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This mock test of GATE Physics Mock Test Series - 2 for GATE helps you for every GATE entrance exam.
This contains 60 Multiple Choice Questions for GATE GATE Physics Mock Test Series - 2 (mcq) to study with solutions a complete question bank.
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QUESTION: 1

Consider a beam of plane polarised light of wavelength λ is incident on a optical component making angle 45° to the optical axis and output light is circularly polarised then the optical component is

Solution:

For quarter wave plate

n = 0,1,2, 3.......

When a plane polarised light is incident on quarter wave plate with angle θ= 45° then output light is circularly polarised.

QUESTION: 2

A complex matrix is given as

then

Solution:

So , A^{+} = A

This implies that A is Hermitian matrix and all eigen values of Hermittian matrix are real.

QUESTION: 3

A particle of mass m, moves under the action of a central force whose potential is V(r) = kmr^{3 }(k > 0 ) , then the angular frequency is _________ .

Solution:

For angular frequency, we have to find out equation of motion. We use Taylor's series expansion

V(r) = Kmr^{3}

V_{e} =

So at r = r_{0},

T =

ω = √ 5Ka

QUESTION: 4

A particle of mass m moves in a central force field defined by if E is the total energy supplied to the particle, then its speed is given by .

Solution:

Given F =

so , V(r) =

Now energy, E =

QUESTION: 5

The space - time coordinates of two events as measured by O are x_{1}= 6 * 10^{4} m ,y_{1} = z_{1}= Om, t_{1}’ = 2 * 10^{4}sec and X_{2},= 12 x 10^{4}m, t_{1} = 1 x 10^{-4}sec. What must be the velocity of O’ with respect to O if O’ measures the two events to occur simultaneously ?

Solution:

Subtracting two Lorentz transformation.

On solving , we obtain,

Therefore , v is in the negative x - direction.

QUESTION: 6

The value of the surface integral where s is the surface of the sphere x^{2}+ y^{2} + z^{2} = 4, n is the unit outward normal and

Solution:

On putting x = x = r sinθ cosφ

y = r sinθ sinφ

z = r cosθ, we get

= π x 8

QUESTION: 7

Fourier series which will represents f(x) = x sinx in the interval -π< x< π, then

will have value

Solution:

The given f(x) Function f(x) = xsin x is an even function. Hence series will be

f(x) = x sin x

If n =1

f(x) = x sin x

Divided each side by 2

QUESTION: 8

A particle moving in a central force located at r = 0 describes the spiral r = e^{-θ}, the magnitude of force is inversely proportional to

Solution:

r = e^{-θ}

Equation of orbit

QUESTION: 9

Rest mass energy of an electron is 0.51 MeV. A moving electron has a kinetic energy of 9.69 MeV. The ratio of the mass of the moving electron to its rest mass is

Solution:

E = mc^{2 }- m_{0}c^{2} ⇒ E =

9.69 =

10.2 =

m =

QUESTION: 10

A system of four particles is x - y plane. Of these, two particles each of mass m are located at (-1, 1) and (1, -1). The remaining two particles each of mass 2m are located at (1, 1) and (-1, 1). The xy component of the moment of inertia tensor of this system of particles is -

Solution:

The xy component of the moment of inertia tensor

Two particles of mass m located at (-1,1) and (1, -1) other particles of mass 2m are located at (1, 1) and (-1, 1)

W = - [m_{1}x_{1}y_{1} + m_{2}x_{2}y_{2} + m_{3}X_{3}y_{3} + m_{4}x_{4}y_{4}]

= - [(mx -1 x 1 ) + (m x 1 x -1 ) + (2m x 1x1) + (2m x 1x-1)]

= - [-m -m + 2m - 2m]

= [2m] ⇒ I_{xy} = 2m

QUESTION: 11

What is the energy of a uniformly charged spherical shell of total charge q and radius R.

Solution:

The self energy stored in the system

Now the potential at the surface of the sphere is

QUESTION: 12

A current I is uniformly distributed over a wire of circular cross section, with radius a, suppose the current density in the wire is proportional to the distance from the axis J = Kx (K is constant)

Find the total current in the wire

Solution:

The area - perpendicular to flow is πa^{2}, so

J =

J varies with x. Then the current I =

QUESTION: 13

For a Gaussian wave packet described by The expectations value of the momentum operator is

Solution:

Expectation value of the momentum operator is

< p > = 0

QUESTION: 14

The wave function of a particle is given by ψ= c exp (-x^{2}α^{2} ) , co where c and a are constants. The probability of finding the particle in the region

Solution:

The probability of finding the particle in the region 0 < x < is

Now using the normalization condition

QUESTION: 15

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: 16

The solution of the differential equation for subject to the initial condition y(O) = 0 and ,is

Solution:

The differential equation is ,

complementary function (C.F) =

(D^{2} - 1 ) y = 0 , so D = ±1

C.F. =

Now particular integral PI =

Complete solution of given differential equation is

Apply boundary conditions, y(0) and

At t = 0 , y = 0 , we have y(0) = A + B

At t = 0 , we have

from those equations A = 0 , B = 0

y(t) = t sin ht

QUESTION: 17

A nozzle in the shape of a trun cated cone, as shown in the figure. The area at the wide end is 25 cm^{2} and and the narrow end has an area of 1 cm^{2}, water enters the wider end at a rate of 500 gm/sec the height of the nozzle is 50 cm and it is kept vertical with the wider end at the bottom. The magnitude of the pressure difference in k Pa between the two ends of the nozzle is

Solution:

According to Bernoulli’s equation

Now given that θ = 500 gm / sec

According to equation of continuity

QUESTION: 18

What is the magnitude of the linear momentum of a photon of radiation having electric field described by e =

Solution:

The equation for the electric field of an electro- magnetic wave can be written as

QUESTION: 19

Given that the magnetic flux through the closed loop PQRSP is along PQR, the value of along PSR is

Solution:

Consider the closed loop PQRS as shown in the figure

It is given that

QUESTION: 20

Two infinitely extended homogeneous isotropic dielectric media (medium - 1 and medium - 2 with dielectric constant respectively) meet at the z = 0 plane as shown in the figure. A uniform electric field is given by The interface separating the two media is charge free. The electric displacement vector in the medium 2 is given by

Solution:

Given that

And the interface separating of the two media is charge free.

Since, there is no free charge at the interface. So, normal component of is continuous at the boundary

QUESTION: 21

Which of the following truth table for the given logic is true ?

Solution:

At point X1

At X_{2} output is X_{2} = X + Y (OR GATE)

So, Z = X_{1} +X_{2}(OR GATE)

QUESTION: 22

Magnetic flux linked with a stationary loop of resistance R varies with respect to time during the time period T as follows:

Find the amount of heat generated in the loop during that time. Inductance of the coil is negligible

Solution:

Given that = at(T - 1)

Induced e.m.f.e =

= dt (0 - 1) + a (T - 1)

= a (T - 2t)

So induced e.m.f. is also a function of time.

∴ Heat enerated in time T is

QUESTION: 23

At t = 0, a particle of mass m having v_{0} starts moving through a liquid kept in a horizontal tube and experiences a drag force It covers a distance L before coming to rest. If the times taken to cover the distances are t_{2} and t_{4}, respectively, then

Solution:

By Applying the second law of motion

The characterstic equation is

General solution

using the condition

C_{1 }+ c_{2} = 0

using the condition V(0) = V_{0}

using equations → x(t)

from the question

QUESTION: 24

The simplified logic expression for the following logic diagram is

Solution:

QUESTION: 25

For the logic circuit shown in the figure, the output y is given by

Solution:

So, y =

By dinorgen theorem

QUESTION: 26

An open pipe is suddenly closed at one end with the result that the frequency of third harmonic of the closed pipe is found to be higher by 200 H_{2} then the fundamental frequency of the open pipe - The fundamental frequency of the open pipe is

Solution:

Length of the organ pipe is same in both the cases, fundamental frequency of open pipe is and frequency of third harmonic of closed pipe will be

Given that f_{2} = f_{1}+ 200

QUESTION: 27

A paint particle of mass M attached to one end of a massless rigid non - conducting rod of length L. Another paint particle of the same mass is attached to the other end of the rod. The two particles carry changes + q and -q respectively. This arrangement is held in a region of a uniform electric field E such that the rod makes a small angle with the field direction. Find the minimum time needed for the rod to become parallel to the field after it is set free

Solution:

A torque will act on the rod, which tries to align the rod in the direction of electric field. This torque will be of restoring nature and has a magnitude PE sinθ. Therefore

since θ is very small so that sin θ Sin θ ≈ θ

As α is proportional to - θ, motion of the rod is s impleharmonic in natural time period of which is given by

the desired time will be

QUESTION: 28

Which of the following statement regarding the electric fields is correct

Solution:

Only field can represent electrostatic field..

QUESTION: 29

A system of four identical distinguishable particles has energy The single particle states are available at energies Find the average number n(E) of particles in energy e =

Solution:

The possible distribution which can give a total energy are shown in table

QUESTION: 30

Consider 10 atoms fixed at lattices sites. Each atom can have magnetic moment in the z - direction. Let f (μ) denote the probability that the magnetic moment of the system is μ. Assuming statistical mechanics to hold. Find the value of

Solution:

The magnetic moment can be if 6 atoms in state and 4 in state. This can happen in n

for magnetic moment to be zero , five atoms should be in state and five in state. This can happen in n_{2} ways where

The probability will be proportional to the no. of microstates. Hence

*Multiple options can be correct

QUESTION: 31

A solid sphere is in pure rolling motion on an inclined surface having inclination θ :

Solution:

In case of Pure rolling

Therefore, as θ decreases force friction will decrease.

*Multiple options can be correct

QUESTION: 32

The linear mass density of a rod of length L varies from one end to the other aswhere x is the distance from one end with tensions T_{1 }and T_{2} in them (see figure) and λ_{0} is a constant. The rod is suspended from a celling by two massless strings. Then, which of the following statements is/are correct

Solution:

The mass of rod is

The Centre of gravity

Force equation T_{1} + T_{2} = torque equation

Putting the value of T_{2} = in equation

*Multiple options can be correct

QUESTION: 33

Let m_{p }be the mass of proton. m_{n} the mass of neutron. M_{1} the mass of nucleus and M_{2} the mass of nucleus. Then :

Solution:

Due to mass defect (which is responsible for the binding energy of the nucleus), mass of nucleus is always less than the sum of masses of its constituent particles.

is made up to 10 protons + 10 neutrons. Therefore, mass of nucleus

M_{1}< 10 (m_{p} + m_{n})

Also, heavier the nucleus, more is the mars defect.

Thus 20 (m_{n}+m_{p})-M_{2} > 10 (m_{p} +m_{n}) - M_{1}

10(m_{p} + m_{n}) > M_{2} - M_{1}

M_{2}<m_{1} +10(m_{n} + m_{p})

Now since M_{1} < 10 (m_{p} + m_{n})

M_{2} < 2m_{1}

*Multiple options can be correct

QUESTION: 34

Consider the following statement regarding radioactive element emitting β - particle :

Which of the following are correct ?

Solution:

We radioactive element emits β- particle then the atomic number of the element increases by one unit where as the mass number of the element remain same : Hence, the equation for β- decay can be written as

Since in β -decay, a neutron is contributed into a proton, the neutron proton ratio decrease.

*Multiple options can be correct

QUESTION: 35

The graph between 1/λ and stopping potential (v) of three metals having work function in an experiment of photoelectric effect is plotted as shown in the figure. Which of the following statement is / are correct ?

Solution:

From the relation eV =

This is equation of straight line

slope is tanθ =

Violet colour has wavelength 4000A^{0}.

So, Violet colour can eject photoelectrons from metal -1 and metal - 2.

*Multiple options can be correct

QUESTION: 36

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

Solution:

Field due to straight part(B_{1}) = out of the page

Field due to circular part (B_{2}) = into the page.

Not field (B) =B_{1}-B_{2} into the page.

*Multiple options can be correct

QUESTION: 37

The doping of a semiconductor is such that the electron density in the conduction band is given by n = (x) = n_{i} (1 +gx). Find incorrect expression/s for the electric field intensity in the conduction band.

Solution:

Due to density gradient electrons would diffuse in conduction band and due to electric field intensity holes drift in a direction opposite to that of electron diffusion. At equilibrium

or

or

*Multiple options can be correct

QUESTION: 38

A simple pendulum performs simple harmonic motion about x = 0 with an amplitude A and time period T. The speed of the pendulum at x = A/2 won’t be:

Solution:

*Multiple options can be correct

QUESTION: 39

A screen is placed 50 m from a single slit, which is illuminated with 6000 light. If distance between the first and third minima in the diffraction pattern is 3,000 mm, which of the following is/are not the width of the slit?

Solution:

In case of diffraction at single slit, the position of minima is given by

d sin θ= nλ

and for small θ sin θ = θ =

so from eq.(1) and (2), we have

so that,

and hence,

*Multiple options can be correct

QUESTION: 40

An unpolarized beam of light is incident on a group of four polarizing sheets which are arranged in such a way that the characteristic direction of each polarizing sheet makes an angle of 30° with the preceding sheet. What fraction of light is/are not transmitted?

Solution:

Here

∴

∴

QUESTION: 41

Calculate the change in the melting point of wax for a pressure of 50 atmospheres from the following data :

melting point = 64°C, Volume at 0°C = 1 cc; volume of the solid at the melting point = 1.161cc; volume of the liquid at the melting point = 1.166 c.c.; density of the solid at 64° C = 0.96 gm. / c.c latent heat = 97 cal. gm.

Solution:

0. 016 - 0.020

Here the melting point (64°C) is given to be at one atmosphere pressure and we are required to find the change in the melting point for a pressure of 50 atmosphere, i.e for a pressure change of (50 - 1) = 49 atmosphere.

dP = 49 x 76 x 13.6 x 981 dynes / cm^{2}

T = 64°C = 64 + 273 = 337 K,

L = 97 cal. = 97 x 4.2 x 10^{7} ergs.

Now mass of the solid at melting point = volume x density

= 1.161 x 0.96 gm.

∴ Specific volume of the liquid at melting point

and specific volume of the liquid at melting point

V_{2} - V_{1} = 1.0461 - 1.0417 = 0.0044 c.c. / gm.

Now Clausius Clapeyron equation is

So, that = 0.018 K = 0.018°C

QUESTION: 42

A bead of mass 0.1 kg solids at rest from x (which is 4m above the ground) along a frictionless wire as shown in the figure given above, (Take g = 1Om/s^{2}) when at B, the wire exerts a force on the bead equal to ______ N.

Solution:

Applying the conservation law of energy

Total energy at A = Total energy at B

(KE)_{A} + (PE)_{A} = (KE)_{B }+ (PE)_{B}

mu^{2} = 4mg

Now let R' be reaction of ring on the bead directed upwardly as shown in figure. Then centripetal force = mg - R'

R' = mg - mu^{2} (R= 1m)

R' = mg -4mg

R' = -3m g ⇒ R' = - 3 x (0.1)x 10

R' = - 3N

|R' |= 3N

QUESTION: 43

10 gm. of steam at 100°C is blown into 90 gms. of water at 0°C contained in a calorimeter of water equivalent 10 gms. The whole of the steam is condensed. Calculate the increase in the entropy of the system.

Solution:

3.6 - 3.7

Given m_{1} = 10 gm., T_{1} = 100°C = 373 K

m_{2} = 90 + 10 = 100 gm., T_{2} = 0°C = 273 K.

Let the final temperature be T K.

∴ 10 x 540 + 10 ( 373 - T) = 100 (T - 273)

T= 331.2 K.

Now (i) Change in entropy when temperature of water and calorimeter increases from 273 K t o 331.2 K.

(ii) Change in entropy when 10 gms. steam at 373 K condenses to water at the same temperature

where -ve sign indicates that the entypy decreases.

(iii) Change in entropy when 10 gm. water is cooled from 373 K to 331.2 K.

or ΔS_{3 }= - 1.188 cal/K

The net change (or increase in entropy)

ΔS = ΔS_{1} + ΔS_{2} + ΔS_{3} = + 19.324 - 14.477 - 1.188 = + 3.659 cals./K.

QUESTION: 44

The sum of the eigen value of the matrix

Solution:

Let A =

cos^{2}θ- 2λcosθ+ λ^{2} + sin^{2} θ = 0

λ^{2} - 2λcosθ+ 1 = 0

at θ = 60°

λ = cos 60 ± sin60

The sum of the eigen value is

= 2 cos 60 = 1

QUESTION: 45

A police car moving at 22 m/s chases a motorcyclist. The police man sounds his horn at 176 Hz, while both of them move towards a stationary siren of frequency 165 Hz. Calculate the speed of the motorcycle. If it is given that the motorcyclist does not observe any beats (speed of sound = 330 m/s)

Solution:

The motorcyclist observes no beats. So, the apparent frequency observed by him from the two sources must be equal.

Solving this equation, we get

v = 22m/s

QUESTION: 46

The period of a disk of radius 10.2 cm executing small oscillation about a pivot at its rim is measured to be _____ sec. { The value o f the all acceleration due to gravity at that location is 9.83 m/sec^{2}}

Solution:

0.6-0.8

The rotational inertia of a disk about an axis through its centre is where R is the radius and M is the mass of the disk. The rotational inertia about the pivot at the rim is, using the parallel axis theorem

The period of this physical pendulum T =

With d = R, is then

independent of the mass of the disk The simple pendulum having the same period has a length

QUESTION: 47

2kg of ice at - 20°C in an insulating vessel having a negligible heat capacity.The final mass of water remaining in the container i s ------kg. It is given that the specific heat of water and ice are 1KCal/kg/°C and 0.5 kcal/kg/°C while the latent heat of fusion of ice is 80 k cal/kg.

Solution:

Heat released by 5 kg of water when its temperature falls from 20°C to 0°C is

Q_{1} = mcΔT = 5x10^{3} (20 - 0) = 10^{5}cal

when 2 kg ice at - 20°C comes to a temperature of 0°C, if takes an energy

Q^{2} = mcΔT = 2 x 500 x 20 = 0.2 x 10^{5}cal

The remaining heat

Q = Q_{1} - Q_{2} = 0.8 x 10^{5 }cal will melt a mass m of the ice, where

So, the temperature of the mixture will be 0°C , mass of water in it is 5+1 = 6kg

QUESTION: 48

A particle of mass m is placed in the ground state of a one- dimensional harmonic oscillator potential of the form

where the stiffness constant K can be varied externally. The ground state wavefunction has the form

where a is a constant. If suddenly, the parameter k is changed to 4K, the probability that the particle will remain in the ground state of new potential is _____ .

Solution:

where A is normalisation constant

when parameter K is suddenly changed to 4K, the wave function associated with ground state of new potential is given by

Probability that particle will be found in ground state p =

Put

From Normalisation equation

Let

Probability P =

QUESTION: 49

The given figure shows a silicon transistor connected as a common emitter amplifier. Calculate the approximate value of quiescent collectorvoltage of circuit.

Solution:

QUESTION: 50

What will be the input independence Z_{i} for the network shown below ?

Solution:

∴

QUESTION: 51

Find out the minimum number of NAND gates required to implement A + AB + ABC.

Solution:

Hence, number of NAND gates required is 0.

QUESTION: 52

For the given circuit calculate the output current. Where R_{1} = 6 kΩ, R_{f} = 24 kΩ, V_{i}= 1 V and load register of 6 kΩ.

Solution:

This is the relation for a non inverting amplifier

and load current can be calculated using

We know that current I_{1} and l_{L} both are passing away from the output terminal

so

but

So l_{0} = 0.17 mA + 0.83 mA = 1 mA and this is flowing towards the output junction.

QUESTION: 53

Calculate the total output voltage for a differential amplifier in which the signal applied to inverting and non inverting terminals are respectively = -0.45 mV and -0.48 mV and A_{diff} =10^{6} and CMRR = 80 dB.

Solution:

CMRR=

So

The differential input is

V_{diff} = V_{2} - V_{1} = - 0.48 - (0.45) mV = -.03 mV and common mode input is

Total output voltage

≈ -30 volt

QUESTION: 54

Two spherical nuclei have mass number 256 and 4 with their radii R_{1} and R_{2} respectively then find out the ratio

Solution:

∵

Given A_{1} = 254

A_{2} = 4

∵

So,

QUESTION: 55

Image of an object approaching a convex mirror of radius 20 m along its optical axis is observed to move from in 20 sec. The speed (km/hr) of the object is ______

Solution:

4.5

Using mirror formula.

and

speed of object =

speed of object =

QUESTION: 56

A ball of mass m , initially at rest , is dropped from a height of 5 meters. If the coefficient of restitution is 0.9, the speed of the ball just before it hits the floor the second time is approximately______ m/sec. (take g = 10 m/sec^{2})

Solution:

9

Given that height = 5m

g = 10 m/sec^{2}

Then the velocity of ball before hitting the surface first time =

coefficient of restitution = 0.9

So, after collision its velocity = 0.9 x 10 = 9 m/sec

So, before hits the floor second time velocity in = 9 m/sec

speed of object =

QUESTION: 57

AB and CD are two slabs. The medium between the slabs has refractive index 2. Find the minimum angle (In degree) of incidence of Q, so that the ray is totally reflected by both the slabs -

Solution:

60

Critical angles at 1 and 2

Therefore, minimum angle of incidence for total internal reflection to take place on both slabs should be 60º

i_{min} = 60º

QUESTION: 58

Screen s is illuminated by two point sources A and B. Another source C sends a parallel beam of light towards point P on the screen and the lines AP, BP and C P are in one plane. The distances AP, BP and CP are in are plane. The distances AP and BP are 3m and 1 ,5m respectively. The radiant powers of sources A and B are 90ω and 180 W respectively. The beam From c is of intensity 20 V\//m2. calculate intensity at P on the seen

Solution:

13.9 - 14.0

Resultant intensity at P

= 0.97+ 3.18+10

l_{p} = 13.97 W/m^{2}

QUESTION: 59

Two beams , A and B, of plane polarised light with through a polaroid, from the position when the beam A has maximum intensity ( and beam B has zero intensity) a rotation of polaroid through 60°makes the two beams appear equally bright. If the initial intensities of the two beams are l_{A} and l_{B} respectively, then l_{A} / l_{B} equals________.

Solution:

3

By law of malus I = I_{0} cos^{2} θ

QUESTION: 60

The change in specific volume when 1 kg of water freezes is 91 x 10^{-6} m^{3}. The pressure at 273 K i.e. Find out the pressure at which the ice would freeze, when it is given that latent heat of ice = 3.66 * 10^{5} J/kg & one atmosphere=10^{-5}

Solution:

136.2

From Maxwell’s relation

As given,

T =273 K

It means the pressure under which the ice would freeze = 1 + 135.2 = 136.2 atm.

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