List of top Mathematics Questions

The range of the function \( f(x) = \frac{1}{7 + 4\sin x + 3\cos x} \).

Evaluate \( \int \frac{x^2 + 6x + 1}{(x+3)^2} \, dx \).

If \( \alpha^2 - \frac{1}{\alpha^2} = 2 \), find \( \left( \alpha + \frac{1}{\alpha} \right)^{16} \).

If \( 2 \cot^{-1} \left( \frac{4}{3} \right) = \cos^{-1} \left( \frac{x}{5} \right) \), find \( x \).

If \( Z = 1 + i \) and \( Z - 24\overline{Z} = \lambda Z^2 \), find \( \lambda \).

Find the value of \( \cot 10^\circ \times \cot 30^\circ \times \cot 45^\circ \times \cot 60^\circ \times \cot 80^\circ \)

Find the locus of \( Z = x + iy \) satisfying \[ \frac{\text{Re}(z)}{2 + i} + \frac{\text{Im}(z)}{1 + 2i} = \frac{3}{1 - 2i} \]

If \( y = \frac{1}{3\sqrt{x}} \left( \frac{2}{x} - 3 \right) \), find the interval in which \( y \) is strictly decreasing.

If the sum of the first two terms of a G.P. is 12 and the third term is 16, find the common ratio \( r \).

Evaluate \( \lim_{x \to 1} \frac{\sqrt{x+3} \cdot \sqrt{x-1}}{x-1} \)

If \( f(x) = x^2 - 10x \), \( g(x) = e^x + 5 \), find \( g(2x) - f(g(x)) \).

The coefficient of \( x^3 \) in \( (2+x)^n \) is 160. Find the coefficient of \( x^6 \) in \( (2-x^2)^n \).

Given the numbers 4, 7, \( x \), 13, 16, with a mean of 10, find the mean deviation about the mean.

Given \( (3\cos x - 2\sec x)^2 = 9\cos^2 x + 4\tan^2 x + k \), find \( k \).

Max of \( Z = 7x + 10y \) subject to \( x + y \geq 3 \), \( x + 2y \geq 4 \), \( x, y \geq 0 \).

If \( |\mathbf{a} - \mathbf{b}| = \frac{\sqrt{3}}{2} \) where \( \mathbf{a} \) and \( \mathbf{b} \) are unit vectors, find the angle between \( \mathbf{a} \) and \( \mathbf{b} \).

If 2 vectors \( 4\hat{i} + \ell \hat{j} - 6 \hat{k} \) and \( -6 \hat{i} + 12 \hat{j} + 9 \hat{k} \) are collinear, find \(\ell\).

Find \(\int \frac{x \cos 2x}{\cos x - \sin x} \, dx\)

Given \( y = \frac{1}{1 + \tan x} \), find \( f^{-1}(x) \), where \( 0<x<\frac{\pi}{2} \).

Evaluate \( \lim_{x \to 0} \frac{x - \tan(3x)}{\sin(2x)} \)

Arithmetic mean & Geometric mean of 2 numbers \( a \) & \( b \) in the ratio 5:3. Find \( \frac{a^2 + b^2}{ab} \).

If the area of the circle \( x^2 + y^2 + 8x - 6y + c = 0 \).

Maximum of \( f(x) = \alpha - 4x - x^2 \), find \(\alpha = ?\)

If \( n(B) = 61 \), \( n(A \cup B) = 99 \), \( n(A \cap B) = 28 \), find \( n(A') \).

If \( (2 - x)^9 = a_0 + a_1x + \dots + a_9x^9 \), find \( a_1 + a_2 + \dots + a_8 \)