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An ac current is represented as \( i = 5\sqrt{2} + 10\cos\!\left(650\pi t + \frac{\pi}{6}\right)\ \text{Amp} \). The r.m.s value of the current is:

An object of mass \( 1000\,\text{g} \) experiences a time dependent force \( \vec{F} = \left(2t\,\hat{i} + 3t^{2}\,\hat{j}\right)\ \text{N} \). The power generated by the force at time \( t \) is:

Two projectiles are fired from ground with same initial speeds from same point at angles \( (45^\circ + \alpha) \) and \( (45^\circ - \alpha) \) with horizontal direction. The ratio of their times of flights is:

The percentage increase in magnetic field \( (B) \) when space within a current carrying solenoid is filled with magnesium \( \text{(magnetic susceptibility } \chi_{\mathrm{Mg}} = 1.2 \times 10^{-5}\text{)} \) is:

A rod of length \( 5L \) is bent right angle keeping one side length as \( 2L \).

The position of the centre of mass of the system:
(Consider \( L = 10\,\text{cm} \)).

Two plane polarized light waves combine at a certain point whose electric field components are
\( E_1 = E_0 \sin(\omega t) \)
\( E_2 = E_0 \sin\!\left(\omega t + \frac{\pi}{3}\right) \)
Find the amplitude of the resultant wave.

Two thin convex lenses of focal lengths \( 30 \, \text{cm} \) and \( 10 \, \text{cm} \) are placed coaxially, \( 10 \, \text{cm} \) apart. The power of this combination is:

Uniform magnetic fields of different strengths \( (B_1 \) and \( B_2) \), both normal to the plane of the paper exist as shown in the figure. A charged particle of mass \( m \) and charge \( q \), at the interface at an instant, moves into the region \( 2 \) with velocity \( v \) and returns to the interface. It continues to move into region \( 1 \) and finally reaches the interface. What is the displacement of the particle during this movement along the interface?

(Consider the velocity of the particle to be normal to the magnetic field and \( B_2 > B_1 \)):

Two harmonic waves moving in the same direction superimpose to form a wave \( x = a \cos(1.5\, t)\cos(50.5\, t) \) where \( t \) is in seconds. Find the period with which they beat. (close to nearest integer):

In the following circuit, the reading of the ammeter will be: (Take Zener breakdown voltage \( = 4 \, \text{V} \))

A lens having refractive index \( 1.6 \) has focal length of \( 12 \, \text{cm} \), when it is in air. Find the focal length of the lens when it is placed in water: (Take refractive index of water as \( 1.28 \))

Match the LIST-I with LIST-II:

Choose the correct answer from the options given below:

If \( \varepsilon_0 \) denotes the permittivity of free space and \( \phi_E \) is the flux of the electric field through the area bounded by the closed surface, then dimensions of \( \left( \varepsilon_0 \, \frac{d\phi_E}{dt} \right) \) are that of:

A particle of charge \( q \), mass \( m \) and kinetic energy \( E \) enters in magnetic field perpendicular to its velocity and undergoes a circular arc of radius \( r \). Which of the following curves represents the variation of \( r \) with \( E \)?

A wire of resistance \( R \) is bent into a triangular pyramid as shown in figure with each segment having same length. The resistance between points \( A \) and \( B \) is \( \frac{R}{n} \). The value of \( n \) is:

In a hydrogen like ion, the energy difference between the \( 2^{\text{nd}} \) excitation energy state and ground is \( 108.8 \, \text{eV} \). The atomic number of the ion is:

For a hydrogen atom, the ratio of the largest wavelength of Lyman series to that of the Balmer series is:

Two charges \( q_1 \) and \( q_2 \) are separated by a distance of \( 30 \, \text{cm} \). A third charge \( q_3 \) initially at ‘C’ as shown in the figure, is moved along the circular path of radius \( 40 \, \text{cm} \) C to D. If the difference in potential energy due to movement of \( q_3 \) from C to D is given by \( \dfrac{q_3 K}{4 \pi \varepsilon_0} \), the value of \( K \) is:

Two wires A and B are made of same material having ratio of lengths \( \dfrac{L_A}{L_B} = \dfrac{1}{3} \) and their diameters ratio \( \dfrac{d_A}{d_B} = 2 \). If both the wires are stretched using same force, what would be the ratio of their respective elongations?

A cubic block of mass \( m \) is sliding down on an inclined plane at \( 60^\circ \) with an acceleration of \( \dfrac{g}{2} \), the value of coefficient of kinetic friction is:

A, B and C are disc, solid sphere and spherical shell respectively with same radii and masses. These masses are placed as shown in figure.

The moment of inertia of the given system about PQ axis is \( \dfrac{x}{15} I \), where \( I \) is the moment.of inertia of the disc about its diameter. The value of x is ____________.

An ideal gas has undergone through the cyclic process as shown in the figure. Work done by the gas in the entire cycle is ________ \( \times 10^{-1} \) \( \text{J} \) (Take \( \pi = 3.14 \) )

A container contains a liquid with refractive index of \( 1.2 \) up to a height of \( 60 \, \text{cm} \) and another liquid having refractive index \( 1.6 \) is added to height \( H \) above first liquid. If viewed from above, the apparent shift in the position of bottom of container is \( 40 \, \text{cm} \). The value of \( H \) is __________ \( \text{cm} \). (Consider liquids are immiscible)

A wire of length \( 10 \, \text{cm} \) and diameter \( 0.5 \, \text{mm} \) is used in a bulb. The temperature of the wire is \( 1727^\circ \text{C} \) and power radiated by the wire is \( 94.2 \, \text{W} \). Its emissivity is \( \frac{x}{8} \) where \( x = \) ________________.$(\text{Given } \sigma = 6.0 \times 10^{-8} \text{ W m}^{-2} \text{ K}^{-4}, \pi = 3.14 \text{ and assume that the emissivity of wire material}$ $\text{is same at all wavelength.})$

For ac circuit shown in figure, \( R = 100 \, \text{k}\Omega \) and \( C = 100 \, \text{pF} \) and the phase difference between \( V_m \) and \( (V_B - V_A) \) is \( 90^\circ \). The input signal frequency is \( 10^{x} \, \text{rad/sec} \), where ‘x’ is __________.

JEE MAIN 2025 7th April Morning Shift

Mathematics Physics Chemistry

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A rod of length \( 5L \) is bent right angle keeping one side length as \( 2L \).

The position of the centre of mass of the system:
(Consider \( L = 10\,\text{cm} \)).