List of top Questions asked in IISER- 2025

An electron in the ground state (with energy $E_1$) of a hydrogen atom, absorbs a photon of energy $E_a$, and gets excited to a higher energy level of principal quantum number $n$. What is the value of $n$?

A source produces a light beam of intensity $I_0$ polarized along the $x$-direction. The beam is sent along the $z$-direction. It enters a polaroid $P_1$ with its polaroid axis aligned along the $y$-direction so that no light exits the polaroid. When another polaroid $P_2$ is placed in between the source and $P_1$, the intensity measured after $P_1$ is $3I_0/16$. Which among the following is a possible value of $\theta$, the angle of the polaroid axis measured from the $x$-axis?

A charged particle is moving in a circular orbit with radius $r$ and orbital angular frequency $\omega$ in the presence of a magnetic field. The orbit is enclosed within a larger circular metallic frame. The frame is concentric and coplanar with the orbit. The radius of the frame is now gradually decreased. Assuming that the particle remains within the frame at all times, what changes to the trajectory of the particle will occur as the frame is being shrunk?

Consider an equilateral prism of refractive index 1.5 and a parallelepiped block of refractive index 2.0 arranged as shown in the figure such that their adjacent faces are parallel. A light ray enters the prism from air at an angle of incidence such that the ray travels through the prism parallel to its base. What is the angle of emergence $\theta$?

A particle of mass $m$ and charge $q$ moving with a velocity $\vec{v} = v_0(\hat{i} + \hat{j} - \hat{k})$ is placed in a uniform magnetic field $\vec{B} = B_0(\hat{i} + \hat{j} + \hat{k})$. It executes a helical trajectory of radius $r$ and pitch $p$. Which of the following options is correct?

What is the potential difference between the points $P$ and $Q$ in the circuit shown below, once the capacitors are fully charged?

A particle of charge $q = 1e$ and mass $m$ with kinetic energy $K$ enters an electric field set up by two parallel plates of length $l$ as illustrated in the figure. The potential difference between the two plates is $1\text{ V}$ and their separation is $d$. What is the minimum value of $K$ (in $\text{eV}$) for which the particle will not hit either of the plates? [$e$ is the charge of the electron.]}

Consider two pipes $A$ and $B$ of identical length. $A$ has one end closed and one end open. $B$ has both ends open. Each tube is immersed in a closed chamber of ideal gas having volume $V$. The chamber containing tube $A$ is at temperature $T_A$ and the chamber containing tube $B$ is at temperature $T_B$. The sound frequencies corresponding to the $n_A$-th harmonic in tube $A$ and the $n_B$-th harmonic in tube $B$ are the same. What is the relation between the temperatures $T_A$ and $T_B$?

Consider two waves, which are given by $y_1(x, t) = A \sin(kx - \omega t)$ and $y_2(x, t) = \sqrt{3}A \cos(kx - \omega t)$, where $k$ is the wave number and $\omega$ is the angular frequency. The amplitude of the resultant waveform obtained by the superposition of the two waves is $A_s$ and its phase difference with $y_1$ is $\phi_s$. What are $A_s$ and $\phi_s$?

A metallic cube initially kept at a temperature $T$ is emitting black body radiation with a power $P$ (energy emitted per unit time). If $T$ is increased by $1\%$, the power being radiated increases by $4.5\%$. What is the approximate percentage increase in the volume of the cube in this process?

A circular disk of mass $M$ and radius $R$ is rotating clockwise with a uniform angular velocity $\omega$ about an axis passing through the centre, normal to the disk. At time $t = 0$, a torque $T$ is applied along the same axis to oppose the rotation of the disk. What is the angular displacement $\theta$ (measured from $t = 0$ in the clockwise direction) that the disk attains before it starts rotating counterclockwise?

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?

A block of mass $M$ lies at rest connected to a massless spring of spring constant $k$ on a frictionless surface. A bullet of mass $m$ hits the block horizontally with speed $v$ as shown in the figure and is completely stuck to the block. What is the maximum compression in the spring resulting from this impact (assuming that at this point the spring is still not fully compressed)?

Consider an elastic collision between two particles $A$ and $B$ of same mass, moving in the same direction. Particle $A$ is moving at speed $v_A$ and particle $B$ is moving at speed $v_B$. In the figures shown, the solid lines represent the motion before the collision and the dotted lines represent the motion after the collision. Which of the following describes the motion of these two particles most accurately?

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 $\int_{0}^{\pi} x |\cos x| \sin x \, dx$?

A cart of mass $M$ is released from $A$, the highest point of a frictionless track, as shown in the figure. The cart travels along the track and enters the semicircular arc $DBC$ of radius $R$. The heights of the points $A$ and $B$ are $h_1$ and $h_2$ from the ground, respectively. Which of the following quantities does not play any role in ensuring that the cart does not leave the track?

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 $(2, 0)$?

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 $S_n$ 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 $a_{13} - a_{10}$?

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?

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

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 $|4z_1 + z_2 + z_3|$?

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

During the charging and discharging of a lead-acid battery (a $\text{Pb}$ anode, a grid of $\text{Pb}$ packed with $\text{PbO}_2$ as cathode, and an aqueous solution of $\text{H}_2\text{SO}_4$ as an electrolyte), which of the following redox reactions does NOT occur?