NTA JEE Main 27th June 2022 Shift 1

A pendulum of length $$2$$ m consists of a wooden bob of mass $$50$$ g. A bullet of mass $$75$$ g is fired towards the stationary bob with a speed $$v$$. The bullet emerges out of the bob with a speed $$\frac{v}{3}$$ and the bob just completes the vertical circle. The value of $$v$$ is ______ m s$$^{-1}$$
(if $$g = 10$$ m s$$^{-2}$$).

The area of cross-section of a large tank is $$0.5$$ m$$^2$$. It has a narrow opening near the bottom having area of cross-section $$1$$ cm$$^2$$. A load of $$25$$ kg is applied on the water at the top in the tank. Neglecting the speed of water in the tank, the velocity of the water, coming out of the opening at the time when the height of water level in the tank is $$40$$ cm above the bottom, will be ______ cm s$$^{-1}$$.
[Take $$g = 10$$ m s$$^{-2}$$]

In a carnot engine, the temperature of reservoir is $$527°$$C and that of sink is $$200$$ K. If the work done by the engine when it transfers heat from reservoir to sink is $$12000$$ kJ, the quantity of heat absorbed by the engine from reservoir is ______ $$\times 10^6$$ J.

A capacitor of capacitance $$50$$ pF is charged by $$100$$ V source. It is then connected to another uncharged identical capacitor. Electrostatic energy loss in the process is ______ nJ.

A $$220$$ V, $$50$$ Hz AC source is connected to a $$25$$ V, $$5$$ W lamp and an additional resistance $$R$$ in series (as shown in figure) to run the lamp at its peak brightness, then the value of $$R$$ (in ohm) will be

A cell, shunted by a $$8$$ $$\Omega$$ resistance, is balanced across a potentiometer wire of length $$3$$ m. The balancing length is $$2$$ m when the cell is shunted by $$4$$ $$\Omega$$ resistance. The value of internal resistance of the cell will be ______ $$\Omega$$.

The current density in a cylindrical wire of radius $$4$$ mm is $$4 \times 10^6$$ A m$$^{-2}$$. The current through the outer portion of the wire between radial distances $$\frac{R}{2}$$ and $$R$$ is ______ $$\pi$$ A.

In Young's double slit experiment the two slits are $$0.6$$ mm distance apart. Interference pattern is observed on a screen at a distance $$80$$ cm from the slits. The first dark fringe is observed on the screen directly opposite to one of the slits. The wavelength of light will be ______ nm.

A beam of monochromatic light is used to excite the electron in $$Li^{++}$$ from the first orbit to the third orbit. The wavelength of monochromatic light is found to be $$x \times 10^{-10}$$ m. The value of $$x$$ is ______.
[Given $$hc = 1242$$ eV nm]

The height of a transmitting antenna at the top of a tower is $$25$$ m and that of receiving antenna is, $$49$$ m. The maximum distance between them, for satisfactory communication in LOS (Line-Of-Sight) is $$K\sqrt{5} \times 10^2$$ m. The value of $$K$$ is ______.
(Assume radius of Earth is $$64 \times 10^5$$ m) [Calculate upto nearest integer value]