List of practice Questions

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 \[ \mathbf{A} = 5\vec{i} + 3\vec{j} + 4\vec{k} \quad \text{and} \quad \mathbf{B} = 6\vec{i} - 8\vec{j} - \vec{k}. \]
If \( y = \sqrt{\sin x + \sqrt{\sin x + \sqrt{\sin x + \ldots \text{ to } \infty}}} \), then \( \frac{dy}{dx} = \):
If \( y = x^{20} \), then \( \frac{d^2y}{dx^2} = \):
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: \[ \int \sqrt{\cos x} \cdot \sin x \, dx = ? \]
Evaluate the following matrix multiplication: \[ \begin{pmatrix} -3 & 5 & 2 \end{pmatrix} \times \begin{pmatrix} 1 \\ 6 \\ -4 \end{pmatrix} \]
Let \( A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \). Then, \( A^5 = \):
If \( A = \begin{bmatrix} 3 & -5 \\ -1 & 2 \end{bmatrix} \), then \( \text{adjoint } A = \):
Find \( \frac{d}{dx} \log (\sec x + \tan x) \):
Find \( \frac{d}{dx} \left( \sec^{-1} x + \csc^{-1} x \right) \):
If \( y = \tan^{-1} \left( \frac{1 - \cos x}{\sin x} \right) \), then \( \frac{dy}{dx} = \):
If \( x = a \sec \theta, \, y = b \tan \theta \), then \( \frac{dy}{dx} = \):
Solve the following expression: \[ \frac{x}{x - 1} \times \frac{x + 1}{x} \]
If the operation \(\ast\) is defined as \(a \ast b = 2a + b\), then \((2 \ast 3) \ast 4\) is:
Evaluate the determinant of the following matrix: \[ \begin{vmatrix} 1 & 2 & 5 \\ 1 & 1 & 4 \\ -2 & -3 & -9 \end{vmatrix} \]
Evaluate the determinant of the following matrix: \[ \begin{vmatrix} 3 & 1 & 2 \\ -4 & -2 & 3 \\ 5 & 1 & 1 \end{vmatrix} \]
Evaluate the following matrix multiplication: \[ 5 \times \begin{pmatrix} 5 & 7 \\ 6 & 8 \end{pmatrix} \]
For a function \( f: A \to B \), the function will be onto if:
The matrix \( A = [a_{ij}] \) of size \( m \times n \) is a square matrix if:
The angle between two planes \( 2x + y - 2z = 5 \) and \( 3x - 6y - 2z = 7 \) is:
The distance of the plane \( x - 2y + 4z = 9 \) from the point \( (2, 1, -1) \) is: