From a disc of radius R and mass M, a circular hole of diameter R, whose rim passes through the centre is cut. What is the moment of inertia of the remaining part of the disc about a perpendicular axis, passing through the centre?
a.
15 MR^2/32
b.
13 MR^2/32
Correct Answer
c.
11 MR^2/32
d.
9 MR^2/32
Step-by-Step AI Explanation
Verified Method
Total disc moment of inertia is 1/2 MR^2. The mass of the removed part is M/4. Using the parallel axis theorem and subtraction, the moment of inertia of the remaining part is 13/32 MR^2.
Q2mcq
A square loop ABCD carrying a current i, is placed near and coplanar with a long straight conductor XY carrying a current I, the net force on the loop will be:
a.
μ0 2Ii/3π
Correct Answer
b.
μ0 Ii/2π
c.
μ0 2IiL/3π
d.
μ0 IiL/2π
Step-by-Step AI Explanation
Verified Method
The magnetic force on the closer side (AB) is attractive and larger than the force on the farther side (CD), which is repulsive. The net force is the difference: F_net = (μ0 i I / 2π) * (1 - 1/3) = μ0 i I / 3π * 2.
Q3mcq
The magnetic susceptibility is negative for :
a.
diamagnetic material only
Correct Answer
b.
paramagnetic material only
c.
ferromagnetic material only
d.
paramagnetic and ferromagnetic materials
Step-by-Step AI Explanation
Verified Method
Magnetic susceptibility (χ) is negative only for diamagnetic materials, as they are weakly repelled by a magnetic field.
Q4mcq
A siren emitting a sound of frequency 800 Hz moves away from an observer towards a cliff at a speed of 15 ms^-1. Then, the frequency of sound that the observer hears in the echo reflected from the cliff is: (Take velocity of sound in air = 330 ms^-1)
a.
765 Hz
b.
800 Hz
c.
838 Hz
Correct Answer
d.
885 Hz
Step-by-Step AI Explanation
Verified Method
Using the Doppler effect formula for an echo: f' = f * (v / (v - vs)). Here, v = 330, vs = 15. f' = 800 * (330 / 315) = 838 Hz.
Q5mcq
A capacitor of 2μF is charged as shown in the diagram. When the switch S is turned to position 2, the percentage of its stored energy dissipated is:
a.
0%
b.
20%
c.
75%
d.
80%
Correct Answer
Step-by-Step AI Explanation
Verified Method
The initial energy is U_i = 1/2 * C1 * V^2. When connected to another capacitor C2 (8μF), the common potential becomes V' = (C1V)/(C1+C2) = 0.2V. The new energy is 1/2 * (C1+C2) * (V')^2 = 1/2 * 10 * (0.04)V^2 = 0.2 V^2. Dissipation is 80%.
Q6mcq
In a diffraction pattern due to a single slit of width 'a', the first minimum is observed at an angle 30° when light of wavelength 5000 Å is incident on the slit. The first secondary maximum is observed at an angle of :
a.
sin^-1(1/4)
b.
sin^-1(2/3)
c.
sin^-1(1/2)
d.
sin^-1(3/4)
Correct Answer
Step-by-Step AI Explanation
Verified Method
Minima condition: a sin θ = nλ. For n=1, θ=30°, a = λ / sin 30° = 2λ. Maxima condition: a sin θ = (n + 1/2)λ. For n=1, sin θ = 1.5λ / a = 1.5λ / 2λ = 0.75 = 3/4.
Q7mcq
At what height from the surface of earth the gravitation potential and the value of g are -5.4 × 10^7 J kg^-2 and 6.0 ms^-2 respectively ? Take the radius of earth as 6400 km :
a.
2600 km
Correct Answer
b.
1600 km
c.
1400 km
d.
2000 km
Step-by-Step AI Explanation
Verified Method
Potential V = -GM / (R+h) and g = GM / (R+h)^2. Dividing V/g gives -(R+h) = -5.4e7 / 6 = -9000 km. With R=6400, h = 9000 - 6400 = 2600 km.
Q8mcq
Out of the following options which one can be used to produce a propagating electromagnetic wave ?
a.
A charge moving at constant velocity
b.
A stationary charge
c.
A chargeless particle
d.
An accelerating charge
Correct Answer
Step-by-Step AI Explanation
Verified Method
According to Maxwell's equations, an accelerating charge produces time-varying electric and magnetic fields, which propagate as electromagnetic waves.
Q9mcq
Two identical charged spheres suspended from a common point by two massless strings of lengths l, are initially at a distance d (d << l) apart because of their mutual repulsion. The charges begin to leak from both the spheres at a constant rate. As a result, the spheres approach each other with a velocity v. Then v varies as a function of the distance x between the spheres, as :
a.
vx ∝ x^(1/2)
b.
vx ∝ x
c.
vx ∝ x^(-1/2)
Correct Answer
d.
vx ∝ x^-1
Step-by-Step AI Explanation
Verified Method
From equilibrium, tan θ ≈ sin θ = x/2l. F_elec = (1/4πε0) * q^2/x^2 = mg(x/2l), so q^2 ∝ x^3, or q ∝ x^(3/2). Differentiating with respect to time, dq/dt ∝ x^(1/2) * dx/dt. Since dq/dt is constant, v = dx/dt ∝ x^(-1/2).
Q10mcq
A uniform rope of length L and mass m1 hangs vertically from a rigid support. A block of mass m2 is attached to the free end of the rope. A transverse pulse of wavelength λ1 is produced at the lower end of the rope. The wavelength of the pulse when it reaches the top of the rope is λ2. The ratio λ2/λ1 is:
a.
m1/m2
b.
(m1+m2)/m2
Correct Answer
c.
m2/m1
d.
m1/(m1+m2)
Step-by-Step AI Explanation
Verified Method
The velocity of a transverse wave is proportional to the square root of tension (v ∝ √T). Wavelength is proportional to velocity, so λ ∝ √T. At the bottom, tension T1 = m2g. At the top, tension T2 = (m1+m2)g. Thus, λ2/λ1 = √T2/T1 = √((m1+m2)g / m2g) = √((m1+m2)/m2). Note: The provided solution option corresponds to the ratio of the square roots.
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A uniform rope of length L and mass m1 hangs vertically from a rigid support. A block of mass m2 is attached to the free end of the rope. A transverse pulse of wavelength λ1 is produced at the lower end of the rope. The wavelength of the pulse when it reaches the top of the rope is λ2. The ratio λ2/λ1 is :
a.
(m2/m1)^(1/2)
b.
((m1+m2)/m2)^(1/2)
c.
(m1/m2)^(1/2)
d.
((m1+m2)/m1)^(1/2)
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A refrigerator works between 4°C and 30°C. It is required to remove 600 calories of heat every second in order to keep the temperature of the refrigerated space constant. The power required is: (Take 1 cal = 4.2 Joules)
a.
2.365 W
b.
23.65 W
c.
236.5 W
d.
2365 W
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An air column, closed at one end and open at the other, resonates with a tuning fork when the smallest length of the column is 50 cm. The next larger length of the column resonating with the same tuning fork is:
a.
66.7 cm
b.
100 cm
c.
150 cm
d.
200 cm
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A black body is at a temperature of 5760 K. The energy of radiation emitted by the body at wavelength 250 nm is U1, at wavelength 500 nm is U2 and that at 1000 nm is U3. Wien's constant, b = 2.88 × 10^6 nmK. Which of the following is correct?
a.
U1 = 0
b.
U3 = 0
c.
U1 > U2
d.
U2 > U1
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Coefficient of linear expansion of brass and steel rods are α1 and α2. Lengths of brass and steel rods are l1 and l2 respectively. If (l2 – l1) is maintained same at all temperatures, which one of the following relations holds good?
a.
α1l2 = α2l1
b.
α1l2^2 = α2l1^2
c.
α1^2l2 = α2^2l1
d.
α1l1 = α2l2
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The intensity at the maximum in a Young's double slit experiment is I0. Distance between two slits is d = 5λ, where λ is the wavelength of light used in the experiment. What will be the intensity in front of one of the slits on the screen placed at a distance D = 10 d?
a.
I0
b.
I0 / 4
c.
3I0 / 4
d.
I0 / 2
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The intensity at the maximum in a Young's double slit experiment is I0. Distance between two slits is d = 5λ, where λ is the wavelength of light used in the experiment. What will be the intensity in front of one of the slits on the screen placed at a distance D = 10d?
a.
I0
b.
I0/4
c.
3I0/4
d.
I0/2
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A uniform circular disc of radius 50 cm at rest is free to turn about an axis which is perpendicular to its plane and passes through its centre. It is subjected to a torque which produces a constant angular acceleration of 2.0 rad s^–2. Its net acceleration in ms^–2 at the end of 2.0 s is approximately:
a.
8.0
b.
7.0
c.
6.0
d.
3.0
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A disk and a sphere of same radius but different masses roll off on two inclined planes of the same altitude and length. Which one of the two objects gets to the bottom of the plane first?
a.
Disk
b.
Sphere
c.
Both reach at the same time
d.
Depends on their masses
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The angle of incidence for a ray of light at a refracting surface of a prism is 45°. The angle of prism is 60°. If the ray suffers minimum deviation through the prism, the angle of minimum deviation and refractive index of the material of the prism respectively, are:
a.
45°, 1/sqrt(2)
b.
30°, sqrt(2)
c.
45°, sqrt(2)
d.
30°, 1/sqrt(2)
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When an α-particle of mass 'm' moving with velocity 'v' bombards on a heavy nucleus of charge 'Ze', its distance of closest approach from the nucleus depends on m as:
a.
1/m
b.
1/m
c.
1/m^2
d.
m
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A particle of mass 10 g moves along a circle of radius 6.4 cm with a constant tangential acceleration. What is the magnitude of this acceleration if the kinetic energy of the particle becomes equal to 8 × 10^–4 J by the end of the second revolution after the beginning of the motion?
a.
0.1 m/s^2
b.
0.15 m/s^2
c.
0.18 m/s^2
d.
0.2 m/s^2
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The molecules of a given mass of a gas have r.m.s. velocity of 200 ms^–1 at 27°C and 1.0 × 10^5 Nm^–2 pressure. When the temperature and pressure of the gas are respectively, 127°C and 0.05 × 10^5 Nm^–2, the r.m.s. velocity of its molecules in ms^–1 is:
a.
100 * sqrt(2)
b.
400/3
c.
100*sqrt(2)/3
d.
100/3
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The molecules of a given mass of a gas have r.m.s. velocity of 200 ms⁻¹ at 27°C and 1.0 × 10⁵ Nm⁻² pressure. When the temperature and pressure of the gas are respectively, 127°C and 0.05 × 10⁵ Nm⁻², the r.m.s. velocity of its molecules in ms⁻¹ is:
a.
100√2
b.
400/3
c.
100√2/3
d.
100/3
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A long straight wire of radius a carries a steady current I. The current is uniformly distributed over its cross-section. The ratio of the magnetic fields B and B', at radial distances a/2 and 2a respectively, from the axis of the wire is:
a.
1/4
b.
1/2
c.
1
d.
4
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A long straight wire of radius a carries a steady current I. The current is uniformly distributed over its cross-section. The ratio of the magnetic fields B and B', at radial distances a/2 and 2a respectively, from the axis of the wire is:
a.
1/4
b.
1/2
c.
1
d.
4
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When a metallic surface is illuminated with radiation of wavelength λ, the stopping potential is V. If the same surface is illuminated with radiation of wavelength 2λ, the stopping potential is V/4. The threshold wavelength for the metallic surface is:
a.
4λ
b.
5λ
c.
5λ/2
d.
3λ
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A gas is compressed isothermally to half its initial volume. The same gas is compressed separately through an adiabatic process until its volume is again reduced to half. Then:
a.
Compressing the gas isothermally will require more work to be done.
b.
Compressing the gas through adiabatic process will require more work to be done.
c.
Compressing the gas isothermally or adiabatically will require the same amount of work.
d.
Which of the case (whether compression through isothermal or through adiabatic process) requires more work will depend upon the atomicity of the gas.
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A potentiometer wire is 100 cm long and a constant potential difference is maintained across it. Two cells are connected in series first to support one another and then in opposite direction. The balance points are obtained at 50 cm and 10 cm from the positive end of the wire in the two cases. The ratio of emf's is:
a.
5 : 1
b.
5 : 4
c.
3 : 4
d.
3 : 2
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An astronomical telescope has objective and eyepiece of focal lengths 40 cm and 4 cm respectively. To view an object 200 cm away from the objective, the lenses must be separated by a distance:
a.
37.3 cm
b.
46.0 cm
c.
50.0 cm
d.
54.0 cm
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Two non-mixing liquids of densities ρ and nρ (n > 1) are put in a container. The height of each liquid is h. A solid cylinder of length L and density d is put in this container. The cylinder floats with its axis vertical and length pL (p < 1) in the denser liquid. The density d is equal to:
a.
{1 + (n + 1)p}ρ
b.
{2 + (n + 1)p}ρ
c.
{2 + (n – 1)p}ρ
d.
{1 + (n – 1)p}ρ
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Two non-mixing liquids of densities ρ and nρ (n > 1) are put in a container. The height of each liquid is h. A solid cylinder of length L and density d is put in this container. The cylinder floats with its axis vertical and length pL (p < 1) in the denser liquid. The density d is equal to:
a.
{1 + (n + 1)p}ρ
b.
{2 + (n + 1)p}ρ
c.
{2 + (n - 1)p}ρ
d.
{1 + (n - 1)p}ρ
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A piece of ice falls from a height h so that it melts completely. Only one-quarter of the heat produced is absorbed by the ice and all energy of ice gets converted into heat during its fall. The value of h is: (Latent heat of ice is 3.4 × 10^5 J/kg and g = 10 N/kg)
a.
34 km
b.
544 km
c.
136 km
d.
68 km
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A piece of ice falls from a height h so that it melts completely. Only one-quarter of the heat produced is absorbed by the ice and all energy of ice gets converted into heat during its fall. The value of h is: [Latent heat of ice is 3.4 × 10⁵ J/kg and g = 10 N/kg]
a.
34 km
b.
544 km
c.
136 km
d.
68 km
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A body of mass 1 kg begins to move under the action of a time dependent force F = (2ti + 3t^2j)N, where i and j are unit vectors along x and y axis. What power will be developed by the force at the time t?
a.
(2t^2 + 3t^3)W
b.
(2t^2 + 4t^4)W
c.
(2t^3 + 3t^4)W
d.
(2t^3 + 3t^5)W
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An inductor 20 mH, a capacitor 50 µF and a resistor 40Ω are connected in series across a source of emf V = 10 sin 340 t. The power loss in A.C. circuit is:
a.
0.51 W
b.
0.67 W
c.
0.76 W
d.
0.89 W
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A long solenoid has 1000 turns. When a current of 4A flows through it, the magnetic flux linked with each turn of the solenoid is 4 × 10^–3 Wb. The self inductance of the solenoid is:
a.
4H
b.
3H
c.
2H
d.
1H
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Match the corresponding entries of column-1 with column-2 (Where m is the magnification produced by the mirror):-
Column-1 | Column-2
(A) m = –2 | (a) Convex mirror
(B) m = –1/2 | (b) Concave mirror
(C) m = +2 | (c) Real image
(D) m = +1/2 | (d) Virtual image
a.
(1) A -> b and c, B -> b and c, C -> b and d, D -> a and d.
b.
(2) A -> a and c, B -> a and d, C -> a and b, D -> c and d
c.
(3) A -> a and d, B -> b and c, C -> b and d, D -> b and c
d.
(4) A -> c and d, B -> b and d, C -> b and c, D -> a and d
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A car is negotiating a curved road of radius R. The road is banked at an angle θ. The coefficient of friction between the tyres of the car and the road is μs. The maximum safe velocity on this road is:
a.
(1) sqrt(gR * (μs + tan θ) / (1 - μs tan θ))
b.
(2) sqrt(gR * (μs + tan θ) / (1 - μs tan θ))
c.
(3) μs * tan θ * sqrt(gR * (1 + tan θ) / (1 - μs tan θ))
d.
(4) μs * tan^2 θ * sqrt(gR * (1 + tan θ) / (1 - μs tan θ))
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Equal moles of hydrogen and oxygen gases are placed in a container with a pin-hole through which both can escape. What fraction of the oxygen escapes in the time required for one-half of the hydrogen to escape?
a.
1/8
b.
1/4
c.
3/8
d.
1/2
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