General Aptitude (GA) Set-4
Q. 1 – Q. 5 carry one mark each.
1. “ When she fell down the _______, she received many _______ but little help.”
The words that best fill the blanks in the above sentence are
(A) stairs, stares
(B) stairs, stairs
(C) stares, stairs
(D) stares, stares
2. “ In spite of being warned repeatedly, he failed to correct his _________ behaviour.”
The word that best fills the blank in the above sentence is
(A) rational
(B) reasonable
(C) errant
(D) good
3. For 0 ≤ 𝑥 ≤ 2𝜋, sin 𝑥 and cos 𝑥 are both decreasing functions in the interval ________.
(A) 
(B) 
(C) 
(D) 
4. The area of an equilateral triangle is √3. What is the perimeter of the triangle?
(A) 2
(B) 4
(C) 6
(D) 8
5. Arrange the following three-dimensional objects in the descending order of their volumes:
(i) A cuboid with dimensions 10 cm, 8 cm and 6 cm
(ii) A cube of side 8 cm
(iii) A cylinder with base radius 7 cm and height 7 cm
(iv) A sphere of radius 7 cm
(A) (i), (ii), (iii), (iv)
(B) (ii), (i), (iv), (iii)
(C) (iii), (ii), (i), (iv)
(D) (iv), (iii), (ii), (i)
Q. 6 – Q. 10 carry two marks each.
6. An automobile travels from city A to city B and returns to city A by the same route. The speed of the vehicle during the onward and return journeys were constant at 60 km/h and 90 km/h, respectively. What is the average speed in km/h for the entire journey?
(A) 72
(B) 73
(C) 74
(D) 75
7. A set of 4 parallel lines intersect with another set of 5 parallel lines. How many parallelograms are formed?
(A) 20
(B) 48
(C) 60
(D) 72
8. To pass a test, a candidate needs to answer at least 2 out of 3 questions correctly. A total of 6,30,000 candidates appeared for the test. Question A was correctly answered by 3,30,000 candidates. Question B was answered correctly by 2,50,000 candidates. Question C was answered correctly by 2,60,000 candidates. Both questions A and B were answered correctly by 1,00,000 candidates. Both questions B and C were answered correctly by 90,000 candidates. Both questions A and C were answered correctly by 80,000 candidates. If the number of students answering all questions correctly is the same as the number answering none, how many candidates failed to clear the test?
(A) 30,000
(B) 2,70,000
(C) 3,90,000
(D) 4,20,000
9. If 𝑥2 + 𝑥 − 1 = 0 what is the value of 
(A) 1
(B) 5
(C) 7
(D) 9
10. In a detailed study of annual crow births in India, it was found that there was relatively no growth during the period 2002 to 2004 and a sudden spike from 2004 to 2005. In another unrelated study, it was found that the revenue from cracker sales in India which remained fairly flat from 2002 to 2004, saw a sudden spike in 2005 before declining again in 2006. The solid line in the graph below refers to annual sale of crackers and the dashed line refers to the annual crow births in India. Choose the most appropriate inference from the above data.
(A) There is a strong correlation between crow birth and cracker sales.
(B) Cracker usage increases crow birth rate.
(C) If cracker sale declines, crow birth will decline.
(D) Increased birth rate of crows will cause an increase in the sale of crackers.
Physics
Q. 1 – Q. 25 carry one mark each.
1. The eigenvalues of a Hermitian matrix are all
(A) real
(B) imaginary
(C) of modulus one
(D) real and positive
2. Which one of the following represents the 3p radial wave function of hydrogen atom? (a0 is the Bohr radius)
3. Given the following table,
which one of the following correctly matches the experiments from Group I to their inferences in Group II?
(A) P-2, Q-3, R-4, S-1
(B) P-1, Q-3, R-2, S-4
(C) P-3, Q-4, R-2, S-1
(D) P-2, Q-1, R-4, S-3
4. In spherical polar coordinates (𝑟,𝜃,𝜙), the unit vector 
at (10,𝜋/4,𝜋/2) is
5. The scale factors corresponding to the covariant metric tensor 𝑔𝑖𝑗 in spherical polar coordinates are
(A) 1,𝑟2,𝑟2sin2𝜃
(B) 1,𝑟2,sin2𝜃
(C) 1, 1, 1
(D) 1,𝑟,𝑟sin𝜃
6. In the context of small oscillations, which one of the following does NOT apply to the normal coordinates?
(A) Each normal coordinate has an eigen-frequency associated with it
(B) The normal coordinates are orthogonal to one another
(C) The normal coordinates are all independent
(D) The potential energy of the system is a sum of squares of the normal coordinates with constant coefficients
7. For the given unit cells of a two dimensional square lattice, which option lists all the primitive cells?
8. Among electric field 
and vector potential 
which is/are odd under parity (space inversion) operation?
9. The expression for the second overtone frequency in the vibrational absorption spectra of a diatomic molecule in terms of the harmonic frequency 𝜔𝑒 and anharmonicity constant 𝑥𝑒 is
(A) 2𝜔𝑒(1−𝑥𝑒)
(B) 2𝜔𝑒(1−3𝑥𝑒)
(C) 3𝜔𝑒(1−2𝑥𝑒)
(D) 3𝜔𝑒(1−4𝑥𝑒)
10. Match the physical effects and order of magnitude of their energy scales given below, where 
is fine structure constant; 𝑚𝑒 and 𝑚𝑝 are electron and proton mass, respectively.
(A) P-3, Q-1, R-2, S-4
(B) P-2, Q-3, R-1, S-4
(C) P-4, Q-2, R-1, S-3
(D) P-2, Q-4, R-1, S-3
11. The logic expression 
can be simplified to
(A) 𝐴 XOR 𝐶
(C) 0
(D) 1
12. A t low temperatures (𝑇), the specific heat of common metals is described by (with 𝛼 and 𝛽 as constants)
(A) 𝛼 𝑇 + 𝛽 𝑇3
(B) 𝛽 𝑇3
(C) exp(−𝛼/𝑇)
(D) 𝛼 𝑇 + 𝛽 𝑇5
13. In a 2-to-1 multiplexer as shown below, the output X = A0 if C = 0, and X = A1 if C = 1.
Which one of the following is the correct implementation of this multiplexer?
14. T he elementary particle 
is placed in the baryon decuplet, shown below, at
(A) P
(B) Q
(C) R
(D) S
15. The intrinsic/permanent electric dipole moment in the ground state of hydrogen atom is (𝑎0 is the Bohr radius)
(A) −3𝑒𝑎0
(B) zero
(C) 𝑒𝑎0
(D) 3𝑒𝑎0
16. The high temperature magnetic susceptibility of solids having ions with magnetic moments can be described by 
with 𝑇 as absolute temperature and 𝜃 as constant. The three behaviors i.e. paramagnetic, ferromagnetic and anti-ferromagnetic are described, respectively, by
(A) 𝜃<0,𝜃>0,𝜃=0
(B) 𝜃>0,𝜃<0,𝜃=0
(C) 𝜃=0,𝜃<0,𝜃>0
(D) 𝜃=0,𝜃>0,𝜃<0
17. Which one of the following is an allowed electric dipole transition?
(A) 1 S0 → 3 S1
(B) 2 P3/2 → 2 D5/2
(C) 2 D5/2 → 2 P1/2
(D) 3 P0 → 5 D0
18. In the decay, 𝜇+→𝑒++𝜈𝑒+𝑋, what is 𝑋?
(A) 𝛾
(C) 𝜈𝜇
19. A spaceship is travelling with a velocity of 0.7c away from a space station. The spaceship ejects a probe with a velocity 0.59c opposite to its own velocity. A person in the space station would see the probe moving at a speed Xc, where the value of X is ________ (up to three decimal places).
20. For an operational amplifier (ideal) circuit shown below,
if V1 = 1 V and V2 = 2 V, the value of V0 is ________V (up to one decimal place).
21. An infinitely long straight wire is carrying a steady current I. The ratio of magnetic energy density at distance 𝑟1 to that at 𝑟2(=2 𝑟1) from the wire is ________.
22. A light beam of intensity 𝐼0 is falling normally on a surface. The surface absorbs 20% of the intensity and the rest is reflected. The radiation pressure on the surface is given by 𝑋𝐼0/c, where X is ________ (up to one decimal place). Here 𝑐 is the speed of light.
23. The number of independent components of a general electromagnetic field tensor is ________.
24. If 𝑋 is the dimensionality of a free electron gas, the energy (𝐸) dependence of density of states is given by 
where 𝑌 is ________.
25. For nucleus 164Er, a 𝐽𝜋=2+ state is at 90 keV. Assuming 164Er to be a rigid rotor, the energy of its 4+ state is ________ keV (up to one decimal place).
Q. 26 – Q. 55 carry two marks each.
26. Given 
which one of the following 
makes 
a complete set for a three dimensional real linear vector space ?
27. An interstellar object has speed 𝑣 at the point of its shortest distance 𝑅 from a star of much larger mass 𝑀. Given 𝑣2=2 𝐺𝑀/𝑅, the trajectory of the object is
(A) circle
(B) ellipse
(C) parabola
(D) hyperbola
28. A particle moves in one dimension under a potential 𝑉(𝑥)=𝛼|𝑥| with some non-zero total energy. Which one of the following best describes the particle trajectory in the phase space?
29. Consider an infinitely long solenoid with 𝑁 turns per unit length, radius 𝑅 and carrying a current 𝐼(𝑡)=𝛼cos𝜔𝑡, where 𝛼 is a constant and 𝜔 is the angular frequency. The magnitude of electric field at the surface of the solenoid is
(C) 𝜇0𝑁𝑅𝜔𝛼sin𝜔𝑡
(D) 𝜇0𝜔𝑁𝑅cos𝜔𝑡
30. A constant and uniform magnetic field 
pervades all space. Which one of the following is the correct choice for the vector potential in Coulomb gauge?
31. If 𝐻 is the Hamiltonian for a free particle with mass 𝑚, the commutator [𝑥,[𝑥,𝐻]] is
(A) ℏ2/𝑚
(B) −ℏ2/𝑚
(C) −ℏ2/(2𝑚)
(D) ℏ2/(2𝑚)
32. A long straight wire, having radius 𝑎 and resistance per unit length 𝑟, carries a current 𝐼. The magnitude and direction of the Poynting vector on the surface of the wire is
(A) 𝐼2𝑟2𝜋𝑎⁄ , perpendicular to axis of the wire and pointing inwards
(B) 𝐼2𝑟2𝜋𝑎⁄ , perpendicular to axis of the wire and pointing outwards
(C) 𝐼2𝑟𝜋𝑎⁄ , perpendicular to axis of the wire and pointing inwards
(D) 𝐼2𝑟𝜋𝑎⁄ , perpendicular to axis of the wire and pointing outwards
33. Three particles are to be distributed in four non-degenerate energy levels. The possible number of ways of distribution: (i) for distinguishable particles, and (ii) for identical Bosons, respectively, is
(A) (i) 24, (ii) 4
(B) (i) 24, (ii) 20
(C) (i) 64, (ii) 20
(D) (i) 64, (ii) 16
34. The term symbol for the electronic ground state of oxygen atom is
(A) 1𝑆0
(B) 1𝐷2
(C) 3𝑃0
(D) 3𝑃2
35. The energy dispersion for electrons in one dimensional lattice with lattice parameter 𝑎 is given by 
where 𝑊 and 𝐸0 are constants. The effective mass of the electron near the bottom of the band is
36. Amongst electrical resistivity (𝜌), thermal conductivity (𝜅), specific heat (𝐶), Young’s modulus (Y), and magnetic susceptibility (𝜒), which quantities show a sharp change at the superconducting transition temperature?
(A) 𝜌,𝜅,𝐶, Y
(B) 𝜌,𝐶, 𝜒
(C) 𝜌,𝜅,𝐶, 𝜒
(D) 𝜅, Y, 𝜒
37. A quarter wave plate introduces a path difference of 𝜆/4 between the two components of polarization parallel and perpendicular to the optic axis. An electromagnetic wave with 
is incident normally on a quarter wave plate which has its optic axis making an angle 135° with the 𝑥-axis as shown.
The emergent electromagnetic wave would be
(A) elliptically polarized
(B) circularly polarized
(C) linearly polarized with polarization as that of incident wave
(D) linearly polarized but with polarization at 90° to that of the incident wave
38. A p-doped semiconductor slab carries a current 𝐼=100 mA in a magnetic field 𝐵=0.2 T as shown. One measures 𝑉𝑦=0.25 mV and 𝑉𝑥=2 mV. The mobility of holes in the semiconductor is ________ m2V–1s–1 (up to two decimal places).
39. An n-channel FET having Gate-Source switch-off voltage VGS(OFF) = – 2 V is used to invert a 0 – 5 V square-wave signal as shown. The maximum allowed value of R would be ________ k (up to two decimal places).
40. Inside a large nucleus, a nucleon with mass 939 MeVc–2 has Fermi momentum 1.40 fm–1 at absolute zero temperature. Its velocity is Xc, where the value of X is ________ (up to two decimal places).
(ℏ𝑐=197 MeV-fm)
41. 4 MeV 𝛾-rays emitted by the de-excitation of 19F are attributed, assuming spherical symmetry, to the transition of protons from 1d3/2 state to 1d5/2 state. If the contribution of spin-orbit term to the total energy is written as 
the magnitude of 𝐶 is ________ MeV (up to one decimal place).
42. An 𝛼 particle is emitted by a 
nucleus. Assuming the potential to be purely Coulombic beyond the point of separation, the height of the Coulomb barrier is ________ MeV (up to two decimal places).
43. For the transformation
(where 𝛼 is a constant) to be canonical, the value of 𝛼 is________.
44. Given
and boundary conditions 𝑓(0)=1 and 𝑓(1)=0, the value of 𝑓(0.5) is ________ (up to two decimal places).
45. The absolute value of the integral
over the circle |𝑧−1.5|=1 in complex plane, is ________ (up to two decimal places).
46. A uniform circular disc of mass 𝑚 and radius 𝑅 is rotating with angular speed 𝜔 about an axis passing through its center and making an angle 𝜃=30∘ with the axis of the disc. If the kinetic energy of the disc is 𝛼𝑚𝜔2𝑅2, the value of 𝛼 is ________ (up to 2 decimal places).
47. The ground state energy of a particle of mass 𝑚 in an infinite potential well is 𝐸0. It changes to 𝐸0(1+𝛼×10−3), when there is a small potential bump of height 
and width 𝑎=𝐿/100, as shown in the figure. The value of 𝛼 is ________ (up to two decimal places).
48. An electromagnetic plane wave is propagating with an intensity 𝐼=1.0×105 Wm–2 in a medium with 𝜖=3𝜖0 and 𝜇=𝜇0. The amplitude of the electric field inside the medium is ________ ×103 Vm–1 (up to one decimal place).
(𝜖0= 8.85×10−12 C2 N–1 m–2, 𝜇0=4𝜋×10−7NA–2, 𝑐 = 3×108 ms–1)
49. A microcanonical ensemble consists of 12 atoms with each taking either energy 0 state, or energy 𝜖 state. Both states are non-degenerate. If the total energy of this ensemble is 4𝜖, its entropy will be ________ 𝑘𝐵 (up to one decimal place), where 𝑘𝐵 is the Boltzmann constant.
50. A two-state quantum system has energy eigenvalues ±𝜖 corresponding to the normalized states 
At time 𝑡=0, the system is in quantum state 
The probability that the system will be in the same state at 𝑡=ℎ/(6𝜖) is ________ (up to two decimal places).
51. An air-conditioner maintains the room temperature at 27°C while the outside temperature is 47°C. The heat conducted through the walls of the room from outside to inside due to temperature difference is 7000 W. The minimum work done by the compressor of the air-conditioner per unit time is ________ W.
52. Two solid spheres A and B have same emissivity. The radius of A is four times the radius of B, and temperature of A is twice the temperature of B. The ratio of the rate of heat radiated from A to that from B is ________.
53. The partition function of an ensemble at a temperature 𝑇 is
where 𝑘𝐵 is the Boltzmann constant. The heat capacity of this ensemble at 
is 𝑋 𝑁𝑘𝐵, where the value of 𝑋 is ________ (up to two decimal places).
54. An atom in its singlet state is subjected to a magnetic field. The Zeeman splitting of its 650 nm spectral line is 0.03 nm. The magnitude of the field is ________ Tesla (up to two decimal places).
( 𝑒=1.60×10−19 C, 𝑚𝑒=9.11×10−31 kg, 𝑐=3.0×108 ms–1)
55. The quantum effects in an ideal gas become important below a certain temperature 𝑇𝑄 when de Broglie wavelength corresponding to the root mean square thermal speed becomes equal to the inter-atomic separation. For such a gas of atoms of mass 2×10−26 kg and number density 6.4×1025 m–3, 𝑇𝑄= __________ ×10−3 K (up to one decimal place).
(𝑘𝐵=1.38×10−23 J/K, ℎ=6.6×10−34 J-s)