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A light cylindrical vessel is kept on a horizontal surface. Area of 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 impart force of the emerging liquid is \( (a \ll A) \).

2.

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

3.

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 \( \dfrac{h}{3} \) in both the directions.

4.

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 \( r_{2} \) \( (r_{2} < r_{1}) \).

5.

These 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 \( 9 \, \text{m}/\text{s} \) and makes an elastic collision with it. Thereafter \( B \) makes completely inelastic collision with \( C \). All motions occurs along same straight line. The final speed of \( C \) is:

6.

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

7.

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

8.

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

9.

Assertion A : If A, B, C and D are four points on a semi-circular arc with centre at 'O' such that \( |\overrightarrow {AB}| = |\overrightarrow {BC}| = |\overrightarrow {CD}| \), 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:

10.

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

11.

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

12.

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]

13.

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

14.

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

15.

A body takes \( 4 \) min. to cool from \( 61^{\circ} \text{C} \) and \( 59^{\circ} \text{C} \). If the temperature of the surroundings is \( 30^{\circ} \text{C} \), the time taken by the body to cool from \( 51^{\circ} \text{C} \) to \( 49^{\circ} \text{C} \) is:

16.

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 solid capacitor will be: (Given area of plate \( = A \))

17.

In the reported figure, there is a cyclic process \( ABCDA \) on a sample of \( 1 \) mole 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.

18.

Choose the correct answer from the options given below:

19.

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

20.

A capacitor of capacitance \( C = 1 \, \mu\text{F} \) is suddenly connected to a battery of \( 100 \, \text{volt} \) through a resistance \( R = 100 \, \Omega \). The time taken for the capacitor to be charged to get \( 50 \, \text{V} \) is: [Take \( \ln 2 = 0.69 \)]

21.

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

22.

Suppose two plants (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 \( \sqrt{\dfrac{a}{7} \, \dfrac{GM}{7 \, R}} \) then the value of \( 'a' \) is ________.
[Given : The two planets are fixed in their position]

23.

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

24.

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

25.

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 _______ \( \text{V} \). (Round off to the Nearest Integer)

26.

A particle of mass \( 9.1 \times 10^{-31} \, \text{kg} \) travels in a medium with a speed of \( 10^{6} \, \text{m} / \text{s} \) and a photon of a radioactive of linear momentum \( 10^{-27} \, \text{kg m} / \text{s} \) travels in vacuum. The wavelength of photon is _______ times the wavelength of the particle.

27.

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

28.

In Bohr’s atomic model, the electron is assumed to revolve in a circular orbit of radius \( \displaystyle 0.5 \, \overset{\circ}{\text{A}}. \). If the speed of electron is \( 2.2 \times 10^{6} \, \text{m} / \text{s} \), then the current associated with the electron will be _______ \( \times 10^{-2} \, \text{mA} \). [Take \( \pi \) as \( \dfrac{22}{7} \)]

29.

The amplitude of upper and lower side bands of A.M. wave where a carrier signal with frequency \( 11.21 \, \text{MHz} \), peak voltage \( 15 \, \text{V} \) is amplitude modulated by a \( 7.7 \, \text{kHz} \) sine wave of \( 5 \, \text{V} \) amplitude are \( \dfrac{a}{10} \, \text{V} \) and \( \dfrac{b}{10} \, \text{V} \) respectively. Then the value of \( \dfrac{a}{b} \) is ________.

30.

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 ________ \( \text{nm} \).

JEE Main 2021 27th July Morning Shift

Physics Chemistry Mathematics

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

JEE MainsJEE AdvanceBITSATJEE Mains | PYQ

JEE Mains JEE Advance BITSAT
JEE Mains | PYQ