If the efficiency of a Carnot cycle when the hot body temperature T1 is increased by x or when the cold body temperature T2 is decreased by y will be the same. Then the relation between x and y should be
The temperature of the hot body = T1
increased temperature = x
When cold body temperature = T2
decreased temperature = y
Coefficient of performance =
putting the values in this equation then find
A disk of radius q/4 having a uniformly distributed charge 6C is placed in the x-y plane with its centre at A rod of length a carrying a uniformly distributed charge 8C is placed on the x-axis from Two point charges - 7C and 3C are placed at respectively. Consider a cubical surface formed by six surfaces The electric flux through this cubical surface is
Total enclosed charge as already shown is
From Gauss-theorem, net flux,
Among the following displacement versus time plots, which ones may represent an damped oscillator?
For damped oscillation amplitude of oscillation is proportional to where γ → damping parameter.And this is exponentially decreases with time.
In a cubic lattice, the closed packed structure of mixed oxides of the lattice is made up of oxide ions, one eight of the tetrahedral voids are occupied by divalent ions (A2+) while one half of the octahedral voids are occupied by trivalent ions (B3+). Then What is the formula of the oxides?
Let thee be 80 O2- in the crystal
Octahedral voids = 80
Tetrahedral voids = 160
= 1 : 2 : 4
So, AB2O4 is best formula o f the oxide
An electron is moving in the positive X direction, in S frame. The relative velocity between the s and S frames is 0.8 c and the velocity of the electron is also 0. 8 c with respect to an observer in S frame. The velocity of the electron for an observer in the S frame is
The relative relativistic velocity is given as
The Clausis - clapeyron equation indicates that the increase of pressure increases the melting point:
Clausis - clapeyron equation is given as
if volume increases the pressure increase with T.
If a ball is dropped from rest, it bounces from the floor repeatedly. The coefficient of restitution is 0.5 and the speed just before the first bounce is 5 m /sec. The total time taken by the ball to come to rest is
u → initial velocity before strike
v → velocity after the strike
t2 = V/g
The ball again strike after
Hence the total time =
Hence, total time = 1.5- 0.5 = 1.0 sec.
If the wave length of k∝ line silicon (z = 47) is 0.57 Å then the value of wavelength for uranium (z = 92) is
We know that
If the amplitude of a damped - oscillator becomes one - third in 2 minutes, what will be the amplitude of oscillation as compared to initial one after 6 minutes?
In case of damped oscillations, amplitude decrease exponentially with time , i.e
Cosmic ray muons generated at the top of the earth's atmosphere decay according to the radioactive decay law
where N(t) is the number of muons at time t, and T1/2 = 1.74 μs is the proper half-life of the muon. Immediately after generation, most of these muons shoot down towards the Earth's surface. Some of these muons decay on the way, but their interactions with the atmosphere is negligible.
An observer on the top of a mountain of height 3.0 km above mean sea level detects muons with the speed 0.96 cover a period of time and counts 1000 muons. The number of muons of the same speed detected by an observer at mean level in the same period of time would be
We know the time
= 6.21428 sec
Then According to question
travelled distance = 3 x 103 m
velocity = 0.96 x 3 x 108 m/sec
Then time taken by the muons to reach the sea level is t = travelled distance /velocity
A one-dimensional box contains a particle whose ground state energy is ∈• It is observed that a small disturbance causes the particle to emit a photon of energy is ground state energy eigenvalue), after which it is stable. Just before emission, a possible state function of the particle in terms of the energy eigenstates would be
As we know that the total energy
E =P1(E4) + P2 (E5)
According to option (C)
We know that the energy eigenstate of one dimensional box is
2000 neutral spinless particles are confined in a one-dimensional box of length 200 nin. At A given instant of time, if 200 of these particle have energy 4∈0 and the remaining 1800 have 225∈0, then the number of particles in the left half of the box will be approximately.
In every energy state the probability of finding a particle is equal in left and right hand portion of the box.
Which of the following statements is/are true for a matric A defined as:
The characteristic equation of A is
multiple by A-1
A2 + 2A - I - 20A-1 = 0
Let (x,t) and (x’ t ’) be the coordinate systems used by the observers 0 and 0 ’ respectively. Observer O’ moves with a velocity V = βC along their common positive x - axis. If x+ = x + Ct and x- = x - Ct are the linear combinations of the coordinates, the lorentz transformation relating 0 and O’ takes the form.
By the lorentz transformation
Evaluate and V is the volume bounded by the cylinder z = x2 and the p la n e s x = 0 , x = 2 , y = 0 , y = 6, z = 4.
The radiation corresponding to 3→2 transition of hydrogen atom falls on a metal surface to produce photo electrons. These electrons are made to enter a magnetic field of 3 x 10-4 T. If the radius of the largest circular path followed of the metal is close to
When an electron moves in a circular path, then
Work function of the metal W = hu - KEmax
A thin uniform rod of mass m and length I pivoted freely at its base, is allowed to fall from a verticle position. What is the angular velocity when it has turned through an angle 600?
Let A0, be a rod of length I. By the energy equation.
Work done = Kinetic energy
A diffraction pattern is obtained by using a beam of red light. If the red light is replaced by blue light then:-
i.e. the diffraction bands become narrower & crowed together.
Springs of constants k , 2 k , 4k ,8k , ............, 2048k are connected in series. A mass ’m' is attached to one end the system is allowed to oscillate. The time period is approximately :-
No. of springs = 12
The distance between an object & its real image formed by a lens is 'D'. If the magnification is 'm' the focal length of the lens is:-
In the transistor circuit shown in the figure below, collector to ground voltage is + 2 0 V. Which of the following is the probable error?
For collector to ground which is the same as emitter terminal to read the full supply voltage at 20V, no current should be drawn & this can happen if emitter to ground connection is open as then there will be neither lB or lc.
A minimum value of the Emitter bypass capacitor (C2), in given figure if the amplifier must operate over a frequency range from 5KHz to 10KHz is:-
Since RE = 660Ω
Xc of the bypass capacitor, C2 should be not greater than
The capacitive value , at the min freq. of 2KHz.
In the given circuit, if the input A = B = 0, then what are the possible logic state of C & D respectively.
If A = 0 & B = 0, c & D will always be 0.
A metal crystallizes in face-centered cubic lattice with a lattice parameter of 4.20 The shortest atom to atom contact distance in the lattice is:-
Shortest distance = AB/2
The atomic radius (in cm) of an element with a body-centered cubic unit cell of volume 75.8 cm3mol-1, molecular weight 137.3 & density 3.62 gm cm-3 is
Density of unit cell,
Diagonal of any face =
The potential difference between the points A & B of a', uniformly polarized infinite slab shown in the figure is:-
∴ Potential difference (V) = Ed =
The region between two concentric right circular cylinders contain a uniform charge density ρ. Then potential V is given by:-
Using Poisson eqn which reduces to:-
The interaction energy of two dipoles of moments is given by :-
As, potential energy of a dipole ’p’ in on Electric field is given by:-
But for a dipole of moment p1
So potential Energy of dipole in the field p1 :-
When both source and observer move in the same direction with a velocity equal to half the velocity of sound. the change in frequency of the sound as detected by the observer is
In this case n’
Both are moving in the direction of sound.
no change in frequency.
An electromagnetic wave is propagating in free space in the z- direction. If the electric field is given by where ω = ck then the magnetic field is given by:-
A train is moving at 30m/s in still air. The frequency of the locomotive whistle is 500Hz & The speed of the sound is 345m/s. Then which of the following statement is correct: -
are only correct option.
In still air :- (1) front of locomotive:-
= λ' = 0.75m
In case of wind :- (1) front of locomotive:-
(ii) behind the locomotive:- c' = c-vwind = 355m
Two bodies of masses m1 and m2 are connected by a light string passing over a smooth light pulley fixed at the end of an inclined plane. The mass m1 lies on the inclined plane and m2 hangs vertically. If the system is at rest. Select the valid sentences.
If the ice on the polar caps of the earth melts then which of the following statement is/are correct:-
If the ice on the polar caps of the earth melts, the water so formed will spread on the surface of earth . This increases the moment of inertia of the earth about its own axis due to change in the distribution of mass of the particles of water going away from the axis of rotation. To conseive angular momentum angular velocity of earth about its own axis will decrease. As
Hence due to decrease in the value of ω, the duration of the day will increase.
When a fast moving neutron strikes a proton at rest. Which of the following are correct.
The collision between a neutron and a proton is perfectly elastic. In a perfectly elastic collision, the two colliding particles interchange their velocities, if their masses are equal.
Since proton and neutron have almost equal masses , they interchange their velocities. Thus practically neutron comes to rest and proton moves with the velocity of neutron. Hence neutron transfers almost whole of its K.E. to proton.
Which of the following statements is/are correct:-
(A) If the object is moving its total energy is E = mc2 =
So, mass and energy of an moving object are dependant entities. When mass is being created or destroyed an equivalent amount of energy Simultaneously vanished or created and vice versa.
(B) We know that
That means the speed of a massless particle is always equals to the speed of light. Massless particle can exists only if they move with the speed of light. (C) we know that inverse lorentz transformation are
So, the relative velocity of two photons approaching each other is equal to the velocity of light.
An ideal gas at pressure pi and volume vi doubles its volume adiabatically. Then
Final volume Vf = 2Vi
for adiabatic process
Work done by the gas
A metal sphere of radius ' a ' carries a charge Q as shown in figure. It is surrounded, out to radius b, by linear dielectric material of permittivity ' ∈ '. Choose the correct statements-
Inside the metal sphere of course E = P = D = 0
The potential at the centre is therefore:-
Option C,D is the only correct option,
The base biased transistor circuit of figure is subjected to increase in junction temperature from 250C to 750. If β increase from 100 to 150 with rising temperature (Assuming that VBE = 0.7V) choose the correct statement:-
Which of the following statements are correct:-
A small signal amplifier goes into saturation once on each input cycle.
Consider the following statements associated with the basic properties of conductors & choose the correct statements:-
(i) The resultant field inside the ideal conductor is zero.
(ii) The tangential component of E.F at the boundary are discontinuous by the surface current density, the surface current flow only in high conductivity media ; therefore (ii) is correct.
(iii) Because of Electrostriction effect the dielectric material be a state of compression.
(iv) The displacement flux densities normal component at the boundary are discontinuous by surface charge density.
The rank of the matrix A is ......
A is square matrix of order 3.
8 - 2 - 6 = 0
Since A is a square matrix and |A| = 0 hence P(A) ≠ 3
Consider a minor of order 2
= 4 - 2 = 2 ≠ 0
Hence P(A) = 2
A collimated beam of pions originate from an accelerator and propagates in vacuum along a long straight beam pipe. The intensity of this beam was measured in the laboratory after a distance of 60 m and found to have dropped to one fourth of its intensity at the point of origin. If the proper half - life of a pion is 1.6 x 10-8 sec, the speed of the pions in the beam, as measured in the laboratory , must be nc. The value of n is _____
we know that
x = VΔt
After cutting off a circular portion of radius R/2 from the centre of a uniform circular disc of radius R, the moment of inertia about an axis passing through its centre and perpendicular to its plane become I. The moment of inertia of the original disc about the same axis is xl. The value of x is
The mass of shaded portion M' = M/4
Moment of inertia of unshaded portion =
(moment of inertia of whole disc)
A bullet loses half of its velocity after penetrating 30cm in to a target. If would come to rest after covering an additional distance (in cm) equal to .........
If F is the resistance applied by the target then change in K.E bullet = Work done against the resistance.
The loss of energy when its velocity become zero from (v/2)
x = 10 cm
A body of mass 2 kg is moving under the influence of a central force whose potential energy is given by U(r)= 2r3 Joule, If the body is moving in a circular orbit of 5m, its energy will be ______ J
By the effective central potential
for the circular orbit
total Energy =
Two identical sonometer wires have a fundamental frequency of 500 vibrations per second when kept under the same tension. The fractional increase in the tension of the one wire that would cause an occurrence of 5 beats/sec. When both wires vibrate together is ______.
let the tension of one wire be increased by ΔT
The new frequency V’ is V’ =
Dividing by eqn (2) by eq (1) :-
In the circuit shown in the given figure , the transistor is in the active mode. VBE= 0.7volt. The best approximation without knowing hfe for Vc as marked in the figure is-
In the circuit shown in figure, assume that the capacitor 'c‘ is almost shorted for the frequency range of interest of the input signal. Under this condition the voltage gain of the amplifier will be approximately:
The current I for the circuit shown will be _______ (in mA) :-
In the temperature range 100-1000 C, the molar specific heat of a metal varies with temperature T (measured in degrees Celsius) according to the formula Cp=(1 + T/5) J - deg C-1 mol-1. If 0.2 kg of the metal at 600c is brought in thermal contact with 0.1 kg of the same metal at 300 c, the final equilibrium temperature, in deg c, will be ________ .
We know that in the system
Heat gain by the system = Heat released by the system
A wire shaped to a regular hexagon of side 2m carries a current of 2amp. The magnetic induction at the centre of the hexagon is ______
Let each side of the hexagon is 'b' & the corners are at a distance 'a' from the centre.
The magnetic field at the centre ‘O’ due to element AB is the given by the relation:-
It is clear from vector product that points away from the reader in the direction normal to the paper.
Resultant mag. field =
B = 69.3 μwb/m2
Given that electric field in a cylindrical coordinates then the magnitude of work needed to move ω point charge G = 1 nc from any radial distance 'r' = 2m to a point at twice that radial distance is_____
Since the field has only a radial component:-
for the limits of integration use 'r1' & 2 ’r1' initially.
Putting the values , we get ; 9 x 109x10-9x0.693
W = 6.238 J Ans.
In a cylindrical conductor of radius 2mm, the current density varies with the distance from the axis according to J = 103e-400r (A/m2) the total current T is _____(in mA)?
The total current
C onsider an ideal gas where entropy is given by Where n = number of moles, R = universal gas constant U = internal energy, V = volume and σ = constant. The internal energy is given by U = nPV. the value of n is _____
From the first law of thermodynamics
Tds = dU - PdV
Let be a matrix with real entries. If the sum and the product of all the eigenvalues of A are 10 and 30, respectively, then a2 + b2 equals ____
Characteristic polynomial of A = t2 - tr(A) + |A| = t2 - ( a + b + 3)t + 3ab
Sum of eigen values = 10
and product of eigenvalues = 30
Then, t2 - ( a + b + 3) t + 3ab = t2 - 10t + 30
⇒ a+b + 3 = 10
a + b = 7 and
3ab = 30 ⇒ ab = 10
a2 + b2 = ( a + b)2 - 2ab
= ( 7)2- 2 x 10 = 49 - 20 = 29
A vertical tube has diameter 0.06 m at its bottom end from which water flows out at the rate of 1.3 kg/minute. The pressure at the end is atmospheric pressure 0.9 mm of Hg. If the diameter of the tube is 0.04 m at a height of 3m from the bottom end. The pressure at that point is
Mass of water flowing in one second
Let V. be the velocity of flow of water at the bottom . then the mass of the water flowing per second =
Let v2 be the velocity of flow of water at the point where the radius of the tube is 0.02m. The mass of the water flowing per second is 0.03
A particle of mass m is moving in a circular path (radius a) passing through the origin 0 as shown in the figure below.
The particle is acted upon by a force F pointing towards the origin. If L is the magnitude of conserved angular momentum of the particle then the total energy of the particle is (assuming that the potential energy v(r) → 0 as r → ∝)
We know that the force equation of the circular path under the central force is
A heat engine is used to drive a heat pump. The heat transfers from the heat engine and from the heat pump are used to heat the water circulating through the radiators of a building The efficiency of heat engine is 27% and the COP of the heat pump is 4. The ratio of the heat transfer to the circulating water to the heat transfer to the heat engine is
for Heat Engine (HE)
for Heat Pump (HP)
Two vessels. A and B, each of volume 3m3 be connected by a tube of negligible volume. Vessel A contains air at 0.7 mPa, 950C. while vessel B contains air at 0.35 mPa, 2050C. The change of entropy when A is connected to B by working from the first principles and assuming the mixing to be complete and adiabatic is _____.[c = 1.005 KJ/kg.k,cv=0.718KJ/kg.k, R = 0.287KJ/kg.k]
Let the final temperature be Tf
for adiabatic mixing of gas Internal Energy must be same uA = mAcVTA
= 19.88335 x 0.718 x 368 KT = 5253.66 KJ
Consider a particle with mass m in a one dimensional harmonic potential. At t = 0 the normalized wave function is
Where is a constant. The probability that the momentum of the particle at t > 0 is positive is
The probability P for a positive momentum is
we can write ψ(P,t) as a linear combination of the eigen functions in the momentum space.