List of top Mathematics Questions

Integrate \( \log_e x \) with respect to \( x \).
Evaluate: \( \int \frac{1}{\sqrt{a^2 - x^2}} \, dx \).
Find the area of the parallelogram whose adjacent sides are represented by the vectors \( \vec{a} = 3\hat{i} + \hat{j} + 2\hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{j} - 2\hat{k} \).
A relation \( R \) is defined in \( \mathbb{N} \times \mathbb{N} \) as follows: \[ (a, b) \, R \, (c, d) { if and only if } ad = bc. \] Prove that \( R \) is an equivalence relation.
Solve the differential equation \( \frac{dy}{dx} + y = 1 \) (\( y \neq 1 \)).

A die is thrown once. If E represents the event ‘the number obtained on the die is a multiple of 3’ and F represents the event ‘the number obtained on the die is even’, then tell whether the events E and F are independent.

Prove that: \( 3 \sin^{-1} x = \sin^{-1} (3x - 4x^3) \), \( x \in \left[ -\frac{1}{2}, \frac{1}{2} \right] \).
Show that the points \( A(2, 3, -4), B(1, -2, 3) \) and \( C(3, 8, -11) \) are collinear.
Solve: \( y \, dx - (x + 2y^2) \, dy = 0 \).
If the lines \[ \frac{x - 1}{3} = \frac{y - 2}{2k} = \frac{z - 3}{2} \quad {and} \quad \frac{x - 1}{3k} = \frac{y - 1}{1} = \frac{z - 6}{-5} \] are perpendicular, find the value of \( k \).
A die was thrown twice and it was found that the sum of the numbers that appeared was 6. Find the conditional probability that the number 4 appeared at least once.
Maximize \( z = 8000x + 12000y \) subject to constraints: \[ 3x + 4y \leq 60, \quad x + y \leq 30, \quad x \geq 0, \quad y \geq 0. \]
Let \[ \vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}, \quad \vec{b} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k}, \quad \vec{c} = c_1 \hat{i} + c_2 \hat{j} + c_3 \hat{k}, \] show that \[ \vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}. \]
Solve: \( \frac{dy}{dx} = \frac{x^2 + y^2}{2xy} \).
Prove that the radius of the right circular cylinder of maximum curved surface inscribed in a cone is half of the radius of the cone.
Prove: \[ \int_0^{\frac{\pi}{4}} \log_e (1 + \tan x) \, dx = \frac{\pi}{8} \log_e 2. \]
Evaluate: \[ \int \frac{x^2 + x + 1}{(x + 2)(x^2 + 1)} \, dx. \]
Global maximum value of the function \( f(x) = \sin x + \cos x, \quad x \in [0, \pi] \) is: ...

If the value of \( \cos \alpha \) is \( \frac{\sqrt{3}}{2} \), then \( A + A = I \), where \[ A = \begin{bmatrix} \sin\alpha & -\cos\alpha \\ \cos\alpha & \sin\alpha \end{bmatrix}. \]

If \( A = \begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{pmatrix} \), find \( I + A^2 \), where \( I \) is the identity matrix.
Evaluate the integral \( \int \frac{1}{\sqrt{4x - x^2}} \, dx \).
Evaluate the integral \( \int e^x \left( \frac{1 + \sin x}{1 - \sin x} \right) \, dx \).
Evaluate \( \int \frac{e^{2x} - 1}{e^{2x} + 1} \, dx \).
If \( A \) is a square matrix such that \( A^2 = A \), then \( (I - A)^3 - (I + A)^2 = \) __________.
Find \( \sin^{-1} \left( \sin \frac{23\pi}{6} \right) = \) __________.