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CLASS XIIMATHS

IMPORTANT QUESTIONS

HEAD OFFICE : B-1/30, MALVIYA NAGARPH. 26675331, 26675333, 26675334, 8527363750

CENTRES AT: H-36 B, KALKAJI PH. : 26228900, 40601840E-555, 1ST FLOOR, NEAR RAMPHAL CHOWK, SEC-7 DWARKA PH. 9560088728-29

BOOK–2

HEAD OFFICE : B-1/30, MALVIYA NAGAR PH. 26675331, 26675333, 26675334ALSO AT : H-36 B, KALKAJI PH. : 26228900, 40601840 AND E-555, 1ST FLOOR, NEAR RAMPHAL CHOWK, SEC-7 DWARKA PH. 9560088728-29 1

INDEFINITE INTEGRATION

1.2 2cos cos

cos cos

x

x

2.

6

10 4.

sin

sin sin

xdx

x x 3.

1. tan (sec tan ) x x dx 4. 4sin xdx

5. dxx4tan 6. dxxxx )sec(tantan21 7. dxxxx 3sin2sinsin

8. dxxxx 3tan2tantan 9.cos

sin cos3

2x dx

x x 10. sin

sin sin2

5 3x

x xdx 11.

2 33 182x

x xdx

.

12.dx

x x4 3 52 13. 3sin2sin

cos2 x

dxx14.

45

76xxdxx

15.

24

2xx

x dx

16. 2 5 2 3 2( )x x x dx 17. x

x xdx4 22 3 . 18.

dxxaxa

19. dxx

xa

20.1

3 2 2 cos.

xdx 21.

11 3 82 2 sin cos

.x x

dx 22. dx

x5 4 cos. 23.

dxx5 4 sin

.

24. xxdx

sincos2 25. xxdx

cossin 26. )3)(2)(1(

2

xxxdxx

27. )4)(1(5

2xxdxx

28.

)3()1()23(

2 xxdxx

29. )1)(1( 2 xxdx

30.dx

x x x1 2 3 . 31. 14xdx

32. 1 1 4

2

2.( )( )( )

x dxx x x . 33. 107 24 xx

dx 34.

x dxx x

2

3 2( )( ) 35. ( )x dxx x

2

2

15 6

36.

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

22

22

xxdxxx

37. )3)(1(2

22 xxx

38. dx

x xsin cos3 2 39. 3.

cos cosdxx x

40. dx

x x x6 7 22

log log 41.

dxx xn 1 42.

tan tantan

.

3

31d 43.

sinsin

.xx

dx4

44. dx

x x x x(sin cos )(cos sin ) 2 2 45. 2. sinx x 46. 2(log )x dx 47. 1. sinx x

48. 1. tanx x dx 49. 2 1. sinx x 50. x xdx3 1tan . 51. dxxsin 52. dxx1sin

53. dxx 21 )(sin 54. dxx3sec 55. dxxe x 2sin3 56. dxxx

2

1

12sin 57. tan

1 11

xx

dx .

58. tan

1

3

2

31 3

x xx 59. sin

1 x

a xdx . 60.

dxxxx

cos1sin

61. x x xdxcos sin3 .

62. cos log x dx 63.

xex

dxx

1 2 . 64. x ex

dxx

11 3 . 65. e

xx

dxx 11 2

2

.

66. e xx

dxx2

21

1

( ) 67. e x

xdxx 1

1

sincos 68. log(log )

(log )x

xdx

1

2 69. dxxxe x

cos1sin12/

70.

xxdxxx

11

11

cossincossin

, x [0, 1] 71. xdx

2cot1 72.

xdxxx

2sin169)cos(sin

73. 2)cos(sincos

xxdxx


74. dxxx tancot 75.3 23 2

sin coscos sin

x xx x

dx 76.

4

3/13 )(x

dxxx 77. 4/342 )1(xx

dx

78. xe

dx21

79. )(cos)(cos bxaxdx

80. dx

x a a bsin sin 81. xxdxx

44 cossin2sin

82. dx

x xsin cos4 4 83. xxdx

cossin 3 84. dx

x xsin cos3 5 85. 3sin sin

dx

x x .

86. 2

2

)cossin( xxxdxx

87. dxxsin

x2cos 88. dx1xeccos 89. dx

xx

)sin()sin(

90. xx

dx2

411

91.

dxx1 4 92. 124 xx

dx 93. tan xdx

94. dx

xxxx

113

24

2

95. 165 24 xxdx

96. dx

xx 1)3(1

97. .

1)4²(1 dx

xx

98. dx

1²x)1x(1

99. dx

x x2 2 1 100.

dxx x3 1 101.

dxx1x1

102. dxx

xxx

4

22 ]log2)1[log(1 103.

11

2

2.

x dxx

104. 3/12/1

2/1

xxdxx

105. )sin(sin

xx

dx = Ax + B log sin (x –) + C, find A and B.

106. Derive formula for i. 22 xadx

ii. 22 ax

dx iii.

22 xadx

iv. dxax 22

INTEGRATION ANSWERS

1. 2(sin x + x cos ) + C 2. Cxx |10sin|log101|4sin|log

41

3. Cxx

44

2

4. Cxxx

324sin2sin

41

183

5. Cxxx tan

3tan3

6. log | sec x + tan x | + log |sec x | + C 7. Cxxx

2cos

214cos

416cos

61

41

8. –1/3 log | cos 3x | + ½ log | cos 2x | + log + cos x | + C

9. log|sin | / log|cos |x x 1 2 2 C 10. Cxx

55sinlog

33sinlog

11. log | x2 + 3x –18 | –2/3 | log Cxx

63

12. Cx

1923sin

31 1

13. Cxxx |3sin2sin)1(sin|log 2 14. Cxxxxx 20929log342096 22


15. – Cxxx

22sin44 12 16. Cxxxxxx

1732sin

41732

232)32(

32 122/32

17. Cx

21tan

221 2

1 18. a sin–1 Cxa

ax

22 19. Caxxaxaaxx 22

2log

2

20. Cx

5tan3tan

151 1

21. Cx

3tan2tan

61 1 22. C

x

32

tantan

32 1

23. C

x

3

42

tan5tan

32 1

24. C

x

2

12

tantan2 1

25. 12

tan2

12

tan2log

21

x

x

+ C

26. Cxxx |3|log

49|2|log

34|1|log

121

27. Cxxx |2|log65|2|log

25|1|log

35

28. Cxx

x

)1(25

31log

411

29. Cxx

x

112

11log

41

30. Cxxx

1

2tan

21

1|1|log

21

31. Cxxx

1tan

21

11log

41

32. Cxxx

2tan

52

11log

101 1

33. Cxx

5tan

531

2tan

231 11

34. 9 log | x -3 | - 4 log | x- 2 | + x + C

35. x - 5 log | x-2 | + 10 log |x-3| + C 36. Cxxx

2tan3

3tan

32 11

37. Cxx

31log

21

2

2

38. Cxxx cos23log52cos1log

101cos1log

21

39. 1/4 [ cosec x –log | secx+ tanx |] + C 40. log | 2 log x + 1 | – log | 3 log x + 2 | + C

41. Cx

xn n

n

1

log1 42. –1/3 log | 1 + tan | + 1/6 log | tan2 – tan + 1| + 3

1 tan–1 C

3

1tan2

43. Cx

x

1tan22tanlog

51

44. Cxx

xx

sin21sin21log

241

sin1sin1log

81

45. Cxxxx 2cos

812sin

44

2

46. x (log x)2 –2 [x log x –x] + C

47. Cxxxx

41

4sin12 212

48. Cxxxx 11

2

tan21

2tan

2

49. Cxxxx 1/3221

3

)1(911

31sin

3 50.

Cxxxx

412tan

41 3

14

51. Cxxx sincos2 52. Cxxxx

21

21sin

212

53. x (sin–1 x)2 – 2 Cxxx 21 1.sin 54. ½ sec x tan x + ½ log | sec x + tan x | + C

55. CxexeIxx

13

2cos22sin3 33

56. 2x tan–1 x –log (1 + x2) + C


57. Cxxx 21 1cos21

58. 3 32

11 2x x x Ctan log ( )

59. (x + a) tan–1 Caxax

60. –cot Cx

2 61. Cxxxxx

44sin2sin23

321

4cos 4

62. Cxxx logsinlogcos

2 63. Cxe x

1 64.

Cx

ex

21 65. C

xe x

21

66. cxxe x

1)1(

67. Cxe x 2

cot 68. x log (log x) Cx

x

log 69. Cxex

2sec2/

70. Cxxxx

21 2sin)12(2 71. Cxxx

|2cos2sin|log

21

21

72. Cxxxx

cos(sin45)cos(sin45

log401

73. Cxx

xxec

)cos(sin2

14

cot4

coslog221

74. Cxx cossinsin2 1

75. Cxxx

sin2cos3log13

51312

76. Cx

3/4

2 1183

77. Cx

4/1

411

78. Cee xx 1log 2 79. C

bxax

ba

)(sin)(sinlog

)(sin1

80. Cbxax

ba

)(cos)(coslog

)(sin1

81. tan–1 (tan2x) + C 82. Cx

x

tan21tantan

21 2

1 83. C

x

tan

2

84. Cxx

2/32/1

3tantan2 85. –2cosec C

xx

sin)(sin

86. Cxxxxxx

cossincossin

87. Cxxxxxx

cos2coscos2coslog

212coscos2log2 88. Cxxx sinsin

21sinlog 2

89. –cos . sin–1 Cxxx

22 sinsinsinlogsin

coscos

90. Cx

x

21tan

21 2

1

91. Cxxxx

xx

1212log

241

21tan

221

2

221

92. Cxxxx

xx

11log

41

31tan

321

2

221

93. Cxxxx

xx

1tan2tan1tan2tan

log22

1tan2

1tantan2

1 1 94. tan–1 Cx

xx

312tan

31

31 2

12

95. cxxxx

xx

134134log

3161

34tan

381

2

221

96. Cxx

2121

log21

97. Cx

xx

11

tan21

3131

log34

1 1 98. C

xx

11

99. Cx

x

21

100. Cx )1(tan32 31 101.– Cxxxx 21cos1

102 Cxx

3211log11

31

2

2/3

2 103. Cxxxx

xx

22

2

1log21

21log

21

104. Cttttttt

)1(log6

354266

35426

where t = x1/6 105. A = cos ; B = sin


DEFINITE INTEGRATION

1. 10

2

tan

/ dxx

Ans. 4

2. 2 20

2

. [ logcos logsin ]/

x x dx

Ans. 2

log 2

3. log( tan )/

10

4

x dx

Ans. 8

log 2 4. 1

0

)1( dxxx nAns. )2)(1(

1 nn 5. 1 2

0

.sin

cosx x dx

x

Ans. 4

2

6.0

tansec tan

x xx x

Ans. 2

( –2) 7. 4 40

2 sin cossin cos

/ x x xx x

Ans. 16

2

8. 10

sinsin

x xx

Ans. ( –1) 9.

4/

02sin21

sec

xdxx

Ans.

4212log

31

10. dxx x3 20

2

sin cos

/

Ans. 4

11.xdx

a x b x2 2 2 20 cos sin

Ans. ab2

2 12. 1 1 2.

( )( )xdx

x xo

Ans. 4

13.

1

021

)1log( dxx

x Ans. Ans. 8

log 2

14. x dx 52

8. Ans. 9 15. 2 32

0

2

| |x x dx Ans. 4 16. |cos |x dx0

2

Ans. 4

17.

0

2cos1 dxx Ans. 22 18.

0

5

)( dxxf where f(x) = |x| + |x + 2| + |x + 5| Ans. 63/2

19.

2/

2/

||cos||sin

dxxx Ans. 4 20. x x dxsin ./

1

3 2Ans. 2

13

21. | cos |x x dx0

32

Ans.Ans. 21

25

22. 5 40 cosdx

x

Ans. 3

23. 9 16 20

4 sin cossin

/

x xx

Ans. 1/40 log 9 24.4

2 20

xa x

a

Ans. 16

3 4a

25.a xa x

dxa

a

Ans. a 26.1

1 3 23

2 cos( cos ) /

/

/ xdxx

Ans. 1 27.

cos

cos sin.

xx x

dx

2 2

24

2

Ans. 22

28. sin tan sin/

2 1

0

2

x xdx

Ans. 2

–1 29. 20

3

0

2 1

sin/

If xdx a xdx find xdxa

a

a

Ans. 9/2 or 1/2

30. 2/

4/

sinlog2cos

dxxx Ans. 41

842log

31. 1

0

21 )1(cot dxxx Ans. 2log2


32. log sin/

xdx0

2

Ans. – 2

log 2 33. 5.1

0

2 .][ dxx Ans. 22 34.

2/

022

2

sin4coscos

xxxdx

Ans. 6

35.

0

sincos1 xxdx

Ans.

sin)(

36.If f(x) = x

xx

cos2101cos2101cos

, then find value of 2/

0

.)(

dxxf Ans.–1/3.

37. Prove the property aa

dxxafdxxf00

)()( . Use the property to solve the following question.

Evaluate : i.

2/

03tan1

xdx

. Ans. /4. ii.

0

2tan1 x

dx

Ans. /2

38. Prove the property

a

a

a

xfxfifdxxf

xfxfif

dxxf

0

)()()(2

)()(0

)( .

Use the property to solve the following questions. i.

dxx5sin Ans. 0 ii.

2/

2/

2cos1

dxx Ans. 2

39. Prove the property b

a

b

a

dxxbafdxxf )()( . Use the p;roperty to solve the following question :

Evaluate :

3

2 5 xxdxx

. Ans. 1/2

40. Show that 2/

0

4/

0

cos)2(cos2sin)2(sin

xdxxfxdxxf .

41. Sketch the graph of

222,22

)( 2 xxxx

xf . Evaluate : 4

0

.)( dxxf What does the value of this integral represent on the

graph? Ans. 62/3. This value represents area bounded by f(x).

42. Evaluate the following integrals as limit of sums.

i. 0

2

e dxx ii. dxxe 2

013

iii. ( )2 52

1

3

x x dx


AREA UNDER THE CURVE

1. Sketch the graph of y x | |1 .Evaluate | |x dx 14

2

.What does this value represent on the graph? Ans. 9

2. Find the area of the region bounded by y = 1+ |x + 1|, x = –3, x = 3 and x-axis. Ans. 16

3. Using the concept of integration find the area of the region bounded by the triangle ABC ,coordinates of the verticesbeing A ( 1,6 ) , B ( 2,8) & C ( 3,4). Ans. 3

4. Using the concept of integration find the area of the region bounded by the lines

x + 2y = 2 ; y –x = 1; 2x + y = 7 Ans. 6

5. Using the concept of integration find the area between y2 = 4ax and the line y = mx. Ans. 8a2/3m3

6. Using integration find the area bounded by 4y = 3x2 and and the straight line 2y = 3x + 12. Ans. 27

7. Find the area of the region included between the parabola y2 = x and the line x + y =2. Ans. 9/2

8. Using concept of integration find the area bounded by parabola y2 = 4ax and its latus rectum. Ans. 8a2/3

9. Using concept of integration find the area bounded by the curves y = x2 + 2, y = x, x = 0 and x = 3. Ans.21/2

10. Find the area bounded by y2 = 4ax, x2 = 4ay. Ans. 16a2/3

11. Find the area bounded by y2 = x + 1, y2 = –x + 1. Ans. 8/3

12. Find the area bounded by the parabolas which is outside 2x2 –y + 9 = 0 but inside 5x2 –y = 0. Ans. 312

13. Find the area enclosed between the two circles x2 + y2 = 4, (x –2)2 + y2 = 4. Ans. 8/3 32

14. Find the area of both the parts into which the parabola y2 = 6 x divides the circle x2 + y2 = 16.

Ans. i. Area sq units

2 2

383

43

163

. . ii. )38(34

15. Find the area of the region {( , ): , }x y y x x y2 2 24 4 4 9 . Ans. 231

31sin

49

89 1

16. Find the area bounded by the curves {x2 + y2 < 2ax, y2 > ax, a > 0, y > 0} Ans. (/4–2/3)a2

17. Find the area of the region {(x, y) : x2 < y < | x |} Ans. 1/6

18. Find the area of the region {(x, y) : 0 < y < x2 + 1, 0 < y < x + 1, 0 < x < 2} Ans. 23/6

19. Find the area of the region {( , ): }x y x y x y2 2 1 . Ans. /4 – ½

20. Find area of smaller region bounded by the ellipse xa

yb

2

2

2

2 1 and the straight line xa

yb

1. Ans. )2(4

ab

.

21. Find the area bounded by the curve y = cosx and y = sinx and x -axis as x varies from 0 to /2. Ans. 22

22. The area between x = y2 & x = 4 is divided into two equal parts by the line x = a, find value of a. Ans. (4)2/3

23. If the area enclosed by y2 = 4ax and line y = ax is 1/3 sq. units, then find the area enclosed by y = 4x with same parabola.Ans. a = 8 , 8/3 sq. units.

24. Prove that the curves y2 = 4x and x2 = 4y divide the area of the square bounded by x = 0, x =4, y = 4 and y = 0 into threeequal parts. Ans. Each = /3

25. Find area bounded by the curve y = | x –1 | and y = 3 – | x | Ans. 4 sq. units.

26. Using integration find area bounded by the curve |x| + |y| = 1. Ans. 2


DIFFERENTIAL EQUATIONS

1. Find the differential equation corresponding to the function y = c(x –c)2, where c is arbitrary constant.

2. Form the differential equation corresponding to y m a x2 2 2 ( ) where m,a are constants.

3. Form differential equation representing the family of ellipses having foci on x-axis and centre at the origin.

4. Form the differential equation of the family of circles touching the x-axis at origin.

5. Form the differential equation of the family of circles having centre on y-axis and radius 3 units.

ANSWERS : 1.

y

dxdyxy

dxdy 24

3

2.

2

22

dxydxy

dxdyx

dxdyy

3. 02

22

dxdyy

dxydxy

dxdyx 4. 22

2yx

xydxdy

5. (x2 –9)

2

dxdy

+ x2 = 0

VARIABLE SEPARABLE: Solve the following differential equations.

6. i. (x3 + x2 + x + 1) dxdy

= 2x2 + x; y = 1 when x = 0. Ans. 1tan21)1()1(log

41 1322 xxxy

ii. (x + 1) dxdy

= 2e– y –1, given that y = 0 w hen x = 0. Ans. 1,112log

xxxy

iii.

dxdyya

dxdyxy 2 Ans. ( x + a) ( 1–ay) = cy iv. sin–1

dxdy

= x+ y Ans. x =tan (x+y) –sec (x+y) +C

v. sec2 x tan y dx +sec2 y tan x dy = 0 Ans. | tan x tan y| = C

HOMOGENEOUS: Show that the following differential equations are homogeneous and also solve them.

7. i. (x –y) dxdy

= x + 2y.. Ans. log | x2 + xy + y2) | = 2 3 tan–1 Cxxy

3

2

ii. (3xy + y2) dx + (x2 + xy) dy = 0 Ans. | y2 + 2xy | = C/x2

iii. (x2 –y2 )dx + 2 xydy =0; given that y(1) = 1. Ans. x2+y2=2x

iv. xcos xxyy

dxdy

xy

cos Ans. sin

xy = log | Cx |

v.

y

xyx

exydxey 22 dy = 0 given that y (0) =1. Ans. 2log2 ye yx

vi. x dy –y dx = dxyx 22 . Ans. y + 22 yx = Cx2

vii.

xyy

xyx sincos dyx

xyx

xyydxy

cossin Ans. xy cos

xy

= C

viii. x2 dy + (xy + y2) dx = 0; y = 1 when x = 1 Ans. y + 2x = 3x2 y

ix. y dx + x log

xy

dy –2x dy = 0 Ans. Cy = log xy

–1

x. (x dy –y dx) y sin

xy

= (y dx + x dy) x cos

xy

. Ans. sec

xy

= C xy


LINEAR DIFFERENTIAL EQUATION

Solve the following differential equations.

8. i. xydxdy cos . Ans. sin x - cos x/2 + Cex.

ii. x log x dxdy

+ y = x2

log x. Ans. y log x = -2(1+ logx) /x + C

iii. dxdy

+ 2y tan x = sin x; y = 0 when x = /3. Ans. y = cos x –2 cos2 x

iv. dxdy

–3 y cot x = sin 2x; y = 2 when x = /2. Ans. y = 4 sin3 x –2 sin2 x

v. dxdy

+ y cot x = 2x + x2 cot x (x 0) given that y = 0 when x = /2 Ans. )0(sinsin4

22 x

xxy

.

vi. dxdy

+ y sec x = tan x. Ans.y (secx + tanx ) = secx + tanx - x +c

vii. ( )2 10 03x ydydx

y viii. (tan–1y –x) dy = (1 + y2) dx. Ans. yceyx1tan1 )1(tan

ix. y dx –(x + 2y2) dy = 0 Ans. yx

= 2y + C x. y dx + (x –y3) dy = 0 Ans. xy = Cy

4

4

9. Find the equation of the curve passing through the point (1, 1) whose differential equation is

x dy = (2x2 + 1) dx (x 0). Ans. y = x2 + log | x |

10. Show that family of curves for which the slope of tangent at any point (x, y) on it is xyyx

2

22 , is given by x2 –y2 = cx.

11. Find the particular solution of the differential equation log

dxdy

= 3x + 4y given that y = 0 when x = 0.

Ans. 4e3x + 3e–4y –7 = 0

12. Find the equation of a curve passing through the point (0, 1). If the slope of the tangent to the curve at any point (x, y) isequal to the sum of the coordinate (abscissa) and the product of the x coordinate and y coordinate (ordinate) of that

point. Ans. 2

2

21x

ey

13. Solve dxdy

= cos ( x + y) Ans. tan ( x + y)/2 = x + c


VECTORS

1. Find projection of b

+ c on a , where a= 2 i –2 j + k , b

= i + 2 j –2 k & c = 2 i – j + 4 k . Ans. 2

2. If

kjibkjia ˆ2ˆ,3ˆ2ˆ2 & jic ˆˆ3

such that

ba is to c , then find value of . Ans. = 8

3. Show that 7/)632(

kji , 7/)263(

kji , 7/)326(

kji are mutually perpendicular unit vectors.

4. The scalar product of the vector

kji with a unit vector along the sum of vectors

k5j4i2 and

k3j2iis equal to one. Find the value of . Ans. = 1

5. Let a = i + 4 j + 2 k , b

= 3 i –2 j + 7 k and c = 2 i – j + 4 k . Find a vector d

which is perpendicular to both a

and b

, c . d

= 15. Ans. 1/3 (160 i –5 j – 70 k )

6. Dot product of a vector with vectors i + j –3

k , i + 3 j – 2

k and 2 i + j + 4

k are 0, 5, 8 respectively. Find thevector. Ans. i + 2 j +

k

7. If a is any vector in space, then show that .k)k.a(j)j.a(i)i.a(a

8. For any two vector a and b, prove the following :

i.

bababa .2|||||| 222 ii.

bababa .2|||||| 222 iii. )|||(|2|||| 2222

bababa

iv. 22 ||||)).((

bababa v.. )...(2|||||||| 2222

accbbacbacba

9. Find |,ba|

if 5||,2||

ba and 8b.a Ans. 13|ba|

.

10. If

c,b,a are unit vector such that

0cba , then find the value of ....

accbba

Ans. )a.cc.bb.a(

= –3/2

11. If a and b are unit vectors inclined at an angle , then prove that(i) sin /2 = 1/2 | a - b| (ii) cos /2 = 1/2 |a + b|

12. If a, b, c are three non-zero vectors such that

0cba , prove that ....

accbba <0.

13. If 4|b|,3|a|

and 5|c|

such that each is perpendicular to sum of the other two, find |cba|

. Ans. 2550 .

14. If the vertices A, B, C of ABC have position vectors (1, 2, 3), (–1, 0, 0), (0, 1, 2), respectively, then find ABC .

Ans.

10210cosABC 1

15. Prove that the points 2 i – j +

k , i – 3 j –5

k and 3 i – 4 j – 4

k are the vertices of a right angled triangle. Find also theother two angles.

16. Prove that )ba).(ba(

= 22 |b||a|

, If and only if

b,a are othogonal.

17. Prove generally that |b||a||ba|

17. Find |ba|

, if

k7j7ia and

k2j2i3b . Verify that a & a × b are to each other.. Ans. |ba|

= 219 .


18. Find a unit vector perpendicular to the vectors a = 2 i – j + k and b = 3 i + 4 j –

k . Also, determine the sine of the angle

between a and b. Ans. Unit Vector = 1551

)1153(

kji , sine of angle = 156155

19. Find a unit vector perpendicular to each of the vector

ba and

ba , where

k2j2i3a and

k2j2ib .

Ans. 3/)22(

kji

20. Find a unit vector perpendicular to the plane ABC where A, B, C are the points A (3, –1, 2), B (1, –1,–3),

C (–4, –3, 1). Ans. 1651

)4710(

kji

21. If a unit vector a that makes angles 3

with 4,i

with

j and an acute angle with

k , then find the components of aand the angle . Ans. a1 = 1/2,a2 = 2/1 ,a3 = 1/2and = /3.

22.

c,b,a are the position vectors of the non-collinear points A, B, C, respectively in space, show that

baaccbis perpendicular of plane ABC.

23. Find the area of parallelogram whose diagonals are d i j k1 3 2

and d i j k2 3 4

. Ans. 35

24. Find the area of the triangle with vertices (1, 1, 2), (2, 3, 5) and (1, 5, 5). Ans. 61 / 2 sq. units

25. Given )k6j3i2(71a

, )k2j6i3(71b

, )k3j2i6(71c

,

k,j,i being a right handed orthogo-

nal system of unit vectors in space, show that

c,b,a is also another such system.

26. Find a unit vector in the plane of the vectors

j2ia and

k2jb , perpendicular to the vector

k2ji2c .

Ans. 551

)k8j6i5(

27. Show that the angle between the diaganols of a cube is cos-1(1/3).

28. If 7||5||,3||,0 candbacba find the angle between a and b

. Ans. 60o

29. a b c

, , are 3 vectors such that a b c b c a

, . Prove a b c

, , are mutually at right angles & | | , | | | |b c a

1 .

30. If a b c d and a c b d , prove that a d

is parallel to b c

provided a d

and b c

.

31. If a b c

, , represent vectors BC CA and AB

, of a triangle ABC then show that a b b c c a

.

32. If with reference to a right handed system of mutually unit vectors

k,j,i ,

k3ji2,ji3 express

in the form

21, where

1 is || to and

2 is to . Ans. )ji3(

21

1

and

k3j23i

21

2

33. For any vector a , prove that | i x a |2 + | j x a |2 + | k x a |2 = 2 | a |2.


34. A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops. Determine the girl’s

displacement from her initial point of departure. Ans. ji ˆ2

33ˆ25

35. If a , b

, c are mutually perpendicular vectors of equal magnitudes, show that the vector a + b

+ c is equallyinclined to a , b

and c .

36. a , b

, c are unit vectors, suppose a . b

= a . c = 0 and angle between b

and c is /6. Prove that a = + 2 ( b

× c ).

37. The length of the altitude through vertex C of a triangle ABC, with position vectors of vertices cba ,, respectively is

abbaaccb

.

38. Show that: | |a b = a b a b2 2 2

( . ) .

39. Show that ( ) . .. .

a x b a a a ba b b b

2

40. Let a = i + 4 j + 2 k , b

= 3 i –2 j + 7 k and c = 2 i – j + 4 k . Find a vector d

which is perpendicular

to both a and b

and c . d

= 15. Ans. k3

70j35i

3160

41. Define scalar triple product and give its geometrical interpretation.

42. Find the volume of a parallelopiped whose sides are given by -3 i +7 j +5 k , –5 i +7 j –3 k and 7 i –5 j - 3 k .

43. If the volume of the parallelopiped whose edges are represented by –12 i +pk, 3 j – k , 2 i + j – 15 k is 546,find the value of p.

44. If a

= 2 i –3 j , b

= i + j – k , c

= 3 i – k , find [a

b

c

]

45. Show that the vectors i –2 j + 3 k , 2 i + 3 j –4 k and i –3 j + 5 k are coplanar..

46. Show that the points with position vectors 6 i –7 j , 16 i –19 j –4 k , 3 j –6 k and 2 i –5 j + 10 k arecoplanar.

47. Find the value of p, so that the vectors i – j + k , 2 i + j – k , p i – j + p k are coplanar..

48. Show that : [ b

× c

c

× a

a

× b

] = [a

b

c

]2

49. If a b x c and pb x c

a b cq

c x a

a b cr

a x b

a b c

.( )[ ]

,[ ]

,[ ]

0 . Show that a

. p

+ b

. q

+ c

. r = 3.

50. Show that [a

+ b

b

+ c

c

+ a

] = 2 [ a

b

c

]

51. Simplify [a

– b

b

–c

c

–a

]. Also Interpret the result.


3-DIMENSIONAL GEOMETRY

1. Find the vector equation of a line passing through a point with position vector 2 i – j + k , and parallel to the line joining

the points – i +4 j + k and i +2 j + 2 k . Also, find the Cartesian equivalent of this equation.

Ans. i. r = 2 i – j + k + (2 i –2 j + k ) ii. 11

21

22

zyx

2. The points A (4, 5, 10), B (2, 3, 4) and C (1, 2, –1) are three vertices of a parallelogram ABCD. Find vector and Cartesianequations for the sides AB and BC and find the coordinates of D.

Ans. Equation of BD x –2 = y –3 = z –4 Cordinates of D (3, 4, 5)

3. Find the equation of the line passing through the point (–1, 3, –2) and perpendicular to the lines

321zyx

and 51

21

32

zyx Ans. 4

273

21

zyx

4. Find the vector equation of the line passing through the point (1, 2, –4) and perpendicular to the two lines :

710

1619

38

zyx and 5

5829

315

zyx

. Ans. r = i + 2 j –4 k + (2 i + 3 j + 6 k )

5. Show that the lines r = ( i + j – k ) + (3 i – j ) and r = (4 i – k ) + (2 i +3 j ) intersect. Find their point of intersection.

Ans. (4, 0, –1)

6. Find the image of the point (1, 6, 3) in the line 32

21

1

zyx . Ans. (1, 0, 7)

7. Find the shortest distance between the lines whose vector equations are r = (1 –t) i + (t –2) j + (3 –2t) k and

r = (s + 1) i + (2s –1) j – (2s + 1) k . Are the lines coplanar. Ans. 29/8 .No.

8. Find the shortest distance and the vector equation of the line of shortest distance between the lines given by :

r = (8 + 3) i – (9 + 16) j + (10 + 7) k and r = 15 i + 29 j + 5 k + (3 i + 8 j –5 k ).

Ans. 14, r = (5 i + 7 j + 3 k ) + (2 i + 3 j + 6 k )

9. Find the shortest distance and the Cartesian equations of the line of shortest distance between the following pair of lines:

31

21

yx = z and 1

23

1

yx; z = 2. Ans. 7

97592323259

125359,

593

yx

10. Show that the points (0,-1,0), (2,1,-1), (1,1,1) & (3,3,0) are coplanar.

11. Find the equation of the planes that passes through three points : (1, 1, –1), (6, 4, –5), (–4, –2, 3)

Ans. The points are collinear. There will be infinite number of planes passing through the given points.

12. Find the equation of the plane through the points (2,2,1) and (9,3,6) and perpendicular to the plane

2x + 6y + 6z–1=0. Ans. 3 x + 4y – 5z = 9.

13. Find the equation of the plane passing through a point (1,1,-1) and perpendicular to two planes:

x + 2y + 3z =7 and 2x–3y + 4z =0. Ans. 17 x + 2y –7z = 26.

14. Find the equation of the plane through the point 5 i + 2 j –3 k and perpendicular to each of the planes.

r (2 i – j + 2

k )=0 and r ( i + 3 j –5

k ) + 3 = 0. Ans. r ( – i +12 j +7

k ) + 2 = 0.

15. Find the equations of the planes parallel to the plane x –2y + 2z –3 = 0 which is at a unit distance from the point (1,2,3).

Ans. x–2y+ 2z =0, x–2y+ 2z =6


16. Find the equation of the plane passing through the intersection of the planes ;

2x + 3y + z =5 and 3x–y + 4z + 3=0 and perpendicular to the plane x + y–3z =6. Ans. 13x + 14y + 9z –22 = 0

17. Find the equation of the plane through the intersection of the planes x+ 3y + 6 = 0 and 3x–y– 4z = 0

and at unit distance from the origin. Ans. 0322;322 zyxzyx

18. Find coordinates of foot of the r drawn from origin to the plane 2x –3y + 4z –6 = 0. Ans.

2924,

2918,

2912

19. Find the image of the point ( 1,2,-1) in the plane r. (3 i –5 j + 4

k ) = 5. Ans. ( 73/25, –6/5, 39/25)

20. Find distance of point (3, 4, 5) from the plane x + y + z = 2, measured parallel to the line 2x = y = z. Ans. 621. Find the equation of plane passing through the points (2, 2 –1) and (3, 4, 2) and parallel to the line

x y z

17

50

46 . Ans. 12x + 15 y – 14 z = 68

22. Find the equation of the plane through the intersection of the planes 2 x + y– z = 3 and 5 x– 3 y + 4z = –9 and parallel to

the line x y z

1

23

42

5 . Also find the perpendicular distance of (1,1,1) from this plane.

Ans. i. 7x + 9y –10z –27 = 0 ii. 230/21

23. Find the equation of the plane through the point ( 2,–1,1) and parallel to the lines

r = i – k + (3 i + j + k ) and r = 2 i –3 j + 7 k + (7 j – k ). Ans. r . (–8 i + 3 j + 21 k ) = 2.

24. Find the equation of the line passing through the point (3,0,1) and parallel to the planes x y y z 2 0 3 0& .

Ans. 31

123

zyx

25. Find equation of plane containing the line x y z

12

21

31 & point ( 2,3,-2). Ans. 2x –3y + z + 7 = 0

26. A line makes angle , , & with diagonals of a cube, prove that cos2 + cos2 + cos2 + cos2 = 4/3.

27. Find the equation of the plane that contains the point (1, –1, 2) and is perpendicular to each of the planes

2x + 3y –2z = 5 and x + 2y –3z = 8. Ans. 5x –4y –z = 7

28. Find the distance between the point P(6, 5, 9) and the plane determined by the points

A (3, –1, 2), B (5, 2, 4) and C(–1, –1, 6). Ans. 17/343

29. Find the equation of the plane passing through the line of intersection of the planes r .( i + j + k ) = 1 and r . (2 i + 3 j

– k ) + 4 = 0 and parallel to x-axis. Ans. y –3z + 6 = 0

30. If a plane has the intercepts a, b, c & is at a distance of p units from the origin, then prove 1/a2+1/b2+1c2=1/p2.

31. Find the equation of the plane passing through the intersection of the planes,

2x + 3y –z + 1 = 0; x + y –2z + 3 = 0 and perpendicular to the plane 3x –y –2z –4 = 0. Also find the inclination of this

plane with the xy plane. Ans. .94137 zyx Inclination with xy plane = 234/4cos 1 .

32. Show that the lines r = ( i + j – k ) + (3 i – j ) and r = (4 i – k ) + (2 i + 3 k ) are coplanar. Also find the plane

containing these two lines. Ans. r . (3 i +9 j –2 k ) = 14


LINEAR PROGRAMMING

1. A person consumes two types of food A and B everyday to obtain 8 units of protein, 12 units of carbohydrates and 9 unitsof fat which is his daily minimum requirements. 1 kg of food A contains 2,6 and 1 units of protein, carbohydrates andfats respectively. 1 kg of food B contains 1,1 and 3 units of protein, carbohydrates and fat respectively. Food A costs Rs.8 per kg, while food B costs Rs. 5 per kg. Determine how many kgs of each food should he buy daily to minimize his costof food and still meets the minimal nutrition requirements. Ans : A=3,B=2,Z=34

2. The ABC company has been a producer of picture tubes for television sets and printed circuits for radios.The companyhas just expanded into full scale and marketing of AM and AM-FM radios. It has built a new plant that can operate 48hours per week. Production of an AM radio in the new plant wilt require 2 hours and production of an AM-FM radio willrequire 3 hours. Each AM radio will contribute Rs.40 to profit while an AM-FM radio will contribute Rs. 80 to profit.The marketing department, after extensive research, has determined that a maximum of 15 AM radios and 10 AM-FMradios can be sold each week. Formulate a linear programming determine the optimal mix of AM and AM-FM radiosthat will maximize profit. Ans: AM= 9 ,AM-FM=10,Z=1160

3. A box manufacturer makes small and large boxes from a large piece of cardboard. The large boxes require 4 sq. ft, perbox, while the small boxes require 3 sq. ft per box. The manufacturer is required to make at least three large boxes andat least twice as many small boxes as large boxes. If 60 sq. ft. of cardboard is in stock, and if the profits on the small andlarge boxes are Rs. 2 and Rs. 3 per box respectively, how many of each should be made in order to maximize the totalprofit ? Ans : Large=6,Small = 12,Z=42

4. An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 400 is made on each first class ticket and a profitof Rs. 300 is made on each economy class ticket. The airline reserves at least 20 tickets for first class seats. However, atleast 4 times as many passengers prefer to travel by economy class as to the first class. Determine how many tickets ofeach type must be sold in order to maximize the total profit for the airline. Ans : 40,160, Rs. 64,000

5. An airline agrees to charter planes for a group. The group needs at least 160 first class seats and at least 300 tourist classseats. The airline must use at least two of its model 314 planes which have 20 first class and 30 tourist class seats. Theairline will also use some of its model 535 planes which have 20 first class seats and 60 tourist class seas. Each flight ofa model 314 plane costs the company Rs. 1 lakh, and each flight of a model 535 plane costs Rs. 1.5 lakh. How many ofeach type of planes should be used to minimize the flight cost? Ans. 6 planes of model 314 and 2 plane of model 535

6. A retired person wants to invest upto an amount of Rs. 20,000. His broker recommends investing in two bonds. Bond Ayielding 10% return on the amount invested and bond B yielding 15% return on the amount invested. After someconsideration, he decides to invest at least Rs. 5000 in bond A and no more than Rs. 8000 in bond B. He also wants toinvest at least as much in bond A as in bond B. What should his broker recommend if he wants to maximize his return oninvestment ? Ans:A=12,000, B=8000,Z=2400

7. A firm produces three items A, B and C at two plants X and Y. The number of items produced and operating cost perhour are : Plant Items Produced per hour Operating Cost per hour

A B C

X 30 60 40 Rs. 1,200

Y 30 20 100 Rs. 800

It is desired to produce at least 400 items of type A, at least 800 of type B and at least 880 of type C per day. Find thenumber of hours each plant be run on a day to have the costs minimum. Ans: 12, 4, Z=17 600

8. The manager of an oil refinery must decide on the optimal mix of two possible blending processes of which the inputsand outputs per production run are as follows:

Inputs Outputs

Process Crude A Crude B Gasoline X Gasoline Y

1 5 3 5 8

2 4 5 4 4


The maximum amount available of crudes A & B are 200 units & 150 units, respectively. Market requirements show thatat least 100 units of gasoline X & 80 units of gasoline Y must be produced. The profit per production run from process1 & process 2 are Rs. 300 & Rs. 400 respectively. Find the number of runs of each process must be made.

Ans : 400/13 ,150/13 ,Z = 1,80,000/13.

9. A firm manufactures two products X and Y, each requiring the use of three machines M1 M2 and M3. The time requiredfor each product in hours and total time available in hours on each machine are as follows:

Machine Product X Product Y Available Time

M1 2 1 70

M2 1 1 40

M3 1 3 90

If the profit is Rs. 40 per unit for product X and Rs. 60 per unit for product Y, how many units of each product should bemanufactured to maximize profit? Ans : X =15,Y=25,Z=2100

10. The XYZ Company Ltd. manufactures two products A and B. These products are processed on the machine. It takes 20minutes to process one unit of product A and 15 minutes for each units of product B and machine operates for amaximum 80 hours in a week. Product A requires 3 kg and product B, 2 kg of the raw material per unit, the supply ofwhich is 1200 kg. per week. Market constraint on product B is known to be 1500 units every week. If the product A costRs. 10 per unit and can be sold at price of Rs. 15, product B costs Rs. 15 per unit and can be sold in the market at a unitprice of Rs. 22, the problem is to find out the number of units of A and B that should be produced per week in order tomaximize the profit.

11. A gardener has a supply of fertilizer of type-1 which consists of 10 % nitrogen and 6% phosphoric acid and type IIconsists of 5% nitrogen 10 % phosphoric acid. After testing the soil conditions, he finds that he needs at least 14 kg ofnitrogen and 14 kg of phosphoric acid for this crop. If the type 1 fertilizer costs 60 paise per kg and type 2 fertilizer costs40 paise per kg ,determine the minimum cost. Ans. Rs. 92.

12. A man owns a field of area 1000 sq m. He wants to plant fruits in it. He has a sum of RS. 1400 to purchase young trees.He has the choice of two types of trees. Type A requires 10 sq m of ground per tree and costs Rs. 20 per tree and Type Brequires 20 sq m of ground per tree and costs Rs. 25 per tree. When fully grown, type A produces an average of 20 kg offruit which can be sold at a profit of Rs. 2 per kg and type B produces an average of 40 kg of fruit which can be sold ata profit of Rs. 1.50 per kg. How many of each type should be planted to achieve maximum profit when trees are fullygrown and what is the maximum profit. Ans : A=20,B= 40,Z=3200

13. A manufacturer of patent medicines is preparing a production plan on machines, A and B. There are sufficient rawmaterials available to make 20,000 bottles of A and 40,000 bottles of B, but there are only 45,000 bottles into whicheither of the medicines can be put. Further it takes 3 hrs to prepare enough material to fill 1000 bottles of A, it takes 1hour to prepare enough material to fill 1000 bottles of B and there are 66 hours available for this operation. The profit isRs. 8 per bottle for A and Rs. 7 per bottle for B. Determine how the production should be scheduled to maximize profit.

Ans: A= 10,500,B=34,500 ,Z= 3,25,500

14. A factory manufactures two types of screws ,A and B, each type requiring the use of two machines-an automatic and ahand operated .It takes 4 minutes on the automatic and 6 minutes on the hand-operated machines to manufacture apackage of screws A , while it takes 6 minutes on the automatic and 3 minutes on the hand operated machine to manu-facture a package of screws B.Each machine is available for at most 4 hrs. on any day. The manufacturer can sell apackage of screws A at a profit of 70 paise and B at a profit of Re. 1, find his maximum profit.

Ans. A=30,B = 20,Z=41


15. A toy company manufactures two types of dolls. A and B. Market tests and available resources have indicated that thecombined production level should not exceed 1200 dolls per week and the demand for dolls of type B is at most half ofthat for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls ofother type by at most 600 units. If the company makes profit of Rs. 12 and Rs. 16 per doll respectively on dolls A and B,how many of each should be produced weekly in order to maximise the profit?

Ans. 800 dolls of type A and 400 dolls of type B; Maximum profit = Rs. 16,000.

16. A toy manufacturer produces two types of dolls: a basic version - doll A and a deluxe version - doll B. Each doll of typeB takes twice as tong to produce as one doll of type A. The company has time to make a maximum of 2,000 dolls of typeA per day. The supply of plastic is sufficient to produce 1,500 dolls per day and each type requires equal amount of it.The deluxe version, i.e., type B requires a fancy dress of which there are 600 per day available. If the company makesa profit of Rs. 3 and Rs. 5 per doll, respectively, on doll A and B, how many of each should be produced per day in orderto maximize profit? Solve it by graphical method.

Ans : A = 1000,B = 500,Z = 55,00

17. A factory owner purchases two types of machines, A and B for his factory. The requirements and the limitations for themachines are as follows :

Machine Area occupied Labour force Daily output (in units)

AB

1000 m2

1200 m212 men8 men

6040

He has maximum area of 9000 m2 available, and 72 skilled labourers who can operate both the machines. How manymachines of each type should he buy to maximise the daily output? Ans. Machine A = 6 and B =0.

18. A publisher sells a hard cover edition of a text book for Rs. 72.00 and a paper back edition of the same text for Rs. 40.00.Costs to the publisher are Rs. 56.00 and Rs. 28.00 per book respectively. Both types require 5 minutes of printing time,although hardcover requires 10 minutes binding time and the paper back requires only 2 minutes. Both the printing andbinding operations have 4,800 minutes available each week. How many of each type of book should be produced eachweek in order to maximize profit? Ans. 360 hard cover and 600 covers of paper back.

19. If a young man rides his motor cycle at 25 kmph, he has to spend Rs.2 per km on petrol, if he rides at a faster speed of40 kmph, the petrol cost increases to Rs.5 per km. He has Rs.100 to spend on petrol and wishes to find what is themaximum distance, he can travel within one hour, Express as L.P.P. and solve it. Ans.30 km

20. P and Q are two places for the factories with out put 8 and 6 units while A, B, C are places for depots with requirement5, 5 and 4 units. The cost of transportation per unit is as under:

From To A B C

P Rs. 16 Rs.10 Rs.15

Q Rs. 10 Rs. 12 Rs. 10.

How many units should be transported from each factory to each depot to minimum the cost ? Formulate L.P.P. and solveit? Ans. (0, 5), Rs. 155)


PROBABILITY

1. Four cards are drawn at a time from a pack of 52 cards. Find the probabilty of getting

i. all the 4 cards of the same suit. ii. all the 4 cards of same number. iii. one card from each suit

iv. two cards of each colour Ans. i. 44/4165 ii.1/20825 iii. 2197/20825 iv. 4225/10829

2. If A and B are two independent events such that the probability of their simultaneous occurence is 1/8 and the probabiltythat neither of them occurs is 3/8. Find P(A) and P(B). Ans. 1/2, 1/4

3. If A & B are 2 independent events such that P A B P A B( ' ) / & ( ') / 2 15 1 6 ,find P(B). Ans. 4/5, 1/6

4. A problem on mathematics is given to 3 students whose chances of solving it are 1/2,1/3,1/4.What is the probability thatexactly one of them will solve the problem if all the three try to solve the problem simultaneously. What is the probabilitythat the problem will be solved? Ans. i. 11/24 ii. 3/4

5. A speaks truth in 75% cases and B in 25% cases.In what % of cases are they likely to contradict each other in stating thesame fact. Ans. 62.5%

6. The odds are 7 to 5 against A, a person who is now 30 years old living up to 70 years and the odds are 2 to 3 in favour ofB who is now 40 years living up to 80 years.What is the chance that at least one of them will be alive 40 years hence.

Ans. 13/20

7. A husband and wife appear in an interview for two vacancies in the same post. The probability of husband's selection is1/7 and that of wife's selection is 1/5. What is the probability that :

i. Only one of them will be selected? ii. Both of them will be selected? iii. None of them will be selected?

iv. At least one of them will be selected? Ans. i. 2/7, ii. 1/35, iii. 24/35, iv. 11/35

8. An anti-aircraft gun can take a maximum of three shots at an enemy plane moving away from it.The probabilities ofhitting the plane at he first, second and third shot are 2/3, 2/5 and 3/8 respectively. What is the probability that the planeis hit? Ans. 7/8

9. X,Y&Z shoot to hit a target. If X hits target 4 times in 7 trials,Y hits 3 times in 5 trials and Z hits in 2 times in 2 trials,if they fire together a volley what is the probabilty that target is hit at least by 2 persons. Ans. 29/35

10. Three coins are tossed simultaneously. Consider the event E ‘ three heads or three tails ,, F ‘ at least two heads , and G ‘at most two heads ,. Of the pairs (E,F),(E,G) and (F,G), which are dependent and which are independent?

Ans. E and F are independent. E and G and F and G are dependent.

11. A four digit number is formed using the digits 1, 2, 3, 5 with no repetitions. Find the probability that the number isdivisible by 5. Ans. 1/4

12. A bag A contains 2 white and 4 black balls. Another bag B contains 5 white and 7 black balls.A ball is transferred frombag A to bag B. Then a ball is drawn from bag B.Find probability that it is white. Ans. 16/39

13. There are three Urns, A, B and C. Urn A contains 4 red balls and 3 black balls. Urn B contains 5 red and 4 black balls.Urn C contains 4 red and 4 black balls. One ball is drawn from each of these urns. What is the probability that the 3 ballsdrawn consist of 2 red balls and a black ball. Ans. 17/42

14. If each element of a second order determinant is either zero or one, what is the probability that the value of he determinantis positive? (Assume that the individual entries of the determinant are chosen independently, each value being assumedwith probability ½). Ans. 3/16

15. A and B throw alternately with a pair of dice.A wins if he throws 6 before B throws a sum of seven and B wins if hethrows a sum of seven before A throws a sum of 6. Find their respective chances of winning assuming that A begins thegame. Ans. 30/61, 31/61


PROBLEMS BASED ON CONDITIONAL PROBABILITY

1. Let A and B be the events such that P A P B and P A B find P A B and P A B( ) , ( ) ( ) , ( / ) ( / ).' ' ' 13

14

15

Ans. i. P (A´/B) = 1/5 ii. P (A´/B´) = 37/452. In a Class, 40% students read Mathematics, 25% Economics and 15% both Mathematics and Economics.

One student is selected at random,find the probability thati. he reads Mathematics, if it is given that he reads Economics.ii. he does not read Mathematics, if it is given that he reads Economics.iii. he does not read Mathematics, if it is given that he does not read Economics. Ans. i. 3/5 ii. 2/5 iii. 2/3

3. 3 coins are tossed. Find probability that all coins show head, if at least one of the coin shows head. Ans. 1/74. A die is thrown twice and the sum of the numbers appearing is observed to be 6.What is the conditional probabilty that

4 has appeared at least once? Ans. 2/55. A pair of fair dice is thrown.Find probabilty that the sum is 10 or greater, if 5 appears on first die. Ans. 1/36. Two numbers are selected at random from the integers 1 through 13. If the sum is even, find the probabilty that both

numbers are odd. Ans. 7/127. A bag contains 3 red and 4 black balls and another contains 4 red and 2 black balls.One bag is selected at random and

from the selected bag a ball is drawn,each ball equally likely to be drawn .Let E be the event the first bag is selected, Fthe event the second bag is selected and G the event the ball is red. Find P(E), P(F), P(G/E), P(G/F), P(G).

Ans. i. 1/2 ii 1/2 iii. 3/7 iv. 23/428. Two persons A and B throw a coin alternately till one of them gets head and wins the game, find their respective

probabilities of winning. What would be the respective probabilities of winning if there are 3 persons A, B and C.Assume that A begins game followed by B. Ans. i. A =2/3, B = 1/3, ii. A =4/7, B=2/7,C=17

9. A bag contains 3 green and 7 white balls. Two balls are selected at random without replacement. If the second selectedball is given to be green, what is the probability the first selected is also green. Ans. 2/9

10. A couple has two children. Find the probability that both children are boys if it is known that at least one of the childrenis a boy. What would be the answer that both children are boys if it is known that the elder child is a boy. Ans.1/3, 1/2

11. Consider the experiment of tossing a coin. If the coin shows head, toss it again but if it shows tail , then throw a die. Findthe conditional probability of the event that the die shows a number greater than 4 given that there is atleast one tail.

Ans. 2/9BINOMIAL DISTRIBUTION

1. An unbiased coin is tossed six times. Find the probability of obtaining :i. exactly 4 heads ii. less than 3 heads iii. more than 4 heads iv. more than 4 heads & less than 6 headsvi. at least 4 heads vii. at most 4 heads Ans. i. 15/64 ii. 22/64 iii. 15/64 iv. 6/64 vi. 22/64 vii. 57/64

2. A die is thrown 6 times. If 'getting an odd number' is a 'success', what is the probability ofi. 5 successes, ii. at least 5 successes iii. at most 5 successes. Ans. i. 6/64 ii. 7/64 iii. 63/64

3. A die is thrown 20 times. Getting a number greater that 4 is considered a success. Find the mean and variance of thenumber of successes. Ans. Mean = 6.67, Variance = 4.44

4. A product is supposed to contain 5% defective items.What is the probability that a sample of 8 items will contain morethan 2 non- defective items? Ans. 1–10261/208.

5. In a hurdle race, a player has to cross 10 hurdles. The probability that he will clear each hurdle is 5/6. What is theprobability that he will knock down fewer than 2 hurdles? Ans. (5/6)9 5/2

6. If on an average one ship in every ten is wrecked, find the probability that out of 5 ships expected to arrive, 4 at least willarrive safely. Ans. 45927/50,000

7. A bag contains 5 white, 7 red and 8 black balls. If 4 balls are drawn on by one with replacement. what is the probabilitythat i. none is white ii. all are whites ? iii. only 2 are whites? Ans. 81/256 ii. 1/256 iii. 27/128


8. 5 cards are drawn successively with replacement from a well shuffled pack of 52 cards.Find the probabilityi. all 5 cards are spades? ii. only 3 cards are spades? iii. none is spade? Ans. i. 1/1024 ii. 90/1024 243/1024

9. The probability that a bulb produced by a factory will fuse after 100 days of use is 0.05.Find the probability that out of 5 such bulbs, i. none, ii. not more than one iii. more than one andi. at least one will fuse after 100 days of use. Ans. i. (19/20)5 ii. 6/5(19/20)4

10. An urn contains 25 balls of which 10 balls bear a mark 'X' and the remaining 15 bear a mark 'Y'. A ball is drawn atrandom from the urn, its mark noted down and it is replaced. If 6 balls are drawn in this way, find the probability that :i. all will bear 'X' marks ii. not more than 2 will bear 'Y' marks.iii. the number of balls with 'X' marks and 'Y' marks will be equal. iv. at least one ball will bear 'Y' marks. Find also themean number of balls with 'X' mark. Ans. i. (2/5)6 ii. 9 (2/5)4 iii. 4320/56 iv. 1-(2/5)6

11. An unbiased dice is thrown again and again until three sixes are obtained. Find the probability of obtaining the third sixin the sixth throw of the dice. Ans. 1250/66

12. Six dice are thrown 729 times. How many times do you expect at least 3 dice to show a five or six. Ans. 233

PROBABILITY DISTRIBUTION

1. Find the probability distribution of Z, the number of sixes, in two tosses, of a die .

2. Two cards are drawn without replacement from a well-shuffled deck of 52 cards. Find the probability distribution of thenumber of aces. What would be the answer if the cards are drawn with replacement.

3. Find the mean, variance and standard deviation of the number of heads in three tosses of a coin.

4. Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as 'anumber greater than 4'. Also find the mean and variance of the distribution.

5. Find the probability distribution of the number of sixes in three tosses of a die.Find also the mean and variance of thedistribution.

6. Three cards are drawn successively with replacement from a well-shuffled deck of 52 cards. A random variable Xdenotes the number of spades in the three cards. Determine the probability distribution.

7. Two cards were drawn without replacement from a well-shuffled deck of 52 cards. Determine the probability distributionof the number of face cards (i.e. Jack, Queen, King and Ace).

8. Find the probability distribution of the number of green balls drawn when 3 balls are drawn, one by one, withoutreplacement from a bag containing 3 green and 5 white balls.

9. A bag contains 3 red and 4 black balls. One ball is drawn and then replaced in the bag and the process is repeated. Everytime the ball drawn is red we say that the draw has resulted in success. Let X be the number of successes in 3 draws.Assuming that at each draw each ball is equally likely to be selected, find the probability distribution of X.

10 A box contains 16 balls out of which 25% are defective. Three balls are drawn one by one from the box withoutreplacement. Find the probability distribution of the number of defective balls drawn. What would be the answer if theballs are drawn with replacement.

11. Find the probability distribution of the number of doublets in four throws of a pair of dice.

12. A class has 15 students whose ages are 14, 17, 15, 14, 21, 19, 20, 16, 18, 20, 17, 17, 16, 19 and 20 years respectively. Onestudent is selected in such a manner that each has the same chance of being selected and the age X of the selected studentis recorded. What is the probability distribution of the random variable X?

13. A die is loaded in such a way that an even number is twice as likely to occur as an odd number. If the die is tossed twice,find the probability distribution of the random variable X representing the number of perfect squares in the two tosses.

14. In a game, a person is paid Rs. 5 if he gets all heads or all tails when three coins are tossed, and he will pay Rs. 3 if eitherone or two heads show. What can he expect to win on the average per game.


15. In a game a man wins a rupee for a six and losses a rupee for any other number when a fair die is thrown. Theman decided to throw a die thrice but to quit as and when he gets a six. Find the expexted value of theamount he wins/loses.

16. A random variable X has the following probability distribution:x1 : 0 1 2 3 4 5 6 7p1 : 0 k 2k 2k 3k k2 2k2 7k2 +k(i) Find the value of k. (ii) P(X<3) (iii) P ( 0 < X< 3)

ANS. 1. Z 0 1 2P(Z) 25/36 10/36 1/36 2.i. X 0 1 2

P(X) 188/221 32/221 1/221 ii. X 0 1 2P(X) 144/169 24/169 1/169

3. Mean = 1.5, Variance = 3/4, Standard deviaion = 0.87 4. X 0 1 2P(X) 4/9 4/9 1/9 Mean = 2/3, variance = 4/9

5. X 0 1 2 3P(X) 125/216 75/216 15/216 1/216 Mean = 19/36, Variance = 2/9 6.

X 0 1 2 3P(X) 27/64 9/64 1/6427/64

7. X 0 1 2P(X) 105/221 96/221 20/221 8. X 0 1 2 3

P(X) 10/56 30 15/ 1//56 56 56 9. X 0 1 2 3P(X) /64 343 144/343 108/343 27/343

10. X 0 1 2 3P(X) / 455 1 0 66/140 18/140 1/140 11. X 0 1 2 3 4

P(X) 625/1296 500/ 150/ 20/ 1/1296 1296 1296 1296

12. X 14 15 16 17 18 19 20 21P(X) 2/15 1/15 2/ 3/ 1/ 2/ 3/ 1/15 15 15 15 15 15 Mean = 17.53, Var(x) = 4.78, SD(x) = 2.19

13. X 0 1 2P(X) 4/9 14/9 /9 14. Lose Re1. 15. He will lose Rs. 364/216. 16. i. 1/10 ii. 3/10 iii.3/10

BAYES THEOREM

1. Three urns are given, each containing red and white balls as indicated. URN 1 : 6 red and 4 white; URN 2 : 2 red and 6white ; URN 3 : 1 red and 8 white. An urn is chosen at random and a ball is drawn from the urn. The ball drawn is red.Find the probability that (i) urn 1 was chosen (ii) the urn chosen was 2 or 3. Ans. i. 108/173 ii. 165/173

2. There are three urns having the following compositions of black and white balls :

Urn I : 7 white and 3 black balls ; Urn II : 4 white and 6 black balls ; Urn III : 2 white and 8 black balls

One of these urns is chosen at random with probabilities .2, .6 and .2 respectively. Form the chosen urn, two balls aredrawn at random without replacement. Both the balls happen to be white. Calculate the probability that the balls drawnwere from Urn III. Ans. 1/40

3. The contents of three urns are as follows :

Urn I : 1 white, 2 black & 3 red balls; Urn II: 2 white, 1 black & 1 red balls & Urn III: 4 white, 5 black & 3 red balls

One urn is chosen at random and two balls are drawn. These happen to be white and red. What is the probability that theycome from urn, I? Ans. 33/118

4. Bag I contains 3 red and 4 black balls and Bags II contains 4 red and 5 black balls. One ball is transferred from Bag I toBag II and then a ball is drawn from Bag II. The ball so drawn is found to be red in colour. Find he probability that thetransferred ball is black. Ans. 16/31

5. An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of anaccident involving a scooter, a car and a truck is 1/100, 3/300 and 3/20 respectively. One of the insured persons meetswith an accident. What is the probability that he is a i. scooter driver, ii. car driver and iii. truck driver?

Ans. i. 1/52 ii. 3/26 iii. 45/52

6. In a bolt factory, machines A,B,C manufacture 25%, 35% and 40% respectively of the total bolts. Of their output, 5%,4% & 2% are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found to bedefective, what is the probability that is manufactured by the machine B? From which machine, the defective bolt isexpected to have been manufactured. Ans. 28/69


7. The probabilities of X,Y,Z becoming a manager are 4/9, 2/9 and 1/3 respectively. The probabilities that the bonusscheme will be introduced if X,Y,Z become managers are 3/10,1/2 and 4/5 respectively.

i. What is the probability that the bonus scheme will be introduced? Ans. i. 23/45 ii. 6/23

ii. If the bonus scheme has been introduced, what is the probability that the manager appointed was X.

8. A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is six. Find the probability that it isactually a six. Ans. 3/8

9. A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to behearts. Find the probability of the missing card to be heart. Ans. 11/50

10. By examining the chest X-ray, the probability that T.B is detected when a person is actually suffering is .99. TheProbability that the doctor diagnoses incorrectly that a person has TB on the basis of X-ray is .001. In a certain city, 1 in1000 persons suffers from TB. A person is selected at random and is diagnosed to have T.B. What is the chance that heactually has T.B.? Ans. 110/221

11. In a competitive examination, an examinee either guesses or copies or known the answer to a multiple choice questionwith four choices. The probability that he makes a guess is 1/3 and the probability that he copies the answer is 1/6. Theprobability that the answer is correct, given that the copies it is 1/8. Find the probability that he

i. guesses ii. copies and iii. knows, the answer to the answer to the question, given that he correctly answered it.

Ans. i. 4/29 ii. 1/29 iii. 24/29

12. A bank finds that the relationship between mortgage defaults and the size of the down payment is given by the table.

Down Payment % 5% 10% 20% 25

Number of Mortgages with this down payment 1260 700 560 280

Probability of Default .05 .03 .02 .01

If a default occurs, what is the probability that it is on a mortgage with a 5% down payment? Ans. 0.643

13. A letter is known to have come from either TATANAGAR or CALCUTTA. On the envelope just two consecutive lettersTA are visible. What is the probability that the letter has from :

i. TATANAGAR ii. CALCUTTA Ans. i. 7/11 ii. 4/11

14. Assume that the chances of a patient having a heart attack is 40%. It is also assumed that a mediation and yoga coursereduce the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patientcan choose any one of the two options with equal probabilities.

i. Find the probability that the person suffers from heart attack given that he chose one of the two options.

ii. If it is given that after going through one of the two options the patient selected at random suffers a heart attack. Findthe probability that patient followed a course of mediation and yoga? Ans. i. 29/100 ii. 14/29

15. Suppose that the reliability of a HIV test is specified as follows :

Of people having HIV, 90% of the test detect the disease but 10% go undetected. Of people free of HIV, 99% of the testare judged HIV–ive but 1% are diagnosed as showing HIV+ive. From a large population of which only 0.1% have HIV,one person is selected at random, given the HIV test, and the pathologist reports him/her as HIV+ive. What is theprobability that the person actually has HIV? Ans. 0.083 Approx.

16. A laboratory blood test is 99% effective in detecting a certain disease when it is in fact, present. However, the test alsoyields a false positive result for 0.5% of the healthy person tested (i.e. if a healthy person is tested, then, with probability0.005, the test will imply he has the disease). If 0.1 percent of the population actually has the disease, what is theprobability that a person has the disease given that his test result is positive? Ans. 1 98/ 1197

17. In a bag there are five balls. Two balls are drawn and found to be found white. Find the probability that all the balls arewhite. Ans. 1/2

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