If an integral curve of the differential equation = 1 passes (0.0) and (α,1), then α is equal to
Given differential equation is
So, the required solution is
y(0) = 0
0 = 0 - e° + c ⇒c = 1
Then the solution becomes
xey = yey - ey + 1
αe = e - e+1
α = e-1
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
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 )
A solid uniform disc of mass m rolls without slipping down a fixed inclined plank with an acceleration α. The frictional force on the disc due to surface of the plane is
We know that, the angular all eleration
and now frictional force
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.
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
A glass plate of thickness 30 mm is placed just below a well-focused microscope for focusing it again the microscope should be moved through a distance of :
The shift is path when an obstacle of refractive index μ and of length is kept in way of a ray is given as path difference
= ( μ - 1 ) t = (1.5 - 1 ) 30 mm = 15 mm
The Clausius-clapeyron equation indicates that the increase of pressure increases the melting point.
Clausius-clapeyron equation is given as
⇒ If volume increases the pressure increases with T.
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 :
According to de-Broglie the wavelength associated with a particle of mass m moving with velocity v is given as
KE x 2m = (mv)2
mv = √2m E
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_______ .
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
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
∵ energy increases.
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) ?
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 ?
The energy eigenvalue, In one dimensional box is
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
We know that
The electrostatic energy
for the spherical shell
for the solid sphere
Le t where c is the boundary (oriented anticlockwise ) of the region in the first quadrant bounded by y = 0, x2 + y2 = 1 and x = 0. The value of I is
By green’s theorem
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 ?
Now probability density at time t
putting t = π
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.
Equation of motion
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 _____ .
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
So, total magnetic field at point P is
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
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
The black body at a temperature of 6000k emits a radiation whose intensity spectrum peaks at 600 nm. If the temperature is reduced to 5% k, the spectrum will peak at
By the kliien’s displacement law
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
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
A 150Ω resister, a l0μf capacitor a 0.1 H inductor are connected in series to an AC source operating at an angular frequency ω, for what ω the combination acts as a pure resistive load.
Consider the LCR circuit of impedance
The combination acts as purely resistive circuit if
Consider two electromagnetic plane waves propagating in vacuum with their electric field vectors
then the averaged pointing vector for the resultant wave
According to the question, electric fields are
Net electric field
E = E1 + E2
A convex lens of focal length 24 cm (μ = 1.5) is totally immersed in water. Its focal length in water is _______ .
When the lens is in air
When the lens is in water
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_______ .
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
What is the maximum range along the dashed line attained by the water stream coming out at B from a thin tube of the water tank assembly shown in the figure ?
Assume h = 10m, L = 2m and θ = 30°
Let V be the velocity of the water coming out from the point B. Applying conservation mechanical energy
PEA =KEB + PEB
The water ejected at point B will fallow the projective motion.
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
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
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.
In the differential mode :-
In differential mode |V1 = V2| but with opposite polarity & in common mode ( V1= V2) but with same polarity.
In the op-amp circuit shown in the figure below, the input voltage V1 is 1V. The value of the output Vo is
due to virtual grand concept V = 0 at point A
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
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
A BJT circuit has β = 50 & Vcc = 20V & uses a potential divider bias circuit with Rc = 2kΩ, R1=100kΩ & R2= 5kΩ. then Assuming Vbe = 0.2V which of the following statements is/are correct :-
In case of potential divider biasing,
Applying kirchoffs voltage law to base - Emitter circuit
Expressing Current in ‘mA ’ & resistances in ‘ kΩ ’ where as.
0.952 = lb (4.76) + (0.2) + (51 x lb x 0.1)
Solving this above Eqn we get ;
[lb = 0.076mA]
Vce = 20 - ( 3.8 x 2 ) - ( 3.876 x 0.1 )
(Vce = 12V)
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 :-
For the given which of the following is/are correct
Let f(x) = x, 0 < x < 2π
and the Fourier series can be expressed as
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
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
for element B
for element C,
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 ?
∴ 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
Time taken to reach earth
t = 2.19 μ sec (just before the life time)
A uniform rod of mass m = 2kg and length L = 0.5 m is sliding along two mutually perpendicular smooth walls with the two ends P and Q having velocities Vp = 4m/sec and VQ = 3m/sec as shown, then
ω = 10 rad/se
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
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
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 -
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.
One mole of an ideal gas in initial state A undergoes a cyclic process ABCA, as shown in the figure.
Its pressure at A is Po. choose the correct options from the following
Information regarding P and T at C cannot be obtaining from the given graph. Unless it is mentioned that line BC passes through origin or not.
The muon has mass and mean lifetime 2.2 ms in its rest frame. The mean distance traversed by a muon of energy before decaying is approximately _____ km.
We know from einstein law
given E = 315 MeV
We know mean distance travelled by muon is
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 __________ .
q(l + λ) = 0
λ = -1
Two particles of identical mass move in circular orbits under a central potential V (r) = Let I1 and I2 be the angular momenta r1 r2 be radii of the orbits respectively. If the value of is
We know that, central force is conservative in central potential field
Here, (-) sign indicates the direction of force which is radially inward
In circular orbit,
Ratio of angular momenta =
Squaring both sides, we have
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.
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.
The Potential is
V = x[x2 - 2x + x]
V = x3 - 2x2 + 4x
Given x0 = 2 then
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 _________ .
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)
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 ).
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.)
The output voltage of a certain op-amp appears as shown in figure in response to a step input calculate its slow rate
The slow rate (SR)
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 ___________ .
for orthogonal condition -
1 + 2 + 3a = 0
3a = - 3
a = -1
The magnetic field (in A m-1 ) inside a long solid cylindrical conductor of radius a
What is the total current ( in A ) in the conductor ?
We know M.F(B) due to long solid cylindrical having radius a=0. 1m is
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 __________ .
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 ________ .
Four boson are placed in energy microstate so that total boson energy become 5E0, for this no. of microstates are
So, possible microstates are ‘5’.
A particle of mass 2 kg is moving such that at time t its position, in metre, is given by The angular momentum of the particle at t = 2 s about the origin, in kg m2 s-1, is _________ .
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 ________ .
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.
The path difference
Intensity at O is original intensity
where is intensity each beam
Also, fringe shift =
For n = 4 ,
Total thickness =
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.
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).
Kinetic energy of electron
KE = hv - W
eVs = hv - W
At v = 0, Vs = - 2V
W = 2eV
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 _______