NTA JEE Main 29th January 2023 Shift 2

A car is moving on a circular path of radius $$600$$ m such that the magnitudes of the tangential acceleration and centripetal acceleration are equal. The time taken by the car to complete first quarter of revolution, if it is moving with an initial speed of $$54$$ km h$$^{-1}$$ is $$t(1-e^{-\frac{\pi}{2}})$$ s. The value of $$t$$ is ______.

A particle of mass $$100$$ g is projected at time $$t = 0$$ with a speed $$20$$ m s$$^{-1}$$ at an angle $$45°$$ to the horizontal as given in the figure. The magnitude of the angular momentum of the particle about the starting point at time $$t = 2$$ s is found to be $$\sqrt{K}$$ kg m$$^2$$ s$$^{-1}$$. The value of $$K$$ is ______.
(Take $$g = 10$$ m s$$^{-2}$$)

A metal block of base area $$0.20$$ m$$^2$$ is placed on a table, as shown in the figure. A liquid film of thickness $$0.25$$ mm is inserted between the block and the table. The block is pushed by a horizontal force of $$0.1$$ N and moves with a constant speed. If the viscosity of the liquid is $$5.0 \times 10^{-3}$$ Pl, the speed of the block is ______ $$\times 10^{-3}$$ m s$$^{-1}$$.

A particle of mass $$250$$ g executes a simple harmonic motion under a periodic force $$F = (-25x)$$ N. The particle attains a maximum speed of $$4$$ m s$$^{-1}$$ during its oscillation. The amplitude of the motion is ______ cm.

For a charged spherical ball, electrostatic potential inside the ball varies with $$r$$ as $$V = 2ar^2 + b$$. Here, $$a$$ and $$b$$ are constant and $$r$$ is the distance from the center. The volume charge density inside the ball is $$-\lambda a\varepsilon$$. The value of $$\lambda$$ is ______.
$$\varepsilon$$ = permittivity of medium.

A null point is found at $$200$$ cm in potentiometer when cell in secondary circuit is shunted by $$5$$ $$\Omega$$. When a resistance of $$15$$ $$\Omega$$ is used for shunting null point moves to $$300$$ cm. The internal resistance of the cell is ______ $$\Omega$$.

An inductor of inductance $$2$$ $$\mu$$H is connected in series with a resistance, a variable capacitor and an AC source of frequency $$7$$ kHz. The value of capacitance for which maximum current is drawn into the circuit is $$\frac{1}{x}$$ F, where the value of $$x$$ is ______. (Take $$\pi = \frac{22}{7}$$)

Unpolarised light is incident on the boundary between two dielectric media, whose dielectric constants are $$2.8$$ (medium $$-1$$) and $$6.8$$ (medium $$-2$$), respectively. To satisfy the condition, so that the reflected and refracted rays are perpendicular to each other, the angle of incidence should be $$\tan^{-1}\left(1 + \frac{10}{\theta}\right)^{\frac{1}{2}}$$, the value of $$\theta$$ is ______.
(Given for dielectric media, $$\mu_r = 1$$)

The ratio of de-Broglie wavelength of an $$\alpha$$-particle and a proton accelerated from rest by the same potential is $$\frac{1}{\sqrt{m}}$$, the value of $$m$$ is:

When two resistance $$R_1$$ and $$R_2$$ connected in series and introduced into the left gap of a meter bridge and a resistance of $$10$$ $$\Omega$$ is introduced into the right gap, a null point is found at $$60$$ cm from left side. When $$R_1$$ and $$R_2$$ are connected in parallel and introduced into the left gap, a resistance of $$3$$ $$\Omega$$ is introduced into the right gap to get null point at $$40$$ cm from left end. The product of $$R_1 R_2$$ is ______ $$\Omega^2$$