List of top Mathematics Questions

Solve the differential equation: \[ x \frac{dy}{dx} - y + x \sin \left(\frac{y}{x}\right) = 0. \]
Prove that: \[ \int \frac{1}{a^2 - x^2} dx = \frac{1}{2a} \log \left( \frac{a + x}{a - x} \right) + C. \]
Find the approximate value of \( \tan^{-1} (1.002) \). \[ {Given: } \pi = 3.1416. \]
If \( y = \sin^{-1} x \), then show that: \[ (1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} = 0. \]
Find the angle between the line \[ \mathbf{r} = ( \hat{i} + 2\hat{j} + \hat{k} ) + \lambda( \hat{i} + \hat{j} + \hat{k} ) \] and the plane \[ \mathbf{r} \cdot (2\hat{i} + \hat{j} + \hat{k}) = 8. \]
Find the shortest distance between the lines \[ \mathbf{r} = (4\hat{i} - \hat{j}) + \lambda( \hat{i} + 2\hat{j} - 3\hat{k} ) \] and \[ \mathbf{r} = ( \hat{i} - \hat{j} - 2\hat{k} ) + \mu ( \hat{i} + \hat{j} - 5\hat{k} ). \]
Prove by vector method, the angle subtended on a semicircle is a right angle.
In \( \triangle ABC \), prove that: \[ \frac{\cos A}{a} + \frac{\cos B}{b} + \frac{\cos C}{c} = \frac{a^2 + b^2 + c^2}{2abc}. \]
Prove that: \[ \tan^{-1} \left(\frac{1}{2}\right) + \tan^{-1} \left(\frac{1}{3}\right) = \frac{\pi}{4}. \]

Express the following switching circuit in symbolic form of logic. Construct the switching table.

If two coins are tossed simultaneously, write the probability distribution of the number of heads.
Solve: \[ 1 + \frac{dy}{dx} = {cosec}(x + y); \quad {put } x + y = u. \]
Find the area of the region bounded by the curve \( y = x^2 \), and the lines \( x = 1 \), \( x = 2 \), and \( y = 0 \).
Evaluate: \[ \int_{0}^{\frac{\pi}{2}} \cos^2 x \,dx. \]
Evaluate: \[ \int \log x \,dx. \]
Find \( \frac{dy}{dx} \), if \( y = (\log x)^x \).
Find the vector equation of the line passing through the point having position vector \[ 4\hat{i} - \hat{j} + 2\hat{k} \] and parallel to the vector \[ -2\hat{i} - \hat{j} + \hat{k}. \]
Find \( k \), if the sum of the slopes of the lines represented by \[ x^2 + kxy - 3y^2 = 0 \] is twice their product.

Check whether the matrix 


 is invertible or not.

A particle is moving along the X-axis. Its acceleration at time \( t \) is proportional to its velocity at that time. Find the differential equation of the motion of the particle.
Evaluate: \[ \int \frac{1}{x^2 + 25} dx. \]
If the vectors \(2\hat{i} - 3\hat{j} + 4\hat{k}\) and \( p\hat{i} + 6\hat{j} - 8\hat{k} \) are collinear, then find the value of \( p \).
Write the compound statement ‘Nagpur is in Maharashtra and Chennai is in Tamilnadu’ symbolically.

If the mean and variance of a binomial distribution are \( 18 \) and \( 12 \) respectively, then the value of \( n \) is __________.

The integrating factor of the linear differential equation \[ x \frac{dy}{dx} + 2y = x^2 \log x \] is __________.