NTA JEE Mains 29th Jan 2024 Shift 1

A ball rolls off the top of a stairway with horizontal velocity $$u$$. The steps are $$0.1$$ m high and $$0.1$$ m wide. The minimum velocity $$u$$ with which that ball just hits the step 5 of the stairway will be $$\sqrt{x}$$ m s$$^{-1}$$, where $$x =$$ _______ [use $$g = 10$$ m s$$^{-2}$$].

A cylinder is rolling down on an inclined plane of inclination $$60°$$. Its acceleration during rolling down will be $$\frac{x}{\sqrt{3}}$$ m s$$^{-2}$$, where $$x =$$ _______ (use $$g = 10$$ m s$$^{-2}$$).

In a test experiment on a model aeroplane in wind tunnel, the flow speeds on the upper and lower surfaces of the wings are $$70$$ m s$$^{-1}$$ and $$65$$ m s$$^{-1}$$ respectively. If the wing area is $$2$$ m$$^2$$, the lift of the wing is _______ N. (Given density of air $$= 1.2$$ kg m$$^{-3}$$)

When the displacement of a simple harmonic oscillator is one third of its amplitude, the ratio of total energy to the kinetic energy is $$\frac{x}{8}$$, where $$x =$$ _______.

An electron is moving under the influence of the electric field of a uniformly charged infinite plane sheet $$S$$ having surface charge density $$+\sigma$$. The electron at $$t = 0$$ is at a distance of $$1$$ m from $$S$$ and has a speed of $$1$$ m s$$^{-1}$$. The maximum value of $$\sigma$$, if the electron strikes $$S$$ at $$t = 1$$ s is $$\alpha \left[\frac{m\varepsilon_0}{e}\right] \frac{C}{m^2}$$. The value of $$\alpha$$ is _______.

A $$16 \Omega$$ wire is bend to form a square loop. A $$9$$ V battery with internal resistance $$1 \Omega$$ is connected across one of its sides. If a $$4 \mu F$$ capacitor is connected across one of its diagonals, the energy stored by the capacitor will be $$\frac{x}{2} \mu J$$, where $$x =$$ _______.

The magnetic potential due to a magnetic dipole at a point on its axis situated at a distance of $$20$$ cm from its center is $$1.5 \times 10^{-5}$$ T m. The magnetic moment of the dipole is _______ A m$$^2$$. (Given: $$\frac{\mu_0}{4\pi} = 10^{-7}$$ T m A$$^{-1}$$)

A square loop of side $$10$$ cm and resistance $$0.7 \Omega$$ is placed vertically in the east-west plane. A uniform magnetic field of $$0.20$$ T is set up across the plane in the north-east direction. The magnetic field is decreased to zero in $$1$$ s at a steady rate. Then, the magnitude of induced emf is $$\sqrt{x} \times 10^{-3}$$ V. The value of $$x$$ is _______.

In a double slit experiment shown in figure, when light of wavelength $$400$$ nm is used, dark fringe is observed at $$P$$. If $$D = 0.2$$ m, the minimum distance between the slits $$S_1$$ and $$S_2$$ is $$\alpha$$ mm. Write the value of $$10\alpha$$ to the nearest integer.

When a hydrogen atom going from $$n = 2$$ to $$n = 1$$ emits a photon, its recoil speed is $$\frac{x}{5}$$ m s$$^{-1}$$. Where $$x =$$ _______. (Use: mass of hydrogen atom $$= 1.6 \times 10^{-27}$$ kg, charge of electron $$e = 1.6 \times 10^{-19}$$ C)