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10x25

. 05x 150

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11. __ .,,.,_�&,gnez·- s·-'.l"�.;;-1-$f1,,.:....,.,.,;:�-,_ ���mts.t · · ·

11: I �,& 8" ..:..3" =8-3 = 5. n == l ��?;J:> gtfic,(:<..3 t.:i:'..,,n d;, ( ;,-·) ··,. .� #

11 = p + 1 e:,eZu�.

jxj < 2-x2

<:>El2nt!! 2ti �'2:5.i

X �U-e,«;,1X>'.fr··

x<2·-rX1 G)

x·:+xc - ·2:.<-0 (i ·, ·n)(\x:'.11f.ti!·�t ��;J:�

ct�g@t{ t.l-I <-"rr<Jt

= (5m+3")(5 + 3)---.J''' I

=8x5m+5x3'' ·l 3"' 1 • 1'" 1

=5(8m+JP) 0 Sm·' :V \' 1;'

x· 2: o C,) ·�ii.) 1,1

. X=c 2, .. x'

x-' +x- 2. · 0(x + 2)(x · I) - 0

x

�_x-�1(0

X·<O'e:)�W:, �x <;-'2"-X

2 0 xz -x....::2 <0 (x+1)t�-i}<,O' �· <:XS'< 0 �

( 0

,· .. ···. - . . �- •..

·.\. \' ? :a: 0

< I. / H , ·' t > = o\' . 1 0

I .. I . l f i?)_;_ �- �

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····-·····. ···-·--------------..,..----------:-----�-.�• rpumff.) (tie)(,:)� �.r:n ft -3 + 411 = 2 � � zc:,� � �l:J.f·�� q

ciiirn·c�•� � vm Cij re-�-�� �&Cl� z � fz+4il e ��l\�.J (ff� q<l)Otn �

,.., ··---�------------------·-..;..___ ___.:.__ ____________,

..... ··----+-----,---

I z -3 + 4lj :; 2· l z - (3 =- 4i) J == 2

. ·--·-··------:..----------------'------�--:--.......-,

., � c z•· isn n � 5 aiiS ��Ji�+ :1)' 6l � � x.W-lO al �'i�· 1ooe,·� qi} '1lt). tJ 5J. . t x qmo ��tli�. . · . · · · · ·

n=5

to :6(; 3 .< �--:ig" )e<>GJ <:1'25'):>@D ·

n:.:.:.:::.s. @.

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r----,-----------------,.-,-· -.� .. -..... " ... --------

ll·m (x+Ji)'�-4 · =hmx� sin4x x�

-----,--------�- ·----·· .

n \2)

(x+.fi.)4 -(Ji)'' I (x+.fi-n} .. si11•lx

4. . ,,x

------------,---�--......... ��·---··-------·------

-y:::t'.i�:,"�Q�..J'S l ccfoe,Q z:QS yazd:.,, t)�ffl -� 4� tm, \ttle�� 111�

.... I __ 2...:;(1...;.•� ..... J.:..J_ao_· _··ite-· dld_· ...;..··-�_· _· -------------�

· -·------

: ··· · tr··-----­,.

ll I

it ·· Jc'. Ix I J ,· 'd.r 5

., . 0 .

�0 I · · e · 1 · • c' · • I· I

r-;'\ '2-)

5

I ·r. ci�lllfe.lt.:ll o � <TIE� .zy-l!l)Cad c � x = 2 + oos 1.8, J ,.. 4 sin 8 � ��. .

d .· .

i!t�hri ,,-;;rn eri>&.; e>B� otFi� � fJ-:::.; em �eda � C � �(

niiic:e¢ik.',d �z:odcao x -./ly+ 2 = 0 00 �mm.

,l.x dy . ···2sin.2B . - =4cose

.10 . dB

4c.osB ---=---

4sin0cos8 sinB

![. d y r;::; . 1) ..•. 58, -=-v2 4· . dx

< ,1, /J2ftfd�"J5@ct!i.i3 qiflec>Q)@� �@2:)'}d�=·-

1· 2Ji =- -1

(x_; 2) !':"\ . .Ji . \.V

Jiy .: 4 =zx-2 �x-Jiy+2=0.

R · 1\(l0�0)�8(0.'l��·C � � �- C(1.�"D{:3,6)"ca.i:i .r:»� CD�

@e...to0do ce"5 ��" a8 aome)=f:ln

ACB!) �qi��� ma 2S ::a 6\0 i:,,$�� «�

CD� ®:ilro c?:Sf�:15�, M (2, �).

A @�&ie-D �®25)6-ci�: y - S = _..!_ x-0 2

==>.x.+2y-]0;=0 5

A (10,0)

. CD-S . 6-2 4

G) D)d�,-- . q��@.€fuc.:,=-=-=2. :. CDJ_AB. s c:..:!O 3�1 . 2.

6

r------__...;. ___________ .. , ..... , ..... , ........ ... ,.--·--·---·--------. ,. 0 �. � Wee! � y::: 1 � f + r-lt- 2:y + J 111 0 �ti! lfr�m ed�� atp

' I��=�� lltJ qdQ Kx:,tmO. .

' --------- ·., ... ,, ... ,' ' ' . "" " - - . ------'

x2 +y2 .:..2x-2y+l +.-1.(y-l)=_O uad7$c.i (v. o fo::�t:, tiidiiH·tl ,.1·;-,1 !.,1J1/.·i

1-1=0=>1=L G)

�va;i)S €la?Sf 2'3)�C6 e5®zm6�� x2 + y2

-2x � y = 0 c:)

( ' ), ( 1]2 1 (J5)2

x-l - + y�2 =1+4= 1

( l) 5 @zmzf "' 1, 2 , q6c., == � _·

G) 0________________ ......... _..;,.,,;.,,.· ............. '!, •••• . � .......... .-... ·----'------

sin_a+sinP = 1 ,

co a+cosP=V3-

_ · .. -( a+ f3) . {. a . .. P ·) __ . · ( ..... ,.,.) 2sm _ _ co. ······:._--·-· ., .. I ., !·1 2 _} -.. -·-

, a+ /J) {a··· jJ) J .,..,. ... .,. 2�os -_ -_ - co. -··--:-··-, . . , I ( ., )

2 2 '-.-�J

ta+/J }'._.-.,_,::. ·-: @· -_ . ,r .,/_·_,' e@�� -tan :-...,.,...' ·: - · ,.:--·.__:...cc'.' -s· :. 0 < a JI � .)/) J,; · - · - . '. - . ·., )i2' · .. ·. :- . E

?? . - - ' ., "i• ..

7

. I

···-·-··-··----------------------

---------,

it. i ,, ) \ &1 �l�i;iw ·1 &.! F(x), G(x) � H(i} c,1>i' � oai.t:t> �td@Dtn � e�m eieS.

F(x) • (.? - cu+ 1}(.c1 - {3x ': 1), �a�, f3m� � ��

G(x)· -6x.;1 - 35.t3 + 62x2 - 35x + 6�.

ll(.x);.;;.x4 +x

2 +l, (i) f\.t) = o ;s}'.) G(�>:: o ci:s") @fioD ® Q� ® §e S@S .m®, a a:r.i {J §e t:i�d (H� 003'e

C)l�Jr..,)()&Q 6x2 - 35.x + 50 = 0 a0 �ffltn. ��. G{x) = o �l!K56�a1 �§ © ®e �irl�.

(ii) F(x} Ila H(x) s-t)8 m®, a ID� {J <.} Baa, �is �,sj � H(.t)-=O�r:sxf�� @c i;rociclo2;5)

«:00 � @D (3-,tj���.

(M {i� /(x)•2x" +r,l +o.x�l c1.8 \!)�; � y � 6 �� � �. t(-1)=0 �/{-2)=21 @c> � tttB oo.JC.x) a ®cl� r:,�-� � �ox.,?rlffi. 2

(ii) t.:k>� ® m:;,�Zi) xe:,��;, (x2+�+ !) P(-t)+.(i2- J)Q(xl= 3x ���a-tee,.�� P(X)ll'.l:>Q(x) (11:S)e s�:l(g� � @1:)Xjfflm .

.... . "' ·········----·--------------------------------1

(iii

F(x) = (x2 - ax+ 1)(x2 - /3� + 1)

= x4 - (a + /J)x3 + (2 + af3)x2

- (a + {J)x + 1 \....:_)

(i\ F(x) = 0 <0:> G(x) = 0 �ts)@ �c �e3t;'l �@;, ��o G(x)=6F(x)-;;:;;,

hx'' � 35x3 + 62x2 - 35x + 6 =.6[x4 - (a+ f3)x3 + (2 + a{3)x2 -(a+ f3)x + 1] 8 '"'""'"'"'"'""",Me.;: a+ p = 3: 0

2 ·+ rvp . _62 a · 62 50 fc\.... ·. 6 =>ap=

6"""' 2=6 \..:.__)

ti i·.,, Ir �-k�- t0u:i0Q2S°! 'tl:isl v�(J)C, e.i®25ld��. x2

- (a + f1)x + a/3 = O

8

G(x) = 0 �®tS)o�ect �e, F(x) = O ®63251 e\� ei�&.

¢ca (x2 -13° x+ 1) (x2 -�x + 1) � 0 ()

� (3r2 - 10x + 3)(2x4 -· Sx + 2) ::::. 0

¢:=} (x-3)(3x-1.)(x-2)(2x-J) ==·o c��--) . ��--. ... ,. ..

¢:=} x=2� 3eio:f� '2' 3

08 __ .-·. 1' I (I i,'.(· ... ·-

1 '�' / - . ('; ,' • ) . L,Y ,., ... / .. . \.

. ')

r-,,.,· ·n -------- -----------------------· -----· -- -- - . -- -- - - - --· \_ 7 .... - ...

· (ii) . . H(x) :�·F(x) Zl®.

I' J (' .'

x4 + x2 + 1 = x4 · - (a+ {3)x3 + (2 + a/i').f:'. .· .· (u I F ).1 1 I

¢::> a+ p = o s . } I

2 + a/3 = 1 => af3 = -1 Q--�...:... . .,...... "'I

(*] =>. a( -a =· - . � a2 = 1

=> a=+� }(5) @] �cic) p = +1

. -x=±l. - 5_,,. •":;..)

. a �_l �3: /J == -1 @(3t3 .CD��.

______ ,, ___ ... -...... , - - .... - .. , ·, ............... .

. - . .

': . ·.. . ... �: ·.: : � ... 0 L---_..;._.;......__.,;_ ________________ •. ,, ___ ...... _ ... __ . ...... ..... ........... _

H(x)-:- 0

�- ;f .. X)·= 0 =} (.x2 - ,v·4)l (x2 + X + 1)= 0 Q ¢:::} .. ,x�'""'.'.'.':X+ 1 = 0 · &rcl xt.+ x+ 1· = 0

A.'··_ .. -.--1-·4(1)(1) .:- 0 c·�1,):=;.,.!!. ., �::e � 1) � .� .4(1).(1J'i-< ,0 .• u --..

9

I,, (111 i',�..,.�-.--:__:��t1) l:l)��:r. �Dirl �cm�� �iw �o&n �m tOt;l)O

I "'·: 11lC\!'-. .itLm,> c::J€1mBzs:, �:£E6t �ed � �.a> � �t:,). �f!J Mg6i �e � (!)!;I)

i i:t11\t1 l'J't.r:lXltd � � �i:,,l:& � �. �vtrl we, ��. � c:rt::S l ,»,�m�t .51('.c:mi ��� iro � tor:nd �ts,�m � ��- gC.bZ) 6�-i, ! �i_rr.i".J1.!'Q ec:o-J ���� �:r � � c:s. � ®�d@d q:,@':mzd � <it� wm a;;e �.m·

&�fl.en t�8;::i,�t:J�! a>id �� �8' qc:,�� �- owc �mDm ibd �!:If qt)� � ��dt•�J,'.�( :!C: 85:�th mm l'.9I,i§ � q:,r:,,;io CD� �¢fflffl..

{I) ·:rfi ::,)Q�V) c,d a»,Sz:si.x)� � � (l)::J� -�eeD 41� .&:> 9tQ ® �.,.ii) o{:1�,::5) Bro�, lfi� ea����� �CO'l� �<:N;� ®�cl9(! sac St;® �s+

I

I

(11d qw:JCI") �:h:53gd-i � fu 9� e} ts)�-

\ t, .1 , 1 .I •·. e:w;too A(r + 5)1- - B(r+ IY, = r+ C � � A. ·B ro:J C .Sc,mOQ �zi ��.

. . 8 �. croe�o;, �� r flffl � U,,. = . . 2 2, C:,mffi /(r) -j{r + :2)

{r+J) (r+3Xr+5) Ott'I;,_..,:, ::J>t=;, �v@ @E> ��; <!P®e f(r) � · iijtSfli&,' c�·� i�. �-

'\ •

l 1 '\' o er\9'� �:i,c, � � U ·�. ":,+-2 �l3c:,D �G �zn.@!> ��· L r · · � T � 15

, � I r !!:l

3

9L:b,:fi 1 + FS 2 +MS 1

9,J:>� 1 + FS 1 + MS 2

5 4 2

0 == 900 +-375 + 300 = 1575. \V

=2040 ® 15

10

{b) A (r + 5)2 - B (r + 1)2 = r + �

A ( r2 + 1 Or + 2 5) - H ( r A + 'b' + 1 ) ·.,.'. 1 I 1 :

e,.;OJ�25) co@:)25) £3e®zi;

r2 :_ A -_ B = 0 0

T : 10A--2B== 10 0 ro : . 2.r.;.11 - . JI .

A=B=i�0 C=24A=.3(J M)i.• (I I 'j) .{/ f I)'. �scr+3)

----- _____ .. _____________ .. __________ ... ____ .. __ _ ., ,. ,_ .� - .. " ...... , .....

1 1

- (r+1)2(r+3)2 (r+3)2(r+S)2

I c,·,

= f(r)-f(r + 2) e®� f(r) = (r + IY(� + �1)J -��)

Ur= f(r)-f(r+2)r == 1, 2, ···� n c.,«;00:1

U1 = /(1) .:.. jj 3) \�

u, =iciyt,4) .V�3 =f7f�);n-,, =/�(:�(n) � U..., = f(n'ZZ.(n+I) Q Un =J(n -f(n+2) · > \J

L�=1 Ur = f (1) + /(2) - f(n + 1) - f (n + 2) 0

01

(n+3)2(n+S)2

25. - ·· ---------------·----. . -

. .• t . .. . . .. . - - ·- - - - ·· ---------------------

11

t ,11

I)

ii)

A"" ( o _2 -3 ). B=·( a h o )· ::,;i;i c;..( ! 4

23-J �et� ec�-

- lJ -1 2 c d O I

(i) AC= Ii=-( � � ) 00 �i,o,. CA �wm.

(ii) BC = ii lltn � a. h. c � dta � C1ttAX01rtm.(iii) (AA +_µB)C=I1 � �- .l1!03µ�aurlo � � e@l 612:rl:tn.

0=(1-� -:)t1)�, AiO:lB ��?:D6',�DC���··

{l,l �- o,,� � z. :cos fJ+isin {) OOC) ��-,c� � 8(-;r < 8 s Jr)��ii. 't.llDmll a:>a,$d@m t �"toe.d no'� e,:M� c OOc) �

1:us O � sin 8� at l'l» l <lt� eS3 1»irlzo.

w '== -fL � I= � -ll c:,t8 m:Sa; e,eB _z QffltO z.;t ;zi fl2!tJ � C 5m �:t +J z + (i) Jm (w);: 0 Im 8£ (1} � 0 ao � c5aCaS. ftl1 qm

�giotJJ (»Ci�.(ii) w;::2��z��et:e�.

(1ii) 1=i�to.1�z�G:lo&.m�Jm.

/1=(0 2 . 0 -1 -3) B= (a

2 I C

!IC = (4-: 3 - 0) = (1 0

0

-3.+.4 0

(0 2 -1)CA=Ol·O� -

0 0 1 \V

b 0) (3 i4) d O m:iC= �

0) =lz

G- . . ------------------------------·-------------7----.-·

G) BC = ea + 2h 4a + 3b)

( )3c + 2d 4c + 3d = 0 1 -

3a + 2b = 1

3c+2d=-O }@ 1c +_3d = 1

- 3a + 2 (-:a) = 1

a=3,b=-4 G) G)

(-3c) ... c+3 2 :-

c=-2,d =3 <v �

12

0· iii) .(M + µB)C =MC+ µBC=: (ii. + p )le; -�· l

:i

. · � (.:l + µ -1)/2 = 0 � .l + ,, ::-:.·., (;')- - . - - - . -- . -,s::)� - -- - - - - - .. - . - . - - - -- - . -- - - ... - . - ·. .

... . . ..� ...I'.>·�\"' D. = (-3 8 -6) = a (0 :l ·:· :t) l· /I(. I_

, � ) 2 -s 4 0 '"' l · l . , .·. J,J_ 1

. '* 0) -··,:) II. \ ,,·

,. \�-----�;�;.�-�0-----"---- ----· �:��0

.Ci) )!( (:). D = 2A - fJ 01:i DC •:;:; (:U\ · l.i)C. -· /.M'

== 2J.. -· 11

-·,.� I • (�)" . � .. :..,, � -------------------------.- . - - --- ·- . ... - ¥ .. ... - •• ,.._ � .. . - � ... . . '

( ' z = cos I}+ i sine

lzl = vcos2 a + sin2"o = :t;

_. • • 1: z = cos() - l sm e = -;(

z + i = 2cos 9 � cos e == 2 (z + 2) . 2 ;(

t":'\ ��:)

z-2= 2isin8�-sin8 = ).,(i �;) (0. ------ ----- -- ... - ---- - .... -- - ' -- . --- - - - .. - - .... - - ........ •, -· ..... -· ), ... ,. . .· . . .

i) 2z 2 1 � .w =-- = -- =-· - = !,eclJ :; :, lm(w} ,.,, O z2 + 1. l . cos O z+;, �

·1 � z2 - 1 z - -:- i sin() · t = i

= ---l = -- � i tiH1-/J ; ,._ I-fr( t J ... 0z + 1 z+! cosO "l.

------------�--------�------· -----�---·--�--- --� .. . . . .

·ii)

iii)

· w2 + t2

. = sec2 e .+ (i tan.8)2 = sec.:1 tJ ,,. 1a11' o "� I (i)

w = 2 ::::? -1-.::::: 2 or . cos8

e� e� c,o:,Q� �s e == ±i 0

t = i::) itan8:::; if'"':\ · 0::) _tanB = 1

I() CllS fJ :,;:,- -· . \ .......

h •. •

, - 1:, ,.,.., c·-_L0·z ;;::: -· + t --- ;: ., ·. .. f . l . ;! 'J. I )

15

. ----------------. -··---

05

--------�--·-----------

13

I lq i, J / ( 1 ) .-:-:· :Zx1 + yx3 + ox+ 1

r(- 1/;) = O@ium,

1 • (' I)'. . ( 1) I ) I }' . -· + c) - :- + 1 = 0· II, . . II · 2 ·

V 11 , 1·. I- b = 0

l I·!,)' .·,J�\...V

'tli,, I;'{ !1)+8(-2)+1=21

llv I .I.,\ :.:.: ·12

:Jj, ·!.,\'._:::() Q r· · I .. 11:, S = 2

· J (x) ::.-:· 2x4

+ x3 + 2x + 1

= (2x+ 1, x3 + ... i (-1) = 0.

= (2x+ l)(x+ 1)(x2 -x+ 1)

·· �------�---- .P --•--·------ . . --•-•-.-. ---------�-------·---- - --. �-----•-- . -------·

1 \iJ (_x_2 + x + 1) P(x) + (x2 � 1) Q(x)· . 3x

l'(x) =ax+ b W3 Q(x) = ex-+. d.a>tS a,�� •.

cJl'Jo (x2 +·x.+ l)(ax + b) + (x.2 -1)(cx + d) = 3x Q C,1 .. �!) (!';,,:!>') �@:>ZS) 2S18@®25f, .

aJ c::::: 0.: ........... (l) 5 _

1i +a+ d = o .. ; ............. (2) Q 1i.-+·a ..:... c = ................ (3) Q /} ,.,.d 1==:0-.·.�············· (4) Q

14

(1)25:i , C = -2 , !5JV� (4)251' d = -1

P(x)"= 2x -1 ro:> Q(x) = -2-x - 1

Q ' Q

(i1, (x2 + x + 1) P(x) + (x2 -· 1) Q (x) ::,. Tr

,f:.(x) = ax+ b lO:> Q(x) =ex+ d t.:iif3 (n::'.ilt-1 -6v,D (x2 + x + l)(ax +b) + (x 2

-- I )(rx + ii J I., ( 'i )· . ,

.

l

i'e=;.-l : d a J b) = 3.:::} a+ b = l

X=-'-1-: -a+b-=-3 0

I · x = 0 :· --.1 - d = 0 Q _ x� 1 .. : { t� 1) (.:-� �-1) =�

x -= 0 . .-:-,::.:_1-:.d = 0 �. d = -.1

-P(x) ..:.. 2x:·7 � -

(]J ?(x) = ,....zx-

��-----------�-I ...... ,. -·- ·<1

'" .... .. . , . . n,. •

15

I

I

J.I (,ii 1 � fJ c:>t;1'1J Y = t" sin} cze 13)��.

dv · i \ /) A ··"- ·: )I -- OOS- l:OJ ·.lit .>;

1 ri;l i: .. ·4+";;:Q

. . dx" .,

- ;!' ' :2s2+1 .... (/,} 1 7 I c.:.��:x, .J tx,, =:. - -1 0,.:.:1 '3)�.

(z- lJ f(x) &> �.ifil t).?3�tl'.X:) �� �01� c�•�o � �r<kS ccl� im dc,e1-�coo� �::i��.

)' =-/(x)@.s��d �s CK)ei)md ��-

(,) P,.:n63di�.Js.BCD��BC!!D:>AD��mtam.�S.�®e��e@tll ����.AB= CD= a. BC= b1!00AD = b + �S@tf!4tfi� et6tQ!; �0<...t<a�. 8.l!w�

. .

CF Cj� eg� BtD:l c �&)AD� &l>o. (11�. el!S.e!>. B b c

A�-'>E F��D

I

I

ARCD�Bs���S(.xhOOQ�� S(i)�(b+xrla1._z1-_ m��··.@f) �2Sl"c)dm. a = J6 e,:«J b = 4 �tn. x S �&6:> � .SOO colt® f>.m Sf> m�

\ ,l ,i V :·a. x. sin(l/x) , X -::j::. f

(i) (i�. = sin(l/x) + x (�) cos(1/x) G)

� x :: = y- cos(l/x)

. . --·----------------------·-------------------- ·.-------------------- ------------··-------

'ii) x·B�e"'� q:DZlilcmC:r�� w:> sin(l/x) = y/x.C:r��G'®�: .. .

�La

, ,t�y ·;gr- !j . . (-1) .. \.ilx2+r-r+s1 (1/x). x2,

.. d2y ... x4-·-+y = (,x2

16

, 2x2 +1 (b) {Lt)=-. --� . X :;c 1· ·

tx-l)-

f'(x) = (x-1)2 .4x-(zx2 +1).2(.r-·1)

' (x-1)-i ·

_ (x-1)4x-2(2x2 +1)

(x-1)3

-· -2 (2�+1)

(.,.,-·-·-.) (x-1)3 ; (x * 1) . -�--

· x = �1 DID �e ((x) = 0 et).

x = .1 e>� ac:r ((x) ;�;CiD�.. .

=> x � 1 i!l� Sda """1leCJ>!zr19a"'m ·'fl"' · .8

...------,------..---�--:-·····----- .......................... . i-----+-·_x_<____,..._· _1.4.-.,.,2 '-1-->--___,1 _ fil::..!.":�.:! ..... :I__:.�::" ..

j'(x) .� ..

j....:C�&-- ---'-'-+----'(..:...-) __ _...._ _ _(+). ------ ........ ( !·._·_""' / "-·,,.,,

.....___...;_______._. ___ \'· -· -� w • . •• • .-·"

-1-- . I'-·- .. ---··

f ·(-!). = :Z(-�/�)2+1 � . 3/2 -� 2/3 .2 (:i 'i· 2 (-3/2)2

f(x)--+. 2

0.

f.(X. }:,� 2.

(·, . ,.:, \O

' '\,,) ).-,

t ,,r

·--- .. --.----

17

F (_·,o ,r--.....5

l

�... . ( c; )

1�··

-------,,,-------;;lloo- ,X" ---------::-"-=--t:::---r:: ·., -0.5 0

b

A x E F x D

- �---•••••v_----�·--�-----•-•-•-•-------------------

. '

------

S(x) = (4+x)v16-x2

dS.

dx

cl.S

dx

dx

=(4+x} � (-2.x)+v6-X2

2-./6-x2

-x (4 + x) + 6-.x2

-2'1/6:....x2

-2x2..'..4x+6

..Jr,-y"2

-2(x2+zx-3) � 'l/6-x2

\J

dS = o f)25) vc> x + 2x - 3 =·O f0dx: \.J (x +3)(x - 1) = 0

18

x w25) @1.B2Sf x = 1 w16i.i!l cdi!ll:i:;�i.,i 0�8. (��--) '·- ... �

S'(x) iJ l--'c:c..;��=-�--4--(:,_.:+}�--1----.H_ _ __ ., .....

,·. x = 1 'cfl� S(x) coB® r:tD. 8 S(x) ifi cab(!) q©�,::, 1) = (4+ l)v'b� ;:-:. s,/T\' Vt>(l) {hr,1:)l'1.

( •, )

19

R lJ

J5. C.<J) f /(x)dx = J /(1r - x)dx @!> CK,mf.)mm.0 0

J�in2 x dx = !!. at>cl e-o�t5> • . 4

0

. 8 . 2

� J x sin2 A" dx = !!__ @� Gta�trlts, ..

0 4

<b> �a 'Fl� a):] _•aoea, fJr»iK:Jd � �®a, ���. J x'je;z dx .�tdm.

5 (:1) y=1

t

-xc.,1@©z3�.

(0 � 0 X . H

f f(x)fh = J j(Jr- y)(-dy) =J /(n--y)dy. = J f(1r-x)dx. �

0 1r O O . ------------------------------------------------ .--------------

� ii () .. f sin2 xdx=

21j (1-cos2x)dx=

2 [xf-o;;, ·.-[sin2x]t= O

0 0

8 � ----------------------------------------------------------- ---------- ------

20

0 11f

• 2 JC l·JC ,rt , ( It ) ,, ., x sin x dx = - - + - -" ;- n · :

··· --

0 2 4· ,J. !. .!. ' ·1--------------------------------- -------· --- - - - - - .

.·.f x3e-��dx=-1-Jte'dt: 0

�!..ft d (e')dt=!..te 1 -�fe'dr (�;�} 2 di 2 2. · ·

-----.-·----------------------------------·---�-. (c) 1 A Bx+ C

x-3---1 = x---1- + x2 + x + l

= A(x2 +x+ 1) + (x -- '/)(U:r 11,)

- x0 33 e.,o<g,&25) ffi®:125) ti3Bee-zs! , l = A -· (.'

x1 i3 C,•(l?&� e.,�� i538ee>2:1i , 0 � A + C --- h'

. 1 A=-

3

I ··.:--,

(.' -�- -· :: ( ' ):1 ... /

------�---�------------ ---------------------···· -

I . dx. ·_:· If dx _JJ x+2) d-----,-- - .- . . .- :x J·.:._ 1 · _.· 3· - �1- - -·3 : .·-·"2 . : . . 1 ..

X.· !. .X,-.+x+.

·12 . 1) .31 ,1.., 1. ,;i>/\i;X+ _. +-

2 · .. '.:f -� - .(;)ii_ ·. :d· jJ�-- i. 3.J Jf+x+l-· x

. -----

21

.

. - - - -. - - - -... -.... - - -. - -- - .... - - - -- - - - - - - - - - - - - - - --- - -

(d) 'P<>1<0o>: t = tanf h) =,> dt =Jc1 + t'J dx � dx = /+d:2 0

% 1 2 .

f dx . 1·(1 + t2) dt .

·5 + 4COsx·+ 3 SiO.X -. S +. 4- (1.- t2)·+ J 2t0 0 . 1 + t2 ' 1 + t2

1 f . .2 dt = . 5(1 � t2) + 4(1- t2) + 6t · o

1 ·J 2dt = t2 + 6t + 9

0

22

� .. , .. ,. ·---· -, .. �,..s,· ·· ,.··n ·�.

1,. O.tlb � c,!�,t& x1 I )'7 ·t ,., .... I 2h f , ..... �. f(l,"I A

1 t / t 2g';1.+2/y+i:'=O � � l.f'99,..tilmc, � eeo !ffl,;� 1!tt.} t.tiii, lg,' t· 1) .J1.,. �i ._ ,:' M) 1!t0ni'.l')rom,.

:Zt, + ·..; - 8..t - 6y + 16:: 0 �0�� .�4J� ('. {.ifl,lb]')(.� ,N7�19Q d'�i:g � &E) &0�� •

. Officea,S � �� 00� �� '�- C1 �,.)� t.io ry<:k.:i R.(>n�·c1

� � ,,A� B �� � C D1� �" �J:4 , t1, }( " (fil\o,;;)�f '° 1t l););i 8 � � � KX>Q��.

s � t·� ct �� •'tr:, • oo•� lt\'Vll •r\,; fl�il mdm 11,t.) y-'J"itl� � i:,dtt, � �a I �. S � � .-:018 rubf..,; ��.1.'111 ot�iil1'..Wln1.

C8»Cl Qm � �znt> 8 iJ e•)!I� � �o 0" �'\1, m1::it.'1¢l� x·q� � � Qg, � · � et,;� c&,�o�d cof�,d,l<.;>. 41 . • )11 ,., A.O �;ipl_ · P(J ait5bh �� �� �; qiBi

I � Q!;tDQ� 3(�1 ·I yl} ·· 30..r ·· 40y ii.(} fW� .QM11tJtt!t1:t.

'-------------··············--. ·--·- ....

xi + y1 + gx + fy + c =. O 't1);) x-" + y1

·i t//t. + /f'y + r' ., . (J t\1;:b:5> e\z:o gc®a 0-ce, G'o�Z)c:.,\':JD Zll!, �vc) (9 - g1)2 + (/' - tr: ''" _q'I + e'" t: I' J/; I / 'i.. <-:'. G)

(>) 0

c u.idi:i)<.'.I x2 + y2 -8x-6y ·,-16{,,0.

(x- 4)2 t (y ·- �{f ""' :i::.

E>.i:csi�ed 6'mi?}"isc.3 (4,3) (�) •jt1�0 ""' 3

e®® Oadrs1c.i (4,0)��- X-'J'llnlfc.o _u-!r::1bw J'DK1t<l, ---------.. ---... '------. ---------·---P--• - # •0# • ... "' , .... ,; . .. . .� . .

C1 E)a15:f25)Q : x2 + y2 ::;: r2, A(r co:-; u , ·t �ln ,r) c:mf.i-1w,1J;� �il.1610 ilt�:-:c., C r.fotfo;i .z:l)d@, e®tfi4 . J

I .

cosa = -,. sma;::: --·. 5 5

r+3=5:::}r=2 G) fl 11· i-;') :. A:"(·- -· )L-' , .. , ,--·-' .) ,

----------------------------------------------------··--··-h,. _____________________ _

1:5

23

S: x2 + y2

+ 2g_x + 2/ y � c = 0

o-83 e?:12:sfr,co; ( � g, -f) tDHfO� = ,/ g2 + f2

- c

·.� S, y- q'W$c.� rdo�m zs:,dzn @il3zrl q6�= lg!, (Dg2 +f2 -c=g2 =>f = ±-/c. 0

0 . ·. $ � c, 9<"�@t) G'l5��� � + 0 = c - 22

�- c ::;: 4. �e zflco:> f = ±2 <D

S too 0-.:;m ei'.i C · e)azi!:15)� 9e®au �'i25)� � 2g (-4) + 2 f ( -3) = 4 + 16 = 20 · Q =:::;, 4JJ + 3 + 1Q = 0 0

f = +2 => 4g =. -1 - � _g = -4 0f =-2 => 49=-10+6 =>g=.-1(D

s a3 253�c, �12£3 ��2.@66 0(iZ5:l e)zrl�l5'!

.x2 +y2 -Bx·+ 4y+4 == 0 �w0 x2 +y2

- -4y+4=00.

.

----- '

. . ---- --

. --. -- - ------------------- .... ---.----------·------------

24

C tn:i C2 D @t,:i, d�O'a;i2S),�d (6©ZID6&� ; + f = 1 0D; e®.:fl P = (a, 0) �:> Q = (0,b) "'-'ID · �c.,c) @�EJ:t•Z5:1 qw� <S'<jz:i) ID�Dm ed�c.,l3.

5 . a,= 8 seca = 8.4 = 10 ::;;> P = (10,0)

. S 40 ( 40)b·= 8 coseca = 8.3 = 3 => Q =. 0,3

-+-==1 5 x 3y 8 10 40

eDmdw@�w

0-4x + 3y= 40

c WO C2 fJazj'?>JDC eo:,, t'd'ol5a;}?.i'J@c& e,®ZiS>6�<:-' {x2_ t y2 �_8x-6y+ 16)- x..! + y2 -64)= o(

· ®�zf 6''i� ciecv.

· =:>8x+6y-80=0 => 4x+3y=40,0

· ei®tJe,, . P .. (10, o) ""' Q = (o, �o

·. 0 0

(;.-0) (y-b 8 · - ·- =-1 S

-a x-:

x(x _.:. a) +y(y-b) = 0 G

· i.e. x2 + y2 - ax - by = 0

x2 + y2 - lOx - 40 y = 0

3

3(x2 + y2 ) - 30x - 40y == 0

25

17.(a) coa2(a+ /3)+cos1 a+cos2 fl-2cos(.a+P)cosa cosp:; l � �mm.

(b) f(x):; cos2x +sin2x + 2(cos.r + sinx) + r�1s Gi�.J(x)�fflZ') k(l+cosx)sitt{x+a)q:>�gtiio(J;l �o'��; e®&3 k� a"� _;53�6'° �� � &t.» �.

g(.r) ib2:rl� f(x). = ..J2{g(x)-1}�ffi se� ��; � _!!..:ix s � · e�. J-,. cosx 2

y:g(x) 61 9clt1"XJ� �s· t&)t()3)ffl «t{� �; �ffl � �tS ��e1 ��f(x)=O �.?.S)o�oo t,z:i,.

c,c.;�� �d qis w ee,m�mzr.i.,

(c} 9E6'ijj q�z:n��. ABC .@e�� ��too c� �Sa ft):,!tme<�.sS.

a(b-c)cosec.iico(A:(b+cl tan{B-C)�(!J.::..£) @fJ �:dm. 2. Z Z. · 2

(a) cos2 (a + .B) + cos2a + cos2{3..,.. 2 cos (a+ J]) cos a cos/J = 1.

= cos(a + ,B)[cos(a + P) - 2 cos a cos P] + cos2 a+ cosl fJ . i:--fJ I

:f.J'1/ ���v

G) 0¥ ,j'

= -f cos a cos fl - si� sin P] { cos a cos fJ + sin a sin/3] + cos2a + cos2 p 0 G) G)

= -cos2a cos2p + (1 � cos2a)(1- cos2f3) + cos2a + cos2{J

=1_ - G .. � - -- ------------. ------------------�-----�--�---------------------------·------------·-------( b) f (x) = cos 2x + sin 2.x + 2(cos x + sin x) + 1

= 2cos2x -1 + 2sinxcosx + 2 cosx + 2 sinx + 1 5

= 2 cosx (cosx+ 1) + 2 sinx (c�sx + 1) G) = 2"2(cosx+ l)sin(x-1��) 0

'i k=2Ji, a�; G)----------------�--�------------------------------�-----------�-----------·----

I (x} .. 2 . t-z . ( . re) r,;;2{g'( ). }·\--:---,·-,... = . v c.. sm x + - = v.:. x - 1 •+ni·:.r f

y = g(x):::: 2sm(x+J) + 10

26

-----'---ttl'-----+------·--····--·- ... - .. ___________ y_=_1 __

-t---F--------------- ��----------1·-------K --------:!'.4----�-- 1--li �; - i

""'""' Q ""11.-o<>.co8e> G; "'j_""'.;;0(0. . e,'£�0x = 0, y = .J2 + 1 0 y = l '�

J(x) = 0 � g(x) = 1 @·o 62m �e,�®Zii oE>�Zl'f oEl�. :. f (x)= 0 o oa.1 v�t;®mf o®-6.zn oD�. G) =}X::-�

. Gu. - -.-- - ... G)

. . . . ( C) A + B + C = 1C DZl vu � e.,8zrl �253� _a_ ::;: _b __ = _c_ 5 sin A sin B sin c

.b..:...c sins.:...sin c.....,._,=-----

b+c sinB:fsinC_G)

<:,:,..,..4+...fr,rt:':'.::-lf--·-

r} . ,.-;;;-�..,.,-..--�- * \• . . ,4 .,.j £.? -1. C. ;-: I\

�V�ill;�ise'xc�f&,i:CD

27

• JI sm:;-,. **

t tB - C) . Aa(b-c) ant� smz (b + c)2 = · A .-.. (-B .:::.-c._ C-.)

. cot 2 cos -. -2-.

cotA· tan (8 - c)

a(b-c) . .j=(b+c)2 � Slnz cos(-2-J

a(b --· c)cot;cos�c1 = (b + c) tan(8;c) sec (8;c)

--·····-····-··-······ -·--------------'-----------------------------�

G) a · b c 2.fl2S1� -· = - = -..

sin A sin B sin C

�,:�!.-�.0_ = sin A (sinB-sinC) G) (IH,:)'! (sin B:+sin C)2 �

\2) sin A.2 cos(� sin(-��) G)

4-sm 2 -2--Jcosz -2- 5 . (B+q'\ (B-C) G) . . A . (H-C)

Sll)/1Slll·�-Stn - G)· -=:·_ -A·· ... 18":c) :5 (':A+B +c ==·n)

2 cos1 -·COS ... I-.2 \ 2

? • 2 A A . (B-£) -- sm -.cos-sm - .

_ 2 2 . 2

2 cos2-cos?. -· . A cB-C)

2 2

. 11 ·· A (B-£). · sin- tan.:....tan -

;: 2 · · 2 • . . '·(B-:C\ cns7;

. .., ,1 .· "(B-C) . . (e:...c) 1,111 : 1.1n · ... t-an -.. sec .-. -.

. • :, 2 · 2

> .,,(I, , ) 1 n·.,·•

11 nit:l ,.::.. (h + C)l tan (B-c) s�c (B-c)./

50 Ic" ? · · 2 · 2

I I

���------�--�---__J

28