NTA JEE Main 27th July 2021 Shift 1 - Physics

Assertion $$A$$ : If $$A, B, C, D$$ are four points on a semi-circular arc with a centre at $$O$$ such that $$\left|\overrightarrow{AB}\right| = \left|\overrightarrow{BC}\right| = \left|\overrightarrow{CD}\right|$$. Then, $$\overrightarrow{AB} + \overrightarrow{AC} + \overrightarrow{AD} = 4\overrightarrow{AO} + \overrightarrow{OB} + \overrightarrow{OC}$$
Reason $$R$$ : Polygon law of vector addition yields $$\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CD} = \overrightarrow{AD} = 2\overrightarrow{AO}$$

In the light of the above statements, choose the most appropriate answer from the options given below.

A ball is thrown up with a certain velocity so that it reaches a height $$h$$. Find the ratio of the two different times of the ball reaching $$\frac{h}{3}$$ in both the directions.

Three objects $$A$$, $$B$$ and $$C$$ are kept in a straight line on a frictionless horizontal surface. The masses of $$A$$, $$B$$ and $$C$$ are $$m$$, $$2m$$ and $$2m$$ respectively. $$A$$ moves towards $$B$$ with a speed of 9 m s$$^{-1}$$ and makes an elastic collision with it. Thereafter $$B$$ makes a completely inelastic collision with $$C$$. All motions occur along the same straight line. The final speed of $$C$$ is:

List-I                                                                                                                                                                     List-II
(a) MI of the rod (length $$L$$, Mass $$M$$, about an axis $$\perp$$ to the rod passing through the midpoint)          (i) $$\frac{8ML^2}{3}$$
(b) MI of the rod (length $$L$$, Mass 2M, about an axis $$\perp$$ to the rod passing through one of its end)        (ii) $$\frac{ML^2}{3}$$
(c) MI of the rod (length 2L, Mass $$M$$, about an axis $$\perp$$ to the rod passing through its midpoint)          (iii) $$\frac{ML^2}{12}$$
(d) MI of the rod (Length 2L, Mass 2M, about an axis $$\perp$$ to the rod passing through one of its end)        (iv) $$\frac{2ML^2}{3}$$

Choose the correct answer from the options given below:

The figure shows two solid discs with radius $$R$$ and $$r$$ respectively. If mass per unit area is the same for both, what is the ratio of MI of bigger disc around axis $$AB$$ (Which is $$\perp$$ to the plane of the disc and passing through its centre) of MI of smaller disc around one of its diameters lying on its plane? Given $$M$$ is the mass of the larger disc.

A light cylindrical vessel is kept on a horizontal surface. Area of the base is $$A$$. A hole of cross-sectional area $$a$$ is made just at its bottom side. The minimum coefficient of friction necessary to prevent sliding the vessel due to the impact force of the emerging liquid is

A body takes 4 min to cool from 61°C to 59°C. If the temperature of the surroundings is 30°C, the time taken by the body to cool from 51°C to 49°C is:

In the reported figure, there is a cyclic process $$ABCDA$$ on a sample of 1 mol of a diatomic gas. The temperature of the gas during the process $$A \rightarrow B$$ and $$C \rightarrow D$$ are $$T_1$$ and $$T_2$$ ($$T_1 > T_2$$) respectively.

Choose the correct option out of the following for work done if processes $$BC$$ and $$DA$$ are adiabatic.

The number of molecules in one litre of an ideal gas at 300 K and 2 atmospheric pressure with mean kinetic energy $$2 \times 10^{-9}$$ J per molecule is:

A particle starts executing simple harmonic motion (SHM) of amplitude $$a$$ and total energy $$E$$. At any instant, its kinetic energy is $$\frac{3E}{4}$$, then its displacement $$y$$ is given by:

Two identical tennis balls each having mass $$m$$ and charge $$q$$ are suspended from a fixed point by threads of length $$l$$. What is the equilibrium separation when each thread makes a small angle $$\theta$$ with the vertical?

The relative permittivity of distilled water is 81. The velocity of light in it will be: (Given $$\mu_r = 1$$)

In the reported figure, a capacitor is formed by placing a compound dielectric between the plates of parallel plate capacitor. The expression for the capacity of the said capacitor will be: (Given the area of the plate = $$A$$)

Two capacitors of capacities $$2C$$ and $$C$$ are joined in parallel and charged up to potential $$V$$. The battery is removed and the capacitor of capacity $$C$$ is filled completely with a medium of dielectric constant $$K$$. The potential difference across the capacitors will now be:

In the given figure, a battery of emf $$E$$ is connected across a conductor $$PQ$$ of length $$l$$ and different area of cross-sections having radii $$r_1$$ and $$r_2$$ ($$r_2 < r_1$$).

Choose the correct option as one moves from $$P$$ to $$Q$$.

A 0.07 H inductor and a 12 $$\Omega$$ resistor are connected in series to a 220 V, 50 Hz AC source. The approximate current in the circuit and the phase angle between current and source voltage are respectively. [Take $$\pi$$ as $$\frac{22}{7}$$]

In Young's double slit experiment, if the source of light changes from orange to blue then:

If $$f$$ denotes the ratio of the number of nuclei decayed $$(N_d)$$ to the number of nuclei at $$t = 0$$, $$(N_0)$$ then for a collection of radioactive nuclei, the rate of change of $$f$$ with respect to time is given as: [$$\lambda$$ is the radioactive decay constant]

Assertion $$A$$ : If in five complete rotations of the circular scale, the distance travelled on the main scale of the screw gauge is 5 mm and there are 50 total divisions on a circular scale, then the least count is 0.001 cm.
Reason $$R$$ : Least Count = $$\frac{\text{Pitch}}{\text{Total divisions on circular scale}}$$
In the light of the above statements, choose the most appropriate answer from the options given below.

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 _________.