NTA JEE Main 27th July 2021 Shift 1

Suppose two planets (spherical in shape) of radii $$R$$ and $$2R$$, but mass $$M$$ and $$9M$$ respectively have a centre to centre separation $$8R$$ as shown in the figure. A satellite of mass $$m$$ is projected from the surface of the planet of mass $$M$$ directly towards the centre of the second planet. The minimum speed $$v$$ required for the satellite to reach the surface of the second planet is $$\sqrt{\frac{a}{7} \frac{GM}{R}}$$, then the value of $$a$$ is
[Given: The two planets are fixed in their position]

A stone of mass 20 g is projected from a rubber catapult of length 0.1 m and area of cross section $$10^{-6}$$ m$$^2$$ stretched by an amount 0.04 m. The velocity of the projected stone is _________ m s$$^{-1}$$. (Young's modulus of rubber = $$0.5 \times 10^9$$ N m$$^{-2}$$)

In a uniform magnetic field, the magnetic needle has a magnetic moment $$9.85 \times 10^{-2}$$ A m$$^{-2}$$ and moment of inertia $$5 \times 10^{-6}$$ kg m$$^2$$. If it performs 10 complete oscillations in 5 seconds then the magnitude of the magnetic field is _________ mT [Take $$\pi^2$$ as 9.85]

Consider an electrical circuit containing a two way switch $$S$$. Initially $$S$$ is open and then $$T_1$$ is connected to $$T_2$$. As the current in $$R = 6 \, \Omega$$ attains a maximum value of steady-state level, $$T_1$$ is disconnected from $$T_2$$ and immediately connected to $$T_3$$. Potential drop across $$r = 3 \, \Omega$$ resistor immediately after $$T_1$$ is connected to $$T_3$$ is _________ V. (Round off to the Nearest Integer)

A prism of refractive index $$n_1$$ and another prism of refractive index $$n_2$$ are stuck together (as shown in the figure). $$n_1$$ and $$n_2$$ depend on $$\lambda$$, the wavelength of light, according to the relation $$n_1 = 1.2 + \frac{10.8 \times 10^{-14}}{\lambda^2}$$ and $$n_2 = 1.45 + \frac{1.8 \times 10^{-14}}{\lambda^2}$$
The wavelength for which rays incident at any angle on the interface $$BC$$ pass through without bending at that interface will be _________ nm.

A particle of mass $$9.1 \times 10^{-31}$$ kg travels in a medium with a speed of $$10^6$$ m s$$^{-1}$$ and a photon of radiation of linear momentum $$10^{-27}$$ kg m s$$^{-1}$$ travels in a vacuum. The wavelength of the photon is _________ times the wavelength of the particle.

In Bohr's atomic model, the electron is assumed to revolve in a circular orbit of radius 0.5 $$\mathring{A}$$. If the speed of electron is $$2.2 \times 10^6$$ m s$$^{-1}$$. Then the current associated with the electron will be _________ $$\times 10^{-2}$$ mA. [Take $$\pi$$ as $$\frac{22}{7}$$]

A radioactive sample has an average life of 30 ms and is decaying. A capacitor of capacitance 200 $$\mu$$F is first charged and later connected with resistor $$R$$. If the ratio of the charge on the capacitor to the activity of the radioactive sample is fixed with respect to time then the value of $$R$$ should be _________ $$\Omega$$.

A transistor is connected in common emitter circuit configuration, the collector supply voltage is 10 V and the voltage drop across a resistor of 1000 $$\Omega$$ in the collector circuit is 0.6 V. If the current gain factor $$(\beta)$$ is 24, then the base current is _________ $$\mu$$A. (Round off to the Nearest Integer)

The amplitude of upper and lower side bands of AM wave where a carrier signal with frequency 11.21 MHz, peak voltage 15 V is amplitude modulated by a 7.7 kHz sine wave of 5 V amplitude are $$\frac{a}{10}$$ V and $$\frac{b}{10}$$ V respectively. Then the value of $$\frac{a}{b}$$ is _________.