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QUESTION: 1

AB : Isochoric heating

BC : adiabatic expansion

CA : isothermal compression

one mole of an ideal gas is taken round the cycle ABCA as in the diagram shown above. What is the efficiency of a heat engine operating on this cycle?

Solution:

Process CA is isothermal compression and hence heat released in this process is given by

further heat absorbed in isochoric heating process AB is given by Q_{2}=c_{v}(T_{A}-T_{B}). Where c_{v} is specific heat capacity at constant volume.

for the adiabatic process

isotherm al process]

QUESTION: 2

The shortest wavelength in lyman series of hydrogen spectra is 91.2 nm, the longest wavelength in this series must be

Solution:

When electron moving in n_{i}^{th} orbit transited to n_{f}^{th} orbit the frequency of radiation and so wavelength is given by

Thus Lyman series is given as

QUESTION: 3

The relationship between the angular frequency co and the wave number k of a particle associated with it is where c is constant. The group velocity for the particle wave is

Solution:

The wave equation y= a sin(ωt - kx)

phase velocity = ω/k

Group velocity = dω/dk

We have ω= ck^{2}

QUESTION: 4

If ε_{0} is the zero-point energy of a harmonic oscillator of frequency v and h is planck's constant then its energy in the n = 5 state will be

Solution:

The energy of it of nth quantum is proportional to (n + 1/2)

zero point energy

Then, the energy for n = 5 is

QUESTION: 5

A particle is moving with one component of constant velocity parallel to the axis of y and another component of velocity parallel to the axis of x proportional to y. It will describe a

Solution:

When a body is moving parallel to the axis of y and moving parallel to x-axis then it's intersecting point is like a parabola.

QUESTION: 6

A chain is used to lower a block of mass m through a distance h at a constant acceleration of g/5. What is the work done by the chain on the block?

Solution:

Work done by the chain = mgh/5

Real work done by the chain = mgh

work done by the chain on the block is

QUESTION: 7

An ideal gas with the adiabatic exponent γ goes through a process P = P_{0} - ∝V, where P_{0} and ∝ are positive constants, and V is volume. At what volume will the gas entropy have the maximum value?

Solution:

Let (P_{1}V_{1}) is point on line

Let (P, V) be any other point

So, entropy difference

for extremum of ΔS

∴

QUESTION: 8

An adiabatic vessel contains 2 kg of water at 25^{0}C by paddle-wheel work transfer, the temperature of water is increased to 30^{0}C. If the specific heat of water is assumed constant at 4.187 KJ/kg.K, the entropy change of the universe is

Solution:

QUESTION: 9

A mass of 8 kg gas expands within a flexible container so that the p-v relationship is of the form constant. The initial pressure is 1000 kPa and the initial volume is 1 m^{3}. The final pressure is 5 kPa. If specific internal energy of gas decreases by 40 KJ/kg, the heat transfer in magnitude is

Solution:

QUESTION: 10

The eiqen values and eigen vectors of the matrix are

Solution:

The characteristic equation is

λ = 1,6

Eigen vector corresponding to the eigen value λ = 1 by putting λ = 1 ,the matrix becomes

⇒ X = K, V= -K

Eigen vector x_{1} is

Now again Eigen vector corresponding to the eigenvalue λ = 6, By putting λ = 6

QUESTION: 11

Consider a cylinder of height h and radius a, closed at both ends, centered at the origin. Let the position vector and unit vector normal to the surface. The surface integral over the closed surface of the cylinder is

Solution:

For calculation of surface integral ,we need surface normal on the top surface of the cylinder shown in adjacent figure

Where is the given position vector. Thus

top surface

(on top surface)

On the bottom surface of the cylinder

vector is normal to the current surface of the cylinder and therefore for this surface we have

Thus a(area of curved surface)

The value of are the whole surface of cylinder is then

QUESTION: 12

The normalized ground state wave function of a hydrogen atom is given by where 'a' is the bohr radius and r is the distance of the electron from the nucleus located at the origin. The expectation value <1/r^{2}> is

Solution:

Expectation value of 1/r^{2} is given by

QUESTION: 13

For a particle of mass m, in a one-dimensional harmonic oscillation potential of the form the first excited energy eigen state wave function is The value of a is

Solution:

The energy eigen value for harmonic oscillation is

Applying Schrodinger equation

QUESTION: 14

Electric field component of an electromagnetic radiation varies with time as E = a [cosω_{0}t + sinωt cosω_{o}t),where a is a constant and the values of the ω and ω_{o} are 2 x 10^{15} sec^{-1} and 1 x10^{16} sec^{-1} respectively. This radiation falls on a metal of work function 4.4eV. The maximum kinetic energy (in eV) of photoelectrons is

Solution:

The maximum frequency of photon =

QUESTION: 15

Light takes 4 hours to travel the distance from sun to nepton. lf you travel in a spaceship at a speed 0.935 c(where c is the speed of light in vacumm), the time(in minutes) required to cover the same distance measured with a clock on the spaceship will be approximately

Solution:

Clock in spaceship will measure proper time which will be dilated to 240 minutes so,let proper time be

QUESTION: 16

A proton and an α-particle. accelerated through the same potential difference, enter a region of uniform magnetic field normally. If the radius of the proton orbit is 10 cm that of the α-orbit , then radius of proton orbit is

Solution:

QUESTION: 17

A long straight conductor, carrying a current I is bent into the shape shown in the figure. The radius of the circular loop is V . The magnetic field at the centre of the loop is:-

Solution:

Field due to straight part out of the page

Field due to circular part into the page.

Not field into the page.

QUESTION: 18

A charged particle of mass'm',charge 'q' and constant velocity enters a uniform magnetic field at an angle θ to the direction of magnetic field. The angle 'θ' (if in one revolution of the helical motion, the particle advances along the direction of the magnetic field at a distance equal to the radius of the helical path) is:-

Solution:

QUESTION: 19

Two blocks A and B each of mass m, are connected by a massless spring of natural length L and spring constant k. The blocks are initially resting on a smooth horizontal floor with the spring at its natural length, as shown in figure. A third identical block C, also of mass m. moves on the floor with a speed v along the line joining A and B, and collides elastically with A. Then

Solution:

After collision between C and A, C stope and A is moved with speed C' i.e. v. At maximum compression A and B will move with same speed V/2. Let x is maximum compression in this position k.e. of A-B system at maximum compression

or

QUESTION: 20

A stone of mass 5 kg is placed at a distance 5m from the axis of rotaion on the rotating table ( angular velocity w with respect to earth) What would be the angular velocity w so that the stone starts slipping? {The frictional coefficient between stone and table is 1.5 }

Solution:

If the stone starts slipping, then the centripetl force will be equal to the frictional force.

QUESTION: 21

The output of the given circuit if the input is a sinusoidal wave given below (Given V_{m}>V), is _____.

Solution:

For forward biased:-D always act as a short circuit (then output wavefrom

For Reverse biased:-D will be short for V_{in} < V & open for V_{in }> V thus output wavefrom is:-

complete output wave from is→

QUESTION: 22

The concentration of donor atoms is given for a semiconductor specimen (2.54cm x 0.2cm x 0.2cm). as N_{D }The ratio of resistance taken along the longer ends with that of taken along the shorter? (Given N_{D} = 10^{16} cm^{3}' μ = 1200).

Solution:

QUESTION: 23

In a typical n-p-n transistor,the doping concentrations in emitter,base & collector regions are C_{E},C_{B }& C_{c }respectively. These satisfy the relation:-

Solution:

In an n-p-n transistor emitter is heavily doped collector is moderately doped & base is very thin & least doped i.e.

C_{E}>C_{C}>C_{B}

QUESTION: 24

In a compound XY_{2}O_{4}, oxide ions are arranged in fee and cations 'Y' are present in octahedral voids, cations T are equally distributed between octahedral & tetrahedral voids. The fraction of the octahedral voids occupied is:-

Solution:

O^{-2 } ions are arranged in ccp so number of unit cell = 4

X are present in oh voids,(Total no. of unit = 4)

Fraction of oh. voids (occupied) = 1/4

Y present ⇒

Fraction o f octahedral voids (occupied = 1/4)

Total fraction of octahedral voids (occupied)

= 1/2

QUESTION: 25

The Boolean expression [B.(A + B) +can be realized using minimum no of

Solution:

QUESTION: 26

Consider a beam of light of wavelength 'λ' incident on a system of a polarizer & an analyzer. The analyzer is oriented at 45^{0} to the polarizer. When an optical component is introduced between them, the output intensity becomes zero. (Light is incident normally on all components).The optical component is:-

Solution:

When half-wave plate is introduced, the plane of polarization is rotated by Since intensity o f light analyzer is zero, plane o f polarization rotates by an angle 45^{0}.

QUESTION: 27

A combination of two thin convex lens of equal focal length is kept separated along the optic axes by a distance of 20 cm between them. The combination behaves as a lens system of infinite focal length. If an object is kept at 10 cm from the first lens, its image will be formed on the other side at a distance x from the second lens. The value of x is:-

Solution:

for first lens;

QUESTION: 28

The black body spectrum of an object P is radiant intensity at maximum wavelength of 100 nm. Another object Q has the maximum radiant intensity at 400 nm. What will be the ratio of power emitted per unit area by P to that of Q?

Solution:

From wein’s displacement law

QUESTION: 29

A boat is travelling in river with a speed 10m/sec along the stream flowing with a speed 2m/sec. From this boat ,a sound transmitter is lowered into the river through a rigid support .The wavelength of the sound emitter from the transmitter inside the water and air is negligible.The frequency detected by a receiver kept inside the river downstream is:- (given that bulk modulus

Solution:

Velocity of sound in water

frequency of sound in water will be:-

frequency of sound detected by receiver (observer) at rest would be:-

QUESTION: 30

A Narrow beam of light of wavelength 589.3nm from a sodium lamp is incident normally on a diffraction grating of transmission type. If the grating constant is 1000000m^{-1}, the number of principle maxima observed in the transmitted light will be:-

Solution:

Grating constant 10^{6} per metre

Highest order

*Multiple options can be correct

QUESTION: 31

Which of the following statements) regarding photo -emission of electrons is/are incorrect?

Solution:

-Kinetic energy of photoelectrons depends on the frequency of incident light but not on its intensity, for photoelectric effect the frequency of incident light should be greater than threshold frequency.

-In the photoelectric effect, electrons are emitted from matter (metals and non-metallic solids, liquids, or gases) as a consequence of their absorption of energy from electromagnetic radiation of high frequency (short wavelength), such as ultraviolet radiation.

-The **Photoelectric effect** is an **instantaneous** phenomenon. There is no time delay between the **incidence of light** and **emission** of photoelectrons.

*Multiple options can be correct

QUESTION: 32

Consider the following in respect of compton scattering and zeeman effect, which of the following statement(s) is/are incorrect?

Solution:

Compton scattering formula holds less and less rigorous to go towards longer wavelength but it has shorter wavelength.collision between photons is inelastic but it is elastic in nature . when an atom is placed in external magnetic field the spectral lines it emits are split into several polarized components.This effect of magnetic field on the atomic spectral lines is called zeeman effect.

A singlet spectral line viewed normal to the field is split into three plane- polarised components, a central unshift line with the electric vector vibrating parallel to the field and two other lines equally displaced one on either side with electric vector perpendicular to the field.

*Multiple options can be correct

QUESTION: 33

If A body initially at rest ,is acted upon by an external force which is constant in magnitude.The gain in kinetic energy of the body at a given instant

Solution:

Gain in kinetic energy by the particle is given by workdone by it from moving point x_{1}, to x_{2}.Thus , the gain in

Now let force constant is k. Then kinetic energy

Thus K E α (x_{2}-x_{1}) in this relation it is clear that the K.E is directly proportional to the distance covered by it and is independent of time elapsed.

*Multiple options can be correct

QUESTION: 34

Consider the following statement for a particle moving in an elliptic orbit under the influence of a central force. Which of the following statement(s) is/are correct?

Solution:

We have angular momentum,

*Multiple options can be correct

QUESTION: 35

Which of the following statement(s)related toBrownian motion is/are incorrect?

Solution:

To and fro motion of colloidal particles in the solution is called Brownian motion.The size of the particles in colloidal solution is of the order of 104 em.This is a chemical and gravitational phenomenon not electrical phenomenon.Brownian motion is independent of the motion of vessel containing gas Displacement is Brownian motion is given by

Where η is viscosity.Thus .Brownian motion is inversely proportional to η hence Brownian motion increase on decreasing viscosity.

*Multiple options can be correct

QUESTION: 36

A series LCR circuit of a resistance 1 kΩ,an impedance 0.4mH, a capacitance 100pF & an alternating voltage source of 2.5V. Which of the following statement(s) is/are true for the defined circuit?

Solution:

Option(B) is only correct option.

Rms voltage = 2 V

Rms current

Now,at resonance,

Rms voltage a cross "R "= Rl_{rms} = 2 V

Rms voltage across " L" = ω L.I_{rms} = 4V

*Multiple options can be correct

QUESTION: 37

The total charge(Z_{e}) of an atomic nucleus is uniformly distributed within a sphere of radivs 'α'. Which of the following statement is correct?

Solution:

*Multiple options can be correct

QUESTION: 38

For the given op-amp circuit (we have given value as A_{01 }= 200000), choose the correct statement:-

Solution:

are only correct.

The attenuation B, of the feed back circuit

*Multiple options can be correct

QUESTION: 39

Which of the following statement(s) is/are correct for the Electronic devices?

Solution:

are only correct options.

Dynamic curve is the one passing through the different static curves.& the curve which relates the o/p voltage to the input voltage is known as transfer curve.

As for Si & Ge:- ec. & their reverse saturation current approximately doubles for every 10^{0}C rise in temp.

*Multiple options can be correct

QUESTION: 40

The instantaneous position of x(t) of a small block performing one-dimensional damped oscillation is Here ω is the angular frequency 'r' the damping coefficient, 'A' the initial amplitude & a the initial phase. If Which of the following statement(s) is/are correct?

Solution:

Thus, option C is the only correct option.

*Answer can only contain numeric values

QUESTION: 41

A rocket of total initial mass mo is expelling exhaust gas at a steady speed 2.88 m /sec with respect to itself, when its mass is reduced to 1/2 m_{0}, the velocity attained by the rocket will be ______ (in m/sec).

Solution:

Formula for rocket is given as

putting in eq.(1) we get

*Answer can only contain numeric values

QUESTION: 42

A certain op - amp has bias currents of 50 μA & 49.3μA,then the input offset current (in nA) is _________.

Solution:

The input offset current is the difference of input bias currents i.e

*Answer can only contain numeric values

QUESTION: 43

A magnetic field induction B = 30 x 10^{-6}T is applied on an electron in a direction perpendicular to its motion. Find the time required for the electron to complete one revolution, (in μs)

Solution:

Time periode (T) for the electron to complete one revolution is

...(i)

*Answer can only contain numeric values

QUESTION: 44

A current of 1 ampere is flowing in the sides of an equilateral triangle of side 4.5 x 10^{-2}m, then the magnetic field at the centroid of the triangle (inμT) is ____.

Solution:

The Resultant magnetic field at '0' due to all sides of the triangle is

*Answer can only contain numeric values

QUESTION: 45

A particle of mass 20 gm is moving in a circle of 4cm radius with contant speed of 10 cm/sec. What is the angular momentum (gm. cm^{2} / sec) about a point on the axis of the circle 3 cm distance from the centre.

Solution:

Angular momentum about a point on the axis of the circle

Since, is perpendocular to the velocity

*Answer can only contain numeric values

QUESTION: 46

A long solenoid of diameter 0.1 m has 2 x 10^{4} turns per meter. At the centre of the solenoid, a 100 turn coil of radius 0.01 m is placed. The current is decreased at a constant rate from +2A to -2A in 0.5sec. The total charge flowing through the coil, if the resistance of the coil is is 10π^{2} Ω, is ____(in μC).

Solution:

Field inside the solenoid (B)=μ_{0}nl

Flux linked with the coil

Induced emf(e) =

charge flowing through the coil

q = 32μc

*Answer can only contain numeric values

QUESTION: 47

The dc voltage(v_{c}) with respect to ground in the amplifier circuit shown in the figure is _________,( If the current gain β for the transistor is 200.(V_{BE}= 0.7V).

Solution:

Thevenin equivalent of the given circuit:

*Answer can only contain numeric values

QUESTION: 48

A spherical piece of radius much less than the radius of a charged spherical shell (change density σ = 1 c/m^{3}) is removed from the shell itself than calculate the electric field intensity at the midpoint of the aperture?

(in the order of 10^{9})

Solution:

The aperfure midpoint will act like the midpoint of a capacitor, thus

*Answer can only contain numeric values

QUESTION: 49

Two coherent plane waves of light of equal amplitude & each of wavelength 20π x 10^{-6}m, propogating at an angle of π/1080 radius with respect to each other, fall almost normally on a screen. The fringe width(in mm) on the screen is ____.

Solution:

( θ is very small)

θ= d/D

Fringe - width :-

*Answer can only contain numeric values

QUESTION: 50

A young's double slit apparatus has a slit separation of 0.5mm & a screen placed at a distance of 100 cm from the slits. When illuminated by a 632nm Helium-neon laser, the position of the first intensity minimum with respect to the central maximum will be____(in μm).

Solution:

Slit separation (d) =0.5mm = 5x10^{-4}m

Distance b/w slit & screen(D) = 100 cm = 1 m

λ = 632nm

position of minima

first intensity minima corresponds to n =0;

*Answer can only contain numeric values

QUESTION: 51

An n-type silicon rod of length 0.7 cm has a cross sectional area of 0.1cm^{2}. A DC bias voltage of 35v across on the rod results in a 5.6 ampere DC current. Under this DC bias, an electric pulse applied at one end takes 10μ sec to traverse the length of silicon rod, then the mobility of the charge carrier is ____.

Solution:

*Answer can only contain numeric values

QUESTION: 52

An electron is moving in the positive x-direction, in S' frame. The relative velocity between the S and S' frames is 0.7C and the velocity of the electron is also 0.7C with respect to an observer in S' frame. The velocity of the electron for an observer in the S frame (in term of C) is _____.

Solution:

The relative relativistic velocity is given as

*Answer can only contain numeric values

QUESTION: 53

The electric field intensity at a point on the surface of a conductor is given by E (v/m) what is the surface change density in (Pc/m^{2}) at the point?

Solution:

in case of conducter all charge resides on the sorface .i.e

*Answer can only contain numeric values

QUESTION: 54

If a black body radiation in a spherical cavity of volume V satisfies the relation, PV^{4} = constant during a quasistatic is entropic process, then the numerical value of η should be ____.

Solution:

For black body radiation, the adiabatic equation is PV^{4/3} = constant

*Answer can only contain numeric values

QUESTION: 55

Two bodies of mass 100 kg and 20 kg are moving with velocities and m/sec respectively. The magnitude of velocity of the centre of mass is______(in m/sec).

Solution:

If r_{1} and r_{2} are position vectors of two masses m_{1} and m_{2} then the position vector of their centre of mass R_{cm} is given as

Differentiating w.r.t. time, we get

*Answer can only contain numeric values

QUESTION: 56

If a capacitor of 1μF charged to a potential of 300 V is discharged through a resistor kept at room temperature, then the entropy change of the universe is equal to __________ x 10^{-5} J/K.

Solution:

The energy stored in capacitor = 1/2 CV^{2}

So, change in entropy is given dS =dQ/T

Change in entropy =

*Answer can only contain numeric values

QUESTION: 57

An unstable particle of rest energy 1000 MeV decays into a μ-meson and a neutrino, with a mean life time of 10^{-8} sec, when at rest. The mean decay distance in meters, when the particle has a momentum of 1000 MeV/C is ____.

Solution:

We know that relativistic energy and momentum is related as

E^{2} = (PC)^{2} + (m_{0}C^{2})^{2}

Here we have

P = 1000 MeV/C

PC = 1000 MeV

m_{0}C^{2} = (1000) MeV

Energy

But by Einstein mass-energy relation

Total distance = time x velocity =

*Answer can only contain numeric values

QUESTION: 58

A = is symmetric matrix, then Evaluate x

Solution:

For symmetric matrix A = A^{T}

*Answer can only contain numeric values

QUESTION: 59

A gas occupying 1 litre at 80 cm. pressure is expanded adiabatically to 1190 c.c. I f the pressure falls to 60cm . in the process, Deduce the value of λ.

Solution:

Here

For an adiabatic change

*Answer can only contain numeric values

QUESTION: 60

Mass of empty recket is 5000 kg in which fuel of mass 40000 kg is filled up. If the exhaust velocity is 2 km/sec, then find the maximum velocity (km/sec) achieved by the rocket.

Solution:

(4.3 TO 4.4)

Mass of rocket including fuel M_{0 }= 45000 kg

Mass of rocket without fuel M = 5000 kg

Exhaust velocity U_{r} = 2 km/sec

U_{r} = 2000 m/sec

Maximum velocity achieved by the rocket

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