List of top Mathematics Questions

Evaluate \( \int_{0}^{\pi/4} \frac{\cos^2 x}{\cos^2 x + 4 \sin^2 x} \, dx \):

Let \( f(x) = \frac{2 - \sqrt{x + 4}}{\sin 2x}, \, x \neq 0 \). In order that \( f(x) \) is continuous at \( x = 0 \), \( f(0) \) is to be defined as:

Evaluate \( \sin \left( \tan^{-1}\frac{4}{5} + \tan^{-1}\frac{4}{3} + \tan^{-1}\frac{1}{9} - \tan^{-1}\frac{1}{7} \right) \):
The line whose vector equations are \(\vec{r}_1 = 2\hat{i} - 3\hat{j} + 7\hat{k} + \lambda(2\hat{i} + p\hat{j} + 5\hat{k})\) and \(\vec{r}_2 = \hat{i} + 2\hat{j} + 3\hat{k} + \mu(3\hat{i} - p\hat{j} + p\hat{k})\) are perpendicular for all values of \(\lambda\) and \(\mu\). The value of \(p\) is:
The solution of the differential equation \( y^2 dx + (x^2 - xy + y^2) dy = 0 \) is
The integral \( \int e^x \frac{2 + \sin 2x}{1 + \cos 2x} \, dx \) is equal to

If \( X \) is a random variable such that \( P(X = -2) = P(X = -1) = P(X = 2) = P(X = 1) = \frac{1}{6} \), and \( P(X = 0) = \frac{1}{3} \), then the mean of \( X \) is

Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k}, \, \vec{b} = \hat{i} + 3\hat{j} + 5\hat{k} \) and \( \vec{c} = 7\hat{i} + 9\hat{j} + 11\hat{k} \). Then the area of a parallelogram having diagonals \( \vec{a} + \vec{b} \) and \( \vec{b} + \vec{c} \) is:
The order and degree of the differential equation \( \sqrt{\frac{dy}{dx}} - 4 \frac{dy}{dx} - 7x = 0 \) are respectively

The value of : \( \int \frac{x + 1}{x(1 + xe^x)} dx \).

The value of \( \int \frac{x \sin^{-1} x}{\sqrt{1 - x^2}} dx \) is equal to:
The tangent at the point \( (x_1, y_1) \) on the curve \( y = x^3 + 3x^2 + 5 \) passes through the origin. Then \( (x_1, y_1) \) does NOT lie on the curve:
If \( x = \frac{1 - t^2}{1 + t^2} \) and \( y = \frac{2t}{1 + t^2} \), then \( \frac{dy}{dx} \) is equal to:
The Boolean expression \( (\sim(p \land q)) \lor q \) is equivalent to:

Define \( f(x) = \begin{cases} x^2 + bx + c, & x< 1 \\ x, & x \geq 1 \end{cases} \). If f(x) is differentiable at x=1, then b−c is equal to

The absolute maximum value of the function \( f(x) = 2x^3 - 3x^2 - 36x + 9 \) defined on \( [-3, 3] \) is
If \( x = a(\cos \theta + \theta \sin \theta), \, y = a(\sin \theta - \theta \cos \theta) \), then \( \frac{d^2 y}{dx^2} \) equals
If one of the lines given by \( 6x^2 - xy + 4cy^2 = 0 \) is \( 3x + 4y = 0 \), then \( c \) equals
Let the foot of the perpendicular from the point \( (1, 2, 4) \) on the line \( \frac{x+2}{4} = \frac{y-1}{2} = \frac{z+1}{3} \) be \( P \). Then the distance of \( P \) from the plane \( 3x + 4y + 12z + 23 = 0 \) is:

Let \( F(\alpha) = \begin{bmatrix} \cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{bmatrix} \), where \( \alpha \in \mathbb{R} \). Then \( [F(\alpha)]^{-1} \) is equal to:

If two numbers \( p \) and \( q \) are chosen randomly from the set \( \{1, 2, 3, 4\} \) with replacement, what is the probability that \( p^2 \geq 4q \)?
Marks of 5 students of a group are \( 8, 12, 13, 15, 22 \). Find the variance.
Evaluate the following limit: \[ \lim_{\theta \to -\frac{\pi}{4}} \frac{\cos\theta + \sin\theta}{\theta + \frac{\pi}{4}}. \]
The equation of a circle with center \( (5, 4) \) and touching the \( Y \)-axis is:
The point on the line \( 4x - y - 2 = 0 \) which is equidistant from the points \( (-5, 6) \) and \( (3, 2) \) is