भाग 2

iz'ukoyh 7-1

1. 1_{cos 2}

2 x

− 2. 1sin 3

3 x 3.

2

1 2

x e

4. 1 ( )3

3a ax b+ 5.

3

1 4

cos 2

2 3

x

x e

− − 6. 4 3 C

3 x e + +x

7. 3

C 3

x x

− + 8.

3 2

C

3 2

ax bx cx

+ + + 9. 2 3 C

3 x x +e +

10. 2

log 2 C

2 x

x x

+ − + 11.

2

4

5 C

2 x

x x

+ + +

12.

7 3

2 2

2

2 8 C

7x + x + x+ 13.

3

C 3

x x + +

14.

3 5

2 2

2 2

C 3x −5x +

15.

7 5 3

2 2 2

6 4

2 C

7x +5x + x +

16. 2

3sin + x C

x − x e + 17.

3

3 2

2 10

3cos C

3x + x+ 3 x +

18. _{tan x + sec x + C} 19. _{tan x – x + C}

20. _{2 tan x – 3 sec x + C} 21. _C

22. _A

iz'ukoyh 7-2

1. _{log (1 + x}2_{) + C 2.} 1_{(log | |)}3 _C

3 x + 3. log 1+logx+C

4. _{cos (cos x) + C} 5. 1 cos 2( ) C

4a ax b

− + +

6.

3 2

2

( ) C

3a ax b+ + 7.

5 3

2 2

2 4

( 2) ( 2) C

5 x+ −3 x+ +

8.

3 2 2

1

(1 2 ) C

6 + x +

9.

3

2 ₂

4

( 1) C

3 x + +x +

10. 2log x− +1 C

11. 2 4 ( 8) C

12.

7 4

3 3 3 3

1 1

( 1) ( 1) C

7 x − +4 x − + 13. 3 2

1

C 18(2 3x )

− +

+

14.

1

(log ) C 1

m x

m

−

+

− 15.

2 1

8log |9 4 | Cx

− − + 16. 1 2 3 C

2 x e + +

17. 2

1 C 2ex

− + _18. 1

tan x _C

e − + 19. log (ex+e−x) + C

20. 1log( 2 2 ) C

2

x x

e +e− + 21. 1tan (2 3) C

2 x− − +x

22. 1_{tan (7 4 ) C}

4 x

− − + 23. 1(sin 1 )2 C

2 x

− ₊

24. 1log 2 sin 3cos C

2 x+ x+ 25.

1

C (1 tan )− x +

26. 2sin x+C 27.

3 2 1

(sin 2 ) C

3 x + 28. 2 1+sinx+C

29. 1(log sin )2 C

2 x + 30. – log 1+cosx+C 31. 1

C 1+cosx+

32. 1log cos sin C

2 2 x

x x

− + + 33. 1log cos sin C

2 2 x

x x

− − +

34. 2 tanx +C 35. 1(1 log )3 C

3 + x + 36.

3

1

( log ) C 3 x+ x +

37. 1cos (tan 1 4) C

4 x

−

− + 38. _D

39. _B

iz'ukoyh 7-3

1. 1sin (4 10) C 2 8

x

x

− + + 2. 1 cos 7 1cos C

14 x 2 x

− + +

3. 1 1 sin12 1sin 8 1sin 4 C

4 12 x x 8 x 4 x

 _{+ + +} ₊

 

4. 1_{cos (2 1)} 1_{cos (2}3 _{1) C}

2 x 6 x

− + + + + 5. 1cos6 1cos4 C

6 x−4 x+

6. 1 1cos6 1cos 4 1cos 2 C

4 6 x 4 x 2 x

 _{− −} ₊

 

 

7. 1 1sin 4 1 sin12 C

2 4 x 12 x

 ₋ ₊

 

  8. 2 tan2 C

x x − +

9. tan C

2 x

x− + 10. 3 1sin 2 1 sin 4 C

8 4 32

x

x x

− + +

11. 3 1_{sin 4} 1_{sin8 C}

8 8 64

x

x x

+ + + 12. _{x – sin x + C}

13. _{2 (sinx + x} cosα) + C 14.

1

C cos +sinx x

− +

15. 1sec 23 1sec 2 C

6 x−2 x+ 16.

3

1

tan tan C

3 x− x+ +x

17. _{sec x – cosec x + C} 18. _{tan x + C}

19. log tan 1tan2 C 2

x+ x+ 20. _{log cosx}+_sinx+_C

21.

2

C 2 2

x x

π

− + 22. 1 log cos ( ) C

sin ( ) cos ( ) x a

a b x b

− ₊

− −

23. _A 24. _B

iz'ukoyh 7-4

1. _tan−1_x3_{+ C 2.} 1_{log 2 1 4} 2 _C

2 x+ + x +

3. ₂

1

log C

2 x x 4x 5 +

− + − + 4.

–1

1 5

sin C

5 3

x +

5. 3 tan 1 2 2 C

2 2 x

− ₊

6.

3 3

1 1

log C

6 1

x x

+ ₊

7. x2− −1 logx+ x2− +1 C 8. 1log 3 6 6 C 3 x + x +a +

9. log tan + tanx 2x+ 4 +C 10. logx+ +1 x2+2x+ +2 C

11. 1tan 1 3 1 C

6 2

x

−  +  +

 

  12. –1

3

sin C

4 x+

 _{ +}

 

 

13. log –3 2 3 2 C

2

x + x − x+ + 14. sin–1 2 3 C

41 x−

 ₊

 

 

15. log – + ( ) ( ) C

2 a b

x + x−a x b− +

16. 2

2 2x +x− +3 C 17. x2− +1 2 logx+ x2− +1 C

18. 5log 3 2 2 1 11 tan 1 3 1 C

6 3 2 2

x x + x+ − − _ + _+

 

19. 6 2– 9 + 20 34 log 9 2 9 20 C

2

x x + x− + x − x+ +

20. – 4 – 2 4 sin 1 2 C

2 x x x + − _ − _+

 

21. x2+2 +3x +logx+ +1 x2+2x+ +3 C

22. 2

1 2 1 6

log 2 5 log C

2 6 1 6

x

x x

x − −

− − + +

− +

23. 5 x2+4 +10x −7 logx+ +2 x2 +4x+10 + C

24. _B 25. _B

iz'ukoyh 7-5

1.

2

( 2)

log C

1 x

x +

+

+ 2.

1 3

log C

6 3

x x −

+ +

4. 1log 1 2 log 2 3log 3 C

2 x− − x− +2 x− +

5. _{4log +2x} −_2logx+ +_{1 C} 6. log 3log 1 2 C

2 4

x

x x

+ − − +

7. 1log 1 1log ( 2 1) 1tan 1 C

2 x 4 x 2 x

−

− − + + +

8. 2log 1 1 C

9 2 3( 1)

x

x x

− _{− +}

+ − 9.

1 1 4

log C

2 1 1

x

x x

+ _{− +}

− −

10. 5log 1 1 log 1 12log 2 3 C 2 x+ −10 x− − 5 x+ +

11. 5log 1 5log 2 5log 2 C

3 x+ −2 x+ +6 x− +

12.

2 _{1 3}

log 1 log 1 C

2 2 2

x

x x

+ + + − +

13. _{– log x}−₁+1

2log (1 + x

2_{) + tan}–1_{x + C}

14. 3log – 2 5 C

2 x

x

− +

− 15.

1

1 1 1

log tan C

4 1 2

x

x x

−

−

− +

+

16. 1log C

1 n n

x

n x + + 17.

2 – sin

log C

1–sin x x +

18. + 2 tan 1 3tan 1 C

2

3 3

x x

x − − − + 19.

2 2

1 1

log C

2 3

x x

 + 

+

 ₊ 

 

20.

4 4

1 1

log C

4 x

x

− ₊

21. log – 1 C

x x e

e

 

+

 

 

22. _B 23. _A

iz'ukoyh 7-6

1. _{– x cos x + sin x + C} 2. cos3 1sin 3 C

3 9

x

x x

− + +

3. _ex _(x2 _{– 2x + 2) + C 4.}

2 2

log C

2 4

x x

5.

2 2

log 2 C

2 4

x x

x− + 6.

3 3

log C

3 9

x x

x− +

7.

2

2 1

1 1

(2 1) sin C

4 4

x x

x − − x+ − + 8_.

2

1 1 1

tan tan C

2 2 2

x

x x x

− _{− +} − ₊

9_.

–1

2 cos 2

(2 1) 1 C

4 4

x x

x − − −x +

10.

(

)

11. – _ 1–x2 cos–1x+x_+C

  12. x tan x + log cosx + C

13. tan 1 1log (1 2) C 2

x − x− +x + 14.

2 2 2

2

(log ) log C

2 2 4

x x x

x − x+ +

15.

3 3

log C

3 9

x x

x x x

 

+ − − +

 

  16. ex sin x + C

17. C

1+ x e

x+ 18. tan2 C

x x

e +

19. C

x

e

x + 20. 2

C ( 1)

x e x− +

21. 2

(2sin cos ) C 5

x e

x− x + 22. _{2x tan}–1_{x – log (1 + x}2_{) + C}

23. _A 24. _B

iz'ukoyh 7-7

1. 1 4 2 2sin 1 C

2 2

x

x −x + − + 2. 1sin 21 1 1 4 2 C

4 x 2x x

− _{+ − +}

3. ( +2) 2 4 6 log 2 2 4 6 C

2 x

x + x+ + x+ + x + x+ +

4. ( +2) 2 4 1 3log 2 2 4 1 C

2 2

x

x + x+ − x+ + x + x+ +

5. 5sin 1 2 2 1 4 2 C

2 5 2

x x

x x

−  +  + + _{− − +}

 

6. ( +2) 2 4 5 9log 2 2 4 5 C

2 2

x

x + x− − x+ + x + x− +

7. (2 3) 1 3 2 13sin 1 2 3 C

4 8 13

x x

x x −

−  − 

+ − + _{ }+

 

8. 2 +3 2 3 9log 3 2 3 C

4 8 2

x

x + x − x+ + x + x +

9. 2 9 3log 2 9 C

6 2

x

x + + x+ x + +

10. _A 11. _D

iz'ukoyh 7-8

1. 1( 2 2)

2 b −a 2.

35

2 3.

19 3

4. 27

2 5.

1 e

e

− 6.

8

15 2

e +

iz'ukoyh 7-9

1. ₂ 2. log3

2 3.

64 3

4. 1

2 5. 0 6. e

4 _{(e – 1)}

7. 1

2log 2 8.

2 1 log

2 3

 ₋ 

 ₋ 

  9.

π 2

10. π

4 11.

1 3

log

2 2 12.

π 4

13. 1

2log 2 14.

1

1 3

log 6 tan 5

5 5

−

+

15. 1

2(e – 1) 16.

5 5 3

5– 9 log log

2 4 2

 ₋ 

 

17.

4

2 1024 2

π π

+ + 18. ₀ 19. 3log 2 3

8 π +

20. _{1 +} 4 2 2

π− π 21. D 22. C

iz'ukoyh 7-10

1. 1log 2

2 2.

64

231 3.

π

2 – log 2

4. 16 2( 2 1)

15 + 5.

π

4 6.

1 21 5 17

log 4 17

+

7. π

8 8.

2 2

( 2) 4 e e −

9. _D

10. _B

iz'ukoyh 7-11

1. π

4 2.

π

4 3.

π

4 4.

π 4

5. ₂₉ 6. ₉ 7.

1 (n+1) (n+2)

8. π log 2

8 9.

16 2

15 10.

π 1

log

2 2 11.

π 2

12. π 13. ₀ 14. ₀ 15. ₀

16. – π_{log 2} 17.

2 a

18. ₅ 20. _C

21. _C

vè;k; 7 ij fofo/ iz'ukoyh

1.

2 2

1

log C

2 1

x x +

− 2.

3 3

2 2

2

( ) ( ) C

3(a b) x a x b

 

+ − + +

 

− _{ }

3. –2 (a x) C

a x

−

+ 4.

1 4 4

1

– 1+ C

x

 _{ +}

 

5.

1 1 1

3 6 6

2 x−3x +6x −6log (1+x ) C+

6. 1log 1 1log ( 2 9) 3tan 1 C

2 4 2 3

x

x x −

− + + + + +

7. _{sin log sin (a x a}− ₎+_xcosa+_C 8. 3

C 3

x

+

9. sin–1 sin C

2 x

 _{ +}

 

  10. −1₂sin 2x+C

11. 1 log cos ( ) C

sin ( – ) cos( ) x b

a b x a

+ +

+ 12.

1 4

1

sin ( ) C

4 x

− ₊

13. log 1+ C

2+ x x e e   +  

  14. 1 1

1 1

tan tan C

3 6 2

x x

− ₋ − ₊

15. 1cos4 C

4 x

− + 16. 1log( 4 1) C

4 x + +

17. +1 [ ( + )] C ( +1) n f ax b

a n + 18.

sin ( ) –2 C sin sin x x α α + + 19. 2 1

2(2 1) 2

sin C

π π

x x x

x x

−

− −

+ − +

20. _{–2 1– x +cos}−1 _{x + x−x}2 _+C

21. _ex _{tan x + C 22. 2log +1} 1 _{3log 2 C}

1

x x

x

− − + + +

+

23. 1 cos 1 1 2 C

2 x x x

−

 _{− −} ₊

 

  24.

3 2

2 2

1 1 1 2

– 1 log 1 C

3 x x 3

   ₊   ₊  _{− +}         _   _ 25. ₂ e π 26. 8 π 27. 6 π

28. 2sin 1( 3 1) 2

− −

29. 4 2

3 30.

1 log 9 40

31. π 1

2− 32.

π (π 2)

33. 19

2 40.

2

1 1

3 e e

 ₋ 

 

 

41. _A 42. _B

43. _D 44. _B

iz'ukoyh 8-1

1. 14

3 2. 16 4 2− 3.

32 8 2 3 −

4. 12π 5. 6π 6. π

3

7. 2

π 1 2 2 a  ₋ 

 

  8.

2 3

(4) 9.

1 3

10. 9

8 11. 8 3 12. A 13. B

iz'ukoyh 8-2

1. 2 9 sin 1 2 2

6 4 3

−

+ 2. 2π 3

3 2

 

−

 

 

 

3. 21

2 4. 4 5. 8

6. _B 7. _B

vè;k; 8 ij fofo/ iz'ukoyh

1. _(i) 7

3 (ii) 624.8

2. 1

6 3.

7

3 4. 9 5. 4

6. 2 3

8 3

a

m 7. 27 8.

3 (π 2)

9. (π 2) 4 ab ₋

10. 9

2 11. 2 12.

1 3

13. ₇ 14. 7

2 15.

1

9π 9 1 1

sin

8 4 3 3 2

−  

− _{ }+

 

16. _D 17. _C 18. _C 19. _B

iz'ukoyh 9-1

1.

dksfV

?kkr ifjHkkf"kr ugha

dksfV

?kkr

3.

dksfV

?kkr

dksfV

?kkr ifjHkkf"kr ugha

dksfV

?kkr

dksfV

?kkr

7.

dksfV

?kkr

dksfV

?kkr

dksfV

?kkr

dksfV

?kkr

11. _D 12. _A

iz'ukoyh 9-2

11. _D 12. _D

iz'ukoyh 9-3

1. _y″ = 0 2. _xyy″ + _{x (y}′)² – _{y y}′ = 0

3. _y″ – _y′– 6_{y = 0} 4. _y″ – 4_y′ + 4_{y = 0} 5. _y″ – 2_y′ + 2_{y = 0} 6. _2xyy′ + _x2 _{= y}2

7. _xy′ – 2_{y = 0} 8. _xyy″ + _x(y′)² – _yy′ = 0

9. _xyy″ + _x(y′)² – _yy′ = 0 10. _{(x² – 9) (y}′)² + _{x² = 0}

11. _B 12. _C

iz'ukoyh 9-4

1. 2 tan C

2 x

y= − +x 2. _{y = 2 sin (x + C)}

3. _{y = 1 + Ae}–x _{4. tan tanx y=C}

5. y = log (ex + e–x) + C 6.

3 –1

tan = + C

3 x

7. _{y = e}cx _{8. x}– 4 _{+ y}– 4 _{= C} 9. _{y = x sin}–1_{x +} 2

1– x + C 10. tan y = C ( 1 – ex)

11. 1log ( 1) (2 2 1)3 1tan–1 1

4 2

y= _ x+ x + _− x+

12.

2 2

1 1 1 3

log – log

2 2 4

x y

x

 − 

= _{ }

  13.

2

cos y a

x −

 _{ =}

 

 

14. _{y = sec x} 15. _{2y – 1 = e}x_{( sin x – cos x)}

16. _{y – x + 2 = log (x}2 _{(y + 2)}2_{) 17. y}2 _{– x}2 _{= 4}

18. _{(x + 4)}2 _{= y + 3 19.}

1 3

(63t+27)

20. _6.93% 21. _{Rs 1648}

22. 2log 2

11 log

10      

23. _A

iz'ukoyh 9-5

1. ( )2 C

y x

x y x e

−

− = 2. y=xlog x +Cx

3. _tan–1 1_log( 2 2_{) C}

2 y

x y

x  

= + +

 

  4. x

2 _{+ y}2 _{= Cx}

5.

1 2

log log C

2 2 2

x y

x

x y

+

= +

− 6. y+ x2+y2 =Cx2

7. _{xy cos} y

x = C

8. x 1 cos y C sin y

x x

 ₋  ₌  

   

 _{ } _{ }

 

9. _{cy = log} y ₁

x−

10. C

x y ye + =x

11. _{log ( x}2 _{+ y}2_{) + 2 tan}–1 y

x = π

log 2 2+

12. _{y + 2x = 3x}2_{y 13. cot} y _logex

x   =    

14. cos y log ex x

  =  

  15.

2

( 0, )

1 log

x

y x x e

x

= ≠ ≠

−

iz'ukoyh 9-6

1. _{y =} 1

5(2sin x – cos x) + C e

–2x _{2. y = e}–2x _{+ Ce}–3x

3.

4

C 4 x

xy= + 4. _{y(sec x + tan x) = sec x + tan x – x + C}

5. y = (tan x – 1) + C e–tanx 6.

2

2

(4 log 1) C 16

x

y= x− + x−

7. ylogx 2(1 log ) Cx x

−

= + + 8. _{y= (1+x}2_{) log sin}−1 _x +_C(1+_x2₎−1

9. 1 cot C

sin

y x

x x x

= − + 10. _{(x + y +1)= C e}y

11.

2 C

3 y

y

x= + 12. _{x= 3y}2 _{+ Cy}

13. _{y = cos x – 2 cos}2_{x 14. y (1 + x}2_{) = tan}–1 _{x –} 4

π

15. _{y = 4 sin}3_{x – 2 sin}2_{x 16. x + y + 1 = e}x

17. _{y = 4 – x – 2 e}x _{18. C 19. D}

vè;k; 9 ij fofo/ iz'ukoyh

1. _(i)

dksfV

?kkr

dksfV

?kkr

(iii)

dksfV

?kkr ifjHkkf"kr ugha

3.

2 2

2 4

y x y

xy −

′ = 5. _{(x + yy}′)² = (_{x – y)}2 _{(1 + (y}′)2₎

6. _sin–1_{y + sin}–1_{x = C 8. cos y =} sec 2

x

9. _tan–1 _{y + tan}–1_(ex_{) =} π

2 10. C

x y e = +y

11. _{log x–y} = + +_{x y 1} 12. _{y e}2 x=_{(2 x}+_C)

13.

2

2 π

sin 2 (sin 0)

2

y x= x − x ≠ 14.

2 1

log , 1

1

x

y x

x +

= ≠ −

15. ₃₁₂₅₀ 16. _C

17. _C 18. _C

iz'ukoyh 10-1

1.

layXu vko`Qfr esa] lfn'k

okafNr foLFkkiu dks fu:fir djrk gSA

2. _(i)

vfn'k

lfn'k

vfn'k

vfn'k

vfn'k

(vi)

lfn'k

3. _(i)

vfn'k

vfn'k

lfn'k

lfn'k

vfn'k

lfn'k

vkSj

r

lg&vfne gSaA

(ii)

lfn'k

r

vkSj

ur

leku gSA

(iii)

lfn'k

vkSj

lajs[k gS ijarq leku ugha gSaA

5. _(i)

lR;

vlR;

vlR;

vlR;

iz'ukoyh 10-2

1. a= 3, b = 62, c =1 r

r r

2.

laHkkfor mÙkjksa dh la[;k vuar gSA

laHkkfor mÙkjksa dh la[;k vuar gSA

4. _{x = 2, y = 3} 5. _{– 7}

vkSj

vkSj

6. ₋₄ˆ_{j k−}ˆ 7. 1 ˆ 1 ˆ 2 ˆ

6i+ 6 j+ 6k

8_. 1 ˆ 1 ˆ 1 ˆ

3i+ 3 j+ 3k 9.

1 _{ˆ 1 ˆ}

10. 40 ˆ 8 ˆ 16 ˆ

30i− 30 j+ 30k 12.

1 2 3 , , 14 14 14

13. 1, 2 2, 3

−

3 3

− 15. _(i) 1ˆ 4 ˆ 1 ˆ

3i 3 j 3k

− + + (ii) _{− +3i}ˆ _3kˆ

16. ₃ˆ_i+2ˆ_{j k+}ˆ 18. _(C) 19. _(C)

iz'ukoyh 10-3

1. π

4 2.

–1 5 cos

7

   

  3. 0

4. 60

114 6.

16 2 2 2 ,

3 7 3 7 7.

2 2

6a +11 . – 35a b b

r r

r r

8. a=1,b =1

r r

9. ₁₃ 10. ₈

12.

lfn'k

r

dksbZ Hkh lfn'k gks ldrk gSA

2

−

14.

dksbZ Hkh nks ½.ksrj vkSj ijLij yacor~ lfn'kksa

vkSj

r

dks yhft,

15. cos–1 10

102

 

 

  18. (D)

iz'ukoyh 10-4

1. _{19 2} 2. 2ˆ 2ˆ 1 ˆ

3i 3j 3k

± m m 3. π; 1, 1 ,1

3 2 2 2

5. 3,27

2 6.

;k

r r

;k

8.

ugha

dksbZ Hkh 'kwU;srj lajs[k lfn'kksa dks yhft,A

9. 61

2 10. 15 2 11. (B) 12. (C)

vè;k; 10 ij fofo/ iz'ukoyh

1. 3ˆ 1ˆ

2 i+2 j

2. 2 2 2

2– 1, 2 – 1, 2 1; ( 2 1) ( 2 1) ( 2 1)

3. 5ˆ 3 3 ˆ

2 i 2 j

− +

4.

ugha

r

vkSj

dks f=kHkqt dh rhuksa Hkqtkvksa dks fu:fir djrs gq, yhft,A

5. 1

3

± 6. 3 ₁₀ˆ 10 ˆ

2 i+ 2 j

7. 3 ˆ 3 ˆ 2 ˆ

22i− 22 j+ 22k

8. _{2 : 3} 9. ₃ar + _5b

r

10. 1(3 – 6ˆ ˆ 2 ); 11 5ˆ

7 i j+ k

12. 1(160 – 5 – 70 )ˆ ˆ ˆ

3 i j k 13. λ = 1 16. (B)

17. _(D) 18. _(C) 19. _(B)

iz'ukoyh 11-1

1. 0, 1 , 1 2 2

−

2. 1 , 1 , 1

3 3 3

± ± ± 3.

11 2 , 11

6 , 11

9 −

−

5. 2 , 2 , 3 ; 2 , 3 , 2 ; 4 , 5 , 1

17

17 17 17 17 17 42 42 42

− − − − − −

iz'ukoyh 11-2

4. rr= +iˆ 2 jˆ+3kˆ+ λ(3iˆ+2 ˆj−2 )kˆ

tgk¡

,d okLrfod la[;k gSA

5. _rr_=2iˆ_{− +}ˆ_{j 4k}ˆ_{+ λ(i}ˆ₊₂ ˆ_j−kˆ₎

vkSj dkrhZ; :i

1 4 2

1 1

2

− − = + =

− y z

x

gSA

6.

6 5 5

4 3

2 +

= − =

+ y z

x

7. _rr_=(5iˆ_{−4 j}ˆ_{+6 )k}ˆ _{+ λ(3i}ˆ₊₇ˆ_{j+2 )k}ˆ

8.

js[kk dk lfn'k lehdj.k %

js[kk dk dkrhZ; lehdj.k %

3 2 5

z y

x ₌

− =

9.

js[kk dk lfn'k lehdj.k %

js[kk dk dkrhZ; lehdj.k %

0 0 11

x− y+ z+

10. _(i) θ = cos 1 19 21

−  

 

  , (ii) θ =

1 8

cos

5 3

−  

 

 

 

11. _(i) θ = cos 1 26 9 38

−  

 

 

  (ii) θ =

1 2

cos 3 −  

   

12. 70

11

p= 14. 3 2

2

15. _{2 29}

16. 3

19 17. 29 8

iz'ukoyh 11-3

1. _{(a) 0, 0, 1; 2 (b)} 1 , 1 , 1 ; 1

3 3 3 3

(c) 2 , 3 , 1 ; 5

14 14 14 14

−

(d) 0, 1, 0; 5 8

2.

ˆ

ˆ ˆ

3 5 6

7 70

i j k

r⋅_ + − _ =

 

r

3. _{(a) x + y – z = 2 (b) 2x + 3y – 4 z = 1}

(c) (s – 2t) x + (3 – t) y + (2s + t) z = 15

4. _(a) 24 , 36, 48

29 29 29

 

 

  (b)

18 24

0, ,

25 25

 

 

 

(c) 

  

 

3 1 , 3 1 , 3 1

(d) 

  



 −

0 , 5

8 , 0

5. _{(a) [r − (i}ˆ_{− 2k}ˆ _{)] (⋅ + −i}ˆ ˆ_{j k}ˆ_{) =0;} x + y – z = 3

(b) _{[r − (i}ˆ_{+ 4 j}ˆ _{+6 k}ˆ_{) ] (⋅} ˆ_{i −2j}ˆ_+kˆ_{) =0;} x – 2y + z + 1 = 0

6. _(a)

fcanq lajs[k gSaA fn, x, fcanqvksa ls tkus okys ryksa dh la[;k vuar gksxhA

(b) 2x + 3y – 3z = 5

7.

2 5

, 5, –5 8. _{y = 3} 9. _{7x – 5y + 4z – 8 = 0}

10. _r ⋅

(

)

12. cos 1 15 731

−  

 

13. _{(a) cos} 1 2

5 −  

 

  (b)

ry vkil esa yacor~ gSaA

(c)

ry vkil esa lekarj gSaA

ry vkil esa lekarj gSaA

14. _(a)

13 3

(b) 13

3

(c) 3 (d) 2

vè;k; 11 ij fofo/ iz'ukoyh

3. _90° 4.

1 0 0

x ₌y ₌ z

5. ₀o

6. 10

7

k=− 7. _{r = +i}ˆ ₂ ˆ_j+3kˆ_{+ λ( i}ˆ_{+ 2 j}ˆ _{−5 )k}ˆ

8. _{x + y + z = a + b + c} 9. ₉

10. 0, 17 , 13

2 2

−

 

 

  11.

17 23

, 0,

3 3

 

 

  12. (1, – 2, 7)

13. _{7x – 8y + 3z + 25 = 0} 14. _{p =} 3

2

vFkok

6

15. _{y – 3z + 6 = 0} 16. _{x + 2y – 3z – 14 = 0}

17. _{33x + 4 5y + 50 z – 41 = 0} 18. ₁₃

19. _{r = +}ˆ_{i 2}ˆ_j+3kˆ _{+ λ −( 3i}ˆ₊₅ˆ_{j+4 )k}ˆ

20. _r = +_iˆ ₂ˆ_j−_4kˆ+ λ_(2iˆ+₃ˆ_j+_{6 )k}ˆ 22. _D 23. _B

iz'ukoyh 12-1

1. _{(0, 4)}

ij vf/dre

ij U;wure

19 19

 

 

 

ij vf/dre

235 19

4.

3 1 , 2 2

 

 

5. _{(4, 3)}

ij vf/dre

6. _{(6, 0)}

vkSj

dks feykus okyh js[kk [kaM ij fLFkr lHkh fcanqvksa ij U;wure

ij U;wure

(120, 0)

vkSj

dks feykus okyh js[kk [kaM ij fLFkr lHkh fcanqvksa ij vf/dre

8. _{(0, 50)}

vkSj

dks feykus okyh js[kk[kaM ij fLFkr lHkh fcanqvksa ij U;wure

(0, 200)

ij vf/dre

9. _Z

dk dksbZ vf/dre eku ugha gSA

10.

pw¡fd dksbZ lqlaxr {ks=k ugha gS vr%

dk vf/dre eku ugha gSA

iz'ukoyh 12-2

1. 8_{, 0} 3

     

vkSj

1 2,

2

   

 

dks feykus okyh js[kk [kaM osQ lHkh fcanqvksa ij U;wure ewY;

= Rs 160 .

2.

osQdksa dh vf/dre la[;k

,d izdkj dh rFkk 10 vU; izdkj dh gSaA

4 Vsful jSdV rFkk 12 fØosQV cYys

(ii)

vf/dre ykHk

4.

uV osQ rhu iSfdV rFkk oksYV osQ rhu iSfdV_ vf/dre ykHk

5.

30 iSfdV

izdkj osQ isap rFkk 20 iSfdV

izdkj dh isapks osQ rFkk vf/dre ykHk

= Rs 410

6.

4 vk/kj ySai vkSj 4 dkB dk <Ddu_ vf/dre ykHk

7. _A

izdkj osQ 8 Le`fr fpÉ rFkk

izdkj osQ 20 Le`fr fpÉ_ vf/dre ykHk

= Rs 160.

8. ₂₀₀

MsLdVkWi osQ uewus rFkk 50 iksVsZcy uewus_ vf/dre ykHk

9. _{Z = 4x + 6y}

dk U;wurehdj.k dhft, tcfd

vkSj

tgk¡

vkSj

Øe'k% HkksT;

vkSj

dh bdkbZ;ksa dks n'kkZrs gSa_ U;wure ewY;

10.

moZjd

vè;k; 12 ij fofo/ iz'ukoyh

1. ₄₀

iSfdV HkksT;

osQ vkSj 15 iSfdV HkksT;

osQ

foVkfeu

dh vf/dre ek=kk

bdkbZ

2. _P

izdkj osQ 3 FkSys vkSj

izdkj osQ 6 FkSys

feJ.k dk U;wure ewY;

feJ.k dk U;wure ewY;

(HkksT;

dk

rFkk HkksT;

dk

5.

izFke Js.kh osQ 40 fVdV rFkk lk/kj.k Js.kh osQ 160 fVdV_ vf/dre ykHk

ls

vkSj

bdkbZ;k¡

ls

vkSj

bdkbZ;k¡ Øe'k%

vkSj

dks Hksth tkrh

gS rFkk U;wure ewY;

7. _A

ls

vkSj

yhVj

ls

vkSj

yhVj rsy Øe'k%

vkSj

dks

Hksth tkrh gS rFkk U;wure ewY;

8. _P

izdkj osQ 40 FkSys vkSj

izdkj osQ

FkSys

ukbVªkstu dh U;wure ek=kk

izdkj osQ 140 FkSys vkSj

izdkj osQ

FkSys

ukbVªkstu dh vf/dre ek=kk

izdkj dh 800 xqfM+;k¡ vkSj

izdkj dh 400 xqfM+;k¡s

vf/dre ykHk

iz'ukoyh 13-1

1. _{P E|F}

( )

(

)

3, P F|E

=

3

= 2. P A|B

(

)

25

=

3. _{(i) 0.32 (ii) 0.64 (iii) 0.98}

4. 11 26

5. _(i) 4

11 (ii)

4

5 (iii)

2 3

6. _(i) 1

2 (ii)

3

7 (iii)

6 7

7. _{(i) 1 (ii) 0}

8. 1

6 9. 1 10. (a)

1 3, (b)

1 9

11. _(i) 1

2, 1

3 (ii)

1 2,

2

3 (iii)

3 4,

12. _(i) 1

2 (ii)

1

3 13.

5 9

14. 1

15 15. 0 16. C 17. D

iz'ukoyh 13-2

1. 3

25 2.

25

102 3.

44 91

4. _A

vkSj

ijLij Lora=k gSaA

vkSj

ijLij Lora=k ugha gSaA

vkSj

ijLij Lora=k ugha gSaA

7. _(i) 1

10

p= (ii) 1

5

p=

8. _{(i) 0.12 (ii) 0.58 (iii) 0.3 (iv) 0.4}

9. 3

8 10. A

vkSj

ijLij Lora=k ugha gSaA

11. _{(i) 0.18 (ii) 0.12 (iii) 0.72 (iv) 0.28}

12. 7

8 13. (i)

16 81, (ii)

20 81, (iii)

40 81

14. _(i) 2

3, (ii) 1

2 15. (i) , (ii) 16. (a)

1 5 , (b)

1 3, (c)

1 2

17. _D 18. _B

iz'ukoyh 13-3

1. 1

2 2.

2

3 3.

9

13 4.

12 13

5. 22

133 6.

4

9 7.

1

52 8.

1 4

9. 2

9 10.

8

11 11.

5

34 12.

11 50

iz'ukoyh 13-4

1. _{(ii), (iii)}

vkSj

gk¡

4. _{(i) X 0 1 2}

P(X) 1

4 1 2

1 4

(ii) X 0 1 2 3

P(X) 1

8 3 8

3 8

1 8

(iii) X 0 1 2 3 4

P(X) 1

16 1 4

3 8

1 4

1 16

5. _{(i) X 0 1 2}

P(X) 4

9 4 9

1 9

(ii) X 0 1

P(X) 25

36 11 36

6. _{X 0 1 2 3 4}

P(X) 256

625 256 625

96 625

16 625

1 625

7. _{X 0 1 2}

P(X) 9

16 6 16

1 16

8. _(i) 1

10

k= (ii) P(X 3) 3

10

< = (iii) P(X 6) 17

100

> =

(iv) P(0 X 3) 3

10

9. _(a) 1

6

k= (b) P(X 2) 1, P(X 2) 1, P(X 2) 1

2 2

< = ≤ = ≥ =

10. _1.5 11. 1

3 12.

14 3 13. _{Var(X) = 5.833, S.D = 2.415}

14. _{X 14 15 16 17 18 19 20 21}

P(X) 2

15 1 15

2 15

3 15

1 15

2 15

3 15

1 15

ekè;

vkSj

15. _{E(X) = 0.7}

vkSj

iz'ukoyh 13-5

1. _(i) 3

32 (ii)

7

64 (iii)

63 64

2. 25

216 3.

9

29 19

20 20

           

4. _(i) 1

1024 (ii)

45

512 (iii)

243 1024

5. _{(i) (0.95)}5 _{(ii) (0.95)}4 _{× 1.2 (iii) 1 – (0.95)}4 _{× 1.2}

(iv) 1 – (0.95)5

6.

4 9 10

   

  7.

20

20 20

12 13 20

1

20C C ... C

2

   + + + 

   

 

9. 11

243

10. _(a)

50 99 1

100

 

−_ _ (b)

49 1 99 2 100

 

 

  (c)

49 149 99 1

100 100

 

− _{ }

11.

5

7 5

12 6

   

  12.

4 35 5 18 6

   

  13.

3

11

22 9

10

×

vè;k; 13 ij fofo/ iz'ukoyh

1. _{(i) 1 (ii) 0}

2. _(i) 1

3 (ii)

1 2

3. 20

21 4.

10

10 10

7

1 C (0.9) (0.1)_r r r

r

−

=

−

∑

5. _(i)

6 2 5

      (ii)

4 2 7

5

      (iii)

6 2 1

5

  −  

  (iv)

864 3125

6.

10

9 5

2 6× 7.

625

23328 8.

2 7

9.

4 31 2

9 3

   

  10. n≥ 4 11.

91 54

−

12. 1 2 8, ,

15 5 15 13.

14

29 14.

3 16

15. _{(i) 0.5 (ii) 0.05} 16. 16

31

17. _A 18. _C 19. _B