List of top Mathematics Questions asked in IISER

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 $f : \mathbb{R} \rightarrow \mathbb{R}$ be the function defined by
\[ f(x) = \begin{cases} x^2 - 4x - 5 & \text{if } x \ge 1 2x & \text{if } x < 1 \end{cases} \]
Which one of the following statements is TRUE?

Which one of the following is the solution of the differential equation
\[ x^2 \frac{dy}{dx} + 9xy = x^4 \quad (\text{for } x > 0) \]
given that $y = 0$ when $x = 1$?

What is the value of $\int_{0}^{\pi} x |\cos x| \sin x \, dx$?

Let $z_1, z_2$, and $z_3$ be complex numbers satisfying the following conditions:
\[ 2 = |2z_1| = |z_2 - 1| = |z_3 + 1| = \left| \frac{1}{z_1} + \frac{1}{z_2 - 1} + \frac{1}{z_3 + 1} \right| \]
What is the value of $|4z_1 + z_2 + z_3|$?

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x) = |x^3 - 3x|[x]$, where $[x]$ denotes the greatest integer less than or equal to $x$. Which one of the following statements is TRUE?

Let $S_n$ denote the sum of the first $n$ terms of a sequence $a_1, a_2, a_3, \dots$. If $S_{n+3} - S_n = 13n + 7$ for all $n$, what is the value of $a_{13} - a_{10}$?

Five fair coins are tossed independently. What is the probability that at least two heads appear?

What is the derivative of $\log(\sin^2 x)$ with respect to $\sin x$?

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a} + \vec{b}| = 15$ and
\( \vec{a} \times (3\hat{i} - 4\hat{j} + 5\hat{k}) = (3\hat{i} - 4\hat{j} + 5\hat{k}) \times \vec{b} \)
What is the value of $|(\vec{a} + \vec{b}) \cdot (2\hat{i} + 3\hat{j} + \hat{k})|$?

Let $\ell$ be the tangent line to the ellipse $x^2 + 16y^2 = 4$ at $\left(1, \frac{\sqrt{3}}{4}\right)$. What is the equation of the line perpendicular to $\ell$ passing through $(2, 0)$?

How many three digit numbers divisible by 5 are there in which no digits are repeated?

Let $n = \sum_{r=0}^{10} (-1)^r \binom{10}{r} \left(\frac{2}{3}\right)^{2r} 3^{20}$. Which one of the following statements is TRUE?

Let $A = \left\{ x \in \mathbb{R} \;\middle|\; -31 < \det \begin{bmatrix} 3x - 1 & 2 \\ -2 & 5 \end{bmatrix} \le 29 \right\}$. Which one of the following statements is TRUE?

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be the function given by $f(x) = \cos(\tan^{-1} x)$. Which one of the following statements is TRUE?

Consider the differential equation
$\cos(y) \frac{dy}{dx} + \frac{1}{x} \sin(y) = x, \quad (x > 0)$;
given that $y = \frac{\pi}{2}$ at $x = \sqrt{3}$. Which one of the following is the value of $y$ at $x = \sqrt{\frac{3}{2}}$ ?

In the given figure, the angles $\angle BAQ = \angle CPQ = \angle CBQ = \frac{\pi}{2}$; and the lengths $QA = 3$ unit, $AB = 4$ unit, and $BC = 1$ unit. What is the length of $PQ$? }

Let $f : \mathbb{Q} \to \mathbb{Q}$ be a function such that $f(x + y) = f(x) + f(y)$ for all $x, y \in \mathbb{Q}$, and $f(1) = 10$. Which one of the following statements is Correct ?

Consider the following subset of the $XY$-plane:
$S = \{(|z - iz|, |z|^2) : z \text{ is a complex number}\}$.
Which one of the following statements is Correct ?

What is the value of $^{23}C_0 + {}^{23}C_2 + {}^{23}C_4 + \dots + {}^{23}C_{22}$ ?