# Maths

A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is:

1. 12/5
2. 6
3. 4
4. 6/25

If two different numbers are taken from the set {0, 1, 2, 3, ..., 10}; then the probability that their sum as well as absolute difference are both multiple of 4, is

1. 6/55
2. 12/55
3. 14/45
4. 7/55

The eccentricity of an ellipse whose centre is at the origin is ½. If one of its directrices is x = -4, then the equation of the normal to it at (1, 3/2) is

1. 2y - x = 2
2. 4x - 2y = 1
3. 4x + 2y = 7
4. x + 2y = 4

For any three positive real numbers a, b and c, 9(25a2 + b2) + 25(c2 – 3ac) = 15b(3a + c). Then

1. b, c and a are in G.P.
2. b, c and a are in A.P.
3. a, b and c are in A.P.
4. a, b and c are in G.P.

If  then adj (3A2 + 12A) is equal to

Let a vertical tower AB have its end A on the level ground. Let C be the midpoint of AB and P be a point on the ground such that AP = 2AB. If ∠BPC = β, then tan β is equal to:

1. 6/7
2. 1/4
3. 2/9
4. 4/9

If (2 + sin x) dy/dx + (y + 1)cos x = 0 and y(0) = 1, then y(π/2) is equal to

1. 1/3
2. -2/3
3. -1/3
4. 4/3

If the image of the point P(1, –2, 3) in the plane, 2x + 3y – 4z + 22 = 0 measured parallel to the line, x/1 = y/4 = z/5 is Q, then PQ is equal to:

1. 3√5
2. 2√42
3. √42
4. 6√5

The area (in sq. units) of the region {(x, y) : x ≥ 0, x + y ≤ 3, x2 ≤ 4y and y ≤ 1 + √x } is

1. 59/12
2. 3/2
3. 7/3
4. 5/2

Let k be an integer such that the triangle with vertices (k, –3k), (5, k) and (–k, 2) has area 28 sq. units. Then the orthocentre of this triangle is at the point

1. 2, -1/2
2. 1, 3/4
3. 1, -3/4
4. 2, 1/2

For three events A, B and C, P(Exactly one of A or B occurs) = P(Exactly one of B or C occurs) = P(Exactly one of C or A occurs) = 1/4 and P(All the three events occur simultaneously) = 1/16. Then the probability that at least one of the events occurs, is

1. 7/32
2. 7/16
3. 7/64
4. 3/16

If 5(tan2 x – cos2x) = 2 cos 2x + 9, then the value of cos 4x is

1. -3/5
2. 1/3
3. 2/9
4. -7/9

The following statement (p → q) → [(~p→q)→q] is:

1. a tautology
2. equivalent to ~p → q
3. equivalent to p → ~q
4. a fallacy

If S is the set of distinct values of 'b' for which the following system of linear equations
x + y + z = 1
x + ay + z = 1
ax + by + z = 0
has no solution, then S is:

1. an empty set
2. an infinite set
3. a finite set containing two or more elements
4. a singleton

If 0 ≤ x < 2π, then the number of real values of x, which satisfy the equation

cos x + cos 2x + cos 3x + cos 4x = 0 is

1. 3
2. 5
3. 7
4. 9

The sum of the radii of inscribed and circumscribed circles for an n sided regular polygon of side a, is

1. a/2 cot(π/2n)
2. a/4 cot(π/2n)
3. a cot(π/n)
4. a cot(π/2n)

Let cos(a + b) = 4/5 and let sin(a - b) = 5/13 where 0 ≤ a,b ≤ π/4. Then tan 2a is

1. 20/17
2. 25/16
3. 56/33
4. 19/12

In a ∆PQR, if 3 sin P + 4 cos Q = 6 and 4 sin Q + 3 cos P = 1, then the angle R is equal to

1. π/4
2. 3π/4
3. 5π/6
4. π/6

If A = sin2 x + cos4 x, then for all real x

1. 3/4 ≤ A ≤ 1
2. 1 ≤ A ≤ 2
3. 3/4 ≤ A ≤ 13/16
4. 13/16 ≤ A ≤ 1

Let fk(x) = 1/k(sink x + cosk x) where x ∈ R and k ≥ 1. Then f4(x) - f6(x) equals

1. 1/12
2. 1/6
3. 1/4
4. 1/3

The mean and variance of a random variable X having binomial distribution are 4 and 2 respectively, then P (X = 1) is

1. 1/4
2. 1/8
3. 1/16
4. 1/32

Consider 5 independent Bernoulli's trials each with probability of success P. If the probability of at least one failure is greater than or equal to 31/32, then P lies in the interval

1. (3/4,11/2]
2. (11/2,1]
3. (1/2,3/4]
4. [0,1/2]

Three numbers are chosen at random without replacement from {1, 2, 3, ...... 8}. The probability that their minimum is 3, given that their maximum is 6, is

1. 2/5
2. 1/4
3. 3/8
4. 1/5

Assuming the balls to be identical except for difference in colours, the number of ways in which one or more balls can be selected from 10 white, 9 green and 7 black balls is

1. 880
2. 879
3. 629
4. 630

One ticket is selected at random from 50 tickets numbered 00, 01, 02, ... , 49. Then the probability that the sum of the digits on the selected ticket is 8, given that the product of these digits is zero, equals

1. 1/7
2. 1/14
3. 5/14
4. 1/50

Five horses are in a race. Mr.A selects two of the horses at random and bets on them. The probability that Mr.A selected the winning horse is

1. 1/5
2. 2/5
3. 3/5
4. 4/5

If the mean deviation about the median of the numbers a, 2a, ..., 50a is 50, then |a| equals

1. 2
2. 3
3. 4
4. 5

The average marks of boys in class is 52 and that of girls is 42. The average marks of boys and girls combined is 50. The percentage of boys in the class is

1. 20
2. 40
3. 60
4. 80

If the mean deviation of the numbers 1, 1 + d, 1 + 2d, ... , 1 + 100d from their mean is 255, then the d is equal to

1. 10
2. 10.1
3. 20
4. 20.2

In a frequency distribution, the mean and median are 21 and 22 respectively, then its mode is approximately

1. 25.5
2. 24.0
3. 22.0
4. 20.5

The angle between the lines whose direction cosines satisfy the equations l + m + n = 0 and l2 + m2 + n2 = 0 is

1. π/2
2. π/3
3. π/4
4. π/6

If the lines (x - 1)/2 = (y + 1)/2 = (z - 1)/4 and (x - 3)/1 = (y - k)/2 = z/1 intersect, then k is equal to

1. 9/2
2. 2/9
3. -1
4. 0

If the angle between the line x = (y - 1)/2 = (z - 3)/λ and the plane x + 2y + 3z = 4 is cos-1(5/14) then λ is equal to

1. 3/2
2. 2/5
3. 2/3
4. 5/3

A line AB in three-dimensional space makes angles 45° and 120° with the positive x-axis and the positive y-axis respectively. If AB makes an acute angle θ with the positive z-axis, then θ equals

1. 30°
2. 45°
3. 60°
4. 75°

Let the line (x - 2)/3 =  (y - 1)/-5 =  (z - 2)/2 lie in the plane x + 3y - αz + β = 0. Then (α, β) equals

1. (-5, 5)
2. (5, -15)
3. (-6, 7)
4. (6, -17)

The distance of the point (1,0,2) from the point of intersection of the line (x - 2)/3 = (y + 1)/4 = (z - 2)/12 and the plane x - y + z = 16 is

1. 13
2. 8
3. 3√21
4. 2√14

Let P be the point on the parabola, y2 = 8x which is at a minimum distance from the centre C of the circle, x2 + (y + 6)2 = 1. Then the equation of the circle, passing through C and having its centre at P is:

1. x2 + y2 – x/4 + 2y – 24 = 0
2. x2 + y2 – 4x + 8y + 12 = 0
3. x2 + y2 – 4x + 9y + 18 = 0
4. x2 + y2 – x + 4y – 12 = 0

The locus of the foot of perpendicular drawn from the center of the ellipse x2 + 3y2 = 6 on any tangent to it is

1. (x2 - y2)2 = 6x2 + 2y2
2. (x2 - y2)2 = 6x2 - 2y2
3. (x2 + y2)2 = 6x2 - 2y2
4. (x2 + y2)2 = 6x2 + 2y2

An ellipse is drawn by taking a diameter of the circle (x - 1) + y = 1 as its semi-minor axis and a diameter of the circle x2 + (y - 2)2 = 4 as its semi-major axis. If the centre of the ellipse is the origin and its axes are the coordinate axes, then the equation of the ellipse is

1. x2 + 4y2 = 8
2. 4x2 + y2 = 4
3. x2 + 4y2 = 16
4. 4x2 + y2 = 8

The slope of the line touching both the parabolas y2 = 4x and x2 = -32y is

1. 1/2
2. 2/3
3. 3/2
4. 1/8

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal half of the distance between its foci, is:

1. 4/√3
2. √3
3. 4/3
4. 2/√3

Let O be the vertex and Q be any point on the parabola, x2 = 8y. If the point P divides the line segment OQ internally in the ratio 1 : 3, then the locus of P is

1. x2 = y
2. y2 = x
3. x2 = 2y
4. y2 = 2x

Let a, b, c and d be non-zero numbers. If the point of intersection of the lines 4ax + 2ay + c = 0 and 5bx + 2by + d = 0 lies in the fourth quadrant and is equidistant from the two axes then

1. 3bc - 2ad = 0
2. 3bc + 2ad = 0
3. 2bc + 3ad = 0
4. 2bc - 3ad = 0

If the pair of straight lines x2 - 2pxy - y2 = 0 and x2 - 2qxy - y2 = 0 be such that each pair bisects the angle between the other pair, then

1. pq = -1
2. p = -q
3. pq = 1
4. p = q

The shortest distance between the line y - x = 1 and curve y = x2 is

1. √3/4
2. 4/√3
3. 8/3√2
4. 3√2/8

If the line 2x + y = k passes through the point which divides the line segment joining the points (1, 1) and (2, 4) in the ratio 3:2, then k equals

1. 5
2. 6
3. 29/5
4. 11/5

The lines 2x - 3y = 5 and 3x - 4y = 7 are diameters of a circle having area as 154 sq. units. Then the equation of the circle is

1. x2 + y2 + 2x + 2y = 47
2. x2 + y2 + 2x - 2y = 47
3. x2 + y2 - 2x + 2y = 62
4. x2 + y2 - 2x + 2y = 47

The two circles x2 + y2 = ax and x2 + y2 = c2 (c > 0) touch each other if

1. a = 2c
2. |a| = 2c
3. |a| = c
4. 2|a| = c

The differential equation which represents the family of curves y = c1ec2x where c1 and c2 are arbitrary constants, is

1. yy'' = y'
2. y'' = y'y
3. y' = y2
4. yy'' = (y')2

Let y(x) be the solution of the differential equation (x log x)(dy/dx) + y = 2x log(x), (x≥1). Then y(e) is equal to

1. 0
2. 2e
3. e
4. 2