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Identify the valid statements relevant to the given circuit at the instant when the key is closed.
A. There will be no current through resistor \( R \).
B. There will be maximum current in the connecting wires.
C. Potential difference between the capacitor plates \( A \) and \( B \) is minimum.
D. Charge on the capacitor plates is minimum.
Choose the correct answer from the options given below:

A solid sphere of mass \( 'm' \) and radius \(' r '\) is allowed to roll without slipping from the highest point of an inclined plane of length \( 'L' \) and makes an angle \( 30^\circ \) with the horizontal. The speed of the particle at the bottom of the plane is \( v_1 \). If the angle of inclination is increased to \( 45^\circ \) while keeping \( L \) constant. Then the new speed of the sphere at the bottom of the plane is \( v_2 \). The ratio \( v_1^2 : v_2^2 \) is:

A gun fires a lead bullet of temperature \( 300 \, \text{K} \) into a wooden block. The bullet having melting temperature of \( 600 \, \text{K} \) penetrates into the block and melts down. If the total heat required for the process is \( 625 \, \text{J} \), then the mass of the bullet is ______ grams.
(Latent heat of fusion of lead \( = 2.5 \times 10^{4} \, \text{J} \, \text{Kg}^{-1} \) and specific heat capacity of lead \( = 125 \, \text{J} \, \text{Kg}^{-1} \, \text{K}^{-1} \))

The electric flux is \( \phi = \alpha \sigma + \beta \lambda \), where \( \lambda \) and \( \sigma \) are linear and surface charge density, respectively. \( \left( \frac{\alpha}{\beta} \right) \) represents:

Refer to the circuit diagram given in the figure, which of the following observations are correct?
A. Total resistance of circuit is \( 6 \, \Omega \).
B. Current in Ammeter is \( 1 \, \text{A} \).
C. Potential across AB is \( 4 \) Volts.
D. Potential across CD is \( 4 \) Volts.
E. Total resistance of the circuit is \( 8 \, \Omega \).
Choose the correct answer from the options given below:

A light hollow cube of side length \( 10 \, \text{cm} \) and mass \( 10 \, \text{g} \), is floating in water. It is pushed down and released to execute simple harmonic oscillations. The time period of oscillations is \( y \pi \times 10^{-2} \, \text{s} \), where the value of \( y \) is: (Acceleration due to gravity, \( g = 10 \, \text{m}/\text{s}^{2} \), density of water \( = 10^{3} \, \text{kg}/\text{m}^{3} \) )

A radioactive nucleus \( n_{2} \) has \( 3 \) times the decay constant as compared to the decay constant of another radioactive nucleus \( n_{1} \). If initial number of both nuclei are the same, what is the ratio of number of nuclei of \( n_{2} \) to the number of nuclei of \( n_{1} \), after one half-life of \( n_{1} \)?

A spherical surface of radius of curvature \( R \), separates air from glass (refractive index \( = 1.5 \)). The centre of curvature is in the glass medium. A point object ‘O’ placed in air on the optic axis of the surface, so that its real image is formed at ‘I’ inside glass. The line \( OI \) intersects the spherical surface at \( P \) and \( PO = PI \). The distance \( PO \) equals to:

The position of a particle moving on \( x \)-axis is given by \( x(t) = A \sin t + B \cos^{2} t + C t^{2} + D \), where \( t \) is time. The dimension of \( \frac{ABC}{D} \) is:

Consider a moving coil galvanometer (MCG):
A. The torsional constant in moving coil galvanometer has dimentions \( [ML^{2}T^{-2}] \).
B. Increasing the current sensitivity may not necessarily increase the voltage sensitivity.
C. If we increase number of turns \( (N) \) to its double \( (2N) \), then the voltage sensitivity doubles.
D. MCG can be converted into an ammeter by introducing a shunt resistance of large value in parallel with galvanometer.
E. Current sensitivity of MCG depends inversely on number of turns of coil.
Choose the correct answer from the options given below:

Match the LIST-I with LIST-II.

Choose the correct answer from the options given below:

A sub-atomic particle of mass \( 10^{-30} \, \text{kg} \) is moving with a velocity \( 2.21 \times 10^{6} \, \text{m} / \text{s} \). Under the matter wave consideration, the particle will behave closely like _____________________. \( (h = 6.63 \times 10^{-34} \, \text{J s}) \)

Given below are two statements:
Statement I: The hot water flows faster than cold water.
Statement II: Soap water has higher surface tension as compared to fresh water.
In the light above statements, choose the correct answer from the options given below:

A point particle of charge \( Q \) is located at \( P \) along the axis of an electric dipole \( 1 \) at a distance \( r \) as shown in the figure. The point \( P \) is also on the equatorial plane of a second electric dipole \( 2 \) at a distance \( r \). The dipoles are made of opposite charge \( q \) separated by a distance \( 2a \). For the charge particle at \( P \) not to experience any net force, which of the following correctly describes the situation?

The motion of an airplane is represented by velocity-time graph as shown below. The distance covered by airplane in the first \( 30.5 \) second is ______ km.

The electric field of an electromagnetic wave in free space is \( \vec{E} = 57 \cos \!\left[ 7.5 \times 10^{6} t - 5 \times 10^{-3}(3x + 4y) \right] (4\hat{i} - 3\hat{j}) \, \text{N} / \text{C} \). The associated magnetic field in Tesla is:

Regarding self-inductance:
A. The self-inductance of the coil depends on its geometry.
B. Self-inductance does not depend on the permeability of the medium.
C. Self-induced e.m.f. opposes any change in the current in a circuit.
D. Self-inductance is electromagnetic analogue of mass in mechanics.
E. Work needs to be done against self-induced e.m.f. in establishing the current.
Choose the correct answer from the options given below:

Consider a circular disc of radius \( 20 \, \text{cm} \) with centre located at the origin. A circular hole of radius \( 5 \, \text{cm} \) is cut from this disc in such a way that the edge of the hole touches the edge of the disc. The distance of centre of mass of residual or remaining disc from the origin will be:

Given a thin convex lens (refractive index \( \mu_2 \)), kept in a liquid (refractive index \( \mu_1 \), \( \mu_1 < \mu_2 \)) having radii of curvatures \( |R_1| \) and \( |R_2| \). Its second surface is silver polished. Where should an object be placed on the optic axis so that a real and inverted image is formed at the same place?

A positive ion \( A \) and a negative ion \( B \) has charges \( 6.67 \times 10^{-19} \, \text{C} \) and \( 9.6 \times 10^{-10} \, \text{C} \), and masses \( 19.2 \times 10^{-27} \, \text{kg} \) and \( 9 \times 10^{-27} \, \text{kg} \) respectively. At an instant, the ions are separated by a certain distance \( r \). At that instant the ratio of the magnitudes of electrostatic force to gravitational force is \( P \times 10^{-13} \), where the value of \( P \) is __________.
(Take \( \dfrac{1}{4\pi \varepsilon_0} = 9 \times 10^{9} \, \text{Nm}^2 \text{C}^{-1} \) and universal gravitational constant as \( 6.67 \times 10^{-11} \, \text{Nm}^2\text{kg}^{-2} \))

An ideal gas initially at \( 0^\circ \text{C} \) temperature, is compressed suddenly to one fourth of its volume. If the ratio of specific heat at constant pressure to that at constant volume is \( \tfrac{3}{2} \), the change in temperature due to the thermodynamic process is __________ \( \text{K} \).

A force \( \mathbf{f} = x^{2}y\,\hat{i} + y^{2}\,\hat{j} \) acts on a particle in a plane \( x + y = 10 \). The work done by this force during a displacement from \( (0, 0) \) to \( (4\,\text{m}, 2\,\text{m}) \) is __________ \( \text{Joule} \) (round off to the nearest integer).

In the given circuit the sliding contact is pulled outwards such that electric current in the circuit changes at the rate of \( 8 \, \text{A/s} \). At an instant when \( R \) is \( 12 \, \Omega \), the value of the current in the circuit will be ________ A.

Two particles are located at equal distance from origin. The position vectors of those are represented by \( \,\vec{A} = 2\hat{i} + 3n\hat{j} + 2\hat{k} \, \) and \( \,\vec{B} = 2\hat{i} - 2\hat{j} + 4p \hat{k} \, \), respectively. If both the vectors are at right angle to each other, the value of \( n^{-1} \) is ________.