List of top Mathematics Questions

If $\int \frac{5\sqrt{x}(4x^2 - 1)}{x} dx = 8\sqrt{x^5} + a\sqrt{x} + C$, where $a$ is a constant and $C$ is the constant of integration, then the value of $a$ is

$\int (\cot 2x + \csc 2x) dx = $

$\int \frac{x \cos 2x}{\cos x - \sin x} dx = $}

Let $f(x) = \sin x \sin(x + \frac{\pi}{3}), x \in \mathbb{R}$. Then the minimum value of $f$ is equal to

Let $f(x) = \frac{1}{3\sqrt{x}}(\frac{2}{x} - 3), x > 0$. Then $f(x)$ is decreasing in}

The surface area of a cube is increasing at the constant rate of $0.5 \text{ cm}^2/\text{s}$. Then the rate at which the volume of the cube is increasing (in $\text{cm}^3/\text{s}$), when its surface area has reached $12 \text{ cm}^2$, is

If the maximum value of the function $f(x) = \alpha - 4x - x^2$ is 1, then the value of $\alpha$ is equal to

Let $f(x) = \frac{1 + \tan^2 x}{1 - \tan^2 x}$ for $0 < x < \frac{\pi}{4}$. Then the value of $f'(\frac{\pi}{8})$ is equal to

If $y = 4\sqrt{x}$, then $\frac{d^2y}{dx^2} =$}

Let $y = 4e^{-x} - 2e^{-2x} - e^{-3x}, x \in \mathbb{R}$. If $\frac{d^2y}{dx^2} = e^{\alpha x}(4e^{2x} - 8e^x - 9)$ for all $x$, then the value of the constant $\alpha$ is

A cubic curve $y = f(x)$ passes through the points $(1, -7)$ and $(2, 11)$. If $\frac{dy}{dx} = 6x^2 + kx - 5$, where $k$ is a constant, then $f(x) =$}

The point $P(x, y)$, where $y = 4\log_e(2)$, lies on the curve with equation $y = \log_e(x^3 + 24)$. Then the value of $\frac{dy}{dx}$ at the point $P$ is

Let $f(x) = \log_e(9x)$ for $x > 0$ and $h(x) = f(x) + f(x^2) + f(x^3)$. Then the value of $h(\frac{1}{3}e^{1/3})$ is equal to

Let $f(x) = \frac{(\sqrt{x}+3)(\sqrt{x}-1)}{x - 1}$ for $x \neq 1$. Then $\lim_{x \to 1} f(x)$ is equal to

Let $f(x) = \frac{1}{1 + \tan x}, 0 < x < \frac{\pi}{2}$. Then $f^{-1}(x) =$}

Let $f(x) = \begin{cases} \frac{1-\sec^2(\alpha x)}{\alpha x^2}, & \text{for } x \neq 0 \\ -3, & \text{for } x = 0 \end{cases}$ be continuous at $x = 0$. Then the value of $\alpha$ is equal to

The value of $\lim_{x \to 0} \frac{x - \tan(3x)}{\sin(2x)}$ is equal to

The mean of the data : 4, 7, $x$, 13, 16 is 10. Then the mean deviation of the data is

Given the data: 5, 7, 9, 11, 13. The variance of the data, is

A die is rolled twice. Let $A$ be an event of getting 1 in the first roll and let $B$ be an event of getting 4 in the second roll. Then $P(A|B)$ is equal to

There are 20 boys and 5 girls in a class. Three students are selected at random. If $E$ is an event of selecting one boy and two girls, then $P(E)$ is equal to

The straight line passing through the points $(3,2,3)$ and $(5,-1,-2)$ is perpendicular to the straight line passing through the points $(1,3,1)$ and $(\alpha, \alpha, \alpha)$. Then the value of $\alpha$ is equal to

The shortest distance from the point $(-10, 10, -10)$ to the $z$-axis, is

A unit vector parallel to the straight line $\vec{r} = -(5 + 4s)\hat{i} + (7 - 2s)\hat{j} + (3 + 4s)\hat{k}$, where $s$ is the parameter of the line, is

If the straight line passing through the points $(1,-1,2)$ and $(3,2,8)$ makes angle $\beta$ with the $y$-axis, then the value of $\cos \beta$ is equal to