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JEE Advance | PYQ

Young’s modulus of elasticity \( Y \) is expressed in terms of three derived quantities, namely, the gravitational constant \( G \), Plank’s constant \( h \) and the speed of light \( c \), as \( Y = c^{\alpha} h^{\beta} G^{\gamma} \). Which of the following is the correct option?

A particle of mass \( m \) is moving in the \( xy \)-plane such that its velocity at a point \( (x, y) \) is given as \( \vec{v} = \alpha(y\hat{x} + 2x\hat{y}) \), where \( \alpha \) is a non-zero constant. What is the force \( \vec{F} \) acting on the particle?

An ideal gas is in thermodynamic equilibrium. The number of degree of freedom of a molecule of the gas is \( n \). The internal energy of one mole of the gas is \( U_{n} \) and the speed of sound in the gas is \( v_{n} \). At a fixed temperature and pressure, which of the following is the correct option?

A monochromatic light wave is incident normally on a glass slab of thickness \( d \), as shown in the figure. The refractive index of the slab increases linearly from \( n_{1} \) to \( n_{2} \) over the height \( h \). Which of the following statement(s) is (are) true about the light wave emerging out of the slab?

An annular disk of mass \( M \), inner radius \( a \) and outer radius \( b \) is placed on a horizontal surface with coefficient of friction \( \mu \), as shown in the figure. At some time, an impulse \( J_0 \, \hat{x} \) is applied at a height \( h \) above the centre of the disk. If \( h = h_m \) then the disk rolls without slipping along the x-axis. Which of the following statement(s) is (are) correct?

The electric field associated with an electromagnetic wave propagating in a dielectric medium is given by \( \vec{E} = 30 ( 2\hat{x} + \hat{y} ) \sin \left[ 2\pi \left( 5\times 10^{14} t - \dfrac{10^{7}}{3} z \right) \right] \, \text{V m}^{-1} \). Which of the following option(s) is (are) correct?
[Given: The speed of light in vacuum, \( c = 3 \times 10^{8} \, \text{m s}^{-1} \)]

A thin circular coin of mass \( 5 \, \text{gm} \) and radius \( \dfrac{4}{3} \, \text{cm} \) is initially in a horizontal xy-plane. The coin is tossed vertically up (\( +z \) direction) by applying an impulse of \( \sqrt{\dfrac{\pi}{2}} \times 10^{-2} \, \text{N- s} \) at a distance \( \dfrac{2}{3} \, \text{cm} \) from its centre. The coin spins about its diameter and moves along the \( +z \) direction. By the time the coin reaches back to its initial position, it completes \( n \) rotations. The value of \( n \) is ________.
[Given the acceleration due to gravity \( g = 10 \, \text{m s}^{-2} \)]

A rectangular conducting loop of length \( 4 \, \text{cm} \) and width \( 2 \, \text{cm} \) is in the xy-plane, as shown in the figure. It is being moved away from a thin and long conducting wire along the direction \( \dfrac{\sqrt{3}}{2} \, \hat{x} + \dfrac{1}{2} \, \hat{y} \) with a constant speed \( v \). The wire is carrying a steady current \( I = 10 \, \text{A} \) in the positive x-direction. A current of \( 10 \, \mu\text{A} \) flows through the loop when it is at a distance \( d = 4 \, \text{cm} \) from the wire. If the resistance of the loop is \( 0.1 \, \Omega \) then the value of \( v \) ________ \( \text{m s}^{-1} \).
[Given: The permeability of the free space \( \mu_0 = 4\pi \times 10^{-7} \, \text{N A}^{-2} \)]

A string of length \( 1 \, \text{m} \) and mass \( 2 \times 10^{-5} \, \text{kg} \) is under tension \( T \). When the string vibrates, two successive harmonics are found to occur at frequencies \( 750 \, \text{Hz} \) and \( 1000 \, \text{Hz} \). The value of tension \( T \) is _______ Newton.

An incompressible liquid is kept in a container having a weightless piston with a hole. A capillary tube of inner radius \( 0.1 \, \text{mm} \) is dipped vertically into the liquid through the airtight piston hole, as shown in the figure. The air in the container is isothermally compressed from its original volume \( V_0 \) to \( \frac{100}{101} V_0 \) with the movable piston. Considering air as an ideal gas, the height (\( h \)) of the liquid column in the capillary above the liquid level in \( \text{cm} \) is ____________.
[Given : Surface tension of the liquid is \( 0.075 \, \text{N m}^{-1} \), atmospheric pressure is \( 10^{5} \, \text{N m}^{-2} \) acceleration due to gravity (\( g \)) is \( 10 \, \text{m s}^{-2} \), density of the liquid is \( 10^{3} \, \text{kg m}^{-3} \) and contact angle of capillary surface with the liquid is zero]

In a radioactive decay process, the activity is defined as \( A = \dfrac{-dN}{dt} \), where \( N(t) \) is the number of radioactive nuclei at time \( t \). Two radioactive sources, \( S_1 \) and \( S_2 \) have same activity at time \( t = 0 \). At a later time, the activities of \( S_1 \) and \( S_2 \) are \( A_1 \) and \( A_2 \), respectively. When \( S_1 \) and \( S_2 \) have just completed their \( 3^{\text{rd}} \) and \( 7^{\text{th}} \) half-lives, respectively, the ratio \( \dfrac{A_1}{A_2} \) is __________.

One mole of an ideal gas undergoes two different cyclic process I and II, as shown in the P–V diagrams below. In cycle I, process \( a, b, c \) and \( d \) are isobaric, isothermal, isobaric and isochoric, respectively. In cycle II, process \( a', b', c' \), and \( d' \) are isothermal, isochoric, isobaric and isochoric, respectively. The total work done during cycle I is \( W_I \) and that during cycle II is \( W_{II} \). The ratio \( \dfrac{W_I}{W_{II}} \) is __________.

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\( S_1 \) and \( S_2 \) are two identical sound sources of frequency \( 656 \, \text{Hz} \). The source \( S_1 \) is located at \( O \) and \( S_2 \) moves anti-clockwise with a uniform speed \( 4\sqrt{2} \, \text{m s}^{-1} \) on a circular path around \( O \), as shown in the figure. There are three points \( P, Q \) and \( R \) on this path such that \( P \) and \( R \) are diametrically opposite while \( Q \) is equidistant from them. A sound detector is placed at point \( P \). The source \( S_1 \) can move along direction \( OP \).
[Given : The speed of sound in air is \( 324 \, \text{m s}^{-1} \) ]

When only \( S_2 \) is emitting sound and it is at \( Q \), the frequency of sound measured by the detector in \( {H_2} \) is_________. 

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\( S_1 \) and \( S_2 \) are two identical sound sources of frequency \( 656 \, \text{Hz} \). The source \( S_1 \) is located at \( O \) and \( S_2 \) moves anti-clockwise with a uniform speed \( 4\sqrt{2} \, \text{m s}^{-1} \) on a circular path around \( O \), as shown in the figure. There are three points \( P, Q \) and \( R \) on this path such that \( P \) and \( R \) are diametrically opposite while \( Q \) is equidistant from them. A sound detector is placed at point \( P \). The source \( S_1 \) can move along direction \( OP \).
[Given : The speed of sound in air is \( 324 \, \text{m s}^{-1} \) ]

Consider both sources emitting sound. When \( S_2 \) is at \( R \) and \( S_1 \) approaches the detector with a speed \( 4 \, \text{m s}^{-1} \), the beat frequency measured by the detector is ______ \( \text{Hz} \).

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A cylindrical furnace has height \( H \) and diameter \( D \) both \( 1 \, \text{m} \). It is maintained at temperature \( 360 \, \text{K} \). The air gets heated inside the furnace at constant pressure \( P_a \) and its temperature becomes \( T = 360 \, \text{K} \). The hot air with density \( \rho \) rises up a vertical chimney of diameter \( d = 0.1 \, \text{m} \) and height \( h = 9 \, \text{m} \) above the furnace and exits the chimney (see figure). As a result, atmospheric air of density \( \rho_a = 1.2 \, \text{kg m}^{-3} \), pressure \( P_a \) and temperature \( T_a = 300 \, \text{K} \) enters the furnace. Assume air as an ideal gas, neglect the variations in \( \rho \) and \( T \) inside the chimney and the furnace. Also ignore the viscous effects.
[Given :The acceleration due to gravity \( g = 10 \, \text{ms}^{-2} \) and \( \pi = 3.14 \)]

Considering the air flow to be streamline, the steady mass flow rate of air exiting the chimney is ______ \( \text{gm s}^{-1} \).

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A cylindrical furnace has height \( H \) and diameter \( D \) both \( 1 \, \text{m} \). It is maintained at temperature \( 360 \, \text{K} \). The air gets heated inside the furnace at constant pressure \( P_a \) and its temperature becomes \( T = 360 \, \text{K} \). The hot air with density \( \rho \) rises up a vertical chimney of diameter \( d = 0.1 \, \text{m} \) and height \( h = 9 \, \text{m} \) above the furnace and exits the chimney (see figure). As a result, atmospheric air of density \( \rho_a = 1.2 \, \text{kg m}^{-3} \), pressure \( P_a \) and temperature \( T_a = 300 \, \text{K} \) enters the furnace. Assume air as an ideal gas, neglect the variations in \( \rho \) and \( T \) inside the chimney and the furnace. Also ignore the viscous effects.
[Given :The acceleration due to gravity \( g = 10 \, \text{ms}^{-2} \) and \( \pi = 3.14 \)]

When the chimney is closed using a cap at the top, a pressure difference \( \Delta P \) develops between the top and the bottom surfaces of the cap. If the changes in the temperature and density of the hot air, due to the stoppage of air flow, are negligible then the value of \( \Delta P \) is ______ \( \text{N m}^{-2} \).