List of top Mathematics Questions

(b) Find the value of: \[ \int_{0}^{\frac{\pi}{2}} \frac{x \, dx}{a^2 \cos^2 x + b^2 \sin^2 x}. \]
(a) Prove that: \[ \int_{0}^{\frac{\pi}{2}} \frac{x \sin x}{1 + \cos^2 x} \, dx = \frac{\pi^2}{4}. \]
(b)(ii) If the coordinates of midpoints of the sides of a triangle are: \[ (1, 5, -1), \quad (0, 4, -2), \quad (2, 3, 4), \] find the coordinates of its vertices.
(e) Show that the semi-vertical angle of the right circular cone of maximum volume and given slant height is \( \tan^{-1} (\sqrt{2}) \):
(a) Solve the system of linear equations by the matrix method: \[ 2x - 3y + 5z = 11, \quad 3x + 2y - 4z = -5, \quad x + y - 2z = -3. \]
(a) Find the shortest distance between the lines: \[ \frac{x+1}{7} = \frac{y+1}{-6} = \frac{z+1}{1}, \quad \frac{x-3}{1} = \frac{y-5}{-2} = \frac{z-7}{1}. \]
(b)(i) Find the angle between the pair of lines: \[ \frac{x+3}{3} = \frac{y-1}{5} = \frac{z+3}{4}, \quad \frac{x+1}{1} = \frac{y-4}{1} = \frac{z-5}{2}. \]
(c) Solve: \[ \int \frac{3x + 5}{x^3 - x^2 - x + 1} \, dx. \]
(d) If \( (\cos x)^y = (\cos y)^x \), find \( \frac{dy}{dx} \):
(a) If a die is thrown three times, find the probability of getting one appearing number in them will be odd:
(c) Solve the differential equation \( (\tan^{-1} y - x) dy = (1 + y^2) dx \):
(b) Minimize \( Z = x + 2y \) under the following constraints: \[ 2x + y \geq 3, \quad x + 2y \geq 6, \quad x \geq 0, \quad y \geq 0. \]
(d) Prove that the function \( f(x) = |x| \) is not differentiable at \( x = 0 \):
(a) Prove that for the two vectors \( \vec{a} \) and \( \vec{b} \), \( |\vec{a} \cdot \vec{b}| \leq |\vec{a}| |\vec{b}| \):
(c) Find the value of \( \int_{-\pi/2}^{\pi/2} \sin^2 x \, dx \):
(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: \[ \int_0^{2a} f(x) dx = \int_0^a f(x) dx + \int_0^a f(2a - x) dx. \] Hence show that: \[ \int_0^\pi \sin x \,dx = 2 \int_0^{\pi/2} \sin x \,dx. \]
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: \[ f(x) = kx^2(1 - x), \quad {for } 0<x<1, \] \[ = 0, \quad {otherwise.} \]