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A problem in statistics is given to five students A, B, C and D. Their chances of solving it are 1/3, 1/4, 1/5 and 1/6 respectively. What is the probability that the problem will be solved?

Probability of A solving = 1/3 P (A not solving) = 1 - 1/3 = 2/3 Similarly, P (B not solving) = 1 - 1/4 = 3/4 P (C not solving) = 1 -... View Article

If two tangents drawn from a point P to the parabola y^2=4x are at right angles, then the locus of P is (a) 2x+1=0 (b) x=-1 (c) 2x-1=0 (d) x=1

Answer: (b) x=-1 If tangents are perpendicular from point P then point P lies on directrix of parabola Locus of P = directrix of y2=4x... View Article

The graph of which of the following equations passes through the origin? (i) y = 2x + c (ii) y = 2x – c (iii) y = 2x

Answer: (iii) y = 2x If x = 0, then y = 2(0) = 0... View Article

Two varieties of oil are mixed in the ratio 4:3 to produce first quality and if they are mixed in the ratio 2:3, Second quality is obtained. How many kg of the first quality be mixed with 10 kg of the second quality so that a third quality having the 2 varieties in the ratio 5:4 may be produced?

Let the common ratio of the first quality oil be x. Given, the ratio is 4:3 Therefore, the total quantity of the first quality oil = 4x+3x =... View Article

Find the range of eccentricity of the ellipse (x^2/a^2) + (y^2/b^2) = 1 (a>b) such that the line segment joining the foci does not subtend a right angle at any point on the ellipse

Let the point on ellipse be P (a cosθ, b sinθ) Let A and B are the diameter of the ellipse If ∠APB = π/2, then P... View Article

A watch loses 3% time during the first week and then gains 2% time during the next one week. If the watch was set right at 12 noon on a Sunday, what will be the time that the watch will show exactly 14 days from the time it was set right ?

Number of hours in a week = 7 x 24 = 168 hours The watch loses 3% time during the first week. Therefore, 3% of 168 = 5.04 hours During the... View Article

If √(4 + √(8 – √(32 + √768))) = a√2 cos(11π/b), where a and b are natural numbers then (b – a) is divisible by:

Suppose, √768 = 32 cos θ 16√3 = 32 cos θ cos θ = √3/2 θ =... View Article

Let A be the set of all 3 × 3 skew-symmetric matrices whose entries are either −1, 0, or 1. If there are exactly three 0s, three 1s, and three (−1)s, then what is the number of such matrices?

Skew symmetric matrix can be written in the form of: Given, λ1, λ2, λ3 ∈{1,−1} Therefore,... View Article

If t_1,t_2 and t_3 are distinct, the points (t_1,2at_1 + at_1^3), (t_2, 2at_2 + at_2^3) and (t_3, 2at_3 + at_3^3) are collinear. Then what is the value of t_1 + t_2 + t_3?

Let the points are A(t1,2at1 + at13), B(t2, 2at2 + at23) and C(t3, 2at3 + at33) Given, A, B and C are collinear Area of triangle ABC = 0... View Article

For natural numbers m and n, if (1-y)^m (1+y)^n = 1 + a1y + a^2y^2+…, and a_1 = a_2 = 10, then (m,n) is:

Given, For natural numbers m and n, if (1-y)m (1+y)n = 1 + a1y + a2y2+..., and a1 = a2 = 10, Therefore, a1 = n - m = 10 and a2 = (m2-... View Article

If A, B where 0° < A, B < 180° and Sin A+ Sin B = √(3/2), cos A + cos B = √(1/2), then what is the value of (A+B) ?

Given, Sin A + Sin B = √(3/2) 2 sin(A+B)/2 cos(A-B)/2 = √(3/2) ..........(i) Also, given, cos A + cos B =... View Article

A cardboard of size 18 cm x 18 cm, two colour papers say pink and blue, pair of dice, empty box, pair of scissors, sketch pens, fevicol, etc.

Given, A cardboard of size 18 cm x 18 cm, two colour papers say pink and blue, pair of dice, empty box, pair of scissors, sketch pens,... View Article

Suppose is a 3 × 3 matrix. If for every column vector X, X^T AX=0 and a_23 =−1008,then the sum of the digits of a_32 is:

XTAX = 0 XTAX = (XTAX)T = XTATX = -XTAX = XTAX = 0 Therefore, XTATX = -XTAX AT = -A a23 = -a32 or a32 = -a23 a32 = - (-1008) = 1008 Sum... View Article

The value of 4(cos² 60° + cos²45°-sin 30°) is:

4(cos² 60° + cos²45°)-sin2 30° = 4(1/2)2 + 4(1/√2)² - (√3/2)² = 1 + 2 - 3/4 = 3 - 3/4 = 9/4... View Article

Find all three-digit numbers with different digits, with each digits odd.

Total number of digits are 10 (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) Total number of odd digits = 5 A three digit number is 100a+10b+c Here, a = 5, b... View Article

Let A and B be two non-singular matrices such that A⁶ = I and AB² = BA (B ≠ I). If B^m = I (where I is the identity matrix), then the least value of m is:

Answer: A square matrix (m = n) that is not invertible is called singular or degenerate. A square matrix is singular if and only if its... View Article

The edges of a triangular board are 3m, 4m and 5m. Find the cost of white washing the walls of the room and the ceiling at the rate of Rs. 10 per sq m.

Length, breadth and height of a room are 5m, 4m and 3m, respectively. Area of four walls of the room = 2h(l+b) = 2 × 3(5+4) =... View Article

If I1, I2, I3 are excentres of the triangle with vertices (0,0), (5,12), (16, 12) then find the orthocentre of ∆I1I2I3.

Given, I1, I2, I3 are the excentres of a triangle ABC. The vertices of ∆ABC are (0,0), (5,12) and (16, 12). Orthocenter... View Article

Two poles of height 6m and 11m stand on a plane ground of the distance between their feet 12m, find the distance between their tops.

Given, Heights of the poles are 6m and 11m respectively. Distance between their foot = 12m Suppose AB and CD are two poles of 11 m and 6 m... View Article

If ƒ(x) = a(x² – 1)(ax b), a ≠ 0 has local maximum and minimum at x1 and x2 respectively, then f'(x) is:

Given, f(x) = a(x2 - 1) (ax + b) f(x) = 0 at x = ±, x = -b/a f'(x) = a[(2x-1) (ax+b) + (x2-1)a] At x = 1 f'(x) = a [a + b + a]... View Article

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