List of top Mathematics Questions

The integrating factor of the differential equation \[ \frac{dy}{dx} + y \tan x - \sec x = 0 \quad \text{is:} \]
If $\left| \begin{array}{ccc} -1 & 2 & 4 \\ 1 & x & 1 \\ 0 & 3 & 3x \end{array} \right| = -57$, the product of the possible values of $x$ is:
The principal branch of $\cos^{-1} x$ is:
If $f(x) = \left\{ \begin{array}{ll} \frac{1 - \sin^3 x}{3 \cos^2 x} & \text{for} \, x \neq \frac{\pi}{2}, \\ k & \text{for} \, x = \frac{\pi}{2}, \end{array} \right. $ is continuous at $x = \frac{\pi}{2}$, then the value of $k$ is:
If the direction cosines of a line are $\lambda, \lambda, \lambda$, then $\lambda$ is equal to:
Find the equation of a line in vector and Cartesian form which passes through the point \( (1, 2, -4) \) and is perpendicular to the lines \[ \frac{x - 8}{3} = \frac{y + 19}{-16} = \frac{z - 10}{7}. \] and \[ \vec{r} = 15\hat{i} + 29\hat{j} + 5\hat{k} + \mu (3\hat{i} + 8\hat{j} - 5\hat{k}). \]
Show that the area of a parallelogram whose diagonals are represented by \( \vec{a} \) and \( \vec{b} \) is given by \[ \text{Area} = \frac{1}{2} | \vec{a} \times \vec{b} |. \] Also, find the area of a parallelogram whose diagonals are \( 2\hat{i} - \hat{j} + \hat{k} \) and \( \hat{i} + 3\hat{j} - \hat{k} \).
Find: \[ \int \frac{\cos x}{(4 + \sin^2 x)(5 - 4 \cos^2 x)} \, dx. \]
The relation between the height of the plant (\(y\) cm) with respect to exposure to sunlight is governed by the equation \[ y = 4x - \frac{1}{2} x^2, \] where \(x\) is the number of days exposed to sunlight.
(i) Find the rate of growth of the plant with respect to sunlight.
(ii) In how many days will the plant attain its maximum height? What is the maximum height?
Find $k$ so that \[ f(x) = \begin{cases} \frac{x^2 - 2x - 3}{x + 1}, & \text{if } x \neq -1 \\ k, & \text{if } x = -1 \end{cases} \] is continuous at $x = -1$.
Evaluate: \[ \int_{\frac{\pi}{2}}^{\pi} \frac{e^{x} \left(1 - \sin x \right)}{1 - \cos x} \, dx. \]
Check the differentiability of the function $f(x) = |x|$ at $x = 0$.
Calculate the area of the region bounded by the curve \[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \] and the x-axis using integration.
Let $A = \{1, 2, 3\}$ and $B = \{4, 5, 6\}$. A relation $R$ from $A$ to $B$ is defined as $R = \{(x, y) : x + y = 6, x \in A, y \in B \}$. (i) Write all elements of $R$.
(ii) Is $R$ a function? Justify.
(iii) Determine domain and range of $R$.
For the curve \( y = 5x - 2x^3 \), if \( x increases at the rate of 2 units/s, then how fast is the slope of the curve changing when x = 2?
If $f : \mathbb{R}^+ \to \mathbb{R}$ is defined as $f(x) = \log_a x$ where $a>0$ and $a \neq 1$, prove that $f$ is a bijection. (R$^+$ is the set of all positive real numbers.)
Find the domain of $\sin^{-1} \sqrt{x - 1}$.
If $f(x) = x + \frac{1}{x}, \, x \geq 1$, show that $f$ is an increasing function.
Let A and B be two square matrices of order 3 such that $\text{det} = 3$ and $\text{det} = -4$. Find the value of $\text{det}(-6AB)$.
Simplify $\sin^{-1} \left( \frac{x}{\sqrt{1 + x^2}} \right)$.
Assertion : If A and B are two events such that $P(A \cap B) = 0$, then A and B are independent events.
Reason (R): Two events are independent if the occurrence of one does not affect the occurrence of the other.
Assertion : In a Linear Programming Problem, if the feasible region is empty, then the Linear Programming Problem has no solution.
Reason (R): A feasible region is defined as the region that satisfies all the constraints.
The area of the region enclosed by the curve $y = \sqrt{x}$ and the lines $x = 0$ and $x = 4$ and the x-axis is :
The order and degree of the differential equation \[ \left( \frac{d^2y}{dx^2} \right)^2 + \left( \frac{dy}{dx} \right)^2 = x \sin \left( \frac{dy}{dx} \right) \] are :
The value of \[ \int_0^1 \frac{dx}{e^x + e^{-x}} \] is :