IAT (IISER) 2026

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

Consider the following sets of points in the complex plane$A = \left\{ \cos\left(\dfrac{2n\pi}{5}\right) + i\sin\left(\dfrac{2n\pi}{5}\right) : n \in \mathbb{Z} \right\}$ and$B = \left\{ \cos\left(\dfrac{2n}{5}\right) + i\sin\left(\dfrac{2n...

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

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be the function given by$f(x) = |x - 2| + 3|x - 1| + ||x - 2| - 1|$.What is the number of points where $f$ is NOT differentiable?

For real numbers $a$ and $b$, consider the function $f : \mathbb{R} \rightarrow \mathbb{R}$ given by$f(x) = \begin{cases} -ax - b & \text{if } x < -1, \\ 5x + 1 & \text{if } -1 \leq x \leq 1, \\ a^2x + 3b & \text{if } x > 1. \end{cases}$How...

For a $2 \times 2$ matrix $A$, whose elements are real numbers, denote by $A^m$ the product $\underbrace{AA \cdots A}_{m \text{ times}}$, where $m$ is a positive integer. Define $x_0 = 0$, $x_1 = 1$, $x_n = x_{n-1} + x_{n-2}$, for all $n \g...

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

Suppose there are two boxes $B_1$ and $B_2$, each having 3 red and 4 black balls. One ball is drawn at random from $B_1$. If it is red, 4 red balls are put into $B_2$, otherwise 3 black balls are put into $B_2$. Then one ball is randomly dr...

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

Let $a_1, a_2, a_3, \ldots$ 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 $a_1 + a_2 + a_3 + a_4 + a_5$?

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?

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?

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

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?