GATE PH 2017 Question Paper & Solutions

Quantitative Aptitude

The ninth and the tenth of this month are Monday and Tuesday _ (A) figuratively (B) retrospectively (C) respectively (D) rightfully

Quantitative Aptitude

It is (A) easier, than Correct: 1 Wrong : -0.33 the last year's. to read this year's textbook (B) most easy, than (C) easier, from (D) easiest. from

Quantitative Aptitude

A rule states that in order to drink beer, one must be over 18 years old. In a bar, there are 4 people. P is 16 years old. Q is 25 years old. R is drinking milkshake and S is drinking a beer. What must be checked to ensure that the rule is being followed? (A) Only P's drink (B) Only P's drink and S's age (C) Only S's age (D) Only P's drink. Q's drink and S's age

Quantitative Aptitude

Fatima starts from point P. goes North for 3 km. and then East for 4 km to reach point Q. She then turns to face point P and goes 15 km in that direction. She then goes North for 6 km. How far is she from point P. and in which direction should she go to reach point P? (A) 8 km, East (B) 12 km, North (C) 6 km. East (D) 10 km. North

Quantitative Aptitude

500 students are taking one or more courses out of Chemistry. Physics, and Mathematics. Registration records indicate course enrolment as follows: Chemistry (329). Physics (186). Mathematics (295). Chemistry and Physics (83). Chemistry and Mathematics (217). and Physics and Mathematics (63). How many students are taking all 3 subjects? (A) 37 (B) 43 (C) 47 (D) 53

Quantitative Aptitude

"If you are looking for a history of India, or for an account of the rise and fall of the British Raj. or for the reason of the cleaving of the subcontinent into two mutually antagonistic parts and the effects this mutilation will have in the respective sections, and ultimately on Asia, you will not find it in these pages; for though I have spent a lifetime in the country. I lived too near the seat of events. and was too intimately associated with the actors. to get the perspective needed for the impartial recording of these matters." Which of the following statements best reflects the author's opinion? (A) An intimate association does not allow for the necessary perspective. (B) Matters are recorded with an impartial perspective. (C) An intimate association offers an impartial perspective. (D) Actors are typically associated with the impartial recording of matters.

Quantitative Aptitude

Each of P. Q. R. S. W. X. Y and Z has been married at most once. X and Y are married and have two children P and Q. Z is the grandfather of the daughter S of P. Further. Z and W are married and are parents of R. Which one of the following must necessarily be FALSE? (A) X is the mother-in-law of R (B) P and R are not married to each other (C) Pis a son of X and Y (D) Q cannot be married to R

Quantitative Aptitude

1200 men and 500 women can build a bridge in 2 weeks. 900 men and 250 women will take 3 weeks to build the same bridge. How many men will be needed to build the bridge in one week? (A) 3000 (B) 3300 (C) 3600 (D) 3900

Quantitative Aptitude

The number of 3-digit numbers such that the digit 1 is never to the immediate right of 2 is (A) 781 (B) 791 (C) 881 (D) 891

Quantitative Aptitude

A contour line joins locations having the same height above the mean sea level. The following is a contour plot of a geographical region. Contour lines are shown at 25 m intervals in this plot. 425 450 550 575 550 500 500 - 475 2 km Which of the following is the steepest path leaving from P? (A) P to Q (B) P to R (C) P to S (D) P to T

Electrostatics and Magnetostatics

Identical charges q are placed at five vertices of a regular hexagon of side a. The magnitude of the electric field and the electrostatic potential at the centre of the hexagon are respectively (A) 0,0 q (B) - (C) 5q 4лЕ а (D)

Electrostatics and Magnetostatics

A parallel plate capacitor with square plates of side 1 m separated by 1 micro meter is filled with a medium of dielectric constant of 10. If the charges on the two plates are 1 C and -1 C, the voltage across the capacitor is KV. (up to two decimal places). (&o = 8.854x10-12 F/m)

Complex analysis

Light is incident from a medium of refractive index n = 1.5 onto vacuum. The smallest angle of incidence for which the light is not transmitted into vacuum is _ _degrees. (up to two decimal places).

Electrostatics and Magnetostatics

A monochromatic plane wave in free space with electric field amplitude of 1 V/m is normally incident on a fully reflecting mirror. The pressure exerted on the mirror is _x10-12 Pa. (up to two decimal places) (& = 8.854x10-12 F/m).

Complex analysis

The best resolution that a 7 bit A/D convertor with 5 V full scale can achieve is to two decimal places). Correct: 1 Wrong : 0 mV. (up

Complex analysis

In the figure given below, the input to the primary of the transformer is a voltage varying sinusoidally with time. The resistor R is connected to the centre tap of the secondary. Which one of the following plots represents the voltage across the resistor R as a function of time? R (A) (B) (C) (D) 0 t

Complex analysis

The atomic mass and mass density of Sodium are 23 and 0.968 g cm', respectively. The number density of valence electrons is x1022 cm*. (Up to two decimal places.) (Avogadro number, N4 = 6.022x1033).

Crystal Structure & Lattice Dynamics

gap at the edge of the Brillouin zoue (A= a) is: U (A) Vo (B) 2

Statistical Ensembles & Partition Functions

(C) 20o (D) 4 Correct: 1 Wrong: -0.33 Consider a triatomic molecule of the shape shown in the figure below in three dimensions. The heat capacity of this molecule at high temperature (temperature much higher than the vibrational and rotational energy scales of the molecule but lower than its bond dissociation energies) is: (A) - KB 2

Complex analysis

(B) 3kg 9 - KB 2 (D) 6kB Correct: 1 Wrong : -0.33 the angian (te mo g's motive 10 = 4, a (4) let one orine following is TRUE? (A) Both the canonical momentum and equation of motion do not change (B) Canonical momentum changes, equation of motion does not change (C) Canonical momentum does not change, equation of motion changes (D) Both the canonical momentum and equation of motion change

Complex analysis

Two identical masses of 10 gm each are connected by a massless spring of spring constant 1 N/m. The non-zero angular eigenfrequency of the system is _rad/s. (up to two decimal places).

Lagrangian & Hamiltonian Dynamics

The phase space trajectory of an otherwise free particle bouncing between two hard walls elastically in one dimension is a (A) straight line (B) parabola (C) rectangle (D) circle

Complex analysis

The Poison bracket [x, xp, + yP, I is equal to (A) -x (B) y (C) 2Px (D) Py

Lagrangian & Hamiltonian Dynamics

The wavefunction of which orbital is spherically symmetric: (A) Px (B) Py (C) s (D) d

Complex analysis

The contour integral O- de 1+2 evaluated along a contour going from -oo to too along the real axis and closed in the lower half-plane by a half circle is equal to _ _ (up to two decimal places).

Complex analysis

The Compton wavelength of a proton is _fm. (up to two decimal places). (mp =1.67x10-27 kg, h = 6.626x10-34Js, e = 1.602×10-19C, c= 3x108 ms~')

Lagrangian & Hamiltonian Dynamics

Which one of the following conservation laws is violated in the decay t* → put m (A) Angular momentum (B) Total Lepton number (C) Electric charge (D) Tau number

Complex analysis

Electromagnetic interactions are : (A) C conserving (B) C non-conserving but CP conserving (C) CP non-conserving but CPT conserving (D) CPT non-conserving

Lagrangian & Hamiltonian Dynamics

A one dimensional simple harmonic oscillator with Hamiltonian Ho = + -ka is subjected to a small perturbation, H, = ax + Bx* +yx* The first order correction to the ground state energy is dependent on (A) only B (B) a and y (C) a and B (D) only y

Linear Algebra & Vector Calculus

For the Hamiltonian H = a I +b. where a, € R, b is a real vector, Z is the 2x2 identity matrix, and ® are the Pauli matrices, the ground state energy is (A) | b| (B) 2ao-|b| (C) ao-b| (D) do

Complex analysis

The coefficient of elkx in the Fourier expansion of u(x) = Asin? (ax) for k = -20 is (A) A/4 (B) -A/4 (C) A/2 (D) -A/2

Schrodinger equation & Wave Mechanics

The degeneracy of the third energy level of a 3-dimensional isotropic quantum harmonic oscillator (A) 6 (B) 12 (C) 8 (D) 10

Schrodinger equation & Wave Mechanics

The electronic ground state energy of the Hydrogen atom is -13.6 eV. The highest possible electronic energy eigenstate has an energy equal to (A) 0 (B) leV (C) +13.6 eV (D) 00

Statistical Ensembles & Partition Functions

A reversible Carnot engine is operated between temperatures I, and I, (I, > [,) with a photon gas as the working substance. The efficiency of the engine is (A) 1-- 4T2 (B) 1- T, 2 3/4

Particle Physics & Fundamental Interactions

In the nuclear reaction 13C6 +Ve → 13N, + X , the particle X is (A) an electron (B) an anti-electron (C) a muon (D) a pion

Electrostatics and Magnetostatics

Three charges (2 C, -1 C, -1 C) are placed at the vertices of an equilateral triangle of side 1m as shown in the figure. The component of the electric dipole moment about the marked origin along the y direction is C m. 2C 3 - 1C/ -1.5 m.. -10 I

Electrostatics and Magnetostatics

An infinite solenoid carries a time varying current I(t) = At* with A # 0. The axis of the solenoid is along the 2 direction. 7 and are the usual radial and polar directions in cylindrical polar coordinates. B= B, 7 + Bo 0 + B, Z is the magnetic field at a point outside the solenoid. Which one of the following statements is true? (A) B, = 0, Ba = 0, B, =0 (B) B, # 0, Bg # 0, B, =0 (C) B, # 0, Ba # 0, B, # 0 (D) B, = 0, B, =0, B, # 0

Complex analysis

A uniform volume charge density is placed inside a conductor (with resistivity 10-22m). The charge density becomes 1/(2.718) of its original value after time _ femto seconds. (up to two decimal places) (& = 8.854x10-12 F/m)

Statistical Ensembles & Partition Functions

Water freezes at 0° C at atmospheric pressure (1.01x105 Pa). The densities of water and ice at this temperature and pressure are 1000 kg/m and 934 kg/m' respectively. The latent heat of fusion is 3.34x10' J/kg. The pressure required for depressing the melting temperature of ice by 10° C is GPa. (up to two decimal places)

Digital Electronics

The minimum number of NAND gates required to construct an OR gate is: (A) 2 (B) 4 (C) 5 (D) 3

Statistical Ensembles & Partition Functions

Consider a 2-dimensional electron gas with a density of 1019 m?. The Fermi energy of the system 1S _ eV (up to two decimal places). (m. =9.31x10-31 kg, h = 6.626x10-34Js, e = 1.602x10-1°C)

Complex analysis

The total energy of an inert-gas crystal is given by E(R) = - po (in eV), where R is the inter-atomic spacing in Angstroms. The equilibrium separation between the atoms is Angstroms. (up to two decimal places).

Statistical Ensembles & Partition Functions

Consider N non-interacting, distinguishable particles in a two-level system at temperature T. The energies of the levels are 0 and &, where & > 0. In the high temperature limit (kg T » 8), what is the population of particles in the level with energy & ? (B) N (A) (C) N 4 (D) 3N 4

Electrostatics and Magnetostatics

A free electron of energy 1 eV is incident upon a one-dimensional finite potential step of height 0.75 eV. The probability of its reflection from the barrier is _ (up to two decimal places).

Electrostatics and Magnetostatics

Consider a one-dimensional potential well of width 3 mm. Using the uncertainty principle (Ar:Ap=½) an estimate of the minimum depth of the well such that it has at least one bound state for an electron is (m = 9.31x10-31 kg, h = 6.626x10-34 Js, e = 1.602×10-19 C): (A) 1 neV (B) 1 meV (C) 1 eV (D) 1 MeV

Complex analysis

Consider a metal with free electron density of 6x1022 cm-3 . The lowest frequency electromagnetic radiation to which this metal is transparent is 1.38x101 Hz. If this metal had a free electron density of 1.8x1023 cm? instead, the lowest frequency electromagnetic radiation to which it would be transparent is _ × 1016 Hz. (up to two decimal places).

Complex analysis

An object travels along the x- direction with velocity c / 2 in a frame O. An observer in a frame O' sees the same object travelling with velocity c / 4. The relative velocity of O' with respect to O in units of cis \cdot (up to two decimal places).

Complex analysis

ite irise fare do is equal ra. _. (up to two decimal places).

Complex analysis

The imaginary part of an analytic complex function is v(x, y) = 2xy + 3y. The real part of the function is zero at the origin. The value of the real part of the function at 1 + i is _. (up to two decimal places).

Linear Algebra & Vector Calculus

Let X be a column vector of dimension n > 1 with at least one non-zero entry. The number of non-zero eigenvalues of the matrix M = XX* is (A) 0 (B) n (C) 1 (D) n-1

Nuclear Models & Decays

JP for the ground state of the 13C% nucleus is (A) 1+ 3- (B) - 2 3+ (C) 2 1- (D) 2

Complex analysis

A uniform solid cylinder is released on a horizontal surface with speed 5 m/s without any rotation (slipping without rolling). The cylinder eventually starts rolling without slipping. If the mass and radius of the cylinder are 10 gm and 1 cm respectively, the final linear velocity of the cylinder is m/s. (up to two decimal places).

Statistical Ensembles & Partition Functions

The energy density and pressure of a photon gas are given by u = aT* and P=u/ 3, where T is the temperature and a is the radiation constant. The entropy per unit volume is given by aaT3 The value of a is \cdot (up to two decimal places).

Complex analysis

Which one of the following gases of diatomic molecules is Raman, infrared, and NMR active? (A) 'H-'H (B) "C-160 (C) 'H-35Cl (D) 160-160

Nuclear Models & Decays

The it decays at rest to M* and V. Assuming the neutrino to be massless, the momentum of the neutrino is MeV/c. (up to two decimal places) (m = 139MeV / c* , M, =105 MeV/ c*).

Lagrangian & Hamiltonian Dynamics

Using Hund's rule, the total angular momentum quantum number J for the electronic ground state of the nitrogen atom is (A) 1/2 (B) 3/2 (C) 0 (D) 1

Schrodinger equation & Wave Mechanics

Which one of the following operators is Hermitian? Correct: 2 Wrong : -0.66 2 2 (C) eiPxa (D) e iPra

Crystal Structure & Lattice Dynamics

The real space primitive lattice vectors are a, = $ anda, - (&+ V5ý) . The reciprocal space unit vectors b, and b, for this lattice are, respectively 4л a (B) - F) and - 2л (C) & and - 4т (- av3 a (D) Â and 47 * - ý) a S

Complex analysis

Consider two particles and two non-degenerate quantum levels 1 and 2. Level 1 always contains a particle. Hence, what is the probability that level 2 also contains a particle for each of the two cases: (i) when the two particles are distinguishable and (ii) when the two particles are bosons? (A) (1) 1/2 and (i1) 1/3 (B) (i) 1/2 and (ii) 1/2 (C) (i) 2/3 and (ii) 1/2 (D) (i) 1 and (ii) 0

Complex analysis

A person weighs W, at Earth's north pole and We at the equator. Treating the Earth as a perfect sphere of radius 6400 km, the value 100xWp - We) / Wais _ _. (up to two decimal places). (Take g = 10 ms3).

Complex analysis

The geometric cross-section of two colliding protons at large energies is very well estimated by the product of the effective sizes of each particle. This is closest to (A) 10 b (B) 10 mb (C) 10 Mb (D) 10 pb

Complex analysis

For the transistor amplifier circuit shown below with R1 = 10 kg, R2=10 kg, R3=1 ks, and B = 99. Neglecting the emitter diode resistance, the input impedance of the amplifier looking into the base for small ac signal is kS. (up to two decimal places). Усс → Vout

Operational Amplifiers & Analog Circuits

Consider an ideal operational amplifier as shown in the figure below with R1 = 5 kg, R2=1 kg, Ri =100 k. For an applied input voltage V = 10 mV, the current passing through R2 is _MA. (up to two decimal places). + V R2

Linear Algebra & Vector Calculus

Consider the differential equation dy / dx + y tan(x) = cos(x). If y(O) =0, y(п/3) is _. (up to two decimal places).

Angular momentum & Hydrogen Atom

Positronium is an atom made of an electron and a position. Given the Bohr radius for the ground state of the Hydrogen atom to be 0.53 Angstroms, the Bohr radius for the ground state of positronium is Angstroms. (up to two decimal places).