NTA JEE Main 20th July 2021 Shift 1

In a spring gun having spring constant 100 N m$$^{-1}$$ a small ball $$B$$ of mass 100 g is put in its barrel (as shown in figure) by compressing the spring through 0.05 m. There should be a box placed at a distance $$d$$ on the ground so that the ball falls in it. If the ball leaves the gun horizontally at a height of 2 m above the ground. The value of $$d$$ is ___ m.
$$(g = 10$$ m s$$^{-2})$$

A body having specific charge 8 $$\mu$$C g$$^{-1}$$ is resting on a frictionless plane at a distance 10 cm from the wall (as shown in the figure). It starts moving towards the wall when a uniform electric field of 100 V m$$^{-1}$$ is applied horizontally towards the wall. If the collision of the body with the wall is perfectly elastic, then the time period of the motion will be ___ s.

A rod of mass $$M$$ and length $$L$$ is lying on a horizontal frictionless surface. A particle of mass $$m$$ travelling along the surface hits at one end of the rod with a velocity $$u$$ in a direction perpendicular to the rod. The collision is completely elastic. After collision, particle comes to rest. The ratio of masses $$\left(\frac{m}{M}\right)$$ is $$\frac{1}{x}$$. The value of $$x$$ will be ___.

A circular disc reaches from top to bottom of an inclined plane of length $$L$$. When it slips down the plane, it takes time $$t_1$$. When it rolls down the plane, it takes time $$t_2$$. The value of $$\frac{t_2}{t_1}$$ is $$\sqrt{\frac{3}{x}}$$. The value of $$x$$ will be ___.

In the reported figure, heat energy absorbed by a system in going through a cyclic process is ___ $$\pi$$ J.

The amplitude of wave disturbance propagating in the positive $$x$$-direction is given by $$y = \frac{1}{(1+x)^2}$$ at time $$t = 0$$ and $$y = \frac{1}{1+(x-2)^2}$$ at $$t = 1$$ s, where $$x$$ and $$y$$ are in metres. The shape of wave does not change during the propagation. The velocity of the wave will be ___ m s$$^{-1}$$.

The frequency of a car horn encountered a change from 400 Hz to 500 Hz. When the car approaches a vertical wall. If the speed of sound is 330 m s$$^{-1}$$. Then the speed of car is ___ km h$$^{-1}$$.

In an $$LCR$$ series circuit, an inductor 30 mH and a resistor 1 $$\Omega$$ are connected to an AC source of angular frequency 300 rad s$$^{-1}$$. The value of capacitance for which the current leads the voltage by 45$$^\circ$$ is $$\frac{1}{x} \times 10^{-3}$$ F. Then the value of $$x$$ is ___.

An object viewed from a near point distance of 25 cm, using a microscopic lens with magnification 6, gives an unresolved image. A resolved image is observed at infinite distance with a total magnification double the earlier using an eyepiece along with the given lens and a tube of length 0.6 m, if the focal length of the eyepiece is equal to ___ cm.

A carrier wave $$V_C(t) = 160 \sin(2\pi \times 10^6 t)$$ volts is made to vary between $$V_{max} = 200$$ V and $$V_{min} = 120$$ V by a message signal $$V_m(t) = A_m \sin(2\pi \times 10^3 t)$$ volts. The peak voltage $$A_m$$ of the modulating signal is ___.