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Physics

The half-life of \( ^{198}\mathrm{Au} \) is \( 3 \, \text{days} \). If atomic weight of \( ^{198}\mathrm{Au} \) is \( 198 \, \text{g/mol} \) then the activity of \( 2 \, \text{mg} \) of \( ^{198}\mathrm{Au} \) is [in disintegration/second]:

2.

Two different metal bodies A and B of equal mass are heated at a uniform rate under similar conditions. The variation of temperature of the bodies is graphically represented as shown in the figure. The ratio of specific heat capacities is:

3.

In the given figure, there is a circuit of potentiometer of length \( AB = 10 \, \text{m} \). The resistance per unit length is \( 0.1 \, \Omega / \text{cm} \). Across \( AB \), a battery of emf \( E \) and internal resistance '\( r \)' is connected. The maximum value of emf measured by this potentiometer is:

4.

Two wires of same length and radius are joined end to end and loaded. The young’s modulii of the materials of the two wires are \( Y_1 \) and \( Y_2 \). The combination behaves as a single wire then its Young’s modulus is:

5.

A particle of mass \( 4M \) at rest disintegrates into two particles of mass \( M \) and \( 3M \) respectively having non zero velocities. The ratio of de-Broglie wavelength of particle of mass \( M \) to that of mass \( 3M \) will be:

6.

In Amplitude Modulation, the message signal \( V_m(t) = 10 \sin(2\pi \times 10^{5} t) \) volts and carrier signal \( V_c(t) = 20 \sin(2\pi \times 10^{7} t) \) volts. The modulates signal now contains the message signal with lower side band and upper side band frequency, therefore the bandwidth of modulated signal is \( a \) kHz. The value of \( a \) is:

7.

In the Young's double slit experiment, the distance between the slits varies in time as \( d(t)=d_0+a_0 \sin \omega t \) where \( d_0, \omega \) and \( a_0 \) are constants. The difference between the largest fringe width and the smallest fringe width obtained over time is given as:

8.

A ray of laser of a wavelength \( 630 \ \text{nm} \) is incident at an angle of \( 30^\circ \) at the diamond–air interface. It is going from diamond to air. The refractive index of diamond is \( 2.42 \) and that of air is \( 1 \). Choose the correct option.

9.

A monoatomic ideal gas, initially at temperature \( T_1 \) is enclosed in a cylinder fitted with a frictionless piston. The gas is allowed to expand adiabatically to a temperature \( T_2 \) by releasing the piston suddenly. If \( l_1 \) and \( l_2 \) are the lengths of the gas column, before and after the expansion respectively, then the value of \( \dfrac{T_1}{T_2} \) will be:

10.

Two billiard of equal mass \(30 g\) strike a rigid wall with same speed of \(108\,\text{ kmph}\) (as shown) but at different angles. if the balls get reflected with the same speed then the ratio of the magnitude of impulses imparted to ball 'a' and ball ‘b’ by the wall along 'X' direction is :

11.

A linearly polarized electromagnetic wave in vacuum is \( \vec{E}=3.1\cos\!\left[(1.8)z-(5.4\times 10^{6}) t\right]\\,\hat{i}\ \text{N}/\text{C} \) is incident normally on a perfectly reflecting wall at \( z=a \). Choose the correct option

12.

Water droplets are coming from an open tap at a particular rate, The spacing between a droplet observed at \( 4^{\text{th}} \) second after its fall to the next droplet is \( 34.3 \ \text{m} \). At what rate the droplets are coming from the tap? (Take \( g=9.8 \ \text{m}/\text{s}^{2} \))

13.

Identify the logic operation carried out.

14.

The minimum and maximum distances of a planet revolving around the Sun are \( x_1 \) and \( x_2 \). If the minimum speed of the planet on its trajectory is \( v_0 \) then its maximum speed will be

15.

Some nuclei of a radioactive material are undergoing radioactive decay. The time gap between the instances when a quarter of the nuclei have decayed and when half of the nuclei have decayed is given as:
(where \( \lambda \) is the decay constant)

16.

For a gas \( C_p - C_v = R \) in a state P and \( C_p - C_v = 1.10\,R \) in a state Q, \( T_P \) and \( T_Q \) are the temperatures in two different states P and Q respectively. Then

17.

A parallel plate capacitor with plate area '\( A \)' and distance of separation '\( d \)' is filled with a dielectric. What is the capacity of the capacitor when permittivity of the dielectric varies as:
\( \displaystyle \varepsilon(x) = \varepsilon_0 + kx, \text{ for } \left(0 < x \le \frac{d}{2}\right) \)
\( \varepsilon(x) = \varepsilon_0 + k(d-x), \text{ for }\left(\dfrac{d}{2} \le x \le d \right)\)

18.

Match List I with List II.

Choose the correct answer from the options given below:

19.

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: Moment of inertia of a circular disc of mass '\( M \)' and radius '\( R \)' about \( X \), \( Y \) axes (passing through its plane) and \( Z \)-axis which is perpendicular to its plane were found to be \( I_x \), \( I_y \) and \( I_z \) respectively. The respective radii of gyration about all the three axes will be the same.
Reason R: A rigid body making rotational motion has fixed mass and shape.
In the light of the above statements, choose the most appropriate answer from the options given below :

20.

What should be the order of arrangement of de-Broglie wavelength of electron \( (\lambda_e) \), an \( \alpha \)-particle \( (\lambda_\alpha) \) and proton \(( \lambda_p) \) given that all have the same kinetic energy?

21.

The value of aluminium susceptibility is \( 2.2 \times 10^{-5} \). The percentage increase in the magnetic field if space within a current carrying toroid is filled with Aluminium is \( \dfrac{x}{10^4} \). Then the value of \( x \) is __________.

22.

A particle of mass '\( m \)' is moving in time '\( t \)' on a trajectory given by
\( \vec r = 10 \alpha t^{2}\,\hat{i} + 5\beta (t-5)\,\hat{j} \)
Where \( \alpha \) and \( \beta \) are dimensional constants.
The angular momentum of the particle becomes the same as it was for \( t = 0 \) at time \( t = \) __________ seconds.

23.

A particle of mass \( 1 \, \text{mg} \) and charge \( q \) is lying at the mid-point of two stationary particles kept at a distance '\( 2 \, \text{m} \)' when each is carrying same charge '\( q \)'. If the free charged particle is displaced from its equilibrium position through distance '\( x \)' (\( x \ll 1 \, \text{m} \)). The particle executes SHM. Its angular frequency of oscillation will be __________ \( \times 10^{5} \, \text{rad/s} \) if \( q^{2} = 10 \, c^{2} \).

24.

A pendulum bob has a speed of \( 3 \, \text{m/s} \) at its lowest position. The pendulum is \( 50 \, \text{cm} \) long. The speed of bob, when the length makes an angle of \( 60^\circ \) to the vertical will be \( (g = 10 \, \text{m/s}^2) \), ____________ \( \text{m/s} \).

25.

An electric bulb rated as \( 200 \, \text{W} \) at \( 100 \, \text{V} \) is used in a circuit having \( 200 \, \text{V} \) supply. The resistance ‘\( R \)’ that must be put in series with the bulb so that the bulb delivers the same power is ____________ \( \Omega \).

26.

In the reported figure, two bodies A and B of masses \( 200 \, \text{g} \) and \( 800 \, \text{g} \) are attached with the system of springs. Springs are kept in a stretched position with some extension when the system is released. The horizontal surface is assumed to frictionless. The angular frequency will be __________ rad/s when \( k = 20 \, \text{N/m} \).

27.

A circular conducting coil of radius \( 1 \, \text{m} \) is being heated by the change of magnetic field \( \vec{B} \) passing perpendicular to the plane in which the coil is laid. The resistance of the coil is \( 2 \, \mu \Omega \). The magnetic field is slowly switched off such that its magnitude changes in time as \( B = \dfrac{4}{\pi} \times 10^{-3} \, \text{T} \left( 1 - \dfrac{t}{100} \right) \). The energy dissipated by the coil before the magnetic field is switched off completely is \( E = \) __________ \( \text{m} \)).

28.

Student A and Student B used two screw gauges of equal pitch and \( 100 \) equal circular divisions to measure the radius of a given wire. The actual value of the radius of the wire is \( 0.322 \, \text{cm} \). The absolute value of the difference between the final circular scale readings observed by the students A and B is __________________ . [Figure shows position of reference ‘\( O \)’ when jaws of screw gauge are closed]
Given pitch = \( 0.1 \, \text{cm} \).

29.

An inductor of \( 10 \, \text{mH} \) is connected to a \( 20 \, \text{V} \) battery through a resistor of \( 10 \, \text{k}\Omega \) and a switch. After a long time, when maximum current is set up in the circuit, the current is switched off. The current in the circuit after \( 1 \, \mu \text{s} \) is \( \dfrac{x}{100} \, \text{mA} \). Then \( x \) is equal to __________________ .
(Take \( e^{-1} = 0.37 \) )

30.

A body of mass \( 2 \, \text{kg} \) moving with a speed of \( 4 \, \text{m/s} \), makes an elastic collision with another body at rest and continues to move in the original direction but with one fourth of its initial speed. The speed of the two body centre of mass is \( \dfrac{x}{10} \, \text{m/s} \). Then the value of \( x \) is ________________ .

JEE Main 2021 25th July Morning Shift

Physics Chemistry Mathematics

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Identify the logic operation carried out.

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