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Statement I: By doping silicon semiconductor with pentavalent material, the electrons density increases.
Statement II: The \( n \)-type semiconductor has net negative charge.
In the light of the above statements, choose the most appropriate answer from the options given below:

The half-life of \( \mathrm{Au}^{198} \) is \( 2.7 \) days. The activity of \( 1.50 \, \text{mg} \) of \( \mathrm{Au}^{198} \) if its atomic weight is \( 198 \, \text{g mol}^{-1} \), is, \( (N_A = 6 \times 10^{23} / \text{mol}) \).

If \( 2.5 \times 10^{-6} \, \text{N} \) average force is exerted by a light wave on a non-reflecting surface of \( 30 \, \text{cm}^2 \) area during \( 40 \) minutes of time span, the energy flux of light just before it falls on the surface is ______ \( \text{W/cm}^2 \). (Round off to the Nearest Integer) (Assume complete absorption and normal incidence conditions are there)

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?

A certain metallic surface is illuminated by monochromatic radiation of wavelength \( \lambda \). The stopping potential for photoelectric current for this radiation is \( 3V_0 \). If the same surface is illuminated with a radiation of wavelength \( 2\lambda \), the stopping potential is \( V_0 \). The threshold wavelength of this surface for photoelectric effect is __________ \( \lambda \).

Match List –I with List II

List I	List II
(a) Source of microwave frequency	(i) Radioactive decay of nucleus
(b) Source of infrared frequency	(ii) Magnetron
(c) Source of Gamma Rays	(iii) Inner shell electrons
(d) Source of X-rays	(iv) Vibration of atoms and molecules
	(v) LASER
	(vi) RC circuit

Choose the correct answer from the options given below:

An electron and proton are separated by a large distance. The electron starts approaching the proton with energy \( 3 \, \text{eV} \). The proton captures the electron and forms a hydrogen atom in second excited state. The resulting photon is incident on a photosensitive metal of threshold wavelength \( 4000 \, \text{Å} \). What is the maximum kinetic energy of the emitted photoelectron?

In a given circuit diagram, a \( 5 \, \text{V} \) Zener diode along with a series resistance is connected across a \( 50 \, \text{V} \) power supply. The minimum value of the resistance required, if the maximum Zener current is \( 90 \, \text{mA} \) will be _______ \( \Omega \).

At time \( t = 0 \), a material is composed of two radioactive atoms A and B, where \( N_{A}(0) = 2N_{B}(0) \). The decay constant of both kind of radioactive atoms is \( \lambda \). However, A disintegrates to B and B disintegrates to C. Which of the following figures represents the evolution of \( \dfrac{N_{B}(t)}{N_{B}(0)} \) with respect to time \( t \)?

10.

A nucleus with mass number \( 184 \) initially at rest emits an \( \alpha \)-particle. If the Q value of the reaction is \( 5.5 \, \text{MeV} \), calculate the kinetic energy of the \( \alpha \)-particle.

11.

In a photoelectric experiment, increasing the intensity of incident light:

12.

Imagine that the electron in a hydrogen atom is replaced by a muon \( (\mu) \). The mass of muon particle is \( 207 \) times that of an electron and charge is equal to the charge of an electron. The ionization potential of this hydrogen atom will be:

13.

The atomic hydrogen emits a line spectrum consisting of various series. Which series of hydrogen atomic spectra is lying in the visible region?

14.

In a photoelectric experiment ultraviolet light of wavelength \( 280 \, \text{nm} \) is used with lithium cathode having work function \( \phi = 2.5 \, \text{eV} \). If the wavelength of incident light is switched to \( 400 \, \text{nm} \), find out the change in the stopping potential.

\( (h = 6.63 \times 10^{-34} \, \text{J s}, \, c = 3 \times 10^{8} \, \text{m s}^{-1}) \).

15.

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]:

16.

A sample of a radioactive nucleus A disintegrates to another radioactive nucleus B, which in turn disintegrates to some other stable nucleus C. Plot of a graph showing the variation of number of atoms of nucleus B versus time is: (Assume that at \( t = 0 \), there are no B atoms in the sample)

17.

A particular hydrogen like ion emits radiation of frequency \( 2.92 \times 10^{15} \, \text{Hz} \) when it makes transition from \( n = 3 \) to \( n = 1 \). The frequency in \( \text{Hz} \) of radiation emitted in transition from \( n = 2 \) to \( n = 1 \) will be:

18.

An X-ray tube is operated at \( 1.24 \) million volt. The shortest wavelength of the produced photon will be:

19.

The decay of a proton to neutron is:

20.

A light beam of wavelength \( 500 \, \text{nm} \) is incident on a metal having work function of \( 1.25 \, \text{eV} \), placed in a magnetic field of intensity \( B \). The electrons emitted perpendicular to the magnetic field \( B \), with maximum kinetic energy are bent into circular arc of radius \( 30 \, \text{cm} \). The value of \( B \) is ________ \( \times 10^{-7} \, \text{T} \).
Given \( hc = 20 \times 10^{-26} \, \text{J m} \), mass of electron \( = 9 \times 10^{-31} \, \text{kg} \).

21.

A radioactive sample disintegrates via two independent decay processes having half lives \( T_{1/2}^{(1)} \) and \( T_{1/2}^{(2)} \) respectively. The effective half-life, \( T_{1/2} \) of the nuclei is:

22.

Consider a situation in which reverse biased current of a particular \( P-\!N \) junction increases when it is exposed to a light of wavelength \( \leq 621 \, \text{nm} \). During this process, enhancement in carrier concentration takes place due to generation of hole-electron pairs. The value of band gap is nearly.

23.

\( X \) different wavelengths may be observed in the spectrum from a hydrogen sample if the atoms are exited to states principal quantum number \( n = 6 \)? The value of \( X \) is _______.

24.

Consider the following statements:
A. Atoms of each element emit characteristics spectrum.
B. According to Bohr’s Postulate, an electron in a hydrogen atom, revolves in a certain stationary orbit.
C. The density of nuclear matter depends on the size of the nucleus.
D. A free neutron is stable but a free proton decay is possible.
E. Radioactivity is an indication of the instability of nuclei.
Choose the correct answer from the options given below:

25.

An electron of mass \( m \) and a photon have same energy \( E \). The ratio of wavelength of electron to that of photon is: (\( c \) being the velocity of light)

26.

Consider two separate ideal gases of electrons and protons having same number of particles. The temperature of both the gases are same. The ratio of the uncertainty in determining the position of an electron to that of a proton is proportional to:

27.

The de-Broglie wavelength associated with an electron and a proton were calculated by accelerating them through same potential of \( 100 \, \text{V} \). What should nearly be the ratio of their wavelengths? \( m_{p} = 1.00727 \, u,\; m_{e} = 0.00055 \, u \)

28.

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:

29.

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]

30.

From the given data, the amount of energy required to break the nucleus of aluminium \( ^{27}_{13}\text{Al} \) is ______ \(x \times 10^{-3} \, \text{J} \).
Mass of neutron = \( 1.00866 \, \text{u} \)
Mass of proton = \( 1.00726 \, \text{u} \)
Mass of Aluminium nucleus = \( 27.18846 \, \text{u} \)
(Assume \( 1 \, \text{u} \) corresponds to \( x \, \text{J} \) of energy)
(Round off to the nearest integer)

31.

A moving proton and electron have the same de-Broglie wavelength. If \( K \) and \( P \) denote the K.E. and momentum respectively. Then choose the correct option:

32.

A nucleus of mass \( M \) emits \( \gamma \)-ray photon of frequency ‘\( v \)’. The loss of internal energy by the nucleus is: [Take ‘\( c \)’ as the speed of electromagnetic wave]

33.

There are \( 10^{10} \) radioactive nuclei in a given radioactive element. Its half-life time is \( 1 \) minute. How many nuclei will remain after \( 30 \) seconds? \( (\sqrt{2} = 1.414) \)

34.

If an electron is moving in the \( n^{th} \) orbit of the hydrogen atom, then its velocity \( (v_{n}) \) for the \( n^{th} \) orbit is given as:

35.

Which level of the single ionized carbon has the same energy as the ground state energy hydrogen atom?

36.

For a certain radioactive process the graph between \( \ln R \) and \( t \, (\text{sec}) \) is obtained as shown in the figure. Then the value of half life for the unknown radioactive material is approximately:

37.

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.

38.

A particle is travelling \( 4 \) times as fast as an electron. Assuming the ratio of de-Broglie wavelength of a particle to that of electron is \( 2 : 1 \), the mass of the particle is:

39.

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 \( \dfrac{22}{7} \)]

40.

A free electron of \( 2.6 \, \text{eV} \) energy collides with a \( H^{+} \) on. This result in the formation of a hydrogen atom in the first excited state and a photon is released. Find the frequency of the emitted photon.
\( (h = 6.6 \times 10^{-34} \, \text{J s}) \)

41.

Two identical photocathodes receive the light of frequencies \( f_1 \) and \( f_2 \) respectively. If the velocities of the photo-electrons coming out are \( v_1 \) and \( v_2 \) respectively, then:

42.

The nuclear activity of a radioactive element becomes \( \left(\dfrac{1}{8}\right)^{\text{th}} \) of its initial value in \( 30 \) years. The half-life of radioactive element is ______ years.

43.

According to Bohr atom model, in which of the following transitions will the frequency be maximum?

44.

The half life period of a radioactive element \( x \) is same as the mean life time of another radioactive element \( y \). Initially they have the same number of atoms. Then:

45.

The \( K_\alpha \) X-ray of molybdenum has wavelength \( 0.071 \, \text{nm} \). If the energy of a molybdenum atom with a K electron knocked out is \( 27.5 \, \text{keV} \), the energy of this atom when an L electron is knocked out will be ______ \( \text{keV} \). (Round off to the nearest integer)
[\( h = 4.14 \times 10^{-15} \, \text{eVs} \), \( c = 3 \times 10^{8} \, \text{ms}^{-1} \)]

46.

The de Broglie wavelength of a proton and \( \alpha \)-particle are equal. The ratio of their velocities is:

47.

An electron moving with speed \( v \) and a photon moving with speed \( c \), have same D-Broglie wavelength. The ratio of kinetic energy of electron to that of photon is:

48.

When radiation of wavelength \( \lambda \) is incident on a metallic surface, the stopping potential of ejected photoelectrons is \( 4.8 \, \text{V} \). If the same surface is illuminated by radiation of double the previous wavelength, then the stopping potential becomes \( 1.6 \, \text{V} \). The threshold wavelength of the metal is:

49.

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)

50.

A particle of mass \( m \) moves in a circular orbit in a central potential field \( U(r) = U_{0} r^{4} \). If Bohr’s quantization conditions are applied, radii of possible orbitals \( r_{n} \) vary with \( n^{1/\alpha} \), where \( \alpha \) is _______.

51.

Identify the logic operation carried out by the given circuit:

52.

An oil drop of radius \( 2 \, \text{mm} \) with a density \( 3 \, \text{g} \, \text{cm}^{-3} \) is held stationary under a constant electric field \( 3.55 \times 10^{5} \, \text{V} \, \text{m}^{-1} \) in the Millikan’s oil drop experiment. What is the number of excess electrons that the oil drop will possess? Consider \( g = 9.81 \, \text{m} \, /\text{s}^{2} \)

53.

A radioactive material decays by simultaneous emissions of two particle with half lives of \( 1400 \, \text{years} \) and \( 700 \, \text{years} \) respectively. What will be the time after which one third of the material remains? [ Take \( \ln 3 = 1.1 \) ]

54.

Calculate the time interval between \( 33\% \) decay and \( 67\% \) decay if half-life of a substance is \( 20 \) minutes.

55.

The electric field intensity produced by the radiation coming from a \( 100\,W \) bulb at a distance of \( 3\,m \) is \( E \). The electric field intensity produced by the radiation coming from \( 60\,W \) at the same distance is \( \sqrt{\dfrac{x}{5}}\,E \). Where the value of \( x = \) ________.

56.

A radioactive substance decays to \( \left( \dfrac{1}{16} \right)^{\text{th}} \) of its initial activity in \( 80 \, \text{days} \). The half life of the radioactive substance expressed in days is __________ .

57.

The de-Broglie wavelength of a particle having kinetic energy \( E \) is \( \lambda \). How much extra energy must be given to this particle so that the de-Broglie wavelength reduces to \( 75\% \) of the initial value?

58.

The temperature of an ideal gas in \( 3 \)-dimensional is \( 300 \, \text{K} \). The corresponding de-Broglie wavelength of the electron approximately at \( 300 \, \text{K} \), is:
[ \( m_{e} = \) mass of electron \( = 9 \times 10^{-31} \, \text{kg} \)
\( h = \) Planck constant \( = 6.6 \times 10^{-34} \, \text{J s} \)
\( k_{B} = \) Boltzmann constant \( = 1.38 \times 10^{-23} \, \text{J K}^{-1} \) ]

59.

An electron of mass \( m_e \) and a proton of mass \( m_p \) are accelerated through the same potential difference. The ratio of the de-Broglie wavelength associated with the electron to that with the proton is:

60.

A monochromatic neon lamp with wavelength of \( 670.5\,\text{nm} \) illuminates a photo-sensitive material which has a stopping voltage of \( 0.48\,\text{V} \). What will be the stopping voltage if the source light is changed with another source of wavelength of \( 474.6\,\text{nm} \)?

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JEE Main 2021 17th March Morning Shift

If an electron is moving in the \( n^{th} \) orbit of the hydrogen atom, then its velocity \( (v_{n}) \) for the \( n^{th} \) orbit is given as:

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