JEE MainsJEE AdvanceBITSAT

JEE Advance | PYQ

Consider a circuit consisting of a capacitor of capacitance \( C \) and a coil with \( N \) turns per unit length, cross sectional area \( S \) and length \( d \), where \( d^{2} \gg S \). There is another coil of length \( \dfrac{d}{2} \), cross sectional area \( \dfrac{S}{2} \) and \( 2N \) turns per unit length completely inside the larger coil, as shown in the figure. The ends of this smaller coil are connected with each other by an insulated conducting wire. The self-inductance of the larger coil is \( L \). Neglecting edge effects and all the Ohmic resistances, the resonant frequency of the circuit is:

A solid cylinder of radius \( R \) rolls without slipping with a center of mass speed \( v_{0} = \sqrt{\dfrac{gR}{3}} \) on a horizontal surface with a vertical edge, as shown in the figure. Here, \( g \) is the acceleration due to the gravity. At the moment when the cylinder loses contact with the surface due to rotation around the corner, the speed of its center of mass is:

A double convex lens made of glass of refractive index \( 1.5 \) and radii of curvature of the curved surfaces \( 20 \text{cm} \) each is immersed in a liquid of refractive index \( n_{L} \). The correct plot showing the variation of the power, in the units of diopter \( (D) \), as a function of \( n_{L} \) is:

Consider a hydrogen atom with \( v_{k} \), \( r_{k} \), and \( K_{k} \) denoting the velocity, orbital radius and kinetic energy of the electron in the \( k^{\text{th}} \) orbit, respectively. The electron undergoes a transition from the \( n^{\text{th}} \) orbit, emitting radiation corresponding to the Lyman series. Considering \( h \) to be the Planck’s constant and \( \epsilon_{0} \) the permittivity of the free space, the correct statement(s) is/are:

A particle is thrown with a speed \( v \) from a point \( O \) at an angle \( \theta \) with the horizontal plane such that it passes through the point \( P \) at a height of \( 1 \, \mathrm{m} \) and horizontal distance of \( 5 \, \mathrm{m} \) from \( O \), as shown in the figure. If acceleration due to gravity is \( g \, \mathrm{m\,s^{-2}} \), then the correct statement(s) is/are:

A quasi-static cycle of a monoatomic ideal gas contains an isothermal process \( (ab) \), followed by an isochoric process \( (bc) \) and an adiabatic process \( (ca) \) as shown in the figure. The volumes of the gas are \( V_{1} \) and \( V_{2} \) at \( a \) and \( b \), respectively. If the cycle has heat input \( Q_{\text{in}} \) and output \( Q_{\text{out}} \), then the efficiency of the cycle is defined as \( \eta = \frac{Q_{\text{in}} - Q_{\text{out}}}{Q_{\text{in}}} \). The correct statement(s) is/are: [Given: \( \ln 2 \approx 0.7 \) ]

The electric field associated with an electromagnetic wave travelling in vacuum is given by \( E_{0}\sin(3y + 4z + \omega t)\,\hat{i} \), where \( \omega \) is the angular frequency. All quantities are in SI units. The correct statement(s) about this wave is/are: [Given: speed of light in vacuum \( c = 3 \times 10^{8} \, \mathrm{ms^{-1}} \) .]

A tank contains two immiscible liquids of densities \( 6\rho \) and \( 2\rho \). The higher density liquid is filled up to a height \( L/2 \) from the bottom. A thin rod of density \( \rho \) and length \( L \) is fully immersed and hinged at the bottom so that it can oscillate freely, as shown in the figure. If the rod is slightly disturbed from its equilibrium, the time period of small oscillations is \( \dfrac{2\pi}{n}\sqrt{\dfrac{L}{g}} \), where \( g \) is the acceleration due to gravity.
The value of \( n \) is:

As shown in the figure, five Carnot engines, each with efficiency \( \eta \) and same number of cycles per unit time, are operating between six heat reservoirs. The amount of heat released per cycle by one engine is completely absorbed by the next engine. Consider \( Q_0 \) to be the amount of heat absorbed per cycle by the first engine and \( W \) as the amount of total work done by all the engines per cycle, then the net efficiency of the system is found to be \( \eta_{\text{net}} = \dfrac{W}{Q_0} = \dfrac{211}{243} \). The value of \( \eta \) is:

As shown in the figure, an insulated container is fitted with a thermally conducting but immovable partition \( (P_1) \) and a freely movable but thermally insulated piston \( (P_2) \). The partition \( P_1 \) with thermal conductivity \( K \), cross sectional area \( A \) and width \( x \) divides the container into two sections, \( S_1 \) and \( S_2 \), each containing one mole of a monoatomic gas. The piston \( P_2 \) moves freely such that the gas in \( S_2 \) is always at the atmospheric pressure. Initially, the difference between the temperatures of \( S_1 \) and \( S_2 \) is \( \Delta T_0 \). The time it takes for the temperature difference to become \( \dfrac{\Delta T_0}{2} \) is \( \dfrac{n x R}{K A} \), where \( R \) is the universal gas constant. The value of \( n \) is: [Given: \( \ln 2 \approx 0.7 \)]

A hollow, right circular cone of base radius \( R \) and height \( h \), with its tip at the origin is rotating about the \( Z \)-axis with an angular velocity \( \omega \), as shown in the figure. The cone carries a total charge \( Q \) uniformly distributed on its curved surface. The magnitude of magnetic field at a point \( (0, 0, z) \), where \( z \gg R \) and \( z \gg h \), is \( \dfrac{n \mu_0}{4\pi} \dfrac{Q R^2 \omega}{z^3} \). The value of \( n \) is: 

List-I shows four configurations made of straight and semi-circular narrow tubes containing air. A sound wave of wavelength \( \lambda = 0.29 \, \text{m} \) enters these structures at the point \( S \) and a sound detector is placed at \( D \). Between the points \( S \) and \( D \), the sound travels only through the tubes. List-II contains the possible smallest values of \( l \) (refer to the figures) for which the detector \( D \) records maximum amplitude. Ignore effects of sharp corners. [Given \( \cos(15^\circ) = 0.97 \)]
Choose the option that best describes the match between the entries in List-I to those in List-II.

In the List-I four optical effects are mentioned. The physical phenomena of light which are essential to describe these optical effects are given in List-II. Choose the option which describes the correct match between the entries in List-I to those in List-II.

List-I contains four conducting loops lying in the \( XY \) plane, as shown in the figures. The loops are rotating about \( Z \) axis passing through the point \( O \) with time period \( T \) in clockwise direction. The region \( x > 0 \) contains a uniform magnetic field \( B \) in the \( +z \) direction. List-II contains the qualitative variation of the induced current \( i(t) \) for each of these loops. Choose the option which describes the correct match between the entries in List-I to those in List-II.

List-I shows four planar structures made of uniform solid rods each of mass \( m \) and length \( l \). In the List-II the possible moment of inertia of these structure about an axis \( OCO' \), which lies in the plane of the structures, are given. Choose the option which describes the correct match between the entries in List-I to those in List-II.