List of top Mathematics Questions

Two lines $L_1:x = 5\, \frac{y}{3-\alpha} = \frac{z}{-2}$ and $L_1:x = \alpha \, \frac{y}{-1} =\frac{z}{2-\alpha}$ are coplaner. Then, $\alpha$ can take value(s)

A multiple choice examination has $5$ questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get $4$ or more correct answers just by guessing is

Circle(s) touching X-axis at a distance 3 from the origin and having an intercept of length $2 \sqrt7$ on Y-axis is/are

The inverse of the proposition $(p \wedge \sim q)\rightarrow r$ is

If \( f(x) = \left\{ \begin{array}{ll} mx + 1, & \text{for } x \leq \frac{\pi}{2}, \\ \sin x + n, & \text{for } x>\frac{\pi}{2}, \end{array} \right. \) is continuous at \( x = \frac{\pi}{2} \), then:

The value of the determinant \[ \begin{vmatrix} \cos \alpha & -\sin \alpha & 1 \\ \sin \alpha & \cos \alpha & 1 \\ \cos(\alpha + \beta) & -\sin(\alpha + \beta) & 1 \end{vmatrix} \] is:

The general solution of the differential equation \[ (1 + y^2) \, dx + (1 + x^2) \, dy = 0 \] is:

The order and degree of the differential equation \[ \rho = \frac{\left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^{3/2}}{\frac{d^2y}{dx^2}}, \] are respectively:

The domain of the function \[ f(x) = \frac{\sqrt{4 - x^2}}{\sin^{-1}(2 - x)} \] is:

The relation \( R \) defined on the set of natural numbers as \( (a, b) : a \) differs from \( b \) by 3 is given by:

The area lying in the first quadrant and bounded by the circle \( x^2 + y^2 = 4 \), the line \( x = \sqrt{3}y \), and the \( x \)-axis is:

The value of \[ \lim_{x \to \infty} \left( \frac{\pi}{2} - \tan^{-1} x \right)^{1/x} \] is:

The sum of the series \[ 1 + \frac{1^2 + 2^2}{2!} + \frac{1^2 + 2^2 + 3^2}{3!} + \frac{1^2 + 2^2 + 3^2 + 4^2}{4!} + \dots \] is:

The coefficient of \( x^n \) in the expansion of \( \log_a(1 + x) \) is:

If a plane meets the coordinate axes at \( A \), \( B \), and \( C \) in such a way that the centroid of \( \triangle ABC \) is at the point \( (1, 2, 3) \), then the equation of the plane is:

There are 5 letters and 5 different envelopes. The number of ways in which all the letters can be put in wrong envelopes is:

The value of the integral \[ I = \int_0^1 \frac{\sqrt{1 - x}}{\sqrt{1 + x}} \, dx \] is:

\[ \int \frac{dx}{\sin x - \cos x + \sqrt{2}} = ? \]

The value of the integral \[ I = \int_0^1 \left| x - \frac{1}{2} \right| \, dx \] is:

The eccentricity of the ellipse, which meets the straight line \( \frac{x}{7} + \frac{y}{2} = 1 \) on the axis of \( x \) and the straight line \( \frac{x}{3} - \frac{y}{5} = 1 \) on the axis of \( y \), and whose axes lie along the axes of coordinates, is:

If \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a>b) \] and \[ x^2 - y^2 = c^2 \] cut at right angles, then:

The equation of the conic with focus at \( (1, -1) \), directrix along \( x - y + 1 = 0 \), and eccentricity \( \sqrt{2} \) is:

The maximum value of \( 4 \sin^2 x - 12 \sin x + 7 \) is:

If \[ A = \begin{pmatrix} 1 & -5 & 7 \\ 0 & 7 & 9 \\ 11 & 8 & 9 \end{pmatrix}, \] then the trace of matrix \( A \) is:

A straight line through the point \( A(3, 4) \) is such that its intercept between the axes is bisected at \( A \). Its equation is: