NTA JEE Main 13th April 2023 Shift 1

Two trains $$A$$ and $$B$$ of length $$l$$ and $$4l$$ are travelling into a tunnel of length $$L$$ in parallel tracks from opposite directions with velocities $$108$$ km h$$^{-1}$$ and $$72$$ km h$$^{-1}$$, respectively. If train $$A$$ take $$35$$ s less time than train $$B$$ to cross the tunnel then, length $$L$$ of tunnel is:
(Given $$L = 60 \ l$$)

A disc is rolling without slipping on a surface. The radius of the disc is $$R$$. At $$t = 0$$, the top most point on the disc is $$A$$ as shown in figure. When the disc completes half of its rotation, the displacement of point $$A$$ from its initial position is

The ratio of powers of two motors is $$\frac{3\sqrt{x}}{\sqrt{x}+1}$$, that are capable of raising $$300$$ kg water in $$5$$ minutes and $$50$$ kg water in $$2$$ minutes respectively from a well of $$100$$ m deep. The value of $$x$$ will be

A body of mass $$(5 \pm 0.5)$$ kg is moving with a velocity of $$(20 \pm 0.4)$$ m s$$^{-1}$$. Its kinetic energy will be

Two bodies are having kinetic energies in the ratio $$16 : 9$$. If they have same linear momentum, the ratio of their masses respectively is:

A bullet of $$10$$ g leaves the barrel of gun with a velocity of $$600$$ m s$$^{-1}$$. If the barrel of gun is $$50$$ cm long and mass of gun is $$3$$ kg, then value of impulse supplied to the gun will be:

A planet having mass $$9 M_e$$ and radius $$4R_e$$, where $$M_e$$ and $$R_e$$ are mass and radius of earth respectively, has escape velocity in km s$$^{-1}$$ given by: (Given escape velocity on earth $$V_e = 11.2 \times 10^3$$ m s$$^{-1}$$)

Under isothermal condition, the pressure of a gas is given by $$P = aV^{-3}$$, where $$a$$ is a constant and $$V$$ is the volume of the gas. The bulk modulus at constant temperature is equal to

The figure shows a liquid of a given density flowing steadily in a horizontal tube of a varying cross-section. Cross-sectional area at A is $$1.5$$ cm$$^2$$, and that at B is $$25$$ mm$$^2$$, if the speed of liquid at B is $$60$$ cm s$$^{-1}$$ then $$(P_A - P_B)$$ is
(Given $$P_A$$ and $$P_B$$ are liquid pressures at A and B points. Density $$\rho = 1000$$ kg m$$^{-3}$$. A and B are on the axis of tube)

The rms speed of oxygen molecule in a vessel at particular temperature is $$\left(1 + \frac{5}{x}\right)^{\frac{1}{2}}v$$, when $$v$$ is the average speed of the molecule. The value of $$x$$ will be:
(take $$\pi = \frac{22}{7}$$)