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𝑫𝑬𝐏𝐀𝐑𝐓𝐌𝐄𝐍𝐓 𝐎𝐅 𝐄𝐃𝐔𝐂𝐀𝐓𝐈𝐎𝐍 . (𝟏𝟎𝟎 % 𝐏𝐀𝐒𝐒 𝐏𝐑𝐎𝐆𝐑𝐀𝐌𝐌𝐄 𝐈𝐍 𝐌𝐀𝐓𝐇𝐄𝐌𝐀𝐓𝐈𝐂𝐒 . )

Weightage to Content: XII Mathematics

Chapter

No Name

Marks Total

Marks Sec-A Sec-B Sec-C

1 mzpfs; kw;Wk; mzpf; Nfhitfspd; gad;ghLfs; 4 12 10 26

2 ntf;lH ,aw;fzpjk; 6 12 20 38

3 fyg;ngz;fß 4 12 10 26

4 gFKiw tbtfzpjk; 4 6 30 40

5 tif Ez;;fzpjk; gad;ghLfs; - I 4 12 20 36

6 tif Ez;;fzpjk; gad;ghLfs;- II 2 6 10 18

7 njif Ez;;fzpjk; : gad;ghLfs; 4 6 20 30

8 tif nfOr; rkd;ghLfs 4 6 20 30

9 jdp epiyf; fzf;fpay; 4 12 10 26

10 epfo;jfTg; guty;fs; 4 12 10 26

Total 40 96 160 296

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xU kjpg;ngΩ tpdhf;fß (30-kjpg;ngΩfß)

njhFjp -𝐈

1. mzpfs; kw;Wk; mzpf; Nfhitfspd; gad;ghLfs;;(3-tpdhf;fß)

1) A vd;w mzpapd; thpir 3 vdpy; det(KA) vd;gJ

(1) k3 det (A) (2) k2 det (A) (3) k det (A) (4) det (A)

2)

01

10

01

vd;w mzpapd; juk; 2 vdpy;> tpd; kjpg;G fhz;f?

(1) 1 (2) 2 (3) 3 (4) vnjDk; xh; nka;vz;

3) myF mzp I ,d; thpir n, k 0 xU khwpyp vdpy; adj(kI)=

(1) kn (adj I) (2) k (adj I) (3) k2 (adj (I) ) (4) kn -1 (adj I)

4) A= 2 0 1 vdpy;, AAT ,d; juk; fhz;f. (1) 1 (2) 2 (3) 3 (4) 0

5)

541

31

231

k vd;w mzpf;F Neh;khW cz;L vdpy; k-,d; kjpg;Gfs;

(1) k VNjDk; xU nka;naz; (2) k= -4 (3) k -4 (4) k 4 6) A,B vd;w VNjDk; ,U mzpfSf;F AB = 0 vd;W ,Ue;J NkYk; A xU G+r;rpakw;w Nfhit mzp vzpy;

(1) 𝐵 = 0 (2) 𝐵xU G+r;rpaf; Nfhit mzp

(3) 𝐵 xU G+r;rpakw;w Nfhit mzp (4) 𝐵 = 𝐴

7) 𝑎𝑥 + 𝑦 + 𝑧 = 0 ; 𝑥 + 𝑏𝑦 + 𝑧 = 0; 𝑥 + 𝑦 + 𝑐𝑧 = 0 Mfpa rkd;ghLfspd; njhFg;ghdJ xU

ntspg;gilaw;w jPh;it ngw;wpUg;gpd; a1

1+

b1

1+

c1

1=

(1) 1 (2) 2 (3) -1 (4) 0

8) A=

3

2

1

vdpy; AAT , d; juk; fhz;f.

(1) 3 (2) 0 (3) 1 (4) 2

9)

0

4

0

2

1

vd;w %iy tpl;l mzpapd; juk; fhz;f?

(1) 0 (2) 2 (3) 3 (4) 5

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10) A=

50

00vdpy;>A12 vd;gJ

(1)

600

00 (2)

1250

00 (3)

00

00 (4)

10

01

11) xU jpirapyp mzpapd; thpir 3> jpirapyp k o vdpy;> A-1 vd;gJ

(1) Ik2

1 (2) I

k 31

(3) Ik

1 (4) KI

12) 3 15 2

vd;gj∂ Neh;khW

(1)

35

12 (2)

31

52 3)

35

13 4)

21

53

13)

001

010

100

vd;w mzpapd; Neh;khW

(1)

100

010

001

(2)

001

010

100

(3)

001

010

100

(4)

100

010

001

14) kjpg;gpl Ntz;ba %d;W khwpfspy; mike;j %d;W Nehpa mrkgbj;jhdrkd;ghl;Lj; njhFg;gpy;

= 0 kw;Wk; ∆𝑥 = 0, ∆𝑦 0, ∆𝑧 = 0 vdpy;> njhFg;Gf;fhdj; jPh;T. (1) xNu xu jPH;T (2) ,uz;L jPh;Tfs;

3) vz;zpf;iff;ifaw;w jPh;Tfs; (4) jPh;T ,y;yhik 15) 𝑎 𝑒𝑥 + 𝑏𝑒𝑦 = 𝑐 ; 𝑝𝑒𝑥 + 𝑞𝑒𝑦 = 𝑑 kw;Wk;

1=qp

ba; 2=

qd

bc; 3=

dp

ca; vdpy; (𝑥, 𝑦) ,d; kjpg;G

(1)

1,

1

32 (2)

1

3

1

2 log,log (3)

2

1

3

1 log,log (4)

3

1

2

1 log,log

16)

844

422

211

vd;w mzpapd; juk; fhz;f?

(1) 1 (2) 2 (3) 3 (4) 4 17) xU rJu mzp A,d; thpir n vdpy; | adj A| vd;gJ

(1)| A|2 (2) | A|n (3)| A|n -1 (4) | A|

18) –2x + y + z = 1; x --2y + z = m; x + y -- 2z = n vd;w rkd;ghLfs; l + m + n = 0 vDkhW mikAkhapd; mj;njhFg;gpd; jPh;T (1) xNu xU G+r;rpakw;w jPH;T (2) ntspg;gilahdj; jPh;T

(3) vz;zpf;iff;ifaw;w jPh;T (4) jPh;T ,y;yhik ngw;W ,Uf;Fk;

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19) A=

43

12 vd;w mzpf;F (adj A)A =

(1)

5

10

05

1

(2)

10

01 (3)

50

05 (4)

50

05

2. ntf;lH ,aw;fzpjk;(5-tpdhf;fß)

20) 𝑥2 + 𝑦2 + 𝑧2 − 6𝑥 + 8𝑦 − 10𝑧 + 1 = 0 vd;w Nfhsj;jpd; ikak; kw;Wk; Muk;

1 −3, 4, −5 , 49 2 −6, 8, −10 , 1 3 3, −4, 5 ,7 (4) 6, −8, 10 , 7

21)𝑟 = −𝑖 + 2 𝑗 + 3 𝑘 + 𝑡 −2 𝑖 + 𝑗 + 𝑘 kw;Wk;𝑟 = 2 𝑖 + 3𝑗 + 5 𝑘 + 𝑠 𝑖 + 2 𝑗 + 3 𝑘

vd;w NfhLfs; ntl;bf; nfhs;Sk; Gs;sp 1 2, 1, 1 2 1, 2, 1 3 1, 1, 2 (4) (1, 1, 1)

22)𝑢 = 𝑎 𝑥 𝑏 𝑥 𝑐 + 𝑏 𝑥 𝑐 𝑥 𝑎 + 𝑐 𝑥 ( 𝑎 𝑥 𝑏 ), vdpy;>

(1)𝑢 xUxuyF ntf;lh; (2) 𝑢 = 𝑎 + 𝑏 + 𝑐 (3) 𝑢 = 0 (4) 𝑢 ≠ 0

23) x− 3

1 =

y + 3

5=

2z – 5

3f;F ,izahfTk; (1>3>5) Gs;sp topahfTk; nry;yf;$ba Nfhl;bd;

ntf;lh; rkd;ghL

1 𝑟 = 𝑖 + 5 𝑗 + 3 𝑘 + 𝑡 𝑖 + 3 𝑗 + 5 𝑘 2 𝑟 = 𝑖 + 3𝑗 + 5 𝑘 + 𝑡 𝑖 + 5 𝑗 + 3 𝑘

3 𝑟 = 𝑖 + 5 𝑗 +3

2𝑘 + 𝑡 𝑖 + 3 𝑗 + 5 𝑘 4 𝑟 = 𝑖 + 3𝑗 + 5𝑘 + 𝑡 𝑖 + 5 𝑗 +

3

2𝑘

24) a + b + c = 0, │a │ = 3, │b │ = 4, │c │ = 5, vdpy; a f;Fk; b f;Fk; ,ilg;gl;l Nfhzk;

(1) 𝜋

6 (2)

2𝜋

3 (3)

5𝜋

3 (4)

𝜋

2

25) (2> 1> -1) vd;w Gs;sp topahfTk; jsq;fs r • i + 3 j − k = 0 ; r • j + 2k = 0

ntl;bf; nfhs;Sk; Nfhl;il cs;slf;fpaJkhd jsj;jpd; rkd;ghL

1 x + 4y – z = 0 2 x + 9y + 11z = 0 (3) 2x + y – z + 5 = 0 (4) 2x – y + z = 0

26) a , b , c vd;gd a,b,c Mfpatw;iw kl;Lf;fshff; nfhz;L tyf;if mikg;gpy; xd;Wf;nfhd;W

nrq;Fj;jhd ntf;lh;fs; vdpy; [a , b , c ] ,d; kjpg;G

1 𝑎 2𝑏 2𝑐 2 2 0 3 1

𝑎𝑏𝑐 4 𝑎𝑏𝑐

27) 𝑥−3

4=

𝑦−1

2=

𝑧−5

−3 kw;Wk;

𝑥−1

4=

𝑦−2

2=

𝑧−3

−3 vd;w ,iz NfhLfSf;fpilNa cs;s kpf

Fiwe;j njhiyT 1 3 (2) 2 (3) 1 (4) 0 28) a xU G+r;rpakw;w ntf;luhfTk; m xU G+r;rpakw;w jpirapypahfTk; ,Ug;gpd; m a MdJ xuyF ntf;lh; vdpy;>

(1) 𝑚 = 1 (2) 𝑎 = l𝑚l (3) 𝑎 =||

1

m (4) 𝑎 = 1

29) 3i + j − k vd;w ntf;liu xU %iy tpl;lkhfTk; i − 3 j + 4 k [ xU gf;fkhfTk; nfhΩl

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,izfuj;jpd; gug;G

1 10 3 2 6 30 3 3 30

2 (4) 3 30

30) x− 6

−6=

y + 4

4=

z −4

−8 kw;Wk;

x+1

2 =

y + 2

4=

z +3

−2vd;w NfhLfs; ntl;bf; nfhs;Sk; Gs;sp

(1) (0> 0> -4) (2) (1> 0> 0) (3) (0> 2> 0) (4) (1> 2> 0)

31) 𝑖 + 𝑗 , 𝑗 + 𝑘 , 𝑘 + 𝑖 ,d; kjpg;G

(1) 0 (2) 1 (3) 2 (4) 4

32) (2>10>1) vd;w Gs;spf;Fkr • 3 i − j + 4k = 2 26vd;w jsj;jpw;Fk; ,ilg;gl;l kpff;

Fiwe;j

J}uk; (1) 2 26 (2) 26 (3) 2 (4) 1

26

33) 𝑎 𝑥 𝑏 𝑥 𝑐 + 𝑏 𝑥 𝑐 𝑥 𝑎 + 𝑐 𝑥 𝑎 𝑥 𝑏 = 𝑋 𝑥 𝑌 , vdpy;>

(1) 𝑥 = 0 (2) 𝑦 = 0 (3) 𝑥 -k;𝑦 -k; ,izahFk;

(4) 𝑥 = 0 my;yJ𝑦 = 0 my;yJ𝑥 -k; 𝑦 -k; ,izahFk;

34) x−1

2=

y−2

3=

z−3

4kw;Wk;

x−2

3=

y−4

3=

z−5

5 vd;w NfhLfSf;fpilNa cs;s kpff; Fiwe;j njhiyT

1 2

3 2

1

6 (3)

2

3 (4)

1

2 6

35) a kw;Wk; b ,uz;L xuyF ntf;lh; kw;Wk; vd;gJ mtw;wpw;F ,ilg;gl;l Nfhzk;

a + b MdJ xuyF ntf;luhapd;>

1 𝜃 =𝜋

3 2 𝜃 =

𝜋

4 3 𝜃 =

𝜋

2 4 𝜃 =

2𝜋

3

36) 2 i + 3 j + 4 k , a i + b j + c k Mfpa ntf;lh;fs; nrq;Fj;J ntf;lh;fshapd;> 1 a = 2, b = 3, c = – 4 2 a = 4, b = 4, c = 5

3 a = 4, b = 4, c = – 5 (4) a = – 2, b = 3, c = 4

37)r = s i + t j vd;w rkd;ghL Fwpg;gJ (1) i kw;Wk; j Gs;spfis ,izf;Fk; Neh;f;NfhL (2) xoy jsk; (3) yoz jsk; (4) zox jsk;

38) p , q kw;Wk;p + q Mfpait vz;zsT nfhz;l ntf;lh;fshapd; | p − q | MdJ (1) 2𝜆 2 3 𝜆 (3) 2 𝜆 (4) 1

39)𝑃𝑅 = 2 𝑖 + 𝑗 + 𝑘 𝑄𝑆 = −𝑖 + 3 𝑗 + 2𝑘 vdpy;> ehw;fuk; PQRS ,d; gug;G

(1) 5 3 (2) 10 3 (3) 5 3

2 (4)

3

2

40) 𝑥− 1

2=

𝑦 − 1

−1=

𝑧

1 kw;Wk;

𝑥 −2

3 =

𝑦 − 1

−5=

𝑧 −1

2Mfpa ,UNfhLfSk;

(1) ,iz (2) ntl;bf; nfhs;git

(3) xU jsk; mikahjit (4) nrq;Fj;J

41) 𝑎 + 𝑏 , 𝑏 + 𝑐 , 𝑐 + 𝑎 = 8 vdpy; [𝑎 , 𝑏 , 𝑐 ] ,d; kjpg;G

1 4 2 16 3 32 4 − 4

42) 𝑖 + 𝑎 𝑗 – 𝑘 vDk; tpir 𝑖 + 𝑗 vDk; Gs;sptopNar; nray;gLfpwJ𝑗 + 𝑘 vDk; Gs;spiag;

nghWj;J mjd; jpUg;Gj; jpwdpd; msT 8 vdpy; a ,d; kjpg;G (1) 1 (2) 2 (3) 3 (4) 4

43) a , b , c vd;gd xU jsk; mikah ntf;lh;fs; NkYk;

a x b , b x c , c x a = a + b , b + c , c + a vdpy; a , b , c ,d; kjpg;G 1 2 2 3 (3) 1 (4) 0

44) xU NfhL x kw;Wk; y mr;RfSld; kpif jpirapy; 45 0 ,60 0 Nfhzq;fis Vw;gLj;JfpwJ

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vdpy; z mr;Rld; mJ cz;lhf;Fk; Nfhzk;.

(1) 30 0 (2) 90 0 (3) 45 0 (4) 60 0

45) 𝑟 = 𝑖 − 𝑘 + 𝑡 3𝑖 + 2𝑗 + 7𝑘 vd;w NfhLk; 𝑟 • 𝑖 + 𝑗 − 𝑘 = 8 vd;w jsKk;

ntl;bf;nfhs;Sk; Gs;sp (1) (8> 6> 22) (2) (-8> -6> -22) (3) (4> 3> 11) (4) (-4> -3> -11)

46) [𝑎 𝑥 𝑏 , 𝑏 𝑥 𝑐 , 𝑐 𝑥 𝑎 ] = 64 vdpy; a , b , c ,d; kjpg;G

(1) 32 (2) 8 (3) 128 (4) 0

47)( a x b ) x ( c x d ) vd;gJ

(1)𝑎 , 𝑏 , 𝑐 kw;Wk; 𝑑 f;F nrq;Fj;J

(2) 𝑎 𝑥 𝑏 kw;Wk; 𝑐 𝑥 𝑑 vd;w ntf;lh;fSf;F ,iz

(3) 𝑎 , 𝑏 [ nfhz;l jsKk; 𝑐 , 𝑑 [ nfhz;l jsKk; ntl;bf; nfhs;Sk; Nfhl;bw;F ,iz

(4) 𝑎 , 𝑏 [ nfhz;l jsKk; 𝑐 , 𝑑 [ nfhz;l jsKk; ntl;bf; nfhs;Sk; Nfhl;bw;Fr;nrq;Fj;J

48) 𝐹 = 𝑖 + 𝑗 + 𝑘 vd;w tpir xU Jfis A(3> 3> 3) vDk; epiyapypUe;J B(4> 4> 4) vDk; epiyf;F efh;j;jpdhy; mt;tpir nra;Ak; NtiyasT (1) 2 myFfs; (2) 3 myFfs; (3) 4 myFfs; (4) 7 myFfs

49)𝑎 = 𝑖 − 2𝑗 + 3𝑘 kw;Wk;𝑏 = 3𝑖 + 𝑗 + 2𝑘 vdpy; 𝑎 f;Fk; 𝑏 f;Fk; nrq;Fj;jhf cs;s xU xuyF ntf;lh;

(1) 𝑖 + 𝑗 + 𝑘

3 (2)

𝑖 − 𝑗 + 𝑘

3 (3)

−𝑖 + 𝑗 + 2𝑘

3 (4)

𝑖 − 𝑗 − 𝑘

3

50) 𝑎 f;Fk; 𝑏 f;Fk; ,ilg;gl;l Nfhzk; 120 0 NkYk; mtw;wpd; vz;zsTfs; KiwNa

2 , 3 vdpy; 𝑎 ∙ 𝑏 MdJ

1 3(2) − 3 (3) 2 (4) − 3

2

51) 𝑏 ,d; kPJ 𝑎 ,d; tPoy; kw;Wk; 𝑎 ,d;kPJ 𝑏 ,d; tPoYk; rkkhapd; 𝑎 + 𝑏 kw;Wk; 𝑎 − 𝑏 f;F ,ilg; gl;l Nfhzk;

(1) 𝜋

2 (2)

𝜋

3 (3)

𝜋

4 (4)

2𝜋

3

52) 𝑎 , 𝑏 , 𝑐 vd;w xU jskw;w ntf;lh;fSf;F 𝑎 𝑥 𝑏 𝑥 𝑐 = ( 𝑎 𝑥 𝑏 ) 𝑥 𝑐 vdpy;

(1) 𝑎 MdJ 𝑏 -f;F ,iz (2) 𝑏 MdJ 𝑐 -f;F ,iz

(3) 𝑐 MdJ 𝑎 -f;F ,iz (4) 𝑎 + 𝑏 + 𝑐 = 0

53) 𝑂𝑄 vd;w myF ntf;lh; kPjhd 𝑂𝑃 ,d; tPoyhdJ OPRQ vd;w ,izfuj;jpd;

gug;ig Nghd;W Kk;klq;fhapd; POQ MdJ

(1) 𝑡𝑎𝑛−11

3 (2) 𝑐𝑜𝑠−1

3

10 (3) 𝑠𝑖𝑛−1

3

10 (4) 𝑠𝑖𝑛−1

1

3

54) │a + b │ = │a – b │ vdpy;

(1) 𝑎 - k; 𝑏 - k; ,izahFk; (2) 𝑎 - k; 𝑏 - k; nrq;Fj;jhFk;

(3) | 𝑎 | = | 𝑏 | (4) 𝑎 kw;Wk; 𝑏 xuyF ntf;lh;

3. fyg;ngz;fß(3-tpdhf;fß)

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55) 𝑥 2 − 6𝑥 + 𝑘 = 0 vd;w rkd;ghl;bd; xU %yk; − 𝑖 + 3 vdpy; k ,d; kjpg;G

1 5 (2) 5 (3) 10 (4) 10 56) vd;gJ 1 ,d; Kg;gb%yk; vdpy;(1- )(1- 2 )(1- 4 )(1- 8 ) ,d;; kjpg;G

1 9 2 − 9 3 16 (4) 32

57)𝐴 + 𝑖𝐵 = 𝑎1 + 𝑖𝑏1 𝑎2 + 𝑖𝑏2 𝑎3 + 𝑖𝑏3 vdpy;𝐴2 + 𝐵2,d; kjpg;G

1) 𝑎1 2 + 𝑏1

2 + 𝑎2 2 + 𝑏2

2 + 𝑎3 2 + 𝑏3

22) 𝑎1 + 𝑎2+𝑎3 2 + 𝑏1 + 𝑏2+𝑏3

2

3) (𝑎1 2 + 𝑏1

2) 𝑎2 2 + 𝑏2

2 𝑎3 2 + 𝑏3

2 4) 𝑎1 2 + 𝑎2

2 + 𝑎3 2 𝑏1

2 + 𝑏2 2 + 𝑏3

2

58) a= 3 + i kw;Wk; z = 2 – 3i vdpy; cs;s az, 3az kw;Wk; – az vd;gd xU Mh;fd; jsj;jpy;

(1) nrq;Nfhz Kf;Nfhzj;jpd; Kidg;Gs;spfs; (2) rkgf;f Kf;Nfhzj;jpd; Kidg;Gs;spfs; (3) ,U rkgf;f Kf;Nfhzj;jpd; Kidg;Gs;spfs; (4) xNu Nfhliktd

59) x 2 + y 2 = 1 vd;y; iyx

iyx

1

1 -d; kjpg;G

(1) 𝑥 – 𝑖𝑦 (2) 2𝑥 (3) – 2𝑖𝑦 (4) 𝑥 + 𝑖𝑦

60) xU fyg;ngz;zpd; tPr;R 2

vdpy; me;j vz;

(1) Kw;wpYk; fw;gid vz; (2) Kw;wpYk; nka; vz;

(3) 0 (4) nka;Aky;y fw;gidAky;y

61) a = cos -- i sin , b = cos -- i sin , c = cos -- i sin vdpy; 𝑎2𝑐2−𝑏2

𝑎𝑏𝑐 vd;gJ

(1) cos2( - + )+ i sin2( - + ) (2) –2cos(( - + ) (3) - 2i sin ( - + ) (4) 2cos( - + )

62) 34/3 ie vd;w fyg;ngz;zpd; kl;L tPr;R KiwNa 1 𝑒 9 ,

𝜋

2 2 𝑒 9, −

𝜋

2 3 𝑒 6 , −

3𝜋

4 4 𝑒 9, −

3𝜋

4

63)𝑧𝑛 = 𝑐𝑜𝑠 𝑛𝜋

3+ 𝑖 𝑠𝑖𝑛

𝑛𝜋

3 vdpy; z 1 z 2 ….. z 6 vd;gJ

(1) 1 (2) − 1 (3) 𝑖 (4) – 𝑖

64) 2 + i 3 vd;w fyg;ngz;zpd; kl;L

(1) 3 (2) 13 (3) 7 (4) 7 65) − 𝑖 + 2vd;gJ ax 2 -bx + c = 0vd;w rkd;ghl;bd; xU %yk; vdpy; kw;nwhU jPh;T (1) – i –2 (2) i - 2 (2) 2 + i (2) 2 i + i

66) 4-3i kw;Wk; 4+3i vd;w %yq;fisf; nfhz;l rkd;ghL

(1) 𝑥 2 + 8𝑥 + 25 = 0 (2) 𝑥 2 + 8𝑥 – 25 = 0 (3) 𝑥 2– 8𝑥 + 25 = 0 (4) 𝑥 2 – 8𝑥 – 25 = 0

67) 𝑖 13 + 𝑖 14 + 𝑖 15 + 𝑖 16 ,d; ,iz fyg;ngz; (1) 1 (2) -1 (3) 0 (4) –i

68) ax 2 +bx+1 =0 vd;w rkd;ghl;bd; xU jPh;T 1−𝑖

1+𝑖 ,a Ak; b Ak; nka; vdpy; (a,b) vd;gJ

1 1,1 2 1 − 1 3 0,1 (4) (1,0)

69) - z %d;whk; fhy;gFjpapy; mike;jhy; z mikAk; fhy; gFjp

(1) Kjy; fhy;gFjp (2) ,uz;lhk; fhy;gFjp

(3) %d;whk; fhy;gFjp (4) ehd;fhk;; fhy;gFjp

70) i 7 vd;w jPh;Tfisf; nfhz;l ,Ugbr; rkd;ghL

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(1) x 2 +7=0 (2) x 2 -7=0 (3) x 2 +x+7=0 (4) x 2 -x-7=0

71) vd;gJ 1 ,d; Kg;gb %yk; vdpy;(1- + 2 ) 4 +(1+ -- 2 ) 4 ,d; kjpg;G (1) 0 (2) 32 (3) -16 (4)-32 72) 𝑖 + 𝑖 22 + 𝑖 23 + 𝑖 24 + 𝑖 25 ,d; kjpg;G vd;gJ 1) i 2) –i 3) 1 4) –1

73)

2

31 i 100 +

2

31 i 100 ,d; kjpg;G

(1) 2 (2) 0 (3) -1 (4) 1 74) 2𝑚 + 3 + 𝑖(3𝑛 − 2) vd;w fyngz;zpd; ,iz vz; ( 𝑚 − 5) + 𝑖 (𝑛 + 4)vdpy;(𝑛, 𝑚) vd;gJ

(1)

8

2

1 (2)

8,

2

1 (3)

8,

2

1 (4)

8,

2

1

75) fy;gngz; ( 𝑖 25)3 ,d; Nghyhh; tbtk;

(1) cos2

+ i sin

2

(2) cos + i sin (3) cos -- i sin (4) cos

2

-- i sin

2

76) 1+𝑒 𝑖𝜃

1+𝑒 𝑖𝜃 = (1) cos + i sin (2) cos - i sin (3) sin - i cos (4) sin + i cos

77) p MdJ fyg;G vz; khwp z-If; Fwpf;fpd;wJ| 2z --1|= 2| z|vdpy; P ,d; epakg;ghij

(1) x =4

1 vd;w Neh;NfhL (2) y =

4

1 vd;w Neh;NfhL

(3) z =2

1 vd;w Neh;NfhL (4) x 2 +y 2 -- 4x--1= 0 vd;w tl;lk;

78) fyg;ngz; jsj;jp;y; 𝑧 1, 𝑧 2, 𝑧 3 , 𝑧 4vd;w Gs;spfs; KiwNa xU ,izfuj;jpd; Kidg; Gs;spfshf ,Ug;gjw;Fk; mjd; kWjiyAk; cz;ikahf ,Ug;gjw;Fk; cs;s epge;jid (1) z 1 + z 4 = z 2 + z 3 (2) z 1 + z 3 = z 2 + z 4

(2) z 1 + z 2 = z 3 + z 4 (4) z 1 – z 2 = z 3 – z 4

79) z xU fyg;ngz;izf; Fwpg;gnjdpy; arg (z) + arg ( z ) vd;gJ

1 𝜋

4 2

𝜋

2 3 0 4

𝜋

4

80) x = cos + i sin vdpy;x n +nx

1 ,d; kjpg;G

(1) 2 cos n (2) 2 i sin n (3) 2 sin n (4) 2i cos n

81) z 1 = 4 + 5 i , z2 = --3 + 2 i vdpy;>

𝑧1

𝑧2 vd;gJ

1 2

13−

22

13𝑖 2 −

2

13+

22

13𝑖 3 −

2

13−

23

13𝑖 4

2

13+

22

13𝑖

82) iz vd;w fyg;ngz;iz Mjpiag; nghWj;J 2

Nfhzj;jpy; fbfhu vjph;jpirapy;

Row;Wk;NghJ me;j vz;zpd; Gjpa epiy (1) iz (2) – iz (3) –z (4) z

83) vd;gJ 1 ,d; n Mk; gb %yk; vdpy;

(1) 1+ 2 + 4 +….= + 3 + 5 + ….. (2) n =0

(3) n =1 (4) = 1n 4. gFKiw tbtfzpjk;(3-tpdhf;fß)

84) 𝑥𝑦 = 72 vd;w jpl;l nrt;tf mjpgutisaj;jpd; kPJs;s VNjDk; xU Gs;spapypUe;J

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tiuag;gLk; njhLNfhL mjd; njhiyj; njhLNfhLfSld; cz;lhf;Fk; Kf;Nfhzj;jpd; gug;G (1) 36 (2) 18 (3) 72 (4) 144

85) 2𝑥 – 𝑦 + 𝑐 = 0 vd;w Neh;NfhL 4 𝑥2 + 8 𝑦2 = 32 vd;w ePs;tl;lj;jpd; njhLNfhL vdpy;

c d; kjp;g;G

(1) 2 3 (2) 6 (3) 36 (4) 4

86) xy = 32 vd;w nrt;tf mjpgutisaj;jpd; nrt;tfj;jpd; ePsk;

(1) 8 2 (2) 32 (3) 8 (4) 16

87) 12

2

2

2

b

y

a

x vd;w mjpgutisaj;jpd; kPJs;s vNjDk; xU Gs;spapypUe;J

Ftpaj;jpw;F ,ilNaAs;s njhiyTfspy; tpj;jpahrk; 24 kw;Wk; ikaj; njhiyTj;jfT 2 vdpy; mjpgutisaj;jpd; rkd;ghL

(1) 1432144

22

yx

(2) 1144432

22

yx

(3) 131212

22

yx

(4) 112312

22

yx

88) 12 𝑦2 − 4 𝑥2 − 24 𝑥 + 48 𝑦 − 127 = 0 vd;w mjpgutisaj;jpd; ikaj; njhiyT jfT

(1) 4 (2) 3 (3) 2 (4) 6

89) 9x 2 +16y 2 =144 vd;w $k;G tistpd; ,af;F tl;lj;jpd; Muk;

(1) 7 (2) 4 (3) 3 (4) 5

90) (8, 0) vd;w Gs;spapypUe;J 13664

22

yx

vd;w mjpgutisaj;jpd; njhiyj;

njhLNfhLfSf;F tiuag;gLk; nrq;Fj;J J}uq;fspd; ngUf;fy; gyd; (1) 25/576 (2) 576/25 (3) 6/25 (4) 25/6

91) 16x 2 + 25y 2 = 400 vd;w tistiuapd; Ftpaj;jpypUe;J xU njhLNfhl;Lf;F tiuag;gLk; nrq;Fj;Jf; NfhLfspd; mbapd; epakg;ghij

(1) x 2 +y 2 =4 (2) x 2 +y 2 =25 (3) x 2 +y 2 =16 (4) x 2 +y 2 =9

92) y 2 -4x+4y+8=0 vd;w gutisaj;jpd; nrt;tfyj;jpd; ePsk; (1) 8 2) 6 3) 4 4) 2 93) xy = 18 vd;w nrt;tf mjpgutisaj;jpd; xU Ftpak; (1) (6, 6) (2) (3, 3) (3) (4, 4) (4) (5, 5)

94) 1916

2

yx

vd;w mjpgutisaj;jpw;F (2>1) vd;w Gs;spapypUe;J tiuag;gLk;

njhLNfhLfspy; njhLehz; (1) 9x -- 8y –72 = 0 (2) 9x + 8y + 72 = 0 3) 8x --9y –72 = 0 4) 8x + 9y – 72 = 0

95) (2, -3) vd;w Kid>x = 4 vd;w ,af;Ftiuiaf; nfhz;l gutisaj;jpd; nrt;tfy ePsk; (1) 2 (2) 4 (3) 6 (4) 8

96) xy = 9 vd;w nrt;tf mjpgutisaj;jpd; kPJs;s 6,3

2 vd;w Gs;spapypUe;J tiuag;gLk;

nrq;Fj;J tistiuia kPz;Lk; re;jpf;Fk; Gs;sp

(1) 3

8, 24 2 −24,

−3

8 3 −

3

8, −24 (4) 24,

3

8

97) 𝑦2 − 2𝑦 + 8𝑥 − 23 = 0 vd;w gutisaj;jpd; mr;R

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(1) y = --1 (2) x = --3 (3) x = 3 (4) y =1 98) xy = c2 vd;w nrt;tf mjpgutisaj;jpd; njhiyj;njhLNfhLfs; (1) x = c, y = c (2) x = 0, y = c (3) x = c, y = 0 (4) x = 0, y =0

99) 2x + 3y + 9 = 0 vd;w NfhL y 2 =8x vd;w gutisaj;ijj; njhLk;Gs;sp

(1) (0, --3) (2) (2, 4) 3) (--6,2

9) 4) (

2

9,--6)

100) y 2 = x + 4 vd;w gutisaj;jpd; ,af;Ftiuapd; rkd;ghL

(1) x =4

15 (2) x =

4

15 (3) x =

4

17 (4) x =

4

17

101) xy = 8 vd;w nrt;tf gutisaj;jpd; miu FWf;fr;rpd; ePsk; (1) 2 (2) 4 (3) 16 (4) 8

102) 1916

22

yx

vd;w mjpgutisaj;jpd; nrq;Fj;Jj; njhLf;NfhLfspd; ntl;Lk; Gs;spapd;

epakg;ghij (1) x2 + y2 = 25 (2) x2 + y2 = 4 3) x2 + y2 = 3 4) x2 + y2 = 7

103) 144

2x+

169

2y=1 vd;w ePs;tl;lj;jpd; miu-nel;lr;R kw;Wk; miu-Fw;wr;R ePsq;fs;

(1) 26,12 (2) 13,24 (3) 12,26 (4) 13,12

104) y 2 = 12x vd;w gutisaj;jpd; Ftpehzpd; ,Wjpg;Gs;spfspy; tiuag;gLk; njhLNfhLfs; re;jpf;Fk; Gs;sp mikAk; NfhL

1 𝑥 – 3 = 0 2 𝑥 + 3 = 0 3 𝑦 + 3 = 0 (4) 𝑦 – 3 = 0 105) xy = 16 vd;w nrt;tf mjpgutisaj;jpd; Kidapd; Maj; njhiyTfs;

(1) (4, 4), (– 4, – 4 ) (2) (2, 8), (– 2, – 8) (3) (4, 0), (– 4, 0) (4) (8, 0), (– 8, 0)

106) 𝑥 2

16 –

𝑦2

9 = 1vd;w mjpgutisaj;jpd; njhiynjhLNfhLfSf;FfpilNaAs;s Nfhzk;

1 𝜋 − 2 tan−1 3

4 2 𝜋 − 2 tan−1

4

3 3 2 tan−1

3

4 4 2 tan−1

4

3

107) 9x 2 + 5y 2 =180 vd;w ePs;tl;lj;jpd; Ftpaq;fSf;fpilNa cs;s njhiyT (1) 4 (2) 6 (3) 8 (4) 2

108) 𝑥 2 = 8𝑦 – 1 vd;w gutisaj;jpd; Kid

(1)

0,

8

1 (2)

0,

8

1 (3)

8

1,0 (4)

8

1,0

109) x + 2y – 5 = 0, 2x – y + 5 = 0 vd;w njhiyj; njhLNfhLfisf; nfhz;l mjpgutisaj;jpd; ikaj; njhiyj; jfT

(1) 3 (2) 2 (3) 3 (4) 2 110) 4 𝑥2 + 9𝑦2 = 36 vd;w ePs;tl;lj;jpd; kPJs;s VnjDk; xU Gs;spapypUe;J 5, 0 kw;Wk;

− 5, 0 vd;w Gs;spfSf;fpilNa cs;s njhiyTfspd; $Ljy; (1) 4 (2) 8 (3) 6 (4) 18 111) 36𝑦2 − 25 x 2+ 900 = 0 vd;w mjpgutisaj;jpd; njhiy njhLNfhLfs;

(1) y=5

6 x (2) y=6

5x (3) y=

25

36x (4) y= x

36

25

112) x 2 =16y vd;w gutisaj;jpd; Ftpak; (1) (4, 0) (2) (0, 4) (3) (--4, 0) (4) (0, --4)

113) 4x 2 -- y 2 =36 f;F 5x -- 2y + 4k = 0 vd;w NfhL xU njhLNfhL vdpy; k ,d; kjpg;G (1) 4/9 (2) 2/3 (3) 9/4 (4) 81/16

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114) 9 𝑥2 + 5𝑦2 − 54𝑥 – 40𝑦 + 116 = 0 vd;w $k;Gtistpd; ikaj; njhiyj;jfT e ,d; kjpg;G

(1)3

1 (2)

3

2 (3)

9

4 (4)

5

2

115) 4 x + 2y = c vd;w NfhL y 2 =16x vd;w gutisaj;jpd; njhLNfhL vdpy; c ,d; kjpg;G (1) -1 (2) -2 (3) 4 (4) - 4 116) XU ePs;tl;lj;jpd; nel;lr;R kw;Wk; mjd; miuFw;wr;Rfspd; ePsq;fs; 8,2 KiwNa mjd; rkd;ghLfs; 𝑦 − 6 = 0 kw;Wk; 𝑥 + 4 = 0 vdpy; ePs;tl;lj;jpd; rkd;ghL

(1)

4

42

x+

16

62

y=1 (2)

16

42

x+

4

62

y=1

(3)

16

42

x-

4

62

y=1 (4)

4

42

x-

16

62

y=1

117) nrt;tfyj;jpd; ePsk;> Jizar;rpd; ePsj;jpy; ghjp vdj; nfhz;Ls;s mjpgutisaj;jpd; ikaj; njhiyT jfT

(1)2

3 (2)

3

5 (3)

2

3 (4)

2

5

118) 16x 2 -- 3y 2 --32x --12y – 44 = 0 vd;gJ

(1) xh; ePs;tl;lkó (2) xh; tl;lk; (3) xh; gutisak; (4) xh; mjpgutisak

119) (− 4, 4) vd;w Gs;spapypUe;J y 2 =16x f;F tiuag;gLk; ,Uj;njhLNfhLfSf;F ,ilNaAs;s

Nfhzk; (1) 45 0 (2) 30 0 (3) 60 0 (4) 90 0

120) 𝑥2 − 4(𝑦 – 3) 2 = 16 vd;w mjpgutisaj;jpd; ,af;Ftiu

1 𝑦 = ± 8

5 2 𝑥 = ±

8

5 3 𝑦 = ±

5

8 4 𝑥 = ±

5

8

121) 𝑦2 = 8𝑥 vd;w gutisaj;jpd; t1

= t kw;Wk; t2 = 3t vd;w Gs;spfspy; tiuag;gl;l

njhLNfhLfs; ntl;bf;nfhs;Sk; Gs;sp

(1) (6t 2 , 8t) (2) (8t, 6t 2 ) (3) (t 2 , 4t) (4) (4t, t 2 )

njhFjp –𝐈𝐈

5. tif Ez;;fzpjk;- gad;ghLfs; - I (3-tpdhf;fß)

122). 0a , 1b vdf; nfhz;L 12)(2 xxxf vd;w rhu;gpw;F nyf;uhQ;rpd;

,ilkjpg;Gj; Njhw;wj;jpd;gbas;s ‘c’ ,d; kjpg;G

(1) -1 (2) 1 (3) 0 (4) 2

1

123) 100kP2 gug;G nfhz;Ls;s nrt;tfj;jpd; kPr;rpW Rw;wsT (1) 10 (2) 20 (3) 40 (4) 60

124) 54)(2 xxxf vd;w rhHG [0,3] ,y; nfhz;Ls;s kPg;ngU ngUk kjpg;G

(1) 2 (2) 3 (3) 4 (4) 5 125) xxy tan vd;w rhh;G

(1) 0, 𝜋

2 ,y; VWk; rhh;G (2) 0,

𝜋

2 ,y; ,wq;Fk; rhh;G

(3) 0, 𝜋

4 ,y; VWk;

𝜋

4,𝜋

2 ,y; ,wq;Fk; (4) 0,

𝜋

4 ,y; ,wq;Fk;

𝜋

4,𝜋

2 ,y; VWk;.

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126) x

x e

x 2

lim

- d; kjpg;G

(1) 2 (2) 0 (3) (4) 1 127) xU tistiuapd; nrq;NfhL x mr;rpd; kpif jpirapy; θ vd;Dk; Nfhzj;ij Vw;gLj;fpwJ. mr;nrq;NfhL tiuag;gl;l Gs;spapy; tistiuapd; rha;T (1)- cot (2) tan (3) - tan (4) cot 128) r Muk; nfhz;l xU tl;lj;jpd; gug;G A ,y; Vw;gLk; khWk; tpjk;>

(1) 2 r (2) 2 rdt

dr (3) r2

dt

dr (4)

dt

dr

129) 1a kw;Wk; 4b vdf; nfhz;L xxf )( vd;w rhu;gpw;F nyf;uhQ;rpd; ,ilkjpg;Gj;

Njhw;wj;jpd;gb mikAk; ‘c’ ,d; kjpg;G

(1) 4

9 (2)

2

3 (3)

2

1 (4)

4

1

130) xU rJuj;jpd; %iy tpl;lj;jpd; ePsk; mjpfupf;Fk; tPjk; 0.1 nr.kP / tpdhb vdpy; gf;f

msT 2

15 nr.kP Mf ,Uf;Fk; NghJ mjd; gug;gsT mjpfupf;Fk; tPjk;

(1) 1.5 nrkP2/ tpdhb (2) 3 nrkP2/ tpdhb

(3) 3 2 nrkP2/ tpdhb (4) 0.15 nrkP2/ tpdhb

131) 45)(2 xxxf vd;w rhh;G VWk; ,ilntsp

(1) 1, (2) 4,1 (3) ,4 (4) vy;yh Gs;spfsplj;Jk;

132) y = 3x2vd;w tistiuf;F x ,d; Maj;njhiyT 2 vdf; nfhz;Ls;s Gs;spapy; nrq;Nfhl;bd; rha;thdJ

(1) 13

1 (2)

14

1 (3)

12

1 (4)

12

1

133) lim𝑥→0𝑎𝑥−𝑏𝑥

𝑐𝑥−𝑑𝑥- d; kjpg;G

(1) (2) 0 (3) log𝑎𝑏

𝑐𝑑 (4)

log 𝑎

𝑏

log𝑐

𝑑

134) t

1 vDk; tistiuf;F Gs;sp (-3, --1/ 13) vd;w Gs;spapy; nrq;Nfhl;;bd; rkd;ghL

(1) 3 =27𝑡-80 (2) 5 =27𝑡-80 (3) 3 =27𝑡+80 (4)𝜃 =1

𝑡

135) xey 3 kw;Wk; 3

3

ea

y vd;Dk; tistiufs; nrq;Fj;jhf ntl;bf; nfhs;fpd;wd vdpy;

‘a’ kjpg;G

(1)-1 (2) 1 (3) 3

1 (4) 3

136) 𝑓(𝑎) = 2, 𝑓’(𝑎) = 1, 𝑔(𝑎) = −1, 𝑔’(𝑎) = 2 vdpy; lim𝑛→∞𝑔 𝑥 𝑓 𝑎 −𝑔 𝑎 𝑓(𝑥)

𝑥−𝑎 ,d;kjpg;G;

(1) 5 (2) -5 (3) 3 (4) -3 137) y = 2x2 -6x-4 vd;w tistiuapy; x-mr;Rf;F ,izahfTs;s njhLNfhl;;bd; njhL Gs;sp

(1)

2

17,

2

5 (2)

2

17,

2

5 (3)

2

17,

2

5 (4)

2

17,

2

3

138) 323232 ayx vd;;w tistiuapd; nrq;NfhL x - mr;Rld;𝜃 vd;Dk;

Nfhzk;Vw;gLj;Jnkdpy;mr; nrq;Nfhl;b∂rha;T

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(1) – cot θ (2) tan θ (3) – tan θ (4) cot θ

139) x = 2 ,y; y = -2x3 + 3x+ 5 vd;w tistiuapd; rha;T (1) -20 (2) 27 (3) -16 (4) -21 140) gpd;tUtdtw;Ws; vJ (0> ) ,y; VWk; rhh;G?

(1) ex (2) x

1 (3) -x2 (4) x-2

141) 1925

22

yx

kw;Wk; 188

22

yx

vDk; tistiufSf;F ,ilgl;l Nfhzk;

(1) 4

(2)

3

(3)

6

(4)

2

142) y = x4vd;w tistiuapd; tisT khw;Wg;Gs;sp (1) x = 0 (2) X = 3 (3) X = 12 (4) vq;Fkpy;iy

143) 2248 xxy vd;w tistiu y-mr;ir ntl;Lk; Gs;spapy; mikAk; njhLf;Nfhl;bd;

rha;T (1) 8 (2) 4 (3) 0 (4) -4

144) y =5

3xvDk; tistiuf;F (-1, - 1/5) vd;w Gs;spapy; njhLNfhl;bd; rkd;ghL

(1) 5y+3x=2 (2) 5y-3x=2 (3) 3x-5y=2 (4) 3x + 3y = 2

145) xU Nfhsj;jpd; fd msT kw;Wk; Muj;jpy; Vw;gLk; khWtPjq;fs; vz;zstpy; rkkhf ,Uf;Fk;NghJ Nfhsj;jpd; tisgug;G

(1) 1 (2) 2

1 (3) 4 (4)

3

4

146) y2 = x kw;Wk; x2 = y vd;w gutisaq;fSf;fpilNa Mjpapy; mikAk; Nfhzk;

(1) 2 tan-1

4

3 (2) tan-1

3

4 (3)

2

(4)

4

147) xU cUisapd; Muk; 2 nr.kP / tpdhB vd;w tPjj;jpy; mjpfupf;fpd;wJ. mjd; cauk; 3 nr.kP / tpdhb vd;w tPjj;jpy; Fiwfpd;wJ. Muk; 3 nr.kP kw;Wk; cauk; 5 nr.kP Mf ,Uf;Fk; NghJ mjd; fd mstpd; khW tPjk; (1) 23 (2) 33 (3) 43 (4) 53

148) 2)( xxf vd;w rhh;G ,wq;Fk; ,ilntsp

(1) , (2) 0, (3) ,0 (4) ,2

149) y =3x2 + 3sin x vd;w tistiuf;F x=0 tpy; njhL Nfhl;;bd; rha;T

(1) 3 (2) 2 (3) 1 (4) -1

150) gpd;tUk; tistiufSs; vJ fPo;Nehf;fP FopT ngw;Ws;sJ (1) y = -x2 (2) y = x2 (3) y = ex (4) y = x2+2x-3

151) 74 23 tts vdpy; KLf;fk; Gr;rpakhFk; NghJs;s jpirNtfk;

(1) 3

32m/sec (2)

3

16m/sec (3)

3

16m/sec (4)

3

32m/sec

152) 2

cos)(x

xf vd;w rhu;gpw;F 3, ,y; Nuhy; Njw;wj;jpd;gb mike;j C,d; kjpg;G

(1) 0 (2) 2 (3) 2

(4)

2

3

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153) mxey kw;Wk;

mxey , m > 1 vd;Dk; tistiufSf;F ,ilgl;l Nfhzk;

1 tan−1 2𝑚

𝑚2 − 1 2 tan−1

2𝑚

1 − 𝑚2 3 tan−1

−2𝑚

1 + 𝑚2 4 tan−1

2𝑚

𝑚2 + 1

154) tex cos t ; tey sin t vd;w tistiuapd; njhLNfhL x - mr;Rf;F ,izahfTs;sJ

vdpy; t ,d; kjpg;G

(1) - 4

(2)

4

(3) 0 (4)

2

155) 36 xxy NkYk; x MdJ tpdhbf;F 5 myFfs; tPjj;jpy; mjpfupf;fpd;wJ. 𝑋 = 3

vDk; NghJ mjd; rha;tpd; khW tPjk; (1) -90 myFfs; / tpdhb (2) 90 myFfs; / tpdhb (3) 180 myFfs; / tpdhb (4) -180 myFfs; / tpdhb 156) nfhLf;fg;gl;Ls;s miu tl;lj;jpd; tpl;lk; 4nr.kP mjDs; tiuag;gLk; nrt;tfj;jpd; ngUk gug;G (1) 2 (2) 4 (3) 8 (4) 16

157) xey vd;w tistiu

(1) X > 0 tpw;F Nky;Nehf;fpf; FopT (2) X > 0 tpw;F fPo;Nehf;ff;p FopT

(3) vg;NghJk; Nky;Nehf;fpf; FopT (4) vg;NghJk; fPo;Nehf;fpf; FopT 158) xU Neu;f;Nfhl;by; efUk; Gs;spapd; jpiNtfkhdJ mf;Nfhl;by; xU epiyg;Gs;spapypUe;J efUk; Gs;spf;F ,ilapy; cs;s njhiytpd; tu;f;fj;jpw;F Neu; tpfpjkhf mike;Js;sJ. vdpy; mjd; KLf;fk; gpd;tUk; xd;wpDf;F tpfpjkhf mike;Js;sJ. (1) s (2) s2 (3) s3 (4) s4

159) y = x2 vd;w rhu;gpw;F [-2 2 ] ,y; Nuhypd; khwpyp

(1) 3

32 (2) 0 (3) 2 (4) -2

160) XU cUFk; gdpf;fl;bg; Nfhsj;jpd; fd msT 1 nr.kP3./ epkplk; vdf; Fiwfpd;wJ. mjd; tpl;lk; 10 nr.kP vd ,Uf;Fk;NghJ tpl;lk; FiwAk; Ntfk; MdJ

(1) 50

1 nrkP / epkplk; (2)

50

1 nrkP / epkplk;

(3) 75

11 nrkP / epkplk; (4)

75

2 nrkP / epkplk;

161) dcxbxaxy 23

vd;w tistiuf;F x = 1 ,y; xU tisT khw;Wg;Gs;sp cz;nldpy;

(1) a + b = 0 (2) a + 3b = 0 (3) 3a + b= 0 (4) 3a + b =1 162) MjpapypUe;J XU Neh;Nfhl;by; x njhiytpy; efUk; Gs;spapy; jpirNtfk; x njhiytpy; efUk; Gs;spapd; jpirNtfk; v vdTk; a + bv2; = x2 vdTk; nfhLf;fg;gl;Ls;sJ. ,q;F a kw;Wk; b khwpypfs;. mjd; KLf;fk; MdJ.

(1) x

b (2)

x

a (3)

b

x (4)

a

x

163) 832 23 xxx mjpfupf;Fk; tPjkhdJ x mjpfupf;Fk; tPjj;ijg; Nghy; ,U klq;F

vdpy; x ,d; kjpg;Gfs;

1 −1

3, −3 2

1

3, −3 3 −

1

3, 3 4

1

3, 3

164) 323232 ayx vDk; tistiuapd; Jiz myFr; rkd;ghLfs;

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1 𝑥 = 𝑎 sin 3 𝜃, 𝑦 = 𝑎 cos 3 𝜃 2 𝑥 = 𝑎𝑐𝑜𝑠 3𝜃, 𝑦 = 𝑎 sin 3 𝜃

3 𝑥 = 𝑎 3 sin 𝜃 , 𝑦 = 𝑎 3 sin 𝜃 4 𝑥 = 𝑎 3 cos 𝜃 , 𝑦 = 𝑎 sin 3 𝜃

165) xU fdr; rJuj;jpd; fd msT 4 nr.kP3 / tpdhb vd;w tPjj;jpy;mjpfupf;fpd;wJ. mf;fdr; rJuj;jpd; fd msT 8 f.nr.kP Mf ,Uf;Fk;NghJ mjd; Gwg;gug;gsT mjpfupf;Fk; tPjk; (1) 8 nrkP 2 / tpdhb (2) 16 nrkP 2 / tpdhb (3) 2 nrkP 2/ tpdhb (4) 4 nrkP 2/ tpdhb

6. tif Ez;;fzpjk; gad;ghLfs;- II. 1-tpdh

166) 22

1

yxu

vdpy;

y

uy

x

ux

(1) u2

1 (2) u (3) u

2

3 (4) - u

167) )1()2(22 xxxy vd;w tistiuf;Fr;

(1) x mr;Rf;F ,izahd xU njhiyj; njhLNfhL cz;L

(2) y mr;Rf;F ,izahd xU njhiyj; njhLNfhL cz;L (3) ,U mr;RfSf;Fk; ,izahd njhiyj; njhLNfhLfß cz;L (4) njhiyj; njhLNfhLfs; ,y;iy.

168)

x

yfu vdpy;

y

uy

x

ux

,d; kjpg;G

(1) 0 (2) 1 (3) 2 u (4) u 169) y2 (a+x)=x2(3a-x) vd;w tistiu gpd;tUtdtw;Ws; ve;jg; gFjpapy; mikahJ? (1) x >0 (2) 0 < x< 3a (3) x ≤--a kw;Wk; x > 3a (4) –a < x < 3a

170) sin,cos ryrx vdpy;

x

r

(1) sec (2) sin (3) cos (4) cosec 171) y2 (a+2x) = x2 (3a-x) vd;w tistiuapd; njhiyj; njhLNfhL

(1) x =3a (2) x = - 2

a (3) x =

2

a (4) x = 0

172)

xy

yxu

22

log vdpy; y

uy

x

ux

vd;gJ

(1) 0 (2) u (3) 2u (4) u-1 173) 28 ,d; 11 Mk; gb %y rjtpfpjg; gpio Njhuhakhf 28,d; rjtpfpj gpioiag; Nghy; ...................... klq;fhFk;

(1) 28

1 (2)

11

1 (3) 11 (4) 28

174) a 2 y 2 = x 2 (a 2 – x 2) vd;w tistiu

(1) x=0 kw;Wk; x=a f;F ,ilapy; xU fz;zp kl;LNk nfhz;Ls;sJ

(2) x=0 kw;Wk; x=a f;F ,ilapy; ,U fz;zpfs; nfhz;Ls;sJ

(3) x= -a kw;Wk; x=a f;F ,ilapy; ,U fz;zpfs; nfhz;Ls;sJ

(4) fz;zp VJkpy;iy

175)

22

441sin

yx

yxu kw;Wk; f = sin u vdpy; rkgbj;jhd rhHG f ,d; gb

(1) 0 (2) 1 (3) 2 (4) 4 176) gpd;tUtdtw;Ws; rupahd $w;Wfs;

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(i) xU tistiu Mjpiag; nghWj;J rkr;rPH ngw;wpUg;gpd;mJ ,U mr;Rfisg; nghWj;Jk; rkr;rPh ngw;wpUf;Fk;

(ii) xU tistiu ,Umr;Rf;fisg; nghWj;J rkr;rPH ngw;wpUg;gpd;mJ Mjpiag; nghWj;Jk; rkr;rPh ngw;wpUf;Fk

(iii) f (x,y)=0 vd;w tistiu 𝑦 = 𝑥 vd;w Nfhl;ilg; nghWj;J rkr;rPH ngw;Ws;sJ

vdpy; f (x,y) = f(y,x)

(iv) f (x,y) = 0 vd;w tistiuf;F f(x,y) = f(-y,-x) cz;ikahapd; mJ Mjpiag; nghWj;J rkr;rPh ngw;wpUf;Fk;.

(1) (𝑖𝑖), (𝑖𝑖𝑖) (2) (𝑖), (𝑖𝑣) (3) (𝑖), (𝑖𝑖𝑖) (4) (𝑖𝑖) (𝑖𝑣)

177) u = y sin x vdpy;

yx

u2

(1) cos x (2) cos y (3) sin x (4) 0 178) ay2 = x2 (3a-x) vd;w tistiu y mr;ir ntl;Lk; Gs;spfs; (1) x = -3a, x = 0 (2) x = 0, x = 3a (3) x = 0, x = a (4) x = 0

179) yxu vdpy; x

u

f;Fr; rkkhdJ

1 𝑦𝑥𝑦−1 (2) U log x (3) u log y (4)𝑥𝑦𝑥−1 180) 9y2 = x2 (4-x2) vd;w tistiu vjw;F rkr;rPH? (1) y mr;R (2) x mr;R (3) y = x (4) ,U mr;Rfs;

7. njif Ez;;fzpjk; : gad;ghLfs;-(3-tpdhf;fß)

181) NfhLfs; y = x, y = 1 kw;Wk; x = 0 Mfpait Vw;gLj;Jk; gug;G y mr;ir nghWj;J Row;wg;gLk; jplg;nghUspd; fd msT

1 π

4 2

π

2 3

π

3 4

2π

3

182) 43232 yx vd;w tistiuapd; tpy;ypd; ePsk;

(1) 48 (2) 24 (3) 12 (4) 96

183) 12

2

2

2

b

y

a

x vd;w ePs;tl;lj;jpd; gug;ig nel;lr;R Fw;wr;R ,tw;iw nghWj;J Row;wg;gLk;

jplg;nghUspd; fd msT tpfpjk; (1) b 2 : a 2 (2) a 2 : b 2 (3) a : b (4) b : a

184) 23 xy vd;w tistiu x = 0 tpypUe;J x = 4 tiu x mr;ir mrrhf

itj;J Row;wg;gLk; jplg;nghUspd; fd msT

(1) 100 (2) 9

100 (3)

3

100 (4)

3

100

185)

2

2cos2

sin

dxx

x ,d; kjpg;G

(1) 0 (2) 2 (3) log 2 (4) log 4

186)

2

0cossin1

cossin

dxxx

xx ,d; kjpg;G.

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(1) 2

(2) 0 (3)

4

(4)

187) gutis y2 = x f;Fk; mjd; nrt;tfyj;jpw;f;Fk; ,ilg;gl;l gug;G

(1) 3

4 (2)

6

1 (3)

3

2 (4)

3

8

188)

0

32sin xdxxcox ,d; kjpg;G

(1) (2) /2 (3) /4 (4) 0

189) 12

2

2

2

b

y

a

x vd;w ePs; tl;lj;jpw;f;Fk; mjd; Jiz tl;lj;jpw;f;Fk; ,ilg;gl;l gug;G

(1) bab (2) 2 baa (3) baa (4) 2 bab

190) x = 0 ,ypUe;J 4

x tiuapyhd y = sin x kw;Wk; y = cos x vd;w tistiufspd;

,ilg;gl;l gug;G (1) 12 (2) 12 (3) 222 (4) 222

191) Muk; 5 cs;s Nfhsj;ij jsq;fs; ikaj;jpypUe;J 2 kw;Wk 4 Jhuj;jpy; ntl;Lk; ,U

,izahd jsq;fSf;F ,ilg;gl;l gFjpapd; tisg;gug;G

(1) 20 (2) 40 (3) 10 (4) 30

192)

2

0

3535

35

sincos

cos

dxxx

x ,d; kjpg;G

(1) 2

(2)

4

(3) 0 (4)

193) 1169

22

yx

vd;w tistiuia Fw;wr;ir nghWj;J Row;wg;gLk; jplg;nghUspd; fd

msT (1) 48 (2) 64 (3) 32 (4) 128 194)y = 2x , x = 0 kw;Wk; x = 2 ,tw;wpf;F ,ilNa Vw;gLk; gug;G x mr;irg; nghWj;J Row;wg;gLk; jplg;nghUspd; tisg;gug;G

(1) 58 (2) 52 (3) 5 (4) 54 195) y = x vd;w Nfhl;bw;f;Fk; x mr;R>NfhLfs; x = 1kw;Wk; x = 2 Mfpatw;wpf;Fk; ,ilg;gl;l gug;G

(1) 2

3 (2)

2

5 (3)

2

1 (4)

2

7

196) (0,0) (3,0) kw;Wk; (3>3) Mfpatw;iw Kidg;Gs;spfshf nfhz;l Kf;Nfhzj;jpd; gug;G x mr;ir nghWj;J Row;wg;gLk; jplg;nghUspd; fd msT

(1) 18 (2) 2 (3) 36 (4) 9

197) dxx4

0

3 2cos

,d; kjpg;G .

(1) 3

2 (2)

3

1 (3) 0 (4)

3

2

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198) 1

0

4)1( dxxx ,d; kjpg;G

(1) 12

1 (2)

30

1 (3)

24

1 (4)

20

1

199)

0

4sin xdx ,d; kjpg;G

1 3𝜋

16 2

3

16 3 0 (4)

3𝜋

8

8. tif nfOr; rkd;ghLfs;.(3-tpdhf;fß)

200) tiff;nfO rkd;ghL Qpydx

dy tpd; njhiff; fhuzp

(1) 𝑝 𝑑𝑥 (2) 𝑄 𝑑𝑥 (3) 𝑒 𝑄 𝑑𝑥 (4) 𝑒 𝑝 𝑑𝑥 201) Mjpg;Gs;spia ikakhf nfhz;l tl;lq;fspd; njhFg;gpd; tiff;nfO rkd;ghL (1) x dy + y dx = 0 (2) x dy – y dx = 0 (3) x dx + y dy = 0 (4) x dx – y dy = 0 202) (D 2 + 1) y = e 2x ,d; epug;Gr; rhh;G (1) (Ax + B)ex (2) A cos x + B sin x (3) (Ax + B)e2x (4) ( Ax + B) e -x

203) 0)(2 dxyxdyx vd;w rkg;gbj;jhd tiff;nfO rkd;ghl;by; y = vx

vdg; gpujpapL nra;Ak; NghJ fpilg;gJ

(1) 0)2(2 dxvvxdv (2) 0)2(

2 dvxxvdx

(2) 0)(22 dvxxdxv (3) 0)2(

2 dxxxvdv

204) yx

yx

dx

dy

vdpy;

(1) 2 x y + 𝑦2 +𝑥2= c (2) x 2 + y 2 – x + y = c (3) x 2 + y 2 – 2 x y = c (4) x 2 – y 2 – 2 x y = c

205) ( D 2 – 4D + 4 ) y = e 2x ,d; rpwg;G jPHT (𝑃𝐼)

(1)xe

x 22

2 (2) xe2x (3) xe-2x (4) xe

x 2

2

206)

32

3

3

3

1

dx

yd

dx

dy

c

tiff;nfO rkd;ghl;bd; gb ,q;F c xU khwpyp

(1) 1 (2) 3 (3) -2 (4) 2

207) 22

.log

1

xy

xxdx

dy ,d; njhiff;fhuzp

(1) e x (2) log x (3) x

1 (4) e-x

208) xxf )(' kw;Wk; 2)1( f vdpy; )(xf vd;gJ

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(1) −3

2 𝑥 𝑥 + 2 (2)

3

2 𝑥 𝑥 + 2 (3)

2

3 𝑥 𝑥 + 2 (4) )2

3

2xx

209) y = ae3x + be –3x vd;w rkd;ghl;by; a iaAk; b iaAk; ePf;fpf; fpilf;Fk; tiff;nfO rkd;ghL

(1) 02

2

aydx

yd (2) 09

2

2

Ydx

yd

(3) 092

2

dx

dy

dx

Yd (4) 09

2

2

xdx

yd

210) QPydx

dy vd;w tiff;nfOr; rkd;ghl;bd; njhiff;fhuzp cos x vdpy; P d; kjpg;G

(1) – cot x (2) cot x (3) tan x (4) – tan x 211) xU fjphpaf;f nghUspd; khWtPj kjpg;G mk;kjpg;gpd; P NeH tpfpjj;jpy; rpijTWfpwJ. ,jw;F Vw;w tiff;nfOr; rkd;ghL (K Fiw vz;)

(1) p

k

dt

dp (2) kt

dt

dp (3) kp

dt

dp (4) kt

dt

dp

212) 2

231

1dx

yd

dx

dy

tiff;nfOr; rkd;ghl;bd; gb

(1) 1 (2) 2 (3) 3 (4) 6

213) 𝑑𝑦

𝑑𝑥+ 2

𝑦

𝑥= 𝑒4𝑥vd;w tiff;nfOr; rkd;ghl;bd; njhiff;fhuzp

(1) log x (2) x2 (3) ex (4) x

214) xkey vdpy; mjd; tiff;nfO rkd;ghL

(1) ydx

dy (2) ky

dx

dy (3) 0 ky

dx

dy (4) xe

dx

dy

215) y = cx – c2 vd;gjidg; nghJj; jPHthfg; ngw;w tiff;nfO rkd;ghL (1) (y’)2 – xy’ + y = 0 (2) y” = 0 (3) y’ = c (4) (y’)2+ x y’+ y = 0 216) x yjsj;jpYs;s vy;yh Neh;f; NfhLfspd; njhFg;gpd; tiff;nfO rkd;ghL

(1) dx

dy xU khwpyp (2) 0

2

2

dx

yd

(3) 0dx

dyy (4) 02

2

ydx

yd

217) ydyexdydxy 2sec ,d;; njhiff;fhuzp

(1) ex (2) e –x (3) ey (4) e –y 218) xU jsj;jpy; cs;s mr;Rf;F nrq;Fj;jy;yhj NfhLfspd; tiff;nfO rkd;ghL

(1) 0dx

dy (2)

2

2

dx

yd= 0 (3) m

dx

dy (4) m

dx

yd

2

2

219)m < 0 Mf ,Ug;gpd; 0mxdy

dx ,d; jPHT

(1)x = cemy (2) x = ce –my (3) x = my + c (4) x = c

220) xydy

dx

312

5 vd;w tiff;nfOtpd;

(1) thpir 2 kw;Wk; gb 1 (2) thpir 1 kw;Wk; gb 2

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(3) thpir 1 kw;Wk; gb 6 (4) thpir 1 kw;Wk; gb 3 221) (3D2 + D – 14 ) y = 13 e 2x ,d; rpwg;Gj; jPHT

(1) 26x e 2x (2) 13x e 2x (3) x e 2x (4)( x 2/ 2 ) e 2x

222) 𝑦 = 𝑒 x(𝐴𝑐𝑜𝑠 𝑥 + 𝐵 𝑠𝑖𝑛 𝑥)vd;w njhlh;gpy; A iaAk; B iaAk; ePf;fp ngwg;gLk; tiff;nfO

rkd;ghL

1 𝑦2 + 𝑦1 = 0 2 𝑦2 − 𝑦1 = 0 3 𝑦2 − 2𝑦1 + 2𝑦 = 0 4 𝑦2 − 2𝑦1 − 2𝑦 = 0

223) 𝑑𝑦

𝑑𝑥 – 𝑦 𝑡𝑎𝑛 𝑥 = 𝑐𝑜𝑠 𝑥 vd;w tiff;nfO rkd;ghl;bd; njhiff;fhuzp

1 sec x 2 cos x 3 etan x (4) cot x

224) 0)(),()()( agDgaDDf vdpy; tiff;nfO rkd;ghL axeyDf )( rpwg;Gj; jPHT

(1) meax (2) )(ag

eax (3) g (a)eax (4)

)(ag

xeax

225) y = mxvd;w NeHNfhLfspd; njhFg;gpd; tiff;nfO rkd;ghL

(1) mdx

dy (2) 0 xdyydx (3) 02

2

dx

yd (4) y 0 xdydx

9. jdp epiyf; fzf;fpay; (3-tpdhf;fß);

226) ),( 99 Z ,y; [7] ,d; thpir (1) 9 (2) 6 (3) 3 (4) 1 227) ngUf;fiyg; nghWj;J Fykhfpa xd;wpd; Kg;gb %yq;fspy; ω 2 ,d; thpir (1) 4 (2) 3 (3) 2 (4) 1 228) xU $l;Lf; $w;W %d;W jdpf;$w;Wfisf; nfhz;ljhf ,Ug;gpd; nka;al;ltizapYs;s

epiufspd; vz;zpf;if

(1) 8 (2) 6 (3) 4 (4) 2

229) )](~[~ qp d; nka; ml;ltizapy; epiufspd; vz;zpf;if

(1) 2 (2) 4 (3) 6 (4) 2 230) KOf;fspy; * vd;w



(1) nnZ , (2) ,Z (3) ,.Z (4) ,R 237) p apd;nka; kjpg;G T kwWk; q ,d; nka; kjpg;G F vdpy; gpd;tUtdtw;wpy; vit nka; kjpg;G T vd ,Uf;Fk; (i) pvq (ii) ~ pvq (iii) qpv ~ (iv) qp ~^

(1) (i), (ii), (iii) (2) (i), (ii), (iv) (3) (i), (iii), (iv). (4) (ii), (iii), (iv)

238) ngUf;fiyg; nghWj;J Fykhfpa xd;wpd; nMk; gb %yq;fspd; ωk ,d; vjpHkiw nk (1) ω1/k (2) 1 (3) kn (4) kn /

239) 653 1111 ,d; kjpg;G

(1) 0 (2) 1 (3) 2 (4) 3 240) qp f;Fr; rkhdkhdJ.

(1) qp (2) pq (3) )()( pqvqp (4) )()^( pqqp

241) ngUf;fy; tpjpiag; nghWj;J Fykhfpa xd;wpd; ehyhk; %yq;fspy;; - i d; thpir (1) 4 (2) 3 (3) 2 (4) 1 242) gpd;tUtdtw;Ws; vJ Kuz;ghlhFk; (1) pvq (2) qp^ (3) ppv ~ (4) pp ~^

243) nka;naz;fspd; fzk; R ,y; * vd;w 4 kw;Wk; 12 Mfpa kjpg;gfs; KiwNa 1

3,

1

4 kw;Wk;

5

12

Mfpa epfo;jfTfisf; nfhs;Snkdpy; 𝐸(𝑋),d; kjpg;G (1) 5 (2) 7 (3) 6 (4) 3

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251) xU ,ay;epiyg; gutypd; epfo;jfT mlHj;jpr; rhHG )(xf ,d;

ruhrhp vdpy;

dxxf )( ,d; kjpg;G

(1) 1 (2) 0.5 (3) 0 (4) 0.25 252) xU ngl;bapy; 6 rptg;G kw;Wk; 4 nts;isg; ge;Jfs; cs;sd. mtw;wpypUe;J 3 ge;Jfs; rktha;g;G Kiwapy; vLf;fg;gl;lhy; 2 nts;isg; ge;Jfs; fpilf;f epfo;jfT

(1) 20

1 (2)

125

18 (3)

25

4 (4)

10

3

253) X vd;w rktha;g;G khwpapd; epfo;jfT epiwr;rhHG guty; gpd;tUkhW λ tpd; kjpg;G (1) 1 (2) 2 (3) 3 (4) 4

254) xU



,uz;Lk; xNu epwj;jpy; ,Uf;f epfo;jfT.

(1) 2

1 (2)

51

26 (3)

51

25 (4)

102

25

263) rktha;g;G khwp X ,d; guty; rhHG F(x) xU (1) ,wq;Fk; rhHG (2) VWk; rhu;G (3) khwpypr; rhHG (4) Kjypy; VWk; rhHG gpd;dH ,wq;Fk; rhHG 264) X vd;w rktha;g;G khwpapd; gutw;gb 4 NkYk; ruhrhp 2 vdpy; ,d; ruhrhpapd; E(X2) kjpg;G (1) 2 (2) 4 (3) 6 (4) 8

265) xU jdp epiy rktha;g;G khwp X f;F 202 NkYk; 2761

2 vdpy; X ,d; ruhrhpapd; kjpg;G (1) 16 (2) 5 (3) 2 (4) 1 266) X vd;w rktha;g;G khwpapd; epfo;jfTg; guty; gpd;tUkhW

X 0 1 2 3 4 5

P(X=x) 1

4

2a 3a 4a 5a 1

4

41 xP ,d; kjpg;G

(1) 21

10 (2)

7

2 (3)

14

1 (4)

2

1

267) xU rktha;g;G khwp X ,d; epfo;jfT epiwr; rhHG (p,d,f) gpd;tUkhW

X 0 1 2 3 4 5 6 7

P(X=x) 0 K 2 k 2 k 3 k k2 2 k2 7 k2+k

K ,d; kjpg;G

(1) 8

1 (2)

10

1 (3) 0 (4) –1 or

10

1

268) xU gfilia 16 Kiwfs; tPRk; NghJ ,ul;ilg;gil vz; fpilg;gJ ntw;wpahFk; vdpy; ntw;wpapd; gutw;gb (1) 4 (2) 6 (3) 2 (4) 256 269) xU



𝑇𝑒𝑠𝑡 𝑁𝑜. ∶ …………………… 𝐷𝑎𝑡𝑒 ∶ ………… …… ..

𝑵𝒂𝒎𝒆 ∶ ………… ……………………… …… . 𝑺𝒕𝒅 & 𝑆𝑒𝑐. : ………………..

𝑽𝒐𝒍𝒖𝒎𝒆 − 𝑰 𝑽𝒐𝒍𝒖𝒎𝒆 − 𝑰𝑰

1 31 61 91 1 31 61 91 121

2 32 62 92 2 32 62 92 122

3 33 63 93 3 33 63 93 123

4 34 64 94 4 34 64 94 124

5 35 65 95 5 35 65 95 125

6 36 66 96 6 36 66 96 126

7 37 67 97 7 37 67 97 127

8 38 68 98 8 38 68 98 128

9 39 69 99 9 39 69 99 129

10 40 70 100 10 40 70 100 130

11 41 71 101 11 41 71 101 131

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12 42 72 102 12 42 72 102 132

13 43 73 103 13 43 73 103 133

14 44 74 104 14 44 74 104 134

15 45 75 105 15 45 75 105 135

16 46 76 106 16 46 76 106 136

17 47 77 107 17 47 77 107 137

18 48 78 108 18 48 78 108 138

19 49 79 109 19 49 79 109 139

20 50 80 110 20 50 80 110 140

21 51 81 111 21 51 81 111 141

22 52 82 112 22 52 82 112 142

23 53 83 113 23 53 83 113 143

24 54 84 114 24 54 84 114 144

25 55 85 115 25 55 85 115 145

26 56 86 116 26 56 86 116 146

27 57 87 117 27 57 87 117 147

28 58 88 118 28 58 88 118 148

29 59 89 119 29 59 89 119 149

30 60 90 120 30 60 90 120 150

121

𝑵𝒐. 𝒐𝒇 𝒄𝒐𝒓𝒓𝒆𝒄𝒕 𝒂𝒏𝒔𝒘𝒆𝒓𝒔. ∶

(gFjp – M – tpdh tpilfß) 1.mzpfs; kw;Wk; mzpf;Nfhitfsp∂ ga∂ghLfs ;

( 2 - tpdhf;fs; - 12 - kjpg;ngΩfs ;)

𝟏) 𝑨 = 𝟏 𝟐𝟑 −𝟓

vd;w mzpapd; Nru;g;G mzpiaf; fhz;f𝐀 𝐚𝐝𝐣 𝐀 = 𝐚𝐝𝐣 𝐀 𝐀 = 𝐀𝐈vd;gijr;

rupghu; jPh;T : (𝑀𝑎𝑟𝑐 − 2007, 𝑀𝑎𝑟𝑐 − 2009)

A = 1 23 −5

𝑎𝑑𝑗 𝐴 = −5 −2−3 1

𝐴 = 1 23 −5

= −11

A (𝑎𝑑𝑗 𝐴) = 1 23 −5

−5 −2−3 1

= −11 0 0 −11

= −11𝐈

(𝑎𝑑𝑗 𝐴) 𝐴 = −5 −2−3 1

1 23 −5

= −11 0 0 −11

= −11𝐈

vdNt A (adj A) = (adj A) A = A𝐈

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gapw;rpf; fzf;F∶ 𝐴 = −1 2 1 −4

vdpy; 𝐴 (𝑎𝑑𝑗𝐴) = (𝑎𝑑𝑗𝐴) 𝐴 = AI2vd;gijr; rupghu;

𝟐) 𝑰𝒇𝑨 = 𝟓 𝟐𝟕 𝟑

kw;Wk;𝑩 = 𝟐 −𝟏−𝟏 𝟏

vdpy; 𝑨𝑩 −𝟏 = 𝑩−𝟏𝑨−𝟏vd;gijr; rupghu; 𝐽𝑢𝑛𝑒 − 2006

(𝐽𝑢𝑛𝑒 − 2012)

jPh;T ; 𝐴 = 5 27 3

𝐵 = 2 −1−1 1

𝐴 = 5 27 3

= 1 𝐵 = 2 −1−1 1

=1

𝑎𝑑𝑗 𝐴 = 3 −2−7 5

𝑎𝑑𝑗 𝐵 = 1 11 2

𝐴−1 = 3 −2−7 5

𝐵−1 = 1 11 2

𝐵−1𝐴−1 = 1 11 2

3 −2−7 5

= −4 3−11 8

𝐴𝐵 = 5 27 3

2 −1−1 1

= 8 −311 −4

𝐴𝐵 = 8 −311 −4

= 1

𝑎𝑑𝑗 𝐴𝐵 = −4 3−11 8

(𝐴𝐵)−1 = −4 3−11 8

vdNt (𝐴𝐵)−1 = 𝐵−1𝐴−1

gapw;rpf; fzf;F: 1) 𝐴 = 1 21 1

kw;Wk;𝐵 = 0 −11 2

vdpy; (𝐴𝐵)−1 = 𝐵−1𝐴−1vd;gijr; rupghu;

(𝐽𝑢𝑛𝑒 − 2009, 𝐽𝑢𝑛𝑒 − 2010)

2) 𝐴 = 1 21 1

vdpy; 𝑨−𝟏 𝑻

= 𝑨𝑻 −𝟏vd;gijr; rupghu; (𝑀𝑎𝑟𝑐 − 2010)

𝟑) 𝐀 = 𝟓 𝟐𝟕 𝟑

kw;Wk; 𝐁 𝟐 −𝟏−𝟏 𝟏

vdpy; 𝐀𝐁 𝐓 = 𝐁 𝐓𝐀 𝐓vd;gijr; rupghu; ( June − 2006)

jPh;T : A = 5 27 3

kw;Wk; B 2 −1−1 1

𝐴𝐵 = 5 27 3

2 −1−1 1

= 8 −311 −4

𝐴𝐵 𝑇 = 8 11−3 −4

𝐴 𝑇 = 5 72 3

𝐵 𝑇 = 2 −1−1 1

𝐵 𝑇𝐴 𝑇 = 2 −1

−1 1

5 72 3

= 8 11−3 −4

vdNt 𝐴𝐵 𝑇 = 𝐵𝑇𝐴𝑇

𝟒) 𝑨 = −𝟒 −𝟑 −𝟑 𝟏 𝟎 𝟏 𝟒 𝟒 𝟑

- ,d; Nru;g;G mzp 𝐀vd epWTf. (March − 2008, March − 2011)

jPh;T∶ 𝐴 𝐶 =

0 14 3

− 1 14 3

1 04 4

− −3 −3 4 3

−4 −3 4 3

− −4 −3 4 4

−3 −3 0 1

− −4 −31 1

−4 −3 1 0

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= −4 1 4

−3 0 4−3 1 3

𝑎𝑑𝑗 𝐴 = −4 −3 −3 1 0 1 4 4 3

= 𝐴

𝟓) Neu;khwzp fhz;f𝑨 = 𝟏 𝟎 𝟑𝟐 𝟏 −𝟏𝟏 −𝟏 𝟏

jPh;T ∶ 𝐴 = 1 0 32 1 −11 −1 1

𝐴 = 1 0 32 1 −11 −1 1

= −9.

𝐴 𝐶 =

1 −1−1 1

− 2 −11 1

2 11 −1

− 0 3−1 1

1 31 1

− 1 01 −1

0 31 −1

− 1 32 −1

1 02 1

= 0 −3 −3

−3 −2 1−3 7 1

𝑎𝑑𝑗 𝐴 = 0 −3 −3

−3 −2 7−3 1 1

𝐴−1 = 1

𝐴 ( 𝑎𝑑𝑗 𝐴) = −

1

9

0 −3 −3 −3 −2 7−3 1 1

gapw;rpf; fzf;Ffs : (1)Neu;khwzp fhz;f

𝑖 1 3 74 2 31 2 1

𝑖𝑖 1 2 −2−1 3 0 0 −2 1

𝑖𝑖𝑖 8 −1 −3−5 1 2 10 −1 −4

𝑖𝑣 3 1 −12 −2 0 1 2 −1

(𝑂𝑐𝑡 − 2009. 𝑂𝑐𝑡 − 2011))

2) 𝐴 = −1 2 −2 4 −3 4 4 −4 5

vdpy; 𝐴 = 𝐴−1vd;gijr; rupghu;. (𝑀𝑎𝑟𝑐 − 2009)

𝟔) mzpj;juk; fhz;f. 𝟏 𝟐 −𝟏 𝟑𝟐 𝟒 𝟏 − 𝟐𝟑 𝟔 𝟑 − 𝟕

(𝑂𝑐𝑡 − 2008)

jPh;T ∶

A = 1 2 −1 32 4 1 − 23 6 3 − 7

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~ 1 2 −1 30 0 3 − 80 0 6 − 16

R 2 R 2 – 2R 1 R 3 R 3 – 3R 1

~ 1 2 −1 30 0 3 − 80 0 0 0

𝑅3 → 𝑅3−2𝑅2

filrp rkhd mzpahdJ VWgb tbtpy; cs;sJ.

,jpy; ,uz;L ©r;rpakw;w epiufs; cs;sjhy; ( A ) = 2

7) mzpj;juk; fhz;f. −𝟐 𝟏 𝟑 𝟒𝟎 𝟏 𝟏 𝟐𝟏 𝟑 𝟒 𝟕

jPh;T∶ 𝐴 = −2 1 3 40 1 1 21 3 4 7

vd;f

~ 1 3 4 70 1 1 2

−2 1 3 4 𝑅1 ↔ 𝑅3

1 3 4 70 1 1 20 7 11 18

𝑅3 → 𝑅3+2𝑅1

1 3 4 70 1 1 20 0 4 4

𝑅3 → 𝑅3−7𝑅2

filrp rkhd mzpahdJ VWgb tbtpy; cs;sJ

,jpy; 3 ©r;rpakw;w epiufs; cs;sjhy; ( 𝐴 ) = 3, gapw;rpf; fzf;Ffs; : mzpj;juk; fhz;f

𝑖 0 1 2 12 −3 0 − 11 1 −1 0

((𝐽𝑢𝑛𝑒 − 2011, 𝐽𝑢𝑛𝑒 12) 𝑖𝑖 1 1 1 3 2 −1 3 45 −1 7 11

𝑖𝑖𝑖 3 1 2 0 1 0 −1 02 1 3 0

(𝐽𝑢𝑛𝑒 − 2008) 𝑣 1 −3 −8 − 103 1 −4 02 5 6 13

(𝑀𝑎𝑟𝑐 − 2006)

𝑣𝑖 1 −2 3 4−2 4 −1 − 3−1 2 7 6

(𝐽𝑢𝑛𝑒 − 2010)

𝟕) mzpKiwapy; Neupa Kiwapy; rkd;ghl;il jPu;f;fTk ; 𝟐𝒙 – 𝒚 = 𝟕, 𝟑𝒙 –𝟐 𝒚 = 𝟏𝟏.

( 𝐽𝑢𝑛𝑒 − 2007, March − 2012)

jPh;T ∶ jug; gl;l rkd;ghLfspd; mzpr; rkd;ghL

2 −13 −2

𝑥𝑦 =

711

𝐴 𝑋 = 𝐵

𝐴 = 2 −13 −2

, 𝑋 = 𝑥𝑦 and 𝐵 =

711

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𝑋 = 𝐴−1𝐵.

𝐴 = 2 −13 −2

= −1𝑎𝑑𝑗 𝐴 = −2 1 3 2

A−1 = −1 −2 1−3 2

= 2 −13 −2

𝑋 = −2 1 3 2

7

11 = 3

−1

jPh;Tfs;: 𝑥 = 3 , 𝑦 = −1 gapw;rpf; fzf;Ffs;∶ mzpKiwapy; Neupa Kiwapy; rkd;ghl;il jPu;f;fTk; 𝑖 7𝑥 + 3𝑦 = −1, 2𝑥 + 𝑦 = 0. 𝑖𝑖 𝑥 + 𝑦 = 3, 2𝑥 + 3𝑦 = 8. 𝐽𝑢𝑛𝑒 − 2008, 𝑂𝑐𝑡 − 2008

𝟖) mzpNfhit Kiwapy; jPu;f;f ; 𝟒𝒙 + 𝟓𝒚 = 𝟗 , 𝟖𝒙 + 𝟏𝟎 𝒚 = 𝟏𝟖. ( 𝑆𝑒𝑝 − 2006, 𝑂𝑐𝑡 −2009))

jPh;T ∶ = 4 58 10

= 0 x = 9 518 10

= 0 y = 4 98 18

= 0

= 0 kw;Wk;x = y = 0vd,Ug;gjhy; vz;zpf;ifaw;w jPu;Tfisg; ngw;wpUf;Fk;.

Nkw;fz;l njhFg;G 4x + 5 y = 9.vd;w xU jdpr; rkd;ghlhf khWk;.

,jid jPh;f;f y = k. vdf; nfhs;f ∴ 4x + 5k = 9 .

4x = 9 – 5k

x =1

4 9 – 5k ; y = k , k R.

jPh;T ∶ [1

4(9 – 5k ) , k ] k R.

gapw;rpf; fzf;F ; mzpf;Nfhit Kiwapy; jPu;f;f. 𝑖 2𝑥 + 3𝑦 = 8 , 4𝑥 + 6𝑦 = 16. 𝐽𝑢𝑛𝑒 − 2006, March − 2011 .

𝑖𝑖 2𝑥 − 3𝑦 = 7, 4𝑥 − 6𝑦 = 14 (𝐽𝑢𝑛𝑒 − 2009) 9) mzpf;Nfhit Kiwapy; gp∂tUk; mrkgbj;jhd rk∂ghl;L;j; njhFg;Gfisj; jPHf;f.

2𝑥 + 2𝑦 + 𝑧 = 5, 𝑥 – 𝑦 + 𝑧 = 1, 3𝑥 + 𝑦 + 2 𝑧 = 4. ( 𝑀𝑎𝑟𝑐 2008, 𝑀𝑎𝑟𝑐 2009 )

jPh;T ∶ = 2 2 11 −1 13 1 2

= 0.

x = 5 2 11 −1 14 1 2

= 5 −2 − 1 − 2 2 − 4 + 1 1 + 4 = −6 0

= 0 kw;Wk;𝑥 0 , njhFg;GxUq;fikT mw;wJ vdNt jPh;Tfs; fpilahJ 10) mzpf;Nfhit Kiwapy; gp∂tUk; mrkgbj;jhd rk∂ghl;L;j; njhFg;Gfisj; jPHf;f.

( March − 2012) 𝑥 + 𝑦 + 2𝑧 = 4, 2𝑥 + 2𝑦 + 4𝑧 = 8, 3𝑥 + 3𝑦 + 6 𝑧 = 10.

jPh;T ∶ = 1 1 22 2 43 3 6

= 0. x = 4 1 28 2 4

10 3 6 = 0. y =

1 4 22 8 43 10 6

= 0. z = 1 1 42 2 83 3 10

= 0.

= 0 kw;Wk;𝑥 = 𝑦 = 𝑧 = 0. ∆ ,∂ vy;yh 2 x 2rpw;wzpf;Nfhitfspd;

kjpg;Gfs; ©r;rpaq;;fs; MFk;. Mdhy; ∆𝑋rpy; xU rpw;wzpf;Nfhit ©r;rpakw;wjhAs;sjhy; njhFg;G xUq;fikT mw;wJ. vdNt jPh;Tfs; fpilahJ.

𝟏𝟏)gp∂tUk; rk∂ghLfsp∂ njhFg;G xUq;fikT cilajh vd;gij ju Kiwapy; Muha;f. 𝑥 + 𝑦 + 𝑧 = 7, 𝑥 + 2𝑦 + 3𝑧 = 18, 𝑦 + 2𝑧 = 6. ( 𝑂𝑐𝑡 – 2007, 𝑀𝑎𝑟𝑐 − 2010)

jPh;T ; 1 1 11 2 30 1 2

xyz =

7186

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𝐴 𝑋 = 𝐵

(𝐴 𝐵) = 1 1 1 71 2 3 180 1 2 6

1 1 1 70 1 2 11 0 1 2 6

𝑅2 → 𝑅2 − 𝑅1

~ 1 1 1 70 1 2 11 0 0 0 − 5

𝑅3 → 𝑅3 − 𝑅2

,J VWgb tbtpy; cs;sJ.

mJ %∂W ©r;rpa kw;w epiufisg; ngw;Ws;sjhy; 𝐴, 𝐵 = 3NkYk; 𝐴 = 2

( 𝐴 ; 𝐵) ( 𝐴 ) njhFg;G xUq;fikT mw;wJ vdNt jPh;Tfs; fpilahJ. 𝟏𝟐)gp∂tUk; rk∂ghLfsp∂ njhFg;G xUq;fikT cilajh vd;gij ju Kiwapy; Muha;f.

2𝑥 − 3𝑦 + 7𝑧 = 5, 3𝑥 + 𝑦 − 3𝑧 = 13, 2𝑥 + 19 𝑦 − 47𝑧 = 32. ( 𝑂𝑐𝑡 – 2007)

jPh;T ; 2 −3 73 1 −32 19 −47

xyz =

51332

𝐴 𝑋 = 𝐵

(𝐴 𝐵) = 2 −3 7 53 1 −3 132 19 −47 32

~ 2 −3 7 51 4 −10 80 22 −54 27

𝑅2 → 𝑅2 − 𝑅1;𝑅3 → 𝑅3 − 𝑅1

~ 1 4 −10 82 −3 7 50 22 −54 27

𝑅1 → 𝑅2

~ 1 4 −10 80 −11 27 − 110 22 −54 27

𝑅2 → 𝑅2 − 2𝑅1

~ 1 4 −10 80 −11 27 − 110 0 0 5

𝑅3 → 𝑅3 + 2𝑅2

,J VWgb tbtpy; cs;sJ.

mJ %∂W ©r;rpa kw;w epiufisg; ngw;Ws;sjhy; 𝐴, 𝐵 = 3NkYk; 𝐴 = 2

( 𝐴 ; 𝐵) ( 𝐴 ) njhFg;G xUq;fikT mw;wJ vdNt jPh;Tfs; fpilahJ. gapw;rpf; fzf;F : gp∂tUk; rk∂ghLfsp∂ njhFg;G xUq;fikT cilajh vd;gij ju

Kiwapy; Muha;f. 𝑥 – 4 𝑦 + 7 𝑧 = 14, 3 𝑥 + 8𝑦 – 2 𝑧 = 13, 7 𝑥 – 8 𝑦 + 26𝑧 = 5. (𝑂𝑐𝑡 − 2011)

3. fyg;ngz;fs; (gFjp-M-2-tpdhf;fs;-12-kjpg;ngz;fs;)

1) a+ib v∂w jpl;lbtp;y; vOJf. ∶ 𝟏 + 𝒊 (𝟏 − 𝟐𝒊)

𝟏+𝟑𝒊

jPu;T : ( 1 + 𝑖 ) ( 1 – 2 𝑖 ) = 1 – 2 𝑖 + 𝑖 − 2𝑖2 = 1 – 𝑖 + 2 = 3 − 𝑖 1 + 𝑖 (1 − 2𝑖)

1+3𝑖 =

3 − 𝑖

1+3𝑖 𝑥

1−3𝑖

1−3𝑖

= 3−9𝑖−𝑖+3 𝑖2

1−9 𝑖2

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= −10 𝑖

10

= − 𝑖.

gapw;rpf; fzf;Ffs; ∶a+ib v∂w jpl;ltbtp;y; vOJf. 𝑖4+𝑖9+𝑖16

3−2𝑖8−𝑖10−𝑖15

𝟐) − 𝟖 – 𝟔 𝐢 . − ,∂ tHf;f %yk; fhΩf(𝑀𝑎𝑟𝑐 − 2009, 𝑆𝑒𝑝. − 2006)

jPu;T ∶ x + iy = −8 − 6i

tHf;fg;gLj;j𝑥2 − 𝑦2 + 2𝑥𝑦 = −8 − 6𝑖

𝑥2 − 𝑦2 = −8 ------ (1) 2𝑥𝑦 = −6 ---- (2)

𝑥2 + 𝑦2 = 𝑥2 − 𝑦2 2 + (2𝑥𝑦)2 = (−8)2 + (−6)2

𝑥2 + 𝑦2 = 10 ------- (3)

(1) + (3) 2𝑥2 = 2 ~𝑥 = ± 1

𝑥 = 1vdpy;𝑦 = −3. 𝑥 = − 1 vdpy; 𝑦 = 3.

1 – 3 𝑖 , − 1 + 3 𝑖.

khw;WKiw :−8 − 6𝑖 = ` − 6𝑖

= 1 + (3𝑖)2 − 6𝑖

= (1 − 3𝑖)2

⟹ −8 − 6𝑖 = ± 1 − 3𝑖

= 1 − 3𝑖 , −1 + 3𝑖.

3) −𝟕 + 𝟐𝟒 𝒊 − ,∂ tHf;f %yk; fhΩf.( 𝑀𝑎𝑟𝑐 − 2007, 𝐽𝑢𝑛𝑒 − 2009)

jPu;T ∶ -7 + 24 i = 9 − 16 + 24i

= 32 + 24i + 4i 2 = (3 + 4i)2

⟹ −7 + 24i = ± 3 + 4i = 3 + 4i , −3 − 4i.

𝟑) MHf∂jsj;jpy;fyg;ngΩfs;𝟏𝟎 + 𝟖𝒊, −𝟐 + 𝟒 𝒊, kw;Wk; −𝟏𝟏 + 𝟑𝟏 𝒊 mikf;Fk; Kf;Nfhzk;xU nrq;Nfhz Kf;Nfhzk; vd epWTf

jPu;T: 𝐴, 𝐵, 𝐶 vDk; Gs;spfs; KiwNa ( 10, 8 ) , (−2 , 4) kw;Wk; ( −11, 31) vDk; fyg;ngΩfis MHf∂jsj;jpy;Fwpf;fl;Lk;

AB = | (10 + 8 i) – (−2 + 4 i ) | = | 12 + 4 i | = 144 + 16 = 160 .

BC = | −2 + 4 i – −11 + 31 i | = | 9 − 27 i | = 81 + 729 = 810

CA = | ( −11 + 31i ) – ( 10 + 8 i ) | = | − 21 + 23 i | = 441 + 529 = 970 .

𝐴𝐵2 + 𝐵𝐶2 = 𝐶𝐴2 𝐵 = 90.

gapw;rpf; fzf;Ffs; : 1) 3 + 3 i , −3 – 3 i, −3 3 + 3 3 ivDk; fyg;ngΩfs;xU rkgf;f Kff;Nfhzj;ij MHf∂ jsj;jpy; cUthf;Fk; v∂W fhl;Lf. 2) 2i, 1 + i, 4 + 4 i, kw;Wk; 3 + 5 i vDk; fyg;ngΩfs;xU nrt;tfj;ij MHf∂ jsj;jpy; xU nrt;tfjij cUthf;Fk; v∂W fhl;Lf. 3) fyg;ngΩfs; 7 + 9i, −3 + 7 i, 3 + 3 i vDk; fyg;ngΩfs;MHf∂ jsj;jpy;;xU nrq;Nfhz Kf;Nfhzj;ij mikf;Fk vd epWTf. (𝐽𝑢𝑛𝑒 − 2009, 𝑀𝑎𝑟𝑐 − 2010)

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4) 7 + 5i, 5 + 2 i, 4 + 7i kw;Wk; 2 + 4 i vDk; fyg;ngΩfs;xU ,izfuj;ij mikf;Fk; vd epWTf.mikf;Fk

𝟒) 𝟑 + 𝒊I xU jPh;thff; nfhΩl𝐱𝟒 − 𝟖𝐱𝟑 + 𝟐𝟒 𝐱𝟐 – 𝟑𝟐𝐱 + 𝟐𝟎 = 𝟎vDk; rk∂ ghl;b∂

jPh;Tfisf; fhΩf. (𝑀𝑎𝑟𝑐 − 2009, 𝑀𝑎𝑟𝑐 − 2012)

jPu;T : 𝐱𝟒 − 𝟖𝐱𝟑 + 𝟐𝟒 𝐱𝟐 – 𝟑𝟐𝐱 + 𝟐𝟎 = 𝟎

3 + i xUjPh;T vdNt3 – i kw;;nwhU %yk;

%yk;q;fsp∂ $Ljy; = 3 + i + 3 − i = 6.

%yk;q;fsp∂ngUf;fk; = (3 + i ) (3 − i ) = 9 – i2 = 10.

\x2 – 6 x + 10 vdgJ XH fhuzpahfpJ

x4 − 8x3 + 24 x2 – 32x + 20 (x2 – 6 x + 10) (x2 + p x + 2)

x nfOit xg;gpl 10 p – 12 = −32 p = − 2.

x2 – 2 x + 2.kw;;nwhU fhzpahfpwJ

jPHf;f x = 1 ± i vdNt%yk;q;fs; 3 ± i , 1 ± i .

gapw;rpf; fzf;Ffs; : 1) 1 + 2𝑖I xU jPh;thff; nfhΩl x4 − 4 x3 + 11x2 − 14x + 10 = 0vDk; rk∂ ghl;b∂ jPh;Tfisf; fhΩf. (June − 2009, 𝑀𝑎𝑟𝑐 − 2011) 2)2 − 𝑖I xU jPh;thff; nfhΩl 6x4 − 25x3 + 32x2 − 3x − 10 = 0 vDk; rk∂ ghl;b∂ jPh;Tfisf; fhΩf.

3) 2 + 3 i I xU jPh;thff; nfhΩl x4 – 4 x2 + 8x + 35 = 0 vDk; rk∂ ghl;b∂ jPh;Tfisf; fhΩf. 4) 1 + 𝑖I xU jPh;thff; nfhΩl x3 − 4 x2 + 6x − 4 = 0vDk; rk∂ ghl;b∂ jPh;Tfisf; fhΩf.

(June − 2007)

𝟓) − 𝟐 + 𝒊 𝟐 −v∂wfyg;ngΩzp∂kl;L tPr;R fhΩf.

jPh;T : − 2 + i 2 = r ( cos + i sin )vd;f.

r cos = − 2 r sin = 2

kl;L r = 2 kw;Wk; tPr;R = 3

4.

∴ − 2 + 𝑖 2 = 2 𝑐𝑜𝑠 3

4 + 𝑖 𝑠𝑖𝑛

3

4

gapw;rpf; fzf;Ffs; : kl;L tPr;R fhΩf. 1 1 + 𝑖 3 2 − 1 − 𝑖 3.

𝟔) ( 𝒂𝟏 + 𝒊 𝒃𝟏 ) ( 𝒂𝟐 + 𝒊 𝒃𝟐 ) ( 𝒂𝟑 + 𝒊 𝒃𝟑 ) . . . ( 𝒂𝒏 + 𝒊 𝒃𝒏 ) = 𝑨 + 𝒊 𝑩, vdpy; epWTf. (𝒊) ( 𝒂𝟏

𝟐 + 𝒃𝟏𝟐 ) ( 𝒂𝟐

𝟐 + 𝒃𝟐𝟐 ) ( 𝒂𝟑

𝟐 + 𝒃𝟑𝟐 ) . . . ( 𝒂𝒏

𝟐 + 𝒃𝒏𝟐 ) = 𝑨𝟐 + 𝑩𝟐 .

(𝒊𝒊) 𝒕𝒂𝒏−𝟏𝒃𝟏𝒂𝟏

+ 𝒕𝒂𝒏−𝟏𝒃𝟐𝒂𝟐

+ 𝒕𝒂𝒏−𝟏𝒃𝟑𝒂𝟑

+ . . . + 𝒕𝒂𝒏−𝟏𝒃𝒏𝒂𝒏

= 𝒌 𝝅 + 𝒕𝒂𝒏−𝟏𝑩

𝑨 , 𝒌 ∈ 𝒁.

jPu;T : (i) Given( 𝑎1 + 𝑖 𝑏1 ) ( 𝑎2 + 𝑖 𝑏2 ) ( 𝑎3 + 𝑖 𝑏3 ) . . . ( 𝑎𝑛 + 𝑖 𝑏𝑛 ) = 𝐴 + 𝑖 𝐵,

| ( 𝑎1 + 𝑖 𝑏1 ) ( 𝑎2 + 𝑖 𝑏2 ) ( 𝑎3 + 𝑖 𝑏3 ) . . . ( 𝑎𝑛 + 𝑖 𝑏𝑛 ) = | A + i B |,

( a1 + i b1 )( 𝑎2 + 𝑖 𝑏2 ) ( 𝑎3 + 𝑖 𝑏3 ) . . . ( 𝑎𝑛 + 𝑖 𝑏𝑛 )= A + i B

a12 + b1

2 a22 + b2

2 a32 + b3

2 . . . an2 + bn

2 = A2 + B2

tHf;fg;gLj;j( a12 + b1

2 ) ( a22 + b2

2 ) ( a32 + b3

2 ) . . . ( an2 + bn

2 ) = A2 + B2 .

(𝑖𝑖) 𝑎𝑟𝑔 { ( 𝑎1 + 𝑖 𝑏1 ) ( 𝑎2 + 𝑖 𝑏2 ) ( 𝑎3 + 𝑖 𝑏3 ) . . . ( 𝑎𝑛 + 𝑖 𝑏𝑛 ) } = 𝑎𝑟𝑔 ( 𝐴 + 𝑖 𝐵 ).

𝑎𝑟𝑔 (𝑎1 + 𝑖𝑏1) + 𝑎𝑟𝑔 𝑎2 + 𝑖𝑏2 + 𝑎𝑟𝑔 𝑎3 + 𝑖𝑏3 + . . . + 𝑎𝑟𝑔 𝑎𝑛 + 𝑖𝑏𝑛 = 𝑎𝑟𝑔 (𝐴 + 𝑖𝐵)

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nghJthftan−1𝑏1

𝑎1+ tan−1

𝑏2

𝑎2+ tan−1

𝑏3

𝑎3+ . . . + tan−1

𝑏𝑛

𝑎𝑛 = 𝑘 𝜋 + tan−1

𝐵

𝐴 , 𝑘 ∈ 𝑍.

𝟕) 𝐳 – 𝟏 ,∂tPr;R =

𝟔, kw;Wk; 𝐳 + 𝟏 ,∂tPr;R =

𝟐𝛑

𝟑, vdpy; 𝐳 = 𝟏.vd epWTf

jPu;T ∶ arg 𝑧 − 1 = 𝜋

6arg 𝑧 + 1 =

2𝜋

3

arg 𝑧 + 1 − arg 𝑧 − 1 =2𝜋

3−

𝜋

6=

𝜋

2

Let Z = x + i y

𝑎𝑟𝑔 ( 𝑥 + 𝑖 𝑦 + 1) – 𝑎𝑟𝑔 ( 𝑥 + 𝑖 𝑦 – 1 ) = 𝜋

2

arg 𝑥 + 1 + 𝑖𝑦 − arg 𝑥 − 1 + 𝑖𝑦 = 𝜋

2

tan−1𝑦

𝑥 + 1− tan−1

𝑦

𝑥 − 1=

𝜋

2

tan−1

𝑦

𝑥+1 −

𝑦

𝑥−1

1 + 𝑦

𝑥+1

𝑦

𝑥−1

= 𝜋

2

𝑦[ 𝑥−1 −(𝑥+1)]

𝑥2 − 1 𝑥+1 𝑥−1 +𝑦2

𝑥2−1

= tan𝜋

2

𝑦[ 𝑥 − 1 − (𝑥 + 1)]

𝑥 + 1 𝑥 − 1 + 𝑦2= ∞

𝑥2 − 1 + 𝑦2 = 0

𝑥2 + 𝑦2 = 1 𝑧 = 1.

gapw;rpf; fzf;Ffs; :𝑷 vDk; Gs;sp fyg;nΩ khwp Z - If; Fwpj;jhy; P-,∂ epakghijiaf; fhΩf.

𝑖 │𝑧 − 5𝑖│ = │ 𝑧 + 5 𝑖│ 𝑖𝑖 │𝑧 − 3𝑖│ = │𝑧 + 3 𝑖│ 𝑀𝑎𝑟𝑐 − 2009

𝑖𝑖𝑖 │2 𝑧 – 3 │ = 2 𝑖𝑣 2𝑧 − 1 = 𝑧 − 2 𝑀𝑎𝑟𝑐 − 2006(𝑣) 𝑅𝑒 𝑧 + 1

𝑧 − 𝑖 = 0. 𝑀𝑎𝑟𝑐 – 2008 )

𝟖) 𝒁𝟏 , 𝒁𝟐vd;w,U fyg;ngΩfSf;F

(i) z1 . z2 = z1z2 ii arg (z1 . z2) = arg z1 + arg z2vd epWTf. (𝑂𝑐𝑡. −2007, 𝐽𝑢𝑛𝑒 −

2008 )

jPu;T : 𝑧1= 𝑟 1 (𝑐𝑜𝑠 𝜃1 + 𝑖 𝑠𝑖𝑛 𝜃1), 𝑧 2 = 𝑟 2(𝑐𝑜𝑠 𝜃 2 + 𝑖 𝑠𝑖𝑛 𝜃2)

z1 = 𝑟 1, arg z1 = θ1 kw;Wk; z2 = 𝑟 2 , arg z2 = θ2

𝑧1 ∙ 𝑧2 = 𝑟 1 cos 𝜃1 + 𝑖 sin 𝜃1 ∙ 𝑟 2 cos 𝜃2 + 𝑖 sin 𝜃2

= 𝑟 1𝑟 2 cos 𝜃1 cos 𝜃2 + cos 𝜃1 𝑖 sin 𝜃2 + 𝑖𝑠𝑖𝑛 𝜃1 cos 𝜃2 + 𝑖𝑠𝑖𝑛 𝜃1𝑖𝑠𝑖𝑛 𝜃2

= 𝑟 1𝑟 2[ cos 𝜃1 cos 𝜃2 − sin 𝜃1 sin 𝜃2) + 𝑖 (𝑠𝑖𝑛 𝜃1 cos 𝜃2 + 𝑐𝑜𝑠 𝜃1𝑠𝑖𝑛 𝜃2 ]

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= 𝑟 1𝑟 2[ cos (𝜃1 + 𝜃2) + 𝑖 sin (𝜃1 + 𝜃2 ]

𝑧1 ∙ 𝑧2= 𝑟 1𝑟 2

= 𝑧1 . 𝑧2

Arg(𝑧1 ∙ 𝑧2) = 𝜃1 + 𝜃2 = arg 𝑧1 + arg 𝑧2

gapw;rpf; fzf;Ffs; : 𝑍1 , 𝑍2vd;w ,U fyg;ngΩfSf;F

𝑖 𝑧1

𝑧2 =

z1

z2 𝑖𝑖 arg

𝑧1

𝑧2= arg z1 − arg z2vd epWTf.

𝟗) 𝒏 −v∂gJ kpif KO vΩ vdpy ; 𝟏 + 𝒊 𝟑 𝒏

+ 𝟏 − 𝒊 𝟑 𝒏

= 𝟐𝒏+𝟏 𝒄𝒐𝒔 𝒏𝝅

𝟑. vd epWTf.

( 𝐽𝑢𝑛𝑒 – 2008 )

jPu;T ∶ 1 + i 3 = r (cos + i sin ) r = 2 and =π

3

1 + i 3 = 2 (cos π

3+ i sin

π

3)

1 + 𝑖 3 𝑛

=2𝑛(cos nπ

3 + i sin

nπ

3) − − − − − − (1)

1 − i 3 = 2 (cos π

3− i sin

π

3)

1 − 𝑖 3 𝑛

= 2𝑛(cos n nπ

3 − i sin

nπ

3) − − − − − − (2)

(1) + (2) 𝟏 + 𝒊 𝟑 𝒏

+ 𝟏 − 𝒊 𝟑 𝒏

= 𝟐𝒏+𝟏 𝒄𝒐𝒔 𝒏𝝅

𝟑.

gapw;rpf; fzf;Ffs; :n−v∂gJ kpif KO vΩ vdpy ; (𝑖)( 3 + 𝑖) 𝑛 + 3 − 𝑖 𝑛

= 2𝑛+1 𝑐𝑜𝑠 𝑛𝜋

6.

(𝑖𝑖)(1 + 𝑖)𝑛 + (1 − 𝑖)𝑛 = 2𝑛 +2

2 𝑐𝑜𝑠 𝑛/4. (Oct – 2007, March – 2008, 𝑀𝑎𝑟𝑐 − 2010)

iii (1 + cos 𝜃 + 𝑖 sin 𝜃)𝑛+ (1 + cos 𝜃 − 𝑖 sin 𝜃)𝑛 = 2𝑛+1 cos𝑛 𝜃

2 𝑐𝑜𝑠

𝑛𝜃

2 (𝐽𝑢𝑛𝑒 – 2012)

𝑖𝑣 1+sin 𝜃+𝑖 cos 𝜃

1+sin 𝜃−𝑖 cos 𝜃 𝑛

= cos 𝑛 𝜋

2− 𝜃 + 𝑖 sin 𝑛

𝜋

2− 𝜃 vd epWTf.(𝑀𝑎𝑟𝑐 − 2011)

𝟏𝟎) 𝒙 = 𝐜𝐨𝐬 𝜶 + 𝒊 𝐬𝐢𝐧𝜶𝒂𝒏𝒅 𝒚 = 𝐜𝐨𝐬 𝜷 + 𝒊 𝐬𝐢𝐧𝜷, vdpy;𝒙𝒎𝒚𝒏 + 𝟏

𝒙𝒎𝒚𝒏= 𝟐 𝐜𝐨𝐬 𝒎𝜶 + 𝒏𝜷 vd

ep&gp. (𝑀𝑎𝑟𝑐 − 2007) 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 ∶ 𝑥𝑚𝑦𝑛 = (cos 𝛼 + 𝑖 sin 𝛼)𝑚 (cos 𝛽 + 𝑖 sin 𝛽)𝑛

= (cos 𝑚𝛼 + 𝑖 sin 𝑚𝛼) (cos 𝑛𝛽 + 𝑖 sin 𝑛𝛽)

= cos(𝑚𝛼 + 𝑛𝛽) + 𝑖 sin(𝑚𝛼 + 𝑛𝛽) 1

𝑥𝑚𝑦𝑛= cos 𝑚𝛼 + 𝑛𝛽 − 𝑖 sin(𝑚𝛼 + 𝑛𝛽)

∴ 𝑥𝑚𝑦𝑛 + 1

𝑥𝑚𝑦𝑛= 2 cos 𝑚𝛼 + 𝑛𝛽

𝟏𝟏)Kf;Nfhzr; rkdpypia vOjp epWTf. (Oct – 2009, 𝐽𝑢𝑛𝑒 − 2010, 𝑀𝑎𝑟𝑐 − 2012) ,U fyg;ngΩfsp∂ $Ljyp∂ kl;L mt;tpU vΩfsp∂ kl;Lfsp∂ $LjYf;Ff; FiwthfNth

my;yJ rkkhfNth ,Uf;Fk; mjhtJ | 𝑧1 + 𝑧2| ≤ |𝑧1| + |𝑧2|.

jPu;T :MHf∂ jsj;jpy; 𝑧1 kw;Wk; 𝑧2 v∂w ,U fyg;ngΩfis A kw;Wk; B Gs;spfshy; Fwpf;f.

OACB v∂w ,izfuj;ij epiwT nra;f. ,q;F C v∂gJ 𝑧1 + 𝑧2 v∂w fyg;ngΩiz Fwpf;fpwJ

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𝑂𝐴 = | 𝑧1| , 𝑂𝐵 = | 𝑧2|. kw;Wk; 𝑂𝐶 = | 𝑧1 + 𝑧2 | xU Kf;Nfzj;jpy; VNjDk; ,U gf;f ePsq;fsp∂ $Ljy; %∂whtJ gf;f ePsj;ij tpl nghpaJ

∆𝑂𝐴𝐶apypUe;JOA + AC > 𝑂𝐶 𝑌 𝐶

my;yJ 𝑂𝐶 < 𝑂𝐴 + 𝑂𝐵

𝑧1 + 𝑧2 < 𝑧1 + 𝑧2 − − − (1)

NkYk; Gs;spfs; xU Nfhliktd vdpy; 𝐵 𝑧1 + 𝑧2 = 𝑧1 + 𝑧2 − − − (2) 𝐴

(1), (2) ,ypUe;J | 𝑧1 + 𝑧2| ≤ |𝑧1| + |𝑧2|.

𝑂 𝑋

𝟏𝟐) RUf;Ff: (𝒄𝒐𝒔 𝜶 + 𝒊 𝒔𝒊𝒏 𝜶 )3

( 𝒔𝒊𝒏 𝜷 + 𝒊 𝒄𝒐𝒔 𝜷)4

jPu;T : 𝒄𝒐𝒔 𝜶 + 𝒊 𝒔𝒊𝒏 𝜶 3

𝒔𝒊𝒏 𝜷 + 𝒊 𝒄𝒐𝒔 𝜷 4=

cos α+ i sin α 3

i cos β− i2 i sin β 4

= cos α + i sin α 3

𝑖 4 cos β − i sin β 4

= cos 3α + i sin 3α

cos 4β – i sin 4β = cos 3α + i sin 3α cos 4β + i sin 4β

= cos (3α + 4β) + i sin (3α + 4β)

gapw;rpf; fzf;Ffß :RUf;Ff (𝑐𝑜𝑠 𝜃 + 𝑖 𝑠𝑖𝑛 𝜃 )4

( 𝑠𝑖𝑛 𝜃 + 𝑖 𝑐𝑜𝑠 𝜃)5(𝑆𝑒𝑝. 2006, 𝑂𝑐𝑡. 2011)

𝟏𝟑) 𝐜𝐨𝐬 𝜶 + 𝐜𝐨𝐬𝜷 + 𝐜𝐨𝐬 𝜸 = 𝟎 = 𝐬𝐢𝐧 𝜶 + 𝐬𝐢𝐧𝜷 + 𝐬𝐢𝐧𝜸 vdpy; gp∂ tUgdtw;iw epWTf :

𝒊 𝐜𝐨𝐬 𝟑𝜶 + 𝐜𝐨𝐬𝟑𝜷 + 𝐜𝐨𝐬 𝟑𝜸 = 𝟑 𝐜𝐨𝐬 (𝜶 + 𝜷 + 𝜸) 𝒊𝒊 𝐬𝐢𝐧 𝟑𝜶 + 𝐬𝐢𝐧𝟑𝜷 + 𝐬𝐢𝐧𝟑𝜸 = 𝟑 𝐬𝐢𝐧(𝜶 + 𝜷 + 𝜸)

jPu;T : cos 𝛼 + cos 𝛽 + cos 𝛾 = 0 sin 𝛼 + sin 𝛽 + sin 𝛾 = 0

𝑎 = 𝑐𝑜𝑠 𝛼 + 𝑖 𝑠𝑖𝑛 𝛼 , 𝑏 = 𝑐𝑜𝑠 𝛽 + 𝑖 𝑠𝑖𝑛 𝛽 , 𝑐 = 𝑐𝑜𝑠 𝛾 + 𝑖 𝑠𝑖𝑛 𝛾v∂f

∴ 𝑎 + 𝑏 + 𝑐 = (cos 𝛼 + cos 𝛽 + cos 𝛾) + 𝑖 (sin 𝛼 + sin 𝛽 + sin 𝛾) = 0 𝑎 + 𝑏 + 𝑐 = 0 ⟹ 𝑎3 + 𝑏3 + 𝑐3 = 3𝑎𝑏𝑐

⟹ cos 𝛼 + 𝑖 sin 𝛼 3 + 𝑐𝑜𝑠 𝛽 + 𝑖 𝑠𝑖𝑛 𝛽 3 + cos 𝛾 + 𝑖 sin 𝛾 3

= 3 cos 𝛼 + 𝑖 sin 𝛼 𝑐𝑜𝑠 𝛽 + 𝑖 𝑠𝑖𝑛 𝛽 cos 𝛾 + 𝑖 sin 𝛾

𝑐𝑜𝑠 3𝛼 + 𝑖 𝑠𝑖𝑛 3𝛼) + 𝑐𝑜𝑠 3𝛽 + 𝑖 𝑠𝑖𝑛 3𝛽 + 𝑐𝑜𝑠 3𝛾 + 𝑖 𝑠𝑖𝑛 3𝛾 = 3 cos(𝛼 + 𝛽 + 𝛾) + 𝑖 sin(𝛼 + 𝛽 + 𝛾)

(cos3 𝛼 + cos 3𝛽 + cos 3𝛾) + 𝑖(sin 3𝛼 + sin 3𝛽 + sin 3𝛾) = 3 cos(𝛼 + 𝛽 + 𝛾) + 3𝑖 sin(𝛼 + 𝛽 + 𝛾) nka; kw;Wk; fw;gidg; gFjpfis xg;gpl>

cos 3𝛼 + cos 3𝛽 + cos 3𝛾 = 3 cos 𝛼 + 𝛽 + 𝛾 sin 3𝛼 + sin 3𝛽 + sin 3𝛾 = 3 sin(𝛼 + 𝛽 + 𝛾)

gapw;rpf; fzf;Ffß :cos 𝛼 + cos 𝛽 + cos 𝛾 = 0 = sin 𝛼 + sin 𝛽 + sin 𝛾 vdpy; gp∂ tUgdw;iw epWTf

: 𝑖 cos 2𝛼 + cos 2𝛽 + cos 2𝛾 = 0 𝑖𝑖 sin 2𝛼 + sin 2𝛽 + sin 2𝛾 = 0 (𝐽𝑢𝑛𝑒 – 2011)

𝑖𝑖𝑖 𝑐𝑜𝑠 2𝛼 + 𝑐𝑜𝑠2 𝛽 + 𝑐𝑜𝑠2𝛾 = 𝑠𝑖𝑛2𝛼 + 𝑠𝑖𝑛2𝛽 + 𝑠𝑖𝑛 2𝛾 =3

2

6-tifEΩfzpjk;: ga∂ghLfs;-II(gFjp-M-1-tpdh-6-kjpg;ngz;fs;)

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𝟏) 𝒖 = 𝒍𝒐𝒈 𝒕𝒂𝒏 𝒙 + 𝒕𝒂𝒏 𝒚 + 𝒕𝒂𝒏 𝒛 vdpy; 𝐬𝐢𝐧 𝟐𝐱 𝛛𝐮

𝛛𝐱= 𝟐. vd ep&gp.

(𝑀𝑎𝑟𝑐 − 2007, 𝐽𝑢𝑛𝑒 − 2008, 𝑂𝑐𝑡 − 2008) jPu;T : u = log ( tan x + tan y + tan z )

𝜕𝑢

𝜕𝑥=

sec2 𝑥

tan 𝑥 + tan 𝑦 + tan 𝑧

𝑆𝑖𝑛 2𝑥 𝜕𝑢

𝜕𝑥=

𝑠𝑖𝑛 2𝑥 𝑠𝑒𝑐2 𝑥

𝑡𝑎𝑛 𝑥 + 𝑡𝑎𝑛 𝑦 + 𝑡𝑎𝑛 𝑧 =

2 tan 𝑥

𝑡𝑎𝑛 𝑥 + 𝑡𝑎𝑛 𝑦 + 𝑡𝑎𝑛 𝑧

𝑆𝑖𝑛 2𝑦 𝜕𝑢

𝜕𝑦=

𝑠𝑖𝑛 2𝑦 𝑠𝑒𝑐2 𝑦


2 tan 𝑦


𝑆𝑖𝑛 2𝑧 𝜕𝑢

𝜕𝑧=

𝑠𝑖𝑛 2𝑧 𝑠𝑒𝑐2 𝑧


2 tan 𝑧


$l;l, sin 2x ∂u

∂x= 2.

𝟐) 𝒖 = (𝒙 – 𝒚 ) (𝒚 – 𝒛 ) ( 𝒛 − 𝒙 ) vdpy; 𝑢𝑥 + 𝑢𝑦 + 𝑢𝑧 = 𝟎.vdf; fhl;Lf. (𝑀𝑎𝑟𝑐 − 2006)

jPu;T :𝑢𝑥 = 𝑦 − 𝑧 𝑥 − 𝑦 −1 + 𝑧 − 𝑥 . 1 = (𝑦 − 𝑧) [−(𝑥 – 𝑦 ) + (𝑧 − 𝑥 ) ]

= (𝑦 − 𝑧)(𝑧 − 𝑥 ) − (𝑦 − 𝑧 ) (𝑥 − 𝑦 ) 𝑢𝑦 =

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