg<!%XOuil</ !Okx<xl< 6: yV!Lg<Ogi{k<kqe<!&e<X!Ogi{r<gtqe<!ntUgtqe<!%Mkz<!180°!NGl</ Okx<xl< 7: yV! Lg<Ogi{k<kqe<! yV! hg<gl<! fQm<mh<hm<M<! d{<miGl<! outqg<Ogi{k<kqe<!ntU!dt<otkqi<!Ogi{r<gtqe<!ntUgtqe<!%MkZg<Gs<!slliGl</ fq'h{l<: ABC yV!Lg<Ogi{l</!hg<gl< BC jb!fQm<Mg/ BC e<! fQm<sqbqz<! X we<x! Ht<tqjbg<!Gxqg<gUl</! hml< 6.21 -z< outqg<!Ogi{liGl<?
ACX∠A∠ Ul< B∠ Bl<! dt<otkqi<!
Ogi{r<gtiGl</ ACX∠m = m A∠ + m B∠ !we! fq'hqk<kz<! Ou{<Ml</ Ul< ACX∠ C∠ Bl<!lqjg!fqvh<Hg<Ogi{r<gtiGl</ ∴ m + m = 180°. ACX∠ C∠Neiz<?! m A∠ + m B∠ + m = 180°. C∠∴ m + m = mACX∠ C∠ A∠ + m B∠ + m C∠ . -Vosb<b!m = mACX∠ A∠ + m B∠ NGl</ osbz<:! hml<! 6.22 -z<!dt<thc!Lg<Ogi{l<! PQNgqbux<xqe<! fQtr<gtigqb! PQ, QR, RP J! ntfPR + PQ!Ngqb!lkqh<Hgjtg<!g{<Mhqcg<gUl</!!-k (i) PQ + QR > PR, (ii) QR + PR > PQ, (iii) PR + weg<!g{<mxqbzil</!!-s<osbzqzqVf<K!hqe<uVl<!h{<H!gqjmg<gqxK/ !h{<H 7: yV! Lg<Ogi{k<kqe<! -V! hg<gr<gtqe<!%Mkz<!&e<xiuK!hg<gk<kqe<!fQtk<jkuqmg<!%Mkzi!Gxqh<H: wf<kouiV!Lg<Ogi{l< ABC bqZl<? (i) AB < B(ii) BC < CA + AB (iii) CA < AB + BC. -ju!Lg<Ogi{!sleqe<jlgt<!weh<hMl</ osbz<: hml< 6.23 -z<! dt<tuiX! Lg<Ogi{l<!ujvg/! Ogi{r<gt<! A∠ , B∠ , C∠ Ngqbu
ntg<gUl</!OlZl<?!AB, BC, CA!Ngqb!hg<gtUgjt!
145
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hml< 6.21
!hg<gLl<!ohiK m jb!fQg<gl<!C∠
R ujvg/! ! hg<gr<gt<! PQ, QR, RP <K! hii<k<K! PQ + QR, QR + PR, qzqVf<K? PQ > QR
hml< 6.22
fQtr<gtqe<!Gl</
C + CA
ABC x<jx!
Bl<!
hml< 6.23
g{<Mhqcg<gUl</! ! Ogi{r<gtqe<! ntUgjt! yh<hqm<Mh<! hii<k<K! nkqz<! lqgh<! ohiqb!Ogi{l<! wK! we<hkjeg<! g{<mxqbUl</! ! -jkh<! Ohie<Ox! hg<gr<gtqe<! ntUgjt!yh<hqm<Mh<hii<k<K!nkqz<!lqgh<ohiqb!hg<gl<!wK!we<hkjeg<!g{<mxqbUl</!!-kqzqVf<K!ohiqb! Ogi{k<kqx<G! wkqOvBt<t! hg<gl<! ohiqbK! we! nxqbzil</! ! Ogi{! ntU!nkqglig!dt<tjk?!ohiqbOgi{l<!weUl<?!hg<g!ntU!nkqglig!dt<tjk?!ohiqb!hg<gl<!weUl<!ogit<tzil</!!-f<k!d{<jljbh<!hqe<uVl<!h{<hig!wPkzil</!!!
h{<H!8: yV!Lg<Ogi{k<kqz<?!lqgh<ohiqb!hg<gk<kqx<G!wkqiqZt<t!Ogi{l<!lqgh<ohiqbK/ !!
Gxqh<H: l we<x! Ogim<jmg<! gVKg/! hml<! 6.24! -z<!dt<tuiX! l e<! lQK! njlbik P we<x! Ht<tqjb!wMk<Kg<ogit<g/! ! P bqzqVf<K! OgiM! ! l g<G!osr<Gk<Kg<OgiM! PL ujvg/ L nz<zik! OuoxiV!Ht<tq!M J!l!e<!lQK!wMk<Kg<!ogit<g/!hg<gl<!PM J!-j{g<gUl</!Lg<Ogi{l< PLM yV osr<Ogi{!!
Lg<Ogi{liGl</!Woeeqz<!hg<gl< PL,!l g<G!osr<Gk<kiGl</weOu? m = 90°. m + mPLM∠ PLM∠ LMP∠ + m LPM∠ = 90° + m LMP∠ + m LPM∠ = 180° nz<zK m LMP∠ ∴ m LMP∠ < 90°, m LPM∠ < 90°, m LMP∠ = 90° −nkiuK?!Ogi{l< LMP∠ Bl< LPM∠ Bl<!GXr<Ogi{r<gweOu? NeK Lg<Ogi{l< PLM!-z<!lqgh<!ohiqb!lqgh<! ohiqb! hg<gl<A! nkiuK?! PL < PM, M we<hK! OHt<tq/!!weOu!P!bqzqVf<K!ujvbh<hMl<!osr<Gk<Kg<OgiMujvbh<hMl<! lx<x! Ogim<Mk<K{<Mgjt!uqm! lqgg<! Gjxu
LM < PM . lqgh<ohiqb! hg<gligqb!PM!J?!osr<Ogi{!we<gqOxil</!
PLM∠
6.1.4 si<usl!Lg<Ogi{r<gt<
!ucuqbjzh<! hx<xq! nxqBl<ohiPK?! si<usl! Lg
Ou{<cBt<tK/!weOu!nux<xqe<!sqz!Lg<gqb!d{<jlgj!
osbz<: -V!oux<Xk<!kit<gjt!wMk<Kg<ogit<tUl</! !nkit<! ye<xqjeh<! ohiVk<kUl</! yV!kitqe<! lQK! yV!Lgkitqz<! njkh<Ohie<x! yk<k! yV!Lg<Ogi{l<!njlf<kqVhcul<*/! -u<uqV!Lg<Ogi{r<gjtBl<! si<usl!Lg<Ogi{Lg<Ogi{r<gtqe<!slhg<gr<gjtBl<?!sl!Ogi{r<gjtBl<!hm<m! slhg<gr<gt<! yk<k! hg<gr<gt<! weUl<?! slOgi{r<g%xh<hMl</! ! yV! Lg<Ogi{k<kqZt<t! njek<K! hg<gr<g
146
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hml< 6.24
!
180°!we<hkiz<? + m LPM∠ = 90°
m LPM∠ . t<!lx<Xl<!fqvh<Hg<Ogi{r<gt</!
Ogi{l</!!weOu!hg<gl< PM gim<ce<! lQKt<t! WkiuK! yV!
PL, P bqzqVf<K!OgiM!l!g<G!ie! fQtLjmbkiGl</! -r<G?!
Lg<Ogi{l<!PLM e<!gI{l<!
<Ogi{r<gjt! ncg<gc! gVk!th<!hx<xq!okiqf<Kogit<Ouil</
ux<xqx<gqjmOb!yV!jlbs<Sk<!<Ogi{l<! ujvbUl</! !nMk<k!h<hjkg<! gi{zil<! )jlbs<Sh<!r<gt<! weg<!%xzil</! -u<uqV!ohiVk<kzil</!ohiVk<kh<hm<m!!t<! yk<k! Ogi{r<gt<! weUl<!Tl<?! Ogi{r<gTl<! lx<oxiV!
Lg<Ogi{k<kqe<! yk<k! hg<gr<gTg<Gl<?! Ogi{r<gTg<Gl<! sllieiz<?!nl<Lg<Ogi{r<gjt!si<usl!Lg<Ogi{r<gt<!we<xjph<Ohil</ ! !!!!!!
wMk<Kg<gim
-kqz<! AB Lg<Ogi{l<!
∆ABC ≡ ∆DXsi<usl
nxqf<K! oLg<Ogi{rogiMg<gh<hmnxqf<Kt<Ot
(i) -)h.Ogi.
(ii) &e(iii) -V
gQp<g<gi[lnl<Lg<Ogi
(i) (ii) (iii)
osbz<: AB ujvbUl</!wMk<K!nux
ohiVk<k!L
hii<g<gUl<). ds<sq A e<!lQKl<?!ds<s
AC Ngqb!Ogi{l<!∠A
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hml< 6.25
<mig?! hml<! 6.25 z<! dt<t! Lg<Ogi{r<gt< ABC, DPX Jg<! gVKg/!!!!!!! = DX, BC = PX, CA = PD; DAPCXB ∠=∠∠=∠∠=∠ ,, we! -Vh<hkiz<!
ABC Bl<! Lg<Ogi{l< DXP Bl<! si<usl! Lg<Ogi{r<gtiGl</! ! -kje P!we!wPkzil</ ! Lg<Ogi{r<gjth<! ohiXk<K!NX! yk<k! sle<hiMgt<! dt<tK! we<hjk!git<g/! lXkjzbig?! NX! yk<k! sle<hiMgt<! ogiMg<gh<hm<cVf<kiz<!<gt<! si<uslliGl</! hqe<uVl<! &e<X! okiGh<Hgtqz<! WOkEl<! ye<X!<cVf<kiz<! yV! Lg<Ogi{l<! ujvbzil<! we! Lf<jkb! uGh<Hgtqz<!il</ V! hg<gr<gtqe<! ntUgt<?! nux<xqx<gqjmOb! dt<t! Ogi{k<kqe<! ntU!!!!!!h!ogit<jg*/ <X!hg<gr<gtqe<!ntUgt<!)h.h.h!ogit<jg*/ !Ogi{!ntUgt<?!yV!hg<gk<kqe<!ntU!)Ogi.h.Ogi!ogit<jg*/ <!ohiVk<kr<gtqz<!WOkEl<!ye<xqje!-V!Lg<Ogi{r<gTg<G!fqI{bqk<kiz<?!{r<gjt!si<usll<!weg<!%xzil</ hg<gl<!.!Ogi{l<!.!hg<gl<! Ogi{l<!.!hg<gl<!.!Ogi{l<! hg<gl<!.!hg<gl<!.!hg<gl<!
= PQ, PA ∠=∠ , AC = PR we!-Vg<GliX!-V!Lg<Ogi{r<gt<!ABC, PQR nux<jxg<! gk<kiqk<K!
<jx!ye<xe<!lQK!ye<xigh<!
bx<sqg<gUl<! (hml< 6.26 Jh<!-l<Lbx<sqbqz<?! ds<sq P, lQKl<?!ds<sq Q, ds<sq B e<!q R, ds<sq C!e<!lQKl<?!AB , hg<gr<gtiz<! dt<tmg<gqb!
NeK PQ, PR Ngqb!!
1
hml< 6.26
47
hg<gr<gtiz<! dt<tmg<gqb! Ogi{l<! P∠ BmEl<! lqgs<! siqbigh<! ohiVf<Kujk!dx<XOfig<gzil</!!-kqzqVf<K!hqe<uVl<!h{<hqje!nxqgqOxil</ h{<H 9: yV!Lg<Ogi{k<kqe<!-V!hg<gr<gTl<!nju!dt<tmg<gqb!Ogi{Ll<?!lx<oxiV!Lg<Ogi{k<kqe<! -V! hg<gr<gTg<Gl<! nju! dt<tmg<gqb! Ogi{k<kqx<Gl<! sllieiz<?!nu<uqV!Lg<Ogi{r<gTl<!si<uslliGl</ Gxqh<H: -f<k!h{<hqje!si<usl!Lg<Ogi{k<kqe<!hg<gl<!.!Ogi{l<!.!hg<gl<!nch<hjmk<!kk<Kul<!nz<zK!SVg<glig!h.Ogi.h!nch<hjmk<!kk<Kul<!weg<!%xzil</ osbz<: RCQRBCQB ∠=∠=∠=∠ ,, we!-Vg<GliX!-V!Lg<Ogi{r<gt<!ABC, PQR ujvbUl</! nux<jxg<! gk<kiqk<okMk<K! ye<xe<lQokie<xig! ohiVk<kqh<hiIg<jgbqz<?!njubqv{<Ml<!siqbigh<!ohiVf<Kgqe<xe!(hml<!6.27 Jh<!hii<g<gUl<).
!-s<osbzqzqVf<K!gqj
Okx<xl< 8: yV! Lglx<oxiV! Lg<Ogi{sllieiz<?!nf<k!-V!Gxqh<H: -f<k!h{<hqjkk<Kul<!nz<zK!Og!osbz<: ∠A = ∠D, ∠ujvbUl< (hml<!6.28 ABC yV!Lg<Ogi{l
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hml< 6.27
mg<Gl<!nch<hjmk<!kk<Kuk<kqje!yV!Okx<xligg<!%Xgqe<Oxil</
<Ogi{k<kqe<! -V! Ogi{r<gTl<! nux<xiz<! -j{f<k! hg<gLl<!k<kqe<! -V! Ogi{r<gTg<Gl<! nux<xiz<! -j{f<k! hg<gk<kqx<Gl<!!Lg<Ogi{r<gTl<!si<usl!Lg<Ogi{r<gtiGl</
e!si<usl!Lg<Ogi{k<kqe<!Ogi{l<!.!hg<gl<!.!Ogi{l<!nch<hjmk<!i.h.Ogi!nch<hjmk<!kk<Kul<!weg<!%xzil</!!
B = ∠E, BC = EF we!-Vg<GliX!-V!Lg<Ogi{r<gt<!ABC, DEF Jh<!hii<g<gUl<).
hml< 6.28</!weOu =∠+∠+∠ CBA 180°. (1)
148
DEF yV!Lg<Ogi{l</!weOu =∠+∠+∠ FED 180° Neiz<? ∠D = ∠A, ∠B = ∠E ∴ =∠+∠+∠ FBA 180° (2) (1), (2)!e<!hc? =∠+∠+∠ CBA FBA ∠+∠+∠ .
∴ FC ∠=∠ . weOu!Lg<Ogi{l< ABC, DEF z< B∠ , C∠ Bl<!nkEjmb!hg<gl< BC Bl<?! E∠ , x<Gl<!
nkx<Giqb! hg<gl<!EF x<Gl<! sllig!-Vh<hkiz<! Ogi.h.Ogi! h{<hqe<hc!∆ABC ≡ ∆DEF. -kqzqVf<K!gQp<g<g{<m!Okx<xk<jk!nxqbzil</
F∠
Okx<xl< 9: -V! Lg<Ogi{r<gtqz<! ye<xqe<! -V! Ogi{r<gTl<?! nke<! WkiuK! yV!hg<gLl<?! lx<oxiV! Lg<Ogi{k<kqe<! -V! Ogi{r<gTg<Gl<?! yk<k! hg<gk<kqx<Gl<!sloleqz<?!nf<k!-V!Lg<Ogi{r<gTl<!si<uslliGl</ !
Gxqh<H: si<usll<!hx<xqb!-f<k!kk<Kuk<kqje!Ogi{l<! .!Ogi{l<! .!hg<gl<!nch<hjmk<!kk<Kul<!nz<zK!!SVg<glig!Ogi.Ogi.h!nch<hjmk<!kk<Kul<!weg<!%xzil</ !
osbz<: BC = EF, CA = FD, AB = DE we! -Vg<GliX! Lg<Ogi{r<gt<! ABC, DEF Ngqbux<jx! ujvbUl< (hml< 6.29Jh<! hii<g<gUl<). -kqz<! Lg<Ogi{l<! DEF J oum<cobMk<K!Lg<Ogi{l<!ABC !e<!lQK!jug<g?!njubqv{<Ml<!lqgs<!siqbigh< !
ohiVf<Kgqe<xe
njlujk!nxnxqbzil</ !Okx<xl< 10: yhg<gr<gTg<Gs Gxqh<H: Olx<%%xh<hMl<A!SVg!
h.h.Ogi!ohiVf<Kli!we
osbz<: ABC wehii<g<gUl<). XY
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hml< 6.29
/! -f<fqjzbqz<?! D, A bqe<! lQKl<?! E, B bqe<! lQKl<?! F, C bqe<! lQKl<!qbzil</!!weOu!∆ABC ≡ ∆DEF. -s<osbzqzqVf<K!hqe<uVl<!Okx<xk<jk!
V! Lg<Ogi{k<kqe<! &e<X! hg<gr<gt<! lx<oxiV! Lg<Ogi{k<kqe<! &e<X!<!sloleqz<?!nu<uqV!Lg<Ogi{r<gTl<!si<usll<!NGl</!
xqb! kk<Kul<! hg<gl<! .! hg<gl<! .! hg<gl<! nch<hjmk<! kk<Kul<! weg<!<glig?!-kje!h.h.h!nch<hjmk<!kk<Kul<!weg<!%xzil</
kk<Kul<!nz<zK!Ogi.Ogi.Ogi!ogit<jg!si<usl!Lg<Ogi{r<gTg<Gh<!!-h<ohiPK!Osikqh<Ohil</!
<x!Lg<Ogi{l<!ujvg/!BC g<G!-j{big!OgiM!XY ujvg (hml< 6.30Jh<! Bl<! AB Bl<!sf<kqg<Gl<!Ht<tqjb!D weg<!Gxqg<gUl<A!XY Bl<! AC Bl<!
149
sf<kqg<Gl<!Ht<tqjb!E we!Gxqg<gUl</!!hg<gl<!AB Bl<?!hg<gl<!AC Bl<!-j{OgiMgtie!BC, XYgTg<G!GXg<G!oum<cgtiGl</!!weOu! CEBD ∠=∠∠=∠ , (yk<k!Ogi{r<gt<). !!!-r<G?!Lg<Ogi{r<gt< ABCnu<uqV! Lg<Ogi{r<gTlsllz<z/! weOu?! Ogi.Ognch<hjmk<!kk<KulibqVg!
osbz<: OhiKlie! ntuqxOfIg<Ogim<Mk<!K{<M!AB Jogit<g/! B jb! jlblignu<um<ml<!AX jb!-V!Htweg<!Gxqg<gUl< (hml<!6.31 !!!Lg<Ogi{r<gt< ABC? ABDweOu ∆ABC NeK! ∆ABLg<Ogi{r<gt<! ! si<uslnxqbzil</ !osbz<: BC = QR = a nzGjlbh<Ht<tqjb X weUl<?
jlbbr<gtigUl<? 2a nzG
jlbr<gtiigUl<?!b(< a) n
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hml< 6.30
, ADE gTg<G! Ogi.Ogi.Ogi! ohiVk<kl<!dt<tK/!Neiz<?!<! si<usl! Lg<Ogi{r<gt<! nz<z/! Woeeqz<?! yk<k! hg<gr<gt<!i.Ogi! ohiVk<klieK! Lg<Ogi{r<gt<! sIusllibqVh<hkx<G!<giK!we!nxqgqOxil</
<G! fQtlie! yV! OgiM! AX ujvg/! a nzGgt<! fQtLjmb!?!Ogi{l<!∠BAX yV!Gxqh<hqm<m!ntU!-Vg<GliX!ujvf<K!Ul<?! b (< a) nzG! NvligUl<! djmb! um<ml<! ujvg/!<tqgtqz<!oum<Mujk!fil<!gi{<gqOxil</!-h<Ht<tqgjt!C, D
Jh<!hii<g<gUl<).
hml< 6.31gTg<G! h.h.Ogi! ohiVk<kl<!dt<tK/! !Neiz<! AC ≠ AD. D g<Gs<! sIusllz<z/ -s<osbzqzqVf<K! h.h.Ogi! ohiVk<kl<?!lig! -Vh<hkx<G! nch<hjmk<! kk<Kulig! njlbiK! we!
gt<!-Vg<GliX!Ogim<M!K{<Mgt<!BC, QR ujvg/! BC bqe<!QR -e<!jlbh<Ht<tqjb!Y weUl<!Gxqg<gUl</!X, Y -ux<jx!
gt<!NvligUl<! ogi{<M!um<mr<gt<!ujvg/!B, Q -ux<jx!
zGgt< NvligUl<!ogi{<M!uqx<gt<!ujvg/!!-f<k!uqx<gt<!
150
Wx<geOu! ujvf<k! um<mr<gjt! oum<Ml<! Ht<tqgjt! LjxOb! A, P weg<! Gxqg<gUl< (hml< 6.32 Jh<!hii<g<gUl<). AB, AC, PQ, PR jb!-j{g<gUl</! AC, PR fQtr<gjt! !!!ntg<gUl</! AC =gQp<g<gi[l<!Okx<x!!Okx<xl< 11: yV!osr<Ogi{! Lg<O-Vh<hqe<?!nu<uqV!Gxqh<H: -k<Okx<xlslk<Kuk<kqx<gie!SVg<glig!os.g.h 6.1.5 -j{gvk<k fie<G! hgwkqi<hg<gr<gt<!-j !!hml<! 6.33! -z< -h<hmk<kqz< PQ || OlZl< PS || QR. h{<H 1: YI!-j{fq'h{l<: ABCD -j{g<gUl< (hmlNgqbux<jxg<!gVCD!Ngqbux<xqx<GmAD, BC Ngqbux<x
m BDABD ∠=∠
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hml< 6.32
PR we<hjk!nxqbzil</!!weOu!∆ABC ≡ ∆PQR . -s<osbzqzqVf<K!k<kqjeg<!gi{<gqOxil</
osr<Ogi{! Lg<Ogi{k<kqe<! gi<{l<?! WOkEl<! yV! hg<gl<?! lx<oxiV!gi{k<kqe<! gI{l<?! yV! hg<gl<! Ngqbux<xqx<G! LjxOb! sllig!!Lg<Ogi{r<gTl<!si<uslliGl</!
<!kVl<!nch<hjmk<!kk<Kuk<kqje!osr<Ogi{!Lg<Ogi{r<gtqe<!sIu!osr<Ogi{l<! .! gi<{l<! .! hg<gl<! nch<hjmk<! kk<Kul<! nz<zK!!nch<hjmk<!kk<Kul<!we!wPkh<hMgqxK/
qe<!h{<Hgt< <gr<gtiz<! njmhMl<! dVul<! fix<gvl<! we<hkje! nxqOuil</!!{big!dt<t!fix<gvl<!-j{gvl<!weh<hMl</!!
hml<!6.33
ABCD yV! fix<gvl</! hml<! 6.34SR (PQ NeK SR g<G! -j{bi
gvk<kqz<?!wkqi<hg<gr<gt<!sl!fQtLYI! -j{gvl<! we<g/ BD j<! 6.35Jh<! hii<g<gUl<). ∆ABD, ∆BKg/!-r<G!AB || CD, BD NeK!A!GXg<Goum<cbig!dt<tK/!weO
-u<uiOx? AD || BC,! BD Neqx<G!GXg<Goum<c!we<hkiz<?!!.C
151
hml< 6.34
. -z<! PQRS yV! -j{gvliGl</!!Gl<! we<hjk! -u<uiX! Gxqh<Ohil<);
jmbju/
hml< 6.35
b!DC
B, u?!K
m hg<gl< BD, Lg<Ogi{r<gt<! ABD, BDCg<G! ohiKuig! dt<tK/!!weOu?!Ogi.Ogi.h<!ogit<jgbqe<hc ∆ABD ≡ ∆BDC. ∴ yk<k!hg<gr<gt<!sll</!!weOu?!AB = CD lx<Xl< AD = BC.
.m DBCADB ∠=∠
!h{<H 2: YI!-j{gvk<kqz<?!wkqi<Ogi{r<gt<<!slntUjmbju/ fq'h{l<: ABCD YI!-j{gvl<!we<g/ BD jb!-j{g<gUl<!(hml< 6.36!Jh<!hiIg<gUl<). ∆ABD, ∆BDC Ngqbux<Xt< AB || DC, BD NeK! AB, DC! Ngqbux<Xg<G! yV!GXg<Goum<c!we<hkiz<?!m .m BDCABD ∠=∠ OlZl<!!AD || BC, BD NeK AD, BC Ngqbux<Xg<G yV!GXg<Goum<c!we<hkiz<?!m .m CBDADB ∠=∠ ∴ m DBCABDABC ∠+∠=∠ mm
= m ADBBDC ∠+∠ m = m .ADC∠
-u<uiOx m BCDBAD ∠=∠ m we!nxqbzil</! h{<H 3: -j{gvk<kqe<!&jzuqm<mr<gt<!ye<jxobie<X!slfq'h{l<: ABCD YI! -j{gvl</! AC, BD we<he!&jzuqm<mr<gt</! -ju! oum<cg<! ogit<Tl<! Ht<tqjb!M !we<g/!-r<G?!∠BAM = ∠DCM, ∠ABM = ∠ CDM, AB = CD.! weOu?! Ogi.h.Ogi! ogit<jgbqe<! hc?!!∆AMB ≡ ∆CMD (hml< 6.37Jh<!hii<g<gUl<). ∴AM = CM, BM = DM. weOu!&jzuqm<mr<gt<!slg<%xqMl</! h{<H 4: yV! fix<gvk<kqe<! wkqi<hg<gr<gt<! sl! ntU! o-j{gvliGl</ fq'h{l<: ABCD we<x!fix<gvk<kqz< AB = CD, AD = BC. AC jb!-j{g<gUl</ Lg<Ogi{r<gt<!ACB, ADC!Ngqbux<Xt<!h.h.h!ogit<jgbqe<hc ∆ABC ≡ ∆CDA (hml< 6.38Jh<!hii<g<gUl<). hqe<H?!m m,m ACDBAC ∠=∠ .m ACBCAD ∠=∠ ∴ AB || CD ; OlZl< AD || BC. weOu!ABCD YI!-j{gvl</ h{<H 5: yV! fix<gvk<kqe<! wkqi<Ogi{r<gt<! sl!ntU! o-j{gvliGl</ fq'h{l<: ABCD we<El<! fix<gvk<kqz< (hml< 6.39 Jh<!hii<g<gUl<). m BCDBAD ∠=∠ m , m ADCABC ∠=∠ m . BD jb! -j{g<gUl</ ABD, CDB! Ngqb!Lg<Ogi{r<gjtg<!gVKg/!-r<G?!m∠1 + m∠2 + m∠BCD = 180°; m∠3 + m∠4 + m∠BAD = 180°. ∴ m∠1 + m∠2 + m∠BCD = m∠3 + m∠4 + m∠BAD, Woeeqz<?!m∠BCD = m∠BAD .
152
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hml< 6.36
g<%xqMl</
hml< 6.37
gi{<cVf<kiz<! nf<k! fix<gvl<!
hml< 6.38
gi{<cVf<kiz<?! nf<k! fix<gvl<!
hml< 6.39
∴ m∠1 + m∠2 = m∠3 + m∠4 (1) Neiz<? ADCABC ∠=∠+∠∠=∠+∠ m4m2m,m3m1m ∴ 4m2m3m1m ∠+∠=∠+∠ nkiuK? 3m4m2m1m ∠−∠=∠−∠ 2) (1) + (2) ⇒ 2m∠1 = 2m∠4 ⇒ m∠1 = m∠4. ∴ AD || BC. (1) − (2) ⇒ 2m∠2 = 2m∠3 ⇒ m∠2 = m∠3. ∴ AB || CD. weOu?!ABCD YI!-j{gvl</ h{<H 6: yV!fix<gvk<kqe<!&jzuqm<mr<gt<!ye<jxobie<X!slg<%xqMoleqz<?!nf<fix<gvl<!YI!-j{gvliGl</ fq'h{l<: ABCD yV!fix<gvl</ AC, BD &jzuqm<mr<gt</ AC Bl< BD Bl<!Ht<tq M -z<!oum<cg<ogit<gqe<xe!(hml< 6.40Jh<!hii<g<gUl<). M NeK!AC, BD gtqe<!fMh<Ht<tq/!!∴AM = CM, BM = DM. Gk<okkqI!Ogi{r<gt<!sll<!we<hkiz<?!
.mm,mm BMCAMDCMDAMB ∠=∠∠=∠ h.Ogi.h!ogit<jgbqe<hc?!!∆AMB ≡ ∆CMD, ∆AMD ≡ ∆CMB. weOu!∠BAM =∠DCMAB, CD gTg<G!AC yV!GXg<Goum<c/!(1) e<!hc?!ye<Xu ∴AB || CD. -jkh<OhizOu? AD || BC. weOu ABCD YI!- Okx<xl< 12: yV! Osic! wkqi<hg<gr<gt<! -j{bigUl<?! s-j{gvliGl</ ogit<jg: ABCD we<x!fix<gvk<kqz< AB || CD, AB = CD. fq'hqg<g: ABCD YI!-j{gvl</! ujvkz<: ACjb!-j{g<gUl<!(hml< 6.41Jh<!hii<g<gUl<). fq'h{l<: ∆ABC, ∆ADC Ngqbux<Xt<?! (i) AB = CD (ogit<jg)A! (ii) AC ohiKA (iii) AB || CD, AB g<Gl< CD g<Gl< AC GXg<Goum<c ⇒ m∠BAC = m∠ACD!!)ye<Xuqm<m!Ogi{r<gt<!sl∴ h.Ogi.h!ogit<jgbqe<!hc?!∆ABC ≡ ∆ADC. ∴ Lg<Ogi{r<gtqz<!yk<khg<gr<gt<? yk<k!Ogi{r<gt<!sl∴ AD = BC, m∠DAC = m∠ACB. ∴ AD || BC. weOu ABCD h{<H 7: &e<Xl<?! &e<xqx<Gl<! Olx<hm<m! -j{OgiMlQOkx<hMk<kh<hMl<! oum<Mk<K{<Mgt<! slfQtLjmGXg<Goum<cbqe<! lQKl<! nf<k! -j{! OgiMgt<!slfQtLjmbeuiGl</!fq'h{l<: OgiMgt< l1, l2, l3 we<he!ye<Xg<ogie<X!-j{!OgiMgtiGl</ PQ, XY Ngqbju l1, l2, l3! g<G! GXg<Goum<cgtiGl</!AC = CE we<g/! BD = DF we! fq'hqg<g Ou{<Ml</!AG || BD NgUl< CH || DF!NgUl<!ujvg (hml< 6.42 Jh<!hiIg<gUl<).
153
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hml< 6.40
, ∠ABM = ∠CDM (1) qm<m!dm<Ogi{r<gt<!sll</!j{gvl</
lligUl<! dt<t! fix<gvl<! YI!
hml< 6.41
liGl<*/!
l</! YI -j{gvl</
gtiz<! yV! GXg<Goum<cbqe<!beoueqz<?! OuX! wf<k!d{<miGl<! oum<Mk<K{<Mgt<!
hml< 6.42
AGDB, CHFD Ngqbju!-j{gvl<!we!nxqgqOxil</ ∴ AG = BD, CH = DF (1) ACG , CEH!Ngqb!Lg<Ogi{r<gtqz<!!CE = AC, m∠GAC = m∠HCE (yk<k!Ogi{r<gt<), m∠ACG = m∠CEH (yk<k!Ogi{r<gt<). weOu!Ogi.h.Ogi!ogit<jgbqe<hc?!!∆ACG ≡ ∆CEH. ∴ AG = CH. (2) (1), (2) e<!hc!BD = DF. h{<H 8: yV! Lg<Ogi{k<kqz<?! -Vhg<gr<gtqe<! fMh<Ht<tqgjts<! Osi<g<Gl<! OgiM!&e<xiuK!hg<gk<kqx<G!-j{bigUl<?!nke<!ntuqz<!hikqbigUl<!-Vg<Gl</ fq'h{l<: Lg<Ogi{l< ABC -z<? D, E LjxOb AB, AC Ngqb! hg<gr<gtqe<!fMh<Ht<tqgtiGl</!fil<!fq'hqg<g!Ou{<cbK?!
DE || BC, DE = ).(21 BC
CF || BD ujvg/! -K DE bqe<! fQm<sqjb F! -z<! sf<kqg<Gl<! (hml< 6.43Jh<! hii<g<gUl<). Lg<Ogi{l< ADE, CFE!Ngqbux<Xt<?!AD || CF, AC GXg<Goum<c/ ∴ m∠DAE = m∠ECF. AD || CF, DF GXg<Goum<c/!∴ m∠ADE = m∠CFE. OlZl< E NeK AC bqe<!fMh<Ht<tq!Nekiz<? AE = EC. weOu!Ogi.Ogi.h!ogit<jgbqe<hc ∆ADE ≡ ∆CFE. ∴ AD = CF OlZl< DE = EF. Neiz< D, AB bqe<!fMh<Ht<tq!we<hkiz<? BD = AD. ∴ BD = CF. Wx<geOu BD || CF.
∴ BCFD YI -j{gvliGl</ ∴ DF = BC lx<Xl< DF || BC n.K DE || BC. DE + EF = BC n.K DE + DE = BC n.K 2DE = BC.
weOu DE || BC lx<Xl<! DE = .BC)(21
h{<H 9: yV!Lg<Ogi{k<kqz<?! yV! hg<gk<kqe<! fMh<Ht<tq!-j{big!ujvbh<hMl<!OgiM?!&e<xiuK!hg<gk<jk!slgfq'h{l<: ABC we<x!Lg<Ogi{k<kqz< D, AB!bqe<!fMh<Ht<tq! we<g/!D upqbig! BC g<G!-j{big DE ujvg/! nK AC jb E z<! oum<mm<Ml<! (hml< 6.44 Jh<!hii<g<gUl<). E NeK!AC bqe<!fMh<Ht<tq!we!fq'hqk<kz<!Ou{<Ml<. -r<G!BC || DE, BC, DE gtqe<!GXg<Goum<c AB NGl</ ∴ m∠ADE = m∠ABC, (1) AD || CF, AD,CF gtqe<!GXg<Goum<c!DF NGl</ ∴ m∠ADE = m∠EFC lx<Xl<! (2) m∠AED = m∠BCE. (3) (1), (2)!e<!hc m∠ABC = m∠EFC. (4) AD || CF, AD, CF gtqe< GXg<Goum<c!AC NGl</ ∴ m∠BAC = m∠ACF. (5) -h<ohiPK?!m∠BCF = m∠BCE + m∠ECF
154
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hml< 6.43
upqOb! lx<oxiV! hg<gk<kqx<G!<%xqMl</
hml< 6.44
= m∠AED + ∠ACF, (3) e<hc!= m∠AED + m∠DAE, (5) e<hc = m∠BDE (outqg<Ogi{l< = dt<otkqi<!Ogi{r<gtqe<!%Mkz<). nkiuK m∠BCF = m∠BDE (6) (4), (6)e<!hc? BCFD YI!-j{gvliGl</ ∴BD = CF, BC = DF. Lg<Ogi{r<gt<! ADE, CFE Ngqbux<jxg<!gVKg/!-ux<xqz<?!CF = BD = AD we<hkiz<?!CF = AD. OlZl<?!m∠EFC = m∠ADE, m∠DAE = m∠BAC = m∠ACF = m∠ECF. ∴ Ogi.h.Ogi!ogit<jgbqe<hc, ∆ADE ≡ ∆CFE. ∴ DE = EF, AE = EC. ∴ E, AC -e<!fMh<Ht<tqbiGl</ 6.1.6 yV!Ht<tq!upqOb!osz<Zl<!OgiMgt<
&e<X!nz<zK!&e<Xg<G!Olx<hm<m!OgiMgt<!P!we<x!yV!Ht<tq!upqOb!osz<Zl<!weqz<?! ng<OgiMgt<! yV! Ht<tqupqg<! OgiMgt<! weUl<! nh<Ht<tq P J! ng<OgiMgt<!sf<kqg<Gl<!Ht<tq!weUl<!njpg<gh<hMl</ !
osbz< : Lg<Ogi{l< ABC!ujvg/!!hg<gl<!BC, CA Ngqbux<Xg<G! LjxOb! jlbg<!Gk<Kg<OgiMgt<!DX, EY ujvg (hml< 6.45 Jh<! hii<g<gUl<). nju!oum<Ml<Ht<tqjb S!we<g/! hg<gl<! AB bqe<! jlbg<Gk<Kg<OgiM!FZ ujvg. FZ NeK S e<!upqOb osz<ujk!nxqbzil</!!-s<osbzqzqVf<K!gQp<g<gi[l<!Okx<xk<jk!nxqgqe<Oxil</ !
Okx<xl< 13: Lg<Ogi{k<kqe<! hg<gr<gtqe<! jlbg<Gk<Kosz<Zl</ !
Gxqh<H: yV!Lg<Ogi{k<kqe<!&e<X!hg<gr<gtqe<!jlbgnl<Lg<Ogi{k<kqe<!Sx<Xum<m!jlbl<!weh<hMl</!!-K S!!
osbz<: ABC!we<x!Lg<Ogi{k<jkg<!gVKg/!-kqz<!SxSA, SB, SC!Ngqbux<jx!-j{g<gUl</!!SA, SB, SC!Nhii<g<gUl</!filxquK?!SA = SB = SC/ S jb!jlbligUyV! um<ml<! ujvg/! -u<um<ml<?! Lg<Ogi{k<kqe<! &eosz<ujkg<! gi{zil</! ! nu<um<ml<! Lg<Ogi{k<kqe<!
SA = (SB = SC) Sx<Xum<m!Nvl<!weh<hMl</ !!
155
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hml< 6.45
OgiMgt<! yV! Ht<tq! upqOb!
<Gk<Kg<OgiMgt<!osz<Zl<!Ht<tq!
we<x!wPk<kiz<!Gxqg<gh<hMl</
<Xum<m!jlbl< S jbg<!gi{<g/ gqbux<xqe<!fQtr<gjt!ntf<K!
l<? SA ju!NvligUl<!ogi{<M!<X! ds<sqh<Ht<tqgtqe<! upqOb!Sx<Xum<ml<! weh<hMl<A! Nvl<
osbz<: ABC!we<x!Lg<Ogi{l<!ujvg/!!hg<gl<!BC g<G!osr<Gk<kig!AD ujvg/!!nK!BC!jb D z< sf<kqg<gm<Ml</!!hg<gl<!CA Ug<G!osr<Gk<kig!BE!ujvg/!!nK!CA jb Ez<! sf<kqg<gm<Ml</! AD Bl<! BE l<!sf<kqg<Gl<! Ht<tqjb! H weg<!
Gxqg<gUl</! ! AB e<! osr<Gk<K! CF ujvg/! nK! AB jb! F z<!
sf<kqg<gm<Ml< (hml< 6.46Jh<! hii<g<gUl<). CF NeK H e<! upqOb! osz<ujk!nxqbzil</!!osr<Gk<K!OgiMgt<!AD, BE, CF we<he! Lg<Ogi{k<kqe!osr<OgiMgt<!weh<hMl</!!!!-s<osbzqzqVf<K!gQp<g<g{<m!Okx<xk<jk!nxqbzil</ !Okx<xl< 14: yV Lg<Ogi{k<kqe<!osr<OgiMgt<!yV!Ht Gxqh<H: yV!Lg<Ogi{k<kqe<! osr<OgiMgt<! sf<kqg<Gosr<Ogim<M!jlbl<!weh<hMl</!!-K H!we<x!wPk<ki!
osbz<: Lg<Ogi{l< ABC! ujvg/! ! Lg<Ogi{k<kqzOgi{l< ∠A, Ogi{l<!∠B!Ngqbux<xqe<!sloum<cgtujvg/ -jugt<! sf<kqg<Gl<! Ht<tqjb! I wegGxqg<gUl</! !∠C e<! sloum<c! ujvg/! ! -K! I!eupqOb! osz<ujk! nxqbzil< (hml< 6.47Jhhii<g<gUl<). -s<osbzqzqVf<K!gQp<g<gi[l<!Okx<xk<jknxqbzil</ Okx<xl< 15: yV!Lg<Ogi{k<kqe<?!&e<X!Ogi{r<gtqosz<Zl</ !Gxqh<H: yV!Lg<Ogi{k<kqz<!&e<X!Ogi{r<gtqe<! snl<Lg<Ogi{k<kqe<!dt<um<m!jlbl<!weh<hMl</!-K osbz<: I -zqVf<K! Lg<Ogi{l< ABC e<!hg<gr<gTg<G! osr<Gk<Kg<! OgiMgt<! ujvg/!!nux<jx! ntf<K! hii<g<gUl</! nju! slfQtl<!
ogi{<meuig! -Vh<hkjeg<! gi{zil</! I jb!jlbligUl<?! slntuie! osr<Gk<Kg<! Ogim<ce<!fQtk<jk! NvligUl<! ogi{<M! yV! um<ml<!
ujvbUl< (hml< 6.48Jh<! hii<g<gUl<). -u<um<ml<!Lg<Ogi{k<kqe<!&e<X!hg<gr<gjtBl<!!okim<Ms<!osz<ujkg<!gi{zil</!-kje!Lg<Ogi{k<kqe<!!
156
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hml< 6.46
<tq!upqOb!osz<gqe<xe/
l<! Ht<tqbieK?! nl<Lg<Ogi{k<kqe<!!z<!Gxqg<gh<hMl</
hml<!6.47!
<!
<!
<!
<!
<!
!
e<!sl!oum<cgt<!yV!Ht<tq!upqOb!
loum<cgt<! sf<kqg<Gl<! Ht<tqbieK!! I!we<x!wPk<kiz<!Gxqg<gh<hMl</
hml< 6.48
dt<um<ml<! we<xjpg<gqOxil</! -u<um<mk<kqe<! Nv! ntU! Lg<Ogi{l<! ABC! -e<!dt<um<m!Nvl<!weh<hMl</ osbz<: Lg<Ogi{l< ABC!ujvg/!!hg<gr<gt< BC,
CA, AB Ngqbux<xqe<!fMh<Ht<tqgjt!LjxOb!D,
E, F weg<! Gxqg<gUl</! ! AD jbBl< BE jbBl<!
-j{g<gUl</!-ju!oum<cg<!ogit<Tl<!Ht<tqjb
G! weg<!Gxqg<gUl</ CF!jb!ujvg/!nK!G e<!
upqOb!osz<ujk!nxqg (hml< 6.49Jh<!hii<g<gUl<).
OgiMgt< AD, BE, CF Ngqbju! Lg<Ogi{l<!
ABCbqe<!fMg<OgiMgt<!weh<hMl</ AG, GD, BG,
GE, CG, GF Ngqbux<xqe<!fQtr<gjt!ntf<K!hiIg<gUl</!
.12
===GFCG
GEBG
GDAG
-s<osbzqzqVf<K!gQp<g<gi[l<!Okx<xk<jk!nxqbzil</
Okx<xl< 16: yV Lg<Ogi{k<kqe<! fMg<OgiMgt<! yV! Htyu<ouiV!fMg<Ogim<cjeBl< 2:1!we<x!uqgqkk<kqz<!hqiqg<G!
Gxqh<H: yV Lg<Ogi{k<kqe<! fMg<OgiMgt<! sf<kqg<Gl<! HfMg<Ogim<M!jlbl<!weh<hMl</!!-K!G!we<x!wPk<kiz<!!
osbz<: AB we<x! Ogim<Mk<K{<jm! ujvg/! AB g<G!-j{big l we<x! OgiM ujvbUl<! (hml< 6.50Jh<!hii<g<gUl<).! l e<!lQK!Ht<tqgt<!C, D!Ngqbux<jx!AB = CD! we!-Vg<GliX!Gxqg<gUl</! -kqzqVf<K!ABCD YI -j{gvl<! we! nxqbzil</! l g<Gs<!osr<Gk<K AL ujvg/ AL J!ntf<K!hii<g<gUl</ -j{gvl<! ABCD e<!hvh<htU!= nch<hg<gl< × dbvl<!= OuX!-V!Ht<tqgt< P, Q Ngqbux<jx PQ = AB!we!-ABPQ!!YI!-j{gvl<!we!nxqbzil</!!!ABPQ!e<!hvh<htU!=!nch<hg<gl<!×!dbvl<!= AB × AL.!!-s<osbzqzqVf<K!gQp<g<gi[l<!Okx<xk<jk!nxqbzil</! Okx<xl< 17: yOv!ncbqe<lQKl<?!-V!-j{OgiMgTg<Glqjslhvh<Hjmbju/
157
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hml<!6.49!
filxquK?
<tq! upqOb! osz<Zl<A! -h<Ht<tq!
l</
t<tqbieK! nl<Lg<Ogi{k<kqe<!
Gxqg<gh<hMl</
hml< 6.50
AB × AL. Vg<GliX! l!e<!lQK!Gxqg<gUl</!!
!
mbqz<!njlBl<!-j{gvr<gt<!
osbz<: hml< 6.51 Jh<! hiIg<gUl<. AB we<x!Ogim<Mk<K{<jm! ujvg/! AB! g<G! -j{big l we<x Ogim<cje!ujvbUl</!l e<!lQK!Ht<tq!C!jbg<!Gxqg<g/!l! we<x! Ogim<Mg<G! A bqzqVf<K! AL! we<x!osr<Gk<KOgiM! ujvg. AL e< fQtk<jk! ntf<K!hii<g<gUl</ filxquK?! ABC! we<x! Lg<Ogi{k<kqe<!
hvh<htU!=! ×21
nch<hg<gl<!×!dbvl< = .21 ALAB ××
l!e<!lQK!lx<oxiV!Ht<tq!P jbg<!Gxqg<gUl</!!!
Lg<Ogi{l<!ABP -e<!hvh<htU!= ALAB ××21 weg<gi{<g
C, l e<!lQK!wr<G!njlf<kqVf<kiZl<!Lg<Ogi{r<gtqe<!hvh!-s<osbzqzqVf<K!gQp<g<gi[l<!Okx<xk<jk!nxqbzil</ Okx<xl< 18: yOv! ncbqe<! lQKl<?! -V! -j{OgiMgLg<Ogi{r<gt<!slhvh<Hjmbju/
hbqx<sq 6.1 hqe<uVl<!%x<Xgt< siqbi,!kuxi!weg<!%xUl</ 1. -V!-j{OgiMgjt!yV!GXg<Goum<c!oum<ceiz<?!y2. -V!-j{OgiMgjt!yV!GXg<Goum<c!oum<ceiz<?!y3. -V!-j{OgiMgjt!yV!GXg<Goum<c!oum<ceiz<?!G!!njlf<k!dm<Ogi{r<gt<!sll</ 6.2 kVg<g!iQkqbig!fq'hqg<g!Ou{<cb!Okx<xr<gt -Kujv! fil<! sqz! osbz<gjts<! osb<K!ogit<jggjtBl<?! Okx<xr<gjtBl<! siq! hiIk<Okil</!Lg<Ogi{r<gt<! Ngqbux<jxh<! hx<xqb! sqz! Okx<xr<gTg
uqkqkV!Ljxbqz<!(Method of Logical Reasoning)!fq'h{r Okx<xl< 19: yV! gkqi<! lx<oxiV! Ogim<ce<! lQK! fqx<GlienMk<kMk<k!Ogi{r<gtqe<!ntUgtqe<!%Mkz< 180°!g<Gs<!kvU: gkqi< PQ , XY!we<x!Ogim<ce<!lQK!njlf<Kt<tK/ fq'hqg<g: m∠QPX + m∠YPQ = 180°. njlh<H: XY g<G!osr<Gk<kig!PE ujvg. fq'h{l<: m∠QPX = m∠QPE + m∠EPX = m∠QPE + 90° (1) m∠YPQ = m∠YPE − m∠QPE = 90° − m∠QPE (2)
158
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hml<!6.51
qOxil</!Lg<Ogi{k<kqe<!ds<sq
<htUgt<!slole!nxqgqOxil</!!
Tg<Gl<! -jmbqz<! njlBl<!
e<Xuqm<m!Ogi{r<gt<!sll</!k<k!Ogi{r<gt<!sll</ Xg<Goum<cbqe<!yOv!Hxl<!!!
<
hiIk<K?! sqz! nch<hjmg<!-h<ohiPK?! Ogi{r<gt<?!
<G! kVg<g! vQkqbig?! nkiuK?!
<gjtk<!kv!Lx<hMOuil</
iz<?! nkeiz<! Wx<hMk<kh<hMl<!
sll</
hml< 6.52
(1) + (2) ⇒ m∠QPX + m∠YPQ = (m∠QPE + 90°) + (90° − m ∠QPE) = 180°. weOu!Okx<xl<!fq'hqg<gh<hm<mK/ Okx<xl< 20: -V!OgiMgt<!ye<jxobie<X!oum<cg<ogi{<miz<?!Gk<okkqi<!Ogi{r<gt<!sl!ntUt<tjubiGl</ kvU: OgiMgt< AB, CD Ht<tq O uqz<!oum<cg<ogit<gqe<xe!(hml< 6.53 Jh<!hii<g<gUl<). fq'hqg<g: m∠AOC = m∠BOD, m∠BOC = m∠AOD. fq'h{l<: gkqi< OB NeK OgiM CD!bqe<!lQK!fqx<gqe<xK/ ∴ m∠BOD + m∠BOC = 180°. (1)
gkqi< OC OgiM AB!bqe<!lQK!fqx<gqe<xK/ ∴ m∠BOC + m∠AOC = 180°. (2) (1), (2)!e<!hc m∠BOD + m∠BOC = m∠BOC + m∠AOC ∴ m∠BOD = m∠AOC. gkqi< OA, CD!bqe<!lQK!fqx<gqe<xK/ m∠AOC + m∠AOD = 180°. (3) (2), (3)!e<!hc?!m∠BOC + m∠AOC = m∠AOC + m∠AOD. ∴ m∠BOC = m∠AOD. weOu!Okx<xl<!fq'hqg<gh<hm<mK/ !Okx<xl< 21: yV!Lg<Ogi{k<kqz<!&e<X!Ogi{k<kqe<!%Mkz< 180°!NGl</ kvU: ABC!yV!Lg<Ogi{l<.
hml< 6.54
fq'hqg<g: ∠A + ∠B + ∠C = 180°. njlh<H: ds<sq!A e<!upqOb XY we<x!Ogim<cje?!BC!g<G!-j{big!ujvg!(hml<!6.54Jh<!hii<g<gUl<)/! fq'h{l<: XY || BC. -h<ohiPK?!XY, BC!gTg<G!!AB!GXg<Goum<c/ ∴ m∠XAB = m∠ABC (ye<Xuqm<m!Ogi{r<gt<) = ∠B. (1) XY, BC gTg<G AC GXg<Goum<c/ ∴ m∠YAC = m∠ACB (ye<Xuqm<m!Ogi{r<gt<) = ∠C. (2) OlZl< m∠BAC = m∠A. (3) (1) + (2) + (3) ⇒ m∠XAB + m∠YAC + m∠BAC = m∠B + m∠C + m∠A ⇒ (m∠XAB + m∠BAC) + m∠CAY = m∠A + m∠B + m∠C ⇒ m∠XAC + m∠CAY = m∠A + m∠B + m∠C ⇒ 180° = m∠A + m∠B + m∠C. weOu?!Okx<xl<!fq'hqg<gh<hm<mK/ Okx<xl< 22: yV! Lg<Ogi{k<kqz<?! slhg<gr<gTg<G! wkqOvBt<t! Ogi{r<gt<!slntUt<tju/! kvU: Lg<Ogi{l<!ABC -z< AB = AC. fq'hqg<g: ∠B = ∠C. njlh<H: BC!e<!jlbh<Ht<tqjb M weg<!Gxqk<K AM J!-j{g<g!(hml<!6.55 Jh<!hii<g<gUl<)/ fq'h{l<: Lg<Ogi{l< AMB, AMC Ngqbux<Xt<?
159
www.kalvisolai.com
hml< 6.55
hml<!6.53!
(i) BM = CM (ii) AB = AC (iii) AM ohiK/!weOu!h.h.h!ogit<jgh<hc? ∆AMB ≡ ∆AMC. ∴ yk<k!Ogi{r<gt<!sll</!Gxqh<hig?! ∠B = ∠C. weOu!Okx<xl<!fq'hqg<gh<hm<mK/ Okx<xl< 23: yV! Lg<Ogi{k<kqe<! -V! Ogi{r<gtqz<?! ohiqb! Ogi{k<kqx<G! wkqiqz<!njlBl<! hg<gl<! sqxqb! Ogi{k<kqx<G! wkqiqz<! njlBl<! hg<gk<jkuqm! nkqg!fQtLjmbK/ kvU: Lg<Ogi{l<! ABC -z<! Ogi{l<! ∠B NeK! Ogi{l<! ∠C! ju! uqm! ohiqbK/!!!! n.K!m∠B > m∠C (hml<!6.56Jh<!hii<g<gUl<). fq'hqg<g: AC we<x! hg<gk<kqe<! fQtl<? AB we<x!hg<gk<kqe<! fQtk<jk! uqmh<! ohiqbkiGl</!nkiuK?!AC > AB.
fq'h{l<: AB, AC we<x! hg<gr<gtqe<! fQtr<gt<!lqjg! w{<gt<! we<hkiz<?! gQp<g<gi[l<! hqiqUgt<!njlBl</ (i) AC < AB (ii) AC = AB (iii) AC > AB !hqiqU (i) AC < AB!we<g/!hqxG?!hg<gl< AB -e<!fQtlieK!hg<gohiqbkiGl</! weOu?! AB g<G! wkqOvBt<t! Ogi{l< ∠C NeOgi{l<!∠B J!uqm!ntuqz<!ohiqbkig!njlb!Ou{<Ml</!nkkvuqz<! dt<t! d{<jl! m∠B > m∠C! g<G! Lv{<himiekiAC < AB NeK!olb<bz<z. !hqiqU (ii) AC = AB we!njlBlieiz<?!-V!hg<gr<gt<!AB Bl< Aslhg<gr<gTg<G! wkqOvBt<t! Ogi{r<gt<! sll<! we! nxqOuil-KUl<!kvuqz<!dt<t!d{<jl!∠B > ∠C!g<G!Lv{<himiekNeK!olb<bz<z.!!!weOu?!lQkLt<t!hqiqU!(iii) AC > AB!we<hOk!olb<biGl</! Okx<xl< 24: &jzuqm<mr<gt<! ye<Xg<ogie<X! osr<Gk<kig! dtsib<sKvliGl</ kvU: ABCD we<hK! YI! -j{gvl</! -kqz<! &jzuqm<mrye<Xg<ogie<X!osr<Gk<kig!-Vg<gqe<xe/! fq'hqg<g: ABCD!yV!sib<sKvl</ njlh<H: &jzuqm<mr<gt< AC, BD!ujvf<K!nju!oum<Ml<!Ht<t(hml<!6.57Jh<!hii<g<gUl<). fq'h{l<: Lg<Ogi{r<gt< AMB, BMC!Ngqbux<xqz<?
(i) ∠AMB = ∠BMC = 90° (ii) AM = MC (iii) BM ohiK/
weOu?!ogit<jg!h.h.Ogi!uqe<!hc ∆AMB ≡ ∆BMC. weOu?!yk<k!hg<gr<gt<!sll</!Gxqh<hig!AB = BC. ABCD!YI!-j{gvl<!we<hkiz<< AB = CD, BC = AD. ∴ AB = BC = CD = AD. weOu ABCD sib<sKvliGl</!!Okx<xl<!fq'hqg<gh<hm<mK/
160
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hml< 6.56
l<!AC!e<!fQtk<jkuqmh<!!K! AC g<G! wkqOvBt<t!iuK? m∠C > m∠B. -K!Gl</! ! weOu?! wMOgit<!
C Bl<!slliGl</!Neiz<?!</! nkiuK?! ∠B = ∠C.!iGl</!!weOu!AC = AB
<t! YI! -j{gvl<! yV!
<gt<! AC Bl< BD Bl<!
qjb!M weg<!Gxqg<gUl<
hml< 6.57
wMk<Kg<gim<M 3: gQp<g<gi[l<!Ogi{l<!yu<ouie<xqe<!fqvh<Hg<Ogi{l<!gi{<g; (i) 30° (ii) 45° (iii) 55° (iv) 81° kQi<U: fqvh<Hg<Ogi{r<gtqe<!%Mkz< 90° . weOu?
(i) 30° bqe<!fqvh<Hg<Ogi{l< = 90° − 30° = 60°. (ii) 45° bqe<!fqvh<Hg<Ogi{l< = 90° − 45° = 45°. (iii) 55° bqe<!fqvh<Hg<Ogi{l< = 90° − 55° = 35°. (iv) 81° bqe<!fqvh<Hg<Ogi{l< = 90° − 81° = 9°.
!wMk<Kg<gim<M 4: gQp<g<gi[l<!Ogi{l<!yu<ouie<xqe<!lqjgfqvh<Hg<Ogi{l<!gi{<g;
(i) 70° (ii) 45° (iii) 120° (iv) 155° kQi<U: lqjgfqvh<Hg<Ogi{r<gtqe<!%Mkz<!180°.!weOu?
(i) 70° e<!lqjgfqvh<Hg<Ogi{l< = 180° − 70° = 110°. (ii) 45° e<!lqjgfqvh<Hg<Ogi{l< = 180° − 45° = 135°. (iii) 120° e<!lqjgfqvh<Hg<Ogi{l< = 180° − 120° = 60°. (iv) 155° e<!lqjgfqvh<Hg<Ogi{l< = 180° − 155° = 25°.
wMk<Kg<gim<M 5: hqe<uVl<!yu<ouie<xqZl<!dt<t!Ogi{r<gjtg<!g{<Mhqcg<gUl<;! (i) -VOgi{r<gt<!lqjgfqvh<Hg<Ogi{r<gt<; -ux<Xt<!ohiqbK?!sqxqbjkh<!Ohiz<!2 !!!lmr<gig!dt<tK/
(ii)!-VOgi{r<gt<!fqvh<Hg<Ogi{r<gt<; -ux<Xt<!ohiqbK?!sqxqbjk!uqm!20° lqjgbig!dt<tK/! (iii) -VOgi{r<gt<!nMk<kMk<k!Ogi{r<gtigUl<? 120° !Ogi{l<!njlh<hkigUl<<!!!!!!!dt<te/!!nux<Xt<!ohiqbK?!sqxqbjkh<!Ohiz<!4!lmr<jg!uqm!!20° Gjxuig!!!!!!!dt<tK/ (iv) -VOgi{r<gt< Gk<okkqi<!Ogi{r<gtigUl<?!fqvh<Hg<Ogi{r<gtigUl<!dt<te/! kQi<U: (i) sqxqb! Ogi{k<jk x°! we<g/! 0! ohiqb!Ogi{l< 2x°! NGl</! -V! Ogi{r<gt<!!!!!lqjgfqvh<Hg<!Ogi{r<gt<!we<hkiz<? x° + 2x° = 180° n.K!3x° = 180° n.K! x° = 60°. ∴ sqxqb!Ogi{l< = 60°, ohiqb!Ogi{l<!2 × 60° = 120°. !
)ii*! sqxqb!Ogi{l< x° we<g/!0!ohiqb!Ogi{l< x° + 20°NGl</!-u<uqV!Ogi{r<gt<!fqvh<Hg<!Ogi{r<gtikziz<? x° + (x° + 20°) = 90° n.K 2x° = 70° n.K x° = 35°. ∴!sqxqb!Ogi{l< = 35°, ohiqb!Ogi{l< = 35° + 20° = 55°. (iii) sqxqb!Ogi{l< x° we<g/!0!ohiqbOgi{l< 3x° − 20°NGl</!-u<uqV!Ogi{r<gt<!120°!jb!njlh<hkiz<? ∴ x° + (3x° − 20°) = 120°. ∴ 4x° = 140° n.K! x° = 35°. ∴ sqxqb!Ogi{l< = 35°,!ohiqb!Ogi{l< = 3 × 35° − 20°
= 105° − 20° = 85°.
161
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hml< 6.58
!!
hml< 6.59
hml< 6.60
(iv) Gk<okkqi<!Ogi{r<gt< yu<ouie<Xl<!x° we<g/ -ju!fqvh<Hg<Ogi{r<gt<!we<hkiz<? x° + x° = 90° nz<zK 2x° = 90° nz<zK x° = 45°. ∴ Ogi{r<gt< yu<ouie<Xl<! 45° NGl<. wMk<Kg<gim<M 6: hml< 6.62 z< OgiM l3 NeK?!-j{OgiMgt< l1, l2!Ngqbux<xqx<G!GXg<G!oum<c!
NGl</!Ogi{r<gt<!x, y!gi{<g/!kQi<U: ye<Xuqm<m!Ogi{r<gt<!sll</!∴ x = 130°. GXg<Goum<cbqe<!yOv!Hxl<!njlf<k!!
dm<Ogi{r<gtqe<!%Mkz<!180o.
∴ y + 130° = 180° nz<zK!y = 180° − 130° = 50°. !
wMk<Kg<gim<M 7: hml<! 6.63 z<? l4 NeK!-j{OgiMgt< l1, l2, l3 Ngqbux<xqx<<G!
GXg<Goum<cbiGl</!Ogi{r<gt<!x, y gi{<g/ kQi<U: yk<k!Ogi{r<gt<!sll</!∴ x = 75°. GXg<Goum<cbqe<! yOvHxl<! njlf<k!
dm<Ogi{r<gtqe<!%Mkz<!180o.
∴ y + 75° = 180° nz<zK y = 180° − 75° = 105°. !wMk<Kg<gim<M 8: hml<!6.64!z<?!OgiM!l3!NeK!-j{OgiMgt<! l1, l2! Ngqbux<xqx<<G!GXg<Goum<cbiGl</!Ogi{r<gt<!x, y!gi{<g/ kQi<U: GXg<Goum<cbqe<! yOvHxl<! njlf<k!dm<Ogi{r<gt<!lqjgfqvh<Hg<!Ogi{r<gt</!0!!4y + 92° = 180° nz<zK! 4y = 180° − 92° = 88° nz<zK y = 22°. yk<k!Ogi{r<gt<!sll<!⇒! x + 2y = 92° ⇒ x + 44° = 92° ⇒ x = 92° − 44° = 48°. !wMk<Kg<gim<M 9: Lg<Ogi{k<kqe<!Ogi{!ntUgtqe<!ugi{<g/ kQi<U: Ogi{r<gjt 3x, 4x, 5x!weg<!ogi{<miz<?!3x + 4n.K x = 15°. ∴ Ogi{r<gt< 3 × 15°, 4 × 15°, 5 × 15° nz wMk<Kg<gim<M 10: hml<!6.65!z<!Gxqh<hqmh<hm<<m!Ogi{r<gx, y!gi{<g/ kQi<U: ∆ABC!-z< x + 65° + 90° = 180° n.K x = 25°. ∆BDC!-z< x + y + 90° = 180°n.K 25° + y + 90° = 180° n.K y + 115° = 180° n.K! y = 180° − 115° = 65°.
162
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hml< 6.61
hml<!6.62qg
x<z
t
hml< 6.63
hml< 6.64qkl< 3 : 4 : 5 weqz<?!nux<jxg<<!
+ 5x = 180° nz<zK 12x = 180° K 45°, 60°, 75°!NGl</
<!
!hml< 6.65
wMk<Kg<gim<M 11: hqe<uVl<<!hmr<gtqz< x, y gi{<g; (i) (ii) !!kQi<U: (i) A∴ 2x = 24°, 3∴ x = 12°, y = (ii) L∴ ∆ADC ≡ ∆∴ yk<k!Ogi{x = 26° − 20° wMk<Kg<gim<Mnch<hg<gk<kq
kQi<U: ABC wsloum<c! AOu{<Ml</!!L
AB = AC, m∴ h.Ogi.h!o∴ yk<k!hg<g∴ BD = DC. ∠ADC = x weLg<Ogi{l<!
Lg<Ogi{l<!
∴ ∠ADC + ∠n.K!x =180∴AD NeK!n.K!AD Ne wMk<Kg<gim<MDA -j{g<gkQi<U: DA = DweOu!Lg<O∴∠DAB = ∠ ∠DAC = ∠
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hml< 6.66
D = BC, AB = CD. ∴ ABCD YI!-j{gvy = 60° (ye<Xuqm<m!Ogi{r<gt<!sll<). 20°. g<Ogi{r<gt<<!ACD, ACB, -ux<xqz<! AD
ABC. r<gt<!sll</!weOu x + 20° = 26°, y − 5°
, y = 42° + 5° nz<zK x = 6°, y = 47°.
12: YI! -Vslhg<g! Lg<Ogi{k<kqex<G!jlbg<Gk<Kg<OgiM!we!fq'hq/ e<hK!YI!-Vslhg<g!Lg<Ogi{l<<! we<g/!
D we<g/! AD, nch<hg<gl<! BC! g<G! jg<Ogi{r<gt<!ADB, ADC!Ngqbux<Xt<!
∠BAD = m∠DAC ( AD NeK!∠A e<!slogit<jgh<hc, ∆ABD ≡ ∆ACD. r<gt<!sll</ nkiuK?!D NeK BC!bqe<!fMh<Ht<tq!NG<g/!hqe<H!∠ADB = 180° − x.
ADC z<?!∠ADC + ∠C + ∠CAD = 180°. ADB z<?!∠ADB + ∠B + ∠BAD = 180°. C + ∠CAD =∠ADB + ∠B + ∠BAD.
° − x n.K! 2 x = 180° n.K!x = 90°.!BC g<Gs<!osr<Gk<kig!dt<tK/!!K!BC bqe<!jlbg<Gk<Kg<!OgimiGl</!
13: ABC we<hK!yV!Lg<Ogi{l</! Ht<th<hMgqxK/! DA = DC!weqz<? ∠BAC yV!osC !wek<!kvh<hm<Mt<tK D NeK!BC!e<!gi{r<gt<!ABD?< ACD !-Vslhg<g!Lg<OgiDBA (1) DCA (2)
163
hml< 6.67
l</
= AB, CD = BC, AC NeK!ohiK/
= 42° nz<zK
<! ds<sqOgi{k<kqe<! sloum<c!
-r<G!AB = AC, Ogi{l< A bqe<!lbg<Gk<Kg<OgiM! we! fq'hqk<kz<!
um<c*?!! AD ohiK/
l</!
q!D Nr<Ogijlbh{r<g
hml< 6.68
eK BC bqe<!jlbh<Ht<tq/!{l<!we!fq'hq/ <Ht<tq!we<hkiz<!BD = DC/!!t<!NGl</
(1) + (2) ⇒ ∠DAB + ∠DAC = ∠DBA + ∠DCA
⇒ ∠BAC = ∠DBA + ∠DCA ⇒ ∠BAC = ∠CBA + ∠BCA (3) Neiz<! ∠BAC + ∠CBA + ∠BCA = 180° (4) (4) ⇒ ∠BAC + (∠CBA + ∠BCA) = 180° ⇒ ∠BAC + ∠BAC = 180° ((3) Jh<!hbe<hMk<k) ⇒ 2∠BAC = 180° ⇒∠BAC = 90°. wMk<Kg<gim<M 14: yV!fix<gvk<kqe<!fie<G!Ogi{r<gtqe<!%Mkz< 360°!we!fq'hq/ kQi<U: ABCD yV! fix<gvl< we<g/!!!!!!!!!!!!∠A + ∠B + ∠C + ∠D = 360° we! fq'hqk<kz<!Ou{<Ml</!-kx<G?!&jzuqm<ml< AC J!ujvbUl</ Lg<Ogi{r<gt<!ACD, ABC!Ngqbux<xqzqVf<K? ∠DAC + ∠D + ∠ACD = 180° (1) ∠CAB + ∠B + ∠ACB = 180° (2) (1) + (2) ⇒ ∠DAC + ∠D + ∠ACD + ∠CAB + ∠B + ∠ACB = 360° ⇒ (∠DAC + ∠CAB) + ∠B + (∠ACD + ∠ACB)+ ∠D = 360° ⇒ ∠A + ∠B + ∠C + ∠D = 360°. wMk<Kg<gim<M 15: AB = AC we<xqVg<g!ABC YI! -Vslhg<g! Lg<Ogi{l<! we<g/ ∠DBC = ∠DCB!we!njlBliX!Ht<tq D? Lg<Ogi{l<! ABC -e<! dt<Ot!njlf<Kt<tK/! AD NeK ∠A -e<!sloum<c!we!fq'hq/ kQi<U: ∠DBC = ∠DCB we<hkiz<?! Lg<Ogi{l< DBC YI!-Vslhg<g!Lg<Ogi{liGl</!!weOu!!BD = DC. Lg<Ogi{r<gt<! ADB, ADC Ngqbux<Xt<, BD = DC,!AB = AC, AD!ohiK/!!weOu! h.h.h! ogit<jgh<hc?! ∆ADB ≡∆ADC. 0!yk<k!Ogi{r<gt<!sll</ Gxqh<hig?!∠BAD = ∠CAD.∴ AD NeK!∠A!e<!sloum<cbiGl</ wMk<Kg<gim<M 16: yV!Lg<Ogi{l<!ABC e<! osr<OgiMgt<!AD, BEwe<xuiXt<te/! AD = BE!we!fq'hq/ kQi<U: Lg<Ogi{r<gt< ADB, AEB Ngqbux<Xt<
(i) ∠ADB = ∠AEB = 90° (ii) AB ohiK (iii) BD = AE.
∴ os.g.h!ogit<jgh<hc!∆ADB ≡ ∆AEB. yk<k!hg<gr<gt<!sll</!AD = BE.
164
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!
hml< 6.70
hml< 6.71
we<he AE = BD
hml< 6.72
hml< 6.69
wMk<Kg<gim<M 17: ABCD!we<El<!osu<ugk<kqz<!BC e<!jlbh<Ht<tq!E,!NGl</ AE = ED we!fq'hq/!!kQi<U: hml<!6.73 Jh<!hii<g<gUl<. Lg<Ogi{r<gt< ABE, DCE Ngqbux<xqz< (i) ∠ABE = ∠DCE = 90° (ii) BE = CE (E, BC e<!jlbh<Ht<tq) (iii) AB = CD (ABCD yV osu<ugl<) ∴h.Ogi.h!ogit<jgh<hc ∆ABE ≡ ∆DCE. 0!yk<k!hg<gr<gt<!sll</!∴AE = ED. wMk<Kg<gim<M 18: yV! sib<sKvk<kqz<?! &jzuqm<mrosr<Gk<kig!-Vslg<!%xqMl<!we!fq'hq/ kQi<U: ABCD yV! sib<sKvl</! ! &jzuqm<mr<gt<! ACoum<Mh<Ht<tqjb!O weg<!Gxqg<g/!!Ht<tq!O!NeK!AC!NeK!BD!g<G!osr<Gk<K!(⊥) weUl<!fq'hqk<kz<!sib<sKvl<! YI! -j{gvl<! we<hkiz<!&jzuqm<mr<gt<! yu<ouie<Xl<! lx<oxie<jx!-Vslg<%xqMl</!!nkiuK?!OA = OC, OB = OD. -h<ohiPK?!Lg<Ogi{r<gt<<! AOB, BOC!Ngqbux<Xt(i) AB = BC (ii) OB ohiK!!(iii) OA = OC ∴ h.h.h!ogit<jgh<hc!∆AOB ≡ ∆BOC. 0!yk<k!Ogi{r<gt<!sll</!!∴ ∠AOB = ∠BOC = x° we<g/. Neiz<?! ∠AOB + ∠BOC = 180°.
∴ x + x = 180° ∴ 2x = 180° n.K! x = 2
180° = 90°.
∴ &jzuqm<mr<gt<!yu<ouie<Xl<!lx<oxie<jxs<!osr<G !wMk<Kg<gim<M 19: yV! sib<sKvk<kqe<! &jzuqm<mr<gds<sqg<!Ogi{r<gjt!slg<%xqMl<!we!fq'hq/ kQi<U: ABCD yV! sib<sKvl</! ! AC, BD!&jzuqm<mr<gt</! ! AB || CD, AC GXg<Goum<c!we<hkiz<? ∠BAC = ∠ACD (ye<Xuqm<m!Ogi{r<gt<). (1) Neiz<! AD = CD (ABCD sib<sKvl<). ∴∆ADC YI!-Vslhg<g!Lg<Ogi{l</ slhg<gk<kqx<G! wkqOvBt<t! Ogi{r<gt<! sll< we<hkiz<?!∠ACD = ∠DAC. (2) (1), (2)!e<!hc ∠BAC = ∠DAC. weOu AC, Ogi{l< ∠A!ju!slg<%xqMl</ -jkh<Ohie<Ox?! AC NeK ∠C! jbBl<? BD NeKslg<%xqMl<!we!fq'hqg<gzil</ !!
165
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hml< 6.73
<gt<! yu<ouie<Xl<! lx<oxie<jxs<!
, BD! ujvg/ AC Bl<! BD! Bl<!AC, BD!-e<!jlbh<Ht<tq!weUl<?!Ou{<Ml</
hml< 6.74
<
k<kig!-Vslg<!%xqMl</
t<! yu<ouie<Xl<! nju! osz<gqe<x!!
hml< 6.75
∠B lx<Xl<< ∠D! Ngqbux<jxBl<!!!!
wMk<Kg<gim<M 20: AB, CD -j{OgiMgt</!AB, CD gTg<G! -jmOb! Ht<tq O (hml< 6.76Jh<! hii<g<gUl<) ∠APO = 45° -Vg<GliXl< ∠OQC = 35°!-Vg<GliXl<!
hml< 6.76
njlf<Kt<tK!weqz<?!∠POQ g{<Mhqc/!kQi<U: PO ju! fQm<Mg/! ! nK! CD jb!sf<kqg<Gl<! Ht<tqjb! X! weg<! Gxqg<g/!!QO ju! fQm<Mg/! nK AB J! sf<kqg<Gl<!Ht<tqjb Y!weg<!Gxqg<g/ AB || CD, PX GXg<Goum<c!we<hkiz<? ∠OXQ = ∠OPY = 45° (ye<Xuqm<m!Ogi{r<gt<). Lg<Ogi{l<!OXQ z<? outqg<Ogi{l<!∠POQ!NeK!nke<!dt<otkqi<!Ogi{r<gtigqb!∠OXQ lx<Xl< ∠OQX!Ngqbux<xqe<!%MkZg<G!sll</ 0!!∠POQ = ∠OXQ + ∠OQX = 45° + 35° = 80°. wMk<Kg<gim<M 21: ∆ABC -z<? ∠B -e<!sloum<cbieK AC jb! Ht<tq D! -z<!sf<kqg<gqe<xK/!!∠ABC = 80°, ∠BDC = 95°!weqz<?!∠A, ∠C!gi{<g/!kQi<U: hml< 6.77 Jh<!hii<g<gUl</!∆BDC!-z<?!40° + 95° + ∠C = 180° ⇒ ∠C = 180° − 135° = 45°. ∆ABC!-z<?! ∠A + ∠B +∠C = 180°. ∴ ∠A + 80° + 45° = 180° ⇒ ∠A =180° − 125° = 55°. wMk<Kg<gim<M 22: ABCD!yV!siqugl</! ! -kqz<!!AB! Bl<! CD! Bl<! -j{OgiMgt</! AD = BC!weqz<?!∠ADC = ∠BCD we!fq'hq/!kQi<U: hml< 6.78 Jh<!hii<g<gUl</!AD g<G!-j{big!BE ujvg/!-h<ohiPK!ABED YI!-j{gvl<!we<hkiz<?!!BE = AD. Neiz<!AD = BC. ∴ BC = BE. weOu!Lg<Ogi{l<!BEC YI -Vslhg<g!Lg<Ogi{l<. ∴ ∠BCE = ∠BEC. (1) Neiz< AD || BE, AD, BE -ux<xqg<G!DEC yV!GXg<Goum∴∠ADC = ∠BEC (yk<k!Ogi{r<gt<). (2) weOu! (1), (2)!e<!hc?!∠BCE = ∠ADC nz<zK ∠BCD = ∠
hbqx<sq 6.2 1. hqe<uVl<!%x<Xgtqz<!wju!siq?!wju!kuX!we!wPK
(i) yV! gkqI<?! yV! Ogim<ce<! lQK!njlukiz<!njlB%Mkz<!180°!NGl</!
(ii) -V!OgiMgt<!oum<cg<ogi{<miz<?!Gk<okkqi<!Ogi{r (iii) yV!Lg<Ogi{l<!-V!uqiqOgi{r<gjth<!ohx<xqg<Gl<</
166
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hml<!6.77!
hml< 6.78
<c.
ADC.
g; l<!nMk<Kt<t! Ogi{r<gtqe<!
<gt<!sll</
(iv) yV!fix<gvk<kqe<!Ogi{r<gtqe<!%Mkz<!180° NGl</! (v) ∆ABC ≡ ∆PQR weqz<?!∠A = ∠Q. (vi) ∆DEF ≡ ∆XYZ weqz<?! DE = XY. (vii) YI!-j{gvk<kqz<?!&jzuqm<mr<gt<!yu<ouie<Xl<!lx<xjk!slg<%xqMl</ 2. gQp<g<g{<m!yu<ouie<xqe<!fqvh<Hg<Ogi{l<!gi{<g/
(i) 20° (ii) 65° (iii) 70° (iv) 78° 3. gQp<g<g{<m!yu<ouie<xqe<!lqjgfqvh<Hg<Ogi{l<!gi{<g/
(i) 50° (ii) 130° (iii) 80° (iv) 152°. 4. hqe<uVl<!yu<ouie<xqZl<!Ogi{r<gjtg<!gi{<g; (i) Ogi{r<gt<!fqvh<Hg<Ogi{r<gt</!sqxqb!Ogi{l<?!ohiqb!Ogi{k<jkuqm! 40°
GjxuiGl</ (ii) Ogi{r<gt<! fqvh<Hg<Ogi{r<gt</! ohiqb! Ogi{l<?! sqxqb! Ogi{k<jkh<Ohiz<! 4 lmr<G. (iii) Ogi{r<gt<!lqjgfqvh<Hg<Ogi{r<gt</!ohiqb!Ogi{l<?!sqxqbjkuqm!58° nkqgl</ (iv) Ogi{r<gt<!lqjgfqvh<Hg<Ogi{r<gt</!ohiqb!Ogi{l<?!sqxqb!Ogi{k<kqe<!3!!!!!!lmr<jguqm!20° GjxuiekiGl</ (v) !-V!nMk<Kt<t!Ogi{r<gt<!140°!Ogi{k<kqje!njlg<gqe<xe/!sqxqb!Ogi{l<!!
!ohiqb!Ogi{k<jkuqm!28° GjxuiGl</ (vi) Ogi{r<gt<! Gk<okkqi<! Ogi{r<gtigUl<?! lqjgfqvh<Hg<Ogi{r<gtigUl<!!!!!!
-Vg<gqe<xe/ 5. hqe<uVl<!hmr<gtqz<!x, y!gi{<g; (i) (ii)
hml< 6.80
(iii)
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hml< 6.79
(iv)
hml< 6.81
167
hml< 6.82
6. hqe<uVl<!hmr<gt<!yu<ouie<xqZl<!x, y!gi{<g; (i) (ii) (iii)
1. kuX
1. (i) siq (v) kuX 2. (i) 70° 3. (i) 130 4. (i) 25°
(v) 56° 5. (i) x = (iii) x = 6. (i) x =
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hml< 6.83
hml< 6.85
!uqjmgt<!
hbqx<sq 6.1
2. siq 3. kuX
hbqx<sq 6.2
(ii) siq (iii) kuX (iv) kuX (vi) kuX (vii) siq
(ii) 25° (iii) 20° (iv
° (ii) 50° (iii) 100° (iv
, 65° (ii) 18° (iii) 61° , 119° (iv, 84° (vi) 90°, 90°
130°, y = 50° (ii) x = 80°, y = 70° 20°, y = 30° (iv) x = 50°, y = 130°
19, y = 8 (ii) x = 48°, y = 12° (iii) x =
168
hml< 6.84
) 12°
) 28°
) 50°, 130°
6, y = 3.
7. hGLjx!ucuqbz<!
yV! OfIg<Ogim<cz<! njlf<k! yu<ouiV! Ht<tqBl<! siqbig! yV! olb<ob{<[me<!ohiVk<kh<hm<Mt<tK! we<hjk! Lkz<! nk<kqbibk<kqz<! g{<Omil</ -f<k! nk<kqbibk<kqz<!yV! ktk<kqz<! njlf<Kt<t! yV! Ht<tqbieK! wu<uiX! olb<ob{<gtiz<! Sm<cg<!gim<mh<hmzil<! we<hjk!Nvib<Ouil</! ovOe!om <̂giIm <̂! we<gqx! Hgp<! ohx<x! hqovR<S!
fim<M! g{qk! uz<ZfI, Lke<Lkzig! ucuqbjz! hGk<kxqb?! -bx<g{qk! Ljxjb!)w{<gt<!lx<Xl<!fie<G!nch<hjms<!osbz<gt<!hbe<hMk<Kl<!Ljx*!nxqLgh<hMk<kqeiI/!weOukie<?! -bx<g{qk! Ljxjbh<! hbe<hMk<kq! ucuqbjz! hGk<kxqBl<! himh<hGkq!-bx<g{qk! ucuqbz<! nz<zK! ! hGLjx! ucuqbz<! weg<! %xh<hMgqe<xK/! -f<k!
hGLjx! ucuqbjz! ovOe! om <̂giIm <̂! nxquqk<kkiz<, nuI! hGLjx! ucuqbzqe<!kf<jk!weh<hMgqxiI/!
7.1 gii<Csqbe<!ns<S!K~vLjx!
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ye<Xg<ogie<X! osr<Gk<kig! njlf<k! -V! fqjzbie! OfIg<OgiMgjt! ujvg/!!!!!!
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nzgqje! -v{<M! OgiMgTg<Gl<! hbe<hMk<KOuil</! -h<ohiPK! -V! osr<Gk<Kg<!!
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gix<hGkqgTg<Gl<! ohiKuieK/! ktk<kqz<! P we<gqx! WOkEl<! yV! Ht<tqjb! wMk<Kg<!ogit<Ouil</!-h<Ht<tq!P NeK!yV!gix<hGkqbqz<!njlf<Kt<tK/!P upqbig!y-ns<Sg<G!-j{big!yV!Ofi<g<OgiM!ujvg/!-g<OgiM!x-ns<js!L we<gqx!Ht<tqbqz<!sf<kqg<gm<Ml</!-u<uiOx, P upqbig!x-ns<Sg<G!-j{big!yV!Ofi<g<OgiM!ujvg/!-g<OgiM!y-ns<js M we<gqx! Ht<tqbqz<! oum<mm<Ml</ x-ns<S! w{<! Ogim<cz<! L we<x! Ht<tqjbg<<! Gxqg<Gl<!olb<ob{< a NgUl<, y-ns<S!w{<!Ogim<cz< M we<x!Ht<tqjbg<!Gxqg<Gl<!olb<ob{<!b NgUl<!-Vg<gm<Ml</!P NeK!x-ns<sqz<!njlf<kiz<, b = 0 weg<!gi{<gqOxil</!P!NeK!y-ns<sqz<!njlf<kiz<, a = 0 weg<!gi{<gqOxil</!P NeK x-ns<sqOzi!nz<zK!y-ns<sqOzi!-z<zilz<!Lkz<!gix<hGkqg<Gt<!njlf<kiz<, a > 0 lx<Xl<!b > 0 NGl</ a < 0 lx<Xl< b > 0 Ng!-Vf<kiz<!P bieK!-v{<mil<! ( II ) gix<hGkqg<Gt<!njlBl</ P bieK!&e<xil<!!! ( III ) gix<hGkqg<Gt< njlf<kiz<, a < 0 lx<Xl< b < 0. a > 0 lx<Xl<!b < 0 Ng!-Vh<hqe<, P bieK!fie<gil<!( IV ) gix<hGkqg<Gt<!njlBl</ P bieK!Nkqh<Ht<tq! O Ng!!
-Vh<hqe<, a = 0 lx<Xl<!b = 0 NGl</ a we<x!w{<!P wEl<! Ht<tqbqe<! x-ns<S! K~vl<! nz<zK!!!!!x-okijzU!(abscissa) we<Xl<, b wEl<!w{<!P wEl<! Ht<tqbqe< y-ns<S! K~vl<! nz<zK!!!!!!
y-okijzU (ordinate) we<Xl<! njpg<gh<hMl< (hml<! 7.3! Jh<! hii<g<gUl<*.! fil<! a, b wEl<!w{<gjt! gix<Ht<tqbiz< ( , ) -u<uiX!
hqiqg<gh<hm<m!hqjx!njmh<Hg<Gt<, (a, b) we<X!wPKOuil</!njk! a, b we<gqx!uiqjs!Osic!we! njph<<Ohil</! gix<Ht<tqbqe<!
-mKhg<gLt<t! w{<, P Ht<tqbqe<! x-ns<S!K~vk<jkBl<, gix<Ht<tqbqe<!uzK!hg<gLt<t!!
hml< 7.3
w{<, P Ht<tqbqe<!y-ns<S!K~vk<jkBl<!Gxqh<hkiz<?!(a, b) NeK!yV!uiqjs!Osic!we!njpg<gh<hMgqxK/ P Ht<tqjbg<!Gxqg<Gl<!uiqjs!Osic (a, b)!we<hK!ye<Ox ye<Xkie</!nkiuK!P wEl<!yV!Ht<tqg<G!OuX!wf<k!uiqjs!OsicBl<!-Vg<giK/ P wEl<!Ht<tq!P(a, b) weUl<!nz<zK!SVg<glig (a, b) weUl<!Gxqg<gh<hMl</!fil< P bieK! (a, b)
170
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ns<SK~vr<gjtg<! ogi{<Mt<tK! weg<!%XOuil</! -u<uiX!ktk<kqz<!dt<t!yu<ouiV!
Ht<tqBl<, olb<ob{<gtiz<! Ne! yV! uiqjs! Osicbiz<! Gxqh<hqmh<hMl</! ovOe!om^<giIm^<^qe<!dbIf<k!osbjz!lkqg<Gl<!ohiVm<M!-k<ktlieK!gii<Csqbe<!ktl<!
we<X!njpg<gh<hMgqxK/! OlZl<!nK!osu<ug!ns<S!ktl<!nz<zK! xy ktl<! we<Xl<!njpg<gh<hMl</! -u<uiX! w{<uiqjs! Osicgtiz<! ktk<kqz<! njlf<k! Ht<tqgjtg<!
Gxqg<Gl<!Ljx!gii<Csqbe<!Ljx!nz<zK!osu<ug!Ljx!nz<zK!xy ns<S!K~v!Ljx!we<X! njpg<gh<hMgqxK/! -v{<M! ns<SgTl<! osu<ug! ns<Sgt<! nz<zK! okijzU!ns<Sgt<!weh<hMgqe<xe/!!!fil<!nxquK, (i) Nkqh<!Ht<tq!O uqe<!ns<S!K~vr<gt< (0, 0). (ii) x-ns<sqe<!lQKt<t!wf<k!Ht<tqg<Gl<! y-ns<SK~vl<!0. (iii) y-ns<sqe<!lQKt<t!wf<k!Ht<tqg<Gl<! x-ns<SK~vl< 0. (iv) olb<ob{<gtizie! uiqjs! Osic! ye<X! ogiMg<gh<hm<miz<! nkx<G! gii<Csqbe<!
ktk<kqz<!keqk<k!yV!Ht<tqbqeiz<!-ml<!fqi<{bqg<gzil</!ktk<kqz<!nke<!siqbie!
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&e<xil<!gix<hGkqg<Gt<<!njlf<kiz<, -V!ns<S!K~vr<gTl<! Gjx/! fie<gil<! gix<hGkqg<Gt<!
njlf<kiz<, x-ns<S! K~vl<! lqjg! lx<Xl<!!
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7.4 z< gim<mh<hm<Mt<te/!(vi) x-ns<Sg<G!-j{bigs<!osz<Zl<!Ogim<cz<!ny-ns<S!K~vk<jkg<!ogi{<cVg<Gl<!)hml< 7.5 Jh<!hi (vii) y-ns<Sg<G!-j{bigs<!osz<Zl<!Ogim<cz<!nx-ns<S!K~vk<jkg<!ogi{<cVg<Gl<!)hml<!7.6 Jh<!h
171
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!!!
hml< 7.4
jlf<k!wz<zih<!Ht<tqgTl<!sllie i<g<gUl<).
hml< 7.6
hml<!7.5
jlf<k!wz<zih<!Ht<tqgTl<!sllie, iIg<gUl<).
A(x1, y1) lx<Xl< B(x2, y2) we<hju!gii<Csqbe<!ktk<kqz<!njlf<k!WOkEl<!-V!Ht<tqgt<!we<g/!-ux<xqx<G!-jmObBt<t!gqjmlm<mk<!okijzU, x2 > x1!weqz<! x2 − x1!weUl<, x1 > x2 weqz< x1 − x2 weUl<, x2 = x1 weqz<! 0 weUl<!ujvbXg<gh<hMgqxK/!-jkh<!OhizOu, A, B -ux<xqx<gqjmObbie!osr<Gk<Kk<!okijzuieK!y2 > y1!weqz<?!y2 − y1!weUl<, !!y1 > y2!weqz<?!y1 – y2!weUl<, y2 = y1!weqz<?!0 weUl<!ujvbXg<gh<hMgqe<xK/!!wMk<Kg<gim<mig, hml< 7.7 z<, A, B -ux<xqx<G!-jmObBt<t!gqjmlm<mk<!okijzju BN l<,!osr<Gk<Kk<! okijzju AN l< Gxqg<gqe<xe/!fil<!nxquK, BN = LM=OL + OM = (−x1) + (x2) = x2 − x1, AN = AL + LN = y1 + MB = y1 + (−y2) = y1 − y2.!-u<uiOx, hml< 7.8 z<! A(x1, y1) lx<Xl<!!!!B(x2, y2) !weg<!ogi{<miz<! A, B -ux<xqx<G -jmObBt<t! gqjmlm<mk<! okijzU, osr<Gk<Kk<!okijzU!Ngqbe!LjxOb,
hml<!7.7
hml< 7.8
BN = ML = OM – OL = −x2 – (−x1) = x1 – x2, AN = LN – AL = BM – (−y1) = (−y2) + y1 = y1 − y2. -k<okijzUgjt!LjxOb!| x1 − x2 |!we<Xl<!!|!y1 − y2 |!we<Xl<!Gxqh<hqMgqe<Oxil</!! !
wMk<Kg<gim<M 1: A (3, 0), B (0, 2 ), C (4, − 4), D (3, 3), E (−2.5, 1), F (−1, −3), G (−1, 0), H (0, −4) Ngqb!Ht<tqgjt!yV!ujvhmk<kitqz<!Gxqg<gUl</!OlZl<!yu<ouiV!Ht<tqBl<!wf<<k!gix<hGkqbqz<!njlgqe<xK!we<hjkBl<!Gxqh<hqmUl</!kQi<U: hml<!7.9 -z< yV!ktk<kqz<!Ht<tqgt<!!
Gxqg<gh<hm<Mt<te/ A we<gqx!Ht<tq!x-ns<sqe<!!lqjgh<!hGkqbqZl<, B !we<gqx!Ht<tq! y-ns<sqe<!!lqjgh<!hGkqbqZl<, C we<gqx!Ht<tq!fie<gil<!!gix<hGkqbqEt<Tl<, D we<gqx!Ht<tq!Lkz< gix<hGkqbqEt<Tl<, E we<gqx!Ht<tq!-v{<mil<!!gix<hGkqbqEt<Tl<, F we<gqx!Ht<tq!&e<xil< gix<hGkqbqEt<Tl<, G we<gqx!Ht<tq x-ns<sqe<!!Gjxh<hg<gk<kqZl<, H we<gqx!Ht<tq< y-ns<sqe<!!Gjxh<hg<gk<kqZl<!LjxOb!njlf<Kt<te/!!
wMk<Kg<gim<M 2: (−3, −4), (−9, 11) we<gqx! -V!gqjmk<okijzU, osr<Gk<Kk<!okijzU!Ngqbux<jxg<!kQi<U: (−3, −4) , (−9, 11) Ngqb!-V!Ht<tqgTg<G!-osr<Gk<Kk<!okijzU!LjxOb!(−3) − (−9) = 9 − 3 = 6 l
!!!!
172
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hml< 7.9
Ht<tqgTg<G! -jmOb! dt<t!g{<Mhqc/!!jmOb!dt<t!gqjmk<okijzU, x<Xl<! (11) − (−4) = 15 NGl</
hbqx<sq 7.1
1. gQOp! ogiMg<gh<hm<Mt<t! Ht<tqgjt! ujvhmk<kitqz<! Gxqk<K! nju! wf<okf<k!gix<hGkqgtqz< njlgqe<xe!we<hjkg<!Gxqh<hqmUl</!
(i) (2, 3) (ii) (7, 6) (iii) (−2, −3) (iv) (6, −2) (v) (−9, 0) (vi) (5, 0) (vii) (0,11) (viii) (−3, 2)
2. siqbi!kuxi!weh<!hkqz<!ntqg<gUl</!
(i) (9, −1), -v{<mil<!gix<hGkqbqEt<!njlBl</! (ii) (1, 0), y-ns<sq<e<!Olz<!njlBl</! (iii) (−3,1), y-ns<sqe<!uzKhg<gl<!njlBl</! (iv) (1, −1), x-ns<sqe<!gQOp!njlBl</! (v) (0, 0) we<hK!x, y ns<Sg<gt<!oum<cg<!ogit<Tl<!Ht<tq/ (vi) (− ,2 2 ), gix<hGkq II z<!njlBl</
(vii) (−π, − 3 ), gix<hGkq III z<!njlBl</! (viii) ( 2 − 3 , −1), gix<hGkq IV z<!njlBl</ (ix) (0, −3), x-ns<sqe<!-mKhg<gl<!njlBl</ (x) (5, 0), x-ns<sqe<!gQOp!njlBl</ (xi) x-ns<Sg<G!-j{big!dt<t!Ogim<ce<!Olz<!WOkEl<!-VHt<tqgt<, sllie!
x-ns<SK~vr<gjt!ogi{<cVg<Gl</ (xii) (a, b), (c, d) Ngqbju! y-ns<Sg<G!-j{bie!Ogim<ce<!Olz<!!
! WOkEl<!-V!Ht<tqgt<!weqz<, a = c. 3. ogiMg<gh<hm<Mt<t!-V!Ht<tqgTg<G!-jmOb!dt<t!gqjmk<okijzU, osr<Gk<Kk<!okijzU!Ngqbux<jxg<!gi{<g/! (i) (1, 4) lx<Xl<! (3, 5). (ii) (−2, 3) lx<Xl<! (4, −6). (iii) (−3, −5) lx<Xl< (7,2). (iv) (−2, −1) lx<Xl<! (−4, −3). 7.2 Ogim<ce<!sib<U
!
Lkzqz<, x-ns<Sg<Ogi! nz<zK y-ns<Sg<Ogi! -j{big! njlbik LL′ we<x!Ofi<g<Ogim<ce<! sib<uqje ujvbXh<Ohil</! -kx<gig! LL′! Ofi<g<Ogim<ce<! lQkjlf<k!!!!!
P(x, y) we<gqx!Ht<tqbig yV!leqkje fqjek<Kg<ogit<Ouil</!-l<leqke<!P Neue<!ng<Ogim<ce<!lQK!YMgqe<xie<!we<X!fqjek<Kh<!hiIg<gUl</!hml<!7.10 nz<zK!hml<!7.11 Ngqbux<xqz<! Gxqh<hqm<Mt<tK! Ohiz<! WOkEl<! -V! kqjsgtqz<! nue<! osz<zzil<!we<hjk!fil<!nxqgqOxil</!
P bieKnkqgiqh<h
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kqjsjbg<! Ogim<ce<! lqjgk<kqjs! weg<! %XgqOxil</! lx<oxiV! kqjsbieK! Ogim<ce<!
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hml<! 7.12
P NeK P1(x1, y1) we<x!fqjzbqzqVfvm<Ml</ P bqe< x-ns<S! K~vlieK! x1! lk
~vlieK y1 lkqh<hqzqVf<K y2 we<gqx! lk
ijzuqz<!Wx<hMl<!lix<xl<!x2 − x1 . -K, fOk! Ohie<X! y-ns<S! okijzuqz<! Wx<hMl<tqbqe<!Wx<xl<!nz<zK!wPs<sq!(rise) weh<ml<! x2 − x1 lqjgbig! -Vh<hjkg<! gi{z<Ofig<gq! fgVgqe<xK/! nkiuK,! nK!s<sq! njmgqxK/! OlZl<! P2 uieK! P1
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nMk<khcbig, x-ns<<Sg<G!-j{bigz<Zl<!Ogim<ce<!sib<uqjeg<!g{<mxqOuill; 7.14 z<, OgiM! LL′ NeK! x-ns<Sg<Gj{big!dt<tK/!Ogim<ce<!Olz<!njlf<zih<! Ht<tqgTl<! yOv y-ns<S! K~vk<jki{<Mt<te!we<Xl<!gi{<gqOxil</ P1(x1, y1
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17
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12
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4
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12
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nMk<khcbig, y-ns<<Sg<G! -j{bigs<!osz<Zl<! Ogim<ce<! sib<uqjeg<! gi{<Ohil</ P1(x1, y1), P2(x2, y2) Ngqbju! -j{g<!Ogim<cz<! njlf<k! -V! Ht<tqgt<! we<g!(hml<!7.15 Jh<!hiIg<gUl<). -r<G, x1 = x2. weOu, Ym<ml< = x2 − x1 = 0. P1, P2 ouu<OuX!Ht<tqgt</!weOu, y1 ≠ y2. NgOu!
sib<U = = ml<Ym<
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12
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= 0
12 yy −.
-K!ujvbXg<gh<hmuqz<jz/!∴ y-ns<<Sg<G! -j{bigs<! osz<Zl<!Ogim<ce<!sib<U ujvbXg<gh<hmikK/!!wPl<Hl<!Ogim<ce<!sib<U!> 0. uQPl<!Ogim<ce<!sib<U < 0. x-ns<Sg<G!-j{bigs<!osz<Zl<!Ogim<ce<!sib<U! y-ns<<Sg<G!-j{bigs<!osz<Zl<!Ogim<ce<!sib<U u!
-r<G! fil<! gueqg<g! Ou{<cbK, 12
12
xxyy
−−
= (( 1
1
xy
−−
(x2, y2) Ngqb!-V!Ht<tqgjts<!Osi<g<Gl<!Ogim<ce<!
= ml<Ym<
xl<Wx<
12
12
xxyy
−−
= xy
-kqzqVf<K, sib<uieK!Ogim<ce<!kqjsjbh<!ohiVk<K! njlukz<z! we<hjk! fil<!nxqgqOxil</ OlZl<! Ogim<ce<! sib<U!
12
12
xxyy
−−
we<gqx! uqgqkl<! P1, P2 we<gqx!
Ht<tqgtqe<! Gxqh<hqm<m! okiqf<okMk<kjzs<!sii<f<kK nz<z/! -f<k! d{<jljbk<!okiqf<K! ogit<t, nf<OfIg<Ogim<ce<! lQK!OuX! -V! Ht<tqgt<! P3 (x3, y3) lx<Xl<!!!!P4 (x4, y4) wMk<Kg<!ogit<Ouil<! (hml<!7.16 Jh<!hiIg<gUl<). P3 zqVf<K!P4 g<Gs<!osz<Zl<!
Ogim<ce<!sib<U!=34
34
xxyy
−−
.
P1 zqVf<K! P2 g<Gs<! osz<Zl<! Ogim<ce<!
sib<U =12
12
xxyy
−−
.
!
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hml< 7.15
0. jvbXg<gh<hmikK/!!
))2
2
xy
−−
= 21
21
xxyy
−−
. -u<uiX (x1, y1),
sib<U!!
21
21
xy
−−
.
hml< 7.16
Neiz< ∆P1AP2 , ∆P3BP4 we<hju!ucouik<k!Lg<Ogi{r<gt</!
∴ BPAP
3
1 = 4
2
BPAP nz<zK!
APAP
1
2 = BP
BP
3
4 . ∴ 12
12
xxyy
−− =
34
34
xxyy
−−
.
nkiuK, sib<uieK!Ogim<ce<!Olzjlf<k!-V!Ht<tqgtqe<!-mk<jks<!sii<f<kkz<z/!! Gxqh<H: ogiMg<gh<hm<Mt<t (x1, y1) lx<Xl< (x2, y2) Ngqb!-V!Ht<tqgtqe<!upqOb!yOv!yV!
Ofi<g<OgiMkie<!ujvb!LcBl</!nf<OfIg<Ogim<ce<!sib<U = = ml<Ym<
xl<Wx< 12
12
xxyy
−−
.
wMk<Kg<gim<M 3: (5, 6) lx<Xl< (15, 9) Ngqb!Ht<tqgtqe<!upqOb!osz<Zl<!Ogim<ce<!sib<U!gi{<g/! OlZl<!ng<OgiM! OlOzx<xl<! ohXgqxki!nz<zK!gQpqxg<gl<! ogit<gqxki! weg<!%Xg/!kQi<U: (5,6)!J (x1, y1) we<Xl<? (15, 9) J (x2, y2) we<Xl<!ogit<g/!Ogim<ce<!sib<U
= ml<Ym<
xl<Wx< 12
12
xxyy
−−
= 103
51569=
−− , yV!lqjg!w{</
weOu?!OgimieK!)hml< 7.17 Jh<!hiIg<gUl<*!Olz<!Wx<xl<!ohx<Xt<tK/!! wMk<Kg<gim<M 4: (−16, 29), (40, −6) we<gqx!Ht<tqgtqe<!ugi{<g/! OgimieK! Olz<! Wx<xl<! ohXgqxki! nz<zK! g%Xg/!kQi<U:
Ogim<ce<!sib<U!= = ml<Ym<
xl<Wx<
)16(40296−−−−
= 85
5635 −
=− yV!Gjx!w{<<.
weOu!OgimieK!)hml<!7.18 Jh<!hiIg<gUl<*!gQp<!-xg<gl<!ohx<Xt<tK/ wMk<Kg<gim<M 5: gQOp!ogiMg<gh<hm<m!Ht<tqgjts<!Osi<g<Guqtg<Gg/!! (i) (6, 4) lx<Xl<!(−7, 4) (ii) (−2, 8) lx<XlkQi<U:
(i) Ogim<ce<!sib<U = 12
12
xxyy
−−
= ,0130
6744
=−
=−−−
∴ OgimieK!x-ns<Sg<G!-j{biGl</!
(ii)!Ogim<ce<!sib<U = 12
12
xxyy
−−
=01
2287 −
=+−− = ujvbXg
∴ OgimieK x-ns<Sg<Gs<!osr<Gk<kig!dt<tK/!
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hml<! 7.17
pqOb!osz<Zl<!Ogim<ce<!sib<U!Qp<! -xg<gl<! ogit<gqxki! weg<!
hml< 7.18
l<!Ogim<ce<!sib<Ugjth<!hx<xq!
< (−2, 7)
<gh<hmikK/!
wMk<Kg<gim<M 6: sib<U!53
− ogi{<M?!(−2, 3) we<x!Ht<tq!upqbigs<!osz<Zl<!Ogim<ce<!
lx<oxiV!Ht<tqjbg<!gi{<g/!!
kQi<U: Lkzqz<?! sib<U =53− we! wPKOuil<! (nkiuK, hGkqjb! lqjgbigg<!
ogit<Ouil<). (−2, 3) we<x! Ht<tqg<G! hml< 7.19 z<! dt<tK! Ohiz< P weh<ohbiqM/ P bqzqVf<K 5 nzGgt< x-ns<sqx<<G!-j{big!uzK!hg<glig!fgIf<K!!)Woeeqz<!Ym<ml<!= 5) Q (−2 + 5, 3) nkiuK, Q (3, 3)! we<x! Ht<tqjbs<! ose<xjmg/ Q uqzqVf<K!3 nzG!y-ns<sqx<<G!-j{big!gQpqxr<gq! )Woeeqz<?! -r<G! Wx<xl<! = −3) R(3, 3 + (−3)) nkiuK R (3, 0) we<x!Ht<tqjbs<! ose<xjmg/! R (3, 0) we<hK!Ogim<ce<! Olzjlf<k! lx<oxiV Ht<tq/!-kjes<!siqhii<g<g?!P, R Js<!Osi<g<Gl<!!
Ogim<ce<!sib<U =)2(3
30−−− =
53− .
!lix<Xupq;!(−2, 3) J!(x1, y1) we<g/!Ogim<cz<!lx
sib<U!= 12
12
xxyy
−−
=23
2
2
+−
xy
/!!Neiz<, sib<U!=53−!
∴23
2
2
+−
xy
=53− nz<zK!−3x2 − 6 = 5y2 −15 nz<z
ogiMg<g!x2 gqjmg<Gl</!wMk<Kg<gim<mig, y2 = 03x2 = 9 nz<zK!x2 = 3. weOu, (3, 0) !NeK!Ogim 7.2.1 Ofi<g<Ogim<ce<!sle<hiM!
P(x,y) we<hK! ogiMg<gh<hm<m! Ogim<c
-Vg<gm<Ml</! lixqgt<! x, y -ux<jxs<! Osi<g<Glsle<hiM! NGl</! Ogim<cz<! njlf<k! yu<ouiV!Ngqbju!Ogim<ce<!sle<him<jm!fqjxU!osb<Blgii<Csqbe<! ktk<kqz<! Gxqg<g! OfIg<Ogim<ce<! usle<hiM, OgiM!we<Ox!njpg<gh<hMl</!Ogim<ce<!Ht<tqbqZl<, y-ns<js B we<gqx!yOv!yV!Ht<tqbnjlukiz<!nke<! y-ns<S!K~vl< 0 NGl</! a wwe<hK!Ogim<ce<!sle<him<jm!fqjxU!osb<Bl</!xsle<him<cz<! hqvkqbqm, a bqe<! lkqh<jhg<! gi{oum<Mk<K{<M! weh<hMl</! nkiuK, yV! Ogim<cens<js! sf<kqg<Gl<! Ht<tqbqe< x-ns<SK~vl<! NGl-Vh<hkiz<!nke< x-ns<S!K~vl<!0 NGl</!weOu(0, b) we<hK!Ogim<ce<!sle<him<jm!fqjxU!osb
177
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hml< 7.19
<oxiV!Ht<tqjb!(x2, y2) we<g/!weOu,
wek<!kvh<hm<Mt<tK/!!
K!3x2 + 5y2 = 9. -kqz<!y2 x<G!lkq<h<H!
weqz<, 3x2 + 5 × 0 = 9 nz<zK! <ce<!lQK!lx<oxiV!Ht<tqbiGl</!
z<! njlBl<! liXgqe<x! Ht<tqbig!<! -bx<g{qks<! sle<hiM! ng<Ogim<ce<!Ht<tqbqe<! x-ns<SK~vl<, y-ns<SK~vl<!</!-f<k!(x, y) we<x!uiqjs!Osicgjt!jvhml<! gqjmg<Gl</! -eq?! Ogim<ce<!ujvhml<? x-ns<js A we<gqx!yOv!yV!qZl<! sf<kqg<gqxK/ A NeK! x-ns<sqz<!e<hK!A e<! x-ns<SK~vl<! weqz<, (a, 0) g<G a JBl<, y g<G 0 JBl<!Ogim<ce<!zil</! a bqe<! lkqh<H, Ogim<ce< x < x oum<Mk<K{<mieK, OgimieK! x-</! -Ok! Ohie<X, B bieK y-ns<sqz<!, b we<hK!B bqe< y-ns<SK~vl<!weqz<? <Bl</!weOu?!x g<Gh<!hkqzig!0 juBl<,
y g<Gh<!hkqzig b jbBl< Ogim<ce<!sle<him<cz<!hqvkqbqm, b e<!kQIjug<!gi{zil</ b e<!-f<k! lkqh<H, Ogim<ce<! y oum<Mk<K{<M! weh<hMl</! weOu?! OgimieK! wf<k! Ht<tqbqz<!!!y-ns<js! oum<MgqxOki, nf<kh<! Ht<tqbqe< y ns<SK~vl<, Ogim<ce< y oum<Mk<K{<M!weh<hMl<. kx<ohiPK! sib<U m we<Xl<, y oum<Mk<K{<M! c we<Xl<! ogi{<m! Ogim<ce<!sle<him<jm!hqe<!uVliX!gi{<Ohil</
hml< 7.20
!Ogim<ce<! y oum<Mk<K{<M! c big!
-Vh<hkiz<! P1(0, c) we<hK, OgimieK y-ns<js! oum<Ml<OhiK (hml< 7.20 Jh<!hiIg<gUl<) Wx<hMl<!Ht<tq/ P(x, y) we<hK!Ogim<ce<! OuX! WOkEl<! yV!Ht<tqbigm<Ml</!hqe<ei< Ogim<ce<!sib<U!!
= 0−
−x
cy (n.K)!x
cy − .
Neiz< Ogim<ce<! sib<U m we<X!ogiMg<gh<hm<Mt<tK/!!
∴ x
cy − = m (n.K) y − c = mx (n.K) y = mx + c.
OlZt<t!sle<himieK!yV!Ogim<ce<!sle<him<ce<!sib<U.oum<Mk<K{<M!$k<kqvliGl</! Gxqh<H: OgimieK!Nkqh<Ht<tq (0, 0) upqbigs<!ose<xiz<, nke< y oum<Mk<K{<M c = 0. weOu?!Ogim<ce<!sle<hiM y = mx + 0 nz<zK y = mx.
wMk<Kg<gim<M 7: sib<U 21 NgUl<, y-oum<Mk<K{<M −3 NgUl<!dt<t!Ogim<ce<!sle<hiM!
gi{<g/ kQi<U: sib<U.oum<Mk<K{<M!$k<kqvk<jkh<!hbe<hMk<k, Ogim<ce<!sle<hiM!!
y = 21 x + (−3) m =
21
c = −3 y = mx + c
nz<zK 2y = x − 6 nz<zK x− 2y − 6 = 0.
wMk<Kg<gim<M 8: 3x + 4y + 5 = 0 we<x!Ogim<ce<!sib<U, y-oum<Mk<K{<M!Ngqbux<jxg<!gi{<g/
kQi<U: sle<him<jm!lix<xq!wPKjgbqz<!gqjmh<hK, 4y = −3x − 5 (n.K) y = .45
43
⎟⎠⎞
⎜⎝⎛ −+
− x
-f<ks<!sle<him<jm, y = mx + c !dme<!yh<hqMjgbqz<?!!
sib<U!m = 43− , y-oum<Mk<K{<M! c =
45− .
hbqx<sq 7.2 1. ogiMg<gh<hm<m!-V!Ht<tqgjts<!OsIg<Gl<!Ogim<ce<!sib<jug<!gi{<g/!
(i) (−4, 1) lx<Xl< (−5, 2). (ii) (4, −8) lx<Xl< (5, −2). (iii) (−5, 0) lx<Xl< (0, −8). (iv) (0, 0) lx<Xl< ( 3 , 3). (v) (2a, 3b) lx<Xl< (a, −b). (vi) (a, 0) lx<Xl< (0, b).
178
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2. ogiMg<gh<hm<m!!Ht<tq!upqbigUl<, kvh<hm<m!sib<juBl<!ogi{<m!Ogim<ce<!lx<oxiV!!!!Ht<tqjbg<!gi{<g/!
(i) Ht<tq!!(5, 6), sib<U 1. (ii) Ht<tq! (0, 4), sib<U 41 .
(iii) Ht<tq! (2, −2), sib<U −1. (iv) Ht<tq! (1, −3), sib<U!4. (v) Ht<tq! (−1, −4), sib<U!
37 .
3. sib<Ul< , y- oum<Mk<K{<Ml<!LjxOb!ogiMg<gh<hm<Mt<te/!Ogim<ce<!sle<him<jmg<!!!!!gi{<g/!
(i) −3 lx<Xl< −7. (ii) 5 lx<Xl< 9. (iii) −2 lx<Xl< 15. (iv) 6 lx<Xl< −11. (v)
53− lx<Xl< 1. (vi)
52− lx<Xl<
58 .
4. ogiMg<gh<hm<<m!Ogim<ce<!sib<U, y- oum<Mk<K{<M!gi{<g/ (i) 3x + 2y = 4 (ii) 2x = y (iii) x − y − 3 = 0 (iv) 5x − 4y = 8 7.3 (x1, y1) lx<Xl< (x2, y2) Ngqb! -V! Ht<tqgTg<G! -jmObBt<t!okijzU!
-V!Ht<tqgTg<G!-jmObBt<t!okijzU!we<hK!ucuqbzqz<!yV!nch<hjmg<!
gVk<KVuiGl</!-h<OhiK!nkx<gie!-bx<g{qkg<!Ogijujbg<!g{<mxqOuil</!!
P1 (x1, y1), P2 (x2, y2) we<hju! gii<Csqbe<! ktk<kqz<! njlf<k! -V!
Ht<tqgtigm<Ml</ P1, P2 -ux<xqx<G!-jmObBt<t! okijzju d(P1, P2) nz<zK P1P2
weg<Gxqh<Ohil</ 21PP we<x!Ogim<Mk<K{<jm!ujvg/!&e<X!ujggt<!wPgqe<xe/
ujg (i): Ogim<Mk<K{<M 21PP NeK! x-ns<Sg<G -j{big!dt<tK! )hml<!7.21 Jh<!hiIg<gUl<). -r<G y1 = y2. P1L, P2M Ngqbux<jx!x-ns<Sg<G!osr<Gk<kig!ujvg/!P1P2ML yV!osu<ugliGl</!weOu!P1P2 = LM. nkiuK, d(P1, P2) we<hK!L lx<Xl< M we<heux<xqx<G! -jmObBt<t!
okijzU/ Neiz<, L lx<Xl< M we<x!Ht<tqgt<!!x-ns<sqe<!lQKt<te/!-h<Ht<<tqgjt!LjxObx1, x2
we<gqx! olb<ob{<gt<! x-ns<sigqb! w{<Ogim<cz<!Gxqg<gqe<xe/! nkeiz<! LM e<! fQtl<!!!
LM = .21 xx − weOu!d (P1, P2) = 21 xx − .
ujg (ii): 21PP we<gqx! Ogim<Mk<K{<M
y-ns<Sg<G! -j{big!dt<tK (hml<! 7.22 Jh<!hiIg<gUl<). -r<G!x1 = x2 . P1L, P2M!Ngqbux<jx!y-ns<Sg<G! osr<Gk<kig! ujvg/ P1P2ML yV!osu<ugliGl</!weOu!P1P2 = LM. nkiuK,
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hml< 7.21
!!!
!!!
hml< 7.22
d(P1, P2) we<hK L lx<Xl< M we<heux<xqx<G!-jmObBt<t!okijzU/!Neiz<, L lx<Xl< M we<x! Ht<tqgt<! y-ns<sqe<! lQKt<te/! -h<Ht<tqgjt! LjxOb! ! y1, y2 we<gqx!
olb<ob{<gt<!y-ns<sigqb!w{<Ogim<cz<!Gxqg<gqe<xe/!nkeiz<?!LM e<!fQtl<!= 21 yy − .
weOu d(P1, P2) = 21 yy − .
ujg!(iii): Ogim<Mk<K{<M 21PP NeK x-ns<Sg<Gl<!-j{bqz<jz; y-ns<Sg<Gl<!!-j{bqz<jz! (hml< 7.23 Jh<!hiIg<gUl<). P1 upqbig x-ns<Sg<G! -j{big! yV!OgiMl<, P2 upqbig!y-ns<Sg<G!-j{big!yV! OgiMl<! ujvg. -g<OgiMgt< P3 z<!
oum<mm<Ml</! hqe<ei<! P3, (x2, y1) NGl</!Ogim<Mk<K{<M!P1P3e<!fQtl< 21 xx − NGl</!
Ogim<Mk<K{<M! P3P2e<! fQtl<! 21 yy − .
∆P1P3P2 we<hK! yV! osr<Ogi{!Lg<Ogi{l<! we<hjk! fil<! gi{<gqOxil</!
weOu!hqkigv <̂!Okx<xk<kqe<!hc,
hml< 7.23
∴ ( )[ ] ( )[ ] ( )[ ] 221
221
223
231
221 ,,, yyxxPPdPPdPPd −+−=+=
= (x1 − x2)2 + (y1 − y2)2 = (x2 − x1)2 + (y2 − y1)2
∴ d(P1, P2) = 212
212 )()( yyxx −+− .
-K!okijzU!gi[l<!$k<kqvl<!NGl</!-K!(x1, y1) lx<Xl< (x2, y2) Ngqb!ogiMg<gh<hm<m!-V! Ht<tqgTg<G! -jmObBt<t! okijzU! d Jk<! kVl</! gqjmg<Ogim<cOzi! nz<zK!Gk<Kg<Ogim<cOzi!njlbik!-V!Ht<tqgTg<gie!$k<kqvl<!ohxh<hm<Mt<tK/!-r<G!fil<!
gueqg<g!Ou{<cbK!d(P1, P2) = d(P2, P1). OlZl<! -f<k! $k<kqvl<! wf<k! -V! Ht<tqgTg<Gl<! ohiVf<Kl</! P1 lx<Xl<! P2
Ngqbju!yOv!gqjmg<Ogim<cz<!njlf<kiz<, y1 = y2 NGl</!Njgbiz<
d(P1, P2) = 212
212 )()( yyxx −+− = 22
12 0+− xx = 12 xx − . P1 lx<Xl<!P2 Ngqbju!yOv!Gk<Kg<Ogim<cz<!njlf<kiz<, x1 = x2 NGl</!Njgbiz<!
d(P1, P2) = 221
221 )()( yyxx −+− = 2
2120 yy −+ = .21 yy −
!Gxqh<H: Nkqh<Ht<tq O NeK (0, 0). P (x, y) WOkEl<!yV!Ht<tq!weqz<, OP = 22 )0()0( −+− yx = .22 yx +
-f<kk<!okijzU 22 yx + we<hK Nkqh<Ht<tqbqzqVf<K, (x, y) we<gqx!Ht<tqbqe<!Njv!oug<mi<! weh<hMl</!!!!
!
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okijzU!$k<kqvk<jkh<!hbe<hMk<kq, fil<! (i) &e<X!Ht<tqgt<!yOv!Ogim<czjlgqe<xeui!nz<zK!yV!osr<Ogi{!Lg<Ogi{l<!
nz<zK!-Vslhg<g!Lg<Ogi{l<!nz<zK!slhg<g!Lg<Ogi{k<jk!dVuig<Gli!!we<X!Nvibzil</!
(ii) fie<G!Ht<tqgt<, -j{gvl<, osu<ugl<, sKvl< nz<zK sib<sKvl<!Ngqbux<jx!dVuig<Gli!we<X!Nvibzil</!
!wMk<Kg<gim<M 9: A(−15, −3) lx<Xl< B (7, 1) -ux<xqx<gqjmObBt<t!okijzjug<!gi{</!kQi<U: A lx<Xl< B -ux<xqx<gqjmObBt<t!okijzU d !we<g/!d (A, B) = 2
122
12 )()( yyxx −+− (x1, y1) (−15, −3)
(x2, y2) (7, 1) = 22 )31()157( +++
= 22 422 + = 16484 + = 500 = 510 .
wMk<Kg<gim<M 10: (−4, −9), (2, 0) lx<Xl< (4, 3) Ngqbju!yOv!Ogim<czjlBl<!Ht<tqgt<!weg<gim<Mg/!kQi<U: A, B lx<Xl< C!we<he!LjxOb?!ogiMg<gh<hm<m!Ht<tqgtig!-Vg<gm<Ml</!AB = 22 )90()42( +++
A (−4, −9)
B (2, 0)
likqiqh<!hml<!
hml<!7.24
= 22 96 + = 8136 + = 117 = 133139 =× .
BC = 22 )03()24( −+− C (4, 3)
= 22 32 + = 94 + = 13 .
AC = 22 )93()44( +++
= 22 128 + = 14464 + = 208 = 1341316 =× . ∴ -r<G 13413133 =+ . AB + BC = AC we<hjkg<!gi{<gqOxil</ ∴ A, B lx<Xl< C yOv!Ogim<czjlBl<!Ht<tqgt</! wMk<Kg<gim<M 11: (3, −2), (2, 5) lx<Xl< (8, −7) Ht<tqgt<!yV!-Vslhg<g!Lg<Ogi{k<jk!njlg<Gl<!weg<!gim<Mg/!kQi<U: ogiMg<gh<hm<m! Ht<tqgt<! LjxOb P, Q lx<Xl< R we<g. ∆PQR yV!-Vslhg<g!Lg<Ogi{l<!we fq'hqg<g! nkEjmb! -V! hg<gr<gt<!slfQtLt<tju!weg<gim<mzil<. -h<ohiPK?!
d(P,Q) = 22 )25()32( ++− = 49171 22 +=+
,2550 ==
d (Q, R) = 1814436126)57()28( 2222 =+=+=−−+−
d (R, P) = 5025255)5()72()83( 2222 =+=+−=+−+−∴ d (P, Q) = d (R, P) ≠ d (Q, R). ∴ ∆PQR yV!-Vslhg<g!hiIg<gUl<).!Neiz<!slhg<gLg<Ogi{lz<z/!
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hml< 7.25
,560 =
.25= Lg<Ogi{l<!(hml<!7.25 Jh<
wMk<Kg<gim<M 12: (0, 3), (0,1) lx<Xl< ( )2,3 Ngqbju!yV!slhg<g!Lg<Ogi{k<kqe<!ds<sqgt<!weg<!gim<Mg/ kQi<U: ogiMg<gh<hm<m!Ht<tqgt<!LjxOb A, B lx<Xl< C Ng!-Vg<gm<Ml<. ∆ABC yV!slhg<gLg<Ogi{l<!we fq'hqg<g!nke<! wz<zih<! hg<gr<gTl<! sl!fQtlieju!weg< gim<mzil</!-r<G d (A, B) = ,24)2(0)31()00( 2222 ==−+=−+−
d (B, C) = ,2413)12()03( 22 ==+=−+−
d (C, A) = .2413)23()30( 22 ==+=−+− ∴ d (A, B) = d (B, C) = d (C, A). ∴∆ABC yV!slhg<g!Lg<Ogi{l<!(hml<!7.26 Jh< hiIg<gUl<*/! wMk<Kg<gim<M 13: P (7, 1), Q (−4, −1) lx<Xl< R (4, 5) !yV!osr<Ogi{!Lg<Ogi{k<kqe<!ds<sqgti!we!Nvib<g/!kQi<U:!∆PQR!yV!osr<Ogi{!Lg<Ogi{l<!weg<gim<m!YI!ds<sqg<Ogi{l<!90°!weg<gim<m!Ou{<Ml</!-kx<G!Lg<Ogi{k<kqe<!hg<gr<gtqe<!fQtr<gt<!hqkigv^<!Okx<xk<jk!fqjxU!!osb<b!Ou{<Ml</!-r<Og!
PQ = ,551254121)11()74( 22 ==+=−−+−−
QR = ,101003664)15()44( 22 ==+=+++
PR = .525169)15()74( 22 ==+=−+− ∴ PQ2 = 125, QR2 = 100 lx<Xl< PR2 = 25. QR2 + PR2 = PQ2 we<hjkg<!gi{<gqOxil</!weOu!hqkigv <̂!$k∴ ∆PQR yV!osr<Ogi{!Lg<Ogi{l<? ∠R = 90°. wMk<Kg<gim<M 14: (1, 2), (2, −1), (5, 3) lx<Xl< (4, 6) we<x!uiYI!-j{gvk<kqje!njlg<Gl<!weg<gim<Mg/!-K!yV!osu<ukQi<U:!ogiMg<gh<hm<m!Ht<tqgt<!LjxOb P1, P2, P3 lx<Xl< P4 NP1 P2 P3 P4 YI!-j{gvl<!weg<gim<Mukx<G!yV!upqLjx?!nfQtr<gt<!sll<!weg<gim<MukiGl</!-r<Og!
P1P2 = 1091)21()12( 22 =+=−−+− ,
P2P3 = ,525169)13()25( 22 ==+=++−
P3P4 = ,1091)36()54( 22 =+=−+−
P4P1 = .525169)62()41( 22 ==+=−+−
∴ P1P2 = P3P4 = 10 lx<Xl< P2P3 = P4P1 = 5. ∴ P1P2 P3P4 we<hK!YI!-j{gvl</!-r<G?!P1P3 = 17116)23()15( 22 =+=−+− lx<Xl< (P1P2 )2 + (P2P3)2 = 10 + 25 = 35, (P1P3)2 = 17, (P1P2 )2 + (P2P3)∴ ∆P1P2 P3 we<hK!osr<Ogi{!Lg<Ogi{lz<z/!∴ ∠P1P2 P3 w∴ P1P2 P3P4 we<hK!yV!osu<ugl<!-z<jz.
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hml<!8.27
hml< 7.26
<kqvl<!siqhiIg<gh<hm<mK/!!
qjsbqz<!njlf<k!Ht<tqgt<!gli@!Nvib<g/!g!-Vg<gm<Ml</ kEjmb!wkqi<hg<gr<gtqe<!
hml< 7.282 ≠ (P1P3)2. e<hK!osr<Ogi{lz<z/
wMk<Kg<gim<M 15: (0, −1), (−2, 3), (6, 7) lx<Xl< (8, 3) we<x! uiqjsbqz<! wMk<Kg<!ogit<th<hm<m!Ht<tqgt<!yV!osu<ugk<jk!njlg<Gl<!weg<!gim<Mg/!kQi<U: ogiMg<gh<hm<m!Ht<tqgt<!A, B, C lx<Xl< D Ng!-Vg<gm<Ml;. ABCD yV!osu<ugl<!weg<gim<Mukx<G! yV! upqLjx?! nke<! wkqIh<hg<gr<gt<! slfQtLt<tju! we<hkiGl</!OlZl<! nke<! YI! ds<sqg<Ogi{! ntU! 90°! we<hkiGl</! YI! ds<sqg<Ogi{l<! 90°!weg<gim<Mukx<G! yV! upqLjx?! ∆ABC e<! hg<gr<gt<! hqkigv <̂! Okx<xk<jk! fqjxU!osb<gqxK!weg<gim<MuOk/!-r<Og!fil<!gi{<hK!
AB = ,5220164)13()02( 22 ==+=++−−
BC = ,54801664)37()26( 22 ==+=−++
CD = ,5220164)73()68( 22 ==+=−+−
AD = .54801664)13()08( 22 ==+=++−
AC = .101006436)17()06( 22 ==+=++−
∴ AB = CD = 52 , BC = AD = 54 lx<Xl<! AB2 + BC2 = 20 + 80 = 100 = AC2. ∴ ABCD !yV!osu<ugl</ wMk<Kg<gim<M 16: (0, −1), (2, 1) (0, 3) lx<Xl< (−2, 1) we<x!uHt<tqgt<!yV!sKvk<kqe<!ds<sqgt<!weg<gim<Mg/!kQi<U:!ogiMg<gh<hm<m!Ht<tqgt<!LjxOb A, B, C, D we<g. ABCD yV!sKvl<!weg<gim<Mukx<G!nkEjmb!hg<gr<gt<! slfQtLjmbju! lx<Xl< &jz!uqm<mr<gt<! slfQtLt<tju!weg<gim<MuK yV!upq!NGl</!-r<G!
AB = ,22844)11()02( 22 ==+=++−
BC = ,22844)13()20( 22 ==+=−+−
CD = ,22844)31()02( 22 ==+=−+−−
AD = ,22844)11()02( 22 ==+=++−−
BD = ,416016)11()22( 22 ==+=−+−−
AC = .416160)13()00( 22 ==+=++−
∴AB = BC = CD = AD = 22 lx<Xl< BD = AC = 4!weg<!gi∴ ABCD yV!sKvl</!!wMk<Kg<gim<M 17: A(2, −3), B(6, 5), C(−2, 1) lx<Xl< D(−6,ogit<th<hm<m!Ht<tqgt<!yV!sib<sKvk<jk!d{<mig<Gl<;!gim<Mg/ kQi<U: ABCD ! yV! sib<sKvliGl<! weg<! gim<Mukx<G!hg<gr<gTl<! sll<! weg<! gim<MuKkie</! yV! sib<sgim<Mukx<G! yV! upq?! nkEjmb! &jzuqm<mr<gtgim<MuKkie</!-r<G
AB = 22 )35()26( ++− = 6416 + = 80 ,
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hml< 7.29
!Neiz<!sKvlz<z/!
iqjsbqz<!wMk<Kg<ogit<th<hm<m!
hml< 7.30
{<gqOxil</!
−7) we<x!uiqjsbqz<!wMk<Kg<!Neiz<!nK!sKvlz<z!weg<!
yV! upq?! nkEjmb! wz<zi!KvlieK?! sKvlz<z! weg<!<! slfQtlz<zikju! weg<!
BC = 22 )51()62( −+−− = 801664 =+ ,
AC = 22 )31()22( ++−− = 321616 =+ ,
BD = 22 )57()66( −−+−− = 144144+ = 288 ,
CD = 22 )17()26( −−++− = 806416 =+ ,
AD = 22 )37()26( +−+−− = 801664 =+ . ∴ AB = BC = CD = AD, AC ≠ BD. ∴ ABCD we<hK!yV!sib<sKvl<;!Neiz<!nK!sKvlz<z/!
hbqx<sq!7.3 1. !gQOp!ogiMg<gh<hm<Mt<t!Osic!Ht<tqgTg<G!-jmObBt<t
(i) (1, 2) lx<Xl< (4, 3) (vi) (a, −b) lx<Xl< (−b, (ii) (3, 4) lx<Xl< (−7, 2) (vii) )1 ,12( + lx<Xl< (
(iii) (−7, 2) lx<Xl< (3, 2) (viii) ⎟⎠⎞
⎜⎝⎛
45,
32 lx<Xl< ( 1−
(iv) (4, −5) lx<Xl< (−4, 5) (ix) (2, 0) lx<Xl< (5, −4 (v) (a, b) lx<Xl< (b, a) (x) (−2, 3) lx<Xl< (−1,
2.!!gQOp!ogiMg<gh<hm<m!Ht<tqgt<!yOv!Ogim<czjlBl<!Ht<tqg
(i) (5, 2), (3, −2) lx<Xl< (8, 8) (ii) (21 , 1), (1, 2) lx
(iii) (1, 4), (3, −2) lx<Xl< (−3, 16) (iv) (−4, 8), (2, −4) l (v) (8, 4), (5, 2) lx<Xl< (9, 6).
3.! gQOp! ogiMg<gh<hm<m! Ht<tqgt<! -Vslhg<g! Lg<Ogi{kNvib<g/!
(i) (5, 4), (2, 0) lx<Xl< (−2, 3). (ii) (6, − 4), (−2, − 4(iii) (2, −1), (− 4, 2) lx<Xl< (2, 5).
4. gQOp!ogiMg<gh<hm<m!Ht<tqgt<!slhg<g!Lg<Ogi{k<jk!d{ (i) (− 3 , 1), (2 3 , −2) lx<Xl< (2 3 , 4).
(ii) ( 3 , 2) , (0, 1) lx<Xl< (0, 3).
(iii) (0, 3) (0, 5) lx<Xl< ( 3 , 4). 5. !gQOp!ogiMg<gh<hm<m!Ht<tqgt<!osr<Ogi{!Lg<Ogi{k<kqe<!
(i) (4, 4), (3, 5) lx<Xl< (−1, −1). (ii) (2, 0), (−2, 3) lx6. !!gQOp!ogiMg<gh<hm<m!ds<sqgjtg<!ogi{<m!Lg<Ogi{l<!wgi{<g/!!
(i) (−3, 7), (−4, 0) lx<Xl< (−10, 8). (ii) (−5, −2), (0, 6) l
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hml<!7.31
!okijzjug<!gi{<g/!a) 1, 3)
)2,
) −5) t<!weg<gim<Mg/!
<Xl< (0, 23 )
x<Xl< (3, 16)
<jk! d{<mig<Gli! we!
) lx<Xl< (2, 10).
<mig<Gli!we!Nvib<g/!
ds<sqgti!we!Nvib<g/!
<Xl< (−2, −5). u<ujg!Lg<Ogi{l<!weg<!
x<Xl< (8, 1).
7. gQOp!ogiMg<gh<hm<m!uiqjsbqz<!njlf<k!Ht<tqgt<!-j{gvk<jk!njlg<Gli!we!Nvib<g/!
(i) (3, −5), (−5, −4), (7, 10) lx<Xl< (15, 9). (ii) (5, 8), (6, 3), (3, 1) lx<Xl< (2, 6). (iii) (6, 1), (5, 6), (−4, 3) lx<Xl< (−3, −2). (iv) (0, 3), (4, 4), (6, 2) lx<Xl< (2, 1).
8. gQOp!ogiMg<gh<hm<m!uiqjsbqz<!njlf<k!Ht<tqgt<!yV!osu<ugk<jk!njlg<Gli!we!Nvib<g/!
(i) (8, 3), (0, −1), (−2, 3) lx<Xl< (6, 7). (iii) (−3, 0), (1,−2), (5, 6) lx<Xl< (1, 8). (ii) (−2, 7), (5, 4), (−1, −10) lx<Xl< (−8, −7). (iv) (−1, 1), (0, 0) (3, 3) lx<Xl< (2, 4). 9. gQOp!ogiMg<gh<hm<m!uiqjsbqz<!njlf<k!Ht<tqgt<!yV!sKvk<kqje!njlg<Gli!we!Nvib<g/!
(i) (1, 2), (2, 2), (2, 3) lx<Xl< (1, 3). (ii) (−1, −8), (4, −6), (2, −1) lx<Xl< (−3, −3). (iii) (1, −1), (0, −4), (7, −3) lx<Xl< (8, −10). (iv) (12, 9), (20, −6), (5, −14) lx<Xl< (−3, 1). (v) (−1, 2), (1, 0), (1, 4) lx<Xl< (3, 2). 10. gQOp!ogiMg<gh<hm<m!uiqjsbqz<!njlf<k!Ht<tqgt<!yV!sib<sKvk<kqe<!ds<sqgti!we!Nvib<g/!
(i) (0, 0), (3, 4), (0, 8) lx<Xl< (−3, 4). (ii) (2, −3), (6, 5), (−2, 1) lx<Xl< (−6, −7). (iii) (1, 4), (5, 1), (1, −2) lx<Xl< (−3, 1)
!!!!!!!!!!!!!!!!!!!!!!!
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uqjmgt<!
hbqx<sq 7.1 1. (i) I (ii) I (iii) III (iv) IV (v) !!wf<k!gix<hGkqBl<!-z<jz! (vi) !wf<k!gix<hGkqBl<!-z<jz (vii) !!wf<k!gix<hGkqBl<!-z<jz! (viii) II 2. (i) !kuX (ii) kuX (iii) !kuX!!(iv) !siq!! (v) siq!!(vi) siq! (vii) !!siq! (viii) !kuX (ix) !!kuX!!(x) !kuX!!(xi) !!kuX (xii)!siq! 3. (i) (2, 1) (ii) (6, 9) (iii) (10, 7) (iv) (2, 2)
hbqx<sq 7.2
1. (i) −1 (ii) 6 (iii) 58− (iv) 3 (v) ⎟
⎠⎞
⎜⎝⎛
ab 4 (vi)
ab−
2. (i) (6, 7) (ii) (4, 5) (iii) (3, −3) (iv) (2, 1) (v) (2, 3) 3. (i) 3x + y + 7 = 0 (ii) 5x − y + 9 = 0 (iii) 2x + y − 15 = 0 (iv) 6x − y − 11 = 0 (v) 3x + 5y − 5 = 0 (vi) 2x + 5y − 8 = 0
4. (i) ⎟⎠⎞
⎜⎝⎛ − 2 ,
23 (ii) (2, 0) (iii) (1, −3) (iv) ⎟
⎠⎞
⎜⎝⎛ − 2 ,
45
hbqx<sq 7.3
1. (i) 10 (ii) 262 (iii) 10 (iv) 412 (v) (a − b) 2
(vi) (a + b) 2 (vii) 6 (viii) 12481
(ix) 5 (x) .65 2. (i) !yOv!Ogim<czjlBl<!!(ii) yOv!Ogim<czjlbiK!!(iii) !!yOv!Ogim<czjlBl<
(iv) yOv!Ogim<czjlbiK (v) !yOv!Ogim<czjlbiK! 3. (i) -Vslhg<g!Lg<Ogi{l<!(ii) !-Vslhg<g!Lg<Ogi{l< (iii) !-Vslhg<g!Lg<Ogi{l<! 4. (i) slhg<g!Lg<Ogi{l< (ii) !!slhg<g!Lg<Ogi{l<
(iii)!!!slhg<g!Lg<Ogi{l<! 5. (i) osr<Ogi{!Lg<Ogi{l< (ii) !osr<Ogi{!Lg<Ogi{l<!nz<z! 6. (i) -Vslhg<g!osr<Ogi{!Lg<Ogi{l<!(ii) -Vslhg<g!osr<Ogi{!Lg<Ogi{l< 7. (i) -j{gvl< (ii) -j{gvl< (iii) -j{gvl< (iv) -j{gvl< 8. (i) osu<ugl<!!!(ii) !osu<ugl< (iii) !!osu<ugl< (iv) !!osu<ugl< 9. (i) !sKvl< (ii) !sKvl<!(iii) sKvlz<z!!(iv) !sKvl<!!(v) !sKvlz<z 10. (i) sib<!sKvl< (ii) sib<!sKvl< (iii) sib<!sKvl<
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8. Lg<Ogi{uqbz<
hz! F~x<xi{<MgTg<G! Le<Oh! uieuqbz<! hx<xq! nxqukx<G?! g{qkuqbzqz<!!
Lg<Ogi{uqbz<!we<El<!hqiqU!Okix<Xuqg<gh<hm<mK/!!aqh<hiIg<g <̂!)Hipparchus*!we<El<!!gqOvg<g! uieuqbz<! lx<Xl<! ! g{qk! uz<Zfi<?! Lg<Ogi{uqbjz! uqiquig<gq! nke<!uqkqgjth<! ohVltU! ! hbe<hMk<kq! uie<outqh<! ohiVm<gtqe<! -br<G! hijkgjtBl<?!fqjzgjtBl<! ! fqi<{bl<! osb<kii</! weOu! -ujvOb?! Lg<Ogi{uqbzqe<! kf<jk!we<xjph<hK! ohiVk<kliekiGl</! ! Lg<Ogi{uqbjzg<! Gxqg<Gl<! Nr<gqzs<! osiz<zie!
‘Trigonometry’ NeK? Lg<Ogi{l<! we<x! ohiVTjmb! ÄTrigon}! lx<Xl<! ntU! we<x!ohiVTjmb! ÄMetra}!Ngqb!-V!gqOvg<gs<! osix<gtqzqVf<K!ohxh<hm<mkiGl</! ! weOu!Lg<Ogi{uqbz<!we<hK!yV!Lg<Ogi{k<kqe<!hg<gr<gtqe<!ntUgTg<Gl<?!Ogi{r<gtqe<!ntUgTg<Gl<! -jmOb! njlf<k! okimi<Hgjth<hx<xq! nxqBl<! himh<hGkqbiGl</!Lg<Ogi{uqbz<!himk<kqje!hcg<gk<!okimr<Gukx<G!Le<?!!fil<!Ogi{r<gt<?!nux<xqe<!ntUgt<!Ngqbju!hx<xq!Wx<geOu!nxqf<kjk!lQ{<Ml<!fqjeuqx<!ogit<Ouil</ Ogi{r<gTl<!!nux<xqe<!ntUgTl<
yV! ohiKh<! Ht<tqbqzqVf<K! okimr<Gl<! -V! gkqi<gtiz<! yV! Ogi{l<!njlg<gh<hMgqxK/! ! yV! gkqi<?! Ogi{k<kqe<! okimg<gg<gkqI! )okimg<gh<hg<gl<*! weUl<?!lx<oxiV!gkqi<?!Ogi{k<kqe<!LcUg<gkqI!)LcUh<hg<gl<*!weUl<!%xh<hMl</!!ohiKh<Ht<tq?!Ogi{k<kqe<!Lje! weh<hMl</!LjebqzqVf<K! okimr<Gl<! gkqovie<X! okimg<gh<! hg<g!fqjzbqzqVf<K!LcUh<!hg<g!fqjz!ujv!SpZukiz<!Ogi{l<!njlgqe<xK/ gkqi
nz
nxq
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hml<!8.1!
qe<! Spx<sq! gcgiv! Lt<! Spx<sqbqe<! wkqI
<zK! gcgiv!Lt<! Spx<sqbqe<<! kqjsbqOzi!
bzil</! ! gkqi<gt<! OA Ul<! OB Bl<! LjUg<jg!weqz<?!!!Ogi{k<jkg<!GxqbQm<cz<!∠
187
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kqjsbqOzi! )hml<! 8.1 Jh<! hiIg<gUl<*!)hml< 8.2! Jh<! ! hiIg<gUl<*! njlujk!xOb?! yV! Ogi{k<kqe<! okimg<gg<jg?!
AOB!weg<!Gxqh<Ohil</!
nu<uh<OhiK?! gii<Csqbe<! ktk<kqz<!Nkqh<Ht<tqjb!!
LjebigUl<?! x-ns<js! okimg<gg<gkqvigUl<!
ogi{<M?! hml<! 8.3! -z<! gi{<hKOhiz<!Ogi{k<kqje! njlh<Ohil</! ! yV! Ogi{lieK!Olx<%xqb! upqbqz<! fqjzh<hMk<kh<hce<?!ng<Ogi{lieK! ! kqm<m! fqjzbqz<! dt<tK!we<Ohil</! yV! Ogi{k<jk! ntg<g! hijg!we<xjpg<gh<hMl<!nzgqjeh<!hbe<hMk<Kgqe<Oxil</ hijg!ntU
yV!gkqi<?!gcgivLt<Spx<sqbqe<!wkqi<!kqjsbqz<!yV!LP!Spx<sqjb!Wx<hMk<Kl<!
ohiPK?! nr<G! 360! hijggt<! )-K! 360o! we! wPkh<hMl<*! ntUt<t! yV! Ogi{l<!
njlukigg<! %Xgqe<Oxil</! lx<x! Ogi{r<gjt! 360o! Ogi{k<jk! nch<hjmbigg<!ogi{<M!ntg<gqe<Oxil</!wMk<Kg<gim<mig?!yV!gkqi<!Spx<sqjb!Wx<hMk<kuqz<jzobeqz<?!
nK! 0o!ntUt<t!Ogi{l<!njlg<gqe<xK!we<Ohil</! yV!gkqi<! gcgivLt<! Spx<sqbqe<!
wkqi<!kqjsbqz<?!LP!Spx<sqbq<z<!¼!higl<!Spx<sqjb!Wx<hMk<kqeiz<?!nK!¼ (360o*!= 90o!ntUt<t!Ogi{l<!njlg<gqe<xK!we<Ohil</!yV!gkqI!gcgivLt<!Spx<sqbqe<!kqjsbqz<?!
LPs<Spx<sqbqz<!¼!higl<!Spx<sqjb!Wx<hMk<Klieiz<?!nK!¼ (−360o*!= −90o!ntUt<t!Ogi{l<!njlg<gqe<xK!we<Ohil</!weOu?!gcgivLt<!Spx<sqbqe<!wkqi<kqjsbqz<!njlBl<!Spx<sqgt<! lqjg! Ogi{r<gjt! d{<mig<Ggqe<xe! weUl<?! gcgivLt<! Spx<sqbqe<!kqjsbqz<! njlBl<! Spx<sqgt<! Gjx! Ogi{r<gjt! d{<mig<Ggqe<xe! weUl<!
nxqgqe<Oxil</! -u<uk<kqbik<kqz<! fil<?! 0o! zqVf<K! 90o! ujv! njlBl<! Ogi{r<gjt!
lm<MOl!gVKgqe<Oxil</ 0o!zqVf<K!90o!g<Gt<!njlBl<!Ogi{l<!GXr<Ogi{l<!weh<hMl</!!90o! ntuqz<! njlBl<! Ogi{l<! osr<Ogi{l<! weh<hMl</! 180o! ntuqz<! njlBl<!
Ogi{l<! Ofi<g<Ogi{l<! weh<hMl</! -V! GXr<Ogi{r<gtqe<! %Mkz<! 90o! weqz<?! nju!
fqvh<Hg<Ogi{r<gt<!weh<hMl</!-V!lqjgOgi{r<gtqe<!%Mkz<!180o!weqz<?!nju!lqjg!fqvh<Hg<Ogi{r<gt<!weh<hMl</
osr<Ogi{!Lg<Ogi{Ll<?!hqkigv^<!Okx<xLl< !
yV! Lg<Ogi{k<kqz<! yV!
Ogi{k<kqe<! ntU! 90o! weqz<?! nl<!Lg<Ogi{l<! yV! osr<Ogi{! Lg<Ogi{l<!weh<hMl</! hml<! 8.4 z<! ABC yV!
osr<Ogi{! Lg<Ogi{l<?! ! -kqz<! ∠ABC we<x! Ogi{k<kqe<! ntU! 90° NGl</!
hg<gl<! AC !gi<{l<!weh<hMl</!!-KOu!!
!
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hml<!8.4
hml<!8.3
lqgh<ohiqb! hg<gl<?! OlZl<! -K! osr<Ogi{k<kqx<G! wkqOv! njlf<Kt<t! hg<gliGl</!!hqkigv <̂!we<x!gqOvg<g!g{qk!uz<Zfi<?!gi<{k<kqe<!lQK!njlBl<!sKvk<kqe<!hvh<htU?!lx<x!-V!hg<gr<gtqe<!lQkjlBl<!sKvr<gtqe<!hvh<htUgtqe<!%MkZg<Gs<!sll<!weg<!
g{<mxqf<kii</!!nkiuK?!hml<!8.4!-z<?! NGl</!!-KOu!hqkigv^<!Okx<xl<!weh<hMl</
222 BCABAC +=
8.1 Lg<Ogi{uqbz<!uqgqkr<gt< WkiuK! yV! GXr<Ogi{l<! ∠AOB J!
wMk<Kg<ogit<Ouil</!!-jk!gqOvg<g!wPk<kie θ Nz<! Gxqh<Ohil</ P! we<El<! Ht<tq! gkqi< OB bqe<! lQK! njlf<k! Ht<tq! we<g/! gkqi<! OA Ug<Gs<!osr<Gk<kig!P bqzqVf<K!Ogim<Mk<K{<M!PQ ujvg/! hqxG! Lg<Ogi{l<! OQP! yV!osr<Ogi{! Lg<Ogi{l<A! Lje! Q uqz<!osr<Ogi{l<! njlf<Kt<tK/! ! ∆OQP -z<!hg<gl<!!OP !!gi<{l<!!NGl</!!!hg<gl<!! PQ !!
hml< 8.5
Ogi{l<! θ uqe<! wkqi<hg<gliGl</! hg<gl<! OQ Ogi{l<! θ uqe<!nMk<Kt<t! hg<gliGl</!OP , PQ , OQNgqb! hg<gr<gtqe<! fQtr<gjt! LjxOb! OP, PQ, OQ! we<X!Gxqh<hqMgqe<Oxil</!-f<fQtr<gjtg<!ogi{<M!hqe<uVl<!NX!Lg<Ogi{uqbz<!uqgqkr<gjt!ujvbXg<gqe<Oxil</
sine θ = OPPQ
=fQtl<!kqe<{k<gi<
fQtl<!kqe<gk<wkqIhg< ,
cosine θ = OPOQ
=fQtl<!kqe<{k<gi<
fQtl<!kqe<gk<hg<!tKt<nMk< ,
tangent θ = OQPQ
=fQtl<!kqe<gk<hg<!tKt<nMk<
fQtl<!kqe<gk<hg<wkqi< ,
cosecant θ = PQOP
=fQtl<!kqe<gk<hg<wkqi<
fQtl<!kqe<{k<gi< ,
secant θ = OQOP
=fQtl<!kqe<gk<hg<!tKt<nMk<
fQtl<!kqe<{k<gi< ,
cotangent θ = PQOQ
=fQtl<!kqe<gk<hg<!wkqi<
fQtl<!kqe<gk<hg<!tKt<nMk< .
Olx<gi[l<!uqgqkr<gjts<?!SVg<glig!LjxOb?!sinθ , cosθ , tanθ , cosecθ , secθ,
cotθ we! wPKOuil</! ! Olx<gi[l<! uqgqkr<gtqe<!ntUgt<?! Ogi{l<! θ ju!lm<MOl!siIf<kjubiGl<A! osr<Ogi{! Lg<Ogi{l<! OQP -e<! ntjus<! sii<f<kkz<z/! -kje!nxqb? P′ we<El<! OuoxiV!Ht<tq! gkqi< OB bqe<! lQK! wMk<Kg<ogit<g;! OlZl< P′Q′ J OA g<G!osr<Gk<kig!ujvg (hml< 8.5 Jh<!hiIg<gUl<). Lg<Ogi{l< OQP l<!Lg<Ogi{l< OQ′P′ l<!ucouik<k!Lg<Ogi{r<gt<!we<hK!oktquiGl</!-kqzqVf<K!fil<?!
POOP
QOOQ
QPPQ
′=
′=
′′
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we!nxqOuil</!-kqzqVf<K!fil<!ohXl<!uqgqkr<gt<
QOQP
OQPQ,
POQO
OPOQ,
POQP
OPPQ
′′′
=′′
=′′′
=
nz<zK QPQO
PQOQ
QOPO
OQOP
QPPO
PQOP
′′′
=′′
=′′′
= ,,
we!nxqbzil</!!weOu!Olx<%xqb!NX!uqgqkr<gTl<!Ht<tq P, gkqi< OB bqe<!lQK!wf<k!fqjzbqz<! njlf<kqVf<kiZl<! lixiK! we! nxqbzil</! Olx<gi[l<! uqgqkr<gtqzqVf<K!filxquK?
1PQOP
OPPQ
θθ =×=cosec×sin , θ
θθ
θeccos1
=sin,sin
1=cosec .
1=×=sec×cosOQOP
OPOQ
θθ , θ
θ,θ
θsec
1coscos
1sec == .
1cotan =×=×PQOQ
OQPQθtθ .
θθ,
θ cot1tan
tan1cot ==θ .
OlZl<! fil<! gueqh<hK? θOQPQ
OQOP
OPPQ
OPOQOPPQ
θθ tan
cossin
==×=⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
= . kjzgQpqgjt! wMg<g?!
fil<!ohXuK! θθθ
θ cottan
1sincos
== . weOu?! θθθ,θ
θθ cot
sincostan
cossin
== NGl</
Gxqh<H: θ yV!GXr<Ogi{ligUl<?!θ uqe<!!Lg<Ogi{uqbz<!NX!uqgqkr<gtqz<!ye<xqe<!lkqh<H! lm<MOl! okiqf<k! fqjzbqz<?! OlOz! %xqb! $k<kqvr<gjth<! hbe<hMk<kq! lx<x!Lg<Ogi{uqbz<!uqgqkr<gjtg<!gi{!-bZl</!fl<!himh<!hGkqbqz<?!GXr<Ogi{r<gjt!lm<MOl!gVKgqe<Oxil</ !wMk<Kg<gim<M! 1:! hml<! 8.6 -z<! kvh<hm<Mt<t! osr<Ogi{!Lg<Ogi{k<kqzqVf<K! θ!uqe<!NX!Lg<Ogi{uqbz<!uqgqkr<gjtg<!gi{<g/!kQi<U;!hmk<kqz<!Ogi{l< θ uqx<G wkqi<hg<g!fQtl< = 6; nMk<Kt<t!hg<g!fQtl< = 8. hqkigv <̂!Okx<xh<hc? (gi<{k<kqe<!fQtl<)2 = 82 + 62 = 64 + 36 = 100. ∴ gi<{k<kqe<!fQtl< = 100 = 10. weOu?!
,=θ53
106sin ==
fQtl<!kqe<{k<gi<
fQtl<!ghg<wkqi< =cosec θ
54
108
==cos =θfQtl<!kqe<{k<gi<
fQtl<!gthg<Kt<nMk<, =secθ
n
43
=86
==tanfQtl<!gthg<Kt<nMk<
fQtl<!ghg<wkqi<θ , cot θ =
nM
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hml<!8.6
35
610
= =fQtl<!ghg<wkqi<
fQtl<!kqe<{k<gi<,
45
810
= =fQtl<!gthg<Kt<Mk<
fQtl<!kqe<{k<gi<,
34
68 ==
fQtl<!gwkqIhg<
fQtl<!gthg<Kt<k< .
wMk<Kg<gim<M!2: ∆ABC bqz< m∠B= 90°, AB = 8 os/lQ/, AC = 17 os/lQ/!weqz<?!Ogi{l< A lx<Xl<! Ogi{l<!C gtqe<!njek<K!Lg<Ogi{uqbz<!uqgqkr<gjtBl<!gi{<g/ kQi<U; hmk<kqz< A = m∠BAC lx<Xl<! C = m∠BCA (hml< 8.7 Jh< hiIg<gUl<). hqkigv <̂!Okx<xh<hc, AC2 = AB2 + BC2
hml< 8.7
∴ BC2 = AC2 – AB2 = 172 – 82 = 289 – 64 = 225.∴ BC = 225 = 15. weOu?
, A
A, A
A, A
A
,ABBC A,
ACAB A,
ACBC A
1517
sin1eccos
817
cos1sec
158
tan1cot
815tan
178cos
1715sin
======
======
, C
C, C
C, C
C
,BCAB C,
ACBC C,
ACAB C
817
sin1eccos
1517
cos1sec
815
tan1cot
158tan
1715cos
178sin
======
======
Gxqh<H: Olx<g{<m! g{g<gqz<? sin C = cos A, cos C = sin A, tan C = cot A,… we!Ofig<Ggqe<Oxil</!-u<uixqVh<hkx<Gg<!giv{l<!A!Bl<!C!Bl<!fqvh<Hg<Ogi{r<gt<. !
wMk<Kg<gim<M!3: 257
=sin θ weqz<?!lx<x!Lg<Ogi{uqbz<!uqgqkr<gjtg<!gi{<g/
kQi<U; 257
==sinfQtl<!kqe<gI{k<
fQtl<!ghg<wkqIh<θ we<hkiz<?!m∠ABC = 90°, m∠ACB = θ? AB = 7,
AC = 25 we<Xt<tuiX yV!osr<Ogi{!Lg<Ogi{l< ABC Jg<!gVkUl<! (hml<! 8.8!Jh<!hiIg<gUl<).!weOu?!hqkigv <̂!Okx<xh<hc?
AC2 = AB2 + BC2 ∴ 252 = 72 + BC2 nz<zK 625 = 49 + BC2. ∴ BC2 = 625 − 49 = 576. ∴ BC= 576 = 24.
weOu? ,ACBCθ
2524cos ==
,θ
θ
,BCABθ
725
sin1cosec
247tan
==
==
.θ
θ
,θ
θ
724
tan1cot
2425
cos1sec
==
==
!
wMk<Kg<gim<M!4: 2cosec =A weqz<?! (i) sin A + cos A lkqh<Hgjtg<!gi{<g/!
kQi<U; cosec A = 2 = fQtl<!ghg<wkqi<
fQtl<!kqe<{k<gi<=
12
.
weOu?!m∠QRP = A, gI{k<kqe<!fQtl<!= PR = 2 ,!!
wkqIh<hg<gk<kqe<! fQtl<!= PQ = 1 we!njlBliX!yV!osr<Ogi{!Lg<Ogi{l< PQR Jg<!gVKOuil<!
(hml< 8.9 Jh<!hiIg<gUl<).
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hml< 8.8
(ii) tan A + cot A Ngqbux<xqe<!
hml<!8.9
hqkigv <̂!Okx<xh<hc, PR2 = PQ2 + QR2 .
∴ ( 2 )2 = (1)2 + QR2 . ∴ 2 = 1 + QR2. ∴ QR2 = 2 – 1 = 1.
∴ QR = 1. weOu?
.111cot,1
11tan
,2
1cos,2
1sin
======
====
PQQRA
QRPQA
PRQRA
PRPQA
∴(i) sin A + cos A = 2
12
1+ = 2 ⎟
⎠
⎞⎜⎝
⎛2
1 = 2 , (ii) tan A + cot A = 1 + 1 = 2.
Gxqh<H; WOkEl<! yV! sle<him<jm! fq'hqg<g! Ou{<Ml<! weqz<?! gQp<g<gi[l<! upqgtqz<!ye<xqjeh<!hbe<hMk<Kkz<!Ou{<Ml</ upq!1; sle<him<ce<!-mKhg<gg<!Ogijujb!nz<zK!uzK!hg<gg<!Ogijujb!SVg<gl<!osb<K!lx<x!hg<gk<kqz<!dt<t!Ogijujbh<!ohXkz<!Ou{<Ml</ upq!2; sle<him<ce<!-mKhg<gk<kqz<!dt<t!Ogijujb!SVg<gl<!osb<K!yV!ucul<!(1)!ogi{i<Ouil</!nMk<kkig?! !uzK!hg<gk<kqz<!dt<t!Ogijujb!SVg<gl<! osb<K!yV!
ucul<!(2)!ogi{i<Ouil</!!!hqe<ei<!ucul<!(1)!=!ucul<!(2)!weg<!gim<MOuil</!
wMk<Kg<gim<M!5: BABABABA
BABA
sinsincoscossincoscossin
tantan1tantan
−+
=−
+ we!fq'hq/
kQi<U;
-mKhg<g!Ogiju = BABA
tantan1tantan
−+ =
BB
AA
BB
AA
cossin
cossin1
cossin
cossin
×−
+ =
BABABA
BABABA
coscossinsincoscos
coscossincoscossin
−
+
=BABA
BABA
BABAsinsincoscos
coscoscoscos
)sincoscos(sin−
×+
= BABABABA
sinsincoscossincoscossin
−+ = uzKhg<g!Ogiju.
wMk<Kg<gim<M!6: BA
BABA
tantan
tancotcottan
=++ we!fq'hqg<gUl</
kQi<U; -mKhg<g!Ogiju = =++
BABA
tancotcottan
1tan
tan1
tan1
1tan
BA
BA
+
+=
ABA
BBA
tantantan1
tan1tantan
+
+
= ) tan tan 1(
tantan
)1tan(tanBA
ABBA
+×
+ = BA
tantan = uzKhg<g!!Ogiju.
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wMk<Kg<gim<M!7: θθθ
θθ
cos1tansin
cot1tan1
++
=++ we!fq'hqg<gUl</
kQi<U; -mKhg<g!Ogiju = =++
θθ
cot1tan1
θ
θ
tan11
tan1
+
+ = ⎟⎠⎞
⎜⎝⎛ +
+
θθθ
tan1tan)tan1( =
θθθ
tan+1tan
×1tan+1
= tan θ (1)
uzKhg<g!Ogiju =θθθ
cos+1tan+sin
= θθθθ
cos1cossinsin
+
+ =
)cos1(1
cos)sincos(sin
θθθθθ
+×
+
= )cos1(
1cos
)1(cossinθθ
θθ+
×+ =
θθ
cossin
= tan θ. (2)
∴ (1) lx<Xl< (2)!e<!hc!-mKhg<g!Ogiju = uzKhg<g!!Ogiju. sqz!Gxqh<hqm<m!Ogi{r<gtqe<!Lg<Ogi{uqbz<!uqgqkr<gt< 30°, 45° lx<Xl< 60°<! ! ntUt<t! Ogi{r<gtqe<! Lg<Ogi{uqbz<! uqgqkr<gtqe<!lkqh<Hgjtg<!gi{<Ohil</!(i) 30° lx<Xl< 60° Ogi{r<gtqe<!Lg<Ogi{uqbz<!uqgqkr<gt<
hg<g!fQtl<!2!nzGgt<!ogi{<m!slhg<g!Lg<Ogi{l< ABC jb!wMk<Kg<!ogit<Ouil< (hml< 8.10Jh<! hii<g<gUl<). hg<gl<! ABg<G! Lje! C bqzqVf<K!osr<Gk<Kg<OgiM!CD!ujvg. Ht<tq!D, hg<gl< AB e<! jlbh<! Ht<tqbiGl</! ! -h<ohiPK AD = 1, AC = 2, m∠DAC = 60°, m∠ACD = 30°. ADC yV!osr<Ogi{!Lg<Ogi{l<! (hml< 8.11Jh<!hii<g<gUl<).hqkigv <̂!Okx<xh<hc,
hml<!8.10
AC2 = AD2 + DC2 n.K 22 = 12 + DC2 . ∴ DC2 = 3 n.K DC = 3 . osr<Ogi{!Lg<Ogi{l<!ADC bqzqVf<K?!!
sin 60° = 23
=ACDC sin 30° =
21
=ACAD
cos 60° = 21
=ACDA cos 30° =
23
=ACDC
tan 60° = 313
==DADC tan 30° =
31
=DCAD
cot 60° = 3
1=
DCDA cot 30° = 3
13
==ADDC
sec 60° = 212
==DAAC sec 30° =
332
32
==DCAC
cosec 60° = 3
323
2==
DCAC cosec 30° = 2
12
==ADAC
hml<!8.11
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(ii) 45° Ogi{k<kqe<!Lg<Ogi{uqbz<!uqgqkr<gt< ABC YI!-Vslhg<g!osr<Ogi{!Lg<Ogi{l<!we<g/!-r<G!m∠B = 90°,
AB = BC = 1 (hml< 8.12!Jh<!!hiIg<gUl<). -r<G!AC = 2 , m∠CAB = 45°?!m∠BCA = 45° we!nxqf<K!ogit<g/ Lg<Ogi{l<!ABC bqzqVf<K!fil<!ohXuK?!
hml<!8.12
cot 45° = 111
==BCAB
sec 45° = 212
==ABAC
cosec 45° = 212
==BCAC
sin 45° = 2
1=
ACAB
cos 45° = 2
1=
ACAB
tan 45° = 111
==ABBC
(iii) 0°, 90° Ogi{r<gtqe<!Lg<Ogi{uqbz<!uqgqkr<gt<
-g<! Ogi{r<gtqe<! Lg<Ogi{uqbz<!uqgqkr<gjth<! ohXukx<G! giICsqbe<! ns<S!ktk<kqz<!Nkqh<!Ht<tqjb!jlbligUl<?!Nv!ntU!r! nzGl<! ogi{<M! yV! um<mk<kqjeg<! gVKg/!giICsqbe<! ktk<kqe<! Lkx<! gix<hGkqbqz<! njlf<k!-u<um<mk<kqe<!!uqz<!ABbqe<!lQK!P!we<hK!WOkEl<!yV!Ht<tq!we<g!)hml<!8.13!Jh<!hii<g<gUl<*/!!!! x!ns<sqx<G! PM we<x!Gk<Kg<OgiM!ujvg/!!P!we<x!Ht<tqbqe<! ns<S! K~vr<gt<! x, y! we<g/! ! hqe<ei<!!!OM = x, PM = y!!NGl</!osr<Ogi{!Lg<Ogi{l<!!
hml<!8.13 OMP -z<?!hqkigv <̂!Okx<xh<hc?! x2 + y2 = r2!weh<!!
ohXgqOxil</ ∴ r = 22 yx + . ∠MOP = θ we<g/!θ yV!GXr<Ogi{liGl</!!
∴ sin[θ =ry
, cos θ = rx
.
uqz<!ABbqe<!lQK!ouu<OuX!fqjzgtqz<!P Jg<!ogi{<miz<?!fil<!gueqh<hK?!gkqi<!OP NeK!fqjz!OA uqzqVf<K!fqjz!OB g<G Spx<sqbjmBl<ohiPK?!Ogi{l<!θ NeK!0° ntuqzqVf<K! 90° ntuqx<G! nkqgiqg<gqe<xKA! fQtl<! x NeK! r zqVf<K! 0 uqx<G!Gjxgqe<xKA!y NeK!0 uqzqVf<K!r x<G!nkqgiqg<gqe<xK/!weOu?!θ NeK!0° zqVf<K!
90° g<G!nkqgiqg<Gl<ohiPK? rx e<!ntU!1zqVf<K!0Ug<G!GjxujkBl<?!
rye<!ntU!
0zqVf<K! 2x<G! nkqgliujkBl<! nxqgqe<Oxil</! nkiuK! θ uqe<! ntU! 0o! zqVf<K!!!!90o g<G!nkqgliGl<ohiPK?!cos θ uqe<!ntU!1!zqVf<K!0!uqx<G!GjxujkBl<?!sin θ uqe<! ntU! 0! zqVf<K! 1x<G! nkqgliujkBl<?! nxqgq<e<Oxil</! OlZl<?! yu<ouiV!GXr<Ogi{l<! θ! uqx<G! x, y! l<?! Lg<Ogi{uqbz<! uqgqkr<gTl<! keqk<kjugtig!njlBl<! we!nxqgqe<Oxil</! OPNeK! OA uqe<! fqjzjb!njmBl<ohiPK?! θ = 0,
x = r, y = 0 NGl</!weOu?! ,rr
y0=
0==0sin o ! 1===0cos
rr
rxo . OPNeK! OBbqe<!
fqjzjb!njmBl<ohiPK?!θ = 90°,!x =0, y = r NGl</!weOu?!
0=0
==90cos1===90sinrr
x,
rr
ry oo .
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-h<ohiPK?!fil<!ohXuK?
,0= 10
= 0cos0sin
=0tan o
oo
01
= 0sin0cos
=0cot o
oo , ujvbXg<g!-bzikK?
1= 11
= 0cos
1 =0sec o
o ;
01
= 0sin
1=0 cosec o
o , ujvbXg<g!-bzikK
01
=90 cos90 sin
=90tan o
oo , ujvbXg<g!-bzikK,
0 = 10
= 90 sin90 cos
=90 cot o
oo ,
01
=90 cos
1=90 sec o
o , ujvbXg<g!-bzikK,
1 = 11
= 90sin
1 =90 cosec o
o .
0°, 30°, 45°, 60°, 90° e<! Lg<Ogi{uqbz<! uqgqkr<gjt! gQp<g<g{<muiX! nm<muj{h<!hMk<kzil</!
θ 0° 30° 45° 60° 90°
sin θ 0 21
21
23 1
cos θ 1 23 2
1 21 0
tan θ 0 3
1 1 3 ujvbXg<g!-bzikK
cot θ ujvbXg<g!-bzikK 3 1 3
1 0
sec θ 1 32 2 2 ujvbXg<g!
-bzikK
cosec θ ujvbXg<g!-bzikK! 2 2 3
2 1
GxqbQM; we<hkje sin2)(sin θ 2θ we! SVg<glig! wPKgqe<Oxil</! -jkh<Ohie<Ox!
lx<xhcgTg<Gl<! wPKgqe<Oxil</! wMk<Kg<gim<mig?! (tanθ )3 = tan3θ. cos4θ we<hK!!!!(cos θ )4 Jg<!Gxqh<hkiGl</!sin2θ J!sinθ 2 we!wPKuK!kuxiGl</ !!
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wMk<Kg<gim<M!9: lkqh<H!gi{<g;! 2 cos2 30° tan260° − sec245° sin260°.
kQi<U; cos 30° = 23 , tan 60° = 3 , sec 45° = 2 , sin 60° =
23 .
∴2 cos230° tan260° − sec2 45° sin2 60° = 2 ( ) ( )2
222
2323
23
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛
= 2 × 43 × 3 – 2 ×
43
= 29
−23 = 3.
wMk<Kg<gim<M!:; tan A = o
o
60cos160sin
+ weqz<?!GXr<Ogi{l<!A!jbg<!gi{<g/
kQi<U; sin 60° = 23 , cos 60° =
21 .
∴ tan A = 3
133
2323
211
23
===+
.
Neiz<! tan 30° = 3
1 . ∴ A = 30°.
wMk<Kg<gim<M!10:! 2 sin (A + B) = 3 , 1cos2 =B , weqz< A, B gi{<g/
kQi<U; 2 sin (A + B) = 3 we<hkiz<?!sin (A + B) = 23 .
Neiz<? sin 60° = 23 .
weOu A + B = 60°. (1)
2 cos B = 1, we<hkiz< cos B = 2
1 .
Neiz< cos 45° = 2
1 .
weOu B = 45°. (2) (1), (2)Jk<!kQi<g<g? A = 15°.
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!hbqx<sq! 8.1
g{g<Ggt<!1 Lkz<!4 -ux<xqZt<t!osr<Ogi{!Lg<Ogi{r<gtqz<!Gxqh<hqm<Mt<t!Lg<Ogi{uqbz<!uqgqkr<gjtg<!g{<Mhqc/! 1. 2.
hml< 8.14
sin B, cos C, tan B
hml<!8.15
sec X, cot Z, cosec Z
3. 4.
hml<! 8.16
cos Q , tan R, cot Q
hml<! 8.17
tan M, sec N, cosec N
g{g<Ggt<! 5! -zqVf<K! 10! ujv! ogiMg<gh<hm<m! uqgqkl<! Ohig! θ! uqe<! lx<x!Lg<Ogi{uqbz<!uqgqkr<gjtg<!gi{<g/!
5. cos θ = 53 6. sin θ =
1312 7. sec θ =
32
8. cosec θ = 10 9. cot θ = 71 10. tan θ =
52
11. 3735cos =A !weqz<?
AAAA
tansectansec
−+ Jg<!g{<Mhqcg<gUl</
12. sin θ = 53 weqz<?
θθθ
coscoteccos
− Jg<!g{<Mhqcg<gUl</
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13. cosec θ = 2 weqz<? cot θ + θθ
cos1sin+
!Jg<!g{<Mhqcg<gUl</
14. cot θ = 3
1!weqz<?
53
sin2cos1
2
2
=−−
θθ weg<!gi{<g/
15. 3 cot θ = 4 weqz<? θθ
θθeccos3sec2
cos2sin3++ lkqh<jhg<!g{<Mhqcg<gUl</
16. lkqh<H!gi{<g;! (i) cosec2 45° cot2 30° + sin2 60° sec2 30° (ii) cos2 30° − sin2 30° − cos 60°
(iii) 8 sin2 60° cos 60° (iv) oo
o
60tan30tan45tan
+
17. hqe<uVueux<jx!siqhii<g<gUl<; (i) sin2 30° + cos2 30° = 1 (ii) sec2 60° − 1 = tan2 60° (iii) 1 + cot2 30° = cosec2 30°
18. sin (A+B) = 2 sin (A – B) = 1 weqz<? A, B gjtg<!g{<Mhqcg<gUl</ 8.2 Lg<Ogi{uqbz<!Lx<oxiVjlgt<
!
nch<hjmbie! &e<X! Lg<Ogi{uqbz<! Lx<oxiVjlgjt! yV! GXr<Ogi{l<!!!!θ uqx<G!ohXOuil</!NbqEl<?!wz<zi!Ogi{r<gTg<Gl<!-ju!ohiVk<kLXl</
θ e<!Ljejb!Nkqh<Ht<tq!we<g/!θ e<!okimg<gg<jgjb x ns<sig!ogit<Ouil</ P (x, y) we<hK!θ e<!LcUg<jgbqe<!lQkjlf<k!WOkEl<!yV!Ht<tq!we<g!(hml< 8.18 Jh<!hii<g<gUl<). P!bqzqVf<K! x!ns<Sg<G?!!
Gk<Kg<OgiM PQ ujvg/ hqe<ei< OQ = x, PQ = y.
OP = r we<g/!hqkigv^<!Okx<xh<hc?!!
osr<Ogi{!Lg<Ogi{l<!OQP -z<?
x2+ y2 = r2!NGl</!!
-V!hg<gLl<! r2 Nz<!uGg<g?
2
2
2
22
=+
rr
ryx
n.K! 12
2
2
2
=+ry
rx
n.K 122
=⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛
ry
rx .
Neiz<? sin θ = ry , cos θ =
rx . ∴ (cos θ )2+ (sin θ
nkiuK? cos2 θ + sin2 θ
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hml< 8.18
)2 = 1.
= 1. (1)
(1) e<!-V!hg<gLl< cos2θ Nz<!uGg<g?
θθ
θθ22
22
cos1
=cos
sin+cos n.K
2
2
2
2
2
cos1
cossin
coscos
⎟⎟⎠
⎞⎜⎜⎝
⎛=+
θθθ
θθ
22
cos1
cossin1 ⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛+
θθθ n.K! 1 +(tan θ )2 = (sec θ )2
nkiuK? 1 + tan2 θ = sec2 θ . (2) (1) e<!-V!hg<gLl<! sin2 θ Nz<!uGg<g?
θθθθ
22
22
sin1
=sin
sin+cos n.K 2
2
2
2
2
2
)(cosecsin
1sinsin
sincos θ
θθθ
θθ
=⎟⎟⎠
⎞⎜⎜⎝
⎛=+
n.K cot2θ + 1 = cosec2θ nkiuK? 1 + cot2θ = cosec2θ. (3) -f<k! &e<X! Lx<oxiVjlgt<! (1), (2) lx<Xl<! (3) Ngqbju! hqkigv <̂! Okx<xk<jk!nch<hjmbigg<! ogi{<mjubiGl</! -ux<xqzqVf<K! OlZl<! sqz! Lx<oxiVjlgjt!nxqf<K!ogit<Ouil</ (1) Jh<!hbe<hMk<kq! (i) sin2 θ = (sin2 θ + cos2 θ )− cos2θ = 1 – cos2 θ (ii) cos2 θ = (cos2 θ + sin2θ ) – sin2 θ = 1 – sin2 θ we!nxqbzil</ (2) Jg<!ogi{<M!!! (i) tan2 θ = (1 + tan2 θ ) – 1= sec2 θ − 1 (ii) sec2 θ − tan2 θ = (1 + tan2 θ ) – tan2 θ = 1 we!nxqbzil</ (3) Jg<!ogi{<M!!!!(i) cot2 θ = (1 + cot2 θ ) – 1 = cosec2 θ − 1 (ii) cosec2 θ − cot2 θ = (1 + cot2 θ ) – cot2 θ = 1 we!nxqbzil</ !Olx<gi[l<!Lx<oxiVjlgjth<!hqe<uVliX!nm<muj{h<hMk<kzil</
sin2 θ + cos2 θ ≡ 1 1 + tan2 θ ≡ sec2 θ 1 + cot2 θ ≡ cosec2 θ sin2 θ ≡ 1 – cos2 θ tan2 θ ≡ sec2 θ − 1 cot2 θ ≡ cosec2 θ − 1 cos2 θ ≡ 1 – sin2 θ sec2 θ − tan2 θ ≡ 1 cosec2 θ − cot2 θ ≡ 1
OlZl<?! -f<k! Lx<oxiVjlgjt! -mh<hg<gk<kqzqVf<K! uzh<hg<gk<kqx<Gl<?! uzh<hg<gk<kq!zqVf<K!-mh<hg<gk<kqx<Gl<!hbe<hMk<KOuil</!nkiuK?!wMk<Kg<gim<mig?!!sec2 θ − tan2θ ≡ 1 we<Xl<! 1! ≡ sec2 θ − tan2θ weUl<! hbe<hMk<KOuil</!-mKhg<g!Ogijujb!-eq!-/h/Ogi!we<Xl<?!uzKhg<g!Ogijujb!u/h/Ogi!we<Xl<!Gxqh<Ohil</!!wMk<Kg<gim<M 11: fq'hqg<g; sin4 θ + cos4 θ = 1 – 2sin2 θ cos2 θ . kQi<U: -/h/Ogi = sin4 θ + cos4 θ = (sin2 θ )2 + (cos2 θ )2
= [sin2 θ + cos2 θ ]2 – 2 (sin2 θ )(cos2 θ ) ( a2 + b2 = (a + b)2 – 2ab) = (1)2 – 2sin2 θ cos2 θ = 1 – 2sin2 θ cos2 θ = u/h/Ogi/ !
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wMk<Kg<gim<M 12: fq'hqg<g; θθ
sin1cos+
= sec θ − tan θ.
kQi<U: -/h/Ogi!= θθ
sin1cos+
=θθ
θθ
sin1sin1
sin1cos
−−
×+
= θθ)(θ
2sin1sin1cos
−− =
θθ)(θ
2cossin1cos −
= θθ
cossin1− =
θθ
θ cossin
cos1
− = sec θ − tan θ = u/h/Ogi.
!
wMk<Kg<gim<M 13: fq'hqg<g; 2)cotec(coscos1cos1 AA
AA
+=−+ .
kQi<U: -/h/Ogi= AA
AA
cos1cos1
cos1cos1
++
×−+
= A
A2
2
cos1)cos1(
−+ =
22
2
2
sincos
sin1
sincos1
sin)cos1(
⎟⎠⎞
⎜⎝⎛ +=⎥⎦
⎤⎢⎣⎡ +
=+
AA
AAA
AA
= (cosec A + cot A)2 = u/h/Ogi. OuX!upqbig?
u/h/Ogi = (cosec A + cot A)2 =2
sincos
sin1
⎟⎟⎠
⎞⎜⎜⎝
⎛+
AA
A =
2
sincos1
⎟⎟⎠
⎞⎜⎜⎝
⎛ +A
A
= AA
2
2
sin)cos1( + =
AA2
2
cos1)cos1(
−+ =
)cos1()cos1()cos1( 2
AAA−+
+ = AA
cos1cos1
−+ = -/h/Ogi/
wMk<Kg<gim<M 14: fq'hqg<g; sin4θ − cos4θ = sin2θ − cos2θ. kQi<U: -/h/Ogi!= sin4 θ − cos4 θ = (sin2 θ )2 – (cos2 θ )2
= (sin2 θ + cos2 θ ) (sin2 θ − cos2 θ ) =(1) (sin2θ − cos2 θ ) = sin2 θ − cos2 θ = u/h/Ogi.
!
wMk<Kg<gim<M 15: fq'hqg<g; sec A – tan A = AA tansec
1+
kQi<U: u/h/Ogi!= AA tansec
1+
= AAAA
AA tansectansec
tansec1
−−
×+
= AA
AA22 tansec
tansec−− =
1tansec AA−
= sec A – tan A = -/h/Ogi/
!wMk<Kg<gim<M 16: fq'hqg<g; (sec θ + cosθ ) (secθ − cosθ ) = tan2 θ + sin2 θ. kQi<U: -/h/Ogi! = (sec θ + cos θ ) (sec θ − cos θ ) = sec2 θ − cos2 θ = (1 + tan2 θ ) – cos2 θ = tan2 θ + (1 – cos2 θ ) = tan2θ + sin2θ
= u/h/Ogi.
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wMk<Kg<gim<M!17: θθ cos1
1cos11
−+
+ = 2 cosec2θ !we!fq'hqg<gUl</
kQi<U: -/h/Ogi = θθ cos1
1cos11
−+
+ =
)cos(1)cos(1)cos(11)cos1(1
θθθθ
−+++−
= θ
θθ2cos1
cos1cos1−
++− = θ2sin
2 = 2cosec2θ = u/h/Ogi.
wMk<Kg<gim<M 18: fq'hq;!sin2A sin2B + cos2A cos2B + sin2A cos2B + cos2A sin2B = 1. kQi<U: -/h/Ogi!= (sin2A sin2B + sin2A cos2B) + (cos2A cos2B + cos2A sin2B) = sin2A (sin2B + cos2B) + cos2A (cos2B + sin2B) = sin2A(1) + cos2A (1) = sin2A + cos2A = 1 = u/h/Ogi/ wMk<Kg<gim<M 19: m = tan A + sin A, n = tan A – sin A! weqz<? m2 – n2 = 4 mn we!fq'hqg<gUl</ kQi<U: -/h/Ogi = m2 – n2 = (tan A + sin A)2 – (tan A – sin A)2
= tan2 A + sin2 A + 2 tan A sin A – (tan2A + sin2A – 2 tan A sin A) = 4 tan A sin A (1) u/h/Ogi = 4 mn = )sin(tan)sin(tan4 AAAA −+
= AA 22 sintan4 − = AAA 2
2
2
sincossin4 −
= A
AAA2
222
coscossinsin4 − =
AAA
2
22
cos)cos1(sin4 −
= AA 22 tansin4 = 4 sin A tan A/! (2) (1), (2) -ux<xqe<hc?!-/h/Ogi!= u/h/Ogi/! wMk<Kg<gim<M 20: cos6θ + sin6θ = 1 – 3 cos2θ sin2θ we!fq'hqg<gUl</ kQi<U : -/h/Ogi = cos6θ + sin6θ
= (cos2 θ )3 + (sin2 θ )3 = (cos2 θ + sin2 θ ) (cos4 θ − cos2 θ sin2 θ + sin4 θ ) = (1) (cos4 θ + sin4 θ − cos2 θ sin2 θ ) = [(cos2 θ )2 + (sin2 θ )2] – cos2 θ sin2 θ = [(cos2 θ + sin2θ )2 – 2 cos2 θ sin2 θ ] – cos2 θ sin2 θ = (1)2 – 3 cos2 θ sin2 θ = 1 – 3 cos2 θ sin2 θ = u/h/Ogi.
wMk<Kg<gim<M 21: fq'hqg<g; θθ cossin21+ = sinθ + cosθ
kQi<U: -/h/Ogi = θθθθ cossin2cossin 22 ++ 1 = sin + cos θ2 θ2
= 2)cos+(sin θθ = sinθ + cosθ = u/h/Ogi/
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hbqx<sq 8.2 1. fq'hqg<gUl<; AAA secsin1sec2 =− . 2. fq'hqg<gUl<; (sin A + cos A)2 + (sin A – cos A)2 = 2.
3. SVg<Gg;θ−θθ−θ
22
22
tanseccoseccot .
4. fq'hqg<gUl<; AAAA
tansectansec
−+ =
AA
sin1sin1
−+ .
5. fq'hqg<gUl<; θθθ
2sec2sin11
sin11
=−
++
.
6. x = r sin A sin B, y = r sin A cos B, z = r cos A weqz<?!x2 + y2 + z2!e<!lkqh<H!gi{<g/!
7. tan A + cot A = cosec A sec A weg<!gi{<hqg<gUl</
8. fq'hqg<gUl<;! 1cos2tan1tan1 2
2
2
−=+− A
AA .
9. fq'hqg<gUl<; θθ cotcosec
1−
= cosec θ + cot θ.
10. (tan A + cot A)2 = sec2 A+ cosec2A we!fq'hqg<gUl</
11. AAA
AA
A cossincot1
sintan1
cos+=
−+
− we!fq'hqg<gUl</
12. fq'hqg<gUl<; A
AAAAA
cossin1
1sectan1sectan +
=+−−+ .
13. fq'hqg<gUl<; (tan A – tan B)2 + (1 + tan A tan B)2 = sec2A sec2B.
14. fq'hqg<gUl<;!θθ
cot1tan−
+ θθ
tan1cot−
= sec θ cosec θ +1.
15. fq'hqg<gUl<; θθ
θθθθ
cos1cos1
cossin1cossin1
2
+−
=⎟⎟⎠
⎞⎜⎜⎝
⎛++−+ .
16. (sin θ + cosec θ )2 + (cos θ + sec θ )2 = 7 + tan2 θ + cot2 θ we!fq'hqg<gUl</
17. 2sincossincos
sincossincos 3333
=−−
+++
θθθθ
θθθθ we!fq'hqg<gUl</
18. fq'hqg<gUl<;! θθθθθ 2
44
44
tancossin1sincos1
=+−+− .
8.3 fqvh<Hg<Ogi{r<gTg<gie!Lg<Ogi{uqbz<!uqgqkr<gt<
Wx<geOu! osr<Ogi{! Lg<Ogi{k<kqe<!fqvh<H!Ogi{r<gjth<!hx<xq!fil<!nxqf<Kt<Otil</!osr<Ogi{!Lg<Ogi{l<!OQP -z< (hml<!8.19 Jh<!hii<g<gUl<), Q yV!osr<Ogi{l</!-r<G!
hml< 8.19
m∠QOP + m∠OPQ = 90° .!
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weOu?!∠QOP, ∠OPQ Ngqb!Ogi{r<gt<!fqvh<Hg<Ogi{r<gtiGl</!m∠QOP = θ we<g/!hqe<H!m∠OPQ = 90° − θ NGl</!Lg<Ogi{uqbz<!uqgqkr<gtqe<!ujvbjxjbh<!hqe<hx<xq!Ogi{l< θ uqx<G!hqe<uVueux<jx!Wx<geOu!nxqf<Kt<Otil</
OQPQθ,
OPOQθ,
OPPQθ === tancossin ,
PQOQθ,
OQOPθ,
PQOPθ === cotseccosec
-h<ohiPK? 90°− θ g<G!Lg<Ogi{uqbz<!uqgqkr<gt<!nxqb!Lx<hMOuil</!hmk<kqzqVf<K?!!
sin (90° − θ ) = ,OPOQ ,
PQOQθ,
OPPQθ =−=− ) 90tan() 90(cos oo
OQPQθ,
PQOPθ,
OQOPθ =−=−=− ) 90(cot) 90(sec)90(cosec ooo
……….. (1)
………..(2)
(1), (2) yh<hqm!gQp<g<g{<mux<jx!nxqbzil</
)90(cossin θθOPPQ
−== o =OPOQ cos θ = sin (90° − θ )
OQPQ = tan θ = cot (90°− θ ) =
PQOP cosec θ = sec (90° − θ )
OQOP = sec θ = cosec (90° − θ )
PQOQ = cot θ = tan (90° − θ ).
-kje!YI!nm<muj{big!njlg<gzil</ sin (90° − θ ) = cosθ cosec (90° − θ ) = sec θ cos (90° − θ ) = sin θ sec (90° − θ ) = cosec θ tan (90° − θ ) = cot θ cot (90° − θ ) = tan θ
wMk<Kg<gim<M 22: lkqh<H!gi{<g; o
o
25cot65tan .
kQi<U : tan 65° = tan (90° − 25°) = cot 25°. ∴ o
o
25cot65tan =
o
o
25cot25cot = 1.
!wMk<Kg<gim<M 23: lkqh<H!gi{<g; sin 20° tan 60° sec 70°
kQi<U : sec 70° = sec(90°−20°) = cosec 20°=o20sin
1
∴sin 20° tan 60° sec 70° = sin 20° tan 60° cosec 20° = sin 20° × 3 × o20sin
1 = 3 .
wMk<Kg<gim<M 24: cosec x° = sec 25° weqz<?!x° e<!lkqh<H!gi{<g/ kQi<U: cosec x° =sec(90°− x°), sec(90°− x°)= sec 25°. ∴ 90°− x°=25°. ∴ x° = 90°− 25°= 65°. Gxqh<H: OlOz!dt<t!g{g<gqz<?!x!NeK?!sle<him<ce<!-V!hg<gLl<!sec!jb!!fQg<gl<!osb<ukiz<! gqjmg<guqz<jzA! Neiz<?! GXr<Ogi{k<kqx<gie! Lg<Ogi{uqbz<!uqgqkr<gtqe<!keqk<ke<jlh<h{<hqjeh<!hbe<hMk<kq!gi{h<hm<mK!we<hjk!nxqg/
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hbqx<sq 8.3
1. lkqh<H!gi{<g; (i) o
o
54cos36sin (ii)
o
o
55cot35tan (iii) sin θ sec (90° − θ )
2. SVg<Gg; (i) o
o
o
o
o
o
39eccos51sec
23
48cos42sin
21
57cot33tan
++ . (ii) o
o
o
o
43eccos47sec4
67cos23sin3 + .
3. x jbg<!gi{<g; (i) sin 60° = cos x° (ii) cosec x° cos 54° = 1 (iii) sec x° = cosec 25° (iv) tan x° tan 35° = 1
uqjmgt<!
hbqx<sq 8.1
1. 125,
135,
135 2.
25,
21,
25 3.
158,
158,
178 4. 1, 2 , 2
5. 43cot
35sec
45cos
34tan
54sin ===== θ,θ,θec,θ,θ .
6. 125cot
513sec
1213cos
512tan
135cos ===== θ,θ,θec,θ,θ .
7. 3cot2cos3
1tan23cos
21sin ===== θ,θec,θ,θ,θ .
8. 3cot310sec
31tan
103cos
101sin ===== θ,θ,θ,θ,θ .
9. 25sec7
25cos7tan25
1cos25
7sin ===== θ,θec,θ,θ,θ .
10. 25cot
529sec
229cos
295cos
292sin ===== θ,θ,θec,θ,θ .
11. 2549 12.
825 13. 15. 2
7534
16. (i) 7 (ii) 0 (iii) 3 (iv) 43
18. A = 60°, B = 30° hbqx<sq 8.2
3. −1 6. r2
hbqx<sq 8.3 1. (i) 1 (ii) 1 (iii) 1
2. (i) 3 (ii) 7
3. (i) 30° (ii) 36° (iii) 65° (iv) 55°
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9. osb<Ljx!ucuqbz<
nxqLjx!ucuqbzqz<!uqkqgjtBl<?!kIg<g!Ljxg<!giv{r<gjtBl<!hbe<hMk<kq!Lg<Ogi{l<!Ohie<x!ucuqbz<!dVur<gtqe<!h{<Hgjtg<!%Xl<!Okx<xr<gjt!fq'h{l<!osb<gqOxil</! -g<g{qkh<hqiquqz<! uVl<! ucu! dVur<gjt! ntUgTg<Gs<! siqbig!ujvukqz<jz/! likqiqh<! hmr<gjt! ujvf<K! njkk<! Okx<xr<gtqe<! kVg<g! Ljx!fq'h{r<gTg<Gk<! Kj{bigg<! ogit<gqe<<<Oxil</! wf<k! uqklie! ucuqbz<! gVuqgTl<!nxqLjx! ucuqbjzh<! hbqz<ukx<Gk<! OkjubqVg<giK/! wMk<Kg<gim<mig?! nxqLjx!
ucuqbzqz<! PQ we<x! OfIg<Ogim<Mk<! K{<cje?! AB we<x! OfIg<Ogim<Mk<! K{<ce<!
jlbg<!Gk<Kg<!Ogimig!-Vg<gqxK!we<hjkk<!okiqbh<hMk<k?!Kz<zqblig! PQ ju! AB
g<Gs<! osr<Gk<kig! fil<! ujvukqz<jz/! Neiz<?! AB g<Gs<! osr<Gk<kig?! PQ ju!likqiqbig!ujvgqOxil</!nh<hch<hm<m!ucuqbz<!dVur<gjtk<!Kz<zqblig!ujvb!nkqg!Hk<kq! %IjlBl<?! kqxjlBl<! Okju/! ucuqbz<! dVur<gjt! ujvb?! hz! ucuqbz<!gVuqgt<!dt<te/!ucuqbz<!gVuqgjtg<!ogi{<M!ucuqbz<!ujvhmr<gjt!ujvuOk!osb<Ljx! ucuqbziGl</! -h<himh<hqiquqz<?! nxqLjx! ucuqbzqz<! g{<Mt<tKOhiz<?!nch<hjmg<! ogit<jggt<?! Okx<xr<gt<?! kIg<g! Ljx! fq'h{r<gt<! WKl<! gqjmbiK/!nkx<Gh<! hkqzig?! nxqLjx! ucuqbz<! upqbig! fq'hqg<gh<hm<m! njlh<H! Ljxgjtg<!ogi{<M!ucuqbz<!dVur<gjt!ujvuK!lm<MOl!-h<hqiquqe<!Gxqg<OgitiGl</!!ntU!
Gxqg<gh<hmik! OfI!uqtql<H! (straight edge)! lx<Xl<! -V! guvibr<gt<! -ux<jx! lm<MOl!ogi{<M! ucuqbz<! dVur<gjt! ujvb! Ou{<Mole<hK?! ucuqbz<! dVur<gjt!dVuig<Gl<! g{g<Ggtqz<! suizig! njlBl</! ucug{qk! dVur<gjt! dVuig<Gl<!g{g<Ggjt! Nvib<f<k! ujgbqz<! lqg! dbIf<k! d{<jlgTl<?! Okx<xr<gTl<!dVuig<gh<hm<Mt<te/! wMk<Kg<gim<mig?! hqkigv <̂! Okx<xl<! nu<uiOx! dVuieK/!Lf<jkb!uGh<Hgtqz<?!fil<!gQp<g<g{<mux<xqx<gie!ujvLjxgjtg<!gx<Xt<Otil<;!!
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Gxqh<H: Olx<g{<m! Ljxbqz<?! G Jg<! gi{! &e<xiuK! fMg<Ogim<jm! ujvbuqz<jz/!Woeeqz<?! oum<Ml<Ht<tqbie! fMg<Ogim<M! jlbl<! G! Jg<! gi{! -V! fMg<OgiMgOt!OhiKlieK/! &e<xiuK! fMg<Ogim<jm! ujvf<kiz<?! nK! G upqbigs<! osz<ujkg<!gi{zil</ wf<kouiV! njlh<H! nz<zK! ujvhm! g{g<gqZl<?! Lkzqz<! fil<! kvh<hm<m!ntUgjtBl<?!OkjujbBl<!okiqf<K!ogit<t!Ou{<Ml</!-kx<Gh<hqe<?!fil<!yV!likqiqh<!hmk<jk! ujvf<K! ogit<t! Ou{<Ml</! -l<likqiqh<! hmk<kqz<?! njlh<H! nz<zK!ujvkZg<G{<mie!upqLjxh<!hcgjts<!Sm<cg<!gim<m!Ou{<Ml</!!wMk<Kg<gim<M 1: AB = 7 os/lQ? AC = 7.5 os/lQ? BC = 5.5 os/lQ!weqz<!∆ABC J!ujvg/!nke<!fMg<Ogim<M!jlbl<!G Jg<!gi{<g/!AD J!G!wf<k!uqgqkk<kqz<!hqiqg<Gl<!we<hjk!wPKg/!!kQIU: ∆ABC bqe<!likqiqh<!hmk<jk!ujvf<K!kvh<hm<m!ntUgjtg<!Gxqg<gUl< (hml< 9.1 Jh<! hiIg<gUl<). -h<OhiK! fMg<Ogim<M! jlbk<jkg<! gi{<Ohil</! hcgt<! gQOp!kvh<hm<Mt<te/ hc 1: BC = 5.5 os/lQ! ntuqz<! BC wEl<! OfIg<!Ogim<Mk<K{<jm!ujvg/!B J!jlbligg<!ogi{<M!7 os/lQ!Nvk<kqz<!yV!uqz<!ujvg/!-OkOhiz<!C J!jlbligg<! ogi{<M! 7.5 os/lQ!Nvk<kqz<! yV!uqz<!ujvg/! -u<uqV! uqx<gTl<! oum<Ml<! Ht<tqjb! A weg<!Gxqg<gUl</! AB lx<Xl< AC ujvg/!-h<OhiK!Lg<Ogi{l< ABC ujvbh<hm<Mt<tK/!
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GDAG = .
12
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1. ∆ABC z<?!BC = 6 os/lQ, m∠B = 40º, m∠C = 60º. 2. ∆ABC z<?!njek<Kh<!hg<gr<gTl<!6.5 os/lQ fQtl<!ogi{<mju/ 3. ∆PQR z<? m∠R = 90º, PQ = 7 os/lQ, PR = 6 os/lQ. 4. ∆LMN z<? LM = 6 os/lQ, m∠L = 95º, MN = 8 os/lQ.
g{g<Ggt<!5 Lkz<! 8 ujv?!kvh<hm<m!Lg<Ogi{k<kqe<!Sx<Xum<ml<!ujvg/!Sx<Xum<m!Nvk<kqe<!ntjug<!gi{<g/
5. ∆ABC z<? AB = 8 os/lQ, BC = 5 os/lQ, AC = 7 os/lQ. 6. ∆PQR z<? PQ = 5 os/lQ, PR = 4.5 os/lQ, m∠P = 100º. 7. ∆XYZ z<? XY = 7 os/lQ, m∠X = 70º, m∠Y = 60º. 8. ∆PQR z<?!yu<ouiV!hg<gLl<! 5.5 os/lQ!fQtLt<tK/
g{g<Ggt< 9 Lkz< 12 ujv?!dt<um<ml<!ujvf<K?!dt<tivk<kqje!ntuqMg/
9. ∆ABC z<? AB = 9 os/lQ, BC = 7 os/lQ, CA = 5 os/lQ. 10. ∆XYZ z<? XY = YZ = ZX = 8 os/lQ. 11. ∆PQR z<? PQ = 10 os/lQ, m∠P = 90º, m∠Q = 60º. 12. ∆ABC z<? AB = 5.4 os/lQ? m∠A = 50º, AC = 5 os/lQ.
g{g<Ggt< 13 Lkz< 16 ujv?!Lg<Ogi{k<kqe<!osr<Ogim<M!jlbl<!gi{<g/!
13. ∆ABC z<? BC = 5.6 os/lQ, m∠B = 55º, m∠C = 65º. 14. ∆PQR z<? m∠P = 90º, m∠Q = 30º, PR = 4.5 os/lQ. 15. ∆LMN z<? LM = 7 os/lQ, m∠M = 130º, MN = 6 os/lQ. 16. ∆XYZ z<? XY = 7 os/lQ, YZ = 5 os/lQ, ZX = 6 os/lQ.
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hc 2: nkqz<? AB = a , BC = b wEl<hc!-V!K{<Mgt<! oum<cg<! ogit<tUl<! (hml<! 9.12 Jh<!hiIg<gUl<)/
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BDAB =
BCBD nz<zK! BD2 = AB × BC nz<zK x2 = ab nz<zK x = .ab
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ujvf<K?! nK ACJ! oum<Ml<! Ht<tqjb! O we<g/ hc 4: O J! jlbligUl<?! OA J! NvligUl<!ogi{<M!yV!um<ml<!ujvg/ hc 5: B upqbig!DE wEl<!fij{! AC g<Gs<!osr<Ghc 6: BD nz<zK! BE we<he! AB , BC g<gqjmbqziBD nz<zK BE e<!fQtr<gjt!ntg<gUl</!-f<k!fQtOBD fQtk<jk!ntg<g? BD = 5.2 os/lQ weg<!gi{zil</
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hml< 9.13
k<kig!ujvg/ e!ohVg<gz<!svisiqjbg<!Gxqg<Gl</!
l!9, 3 e<!ohVg<gz<!svisiq!NGl</!!
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12 ≈ 3.4.!
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hml<! 9.14
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hbqx<sq 9.2 1. 5 2. 7.5 3. 6 4. 4.5
5. 2.4 6. 3.3 7. 2.5 8. 4.9
9. 3.9 10. 4.2 11. 4.6 12. 4.9
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kvh<hm<Mt<t!30 li{uIgtqe<!g{qk!lkqh<oh{<gjtg<!gVKOuil</ 31 39 37 46 39 49 42 31 31 40 43 46 48 42 30 43 42 42 46 48 40 56 56 50 37 50 45 37 45 48
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Olx<gi[l<!Ht<tq!uquvl<!okiGg<gh<hmik!fqjzbqz<!dt<tK/!kvh<hm<m!lkqh<oh{<gjt!WX!uiqjsbqz<!wPk?!flg<Gg<!gqjmh<hK?!30, 31, 31, 31, 37, 37, 37, 39, 39, 40, 40, 42, 42, 42, 42, 43, 43, 45, 45, 46, 46, 46, 48, 48, 48, 49, 50, 50, 56, 56. Olx<%xqb! uiqjsbqz<! lkqh<oh{<! 30 yV! LjxBl<?! 31 &e<X! LjxBl<?! 37 &e<X!LjxBl<! lx<Xl<! -u<uiOx! lx<x! lkqh<oh{<gTl<! njlf<Kt<te/! nf<k! Ljxbqz<!kvh<hm<m! lkqh<oh{<gjt! w{<[ukqeiz<! flg<Gg<! gqjmg<Gl<! fqgp<ou{<gt<!nm<muj{jb! okiGg<gh<hmik! Ht<tq! uquvr<gtqe<! fqgp<Uh<! hm<cbz<! we<gqOxil</!!!!!!x we<hK! yV! lkqh<oh{<{qjeBl<?! f we<hK! x lkqh<oh{<! ohx<x! li{uIgtqe<!w{<{qg<jg!nz<zK! x e<! fqgp<ou{<{qjeg<! Gxqh<hkigg<! ogit<g/! fqgp<ou{<{qje!njzou{<!we<Xl<!fil<!%XOuil</!
x 30 31 37 39 40 42 43 45 46 48 49 50 56 f 1 3 3 2 2 4 2 2 3 3 1 2 2
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uQs<S = lqgh<ohiqb!lkqh<H − lqgs<sqxqb!lkqh<H = 56 − 30 = 26. fil<?!kvh<hm<cVg<Gl<! lkqh<<oh{<gjts<! OsIk<Kg<ogit<t!hqiqU!-jmoutqgt<! weh<hMl<!
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w{<{qg<jgbieK! nu<uGh<hqe<! fqgp<ou{<! (frequency) weh<hMgqxK/! -u<um<muj{?!okiGg<gh<hm<m! Ht<tq! uquvr<gtqe<! fqgp<ou{<hm<cbz<! weh<hMgqxK/! Olx<%xqb!
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wz<jzjb!nf<kf<k!-jmoutq<!juk<kqVg<giK!we<x!gVk<Kme< 29.5-34.5, 34.5-39.5, …, 54.5-59.5 we lix<xq!njlg<gzil</!-k<kjgb!lix<xk<jk!djmb!nm<muj{jb!gQOp!gi{zil</
hqiqU!-jmoutq! w{<{qg<jg!Gxq! fqgp<ou{<!29.5-34.5 |||| 4 34.5-39.5 |||| 5 39.5-44.5 |||| ||| 8 44.5-49.5 |||| |||| 9 49.5-54.5 || 2 54.5-59.5 || 2 olik<kl<! 30
-r<G?! fqgp<ou{<! hm<cbzieK! x lixqbqe<! okimIs<sqbie! lix<xk<jkk<! okiquqg<gqxK/!yV! hqiqU! -jmoutqbqe<! Olz<! wz<jz! lx<Xl<! gQp<! wz<jzgtqe<! uqk<kqbisk<jk!nu<uqjmoutqbqe<! ntU (size) we<Xl<?! Olz<! wz<jz! lx<Xl<! gQp<! wz<jzgtqe<!svisiqjb! ! hqiquqe<! lkqh<H! (class mark) we<Xl<! njpg<gqOxil</! -r<G 34.5-39.5! hqiqU!-jmoutqbqe< ntU = 5 lx<Xl<!hqiquqe<!lkqh<H!= 37?!-K!-jmoutqbqe<!jlblkq<h<H!NGl</!OlZt<t!nm<muj{bqzqVf<K?!34.5 g<Gg<gQp<!lkqh<oh{<!ohx<xuIgt< 4, 39.5 g<Gg<!gQp<!lkqh<oh{<gt<!ohx<xuIgt<!4 + 5 = 9, 44.5 g<Gg<gQp<!lkqh<oh{<!ohx<xuIgt< 4 + 5 + 8 = 17, 49.5 g<Gg<gQp<! lkqh<oh{<gt<! ohx<xuIgt<! 4 + 5 + 8 + 9 = 26, 54.5 g<Gg<gQp<!lkqh<oh{<gt<!ohx<xuIgt<! 4 + 5 + 8 + 9 + 2 = 28 lx<Xl< 59.5 g<Gg<gQp<!lkqh<oh{<gt<!ohx<xuIgt<!4 + 5 + 8 + 9 + 2 + 2 = 30. -k<kjgb!fqgp<Ugtqe<!w{<{qg<jggjt!GuqU!fqgp<ou{<gt<!(cumulative frequencies (c.f)) nz<zK!GuqU!njzou{<gt<!we<gqOxil</!GuqU!fqgp<ou{<gtqe<!nm<muj{!gQOp!kvh<hm<Mt<tK!!
hqiqU!-jmoutq! jlblkqh<H x fqgp<ou{<!f GuqU!fqgp<ou{<!c.f 29.5-34.5 32 4 4 34.5-39.5 37 5 9 39.5-44.5 42 8 17 44.5-49.5 47 9 26 49.5-54.5 52 2 28 54.5-59.5 57 2 30 olik<kl<! 30
10.1 jlbfqjzh<!Ohig<G!ntjugt<!!
!osh<heqmh<hmik! Ht<tququvr<gtqe<! fqgp<ou{<hm<cbjz! Olx<%xqbuiX! njlh<hke<!&zl<?! nh<Ht<tq! uquvr<gtqe<! yV! oktquie! njlh<hqjeg<! gi{LcgqxK/! yV!Gxqh<hqm<m! lkqh<hqzqVf<K! Ht<tququv! lkqh<Hgt<! njmBl<! lix<xk<kqe<! Ohig<gqjeh<hx<xq!
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OlZl<!nxqf<Kogit<t?!Ht<tq!uquvr<gjt!yVlqk<klig!nxqf<Kjvg<Gl<!kqm<mum<mlie!
ntjugt<!dt<te/!nk<kjgb!ntjugt<!jlbfqjzh<Ohig<G!ntjugt<!(Measures of Central Tendency) nz<zK! jlb=Ih<H! ntjugt<! (Measures of Location) weh<hMgqe<xe/!nk<kjgb?!sqz!ntjugtiue:
1. %m<Ms<!svisiq!(Average) 2. -jmfqjz!(Median) 3. LgM!(Mode)
10.1.1 %m<Ms<!svisiq!
11, 22, 7, 33, 27 we<x! lkqh<Hgjtg< gVKOuil</! Olx<%xqb! yu<ouiV!
lkqh<hqzqVf<Kl<?! 20 Jg<! gpqg<g?! flg<Gg<! gqjmh<hK! −9, 2, −13, 13, 7. -f<k!uqk<kqbisr<gjtg<! %m<m! flg<G! gqjmh<hK! 0. -kqzqVf<K?! w{<! 20 kvh<hm<cVg<Gl<! 5 lkqh<Hgtqe<!fMuqz<!njlf<kqVg<gqxK!we!nxqgqOxil</!-K!kvh<hm<cVg<Gl<!lkqh<Hgtqe<!
svisiq!nz<zK!%m<Ms<!svisiq!weh<hMl</!ohiKuig?!x1, x2, …, xn we<gqx!n lkqh<Hgtqe<!%m<Ms<!svisiq!(average) nz<zK!w{<g{qks<!svisiq!(arithmetic mean) nz<zK!SVg<glig!svisiq! (mean) x we<x!w{<?! x !zqVf<K! n lkqh<Hgt<! x1, x2, …, xn e<!uqzg<gr<gtqe<!
%Mkz<! 0 we<xqVg<GliX! ujvbXg<gh<hMgqe<xK/! nkiuK?! x1, x2, …, xn wEl< n lkqh<Hgtqe<!svisiq! x NeK
(x1 − x ) + (x2 − x ) + ... + (xn − x ) = 0 nz<zK! ( nxxx +++ ...21 ) − n × x = 0 we<x!sle<him<miz<!gqjmg<gh<ohXl</ weOu
.n
x...xxx n21 +++=
g{qkk<kqz<?!GxqbQM!∑, sqg<li!)sigma*!NeK?!%Mkjzg<!Gxqg<gh<!hbe<hMk<kh<hMgqxK/!
-f<kg<!GxqbQm<ce<!&zl<?!%Mkz< nxxx +++ ...21 !NeK! nz<zK wtqkig
we!Gxqg<gh<hMl</!weOu?!fil<!ujvbXh<hK?
∑=
n
iix
1∑ ix
.nx
x i∑=
Ht<tq!uquv!lkqh<Hgt<!fqgp<ou{<!hm<cbzqz<!njlg<gh<hm<cVf<kiz<?!svisiq! x NeK?
n
nn
fffxfxfxf
x++++++
=....
...
21
2211
we<X! ohxh<hMgqe<xK/! -r<G x1, x2, …, xn NeK! LjxOb f1, f2,…, fn we<x!fqgp<ou{<gjtg<! ogi{<m! keqh<hm<m! lkqh<HgtigOui! nz<zK! f1, f2, …, fn we<x!fqgp<ou{<gjtg<! ogi{<m! hqiqU! -jmoutqgtqe<! -jmlkqh<H! ntUgtigOui!-Vg<Gl</!sqg<li!GxqbQm<cz<!wPk?
x = N
xf ii∑ , -r<G!N = nf....ff +++ 21 .
kvh<hm<cVg<Gl<!lkqh<Hgtqe<!lm<Mlkqh<Hgt<!lqgh<!ohiqbjubig!-Vf<kiz<?!svisiqbieK!SVg<G.upqLjxobie<xqeiz<!g{g<gqmh<hMl</!A we<hK!ohiVk<klig!OkIf<okMg<gh<hm<m!
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YI! w{<! we<g/! fil<? x1− A, x2 − A, …, xn − A we<gqx! lixz<gjt! njlh<Ohil</!-l<lixz<gTg<G c we<x! ohiKuie! giv{q! -Vh<hkigg<! ogi{<miz<? hqe<uVl<!uqgqkr<gjt!njlh<Ohil<;!
.c
Ax...,,
cAx,
cAx n −−−
21
-ux<jx!LjxOb! d1, d2, …, dn we<g/ -h<ohiPK?!
∑ ii df = f1 × d1 + f2 × d2 + …..+ fn × dn = c
Axf...
cAxf
cAxf n
n−
×++−
×+−
× 22
11
= ( ) ( ) ([ ]AfxfAfxfAfxfc nnn −++−+− ...1
222111 )
= ( ) ( )[ ]nnn fffAxfxfxfc
+++−+++ ......1212211 = [ ] 1 NAxf
c ii ×−∑ .
∴ NAxf ii ×−∑ = c × ∑ n.K ii dfiiii dfcNAxf ∑∑ +×=
n.K x = N
xf ii∑ = A + c × N
df ii∑ .
Gxqh<H;! x1− A, x2 − A, …, xn − A gjt! nxqf<khqe<Hkie<?! ohiKg<giv{q! c Jg<!Gxqh<hqmLcBl</!!wMk<Kg<gim<M 1: 9, 11, 13, 15, 17, 19 we<gqx!Ht<tq!uquvr<gtqe<!svisiqjbg<!gi{<g/
kQIU : x = N
xi∑ = 6
19 17 15 13 11 9 +++++ = 6
84 = 14.
wMk<Kg<gim<M 2: hqe<uVl<!Ht<tq!uquvr<gTg<G!%m<Ms<!svisiqjbg<!gi{<g/!
x 10 11 13 15 16 19 f 4 5 8 6 4 3
kQIU: SVg<G.upq!Ljx;!A = 14, c = 1 weg<!ogit<g/!hqe<H! d = x − A
x f d f × d 10 4 −4 −16 11 5 −3 −15 13 8 −1 − 8 15 6 1 6 16 4 2 8 19 3 5 15
olik<kl<! N=30 ∑ fd = 29 − 39 = −10
x = A+ c × N
fd∑ =14 + 1 × 3010−
≈14 − 0.33 = 13.67.
Ofvc!Ljx;!
∴ x = N
xf ii∑ = 30410 ≈ 13.67.
x f f × x 10 4 40 11 5 55 13 8 104 15 6 90 16 4 64 19 3 57
olik<kl< N = 30 ∑ fx = 410
!
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wMk<Kg<gim<M 3: hqe<uVl<!Ht<tq!uquvr<gTg<G!%m<Ms<!sivsiq!gi{<g/
lkqh<oh{<! 80 85 90 95 100 li{uIgtqe<!
w{<{qg<jg!5 6 6 2 1
kQIU: A = 90 , c = 5 ,c
Axd −= we<g/ x− A gjtg<!g{g<gqm<M!c = 5 we!nxqg/!
x f d f × d
80 5 −2 −10
85 6 −1 −6
90 6 0 0
95 2 1 2
100 1 2 2
N = 20 ∑ fd = −12
∴ x = A+ c × N
fd∑
= 90 + 5 ×2012− = 90 − 3 = 87.
wMk<Kg<gim<M 4: hqe<uVl<!Ht<tq!uquvr<gTg<G!%m<Ms<!svisiq!gi{<g/
hqiqU!-jmoutq! 0-10 10-20 20-30 30-40 40-50 50-60 lkqh<oh{<! 12 18 27 20 17 6
kQIU: Ofvc!Ljx;!
hqiqU!-jmoutq! jlb!lkqh<H x fqgp<ou{<! f f × x
0-10 5 12 60
10-20 15 18 270
20-30 25 27 675
30-40 35 20 700
40-50 45 17 765
50-60 55 6 330
N = 100 ∑ fx = 2800
nm<<muj{bqzqVf<K?!N = olik<k!fqgp<ou{< = 100, ∑ fx = 2800.
∴ x =N
fx∑ = 1002800 = 28.
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SVg<G.upqLjx;!A = 30, c = 5, c
Axd −= we<g/
hqiqU!-jmoutq! jlb!lkqh<H! x d fqgp<ou{<! f f × d
0-10 5 −5 12 −60 10-20 15 −3 18 −54 20-30 25 −1 27 −27 30-40 35 1 20 20 40-50 45 3 17 51 50-60 55 5 6 30
N = 100 ∑ fd = 101−141= −40
∴ x = A+ c × N
fd∑ = 30 + 5 ×100
40− = 30 − 2 = 28.
hbqx<sq 10.1.1 1. svisiqjbg<!gi{<g/! 7, 12, 18, 14, 19, 20. 2. yV!uGh<hqZt<t!15 li{uIgtqz<? 4 li{uIgt<!66 lkqh<oh{<gTl<?!5 li{uIgt<!
67 lkqh<oh{<gTl<?!6 li{uIgt<!68 lkqh<oh{<gTl<!wMk<kiz<?!nu<uGh<hqe<!%m<Ms<!svisiqjbg<!gi{<g/
3. hqe<uVl<!Ht<tq!uquvr<gTg<G!%m<Ms<!svisiqjbg<!g{g<gqMg/!
x 5 10 15 20 25 30 f 4 5 7 4 3 2
4. hqe<uVl<!Ht<tq!uquvr<gTg<G!%m<Ms<!svisiqjbg<!g{g<gqMg/!
lkqh<oh{<gt<! 65 70 75 80 85 90 100 li{uIgtqe<!w{<{qg<jg!
11 6 3 6 4 10 4
5. hqe<uVl<!Ht<tq!uquvr<gtqzqVf<K!%m<Ms<!svisiqjbg<!gi{<g/!
lixq! 15 25 35 45 55 65 75 85 fqgp<ou{<! 12 20 15 14 16 11 7 8
6. !hqe<uVl<!nm<muj{jbg<!ogi{<M!%m<Ms<!svisiqg<!gi{<g!!
hqiqU!-jmoutq! 10-20 20-30 30-40 40-50 50-60 fqgp<ou{<! 8 14 7 10 11
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10.1.2 -jmfqjz!ntU
kvh<hm<cVg<Gl<! osh<heqmh<hmik! Ht<tq! uquvr<gjt! WX! nz<zK! -xr<G!uiqjsbqz<! wPKl<OhiK?! fl<liz<! yPr<Gh<hMk<kq! -Vg<Gl<! uiqjsbqz<! fMfqjzbig!njlf<kqVg<Gl<! yV!lkqh<jhg<! g{<mxqb!LcBl</!-f<k! fM!lkqh<H!nz<zK!lk<kqbqz<!dt<t!lkqh<H!Ht<tq!uquvr<gtqe<!-jmfqjz!ntU!weh<hMl</!
!wMk<Kg<gim<mig?!14, 28, 20, 29, 18, 25, 26, 17, 36 we<gqx Ht<tq!uquvr<gjtg<!gVKOuil</!nux<jx!WX!uiqjsbqz<!wPk?!14, 17, 18, 20, 25, 26, 28, 29, 36 we!flg<Gg<!gqjmg<gqxK/! -u<uiqjsbqz<?! 25 NeK?! Ht<tq! uquvr<gtqe<! fMfqjzbig!
njlf<kqVg<gqxK/! weOu! 25 NeK?! Ht<tq! uquvr<gtqe<! -jmfqjz! ntU! NGl</!-r<G?! kvh<hm<cVg<Gl<! lkqh<Hgtqe<! w{<{qg<jg! yx<jxh<hjmbqz<! njlf<Kt<tkiz<?!uiqjsbqz<!njlf<Kt<t!lkqh<hig!-jmfqjz!ntU!-Vg<gqxK/
85, 79, 57, 59, 66, 26, 40, 33, 48, 53 we<gqx! Ht<tq! uquvr<gjtg<! gVKOuil</!
nux<jx!yPr<GhMk<kq!WX!uiqjsbqz<!wPk? 26, 33, 40, 48, 53, 57, 59, 66, 79, 85 weg<!gqjmg<gqxK/! -r<G! -vm<jmh<hjm! w{<{qg<jgbqz<! lkqh<Hgt<! njlf<Kt<tkiz<?!
yPr<GhMk<kqb!uiqjsbqe<! fMuqz<! 53 lx<Xl<! 57 njlf<Kt<te/!weOu?!fil<!-u<uqV!
w{<gtqe<! svisiqjbg<!g{g<gqm?! 552
1102
5753==
+ NeK!uiqjsg<G!jlb!lkqh<hig!
-Vg<gqxK/! lkqh<H! 55 NeK?! kvh<hm<m! Ht<tq! uquvr<gtqz<! yV! lkqh<hig!-z<jzobe<xiZl<?! nKOu! -jmfqjz! lkqh<hig! njlgqxK/! NgOu?! kvh<hm<m!osh<heqmik! Ht<tq! uquvr<gtqe<! -jmfqjz! ntuqjeh<! ohx! hqe<uVl<! upqgjt!Olx<ogit<Ouil</ Lkzqz<?! kvh<hm<m! olik<k! Ht<tq! uquvr<gjt! WXuiqjsbqOzi! nz<zK!-xr<GuiqjsbqOzi! wPk! Ou{<Ml</! N we<hK! lkqh<Hgtqe<! w{<{q<g<jg! we<g/! N yx<jxh<hjm! w{<{ig! -Vf<kiz<?! Olx<g{<m! uiqjsbqz<! yV! lkqh<H! fMfqjzbig!
-Vg<Gl</! -K! kvh<hm<m! lkqh<Hgtqe<! WX! nz<zK! -xr<Guiqjsbqz<! ⎟⎠⎞
⎜⎝⎛ +
21NuK
dXh<hig! njlBl</! -KOu! -jmfqjz! lkqh<H! NGl</! N -vm<jmh<hjm! w{<{ig!njlf<kiz<?! nr<G! -V! lkqh<Hgt<! fMfqjzbig! njlBl</! nju! Ht<tq! uquv!
lkqh<Hgtqe<! WX! nz<zK! -xr<Guiqjsbqz<!2N uK! lx<Xl<! ⎟
⎠⎞
⎜⎝⎛ +1
2N uK!
dXh<HgtiGl</!weOu?!-u<uqV!dXh<Hgtqe<!svisiqOb!-jmfqjz!NGl</! kvh<hm<m!Ht<tq!uquvr<gt<?!fqgp<ou{<!hm<cbzqz<!njlf<kqVf<kiz<!-jmfqjzjb!gi{h<!hqe<uVl<!Ljxjbg<!jgbitzil</!
Lkzqz<?!GuqU!fqgp<ou{<!fqvjz!njlk<K?!2N e<!lkqh<jhg<!gi{!Ou{<Ml</!
-r<G! N we<hK! olik<k! fqgp<ou{<! (N = nf....ff +++ 21 )! NGl</! hqxG!
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nm<muj{bqzqVf<K?!2N g<Gs<! sllie! nz<zK!
2N g<G! sx<Ox! nkqglie! GuqU!
fqgp<ou{<! lkqh<hqje!djmb! lixqbqe<! lkqh<H!nz<zK! hqiqU! -jmoutqjbg<! gi{!Ou{<Ml<! )-K! hvuzqe<! -jmfqjz! hqiqU! -jmoutq! weh<hMl<*/! kvh<hm<m! Ht<tq!uquvr<gTg<G?! -jmfqjz! hqiqU! -jmoutqg<Giqb! lixqbqe<! lkqh<Oh! kvh<hm<m! Ht<tq!uquvk<kqx<gie!-jmfqjz!lkqh<hiGl</ ! wMk<Kg<gim<M 5: 23, 25, 29, 30, 39 Ngqbux<xqe<!-jmfqjz!gi{<g/!kQIU: kvh<hm<cVg<Gl< lkqh<Hgt<!Wx<geOu!WX!uiqjsbqz<!njlf<Kt<te/!!N = lkqh<Hgtqe<!w{<{qg<jg!= 5. -K!YI!yx<jxh<hjm!w{</!NgOu?!
-jmfqjz = ⎟⎠⎞
⎜⎝⎛ +
21NuK!dXh<H!= ⎟
⎠⎞
⎜⎝⎛ +
215uK!dXh<H = 3!uK!dXh<H!= 29.
∴ -jmfqjz = 29. !wMk<Kg<gim<M 6: 3, 4, 10, 12, 27, 60, 55, 49, 50, 41, 32, 63, 71, 75, 80!e<! -jmfqjz!gi{<g/ kQIU: kvh<hm<cVg<Gl<!lkqh<Hgjt!WXuiqjsbqz<!njlg<g?!
3, 4, 10, 12, 27, 32, 41, 49, 50, 55, 60, 63, 71, 75, 80. N = lkqh<Hgtqe<!w{<{qg<jg = 15, YI!yx<jxh<hjm!w{</!
∴ -jmfqjz = ⎟⎠⎞
⎜⎝⎛ +
21N uK!dXh<H = ⎟
⎠⎞
⎜⎝⎛ +
2115uK!dXh<H = 8 uK!dXh<H= 49.
wMk<Kg<gim<M 7:!29, 23, 25, 29, 30, 25, 28 -ux<xqe<!-jmfqjz!gi{<g/ kQIU: kvh<hm<cVg<Gl<!lkqh<Hgjt!WXuiqjsbqz<!wPk?!!
23, 25, 25, 28, 29, 29, 30. N = lkqh<Hgtqe<!w{<{qg<jg!= 7, YI!yx<jxh<hjm!w{</
∴ -jmfqjz = ⎟⎠⎞
⎜⎝⎛ +
21N!uK!dXh<H = ⎟
⎠⎞
⎜⎝⎛ +
217 uK!dXh<H = 4!uK!dXh<H = 28.
wMk<Kg<gim<M 8:!26, 25, 29, 23, 25, 29, 30, 25, 28, 30 e<!-jmfqjz!gi{<g/ kQIU: kvh<hm<cVg<Gl<!Ht<tq!uquvr<gjt!WXuiqjsbqz<!wPk?
23, 25, 25, 25, 26, 28, 29, 29, 30, 30. N = lkqh<Hgtqe<!w{<{qg<jg!= 10, YI!-vm<jmh<hjm!w{</
∴ -jmfqjz = 2NuK!dXh<H!lx<Xl< ⎟
⎠⎞
⎜⎝⎛ +1
2N uK!dXh<H!Ngqbux<xqe<!svisiq!
= 5 uK!lx<Xl<!6 uK!dXh<Hgtqe<!svisiq!
= 26 lx<Xl<!28 gtqe<!svisiq = 2
2826 + = 27.
!!!!
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wMk<Kg<gim<M 9: hqe<uVl<!nm<muj{g<G!-jmfqjz!ntju!g{g<gqMg/
lixq ( x) 5 10 15 20 25 30 fqgp<ou{<!( f ) 3 6 10 8 2 3
!kQIU:
x f GuqU!fqgp<ou{<!
5 3 3 10 6 9 15 10 19 20 8 27 25 2 29 30 3 32
olik<k!fqgp<ou{<!= N = ∑f = 32 . Njgbiz<!2N = 16.
-jmfqjz!=! ⎟⎠⎞
⎜⎝⎛
2N !uK!lkqh<H = 16 uK!dXh<hqe<!lkqh<H/!Neiz<!16!uK!lkqh<hieK?!
GuqU! fqgp<ou{<! 19 dt<t! hqiqU! -jmoutqbqz<! njlf<Kt<tK/! -kx<Gh<!ohiVk<kLx<x!lixqbqe<!lkqh<H!15!NGl</!!weOu?!-jmfqjz = 15.
!hbqx<sq 10.1.2
1. hqe<uVl<!lixqgtqe<!okiGh<hqx<G!-jmfqjz!gi{<g;
(i) 66, 63, 55, 60, 46, 10 (ii) 35, 39, 36, 34, 28, 27, 45, 41 (iii) 60, 61, 60, 58, 57, 59, 70 (iv) 41, 45, 36, 37, 43, 45, 41, 36
2. 40 li{uIgtqe<!lkqh<oh{<gTg<gie!-jmfqjz!gi{<g;!
lkqh<oh{<gt<! 24 20 35 52 50 48 li{uIgtqe<!w{<{qg<jg!
4 7 3 9 5 12
3.!hqe<uVl<!Ht<tq!uquvr<gTg<G!-jmfqjz!gi{<g;!
x 1 2 3 4 5 6 f 4 6 5 3 2 5
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4. 43 okipqzitIgtqe<! %zqgt<! kvh<hm<Mt<te/! -ux<xqe<! -jmfqjz! lkqh<hqjeg<!gi{<g/
%zq! 25 35 45 55 65 okipqzitIgtqe<!w{<{qg<jg!
3 5 20 10 5
10.1.3 LgM!
jlbfqjz!ntjugtqz<!LgMl<!YI!ntuiGl</ (i) keqk<okiGkqbig! njlf<Kt<t! lkqh<Hgtqe<! g{k<kqz<! wf<k! yV! lkqh<hieK!
nkqg! w{<{qg<jgbqz<! -Vg<gqxOki! nK! kvh<hm<m! Ht<tq! uquvr<gtqe<! LgM!weh<hMl</
(ii) kvh<hm<m! Ht<tq! uquvr<gjt! yPr<GhMk<kq! yV! fqgp<ou{<! hm<cbzqz<!njlk<kiz<?! nkqg! fqgp<ou{<j{! ogi{<m! hqiqU?! LgM! hqiqU! weh<hMgqxK/!-h<hqiquqz<!dt<t!lixqbqe<!-jmh<hm<m!lkqh<H?!kvh<hm<m!Ht<tq!uquvk<kqe<!LgM!weh<hMl</!
wMk<Kg<gim<M 10: 7, 4, 5, 1, 7, 3, 4, 6,7 e<!LgM!gi{<g/ kQIU: Ht<tququvr<gjt!WXuiqjsbqz<!wPk?
1, 3, 4, 4, 5, 6, 7, 7, 7. Olx<gi[l<!Ht<tq!uquvr<gtqz<!7, nkqg!kmju!gi{h<hMgqxK/!weOu?!LgM!= 7. !
wMk<Kg<gim<M 11: 19, 20, 21, 24, 27, 30 e<!LgM!gi{<g/ kQIU: Ht<tq!uquvr<gt<! Wx<geOu!WXuiqjsbqz<!dt<te/!yu<ouiV!lkqh<Hl<!yOv!yV!kmju!Olx<g{<m!uiqjsbqz<!njlf<kqVg<gqxK/!weOu?!kvh<hm<m!Ht<tq!uquvr<gTg<G!LgM!-z<jz/ wMk<Kg<gim<M 12: 12, 15, 11, 12, 19, 15, 24, 27, 20, 12, 19, 15 e<!LgM!gi{<g/! kQIU: kvh<hm<m!Ht<tq!uquvr<gjt!WXuiqjsbqz<!wPk?
11, 12, 12, 12, 15, 15, 15, 19, 19, 20, 24, 27. -r<G!12!&e<X!LjxBl<?!15 &e<X!LjxBl<!njlf<kqVg<gqxK/!!∴ 12 lx<Xl< 15 -v{<Ml<!kvh<hm<m!Ht<tq!uquvr<gTg<G!LgMgtig!njlgqe<xe/!-r<G?!kvh<hm<m!Ht<tq!uquvr<gTg<G!-V!LgMgt<!-Vg<gqe<xe!we<X!%Xgqe<Oxil</ wMk<Kg<gim<M 13: hqe<uVl<!fqgp<ou{<!hm<cbzqzqVf<K!LgM!gi{<g/!
%zq! 45 50 55 60 65 70 75 okipqzitIgtqe<!w{<{qg<jg!
12 11 14 13 12 10 9
kQIU: hm<cbzqzqVf<K! fil<! nxquK?! lqg! dbIf<k! fqgp<ou{<! 14 NGl</! -kx<G!-jsuie! lixq! %zqbqe<! lkqh<H! 55 NGl</! -KOu! kvh<hm<m! uquvk<kqe<! LgmiGl</!nkiuK?!LgM! = 55.
!
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!hbqx<sq!10.1.3
1. hqe<uVl<!Ht<tq!uquvr<gTg<G!LgM!gi{<g;! (i) 84, 91, 72, 68, 87, 84 (ii) 65, 61, 72, 81, 51, 31 (iii) 38, 31, 22, 20, 31, 61, 15, 20 (iv) 15, 11, 18, 23, 11, 19, 11
2. hqe<uVl<!hvuzqx<G!LgM!gi{<g;!x 10 20 30 40 50 60 f 8 15 12 10 9 6
3. hqe<uVl<!nm<muj{bqzqVf<K!LgM!gi{<g/!
x 60 61 62 63 64 65 f 5 8 14 16 10 7
!
uqjmgt<!
hbqx<sq 10.1.1
1. 15 2. 67.1 3. 15.6 4. 79.1 5. 45 6. 35.4
hbqx<sq 10.1.2
1. (i) 57.5 (ii) 35.5 (iii) 60 (iv) 41 2. 48 3. 3 4. 45
hbqx<sq!10.1.3
1. (i) 84 (ii) LgM!-z<jz! (iii) 20, 31 (iv) 11 2. 20 3. 63!
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11. ujvhmr<gt<
nxquqbz<?! ohixqbqbz<! lx<Xl<! uIk<kgl<! Ohie<x! Kjxgtqz<! fil<! olb<ob{<!lkqh<Hgt<! ohXl<! hz<OuX! lixqgjtg<! g{<cVg<gqOxil</! wMk<Kg<gim<mig?! u{qgk<kqz<?!
yV! svg<gqe<! kvU! )s*! lx<Xl<! uqjz! )p*! -V! olb<lixqgtiGl</! -l<lixqgjt! yV!sle<him<ce<! upqbig! -j{g<gzil</! -s<sle<him<jmh<! hbe<hMk<kq?! yu<ouiV! s -e<!lkqh<hqx<Gl<! p -e<! lkqh<jh! fil<! ohxzil<<;! lx<Xl<! olb<ob{<gtiz<! Ne! uiqjs!Osicgt<! )s, p*! -e<! g{k<jkh<! ohxzil</! wz<zi! uiqjs! OsicgjtBl<? gqjmbs<S!!!!!!!!!!s-ns<sigUl<?! Gk<ks<S! p-ns<sigUl<! ogi{<m! giICsqbe<! ktk<kqz<! Ht<tqgtigg<!Gxqg<gzil</! -h<ohiPK!-h<Ht<tqgt<?! lixqgTg<gqjmObBt<t!dxuqe<! ujvhmk<kqje!!ujvbXg<gqxK! we<X! osiz<gqOxil</! -u<ujvhml<! lixqgtqe<! dxuqe<! ke<jljbg<!gim<MgqxK/! lqgLg<gqblie?! fil<! ncg<gc! hbe<hMk<Kl<! yV! ujvhml<! OfIg<OgiM!ujvhmliGl</! fil<! -h<ohiPK! OfIg<OgiM! ujvhmr<<gjth<! hx<xqBl<?! nju! wu<uiX!ujvbh<hMgqe<xe! we<hjkh<! hx<xqBl<?! nux<jxh<! hbe<hMk<kqs<! sqz! sle<hiMgjtk<!kQIh<hK!Gxqk<Kl<!gi{!Lx<hMOuil</!
11.1 OfIg<OgiM!ujvhmr<gt< x lx<Xl< y -V!lixqgt<! we<g/!nju! y = mx + c we<x! sle<him<M!ucuk<kqz<!
-j{g<gh<hm<miz<?!nju!Ofiqb!Ljxbqz<!-j{g<gh<hm<Mt<te!we<X!%XOuil</!fil<!
Wx<geOu!hGLjx!ucuqbz<!himk<kqz<?!giICsqbe<!ktk<kqz<!y = mx + c!we<<x!sle<hiM!yV! OfIg<Ogicjeg<! Gxqg<gqxK! weg<! g{<Mt<Otil</! -g<giv{k<kiz<kie<! x lx<Xl< y -ux<xqx<gqjmObbie dxju! yV! Ofiqb!dxU! we<gqOxil</! yu<ouiV! x lkqh<hqx<Gl<?!!!y = mx + c!we<x!sle<hiM!y!-e<!yV!lkqh<jhg<!ogiMg<gqxK;!lx<Xl<!x, y w{<gtiz<!njlbh<ohx<x yV! uiqjs! Osic! )x, y*! gqjmg<gqxK/! -u<uixigh<! ohxh<hm<m! uiqjs!!Osicgtqe<!g{l<?!y = mx + c !-e<!ujvhmk<kqje!ujvbjx!osb<gqxK/!OlZl<!-jk!Ofiqb!ujvhml<!we!njpg<gqOxil</!sle<hiM!y = mx + c!z<?!m NeK!nf<OfIg<Ogice<!sib<U?! c! NeK! nke<! y-oum<Mk<K{<M! we<hjk! fil<! fqjeU! %IOuil</! x=0!wEl<OhiK? y -e<! lkqh<H! y-oum<Mk<K{<miGl</! sqz! slbr<gtqz<! y-oum<Mk<K{<ce<!lkqh<H! 0 Ng! -Vg<Gl</! -s<$p<fqjzbqz<?! sle<hiM! y = mx! we<xigqxK;! lx<Xl<!-f<OfIg<OgiM! Nkqh<Ht<tq! upqs<osz<gqxK/! fil<! -h<ohiPK! hz<OuX! $p<fqjzgtqz<!OfIg<Ogice<! ujvhml<! ujvuK! Gxqk<Kh<! hiIh<Ohil</! yV! OfIg<OgiM! ujvukx<gie!nch<hjmg<!Gxqg<Ogit<?!nf<OfIg<Ogice<!lQK!-V!Ht<tqgt<!flg<Gk<!Okju!we<hkiGl</!hqe<uVl<<!Ljx?!OfIg<OgiM!ujvukx<gigh<!hqe<hx<xh<hMgqe<xK; hc 1: x -e<! -V! ouu<OuX! lkqh<Hgt<! x1, x2 -ux<jxs<! sle<hiM! y = mx + c -z<<!
hqvkqbqm<M?!y g<G!-V!lkqh<Hgt<!y1, y2!fil<!ohXgqe<Oxil</!-h<ohiPK!fil<!-V!Ht<tqgt<!)x1, y1*!lx<Xl< (x2, y2) -ux<jx!nf<OfIg<Ogice<!lQK!!ohx<Xt<Otil</!
229
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hc 2: x-ns<S lx<Xl< y-ns<S -ux<jx! yV! ujvhmk<kitqz<! ujvf<K! nux<xqe<! lQK!Okjug<Ogx<h! ntuQMgjtg<! Gxqk<Kg<ogit<Ouil</! hc! 1 -z<! ohxh<hm<m! ns<S!K~vr<gtqe<!ntjuh<!ohiXk<Ok!-V!ns<Sgjt!nzgqMkz<!osb<gqOxil</!ns<S!K~vr<gt<!nkqg!lkqh<Hgt<!djmbe!weqz<?!ns<Sgtqz<!1!os/lQ/!ntju!ohiqb!lkqh<jhg<!Gxqg<Gl<!nzgigg<!ogit<gqOxil</
hc 3: -V! Ht<tqgt< (x1, y1) lx<Xl<! (x2, y2) J! ujvhmk<kitqz<! gVkqb! giICsqbe<!ktk<kqz<!Gxqg<gUl</
hc 4: -V!Ht<tqgjtBl<!yV!OfIg<Ogim<Mk<K{<miz<!-j{k<K!-V!hg<gLl<!fQm<mUl</!-KOu!Okjubie!OfIg<OgiM!ujvhmliGl</!
wMk<Kg<gim<M 1: (2, 3) lx<Xl< (−4, 1) J!-j{g<Gl<!Ogim<ce<!ujvhml<!ujvg. kQIU:! ujvhmk<kitqz<! x, y-ns<Sgjt ujvf<K!nux<xqe<! lQK! 1 os/lQ/ = 1 nzG!weg<! Gxqg<gUl</!A(2, 3) lx<Xl<! B (−4, 1) we<he! kvh<hm<m! -V!Ht<tqgt<! we<g. fil<! -u<uqV! Ht<tqgjtBl<!ujvhmk<kitqz<! Gxqg<gqOxil</! fil<?! A lx<Xl< B J!OfIg<Ogimiz<! -j{k<K! -V! HxLl<! fQm<MOuil</!-h<ohiPK!Okjubie!ujvhml<!ohxh<hm<mK (hml<!11.1 Jh<!hiIg<gUl<). wMk<Kg<gim<M 2: y = 2x -e<!ujvhml<!ujvg. kQIU: y = 2x we<hK!OfIg<Ogice<!sle<hiM!we<hkiz<?!OfIg<OgiM!Nkqh<Ht<tq!upqbigs<!osz<gqxK/ x = −1, 0, 1 wes<! sle<him<cz<! hqvkqbqm?! LjxOb! fil<! y = −2, 0, 2 weh<! ohXgqOxil</! -kjeg<! gQOp!nm<muj{bigk<!kf<Kt<Otil<.
x −1 0 1 y −2 0 2
ujvhmk<kitqz<! x-ns<S?! y-ns<S! ujvf<K?! nux<xqe<! lQKogit<gqe<Oxil</! fil<! -h<ohiPK! (−1, −2), (0, 0), ujvhmk<kitqz<! Gxqg<gqOxil<?! -h<Ht<tqgjt! -j{k<K-h<ohiPK! flg<Gk<! Okjubie! OfIg<OgiM! ujvhml<! ghiIg<gUl<). !
wMk<Kg<gim<M 3: y = 3x −1!-e<!ujvhml<!ujvg/ kQIU: x = −1, 0, 1 wes<! sle<him<cz<! hqvkqbqm!LjxOb! fil<! y = −4, −1, 2 weh<! ohXgqOxil</!ujvhmk<kqz<! (−1, −4), (0, −1) lx<Xl< (1, 2) Jg<!Gxqg<gUl</!Ht<tqgjt!OfIg<Ogimiz<!-j{k<K!-V!HxLl<! fQm<mUl</! flg<Gk<! Okjubie! OfIg<Ogice<!ujvhml<!gqjmk<Kt<tK!(hml<!11.3 Jh<!hiIg<gUl<).
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hml< 11.1
hml< 11.2
! 1 os/lQ/ = 1 nzG! weg<!(1, 2) ! Ngqb! Ht<tqgjt!! -Vhg<gLl<! fQm<MgqOxil</!qjmk<Kt<tK (hml<! 11.2 Jh<!
hml< 11.3
wMk<Kg<gim<M 4: sib<U 23− NgUl<?! y-oum<Mk<K{<M −3 NgUl<!ogi{<m!OfIg<Ogice<!
ujvhml<!ujvg. kQIU: Ogice<!sle<hiM! y = mx + c nz<zK
y = 23− x + (−3) nz<zK! y =
23− x −3.
x = −2, 0, 2 weh<! hqvkqbqm! LjxOb!!!!y = 0, −3, −6 we!fil<!ohXgqOxil</ (−2, 0), (0, −3) lx<Xl< (2, −6) Ht<tqgjt!ujvhmk<kitqz<!Gxqg<gUl</!Ht<tqgjt!!
x −2 0 2 y 0 −3 −6
-j{k<K! -V! HxLl<! fQm<mUl</!-h<ohiPK! flg<Gk<! Okjubie!OfIg<Ogice<!ujvhml<!gqjmg<gqxK!)hml<!11.4!Jh<!hiIg<gUl<*/ !wMk<Kg<gim<M 5:! 2x + 3y = 12 we<x!OfIg<Ogim<cekQIU: ogiMg<gh<hm<ms<!sle<him<jm!hqe<uVliX!w
!!3y = −2x + 12 nz<<zK!y = ⎟⎠⎞
⎜⎝⎛ −
32 x + 4.
x = −3, 0, 3 weh<! hqvkqbqm! LjxOb y = 6, 4, 2 we fil<!ohxzil</ (−3, 6), (0, 4) lx<Xl<! (3, 2) Ngqb! Ht<tqgjt!ujvhmk<kitqz<!Gxqg<gUl</
x −3 0 3 y 6 4 2
Ht<tqgjt! -j{k<K! -V! HxLl<!fQm<mUl</! -KOu! Okjubie! OfIg<OgiM!ujvhmliGl<!)hml<!11.5!Jh<!hiIg<gUl<*/ !
wMk<Kg<gim<M 6: x = 3 !-e<!ujvhml<!ujvg. kQIU: x = 3! we<x! sle<him<cz< y Jh<! hx<xGxqh<hqmh<hmuqz<jz! we<hjk! fil
gueqg<gqOxil</! weOu?! y! -e<! wf<k! yV
lkqh<hqx<Gl<!x = 3 NGl<. y g<G!1!lx<Xl<!2!wekOkIf<okMg<g?!-V!Ht<tqgt<!(3, 1) lx<Xl< (3, 2)x = 3 we<x! Ogim<ce<! lQK! -Vg<gqxK! wehohXgqOxil<. -h<Ht<tqgjt! ujvhmk<kitqzGxqg<gUl</!-V!Ht<tqgjt!OfIg<Ogimiz<!!
231
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hml< 11.4
<!ujvhml<!ujvg. Pkzil</!hml< 11.5
q!<!
!
<!
, <!
<!
hml< 11.6
-j{k<K!!-VHxLl<!fQm<mUl</!flg<Gk<!Okjubie!OfIg<OgiM!ujvhml<!gqjmg<gqxK!
)hml<! 11.6! Jh<! hiIg<gUl<*/! -r<G! OfIg<O<giM! y-ns<Sg<G! -j{big! dt<tjk!gueqg<gUl</ wMk<Kg<gim<M 7: y = − 4 -e<!ujvhml<!ujvg. !kQIU: y -e<! lkqh<H! lixilz<! −4 weUl<?!sle<him<cz< x -e<! lkqh<H! Gxqh<hqmh<hmilZl<!-Vh<hjkg<!gi{<gqOxil</!weOu?!fil<!x g<G!-V!lkqh<Hgt< −2, 2 wek< OkIf<okMg<gqOxil<. -h<ohiPK!flg<G!-V!Ht<tqgt< (−2, − 4) lx<Xl< (2, − 4), OfIg<OgiM y = − 4 -e<! lQkqVh<hkigh<!ohXgqOxil</!!
x − 2 2 y − 4 − 4
!!!!-u<uqV!Ht<tqgjtBl<!ujvhmk<kqz<!Gxqk<K!yV!Of-VHxLl<! fQm<mUl</! -KOu! flg<Gk<! Okjubie!hiIg<gUl<*/!-r<G!OfIg<OgimieK!x-ns<Sg<G!-j{b
hbqx<sq 11.1 1. gQp<g<g{<m!Ht<tqgt<!upqs<osz<Zl<!OfIg<OgiM!uj (i) (2, 3) lx<Xl<! (4, −6) (ii) ( (iii) (−3, 2) lx<Xl<!(5, −1) (iv) (2. hqe<uVueux<xqe<<!ujvhml<!ujvg: (i) y = −2x (ii) y = 3x (iii) x = 53. hqe<uVueux<xqe<<!ujvhml<!ujvg: (i) x = −3 (ii) y = 5 (iii) (iv) y = − 4 (v) 2x + 3 = 0 (vi) 4. y = mx + c we<hkx<gie!ujvhmk<kqjeh<!hqe<uVl<!
(i) m = 3 lx<Xl<!c = 4 (ii) m
(iii) m = −3 lx<Xl<!c = − 4 (iv) m5. hqe<uVl<!sle<hiMgtqe<!ujvhml<!ujvg: (i) 2x + 3y = 12 (ii) (iii) y + 2x −5 = 0 (iv)
232
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hml< 11.7
Ig<Ogiceiz< -j{g<gUl</!-kje!ujvhmliGl<! )hml<! 11.7! Jh<!ig!dt<tjk!gueqg<gUl</!!!!!!!!!
vhml<!ujvg; −1, 0) lx<Xl<! (−2, −5) −2, −3) lx<Xl<! (5, − 4)
y (iv) x = − 4y
x = 5 1 + 2y = 0 lkqh<Hgjtg<!ogi{<M!ujvg;
= 32− lx<Xl<! c = 3
= 2 lx<Xl<! c = −5
x −5y = 10 x −2y + 1 = 0
11.2 OfIg<OgiM!ujvhmr<gtqe<!hbe<hiM -bx<g{qks<! osbz<hiMgtqe<xq! -V! yVr<gjl! OfIg<OgiMs<! sle<hiMgTg<G?!
nux<xqe<! ujvhmr<gjt! ye<xig! ujvf<K! kQIU! gi{zil</! x, y -z<! njlf<k! yV!Ofiqb!sle<hiM ax + by + c = 0 we<x!ucuqz<!-Vg<Gl<. -s<sle<hiM!giICsqbe<!ktk<kqz<!yV!OfIg<Ogijmg<!Gxqg<Gl</!weOu?!-V!yVr<gjls<!sle<hiMgtqe<!kQIU!gi{<hK!-V!OfIg<OgiMgtqe<!ohiKh<Ht<tqjbg<!g{<Mhqch<hkx<Gs<!slliGl</!-r<G?!&e<X!ujggt<!d{<migqe<xe/
Lkziukig?!-V!OfIg<OgiMgTl<!ye<Oxiomie<xig!njlgqe<xe;!nkiuK?!-V!ujvhmr<gTl<! ye<xigqe<xe/! -s<$p<fqjzbqz<! Lcuqzi! Ht<tqgt<! -V!
ujvhmr<gTg<Gl<! ohiKuieju! we! nxqgqOxil</! weOu, ogiMg<gh<hm<m!sle<hiMgTg<G!w{<{x<x!kQIUgt<!d{<M/
-v{<miukig?! -V! OfIg<OgiM! ujvhmr<gt<! ye<Oxiomie<xig! njlukqz<jz;!Neiz<?! nju! -j{bieju/! -s<$p<fqjzbqz<! -V! OfIg<OgiM! ujvhmr<gTl<!ye<jxobie<X!oum<cg<! ogit<ukqz<jz/!weOu?!-V!OgiMgTg<Gl<!ohiKuie!Ht<tq!gqjmbiK/!weOu?!-u<ouiVr<gjls<!sle<hiMgTg<Gk<!kQIU!gqjmbiK/
&e<xiukig?! -V! OfIg<OgiMgt<! yOv! yV! Ht<tqbqz<! oum<cg<ogit<gqe<xe/!-s<$p<fqjzbqz<?! ogiMg<gh<hm<Mt<t! yVr<gjls<! sle<hiMgTg<Gk<! keqk<k! kQIUgt<!d{<M/! -ju?!nf<OfIg<OgiMgt<! ! oum<cg<ogit<Tl<! ! Ht<tqbqe<!ns<S!K~vr<gtiGl</!weOu?!-u<ouiVr<gjls<!sle<hiMgTg<G!yOvobiV!kQIU!d{<M/ wMk<Kg<gim<M 8: 2x + y = 1 lx<Xl<! 4x + 2y = 2 we<x! yVr<gjls<! sle<hiMgjt!ujvhml<!&zl<!kQI. kQIU: OgiM 1: y = −2x + 1 yOv!Ht<t)OgifQm<mU-j{)hmllQKtOgimHt<tohiKkQIUgsle !
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x −1 1 y 3 −1
ujvhmk<kitqz<! -V! sle<hiMgtqe<!qgjtg<! Gxqg<gUl</! Ht<tqgjt!m<Mk<K{<miz<*! -j{k<K! -VHxLl<!l</! -h<ohiPK! -V! ye<Oxiomie<X!f<k! OfIg<OgiMgt<! gqjmg<gqe<xe <!11.8!Jh<!hiIg<gUl<*/!yV!Ogim<ce<!<t! wf<k! yV! Ht<tqBl<! lx<oxiV!<ce<!lQK!dt<tK/!weOu?!w{<{x<x!qgt<! -V! OfIg<OgiMgTg<Gl<!uig! dt<te/! weOu?! w{<{x<x!t<?! ogiMg<gh<hm<m! -V! yVr<gjls<!<hiMgTg<G!d{<M/
233
hml<!11.8!
OgiM 2: 2y = −4x + 2 i.e., y = −2x + 1
x −1 1 y 3 −1
wMk<Kg<gim<M 9: x − 2y = 4 lx<Xl<! x − 2y = −6 -e<!ujvhml<!ujvf<K!nke<!&zl<!-s<sle<hiMgtqe<!kQIU!gi{<g/! kQIU: OgiM 1: x − 2y = 4 nz<zK!
! 2y = x − 4 nz<zK!y = ⎟⎠⎞
⎜⎝⎛
21 x − 2.
x 0 2 y −2 −1
OgiM 2: x −2y = − 6 nz<zK!
2y = x + 6 nz<zK y = ⎟⎠⎞
⎜⎝⎛
21 x + 3.
x 0 2 y 3 4
fil<! (0, −2) lx<Xl< (2, −1) we<x!Ht<tqgjt!ujvhOfIg<OgiM! ujvOuil</! nMk<K?! fil<! (0, 3) lx<Xujvhmk<kitqz<!Gxqk<K!nux<xqe<!upqOb!OfIg<Og-j{bigs<!osz<ujk!fil<!gi{zil</!weOu?!nj
-u<ouiVr<gjls<!sle<hiMgTg<Gk<!kQIU!gqjmbiK
!wMk<Kg<gim<M 10: yVr<gjls<!sle<hiMgt< x + y =&zl<!kQIg<gUl<. kQIU: !OgiM 1: x + y = 5 !!!!nz<zK y = −x + 5 (1)
x −2 −1 3 y 7 6 2
n
(−2, 7), (−1, 6) lx<Xl<! (3, 2) Ngqb!Ht<tqgjt!ujvhmk<kitqz<! Gxqg<gUl</! -h<Ht<tqgt<! upqOb!OfIg<OgiM!ujvbUl</!-KOu!sle<hiM!)2*!-e<!Ofi<g<OgiM!ujvhmliGl</!nMk<K! (1, −2), (0, −3) lx<Xl< (3, 0) Ngqb! Ht<tqgjt! nOk!ujvhmk<kitqz<! Gxqg<gUl</! -h<Ht<tqgt<! upqOb!OfIg<OgiM! ujvbUl</! -KOu! ! sle<hiM! (2)!-e<! OfIg<<OgiM! ujvhmliGl</! -u<uqV!OfIg<OgiM! ujvhmr<gTl<! yV! Ht<tq! ! P(4, 1)!-z<! oum<MgqxK! (hml<! 11.10! Jh<! hiIg<gUl<*/!-h<Ht<tq?! -V! OgiMgtqe<<! lQKl<! njlukiz<?!yVr<gjls<!sle<hiMgtqe<!kQIU x = 4, y = 1 we!nxqgqe<Oxil<.
!
234
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hml< 11.9
mk<kitqz<!Gxqk<K!nux<xqe<!upqOb!
l< (2, 4) Ngqb! Ht<tqgjt! nOk!iM!ujvOuil</!-r<G!OfIg<OgiMgt<!u!oum<cg<!ogit<tuqz<jz/!weOu?!
!(hml<!11.9!Jh<!hiIg<gUl<*/!
5, x − y = 3 -ux<jx ujvhml<!
z
x y
hml< 11.10
OgiM 2 : x −y = 3 <zK!!!y = x −3 (2)
1 0 3 −2 −3 0
hbqx<sq 11.2 gQOp!ogiMg<gh<hm<Mt<t! 1 Lkz<! 10 ujvbqzie!sle<hiMgtqe<! okiGkqg<G!ujvhml<!!!&zl<!kQIU!gi{<g;
1. x + y = 0, x = 4. 6. 2x + y = 1, 4x + 2y = 2.
7. x + 2y = 4, x + 2y = 6. 8. x − 3y = 4, x + 2y = −1. 9. 3x + y = 2, 6x − y = 7. 10. 2x + 3 = 0, 4x + y + 4 = 0.
2. x − y = 0, y = −3. 3. x + y = 2, x − y = 2. 4. x − y = 6, 2x + y = 9. 5. x + y = 5, x − y = 1.
uqjmgt<!hbqx<sq!11.1!
1. (i)
1. (ii)
1. (iii)
1. (iv)
2 (i)
x −1 0 1 y 2 0 −2
2. (ii)
235
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x −1 0 1 y −3 0 3
2. (iii) 2. (iv) x 0 5 −5
y 0 1 −1
3. (i) (ii) x −3 −3 −3
y 1 2 −1
(iii) (iv) x 5 5 5
y 0 1 2
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x 0 4 −4 y 0 −1 1
x 1 2 −1 y 5 5 5 x 0 1 −1 y −4 −4 −4236
(v) (vi) x −1.5 −1.5 −1.5
y 0 1 −1
4. (i) x −1 0 1
y 1 4 7
(iii) x 0 1 −1
y −4 −7 −1
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x 0 1 −1 y −0.5 −0.5 −0.5
(ii)
x −3 0 3 y 5 3 1(iv)
x 0 1 2 y −5 −3 −1237
5. (i) (ii) x −3 0 3
y 6 4 2
(iii) (iv) x −1 0 1
y 7 5 3
hbqx<sq 11.2 1.
x −1 0 1
y 1 0 −1
kQIU!x = 4; y = −4.
2.
x −1 0 1
y −1 0 1
kQIU x = −3; y = −3.
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x 4 4 4
y 1 0 −1
x 0 1 2
y −3 −3 −3
238
x 5 0 −5 y −1 −2 −3
x 1 3 5 y 1 2 3
3.
x 0 1 2
y 2 1 0
kQIU x = 2; y = 0.
4.
x 3 4 5
y −3 −2 −1
kQIU x = 5; y = −1. 5.
x 2 3 4
y 3 2 1
kQIU x = 3; y =2. 6.
x −1 0 1 y 3 1 −1
w{<{x<x!kQIUgt/<! 7.
x 0 2 4
y 2 1 0
kQIU!-z<jz/
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x 3 4 5
y 1 2 3
x 2 3 4
y 5 3 1
x 2 3 4
y 1 2 3
x −1 0 1 y 3 1 −1
x 0 2 4
y 3 2 1
239
8.
x −1 1 3
y 0 −1 −2
x −2 1 4
y −2 −1 0
kQIU x = 1; y = −1. 9.
x 0 1 2
y −7 −1 5
x 0 1 2
y 2 −1 −4
kQIU x = 1; y = −1. 10.
x −2 −1 0
y 4 0 −4 x 2
3−
23−
23−
y −2 0 2
kQIU! x =
23− ; y = 2.
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