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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 = cos-1 √3/2 θ = π/6 Therefore, \(\begin{array}{l}\sqrt{4+\sqrt{8-\sqrt{32+32 \cos \frac{\pi}{6}}}} \end{array}... 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: \(\begin{array}{l}\left[\begin{array}{ccc}0 & \lambda_{1} & \lambda_{2} \\-\lambda_{1} & 0 & \lambda_{3}... 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... 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... 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,... 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 \(\begin{array}{l}A=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33}\end{array}\right]\end{array} \) XTAX... 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 –... 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... 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... 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... 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).... 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... 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) =... View Article

arctan x/√(a² – x²) = ?

Given, \(\begin{array}{l}\tan ^{-1} \frac{x}{\sqrt{a^{2}-x^{2}}}\end{array} \) Let us say, x = a sin θ \(\begin{array}{l}=\tan ^{-1}\left(\frac{\mathrm{a} \sin \theta}{\sqrt{\mathrm{a}^{2}-(\mathrm{a} \sin \theta)^{2}}}\right)\end{array} \) \(\begin{array}{l}=\tan... View Article

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