GATE PH 2011 Question Paper & Solutions

Quantitative Aptitude

Two matrices A and B are said to be similar if B = P-'AP for some invertible matrix P. Which of the following statements is NOT TRUE? (A) Det A = Det B (B) Trace of A = Trace of B (C) A and B have the same eigenvectors (D) A and B have the same eigenvalues

Quantitative Aptitude

If a force F is derivable from a potential function V(r), where r is the distance from the origin of the coordinate system, it follows that (A) ЙxF=0 (В) V.F=0 (C) VV=0 (D) V'V=0

Quantitative Aptitude

The quantum mechanical operator for the momentum of a particle moving in one dimension is given by (A) ih- d (B) -ih- d (C) ih (D) n d' dx dx дt 2m dx?

Quantitative Aptitude

A Carnot cycle operates on a working substance between two reservoirs at temperatures T, and 7,, with T, > T,. During each cycle, an amount of heat &, is extracted from the reservoir at T, and an amount & is delivered to the reservoir at T,. Which of the following statements is INCORRECT? (A) work done in one cycle is Q, - Q2 (B) (C) entropy of the hotter reservoir decreases (D) entropy of the universe (consisting of the working substance and the two reservoirs) increases

Quantitative Aptitude

In a first order phase transition, at the transition temperature, specific heat of the system (A) diverges and its entropy remains the same (B) diverges and its entropy has finite discontinuity (C) remains unchanged and its entropy has finite discontinuity (D) has finite discontinuity and its entropy diverges PH 2/20 2011 PH

Quantitative Aptitude

The semi-empirical mass formula for the binding energy of nucleus contains a surface correction term. This term depends on the mass number A of the nucleus as (A) A-13 (B) A''3 (C) A'/3 (D) A The population inversion in a two level laser material CANNOT be achieved by optical pumping because (A) the rate of upward transitions is equal to the rate of downward transitions (B) the upward transitions are forbidden but downward transitions are allowed (C) the upward transitions are allowed but downward transitions are forbidden (D) the spontaneous decay rate of the higher level is very low The temperature ( T ) dependence of magnetic susceptibility (X) of a ferromagnetic substance with a Curie temperature (T,) is given by (A) for T <T. (B) C forT > T. T-T. T-T. (C) C for I > 1, (D) C for all temperatures T+T, T+T, where C is constant.

Quantitative Aptitude

The population inversion in a two level laser material CANNOT be achieved by optical pumping because (A) the rate of upward transitions is equal to the rate of downward transitions B) the upward transitions are forbidden but downward transitions are allowe C) the upward transitions are allowed but downward transitions are forbidde (D) the spontaneous decay rate of the higher level is very low The temperature ( T) dependence of magnetic susceptibility (X) of a ferromagnetic substance with a Curie temperature (T.) is given by (A) T- I. for I < I, (B) (C) for I > T, (D) T+T, C T-I. C =, T+T, for T > T, for all temperatures where Cis constant.

Quantitative Aptitude

The order of magnitude of the energy gap of a typical superconductor is (A) 1 MeV (B) 1 KeV (C) 1 eV (D) 1 meV

Quantitative Aptitude

The order of magnitude of the energy gap of a typical superconductor is (A) 1 MeV (B) 1 KeV (C) 1eV (D) 1 meV

Quantitative Aptitude

Which of the following statements is CORRECT for a common emitter amplifier circuit? (A) The output is taken from the emitter (B) There is 180° phase shift between input and output voltages (C) There is no phase shift between input and output voltages (D) Both p-n junctions are forward biased

Linear Algebra & Vector Calculus

A 3X3 matrix has elements such that its trace is 11 and its determinant is 36. The eigenvalues of the matrix are all known to be positive integers. The largest eigenvalue of the matrix is (A) 18 (B) 12 (C) 9 (D) 6

Small oscillations

A heavy symmetrical top is rotating about its own axis of symmetry (the z-axis). If 1,,l, and 13 are the principal moments of inertia along x, y and z axes respectively, then (A) 12 = 1z; 1, #12 (B) 4, =13; 1, #12 (C) 4, =1z;1, #13 (D) 1, 712 #13 PH 3/20 2011 PH

Electrostatics and Magnetostatics

An electron with energy E is incident from left on a potential barrier, given by V(x) =0 forx<0 =V forx> 0 as shown in the figure. V(x) --->E 0 For E < Vo, the space part of the wavefunction for x > O is of the form (A) eax (B) eax (c) iax (D) é1ax where a is a real positive quantity.

Lagrangian & Hamiltonian Dynamics

If L,L, , and L, are respectively the x, y and z components of angular momentum operator L, the commutator [L,L,, L,] is equal to (A) it(L + L}) (B) 2ihL. (C) in([, -[,) (D) 0 The normalized ground state wavefunction of a hydrogen atom is given by y(r) = abze , where a is the Bohr radius and r is the distance of the electron from the nucleus, located at the origin. The expectation value (=) is (A) (B) 4П (C) 4 (D) 2 a

Complex analysis

Two charges q and 2q are placed along the x-axis in front of a grounded, infinite conducting plane, as shown in the figure. They are located respectively at a distance of 0.5 m and 1.5 m from the plane. The force acting on the charge q is - 0.5 m →g 1.5 m 29. >x → 1 79 (A) ATLE, 2 (B) - 2q? ATEO 1 (C) -q? ATE. (D) 1 q ATLE, 2 PH 4/20 2011

Electrostatics and Magnetostatics

Two charges q and 2q are placed along the x-axis in front of a grounded, infinite conducting plane, as shown in the figure. They are located respectively at a distance of 0.5 m and 1.5 m from the plane. The force acting on the charge q is -0.5 m →g 29.>x 1.5 m - → (A) (B) -2q 1 (C) 1 (D) 1 q 4TE, 2 ATEO ATEO 4TE, 2 PH 4/20 PH A uniform surface current is flowing in the positive y-direction over an infinite sheet lying in x-y plane. The direction of the magnetic field is (A) along Î for z > 0 and along -i for z < 0 (B) along k for z > 0 and along :-k for z <0 (C) along - Î for z > 0 and along Î for z <0 (D) along -k for z > 0 and along k for z <0

Lagrangian & Hamiltonian Dynamics

PH A uniform surface current is flowing in the positive y-direction over an infinite sheet lying in x-y plane. The direction of the magnetic field is (A) along ( for z > 0 and along -i for z<0 (B) along * for z > 0 and along -k for z <0 (C) along -i for z > 0 and along i for 2 < 0 (D) along -k for z > 0 and along k for 2. <0 A magnetic dipole of dipole moment in is placed in a non-uniform magnetic field B. If the position vector of the dipole is F, the torque acting on the dipole about the origin is (C) m×B (D) mx B+7x0(m. B)

Lagrangian & Hamiltonian Dynamics

A magnetic dipole of dipole moment m is placed in a non-uniform magnetic field B. If the position vector of the dipole is 7, the torque acting on the dipole about the origin is (A) FX(m×B) (B) Fxй(m\cdotB) (C) mхВ (D) mxB+rxV(m\cdotB)

Electrostatics and Magnetostatics

Which of the following expressions for a vector potential Ä DOES NOT represent a uniform magnetic field of magnitude B, along the z-direction? (A) Ä =(0,B,x,O) (B) А =(-B,y,0,0) (C) Ä= B,x Boy 2 (D) Ä=|- BoY 2 Box 2 ч, о) 2

Complex analysis

A neutron passing through a detector is detected because of (A) the ionization it produces (B) the scintillation light it produces (C) the electron - hole pairs it produces (D) the secondary particles produced in a nuclear reaction in the detector medium

Lagrangian & Hamiltonian Dynamics

An atom with one outer electron having orbital angular momentum l is placed in a weak magnetic field. The number of energy levels into which the higher total angular momentum state splits, is (A) 21+2 (B) 21 + 1 (C) 21 (D) 21 - 1

Lagrangian & Hamiltonian Dynamics

For a multi-electron atom, I, L and S specify the one-electron orbital angular momentum, total orbital angular momentum and total spin angular momentum, respectively. The selection rules for electric dipole transition between the two electronic energy levels, specified by I, L and S are (A) AL = 0, +1; AS = 0; A/ = 0, $1 (B) AL = 0, 1; AS = 0; Д/ = $1 (C) AL = 0, 1; AS = #l; A/ = 0, $1 (D) AL = 0, ≤l; AS = ‡1; A/= £l

Crystal Structure & Lattice Dynamics

For a three-dimensional crystal having N primitive unit cells with a basis of p atoms, the number of optical branches is (A) 3 (B)3p (С) 3p-3 (D) 3N-3p PH 5/20 2011 PH

Statistical Ensembles & Partition Functions

For an intrinsic semiconductor, m, and m" are respectively the effective masses of electrons and holes near the corresponding band edges. At a finite temperature, the position of the Fermi level (A) depends on m, but not on m" (B) depends on m", but not on m" (C) depends on both m° and mi (D) depends neither on m, nor on m,"

Complex analysis

In the following circuit, the voltage across and the current through the 2 kS resistance are 500 S2 1kQ W + 20 V 10 V 30 V ,2 kg (A) 20 V, 10 mA (B) 20 V, 5 mA (C) 10 V, 10 mA (D) 10 V, 5 mA

Complex analysis

The unit vector normal to the surface x' + y2 - z =1 at the point P(1,1,1) is (A) I+]-k (B) 2ĩ + ]-K (0) 1+2] -k (D) 21+27-K V3 3

Linear Algebra & Vector Calculus

Consider a cylinder of height h and radius a, closed at both ends, centered at the origin. Let F=ix+ jy+kz be the position vector and îî a unit vector normal to the surface. The surface integral fr.ii ds over the closed surface of the cylinder is Z (A) 2nả'(a+h) (B) 3nảh (C) 2na'h (D) zero PH 6/20 2011 PH

Linear Algebra & Vector Calculus

The solutions to the differential equation =- X dx y+1 are a family of (A) circles with different radii (B) circles with different centres (C) straight lines with different slopes (D) straight lines with different intercepts on the y-axis

Electrostatics and Magnetostatics

A particle is moving under the action of a generalized potential V(q,9) = (I+4) The magnitude of the generalized force is (A) . 2(1+q) (в) 2(1-4) (C) 9 (D)

Small oscillations

Two bodies of mass m and 2m are connected by a spring of spring constant k. The frequency of the normal mode is (A) V3k/2m (B) Vk/m (C) V2k/3m (D) Vk/2m

Complex analysis

Let (P,9) and (P,Q) be two pairs of canonical variables. The transformation e=9ª cos(Bp) P=qª sin(Bp) is canonical for (A) a = 2, B =1/2 (B) a=2, B=2 (C) a =1, B=1 (D) 0=1/2, B=2

Special theory of relativity

Two particles, each of rest mass m collide head-on and stick together. Before collision, the speed of each mass was 0.6 times the speed of light in free space. The mass of the final entity is (A) 5m/4 (B) 2m (C) 5m/2 (D) 25m/8

Perturbation theory & Approximation Methods

The normalized eigenstates of a particle in a one-dimensional potential well V(x) = 0if0≤x≤a otherwise are given by 4,(x) = where n = 1,2,3, sin(nx). The particle is subjected to a perturbation V(+)=v,cos (*r) for 05x5a =0 otherwise The shift in the ground state energy due to the perturbation, in the first order perturbation theory, is (A) 2Ve (B) (c)-Ve (D) _2V. PH 1/20 2011 PH

Classical Thermodynamics

If the isothermal compressibility of a solid is K, = 10-1º (Pa)'' , the pressure required to increase its density by 1% is approximately (A) 10* Pa (B) 10° Pa (C) 10% Ра (D) 10'º Pa

Statistical Ensembles & Partition Functions

A system of N non-interacting and distinguishable particles of spin 1 is in thermodynamic equilibrium. The entropy of the system is (A) 2kg In N (B) 3kg In N (C) Nkg In 2 (D) Nkg In3

Statistical Ensembles & Partition Functions

A system has two energy levels with energies & and 2E. The lower level is 4-fold degenerate while the upper level is doubly degenerate. If there are N non-interacting classical particles in the system, which is in thermodynamic equilibrium at a temperature T, the fraction of particles in the upper level is (A) 1 (B) 1 1+eElkeT I+2eE1kgT (C) 1 ZeElknT +4eZElKBT (D) ZeElksT - 4eZElKBT

Electrostatics and Magnetostatics

A spherical conductor of radius a is placed in a uniform electric field E = E,k . The potential at a point P(r,0) for r > a, is given by p(r,0) = constant- E,rcos0+- E,a' p?COSO where r is the distance of P from the centre O of the sphere and @ is the angle OP makes with the z-axis. The charge density on the sphere at 0 = 30° is (A) 3V3E, E, 12 (B) 3E,EQ12 (C) J3E,E,12 (D) E,E,12

Nuclear Models & Decays

According to the single particle nuclear shell model, the spin-parity of the ground state of '?O is (A) } (В) 3 (0) 3* (D) 2 PH 8/20 2011 PH

Nuclear Models & Decays

In the B-decay of neutron n → p+è +V., the anti-neutrino V, escapes detection. Its existence is inferred from the measurement of (A) energy distribution of electrons (B) angular distribution of electrons (C) helicity distribution of electrons (D) forward-backward asymmetry of electrons

Nuclear Models & Decays

The isospin and the strangeness of S baryon are (A) 1,-3 (B) 0,-3 (C) 1,3 (D) 0,3

Complex analysis

The lifetime of an atomic state is 1 nanosecond. The natural line width of the spectral line in the emission spectrum of this state is of the order of (A) 10-1° eV (B) 10' eV (C) 10° eV (D) 10-* eV

Complex analysis

The degeneracy of an excited state of nitrogen atom having electronic configuration Is'2s'2p'3d' is (A) 6 (B) 10 (C) 15 (D) 150

Molecular Spectroscopy

The far infrared rotational absorption spectrum of a diatomic molecule shows equidistant lines with spacing 20 cm*' . The position of the first Stokes line in the rotational Raman spectrum of this molecule is (A) 20 cm"' (B) 40 cm"' (C) 60 cm*' (D) 120 cm"'

Crystal Structure & Lattice Dynamics

A metal with body centered cubic (bcc) structure shows the first (i.e. smallest angle) diffraction peak at a Bragg angle of 0=30°. The wavelength of X-ray used is 2.1 À. The volume of the PRIMITIVE unit cell of the metal is (A) 26.2 (Ä)' (B) 13.1 (Ả)' (C) 9.3 (Ả) (D) 4.6 (Ả)3

Complex analysis

In the following circuit, Trl and Tr2 are identical transistors having VBE = 0.7 V. The current passing through the transistor Tr2 is S100 Tr2 Trl (A) 57 mA (B) 50 mA (C) 48 mА (D) 43 mA PH 9/20 2011 PH

Complex analysis

The following Boolean expression can be simplified to Y=A\cdotB\cdotC\cdotD+Ã\cdotB\cdotC\cdotD+Ã\cdotB\cdotC\cdotD+Ã\cdotB\cdotC\cdotD+Ã\cdotB\cdotC\cdotD+A\cdotB\cdotC\cdotD (A) Ã\cdotB\cdotC+A\cdotD (B) A\cdotB\cdotC+A\cdotD (C) A\cdotB\cdotC+Ã\cdotD (D) A\cdotB\cdotC+Ã\cdotD

Complex analysis

Consider the following circuit. 1 kS 4 kg www ww +10 V Vin -10 V Which of the following correctly represents the output Vou corresponding to the input Vin? (A) (B) +SVL +5V +2V] +2V Vin -2V - time Vin -2V time -Sv/ -5V +10V +10V Vout Yout -10V time time → -10V (C) (D) +5V1 +SVI +2V +2V Vin -2V time Vin -2V time -5V -SV +10V +10V Vout Yout -10V time time → -10V PH 10/20 2011 PH Common Data Questions Common Data for Questions 48 and 49: Consider a function f (z) =- zsin z - of a complex variable z. (2-ПT)'

Complex analysis

Which of the following statements is TRUE for the function f (z) ? (A) f(z) is analytic everywhere in the complex plane (B) f(z) hasa zero at z = П (C) f(z) has a pole of order 2 at z = П (D) f (z) has a simple pole at z = П

Linear Algebra & Vector Calculus

Consider a counterclockwise circular contour (z| = 1 about the origin. The integral ff (z) dz over this contour is (A) -іл (B) zero (С) іл (D) 2int Common Data for Questions 50 and 51: The tight binding energy dispersion (E-k) relation for electrons in a one-dimensional array of atoms having lattice constant a and total length L is E = Eo-B-2ycos(ka), where Eo,B and y are constants and k is the wave-vector.

Complex analysis

The density of states of electrons (including spin degeneracy) in the band is given by (A) L (B) L toyasin(ka) 2mya sin(ka) (C) L (D) L 2nyacos(ka) tyacos(ka)

Schrodinger equation & Wave Mechanics

The effective mass of electrons in the band is given by (A) rả cos(ka) (B) Zya cos(ka) (C) tỉ (D) rả sin(ka) 2rả sin(ka) PH 11/20 2011 PH Linked Answer Questions Statement for Linked Answer Questions 52 and 53: In a one-dimensional harmonic oscillator, Po,4, and P, are respectively the ground, first and the second excited states. These three states are normalized and are orthogonal to one another. 4, and W2 are two states defined by 4,=Ф0-201 +342 where a is a constant. V2=Ф0-Ф,+аФ2

Complex analysis

The value of a for which y, is orthogonal to v, is (A) 2 (B) 1 (C) -1 (D) -2

Electrostatics and Magnetostatics

For the value of a determined in Q. 52, the expectation value of energy of the oscillator in the state W, is (A) h0 (B) 3hw/2 (C) 3h0 (D) 9h0/2 Statement for Linked Answer Questions 54 and 55: A plane electromagnetic wave has the magnetic field given by B(x, y, z,t) = B, sin (*+») ½+ ur lê respectively. where k is the wave number and Î,] and k are the Cartesian unit vectors in x, y and z directions,

Electrostatics and Magnetostatics

The electric field Ë(x, y, z,t) corresponding to the above wave is given by (A) cB, sin (x+y) k + wt (-]) V2 (B) CB sin (x+») z + ar |"13) (C) cB, sin/ (x+y)- K+ ur i D) cB, sin (x+)) Fy+ux)1

Electrostatics and Magnetostatics

The average Poynting vector is given by (A) ,CB? (Î- ]) (B) _CBẻ (Î -]) 2M. 2M. V2 (C) CBẻ (+ i) 52 (D)_CB: (Î +]) 2M0 V2 PH 12/20 2011 PH General Aptitude (GA) Questions

Complex analysis

Choose the most appropriate word from the options given below to complete the following sentence. If you are trying to make a strong impression on your audience, you cannot do so by being understated, tentative or (A) hyperbolic (B) restrained (C) argumentative (D) indifferent

Complex analysis

Choose the most appropriate word(s) from the options given below to complete the following sentence. I contemplated Singapore for my vacation but decided against it. (A) to visit (B) having to visit (C) visiting (D) for a visit

Complex analysis

If Log (P) = (1/2)Log (Q) = (1/3) Log (R), then which of the following options is TRUE? (A) P' = Q'R' (B) Q'= PR (C) Q'=R'P (D) R= P'Q'

Complex analysis

Which of the following options is the closest in the meaning to the word below: Inexplicable (A) Incomprehensible (B) Indelible (C) Inextricable (D) Infallible

Complex analysis

Choose the word from the options given below that is most nearly opposite in meaning to the given Amalgamate (A) merge (B) split (C) collect (D) separate

Complex analysis

A transporter receives the same number of orders each day. Currently, he has some pending orders (backlog) to be shipped. If he uses 7 trucks, then at the end of the 4th day he can clear all the orders. Alternatively, if he uses only 3 trucks, then all the orders are cleared at the end of the 10th day. What is the minimum number of trucks required so that there will be no pending order at the end of the 5th day? (A) 4 (B) 5 (C) 6 (D) 7

Complex analysis

The variable cost (V) of manufacturing a product varies according to the equation V= 4q, where q is the quantity produced. The fixed cost (F) of production of same product reduces with q according to the equation F = 100/q. How many units should be produced to minimize the total cost (V+F)? (A) 5 (B) 4 (C) 7 (D) 6 PH 13/20 P, Q, R and S are four types of dangerous microbes recently found in a human habitat. The area of each circle with its diameter printed in brackets represents the growth of a single microbe surviving human immunity system within 24 hours of entering the body. The danger to human beings varies proportionately with the toxicity, potency and growth attributed to a microbe shown in the figure 1000 P (50 mm) 800 600 Toxicity 400 Q (40 mm) /R (30 mm) S (20 mm) 200 half of the body mass in kilograms) 0.2 0.4 0.6 0.8 1 Potency (Probability that microbe will overcome human immunity system) A pharmaceutical company is contemplating the development of a vaccine against the most dangerous microbe. Which microbe should the company target in its first attempt? (A) P (B) Q (C)R (D)S

Digital Electronics

P, Q, R and S are four types of dangerous microbes recently found in a human habitat. The area of each circle with its diameter printed in brackets represents the growth of a single microbe surviving human immunity system within 24 hours of entering the body. The danger to human beings varies proportionately with the toxicity, potency and growth attributed to a microbe shown in the figure (Toxicity: P=800, Q=600, R=300, S=200; Potency: P=0.4, Q=0.5, R=0.6, S=0.8; Diameter: P=50 mm, Q=40 mm, R=30 mm, S=20 mm). A pharmaceutical company is contemplating the development of a vaccine against the most dangerous microbe. Which microbe should the company target in its first attempt? (A) P (B) Q (C) R (D) S

Complex analysis

Few school curricula include a unit on how to deal with bereavement and grief, and yet all students at some point in their lives suffer from losses through death and parting. Based on the above passage which topic would not be included in a unit on bereavement? (A) how to write a letter of condolence (B) what emotional stages are passed through in the healing process (C) what the leading causes of death are (D) how to give support to a grieving friend

Complex analysis

A container originally contains 10 litres of pure spirit. From this container 1 litre of spirit is eplaced with 1 litre of water. Subsequently, I litre of the mixture is again replaced with 1 litre c ater and this process is repeated one more time. How much spirit is now left in the container (A) 7.58 litres (B) 7.84 litres (C) 7 litres (D) 7.29 litres END OF THE QUESTION PAPER PH 14/20