GATE PH 2009 Question Paper & Solutions

Complex analysis

The value of the contour integral, fix dãl , fora cirele C of radius r with center at the origin is (A) 2mr (в) ½ (C) тr? (D)г

Electrostatics and Magnetostatics

An electrostatic field È exists in a given region R. Choose the WRONG statement. (A) Circulation of E is zero (B) E can always be expressed as the gradient of a scalar field (C) The potential difference between any two arbitrary points in the region R is zero (D) The work done in a closed path lying entirely in R is zero

Lagrangian & Hamiltonian Dynamics

The Lagrangian of a free particle in spherical polar co-ordinates is given by L = Em(, +rở +r$ sin' 0 ). The quantity that is conserved is (B) OL (A) ai ao (C) aL (D) độ +iò

Electrostatics and Magnetostatics

A conducting loop L of surface area S is moving with a velocity in a magnetic field B(F,t) = B,?, B, is a positive constant of suitable dimensions. The emf induced, Vemf , in the loop is given by \cdotоB (B) ф(тxB). đi (A) - S. đt đ8 (С)-.01 \cdotđs-ф(vxB) di (D) - dt -đs + §(vxB) đĩ 2/16 PH 2009 PH

Linear Algebra & Vector Calculus

The eigenvalues of the matrix A=° 1 i 0 are (A) real and distinct (B) complex and distinct (C) complex and coinciding (D) real and coinciding

Linear Algebra & Vector Calculus

o, (i =1, 2, 3) represent the Pauli spin matrices. Which one of the following is NOT true ? (A) 0,0, +0,0, = 20 (B) Tr(o,) =0 (C) The eigenvalues of o, are $1 (D) det (o,) =1

Linear Algebra & Vector Calculus

Which one of the functions given below represents the bound state eigenfunction of the operator dx in the region, 0 ≤ x < oo, with the eigenvalue -4 ? (A) A,e2* (B) A, cosh 2x (C) 4,e2x (D) A, sinh 2x

Nuclear Properties & Nuclear Forces

Pick the WRONG statement. (A) The nuclear force is independent of electric charge (B) The Yukawa potential is proportional to r' exp "), where r is the separation between two nucleons (C) The range of nuclear force is of the order of 10-15m - 10-14m (D) The nucleons interact among each other by the exchange of mesons

Complex analysis

If p and q are the position and momentum variables, which one of the following is NOT a canonical transformation ? (A) Q= aq and P =- P, for a # 0 (B) Q=aq+ Bp and P = Bq+ap for a,ß real and a' - ß* =1 (C) Q = p and P = q (D) Q = p and P = -q

Operational Amplifiers & Analog Circuits

The Common Mode Rejection Ratio (CMRR) of a differential amplifier using an operational amplifier is 100 dB. The output voltage for a differential input of 200 uV is 2 V. The common mode gain is (A) 10 (B) 0.1 (C) 30 dB (D) 10 dB

Complex analysis

In an insulating solid which one of the following physical phenomena is a consequence of Pauli's exclusion principle ? (A) Ionic conductivity (B) Ferromagnetism (C) Paramagnetism (D) Ferroelectricity PH 3/16 PH 2009

Linear Algebra & Vector Calculus

Which one of the following curves gives the solution of the differential equation dx + k,x = k,,where k,,kz and k, are positive constants with initial conditions x =0 at / = 0? (A) (B) t→ t → (C) (D)

Blackbody radiation & Phase transitions

Identify which one is a first order phase transition ? (A) A liquid to gas transition at its critical temperature. (B) A liquid to gas transition close to its triple point. (C) A paramagnetic to ferromagnetic transition in the absence of a magnetic field. (D) A metal to superconductor transition in the absence of a magnetic field

Electromagnetic Waves in Media

Group I lists some physical phenomena while Group Il gives some physical parameters. Match the phenomena with the corresponding parameter. Group I Group II P. Doppler Broadening 1. Moment of inertia Q. Natural Broadening 2. Refractive index R. Rotational spectrum 3. Lifetime of the energy level S. Total internal reflection 4. Pressure (A) P - 4, Q- 3, R - 1, S- 2 (B) P-3, Q- 2, R - 1, S-4 (C) P - 2, Q-3, R- 4, S-1 (D) P- 1, Q-4, R -2, S - 3

Molecular Spectroscopy

The separation between the first Stokes and corresponding anti-Stokes lines of the rotational Raman spectrum in terms of the rotational constant, B is (A) 2B (B) 4B (C) 6B (D) 12B

Electrostatics and Magnetostatics

A superconducting ring is cooled in the presence of a magnetic field below its critical temperature (Tc). The total magnetic flux that passes through the ring is (A) zero (B) 2e C) nh (D) ne 4ле hc 4/16 PH 2009 PH

Crystal Structure & Lattice Dynamics

In a cubic crystal, atoms of mass M, lie on one set of planes and atoms of mass M2 lie on planes interleaved between those of the first set. If C is the force constant between nearest neighbour planes, the frequency of lattice vibrations for the optical phonon branch with wavevector k = 0 is 2C (A) V M, -+ (B) с 2M, 1 + 1 M, 2M2 (D) 0

Nuclear Models & Decays

In the quark model which one of the following represents a proton ? (A) udd (B) uud (C) ub (D) cc

Semiconductor Devices & Diodes

\cdot Vec Vin TI C, The circuit shown above (A) is a common-emitter amplifier (B) uses a pnp transistor (C) is an oscillator (D) has a voltage gain less than one

Nuclear Properties & Nuclear Forces

Consider a nucleus with N neutrons and Z protons. If mp, m, and BE represent the mass of the proton, in free space, the mass of the nucleus is given by the mass of the neutron and the binding energy of the nucleus respectively and c is the velocity of light (A) Nm, + Zm, (B) Nm, + Zm, BE (C) Nm, + Zm, + BE (D) Nm, + Zm, + PH 5/16 2009 PH

Electrostatics and Magnetostatics

The magnetic field (in A m') inside a long solid cylindrical conductor of radius a = 0.1 m is, H = 10* , sin(ar) - [ cos(ar) ₫, where a =- \cdot. What is the total current (in A) in the 2a conductor ? (A) (B) 800 (C) 400 (D) 300 2a

Electrostatics and Magnetostatics

Which one of the following current densities, J, can generate the magnetic vector potential Ä=(v{+x']) ? (A) 2 (xỉ + xộ) (B) - 2 (i+j) (C) - 2 (i-3) (D) 2(xỉ -xộ) e*

Linear Algebra & Vector Calculus

The value of the integral z-3z+2 - dz, where the contour C is the cirele |2| = 3/2 is (A) 2mie (В) пie (C) -2nie (D) -nie

Electrostatics and Magnetostatics

In a non-conducting medium characterized by & = 6., M= M. and conductivity o = 0, the electric field (in V m') is given by Ë = 20sin [10%/ - kz ]}. The magnetic field, F (in A m'), is given by (A) 20k cos 10°t-kz i (B) 20k sin [10°t -kz_j 10° M. (С)- 20k 10° M. K sin [10°t - kz ]i (D)-20k cos [10°г - kz ]/

Electrostatics and Magnetostatics

A cylindrical rod of length L and radius r, made of an inhomogeneous dielectric, is placed with its axis along the z direction with one end at the origin as shown below. If the rod carries a polarization, P = (5z' + 7)k, the volume bound charge inside the dielectric is (A) Zero (B) 10xr'L (C) -5Tr?L (D) -5mr' L PH| 6/16 2009 PH Let T, = _Eika, and Px =2 i,j _Ejk-, where Eyk is the Levi-Civita density, defined to be zero if two of the indices coincide and +1 and -1 depending on whether ijk is even or odd permutation of 1,2,3. Then B, is equal to (A) 2a, (B) -2a, (C) a, (D) -a,

Statistical Ensembles & Partition Functions

The dependence of the magnetic susceptibility (X) of a material with temperature (T) can be represented by X oc T- 0, where ® is the Curie-Weiss temperature. The plot of magnetic susceptibility versus temperature is sketched in the figure, as curves P, Q and R with curve Q having 0 = 0. Which one of the following statements is correct ?

Statistical Ensembles & Partition Functions

The dependence of the magnetic susceptibility (X) of a material with temperature (T) can be represented by X oc ,, where 0 is the Curie-Weiss temperature. The plot of magnetic susceptibility one of the following statements is correct ? versus temperature is sketched in the figure, as curves P, Q and R with curve Q having 0= 0. Which R -0 = 0 (A) Curve R represents a paramagnet and Q a ferromagnet (B) Curve Q represents a ferromagnet and P an antiferromagnet (C) Curve R represents an antiferromagnet and Q a paramagnet (D) Curve R represents an antiferromagnet and Q a ferromagnet

Dielectric, Magnetic Properties & Superconductivity

The dielectric constant of a material at optical frequencies is mainly due to (A) ionic polarizability (B) electronic polarizability (C) dipolar polarizability (D) ionic and dipolar polarizability

Band Theory of Solids & Semiconductors

An electron of wavevector k e, velocity V, and effective mass m, is removed from a filled energy band The resulting hole has wavevector k,, velocity Vn, and effective mass m. Which one of the following statements is correct ? (A) kn = -k; Vn =-Vo; m, =-m, (B) kn= k.; V, = Ve; m, =m, (C) K4=k.; Vn =-V; m,=-m. (D) Kn = -k ; Vn = V: m,=-m, PHI 7/16 2009 PH

Complex analysis

In a diatomic molecule, the internuclear separation of the ground and first excited electronic state are the same as shown in the figure. If the molecule is initially in the lowest vibrational state of the ground state, then the absorption spectrum will appear as r internuclear distance → (A) (B) cm™' → cm™' → continuum cm™' → cm'

Atomic Spectra & Multi-Electron Atoms

Five energy levels of a system including the ground state are shown below. Their lifetimes and the allowed electric dipole transitions are also marked. 10-7 s 4 2 x 10-8s 3 10-6g 2 10-8 s 1 0 (ground state) Which one of the following transitions is the most suitable for a continuous wave (CW) laser ? (A) 1 → 0 (B) 2 → 0 (C) 4 →2 (D) 4 →3

Complex analysis

Assuming the mean life time of a muon (in its rest frame) to be 2x10's, its life time in the laboratory frame, when it is moving with a velocity 0.95c is (A) 6.4x10€s (B) 0.62x10-6 s (C) 2.16x10°s D) 0.19x10€s

Nuclear Properties & Nuclear Forces

Cesium has a nuclear spin of 7/2. The hyperfine spectrum of the D lines of the cesium atom will consist of (A) 10 lines (B) 4 lines (C) 6 lines (D) 14 lines 8/16 2009 PH

Electrostatics and Magnetostatics

The probability that an energy level e at a temperature T is unoccupied by a fermion of chemical potential y is given by (A) đe-A) aT +T (B) TO-A NAT -T CO eMONtOT +T ) a-oa -

Nuclear Models & Decays

Consider the following expression for the mass of a nucleus with Z protons and A nucleons: M (4,2) = z(J(1) + yZ +z7'). Here J (4) is a function of A, y=-4а,, z =a,A"" + 4а,А", a, and a, are constants of suitable dimensions. For a fixed A, the expression of Z for the most stable nucleus is (A) Z = A/2 (B) Z = A/2 1+ ac A2/3 1+(%a,1 (C) Z = (D) Z = A 1+ ac A2/3 1+ A2/3 4а,

Statistical Ensembles & Partition Functions

The de Broglie wavelength of particles of mass m with average momentum p at a temperature T in three dimensions is given by (A) 2 = (B) 2= (C) 2= (D) 2=- h J2mk,T J3mk,T V2k,T J3m

Complex analysis

10 92 15 V \cdot10 g RL above circuit are Assuming an ideal voltage source, Thevenin's resistance and Thevenin's voltage respectively for the (A) 152 and 7.5 V (B) 2052 and 5 V (C) 1052 and 10 V (D) 302 and 15 V PH 9/16 2009 PH

Angular momentum & Hydrogen Atom

Let n)and P) denote the isospin states with 1 = ½, Is = ½/ and 1 = ½/ , lz = -½ of a nucleon respectively. Which one of the following two-nucleon states has I= 0, I,= 0 ? (A) ((nn) -|pp)) (B) [(mn) +|pp)) (C) F(mp)-|pn)) (D) [(пр) +|pm))

Complex analysis

An amplifier of gain 1000 is made into a feedback amplifier by feeding 9.9 % of its output voltage in series with the input opposing. If fi = 20 Hz and fu = 200 kHz for the amplifier without feedback, then due to the feedback (A) the gain decreases by 10 times (B) the output resistance increases by 10 times (C) the fu increases by 100 times (D) the input resistance decreases by 100 times

Complex analysis

Rs=1 kQ Vin 15-25 V Vz = 10 V RL /Iz Pick the correct statement based on the above circuit. (A) The maximum Zener current, Iz(max), when R_ = 10 kS is 15 mA (B) The minimum Zener current, Iz(min), when R_ = 10 kJ is 5 mA (C) With Vin = 20V, L= Iz, when R, = 2 kQ (D) The power dissipated across the Zener when R_ = 10 kJ and Vin = 20 V is 100 mW

Special theory of relativity

The disintegration energy is defined to be the difference in the rest energy between the initial and final states. Consider the following process : 240 Pu →336 U +* He. The emitted a particle has a kinetic energy 5.17 MeV. The value of the disintegration energy is (A) 5.26 MeV (B) 5.17 MeV (C) 5.08 MeV (D) 2.59 MeV PH 10/16 2009 PН

Linear Algebra & Vector Calculus

A classical particle is moving in an external potential field V (x,y,z) which is invariant under the following infinitesimal transformations x → x = x+бx, y→ y = у+бу, Rz where R,is the matrix corresponding to rotation about the z axis. The conserved quantities are (the symbols have their usual meaning) (A) Px\cdot P2, L, (B) Px, Py,L,,E (C) Py,L,,E (D) Py, P,, L,, E

Linear Algebra & Vector Calculus

The spin function of a free particle, in the basis in which S, is diagonal, can be written as and 1 with eigenvalues + 2 - and -- , respectively. In the given basis, the normalized (g) eigenfunction of S, with eigenvalue - (A) Z0) B (i) (C) D) Z()

Complex analysis

 and B represent two physical characteristics of a quantum system. If  is Hermitian, then for the product Âß to be Hermitian, it is sufficient that (A) Ô is Hermitian (B) B is anti-Hermitian (C) B is Hermitian and  and B commute (D) B is Hermitian and  and B anti-commute

Complex analysis

Consider the set of vectors in three-dimensional real vector space R3, S = {(1,1,1), (1,-1,1), (1,1,-1)} . Which one of the following statements is true ? (A) S is not a linearly independent set. (B) S is a basis for R3. (C) The vectors in S are orthogonal. (D) An orthogonal set of vectors cannot be generated from S.

Statistical Ensembles & Partition Functions

For a Fermi gas of N particles in three dimensions at T = 0 K, the Fermi energy, Ef is proportional to (A) N2/3 (B) N3/2 (C) (D) Nº 11/16 2009 PH

Lagrangian & Hamiltonian Dynamics

The Lagrangian of a diatomic molecule is given by L = - +x2)-=x,x,, where m is the mass of each of the atoms and x, and x, are the displacements of atoms measured from the equilibrium position and k > 0. The normal frequencies are 1/2

Lagrangian & Hamiltonian Dynamics

A particle is in the normalized state (v) which is a superposition of the energy eigenstates (E. =10eV) and| E, = 30eV). The average value of energy of the particle in the state (y) is 20 eV. The state (v) is given by (A)HIE. =10er) + 3/E, = 30eг) (ВЕ.=10eV)+ 315, = 300V) (c)}E, =10ev) -V3 - E, =30eV) 10с)-=30еv) Q,19 The Lagrangian of a particle of mass m moving in one dimension is I = exp (at) mi ka k2'] where a and k are positive constants. The equation of motion of the particle is (A) x+ax=0 (В) *+ Ex=0 (C) *-ax+-x=0 (D) x+ax+-x=0 m

Electromagnetic Waves in Media

Two monochromatic waves having frequencies a and a + A∞ (Aa < w) and corresponding wavelengths 2 and 2 - 2(A2 << 1) of same polarization, traveling along x-axis are superimposed on each other. The phase velocity and group velocity of the resultant wave are respectively given by c2 Д002? (А) 2т' 2тл2 4022 (B) ∞2, (C) WAR AWAR 2т ' (D) ФA2, 0A2 PH 12/16 PH 2009 Common Data Questions Common Data for Questions 51 and 52 : Consider a two level quantum system with energies 8, = 0 and &, = & .

Electromagnetic Waves in Media

Two monochromatic waves having frequencies @ and w + Aw (Aw < w) and corresponding wavelengths 2 and 2 - Д2 (A2 << 2) of same polarization, traveling along x-axis are superimposed on each other. The phase velocity and group velocity of the resultant wave are respectively given by A) 2n' 2TA wл дол (B) cл, 4012 (C) OA1 ДOAL 2m ' (D) WA2, WA2 PH 12/16 2009 PH Common Data Questions Common Data for Questions 51 and 52 : Consider a two level quantum system with energies &, = 0 and &, = & .

Statistical Ensembles & Partition Functions

The Helmholtz free energy of the system is given by (A) -KgTln(1+e &/kBT) (B) kgTIn(1+e6/kgT) (D) E-kgT (о) -к,T

Statistical Ensembles & Partition Functions

The specific heat of the system is given by e E/kgl e-elkgT (A) KgT (1+e-e/kT)3 KgTa (Ite-olkst) (C) se-ElkgT (D) e-E/kgT (1+e-81KgT)} KgT' (1+e-8/kgT)} Common Data for Questions 53 and 54 : A free particle of mass m moves along the x direction. At (=0, the normalized wave function of the particle is given by y (x, 0) = (2та), еxр 4o + ix , where a is a real constant.

Schrodinger equation & Wave Mechanics

The expectation value of the momentum, in this state is (A) ha (B) nVa (C) a (D)

Atomic Spectra & Multi-Electron Atoms

The expectation value of the particle energy is 1| tỉ 4a2 +1 tỉ (A) 2m 2a3/2 B) 2m a? (C) 2m 4a3/2 (D) 8mа312 Common Data for Questions 55 and 56 : Consider the Zeeman splitting of a single electron system for the 3d →3p electric dipole transition.

Atomic Spectra & Multi-Electron Atoms

The Zeeman spectrum is (A) randomly polarized (B) only polarized (C) only o polarized (D) both i and o polarized

Atomic Spectra & Multi-Electron Atoms

The fine structure line having the longest wavelength will split into (A) 17 components (B) 10 components (C) 8 components (D) 4 components PH 13/16 2009 PH Linked Answer Questions Statement for Linked Answer Questions 57 and 58 : The primitive translation vectors of the face centered cubic (fcc) lattice are a, = ₴(2+k); â, = ₴ (1+k); a, = §(i+]).

Crystal Structure & Lattice Dynamics

The primitive translation vectors of the fcc reciprocal lattice are (A) 6, (2) (-1+ 3 +h); 6, = (2) (-2+8):6, -(2) (+3-k)) (8) 6, =(5)(-1+3+6); 6 =(5)(2-3+6); 6, = (5)(1+3-k) 10) 16=(2) (2+3+6);6 - za) (-3+h); 6 = (za)(+3-4) (0) 6 () (-1+3+8):63 -(87) (1-3+6); 6,- (%)(2+2-k)

Crystal Structure & Lattice Dynamics

The volume of the primitive cell of the fcc reciprocal lattice is (A) A (2) (OA() () 4(2) Statement for Linked Answer Questions 59 and 60 : The Karnaugh map of a logic circuit is shown below: R R FO 1 FO 1 PQ 1 1

Complex analysis

The minimized logic expression for the above map is (A) Y = PR+@ (B) Y = Q\cdotPR (C) Y =Q+PR (D) Y =Q\cdotPR

Complex analysis

The corresponding logic implementation using gates is given as: P P (A) R- -Y (B) R -Y Q- P P (C) Q (D) Q R R - Y PH END OF THE QUESTION PAPER 14/16

Complex analysis

Official Question 61 from GATE PH 2009 Examination.

Complex analysis

Official Question 62 from GATE PH 2009 Examination.

Complex analysis

Official Question 63 from GATE PH 2009 Examination.

Complex analysis

Official Question 64 from GATE PH 2009 Examination.

Complex analysis

Official Question 65 from GATE PH 2009 Examination.