Γενεραλ Απτιτυδε ΓΑ Σετ 2

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1 ΓΑΤΕ 6 Γενεραλ Απτιτυδε ΓΑ Σετ Q. Q. carry one mar each. Θ. Τηε ϖολυmε οφ α σπηερε οφ διαmετερ υνιτ ισ τηαν τηε ϖολυmε οφ α χυβε οφ σιδε υνιτ. (Α) λεαστ (Β) λεσσ (Χ) λεσσερ λοω Θ. Τηε υνρυλψ χροωδ δεmανδεδ τηατ τηε αχχυσεδ βε ωιτηουτ τριαλ. (Α) ηανγεδ (Β) ηανγινγ (Χ) ηανκερινγ ηυνγ Θ. Χηοοσε τηε στατεmεντ(σ) ωηερε τηε υνδερλινεδ ωορδ ισ υσεδ χορρεχτλψ: (ι) (ιι) (ιιι) Α προνε ισ α δριεδ πλυm. Ηε ωασ λψινγ προνε ον τηε φλοορ. Πεοπλε ωηο εατ α λοτ οφ φατ αρε προνε το ηεαρτ δισεασε. (Α) (ι) ανδ (ιιι) ονλψ (Β) (ιιι) ονλψ (Χ) (ι) ανδ (ιι) ονλψ (ιι) ανδ (ιιι) ονλψ Θ.4 Fact: Ιφ ιτ ραινσ τηεν τηε φιελδ ισ ωετ. Ρεαδ τηε φολλοωινγ στατεmεντσ: (ι) Ιτ ραινσ (ιι) Τηε φιελδ ισ νοτ ωετ (ιιι) Τηε φιελδ ισ ωετ (ιϖ) Ιτ διδ νοτ ραιν Wηιχη ονε οφ τηε οπτιονσ γιϖεν βελοω ισ NO λογιχαλλψ ποσσιβλε βασεδ ον τηε γιϖεν φαχτ? (Α) Ιφ (ιιι) τηεν (ιϖ). (Χ) Ιφ (ι) τηεν (ιι). (Β) Ιφ (ι) τηεν (ιιι). Ιφ (ιι) τηεν (ιϖ). Θ. Α ωινδοω ισ mαδε υπ οφ α σθυαρε πορτιον ανδ αν εθυιλατεραλ τριανγλε πορτιον αβοϖε ιτ. Τηε βασε οφ τηε τριανγυλαρ πορτιον χοινχιδεσ ωιτη τηε υππερ σιδε οφ τηε σθυαρε. Ιφ τηε περιmετερ οφ τηε ωινδοω ισ 6 m τηε αρεα οφ τηε ωινδοω ιν m ισ. (Α).4 (Β).6 (Χ).68.88

2 ΓΑΤΕ 6 Q. 6 Q. carry two mars each. Γενεραλ Απτιτυδε ΓΑ Σετ Θ.6 Στυδεντσ τακινγ αν εξαm αρε διϖιδεδ ιντο τωο γρουπσ P ανδ Q συχη τηατ εαχη γρουπ ηασ τηε σαmε νυmβερ οφ στυδεντσ. Τηε περφορmανχε οφ εαχη οφ τηε στυδεντσ ιν α τεστ ωασ εϖαλυατεδ ουτ οφ mαρκσ. Ιτ ωασ οβσερϖεδ τηατ τηε mεαν οφ γρουπ P ωασ ωηιλε τηατ οφ γρουπ Q ωασ 8. Τηε στανδαρδ δεϖιατιον οφ γρουπ P ωασ ωηιλε τηατ οφ γρουπ Q ωασ. Ασσυmινγ τηατ τηε mαρκσ ωερε διστριβυτεδ ον α νορmαλ διστριβυτιον ωηιχη οφ τηε φολλοωινγ στατεmεντσ ωιλλ ηαϖε τηε ηιγηεστ προβαβιλιτψ οφ βεινγ RU? (Α) Νο στυδεντ ιν γρουπ Q σχορεδ λεσσ mαρκσ τηαν ανψ στυδεντ ιν γρουπ P. (Β) Νο στυδεντ ιν γρουπ P σχορεδ λεσσ mαρκσ τηαν ανψ στυδεντ ιν γρουπ Q. (Χ) Μοστ στυδεντσ οφ γρουπ Q σχορεδ mαρκσ ιν α ναρροωερ ρανγε τηαν στυδεντσ ιν γρουπ P. Τηε mεδιαν οφ τηε mαρκσ οφ γρουπ P ισ. Θ.7 Α σmαρτ χιτψ ιντεγρατεσ αλλ mοδεσ οφ τρανσπορτ υσεσ χλεαν ενεργψ ανδ προmοτεσ συσταιναβλε υσε οφ ρεσουρχεσ. Ιτ αλσο υσεσ τεχηνολογψ το ενσυρε σαφετψ ανδ σεχυριτψ οφ τηε χιτψ σοmετηινγ ωηιχη χριτιχσ αργυε ωιλλ λεαδ το α συρϖειλλανχε στατε. Wηιχη οφ τηε φολλοωινγ χαν βε λογιχαλλψ ινφερρεδ φροm τηε αβοϖε παραγραπη? (ι) (ιι) (ιιι) (ιϖ) (Α) (ι) ανδ (ιϖ) ονλψ (Χ) (ιϖ) ονλψ Αλλ σmαρτ χιτιεσ ενχουραγε τηε φορmατιον οφ συρϖειλλανχε στατεσ. Συρϖειλλανχε ισ αν ιντεγραλ παρτ οφ α σmαρτ χιτψ. Συσταιναβιλιτψ ανδ συρϖειλλανχε γο ηανδ ιν ηανδ ιν α σmαρτ χιτψ. Τηερε ισ α περχεπτιον τηατ σmαρτ χιτιεσ προmοτε συρϖειλλανχε. Θ.8 Φινδ τηε mισσινγ σεθυενχε ιν τηε λεττερ σεριεσ. Β ΦΗ ΛΝΠ. (Β) (ιι) ανδ (ιιι) ονλψ (ι) ονλψ (Α) ΣΥWΨ (Β) ΤΥςW (Χ) ΤςΞΖ ΤWΞΖ Θ.9 Τηε βιναρψ οπερατιον ισ δεφινεδ ασ a b = ab+(a+b) ωηερε a ανδ b αρε ανψ τωο ρεαλ νυmβερσ. Τηε ϖαλυε οφ τηε ιδεντιτψ ελεmεντ οφ τηισ οπερατιον δεφινεδ ασ τηε νυmβερ x συχη τηατ a x = a φορ ανψ a ισ. (Α) (Β) (Χ)

3 ΓΑΤΕ 6 Γενεραλ Απτιτυδε ΓΑ Σετ Θ. Wηιχη οφ τηε φολλοωινγ χυρϖεσ ρεπρεσεντσ τηε φυνχτιον ln φορ? Ηερε ρεπρεσεντσ τηε αβσχισσα ανδ ρεπρεσεντσ τηε ορδινατε. (Α) (Β) (Χ) ND OF H QUSION PPR

4 Q. Q. carry one mar each. Q. Consider the linear differential equation dy xy. If y at x then the value of y at x= is dx given by e () e e e Q. Which of the following magnetic vector potentials gives rise to a uniform magnetic field ˆ? z ˆ () xˆ j y iˆ x ˆ j y iˆ x ˆj Q. he molecule 7 O is Raman active but not NMR (nuclear magnetic resonance) active. () Infrared active and Raman active but not NMR active. Raman active and NMR active. Only NMR active. Q.4 here are four electrons in the d shell of an isolated atom. he total magnetic moment of the atom in units of ohr magneton is. Q. Which of the following transitions is NO allowed in the case of an atom according to the electric dipole radiation selection rule? s-s () p-s p-s d-p Q.6 In the SU() quar model the triplet of mesons Isospin = Strangeness = () Isospin = Strangeness = Isospin = Strangeness = + Isospin = Strangeness = - Q.7 he magnitude of the magnetic dipole moment associated with a square shaped loop carrying a steady current I is m. If this loop is changed to a circular shape with the same current I passing pm through it the magnetic dipole moment becomes. he value of p is. Q.8 he total power emitted by a spherical blac body of radius R at a temperature is P. Let P be the total power emitted by another spherical blac body of radius R ept at temperature. he ratio P P is. (Give your answer upto two decimal places) has PH

5 Q.9 he entropy of a system of spins which may align either in the upward or in the downward direction is given by S N p ln p ( p) ln( p). Here is the oltzmann constant. he probability of alignment in the upward direction is p. he value of p at which the entropy is maximum is. (Give your answer upto one decimal place) Q. For a system at constant temperature and volume which of the following statements is correct at equilibrium? he Helmholtz free energy attains a local minimum. () he Helmholtz free energy attains a local maximum. he Gibbs free energy attains a local minimum. he Gibbs free energy attains a local maximum. Q. atoms of an ideal gas are enclosed in a container of volume. he volume of the container is changed to 4 while eeping the total energy constant. he change in the entropy of the gas in units of N ln is where is the oltzmann constant. Q. Which of the following is an analytic function of z everywhere in the complex plane? z * () z z z Q. In a oung s double slit experiment using light the apparatus has two slits of unequal widths. When only slit- is open the maximum observed intensity on the screen is 4I. When only slit- is open the maximum observed intensity is I. When both the slits are open an interference pattern appears on the screen. he ratio of the intensity of the principal maximum to that of the nearest minimum is. Q.4 Consider a metal which obeys the Sommerfeld model exactly. If F is the Fermi energy of the metal at = K and R H is its Hall coefficient which of the following statements is correct? H F R () H F R R H F F. R H is independent of Q. one-dimensional linear chain of atoms contains two types of atoms of masses m and m (where m > m ) arranged alternately. he distance between successive atoms is the same. ssume that the harmonic approximation is valid. t the first rillouin zone boundary which of the following statements is correct? he atoms of mass m are at rest in the optical mode while they vibrate in the acoustical mode. ( he atoms of mass m are at rest in the optical mode while they vibrate in the acoustical mode. oth types of atoms vibrate with equal amplitudes in the optical as well as in the acoustical modes. oth types of atoms vibrate but with unequal non-zero amplitudes in the optical as well as in the acoustical modes. Q.6 Which of the following operators is Hermitian? d dx d () dx d d i dx dx PH

6 Q.7 he inetic energy of a particle of rest mass m is equal to its rest mass energy. Its momentum in units of c where c is the speed of light in vacuum is. (Give your answer upto two m decimal places) Q.8 he number density of electrons in the conduction band of a semiconductor at a given temperature 9 is m. Upon lightly doping this semiconductor with donor impurities the number density of conduction electrons at the same temperature becomes 4 m. he ratio of majority to minority charge carrier concentration is. Q.9 wo blocs are connected by a spring of spring constant. One bloc has mass m and the other bloc has mass m. If the ratio m = 4 s the angular frequency of vibration ω of the two bloc spring system in is. (Give your answer upto two decimal places) s Q. particle moving under the influence of a central force F( r) r (where r is the position vector of the particle and is a positive constant) has non-zero angular momentum. Which of the following curves is a possible orbit for this particle? straight line segment passing through the origin. () n ellipse with its center at the origin. n ellipse with one of the foci at the origin. parabola with its vertex at the origin. Q. 4 Consider the reaction Mn γ + 4 e Cr 4 +. he particle is () n π e Q. he scattering of particles by a potential can be analyzed by orn approximation. In particular if the scattered wave is replaced by an appropriate plane wave the corresponding orn approximation is nown as the first orn approximation. Such an approximation is valid for large incident energies and wea scattering potentials. () large incident energies and strong scattering potentials. small incident energies and wea scattering potentials. small incident energies and strong scattering potentials. Q. Consider an elastic scattering of particles in l states. If the corresponding phase shift is 9 and the magnitude of the incident wave vector is equal to cross section in units of fm is. fm - then the total scattering Q.4 hydrogen atom is in its ground state. In the presence of a uniform electric field z ˆ the n leading order change in its energy is proportional to( ). he value of the exponent n is. PH

7 Q. solid material is found to have a temperature independent magnetic susceptibility C. Which of the following statements is correct? If C is positive the material is a diamagnet. () If C is positive the material is a ferromagnet. If C is negative the material could be a type I superconductor. If C is positive the material could be a type I superconductor. Q. 6 Q. carry two mars each. Q.6 n infinite conducting slab ept in a horizontal plane carries a uniform charge density. nother infinite slab of thicness t made of a linear dielectric material of dielectric constant is ept above the conducting slab. he bound charge density on the upper surface of the dielectric slab is ( ) () ( ) Q.7 he number of spectroscopic terms resulting from the L S coupling of a p electron and a d electron is. Q.8 Which of the following statements is NO correct? deuteron can be disintegrated by irradiating it with gamma rays of energy 4 MeV. () deuteron has no excited states. deuteron has no electric quadrupole moment. he S state of deuteron cannot be formed. Q.9 If s and s are the spin operators of the two electrons of a He atom the value of s s ground state is () 4 4 for the Q. two-dimensional square rigid box of side L contains six non-interacting electrons at = K. he mass of the electron is m. he ground state energy of the system of electrons in units of is ml. Q. n alpha particle is accelerated in a cyclotron. It leaves the cyclotron with a inetic energy of 6 MeV. he potential difference between the D electrodes is ilovolts. he number of revolutions the alpha particle maes in its spiral path before it leaves the cyclotron is. Q. Let be the component of a vector field which has zero divergence. If the expression for is equal to j Vi () j Vi V j i V j i PH 4

8 Q. he direction of for a scalar field () = + at the point () is ˆj ˆ ˆj ˆ ˆj ˆ ˆj ˆ () Q.4 and is equal to x y z are the Pauli matrices. he expression x y y x i () z i z i z i z Q. particle of mass =. g is initially at rest at origin. It starts moving with a uniform acceleration ˆ a i ms at =. he action of the particle in units of J-s at = is. (Give your answer upto two decimal places) Q.6 periodic function () of period is defined in the interval ( < < ) as: < < () = < < he appropriate Fourier series expansion for () is f ( x) 4 [sin x (sin x) (sin x)...] () f ( x) 4 [sin x (sin x) (sin x)...] f ( x) 4 [cos x (cos x) (cos x)...] f ( x) 4 [cos x (cos x) (cos x)...] Q.7 toms which can be assumed to be hard spheres of radius R are arranged in an fcc lattice with lattice constant a such that each atom touches its nearest neighbours. ae the center of one of the atoms as the origin. nother atom of radius r (assumed to be hard sphere) is to be accommodated at a position ( a ) without distorting the lattice. he maximum value of rr is. (Give your answer upto two decimal places) Q.8 In an inertial frame of reference S an observer finds two events occurring at the same time at coordinates x and x d. different inertial frame S moves with velocity v with respect to S along the positive x-axis. n observer in S also notices these two events and finds them to occur at times t and t and at positions x and x respectively. If t t t x x x and v c which of the following statements is true? t x d () t x d t vd c x d t vd c x d PH

9 Q.9 he energy vs. wave vector approximated as 4 ( a) ( a) relationship near the bottom of a band for a solid can be where the lattice constant a =. Å. he values of and 9 are 6. J and. J respectively. t the bottom of the conduction band the ratio of the effective mass of the electron to the mass of free electron is. (Give your answer upto two decimal places) (ae =.x -4 J-s mass of free electron= 9.x - g) Q.4 he electric field component of a plane electromagnetic wave travelling in vacuum is given by ( z t) cos( z t) iˆ. he Poynting vector for the wave is c cos ( z t) j c cos ( z t) ˆ j c cos ( z t) ˆ () c cos ( z t) ˆ ˆ Q.4 Consider a system having three energy levels with energies and with respective degeneracies of and. Four bosons of spin zero have to be accommodated in these levels such that the total energy of the system is. he number of ways in which it can be done is. Q.4 he Lagrangian of a system is given by sin mgl cos where m l and g are constants. L ml Which of the following is conserved? sin () sin sin sin Q.4 Protons and α-particles of equal initial momenta are scattered off a gold foil in a Rutherford scattering experiment. he scattering cross sections for proton on gold and α-particle on gold are and respectively. he ratio p is. p PH 6

10 PH 7 Q.44 For the digital circuit given below the output is C () C) ( C) ( C) ( Q.4 he Fermi energies of two metals and are ev and 7 ev and their Debye temperatures are 7 K and 4 K respectively. he molar specific heats of these metals at constant volume at low temperatures can be written as C V and C V where γ and are constants. ssuming that the thermal effective mass of the electrons in the two metals are same which of the following is correct? 8 7 () Q.46 two-level system has energies zero and. he level with zero energy is non-degenerate while the level with energy is triply degenerate. he mean energy of a classical particle in this system at a temperature is e e () e e e e e e C

11 Q.47 particle of rest mass M is moving along the positive x-direction. It decays into two photons and as shown in the figure. he energy of is GeV and the energy of is.8 GeV. he value of M (in units of GeVc ) is. (Give your answer upto two decimal places) Q.48 If x and p are the x components of the position and the momentum operators of a particle respectively the commutator [ x p ] is i ( xp px) () i ( xp px) i ( xp px) i ( xp px) Q.49 he x-y plane is the boundary between free space and a magnetic material with relative permeability r. he magnetic field in the free space is iˆ x zˆ. he magnetic field in the magnetic material is i ˆ x zˆ () iˆ x ˆ r z iˆ x zˆ iˆ r x zˆ r Q. Let l m be the simultaneous eigenstates of x y z L and L z. Here L is the angular momentum operator with Cartesian components ( L L L ) l is the angular momentum quantum number and m is the azimuthal quantum number. he value of ( L il ) () Q. For the parity operator P which of the following statements is NO true? = () P P x y P I = is

12 Q. For the transistor shown in the figure assume V =.7 V and β dc =. If V in = V V out (in Volts) is. (Give your answer upto one decimal place) Q. he state of a system is given by where and form an orthonormal set. he probability of finding the system in the state is. (Give your answer upto two decimal places) Q.4 ccording to the nuclear shell model the respective ground state spin-parity values of 8 O and 8O nuclei are 7 ()

13 Q. particle of mass m and energy moving in the positive x direction is incident on a step potential at x = as indicated in the figure. he height of the potential is V where V >. t x x where x the probability of finding the electron is e times the probability of finding m( V it at x =. If ) the value of x is V x = x = x () ND OF H QUSION PPR 4 PH