List of top Mathematics Questions

If a line makes an angle of \( \frac{\pi}{4} \) with the positive directions of both \( x \)-axis and \( z \)-axis, then the angle which it makes with the positive direction of \( y \)-axis is:
If \( |a| = 2 \) and \( -3 \leq k \leq 2 \), then \( |a| |k| \in: \)
The derivative of \( 2^x \) w.r.t. \( 3^x \) is:
If \[ \begin{bmatrix} 1 & 3 & 1 \\ k & 0 & 1 \\ 1 & 0 & 1 \end{bmatrix} \] has a determinant of \( \pm 6 \), then the value of \( k \) is:
Let \( E \) and \( F \) be two events such that \( P(E) = 0.1, P(F) = 0.3, P(E \cup F) = 0.4 \). Then \( P(F \,|\, E) \) is:
The restrictions imposed on decision variables involved in an objective function of a linear programming problem are called:
If \( \alpha, \beta \), and \( \gamma \) are the angles which a line makes with the positive directions of \( x, y, z \) axes respectively, then which of the following is not true?
If \( \vec{a} = 2\hat{i} - \hat{j} + \hat{k} \) and \( \vec{b} = \hat{i} - 2\hat{j} + \hat{k} \), then \( \vec{a} \) and \( \vec{b} \) are:
\( x \log x \frac{dy}{dx} + y = 2 \log x \) is an example of a:
\( \int_{-a}^a f(x) \, dx = 0 \), if:
If the sides of a square are decreasing at the rate of \( 1.5 \, \mathrm{cm/s} \), the rate of decrease of its perimeter is:
A function \( f(x) = |1 - x + |x|| \) is:
If \(A = \begin{bmatrix} 1 & -2 & 0 \\ 2 & -1 & -1 \\ 0 & -2 & 1 \end{bmatrix}\), find \(A^{-1}\) and use it to solve the following system of equations: \[ x - 2y = 10, \quad 2x - y - z = 8, \quad -2y + z = 7. \]
Find: \[ \int_{\pi/6}^{\pi/3} \frac{\sin x + \cos x}{\sqrt{\sin 2x}} \, dx. \]
(a) Evaluate: \[ \int_{0}^{\pi/2} e^x \left( \frac{1 + \sin x}{1 + \cos x} \right) dx. \]
(b) If \(y = (\tan x)^x\), then find \(\frac{dy}{dx}\).
(a) If \(\sqrt{1 - x^2} + \sqrt{1 - y^2} = a(x - y)\), prove that \(\frac{dy}{dx} = \sqrt{\frac{1 - y^2}{1 - x^2}}\).
Solve the following linear programming problem graphically: \[ \text{Maximise } z = 5x + 4y \] subject to the constraints: \[ x + 2y \geq 4, \quad 3x + y \leq 6, \quad x + y \leq 4, \quad x, y \geq 0. \]
Find the general solution of the differential equation: \[ \frac{dy}{dx} = \frac{x^2 + y^2}{2xy}. \]
(b) Evaluate: \[ \int_{1}^{3} \left(|x - 1| + |x - 2| + |x - 3|\right) \, dx. \]
(a) Find: \[ \int \frac{x^2}{(x^2 + 4)(x^2 + 9)} \, dx. \]
Find the absolute maximum and minimum values of the function: \[ f(x) = 12x^{4/3} - 6x^{1/3}, \quad x \in [0, 1]. \]
(b) Show that the function \( f(x) = |x|^3 \) is differentiable at all points of its domain.
(a) If \( y = \sqrt{\cos x + y} \), prove that \[ \frac{dy}{dx} = \frac{\sin x}{1 - 2y}. \]
If \( \vec{a} \) and \( \vec{b} \) are two non-zero vectors such that \( (\vec{a} + \vec{b}) \perp \vec{a \) and \( (2\vec{a} + \vec{b}) \perp \vec{b} \), then prove that \( |\vec{b}| = \sqrt{2} |\vec{a}| \).}