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Moment of inertia of a square plate of side \( l \) about the axis passing through one of the corner and perpendicular to the plane of square plate is given by:

A uniformly charged disc of radius \( R \) having surface charge density \( \sigma \) is placed in the \( xy \) plane with its centre at the origin. Find the electric field intensity along the \( z \)-axis at a distance \( Z \) from origin:

Find the distance of the image from object \( O \), formed by the combination of lenses in the figure:

If \( E \) and \( H \) represents the intensity of electric field and magnetising field respectively, then the unit of \( E/H \) will be:

Five identical cells each of internal resistance \( 1\,\Omega \) and emf \( 5\,\text{V} \) are connected in series and in parallel with an external resistance \( R \). For what value of \( R \), current in series and parallel combination will remain the same?

An ideal gas is expanding such that \( PT^{3} = \text{constant} \). The coefficient of volume expansion of the gas is:

An object is placed beyond the centre of curvature \( C \) of the given concave mirror. If the distance of the object is \( d_1 \) from \( C \) and the distance of the image formed is \( d_2 \) from \( C \), the radius of curvature of this mirror is:

Two ions of masses \( 4 \, \text{amu} \) and \( 16 \, \text{amu} \) have charges \( +2e \) and \( +3e \) respectively. These ions pass through the region of constant perpendicular magnetic field. The kinetic energy of both ions is same. Then:

In a photoelectric experiment, increasing the intensity of incident light:

The balloon carries a total load of \( 185 \, \text{kg} \) at normal pressure and temperature of \( 27^\circ \text{C} \). What load will the balloon carry on rising to a height at which the barometric pressure is \( 45 \, \text{cm} \) of \( \text{Hg} \) and the temperature \( -7^\circ \text{C} \). Assuming the volume constant?

The variation of displacement with time of a particle executing free simple harmonic motion is shown in the figure.

The potential energy \( U(x) \) versus time \( (t) \) plot of the particle is correctly shown in figure:

Electric field in a plane electromagnetic wave is given by
\( E = 50 \sin(500x - 10 \times 10^{10} t) \ \text{V}/\text{m} \)
The velocity of electromagnetic wave in this medium is: (Given \( C \) = Speed of light in vacuum)

Which of the following is not a dimensionless quantity?

In Millikan’s oil drop experiment, what is viscous force acting on an uncharged drop of radius \( 2.0 \times 10^{-5} \, \text{m} \) and density \( 1.2 \times 10^{3} \, \text{kg m}^{-3} \)? Take viscosity of liquid \( = 1.8 \times 10^{-5} \, \text{N s m}^{-2} \). (Neglect buoyancy due to air).

A huge circular arc of length \( 4.4 \, \text{ly} \) subtends an angle ‘\( 4s \)’ at the centre of the circle. How long it would take for a body to complete \( 4 \) revolution if its speed is \( 8 \, \text{AU per second} \)?
Given:
\( 1 \, \text{ly} = 9.46 \times 10^{15} \, \text{m} \)
\( 1 \, \text{AU} = 1.5 \times 10^{11} \, \text{m} \)

There are \( 10^{10} \) radioactive nuclei in a given radioactive element. Its half-life time is \( 1 \) minute. How many nuclei will remain after \( 30 \) seconds? \( (\sqrt{2} = 1.414) \)

A bar magnet is passing through a conducting loop of radius \( R \) with velocity \( v \). The radius of the bar magnet is such that it just passes through the loop. The induced e.m.f. in the loop can be represented by the approximate curve:

For a transistor in CE mode to be used as an amplifier, it must be operated in:

The resultant of these force $\overrightarrow{OP},\overrightarrow{OQ},\overrightarrow{OR},\overrightarrow{OS} \text{ and } \overrightarrow{OT}$ is approximately_______ N.
$\text{[Take } \sqrt{3}=1.7,\sqrt{2}=1.4 \text{ Given }\hat{i} \text{ and } \hat{j} \text{ unit vectors along } x, y \text{ axis]}$

Calculate the amount of charge on capacitor of \( 4\,\mu\mathrm{F} \). The internal resistance of battery is \( 1\,\Omega \):

A uniform conducting wire of length \( 24a \), and resistance \( R \) is wound up as a current carrying coil in the shape of an equilateral triangle of side \( a \) and then in the form of a square of side \( a \). The coil is connected to a voltage source \( V_0 \). The ratio of magnetic moment of the coils of equilateral triangle to that for square is \( 1 : \sqrt{y} \) where \( y \) is __________.

If the velocity of a body related to displacement \( x \) given by \( v=\sqrt{5000+24x} \, \mathrm{m/s} \), then the acceleration of the body is __________ \( \mathrm{m/s^2} \).

A rod \( CD \) of thermal resistance \( 10.0\,\mathrm{kW^{-1}} \) is joined at the middle of an identical rod \( AB \) as shown in figure. The ends \( A \), \( B \) and \( D \) are maintained at \( 200^\circ\mathrm{C} \), \( 100^\circ\mathrm{C} \) and \( 125^\circ\mathrm{C} \) respectively. The heat current in \( CD \) is \( P \) watt. The value of \( P \) is _______.

A circuit is arranged as shown in figure. The output voltage \( V_0 \) is equal to ______ V.

First, a set of \( n \) equal resistors of \( 10 \, \Omega \) each are connected in series to a battery of emf \( 20 \, \text{V} \) and internal resistance \( 10 \, \Omega \). A current \( I \) is observed to flow. Then, the \( n \) resistors are connected in parallel to the same battery. It is observed that the current is increased \( 20 \) times, then the value of \( n \) is ______.

The alternating current is given by
\( i = \left( \sqrt{42} \sin\!\left( \dfrac{2\pi}{T} t \right) + 10 \right) \, \text{A} \).
The r.m.s. value of this current is__________ \( \text{A} \).

A body of mass \( (2M) \) splits into four masses \( (m, \, M - m, \, m, \, M - m) \), which are rearranged to form a square as shown in the figure. The ratio of \( \dfrac{M}{m} \) for which, the gravitational potential energy of the system becomes maximum is \( x : 1 \). The value of \( x \) is ______.

A transmitting antenna has height of \( 320 \, \text{m} \) and that of receiving antenna is \( 2000 \, \text{m} \). The maximum distance between them for satisfactory communication in line of sight mode is \( d \). The value of \( d \) is ______ \( \text{km} \).

Two person A and B perform same amount of work in moving a body through a certain distance \( d \) with application of forces acting at angle \( 45^\circ \) and \( 60^\circ \) with the direction of displacement respectively. The ratio of force applied by person A to the force applied by person B is \( \dfrac{1}{\sqrt{x}} \). The value of \( x \) is ______.

The cars X and Y are approaching each other with velocities \( 36 \, \text{km/h} \) and \( 72 \, \text{km/h} \) respectively. The frequency of a whistle sound as emitted by a passenger in car X, heard by the passenger in car Y is \( 1320 \, \text{Hz} \). If the velocity of sound in air is \( 340 \, \text{m/s} \), the actual frequency of the whistle sound produced is ______ \( \text{Hz} \).

JEE Main 2021 27th August Morning Shift

Physics Chemistry Mathematics

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Electric field in a plane electromagnetic wave is given by
\( E = 50 \sin(500x - 10 \times 10^{10} t) \ \text{V}/\text{m} \)
The velocity of electromagnetic wave in this medium is: (Given \( C \) = Speed of light in vacuum)