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Electron beam used in an electron microscope, when accelerated by a voltage of \( 20 \, \text{kV} \), has a de-Broglie wavelength of \( \lambda_0 \). If the voltage is increased to \( 40 \, \text{kV} \), then the de-Broglie wavelength associated with the electron beam would be:

2.

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: Photodiodes are used in forward bias usually for measuring the light intensity.
Reason R: For a \( p - n \) junction diode, at applied voltage \( V \) the current in the forward bias is more than the current in the reverse bias for \( |V_z| \ge \pm V \ge |V_0| \) where \( V_0 \) is the threshold voltage and \( V_z \) is the breakdown voltage.
In the light of the above statements, choose the correct answer from the options given below.

3.

Assume that the earth is a solid sphere of uniform density and a tunnel is dug along its diameter throughout the earth. It is found that when a particle is released in this tunnel, it executes a simple harmonic motion. The mass of the particle is \( 100 \, \text{g} \). The time period of the motion of the particle will be (approximately):
(Take \( g = 10 \, \text{ms}^{-2} \), radius of earth = \( 640 \, \text{km} \))

4.

In Young’s double slits experiment, the position of \( 5^{\text{th}} \) bright fringe from the central maximum is \( 5 \, \text{cm} \). the distance between slits and screen is \( 1 \, \text{m} \) and wavelength of used monochromatic light is \( 600 \, \text{nm} \). The separation between the slits is:

5.

A bowl filled with very hot soup cools from \( 98^\circ \text{C} \) to \( 86^\circ \text{C} \) in \( 2 \) minutes when the room temperature is \( 22^\circ \text{C} \). How long it will take to cool from \( 75^\circ \text{C} \) to \( 69^\circ \text{C} \)?

6.

A uniform metallic wire carries a current \( 2 \, \text{A} \), when \( 3.4 \, \text{V} \) battery is connected across it. The mass of uniform metallic wire is \( 8.92 \times 10^{-3} \, \text{kg} \), density is \( 8.92 \times 10^{3} \, \text{kg}/\text{m}^{3} \) and resistivity is \( 1.7 \times 10^{-8} \, \Omega\text{-m} \). The length of wire is:

7.

A car is moving with a constant speed of \( 20 \, \text{m/s} \) in a circular horizontal track of radius \( 40 \, \text{m} \). a bon is suspended from the roof of the car by a massless string. The angle made by the string with the vertical will be: (Take \( g = 10 \, \text{m/s}^2 \))

8.

A car travels a distance of \( 'x' \) with speed \( v_{1} \) and then same distance \( 'x' \) with speed \( v_{2} \) in the same direction. The average speed of the car is:

9.

A solenoid of \( 1200 \) turns is wound uniformly in a sample layer on a glass tube \( 2 \, \text{m} \) long and \( 0.2 \, \text{m} \) in diameter. The magnetic intensity at the center of the solenoid when a current of \( 2 \, \text{A} \) flows through it is:

10.

Match List I with List II.

Choose the correct answer from the options given below:

11.

A message signal of frequency \( 5 \, \text{kHz} \) is used to modulate a carrier signal of frequency \( 2 \, \text{MHz} \). The bandwidth for amplitude modulation is:

12.

In an LC oscillator, if values of inductance and capacitance become twice and eight times, respectively, then the resonant frequency of oscillator becomes \( x \) times its initial resonant frequency \( \omega_{0} \). The value of \( x \) is:

13.

The root mean square velocity of molecules of gas is:

14.

Match List I with List II.

Choose the correct answer from the options given below:

15.

An object of mass \( 8 \, \text{kg} \) is hanging from one end of a uniform rod CD of mass \( 2 \, \text{kg} \) and length \( 1 \, \text{m} \) pivoted at its end C on a vertical wall as shown in figure. It is supported by a cable AB such that the system is in equilibrium. The tension in the cable is: (Take \( g = 10 \, \text{m}/\text{s}^2 \))

16.

The ratio of the density of oxygen nucleus \(( ^{16}_{8}\text{O} )\) and helium nucleus \(( ^{4}_{2}\text{He} )\) is:

17.

A Carnot engine with efficiency \( 50\% \) takes heat from a source at \( 600 \, \text{K} \). In order to increase the efficiency to \( 70\% \), keeping the temperature of sink same, the new temperature of the source will be:

18.

An electromagnetic wave is transporting energy in the negative \( z \) direction. At a certain point and certain time the direction of electric field of the wave is along positive \( y \) direction. What will be the direction of the magnetic field of the wave at that point and instant?

19.

A parallel plate capacitor has plate area \( 40 \, \text{cm}^2 \) and plates separation \( 2 \, \text{mm} \). The space between the plates is filled with a dielectric medium of a thickness \( 1 \, \text{mm} \) and dielectric constant \( 5 \). The capacitance of the system is:

20.

\( T \) is the time period of simple pendulum on the earth’s surface. Its time period become \( x \, T \) when taken to a height \( R \) (equal to earth’s radius) above the earth’s surface. Then, the value of \( x \) will be:

21.

A ray of light is incident from air on a glass plate having thickness \( \sqrt{3} \, \text{cm} \) and refractive index \( \sqrt{2} \). The angle of incidence of a ray is equal to the critical angle for glass-air interface. The lateral displacement of the ray when it passes through the plate is ________ \( \times 10^{-2} \, \text{cm} \). (Given \( \sin 15^\circ = 0.26 \))

22.

As shown in the figure, in an experiment to determine young’s modulus of a wire, the extension-load curve is plotted. The curve is a straight line passing through the origin and makes an angle of \( 45^\circ \) with the load axis. The length of wire is \( 62.8 \, \text{cm} \) and its diameter is \( 4 \, \text{mm} \). The Young’s modulus is found to be \( x \times 10^{4} \, \text{N} \, \text{m}^{-2} \). The value of \( x \) is ___________.

23.

In the given circuit, the equivalent resistance between the terminal \( A \) and \( B \) is ____________ \( \Omega \).

24.

An object of mass \( 'm' \) initially at rest on a smooth horizontal plane starts moving under the action of force \( F = 2 \, N \). In the process of its linear motion, the angle \( \theta \) (as shown in figure) between the direction of force and horizontal varies as \( \theta = kx \), where \( k \) is a constant and \( x \) is the distance covered by the object from its initial position. The expression of kinetic energy of the object will be \( E = \dfrac{n}{k} \sin \theta \). The value of \( n \) is ___________.

25.

A uniform electric field of \( 10 \, \text{N/C} \) is created between two parallel charged plates (as shown in figure). An electron enters the field symmetrically between the plates with a kinetic energy \( 0.5 \, \text{eV} \). the length of each plate is \( 10 \, \text{cm} \). The angle (\( \theta \)) of deviation of the path of electron as it comes out of the field is ________ (in degree).

26.

An LCR series circuit of capacitance \( 62.5 \, \text{nF} \) and resistance of \( 50 \, \Omega \), is connected to an A.C. source of frequency \( 2.0 \, \text{kHz} \). For maximum value of amplitude of current in circuit, the value of inductance is ______________ \( \text{mH} \). (Take \( \pi^{2} = 10 \))

27.

The distance between two consecutive points with phase difference of \( 60^{\circ} \) in a wave of frequency \( 500 \, \text{Hz} \) is \( 6.0 \, \text{m} \). the velocity with which wave is traveling is ___________ \( \text{km/s} \).

28.

If \( \vec{P} = 3\hat{i} + \sqrt{3}\hat{j} + 2\hat{k} \) and \( \vec{Q} = 4\hat{i} + \sqrt{3}\hat{j} + 2.5\hat{k} \) then, The unit vector in the direction of \( \vec{P} \times \vec{Q} \) is \( \frac{1}{x}\left( \sqrt{3}\hat{i} + \hat{j} - 2\sqrt{3}\hat{k} \right) \). The value of \( x \) is:

29.

The wavelength of the radiation emitted is \( \lambda_{0} \) when an electron jumps from the second excited state to the first excited state of hydrogen atom. If the electron jumps from the third excited state to the second orbit of the hydrogen atom, the wavelength of the radiation emitted will be \( \frac{20}{x} \lambda_{0} \). the value of \( x \) is ________

30.

\( I_{CM} \) is the moment of inertia of a circular disc about an axis (CM) passing through its center and perpendicular to the plane of disc. \( I_{AB} \) is it’s moment of inertia about an axis AB perpendicular to plane and parallel to axis CM at a distance \( \frac{2}{3} R \) from center. Where \( R \) is the radius of the disc. The ratio of \( I_{AB} \) and \( I_{CM} \) is \( x:9 \). The value of \( x \) is ___________.

JEE Main 2023 25th Jan Morning Shift

Physics Chemistry Mathematics

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A Carnot engine with efficiency \( 50\% \) takes heat from a source at \( 600 \, \text{K} \). In order to increase the efficiency to \( 70\% \), keeping the temperature of sink same, the new temperature of the source will be:

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