List of top Mathematics Questions

Let $n = 20^{26}$. What is the remainder when $49^n + 41^n + 10n$ is divided by 100?

Let $a_1, a_2, a_3, \dots$ be a geometric progression of positive integers such that $a_1 = 3$ and $a_{n+2} - 2a_n = a_{n+1}$ for all positive integers $n$. What is the value of $a_1 + a_2 + a_3 + a_4 + a_5$?

For a $2 \times 2$ matrix $A$, whose elements are real numbers, denote by $A^m$ the product $AA\dots A$ ($m$ times), where $m$ is a positive integer. Define $x_0 = 0$, $x_1 = 1$, $x_n = x_{n-1} + x_{n-2}$, for all $n \ge 2$ and \[ A_n = \begin{bmatrix} x_{n+1} & x_n\\ x_n & x_{n-1} \end{bmatrix}, \text{ for all } n \ge 1. \] Which of the following statements is TRUE for all $m \ge 3$?

For real numbers $a$ and $b$, consider the function $f : \mathbb{R} \to \mathbb{R}$ given by \[ f(x) = \begin{cases} -ax - b ;& \text{if } x \\ 5x + 1 ;& \text{if } -1 \le x \le 1, \\ a^2x + 3b ;& \text{if } x > 1 . \end{cases} \] How many pairs $(a, b)$ are there for which $f$ is continuous at every point of $\mathbb{R}$?

Consider the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \sin^2(7x) - \sin^2(5x)$. Which of the following statements is NOT TRUE?

Consider the data of scores obtained by students in an examination. If the score of every student is increased by 2 marks, then which of the following statements is TRUE?

What is the value of $\int_{-1}^{2} \min\{1 - x, 1 - x^3\} \, dx$?

Let $\mathcal{C}$ be the set of all the circles in a plane. If \[ \mathcal{R} = \{(C_1, C_2) \in \mathcal{C} \times \mathcal{C} \mid C_1 \text{ and } C_2 \text{ intersect}\} , \] then which of the following statements is TRUE?

Let $r, l$ be two integers such that $r \ge l \ge 3$. What is the total number of functions \[ f : \{1, 2, \dots, r\} \to \{1, 2, \dots, r\} \] such that $f(1), f(2), \dots, f(l)$ are all distinct?

Let $l_1$ be the line joining $(1, 1, 1)$ and $(3, 1, 3)$ and let $l_2$ be the line joining $(0, 2, -1)$ and $(2, 0, 3)$. What is the angle between $l_1$ and $l_2$?

Consider the points $A(4\hat{i} + \hat{j} + 3\hat{k})$, $B(2\hat{j})$ and $C(-4\hat{i} + 3\hat{j} - 3\hat{k})$. Which of the following statements is TRUE?

Consider the following sets of points in the complex plane \[ A = \left\{ \cos \left( \frac{2n\pi}{5} \right) + i \sin \left( \frac{2n\pi}{5} \right) : n \in \mathbb{Z} \right\} \text{ and} \] \[ B = \left\{ \cos \left( \frac{2n}{5} \right) + i \sin \left( \frac{2n}{5} \right) : n \in \mathbb{Z} \right\} . \] Which of the following statements is TRUE?

Let $p(x)$ be a quadratic polynomial such that $p(1) = p(-1) = 0$. What is the coefficient of $x$ in $p(x)$?

Let \[ f(x)=\int_{1}^{4}\log[x]\ dx, \] where \([x]\) denotes the greatest integer function. Then the value of \(f(x)\) is:

Evaluate $\int_0^{\pi/2} \sin^6 x \cos^4 x dx$

Solve $(1+y^2)+(x-e^{-\tan^{-1}y})\frac{dy}{dx}=0$

Solve $\cos(x+y)dy=dx$

Eliminate constants from $y=A(x+B)^2$

Evaluate $\int_{-\pi}^{\pi} \frac{\cos^2 x}{1+a^x} dx$

If $\int x^5 e^{-4x^3} dx = \frac{1}{48}e^{-4x^3} f(x) + c$, then $f(x) =$

If $\int \frac{\cos 8x + 1}{\cot 2x - \tan 2x} dx = A\cos 8x + c$, then $A =$

Evaluate $\lim_{n\to\infty} \frac{1}{n^2}\sum_{r=1}^n r e^{r/n}$

Area between $y^2=x$ and $y=|x|$ is:

Evaluate $\int \frac{x-1}{(x+1)^3}e^x dx$:

The evaluation of the indefinite integral $\int \frac{dx}{\sin x + \sin 2x}$ is: