GATE PH 2012 Question Paper & Solutions

Quantitative Aptitude

Identify the CORRECT statement for the following vectors ˆ ˆ 3 2 a i j   and ˆ ˆ 2 b i j 

Analytical & Spatial Aptitude

Two uniform thin rods of equal length, L, and masses M1 and M2 are joined together along the length. The moment of inertia of the combined rod of length 2L about an axis passing through the mid-point and perpendicular to the length of the rod is,

Quantitative Aptitude

The space-time dependence of the electric field of a linearly polarized light in free space is given by 0 cos( ) xE t kz - where E0,  and k are the amplitude, the angular frequency and the wavevector, respectively. The time averaged energy density associated with the electric field is

Quantitative Aptitude

If the peak output voltage of a full wave rectifier is 10 V, its d.c. voltage is

Quantitative Aptitude

A particle of mass m is confined in a two dimensional square well potential of dimension a. This potential V(x,y) is given by ( , ) 0 V x y  for a x a - and a y a -   elsewhere The energy of the first excited state for this particle is given by,

Quantitative Aptitude

The isothermal compressibility,  of an ideal gas at temperature T0 and volume V0, is given by

Quantitative Aptitude

The ground state of sodium atom (11Na) is a 2S1/2 state. The difference in energy levels arising in the presence of a weak external magnetic field B, given in terms of Bohr magneton, µB, is

Quantitative Aptitude

For an ideal Fermi gas in three dimensions, the electron velocity vF at the Fermi surface is related to electron concentration n as,

Quantitative Aptitude

Which one of the following sets corresponds to fundamental particles?

Quantitative Aptitude

In case of a Geiger-Muller (GM) counter, which one of the following statements is CORRECT?

Electromagnetic Waves in Media

A plane electromagnetic wave traveling in free space is incident normally on a glass plate of refractive index 3/2. If there is no absorption by the glass, its reflectivity is

Crystal Structure & Lattice Dynamics

A Ge semiconductor is doped with acceptor impurity concentration of 1015 atoms/cm3. For the given hole mobility of 1800 cm2/V-s, the resistivity of this material is

Statistical Ensembles & Partition Functions

A classical gas of molecules, each of mass m, is in thermal equilibrium at the absolute temperature, T. The velocity components of the molecules along the Cartesian axes are , x y v v and zv . The mean value of   2 x y v v  is

Lagrangian & Hamiltonian Dynamics

In a central force field, the trajectory of a particle of mass m and angular momentum L in plane polar coordinates is given by, where,  is the eccentricity of the particle’s motion. Which one of the following choices for  gives rise to a parabolic trajectory?

Statistical Ensembles & Partition Functions

Identify the CORRECT energy band diagram for Silicon doped with Arsenic. Here CB, VB, ED and EF are conduction band, valence band, impurity level and Fermi level, respectively.

Molecular Spectroscopy

The first Stokes line of a rotational Raman spectrum is observed at 12.96 cm-1. Considering the rigid rotor approximation, the rotational constant is given by

Statistical Ensembles & Partition Functions

The total energy, E of an ideal non-relativistic Fermi gas in three dimensions is given by 53 23 N E V  where N is the number of particles and V is the volume of the gas. Identify the CORRECT equation of state (P being the pressure),

Schrodinger equation & Wave Mechanics

Consider the wavefunction   1 2 , s r r    for a fermionic system consisting of two spin-half particles. The spatial part of the wavefunction is given by,     1 2 1 1 2 2 2 1 1 2 1 , 2 r r r r r r            where 1 2 and  are single particle states. The spin part s  of the wavefunction with spin states (+½) and (-½) should be

Electrostatics and Magnetostatics

The electric and the magnetic fields ( , ) E z t and ( , ) B z t , respectively corresponding to the scalar potential (z,t) = 0 and vector potential ( , ) A z t itz  are

Complex analysis

Consider the following OP-AMP circuit. Vin + 5 V 4 k 1 k + - + 10 V - 10 V Vout Which one of the following correctly represents the output Vout corresponding to the input Vin? (A) Vin + 5 V Vout - 10 V + 10 V + 1 V t ® t ® 0 V (B) Vin + 5 V Vout - 10 V + 10 V + 1 V t ® t ® 0 V (C) Vin + 5 V Vout - 10 V + 10 V 0 V t ® t ® (D) Vin + 5 V Vout - 10 V + 10 V 0 V t ® t ®

Nuclear Properties & Nuclear Forces

Deuteron has only one bound state with spin parity 1+, isospin 0 and electric quadrupole moment 0.286 efm2. These data suggest that the nuclear forces are having (A) only spin and isospin dependence (B) no spin dependence and no tensor components (C) spin dependence but no tensor components (D) spin dependence along with tensor components 2012 PHYSICS - PH PH 6/20

Electrostatics and Magnetostatics

A particle of unit mass moves along the x-axis under the influence of a potential,   2 ( ) 2 V x x x  - . The particle is found to be in stable equilibrium at the point 2 x . The time period of oscillation of the particle is

Crystal Structure & Lattice Dynamics

Which one of the following CANNOT be explained by considering a harmonic approximation for the lattice vibrations in solids?

Electrostatics and Magnetostatics

A particle is constrained to move in a truncated harmonic potential well (x > 0) as shown in the figure. Which one of the following statements is CORRECT?

Linear Algebra & Vector Calculus

The number of independent components of the symmetric tensor Aij with indices i, j = 1, 2, 3 is

Perturbation theory & Approximation Methods

Consider a system in the unperturbed state described by the Hamiltonian, 0 1 0 0 1 H       . The system is subjected to a perturbation of the form H         , where 1   . The energy eigenvalues of the perturbed system using the first order perturbation approximation are

Statistical Ensembles & Partition Functions

Inverse susceptibility (1/) as a function of temperature, T for a material undergoing paramagnetic to ferromagnetic transition is given in the figure, where O is the origin. The values of the Curie constant, C, and the Weiss molecular field constant, , in CGS units, are

Electrostatics and Magnetostatics

A plane polarized electromagnetic wave in free space at time 0 t  is given by ( , ) 10 exp[ (6 8 )] E x z j i x z   . The magnetic field ( , , ) B x z t is given by

Linear Algebra & Vector Calculus

The eigenvalues of the matrix 0 1 0 1 0 1 0 1 0           are

Atomic Spectra & Multi-Electron Atoms

Match the typical spectroscopic regions specified in Group I with the corresponding type of transitions in Group II. Group I Group II (P) Infra-red region (i) electronic transitions involving valence electrons (Q) Ultraviolet-visible region (ii) nuclear transitions (R) X-ray region (iii) vibrational transitions of molecules (S) -ray region (iv) transitions involving inner shell electrons

Complex analysis

In the following circuit, for the output voltage to be Vo = ( - V1 + V2 /2), the ratio R1/R2 is V2 + - + VCC - VCC Vo R R V1 R2 R1 (A) 1/2 (B) 1 (C) 2 (D) 3

Atomic Spectra & Multi-Electron Atoms

The terms   1 2 j , j J arising from 2s13d1 electronic configuration in j-j coupling scheme are (A) 2,1 3,2 1 3 1 5 , and , 2 2 2 2             (B) 1,0 2,1 1 1 1 3 , and , 2 2 2 2             (C) 1,0 3,2 1 1 1 5 , and , 2 2 2 2             (D) 2,1 3,2 3 1 1 5 , and , 2 2 2 2            

Complex analysis

In the following circuit, the voltage drop across the ideal diode in forward bias condition is 0.7 V. 24 Volt 12 k 6 k 3.3k The current passing through the diode is (A) 0.5 mA (B) 1.0 mA (C) 1.5 mA (D) 2.0 mA

Nuclear Models & Decays

Choose the CORRECT statement from the following. (A) Neutron interacts through electromagnetic interaction (B) Electron does not interact through weak interaction (C) Neutrino interacts through weak and electromagnetic interaction (D) Quark interacts through strong interaction but not through weak interaction

Special theory of relativity

A rod of proper length l0 oriented parallel to the x-axis moves with speed 2c/3 along the x-axis in the S-frame, where c is the speed of the light in free space. The observer is also moving along the x-axis with speed c/2 with respect to the S-frame. The length of the rod as measured by the observer is (A) 0.35 l0 (B) 0.48 l0 (C) 0.87 l0 (D) 0.97 l0 + - 2012 PHYSICS - PH PH 9/20

Statistical Ensembles & Partition Functions

A simple cubic crystal with lattice parameter ac undergoes transition into a tetragonal structure with lattice parameters at = bt = 2 ac and ct = 2ac, below a certain temperature. The ratio of the interplanar spacings of (1 0 1) planes for the cubic and the tetragonal structures is (A) 1 6 (B) 1 6 (C) 3 8 (D) 3 8

Semiconductor Devices & Diodes

Consider the following circuit in which the current gain dc  of the transistor is 100. + 15 V 900  100  100 k Which one of the following correctly represents the load line (collector current IC with respect to collector-emitter voltage VCE) and Q-point of this circuit? (A) Q-point (2 V, 13 mA) 15 mA VCE ® 15 V IC ® (B) Q-point (2 V, 10 mA) 13 mA VCE ® 15 V IC ® (C) Q-point (7.5 V, 7.5 mA) 15 mA VCE ® 15 V IC ® (D) Q-point (7.5 V, 6.5 mA) 13 mA VCE ® 15 V IC ®

Statistical Ensembles & Partition Functions

Consider a system whose three energy levels are given by 0,  and 2. The energy level  is two-fold degenerate and the other two are non-degenerate. The partition function of the system with 1 B k T  is given by (A) 1 2e  -  (B) 2 2e e   - -  (C) 2 (1 ) e  -  (D) 2 1 e e   - -   2012 PHYSICS - PH PH 10/20

Electrostatics and Magnetostatics

Two infinitely extended homogeneous isotropic dielectric media (medium-1 and medium-2 with dielectric constants 1 0 2   and 2 0 5   , respectively) meet at the z = 0 plane as shown in the figure. A uniform electric field exists everywhere. For z  0, the electric field is given by 1 2 3 5 E i j k  -  . The interface separating the two media is charge free. The electric displacement vector in the medium-2 is given by (A) 2 0[10 15 10 ] D i j k     (B) 2 0[10 15 10 ] D i j k   -  (C) 2 0[4 6 10 ] D i j k   -  (D) 2 0[4 6 10 ] D i j k    

Schrodinger equation & Wave Mechanics

The ground state wavefunction for the hydrogen atom is given by 0 3/2 / 100 0 1 1 4 r a e a   -        , where a0 is the Bohr radius. The plot of the radial probability density, P(r) for the hydrogen atom in the ground state is (A) (B) (C) (D)

Complex analysis

Total binding energies of O15, O16 and O17 are 111.96 MeV, 127.62 MeV and 131.76 MeV, respectively. The energy gap between 1p1/2 and 1d5/2 neutron shells for the nuclei whose mass number is close to 16, is (A) 4.1 MeV (B) 11.5 MeV (C) 15.7 MeV (D) 19.8 MeV z = 0 medium-1 medium-2 2012 PHYSICS - PH PH 11/20

Lagrangian & Hamiltonian Dynamics

A particle of mass m is attached to a fixed point O by a weightless inextensible string of length a. It is rotating under the gravity as shown in the figure. The Lagrangian of the particle is   2 2 2 2 1 ( , ) sin cos 2 L ma mga       - where and   are the polar angles. The Hamiltonian of the particle is (A) 2 2 2 2 1 cos 2 sin p H p mga ma         -       (B) 2 2 2 2 1 cos 2 sin p H p mga ma                (C)   2 2 2 1 cos 2 H p p mga ma      - (D)   2 2 2 1 cos 2 H p p mga ma      

Complex analysis

Given B r F      , where   k j i B B ˆ ˆ ˆ 0     is a constant vector and r is the position vector. The value of  C r d F   , where C is a circle of unit radius centered at origin is, (A) 0 (B) 0 2 B  (C) 0 2 B  - (D) 1

Linear Algebra & Vector Calculus

The value of the integral  C zdz e1 , using the contour C of circle with unit radius 1  z is (A) 0 (B) 1-2 i (C) 1+2 i (D) 2 i

Electrostatics and Magnetostatics

A paramagnetic system consisting of N spin-half particles, is placed in an external magnetic field. It is found that N/2 spins are aligned parallel and the remaining N/2 spins are aligned antiparallel to the magnetic field. The statistical entropy of the system is, (A) 2 ln2 B Nk (B) ln 2 2 B N k (C) 3 ln 2 2 B N k (D) ln 2 B Nk C x y z g  m a  2012 PHYSICS - PH PH 12/20

Complex analysis

The equilibrium vibration frequency for an oscillator is observed at 2990 cm-1. The ratio of the frequencies corresponding to the first and the fundamental spectral lines is 1.96. Considering the oscillator to be anharmonic, the anharmonicity constant is (A) 0.005 (B) 0.02 (C) 0.05 (D) 0.1

Schrodinger equation & Wave Mechanics

At a certain temperature T, the average speed of nitrogen molecules in air is found to be 400 m/s. The most probable and the root mean square speeds of the molecules are, respectively, (A) 355 m/s, 434 m/s (B) 820 m/s, 917 m/s (C) 152 m/s, 301 m/s (D) 422 m/s, 600 m/s Common Data Questions Common Data for Questions 48 and 49: The wavefunction of a particle moving in free space is given by, x k i x k i e e -   2  Q.48 The energy of the particle is (A) m k 2 5 2 2  (B) m k 4 3 2 2  (C) m k 2 2 2  (D) m k 2 2 

Complex analysis

The energy of the particle is (A) m k 2 5 2 2 (B) m k 4 3 2 2 (C) m k 2 2 2 (D) m k 2 2

Electrostatics and Magnetostatics

The probability current density for the real part of the wavefunction is (A) 1 (B) k m (C) 2 k m (D) 0 Common Data for Questions 50 and 51: The dispersion relation for a one dimensional monatomic crystal with lattice spacing a , which interacts via nearest neighbour harmonic potential is given by sin 2 Ka A  , where A is a constant of appropriate unit. Q.50 The group velocity at the boundary of the first Brillouin zone is (A) 0 (B) 1 (C) 2 2 Aa (D) 2 1 2 2 Aa

Crystal Structure & Lattice Dynamics

( The group velocity at the boundary of the first Brillouin zone is (A) 0 (B) 1 (C) 2 2 Aa (D) 21 22 Aa

Electrostatics and Magnetostatics

The force constant between the nearest neighbour of the lattice is (M is the mass of the atom) (A) 2 4 MA (B) 2 2 MA (C) 2 MA (D) 2 2 MA 2012 PHYSICS - PH PH 13/20 Linked Answer Questions Statement for Linked Answer Questions 52 and 53: In a hydrogen atom, consider that the electronic charge is uniformly distributed in a spherical volume of radius a (= 0.5  10-10 m) around the proton. The atom is placed in a uniform electric field E = 30  105 V/m. Assume that the spherical distribution of the negative charge remains undistorted under the electric field.

Complex analysis

In the equilibrium condition, the separation between the positive and the negative charge centers is (A) 8.66  10-16 m (B) 2.60  10-15 m (C) 2.60  10-16 m (D) 8.66  10-15 m

Angular momentum & Hydrogen Atom

The polarizability of the hydrogen atom in unit of (C2m/N) is (A) 2.0  10-40 (B) 1.4  10-41 (C) 1.4  10-40 (D) 2.0  10-39 Statement for Linked Answer Questions 54 and 55: A particle of mass m slides under the gravity without friction along the parabolic path 2 y ax  as shown in the figure. Here a is a constant.

Lagrangian & Hamiltonian Dynamics

The Lagrangian for this particle is given by, (A) 2 2 1 2 L mx mgax  - (B) 2 2 2 2 1 (1 4 ) 2 L m a x x mgax   - (C) 2 2 1 2 L mx mgax   (D) 2 2 2 2 1 (1 4 ) 2 L m a x x mgax   

Atomic Spectra & Multi-Electron Atoms

The Lagranges equation of motion of the particle is (A) 2 x gax  (B) 2 2 2 2 (1 4 ) 2 4 m a x x mgax ma xx  - - (C) 2 2 2 2 (1 4 ) 2 4 m a x x mgax ma xx    (D) 2 x gax - y x m 2012 PHYSICS - PH PH 14/20 General Aptitude (GA) Questions Q. 56 - Q. 60 carry one mark each. Q.56 Choose the grammatically INCORRECT sentence: (A) They gave us the money back less the service charges of Three Hundred rupees. (B) This country’s expenditure is not less than that of Bangladesh. (C) The committee initially asked for a funding of Fifty Lakh rupees, but later settled for a lesser sum. (D) This country’s expenditure on educational reforms is very less. Q.57 Which one of the following options is the closest in meaning to the word given below? Mitigate (A) Diminish (B) Divulge (C) Dedicate (D) Denote Q.58 Choose the most appropriate alternative from the options given below to complete the following sentence: Despite several --------- the mission succeeded in its attempt to resolve the conflict. (A) attempts (B) setbacks (C) meetings (D) delegations Q.59 The cost function for a product in a firm is given by 5q2, where q is the amount of production. The firm can sell the product at a market price of 50 per unit. The number of units to be produced by the firm such that the profit is maximized is (A) 5 (B) 10 (C) 15 (D) 25 Q.60 Choose the most appropriate alternative from the options given below to complete the following sentence: Suresh’s dog is the one --------- was hurt in the stampede. (A) that (B) which (C) who (D) whom Q. 61 - Q. 65 carry two marks each. Q.61 Which of the following assertions are CORRECT? P: Adding 7 to each entry in a list adds 7 to the mean of the list Q: Adding 7 to each entry in a list adds 7 to the standard deviation of the list R: Doubling each entry in a list doubles the mean of the list S: Doubling each entry in a list leaves the standard deviation of the list unchanged (A) P, Q (B) Q, R (C) P, R (D) R, S 2012 PHYSICS - PH PH 15/20 Q.62 An automobile plant contracted to buy shock absorbers from two suppliers X and Y. X supplies 60% and Y supplies 40% of the shock absorbers. All shock absorbers are subjected to a quality test. The ones that pass the quality test are considered reliable. Of X’s shock absorbers, 96% are reliable. Of Y’s shock absorbers, 72% are reliable. The probability that a randomly chosen shock absorber, which is found to be reliable, is made by Y is (A) 0.288 (B) 0.334 (C) 0.667 (D) 0.720 Q.63 A political party orders an arch for the entrance to the ground in which the annual convention is being held. The profile of the arch follows the equation y = 2x - 0.1x2 where y is the height of the arch in meters. The maximum possible height of the arch is (A) 8 meters (B) 10 meters (C) 12 meters (D) 14 meters Q.64 Wanted Temporary, Part-time persons for the post of Field Interviewer to conduct personal interviews to collect and collate economic data. Requirements: High School-pass, must be available for Day, Evening and Saturday work. Transportation paid, expenses reimbursed. Which one of the following is the best inference from the above advertisement? (A) Gender-discriminatory (B) Xenophobic (C) Not designed to make the post attractive (D) Not gender-discriminatory Q.65 Given the sequence of terms, AD CG FK JP, the next term is (A) OV (B) OW (C) PV (D) PW END OF THE QUESTION PAPER

Complex analysis

Choose the grammatically INCORRECT sentence:

Complex analysis

Which one of the following options is the closest in meaning to the word given below? Mitigate (A) Diminish (B) Divulge (C) Dedicate (D) Denote

Complex analysis

Choose the most appropriate alternative from th e options given below to complete the following sentence: Despite several --------- the mission succeeded in its attempt to resolve the conflict. (A) attempts (B) setbacks (C) meetings (D) delegations

Complex analysis

The cost function for a product in a firm is gi ven by 5q2, where q is the amount of production. The firm can sell the product at a market price of 50 per unit. The number of units to be produced by the firm such that the profit is maximized is (A) 5 (B) 10 (C) 15 (D) 25

Complex analysis

Choose the most appropriate alternative from the options given below to complete the following sentence: Suresh's dog is the one --------- was hurt in the stampede. (A) that (B) which (C) who (D) whom Q. 61 - Q. 65 carry two marks each.

Complex analysis

Which of the following assertions are CORRECT? P: Adding 7 to each entry in a list adds 7 to the mean of the list Q: Adding 7 to each entry in a list adds 7 to the standard deviation of the list R: Doubling each entry in a list doubles the mean of the list S: Doubling each entry in a list leaves the standard deviation of the list unchanged (A) P, Q (B) Q, R (C) P, R (D) R, S 2012 PHYSICS - PH PH 15/20

Complex analysis

An automobile plant contracted to buy shock absorbers from two suppliers X and Y. X supplie s 60% and Y supplies 40% of the shock absorbers. All shock absorbers are subjected to a quality test. The ones that pass the quality test are considered reliable. Of X's shock absorbers, 96% are reliable. Of Y's shock absorbers, 72% are reliable. The probability that a randomly chosen shock absorber, which is found to be reliable, is made by Y is (A) 0.288 (B) 0.334 (C) 0.667 (D) 0.720

Complex analysis

A political party orders an arch for the entrance to the ground in which the annual convention is being held. The profile of the arch follows the equation y = 2x - 0.1x2 where y is the height of the arch in meters. The maximum possible height of the arch is (A) 8 meters (B) 10 meters (C) 12 meters (D) 14 meters

Complex analysis

Wanted Temporary, Part-time persons for the post of Field Interviewer to conduct personal interviews to collect and collate economic data. Requirements: High School -pass, must be available for Day, Evening and Saturday work. Transportation paid, expenses reimbursed. Which one of the following is the best inference from the above advertisement?

Complex analysis

Given the sequence of terms, AD CG FK JP, the next term is (A) OV (B) OW (C) PV (D) PW END OF THE QUESTION PAPER Paper Question no. Key Paper Question no. Key PH 1A PH 41 B PH 2C PH 42 B PH 3A PH 43 C PH 4C PH 44 D PH 5 Marks to All PH 45 D PH 6A PH 46 B PH 7B PH 47 A PH 8D PH 48 C PH 9C PH 49 D PH 10 A PH 50 A PH 11 A PH 51 A PH 12 C PH 52 C PH 13 D PH 53 B PH 14 B PH 54 B PH 15 Marks to All PH 55 B PH 16 C PH 56 D PH 17 B PH 57 A PH 18 B PH 58 B PH 19 D PH 59 A PH 20 A PH 60 Marks to All PH 21 D PH 61 C PH 22 B PH 62 B PH 23 D PH 63 B PH 24 D PH 64 D PH 25 C PH 65 A PH 26 A PH 27 C PH 28 A PH 29 B PH 30 C PH 31 D PH 32 A PH 33 B PH 34 A PH 35 D PH 36 C PH 37 A PH 38 C PH 39 B PH 40 D GATE 2012 ‐ Answer Key ‐ Paper : PH