NTA JEE Main 1st February 2023 Shift 2

For a train engine moving with speed of $$20$$ ms$$^{-1}$$, the driver must apply brakes at a distance of $$500$$ m before the station for the train to come to rest at the station. If the brakes were applied at half of this distance, the train engine would cross the station with speed $$\sqrt{x}$$ ms$$^{-1}$$. The value of $$x$$ is ______. (Assuming same retardation is produced by brakes)

A block is fastened to a horizontal spring. The block is pulled to a distance $$x = 10$$ cm from its equilibrium position (at $$x = 0$$) on a frictionless surface from rest. The energy of the block at $$x = 5$$ cm is $$0.25$$ J. The spring constant of the spring is ______ N m$$^{-1}$$.

A force $$F = (5 + 3y^{2})$$ acts on a particle in the $$y$$-direction, where $$F$$ is newton and $$y$$ is in meter. The work done by the force during a displacement from $$y = 2$$ m to $$y = 5$$ m is ______ J.

Moment of inertia of a disc of mass $$M$$ and radius '$$R$$' about any of its diameter is $$\frac{MR^2}{4}$$. The moment of inertia of this disc about an axis normal to the disc and passing through a point on its edge will be, $$\frac{x}{2}MR^2$$. The value of $$x$$ is ______.

The surface of water in a water tank of cross section area $$750$$ cm$$^2$$ on the top of a house is $$h$$ m above the tap level. The speed of water coming out through the tap of cross section area $$500$$ mm$$^2$$ is $$30$$ cm s$$^{-1}$$. At that instant, $$\frac{dh}{dt}$$ is $$x \times 10^{-3}$$ m s$$^{-1}$$. The value of $$x$$ will be ______.

A cubical volume is bounded by the surfaces $$x = 0$$, $$x = a$$, $$y = 0$$, $$y = a$$, $$z = 0$$, $$z = a$$. The electric field in the region is given by $$\vec{E} = E_0 x \hat{i}$$. Where $$E_0 = 4 \times 10^4$$ NC$$^{-1}$$ m$$^{-1}$$. If $$a = 2$$ cm, the charge contained in the cubical volume is $$Q \times 10^{-14}$$ C. The value of $$Q$$ is ______. (Take $$\epsilon_0 = 9 \times 10^{-12}$$ C$$^2$$ N$$^{-1}$$m$$^{-2}$$)

As shown in the figure, a long straight conductor with semicircular arc of radius $$\frac{\pi}{10}$$ m is carrying current $$I = 3A$$. The magnitude of the magnetic field at the center $$O$$ of the arc is: (The permeability of the vacuum $$= 4\pi \times 10^{-7}$$ NA$$^{-2}$$)

Answer in $$\mu$$T.

A square shaped coil of area $$70$$ cm$$^2$$ having $$600$$ turns rotates in a magnetic field of $$0.4$$ Wb m$$^{-2}$$, about an axis which is parallel to one of the side of the coil and perpendicular to the direction of field. If the coil completes $$500$$ revolution in a minute, the instantaneous emf when the plane of the coil is inclined at $$60°$$ with the field, will be ______ V. (Take $$\pi = \frac{22}{7}$$)

As shown in the figure, in Young's double slit experiment, a thin plate of thickness $$t = 10$$ $$\mu$$m and refractive index $$\mu = 1.2$$ is inserted infront of slit $$S_1$$. The experiment is conducted in air ($$\mu = 1$$) and uses a monochromatic light of wavelength $$\lambda = 500$$ nm. Due to the insertion of the plate, central maxima is shifted by a distance of $$x\beta_0$$. $$\beta_0$$ is the fringe-width before the insertion of the plate. The value of $$x$$ is ______.

Nucleus A having $$Z = 17$$ and equal number of protons and neutrons has $$1.2$$ MeV binding energy per nucleon. Another nucleus $$B$$ of $$Z = 12$$ has total 26 nucleons and $$1.8$$ MeV binding energy per nucleons. The difference of binding energy of $$B$$ and $$A$$ will be ______ MeV.