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Concept:

There are two common time measures of how long any given type of radionuclide lasts. One measure is the half-life T_1/2 of a radionuclide, which is the time at which both N and R have been reduced to one-half their initial values.

The other measure is the mean life τ, which is the time at which both N and R have been reduced to e^–1of their initial values. To relate T_1/2 to the disintegration constant λ, we put R = (1/2) R₀ and t = T_1/2 in and solve for T_1/2, we find T_1/2 = ln2/λ = 0.693/λ .The average life or mean life, τ can also be obtained.

Calculation:

Given,

At t = 0, count rate or initial activity is A₀ = 1600 s^-1

At t = 8 s, count rate or activity is A = 100 s^-1

So, decay scheme for given sample is

\(1600\mathop \to \limits^{{T_{1/2}}} 800\mathop \to \limits^{{T_{1/2}}} 400\mathop \to \limits^{{T_{1/2}}} 200\mathop \to \limits^{{T_{1/2}}} 100\)

So,

8 s = 4T_1/2

Where, T_1/2 = Half-life time.

\(\Rightarrow {T_{1/2}} = \frac{8}{4}\;s\) 

∴ T_1/2 = 2 s

∴ From above decay scheme, we see that activity after 6s is 200 counts per second i.e., after every two seconds, it reduced to half of its initial value.

I.e.,

After first 2 s, it reduced from 1600 to 800

After next 2 s i.e., t = 4 s, it reduced from 800 to 400.

After next 2 s i.e., t = 6 s, it reduced from 400 to 200.

Concept:

Quantization of charge: It is a property of charge by which charge on any object can be expressed in an integral multiple of unit charge.

• Charge on an object is given by Q = n x e

Where Q is the total charge

n is an integer value

e is a charge on an electron (1.6 x 10^-19 C)

• In 1833 physicist Micheal Faraday first suggested the quantization of charge in experimental laws of electrolysis.

•  In 1912 Millikan demonstrated quantization of charge by oil drop experiment.
• Which concluded that the charge is an integral multiple of unit value which is 1.6 x 10-19 Columb.

Concept:

The momentum, p of an object is defined as the product of its mass, m and velocity, v.

p = mv

Momentum has both direction and magnitude. Its direction is the same as that of velocity, v. The SI unit of momentum is kilogram-metre per second.

Calculation:

Linear momentum = mv = 9.1 × 10^-31 × 2.2 × 10⁶ = 2.0 × 10^-24 kg-m/s

A laser makes monochromatic light of a single, very precise frequency and colour. The laser line is also very coherent as the crest of every wave is lined up with the crest of other waves.

First working model of laser based on the theoretical principles was made in 1960 by shining a high power flash lamp on a ruby rod with silver coated surfaces.

Note:

For this question in Jee paper, there was no correct option given, hence one of the options is replaced by the correct one.

Concept:

Energy absorbed or released in a nuclear reaction is given by;

ΔQ = Binding energy of products – Binding energy of reactants.

If energy is absorbed, ΔQ is negative and if it is positive then energy is released.

Also, Binding energy = Binding energy per nucleon × number of nucleons.

Calculation:

Given reaction is

Ne²⁰ → 2He⁴ + C¹²

Binding energy per nucleon of Ne²⁰ is 8.03 MeV

Number of nucleon in Ne²⁰ is 20

Binding energy per nucleon of He⁴ is 7.07 MeV

Number of nucleon in He⁴ is 4

Binding energy per nucleon of C¹² is 7.86 MeV

Number of nucleon in C¹² is 12

Here, binding energy of products i.e., 2(He)⁴ + C¹² is

= 2 × (B.E of He⁴) + (B.E of C¹²)

= 2(4 × 7.07) + (12 × 7.86)

= 2 × 28.28 + 94.32

= 56.56 + 94.32

= 150.88 MeV

And binding energy of reactants, i.e., Ne²⁰ is

= 20 × 8.03 = 160.6 MeV

So, ΔQ = (B.E)_Products – (B.E)_reactants

= 150.88 – 160.6 = -9.72 MeV

As ΔQ is negative,

∴ Energy of 9.72 MeV is absorbed in the reaction.

Concept:

When a source is moving towards a stationary observer, observed frequency is given by

\({f_{observed}} = f\left( {\frac{v}{{v + {v_s}}}} \right)\)

Where, f = frequency of sound from the source,

v = speed of sound and

v_s = speed of source.

Calculation:

Given,

Speed of the train = 34 m/s

Frequency when the whistle blows is f₁

Speed of the train is reduced to 17 m/s after sometime

Speed of the sound = 340 m/s

Frequency registered after = f₂

Now applying above formula to two different conditions given in problem, we get

f₁ = Observed frequency initially

\(\Rightarrow {f_1} = f\left( {\frac{{340}}{{340 - 34}}} \right)\)

\(\therefore {f_1} = f\left( {\frac{{340}}{{306}}} \right)\)

f₂ = Observed frequency when speed of source is reduced 

\(\Rightarrow {f_2} = f\left( {\frac{{340}}{{340 - 17}}} \right)\)

\(\therefore {f_2} = f\left( {\frac{{340}}{{323}}} \right)\)

So, the ratio f₁ : f₂ is

\(\Rightarrow \frac{{{f_1}}}{{{f_2}}} = \frac{{f\left( {\frac{{340}}{{306}}} \right)}}{{f\left( {\frac{{340}}{{323}}} \right)}} = \frac{{340}}{{306}} \times \frac{{323}}{{340}}\)

\(\Rightarrow \frac{{{f_1}}}{{{f_2}}} = \frac{{323}}{{306}} = \frac{{19}}{{18}}\)

\(\therefore \frac{{{f_1}}}{{{f_2}}} = \frac{{19}}{{18}}\)

Concept:

Volume of a cylinder of radius ‘r’ and height h is given by

V = πr² h

Or

\(V = \pi {\left( {\frac{D}{2}} \right)^2}h\)

\(\left[\because {d = 2r \Rightarrow r = \frac{d}{2}} \right]\) 

\(V = \frac{1}{4}\left( {\pi {D^2}h} \right)\)

Where D is the diameter of circular surface

Calculation:

Given,

Diameter of the cylinder, D = 12.6 ± 0.1 cm

Height of the cylinder, h = 34.2 ± 0.1 cm

Here, D = 12.6 cm and h = 34.2 cm

\(V = \frac{\pi }{4} \times {\left( {12.6} \right)^2} \times \left( {34.2} \right)\)

\(V = \frac{\pi }{4} \times 158.76 \times 34.2\)

\(V = \frac{\pi }{4} \times 5429.592\)

V = 1357.398 × π

V = 1357.398 × 3.14

V = 4262.22 cm³

V = 4260 (in three significant numbers)

We know that,

ΔD = ± 0.1 and Δh = ± 0.1

Now, error calculation can be done as

\(\frac{{{\rm{\;\Delta }}V}}{V} = 2\left( {\frac{{{\rm{\Delta }}D}}{D}} \right) + \frac{{{\rm{\Delta }}h}}{h} = \frac{{2 \times 0.1}}{{12.6}} + \frac{{0.1}}{{34.2}}\) 

\(\frac{{{\rm{\Delta }}V}}{V} = 0.0158 + 0.0029\) 

\(\frac{{{\rm{\Delta }}V}}{V} = 0.01879\) 

ΔV = (0.01879) × (4262.22)

ΔV = 79.7 ≈ 80 cm³

∴ For proper significant numbers, volume reading will be V = 4260 ± 80 cm³

Explanation:

As the levels are non-degenerate, there is only one state for each energy.

Let the number of particles occupying the 3 energy states be N1, N2, and N3 respectively

Where 

N₁ + N₂ + N₃ = 5
 

The particles are identifiable, the number of ways of choosing the particles is
\(W = \frac{{5!}}{{{N_1}!\;{N_2}!{N_3}!}}\)

The energy of the system is

0 × N₁ + E × N₂ + 2 E × N₃ = 3E (given)

N2 + 2 N3 = 3  -----(1)
 

Now, the most probable distribution is the one in which W is a maximum, subject to constraint given by equation (1)

Thus if

N₂ = 1

N₃ = (3 -1) /2 = 1

and

N₁ = 5 - (N₁ + N₂) = 5 - 2 = 3

If N2 = 3, N3 = 0

and N1 = 5 – (3 + 0) = 2

No other distribution are possible

For

N₁ = 3, N₃ = 1 and N₃ = 1  

\(W = \frac{{5!}}{{3!}} = 20\)

So must probable distribution is 

N₁ = 3, N₂ = 1 and N₃ = 1

The correct answers are: 

N1 = 3, N2 = 1 and N3 = 1

Concept:

The linear expansion coefficient is an intrinsic property of every material. Hence it varies from one material to another. The rate at which a material expands purely depends on the cohesive force between the atoms. Cohesive force is the force that binds two or more atoms. In other words, the cohesive force resists the separation between the atoms. However, the greater the cohesive force, the expansion will be low for a given increase in temperature.

Thermal expansion in length of rod due to heating is given by the relation.
Δl = l₀ α(ΔT) = l₀ α(T₂ – T₁)

Calculation:

Let initial length of identical rods is l₀

Thermal expansion in length of rod due to heating is given by the relation

Δl = l₀ α(ΔT) = l₀ α(T₂ – T₁ )

Here, α is coefficient of linear expansion.

So, change in length of rods are

Δl₁ = l₀ α₁ (180 – 30)

Δl₂ = b₀ α₂ (T – 30)

Because new lengths are same, so change in lengths of both rods are equal.

i.e. Δl₁ = Δl₂

⇒ l₀ α₁ (180 – 30) = l₀ α₂ (T – 30)

\({\rm{or\;}}\frac{{{\alpha _1}}}{{{\alpha _2}}} = \frac{{\left( {T - 30} \right)}}{{150}}\) 

Given α₁ : α₂ = 4 : 3

\(\therefore \;\frac{{T - 30}}{{150}} = \frac{4}{3} \Rightarrow T - 30 = \frac{4}{3} \times 150 = 200\)

If an atom is excited into a metastable state it can stay there long enough for a photon of correct frequency to arrive. Such a situation promotes stimulated emission at the expense of spontaneous emission and leads to laser action.

Concept:

From question, the wave equation given is:

Y = 10^-3 sin (50t + 2x)

Speed of wave is obtained by differentiating phase of wave.

Calculation:

Now, phase of wave from given equation is:

ϕ = 50t + 2x = constant

On differentiation ϕ with respect to ‘t’,

\(\Rightarrow \frac{d}{{dt}}\left( {50t + 2x} \right) = \frac{d}{{dt}}{\rm{\;(constant)}}\)

\(\Rightarrow 50 + 2\left( {\frac{{dx}}{{dt}}} \right) = 0\)

\(\Rightarrow \frac{{dx}}{{dt}} = \frac{{ - 50}}{2}\)

\(\therefore \frac{{dx}}{{dt}} = - 25{\rm{\;m}}/{\rm{s}}\)

The correct answer is 344 m/sec.

• The speed of sound is the distance traveled per unit of time by a sound wave as it propagates through an elastic medium.
• At 20 °C (68 °F), the speed of sound in air is about 344 meters per second.
• It depends strongly on temperature as well as the medium through which a sound wave is propagating.
• The speed of sound in an ideal gas depends only on its composition and temperature. The speed has a weak dependence on pressure and frequency in ordinary air, deviating slightly from ideal behavior.

The correct answer is Fleming's right-hand rule.

• Fleming's right-hand rule is used to determine the direction of the induced current in a generator.

• Fleming's right-hand rule:
  • According to Fleming's right-hand rule, the thumb, forefinger, and middle finger of the right hand are stretched to be perpendicular to each other and if the thumb represents the direction of the movement of the conductor, the fore-finger represents the direction of the magnetic field, then the middle finger represents the direction of the induced current.
  • It is used used to determine the direction of current in a generator's windings.

• Fleming's left-hand rule:
  • According to Fleming's left-hand rule, if the thumb, forefinger, and middle finger of the left hand are stretched to be perpendicular to each other and if the forefinger represents the direction of the magnetic field, the middle finger represents the direction of the current, then the thumb represents the direction of the force.
  • Fleming's left-hand rule is applicable for motors.
• Maxwell's corkscrew rule:
  • ​The Maxwell Screw Rules is also called Maxwell's Corkscrew Rule.
    • Imagine a right-handed screw being turned so that it bores its way in the direction of the current in the wire.
    • The direction of rotation gives the direction of the magnetic field and the direction of thumb give the direction of current
• Ampere's swimming rule:
  • Ampere's swimming rule states that "If a man swims along the wire carrying current such that his face is always towards the magnetic needle with current entering his feet and leaving his head, then the north pole of the magnetic needle is always deflected towards his left hand."

One of the properties of laser is its coherence. There are lasers that emit radiation in different wavelengths in all spectrums. Also, laser shows the wave behaviour.

The correct answer is 152 million km.

• It is approximately the average distance between the Earth and the Sun about 150 million km.
• The Earth-Sun orbit distance is defined to be exactly 1.00, and we call this distance 1.00 AUs.
• An Astronomical unit is a unit of length it is used to measure distances within the Solar System.
• 1 A.U. represents the mean distance between the Earth and our sun.
• The mean distance between the sun and the earth is 149,598,000 km.
• In terms of light-years, it is 8.311 minutes.
  A light-year is a measure of distance and not of time.
• Light travels at a speed of 300,000 km/second.
• It is the distances that light will travel in one year that is taken to be one light year.
• This equals to 9.461×1012 km. 

Concept:

Internal energy of ‘n’ moles of a gas with degree of freedom f (=3 for an ideal gas), at temperature T is given as:

\({\rm{E}} = \frac{{\rm{f}}}{2}{\rm{nRT}} = \frac{3}{2}{\rm{nRT}}\) 

So, for an ideal gas, the internal energy is given as:

\({\rm{E}} = \frac{3}{2}{\rm{nRT}}\) 

∵ [pV = nRT]

\({\rm{E}} = \frac{3}{2}pV\) 

Where,

p = Pressure of the gas

V = Volume of the gas

Calculation:

On substituting the given values,

\({\rm{E}} = \frac{3}{2} \times 3 \times {10^6} \times 2\) 

Sapphire, ruby and pure corundum are α-alumina, which is most stable form of AI₂O₃.

Hologram used the technique holography which records scattered light and presents in 3 dimensional form. It makes use of laser beam for the purpose.

(A) प्राकृतिक स्पेक्ट्म

Correct answer is (D). Stock value

Answer- टंगस्टन​

Answer-  (A) प्रकाश वर्ष

Concept:

According to the given condition in the question, after collision the mass of combined system is doubled. Also, this system would be displaced from its circular orbit.

So, the combined system revolves around centre of mass of system + earth’ under action of a central force.

Concept:

In a screw gauge,

\({\rm{Least\;count}} = \frac{{{\rm{\;Measure\;of\;}}1{\rm{\;main\;scale\;division\;(MSD)\;}}}}{{{\rm{\;Number\;of\;division\;on\;circular\;scale\;}}}}\) 

Calculation:

Here, minimum value to be measured/least count is 5 μm = 5 × 10^-6 m

From the values given in the question,

\(\Rightarrow 5 \times {10^{ - 6}} = \frac{{1 \times {{10}^{ - 3}}}}{{\rm{N}}}\) 

\(\Rightarrow {\rm{N}} = \frac{{{{10}^{ - 3}}}}{{5 \times {{10}^{ - 6}}}} = \frac{{1000}}{5}\) 

The resistance of the mica between the two faces is

\(\rho \frac{l}{A} = \frac{{\left( {1 \times {{10}^{13}}} \right) \times {{10}^{ - 3}}}}{{10.0 \times {{10}^{ - 4}}}}\) 

= 1 × 10¹³ Ω.

X rays are classified into two types, soft and hard X-rays. Soft X-rays have less energy than hard X-rays, hence they have longer wavelength.