Plus Two Maths Question Bank Tm

7/26/2019 Plus Two Maths Question Bank Tm

1/34

XII COME BOOK 3 marks reserved with Miss N. Mahalakshmi

PGT (Mathematics)

XII MATHS COME BOOK MODEL QPs3 MARK QUESTIONS (14)

2,2,2,2, YdPo CVtLRmYdPo CVtLRmYdPo CVtLRmYdPo CVtLRm

1. ,ar

,br

cr

GuT] JudLu Nej YdPoLs Gp a

r

br

c

r

= abc

G]d Lh, CRu URXm EiU G]m LhL, Ex. 2.5 (3)

2. 2 (3 4 ) 4r i j k + + =r r r

Gu\L[ju UVm. BWmLiL, Ex. 2.11 (5) (ii)

5, YL iLRm : TVuTLsYL iLRm : TVuTLsYL iLRm : TVuTLsYL iLRm : TVuTLs I1, 2 2 1 0y x+ + = Gu\ TWY[Vjt (-1. 1) Gu\ sp RLhu

NUuT LiL, OBQ

2, xy e= . xy e= Y[YWLd CPlThP LQjRd LiL, OBQ

3. f(x) = 2x3 5x2 4x + 3,1

2 x 3 Wu Rt\jRl TVuTj. cCu

UlL[d LidL, Eg. 5.21 (iii)

6, YL iLRm : TVuTLsYL iLRm : TVuTLsYL iLRm : TVuTLsYL iLRm : TVuTLs II1, YL dy LiL, Um LdLlThPxUtmdx-u UlLddy-u

UlL[ LQdL, y =x4 3x3+x 1,x = 2, dx = 0.1. Ex 6.1 (2) (ii)

2, u = sin cosx y

y xx y

e ey x

+ Gp.u u

x yx y

+

= 0 G]d LhL, Ex 6.3 (2) (ii)

8,8,8,8, YLdLf NUuTLsYLdLf NUuTLsYLdLf NUuTLsYLdLf NUuTLs1, odL: ( )2 6 9 0+ + =D D y Eg 8.23

9,9,9,9, RX LQdVpRX LQdVpRX LQdVpRX LQdVp1, uYm td Un AhPYQV AUdL : p ( q) Ex 9.2 (1)

2, uYm td Un AhPYQV AUdL :( p) ( q) Eg 9.4 (i)

3. uYm td Un AhPYQV AUdL : (( p) q) Eg 9.4 (ii)

4, uYm td Un AhPYQV AUdL : (p q) ( q) Ex 9.2 (8)

10,10,10,10, LrLrLrLrRLl TRLl TRLl TRLl TWYpWYpWYpWYp1, 5 VtLs[. J Dll TWYu NWN Utm TWYtTu Rp 4,8

Gp TWYXd LiL, Eg 10.20

2. J Dll TWYu NWN 6. Utm hP XdLm 3, Cdt UnVApX RY\? Y, Ex 10.3 (1)

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2/34

XII PUBLIC 19 QPs 3 marks reserved with Miss N. Mahalakshmi

PGT (Mathematics)

XII PUBLIC EXAM - 3 MARK QUESTIONS (51)

1,1,1,1, ALs Utm AdLYLuALs Utm AdLYLuALs Utm AdLYLuALs Utm AdLYLu TVuTLsTVuTLsTVuTLsTVuTLs

1.1 2

1 4

Gu\ Au SoU AVd LiL, Eg.1.5 (i)

2. A J fVUt\ LY AVu(AT) 1= (A 1) TGuTR L, Page 5 (2)


1. a,b

GuT] C AX YdPoLs Utm AYtu CPlThP LQmGp

1

2 2sin a b

= G] , Eg. 2.9

2, 2a i j k = + +rr rr

, 3 2b i j k = + r rr r

Gp ( ) ( )3 . 2a b a b+ r r r r

Id LiL,Ex. 2.1 (2)

3, HRm Ko YdPo

rr

d ( ) ( ) ( ). . .r r i i r j j r k k = + +r urr r ur r r ur r r

G] L, Eg. 2.6

4. 2i j k +rr r. 3 5i j k

rr r, 3 4 4i j k + +

rr rGu\ YdPoLs Ko NeLQ

dLQju TdLeL[L AUm G] , Eg. 2.11

5. 3 2i j k +rr r

, 3 5i j k +rr r

, 2 4i j k+ rr rGuTY J NeLQ dLQjR

EYdm G]d LhL, Ex. 2.1 (12)

6. 2 2i j k +rr rGm YdPd CQV]m GiQ[ 5 EPVU] N.

J L[(1, 2, 3) Gu\ sp Ck (5, 3, 7) Gu\ sd SLojUuAqN Nnm YXVd LQdL, Ex. 2.2 (6)

7. ,a br r

GuT] CWi YdPoLs Gp2 2 2 2

.a b a b a b + =r r r r r r

G] L,

Eg. 2.20

8. 13,a =r

5,b =r

Utm . 60,a b =urr

Gp a br r

Id LiL, Eg. 2.22

9. 4i j 3k +r r r

, 2i j 2k + r r r

Gm YdPoLd NejR]m Gi A[ 6

EPVU] YdPoL[d LiL, Eg. 2.21

10. 3 2 4i j k+ rr r

Gu\ YdPWp RWlTm NV] (1. 1. 2) Gu\ sp

NjRlT\, (2, -1, 3) Gu\ sVl Tj Nu lj\uLiL, Eg. 2.31

11. B(5, 2, 4) Gu\ s Yf NVpTm N 4 2+ +rr r

i j ku YdPo lj

\uA(3, 1, 3) Gu\ sVl Tj 2 8+ rr r

i j kG]d LhL,Ex. 2.4 (9)

12. 12 3 2 15. .i k j k i j k + + r r rr r r r

Gu\ YdPoL[ ]lsL[Ld LiP

CQLWjiUju L] A[ 546 Gp -Cu Ul LiL, Ex. 2.5 (2)

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3/34


PGT (Mathematics)

13. ,ar

,br

crGuT] JR[ AU YdPoLs Gp ,a b+

r r ,b c+

r r c a+

r rGuTYm

JR[ AU YdPoLs Bm, CRu URXm EiU GuTR]dLhL, Ex. 2.5 (1)

14. . 0,x a =r r

&& . 0,x b =

rr . 0x c =

r r Um 0x

rrGp ,a

r ,b

r c

rJW R[ AU

YdPoLs G]d LhL, Eg. 2.34

15, GkR J ar-dm ( ) ( ) ( ) 2i a i j a j k a k a + + =

r r r r r r r r r rG] , Ex. 2.5 (9)

16. 5 7 4 2( ) ( )r i j i j k = + + +rr r r rr

Utm 2 3 4( ) ( )r i k i k = + + +r rr rr

Gu\

LLu CPlThPd LQm LiL, Ex. 2.6 (9)

17.1 4

2 3 6

y z= + = x-1Utm

2 41

2 2

y zx

+ + = =

Gu\ LLu

CPlThPd LQm LiL, Ex. 2.6 (8)

18. (3, 2, 4), (9, 8, 10) Utm (, 4, 6) JW LhPUl sLs Gp -CuUl LiL, Eg. 2.47

19. 2x y + z = 4 Utmx + y + 2z = 4 Gu\ R[eLd CPlThP LQm LiL,Eg. 2.60

20. 2x + y - z = 9 Utmx + 2y + z = 7 Gu\ R[eLd CPlThP LQm LiL,Ex. 2.10 (1) (i)

21. ( ). 2 3 10r i j k + =r r r r

Utm ( ). 3 5r i j k + + =r r r r

Gu\ R[eLs Nej

GpLiL, Ex. 2.10 (3)

22.2

.(4 + 2 - 6 ) - 11 0r r i j k =

r r r r r

Gu\L[ju UVm. BWmLiL, Ex. 2.11 (5) (iv)

23. x2+ y2+ z2 3x 2y + 2z 15 = 0 Gu\ L[ju hPmAB UtmA-CuBVjRXLs( 1, 4, 3)GpB-Cu BVjRXL[d LiL,

Ex. 2.11 (4)

3, LXlTiLsLXlTiLsLXlTiLsLXlTiLs

1.1

1 i+u Un. LtT]l TL[d LiL, Ex. 3.1 (2)(i)

2, 3 1 = Gp.

5 5

1 i 3 1 i 31

2 2

+ + =

G] L, Ex. 3.5 (3)(ii)

5,5,5,5, YL iLRm : TVuTLsYL iLRm : TVuTLsYL iLRm : TVuTLsYL iLRm : TVuTLs I1. f(x) = 21 x , 1 x 1 Gu\ Nod WuRt\jRl TVuTj.cCu

Ul LiL, Eg. 5.21 (i)

2, f(x) = sin x, 0 x Gm Nod WuRt\jRf NTdL, Ex. 5.3 (1)(i)

3, uYtt Wu Rt\jRf NTdL: f(x) = x3 3x + 3; 0 x 1 Eg. 5.22 (i)

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4/34


PGT (Mathematics)

4. ex Gu\ NodUdXu LiL, Eg. 5. 28 (1)

5, Ul LiL: limx

2

x

x

e Eg. 5.32

6. R pexhPUL Hm No G] dL, Ex. 5.7 (1)

7, x3/5(4 x)u UX GiL[d LiL, Eg. 5.47

8. y = 2 x2 Gu\ Y[YWu ()-u NoTLjRd LiL, Eg. 5.59

7,7,7,7, RL iLRm : TVuTLsRL iLRm : TVuTLsRL iLRm : TVuTLsRL iLRm : TVuTLsUl LiL:

12

20

sin xdx

1 cos x

+ Eg 7.1 2.

/23x

0e cos xdx

Ex 7.1 (11)

31

20 4

dx

x Ex 7.1 (5) 4.

a2 2

0a x dx Eg 7.3

5.2 6

0sin x dx

Ex 7.3 (2) (i) 6.

/43 3

/4

x cos x dx

Ex 7.2 (2)

7.1

1

3 xlog dx

3 x

+ Eg 7.6

8,8,8,8, YLdLf NUuTLsYLdLf NUuTLsYLdLf NUuTLsYLdLf NUuTLs1, 2xy = e (A + Bx) Gu\ NUuThtL] YLdLf NUuThP AUdL,

Eg 8.2 (i)

2, odL: ( )2D + D +1 y = 0 Eg 8.24

9,9,9,9, RX LQdVpRX LQdVpRX LQdVpRX LQdVp1. (p q) ( q) Gu\ td Un AhPYQ AUdL, Eg 9.4 (iii)

2, p ( p) J WiT G] , Eg 9.9 (ii)

10,10,10,10, LrRLl TWYpLrRLl TWYpLrRLl TWYpLrRLl TWYp1, uYTY] LrRL APojf NoT G] NTodLm:

2x 9 0 x 3f(x)

0 \emUt

=

Ex 10.1 (5) (a)

2.11

F(x) tan x2

= +

, x < < GuT J NUYnl UXCu TWYp

No GpP(0 x 1) I LiL, Eg 10.7

3, J Dll TWYu NWN Utm TWYtTu jVNm 1 Bm, UmAYtu YodLeLu jVNm 11 Gp nCu Ul LiL, Eg 10.21

4, J RP RiRp TkRVjp J [Vh Wo 10 RPL[j RiPYim, JYo JqY RPVj RiYu LrRL 5/6 Gp. AYoCWitm \Y] RPL[ rjYu LrRL LiL, Ex 10.3 (6)

5, Tn^u TWYX TVuTj. LrRLu Rp Ju G] L, Eg 10.22

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5/34


PGT (Mathematics)

XII MATHS COME BOOK MODEL QPs

6 MARK QUESTIONS (30)

1,1,1,1, ALs Utm AdLYLuALs Utm AdLYLuALs Utm AdLYLuALs Utm AdLYLu TVuTLsTVuTLsTVuTLsTVuTLs

1, A =1 2

1 4

Gp. A (adj A) = (adj A) A = | A | . I2GuTRf NTodL, Eg. 1.3

2, SoU A LQp \p uYm SV NUuThj Rl]jodLm,7x + 3y = -1, 2x +y = 0. Ex. 1.2 (2)

3, uYm ANUTjR] NUuThj Rl] AdLY]lTVuTjj odL: x +y + 2z = 4, 2x + 2y + 4z = 8, 3x + 3y + 6z = 10. Eg. 1.18 (5)


1, Jo AWYhPjp Es[ LQm J NeLQm, CR] YdPo \pdL, Eg. 2.14

2, L[ju hPm. UtTWlp HRm J sp HtTjm LQmNeLQm G]d Lh, Ex. 2.11 (6)

3,3,3,3, LXlTiLsLXlTiLsLXlTiLsLXlTiLs1. z1, z2Gu\ HRm C LXlTiLd

(i) 11

2 2

zz

z z

= (ii) 1

2

argz

z

= arg z1 arg z2 G]d LhL, Page 128

2, 2i, 1 + i, 4 + 4i Utm 3 + 5i Gm LXlTiLs BoLu R[jp JNqYLjR EYdm G]d LhL, Eg. 3.14

3. (2+ 3 i) I J oYLd LiP x4 - 4x2+ 8x + 35 = 0 Gm NUuThPj o,Eg. 3.17

4, cos + cos + cos = 0 = sin + sin + sin Gpcos 3+ cos 3+ cos 3= 3 cos (+ + ) Utmsin 3+ sin 3+ sin 3= 3 sin (+ + ) G]d LhL, Ex. 3.4 (3) (i)(ii)

5. (1 + cos + i sin )n+ (1+cos i sin )n= 2n + 1cosn (/ 2) cos

2

nG] L,

Ex. 3.4 (4) (iii)

4,4,4,4, TTTT\\\\ YYdLRmYYdLRmYYdLRmYYdLRm1, ATWY[Vju HRm J sk ARu RXj

RLLu Nejj WeLu TdjRL J UGum

ARu Ul2 2

2 2

a b

a b+G]m LhL, Eg. 4.68

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6/34


PGT (Mathematics)

2, xy = c2 Gu\ NqYL ATWY[Vju HRm J sp YWVlTmRL x, y AfdLp Yhm iLs a, b G]m ClspNeLhu Yhm iLsp, q G]m Cluap + bq = 0 G]d LhL,

Eg. 4.70

5,5,5,5, YL iLRm : TVuTLsYL iLRm : TVuTLsYL iLRm : TVuTLsYL iLRm : TVuTLs I

1, J dLQju CWi TdLeLu [eLs \V 4. 5 Bm,Utm AYtt CPlThP LQ A[u Hm Rm ]d 0,06WVu Gp. XV] [eL[ EPV AkR TdLeLd CPVLQ A[/3BL Cdm T. ARu TWlp HtTm Ht\ Rm LiL, Ex 5. 1 (7)

2, J o NR]l Thk Gj. Ldm p YjR EPu JYlTU 19CCk 100 BL U\ 14 ]Ls Gjd Ls\,

CPp HRm J SWjp TRWNm NVL 8.5C/sec. Gu\ RjpH\ Gu LidL, Eg. 5.27

3. f(x) = 2x3+x220xGu\ Nou Hm Utm C\em CPYL[d LiL,

Eg. 5.38

4. xCu GkR Ultf(x) = 2x3 15x2 + 36x + 1 Gu\ No Hm Um GkRUlt C\em? Um GkRl sLp Nou Y[YWdYWVlTm RLLsxAfd CQVL Cdm? Eg. 5.42

5, LdLlThP CPYd fCu lT TU Utm f U

UlL[d LiL,, ( )1

xf x

x=

+ [1, 2] Ex. 5. 9 (2) (v)

6, YL iLRm : TVuTLs6, YL iLRm : TVuTLs6, YL iLRm : TVuTLs6, YL iLRm : TVuTLs II

1. x = r cos , y = r sin Gu CdUw = log (x2+ y2) G] YWVdLlT\

Gpw

r

Utm

w

Id LiL, Ex 6.3 (4)(i)

2. x = u + v, y = u v Gu CdUw = sin1(xy) G] YWVdLlT\ Gp.w

u

Utm

w

v

Id LiL, Ex 6.3 (4)(iii)

7,7,7,7, RL iLRm : TVuTLsRL iLRm : TVuTLsRL iLRm : TVuTLsRL iLRm : TVuTLs

1, UlL :1

4

0

xxe dx

Eg 7.17 (ii)

2. y2= 4ax Gu\ TWY[Vjtm ARu NqYLXjtm CPlThP TWl]dLiL, Ex 7.4 (6)

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7/34


PGT (Mathematics)

3,2 2

2 21

x y

a b+ = (a > b > 0) Gu\ sYhPm HtTjm TWl]. t\fNl

Tjf Zt]p HtTm PlTu L] A[ LiL, Eg 7.35

8,8,8,8, YLdLf NUuTLsYLdLf NUuTLsYLdLf NUuTLsYLdLf NUuTLs

1. x-Afu UVm Utm KWX BWm LiP YhPj RluYLdLf NUuThP AUdL, Ex 8.1 (4)

2, odL: x x 23e tany dx +(1+e )sec y dy = 0 Eg 8.4

3, odL: tandy y y

dx x x= + Eg 8.12

4, odL: ( )2 2 3 sin cosD D y x x = Ex 8.5 (8)

10,10,10,10, LrLrLrLrRLl TWYpRLl TWYpRLl TWYpRLl TWYp1, Su\Ld LXdLlThP 52 hdL[PeV hdLhk C hLs

mT Ydm \p GdLlTu\], Hv (ace) hLuGidLd NWNm. TWYtTm LiL, Ex 10.2 (4)

2, J RtNXp EtTjVm RrlTsLp 20% \PVYVLEs[], 10 RrlTsLs NUYnl \p GdLlTm T NVL 2RrlTsLs \PVYVL CdL (i) Dll TWYp (ii) Tn^u TWYpXUL LrRL LiL, [e2= 0.1353]. Ex 10.4 (3)

3, J Rl Fu TdL [Yp TdLlTYRtL] LrRL 0,005Bm, 1000 SToLd Rl F Tm T (i) ALThNm 1 SToTdLlTP ii) 4. 5 ApX 6 SToLs TdLlTP LrRL LiL, [e5= 0.0067]

Eg 10.24

4, To WoLu LXLu BhLXm CVpXl TWYX Jjd\,CkRl TWYu NWN 8 URULm. hPXdLm 2 URULm AU\,5000 N LXLs AdLlThP T. GjR] NL[ 12UReLds[L Ut\lTP YiU] GoTodLXm? Ex 10.5 (4)

************ NO SUBSTITUTE FOR HARD WORK ***********

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8/34


PGT (Mathematics)


1, ALs Utm AdLYLu TVuTLs

1. A =

4 3 3

1 0 1

4 4 3

Cu Nol AA G] L, Ex. 1.1 (8)

2. A =1 2

3 5

Gu\ Au NolTd Li. A (adj A) = (adj A) A = | A | . I

GuTRf NTodL, Ex. 1.1 (2)

3.

3 1 1

2 2 0

1 2 1

Gu\ Au SoU A LiL, Eg. 1.5 (iv)

(AB)1 = B1A1 GuTRf NTo,

4, A =1 2

1 1

UtmB =0 1

1 2

Eg. 1.6

5, A =5 2

7 3

UtmB =2 1

1 1

Ex. 1.1 (5)(i)

6. A =

1 2 2

4 3 4

4 4 5

Gp A = A1G]d LhL, Ex. 1.1 (10)

7.2 3

5 6A

=

Gp. ( ) ( )

11

TTA A

= GuTRf NTodL, OBQ

8. A, BCWi fVUt\ LY ALsGp, (AB)1 = B1A1 G] L,

SoU A LQp \p odL:9. x + y = 3, 2x + 3y = 8 Eg. 1.7

10. 2x y = 7, 3x 2y = 11 Ex. 1.2 (1)

Au RWm LiL,

11.1 2 3 12 4 6 2

3 6 9 3

Eg. 1.14 12.1 2 1 32 4 1 2

3 6 3 7

Ex 1.3 (5)

13.

1 2 3 4

2 4 1 3

1 2 7 6

Ex. 1.3 (6) 14.

3 1 5 1

1 2 1 5

1 5 7 2

Eg. 1.16

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9/34


PGT (Mathematics)

15.

3 1 2 0

1 0 1 0

2 1 3 0

Ex. 1.3 (3) 16.

0 1 2 1

2 3 0 1

1 1 1 0

Ex. 1.3 (4)

17.

2 1 3 4

0 1 1 2

1 3 4 7

OBQ

uYm ANUTjR] SV NUuTh Rl] AdLY \p odL:

18, 2x + 3y = 8 4x + 6y = 16 Eg 1.17 (2)19. 4x + 5y = 9 8x + 10y = 18 Ex 1.4 (3)20. 2x - 3y = 7 4x - 6y = 14 OBQ

21. 2x + 2y + z = 5 x y + z = 1 3x + y + 2z = 4 Eg 1.18 (3)

uYm NUuTLu Rlu JeLUj RuUVj RW \VlTVuTj BWnL,

22. x + y + z = 7 x + 2y + 3z = 18 y + 2z = 6 Ex. 1.5 (1)(iii)23. x 4y +7z = 14 3x + 8y 2z = 13 7x 8y + 26z = 5 Ex. 1.5 (1)(iv)

2, YdPo CVtLRm

1. K Nn NWju X hPeLs Ju\ Ju NejRL YhdLsm GuTR] YdPo \p L, Eg. 2.15

2, YdPo \psin sin sin

a b c

A B C= = Gu dL, Eg. 2.28

3. 4 3 2i j k

r r r

IX YdPWLd LiP s PCp NVpTm NLs2 7i j+

r r, 2 5 6i j k +

r r rUtm 2i j k +

r r rBm, CYLu [ Nu

lj \]6 3i j k+ r r r

-I X YdPWLd LiPQGu\ sVl

Tjd LiL, Ex. 2.4 (8)

4. [ a br r

b cr r

c ar r

] = 0G] L, OBQ

5. A(1, 2, 3), B(3, -1, 2), C(2, 3, 1), D(6, -4, 2) BV sLsJW R[jpAUm G]d LhL, OBQ

6 .GpX YdPoLs , ,

ur ur r

a b cdm.

2

, , =

r r r r r r r r r

a b b c c a a b c G] L, Eg. 2.38

7. (3, 4, 2) Gu\ sYf NpYm9 6 2i j k+ +r r r

Gu\ YdPdCQV]U] Lhu YdPo Utm LoVu NUuTL[d LiL,

Ex. 2.6 (6)

8. ( ) ( )2= + +r r r r rrr i j t i j k Utm ( ) ( )2 2= + + + +

r r r r r rrr i j k s i j k Gu\ CQ

LLu CPlThP WjRd LiL, Eg. 2.42

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10/34


PGT (Mathematics)

9. ( ) ( )3 5 7 2r i j k t i j k = + + + +r r r r r r r

Utm ( ) ( )s 7 6 7r i j k i j k = + + + + +r r r r r r r

GuT]

J R[jp AUVR LLs G]d LhL, Ex. 2.7 (2)

10. ( ) ( )2r i j t i k = + +r r r r r

Utm ( ) ( )2r i j s i j k = + + r r r r r r

Gu\ C LLs JW

R[ AUVd LLs G]d Lh. AYtt CPlThP WjRm LiL,Eg. 2.43

11. (3, 1, 1), (1, 0, 1) Utm(5, 2, 1) Gu\ sLs J LhPUl sLsG]d LhL, Eg. 2.46

12. ( ) ( )2 3 2r i j k t i j k = + + r r r r r r r

Gu\ Lm x 2y + 3z + 7 = 0 Gu\ R[m

Nkdu\ sVd LiL, Ex. 2.9 (4)

13. 2 3i j k +r r r

Gm X YdPW EPV sV UVULm 4 AXL[BWULm LiP L[ju YdPo Utm LoVu NUuTL[dLiL, Ex. 2.11 (1)

14, (1. 1. 1)I UVULm r (i j 2k 5 + + =r r r r

Gu\ L[ju BWjt NUU]

UlT BWjRd LiP L[ju YdPo Utm LoVuNUuTL[j RL, Ex. 2.11 (3)

15. (5. 5. 3) Gu\ s Yf NpYm (1. 2. 3) UVULm AUm L[juYdPo Utm LoVu NUuTL[d LiL, Eg. 2.63

16. 2 6 7i j k+ r r r

Utm 2 4 3i j k +r r r

Gm YdPoL[ X YdPoL[Ld

LiP sLs \V A, B,CR] CQdm sL[ hPULdLiP L[ju NUuT RL, Eg. 2.64

17. 2 6 7i j k+ r r r

Utm 2 4 3i j k + r r r

Gm X YdPoL[PV sLs

\V A, B Bm, CYt\ CQdm LhP hPULd LiPL[ju YdPo Utm LoVu NUuTL[d LiL. Um UVmUtm BWm LiL, Ex. 2.11 (2)

18,2

.(8 - 6 +10 ) - 50 0r r i j k =r r r r r

Gu\ YdPo NUuThPPV L[ju

UVjRm BWjRmLiL, Eg. 2.65

3, LXlTiLs

1, uYm NUuThP \ Nnm x Utm y-u Un UlL[dLiL, 2x + 3x + 8 + (x + 4)i = y(2 + i) . Ex. 3.1 (4) (iii)

2. ( 8 6i) -Cu YodL XeLs LiL, Ex. 3.2 (2)

3. ( 7 + 24i) -Cu YodL XeLs LiL, Eg. 3.16

4, LXlTiLu dLQ NUV G , Page 124

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11/34


PGT (Mathematics)

5. z1Utmz2Gu\ C LXlTiLd(i) | z1z2| = | z1|.| z2| (ii) arg (z1.z2) = arg z1+ arg z2 GuTR , Page 127

6, dL:( )

4

5

cos sin

(sin cos )

i

i

+

+ Eg. 3.19

P Gum s LXlTi Uz Id jRpP-Cu VUlTRV uYmTkR]d EhTh LiL,

7. 2 1 2z z = OBQ

8. 3 3z i z i = + OBQ

9. 3 5 3 1 = +z z OBQ

10.1

Re 0z

z i

+ =

OBQ

11. (7 + 9i), ( 3 + 7i), (3 + 3i) Gm LXlTiLs BoLu R[jp J NeLQdLQjR AUdm G] L, Eg. 3.15

12, LXlTi R[jp(10 + 8i), ( 2 + 4i), (-11 + 31i) Gm LXlTiLs AUdmdLQm J NeLQ dLQm G] L, Ex. 3.2 (4)

13. UnVi QLeL[d LiP P(x) = 0 Gu\ Tpld LYfNUuThu LXlTi XeLs CQVi CWhPVLjRu CPmTmG] L, Page 140

14. 3 + i I J oYLd LiPx4 8x3+ 24x2 32x + 20 = 0 Gm NUuThu \oL[d LiL, Ex. 3.3 (1)

15. 1 + 2i J XULd LiPx4 4x3+ 11x2 14x + 10 = 0 Gm NUuThuoL[d LiL, Ex. 3.3 (2)

16. (1+i) I J oYLd LiP x4 + 4 = 0 Gm NUuThu oL[d LiL,OBQ

17. (1 - i) I J oYLd LiP x3 - 4x2+6x 4= 0 Gm NUuThu oL[dLiL, OBQ

18. x = cos + i sin , y = cos + i sin Gp1m n

m nx y

x y+ = 2 cos (m+ n) G]

, Ex. 3.4 (9)

19. n GuT L Gi Gp1 sin cos

cos sin

1 sin cos 2 2

+ + = +

+

ni

n i n

i

G] dL, Eg. 3.20

20. cos + cos + cos = 0 = sin + sin + sin Gpcos 2+ cos 2+ cos 2= 0Utm sin 2+ sin 2+ sin 2= 0 G]d LhL, Ex. 3.4 (3)(iii)(iv)

21. n GuTL Gi Gp (1 + i)n+ (1 i)n=2

22n+

cos4

nG] ,

Ex. 3.4 (4) (i)

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12/34


PGT (Mathematics)

22. n N Gp(1 + i 3 )n+ (1 i 3 )n= 12n+ cos3

nG] L, Ex. 3.4 (4) (ii)

23,1

x + = 2cosx

Gp (i) nn

1x + = 2cosn

x(ii) n

n

1x = 2isinn

x Ex. 3.4 (7)

24, odL : x4+4 = 0. Ex. 3.5 (4) (i)

4, T\ YYdLRm

1, J CfNdLW YL]ju Ll [dp Es[ WTlTu JTWY[V AUlp Es[, ARu hPm 12 N,. BZm 4 N,Gp ARu Afp GqPjp Tp] (bulb) Tj]p Ll [dLf \kR \p JVj RWm G]d LQdL, Eg. 4.9

2, J sYhPju VeLs (2. 1). (2. 1) Utm NqYLXju Xm 6 GpARu NUuThPd LiL, Eg. 4.24

3, CVdYW2x + y 1 = 0, Vm(1, 2) Um UVj RXjRL 3 Gp

ATWY[Vju NUuThPd LiL, Eg. 4.36

4, UVm: (0. 0); AWd dLfu [m 6; e = 3, Um dLf. y-AfdCQVL Es[, CqTWeLdV ATWY[Vju NUuThPdLiL, Ex. 4.3 (1) (iii)

5, UVm (2. 1) Um J Vm (8. 1) G]m CRtLjR CVdYWx = 4G]m EPV ATWY[Vju NUuThPd LiL, Eg. 4.43

6, x2+ 2x 4y + 4 = 0Gu\ TWY[Vjt(0, 1) Gu\ sp RL.NeL CYtu NUuTLs LiL, Ex. 4.4 (1) (iii)

7,(1, 2)

k2x2 3y2 = 6

Gu\ ATWY[Vjt YWVjRdL CRLLu NUuTL[d LiL, Ex. 4.4 (4) (iii)

8. 3x2 5xy 2y2 + 17x + y + 14 = 0 Gu\ ATWY[Vju RXjRLLu RjRf NUuTL[d LiL, Eg. 4.64

9. 2x + 3y 8 = 0 Utm 3x 2y + 1 = 0 GuTYt\j RXj RLL[Lm.(5, 3) Gu\ s YVLf Npm ATWY[Vju NUuThPd LiL,

Ex. 4.5 (2) (i)

10, 4x2 5y2 16x + 10y + 31 = 0 Gu\ ATWY[Vju RXjRLLLd CPlThP LQjRd LiL, Ex. 4.5 (3) (iii)

11, 3x2 5xy 2y2 + 17x + y + 14 = 0 Gu\ ATWY[Vju RXj

RLLd CPlThP LQjRd LiL, Eg. 4.67

12, J hP NqYL ATWY[Vju ]Ls (5. 7) Utm (3.1) BLmClu. ARu NUuThPm. RXj RLLu NUuTL[mLiL, Ex. 4.6 (4)

13, NqYL ATWY[Vju HRm J sPj YWVlTm RL.RXj RLLPu AUdm dLQju TWl J U G]

L, Ex. 4.6 (7)

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13/34


PGT (Mathematics)

5, YL iLRm : TVuTLs I

1. KWX \PV J Lst ] SWjp HtTjm CPlTVofx = 3 cos (2t 4) Gp. 2 ]Lu p ARu dLm Utm ARu

CVdL Bt\p (K.E.) RVYt\d LiL, Ex. 5.1 (2)

2. y = x3

Gm Y[YWd (1,1) Gu\ sp YWVlTm RL.NeL BVYtu NUuTL[d LiL, Eg. 5. 10

3. y = x2 x 2 Gm Y[YWd(1, 2) Gu\ sp YWVlTm RL.NeL BVYtu NUuTL[d LiL, Eg. 5. 11

4. 2x2+ 4y2= 1 Utm 6x2 12y2= 1 Gm Y[YWLs Ju\ Ju NejRLYhd Lsm G]d LhL, Ex. 5.2 (8)

5. f(x) = x3Gu\ Not [2,2] Gu\ CPYp XdWgu CPUljRt\jR NTodLm, Eg. 5. 24

6. f(x) = x3 5x2 3x , [1,3] Gu\ NodXdWgu CPUlj Rt\jRNTodLm, Ex. 5.4 (1) (v)

7. f(x) = 2x3+ x2 x - 1 , [0, 2] Gu\ NodXdWgu CPUlj Rt\jRNTodLm, Ex. 5.4 (1) (iii)

8, UVm 2,00 Ud J tku YLU 30 UpLs/U G]m 2,10Ud YLU 50 UpLs/U G]m Lh\, 2,00 Udm 2,10Udm CPlThP HR J NUVjp dLm NVL 120 UpLs/U2BL Ckdm G]d LhL, Ex. 5. 4 (3)

9.1

1 x+Gu\ NodUdXu LiL, Ex. 5. 5 (3)

10. log e (1 + x) Gu\ NodUdXu LiL, Eg. 5. 28 (2)

11. tan x,2

< x b > 0) Gu\ sYhPm HtTjm TWl]. ShPfNl

Tjf Zt]p HtTm PlTu L] A[ LiL, Ex 7.4 (14)

8, YLdLf NUuTLs

1, odL:4

5( 4 )xx dy y x e dx= + Eg 8.8

2, odL: 2 2( )x y dy xy dx+ = Ex 8.3 (3)

3, odL: 2 2 2dy

x y xydx

= + ; x = 1 Gpy = 1 Ex 8.3 (4)

4, odL: cot 2cosdy

y x xdx

+ = Eg 8.17

5. odL: 2 tan x = sin xdy

ydx

+ Eg 8.21

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16/34


PGT (Mathematics)

6, odL:2 2 2

4 1

1 ( 1)

dy xy

dx x x+ =

+ + Ex 8.4 (2)

7. odL:dy2(1+ x ) + 2xy = cosxdx

Ex 8.4 (4)

8, odL: dy y xdx

+ = Ex 8.4 (1)

9 o:dy

xy xdx

+ = Ex 8.4 (6)

10, odL :1

x2 2(2D 5D 2)y e

+ + = Eg 8.28

11, odL: ( )2 4 13 cos3+ + =D D y x Eg 8.30

12, odL: ( )2 5 4 sin 5+ + =D D y x OBQ

13, odL: ( )2 3 2 + =D D y x Eg 8.32

14, odL: ( )2 714 49 4xD D y e+ + = + Ex 8.5 (3)

15, odL: ( )2 23 14 13 10x xD D y e e+ = + OBQ

9, RX LQdVp

1. ( ) ( )p q Gu\ td Un AhPYQ AUdL, Eg 9.4 (iv)

2. ( ) ( )p q r dV Un AhPYQV AUdL, Eg 9.6

3. ( ) ( )p q r u Un AhPYQV AUdL, Ex 9.2 (10)

4. ( ) ( )p q r dV Un AhPYQV AUdL, Eg 9.5

5. ( ) ( ) ( )p q p q G]d LhL, Eg 9.7

6, ( ) ( ) ( )p q p q G]d LhL, Ex 9.3 (5)

7. p q (( p) q) (( q) p) G]d LhL, Ex 9.3 (4)

8. p q (p q) (q p) G]d LhL, Ex 9.3 (3)

9. p qUtmq p NU]Ut\Y G]d LhL, Ex 9.3 (6)

10. [( p) ] p( )q J UnU G]d LhL, Eg 9.10 (i)

11. (( p) q) (p ( p)) J UnUV GuTR] Un AhPYQVdLi oUdL, OBQ

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17/34


PGT (Mathematics)

12. (p ( q)) (( p) q) Gu\ t UnUV. WiTP GuTR]oUdL, Ex 9.3 (1) (iii)

13. ( ) ( )p q p q GuT J UnU G]d LhL, Ex 9.3 (7)

14, (( q) p) q J WiT G]d LhL, Eg 9.10 (ii)

15, Un AhPYQVd Li uYm t UnUV ApX WiTPG]j oUdLm: (p ( p)) (( q) p) Ex 9.3 (1) (v)

16, YWVdLlThP hu T (Z5 {[0]}, .5) J Xm G] , OBQ

17, 1Cu 4Bm T XeLsJ Y] GVu XjR TdLu rAUdm G]d LhL, Eg 9.15

18,1 0

0 1

,1 0

0 1

,1 0

0 1

,

1 0

0 1

BV Su ALm APeV Eg 9.20

LQm AlTdLu r J GVu XjR AUdm G]d LhL,

19. 2 2 YN LiP fVUt\ LY ALs Vm Yt\ GVuApXR XjR A TdLu r AUdm G]d LhL, (CeAu ElLs VmRIf NokRY) Eg 9.19

20. (Z, +) J Yt\ GVu Xm G] L, Eg 9.12

21. (Z6, +6) Gu\ Xju JqY Elu YN]d LiL, OBQ

22. (Z7-{[0]} .7) Gu\ Xju JqY Elu YN]d LiL, OBQ

23, Xju dLp L[ G L, Page 181

24. GJ Xm GuL, a, b G Gp (a * b) 1= b1* a1G] , (ApX)ApX)ApX)ApX)Xjp GoU\u R] lRp ] G , Page 182

10,10,10,10, LrRLl TWYpLrRLl TWYpLrRLl TWYpLrRLl TWYp

1, u TLPL[ J \ m T 6 -Ls PlTRtL] LrRLlTWYXd LiL, Ex 10.1 (1)

2, J RjR NUYnl U X-u LrRLl TWYp (\fNo) ZLdLlThs[,

X 0 1 2 3 4 5 6 7 8

P(X = x) a 3a 5a 7a 9a 11a 13a 15a 17a

(i) a-u Ul LiL,(ii) P(x < 3)

(iii) P(3 < x < 7)CYt\d LiL,, Ex 10.1 (4)

3,3cx(1 x) ; 0 x 1

f(x)0 ;

< =

\ emUt

Gu\ LrRL APojf Nod NWN Utm TWYtT LiL, Ex 10.2 (7)(iii)

11, uYm LrRL APojf Not NWNVm. TWYtTVm LiL,3x3e ; 0 x

f(x)

0 ; \emUt

< < =

Eg 10.16

12, J Dll TWYu UXu NWN 2. hP XdLm2

3

Gp. LrRLf

NoTd LiL, Eg 10.17

13, J LsLXp 4 Ys[m 3 Yll TkLm Es[], mTYdU NUYnl \p u \ TkL[ Ju\u u Ju\LGdm T Pdm Yll TkLu GidLu LrRLlTWYXd LiL, Um NWN. TWYtT BVYt\d LiL, Eg 10.13

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19/34


PGT (Mathematics)

14, JW NUVjp 4 SQVeLs iPlTu\], (a) NVL 2 RXLs(b) \kRThNm 2 RXLs (c) ALThNm 2 RXLs PdL LrRLLiL, Ex 10.3 (4)

15, J _l TLPLs 10 \ EhPlTu\], C TLPLm JW GiLhYR Yt G]d LiPp (i) 4 YtLs (ii)fV Yt -

CYtu LrRL LiL,Eg 10.18

16, J lhP Rop. Rof Tt\YoLu NRRm 80 Bm, 6 SToLsRo G]p. \kRThNm 5 SToLs Rof T\ LrRL LiL,

Ex 10.3 (5)17, J Tnû TWYpP(X = 2) = P(X = 3) GpP(X =5) I LiL,

[e3= 0.050 G]d LdLlThs[]. Eg 10.25

18, J Tnû UXu NWN 4 Bm,(i) P(X 3) (ii) P(2 X < 5) LiL,[e4= 0.0183]. Ex 10.4 (1)

19, J LVdLl Tk Bp*T LsLs NWNVL 20 P LXCPYp 5 G] EZlT\, Tnû TWYXl TVuTj lhP

20 P CPYp (i) 2 EZpLs(ii) \kRThNm 2 EZpLdL]LrRLYd LiL,[e5= 0.0067]. Ex 10.4 (4)

20, J CVpXl TWYu LrRL APojf No ( )2

2x 4x 2f x k e

+ = , Gp

k, UtmCYt\d LiL, OBQ

21, AUdL LiPjp _h U]jp TVQm Nnm J STo LvdLVdLj]p TdLlTY J CVpX TWYXm, CRu NWN4.35m rem BLm. hP XdLm0.59 m rem BLm AUks[, J STo 5.20 mrem d Up Lvd LVdLj]p TdLlTYo GuTRt LrRLLiL, [P (0 < z < 1.44 = 0.4251] Ex 10.5 (3)

22, 300 UQYoLu EVWeLs CVpXl TWYX Jjd\, CRu NWN64,5 AeXeLs, Um hP XdLm 3,3 AeXeLs, GkR EVWjtdr 99% UQYoLu EVWm APedm? Ex 10.5 (6)

23, J Tsu 800 UQYoLdd LdLlThP \]nj RouUlTiLs CVpXl TWYX Jjd\, 10% UQYoLs 40UlTiLdd Zm. 10% UQYoLs 90 UlTiLd UmT\oLs, 40 UlTiLdm 90 UlTiLdm CPVUlTiLs Tt\ UQYoLu GidLVd LiL, Ex 10.5 (7)

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20/34


P.G. T. (Mathematics)

XII MATHS COME BOOK MODEL QPs

10 MARK QUESTIONS (16)


1, YdPo \pcos (A -B) = cosA cosB +sinA sin B G] L, Eg. 2.17

2, (-1. 3. 2)Gu\ s Yf NpYm x + 2y + 2z =5Utm 3x + y +2z = 8BVR[eLd NejR]U] R[ju YdPo Utm LoVuNUuTL[d LiL, Ex. 2.8 (9)

3,3,3,3, LXlTiLsLXlTiLsLXlTiLsLXlTiLs

1, P Gum s LXlTi Uz Id jRpP-Cu VUlTRV

1arg

1 3

z

z

= +

Gu\ TkR]d EhTh LiL, Eg. 3.11 (ii)

2. P Gum s LXlTi Uz Id jRpP-Cu VUlTRV

1arg

3 2

z

z

=

+ Gu\ TkR]d EhTh LiL, Ex. 3.2 (8) (v)

5,5,5,5, YL iLRm : TVuTLsYL iLRm : TVuTLsYL iLRm : TVuTLsYL iLRm : TVuTLs - I

1, J N ClTu Xm NjRlTm LeLp _pLs. Pjd 30

L,A Rm Uk Z LhPlTmT AY m YYjRdLd\, GkSWjm Admu hPm. EVWm NUULY

CdU]p. mu EVWm 10 AVL Cdm T EVWm Gu]Rjp EVo\ GuTRd LiL, Ex. 5.1 (9)

2, Ul LiL:0

limx +

sinxx Eg. 5.35

3, J YN NqYL YYU] YVd YP Yis[, AqYVuJ TdLjp B Ju SodLhp K\, AlTdLjt YRYpX, AYo 2400 Ad YP Ls[o, AqYLp TUTWlT[ LsU Es[ [. ALX A[Ls Gu]? Eg. 5.52

4, LVu Y[YW

2

xy e

= . GkR CPYLp . AP\GuTRm. Y[ Utl sL[m LiL, Eg. 5.64

6,6,6,6, YL iLRm : TVuTLsYL iLRm : TVuTLsYL iLRm : TVuTLsYL iLRm : TVuTLs II

1,2 2

x yu

y x= Gu\ Nod

2 2u u

x y y x

=

GuTR NTodL, Ex. 6.3 (1) (ii)

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21/34


P.G. T. (Mathematics)

7,7,7,7, RL iLRm : TVuTLsRL iLRm : TVuTLsRL iLRm : TVuTLsRL iLRm : TVuTLs

1, Y[YWy2= x Utmy = x 2 Gu\ Lh]p APTm TWl]d LiL,Eg 7.28

8,8,8,8, YLdLf NUuTLsYLdLf NUuTLsYLdLf NUuTLsYLdLf NUuTLs

1. 2secydx xdy e y dy+ = o LiL, Ex 8.4 (7)

2. ( )3 2 21 3 secdy

x x y xdx

= Gu\ YLdLf NUuTh]j odL: OBQ

3. o: ( )2 13 12 5 xD D y x e + = + OBQ

9,9,9,9, RX LQdVpRX LQdVpRX LQdVpRX LQdVp

1.2 2

2 2

0 01 0 0 10 0, , , , ,

0 1 1 00 0 0 0

Gu\ LQm Al

TdLu r J XjR AUdm G]d LhL, ( 3=1) Ex 9.4 (6)

2.x x

x x

,x R {0} Gu\ AUlp Es[ ALs Vm APeV LQm G

B] AlTdLu r J Xm G]d LhL, Eg 9.21

10,10,10,10, LrRLl TWYpLrRLl TWYpLrRLl TWYpLrRLl TWYp

1, S] tkLp TjRlTm NdLWeLk NUYnl \pRokRdLlTm NdLWju Lt\jRm CVpXl TWYX Jjd\,Lt\jR NWN31 psi. Um hP XdLm0.2 psi Gp

(i) (a) 30.5 psi dm31.5 psidm CPlThP Lt\jRm(b) 30 psi dm32 psi dm CPlThP Lt\jRm G] CdmTVLNdLWj] RokRdL LrRL LiL,

(ii) NUYnl \p RokRdLlTm NdLWju Lt\jRm 30.5 psid ALUL CdL LrRL LiL, Eg 10.32

************ NO SUBSTITUTE FOR HARD WORK ***********

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22/34


PGT (Mathematics)


1, ALs Utm AdLYLuALs Utm AdLYLuALs Utm AdLYLuALs Utm AdLYLu TVuTLsTVuTLsTVuTLsTVuTLs

1, SoU A LQp \p odL:

2x y + 3z = 9, x +y +z = 6, x y +z = 2 Eg. 1. 8

2, WUu lT odL:

1 2 11

x y z+ =

2 4 15

x y z+ + =

3 2 20

x y z = Ex 1. 4 (9)

3, J Tp , 1. Utm , 2. Utm ,5 SQVeLs Es[], Tn 100Ult UjRm 30 SQVeLs Es[], AqY\u JqY YLmEs[ SQVeLu GidLV LiL, Eg .1.19

4, J V LjRWe A\p 100 StLLs YlTRt TU]CPs[, u YqY\] \eLp StLLs YeL Yis[,(Ll. Xm Utm TfN), Ll YiQ StLu X ,240.

XYiQ StLu X .260. TfNYiQ StLu X .300,UjRm .25.000 Uls[ StLLs YeLlThP, AqY\u JqYYiQjm YeLjRdL StLLu GidLd \kRThNm uoL[d LiL, Ex. 1. 4 (10)

5, AdLY \ TVuTj x + 2y + z = 2 2x +4y +2z = 4 x 2y z = 0Gu\ NUuTLu Rl]j odL, OBQ

6, RW \]l TVuTj 2x + 5y + 7z = 52,x +y +z = 9, 2x +y z = 0 Gu\NUuTLu Rl JeLUY] G] j. o LiL, Eg. 1.22

7, RW \]l TVuTjx +y +z = 6, x + 2y + 3z = 14, x + 4y + 7z = 30 Gu\

NUuTLu Rl JeLUY] G] j. o LiL, Eg. 1.24

8, uYm NUuTj Rl JeLU EPVR GuTR BWnL, AqYJeLU EPVRu AR]j odLm (RW \Vl TVuTjRm):x +y z = 1 2x + 2y 2z = 2 3x 3y + 3z = 3 Ex. 1. 5 (1) (v)

9. -u GpX UlLdm uYm NUuThj Rlu oL[jRWj]l TVuTj BWnL,x +y +z = 2, 2x +y 2z = 2, x +y + 4z = 2

Ex. 1. 5 (2)

10. k-u GmUlLd uYm NUuThj Rl kx + y + z = 1, x + ky + z =1,x +y + kz = 1 (i) JW J o (ii) Jud UtThP o (iii) o CpXUTm? Ex. 1.5 (3)

11. , -Cu GmUlLd x + y + z = 6, x + 2y + 3z = 10, x + 2y + z = Gu\NUuTLs (i) VR om TtW (ii) JW J oY Ttdm (iii)GidLVt\ oL[l Ttdm GuTR] BWnL, Eg. 1.26

12. u GmUlt x + y + 3z = 0, 4x + 3y + z = 0, 2x + y + 2z = 0 Gu\Rlt (i) YlTPj o (ii) YlTPVt\ o Pdm?(RWj]l TVuTj) Eg. 1.28

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23/34


PGT (Mathematics)

2, YdPo CVtLRmYdPo CVtLRmYdPo CVtLRmYdPo CVtLRm

1, J dLQju jdLLs JW sp Nkdm GuTR]YdPo \p L, Eg. 2. 16

2, YdPo \pcos (A +B) = cosA cosB sinA sin B G] L, Ex. 2. 2 (4)

3. Sin (A B) = sinA cosB cosA sinB G] YdPo \p , Ex. 2. 4 (7)

4. YdPo \p Sin (A +B) = sinA cosB + cosA sinB G] L, Eg. 2. 29

5. 2 3a i j k = + rr rr

, 2 5b i k= +r rr

, 3c j k= rrr

, Gp ( ) ( . ) ( . )a b c a c b a b c = r r rr r r r r r

G] NTodL, Ex. 2. 5 (5)

6, a i j k = + +rr rr

, 2b i k= +r rr

, 2c i j k = + +rr rr

, 2d i j k = + +r rr r

Gp

( ) ( ) [ ] [ ]a b c d a b d c a b c d = r r r r r rr r r r r r

GuTRf NTodL, Ex. 2.5 (12)

7. 1 13 1 0

y z= = +

x - 1 Utm 4 12 0 3

x y z = = +uu uuuuuu u Gu\ LLs Yhd Lsm

G]d LhL, Um AY Yhm sVd LiL, Eg. 2. 44

8. 1

1 1 3

y z= + =

x -1 Utm 2 1 11 2 1

x y z = = uuuuuuuuu uuuu Gu\ LLs Yhd Lsm

G]d LhL, Um AY Yhm sVd LiL, Ex. 2. 7 (3)

9. (2, -1, -3) Gu\ s Yf NpYm- 2 = - 1 - 3

3 2 4

x y z=

uuuu uuuuuuuu Utm

- 1 = 1 - 2

2 3 2

+ =

uuuuuuuu uuuuuux y z Gu\ LLd CQV]U] R[ju

YdPo Utm LoVu NUuTL[d LiL, Eg. 2. 50

10, (-1. -2. 1)Gu\ s Yf NpYm x + 2y + 4z +7= 0Utm 2x - y + 3z +3= 0BV R[eLd NejRLm Es[ R[ju YdPo Utm LoVuNUuTL[d LiL, OBQ

11, (1, 2, -2) YVf NpXd Vm+ 2 = + 1 - 4

3 2 4

=

uuuuuuuuu uuuuux y zGu\ Lht

CQVLm 2x + 3y + 3z = 8Gu\ R[jt NejRLm Es[ R[juYdPo Utm LoVu NUuTL[d LiL, OBQ

12, (1. 2. 3) Utm (2. 3. 1) Gu\ sLs YVf NpXd Vm3x 2y + 4z 5 = 0Gu\ R[jtf NejRLm AUkR R[ju YdPoUtm LoVu NUuTL[d LiL, Ex. 2. 8 (11)

13, (1. 1. 1) Utm (1. 1. 1) BV sLs YVf NpXd Vmx + 2y + 2z = 5 Gu\ R[jt NejRL AUYU] R[ju YdPoUtm LoVu NUuThPd LiL, Eg. 2. 51

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24/34


PGT (Mathematics)

14. 2 2 12 3 2

x y z = =

uu u uu u u u u uu uu Gu\ LhP Es[PdVm ( 1. 1. 1) Gu\

s YVf NpXd VU] R[ju YdPo Utm LoVuNUuTL[d LiL, Ex. 2.8 (12)

15. (2, 2, 1), (3, 4, 2) Utm (7, 0, 6) BV sLs YVf NpXdV R[ju

YdPo Utm LoVu NUuThPd LiL, Eg. 2.52

16, 3 4 2 2 2.i j k i j k + + r rr r r r

Utm 7i k+rr

BVYt\ X YdPoL[Ld

LiP sLs YVf Npm R[ju YdPo Utm LoVuNUuTL[d LiL, Ex. 2. 8 (13)

17, Yhji Yp J R[ju NUuThP YdPo \mLoVu \m RdL, Ex. 2.8 (14)

3, LXlTiLsLXlTiLsLXlTiLsLXlTiLs

1. P Gm s LXlTi U zId jRp2 1

Im 21

z

iz

+ =

+ d P-Cu

VUlTRVd LiL, Ex. 3. 2 (8) (i)

2. a = cos2+ i sin 2, b = cos2+ i sin 2 Utmc = cos 2+ i sin 2Gp

(i)1

abcabc

+ = 2 cos (+ + )

(ii)2 2 2a b c

abc

+= 2 cos 2(+ ) G] , Ex. 3.4 (10)

3. , GuTYx2 2x + 2 = 0-u XeLs Utm cot =y + 1 Gp

( ) ( )

n ny y

+ +

uuuuuuuuuuuuuuuuuuuuuuuu = sinsinn

n

uuuuuuG]d LhL, Eg. 3. 22

4. x2 2px + (p2+ q2) = 0 Gu\ NUuThu XeLs . Utm tan = q/(y + p)

Gp( ) ( )

n ny y

+ +

uuuuuuuuuuuuuuuuuuuuuuuu = q n 1 sinsinn

n

uuuuuu G] L, Ex. 3. 4 (5)

5, x2 2x + 4 = 0-u XeLsUtmGp n n= i2n + 1sin n/3 Ak9 9-u UlT TL, Ex. 3. 4 (6)

6.2/ 3

( 3 )i+ u GpX UlL[m LiL, Eg. 3. 25

7. ( )

3/431

2 2i u GpX UlL[m LiL Utm ARu UlLuTdLtTXu 1 G]m LhL, Ex. 3. 5 (5)

8, odL: x4 x3+ x2 x + 1 = 0. Ex. 3. 5 4(ii)

9. x7+ x4+ x3+ 1 = 0 Gu\ NUuThPj odL, Eg. 3.24

10. x9+ x5 x4 1 = 0 Gu\ NUuThPj odL, Eg. 3.23

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25/34


PGT (Mathematics)

4, TTTT\\\\ YYdLRmYYdLRmYYdLRmYYdLRm

TWY[Vju Af. ]. Vm. CVdYWu NUuT. NqYLXjuNUuT. NqYLXju [m BVYt\d LiL, Um ARu Y[YWVYWL,1, y2- 8x+ 6y+ 9 = 0 Eg 4.7 (iv)

2. y2

- 8x2y + 17 = 0 OBQ3. y

2+ 8x6y+ 1 = 0 Ex 4.1 (2 iv)

4. y2+ 4y+ 4x + 8 = 0 OBQ

5. x2 6x 12y 3 = 0 Ex 4.1 (2 v)

6. 2x 4x + 4y 0 = OBQ

7, J Yp iu (comet)B] V]f (sun) t TWY[Vl TRpNp\, Utm Vu TWY[Vju Vjp AU\, Yp iuVk 80 pVu ,, RXp AUk Cdm T Ypi]m V]m CQdm L TRu AfPu /3LQj] HtTjU]p (i) Yp iu TRu NUuThPdLiL (ii) Yp iu Vd GqY[ Ap YWm GuTRm

LiL, (TR YX\m \lPVRL LsL),Eg. 4. 13

8, RWUhPjk 7,5 EVWjp RWd CQVL TjRlThP J

Zk YVm o RWVj Rm TR J TWY[VjRHtTj\, Um CkR TWY[Vl TRu ] Zu YpAU\, Zn UhPjt 2,5 Z u TnY] Zu ]YVLf Npm X jdLht 3 hPo Wjp Es[ GpjdLhk GqY[ Wjt AlTp W] RWp mGuTRd LiL, Eg. 4.12

9, J Re TXju Lm YPm TWY[V Ys[, ARu TWmPUhPUL WL TWs[, ARj Rem C iLd

CPVs[ Wm 1500 A, Lm YPjRj Rem sLs pRWk 200 A EVWjp AUks[], Um RWk LmYPju RrY] su EVWm 70 A. LmYPm 122 A EVWjpRem LmTjt CPV Es[ Nej [m LiL, Eg. 4. 14

10, J Re TXju Lm YPm TWY[V Ys[, ARu [m40 hPo Bm, YlTRV] Lm YPju rUhPl sk 5hPo Z Es[, Lm YPjRj Rem iLu EVWeLs 55 hPoGp. 30 hPo EVWjp Lm YPjt J Q Re RXLdLdLlThPp AjQjReu [jRd LiL, Ex. 4. 1 (5)

11, J WpY TXju Up Y[ TWY[Vju AUlTd Lis[,AkR Y[u ALXm 100 AVLm AqY[u Eflsu EVWmTXjk 10 AVLm Es[ Gp. TXju Ujk CPl\m

ApX YXl\m 10 A Wjp TXju Up Y[ GqY[ EVWjpCdm G]d LiL, Eg. 4.8

12, J WdLh YV] LjmT A J TWY[Vl TRpNp\, ARu EfN EVWm 4 -I GhmT A LjRlThP

CPjk PUhP Wm 6 RXs[, CVL PUhPUL12 RXp RWV YkRP\ Gp \lThP CPjp RWPuHtTjRlTm GLQm LiL, Eg. 4. 10

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26/34


PGT (Mathematics)

sYhPjt UVj RXj RL. UVm. VeLs. Utm EfLsBVYt\dLiL, Um ARu Y[YWVd LiL,

13. 36x2 + 4y272x+ 32y44 = 0 Eg 4.31 (iv)14. 16x

2+ 9y

2- 32x+ 36y- 92 = 0 OBQ

15. 9x2+25y2- 18x -100y -116 = 0 OBQ

16, J Y[ AW-sYhP YYjp Es[, ARu ALXm 48 A. EVWm 20A, RWk 10 A EVWjp Y[u ALXm Gu]? Eg. 4. 32

17, J TXju Y[Y] AW sYhPju Yp Es[, PUhPjpARu ALXm 40 AVLm UVjk ARu EVWm 16 AVLmEs[ Gp UVjk YX ApX CPl\jp 9 A WjpEs[ RWlsk TXju EVWm Gu]? Ex. 4.2 (10)

18, J Z Yu UtWV] AW-sYhP YYjp Es[, CRuALXm 20A UVjk ARu EVWm 18 A Utm TdLf YoLuEVWm 12 A Gp HRm J TdLf Yk 4 A Wjp

UtWu EVWm Gu]YL Cdm?Eg. 4. 33

19, Vu VjdU Uod WLU] V] J sYhPl

TRp t Y\, ARu AW ShPfu [m 36 pVu UpLsBLm UVj RXj RL 0.206 BLm CdUu (i) UodWLU] Vd L ALUp YmT Es[ Wm (ii) UodWLU] Vd Lj RXp CdmT Es[ WmBVYt\d LiL, Ex. 4.2 (9)

20, J sYhPl TRu Vjp CdU J QdLs tY\, CRu UVj RXj RL BLm dm QdLdm CPlThP f Wm 400 X hPoLs BLm

CdU]p dm QdLdm CPlThP ALThN Wm Gu]?

Ex. 4. 2 (8)21, J L-L [Vh Wo [Vhl TtuT AYdm L-L fLdm CPVs[ Wm GlTm 8 BL CdUEQo\o, Aq fLd CPlThP Wm 6 Gp AYo KmTRu NUuThPd LiL, Ex. 4.2 (7)

22, J NUR[ju Up NejRL AUks[ Yu 15 [s[ JHV] R[j]m Yt]m RU SLok Li Cd\Gp. Hu rUhP ]k 6 Wjp Hp AUks[ PGu\ su VUlTRVd LiL, Eg. 4. 35

ATWY[Vju UVj RXj RL. UVm. VeLs. EfLs BVYt\dLiL, Um ARu Y[YWV YWL,

23, 12x2 4y2- 24x + 32y - 127 = 0 OBQ24. 9x2 16y2- 18x - 64y - 199 = 0 Eg 4.56

25. x2 4y2+ 6x + 16y - 11 = 0 Ex 4.3 (5 iii)

26. x2 3y2+ 6x + 6y + 18 = 0 Ex 4.3 (5 iv)

27. 5x + 12y = 9 Gu\ SodL ATWY[Vm x2 9y2= 9 Ij R\ G]dL, Um Rm sVm LiL, Ex. 4.4 (5)

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27/34


PGT (Mathematics)

28, s (2. 0) YVLf Npm J ATWY[Vju UVm (2. 4) Bm,CRu RXj RLLs x + 2y 12 = 0 Utmx 2y + 8 = 0 Gu\LLd CQVL Clu. AqYTWY[Vju NUuThPd LiL,

Ex. 4. 5 (2) (ii)

29. x + 2y 5 = 0 I J RXj RLPLm. (6. 0) Utm (3. 0) Gu\sLs YV NpXdVU] NqYL ATWY[Vju NUuTLiL, Ex. 4. 6 (3)

5, YL iLRm : TVuTLsYL iLRm : TVuTLsYL iLRm : TVuTLsYL iLRm : TVuTLs I

1, J oXj RhV] RXZn YdLlThP J SoYhP muYp Es[, ARu BWm 2 hPo. ARu BZm 4 hPo Bm, Pjt2 L,hPo Rm Rhp o TnfNlT\, Rhp u BZm3hPWL Cdm T. o UhPju EVWm ALdm RjRd LiL,

Eg. 5. 9

2. J dLQju CWi TdL A[Ls \V 12. 15 Utm CYtuCPlThP LQju Hm Rm Pjt 2 Gp XV] [eLs

LiP TdLeLd CPlThP LQm 60BL Cdm T. ARuu\Y TdLju [m GqY[ WYL ALdm GuTRd LiL, Ex. 5. 1 (8)

3. 10 hPo [s[ J H NejR] Yp Nnj YdLlThs[,Hu AlTdLm Ytk Xf Npm Rm 1 /] Gp.Hu AlTdLm Ytk 6 RXp Cdm T. ARuEf GqY[ Rjp rSd C\em GuTRd LiL, Eg. 5. 7

4, J tk A B] Ud 50 ,, YLjp Utk Zd SdfNp\, Ut\ tk B B] Ud 60 ,, YLjp YPdSdf Np\, CY CWim NXLs Nkdm CPjR SdfNpu\], NXLs Nkdm ]k tk A B] 0,3 ,,

Wjm tk B B] 0,4 ,, Wjm CdmT Ju\ JuSem YL RjRd LQdL, Eg. 5. 8

5, SiTLp A Gu\ LlTp. B Gu\ LlTd Utl \UL 100 ,, WjpEs[, LlTpA B] Ud 35 ,, YLjp Zd Sdf Np\,LlTpBB] Ud 25 ,, YLjp YPd Sdf Npu\ Gp.UX 4.00 Ud CWi LlTpLdm CPlThP Wm GqY[ YLULUm GuTRd LiL, Ex. 5.1 (6)

6, J dLQju jVWm 1 N, / Pm Rjp ALdm T.ARu TWl 2 N,N,, / Pm Gm Rjp ALd\, jVWm 10N,, BLm TWl 100 N,N, BLm Cdm T dLQju

AlTdLm Gu] Rjp Um GuTRd LiL, Ex. 5. 1 (5)

7, x =acos3;y = a sin3Gm Q AX NUuTL[d LiP Y[YWdCp YWVlTm NeLhu NUuT x cos y sin = a cos 2G]dLhL, Ex. 5. 2 (10)

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28/34


PGT (Mathematics)

8. x = a cos4, y = a sin4, 0 /2 Gu\ Q AX NUuTL[d LiPY[YWd YWVlThP GkRY RLm HtTjm BV Afj

iLu Rp a G]d LhL, Eg. 5. 20

9. y = x3 Gu\ Y[YWu s[ J s P GuL, PCp YWVlThPRLP] Y[YWV UTm Q Cp NkdU]p. QCp

RLhu Nn.PCp Es[ NnYl Tp 4 UPe G]d LhL,Ex. 5. 2 (7)

10, y = x2 Utm y = (x 2)2 Gu\ Y[YWLs Yhd Lsm spAYLd CPlThP LQjRd LiL, Eg. 5. 17

11. .y2= x Utm xy = k Gm Y[YWLs Ju\Vu NejRL YhdLiPp.8k2= 1 G] dL, Ex. 5.2 (11)

12, Ul LiL:

2

lim x

cos(tan ) xx Ex. 5. 6 (11)

13, Ul LiL:0

limx

sin(cot ) xx Eg. 5. 34

14. 3 2( ) 2 3 36 10f x x x x= + + Gu\ Nou CPgNokR TU Utm UUlL[d LiL, OBQ

15, LdLlThP J TWlT[]d LiP NqYLeLs NWm UhUUf t\[ Ttdm G]d LhL, Ex. 5.10 (3)

16, LdLlThP J t\[]d LiP NqYLeLs NWm UhUTU TWlT[Yd Lidm G]d LhL, Ex. 5.10 (4)

17, J YWhu Up Utm Au KWeLs 6 N, UtU ARu TdLKWeLs 4 N,, Bm, AfYWhp AfNdLlThP YNLeLu TWl384 N,2 G] YWVdLlThPp ARu TWl U A[ LsU

Es[ [ ALXeL[d LiL, Eg. 5. 55

18, J hP NW AlTLm Lis[ (L]f NqYLju) ThuLs[[ 2000 L,N,. AlThu AlTLm Utm UpTLjtL]Xl ThLu X J N,N,d , 3 Utm ARu TdLeLdL]Xl ThLu X J NW N,,d , 1,50. Xl ThLuX U A[ LsU Es[ Thu [m. EVWm LiL, Eg. 5. 57

19. aAXBWs[ L[js TU A[ LsU LQlTm muLs[[. L[ju Ls[[u 8/ 27 UPe G]d LhL, Eg. 5. 56

20, 3,, ALXjp SWL Km Btu J LWp P Gu\ sp JYotu\o, AYo WhP Np. LWu GoTdLm 8 ,, RXs[

QY Sd YLULf Nu APV Yis[, AYo TPL SWLGojNRd Khf Nu Aek Qd KfNpXXm ApX QdSWL TPL Khf NpXXm ApXQ UtmRd CPVs[ Sd KhfNu Aek Qd Kf NpXXm AYo TP Khf Npm YLm 6,/U. Km YLm 8 ,/U Gp QY YLULf Nu\PV AYoTPL GeL LW NodL Yim? Eg. 5.58

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29/34


PGT (Mathematics)

21. r AX BWs[ AWYhPjs TU A[ LsU YWVlTmNqYLju TWld LiL, Eg. 5. 54

22, r BWs[ YhPjs YWVlTm Ll TV TWlT[ LiPNqYLju [ ALXeLs Gu]YL Cdm? Ex. 5.10 (5)

23. f(x) = x4 6x2

Gu\ No GkR CPYLp AP\ GuTRmUtm Y[ Utl sL[m LiL, Ex. 5. 11 (4)

24. y = 12x2 2x3 x4Gu\ No GkR CPYLp AP\ GuTRmUtm Y[ Utl sL[m LiL, Ex. 5. 11 (6)

6, YL iLRm : TVuTLsYL iLRm : TVuTLsYL iLRm : TVuTLsYL iLRm : TVuTLs II

1. y =x3Gu\ Y[YWV YWL, Ex 6. 2 (1)

2. y=x3+1 Gu\ Y[YWV YWL, Eg 6. 9

3. 2 32y x= Gu\ Y[YWV YWL, Eg 6. 10

4, 1tanx

uy

=

Gu\ Nod2 2u u

x y y x

=

GuTR NTodL, Ex. 6.3 (1) (iv)

5,2 2

1( )f x

x y=

+Gu\ Nod Xu Rt\jR NTodL, Eg 6. 20

6. 1sinx y

ux y

= +

Gp Xu Rt\jRl TVuTj1

tan2

u ux y u

x y

+ =

G]d LhL, Eg 6. 22

7, Xu Rt\jRl TVuTj.

3 3

1tanx y

ux y

+=

Gp. sin 2u ux y ux y

+ =

G] dL, Ex. 6.3 (5) (i)

7, RL iLRm : TVuTLsRL iLRm : TVuTLsRL iLRm : TVuTLsRL iLRm : TVuTLs

1. x2+ y2= 16 Gu\ YhPjtm y2= 6x Gu\ TWY[Vjtm TY] TWlTdLiL, Eg 7.29

2. y = x2 x 2 Gu\ Y[YW x = 2, x = 4 Gu\ LLs Utm x-AfBVYt\p APTm AWeLju TWlTd LiL, Eg 7.25

3. y = 3x2 x Gu\ Y[YW x-Af x = 1 Utm x = 1 Gu\ LL[pAPTm AWeLju TWl]d LiL, Ex 7.4 (4)

4. y = x2 2x 3 Gu\ Y[YW x = 3, x = 5 Gu\ LLs Utm x-AfBVYt\p APTm Tu TWl LiL, OBQ

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30/34


PGT (Mathematics)

5.2 2

19 5

x y+ = Gu\ sYhPjp Es[ CWi NqYLXjt CPlThP

TWl]d LiL, Ex 7.4 (7)

6. 3ay2=x(xa)2Gu\ Y[YWu Liu(loop) TWlTd LiL, Eg 7.33

7. 4y2

= 9x Utm3x2

= 16y Gu\ TWY[VeLd CPlThP TWl]d LiL,Ex 7.4 (9)

8. 2y x= Utm 2x y= BV C TWY[VeLd CPlThP TWlTd LiL,OBQ

9. y = x3Gu\ Y[YWdm y = x Gu\ Lhtm CPlThP AWeLjuTWlTd LiL, Eg 7.27

10. x = a (2t sin 2t), y = a (1 cos 2t) Gu\ YhP EsY[ (cycloid)u JY[tm. x-Aftm CPVs[ AWeLju TWlTd LiL,Eg 7.34

11, BWm

r, jWVm h EPV mu L]A[Yd Lm jWj]RL]l TVuTj LiL, Ex 7.4 (15)

12. y=0, x=4 Utm 3x-4y = 0 Gu\ NUuTL[ TdLeL[Ld LiPdLQju TWlT x-AfNl Tj ZtYRpHtTm PlTuL]A[ LiL, (ApX)ApX)ApX)ApX)

( )0, 0 , ( )4,0 Utm ( )4,3 BV ]L[d LiP dLQju TWlT[

x-Af ZtYRp HtTm Tu L] A]]d LiL, OBQ

13. 2x t= ;3

3

ty t= Gu\ QVX NUuTL[d LiP LiVp

ZlThP TWlu x-AfNl Tj Ztm T HtTjmY[YWs[ PlTu L] A[Yd LiL, OBQ

14. 4y2= x3Gu\ Y[YWp x = 0 k x = 1 YWs[ pu [jRdLiL, Eg 7.37

15, BWm a EPV YhPju t\[Y RLh \p LiL, Ex 7.5 (1)

16. x = a (t sin t), y = a (1 cos t) Gu\ Y[YWu [j] t = 0 Rp t = YW LQdL, Ex 7.5 (2)

17.

2 2

3 31

x y

a a

+ =

Gu\ Y[YWu [jRd LiL, Eg 7.38

18. y = sin x Gu\ Y[YW x = 0, x = Utm x-Af BVYt\p HtTmTWl] x-Af]l Tj Ztm T Pdm PlTu

Y[TWl2[ 2 + log (1 + 2 )] G] L, Eg 7.39

19. y2= 4ax Gu\ TWY[Vju ARu NqYLXm YWX] TWl] x-Afu ZtmT Pdm PlTu Y[TWlTd LiL, Ex 7.5 (3)

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31/34


PGT (Mathematics)

20, BWm r AXLs Es[ L[ju UVjk a Utm b AXLsRXp AUkR C CQV] R[eLs L[jR YhmT

CPlTm Tu Y[TWl 2r (b a) G] L, Ck L[juY[TWlT Y,(b > a). Ex 7.5 (4)

21. x = a (t + sin t), y = a (1 + cos t) Gu\ YhP Es Y[ (cycloid) ARu

AlTdLjRl (x-Af)

Tj ZtYRp HtTm Pl TuY[lTWlTd LiL, Eg 7.40

8, YLdLf NUuTLYLdLf NUuTLYLdLf NUuTLYLdLf NUuTLssss

1, odL: (x3+ 3xy2) dx + (y3+ 3x2y) dy = 0 Eg 8.14

2, o LiL: ( )2 2+ =

dyx y a

dx Eg 8.7

3, o LiL: ( )2

1dy

x ydx

+ = Ex 8.2 (7)

4, J lTl Tpld LYx = 1 Gm T TU Ul 4 BLmx = 1

Gm T U Ul 0 BLm Clu. AdLYVd LiL, Eg 8.10

5, odL: (x2+ y2) dx + 3xy dy = 0 Ex 8.3 (5)

6, odL : 2 2(1 ) 2 (1dy

x xy x xdx

+ = Eg 8.18

7, odL: (1 + y2) dx = (tan1 y x) dy Eg 8.19

8.3 2 2dy(1 + 2x ) + 6x y = cosec x

dx OBQ

9,, GkRY sm Nn y+2x G]d Li BYVLf Npm

Y[YWu NUuTy = 2(exx 1)

G]d LhL, Ex 8.4 (9)

10, odL:2

3

23 2 2 x

d y dyy e

dxdx + = ;Cex = log2 Gpy = 0 Utmx = 0 Gpy = 0.

Ex 8.5 (6)

11, odL: (D2 1) y = cos 2x 2 sin 2x Ex 8.5 (11)

12, odL: ( )2 35 6 sin 2 xD D y x e + = + OBQ

13, odL: ( )2 2 2 sin 2 5D D y x + = + OBQ

14, odL: (D2 6D + 9) y = x + e2x Ex 8.5 (10)

15, ioLu TdLjp. TdVu TdLRU] ApLQlTm TdVu GidLd RUL AUks[,

ClTdLjRp TdVu GidL 1 U SWjp mUPeL\Gp Ik U SW p TdVu GidL BWmT XVdLhm 35UPeLm G]d LhL, Eg 8.39

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32/34


PGT (Mathematics)

16, YlT X 15C Es[ J A\p YdLlThs[ Ru YlT X100C Bm, A 5 PeLp 60C BL \k \, Um 5 PmLj Ru YlT X]d LiL, Ex 8.6 (3)

17, J SLWjp Es[ UdLs RLu Y[ofRm AkSWjp Es[ UdLsRLd RUL AUks[, 1960Bm Bip UdLs RL 1,30,000

G]m 1990Cp UdLs RL1,60,000

BLm Clu 2020Bm BipUdLs RL GqY[YL Cdm? .42

16log .2070;e 1.52

13e

= =

Ex 8.6 (4)

18, WVm (Radium) Rm URU]. Ap LQlTm A[t RULAUks[, 50 YPeLp BWmT A[k 5 NRRmRkd\ Gp 100 YP p dm A[ Gu]? [A0 IBWmT A[ G]d LsL]. Ex 8.6 (1)

19, J CWNV] [p. J Ts Ut\m APm U RU] tSWjp Ut\UPVR AlTu A[t RUL Es[, J USW p 60 Wm Utm 4 U SW p 21 Wm RkRp.BWmT Xp. AlTu GP]d LiL, Eg 8.34

20, J YeV] RPo h Yh \p YhVd LQd\,ARY Yh RjR AkRkR SWjp ANu U RRjpLQd\, JYW Ye Clp RPofV] h Yh XmBiPud 8% Yh T\ Gp. AYW Yelu J YPLX ALlu NRRjRd LQdL,[e.081.0833 Gjd LsL,]Eg 8.35

21, J C\kRYo EPX UjYo TNdm T. C\kR SWjRRWULLQdP Yis[, C\kRYu EPu YlT X LX 10.00UV[p 93.4F G] jd Ls\o, Um 2 U SWm Lj YlT

X A[Y 91.4F G]d Li\o, A\u YlT X A[(XV]) 72F Gp. C\kR SWjRd LQdL,(J UR EPu

NRWQ ExQ X 98.6F G]d LsL).19.4 26.6

log 0.0426 2.303 log 0.0945 2.30321.4 21.4

e e

= =

Utm Eg 8.37

22, J LVdLl Ts Rm URU]. ARu GPd RULAUks[, ARu GP 10 ,Wm BL Cdm T Rm URm

S[ud 0.051 ,Wm Gp ARu GP 10 Wk 5 WULd\V Gjd Lsm LX A[Yd LiL,[loge2 = 0.6931] Ex 8.6 (5)

23, 1000 Gu\ RLd RPof h Yh LQdPlT\, YhRmBiPud 4 NRRUL Clu, AjRL GjR] BiLpBWmTj RLVl Tp C UPeLm?(loge2 = 0.6931). Ex 8.6 (2)

9, RX LQdVpRX LQdVpRX LQdVpRX LQdVp

1, fVUt\ LXlTiLu LQU] C {0} p YWVdLlThPf1 (z) = z,

f2 (z) = z, 31

( )f zz

= , 41

( )f zz

= z C {0}Gu\ NoLs Vm APeV

LQm {f1, f2, f3, f4} B] NoLu Nolu r J GVu Xm AUdmG] L, Eg 9.24

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PGT (Mathematics)

2, (Z7 {[0]}, .7) J XjR AUdm G]d LhL, Eg 9.26

3. 11-u Uhd LQlTt\ TdLur {[1], [3], [4], [5], [9]} Gu\ LQmJ GVu XjR AUdm G]d LhL, Ex 9.4 (9)

4. (Zn, +n)J Xm G]d LhL, Eg 9.25

5, YZdLU] TdLu r 1u nm T XeLs Y] GVu XjRAUdm G]d LhL, Eg 9.27

6.0

0 0

a

,a R {0} AUlp Es[ GpX ALm APeV LQm Al

TdLu r J GVu XjR AUdm G]d LhL, Ex 9.4 (11)

7. G GuT L R Gi LQm GuL, , a * b =3

ab,a b G GU

YWVdLlThP NV*u r J XjR AUdm G]dLhL, Ex 9.4 (5)

8. (Z, *) J Yt\ GVu Xm G]d LhL, Ce * B] a * b = a + b + 2GU YWVdLlThs[, Eg 9.18

9, 1 I RW Ut\ GpX R GiLm APeV LQmG GuL,G p* Ia * b = a + b ab, ,a b G GU YWVlTm, (G, *) J Yt\ GVuXm G]d LhL, Eg 9.23

10. 1 I RW Ut\ GpX R GiLm Es[PdV LQm G B] GpX,a b G a * b = a + b + ab GU YWVdLlThP NV *-Cu r J

GVu XjR AUdm G]d LhL, Ex 9.4 (8)

11. | z | = 1 GU Es[ LXlTiLs Vm APeV LQm M B]

LXlTiLu TdLu r J XjR AUdm G]d LhL,Ex 9.4 (7)

10,10,10,10, LrLrLrLrRLl TWYpRLl TWYpRLl TWYpRLl TWYp

1, J LsLXjp 4 Ys[ Utm 3 Yll TkLm Es[], 3TkL[ JqYu\L Gdm T. Yl \lTkLu GidLu

LrRLl TWYp (\fNo) LiL,(i) mT Ydm \p(ii) mT YdL \p Eg 10.3

2, J NUYnl UX-Cu LrRL \fNo TWYp uYU Es[ :

X 0 1 2 3 4 5 6

P(X = x) k 3k 5k 7k 9k 11k 13k

(1) k-Cu Ul LiL, (2) P(X < 4), P(X 5) P(3< X 6) Ul LiL,

(3) P (X x) >1

2BL CdLxCu f Ul LiL, Eg 10.2

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XII PUBLIC 19 QP 10 k d ith Mi N M h l k h i

3, J NUYnl UxCu LrRL APojf No

1 xkx e ; x, , 0f(x)

0 ; \emUt

>=

Gp(i) kCu Ul LiL (ii) P(X > 10) LiL, Ex 10.1 (7)

4, J SLWjp YPL Yi Kh]oL[p HtTm TjLu GidLTn^u TWYX Jjd\, CRu TiT[Y 3 Gp. 1000 KhSoLp(i) J YPjp J Tjm HtTPUp (ii) J YPjp uTjLd Up HtTjm Kh]oLu GidLVd LiL,[e3= 0.0498] Ex 10.4 (5)

5, CVpX U Xu NWN 6 Utm hP XdLm 5 Bm,(i) P(0 X 8) (ii) P( | X 6 | < 10)BVYt\d LiL,

P [0 < z < 1.2] = 0.3849 P [0 < z < 0.4] = 0.1554

P [0 < z < 1] = 0.3413 P [0 < z < 2] = 0.4772 Eg 10.29

6, J lhP Lpp 500 UQYoLu GPLs J CVpXl TWYXJjlTRLd Ls[lT\, Cru NWN 151 TiL[Lm hPXdLm 15 TiL[Lm Es[] Gp(i) GP120 Tidm 155 Tidm CPVs[ UQYoLs

(ii)GP185 Tid Up \s[ UQYoLu GidL LiL,

P [0 < z < 2.067] = 0.4803, P [0 < z < 0.2667] = 0.1026,

P [0 < z < 2.2667] = 0.4881 Ex 10.5 (5)

7, J Rop 1000 UQYoLu NWN UlTi 34 Utm hP XdLm 16Bm, UlTi CVpXl TWYX Ttlu (i) 30Ck 60UlTiLdPV UlTi Tt\ UQYoLu GidL (ii) UjV70% UQYoLs Tm UlTiLu GpXLs CYt\d LiL, Eg 10.30

8, CVpXl TWYu LrRL APojf No ( )22x 4xf x k e += , < x <

Gpk, Utm2Cu Ul LiL, Eg 10.31

9, J CVpXl TWYu LrRLl TWYp ( )2

3xf x c e += , < x < Gp.

c, , 2CYt\d LiL, Ex 10.5 (8)

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