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Consider the efficiency of carnot’s engine is given by \( \eta = \displaystyle\frac{\alpha \beta}{\sin \theta} \log_e \frac{\beta x}{k T} \), where \( \alpha \) and \( \beta \) are constants If \( T \) is temperature, \( k \) is Boltzmann constant, \( \theta \) is angular displacement and \( x \) has the dimensions of length. Then, choose the incorrect option.

2.

At time \( t = 0 \) a particle starts travelling from a height \( 7\hat{z}\,\text{cm} \) in a plane keeping \( z \) coordinate constant. At any instant of time it’s position along the \( \hat{x} \) and \( \hat{y} \) directions are defined as \( 3t \) and \( 5t^{3} \) respectively. At \( t = 1\,\text{s} \) acceleration of the particle will be:

3.

A pressure-pump has a horizontal tube of cross sectional area \( 10\,\text{cm}^{2} \) for the outflow of water at a speed of \( 20\,\text{m/s} \). The force exerted on the vertical wall just in front of the tube which stops water horizontally flowing out of the tube, is:
[given : density of water \( =1000\,\text{kg}/\text{m}^{3} \) ]

4.

A uniform metal chain of mass \( m \) and length ‘\( L \)’ passes over a massless and frictionless pulley. It is released from rest with a part of its length ‘\( l \)’ is hanging on one side and rest of its length ‘\( L-l \)’ the other side of the pully. At a certain point of time, when \( l=\dfrac{L}{x} \), the acceleration of the chain is \( \dfrac{g}{2} \). The value of \( x \) is ______.

5.

A bullet of mass \( 200\,\text{g} \) having initial kinetic energy \( 90\,\text{J} \) is shot inside a long swimming pool as shown in the figure. If it’s kinetic energy reduces to \( 40\,\text{J} \) within \( 1\,\text{s} \), the minimum length of the pool, the bullet has to travel so that it completely comes to rest is:

6.

Assume there are two identical simple pendulum clocks. Clock-\( 1 \) is placed on the earth and Clock-\( 2 \) is placed on a space station located at a height \( h \) above the earth surface. Clock-\( 1 \) and Clock-\( 2 \) operate at time periods \( 4\,\text{s} \) and \( 6\,\text{s} \) respectively. Then value of \( h \) is: \( \big(\text{consider radius of earth } R_{E}=6400\,\text{km} \text{ and } g \text{ on earth } 10\,\text{m}/\text{s}^{2}\big) \)

7.

Consider a cylindrical tank of radius \( 1 \, \text{m} \) is filled with water. The top surface of water is at \( 15 \, \text{m} \) from the bottom of the cylinder. There is a hole on the wall of cylinder at a height of \( 5 \, \text{m} \) from the bottom. A force of \( 5 \times 10^{5} \, \text{N} \) is applied on the top surface of water using a piston. The speed of ifflux from the hole will be:
(given atmospheric pressure \( P_A = 1.01 \times 10^{5} \, \text{Pa} \), density of water \( \rho_w = 1000 \, \text{kg/m}^{3} \) and gravitational acceleration \( g = 10 \, \text{m/s}^{2} \))

8.

A vessel contains \( 14 \, \text{g} \) of nitrogen gas at a temperature of \( 27^\circ \text{C} \). The amount of heat to be transferred to the gas to double the r.m.s. speed of its molecules will be:
(Take \( R = 8.32 \, \text{J mol}^{-1} \, \text{K}^{-1} \).)

9.

A slab of dielectric constant \( K \) has the same cross-sectional area as the plates of a parallel plate capacitor and thickness \( \dfrac{3}{4} d \), where \( d \) is the separation of the plates. The capacitance of the capacitor when the slab is inserted between the plates will be:
(Given \( C_{0} = \) capacitance of capacitor with air as medium between plates.)

10.

A uniform electric field \( E = (8m/e) \, \text{V/m} \) is created between two parallel plates of length \( 1 \, \text{m} \) as shown in figure, (where \( m = \) mass of electron and \( e = \) charge of electron). An electron enters the field symmetrically between the plates with a speed of \( 2 \, \text{m/s} \). The angle of the deviation \( ( \theta ) \) of the path of the electron as it comes out of the field will be ________.

11.

Given below are two statements:

In the light of the above statements, choose the most appropriate answer from the option given below:

12.

A triangular shaped wire carrying \( 10 \, \text{A} \) current is placed in a uniform magnetic field of \( 0.5 \, \text{T} \), as shown in figure. The magnetic force on segment \( CD \) is:
(Given \( BC = CD = BD = 5 \, \text{cm} \))

13.

The magnetic field at the centre of current carrying circular loop is \( B_1 \). The magnetic field at a distance of \( \sqrt{3} \) times radius of the given circular loop from the centre on its axis is \( B_2 \). The value of \( {B_1}/{B_2} \) will be:

14.

A transformer operating at primary voltage \( 8 \, \text{kV} \) and secondary voltage \( 160 \, \text{V} \) serves a load of \( 80 \, \text{kW} \). Assuming the transformer to be ideal with purely resistive load and working on unity power factor, the loads in the primary and secondary circuit would be:

15.

Sun light falls normally on a surface of area \( 36 \, \text{cm}^2 \) and exerts an average force of \( 7.2 \times 10^{-9} \, \text{N} \) within a time period of \( 20 \, \text{minutes} \). Considering a case of complete absorption. The energy flux of incident light is:

16.

The power of a lens (biconvex) is \( 1.25 \, \text{m}^{-1} \) in particular medium. Refractive index of the lens is \( 1.5 \) and radii of curvature are \( 20 \, \text{cm} \) and \( 40 \, \text{cm} \) respectively. The refractive index of surrounding medium:

17.

Two streams of photons, possessing energies equal to five and ten times the work function of metal are incident on the metal surface successively. The ratio of maximum velocities of the photoelectron emitted, in the two cases respectively, will be:

18.

A radioactive sample decays \( \dfrac{7}{8} \) times its original quantity in \( 15 \, \text{min} \). The half-life of the sample is:

19.

An n.p.n. transistor with current gain \( \beta = 100 \) in common emitter configuration is shown in figure. The output voltage of the amplifier will be:

20.

A FM Broad cast transmitter, unsing modulating signal of frequency \( 20 \, \text{kHz} \) has a deviation ratio of \( 10 \). The Bandwidth required for transmission is:

21.

A ball is thrown vertically upwards with a velocity of \( 19.6 \, \text{m s}^{-1} \) from the top of a tower. The ball strikes the ground after \( 6 \, \text{s} \). The height from the ground up to which the ball can rise will be \( \left(\dfrac{k}{5}\right) \, \text{m} \). The value of \( k \) is _______ (use \( g = 9.8 \, \text{m/s}^2 \) ).

22.

The distance of centre of mass from end A of a one dimensional rod (AB) having mass density \( \rho = \rho_0 \left(1 - \dfrac{x^2}{L^2}\right) \, \text{kg/m} \) and length \( L \) (in meter) is \( \dfrac{3L}{\alpha} \, \text{m} \). The value of \( \alpha \) is ________. (where \( x \) is the distance from end A)

23.

A string of area of cross-section \( 4 \, \text{mm}^2 \) and length \( 0.5 \, \text{m} \) is connected with a rigid body of mass \( 2 \, \text{kg} \). The body is rotated in a vertical circular path of radius \( 0.5 \, \text{m} \). The body acquires a speed of \( 5 \, \text{m/s} \) at the bottom of the circular path. Strain produced in the string when the body is at the bottom of the circle is _______ \( \times 10^{-5} \).
(use young’s modulus \( 10^{11} \, \text{N/m}^2 \) and \( g = 10 \, \text{m/s}^2 \) )

24.

At a certain temperature, the degrees of freedom per molecule for gas is \( 8 \). The gas performs \( 150 \, \text{J} \) of work when it expands under constant pressure. The amount of heat absorbed by the gas will be _____ \( \text{J} \).

25.

The potential energy of a particle of mass \( 4 \, \text{kg} \) in motion along the \( x \)-axis is given by \( U = 4(1 - \cos 4x) \, \text{J} \). The time period of the particle for small oscillation \( (\sin \theta \simeq \theta) \) is \( \left( \dfrac{\pi}{K} \right) \, \text{s} \). The value of \( K \) is __________.

26.

An electrical bulb rated \( 220 \, \text{V} \), \( 100 \, \text{W} \), is connected in series with another bulb rated \( 220 \, \text{V} \), \( 60 \, \text{W} \). If the voltage across combination is \( 220 \, \text{V} \), the power consumed by the \( 100 \, \text{W} \) bulb will be about ______ \( \text{W} \).

27.

For the given circuit the current through battery of \( 6 \, \text{V} \) just after closing the switch ‘\( S \)’ will be _____ \( \text{A} \).

28.

An object ‘\( o \)’ is placed at a distance of \( 100 \, \text{cm} \) in front of a concave mirror of radius of curvature \( 200 \, \text{cm} \) as shown in the figure. The object starts moving towards the mirror at a speed \( 2 \, \text{cm/ s}\). The position of the image from the mirror after \( 10 \, \text{s} \) will be at __________ \( \text{cm} \).

29.

In an experiment with a convex lens. The plot of the image distance \( (v') \) against the object distance \( (u') \) measured from the focus gives a curve \( v' u' = 225 \). If all the distances are measured in \( \text{cm} \). The magnitude of the focal length of the lens is __________ \( \text{cm} \).

30.

In an experiment to find acceleration due to gravity \( (g) \) using simple pendulum, time period of \( 0.5 \, \text{s} \) is measured from time of \( 100 \) oscillation with a watch of \( 1 \, \text{s} \) resolution. If measured value of length is \( 10 \, \text{cm} \) known to \( 1 \, \text{mm} \) accuracy. The accuracy in the determination of \( g \) is found to be \( x\% \). the value of \( x \) is:

JEE Main 2022 28th July Evening shift

Physics Chemistry Mathematics

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An object ‘\( o \)’ is placed at a distance of \( 100 \, \text{cm} \) in front of a concave mirror of radius of curvature \( 200 \, \text{cm} \) as shown in the figure. The object starts moving towards the mirror at a speed \( 2 \, \text{cm/ s}\). The position of the image from the mirror after \( 10 \, \text{s} \) will be at __________ \( \text{cm} \).

JEE MainsJEE AdvanceBITSATJEE Mains | PYQ

JEE Mains JEE Advance BITSAT
JEE Mains | PYQ