GATE 2026 Physics (PH) Question Paper with Solutions: Free PDF Download with Answer Key and Exam Pattern

"He often _____ the numbers. False claims are not going to help. Honesty _____ trust", said the manager.
Choose the option with the correct order of words to fill the blanks.

• (A) exaggerates; engenders
• (B) excels; encourages
• (C) aggravates; alleviates
• (D) diminishes; eliminates

In the sequence of tiles shown below, the missing tile indicated by the question mark should be:

• (A)

• (B)

• (C)

• (D)

A school has 100 students distributed among 1st to 10th standards.
Based on this, which one of the following statements is always correct?

• (A) There are at least 10 students who belong to the same standard.
• (B) There is at least one student in each standard.
• (C) There are at most 10 students in 10th standard.
• (D) The total number of students from 1st to 5th standards is at least 50.

How many 3-digit numbers can be formed using three distinct single digit prime numbers?

• (A) 64
• (B) 24
• (C) 12
• (D) 4

In a group of students, 10 students like Mathematics, 12 students like English, 4 students like both Mathematics and English, and 6 students like neither Mathematics nor English. The number of students in the group is ____

• (A) 18
• (B) 20
• (C) 24
• (D) 32

Charity : P :: Retaliation : Q

Choose the appropriate pair of words P and Q that fit the analogy.

• (A) P = Parsimonious; Q = Vengeful
• (B) P = Altruistic; Q = Amicable
• (C) P = Resentful; Q = Spiteful
• (D) P = Magnanimous; Q = Vindictive

A paper shown in Panel I is folded along the dashed lines (- - -) to construct a cube. The shaded regions shown in Panel I appear on the outer surface of the cube.

• (A) Only (i) can correspond to the unfolded cube in Panel I.
• (B) Only (ii) can correspond to the unfolded cube in Panel I.
• (C) Both (i) and (ii) can correspond to the unfolded cube in Panel I.
• (D) Neither (i) nor (ii) can correspond to the unfolded cube in Panel I.

Consider the cube shown below with its 8 corners labelled a, b, c, d, e, f, g, and h. The figure is representative.

• (A) 8
• (B) 4
• (C) 3
• (D) 2

Four hills H1, H2, H3, and H4 are present in an area. The following observations are made about them:

i. Neither H2 nor H3 is the easternmost hill.
ii. Neither H2 nor H3 is the westernmost hill.
iii. Neither the easternmost hill nor the westernmost hill is the southernmost hill.
iv. Two hills are located to the west of H2.
v. The southernmost hill has at least two hills to its east.

The southernmost hill is ________.

• (A) H1
• (B) H2
• (C) H3
• (D) H4

As shown in the figure, circle \(C_1\) with center \(O_1\) and radius \(r_1\) touches the square \(VWXY\) at points \(P\) and \(Q\) while circle \(C_2\) with center \(O_2\) and radius \(r_2\) touches the square \(VWXY\) at points \(R\) and \(S\). The two circles touch each other at \(T\).

• (A) \(4 - 3\sqrt{2}\)
• (B) \(1 + 2\sqrt{2}\)
• (C) \(7 - 4\sqrt{2}\)
• (D) \(5 + 3\sqrt{2}\)

In free space, an electromagnetic wave is travelling whose wavevector is \(\vec{k} = 10(\hat{x} + \sqrt{3}\hat{y})\) m\(^{-1}\). The electric field component of this electromagnetic wave is given by \(\vec{E}(\vec{r},t) = \hat{z}\, 600\cos(\vec{k}\cdot\vec{r} - \omega t)\) V.m\(^{-1}\). The speed of light in free space is \(c = 3.0 \times 10^{8}\) m.s\(^{-1}\). The corresponding magnetic field \(\vec{B}(\vec{r},t)\) is

• (A) \(\vec{B}(\vec{r},t) = 2\times10^{-6}(\sqrt{3}\hat{x} - \hat{y})\cos(\vec{k}\cdot\vec{r}-\omega t)\) V.m\(^{-2}\).s
• (B) \(\vec{B}(\vec{r},t) = 10^{-6}(\sqrt{3}\hat{x} - \hat{y})\cos(\vec{k}\cdot\vec{r}-\omega t)\) V.m\(^{-2}\).s
• (C) \(\vec{B}(\vec{r},t) = 2\times10^{-5}(-\sqrt{3}\hat{x} + \hat{y})\cos(\vec{k}\cdot\vec{r}-\omega t)\) V.m\(^{-2}\).s
• (D) \(\vec{B}(\vec{r},t) = 10^{-5}(\sqrt{3}\hat{x} - \hat{y})\cos(\vec{k}\cdot\vec{r}-\omega t)\) V.m\(^{-2}\).s

An infinitely large non-conducting thin sheet in the \(xy\) plane (\(z=0\)) carries a uniform surface charge density \(\sigma = 17.70\times10^{-12}\) C.m\(^{-2}\). The electric field in the region \(z < 0\) is \(\vec{E}_2 = \hat{x}+2\hat{y}+3\hat{z}\). Then, the electric field \(\vec{E}_1\) in the region \(z > 0\) will be (\(\epsilon_0 = 8.85\times10^{-12}\) C\(^2\).N\(^{-1}\).m\(^{-2}\))

• (A) \(\vec{E}_1 = \hat{x}+2\hat{y}+5\hat{z}\)
• (B) \(\vec{E}_1 = \hat{x}+2\hat{y}+4\hat{z}\)
• (C) \(\vec{E}_1 = \hat{x}+2\hat{y}+3\hat{z}\)
• (D) \(\vec{E}_1 = \hat{x}+4\hat{y}+\hat{z}\)

Consider an operator \(\hat{A}\) which is not Hermitian. Find the possible values of \(c\) and \(d\) such that the operator \((c\hat{A} - d\hat{A}^{\dagger})\) is Hermitian.

• (A) \(c = i\) and \(d = i\)
• (B) \(c = 1\) and \(d = 1\)
• (C) \(c = -1\) and \(d = i\)
• (D) \(c = i\) and \(d = -i\)

For a scalar field \(\psi(\vec{r})\) and a vector field \(\vec{A}(\vec{r})\), \(\nabla\times(\vec{A}\,\psi)\) is equivalent to the expression

• (A) \(\psi(\nabla\times\vec{A}) - \vec{A}\times(\nabla\psi)\)
• (B) \(\psi(\nabla\times\vec{A}) + \vec{A}\times(\nabla\psi)\)
• (C) Null vector
• (D) \(\vec{A}\times(\nabla\psi) - \psi(\nabla\times\vec{A})\)

Which of the following options is correct for transformation of electric field \(\vec{E}\) and magnetic field \(\vec{B}\) under time reversal, i.e., \(t \to -t\)?

• (A) \(\vec{E} \to \vec{E}\) and \(\vec{B} \to \vec{B}\)
• (B) \(\vec{E} \to -\vec{E}\) and \(\vec{B} \to \vec{B}\)
• (C) \(\vec{E} \to \vec{E}\) and \(\vec{B} \to -\vec{B}\)
• (D) \(\vec{E} \to -\vec{E}\) and \(\vec{B} \to -\vec{B}\)

On a horizontal plane, a projectile of mass \(m\) is launched from the ground with speed \(v_0\) at an angle \(\theta_0\) with the horizontal. In addition to the gravitational force (\(mg\)), it also experiences a drag force \(\vec{F}_{drag} = -\gamma \vec{v}\), where \(\vec{v}\) is its velocity and \(\gamma\) is a constant. It hits the ground at a distance \(R\) from the point of launch with its velocity making an angle \(\theta\) with the horizontal, as shown schematically in the figure below.

• (A) \(R = \dfrac{v_0^2 \sin 2\theta}{g}\), \(\theta \lt \theta_0\)
• (B) \(R \lt \dfrac{v_0^2 \sin 2\theta_0}{g}\), \(\theta \lt \theta_0\)
• (C) \(R \lt \dfrac{v_0^2 \sin 2\theta_0}{g}\), \(\theta \gt \theta_0\)
• (D) \(R = \dfrac{v_0^2 \sin 2\theta}{g}\), \(\theta \gt \theta_0\)

Consider the Otto cycle for an ideal gas engine consisting of two quasistatic adiabatic and two quasistatic isochoric processes. The correct temperature-entropy (T-S) phase diagram for the cycle is:

• (A)

• (B)

• (C)

• (D)

The formula for energy \(E\) of a photon gas at temperature \(T\) in a two-dimensional box at equilibrium with \(g_{2d}(\nu)\) denoting the density of states of photons is given below, where symbols \(\nu\), \(h\) and \(k_B\) have their standard meaning. The specific heat (\(C_V\)) of this photon gas obeys
\[ E = \int_0^{\infty} d\nu \, g_{2d}(\nu) \, \frac{h\nu}{\exp\left(\dfrac{h\nu}{k_B T}\right) - 1} \]

• (A) \(C_V \propto T\)
• (B) \(C_V \propto T^2\)
• (C) \(C_V \propto T^3\)
• (D) \(C_V \propto T^4\)

For the electric field of an electromagnetic wave given below, which of the following statements is correct?
\[ \vec{E} = \hat{x}\, E_0 \cos(\omega t) + \hat{y}\, 2E_0 \cos\left(\omega t + \frac{\pi}{2}\right) \]

• (A) The electric field is linearly polarised with slope 2.
• (B) The electric field is circularly polarised with radius \(E_0\).
• (C) The electric field is elliptically polarised with a ratio of major to minor axis being 2.
• (D) The electric field is unpolarised with the two components being phase shifted by \(\pi/2\).

A gas of non-interacting \(^4\text{He}\) atoms (of mass \(m\)) is in a three-dimensional trap whose energy levels can be approximated by those of a harmonic oscillator potential \(V(x,y,z) = \dfrac{1}{2} m \omega^2 (x^2+y^2+z^2)\). The chemical potential of the gas at \(T=0\) K is

• (A) 0
• (B) \(\dfrac{1}{2}\hbar\omega\)
• (C) \(\dfrac{3}{2}\hbar\omega\)
• (D) \(3\hbar\omega\)

Given \( |v_1\rangle = \dfrac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ i \end{pmatrix} \) and \( |v_2\rangle = \dfrac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -i \end{pmatrix} \), the tensor product \( |v_1\rangle \otimes |v_2\rangle \) is

• (A) \( \dfrac{1}{2}\begin{pmatrix} 1 \\ -i \\ i \\ 1 \end{pmatrix} \)
• (B) \( \dfrac{1}{2}\begin{pmatrix} 1 \\ i \\ -i \\ 1 \end{pmatrix} \)
• (C) \( \dfrac{1}{2}\begin{pmatrix} 1 & i \\ i & -1 \end{pmatrix} \)
• (D) \( \dfrac{1}{2}\begin{pmatrix} 1 & i \\ -i & 1 \end{pmatrix} \)

Sketch of a two-dimensional vector field \(\vec{V}\) is shown below. Here, length and arrow head of the arrows denote magnitude and direction of the vector field, respectively.

• (A) It is zero everywhere in the two-dimensional space.
• (B) Its magnitude is non-zero and its direction is out of the two-dimensional plane.
• (C) Its magnitude is non-zero and its direction is into the two-dimensional plane.
• (D) It points in opposite directions above and below the x-axis.

Which one of the following is an allowed process?

• (A) \( \pi^- + p \to \pi^0 + n \)
• (B) \( \pi^0 \to \gamma + \gamma + \gamma \)
• (C) \( p + \bar{p} \to \Lambda^0 + \Lambda^0 \)
• (D) \( p + \bar{p} \to \gamma \)

Given \(Q\) is the electromagnetic charge and \(S\) is the strangeness quantum number, identify the particle(s) for which \((Q - S) = 0\) is satisfied.

• (A) \(\Sigma^{*-}\)
• (B) \(K^+\)
• (C) \(\Omega^-\)
• (D) \(\Delta^{++}\)

Schematic variation of the specific heat \(C_p\) of an ideal gas of diatomic molecules with temperature \(T\) is shown in the figure below. For rotational energy \(E_R\) and vibrational energy \(E_v\) of the molecule, which of the following options is/are correct? Here \(k_B\) is the Boltzmann constant.

• (A) \(E_R \cong k_B T_1\)
• (B) \(E_R \cong k_B T_2\)
• (C) \(E_v \cong k_B T_1\)
• (D) \(E_v \cong k_B T_2\)

If the perturbation \(V = \lambda x^3\) is added to the Hamiltonian of a one dimensional harmonic oscillator, the matrix element \(\langle m|V|0\rangle\) is/are non-zero for which of the following states? Here, the eigenstates of the harmonic oscillator are denoted by \(|n\rangle\).

• (A) \(|m=3\rangle\)
• (B) \(|m=1\rangle\)
• (C) \(|m=2\rangle\)
• (D) \(|m=5\rangle\)

For which of the following functions does the Laplacian vanish?

• (A) \(xe^{y} - ye^{x}\)
• (B) \(x\cos(y) - y\cos(x)\)
• (C) \(e^{x+iy}\)
• (D) \(yx^{2} - \dfrac{y^{3}}{3} - xy\)

A projectile of mass \(m\) is launched from the ground with the initial speed \(v_0\) at an angle \(30^{\circ}\) from the horizontal. Take the ground to be horizontal. Ignoring the drag, the magnitude of Hamilton's action \(\int L\,dt\) for the particle from the beginning till it hits the ground is \(f \times \left(\dfrac{mv_0^{3}}{g}\right)\). The value of \(f\) (rounded off to two decimal places) is .

Consider an electron in the energy eigenstate \(\psi_{211}(\vec{r})\) of the hydrogen atom. Given that the radial probability distribution of the electron in such a state takes its maximum value at \(r = n_0 a\), where \(a\) is the Bohr radius, and \(n_0\) is an integer. The value of \(n_0\) (in integer) is . The radial part of the wavefunction \(\psi_{211}(\vec{r})\) is given by \(R_{21}(r) = \dfrac{1}{\sqrt{24a^{5}}}\,re^{-r/2a}\).

A dielectric sphere carries a uniform polarization \(P = 26\ \mu\text{C}.\text{cm}^{-2}\). The magnitude of the electric field at the center of the sphere is \(E \times 10^{9}\ \text{N}.\text{C}^{-1}\). The value of \(E\) (rounded off to one decimal place) is . \((\epsilon_0 = 8.85 \times 10^{-12}\ \text{C}^{2}.\text{N}^{-1}.\text{m}^{-2})\)

Consider a metal-superconductor junction connected to a dc voltage \(V\). At \(T < T_c\), where \(T_c\) is the superconductor's transition temperature, the current \(I\) versus \(V\) behavior of this junction is shown schematically in the figure below.

For the energy dispersion of an electron in a one-dimensional solid \(E(k) = E_0 - 2\gamma \cos(ka)\), the ratio of the effective mass of the electron in the solid to the free electron mass (\(m_e\)) at \(k = 0\) is \(R_0\). Taking \(\gamma = 0.5\) eV and \(a = 0.5\) nm, the value of \(R_0\) (rounded off to two decimal places) is
(\(\hbar = 1.054 \times 10^{-34}\) J.s, \(m_e = 9.1 \times 10^{-31}\) kg, electron charge \(= 1.6 \times 10^{-19}\) C)

The specific heat \(C_p(T)\) of one mole of a material as a function of temperature \(T\) is given as \(C_p(T) = AT + BT^3\), where \(A = 0.695\) mJ.mol\(^{-1}\).K\(^{-2}\) and \(B = 0.045\) mJ.mol\(^{-1}\).K\(^{-4}\). When \(T\) is changed from 1 K to 10 K at constant pressure, the change in entropy \(\Delta S\) in mJ.mol\(^{-1}\).K\(^{-1}\) (rounded off to one decimal place) is

Raman spectrum of a molecule was recorded using a source of wavelength 5000 Angstrom. The first Stokes line is observed at 5100 Angstrom. The first anti-Stokes line will appear at a wavelength \(L\) (in Angstrom). The value of \(L\) (rounded off to nearest integer) is

A rocket of length 18.0 m is moving at speed \(0.9c\) (where \(c\) is the speed of light) parallel to its own length, relative to the earth. The length of the rocket measured in meters by an observer on earth (rounded off to two decimal places) is

The function \(f(z)\) of the complex variable \(z\) given below,\[ f(z) = \dfrac{z^{2} - 5z + 4}{z^{3} + 4z - z^{2} - 4} \]has singular points at \(z =\)

• (A) 1 and \((2 - i)\)
• (B) \(2i\) and \(-2i\)
• (C) 1 and \((2 + i)\)
• (D) \((2 + i)\) only

Consider the Pauli matrices \(\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\), \(\sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}\), \(\sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\).
The value of \(\text{Tr}\big(\sigma_z [\sigma_x, \sigma_y]\big)\) is

• (A) \(2i\)
• (B) \(i\)
• (C) \(4i\)
• (D) \(\dfrac{i}{2}\)

Which one of the following statements is true?

• (A) In the decay \(\mu^+ \rightarrow e^+ + \nu_e + \bar\nu_\mu\), CPT is violated.
• (B) The decay \(\Lambda \rightarrow p^+ + \pi^-\) is allowed and strangeness is violated.
• (C) The decay \(p^+ \rightarrow e^+ + \gamma\) is allowed.
• (D) The decay \(\Omega^- \rightarrow \Xi^0 + K^-\) is allowed.

The Hamiltonian for a quantum particle of mass \(m\) is given below, where \(\omega < \Omega\). The Schrodinger equation for this system can be solved exactly using the orthogonal transformation \(x = \dfrac{x_1 - x_2}{\sqrt2}\) and \(y = \dfrac{x_1+x_2}{\sqrt2}\).
\[ H = -\dfrac{\hbar^2}{2m}\left[\dfrac{\partial^2}{\partial x^2} + \dfrac{\partial^2}{\partial y^2}\right] + \dfrac12 m\Omega^2(x^2+y^2) + m\omega^2 xy \]The ground state energy of this system is

• (A) \(\dfrac{\hbar}{2}\Big[\sqrt{\Omega^2-\omega^2} + \sqrt{\Omega^2+\omega^2}\Big]\)
• (B) \(\hbar\Big[\sqrt{\Omega^2-\omega^2} + \sqrt{\Omega^2+\omega^2}\Big]\)
• (C) \(\dfrac{\hbar}{2}\Big[\sqrt{\Omega^2+\Omega\omega} + \sqrt{\Omega^2-\Omega\omega}\Big]\)
• (D) \(\hbar\Big[\sqrt{\Omega^2+\Omega\omega} - \sqrt{\Omega^2-\Omega\omega}\Big]\)

Consider two particles with angular momenta \(j_1 = 2\hbar\) and \(j_2 = \hbar/2\). If the expression
\[ |j=\tfrac52, m=\tfrac32\rangle = c_1|j_1=2,m_1=1\rangle|j_2=\tfrac12,m_2=\tfrac12\rangle + c_2|j_1=2,m_1=2\rangle|j_2=\tfrac12,m_2=-\tfrac12\rangle \]gives an eigenstate of the total angular momentum of the two particles, using standard notation, which of the following is true?
(Hint: \(\hat J_{\pm}|j,m\rangle = \sqrt{j(j+1)-m(m\pm1)}\,|j,m\pm1\rangle\))

• (A) \(c_1 = \dfrac{2}{\sqrt5},\quad c_2 = \dfrac{1}{\sqrt5}\)
• (B) \(c_1 = \dfrac{1}{\sqrt5},\quad c_2 = \dfrac{2}{\sqrt5}\)
• (C) \(c_1 = \dfrac{1}{\sqrt2},\quad c_2 = \dfrac{1}{\sqrt2}\)
• (D) \(c_1 = 0,\quad c_2 = 1\)

The energy \(E\) and degeneracy \(d\) of the second excited state of a three-dimensional, isotropic quantum harmonic oscillator with angular frequency \(\omega\) are

• (A) \(E = \dfrac{7}{2}\hbar\omega,\ d = 6\)
• (B) \(E = \dfrac{7}{2}\hbar\omega,\ d = 3\)
• (C) \(E = \dfrac{5}{2}\hbar\omega,\ d = 3\)
• (D) \(E = \dfrac{5}{2}\hbar\omega,\ d = 6\)

Two identical particles with a fixed total energy \(E = 2\hbar\omega\) are in thermal equilibrium in a one-dimensional harmonic oscillator potential \(\frac{1}{2}m\omega^2x^2\). Let the entropy of the particles be denoted by \(S_F\) if they are fermions with spin \(\frac{1}{2}\) (\(S_z = \pm\frac{\hbar}{2}\)) and by \(S_B\) if they are bosons with spin 0. Then, which of the following options is correct? (\(k_B\) is the Boltzmann constant)

• (A) \(S_F = k_B\ln 2,\ S_B = k_B\ln 2\)
• (B) \(S_F = 2k_B\ln 2,\ S_B = 0\)
• (C) \(S_F = 4k_B\ln 2,\ S_B = 0\)
• (D) \(S_F = 2k_B\ln 2,\ S_B = k_B\ln 2\)

In the circuit shown, \(V(t) = 2\sin(2000\pi t)\) Volts, where \(t\) is in seconds. The source \(V(t)\) drives the non-inverting (+) input of an opamp directly, the inverting (-) input is tied to ground (0 V), and the opamp is powered from +15 V and -15 V rails with no feedback network between its output and either input. Take the opamp to be ideal. Which of the following options is correct?

• (A) \(V_{out}(t)\) is square wave with peak-to-peak voltage = 30 V and time period is 1 ms.
• (B) \(V_{out}(t)\) is a sine wave with peak-to-peak voltage = 4 V and time period is 1 ms.
• (C) \(V_{out}(t)\) is sine wave with peak-to-peak voltage = 30 V and time period of 1 ms.
• (D) \(V_{out}(t)\) is square wave with peak-to-peak voltage = 4 V and time period is 1 ms.

• (A) NOT gate between pins 1 and 2 is faulty.
• (B) NOR gate is faulty.
• (C) NOT gate between pins 4 and 5 is faulty.
• (D) The NOR and output NOT gates, are both faulty.

A gas of \(N\) classical particles that can occupy energy levels \(\epsilon_1\) and \(\epsilon_2 = \epsilon_1+\Delta\) is in equilibrium with a reservoir at temperature \(T\). From the schematics shown below, choose the correct dependence of the internal energy \(U\) on \(T\).

• (A) A curve that starts flat at a low value for small \(T\), rises smoothly through an S-shaped (sigmoid) transition, and saturates to a higher flat value at large \(T\).
• (B) A curve that starts at zero and keeps rising with an ever-increasing slope as \(T\) grows, with no upper flat limit.
• (C) A curve that starts high at small \(T\) and falls off smoothly to a lower flat value as \(T\) increases.
• (D) A curve that starts at a small nonzero value and keeps rising with an ever-increasing slope as \(T\) grows, with no upper flat limit.

The Lagrangian \(L_0 = \frac{1}{2}m\dot{q}^2 - \frac{1}{2}m\omega^2q^2\) with the generalized coordinate \(q\) is transformed to \(L = L_0 + \alpha \frac{df(q)}{dt}\). Consider the following statements:
(i) Expression for the canonical momentum does not change.
(ii) The equation of motion does not change.
Which of the following options is correct for the above statements?

• (A) Both (i) and (ii) are correct.
• (B) Both (i) and (ii) are not correct.
• (C) (i) is correct and (ii) is not correct.
• (D) (i) is not correct and (ii) is correct.

An infinitely large thin sheet in the \(xy\)-plane carries uniform positive charge density and is moving with constant velocity \(\vec{v}\) in the \(+x\) direction (see figure below). The direction of the corresponding Poynting vector is

• (A) \(+x\) for both \(z < 0\) and \(z > 0\)
• (B) \(+x\) for \(z < 0\) and \(-x\) for \(z > 0\)
• (C) \(-x\) for \(z < 0\) and \(+x\) for \(z > 0\)
• (D) \(-x\) for both \(z < 0\) and \(z > 0\)

A positive point charge is fixed at the origin. At some distance from it on the \(x\) axis, a point dipole is kept pointing in the \(+y\) direction. The force on the dipole is

• (A) 0
• (B) in the \(+y\) direction
• (C) in the \(-y\) direction
• (D) in the \(+x\) direction

Two frames \(S\) (solid lines) and \(S'\) (dashed lines) with common origin are shown in the figure below. Frame \(S\) is inertial while \(S'\) is rotating about the common \(z\)-axis. There is a point mass fixed at \(P\) on the \(x\)-axis of the \(S\) frame. The magnitude of the centrifugal force and the Coriolis force experienced by the mass in the \(S'\) frame is \(F_{cen}\) and \(F_{cor}\), respectively. Which of the following options is correct for these forces?

• (A) \(F_{cen} = 0\) and \(F_{cor} = 0\)
• (B) \(F_{cen} \neq 0\) and \(F_{cor} \neq 0\) and \(F_{cen} = \dfrac{F_{cor}}{2}\)
• (C) \(F_{cen} \neq 0\) and \(F_{cor} \neq 0\) and \(F_{cen} = 2F_{cor}\)
• (D) \(F_{cen} \neq 0\) and \(F_{cor} \neq 0\) and \(F_{cen} = F_{cor}\)

Which of the following operators is/are self-adjoint?

• (A) \(x^2\dfrac{d^2}{dx^2} + 3x\dfrac{d}{dx} + x^2\)
• (B) \((1-x^2)\dfrac{d^2}{dx^2} - 2x\dfrac{d}{dx} + 3x\)
• (C) \((3x-4x^3)\dfrac{d^2}{dx^2} + (3-12x^2)\dfrac{d}{dx} + 12\)
• (D) \(x\dfrac{d^2}{dx^2} + x^2\dfrac{d}{dx} + \dfrac{5x}{3}\)

Consider two operators \(\hat{A}\) and \(\hat{B}\) which are related as \(\hat{A} = \exp(i\theta \hat{B})\). If \(\theta\) is a non-zero real number, which of the following statements is/are true?

• (A) If \(\hat{B}\) is Hermitian, then \(\hat{A}\) is unitary.
• (B) If \(\hat{B}\) is anti-Hermitian, then \(\hat{A}\) is unitary.
• (C) If \(\hat{B}\) is Hermitian, then \(|\text{Det}(\hat{A})| = 1\).
• (D) If \(\hat{B}\) is anti-Hermitian, then \(\hat{A}\) is Hermitian.

Consider operators \(\hat{A}\), \(\hat{B}\), and \(\hat{C}\) for three observables of a quantum system satisfying \([\hat{A},\hat{B}] = 0\), \([\hat{B},\hat{C}] = 0\), and \([\hat{A},\hat{C}] \neq 0\), with uncertainties \(\Delta A, \Delta B, \Delta C\), respectively. From the options given below, which is/are implied by the commutation relations among \(\hat{A}\), \(\hat{B}\), and \(\hat{C}\)?

• (A) \(\Delta A \, \Delta B > 0\)
• (B) \(\Delta A \, \Delta C > 0\)
• (C) \(\hat{A}, \hat{B}\) can be simultaneously diagonalized.
• (D) \(\hat{A}, \hat{B}, \hat{C}\) can be simultaneously diagonalized.

Consider the distribution of outcomes generated by \(N\) (\(N \gg 1\)) independent throws of (i) a coin or (ii) a six-sided dice. A coin (dice) is unbiased if both (all) its sides have equal probability to show up in a throw; it is biased otherwise. For the cases (i) and (ii) above, which of the following statements is/are true?

• (A) The entropy of an unbiased coin is smaller than that of an unbiased dice.
• (B) The entropy of an unbiased coin is greater than that of an unbiased dice.
• (C) The entropy of a biased dice is smaller than that of an unbiased dice.
• (D) The entropy of a biased coin is greater than that of an unbiased coin.

The dispersion (\(E(k)\)) of the conduction band (CB) and valence band (VB) for a semiconductor are shown schematically in the figure. Considering the possibility of an electron making a transition from the bottom of the CB to the top of the VB, which of the following options is/are correct?

• (A) The transition is forbidden.
• (B) A photon can be emitted with an energy exactly equal to \(E_g\).
• (C) A photon can be emitted with an energy less than \(E_g\).
• (D) A phonon can be created with a crystal momentum \(\hbar q\).

A symmetric rigid body has moment of inertia \(I_1, I_2, I_3\) about its principal axes 1, 2, and 3, respectively, with \(I_1 = I_3 = I_\perp\) and \(I_2 \neq I_\perp\). It is rotating in space with no torque on it so that its angular momentum \(\vec{L}\) is constant. Let \(\omega_1, \omega_2, \omega_3\) be the components of its angular velocity along the principal axes 1, 2, and 3, respectively. Which of the following quantities is/are constant during the motion of this rigid body?

• (A) \(\omega_1 + \omega_3\)
• (B) \(\omega_1^2 + \omega_3^2\)
• (C) Angle between axis 2 and \(\vec{L}\)
• (D) \(\omega_2\)

An alpha particle moves towards a fixed nucleus carrying charge \(Ze\), with initial speed \(v_0\) and impact parameter \(b\). Starting from a large distance from the nucleus, its distance of closest approach is \(r_m\) and its speed there is \(v_m\). Then which of the following options is correct?

Consider a particle of mass \(m = 9.0 \times 10^{-5}\) kg and charge \(q = 3.0 \times 10^{-4}\) C in a uniform electromagnetic field \(\vec{E} = 2\,\hat{x}\) V.m\(^{-1}\), \(\vec{B} = 3\,\hat{z}\) V.m\(^{-2}\).s. The particle is released from the coordinates \((0, 5\text{ m}, 0)\) at time \(t = 0\). Starting from initial speed zero, it comes back to the \(y\)-axis for the first time at time \(t\). The value of \(t\) in seconds (rounded off to two decimal places) is ______

An atom has two energy levels with energy difference 2.2 eV between them. A gas of these atoms has \(8 \times 10^{20}\) atoms in the upper state and \(5 \times 10^{20}\) atoms in the lower state. Ignoring spontaneous emission, the maximum possible energy released by this gas of atoms by stimulated emission is \(E\) Joules. The value of \(E\) (rounded off to one decimal place) is ______ \((e = 1.6\times10^{-19}\text{ C})\)

The Hamiltonian of two interacting spin-1/2 particles is \(H = \dfrac{A}{\hbar^2}\vec{S}_1\cdot\vec{S}_2\), where \(\vec{S}_1\) and \(\vec{S}_2\) are the spin angular momenta of particles 1 and 2, respectively. Here, \(A = 10.56\) eV. The energy in eV required to induce an excitation from the ground state to the excited state (rounded off to two decimal places) is ______

The output signal (current \(I_d\)) of a reversed biased (with a voltage \(V_b\)) photodiode, on which light is incident, is fed to an amplifier (see figure). The output voltage is digitized by a 10 bit Analogue to Digital convertor (ADC) which has a reference voltage of 5 V. The smallest current which can be measured by the circuit in nano-Amperes (rounded off to one decimal place) is ______

A capacitor is made of two circular metal plates of radius 1 m, separated by a distance \(d = 1\text{ mm}\). The space between the plates is filled with a dielectric of permittivity \(\epsilon_r = 5\). The capacitor is connected to a voltage \(V = 10\sin(2\pi \times 10^6 \times t)\) volts, where \(t\) is in seconds. The maximum value of the magnetic field between the plates, at a radial distance \(r = 0.5\text{ m}\) from the centre, is \(B \times 10^{-6}\text{ T}\). The value of \(B\) (rounded off to two decimal places) is ______.
(Speed of light in vacuum \(c = 3 \times 10^8\text{ m.s}^{-1}\))

Rotational spectrum of a diatomic molecule consists of lines of equal spacing with an interval of \(20.0\text{ cm}^{-1}\). Its moment of inertia is found to be \(I_0 \times 10^{-47}\text{ kg.m}^2\), the value of \(I_0\) (rounded off to one decimal place) is ______.
(\(h = 6.6 \times 10^{-34}\text{ J.s}\), speed of light in vacuum \(c = 3 \times 10^8\text{ m.s}^{-1}\))

Copper has an electron number density of \(8.3 \times 10^{28}\text{ m}^{-3}\). Its Fermi energy in eV (rounded off to one decimal place) is ______.
(\(\hbar = 1.06 \times 10^{-34}\text{ J.s}\), mass of electron \(m_e = 9.10\times 10^{-31}\text{ kg}\), charge of electron \(= 1.60\times 10^{-19}\text{ C}\))

Two 1 kg blocks are connected to two massless springs of spring constants \(8\text{ N.m}^{-1}\) and \(4\text{ N.m}^{-1}\). The system is kept on a frictionless horizontal floor with one end of a spring attached to a wall (see figure below). They are performing oscillatory motion along the x-axis with the normal mode frequencies \(\omega_H\) and \(\omega_L\) (\(\omega_H > \omega_L\)). The ratio \(\dfrac{\omega_H}{\omega_L}\) (rounded off to two decimal places) is ______

A 15 cm long scale is held horizontally with one of its ends on the edge of a 1 m high table and the other end resting on one's index finger. As the finger is removed (see figure below), the scale starts rotating about its end on the table. After 0.1 s, during which it has rotated by a negligibly small angle but has gained a rotational speed, it leaves the table and falls vertically towards the ground. When its centre of mass has fallen by 0.5 m, it has rotated by an angle \(\theta\). The value of \(\theta\) in degrees (rounded off to one decimal place) is ______
(\(g = 9.8\text{ m.s}^{-2}\))