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The potential φ in Joule of a particle of mass 1 kg moving in x - y plane obey the law of φ = 3x + 4y. Here x and y are in meters. If the particle is at rest at (6m, 8m) at time t = 0, then the work done by conservative force on the particle from the initial position to the instant when it crosses the x - axis is
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 2
OR, the direction of motion of the particle is along the origin. The particle will cross the X-axis at origin.
So, by the energy conservation at time t = 0, (6m, 8m) and at origin
Ki+ Ui = Kf + Uf
0 + ( 6 x 3 + 4 x 8 ) = K f + ( 0 x 3 + 0 x 4 )
Consider the earth as a uniform sphere of mass M and radius R. Imagine a straight smooth tunnel made through the earth which connects any two points on its surface. The motion of a particle of mass m along this tunnel under the action of gravitation simple harmonic. Determine the time that a particle would take to go from one end to the other through the tunnel.
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 4
Suppose at some instant the particle is at radial distance r from centre of earth O. since the particle is constrained to move along the tunnel, we define its position as distance x from c. Hence, equation of motion of the particle is max = fx
The gravitational force on mass m at distance r is F =
Since, Fx α- x , motion is simple harmonic in nature.
Time period of oscillation
Hence, the time taken by the particle to go from one end to another end is
The magnitude of the de-Broglie wavelength λ of electron (e), proton (p), neutron (n) and α - particle (α) all having the some energy of 1 MeV, in the increasing order will follow which of the following sequence :
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 7
According to de-Broglie the wavelength associated with a particle of mass m moving with velocity v is given as
The frequency of a cyclotron oscillator is 107 Hz. The cyclotron is accelerating protons. If the radius of the does of the cyclotron be 0.6 m, the kinetic energy of the proton beam produced by the accelerator will be nearly_______ .
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 8
The frequency of cyclotron f 107 Hz
mv = Bqr
Two identical capacitors A and B shown in the given circuit are joined in series with a battery. If a dielectric slab of dielectric constant K is slipped between the plates of capacitor B and battery remain connected then the energy of capacitor A will
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 9
The energy stored in capacitor of capacity C and potential V is given as
Now when slab of dielectric constant K is introduced without disconnecting the battery the equivalent capacities becomes
Consider a particle with three possible spin state S = 0 and ±1. There is a magnetic field h present and the energy for a spin state S is -hS. The system is at a temperature T. Which of the following statements is true about the entropy S(T) ?
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 10
we know that
Entropy s = K In Ω at high temperature T, microstates are 3.
So, s = K In 3.
at law temp T= 0 K, microstate = 1
(at T = 0 K; system is perfect ordered )
s = K In 1 = 0
Consider three situations of 6 particles in one dimensional box of width L with hard walls. In case (i), the particles are fermions, in case (ii) they are bosons and in case (iii) they are classical. If the total ground state energy of the six particles in these three cases are EF, EB and Ecl respectively, which of the following is correct ?
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 11
The electric fields inside (r < R) and outside (r > R) a solid sphere with a uniform volume charge density are given by , respectively, while the electric field outside a spherical shell with a uniform surface charge density is given by being the total charge. The correct ratio of the electrostatic energies for the first case to second case is
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 12
We know that
The electrostatic energy
for the spherical shell
A quantum mechanical particle in a harmonic oscillator potential has the initial wave function , where ψ0 and ψ1 are the real wave functions in the ground and first excited state of the harmonic oscillator. For convenience we take m = ℏ = ψ= 1 for the oscillator. What is the probability density of finding the particle at x at time t = π ?
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 14
A particle of mass m is thrown upward with velocity v and there is retarding air resistance proportional to the square of the velocity with proportionality constant K. If the particle attains a maximum height after time t, and g is the gravitational acceleration, what is the velocity.
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 15
A monochromatic wave propagating in a direction making an angle 30° with the y-axis in the reference frame of source. The source moves at speed v towards the observer. The direction of the (cosine of angle) wave as seen by the observer’s _____ .
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 16
A system of two circular co-axial coils carrying equal currents i along same direction having equal radius R and separated by a distance 2R (as shown in figure below). The magnitude of magnetic field at the midpoint P is given by
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 17
Consider a system of 4N non-interacting spin particles each fixed in position and carrying a magnetic moment μ . The system is immersed in a uniform magnetic field B. The number of spin down particle for which the entropy of the system will be maximum is
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 18
Let us consider n number of spin out of 4N particle have spin down remaining 4N-n is up.
Number of ways
for entropy maximum at equilibrium for spin down particle
Let be a matrix with real entries. If the sum and the product of all the eigen values of A are 10 and 30, respectively, then a2+b2 equals
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 20
Characteristic polynomial of A = t2 - tr(A)t + |A|
= t2 - (a + b + 3) t + 3ab
Sum of eigen values = 10
and product of eigen values = 30
⇒ a + b + 3 = 10⇒ a + b = 7
3ab = 30 ⇒ ab = 10
Now, a2 + b2 = (a + b)2 - 2ab
= (7)2-2x10 = 29
Unpoiarized light falls on two polarizing sheets placed on top of the other. What be the angle between the characteristic directions of the sheets if the intensity of the transmitted light is one third intensity of the incident beam_______ .
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 24
Intensity of light transmitted through the first polarizer
where I0 is the intensity of the incident unpolarized light.
Intensity of the light transmitted through the second polarizer is I2 = I0 cos2 θ
Consider the digital circuit shown below in which the input C is always high (1)
The truth table for the circuit can be written as
The entries in the Z column (vertically) are
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 26
This is X-OR-GATE in which when both inputs are same
then output will be zero if both inputs are not same then output will be high so for
the Ist set
(i ) A = 0, B = 0 then A . B = 0 = y1
and B = 0, C = 1 then Y2 = 1
(ii) A = 0, B = 1 them Y1 = A . B = 0
B = 1, C = 1 then Y2 = 0
(iii) A = 1, B = 0 then Y1= A .B = 0
B = 0, C = 1 then Y2 = 1
(iv) A = 1, B = 1 then Y1 = A . B = 1
B = 1, C = 1 then Y2 = 0
so the output is 1011
In the operational amplifier circuit below, the voltage at point A is
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 27
By the help of virtual ground’s condition then potential at A point of inverting terminal is equal to at point O of non-inverting terminal. Let potential at 0 is considered as VA then q by the help of node analysis at point 0.
The output, 0, of the given circuit is cases I and II, where
Case I : A, B = 1; C, D = 0; E, F = 1 and G = 0
Case II : A, B = 0; C, D = 0; E, F = 0 and G = 1
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 30
Let o/p of AB is x, of CD is y and of x and y is z and o/p or z and E is x’ of x’ and f is y’ and o/p of y’ and G is z’. And z’ is the final output given that
for Case I A, B = 1 ; C,D = 0 ; E, F = 1 and G = 0
Case II A, B = 0 ; C,D = 0 ; E, F = 0 and G = 1
For Case I
⇒ z = x + y = 0 + 0 = 0
y = 0.0 = 0
x’ = z. E = 0
y’ = x’ + F = 0 + 1 =1
For case II
⇒ z = x = y = 1
y = 0.0 = 0
x’ = Z.E= 1.0 = 0
y’ = x’ + F = 0 + 0 = 0
The Mobility of e- in copper assuming that each atom contribute one free e- for conduction (Given : - Resistivity of copper is 1.7x1-6 ohm - cm , Atomic weight = 63.54 , density = 8.96 gm /c .c ) will be :-
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 32
Consider the radioactive transformation a → B →C with decay constant λA and λB for elements A and B . C is a stable element . Assume that at t = 0, NA = N , NB = 0 and Nc = 0 where NA, NBand NC are the number of atoms of A, B and C respectively. Then which of the following is/are correct
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 34
Rate of disintegration of the element
According to question A → B → C
at t = 0, NA = N0 and NB = Nc = 0
At any time t, for element A
Muons are elementary particles produced in the upper atmosphere. The have a life time of 2.2 us. Consider muons which are travelling vertically towards the earth’s surface at a speed of 0.998 C. For an observer on earth the height of the atmosphere above the surface of the earth is 10.4 km. Which of the following statements are true ?
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 35
∴ Muon’s life time in earth’s frame
Muon’s travelling at speed grater than 0.998C reach the earth’s surface . Muon’s reach the surface of earth because : the appearance thickness of atmosphere
A satellite of mass m is just placed over the surface of earth. In this position mechanical energy of satellite is E1 Now it starts orbiting round the earth in a circular path at height h = radius of earth. In this position, kinetic energy, potential energy and total mechanical energy of satellite are K2, U2 and E2 respectively. Then
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 37
Two charges + Q each are fixed at points C and D. Line AB is the bisector line of CD. A third charge +q is moved from A to B, then from B to C
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 38
Along the line AB, charge q is at unstable equilibrium position at B (when displaced from B along AB, net force on it is away from B, where as force at B is zero) Hence potential energy at B is maximum.
Along CD equilibrium of q is stable. Hence potential energy at B is minimum along CD.
A charged particle is moving along positive y-axis in uniform electric and magnetic fields.
Here E0 and B0 are positive constants, choose the correct options -
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 39
Depending on sign of q, may be along positive z-axis or along negative z- axis.
Again, depending on the value q it may be along positive z-axis or along negative z-axis.
If q is positive, comes along negative z-axis also.
But comes along positive z-axis. so it may also path undeflected.
Consider two point charges q and λq located at points x = a and x = μa respectively. Assuming that sum of two charges is constant than what is value of λ for which the magnitude of electrostatic for is max __________ .
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 42
A battery powers two circuits C1 and C2 as shown in the figure
The total current I drawn from the battery is estimated by measuring the currents I1 and l3 through the individual circuits. If I1 and l2 are both 200 mA and if the errors in their measurement are 3 mA and 4mA respectively, the error in the estimate of I is mA.
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 44
Let a and b be any two measured values and the error in this measurement are da and db then
if y = a + b
then y + dy = (a + da) + (b + db)
⇒ y + Δy = a + b + Δa + Δb
y + Δy = y + Δa + Δb
Δi = 3 + 4 mA
ΔI = 7mA
A particle, of unit mass moves along the x-axis under the influence of a potential, V (x) = x (x - 2 )2 . The particle is found to be in stable equilibrium at the point x = 2. Find out the time period of oscillation of the particle.
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 45
The Potential is
V = x[x2 - 2x + x]
V = x3 - 2x2 + 4x
A 3 x 3 matrix has elements such that its trace is 11 and its determinant is 36. The eigenvalues of the matrix are all known to be positive integers. The largest eigenvalue of the matrix is _________ .
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 46
A 3 x 3 matrix have e eigen value.
Let L1 l2l3 are three eigen values.
So, I1 l2l3 = 36
and Trac A = 11
So, l2 + l2 + l3 =11
If I1 = 6, l2 = 3 and l3 = 2
I1 + l2 + l3= 11
So , largest eigen value of matrix is(6)
A p - n junction is formed from germanium of conductivity 0.8 ohm-1 cm-1on p- side & 1.6 ohm-1 cm-1 on n-side. Calculate potential barrier at 300K (in V). (ni = 2.1 x 1013 cm-1) (Given μp = 2000 cm2/volt-sec and μn = 4000 cm2/volt-sec) (K = 1.38 x 10-16 erg per °K)
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 47
T h e current gain 'α’ of NPN transistor is 0.98. It is connected in the CB mode & gives a reverse saturation current Ico =12μA. Then the base current for an emitter current of 2mA will be _______(in μA ).
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 48
For NPN transistor,
= 1.972 mA
again ; Ib - Ie - Ic = ( 2 - 1.972 ) mA = 28 μA
A gain transistor has a circuit gain of β = 60. If it is connected in a grounded base configuration, what theoretical a.c collector current (in mA) will flow when an a.c current of 2 mA flows through the emitter (Assume the collector potential to be const.)
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 49
In a one-dimensional harmonic oscillator, are respectively the ground, first and the second existed states. These three states are normalized and are orthogonal to one another.
ψ1 and ψ2 are two states defined by
where a is a constant.
The value of a for which y2 is orthogonal to y1 is ___________ .
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 51
The vapour pressure p (in mm of Hg) of a solid, at temperature T, is expressed by in p = 23 - 3863/T and that of its liquid phase by In p = 19 - 3063/T. The triple point (in Kelvin) o f the material is __________ .
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 53
Vapour pressure p of solid is
logp = 23 - 3863/T ...(1)
Vapour pressure P of liquid phase is -
logp = 19 - 3063/T ...(2)
At triple point (logp)solid = (logp)liquid
A system has energy level E0, 2E0, 3E0...... . where the excited states are triply degenerate. Four non-interacting bosons are placed in this system. If the total energy of these bosons is 5E0, the number of microstates is ________ .
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 54
Four boson are placed in energy microstate so that total boson energy become 5E0, for this no. of microstates are
A car is moving with constant linear acceleration a along horizontal x-axis. A solid sphere of mass M and radius R is found rolling without slipping on the horizontal floor of the car in the same direction as seen from an inertial frame outside the car. The acceleration of the sphere in the inertial frame is ________ .
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 56
The torque about the point O is,
where a’ is the acceleration of solid sphere during its motion,
we know that,
The net force in the inertial frame is,
Put the value of in equation (1), we get
The acceleration of the sphere in the inertial frame is,
In a Young’s experiment the upper slit is covered by a thin transparent plate of refractive index 1.1 and the lower slit is covered by another thin transparent plate of refractive index 1.9 but of thickness half the thickness of the first one. It is observed that the point O on the screen, where the central maxima fall before the insertion of the plates, now has 3/4th of the original intensity. It is also observed that what used to be the 4th maxima earlier lies below O whereas the 4th minimum is above O. Calculate the thickness (μm) of the upper plate. Take the wavelength of light to be 7200 A° and neglect any absorption of light.
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 57
The path difference
Intensity at O is original intensity
where is intensity each beam
A beam of plane-polarized light falls on a polarizer which rotates about the axis of the ray with angular velocity ω= 21 rad/s. Find the energy (mJ) of light passing through the polarizer per one revolution if the flux of energy of the incident ray is equal to φ = 4.0 m W.
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 58
When the polarizer rotates with angular velocity ω its instantaneous principal direction makes angle ωt from a reference direction which we choose to be along the direction of vibration of the plane polarized incident light. The transmitted flux at this instant is ωt and the total energy passing through the polarizer per revolution is
The figure shows the results obtained during a photoelectric effect experiment where the stopping potential is plotted against the frequency of the incident radiation. The work function of the metal is close to _______ (in eV).
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 59
Kinetic energy of electron
KE = hv - W
eVs = hv - W
Consider a maxwellian distribution of the energy of the molecules of an ideal gas in three dimension . let Eav and Erms denote the average energy and the root mean square energy, respectively. The magnitude of ratio is _______
Detailed Solution for GATE Physics Mock Test Series - 7 - Question 60
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