Q4
MCQ • 1M
Quantitative & Analytical Aptitude
In 2022, June Huh was awarded the Fields medal, which is the highest prize in Mathematics. When he was younger, he was also a poet. He did not win any medals in the International Mathematics Olympiads. He dropped out of college. Based only on the above information, which one of the following statements can be logically inferred with certainty?
Q6
MCQ • 2M
Quantitative & Analytical Aptitude
Human beings are one among many creatures that inhabit an imagined world. In this imagined world, some creatures are cruel. If in this imagined world, it is given that the statement " Some human beings are not cruel creatures" is FALSE, then which of the following set of statement(s) can be logically inferred with certainty? (i) All human beings are cruel creatures. (ii) Some human beings are cruel creatures. (iii) Some creatures that are cruel are human beings. (iv) No human beings are cruel creatures.
Q8
MCQ • 2M
Quantitative & Analytical Aptitude
The World Bank has declared that it does not plan to offer new financing to Sri Lanka, which is battling its worst economic crisis in decades, until the country has an adequate macroeconomic policy framework in place. In a statement, the World Bank said Sr i Lanka needed to adopt structural reforms that focus on economic stabilisation and tackle the root causes of its crisis. The latter has starved it of foreign exchange and led to shortages of food, fuel, and medicines. The bank is repurposing resources under existing loans to help alleviate shortages of essential items such as medicine, cooking gas, fertiliser, meals for children, and cash for vulnerable households. Based only on the above passage, which one of the following statements can be inferred with certainty?
Q41
MCQ • 2M
Nuclear & Particle Physics
A symmetric top has principal moments of inertia I1 = I2 = 2𝛼 3 , I3 = 2𝛼 about a set of principal axes 1, 2, 3 respectively, passing through its center of mass, where 𝛼 is a positive constant. There is no force acting on the body and the angular speed of the body about the 3 -axis is 𝜔3 = 1 8 rad/s. With what angular frequency in rad/s does the angular velocity vector 𝜔⃗⃗ 1 precess about the 3-axis?
Q42
MCQ • 2M
Nuclear & Particle Physics
A particle of mass m is free to move on a frictionless horizontal two dimensional (r, 𝜃) plane, and is acted upon by a force F = - k 2r3 r̂ with k being a positive constant. If pr and p𝜃 are the generalised momenta corresponding to r and 𝜃 respectively, then what is the value of dpr dt ?
Q43
MCQ • 2M
Nuclear & Particle Physics
Consider two real functions U(x, y) = xy(x2 - y2) , V(x, y) = ax4 + by4 + cx2y2 + k , where k is a real constant and a, b, c are real coefficients. If U(x, y) + i V(x, y) is analytic, then what is the value of a × b × c ?
Q44
MCQ • 2M
Nuclear & Particle Physics
Young's double slit experiment is performed using a beam of C60 (fullerene) molecules, each molecule being made up of 60 carbon atoms. When the slit separation is 50 nm, fringes are formed on a screen kept at a distance of 1 m from the slit s. Now, t he experiment is repeated with C70 molecules with a slit separation of 92.5 nm. The kinetic energies of both the beams are the same. The position of the 4th bright fringe for C60 will correspond to the nth bright fringe for C70. What is the value of n (rounded off to the nearest integer) ?
Q48
NAT • 2M
Nuclear & Particle Physics
In a hadronic interaction, 𝜋0's are produced with different momenta, and they immediately decay into two photons with an opening angle 𝜃 between them. Assuming that all these decays occur in one plane, which one of the following figures depicts the behaviour of 𝜃 as a function of the 𝜋0 momentum p? (A) (B) (C) (D)
Q49
MCQ • 2M
Quantum Mechanics
A particle has wavefunction 𝜓(x, y, z) = N z e-𝛼(x2+y2+z2) , where N is a normalization constant and 𝛼 is a positi ve constant. In this state, which one of the following options represent s the eigenvalues of L2 and Lz respectively? Some values of Yℓ m are : Y0 0 = √ 1 4𝜋 , Y1 0 = √ 3 4𝜋 cos 𝜃 , Y1 ±1 = ∓√ 3 8𝜋 sin 𝜃 e±i𝜙
Q52
MCQ • 2M
Classical Mechanics
Consider two non-identical spin 1 2 particles labelled 1 and 2 in the spin product state | 1 2 , 1 2〉 | 1 2 , - 1 2〉. The Hamiltonian of the system is H = 4λ ℏ2 S 1 ⋅ S 2 , where S 1 and S 2 are the spin operators of particles 1 and 2, respectively, and 𝜆 is a constant with appropriate dimensions. What is the expectation value of H in the above state?
Q57
MCQ • 2M
Nuclear & Particle Physics
Consider the vector field V⃗ consisting of the velocities of points on a thin horizontal disc of radius R = 2 m, moving anticlockwise with uniform angular speed 𝜔 = 2 rad/sec about an axis passing through its center. If V = |V⃗ | , then which of the following options is(are) CORRECT ? (In the options, r̂ and 𝜃̂ are unit vectors corresponding to the plane polar coordinates r and 𝜃). You may use the fact that in cylindrical coordinates (s, 𝜙, z) (s is the distance from the z-axis), the gradient, divergence, curl and Laplacian operators are: ∇⃗⃗ f = 𝜕f 𝜕s ŝ + 1 s 𝜕f 𝜕𝜙 𝜙̂ + 𝜕f 𝜕z ẑ ; ∇⃗⃗ . A = 1 s 𝜕 𝜕s (sAs) + 1 s 𝜕A𝜙 𝜕𝜙 + 𝜕Az 𝜕z ; ∇⃗⃗ × A = ( 1 s 𝜕Az 𝜕𝜙 - 𝜕A𝜙 𝜕z ) ŝ + ( 𝜕As 𝜕z - 𝜕Az 𝜕s ) 𝜙̂ + 1 s ( 𝜕 𝜕s (sA𝜙) - 𝜕As 𝜕𝜙 ) ẑ ; ∇⃗⃗ 2f = 1 s 𝜕 𝜕s (s 𝜕f 𝜕s) + 1 s2 𝜕2f 𝜕𝜙2 + 𝜕2f 𝜕z2 .
Q59
NAT • 2M
Electromagnetic Theory
Two independent electrostatic configurations are shown in the figure. Configuration (I) consists of an isolated point charge q = 1 C, and configuration (II) consists of another identical charge surrounded by a thick conducting shell of inner radius R1 = 1 m and outer radius R2 = 2 m, with the charge being at the center of the shell. WI = 𝜖0 2 ∫ EI 2 dV and WII = 𝜖0 2 ∫ EII 2 dV , where EI and EII are the magnitudes of the electric fields for configurations (I) and (II) respectively, 𝜖0 is the permittivity of vacuum, and the volume integrations are carried out over all space. If 8𝜋 𝜖0 |WI - WII| = 1 n , what is the value of the integer n?
Q60
NAT • 2M
Nuclear & Particle Physics
In pion nucleon scattering, the pion and nucleon can combine to form a short lived bound state called the Δ particle (𝜋 + N → Δ). The masses of the pion, nucleon and the Δ particle are 140 MeV/c2, 938 MeV/c2 and 1230 MeV/c2, respectively. In the lab frame, where the nucleon is at rest, what is the minimum energy (in MeV/c2, rounded off to one decimal place) of the pion to produce the Δ particle?
Q62
NAT • 2M
Nuclear & Particle Physics
An 𝛼-particle is emitted from the decay of Americium ( Am) at rest, i.e., Am94 241 → U92 237 + 𝛼. The rest mass es of Am94 241 , U92 237 and 𝛼 are 224.544 GeV/c2, 220.811 GeV/ c2 and 3.728 GeV/ c2 respectively. What is the kinetic energy (in MeV/c2, rounded off to two decimal places) of the 𝛼-particle?
Q64
NAT • 2M
Nuclear & Particle Physics
Two identical, non-interacting 4He2 atoms are distributed among 4 different non- degenerate energy levels. The probability that they occupy different energy levels is p. Similarly, two 3He2 atoms are distributed among 4 different non-degenerate energy levels, and the probability that they occupy different levels is q. What is the value of p q (rounded off to one decimal place)?