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JEE Mains | PYQ

Consider the equation \( H=\dfrac{x^{p}\,\varepsilon^{q}\,E^{r}}{t^{s}} \).
Where \( H=\) magnetic field; \( E=\) electric field, \( \varepsilon=\) permittivity, \( x=\) distance, \( t=\) time
The values of \( p, q, r \) and \( s \) respectively are:

A car moving with a speed of \( 54 \,\text{km/h} \) takes a turn of radius \( 20 \,\text{m} \). A simple pendulum is suspended from the ceiling of the car. Determine the angle made by the string of the pendulum with the vertical during the turning.
\([ \text{Take } g=10 \,\text{m/s}^{2} ]\)

A gas balloon is going up with a constant velocity of \( 10 \,\text{m/s} \). When this balloon reached a height of \( 75 \,\text{m} \), a stone is dropped from it and balloon keeps moving up with the same velocity. The height of the balloon when the stone hits the ground is ______ \( \text{m} \). \([ \text{Take } g=10 \,\text{m/s}^{2} ]\)

A thin biconvex lens is prepared from the glass \( (\mu=1.5) \) both curved surfaces of which have equal radii of \( 20 \,\text{cm} \) each. Left side surface of the lens is silvered from outside to make it reflecting. To have the position of image and object at the same place, the object should be placed, from the lens at a distance of ______ \( \text{cm} \).

Two identical bodies, projected with the same speed at two different angles cover the same horizontal range \( R \). If the time of flight of these bodies are \( 5 \) and \( 10 \) \( \text{s} \), respectively, then the value of \( R \) is ______ \( \text{m} \). [Take \( g = 10 \, \text{m} \, \text{s}^{-2} \)]

A solid cylinder having radius \( R \) and length \( L \) is slipping on a rough horizontal plane. At time \( t = 0 \) the cylinder has a translational velocity \( v_0 = 49 \, \text{m} \, \text{s}^{-1} \), perpendicular to its axis and a rotational velocity \( \dfrac{v_0}{4R} \) about the centre. The time taken by the cylinder to start rolling is ______ seconds. [Coefficient of kinetic friction \( \mu_K = 0.25 \) and \( g = 9.8 \, \text{m} \, \text{s}^{-2} \)]

A liquid of density \( 600 \, \text{kg} \, \text{m}^{-3} \) flowing steadily in a tube of varying cross-section. The cross-section at a point A is \( 1.0 \, \text{cm}^2 \) and that at B is \( 20 \, \text{mm}^2 \). Both the points A and B are in same horizontal plane, the speed of the liquid at A is \( 10 \, \text{cm} \, \text{s}^{-1} \). The difference in pressures at A and B points is ______ \( \text{Pa} \).

A spherical liquid drop of radius \( R \) acquires the terminal velocity \( v_1 \) when falls through a gas of viscosity \( \eta \). Now the drop is broken into \( 64 \) identical droplets and each droplet acquires terminal velocity \( v_2 \) falling through the same gas. The ratio of terminal velocities \( \dfrac{v_1}{v_2} \) is ______.

One mole of diatomic gas having rotational modes only is kept in a cylinder with a piston system. The cross-section area of the cylinder is \( 4 \, \text{cm}^2 \). The gas is heated slowly to raise the temperature by \( 1.2^\circ \text{C} \) during which the piston moves by \( 25 \, \text{mm} \). The amount of heat supplied to the gas is ______ \( \text{J} \). (Neglect mass of the piston) [Atmospheric pressure = \( 100 \, \text{kPa} \), \( R = 8.3 \, \text{J} \, \text{mol}^{-1} \, \text{K}^{-1} \)]

Initial pressure and volume of a monoatomic ideal gas are \( P \) and \( V \). The change in internal energy of this gas in adiabatic expansion to volume \( V_{\text{final}} = 27V \) is ______ \( \text{J} \).

The frequency of oscillation of a mass \( m \) suspended by a spring is \( v_1 \). If the length of the spring is cut to half, the same mass oscillates with frequency \( v_2 \). The value of \( \frac{v_2}{v_1} \) is ______.

A monochromatic source of light operating at \( 15 \ \text{kW} \) emits \( 2.5 \times 10^{22} \) photons/s. The region of an electromagnetic spectrum to which the emitted electromagnetic radiation belongs to ______.
[Take \( h = 6.6 \times 10^{-34} \ \text{J}\!\cdot\!\text{s} \) and \( c = 3 \times 10^{8} \ \text{m/s} \)]

A current carrying circular loop of radius \( 2 \ \text{cm} \) with unit normal \( \hat{n}=\dfrac{\hat{k}+\hat{i}}{\sqrt{2}} \) is placed in a magnetic field, \( \vec{B}=B_{0}(3\hat{i}+2\hat{k}) \). If \( B_{0}=4\times 10^{-3} \ \text{T} \) and current \( I=200\sqrt{2} \ \text{A} \), the torque experienced by the loop is ______ \( \text{Wb}\!\cdot\!\text{A} \).
[\( \pi = 3.14 \)]

A \( 30 \ \text{cm} \) long solenoid has \( 10 \) turns per \( \text{cm} \) and area of \( 5 \ \text{cm}^2 \). The current through the solenoid coil varies from \( 2 \ \text{A} \) to \( 4 \ \text{A} \) in \( 3.14 \ \text{s} \). The e.m.f. induced in the coil is \( \alpha \times 10^{-5} \ \text{V} \). The value \( \alpha \) is ______.

Two point charges \( q_1 = 3\,\mu\text{C} \) and \( q_2 = -4\,\mu\text{C} \) are placed at points \( (2\hat{i}+3\hat{j}+3\hat{k}) \) and \( (\hat{i}+\hat{j}+\hat{k}) \) respectively. Force on charge \( q_2 \) is ______ \( \text{N} \). [Take \( \dfrac{1}{4\pi \varepsilon_0} = 9\times 10^{9} \) SI units]

Light ray incident along a vector \( \overrightarrow{AO} \, ( \overrightarrow{AO} = 2\hat{i} - 3\hat{j} ) \) emerges out along vector \( \overrightarrow{OB} \, ( \overrightarrow{OB} = C\hat{i} - 4\hat{j} ) \) as shown in the figure below. The value of \( C \) is _______.

\( K_1 \) and \( K_2 \) be the maximum kinetic energies of photoelectrons emitted from a surface of a given material for the light of wavelength \( \lambda_1 \) and \( \lambda_2 \), respectively. If \( \lambda_1 = 2\lambda_2 \) then the work function of material is given by:

Two radioactive substances \( A \) and \( B \) of mass numbers \( 200 \) and \( 212 \) respectively, shows spontaneous \( \alpha \)-decay with same \( Q \) value of \( 1 \, \text{MeV} \). The ratio of energies of \( \alpha \)-rays produced by \( A \) and \( B \) is ______.

The output \( Y \) for the given inputs \( A \) and \( B \) to the circuit is:

A parallel plate capacitor is having separation between plates \( 0.885 \, \text{mm} \). It has a capacitance of \( 1 \, \mu\text{F} \) when the space between the plates is filled with an insulating material of resistivity \( 1 \times 10^{13} \, \Omega \text{m} \) and resistance \( 17.7 \times 10^{14} \, \Omega \). Relative permittivity of the insulating material is \( \alpha \times 10^{7} \). The value of \( \alpha \) is ______.
[Take permittivity of free space \( = 8.85 \times 10^{-12} \, \text{F/m} \)]

Some distant star is to be observed by some telescope of diameter of objective lens \( a \), at an angular resolution of \( 3.0 \times 10^{-7} \) radian. If the wavelength of light from the star reaching the telescope is \( 500 \, \text{nm} \), the minimum diameter of the objective lens of the telescope is ______ \( \text{cm} \). (Nearest integer)

A \( 5 \, \mathrm{mg} \) particle carrying a charge of \( 5\pi \times 10^{-6} \, \mathrm{C} \) is moving with velocity of \( (3\hat{\imath} + 2\hat{k}) \times 10^{-2} \, \mathrm{m\, /s} \) in a region having magnetic field \( |\vec{B}| = 0.1 \hat{k} \, \mathrm{Wb\, /m^{2}} \). It moves a distance of \( \alpha \) meter along \( \hat{k} \) when it completes \( 5 \) revolutions. The value of \( \alpha \) is ______.

The stored charge in the capacitor in steady state of the following circuit is ______ \( \mu \mathrm{C} \).

Two masses of \( 3.4 \, \mathrm{kg} \) and \( 2.5 \, \mathrm{kg} \) are accelerated from an initial speed of \( 5 \, \mathrm{m\, /s} \) and \( 12 \, \mathrm{m\, /s} \), respectively. The distances traversed by the masses in the \( 5^{\text{th}} \) second are \( 104 \, \mathrm{m} \) and \( 129 \, \mathrm{m} \), respectively. The ratio of their momenta after \( 10 \, \mathrm{s} \) is \( \dfrac{x}{8} \). The value of \( x \) is ______.