GATE PH 2007 Question Paper & Solutions

Linear Algebra & Vector Calculus

The eigenvalues of a matrix are $i$, $-2i$ and $3i$. The matrix is

Lagrangian & Hamiltonian Dynamics

A space station moving in a circular orbit around the Earth goes into a new bound orbit by firing its engine radially outwards. This orbit is (A) a larger circle. (B) a smaller circle. (C) an ellipse. (D) a parabola.

Complex analysis

A power amplifier gives 150W output for an input of 1.5W. The gain, in dB, is (A) 10 (B) 20 (C) 54 (D) 100

Electrostatics and Magnetostatics

Four point charges are placed in a plane at the following positions: +Q at (1, 0), -Q at (-1, 0), +Q at (0, 1) and -Q at (0, -1). At large distances the electrostatic potential due to this charge distribution will be dominated by the (A) monopole moment. (B) dipole moment. (C) quadrupole moment. (D) octopole moment.

Electrostatics and Magnetostatics

A charged capacitor (C) is connected in series with an inductor (L). When the displacement current reduces to zero, the energy of the LC circuit is (A) stored entirely in its magnetic field. (B) stored entirely in its electric field. (D) radiated out of the circuit. (C) distributed equally among its electric and magnetic fields.

Lagrangian & Hamiltonian Dynamics

Match the following P. Franck-Hertz experiment 1. electronic excitation of molecules 2. wave function of atoms Q. Hartree-Fock method R. Stern-Gerlach experiment 3. spin angular momentum of atoms S. Franck-Condon principle 4. energy levels in atoms (A) P-4 (B) P-1 (C) P-3 P-4 (D) Q-2 R-3 R-3 Q-4 Q-2 R-4 Q-1 R-3 S-1 S-2 S-1 S-2

Electrostatics and Magnetostatics

The wavefunction of a particle, moving in a one-dimensional time-independent potential V(x), is given by y(x) = e~lar+b , where a and b are constants. This means that the potential V(x) is of the form (A) V(x) oc x (B) V(x) ocx (C) V(x) = 0 (D) V(x) oe*"

Atomic Spectra & Multi-Electron Atoms

The $D_1$ and $D_2$ lines of Sodium ($3^2P_{1/2} \rightarrow 3^2S_{1/2}$ and $3^2P_{3/2} \rightarrow 3^2S_{1/2}$) will split on the application of a weak magnetic field into

Atomic Spectra & Multi-Electron Atoms

In a He-Ne laser, the laser transition takes place in (A) He only. (B) Ne only. (C) Ne first, then in He. (D) He first, then in Ne.

Statistical Ensembles & Partition Functions

The partition function of a single gas molecule is Z. . The partition function of N such non-interacting gas molecules is then given by (A) (B) (za)" (C) N(Za) (D) (Z.

Electrostatics and Magnetostatics

A solid superconductor is placed in an external magnetic field and then cooled below its critical temperature. The superconductor (A) retains its magnetic flux because the surface current supports it. (B) expels out its magnetic flux because it behaves like a paramagnetic material. (C) expels out its magnetic flux because it behaves like an anti- ferromagnetic material. (D) expels out its magnetic flux because the surface current induces a field in the direction opposite to the applied magnetic field.

Electrostatics and Magnetostatics

A particle with energy E is in a time-independent double well potential as shown in the figure. correct? Which of the following statements about the particle is NOT (A) The particle will always be in a bound state. (B) The probability of finding the particle in one well will be time-dependent. (C) The particle will be confined to any one of the wells. (D) The particle can tunnel from one well to the other, and back.

Schrodinger equation & Wave Mechanics

It is necessary to apply quantum statistics to a system of particles if (A) there is substantial overlap between the wavefunctions of the particles. (C) the particles have identical mass and charge. (B) the mean free path of the particles is comparable to the inter-particle separation. (D) the particles are interacting.

Complex analysis

When liquid oxygen is poured down close to a strong bar magnet, the oxygen stream (A) repelled towards the lower field because it is diamagnetic. (B) attracted towards the higher field because it is diamagnetic. (C) repelled towards the lower field because it is paramagnetic. (D) attracted towards the higher field because it is paramagnetic.

Complex analysis

Fission fragments are generally radioactive as (A) they have excess of neutrons. (B) they have excess of protons. (C) they are products of radioactive nuclides. (D) their total kinetic energy is of the order of 200 MeV.

Semiconductor Devices & Diodes

In a typical npn transistor the doping concentrations in emitter, base and collector regions are Ce, Cв and C. respectively. These satisfy the relation, (А) Св > Сс> Св (В) Сд >Св> Сс (С) Сс >Св>Св (D) Сд= Сс > Св

Complex analysis

The allowed states for He (2p') configuration are (A) 'So, 'S1, 'P1, 3Po,1,2, 'D2 and 'D1,2; (B) 'So, 3Po.1,2 and 'D2 (C) 'P, and 'Po,13 (D) 'So and 'P;

Electrostatics and Magnetostatics

The energy levels of a particle of mass m in a potential of the form V(x) = oo, xS0 =mw'x, x>0 are given, in terms of quantum number n = 0,1,2,3,..., by (A) (n+}) ta (B) (2n +}) tra (C) (2n + }) ha (D) (n+})ha

Electrostatics and Magnetostatics

The electromagnetic field due to a point charge must be described by Liénard- Weichert potentials when (A) the point charge is highly accelerated. (B) the electric and magnetic fields are not perpendicular. (C) the point charge is moving with velocity close to that of light. (D) the calculation is done for the radiation zone, i.e. far away from the charge.

Particle Physics & Fundamental Interactions

The strangeness quantum number is conserved in (A) strong, weak and electromagnetic interactions. (B) weak and electromagnetic interactions only. (C) strong and weak interactions only. (D) strong and electromagnetic interactions only.

Linear Algebra & Vector Calculus

The eigenvalues and eigenvectors of the matrix are (A) 6, 1 and [1721) (B) 2, 5 and (7(4] (C) 6, 1 and (D) 2, 5 and (4121]

Linear Algebra & Vector Calculus

A vector field is defined everywhere as $\vec{F} = 2x^2\hat{i} + 3y^2\hat{j} + z^2\hat{k}$. The net flux of $\vec{F}$ associated with a cube of side $L$, with one vertex at the origin and sides along the positive $X, Y,$ and $Z$ axes, is

Linear Algebra & Vector Calculus

If $\vec{F} = x\hat{i} + y\hat{j}$, then

Linear Algebra & Vector Calculus

Consider a vector $\vec{p} = 2\hat{i} + 3\hat{j} + 2\hat{k}$ in the coordinate system $(\hat{i}, \hat{j}, \hat{k})$. The axes are rotated anti-clockwise about the $Y$-axis by an angle of $60^\circ$. The vector $\vec{p}$ in the rotated coordinate system $(\hat{i}', \hat{j}', \hat{k}')$ is

Complex analysis

The contour integral g z* + a' is to be evaluated on a cirele of radius 2a centered at the origin. It will have contributions only from the points (A) 1+i -a and - 1+1 a (B) ia and -ia V2 (C) ia, -ia, 1-i -a and - 1-i (D) 1+ i /2 J2",- and

Fourier and Laplace transforms

The inverse Laplace transform $\mathcal{L}^{-1}\left\{ \frac{s}{s^2 + 4} \right\}$ is

Linear Algebra & Vector Calculus

The points, where the series solution of the Legendre differential equation (1-x)d'y dx? -2xdy + 3₴+1)y=0 will diverge, are located at (A) 0 and 1 (B) 0 and -1 (C) -1 and 1 (D) } and g

Linear Algebra & Vector Calculus

Solution of the differential equation x- -+ y=x* , with the boundary condition that y = 1, at x = 1, is (A) y = 5x* - 4 (В) y = x' (C) y = 4x' (D) y= x* 5x

Electrostatics and Magnetostatics

Match the following P. rest mass 1. timelike vector Q. charge 2. Lorentz invariant R. four-momentum 3. tensor of rank 2 S. electromagnetic field 4. conserved and Lorentz invariant (A) (B) (C) (D) P-2 P-4 P-2 P-4 Q -4 Q-2 Q-4 Q-2 R-3 R-1 R-1 R-3 S-1 S-3 S-3 S-1

Small oscillations

The moment of inertia of a uniform sphere of radius r about an axis passing through its centre 21 4 7t is given by 3 -rр) A rigid sphere of uniform mass density p and radius R has two smaller spheres of radius R/2 hollowed out of it, as shown in the figure. The moment of inertia of the resulting body about the Y axis is (A) пp R° 4 (B) 5mp R$ 12 (C) 7tp R$ 12 (D) 3пp R$ 4

Lagrangian & Hamiltonian Dynamics

The Lagrangian of a particle of mass m is 1= (d) + (d) +(d) -[( +y*)+W sinur, where V. W and o are constants. The conserved quantities are (A) energy and z-component of linear momentum only. (B) energy and z-component of angular momentum only. (C) z-components of both linear and angular momenta only. (D) energy and z-components of both linear and angular momenta.

Lagrangian & Hamiltonian Dynamics

Three particles of mass m each situated at x, (1), x,(t) and x,(t) respectively are connected by two springs of spring constant k and un-stretched length l . The system is free to oscillate only in one dimension along the straight line joining all the three particles. The Lagrangian of the system is (A) L- (d) -(dai) - (dài) - (x, - x,-1)+ k (x, - x2 -l) (B) L - ] (da) (đa) -(da) -{(a -x-07 -5(5 -x,-07 (0) 2 - *(ca) - (da) -(da) -{(a -x +0) -3(5 -x, +0) (D) L = (da) (da) (dy) - =(5 -x-1) -{(5-x--1)

Lagrangian & Hamiltonian Dynamics

The Hamiltonian of a particle is H = P' - + pq, where q is the generalized coordinate and p is the corresponding canonical momentum. The Lagrangian is (A) m(da + g) 2 (B) m 2 di (0) 7 (da) +qda -q dt (D) m 2 ( da (ất) -q dg dt -+ q?

Complex analysis

A toroidal coil has N closely-wound turns. Assume the current through the coil to be I and the toroid is filled with a magnetic material of relative permittivity H,. The magnitude of magnetic induction B inside the toroid, at a radial distance r from the axis, is given by (A) H,M,NIr (B) 4,H.NI (C) H,H6NI (D) 2тн,H.NIr 2r

Complex analysis

An electromagnetic wave with E(z,f) = E, cos(w t - kz) / is traveling in free space and crosses a disc of radius 2 m placed perpendicular to the z-axis. If E, = 60 V m'' the average power, in Watt, crossing the disc along the z-direction is (A) 30 (B) 60 (C) 120 (D) 270

Electrostatics and Magnetostatics

Can the following scalar and vector potentials describe an electromagnetic field? Ф(7,t) = 3xyz - 4t Ä(F,1) = (2x- at)Î + (y- 22) ] + (2-2xe'*)k where w is a constant. (A) Yes, in the Coulomb gauge. (B) Yes, in the Lorentz gauge. (C) Yes, provided w = 0. (D) No. - Q.37 For a particle of mass m in a one-dimensional harmonic oscillator potential of the form V(x) ==mc'x , the first excited energy eigenstate is 4 (x) = xe*™* The value of a is (A) ma/ 4h (B) ma/3h (C) ma/2h (D) 2ma/3h

Lagrangian & Hamiltonian Dynamics

The Hamiltonian of a particle is H = P' - + pq, where q is the generalized coordinate and p is the corresponding canonical momentum. The Lagrangian is (A) m(da + g) 2 (0) 7 (da) +qda -q dt m (B) 2 di m (D) 2 ( da dg (ất) -q -+ q? dt A toroidal coil has N closely-wound turns. Assume the current through the coil to be I and the toroid is filled with a magnetic material of relative permittivity H,. The magnitude of magnetic induction B inside the toroid, at a radial distance r from the axis, is given by (A) H,M,NIr (B) 4,H.NI (C) H,H6NI 2r (D) 2тн,H.NIr An electromagnetic wave with E(z,f) = E, cos(w t - kz) / is traveling in free space and crosses a disc of radius 2 m placed perpendicular to the z-axis. If E, = 60 V m'' the average power, in Watt, crossing the disc along the z-direction is (A) 30 (B) 60 (C) 120 (D) 270 Can the following scalar and vector potentials describe an electromagnetic field? Ф(7,t) = 3xyz - 4t Ä(F,1) = (2x- at)Î + (y- 22) ] + (2-2xe'*)k where w is a constant. (A) Yes, in the Coulomb gauge. (C) Yes, provided w = 0. (B) Yes, in the Lorentz gauge. (D) No. For a particle of mass m in a one-dimensional harmonic oscillator potential of the form V(x) ==mc'x , the first excited energy eigenstate is 4 (x) = xe*™* The value of a is (A) ma/ 4h (B) ma/3h (C) ma/2h (D) 2ma/3h If [x,p ]= ih, the value of [x',p] is (A) 2ihx? (B) - 2ihx? (C) 3ihx' (D) - 3ihx? PH-8/20 R

Complex analysis

If [x,p ]= ih, the value of [x',p] is (A) 2ihx? (B) - 2ihx? (C) 3ihx' (D) - 3ihx?

Electrostatics and Magnetostatics

There are only three bound states for a particle of mass m in a one-dimensional -a/2 +a/2 potential well of the form shown in the \cdot x figure. The depth Vo of the potential satisfies --Vo (A) 2xỉ n° -<%< 9n'n (B) ma? -<V< 2n'h ma* 2л'h 2ma? 8л h 2n h 507'h (C) та' -<%< ma? (D) ma -<V < ma

Angular momentum & Hydrogen Atom

An atomic state of hydrogen is represented by the following wavefunction: $\psi(r, \theta, \phi) = \frac{1}{4\sqrt{2\pi}} \left(\frac{1}{a_0}\right)^{3/2} \left(2 - \frac{r}{a_0}\right) e^{-r/(2a_0)} \cos\theta$ where $a_0$ is the Bohr radius. The quantum numbers of the state are

Linear Algebra & Vector Calculus

Three operators X, Y and Z satisfy the commutation relations [X,Y]=ihz, [Y,Z)=ihX and [Z,X]=ihY. The set of all possible eigenvalues of the operator Z, in units of h, is (A) {0, $1, +2, t3,... ) (B) 2 \cdot, 1, 3 (D) 2 (00 (0. + 2 +1 + 2 +2.+2)

Statistical Ensembles & Partition Functions

A heat pump working on the Carnot cycle maintains the inside temperature of a house at 22° C by supplying 450 kJ s'. If the outside temperature is 0° C, the heat taken, in ', from the óutside air is approximately (A) 487 (B) 470 (C) 467 (D) 417

Statistical Ensembles & Partition Functions

The vapour pressure p (in mm of Hg) of a solid, at temperature T, is expressed by In p = 23 -3863/T and that of its liquid phase by In p = 19 -3063/T . The triple point (in Kelvin) of the material is (A) 185 (B) 190 (C) 195 (D) 200

Statistical Ensembles & Partition Functions

The free energy for a photon gas is given by F = VT', where a is a constant. The entropy S and the pressure P of the photon gas are (A)S = 4 3 -aVT, P== 3 -T" (B) S= 3 -aVT', P= 4a T3 (C) S= 4 -aVT', P=T (D) S = зavT, 4а T* 3

Complex analysis

A system has energy levels Eo, 2Eo, 3Eo,..., where the excited states are triply degenerate. Four non-interacting bosons are placed in this system. If the total energy of these bosons is 5Eo , the number of microstates is (A) 2 (B) 3 (C) 4 (D) 5

Atomic Spectra & Multi-Electron Atoms

In accordance with the selection rules for electric dipole transitions, the 4'P, state of helium can decay by photon emission to the states (A) 2'So, 2'P, and 3'D2 (B) 3'P1, 3'D2 and 3'So (C) 3'P2, 3'D3 and 3°Po (D) 2'S1, 3'D2 and 3'D,

Lagrangian & Hamiltonian Dynamics

If an atom is in the 'D, state, the angle between its orbital and spin angular momentum vectors (Land S) is (A) cos* (B) cos* J3 2 (C) cos" ! (D) COs

Atomic Spectra & Multi-Electron Atoms

The hyperfine structure of Na (3?P32) with nuclear spin / = 3/2 has (A) 1 state. (B) 2 states. (C) 3 states. (D) 4 states.

Atomic Spectra & Multi-Electron Atoms

The allowed rotational energy levels of a rigid hetero-nuclear diatomic molecule are expressed as e, = BJ(J +1), where B is the rotational constant and J is a rotational quantum number. In a system of such diatomic molecules of reduced mass j, some of the atoms of one element are replaced by a heavier isotope, such that the reduced mass is changed to 1.05u. In the rotational spectrum of the system, the shift in the spectral line, corresponding to a transition J = 4 → J = 5, is (A) 0.475 B (B) 0.50 B (C) 0.95 B (D) 1.0 B

Molecular Spectroscopy

The number of fundamental vibrational modes of COz molecule is (A) four: 2 are Raman active and 2 are infrared active. (B) four : 1 is Raman active and 3 are infrared active. (C) three: 1 is Raman active and 2 are infrared active. (D) three: 2 are Raman active and 1 is infrared active.

Electrostatics and Magnetostatics

A piece of paraffin is placed in a uniform magnetic field Ho. The sample contains hydrogen nuclei of mass mp. which interact only with external magnetic field. An additional oscillating magnetic field is applied to observe resonance absorption. If gi is the g-factor of the hydrogen nucleus, the frequency, at which resonance absorption takes place, is given by 3g,eHo 3g,eHo g,eH, g,eHo 2mmp 47m, 27mр 4лmр

Electrostatics and Magnetostatics

The solid phase of an element follows van der Waals bonding with inter-atomic potential V(r) = -- ,o+ ,iz, where P and Q are constants. The bond length can be expressed as (A) (22) (В) (7) (C) (D) (6)

Crystal Structure & Lattice Dynamics

Consider the atomic packing factor (APF) of the following crystal structures: P. Simple Cubic Q. Body Centred Cubic R. Face Centred Cubic S. Diamond T. Hexagonal Close Packed Which two of the above structures have equal APF? (A) P and Q (B) S and T (C) R and S (D) R and T

Crystal Structure & Lattice Dynamics

In a powder diffraction pattern recorded from a face-centred cubic sample using x- rays, the first peak appears at 30°. The second peak will appear at (A) 32.80 (B) 33.7º (C) 34.8° (D) 35.3°

Statistical Ensembles & Partition Functions

Variation of electrical resistivity p with temperature T of three solids is sketched (on and R. different scales) in the figure, as curves P, Q Which one of the following statements describes the variations most appropriately? R Temperature (A) P is for a superconductor, and R for a semiconductor. (B) Q is for a superconductor, and P for a conductor. (C) Q is for a superconductor, and R for a conductor. (D) R is for a superconductor, and P for a conductor.

Crystal Structure & Lattice Dynamics

An extrinsic semiconductor sample of cross-section A and length L is doped in such a way that the doping concentration varies as N,(x) = N, exp| , where No is a constant. Assume that the mobility fof the majority carriers remains constant . The resistance R of the sample is given by (A) R =\cdot AueNo - [exp(1.0) - 1] (B) R=- HeNo -[exp(1.0)-1] (C) R= [exp(-1.0) - 1] (D) R= L AueN® ApeN®

Complex analysis

A ferromagnetic mixture of iron and copper having 75% atoms of Fe exhibits a saturation magnetization of 1.3 x10° A m''. Assume that the total number of atoms per unit volume is 8x 10'8 m3 . The magnetic moment of an iron atom, in terms of the Bohr Magneton, is (A) 1.7 (B) 2.3 (C) 2.9 (D) 3.8

Particle Physics & Fundamental Interactions

The half-life of a radio-isotope is $4 \times 10^4$ years. If there are $10^9$ radioactive nuclei in a sample initially, the number of nuclei remaining undecayed after $8 \times 10^4$ years is

Complex analysis

In the deuterium + tritium (d + t) fusion more energy is released as compared to deuterium + deuterium (d + d) fusion because (A) tritium is radioactive. (B) more nucleons participate in fusion. (C) the Coulomb barrier is lower for the d + t system than d + d system. (D) the reaction product *He is more tightly bound

Nuclear Models & Decays

According to the shell model the ground state spin of the 'O nucleus is (A) (B) (C) 2 3 (D) 2 5

Special theory of relativity

A relativistic particle travels a length of $3 \times 10^{-3}$ m in air before decaying. The proper lifetime of the particle is $10^{-11}$ s. The speed of the particle is approximately

Nuclear Models & Decays

The strange baryon Z* has the quark structure (A) uds (B) und (C) uus (D) us

Lagrangian & Hamiltonian Dynamics

A neutron scatters elastically from a heavy nucleus. The initial and final states of the neutron have the (A) same energy. (B) same energy and linear momentum. (C) same energy and angular momentum. (D) same linear and angular momenta.

Operational Amplifiers & Analog Circuits

The circuit shown is based on ideal operational amplifiers. It acts as a v,o m (A) subtractor. (B) buffer amplifier. (C) adder. (D) divider.

Complex analysis

Identify the function F generated by the logic network shown → F x. (A) F = (X + Y) Z (B) F =Z+Y+ YX (C) F = ZY (Y + X) (D) F = XYZ