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A hemispherical vessel is completely filled with a liquid of refractive index \( \mu \). A small coin is kept at the lowest point \( (O )\) of the vessel as shown in figure. The minimum value of the refractive index of the liquid so that a person can see  the coin from point \( E \) (at the level of the vessel) is ______.

Consider a long thin conducting wire carrying a uniform current \( I \). A particle having mass \( ''M'' \) and charge \( ''q ''\) is released at a distance \( ''a'' \) from the wire with a speed \( v_0 \) along the direction of current in the wire. The particle gets attracted to the wire due to magnetic force. The particle turns round when it is at distance \( x \) from the wire. The value of \( x \) is: [\( \mu_0 \) is vacuum permeability]

Three infinitely long wires with linear charge density \( \lambda \) are placed along the \( x \)-axis, \( y \)-axis and \( z \)-axis respectively. Which of the following denotes an equipotential surface?

For a particular ideal gas which of the following graphs represents the variation of mean squared velocity of the gas molecules with temperature?

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : In a central force field, the work done is independent of the path chosen.
Reason R : Every force encountered in mechanics does not have an associated potential energy.
In the light of the above statements, choose the most appropriate answer from the options given below:

A wire of resistance \( R \) is bent into an equilateral triangle and an identical wire is bent into a square. The ratio of resistance between the two end points of an edge of the triangle to that of the square is:

In the experiment for measurement of viscosity '\( \eta \)' of given liquid with a ball having radius \( R \), consider following statements.
A. Graph between terminal velocity \( V \) and \( R \) will be a parabola.
B. The terminal velocities of different diameter balls are constant for a given liquid.
C. Measurement of terminal velocity is dependent on the temperature.
D. This experiment can be utilized to assess the density of a given liquid.
E. If balls are dropped with some initial speed, the value of \( \eta \) will change.
Choose the correct answer from the options given below:

A bead of mass '\( m \)'slides without friction on the wall of a vertical circular hoop of radius '\( R \)' as shown in figure. The bead moves under the combined action of gravity and a massless spring \( (k) \) attached to the bottom of the hoop. The equilibrium length of the spring is '\( R \)'. If the bead is released from top of the hoop with (negligible) zero initial speed, velocity of bead, when the length of spring becomes '\( R \)', would be (spring constant is '\( k \)', \( g \) is acceleration due to gravity).

A Carnot engine \( (E) \) is working between two temperatures \( 473 \, \text{K} \) and \( 273 \, \text{K} \). In a new system two engines - engine \( E_1 \) works between \( 473 \, \text{K} \) to \( 373 \, \text{K} \) and engine \( E_2 \) works between \( 373 \, \text{K} \) to \( 273 \, \text{K} \). If \( \eta_{12} \), \( \eta_1 \) and \( \eta_2 \) are the efficiencies of the engines \( E \), \( E_1 \) and \( E_2 \), respectively, then:

A particle of mass \( 'm' \) and charge \( 'q' \) is fastened to one end ‘A’ of a massless string having equilibrium length \( l \), whose other end is fixed at point \( 'O' \). The whole system is placed on a frictionless horizontal plane and is initially at rest. If uniform electric field is switched on along the direction as shown in figure, then the speed of the particle when it crosses the \( x \)-axis is: 

Consider following statements:
A. Surface tension arises due to extra energy of the molecules at the interior as compared to the molecules at the surface, of a liquid.
B. As the temperature of liquid rises, the coefficient of viscosity increases.
C. As the temperature of gas increases, the coefficient of viscosity increases.
D. The onset of turbulence is determined by Reynold’s number.
E. In a steady flow two stream lines never intersect.

Find the equivalent resistance between two ends of the following circuit:

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : A sound wave has higher speed in solids than gases.
Reason R : Gases have higher value of Bulk modulus than solids.
In the light of the above statements, choose the correct answer from the options given below:

Due to presence of an em-wave whose electric component is given by \( E = 100 \sin (\omega t - k x) \, \mathrm{N\,C}^{-1} \), a cylinder of length \( 200 \, \text{cm} \) holds certain amount of em-energy inside it. If another cylinder of same length but half diameter than previous one holds same amount of em-energy, the magnitude of the electric field of the corresponding em-wave should be modified as:

A thin prism \( P_{1} \) with angle \( 4^\circ \) made of glass having refractive index \( 1.54 \), is combined with another thin prism \( P_{2} \) made of glass having refractive index \( 1.72 \) to get dispersion without deviation. The angle of the prism \( P_{2} \) in degrees is:

The center of mass of a thin rectangular plate (fig - $x$) with sides of length $a$ and $b$, whose mass per unit area ($\sigma$) varies as $\sigma = \frac{\sigma_0 x}{ab}$ (where $\sigma_0$ is a constant), would be __________.

A proton of mass '$m_p$' has same energy as that of a photon of wavelength '$\lambda$'. If the proton is moving at non-relativistic speed, then ratio of its de Broglie wavelength to the wavelength of photon is:

Choose the correct nuclear process from the below options:

[$p$ : proton, $n$ : neutron, $e^-$ : electron, $e^+$ : positron, $\nu$ : neutrino, $\bar{\nu}$ : antineutrino]

Two capacitors $C_1$ and $C_2$ are connected in parallel to a battery. Charge-time graph is shown below for the two capacitors. The energy stored with them are $U_1$ and $U_2$, respectively. Which of the given statements is true?

A tiny metallic rectangular sheet has length and breadth of $5\text{ mm}$ and $2.5\text{ mm}$, respectively. Using a specially designed screw gauge which has pitch of $0.75\text{ mm}$ and $15$ divisions in the circular scale, you are asked to find the area of the sheet. In this measurement, the maximum fractional error will be $\frac{x}{100}$ where $x$ is _________.

In a measurement, it is asked to find modulus of elasticity per unit torque applied on the system.

The measured quantity has dimension of $[M^a L^b T^c]$. If $b = 3$, the value of $c$ is ________.

A double slit interference experiment performed with a light of wavelength \( 600 \, \text{nm} \) forms an interference fringe pattern on a screen with \( 10^{\text{th}} \) bright fringe having its centre at a distance of \( 10 \, \text{mm} \) from the central maximum. Distance of the centre of the same \( 10^{\text{th}} \) bright fringe from the central maximum when the source of light is replaced by another source of wavelength \( 660 \, \text{nm} \) would ______ be \( \text{mm} \).

Two iron solid discs of negligible thickness have radii \( R_{1} \) and \( R_{2} \) and moment of inertia \( I_{1} \) and \( I_{2} \), respectively. For \( R_{2} = 2 R_{1} \), the ratio of \( I_{1} \) and \( I_{2} \) would be \( \frac{1}{x} \), where \( x = \) ______.

The moment of inertia of a solid disc rotating along its diameter is \( 2.5 \) times higher than the moment of inertia of a ring rotating in similar way. The moment of inertia of a solid sphere which has same radius as the disc and rotating in similar way, is \( n \) times higher than the moment of inertia of the given ring. Here, \( n = \) ______. Consider all the bodies have equal masses.