List of top Mathematics Questions

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 $x = 0$.

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.)

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 $R$.

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.

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 :

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 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 :

If the sides $AB$ and $AC$ of $\triangle ABC$ are represented by vectors $\hat{i} + \hat{j} + 4 \hat{k}$ and $3 \hat{i} - \hat{j} + 4 \hat{k}$ respectively, then the length of the median through A on BC is :

The function $f$ defined by \[ f(x) = \begin{cases} x, & \text{if } x \leq 1 \\ 5, & \text{if } x>1 \end{cases} \] is not continuous at :

If $f(x) = 2x + \cos x$, then $f(x)$ :

If $f(x) = \begin{cases} 3x - 2, & 0 \leq x \leq 1\\ 2x^2 + ax, & 1<x<2 \end{cases}$ is continuous for $x \in (0, 2)$, then $a$ is equal to :

The matrix \[ \begin{pmatrix} 0 & 1 & -2 \\ -1 & 0 & -7 \\ 2 & 7 & 0 \end{pmatrix} \] is a :

If $f : \mathbb{N} \rightarrow \mathbb{W}$ is defined as \[ f(n) = \begin{cases} \frac{n}{2}, & \text{if } n \text{ is even} \\ 0, & \text{if } n \text{ is odd} \end{cases} \] then $f$ is :

If $M$ and $N$ are square matrices of order 3 such that $\det(M) = m$ and $MN = mI$, then $\det(N)$ is equal to :

The values of $\lambda$ so that $f(x) = \sin x - \cos x - \lambda x + C$ decreases for all real values of $x$ are :

If P is a point on the line segment joining $(3, 6, -1)$ and $(6, 2, -2)$ and the $y$-coordinate of P is 4, then its $z$-coordinate is :

If A and B are square matrices of same order such that AB = BA, then $A^2 + B^2$ is equal to :