NTA JEE Main 25th July 2021 Shift 1

A body of mass 2 kg moving with a speed of 4 m s$$^{-1}$$ makes an elastic collision with another body at rest and continues to move in the original direction but with one fourth of its initial speed. The speed of the two body centre of mass is $$\frac{x}{10}$$. Find the value of x.

A particle of mass $$m$$ is moving in time $$t$$ on a trajectory given by,
$$\vec{r} = 10\alpha t^2 \hat{i} + 5\beta (t - 5)\hat{j}$$
where $$\alpha$$ and $$\beta$$ are dimensional constants. The angular momentum of the particle becomes the same as it was for $$t = 0$$ at time $$t =$$ ___ seconds.

In the reported figure, two bodies A and B of masses 200 g and 800 g are attached with the system of springs. Springs are kept in a stretched position with some extension when the system is released. The horizontal surface is assumed to be frictionless. The angular frequency will be ___ rad s$$^{-1}$$ when $$k = 20$$ N m$$^{-1}$$.

A pendulum bob has a speed of 3 m s$$^{-1}$$ at its lowest position. The pendulum is 50 cm long. The speed of bob, when the length makes an angle of 60$$^\circ$$ to the vertical will be ___ m s$$^{-1}$$. ($$g = 10$$ m s$$^{-2}$$)

A particle of mass 1 mg and charge $$q$$ is lying at the mid-point of two stationary particles kept at a distance 2 m when each is carrying same charge $$q$$. If the free charged particle is displaced from its equilibrium position through distance $$x$$ ($$x << 1$$ m). The particle executes SHM. Its angular frequency of oscillation will be ___ $$\times 10^5$$ rad s$$^{-1}$$ (if $$q^2 = 10C^2$$)

An electric bulb rated as 200 W at 100 V is used in a circuit having 200 V supply. The resistance R that must be put in series with the bulb so that the bulb delivers the same power is ___ $$\Omega$$.

The value of aluminium susceptibility is $$2.2 \times 10^{-5}$$. The percentage increase in the magnetic field if space within a current carrying toroid is filled with aluminium is $$\frac{x}{10^4}$$. Then the value of $$x$$ is ___.

A circular conducting coil of radius 1 m is being heated by the change of magnetic field $$\vec{B}$$ passing perpendicular to the plane in which the coil is laid. The resistance of the coil is 2 $$\mu\Omega$$. The magnetic field is slowly switched off such that its magnitude changes in time as
$$B = \frac{4}{\pi} \times 10^{-3} T\left(1 - \frac{t}{100}\right)$$
The energy dissipated by the coil before the magnetic field is switched off completely is $$E =$$ ___ mJ.

An inductor of 10 mH is connected to a 20 V battery through a resistor of 10 k$$\Omega$$ and a switch. After a long time, when maximum current is set up in the circuit, the current is switched off. The current in the circuit after 1 $$\mu$$s is $$\frac{x}{100}$$ mA. Then $$x$$ is equal to ___. (Take $$e^{-1} = 0.37$$)

Student A and student B used two screw gauges of equal pitch and 100 equal circular divisions to measure the radius of a given wire. The actual value of the radius of the wire is 0.322 cm. The absolute value of the difference between the final circular scale readings observed by the students A and B is ___.

Given pitch = 0.1 cm.