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QUESTION: 1

For n-p-n transistor connected as shown in figure V_{BE }= 0.7 volts. Given that reverse saturation current of the .junction at room temperature 300^{0} K is 10^{-13} A, the emitter current is

Solution:

*Answer can only contain numeric values

QUESTION: 2

In the bistable circuit shown in the figure, the ideal OP- Amp lias saturation levels of ±5 V. The value of R_{1} (in kΩ) that gives a hysteresis width of 500 mV is______

Solution:

QUESTION: 3

The circuit shown in the figure is

Solution:

QUESTION: 4

Let the magnitude of the gain in the inverting Op-Amp circuit shown be x with switch S_{1 }open. When the switch is closed, the magnitude of gain becomes.

Solution:

*Answer can only contain numeric values

QUESTION: 5

Consider the circuit shown in the figure. Assuming the value of the dc voltage V_{C2} (in volt) is _____.

Solution:

QUESTION: 6

A Weiii bridge oscillator is shown in figure, which of the following statements are true, if f " is the ffeequency of oscillation?

Solution:

QUESTION: 7

In a voltage follower op-amp circuit, find the closed loop gain V_{0}/V_{s} if the open loop gain = 2

Solution:

QUESTION: 8

The circuit shown in figure is best described as a

Solution:

CKT is voltage doubles circuit

⇒ Once capacitor charge it can't discarge

QUESTION: 9

In the shunt regulator shown below, the Vz = 8.2 V and VBE = 0.7 V. The regulated output voltage V_{0} is

Solution:

KVL

QUESTION: 10

Let denote the Exclusive OR (XOR) operation, Let '1' and '0' denote the binary constants. Consider the following Boolean expression for F over two variables P and Q:

The equivalent expression for F is

Solution:

QUESTION: 11

Suppose one Si and one Ge-diodes are connected in parallel and then this combination is connected to a power supply of 5 V together with a series resistance 220 Ω. The value of the current through the Si(I_{1}) and Ge(I_{2}) are

[Given the cut in voltage of Si diode is 0.7 V and Ge diode is 0.3 V]

Solution:

Since the cut in voltage of Ge-diode is less than the cut in voltage of Si diode, so first Ge-diode will be on and all current will pass through it and Si diode will never on.

QUESTION: 12

In a certain CE amplifier with emitter resistance and emitter by-pass capacitor, if the by-pass capacitor is removed the voltage gain of the amplifier.

Solution:

The r_{e} model of CE transistor.

Therefore, gain will decrease if we remove capacitor.

QUESTION: 13

In the given circuit the time constant is larger than the period of the signal and the zener breakdown voltage of zener diode is 6V. A square wave is applied to the input as shown in figure. Which of the following graph represent the nature of the output.

Solution:

During positive half cycle diocle will be conducting so output will be zero and capacitor will changed at -10V. During negative half cycle effective input will -20 V. and diode will be reverse biased and input voltage is less than V_{th}, so zener diode will be behave like voltage regulator. So output will be -6V.

QUESTION: 14

For the Karnaugh map shown in the given figure, the minimum Boolean function is:

Solution:

QUESTION: 15

The given circuit will behave like

Solution:

QUESTION: 16

If volume and number of particles (F, N) of a thermodynamic system are doubled, which of the following quantities is bound to be doubled?

Solution:

Pressure (P), Temperature (T) and Chemical potential (//) are intensive variables whereas entropy (S) is an extensive variable. On increasing the size of a system, extensive variables are bound to increase proportionally whereas intensive variables are not bound to change just because the size of the system has increased.

QUESTION: 17

An ideal bose gas in 5-dimensions has the energy dispersion relation ∈ = Ap^{s}. For the Bose gas to undergo Bose-Einstein condensation, which of the following should hold?

Solution:

For Bose-Einstein condensation o f an ideal bose gas is d-dimensions with dispersion relation (S > 0 because increasing momentum should not decrease energy), we have

*Answer can only contain numeric values

QUESTION: 18

10 bosons are to be distributed in 3 different energy levels with the 2nd energy level being doubly degenerate and the other two are non-degenerate. The number of ways of achieving this is _____

Solution:

3 energy levels with one of them being doubly degenerate effectively means 4 energy levels.

∴ Number of ways of distributing 10 bosons in 4 energy levels

QUESTION: 19

For a gas confined in a box of volume V having total energy E and N number of particles, the number of microstates is Then the equation of state is

Solution:

QUESTION: 20

A real gas expands into vacuum in thermally insulated container. Winch of the following properties of the gas will not change?

Solution:

The given case is of adiabatic free expansion. For this, the internal energy of all the gases remains conserved. Moreover, the expression of internal energy of a real gas is given by

∴

Since, dV > 0, we have dT < 0. So, the temperature of the real gas will decrease.

*Answer can only contain numeric values

QUESTION: 21

Consider a system of 5 non-interacting distinguishable particles. Each particle can occupy only two energy levels ∈ and 2∈. If the total energy of the system 8∈. Then, the entropy of the system is _____ k_{B}. (answer must be given upto 2 decimal places)

Solution:

Let N_{1} and N_{2} be the number of particles occupying energy levels ∈ and 2∈ respectively.

*Answer can only contain numeric values

QUESTION: 22

Bulk modulus of a substance is defined as where k_{T }is tlie isothermal compressibility. If instead kT of 1 mole of an ideal gas, we take 5 moles of the same gas, then the ratio of bulk modulus to pressure B:P =_____. (Correct answer must be given upto two place decimal)

Solution:

For ideal gas, we have

QUESTION: 23

Water has the anomalous property that it contracts on melting. It means that

Solution:

By Clausius - Clapeyron equation

where L = Latent heat of fusion, V_{2} = Volume of water and V_{1} = Volume of ice

It means that with increase in pressure, there is decrease in temperature.

Hence, melting point decreases with increase of pressure.

QUESTION: 24

Below a certain temperature θ, the molecular specific heat of a certain gas is 5/2 k_{B}. As the temperature is raised above θ , the specific heat becomes 7/2 k_{B. }Which of the following can be the gas in this experiment?

Solution:

The specific heat of a diatomic molecule looks like

This jump of C_{V }from comes from vibrational degrees of freedom.

Hence, the gas to be nitrogen.

QUESTION: 25

For a 2nd order phase transition, which of the following remains continuous?

Solution:

For a 2nd order phase transition, the quantities which involves second order derivative of free energy are discontinuous whereas the quantities which involves first order derivative of free energy are continuous .Now, we have

As are 2nd order derivatives of free energy A, they are discontinuous. Moreover, entropy is continuous across 2nd order phase transition.

QUESTION: 26

In the Op-Amp circuit shown, assume that the diode current follows the equation The relationship between

Solution:

Logarithmic Amplifies

Case-1

QUESTION: 27

A regulated power supply, shown in figure below, lias an unregulated input (UR) of 15 volts and generates a regulated ouput. Use the component values shown in the figure.

The power dissipation across the transistor Q_{1} shown in the figure is

Solution:

QUESTION: 28

The output voltage of the regulated power supply shown in figure is

Solution:

*Answer can only contain numeric values

QUESTION: 29

Ill the circuit shown, assume that diodes D_{1 }and D_{2} are ideal. In the steady-state condition, the average voltage V_{ab }( in Volts) across the 0.5 μF capacitor is ______

Solution:

*Answer can only contain numeric values

QUESTION: 30

The figure shows a half-wave rectifier. The diode D is ideal. The average steady state current (in Amperes) through the diode is approximately______

Solution:

Positive peak detector circuit

*Answer can only contain numeric values

QUESTION: 31

The amplifier circuit shown in the figure is implemented using a compensated operational amplier (OP-Amp). and has an open-loop voltage gain, and an open - loopcut - off frequency. f_{c }= 8Hz.

The voltage gain of the amplifier at 15 kHz. in V/V, is ______

Solution:

*Answer can only contain numeric values

QUESTION: 32

Consider the circuit shown in the figure Assume base-to-emitter voltage V_{BE} = 0.8 V. and common base current gain(α) of the transistor is unity.

The value of the collector-to -emitter voltage V_{CE }(in volts) is ________.

Solution:

*Answer can only contain numeric values

QUESTION: 33

For the given circuit, value of the base current ( I_{b} ) of the npn transistor will be _____________ mA. ( β is the current gain and assume Op-Amp as ideal).

Solution:

QUESTION: 34

Transform the following logic circuit (without expressing its switching function) into an equivalent logic circuit that employs only NAND gates each with 2-inputs.

Solution:

*Answer can only contain numeric values

QUESTION: 35

The number of NAND gate required to implement _______

Solution:

QUESTION: 36

Non-interacting bosons trapped in a 3-D cubical box of side α (α is very small). For Bose-Einstein condensation, the chemical potential must be equal to

Solution:

For Bose-Einstein condensation to occur, the chemical potential must be equal to the ground state energy.

QUESTION: 37

Our Sun is projected to be a red-giant star after approximately 5 billion years from now. In the red giant phase, the Light that will be emitted will be mostly infrared around a wavelength of 1μw. The Sun is now at 10000K and emitting mostly around light of 0.5 μm wavelength. From tins information, the surface temperature of the red-giant sun is found to be

Solution:

From Wein’s displacement law, we have

Therefore, the surface temperature of the red-giant sun will be around 5000k.

QUESTION: 38

Two identical fermions can occupy any of the four energy levels E, 3E, 5E, 7E. The energy' levels 3E and 5E are doubly degenerate whereas the energy' levels E and IE are non-degenerate. Assuming no spin degeneracy; the partition function is

Solution:

The possible configurations are

QUESTION: 39

Given a microstate of a system specified by (E, V, N). Which of the following is correct with respect to the given microstate? (The symbols have their usual meaning)

Solution:

Given system can be treated as a micro-canonical ensemble in which (E, V, N) are fixed.

There can be many nucrostates complying with the given inicrostate. The switching of the members of the microcanonical ensemble among the nucrostates is governed by quantum evolution of states under the relevant Hamiltonian and has nothing to do with increase or decrease of entropy of the system. Given the state variables (£, V} AO, entropy is already fixed. The system described by a micro- canonical ensemble, has equal probability to be in any inicrostate.

Therefore , the system over along span of tune, spend sequal tune in each microstate .

QUESTION: 40

The entropy of a body of fixed volume which radiates a blackbody varies with temperature as:

Solution:

The internal energy of the blackbody in thermal equilibrium at temperature T energy radiated by blackbody is given by

*Answer can only contain numeric values

QUESTION: 41

A system is in contact with a heat both at temperature T and in contact with a particle reservoir as well. The system contains three accessible energy level 0, E and 2E and it may or may not contain one particle at a tune one of the given energy levels. How many accessible states are possible?

Solution:

The possible configurations are

QUESTION: 42

The excitations of ferromagnetic are bosonic in nature and are described by a dispersion relation E ∝ k^{2}, where 'E' is the energy and 'k' is the wavevector of die excitation. The average energy of the 3 -dimensional ferromagnet vanes with temperature as (assuming die bosonic excitations to be non-interacting)

Solution:

The average energy is given by

QUESTION: 43

Given a system of two spins s_{1} and s_{2} each of which can take values +1 and -1. The energy of the system is E = - Js_{1}s_{2}, where J is some positive constant denoting the strength of interaction between the spurs. The minimum energy and the corresponding number of spin configuration are given by

Solution:

QUESTION: 44

Given a N classical harmonic oscillators in 3-dimension such that the total energy of the system is E ≤ E_{0}. The available phase space volume depends on E_{0} as

Solution:

For N classical harmonic oscillators, we have

Therefore, available phase space volume

QUESTION: 45

A box of volume V contains N molecules of an ideal gas. It is in contact with an empty box of volume 2 V through a permeable wall allowing exchange of molecules. If a measurement is made for the number of particles in the empty' box after a long time of contact. What is the standard deviation seen in the measurement?

Solution:

Given no other variables involved in the problem the probability of a particle (as a molecule) to be m one of the box depends on the volume of the particular box vis-a-vis the total volume available to the system.

Therefore, probability of occupying the first box.

and that of occupying the second box,

Tins is a binary distribution whose standard deviation is given by

QUESTION: 46

Two classical particles are distributed on the vertices of a N-sided polygon. Each site can accommodate one particle at a tune. Iftlie adjacent vertices of the polygon are occupied by the particles, the energy of the system increases by E. The mean energy of the system is

Solution:

Let we have two classical particles A and B. Let us fix particle A at one vertex and write the partition function for it. The corresponding partition function is

The partition function of the syste m is

*Answer can only contain numeric values

QUESTION: 47

Consider a mole of an ideal diatomic gas undergoing a cyclic process as shown in the figure:

The magnitude of the heat released in the cyclic process is_____ P_{0}V_{0}. (answer must be given upto two place after decimal)

Solution:

Along isothermal process AB,

Along Lsobaric process BC,

Along adiabatic process CA, ΔQ = 0

Therefore, heat supplied to the system in the cycle =

*Answer can only contain numeric values

QUESTION: 48

1 kg of water at 27^{0}C is mixed with 2 kg of water at 77^{0}C. After thermal equilibrium is reached, the change in

entropy of the system is_____J/K. (Answer must be given upto two place after decimal) (Take specific heat of water to be 4.181 kJ/K)

Solution:

Using principle of calorimetry, we have

Heat gamed = Heat lost

QUESTION: 49

A classical ideal gas. consisting of N particles (.V → ∝ ) is confined in a box of volume F at a certain temperature T and pressure P. In a part of the box, which has volume V_{0} a measurement is made to count the number of particles present in it. What is the probability that no particle are present inside?

Solution:

Probability that no particle is inside the volume V_{0} is

[∴ All the paiticles then occupy the remaining volume]

*Answer can only contain numeric values

QUESTION: 50

In an experiment to understand why a two-level system cannot act as a laser, a student heated the gas inside began looking at a spin system with only two energy states -μH and μH corresponding to parallel alignment and anti-parallel alignment respectively with the magnetic field H. In Ins measurements of ratio of number of spin up to spin down particles as he increased the magnetic field he finds that it never exceeds a number α. What is α? ( answer must be given upto 2 decimal places)

Solution:

For a laser action there is a necessary condition of population inversion to be achieved. In a two-level system, Maxwell-Boltzmann statistics does not allow the upper level to have more N/2 atoms in it even at large excitation energies.

QUESTION: 51

The entropy of a thermodynamic system is given as then, temperature of the system α, E_{0}, are positive constants respectively)

Solution:

So, T is always greater than or equal to zero where, T= 0 when E = E_{0}/2

*Answer can only contain numeric values

QUESTION: 52

3 non-interacting fermions are confined in a 2-D box of area ‘a^{2'}. If the energy' of the 1st excited state of the system is

Solution:

Energy of a particle in a 2-D box is

where, is ground state, is the 1st excited state.

Therefore, 1st excited state has one fermion each in (1,1), (1,2), (2,1) levels.

*Answer can only contain numeric values

QUESTION: 53

A particle of mass 2 x l0^{-15} kg is constrained to move in two-dimensional x-y plane. The particle ls in equilibrium with a thermal bath (k_{B}T = 10^{-13}J). What is the RMS value of the position (in units of metres) of the particle after a time 10 sec if the particle starts from origin at r = 0 seconds?

(answer must be given in integers)

Solution:

By Maxwell-Boltman distribution of velocities, we have

QUESTION: 54

There is a particle moving in a liquid in 3-D following Brownian motion.What is the root-mean squared displacement o f the particle after 5 minutes if the diffusion constant is 10^{-2} cm^{2}/s?

Solution:

The root-mean squared displacement is given by

QUESTION: 55

A three-level system at thermal equilibrium at temperature T is governed by the Hamiltonian

The average energy of the system is

Solution:

The given Hamiltonain is an upper triangular matrix and hence its eigen values are its diagonal elements. So the corresponding eigen values are

∴ The average energy of the system is

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