**GATE 2019 General Aptitude (GA) Set-8**

**Q. 1 – Q. 5 carry one mark each.**

1. The fishermen, ________ the flood victims owed their lives, were rewarded by the government.

(A) whom

(B) to which

(C) to whom

(D) that

2. Some students were not involved in the strike.

If the above statement is true, which of the following conclusions is/are logically necessary?

1. Some who were involved in the strike were students.

2. No student was involved in the strike.

3. At least one student was involved in the strike.

4. Some who were not involved in the strike were students.

(A) 1 and 2

(B) 3

(C) 4

(D) 2 and 3

3. The radius as well as the height of a circular cone increases by 10%. The percentage increase in its volume is .

(A) 17.1

(B) 21.0

(C) 33.1

(D) 72.8

4. Five numbers 10, 7, 5, 4 and 2 are to be arranged in a sequence from left to right following the directions given below:

1. No two odd or even numbers are next to each other.

2. The second number from the left is exactly half of the left-most number.

3. The middle number is exactly twice the right-most number.

Which is the second number from the right?

(A) 2

(B) 4

(C) 7

(D) 10

5. Until Iran came along, India had never been in kabaddi.

(A) defeated

(B) defeating

(C) defeat

(D) defeatist

**Q. 6 – Q. 10 carry two marks each.**

6. Since the last one year, after a 125 basis point reduction in repo rate by the Reserve Bank of India, banking institutions have been making a demand to reduce interest rates on small saving schemes. Finally, the government announced yesterday a reduction in interest rates on small saving schemes to bring them on par with fixed deposit interest rates.

Which one of the following statements can be inferred from the given passage?

(A) Whenever the Reserve Bank of India reduces the repo rate, the interest rates on small saving schemes are also reduced

(B) Interest rates on small saving schemes are always maintained on par with fixed deposit interest rates

(C) The government sometimes takes into consideration the demands of banking institutions before reducing the interest rates on small saving schemes

(D) A reduction in interest rates on small saving schemes follow only after a reduction in repo rate by the Reserve Bank of India

7. In a country of 1400 million population, 70% own mobile phones. Among the mobile phone owners, only 294 million access the Internet. Among these Internet users, only half buy goods from e-commerce portals. What is the percentage of these buyers in the country?

(A) 10.50

(B) 14.70

(C) 15.00

(D) 50.00

8. The nomenclature of Hindustani music has changed over the centuries. Since the medieval period dhrupad styles were identified as baanis. Terms like gayaki and baaj were used to refer to vocal and instrumental styles, respectively. With the institutionalization of music education the term gharana became acceptable. Gharana originally referred to hereditary musicians from a particular lineage, including disciples and grand disciples.

Which one of the following pairings is NOT correct?

(A) dhrupad, baani

(B) gayaki, vocal

(C) baaj, institution

(D) gharana, lineage

9. Two trains started at 7AM from the same point. The first train travelled north at a speed of 80km/h and the second train travelled south at a speed of 100 km/h. The time at which they were 540 km apart is AM.

(A) 9

(B) 10

(C) 11

(D) 11.30

10. “I read somewhere that in ancient times the prestige of a kingdom depended upon the number of taxes that it was able to levy on its people. It was very much like the prestige of a head-hunter in his own community.”

Based on the paragraph above, the prestige of a head-hunter depended upon

(A) the prestige of the kingdom

(B) the prestige of the heads

(C) the number of taxes he could levy

(D) the number of heads he could gather

**Physics**

**Q.1 – Q. 25 carry one mark each.**

1. The relative magnetic permeability of a type-I superconductor is

(A) 0

(B) −1

(C) 2𝜋

(D) 1/4π

2. Considering baryon number and lepton number conservation laws, which of the following processes is/are allowed?

(i) 𝑝 → 𝜋^{0} + 𝑒‑ + 𝜈_{𝑒}

(ii) 𝑒^{+} + 𝜈_{𝑒} → 𝜇^{+} + 𝜈_{𝜇}

(A) both (i) and (ii)

(B) only (i)

(C) only (ii)

(D) neither (i) nor (ii)

3. For the following circuit, what is the magnitude of V_{out} if V_{in} = 1.5 V?

(A) 0.015 V

(B) 0.15 V

(C)15 V

(D)150 V

4. For the differential equation where 𝑛 is a constant, the product of its two independent solutions is

(A) 1/x

(B) x

(C) x^{n}

(D)1/x^{n+1}

5. Consider a one-dimensional gas of 𝑁 non-interacting particles of mass 𝑚 with the Hamiltonian for a single particle given by,

The high temperature specific heat in units of 𝑅 = 𝑁𝑘_{𝐵} (𝑘_{𝐵} is the Boltzmann constant) is

(A) 1

(B) 1.5

(C) 2

(D) 2.5

6. An electric field is applied to a Hydrogen atom in 𝑛 = 2 excited state. Ignoring spin, the 𝑛 = 2 state is fourfold degenerate, which in the |𝑙, 𝑚⟩ basis are given by |0,0〉, |1,1⟩, |1,0⟩ and |1, −1⟩. If 𝐻′ is the interaction Hamiltonian corresponding to the applied electric field, which of the following matrix elements is nonzero?

(A) ⟨0,0|𝐻′|0,0⟩

(B) ⟨0,0|𝐻′|1,1⟩

(C) ⟨0,0|𝐻′|1,0⟩

(D) ⟨0,0|𝐻′|1, −1⟩

7. A large number 𝑁 of ideal bosons, each of mass m, are trapped in a three-dimensional potential The bosonic system is kept at temperature 𝑇 which is much lower than the Bose-Einstein condensation temperature 𝑇_{𝑐}. The chemical potential (𝜇) satisfies

8. During a rotation, vectors along the axis of rotation remain unchanged. For the rotation matrix the unit vector along the axis of rotation is

9. For a spin denote its spin up and spin down states, respectively. If are composite states of two such particles, which of the following statements is true for their total spin S ?

10. Consider a transformation from one set of generalized coordinate and momentum (𝑞, 𝑝) to another set (𝑄, 𝑃) denoted by,

𝑄 = 𝑝𝑞^{𝑠}; 𝑃 = 𝑞^{𝑟}

where 𝑠 and 𝑟 are constants. The transformation is canonical if

(A) 𝑠 = 0 and 𝑟 = 1

(B) 𝑠 = 2 and 𝑟 = −1

(C) 𝑠 = 0 and 𝑟 = −1

(D) 𝑠 = 2 and 𝑟 = 1

11. In order to estimate the specific heat of phonons, the appropriate method to apply would be

(A) Einstein model for acoustic phonons and Debye model for optical phonons

(B) Einstein model for optical phonons and Debye model for acoustic phonons

(C) Einstein model for both optical and acoustic phonons

(D) Debye model for both optical and acoustic phonons

12. The pole of the function f(z) = cot z at z = 0 is

(A) a removable singularity

(B) an essential singularity

(C) a simple pole

(D) a second order pole

13. A massive particle X in free space decays spontaneously into two photons. Which of the following statements is true for X?

(A) X is charged

(B) Spin of X must be greater than or equal to 2

(C) X is a boson

(D) X must be a baryon

14. The electric field of an electromagnetic wave is given by The wave is

(A) linearly polarized at an angle tan^{−1} (4/3) from the x-axis

(B) linearly polarized at an angle tan^{−1} (3/4) from the x-axis

(C) elliptically polarized in clockwise direction when seen travelling towards the observer

(D) elliptically polarized in counter-clockwise direction when seen travelling towards the observer

15. The nuclear spin and parity of in its ground state is

(A) 0^{+}

(B) 0^{−}

(C) 1^{+}

(D) 1^{−}

16. An infinitely long thin cylindrical shell has its axis coinciding with the z-axis. It carries a surface charge density 𝜎_{0}cos𝜙, where 𝜙 is the polar angle and 𝜎_{0} is a constant. The magnitude of the electric field inside the cylinder is

(A) 0

(B) 𝜎_{0}/2𝜖_{0}

(C) 𝜎_{0}/3𝜖_{0}

(D) 𝜎_{0}/4𝜖_{0}

17. Consider a three-dimensional crystal of 𝑁 inert gas atoms. The total energy is given by where 𝑝 = 12.13, 𝑞 = 14.45, and 𝑅 is the nearest neighbour distance between two atoms. The two constants, 𝜖 and 𝑅, have the dimensions of energy and length, respectively. The equilibrium separation between two nearest neighbour atoms in units of 𝜎 (rounded off to two decimal places) is ______

18. The energy-wavevector (𝐸-𝑘) dispersion relation for a particle in two dimensions is 𝐸 = 𝐶𝑘, where 𝐶 is a constant. If its density of states 𝐷(𝐸) is proportional to 𝐸𝑝 then the value of 𝑝 is________

19. A circular loop made of a thin wire has radius 2 cm and resistance 2 Ω. It is placed perpendicular to a uniform magnetic field of magnitude Tesla. At time 𝑡 = 0 the field starts decaying as The total charge that passes through a cross section of the wire during the decay is Q. The value of Q in 𝜇𝐶 (rounded off to two decimal places) is _______

20. The electric field of an electromagnetic wave in vacuum is given by

The wave is reflected from the 𝑧 = 0 surface. If the pressure exerted on the surface is 𝛼𝜖_{0}𝐸^{2}_{0}, the value of 𝛼 (rounded off to one decimal place) is ______

21. The Hamiltonian for a quantum harmonic oscillator of mass 𝑚 in three dimensions is

where 𝜔 is the angular frequency. The expectation value of r^{2} in the first excited state of the oscillator in units of h/mω (rounded off to one decimal place) is_______

22. The Hamiltonian for a particle of mass m is where 𝑞 and 𝑝 are the generalized coordinate and momentum, respectively, 𝑡 is time and 𝑘 is a constant. For the initial condition, 𝑞 = 0 and 𝑝 = 0 at 𝑡 = 0, 𝑞(𝑡) ∝ 𝑡^{α}. The value of α is_______

23. At temperature T Kelvin (K), the value of the Fermi function at an energy 0.5 eV above the Fermi energy is 0.01. Then T, to the nearest integer, is ________

(𝑘_{B} = 8.62 × 10^{−5} eV/K)

24. Let represent two possible states of a two-level quantum system. The state obtained by the incoherent superposition of |𝜓_{1}〉 and |𝜓_{2}〉 is given by a density matrix that is defined as 𝜌 ≡ 𝑐_{1}|𝜓_{1}〉〈𝜓_{1}| + 𝑐_{2}|𝜓_{2}〉〈𝜓_{2}|. If 𝑐_{1} = 0.4 and 𝑐_{2} = 0.6, the matrix element 𝜌_{22} (rounded off to one decimal place) is _________

25. A conventional type-I superconductor has a critical temperature of 4.7 K at zero magnetic field and a critical magnetic field of 0.3 Tesla at 0 K. The critical field in Tesla at 2 K (rounded off to three decimal places) is ______

** Q.26 – Q. 55 carry two marks each.**

26. Consider the following Boolean expression:

It can be represented by a single three-input logic gate. Identify the gate.

(A) AND

(B) OR

(C) XOR

(D) NAND

27. The value of the integral where 𝑘 > 0 and 𝑎 > 0, is

28. The wave function 𝜓(𝑥) of a particle is as shown below

Here 𝐾 is a constant, and 𝑎 > 𝑑. The position uncertainty 〈Δ𝑥〉 of the particle is

29. A solid cylinder of radius 𝑅 has total charge 𝑄 distributed uniformly over its volume. It is rotating about its axis with angular speed 𝜔. The magnitude of the total magnetic moment of the cylinder is

30. Consider the motion of a particle along the x-axis in a potential 𝑉(𝑥) = 𝐹|𝑥|. Its ground state energy 𝐸_{0} is estimated using the uncertainty principle. Then 𝐸0 is proportional to

(A) F^{1/3}

(B) F^{1/2}

(C) F^{2/5}

(D) F^{2/3}

31. A 3-bit analog-to-digital converter is designed to digitize analog signals ranging from 0 V to 10 V. For this converter, the binary output corresponding to an input of 6 V is

(A) 011

(B) 101

(C) 100

(D) 010

32. The Hamiltonian operator for a two-level quantum system is If the state of the system at t = 0 is given by at a later time 𝑡 is

33. A particle of mass m moves in a lattice along the x-axis in a periodic potential 𝑉(𝑥) = 𝑉(𝑥 + 𝑑) with periodicity d. The corresponding Brillouin zone extends from −𝑘0 to 𝑘0 with these two k-points being equivalent. If a weak force F in the x-direction is applied to the particle, it starts a periodic motion with time period T. Using the equation of motion for a particle moving in a band, where p_{crystal} is the crystal momentum of the particle, the period T is found to be (h is the Planck constant)

34. Consider a potential barrier 𝑉(𝑥) of the form:

where 𝑉_{0} is a constant. For particles of energy 𝐸 < 𝑉0 incident on this barrier from the left, which of the following schematic diagrams best represents the probability density |𝜓 (𝑥)|^{2} as a function of 𝑥.

35. The spin-orbit interaction term of an electron moving in a central field is written as where r is the radial distance of the electron from the origin. If an electron moves inside a uniformly charged sphere, then

(A) 𝑓(𝑟) = constant

(B) 𝑓(𝑟) ∝ 𝑟^{−1}

(C) 𝑓(𝑟) ∝ 𝑟^{−2}

(D) 𝑓(𝑟) ∝ 𝑟^{−3}

36. For the following circuit, the correct logic values for the entries X_{2} and Y_{2} in the truth table are

(A) 1 and 0

(B) 0 and 0

(C) 0 and 1

(D) 1 and 1

37. In a set of 𝑁 successive polarizers, the 𝑚th polarizer makes an angle (𝑚𝜋/2N) with the vertical. A vertically polarized light beam of intensity 𝐼_{0} is incident on two such sets with 𝑁 = 𝑁_{1} and 𝑁 = 𝑁_{2}, where 𝑁_{2} > 𝑁_{1}. Let the intensity of light beams coming out be 𝐼(𝑁1) and 𝐼(𝑁2) , respectively. Which of the following statements is correct about the two outgoing beams?

(A) 𝐼(𝑁_{2}) > 𝐼(𝑁_{1}) ; the polarization in each case is vertical

(B) 𝐼(𝑁_{2}) < 𝐼(𝑁_{1}) ; the polarization in each case is vertical

(C) 𝐼(𝑁_{2}) > 𝐼(𝑁_{1}) ; the polarization in each case is horizontal

(D) 𝐼(𝑁_{2}) < 𝐼(𝑁_{1}) ; the polarization in each case is horizontal

38. A ball bouncing off a rigid floor is described by the potential energy function

V(x) = mgx for x > 0

= ∞ for x ≤ 0

Which of the following schematic diagrams best represents the phase space plot of the ball?

39. An infinitely long wire parallel to the x-axis is kept at 𝑧 = 𝑑 and carries a current I in the positive x direction above a superconductor filling the region 𝑧 ≤ 0 (see figure). The magnetic field inside the superconductor is zero so that the field just outside the superconductor is parallel to its surface. The magnetic field due to this configuration at a point (𝑥, 𝑦, 𝑧 > 0) is

40. The vector potential inside a long solenoid, with n turns per unit length and carrying current 𝐼, written in cylindrical coordinates If the term where α≠ 0, β ≠ 0, is added to the magnetic field remains the same if

(A) 𝛼 = 𝛽

(B) 𝛼 = −𝛽

(C) 𝛼 = 2𝛽

(D) 𝛼 = 𝛽/2

41. Low energy collision (s-wave scattering) of pion (π+) with deuteron (d) results in the production of two protons (𝜋+ + 𝑑 → 𝑝 + 𝑝). The relative orbital angular momentum (in units of ℏ) of the resulting two-proton system for this reaction is

(A) 0

(B) 1

(C) 2

(D) 3

42. Consider the Hamiltonian where 𝛼 and 𝛽 are parameters with appropriate dimensions, and 𝑞 and 𝑝 are the generalized coordinate and momentum, respectively. The corresponding Lagrangian 𝐿(𝑞, 𝑞̇) is

43. For a given load resistance RL = 4.7 ohm, the power transfer efficiencies of a dc voltage source and a dc current source with internal resistances R1 and R2, respectively, are equal. The product R1R2 in units of ohm2 (rounded off to one decimal place) is _______

44. The ground state electronic configuration of the rare-earth ion (Nd^{3+}) is [Pd] 4f^{3}5s^{2}5p^{6}. Assuming LS coupling, the Land𝑒́ 𝑔-factor of this ion is 8/11 . The effective magnetic moment in units of Bohr magneton 𝜇_{B} (rounded off to two decimal places) is__________

45. A projectile of mass 1 kg is launched at an angle of 30° from the horizontal direction at 𝑡 = 0 and takes time 𝑇 before hitting the ground. If its initial speed is 10 ms−1 , the value of the action integral for the entire flight in the units of kg m2s^{−1} (rounded off to one decimal place) is________

[Take 𝑔 = 10 ms^{−2}]

46. Let 𝜃 be a variable in the range − 𝜋 ≤ 𝜃 < 𝜋. Now consider a function

𝜓(𝜃) = 1 for − 𝜋/2 ≤ 𝜃 < 𝜋/2

= 0 otherwise.

If its Fourier-series is written as then the value of|𝐶_{3}|^{2} (rounded off to three decimal places) is ____

47. Two spaceships A and B, each of the same rest length 𝐿, are moving in the same direction with speeds 4𝑐/5 and 3𝑐/5, respectively, where 𝑐 is the speed of light. As measured by B, the time taken by A to completely overtake B [see figure below] in units of 𝐿/𝑐 (to the nearest integer) is _________

48. A radioactive element X has a half-life of 30 hours. It decays via alpha, beta and gamma emissions with the branching ratio for beta decay being 0.75. The partial half-life for beta decay in unit of hours is_______

49. In a thermally insulated container, 0.01 kg of ice at 273 K is mixed with 0.1 kg of water at 300 K. Neglecting the specific heat of the container, the change in the entropy of the system in J/K on attaining thermal equilibrium (rounded off to two decimal places) is ________

(Specific heat of water is 4.2 kJ/kg-K and the latent heat of ice is 335 kJ/kg).

50. Consider a system of three charges as shown in the figure below:

For 𝑟 = 10 m; 𝜃 = 60 degrees; 𝑞 = 10^{−6} Coulomb, and 𝑑 = 10^{−3} m, the electric dipole potential in volts (rounded off to three decimal places) at a point (𝑟, 𝜃) is ________

51. Consider two systems A and B each having two distinguishable particles. In both the systems, each particle can exist in states with energies 0, 1, 2 and 3 units with equal probability. The total energy of the combined system is 5 units. Assuming that the system A has energy 3 units and the system B has energy 2 units, the entropy of the combined system is 𝑘B ln 𝜆. The value of 𝜆 is _________

52. Electrons with spin in the 𝑧-direction are passed through a Stern-Gerlach (SG) set up with the magnetic field at 𝜃 = 60o from . The fraction of electrons that will emerge with their spin parallel to the magnetic field in the SG set up (rounded off to two decimal places) is _________

53. The Hamiltonian of a system is with 𝜀 ≪ 1. The fourth order contribution to the ground state energy of 𝐻 is 𝛾𝜀^{4}. The value of 𝛾 (rounded off to three decimal places) is _________

54. Two events, one on the earth and the other one on the Sun, occur simultaneously in the earth’s frame. The time difference between the two events as seen by an observer in a spaceship moving with velocity 0.5c in the earth’s frame along the line joining the earth to the Sun is Δ𝑡, where c is the speed of light. Given that light travels from the Sun to the earth in 8.3 minutes in the earth’s frame, the value of |Δ𝑡| in minutes (rounded off to two decimal places) is__________

(Take the earth’s frame to be inertial and neglect the relative motion between the earth and the sun)

55. In a certain two-dimensional lattice, the energy dispersion of the electrons is

where denotes the wave vector, 𝑎 is the lattice constant and 𝑡 is a constant in units of eV. In this lattice the effective mass tensor 𝑚𝑖𝑗 of electrons calculated at the center of the Brillouin zone has the form The value of 𝛼 (rounded off to three decimal places) is _________

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