IAT (IISER) 2025

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

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

What is the value of $\int_0^\pi x|\cos x| \sin x d x$ ?

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

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

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

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

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

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

Let $A$ be a $3 \times 3$ matrix with real entries such that
$$ A=\left[\begin{array}{ccc} 4 & -1 & \cos x \\ -1 & 5 x & 25 \\ x^2+1 & 25 & 7 \end{array}\right] $$
For how many values of $x$, the matrix $A$ is symmetric?

Let
$$ A=\left\{x \in \mathbf{R} \left\lvert\,-31<\operatorname{det}\left[\begin{array}{cc} 3 x-1 & 2 \\ -2 & 5 \end{array}\right] \leq 29\right.\right\} $$
Which one of the following statements is TRUE?

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

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

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

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 \h...