1

wuúokdvf0ifpmar;yGJ

ocsFm bmom&yf ydkYcscsuf

r*FvmygwynfhwdkY-

2016 ckESpf wuúodkvf0ifwef;pmar;yGJ ocsFmbmom&yf ajzqdkMurnfh ausmif;om;?

ausmif;olrsm;taejzifh pmar;yGJajzqdk&mwGif rSefuefonfhenf;pepfrsm;jzifh pepfwus

wGufcsuf a&;om;ajzqdkwwfap&ef? rSm;wwfonfh trSm;rsm;udk owdjyKa&SmifMuOf EkdifMuap&ef

ESifh pmar;yGJajzqdk&mwGif owdjyK&rnfh tcsuftvufrsm;udk od&SdMuap&efpaom &nf&G,fcsuf

rsm;jzifh þpmrludk a&;om;&jcif;jzpfygonf/

yxrOD;pGm ocsFmar;cGef;vTmwGif yg0ifonfh Section(A), Section(B) ESifh Section (C)

tydkif;toD;oD;udk ajzqdkcsdef(3)em&DNzifh ajzqdk&mwGif wpfydkif;pDtwGuf ,loifhaom

ajzqdkcsdefudk tMuHjyKygrnf/

Section (A) rS multiple choice question (1) rSwfwef (25)yk'fudk rdepf (40)cefY trsm;

qHk;xm;ívnf;aumif;? Section (B) rS (3)rSwfwef ykpäm (5)yk'fudk rdepf (30)cefY trsm;qHk;

xm;ívnf;aumif;? Section (C) rS (10)rSwfwef ykpäm (6)yk'fudk rdepf (90)cefY trsm;qHk;

xm;ívnf;aumif; NyD;atmifajzqdk&ef vdktyfygonf/ tMurf;tm;jzifh xdkodkY ajzqdkEkdifrnf

qdkygu rdrdwdkY\ ajzqkdcsufrsm;udk jyefvnfppfaq;&ef tenf;qHk;rdepf (20)cefY tcsdef&&Sdrnf

jzpfygonf/ jyefvnfppfaq;cGifh&&Sdjcif;onf ocsFmbmom&yf ajzqkdrItwGuf rsm;pGm

taxmuftuljyKapygonf/

Section (A) ar;cGef; No. (1) rSm multiple choice ar;cGef; (25) yk'f yg0ifNyD; ar;cGef;

tm;vkH;ajzqdk&rSm jzpfygonff/ ajzqdkonfhtcgrSm ay;xm;aom A, B, C, D, E tajz(5)ck

teuf trSefwpfckudk a&G;NyD; A (or) B (or) C (or) D (or) E udk ar;cGef; No.ESifhwGJí xif&Sm;pGm

a&;ay;&rSm jzpfygonff/

Oyrm Section (A) ar;cGef;rS ykpämeHygwf (3) \erlemajzqdkcsufudkMunfhygrnf/

(3) When 2x2–5x +1 is divided by x–1, the remainder is

A.2 B.–2 C.1 D.–1 E.0

2

,if;twGuf tajzyHkpHHrSm (3) B jzpfonf/ capital letter A, B, C, D, E jzifhom azmfjy&rSmjzpfjyD;

small letter a, b, c, d, e jzifh azmfjyjcif; rjyK&ef? tajzcsnf; oufoufom a&;csjcif; rjyK&ef

owdxm;&ygrnf/

Section (B) rSm ar;cGeff; No. (2) rS No. (6) txd ar;cGef; (5)yk'f yg0ifNyD; ar;cGef;

tm;vHk;udk ajzqdk&rSm Nzpfygonff/ (OR) cHí ar;xm;aom ar;cGef;rS BudKuf&mar;cGef; wpfckudk

ajzqdk&rSmNzpfygonf/

Section (C) rSm ar;cGeff; No. (7) rS No. (15) txd ar;cGef; (9)yk'f yg0ifNyD; BudKuf&m

(6)yk'fudk ajzqdk&rSmNzpfygonf/ (6)yk'fxuf ydkí ajzqdkxm;aom tyk'fydkrsm;udk xnfhoGif;

pOf;pm;rnf r[kwfaMumif; owdjyK&ygrnf/

qufvufí jyXmef;pmtkyfwGifyg0ifaom Chapter wpfckcsif;tvkduf owdjyK&rnfh

tcsufrsm;udk aqG;aEG;wifjyoGm;ygrnf/

Chapter (1) Functions tcef;rSm zef½Sifqdkif&m oauFwrsm; rSefuefpGm a&;om;&ef

owdjyK&ygrnff/ Oyrm f ESifh g zef½SifESpfcktwGuf ESifh composite vkyfygu

( )( ) ( ( ))f g x f g x= [k a&;om;&ygrnf/ vuf,mbufwGif ( ( ))f g x [ka&;&rnfh tpm;

( ( ))f g x ra&;rdap&ef? uGif;rsm; jynfhpHkap&ef owdjyK&ygrnf/ qufvufí composite

function rsm;\ yHkaoenf;rsm;½Sm&mwGif owdjyK&rnfhtcsufudk azmfjyvdk ygonf/ f ESifh g

wdkU\ composite function f g twGuf 2 3( )( )

1xf g xx+

=−

[lí wGuf,l&½SdcJhygvQif zef½Sif

f g t"dyÜm,f½Sdap&ef 1x ≠ uefYowfcsufudk xnfhoGif; ajzqdk&efvdktyfygonf/

zef&Sifqdkif&m qifwl,dk;rSm; oauFwrsm;udkvnf; owdxm;&ygrnf/ Oyrm zef&Sif f \

inverse function udk azmfjy&mü oauFw f –1 tpm; derivative oauFw f ′ESihf

rSm;,Gif;ra&;om;rdap&ef owdxm;&ygrnf/

pmar;yGJ ajzqdkolrsm; taejzifh ar;cGef;rS awmif;qdkcsuf jynfhpHkonftxd ajzqdk

ay;&ef vdktyfygonf/

3

Oyrm onf binary operation jzpfaMumif; jyvdkygu ay;xm;aom set (qdkygpdkU set

A) xJ½Sd tpk0if a, b wdkif;twGuf a b € A azmfjyjyD; onf binary operation jzpfonf[k

a&;ay;&rnf/ onf binary operation rjzpfaMumif; jyvdkygu a b onf set A xJwGif

ryg0ifonfh A \ tpk0if a ESifh b wdkU½SdaMumif; azmfjyjyD; onf binary operation rjzpfyg[k

a&;om;ay;&rnf/

Oya'o (Law) rsm; rSefuefaMumif;jyvdkvQif tpk0iftm;vHk;twGuf rSefuefaMumif;

jyay;&ef vdktyfygonf/

Oyrmtaejzifh ay;xm;aom binary operation onf commutative law ESifh

associative law rsm;rSefaMumif; oufaojyvdkygu rnfonfhtpk0iftwGufrqdk Oya'o (law) rSefuefaMumif; oufaojy&ef vdktyfygonf/ tpk0iftcsKdUtwGufom qifjcifNyD; Oya'o

rSefaMumif; oufaojyygu jynfhpHkaomtajz r[kwfaMumif; owdjyK&ygrnf/

Chapter (2) The Remainder Theorem and the Factor Theorem qdkif&m ykpämrsm;

wGif x jzifh jyaom ydkvDEdkrD,,f (polynomial in x) udk f(x), g(x), h(x) [kvnf;aumif;?

z jzifh jyaom ydkvDEdkrD,,fudk f(z), g(z), h(z) [kvnf;aumif; udef;&Sif\trnfudk trSDjyKNyD;

a&;om;&rnfjzpfygonf/

ykpämwpfyk'fudk ajzqdk&mwGif ar;cGef;vTm&Sd ay;&if;tcsuftvufrsm;ESifh oDtdk&rfrsm; \

rSefuefcsuftqdkrsm;udk qufpyfwGufcsufwwf&efvdktyfygonf/

Oyrmtaejzifh y jzifhjyaom ydkvDEdkrD,,f y2+2y–5 udk (y–1) jzifh pm;vQif &&Sdrnfh

t<uif; (the remainder)udk &Smvdkonf qdkygpdkY/ atmufygerlem ajzqdkcsufudk avhvmyg/

Let f(y) = y2 + 2y – 5.

When f(y) is divided by (y–1),

the remainder = f(1).

But f(1) = 1 + 2–5 = – 2

∴the remainder = – 2

þae&mwGif pm;<uif;udk azmfxkwfa&;om;jy&efvdktyfaMumif; owdjyKyg/

4

'kwd, Oyrm taejzifh x jzifhjyaom ydkvDEdkrD,,f x2+2x+p twGuf (x+1) onf qcGJudef;

(factor) wpfckjzpfcJhvQif p udk&Smyg[laom ykpämudk wGufcsufvdkonf qdkygpdkY/ erlem ajzqdkcsuf

tjzpf atmufygajzqdkcsufudk avhvmyg/

Let f(x) = x2 + 2x + p.

If (x + 1) is a factor of f(x), then

f(–1) = 0.

But f(–1) = (–1)2 + 2(–1) + p

Hence –1 + p = 0

p = 1.

þwGif f(–1) = 0 [laomtcsufudk xifxif&Sm;&Sm;xnfhoGif;azmfjy&ef vdktyfygonf/

Chapter (3) The Binomial Theorem tcef;rSm bdkifEdkrD,,f tus,fjzefYcsuf (The

binomial expansion) wGif (r+1) Budrfajrmuf udef;vHk; ((r+1)th term) udk a&;om;&mwGif

a&;om;csuf rSm;,Gif;jcif;? rjynfhpHkjcif; rjzpfap&ef owdjyK&ygrnf/

Oyrmtaejzifh 5( )2ba + \ (r + 1) Budrfajrmufudef;vHk; a&;om;&mwGif

(r + 1)th term qkdonfh pum;vHk;rygbJ 5 5 5( ) ( )2 2

r rr

b ba C a −+ = [laoma&;om;csufrsKd;

ESifh (r + 1)th term of 5 5 5( )2 2

rr

rb ba C a −+ = [lí

2b udk uGif;rcwfbJ a&;om;azmfjyjcif;rsKd;

rjzpfap&ef *½kjyK&rnfjzpfygonf/ 5 5 5( 1) term of ( ) ( )2 2

th r rr

b br a C a −+ + = [k rSefuef

jynfhpHkpGm a&;om;&ef vdktyfygonf/

bdkifEdkrD,,f tus,fjzefYcsufwGif ar;cGef;rS awmif;qdkcsufudk owdxm;&ygrnf/

bdkifEkdrD,,f tus,fjzefYcsufwGif yxrudef;oHk;vHk; (the first three terms) udk &Smckdif;ygu

tus,fjzefYcsufwGif aemufxyfudef;vHk;rsm; usefonfudk udk,fpm;jyKazmfjy&ef rsOf;puf ....

xnfhoGif;NyD; atmufyguJhodkY ajzqdkEkdifonf/

Oyrm 8 8 8 8 7 8 6 20 1 2(2 ) 2 2 ( ) 2 ( )x C C x C x− = + − + − +

5

þwGif rsOf;puf ... xnfhí ajzqdk&ef vdktyfaMumif; owdjyKyg/ tu,fí

yxrudef; oHk;vHk;udkom azmfjyvdkygu

the first 3 terms of 8 8 8 8 7 8 6 20 1 2(2 ) 2 2 ( ) 2 ( )x C C x C x− = + − + − [lí tpufxnfhra&;bJ

ajzqdk&rnf/

qufvufí owdjyK&rnfhtcsufrSm ar;cGef;vTmwGif awmif;qdkxm;aom tajzudk

twdtus azmfjy&efjzpfonf/ tu,fí x2 \ ajr§mufazmfudef; (the coefficient of x2) [k

awmif;qdkygu x2 rygbJ ajr§mufazmfudef;csnf; oufoufudk ajzqdk&ef ESifh x2 ygaomudef;

(the term in x2) [k awmif;qkdygu x2 udkyg xnfhíajzqdk&ef jzpfygonf/ qdkvdkonfrSm

(1+2x)5 \ tus,fzGifhcsufrSm (1+2x)5 = 15+ 5(1)4(2x)+10(1)3(2x)2+…

=1+10x+40x2+… jzpfonfhtwGuf

x2 \ ajrSmufazmfudef; (The coefficient of x2) rSm 40 jzpfNyD; x2 ygaomudef; (The term in x2)

rSm 40x2 jzpfonff/

Chapter (4) Inequations tcef;rSm x2 ygaom rnDrQjcif; (quadratic inequation) udk

ajz&Sif;&mü ar;cGef;vTmwGif owfrSwfay;xm;aom wGufenf;jzifhom wGuf&rnf jzpfygonf/

Oyrm algebraic method jzifh wGufyg[laom ykpämudk graphical method jzifh ajzqdkjcif; rjyKEkdif

aMumif; owdjyKyg/

Chapter (5) Sequences and Series tcef;ü t"duodxm;&rnfhtcsufrSm A.P. ESifh G.P. wdkY\ n Budrfajrmufudef; un ESifh udef;vHk;a& n txd aygif;v'f Sn yHkaoenf;rsm;

jzpfonf/

Chapter (6) Matrices tcef;ü arMwpfnDrQjcif;rsm; ay;xm;NyD; rodudef;&Smckdif;aom

ykpämrsm;wGif Equality of matrices rS&&Sdvmaom wpfNydKifeufnDrQjcif; (simultaneous equation)

tm;vHk;udk toHk;csí tajzudk &,l&rnfjzpfygonf/

Oyrmtaejzifh2

2

1 11 1

x yx y

− =

udk½Sif;&mwGif

6

x2=1, y = –1, x = 1, y2 = 1 [laom nDrQjcif; av;ckvHk; wpfNydKifeufrSefaponfh tajzonf

x = 1, y = –1 jzpfaMumif; awGUEkdifonf/

oauFwrsm;oHk;&mwGifvnf; additive inverse of matrix A udk –A, multiplicative

inverse of matrix A udk A–1 ESifh transpose of matrix A udk A′ [k rSefuefpGma&;om;&ef

jzpfygonf/

Chapter (7) Introduction to Probability tcef;wGif jzpf&yf (event) wpfck A \

jzpfEdkifpGrf; (Probability of an event A), P(A) &JU wefzdk;u 0 ≤ P(A) ≤ 1 jzpfonfudk rSwfom;

xm;&ygrnf/

usbrf;vkyfaqmifcsuf (Random experiment) wpfckrS xGufay:vmEdkifaom

jzpf&yfrsm;onf wpfjydKifeuf rjzpfEdkifvQif ¤if;jzpf&yfrsm;udk mutually exclusive event rsm; [k

ac:onf/ Oyrm tHpmwHk;wpfck ypfaom vkyfaqmifcsufwGif A onf pHkudef;usaom jzpf&yf?

B onf rudef;usaomjzpf&yf [k owfrSwfcJhvQif jzpf&yf A ESifh B onf wpfjydKifeuf rjzpfEdkif

aMumif; xif½Sm;í jzpf&yf A ESifh B wdkUonf mutually exclusive event rsm; jzpfMuonf/

mutually exclusive event A ESifh B twGuf Probability wefzdk;rSm

P(A or B) = P(A) + P(B) jzpfonf/

tu,fí Random experiment wpfckrSxGufay:vmEdkifaom jzpf&yfrsm;onf wpfckay:

wpfck trSDtcdkuif;pGm jzpfay:EdkifvQif ¤if;jzpf&yfrsm;udk independent events [kac:onf/

Oyrm - tHpmwHk;ESpfck ypfaomvkyfaqmifcsufwGif A onf yxr tHpmwHk; pHkudef;usaom

jzpf&yf? B onf 'kwd, tHpmwHk; rudef;usaomjzpf&yf [k owfrSwfcJhvQif jzpf&yf A ESifh B onf trSDtcdkuif;pGm jzpfay:EdkifaMumif; xif½Sm;í jzpf&yf A ESifh B wdkUonf independent

event rsm;jzpfMuonf/

Independent event A ESifh B twGuf Probability wefzdk;rSm P(A and B) = P(A) × P(B)

jzpfonf/ ,ckaz:jycJhaom mutually exclusive events ESifh Independent events wdkU\

Probability ½Smaom yHkaoenf;rsm;udk ra&maxG;&ef owdxm;&ygrnf/

7

Chapter (8) Circles ESifh Chapter (9) Areas of Similar Triangles tcef;wGif yg0if

aom Geometry ykpämrsm; ajzqdk&mwGif yHkrsm; rSefuefatmif qGJom;í a&;om;ajzqdk&ef

vdktyfygonf/ ajzqdkcsufwGif xnfhoGif;ajzqdkxm;aom tem;? axmifh tnTef;rsm;udk yHkxJwGif

rjznfhxm;rdygu tajzrSef[k ,lqrnf r[kwfaMumif; odMu&ygrnf/

Areas of Similar Triangles tcef;wGif oP²mefwljcif; (Similarity oauFw ~) ESifh

xyfwlnDjcif; (congruence oauFw ≅) wdkYudk vGJrSm;ra&;rdap&ef owdjyK&ygrnf/

Chapter (10) Introduction to Vectors and Transformation Geometry tcef;rS

Vector ykpämrsm; ajzqdk&mwGif Vector oauFwjr§m; (→) rusefap&ef ESifh yHkwGif jr§m;acgif;rsm;

rSefuefpGm jznfhpGuf&ef owdjyK&ygrnf/

Transformation Geometry qdkif&m Matrix rsm; jzpfonfh Reflexion Matrix,

Translation Matrix ESifh Rotation Matrix rsm;udkvnf; rSefuefpGm az:jyEdkif&ygrnf/

Chapter (11) Trigonometry ykpämrsm; ajz&Sif;&mwGif vdktyfaom yHkrsm;udk rSefuefpGm

qGJom;í Logarithm Z,m;oHk; wGufcsuf&ef vdktyfygu xdkZ,m;oHk; wGufcsufrIrsm;udkyg

xnfhoGif; ajzqdkay;&ef vdktyfygonf/

axmifhrsm;qdkif&m yHkaoenf;rsm; (Formula) ESifh Oya'o (Law) rsm;udk

vnf;rSefuefpGm usufrSwfxm;&ef vdktyfygonf/

Chapter (12) Calculus oifcef;pm\ limit &Smaom ykpämrsm;wGif a&;om;csufrsm;

rSefuef&ef owdjyK&ygrnf/

8

Find the limit of 2

2

2( )4

x xf xx−

=−

when 2x → ykpäm &JU erlemajzqdkcsufudk Munfhygrnf/

2

22 2

2 ( 2)lim lim4 ( 2)( 2)x x

x x x xx x x→ →

− −=

− + −

2

lim2x

xx→

=+

2

2 2=

+ 1

2=

'Dae&mrSm 2

2lim( 2) (2 2)x

xx→

=+ +

[k a&;&rnfh tpm; 2 2

2lim lim( 2) (2 2)x x

xx→ →

=+ +

[k

ra&;rd ap&ef ESifh nDrQjcif;oauFwrsm; usefcJhjcif; r½Sdap&ef owdxm;&ygrnf/

pmajzoltm;vHk;twGuf tusOf;csKyfrSmMum;vdkonfrSm

ajzqdkcsuf jyefvnfppfaq;csdef tenf;qHk; rdepf (20)&&Sdatmif avhusifh&ef

ocsFm oauFwrsm;udk rSefrSefuefuefoHk;pGJ&ef

ta&;BuD;aom? rygrjzpfaom tqifhrsm;udk rausmfbJ jynfhjynfhpHkpHk wGufcsuf&ef

ar;cGef;vTmwGif owfrSwfay;xm;aom wGufenf;jzifhom wGufcsuf&ef

jy|mef;pmtkyftcef; (12)cef;vHk;wGif yg0ifonfh rSefuefcsufrsm;? yHkaoenf;rsm; ESifh

ocsFmoabmw&m;rsm;udk usKd;aMumif;qufpyf awG;ac:wwfatmif avhusifhxm;&ef

wkdYjzpfygonf/

,ckaqG;aEG;cJhaom tBuHjyKcsufrsm;udk vdkufemjyD; wuúodkvf0ifwef;pmar;yGJ ajzqdkMu

rnfh ausmif;om;? ausmif;oltm;vHk; *kPfxl;rSwf? &mjynfhrSwfrsm;jzifh xl;xl; cRefcRef

atmifjrifrI &&SdEkdif&ef BudK;pm;MuapvdkaMumif; wkdufwGef;vdkuf&ygonf/