Question Bank

Probability
1.

2. A family has two children. What is the probability that both the children are boys
given that at least one of them is a boy?
3. In a school, there are 1000 students, out of which 430 are girls. It is known that out of
430, 10% of the girls study in class XII. What is the probability that a student chosen
randomly studies in Class XII given that the chosen student is a girl?
4. An instructor has a question bank consisting of 300 easy True / False questions, 200
difficult True / False questions, and 500 easy multiple choice questions and 400
difficult multiple choice questions. If a question is selected at random from the
question bank, what is the probability that it will be an easy question given that it is a
multiple choice question?
5. Given that the two numbers appearing on throwing two dice are different. Find the
probability of the event ‘the sum of numbers on the dice is 4’.
6. Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the die
again and if any other number comes, toss a coin. Find the conditional probability of
the event ‘the coin shows a tail’, given that ‘at least one die shows a 3’.
7. Consider the experiment of tossing a coin. If the coin shows head, toss it again but if
it shows tail, then throw a die. Find the conditional probability of the event that ‘the
die shows a number greater than 4’ given that ‘there is at least one tail’.
8. Three cards are drawn successively, without replacement from a pack of 52 well
shuffled cards. What is the probability that first two cards are kings and the third card
drawn is an ace?
9. An unbiased die is thrown twice. Let the event A be ‘odd number on the first throw’
and B the event ‘odd number on the second throw’. Check the independence of the
events A and B.
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 independent? Which are dependent?
11. Two balls are drawn at random with replacement from a box containing 10 black and
8 red balls. Find the probability that
(i) Both balls are red.
(ii) First ball is black and second is red.
(iii) One of them is black and other is red.
12. Probability of solving specific problem independently by A and B are1/2 and 1/3
respectively. If both try to solve the problem independently, find the probability that
(i) the problem is solved (ii) exactly one of them solves the problem.
13. One card is drawn at random from a well shuffled deck of 52 cards. In which of the
following cases are the events E and F independent?
(i) E: ‘the card drawn is a spade’ ; F: ‘the card drawn is an ace’
(ii) E: ‘the card drawn is black’ ; F: ‘the card drawn is a king’
(iii) E: ‘the card drawn is a king or queen’ ; F: ‘the card drawn is a queen or
jack’.
14. If A and B are two independent events, then the probability of occurrence of at least
one of A and B is given by 1– P(A’) P(B’)
15. A person has undertaken a construction job. The probabilities are 0.65 that there will
be strike, 0.80 that the construction job will be completed on time if there is no strike,
and 0.32 that the construction job will be completed on time if there is a strike.
Determine the probability that the construction job will be completed on time.
16. Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes
the number of heads. If she gets 1, 2, 3 or 4, she tosses a coin once and notes whether
a head or tail is obtained. If she obtained exactly one head, what is the probability that
she threw 1, 2, 3 or 4 with the die?
17. A manufacturer has three machine operators A, B and C. The first operator A
produces 1% defective items, where as the other two operators B and C produce 5%
and 7% defective items respectively. A is on the job for 50% of the time, B is on the
job for 30% of the time and C is on the job for 20% of the time. A defective item is
produced, what is the probability that it was produced by A?
18. 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 be both diamonds. Find the probability of the lost
card being a diamond.
19. A laboratory blood test is 99% effective in detecting a certain disease when it is in
fact, present. However, the test also yields a false positive result for 0.5% of the
healthy person tested (i.e. if a healthy person is tested, then, with probability 0.005,
the test will imply he has the disease). If 0.1 percent of the population actually has the
disease, what is the probability that a person has the disease given that his test result is
positive ?
20. There are three coins. One is a two headed coin (having head on both faces), another
is a biased coin that comes up heads 75% of the time and third is an unbiased coin.
One of the three coins is chosen at random and tossed, it shows heads, what is the
probability that it was the two headed coin ?
21. An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck
drivers. The probability of an accidents are 0.01, 0.03 and 0.15 respectively. One of
the insured persons meets with an accident. What is the probability that he is a scooter
driver?
22. Two cards are drawn simultaneously (or successively without replacement) from a
well shuffled pack of 52 cards. Find the mean, variance and standard deviation of the
number of kings.
23. Find the variance of the number obtained on a throw of an unbiased die.
24. Let a pair of dice be thrown and the random variable X be the sum of the numbers that
appear on the two dice. Find the mean or expectation of X.
25. Find the probability distribution of number of doublets in three throws of a pair of
dice.
26. Two cards are drawn successively with replacement from a well-shuffled deck of 52
cards. Find the probability distribution of the number of aces.
27. Find the probability distribution of
(i) number of heads in two tosses of a coin.
(ii) number of tails in the simultaneous tosses of three coins.
(iii) number of heads in four tosses of a coin.
28. Find the probability distribution of the number of successes in two tosses of a die,
where a success is defined as
(i) number greater than 4
(ii) six appears on at least one die
29. From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at
random with replacement. Find the probability distribution of the number of defective
bulbs.
30. A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is
tossed twice, find the probability distribution of number of tails.
31. Two numbers are selected at random (without replacement) from the first six positive
integers. Let X denote the larger of the two numbers obtained. Find E(X).
32. Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the
variance and standard deviation of X.
33. A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17,
16, 19 and 20 years. One student is selected in such a manner that each has the same
chance of being chosen and the age X of the selected student is recorded. What is the
probability distribution of the random variable X? Find mean, variance and standard
deviation of X.
34. In a meeting, 70% of the members favour and 30% oppose a certain proposal. A
member is selected at random and we take X = 0 if he opposed, and X = 1 if he is in
favour. Find E(X) and Var (X).

3D Geometry

35. If a line makes angles 90°, 135°, 45° with the x, y and z-axes respectively, find its
direction cosines.
36. Find the direction cosines of a line which makes equal angles with the coordinate
axes.
37. If a line has the direction ratios –18, 12, – 4, then what are its direction cosines?
38. Show that the points (2, 3, 4), (– 1, – 2, 1), (5, 8, 7) are collinear.
39. Find the direction cosines of the sides of the triangle whose vertices are (3, 5, – 4), (–
1, 1, 2) and (– 5, – 5, – 2).
40. Find the vector and the Cartesian equations of the line through the point (5, 2, – 4)
and which is parallel to the vector 3iˆ 2 ˆj −8kˆ.
41. Find the vector equation for the line passing through the points (–1, 0, 2) and (3, 4, 6).
42. The Cartesian equation of a line is

Find the vector equation for the line.

43. Find the angle between the pair of lines given by

44. Find the angle between the pair of lines

45. Find the shortest distance between the lines l1 and l2 whose vector equations are
46. Find the distance between the lines l1 and l2 given by

47. Show that the line through the points (1, – 1, 2), (3, 4, – 2) is perpendicular to the line
through the points (0, 3, 2) and (3, 5, 6).
48. Show that the line through the points (4, 7, 8), (2, 3, 4) is parallel to the line through
the points (– 1, – 2, 1), (1, 2, 5).
49. Find the equation of the line which passes through the point (1, 2, 3) and is parallel to
the vector 3 iˆ 2 ˆj –2 kˆ .
50. Find the equation of the line in vector and in Cartesian form that passes through the
point with position vector 2 iˆ−j 4 kˆ and is in the direction iˆ 2 ˆj −kˆ.
51. Find the vector and the Cartesian equations of the line that passes through the points
(3, – 2, – 5), (3, – 2, 6).
52. Find the vector equation of the plane which is at a distance of 6/√29 from the origin
and its normal vector from the origin is 2iˆ −3 ˆj 4kˆ . Also find its cartesian
form.
53. Find the direction cosines of the unit vector perpendicular to the plane r (6 iˆ −3
ˆj −2 kˆ) 1 = 0 passing through the origin.
54. Find the distance of the plane 2x – 3y + 4z – 6 = 0 from the origin.
55. Find the coordinates of the foot of the perpendicular drawn from the origin to the
plane 2x – 3y + 4z – 6 = 0.
56. Find the vector and cartesian equations of the plane which passes through the point (5,
2, – 4) and perpendicular to the line with direction ratios 2, 3, – 1.
57. Find the vector equations of the plane passing through the points R(2, 5, – 3), S(– 2, –
3, 5) and T(5, 3,– 3).
58. Find the equation of the plane with intercepts 2, 3 and 4 on the x, y and z-axis
respectively.
59. Find the vector equation of the plane passing through the intersection of the planes r
(iˆˆj kˆ)6 and r (2iˆ3 ˆj 4kˆ) −5, and the point (1, 1, 1).
60. Show that the lines are coplanar.

61. Find the angle between the two planes 2x + y – 2z = 5 and 3x – 6y – 2z = 7 using
vector method.
62. Find the angle between the two planes 3x – 6y + 2z = 7 and 2x + 2y – 2z =5.
63. Find the distance of a point (2, 5, – 3) from the plane r . ( 6 iˆ −3 ˆj 2 kˆ ) = 4
64. Find the angle between the line and the plane 10 x + 2y – 11 z = 3

65. Find the coordinates of the point where the line through (5, 1, 6) and (3, 4,1) crosses
the YZ-plane.
66. Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses
the ZX-plane.
67. Find the coordinates of the point where the line through (3, – 4, – 5) and (2, – 3, 1)
crosses the plane 2x + y + z = 7.
68. Find the equation of the plane passing through the point (– 1, 3, 2) and perpendicular
to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.
69. If the points (1, 1, p) and (– 3, 0, 1) be equidistant from the plane r.(3 iˆ 4 jˆ
12 kˆ) 13 0, then find the value of p.
70. Find the equation of the plane passing through the line of intersection of the planes r .
(iˆ + ˆj + kˆ) =1  and r . (2 iˆ + 3 ˆj − kˆ) + 4 = 0 and parallel to x-axis.
71. If O be the origin and the coordinates of P be (1, 2, – 3), then find the equation of the
plane passing through P and perpendicular to OP.
72. Find the equation of the plane which contains the line of intersection of the planes
r (iˆ 2 ˆj 3 kˆ) −4 0 , r . (2 iˆ ˆj −kˆ) 5 0 and which is
perpendicular to the
plane r (5 iˆ 3 ˆj −6kˆ) 8 0 .

VECTOR ALGEBRA

73. Represent graphically a displacement of 40 km, 30° west of south.

74. which of the vectors are: (i) Collinear (ii) Equal (iii) Coinitial

75. Find the values of x, y and z so that the vectors and are equal. ;

76.

77. Find unit vector in the direction of vector

78. Find a vector in the direction of vector that has magnitude 7 units.

79. Find the unit vector in the direction of the sum of the vectors,

80. Write the direction ratio’s of the vector and hence calculate its direction cosines.

81. Find the vector joining the points P(2, 3, 0) and Q(– 1, – 2, – 4) directed from P to Q.
82. Consider two points P and Q with position vectors . Find the position vector of a point
R which divides the line joining P and Q in the ratio 2:1, (i) internally, and (ii)
externally.

83. Show that the points A(2iˆ −ˆj kˆ), B(iˆ −3 ˆj −5kˆ), C(3iˆ −4 j −4kˆ) are
the vertices of a right angled triangle.
84. Show that the vectors 2iˆ −3 ˆj 4kˆ and −4iˆ 6 ˆj −8kˆ are collinear
85. Show that the points A, B and C with position vectors, a = 3iˆ − 4 ˆj − 4kˆ, b=2iˆ − ˆj
+ kˆ and c=iˆ − 3 ˆj − 5kˆ , respectively form the vertices of a right angled triangle.
86. Find the projection of the vector iˆ −ˆj on the vector iˆ ˆj .
87. Find the projection of the vector iˆ 3 ˆj 7kˆ on the vector 7iˆ −ˆj 8kˆ .
88. Show that each of the given three vectors is a unit vector: Also, show that they are
mutually perpendicular to each other.

89.
90.

91. If the vertices A, B, C of a triangle ABC are (1, 2, 3), (–1, 0, 0), (0, 1, 2), respectively,
then find ABC. [ABC is the angle between the vectors BA and BC ] .
92. Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, –1) are collinear.
93. Show that the vectors 2iˆ −ˆj + kˆ, iˆ −3 ˆj −5kˆ and 3iˆ −4 ˆj −4kˆ form the vertices of
a right angled triangle.
94. Find the area of a parallelogram whose adjacent sides are given by the vectors

95. Find a unit vector perpendicular to each of the vectors(a + b) and (a-b)

96. Find the area of a triangle having the points A(1, 1, 1), B(1, 2, 3) and C(2, 3, 1) as its
vertices.

97. Find and µif

98. Find the area of the parallelogram whose adjacent sides are determined by the vectors

99. Write all the unit vectors in XY-plane.

100. Write down a unit vector in XY-plane, making an angle of 30° with the positive
direction of x-axis.
101. Find the scalar components and magnitude of the vector joining the points P(x1, y1,
z1) and Q(x2, y2, z2).
102. 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.
103. Find the value of x for which x(iˆ ˆj kˆ) is a unit vector.
104. Find a vector of magnitude 5 units, and parallel to the resultant of the vectors
105. Show that the points A(1, – 2, – 8), B(5, 0, –2) and C(11, 3, 7) are collinear, and find
the ratio in which B divides AC.
106. Find the position vector of a point R which divides the line joining two points P and
Q whose position vectors (2a+b) and (a-3b) are externally in the ratio 1 : 2. Also,
show that P is the mid point of the line segment RQ.
107. The two adjacent sides of a parallelogram are 2iˆ − 4 ˆj + 5kˆ and iˆ − 2 ˆj − 3kˆ .
Find the unit vector parallel to its diagonal. Also, find its area.
108. Show that the direction cosines of a vector equally inclined to the axes OX, OY and
OZ are

109. The scalar product of the vector iˆ + ˆj + kˆ with a unit vector along the sum of
vectors 2iˆ + 4 ˆj − 5kˆ and _iˆ + 2 ˆj + 3kˆ is equal to one. Find the value of .
110. If a, b, c are mutually perpendicular vectors of equal magnitudes, show that the c . d
=15
vector is equally inclined to a , b, c.
111.

Differential Equation

112. Determine order and degree (if defined) of differential equations

113. For each of the exercises given below, verify that the given function (implicit or
explicit) is a solution of the corresponding differential equation.

114.
115.

116.

117.

118.

119.

120.

121.

122. 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) is equal to the sum of the x coordinate
(abscissa) and the product of the x coordinate and y coordinate (ordinate) of that
point.

123. Find the particular solution of the differential equation given that y = 0 when x =π/2
 124. Find the general solution of the differential equation y dx – (x + 2y2) dy = 0.
125. Find the general solution of the differential equation

126. Find the general solution of the differential equation

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

128. Show that the differential equation is homogeneous and find its particular solution,
given that, x = 0 when y = 1.

129. Show that the differential equation is homogeneous and solve it.

130. Show that the differential equation (x – y) dy/dx= x + 2y is homogeneous and solve
it.
131. Find the general solution of the differential equation

132. Find the general solution of the differential equation

133. Find the particular solution of the differential equation dy/dx= −4xy2 given that y =
1, when x = 0.
134. Find the equation of the curve passing through the point (1, 1) whose differential
equation is x dy = (2x2 + 1) dx.
135. Find the equation of a curve passing through the point (–2, 3), given that the slope of
the tangent to the curve at any point (x, y) is 2x/y2.
136. In a bank, principal increases continuously at the rate of 5% per year. In how many
years Rs 1000 double itself?

Application of Integral
137. Find the area under the given curves and given lines: (i) y = x2,
x = 1, x = 2 and x-axis (ii) y = x4, x = 1, x = 5 and x-axis
138. Find the area between the curves y = x and y = x2.
139. Find the area of the region lying in the first quadrant and
bounded by y = 4x2, x = 0, y = 1 and y = 4.
140. Sketch the graph of y = x 3 and evaluate .
141. Find the area bounded by the curve y = sin x between x = 0
and x = 2π.
142. Find the area enclosed between the parabola y2 = 4ax and the
line y = mx.
143. Find the area enclosed by the parabola 4y = 3x2 and the line 2y
= 3x + 12.
144.Find the area of the smaller region bounded by the ellipse

and the line

145.Find the area of the smaller region bounded by the ellipse

and the line

146.Find the area of the region enclosed by the parabola x2 = y, the line y = x
+ 2 and the x-axis.
147.Using the method of integration find the area bounded by the curve |x|
y| 1.
[Hint: The required region is bounded by lines x + y = 1, x– y = 1, – x + y = 1
and – x – y = 1].
148.Find the area bounded by curves {(x, y) : y x2 and y = | x |}.
149.Using the method of integration find the area of the triangle ABC,
coordinates of whose vertices are A (2, 0), B (4, 5) and C (6, 3).
150.Using the method of integration find the area of the region bounded by
lines: 2x + y = 4, 3x – 2y = 6 and x – 3y + 5 = 0
151.Find the area of the region {(x, y) : y2 4x, 4x2 + 4y2 9}