List of practice Questions

Let $A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$, then $A^{10}$ is equal to:

If $f(x) = \begin{vmatrix} x - 3 & 2x^2 - 18 & 2x^3 - 81 \\ x - 5 & 2x^2 - 50 & 4x^2 - 500 \\ 1 & 2 & 3 \end{vmatrix}$, then $f(1) \cdot f(3) \cdot f(5) + f(5) \cdot f(1)$ is:

Let $A = \begin{pmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{pmatrix}$ is the adjoint of a $3 \times 3$ matrix $A$ and $|A| = 4$, then $\alpha$ is equal to:

If $A = \begin{bmatrix} x & 1 \\ 1 & x \end{bmatrix}$ and $B = \begin{bmatrix} x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x \end{bmatrix}$, then $\frac{dB}{dx}$ is:

If \(f(x) = \begin{bmatrix} \cos x & x &1 \\ 2 \sin x & x & 2x \\ \sin x & x & x \end{bmatrix}\). Then \(\lim_{x \to 0} \frac{f(x)}{x^2}\) is:

Which one of the following observations is correct for the features of the logarithm function to any base $b > 1$?

The function $f(x) = |\cos x|$ is:

If $y = 2x^{3x}$, then $\frac{dy}{dx}$ at $x = 1$ is:

Let the function satisfy the equation $f(x + y) = f(x)f(y)$ for all $x, y \in \mathbb{R}$, where $f(0) \neq 0$. If $f(5) = 3$ and $f'(0) = 2$, then $f'(5)$ is:

The value of $C$ in $(0, 2)$ satisfying the mean value theorem for the function $f(x) = x(x - 1)^2$, $x \in [0, 2]$ is equal to:}

$\frac{d}{dx} \left[ \cos^2 \left( \cot^{-1} \sqrt{\frac{2 + x}{2 - x}} \right) \right]$ is:}

For the function $f(x) = x^3 - 6x^2 + 12x - 3$, $x = 2$ is:}

The function $x^x$, $x > 0$ is strictly increasing at:

The maximum volume of the right circular cone with slant height $6$ units is:

If $f(x) = x e^{x^{1-x}}$, then $f(x)$ is:

$\int \frac{\sin x}{3 + 4\cos^2 x} \, dx =$

$\int_{-\pi}^\pi (1 - x^2)\sin x \cos^2 x \, dx =$

$\int \frac{1}{x \left(6(\log x)^2 + 7\log x + 2\right)} \, dx =$

$\int \frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}} \, dx =$

$\int_{1}^{5} \left(|x - 3| + |1 - x|\right) \, dx =$

$\lim_{n \to \infty} \left(\frac{n}{n^2 + 1^2} + \frac{n}{n^2 + 2^2} + \dots + \frac{n}{n^2 + 3^2 + \dots + \frac{1}{5n}} \right) =$

The area of the region bounded by the line $y = 3x$ and the curve $y = x^3$ in sq. units is:

The area of the region bounded by the line $y = x$ and the curve $y = x^3$ is:

The solution of $e^{\frac{dy}{dx}} = x + 1, \, y(0) = 3$ is:

The family of curves whose $x$ and $y$ intercepts of a tangent at any point are respectively double the $x$ and $y$ coordinates of that point is: