NTA JEE Main 31st January 2023 Shift 1

The speed of a swimmer is 4 km h$$^{-1}$$ in still water. If the swimmer makes his strokes normal to the flow of river of width 1 km, he reaches a point 750 m down the stream on the opposite bank. The speed of the river water is ______ km h$$^{-1}$$.

A lift of mass $$M = 500$$ kg is descending with speed of 2 m s$$^{-1}$$. Its supporting cable begins to slip thus allowing it to fall with a constant acceleration of 2 m s$$^{-2}$$. The kinetic energy of the lift at the end of fall through to a distance of 6 m will be ______ kJ.

A solid sphere of mass 1 kg rolls without slipping on a plane surface. Its kinetic energy is $$7 \times 10^{-3}$$ J. The speed of the centre of mass of the sphere is ______ cm s$$^{-1}$$.

A thin rod having a length of 1 m and area of cross-section $$3 \times 10^{-6}$$ m$$^2$$ is suspended vertically from one end. The rod is cooled from 210°C to 160°C. After cooling, a mass $$M$$ is attached at the lower end of the rod such that the length of rod again becomes 1 m. Young's modulus and coefficient of linear expansion of the rod are $$2 \times 10^{11}$$ N m$$^{-2}$$ and $$2 \times 10^{-5}$$ K$$^{-1}$$, respectively. The value of $$M$$ is ______ kg. (Take $$g = 10$$ m s$$^{-2}$$)

In the figure given below, a block of mass $$M = 490$$ g placed on a frictionless table is connected with two springs having same spring constant ($$K = 2$$ N m$$^{-1}$$). If the block is horizontally displaced through $$X$$ m then the number of complete oscillations it will make in $$14\pi$$ seconds will be ______.

Expression for an electric field is given by $$\vec{E} = 4000 \; x^2 \; \hat{i}$$ V m$$^{-1}$$. The electric flux through the cube of side 20 cm when placed in electric field (as shown in the figure) is ______ V cm.

Two identical cells, when connected either in parallel or in series gives same current in an external resistance 5 $$\Omega$$. The internal resistance of each cell will be ______ $$\Omega$$.

An inductor of 0.5 mH, a capacitor of 20 $$\mu$$F and resistance of 20 $$\Omega$$ are connected in series with a 220 V ac source. If the current is in phase with the emf, the amplitude of current of the circuit is $$\sqrt{x}$$ A. The value of $$x$$ is-

In a medium the speed of light wave decreases to 0.2 times to its speed in free space. The ratio of relative permittivity to the refractive index of the medium is $$x$$:1. The value of $$x$$ is ______.
(Given speed of light in free space $$= 3 \times 10^8$$ m s$$^{-1}$$ and for the given medium $$\mu_r = 1$$)

For hydrogen atom, $$\lambda_1$$ and $$\lambda_2$$ are the wavelengths corresponding to the transitions 1 and 2 respectively as shown in figure. The ratio of $$\lambda_1$$ and $$\lambda_2$$ is $$\dfrac{x}{32}$$. The value of $$x$$ is ______.