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An expression of energy density is given by \( u = \dfrac{\alpha}{\beta}\sin\!\left(\dfrac{\alpha x}{k t}\right) \), where \( \alpha, \beta \) are constants, \( x \) is displacement, \( k \) is Boltzmann constant and \( t \) is the temperature. The dimensions of \( \beta \) will be:

2.

A body of mass \( 10 \, \text{kg} \) is projected at an angle of \( 45^\circ \) with the horizontal. The trajectory of the body is observed to pass through a point \( (20, 10) \). If \( T \) is the time of flight, then its momentum vector, at time \( t = \dfrac{T}{\sqrt{2}} \), is __________ [Take \( g = 10 \, \text{m}/\text{s}^{2} \)]

3.

A block of mass \( M \) slides down on a rough inclined plane with constant velocity. The angle made by the incline plane with horizontal is \( \theta \). The magnitude of the contact force will be:

4.

A block ‘A’ takes \( 2 \, \text{s} \) to slide down a frictionless incline of \( 30^\circ \) and length '\( l \)', kept inside a lift going up with uniform velocity ‘\( v \)’. If the incline is changed to \( 45^\circ \), the time taken by the block, to slide down the incline, will be approximately.

5.

The velocity of the bullet becomes one third after it penetrates \( 4 \, \text{cm} \) in a wooden block. Assuming that bullet is facing a constant resistance during its motion in the block. The bullet stops completely after travelling at \( (4 + x) \, \text{cm} \) inside the block. The value of \( x \) is:

6.

A body of mass \( m \) is projected with velocity \( \lambda v_{e} \) in vertically upward direction from the surface of the earth into space. It is given that \( v_{e} \) is escape velocity and \( \lambda < 1 \). If air resistance is considered to be negligible, the maximum height from the centre of earth, to which the body can go, will be: (\( R \): radius of earth)

7.

A steel wire of length \( 3.2 \, \text{m} \) \( (Y_s = 2.0 \times 10^{11} \, \text{N m}^{-2}) \) and a copper wire of length \( 4.4 \, \text{m} \) \( (Y_C = 1.1 \times 10^{11} \, \text{N m}^{-2}) \), both of radius \( 1.4 \, \text{mm} \) are connected end to end. When stretched by a load, the net elongation is found to be \( 1.4 \, \text{mm} \). The load applied, in Newton, will be: (Given \( \pi = \dfrac{22}{7} \))

8.

In \( 1^{\text{st}} \) case, Carnot engine operates between temperature \( 300 \, \text{K} \) and \( 100 \, \text{K} \). In \( 2^{\text{nd}} \) case, as shown in the figure, a combination of two engines is used. The efficiency of this combination (in \( 2^{\text{nd}} \) case) will be:

9.

Which statements are correct about degrees of freedom?
\( (A) \) A molecule with \( n \) degrees of freedom has \( n^{2} \) different ways of storing energy.
\( (B) \) Each degree of freedom is associated with \( \dfrac{1}{2} RT \) average energy per mole.
\( (C) \) A monatomic gas molecule has \( 1 \) rotational degree of freedom where as diatomic molecule has \( 2 \) rotational degrees of freedom.
\( (D) \) \( \mathrm{CH}_{4} \) has a total of \( 6 \) degrees of freedom.
Choose the correct answer from the options given below:

10.

A charge of \( 4 \, \mu \text{C} \) is to be divided into two. The distance between the two divided charges is constant. The magnitude of the divided charges so that the force between them is maximum, will be:

11.

\( (A) \) The drift velocity of electrons decreases with the increase in the temperature of conductor.
\( (B) \) The drift velocity is inversely proportional to the area of cross-section of given conductor.
\( (C) \) The drift velocity does not depend on the applied potential difference to the conductor.
\( (D) \) The drift velocity of electron is inversely proportional to the length of the conductor.
\( (E) \) The drift velocity increases with the increase in the temperature of conductor.
Choose the correct answer from the options given below:

12.

A compass needle of oscillation magnetometer oscillates \( 20 \) times per minute at a place \( P \) of dip \( 30^\circ \). The number of oscillations per minute become \( 10 \) at another place \( Q \) of \( 60^\circ \) dip. The ratio of the total magnetic field at the two places \( (B_Q : B_P) \) is:

13.

A cyclotron is used to accelerate protons. If the operating magnetic field is \( 1.0 \, \text{T} \) and the radius of the cyclotron ‘dees’ is \( 60 \, \text{cm} \), the kinetic energy of the accelerated protons in \( \text{MeV} \) will be:
[use \( m_p = 1.6 \times 10^{-27} \, \text{kg} , \ e = 1.6 \times 10^{-19} \, \text{C} \)]

14.

A series LCR circuit has \( L = 0.01 \, \text{H} , \ R = 10 \, \Omega \) and \( C = 1 \, \mu \text{F} \) and it is connected to ac voltage of amplitude \( (V_m) \) \( 50 \, \text{V} \). At frequency \( 60 \% \) lower than resonant frequency, the amplitude of current will be approximately:

15.

Identify the correct statements from the following descriptions of various properties of electromagnetic waves.
(A) In a plane electromagnetic wave electric field and magnetic field must be perpendicular to each other and direction of propagation of wave should be along electric field or magnetic field.
(B) The energy in electromagnetic wave is divided equally between electric and magnetic fields
(C) Both electric field and magnetic field are parallel to each other and perpendicular to the direction of propagation of wave.
(D) The electric field, magnetic field and direction of propagation of wave must be perpendicular to each other.
(E) The ratio of amplitude of magnetic field to the amplitude of electric field is equal to speed of light.
Choose the most appropriate answer from the options given below:

16.

Two coherent sources of light interfere. The intensity ratio of two sources is \( 1:4 \). For this interference pattern if the value of \( \dfrac{I_{\max}+I_{\min}}{I_{\max}-I_{\min}} \) is equal to \( \dfrac{2\alpha+1}{\beta+3} \), then \( \dfrac{\alpha}{\beta} \) will be:

17.

With reference to the observations in photo-electric effect, identify the correct statements from below:
(A) The square of maximum velocity of photoelectrons varies linearly with frequency of incident light.
(B) The value of saturation current increases on moving the source of light away from the metal surface.
(C) The maximum kinetic energy of photo-electrons decreases on decreasing the power of LED (light emitting diode) source of light.
(D) The immediate emission of photo-electrons out of metal surface can not be explained by particle nature of light/electromagnetic waves.
(E) Existence of threshold wavelength can not be explained by wave nature of light/electromagnetic waves.
Choose the correct answer from the options given below:

18.

The activity of a radioactive material is \( 6.4 \times 10^{-4} \, \text{curie} \). Its half life is \( 5 \, \text{days} \). The activity will become \( 5 \times 10^{-6} \, \text{curie} \) after:

19.

For a constant collector-emitter voltage of \( 8 \, \text{V} \), the collector current of a transistor reached to the value of \( 6 \, \text{mA} \) from \( 4 \, \text{mA} \), whereas base current changed from \( 20 \, \mu\text{A} \) to \( 25 \, \mu\text{A} \) value. If transistor is in active state, small signal current gain (current amplification factor) will be:

20.

A square wave of the modulating signal is shown in the figure. The carrier wave is given by \( C(t) = 5 \sin(8\pi t)\, \text{Volt} \). The modulation index is:

21.

In an experiment to determine the Young’s modulus, steel wires of five different lengths \(( 1, 2, 3, 4 \) and \( 5 \, \text{m}) \) but of same cross section \(( 2 \, \text{mm}^2 )\) were taken and curves between extension and load were obtained. The slope (extension/load) of the curves were plotted with the wire length and the following graph is obtained. If the Young’s modulus of given steel wires is \( x \times 10^{11} \, \text{N m}^{-2} \), then the value of \( x \) is ______.

22.

In the given figure of meter bridge experiment, the balancing length \( AC \) corresponding to null deflection of the galvanometer is \( 40 \, \text{cm} \). the balancing length, if the radius of the wire \( AB \) is doubled, will be _____ \( \text{cm} \).

23.

A thin prism of angle \( 6^\circ \) and refractive index for yellow light \( (n_\gamma) \, 1.5 \) is combined with another prism of angle \( 5^\circ \) and \( n_\gamma = 1.55 \). The combination produces no dispersion. The net average deviation \( (\delta) \) produced by the combination is \( \left(\dfrac{1}{x}\right)^\circ \). The value of \( x \) is ______.

24.

A conducting circular loop is placed in X–Y plane in presence of magnetic field \( \vec{B} = (3 t^3 \, \hat{j} + 3 t^2 \, \hat{k}) \) in SI unit. If the radius of the loop is \( 1 \, \text{m} \), the induced emf in the loop, at time \( t = 2 \, \text{s} \) is \( n\pi \, \text{V} \). The value of \( n \) is _____.

25.

As shown in the figure, in steady state, the charge stored in the capacitor is ________ \( \times 10^{-6} \, \text{C} \).

26.

A parallel plate capacitor with width \( 4 \, \text{cm} \), length \( 8 \, \text{cm} \) and separation between the plates of \( 4 \, \text{mm} \) is connected to a battery of \( 20 \, \text{V} \). A dielectric slab of dielectric constant \( 5 \) having length \( 1 \, \text{cm} \), width \( 4 \, \text{cm} \) and thickness \( 4 \, \text{mm} \) is inserted between the plates of parallel plate capacitor. The electrostatic energy of this system will be ________ \( \varepsilon_{0} \, \text{J} \).
(Where \( \varepsilon_{0} \) is the permittivity of free space).

27.

A wire of length \( 30 \, \text{cm} \), stretched between rigid supports, has its \( n^{\text{th}} \) and \( (n+1)^{\text{th}} \) harmonics at \( 400 \, \text{Hz} \) and \( 450 \, \text{Hz} \), respectively. If tension in the string is \( 2700 \, \text{N} \), its linear mass density is ______ \( \text{kg/m} \).

28.

A spherical soap bubble of radius \( 3 \, \text{cm} \) is formed inside another spherical soap bubble of radius of \( 6 \, \text{cm} \). If the internal pressure of the smaller bubble of radius \( 3 \, \text{cm} \) in the above system is equal to the internal pressure of the another single soap bubble of radius \( r \, \text{cm} \). The value of \( r \) is _____

29.

A solid cylinder length is suspended symmetrically through two massless strings, as shown in the figure. The distance from the initial rest position, the cylinder should be unbinding the strings to achieve a speed of \( 4 \, \text{ms}^{-1} \), is ________ \( \text{cm} \). (Take \( g = 10 \, \text{ms}^{-2} \)).

30.

Two inclined planes are placed as shown in figure. A block is projected from the point \( A \) of inclined plane \( AB \) along its surface with a velocity just sufficient to carry it to the top point \( B \) at a height \( 10 \, \text{m} \). After reaching the Point \( B \) the block slides down on inclined plane \( BC \). Time it takes to reach to the point \( C \) from point \( A \) is \( t(\sqrt{2}+1) \, \text{s} \). The value of \( t \) is _______. (use \( g = 10 \, \text{m/s}^{2} \)).

JEE Main 2022 27th July Evening Shift

Physics Chemistry Mathematics

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In \( 1^{\text{st}} \) case, Carnot engine operates between temperature \( 300 \, \text{K} \) and \( 100 \, \text{K} \). In \( 2^{\text{nd}} \) case, as shown in the figure, a combination of two engines is used. The efficiency of this combination (in \( 2^{\text{nd}} \) case) will be:

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