List of top Mathematics Questions

A relation \( R \) is defined on \( \mathbb{N} \times \mathbb{N} \) (where \( \mathbb{N} \) is the set of natural numbers) as: \[ (a, b) \, R \, (c, d) \iff a - c = b - d. \] Show that \( R \) is an equivalence relation.
Find the particular solution of the differential equation given by: \[ 2xy + y^2 - 2x^2 \frac{dy}{dx} = 0; \quad y = 2, \, \text{when } x = 1. \]
Find the general solution of the differential equation: \[ y \, dx = (x + 2y^2) \, dy. \]
The position vectors of vertices of \( \Delta ABC \) are \( A(2\hat{i} - \hat{j} + \hat{k}) \), \( B(\hat{i} - 3\hat{j} - 5\hat{k}) \), and \( C(3\hat{i} - 4\hat{j} - 4\hat{k}) \). Find all the angles of \( \Delta ABC \).
A pair of dice is thrown simultaneously. If \( X \) denotes the absolute difference of the numbers appearing on top of the dice, then find the probability distribution of \( X \).
Find: \[ \int \frac{1}{x \left[(\log x)^2 - 3 \log x - 4\right]} \, dx. \]
Evaluate: \[ \int_{-2}^{2} \sqrt{\frac{2 - x}{2 + x}} \, dx. \]
Show that: \[ \frac{d}{dx} \left(|x|\right) = \frac{x}{|x|}, \quad x \neq 0. \]
If \( x = e^{\cos 3t} \) and \( y = e^{\sin 3t} \), prove that \( \frac{dy}{dx} = -\frac{y \log x}{x \log y} \).
Find: \[ \int \frac{e^{4x} - 1}{e^{4x} + 1} \, dx. \]
If \( y = \csc(\cot^{-1} x) \), then prove that \( \sqrt{1 + x^2} \frac{dy}{dx} - x = 0 \).
Find the domain of the function \( f(x) = \sin^{-1(x^2 - 4) \). Also, find its range.}
If \( f(x) = |\tan 2x| \), then find the value of \( f'(x) \) at \( x = \frac{\pi}{3} \).
Assertion (A): For two non-zero vectors \( \vec{a} \) and \( \vec{b} \), \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} \).
Reason (R): For two non-zero vectors \( \vec{a} \) and \( \vec{b} \), \( \vec{a} \times \vec{b} = -\vec{b} \times \vec{a} \).
If a line makes an angle of \( 30^\circ \) with the positive direction of \( x \)-axis, \( 120^\circ \) with the positive direction of \( y \)-axis, then the angle which it makes with the positive direction of \( z \)-axis is:
Assertion (A): For any symmetric matrix \( A \), \( B'A B \) is a skew-symmetric matrix.
Reason (R): A square matrix \( P \) is skew-symmetric if \( P' = -P \).
Direction ratios of a vector parallel to the line \( \frac{x - 1}{2} = -y = \frac{2z + 1}{6} \) are:
The unit vector perpendicular to both vectors \( \hat{i} + \hat{k} \) and \( \hat{i} - \hat{k} \) is:
The degree of the differential equation \( (y'')^2 + (y')^3 = x \sin(y') \) is:
The derivative of \( \tan^{-1}(x^2) \) w.r.t. \( x \) is:
If \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix, where \( a_{ij} = i - 3j \), then which of the following is \textbf{false?}
Let \( E \) be an event of a sample space \( S \) of an experiment, then \( P(S | E) \) is:
The differential equation \( \frac{dy}{dx} = F(x, y) \) will not be a homogeneous differential equation, if \( F(x, y) \) is:
The common region determined by all the constraints of a linear programming problem is called:
For any two vectors \( \vec{a} \) and \( \vec{b} \), which of the following statements is always true?