List of top Mathematics Questions

Solve the following system of equations by the method of reduction: \[ x + y + z = 6, \quad y + 3z = 11, \quad x + z = 2y. \]

Find \( k \), if the probability density function is given by: \[ f(x) = kx^2(1 - x), \quad {for } 0<x<1, \] \[ = 0, \quad {otherwise.} \]

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 \[ \mathbf{r} = ( \hat{i} - \hat{j} - 2\hat{k} ) + \mu ( \hat{i} + \hat{j} - 5\hat{k} ). \]

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: \[ \tan^{-1} \left(\frac{1}{2}\right) + \tan^{-1} \left(\frac{1}{3}\right) = \frac{\pi}{4}. \]

Solve: \[ 1 + \frac{dy}{dx} = {cosec}(x + y); \quad {put } x + y = u. \]

Evaluate: \[ \int_{0}^{\frac{\pi}{2}} \cos^2 x \,dx. \]

Evaluate: \[ \int \log x \,dx. \]

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.

Evaluate: \[ \int \frac{1}{x^2 + 25} dx. \]