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JEE Mains | PYQ

Statement I:
If three forces \( \vec{F_1}, \vec{F_2} \) and \( \vec{F_3} \) are represented by three sides of a triangle and \( \vec{F_1} + \vec{F_2} = -\vec{F_3} \), then these three forces are concurrent forces and satisfy the condition for equilibrium.
Statement II:
A triangle made up of three forces \( \vec{F_1}, \vec{F_2} \) and \( \vec{F_3} \) as its sides taken in the same order, satisfy the condition for translatory equilibrium.
In the light of the above statements, choose the most appropriate answer from the options given below:

The magnetic field vector of an electromagnetic wave is given by \( \vec{B_0} = B_0 \dfrac{\hat{i} + \hat{j}}{\sqrt{2}} \cos(k z - \omega t) \); where \( \hat{i}, \hat{j} \) represents unit vector along \( x \) and \( y \)-axis respectively. At \( t = 0 \, \text{s} \), two electric charges \( q_1 \) of \( 4\pi \) coulomb and \( q_2 \) of \( 2\pi \) coulomb located at \( \left(0, 0, \dfrac{\pi}{k}\right) \) and \( \left(0, 0, \dfrac{3\pi}{k}\right) \), respectively, have the same velocity of \( 0.5\,c\,\hat{i} \), (where \( c \) is the velocity of light). The ratio of the force acting on charge \( q_1 \) to \( q_2 \) is:

A coil is placed in a magnetic field \( \vec{B} \) as shown below:

A current is induced in the coil because \( \vec{B} \) is:
A current is induced in the coil because \( \vec{B} \) is:

A bob of mass ‘\( m \)’ suspended by a thread of length \( l \) undergoes simple harmonic oscillations with time period \( T \). If the bob is immersed in a liquid that has density \( \dfrac{1}{4} \) times that of the bob and the length of the thread is increased by \( \dfrac{1}{3}^{rd} \) of the original length, then the time period of the simple harmonic oscillations will be:

A block moving horizontally on a smooth surface with a speed of \( 40 \,\text{m/s} \) splits into two parts with masses in the ratio of \( 1:2 \). If the smaller part moves at \( 60 \,\text{m/s} \) in the same direction, then the fractional change in kinetic energy is:

Choose the incorrect statement:
(a) The electric lines of force entering into a Gaussian surface provide negative flux.
(b) A charge '\( q \)' is placed at the centre of a cube. The flux through all the faces will be the same.
(c) In a uniform electric field net flux through a closed Gaussian surface containing no net charge, is zero.
(d) When electric field is parallel to a Gaussian surface, it provides a finite non-zero flux.
Choose the most appropriate answer from the options given below:

If velocity \( [V] \) time \( [T] \) and force \( [F] \) are chosen as the base quantities, the dimensions of the mass will be:

A free electron of \( 2.6 \, \text{eV} \) energy collides with a \( H^{+} \) on. This result in the formation of a hydrogen atom in the first excited state and a photon is released. Find the frequency of the emitted photon.
\( (h = 6.6 \times 10^{-34} \, \text{J s}) \)

If \( V_{A} \) and \( V_{B} \) are the input voltages (either \( 5 \, \text{V} \) or \( 0 \, \text{V} \)) and \( V_{0} \) is the output voltage then the two gates represented in the following circuits (A) and (B) are:

A system consists of two identical spheres each of mass \( 1.5 \, \text{kg} \) and radius \( 50 \, \text{cm} \) at the ends of a light rod. The distance between the centres of the two spheres is \( 5 \, \text{m} \). What will be the moment of inertia of the system about an axis perpendicular to the rod passing through its midpoint ?

The equivalent resistance of the given circuit between the terminals A and B is:

Statement I:  Two forces $(\vec{P} + \vec{Q})$ and $(\vec{P} - \vec{Q})$ where $\vec{P} \perp \vec{Q}$, when act at an angle $\theta_1$ to each other, the magnitude of their resultant is $\sqrt{3 \left(P^2 + Q^2\right)}$, when they act at an angle $\theta_2$, the magnitude of their resultant becomes $\sqrt{2 \left(P^2 + Q^2\right)}$. This possible only when $\theta_1 < \theta_2$.
Statement II: In the situation given above $\theta_1 = 60^\circ \text{ and } \theta_2 = 90^\circ$.
In the light of the above statements, choose the most appropriate answer from the options given below:

A current of \(\ 1.5 \, \text{A} \) is flowing through a triangle, of side \(\ 9 \, \text{cm} \) each. The magnetic field at the centroid of the triangle is: (Assume that the current is flowing in the clockwise direction.)

Two thin metallic spherical shells of radii \(\ r_1 \) and \(\ r_2 \) \(\ (r_1 < r_2) \) are placed with their centres coinciding. A material of thermal conductivity \(\ K \) is filled in the space between the shells. The inner shell is maintained at temperature \(\ \theta_1 \) and the outer shell at temperature \(\ \theta_2 \, (\theta_1 < \theta_2) \). The rate at which heat flows radially through the material is:

Consider two separate ideal gases of electrons and protons having same number of particles. The temperature of both the gases are same. The ratio of the uncertainty in determining the position of an electron to that of a proton is proportional to:

For a body executing S.H.M.:
(a) Potential energy is always equal to its K.E.
(b) Average potential and kinetic energy over any given time interval are always equal.
(c) Sum of the kinetic and potential energy at any point of time is constant.
(d) Average K.E. in one time period is equal to average potential energy in one time period.
Choose the most appropriate option from the options given below:

A mixture of hydrogen and oxygen has volume \( 500 \, \text{cm}^3 \), temperature \( 300 \, \text{K} \), pressure \( 400 \, \text{kPa} \) and mass \( 0.76 \, \text{g} \). The ratio of masses of oxygen to hydrogen will be:

Four identical hollow cylindrical columns of mild steel support a big structure of mass \( 50 \times 10^{3} \, \text{kg} \). The inner and outer radii of each column are \( 50 \, \text{cm} \) and \( 100 \, \text{cm} \) respectively. Assuming uniform load distribution, calculate the compression strain of each column, [use \( Y = 2.0 \times 10^{11} \, \text{Pa} \), \( g = 9.8 \, \text{m} / \text{s}^{2} \)]

Statement I :
To get a steady dc output from the pulsating voltage received from a full wave rectifier we can connect a capacitor across the output parallel to the load \( R_L \).
Statement II :
To get a steady dc output from the pulsating voltage received from a full wave rectifier we can connect an inductor in series with \( R_L \).
In the light of the above statements, choose the most appropriate answer from the options given below:

A parallel plate capacitor of capacitance \( 200 \, \mu \text{F} \) is connected to a battery of \( 200 \, \text{V} \). A dielectric slab of dielectric constant \( 2 \) is now inserted into the space between plates of capacitor while the battery remain connected. The change in the electrostatic energy in the capacitor will be _______ \( \text{J} \).

A bandwidth of \( 6 \, \text{MHz} \) is available for A.M. transmission. If the maximum audio signal frequency used for modulating the carrier wave is not to exceed \( 6 \, \text{kHz} \). The number of stations that can be broadcasted within this band simultaneously without interfering with each other will be _______.

Cross-section view of a prism is the equilateral triangle \( ABC \) shown in the figure. The minimum deviation is observed using this prism when the angle of incidence is equal to the prism angle. The time taken by light to travel from \( P \) (midpoint of \( BC \)) to \( A \) is _______ \( \times 10^{-10} \, \text{s} \). (Given, speed of light in vacuum \( = 3 \times 10^{8} \, \text{m/s} \) and \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \))

A sample of gas with \( \gamma = 1.5 \) is taken through an adiabatic process in which the volume is compressed from \( 1200 \, \text{cm}^3 \) to \( 300 \, \text{cm}^3 \). If the initial pressure is \( 200 \, \text{kPa} \). The absolute value of the workdone by the gas in the process = ______ J .

In a Young's double slit experiment, the slits are separated by \( 0.3 \, \text{mm} \) and the screen is \( 1.5 \, \text{m} \) away from the plane of slits. Distance between fourth bright fringes on both sides of central bright fringe is \( 2.4 \, \text{cm} \). The frequency of light used is ______ \( \times 10^{14} \, \text{Hz} \) .

A resistor dissipates \( 192 \, \text{J} \) energy in \( 1 \, \text{s} \) when a current of \( 4 \, \text{A} \) is passed through it. Now, when the current is doubled, the amount of thermal energy dissipated in \( 5 \, \text{s} \) is ______ J.

At very high frequencies, the effective impedance of the given circuit will be ______ \( \Omega \) .

A long solenoid with \( 1000 \) turns/m has a core material with relative permeability \( 500 \) and volume \( 10^{3} \, \text{cm}^{3} \). If the core material is replaced by another material having relative permeability of \( 750 \) with same volume maintaining same current of \( 0.75 \, \text{A} \) in the solenoid, the fractional change in the magnetic moment of the core would be approximately \( \left( \frac{x}{499} \right) \). Find the value of \( x \).

The diameter of a spherical bob is measured using a vernier callipers. \( 9 \) divisions of the main scale, in the vernier callipers, are equal to \( 10 \) divisions of vernier scale. One main scale division is \( 1 \, \text{mm} \). The main scale reading is \( 10 \, \text{mm} \) and \( 8^{\text{th}} \) division of vernier scale was found to coincide exactly with one of the main scale division. If the given vernier calliper has positive zero error of \( 0.04 \, \text{cm} \), then the radius of the bob is ______ \( \times 10^{-2} \, \text{cm} \) .

A particle is moving with constant acceleration ‘\( a \)’. Following graph shows \( v^2 \) versus \( x \) (displacement) plot. The acceleration of the particle is ______ \( \mathrm{m/s^2} \).