List of practice Questions

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:
If two planes \( 2x - 4y + 3z = 5 \) and \( x + 2y + \lambda z = 12 \) are mutually perpendicular to each other, then \(\lambda =\):
If the line \[ \frac{x - 3}{a} = \frac{y - 4}{b} = \frac{z - 5}{c} \] is parallel to the line \[ \frac{x}{5} = \frac{y}{3} = \frac{z}{2} \] then:
If the line \[ \frac{x - x_1}{a_1} = \frac{y - y_1}{b_1} = \frac{z - z_1}{c_1} \] is parallel to the plane \[ a_2 x + b_2 y + c_2 z + d = 0 \] then:
If \[ \frac{x}{18} = \frac{6}{18} \] then \(x\) is equal to:
Evaluate the integral: \[ \int \frac{1 - \sin 2x}{dx} \]
Given that \[ |\vec{a} + \vec{b}| = |\vec{a} - \vec{b}| \] which of the following is true?
The projection of the vector \[ \vec{a} = \hat{i} - 2\hat{j} + \hat{k} \] on the vector \[ \vec{b} = 4\hat{i} - 4\hat{j} + 7\hat{k} \] is:
The minimum value of \( Z = 3x + 5y \) subject to the constraints: \[ x + y \leq 2, \quad x \geq 0, \quad y \geq 0 \] is:
The maximum value of \( Z = 3x + 2y \) subject to the constraints: \[ 3x + y \leq 15, \quad x \geq 0, \quad y \geq 0 \] is: