GA – General Aptitude
Q1 – Q5 carry one mark each.
1. He is known for his unscrupulous ways. He always sheds______ tears to deceive people.
2. Jofra Archer, the England fast bowler, is ____ than accurate.
(A) more fast
(C) less fast
(D) more faster
3. Select the word that fits the analogy:
Build : Building :: Grow : ______
4. I do not think you know the case well enough to have opinions. Having said that, I agree with your other point.
What does the phrase “having said that” mean in the given text?
(A) as opposed to what I have said
(B) despite what I have said
(C) in addition to what I have said
(D) contrary to what I have said
5. Define [x] as the greatest integer less than or equal to x, for each x ∈ (−∞, ∞). If y = [x], then area under y for x ∈ [1, 4] is _______.
Q6 – Q10 carry two marks each.
6. Crowd funding deals with mobilization of funds for a project from a large number of people, who would be willing to invest smaller amounts through web-based platforms in the project.
Based on the above paragraph, which of the following is correct about crowd funding?
(A) Funds raised through unwilling contributions on web-based platforms.
(B) Funds raised through large contributions on web-based platforms.
(C) Funds raised through coerced contributions on web-based platforms.
(D) Funds raised through voluntary contributions on web-based platforms.
7. P, Q, R and S are to be uniquely coded using α and β. If P is coded as αα and Q as αβ, then R and S, respectively, can be coded as ______.
(A) βα and αβ
(B) ββ and αα
(C) αβ and ββ
(D) βα and ββ
8. The sum of the first n terms in the sequence 8, 88, 888, 8888, … is ______.
9. Select the graph that schematically represents BOTH y = xm and y = x1/m properly in the interval 0 ≤ x ≤ 1, for integer values of m, where m > 1.
10. The bar graph shows the data of the students who appeared and passed in an examination for four schools P, Q, R and S. The average of success rates (in percentage) of these four schools is ________.
Q1 – Q25 carry one mark each.
1. Which one of the following is a solution of for k real?
(B) sin kx
(C) cos kx
(D) sin hx
2. A real, invertible 3 × 3 matrix M has eigenvalues λi, (i = 1, 2, 3) and the corresponding eigenvectors are Which one of the following is correct?
3. A quantum particle is subjected to the potential
The ground state wave function of the particle is proportional to
4. Let respectively denote the lowering and raising operators of a one-dimensional simple harmonic oscillator. Let be the energy eigenstate of the simple harmonic oscillator. Given that is also an eigenstate of the corresponding eigenvalue is
(A) n(n – 1)
(B) n(n + 1)
(C) (n + 1)2
5. Which one of the following is a universal logic gate?
6. Which one of the following is the correct binary equivalent of the hexadecimal F6C?
(A) 0110 1111 1100
(B) 1111 0110 1100
(C) 1100 0110 1111
(D) 0110 1100 0111
7. The total angular momentum j of the ground state of the nucleus is
8. A particle X is produced in the process π+ + p → K+ + X via the strong interaction. If the quark content of the K+ is , the quark content of X is
(B) u u d
(C) u u s
9. A medium (εr > 1, μr = 1, σ > 0) is semi-transparent to an electromagnetic wave when
(A) Conduction current >> Displacement current
(B) Conduction current << Displacement current
(C) Conduction current = Displacement current
(D) Both Conduction current and Displacement current are zero
10. A particle is moving in a central force field given by is the unit vector pointing away from the center of the field. The potential energy of the particle is given by
11. Choose the correct statement related to the Fermi energy (EF) and the chemical potential (μ) of a metal.
(A) μ = EF only at 0 K
(B) μ = EF at finite temperature
(C) μ < EF at 0 K
(D) μ > EF at finite temperature
12. Consider a diatomic molecule formed by identical atoms. If EV and Ee represent the energy of the vibrational nuclear motion and electronic motion respectively, then in terms of the electronic mass m and nuclear mass M, EV/Ee is proportional to
13. Which one of the following relations determines the manner in which the electric field lines are refracted across the interface between two dielectric media having dielectric constants ε1 and ε2 (see figure) ?
(A) ε1 sinθ1 = ε2 sinθ2
(B) ε1 cosθ1 = ε2 cosθ2
(C) ε1 tanθ1 = ε2 tanθ2
(D) ε1 cotθ1 = ε2 cotθ2
14. If are the electric and magnetic fields respectively, then is
(A) odd under parity and even under time reversal
(B) even under parity and odd under time reversal
(C) odd under parity and odd under time reversal
(D) even under parity and even under time reversal
15. A small disc is suspended by a fiber such that it is free to rotate about the fiber axis (see figure). For small angular deflections, the Hamiltonian for the disc is given by
where I is the moment of inertia and α is the restoring torque per unit deflection. The disc is subjected to angular deflections (θ) due to thermal collisions from the surrounding gas at temperature T and p0 is the momentum conjugate to θ. The average and t he root-mean-square angular deflection, θavg and θrms, respectively are
16. As shown in the figure, an ideal gas is confined to chamber A of an insulated container, with vacuum in chamber B. When the plug in the wall separating the chambers A and B is removed, the gas fills both the chambers. Which one of the following statements is true?
(A) The temperature of the gas remains unchanged
(B) Internal energy of the gas decreases
(C) Temperature of the gas decreases as it expands to fill the space in chamber B
(D) Internal energy of the gas increases as its atoms have more space to move around
17. Particle A with angular momentum j = 3/2 decays into two particles B and C with angular momenta j1 and j2, respectively. If the value of α is ______.
18. Far from the Earth, the Earth’s magnetic field can be approximated as due to bar magnet of magnetic pole strength 4 × 1014 Assume this magnetic field is generated by a current carrying loop encircling the magnetic equator. The current required to do so is about 4 × 10n A, where n is an integer. The value of n is _____.
(Earth’s circumference: 4 × 107 m)
19. The number of distinct ways the primitive unit cell can be constructed for the two dimensional lattice as shown in the figure is_____.
20. A hydrogenic atom is subjected to a strong magnetic field. In the absence of spin-orbit coupling, the number of doubly degenerated states created out of the d-level is _______.
21. A particle Y undergoes strong decay Y → π− + π−. The isospin of Y is ______.
22. For a complex variable z and the contour c: |z| = 1 taken in the counter clockwise direction, _______.
23. Let p be the momentum conjugate to the generalized coordinate q. If the transformation
Q = √2qm cos p
P = √2qm sin p
is canonical, then m = ______.
24. A conducting sphere of radius 1 m is placed in air. The maximum number of electrons that can be put on the sphere to avoid electrical breakdown is about 7 × 10n, where n is integer. The value of n is ______.
Breakdown electric field strength in air is
Permittivity of free space ε0 = 8.85 × 10−12 F/m
Electron charge e = 1.60 × 10−19 C
25. If a particle is moving along a sinusoidal curve, the number of degrees of freedom of the particle is ______.
Q26 – Q55 carry two marks each.
26. The product of eigenvalues of is
27. Let Let ℝ3 denote the three-dimensional real vector space. Which one of the following is correct?
(A) S is an orthonormal set
(B) S is a linearly dependent set
(C) S is a basis for ℝ3
28. denotes the spin operator defined as Which one of the following is correct?
(A) The eigenstates of spin operator
(B) The eigenstates of spin operator
(C) In the spin state upon the measurement of , the probability for obtaining
(D) In the spin state upon the measurement of , the probability for obtaining
29. The input voltage (Vin) to the circuit shown in the figure is 2cos(100t) V. The output voltage (Vout) is If R = 1 kΩ, the value of C (in μF) is
30. Consider a 4-bit counter constructed out of four flip-flops. It is formed by connecting the J and K inputs to logic high and feeding the Q output to the clock input of the following flip-flop (see the figure). The input signal to the counter is series of square pulses and the change of state is triggered by the falling edge. At time t = t0 the outputs are in logic low state (Q0 = Q1 = Q2 = Q3 = 0). Then at t = t1, the logic state of the outputs is
(A) Q0 = 1, Q1 = 0, Q2 = 0 and Q3 = 0
(B) Q0 = 0, Q1 = 0, Q2 = 0 and Q3 = 1
(C) Q0 = 1, Q1 = 0, Q2 = 1 and Q3 = 0
(D) Q0 = 0, Q1 = 1, Q2 = 1 and Q3 = 1
31. Consider the Lagrangian where a, b and c are constants. If px and py are the momenta conjugate to the coordinates x and y respectively, then the Hamiltonian is
32. Which one of the following matrices does NOT represent a proper rotation in a plane?
33. A uniform magnetic field exists in an inertial frame K. A perfect conducting sphere moves with a constant velocity with respect to this inertial frame. The rest frame of the sphere is K’ (see figure). The electric and magnetic fields in K and K’ are related as
The induced surface charge density on the sphere (to the lowest order in v/c) in the frame K’ is
(A) maximum along z’
(B) maximum along y’
(C) maximum along x’
(D) uniform over the sphere
34. A charge q moving with uniform speed enters a cylindrical region in free space at t = 0 and exits the region at t = τ (see figure). Which one of the following options best describes the time dependence of the total electric flux φ(t), through the entire surface of the cylinder?
35. Consider a one-dimensional non-magnetic crystal with one atom per unit cell. Assume that the valence electrons (i) dot not interact with each other and (ii) interact weakly with the ions. If n is the number of valence electrons per unit cell, then at 0 K,
(A) the crystal is metallic for any value of n
(B) the crystal is non-metallic for any value of n
(C) the crystal is metallic for even values of n
(D) the crystal is metallic for odd values of n
36. According to the Fermi gas model of the nucleus, the nucleons move in a spherical volume of radius R(=R0A1/3, where A is the mass number and R0 is an empirical constant with the dimensions of length). The Fermi energy of the nucleus EF is proportional to
37. Consider a two dimensional crystal with 3 atoms in the basis. The number of allowed optical branches (n) and acoustic branches (m) due to the lattice vibrations are
(A) (n, m) = (2, 4)
(B) (n, m) = (3, 3)
(C) (n, m) = (4, 2)
(D) (n, m) = (1, 5)
38. The internal energy U of a system is given by U(S, V) = λV−2/3S2, where λ is a constant of appropriate dimensions; V and S denote the volume and entropy, respectively. Which one of the following gives the correct equation of state of the system?
39. The potential energy of a particle of mass m is given by U(x) = a sin(k2x – π/2), a > 0, k2 > 0. The angular frequency of small oscillations of the particle about x = 0 is
40. The radial wave function of a particle in a central potential is given by where A is the normalization constant and a is positive constant of suitable dimensions. If γa is the most probable distance of the particle from the force center, the value of γ is _________.
41. A free particle of mass M is located in a three-dimensional cubic potential well with impenetrable walls. The degeneracy of the fifth excited state of the particle is __________.
42. Consider the circuit given in the figure. Let the forward voltage drop across each diode be 0.7 V. The current I (in mA) through the resistor is ____________.
43. Let uu denote the 4-velocity of a relativistic particle whose squre uuuμ = 1. If εμvρσ is the Levi-Civita tensor then the value of εμvρσuμuvuρuσ is ______.
44. Consider a simple cubic monoatomic Bravias lattice which has a basis with vectors a is the lattice parameter. The Bragg reflection is observed due to t he change in the wave vector between the incident and the scattered beam as given by where are primitive reciprocal lattice vectors. For n1 = 3, n2 = 3 and n3 = 2, the geometrical structure factor is _______.
45. A plane electromagnetic wave of wavelength λ is incident on a circular loop of conducting wire. The loop radius is a(a << λ). The angle (in degrees), made by the Poynting vector with t he normal to the plane of the loop to generate a maximum induced electrical signal, is _______.
46. An electron in a hydrogen atom is in the state n = 3, l = 2, m = − Let denote the y-component of the orbital angular momentum operator. If the value of α is ________.
47. A sinusoidal voltage of the form V(t) = V0 cos(ωt) is applied across a parallel plate capacitor placed in vacuum. Ignoring the edge effects, the induced emf within the region between the capacitor plates can be expressed as a power series in ω. The lowest non-vanishing exponent in ω is ________.
48. If for –π ≤ x ≤ π, the value of a2 is ______.
The value of is _______.
50. Consider the Hamiltonian where
is the time independent penturbation given by
where k > 0. If the maximum energy eigenvalue of is 3 eV corresponding to E = 2 eV, the value of k (rounded off to three decimal places) in eV is ______.
51. A hydrogen atom is in an orbital angular momentum state lies on a cone which makes a half angle 30° with respect to the z-axis, the value of l is _______.
52. In the center of mass frame, two protons each having energy 7000 GeV, collide to produce protons and anti-protons. The maximum number of anti-protons produced is_______.
(Assume the proton mass to be 1 GeV/c2)
53. Consider a gas of hydrogen atoms in the atmosphere of the Sun where the temperature is 5800 K. If a sample from this atmosphere contains 6.023 × 1023 of hydrogen atoms in the ground state, the number of hydrogen atoms in the first excited state is approximately 8 × 10n, where n is an integer. The value of n is ______.
(Boltzmann constant: 8.617 × 10−5 eV/K)
54. For a gas of non-interacting particles, the probability that a particle has a speed v in the interval v to v + dv is given by
If E is the energy of a particle, then the maximum in the corresponding energy distribution in units of E/kBT occurs at _______ (rounded off to one decimal place).
55. The Planck’s energy density distribution is given by At long wavelengths, the energy density of photons in thermal equilibrium with a cavity at temperature T varies as Tα, where α is _______.
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