List of practice Questions

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.

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

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

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

Maximize \[ Z = 20x + 3y, \] subject to the constraints: \[ 3x + 2y \leq 10, \quad x \geq 0, \quad y \geq 0. \]

Solve the differential equation: \[ x^2 \frac{dy}{dx} = 2xy. \]

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

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 \(\frac{dy}{dx}\).

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

If \( y = \sin(xy) \), then find \( \frac{dy}{dx} \):

Evaluate the integral: \[ \int (x+2)^2 \, dx = ? \]

Evaluate \( P(A \cup B) \) if \( 2P(A) = P(B) = \frac{5}{13} \) and \( P(A|B) = \frac{2}{5} \):

Prove that \[ 4\left(\cot^{-1} 3 + \cot^{-1} 2 \right) = \pi. \]

Prove that \[ \tan^{-1} x + \tan^{-1} y = \tan^{-1} \left( \frac{x+y}{1 - xy} \right), \] provided \( xy<1 \).

Find \( \frac{dy}{dx} \) if \[ y = \sqrt{\sin x^2}. \]

Find \( f \circ g \) and \( g \circ f \) if \[ f(x) = 8x^3 \quad \text{and} \quad g(x) = x^{1/3}. \]

Find the angle between the vectors \[ \mathbf{A} = 5\vec{i} + 3\vec{j} + 4\vec{k} \quad \text{and} \quad \mathbf{B} = 6\vec{i} - 8\vec{j} - \vec{k}. \]

Evaluate the integral: \[ \int \sqrt{1 + \cos 2x} \, dx = ? \]

Evaluate the integral: \[ \int \frac{\log x}{x} \, dx = ? \]

Evaluate the integral: \[ \int \frac{\cos \sqrt{x}}{\sqrt{x}} \, dx = ? \]

Evaluate the integral: \[ \int \sqrt{\cos x} \cdot \sin x \, dx = ? \]