Q2
MCQ • 1M
Linear Algebra & Vector Calculus
ae'" An unitary matrix ce' is given, where a,b,c,d,a and B are real. The inverse of the matrix is ae"" -ce'ß (B) ae'" ce'P (A) b d b d ae"a ce"p (C) се"
Q7
MCQ • 1M
Electrostatics and Magnetostatics
Unpolarized light falls from air to a planar air-glass interface (refractive index of glass is 1.5) and the reflected light is observed to be plane polarized. The polarization vector and the angle of incidence 0, are (A) perpendicular to the plane of incidence and 0,= 42°. (B) parallel to the plane of incidence and 0, = 56'. (C) perpendicular to the plane of incidence and 0, = 56. (D) parallel to the plane of incidence and 0,= 42 .
Q8
MCQ • 1M
Linear Algebra & Vector Calculus
A finite wave train, of an unspecified nature, propagates along the positive x axis with a constant speed v and without any change of shape. The differential equation among the four listed below, whose solution it must be, is (A) y (x,t) = 0 1 a2 (B) тº - v đr V (r,l) = 0 (C) n 22 o y (F,1) =0 2m дx - in ° (x1) = 0 (D) V +a-
Q10
MCQ • 1M
Statistical Ensembles & Partition Functions
Thermodynamic variables of a system can be volume V, pressure P, temperature T, number of particles N, internal energy E and chemical potential j, etc. For a system to be specified by Microcanonical (MC), Canonical (CE) and Grand Canonical (GC) ensembles, the parameters required for the respective ensembles are: (A) MC: (N,V,T); CE: (E,V,N); GC: (V, T,4) (B) MC: (E,V,N); CE: (N,V,T); GC: (V,T,u) (C) MC: (V,T,4); CE: (N,V,T); GC: (E,V,N) (D) MC: (E,V,N); CE: (V,T,u); GC: (N,V,T)
Q12
MCQ • 1M
Angular momentum & Hydrogen Atom
The radial wave function of the electrons in the state of n=1 and /=0 in a hydrogen atom is R,o =\cdot ao 2, еxр , ao is the Bohr radius. The most probable value of r for an electron is (A) ao (B) 2a, (C) 4ao (D) 8a, pH 3/24 2008 MAIN PAPER - PH
Q20
MCQ • 1M
Electrostatics and Magnetostatics
Under normal operating conditions, the gate terminal of an n-channel junction field effect transistor (JFET) and an n-channel metal oxide semiconductor field effect transistor (MOSFET) are (A) both biased with positive potentials. (B) both biased with negative potentials. (C) biased with positive and negative potentials, respectively. (D) biased with negative and positive potentials, respectively.
Q21
MCQ • 1M
Complex analysis
to Q.75, 0.5 mark will be deducted for each wrong answer. For the pairs of questions with linked answers, there will be negative marks only for wrong answer to the first question, i.e. for Q.76, 9.78, Q.80, Q.82 and Q.84, 0.5 mark will be deducted for each wrong answer. There is no negative marking for Q.77, Q.79, Q.81, Q.83 and Q.85. 10. Calculator without data connectivity is allowed in the examination hall. 11. Charts, graph sheets and tables are NOT allowed in the examination hall. 12. Rough work can be done on the question paper itself. Additional blank pages are given at the end of the question paper for rough work. PH 1/24 2008 MAIN PAPER - PH
Q25
MCQ • 1M
Complex analysis
Under a certain rotation of coordinate axes, a rank-1 tensor Va (a=1,2,3) transforms according to the orthogonal transformation defined by the relations vi (v, + ½2) ; ½2= VE (-v, +v2) ; V =V3. Under the same rotation a rank-2 tensor Tas would transform such that (B) Tu = Tu (A) Tu = 1,112 (C) T = Tu + 2722 - 121 (D) Tu F(T + 722 + T,2 + T21)
Q26
MCQ • 2M
Complex analysis
Under a certain rotation of coordinate axes, a rank-1 tensor va (a=1,2,3) transforms according to the orthogonal transformation defined by the relations vj (v, +½) : ½= Tz(-v, +v2) ; Vj =v,. Under the same rotation a rank-2 tensor Ta» would transform such that (A) T =T, Ti2 (B) Tu =Tu (C) T = Tu +2722 - T21 (D) 7 =- 1 (Tiu + 72,2 + 71,2 + 12,1) 2
Q31
MCQ • 2M
Lagrangian & Hamiltonian Dynamics
For a simple harmonic oscillator the Lagrangian is given by =q?. If 4(p,q) = P+iq and H(p,q) is the Hamiltonian of the system, the Poisson bracket {A(p,q), H (p,q)} is given by (A) iA(p,9) (B) A°(p,q) (C) -iA" (p,q) (D) -iA(p,q)
Q32
MCQ • 2M
Electromagnetic Waves in Media
A plane electromagnetic wave is given by E,(ê + e" f)expfi(kz-wt)}. At a given location, the number of times È vanishes in one second is (A) An integer near when S = nл and zero when б # nлt , n is integer (B) An integer near and is independent of & (C) An integer near 2л when б = n and zero when & # nzt , n is integer (D) An integer near and is independent of & PH 7/24 MAIN PAPER - PH 2008
Q34
MCQ • 2M
Electrostatics and Magnetostatics
A rod of length L with uniform charge density & per unit length is in the xy-plane and rotating about z-axis passing through one of its edge with an angular velocity c as shown in the figure below. (f,ф,2) refer to the unit vectors at Q, Ä is the vector potential at a distance d from the origin O along z-axis for d > L and J is the current density due to the motion of the rod. Which one of the following statements is correct? P (A) J along f ; along 2 ; |f/ c - (B) J along ở; A along 8 ; 17/ o d (C) J along i: A along 2 ; |ã/ c - (D) J along ở: Ă along $; 17/ oc -
Q35
MCQ • 2M
Electrostatics and Magnetostatics
Corcos0 -. The A circular disc of radius a on the xy plane has a surface charge density o = a electric dipole moment of this charge distribution is о ла" oora' (D) -- Cora* 4 (С) - Гora 8/24 PH 2008 MAIN PAPER - PH| 0.36 At time t =0, a charge distribution p(F,0) exists within an ideal homogeneous conductor of permittivity & and conductivity o . At a later time p(F,t) is given by (B) p(F,t)= P(F,0) (1) p(T.1)= p(7,0)exp(-°) 1+(ot/ &) (D) p(F,t) = p(F,O)≤ sin ot (C) p(F,t) = p(F,0)exp
Q38
MCQ • 2M
Schrodinger equation & Wave Mechanics
For a physical system, two observables O, and O, are known to be compatible. Choose the correct implication from amongst those given below: (A) Every eigenstate of O, must necessarily be an eigenstate of O2. (B) Every non-degenerate eigenstate of O, must necessarily be an eigenstate of O2. (C) When an observation of O, is carried out on an arbitrary state |V) of the physical system, a subsequent observation of O, leads to an unambiguous result. (D) Observation of O, and Oz, carried out on an arbitrary state |*') of the physical system, lead to the identical results irrespective of the order in which the observations are made.
Q39
MCQ • 2M
Schrodinger equation & Wave Mechanics
An exact measurement of the position of a simple harmonic oscillator (SHO) is made with the result x =Xo. [The SHO has energy levels E, (n =0,1,2,...) and associated normalized wave- functions Vn]. Subsequently, an exact measurement of energy E is made. Using the general notation Pr(E=E') denoting the probability that a result E' is obtained for this measurement, the following statements are written. Which one of the following statements is correct? (A) Pr(E = E.)= 0 (B) Pr(E = E,) = 1 for some value of n. (C) Pr(E = E,) oc V„(x) (D) Pr( E > E") > 0 for any E".
Q40
MCQ • 2M
Lagrangian & Hamiltonian Dynamics
Consider the combined system of proton and electron in the hydrogen atom in its (electronic) ground state. Let I denote the quantum number associated with the total angular momentum and let <9> denote the magnitude of the expectation value of the net magnetic moment in the state. Which of the following pairs represents a possible state of the system (Hg is Bohr magneton)? (A) 1=0, <9> = 0 (B)I = ½, <9*>=14в (C)I=1, <9> ≥ 1 Нв (D) I=0, <Ф> =2 Нв PHI 9/24 2008 MAIN PAPER - PH
Q41
MCQ • 2M
Complex analysis
A particle is placed in a one dimensional box of size L along the x-axis (0< x < L). Which of the following is true? (A) In the ground state, the probability of finding the particle in the interval (L/4,3L/4) is half. (B) In the first excited state, the probability of finding the particle in the interval (L/4,3L/4) is half. This also holds for states with n =4, 6, 8,... (C) For an arbitrary state |V'), the probability of finding the particle in the left half of the well is half. (D) In the ground state, the particle has a definite momentum.
Q43
MCQ • 2M
Electrostatics and Magnetostatics
A system containing N non-interacting localized particles of spin ½ and magnetic moment u each is kept in constant external magnetic field B and in thermal equilibrium at temperature T. The magnetization of the system is, (A) Nucoth HB (B) Nu tanh uB k,T) K„T (C) Nu sinh uB (D) Nucosh uB k,T) k,T
Q44
MCQ • 2M
Statistical Ensembles & Partition Functions
Two identical particles have to be distributed among three energy levels. Let rg, Ip and re represent the ratios of probability of finding two particles to that of finding one particle in a given energy state. The subscripts B, F and C correspond to whether the particles are bosons, fermions and classical particles, respectively. Then, tg : Tp : rc is equal to (C) 1: (D) 1:0: 1 (A) {:0:1 (B) 1:=:1 2 2 PH 10/24 2008 MAIN PAPER - PH
Q47
MCQ • 2M
Complex analysis
Consider a system of two non-interacting classical particles which can occupy any of the three energy levels with energy values E =0,e and 2e having degeneracies g(E)=1,2 and 4 respectively. The mean energy of the system is (A) (B) 4 expl %/k„T + 8exp( - L8 k,T) 2еxр( -%k„T)+8exр| -28/k,т) 1+2 exp 6k,T) + 4 exp -28 1+2exp %K„T) )+4exр -28 /k,T (C) (D) 2exp( -%k„T +4еxр( - 2% /k,7) exp(-%/k,7) )+2ехр( - 2%/,т) 1+2exp| %/k,T, + 4exp 1+exp - %k,7 ) + exр( -28, k,T)
Q53
MCQ • 2M
Crystal Structure & Lattice Dynamics
The primitive translation vectors of the body centered cubic lattice are ä =ª(ê + j-2), 6 ={(-f+8+2) and & = f(i - f +2). The primitive translation vectors Ä, B and C of the reciprocal lattice are (A) A = 27 (f - f); B= 27 (0+z); C=27 (i+2) (B) Й= 2П (i+f); B=2л (0-2); С=2л (f+z) (C) й = 27 (&+f); B= 2* (0+z); C=2n (1-z) (\cdot) = 27 (2+$): B = 27 (0 +*); C=2# (8+*) The structure factor of a single cell of identical atoms of form factor f is given by Smu = J Lexp(-i2z(x,h+y,k +z,l)) where (x,,Y,,z,) is the coordinate of an atom, and hkl are the Miller indices. Which one of the following statement is correct for the diffraction peaks of body centered cubic (BCC) and face centered cubic (FCC) lattices? (A)BCC : (200); (110); (222) (B) BCC : (210); (110); (222) FCC: (111); (311); (400) FCC : (111); (311); (400) (C) BCC : (200); (110); (222) (D) BCC : (200); (210); (222) FCC : (111); (211); (400) FCC: (111); (211); (400) PH 12/24 2008 MAIN PAPER - PH
Q54
MCQ • 2M
Crystal Structure & Lattice Dynamics
The structure factor of a single cell of identical atoms of form factor f is given by SuH = fLexp|-i2z(x,h+ y,k + z,1)) where (x,,Y,,z,) is the coordinate of an atom, and hkl are the Miller indices. Which one of the following statement is correct for the diffraction peaks of body centered cubic (BCC) and face centered cubic (FCC) lattices? (A)BCC : (200); (110); (222) (B) BCC : (210); (110); (222) FCC : (111); (311); (400) FCC : (111); (311); (400) (C) BCC : (200): (110); (222) (D) BCC : (200); (210); (222) FCC : (111); (211); (400) FCC: (111); (211); (400) PH 12/24 2008 MAIN PAPER - PH
Q55
MCQ • 2M
Statistical Ensembles & Partition Functions
The lattice specific heat C of a crystalline solid can be obtained using the Dulong Petit model, Einstein model and Debye model. At low temperature hw >> kT, which one of the following statements is true ( a and A are constants) (A)Dulong Perit: C &x exp(a/7); Einsitein : C =, constant; Debye : C* ($) (B) Dulong Petit : C = . constant ; Einstein : C o - ; Debye : C oc exp(-a/T) (C) Dulong Petit : C . constan ; Einssein : C *T : Debye : C a (E) (D) Dulong Petit : C o ; Einstein : C oc - T2 ; Debye: C=. constant
Q58
MCQ • 2M
Statistical Ensembles & Partition Functions
An intrinsic semiconductor with mass of a hole m,and mass of an electron m, is at a finite temperature T. If the top of the valence band energy is E, and the bottom of the conduction band energy is E., the Fermi energy of the semiconductor is (A) Ep = E, + E -kgT In 3 m, 2 4 3 (B) E, = (*2)+Z (E, + E.)m(m.) (C) Ep = E, + E. 2 + -kgT In 4 m, \cdot) E, -(*2)-Z(E,+E.)(M.)