List of top Mathematics Questions

Evaluate: \[ (\vec{i} - 3\vec{j} + 4\vec{k}) \cdot \big[(2\vec{i} - \vec{j}) \times (\vec{j} + \vec{k})\big]. \]

Evaluate the determinant \[ \begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{vmatrix}. \]

Minimize \[ Z = 2x + y, \] subject to \[ 5x + 10y \le 50, \quad x + y \ge 1, \quad y \le 4, \quad x \ge 0, \quad y \ge 0. \]

Find \(\frac{dy}{dx}\) when \[ (\sin y)^x = (\cos x)^y. \]

In four throws, with a pair of dice, what is the probability of occurrence of doublets at least twice?

Find the value of \[ \int_0^{\pi/2} x \cos x \, dx. \]

If \(\vec{a} = \vec{i} - \vec{j} + \vec{k}\) and \(\vec{b} = 2\vec{i} + \vec{j} + 3\vec{k}\), then find the value of \(|\vec{a} + \vec{b}|\).

Find the direction cosines of the vector \(3\vec{i} - 4\vec{j} + 12\vec{k}\).

Evaluate: \[ \int_0^{\pi/2} \log (\cos x) \, dx. \]

Solve: \[ (1+x^2) \frac{dy}{dx} + y = \tan^{-1} x. \]

Prove that \[ \sin^{-1} \frac{4}{5} + \sin^{-1} \frac{5}{13} + \sin^{-1} \frac{16}{65} = \frac{\pi}{2}. \]

Integrate \[ \int \sin^3 x \, dx. \]

Integrate \[ \int \frac{dx}{\sqrt{x+1} + \sqrt{x+2}}. \]

Find the value of \[ \int_0^{\pi/2} \frac{dx}{1 + \sqrt{\tan x}}. \]

Find the value of \(p\) so that the lines \[ \frac{x-1}{2} = \frac{y-2}{3} = \frac{z+17}{p} \quad \text{and} \quad \frac{x+4}{2} = \frac{y+9}{2} = \frac{z-1}{2} \] are mutually perpendicular.

Find the equation of the plane whose intercepts on the axes \(x, y, z\) are respectively \(2, 3\), and \(-4\).

Evaluate the integral \[ \int \cos^3 x \cdot \sin x \, dx. \]

Find the distance between the planes \[ x - 2y + 2z = 6 \quad \text{and} \quad 3x - 6y + 6z = 2. \]

Find the value of \[ \int_0^a \sqrt{a^2 - x^2} \, dx. \]

Solve the differential equation \[ \frac{dy}{dx} = e^{x + y}. \]

Evaluate the integral \[ \int \frac{x^2 - 1}{x^2 + 4} \, dx. \]

If \[ x \cos y = \sin(x + y), \] find \(\frac{dy}{dx}\).

Differentiate \[ \tan^{-1} \left( \frac{2x}{1 - x^2} \right) \] with respect to \[ \sin^{-1} \left( \frac{2x}{1 + x^2} \right). \]

If \[ x = \sqrt{1 + t^2}, \quad y = \sqrt{1 - t^2}, \] then find \(\frac{dy}{dx}\).

Evaluate the determinant: \[ \begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^3 & b^3 & c^3 \end{vmatrix}. \]