List of top Mathematics Questions

(c) Find the value of \( \int_{-\pi/2}^{\pi/2} \sin^2 x \, dx \):
(d) Prove that the function \( f(x) = |x| \) is not differentiable at \( x = 0 \):
(a) Prove that for the two vectors \( \vec{a} \) and \( \vec{b} \), \( |\vec{a} \cdot \vec{b}| \leq |\vec{a}| |\vec{b}| \):
(b) Find the differential coefficient of the function \( x^x \) with respect to \( x \):
(c) If the vectors \( \vec{v_1} = 3\hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{v_2} = \hat{i} - 4\hat{j} + \lambda \hat{k} \) are perpendicular, find the value of \( \lambda \):
(a) Prove that the function \( f(x) = |x - 1| \) is continuous at \( x = 1 \):
(c) The value of the integral: \[ \int_{\frac{1}{\sqrt{3}}}^{\sqrt{3}} \frac{dx}{1+x^2} \]
(d) The value of the expression: \[ \hat{i} \cdot \hat{i} + \hat{j} \cdot \hat{j} + \hat{k} \cdot \hat{k} \]
Prove that the acute angle \( \theta \) between the lines represented by the equation \[ ax^2 + 2hxy + by^2 = 0 \] is \[ \tan \theta = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right|. \] Hence find the condition that the lines are coincident.
Solve the following system of equations by the method of reduction: \[ x + y + z = 6, \quad y + 3z = 11, \quad x + z = 2y. \]
A die is thrown 6 times. If "getting an odd number" is a success, find the probability of 5 successes.
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: \[ \int \frac{1}{a^2 - x^2} dx = \frac{1}{2a} \log \left( \frac{a + x}{a - x} \right) + C. \]
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: \[ \tan^{-1} \left(\frac{1}{2}\right) + \tan^{-1} \left(\frac{1}{3}\right) = \frac{\pi}{4}. \]
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 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.