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Calculation:

In an isothermal process, the temperature of the system remains constant. According to the first law of thermodynamics:

ΔU = Q - W

Where:

ΔU is the change in internal energy

Q is the heat absorbed by the system

W is the work done by the system

For an isothermal process, the change in internal energy (ΔU) is zero because the temperature does not change. Thus:

ΔU = 0

Therefore, the heat given to the system (Q) is entirely converted into work done by the system (W). Hence, in an isothermal process:

Q = W

Thus, the correct answer is Option 2: net work done by the system.

Calculation:

If E is the translational kinetic energy, then the correct relation can be derived using the ideal gas law and the kinetic theory of gases.

From the kinetic theory of gases, the translational kinetic energy (E) is related to the pressure (P), volume (V), and temperature (T) by the equation:

For an ideal gas, the internal energy E is given by:

E = (3/2)RT = (3/2) PV

Thus, the correct relation that holds good is:

PV = (2/3) E

Correct Answer: Option 4 - PV = (2/3) E

Given:

For monoatomic gas: γ = 5/3

For diatomic gas: γ = 7/5

One mole of each gas is mixed.

For monoatomic gas:

Cv = 3R/2

Cp = 5R/2

For diatomic gas:

Cv = 5R/2, Cp = 7R/2

Total heat capacities of mixture:

Cv(total) = 3R/2 + 5R/2

⇒ Cv(total) = 8R/2 = 4R

Cp(total) = 5R/2 + 7R/2

⇒ Cp(total) = 12R/2 = 6R

Now,

γ(mixture) = Cp(total) / Cv(total)

⇒ γ = 6R / 4R = 3/2 = 1.50

Hence, the correct answer is 1.50.

Calculation:

The displacement of a particle in Simple Harmonic Motion (SHM) is given by the equation:

x = 10sin(2t - π/6) metres

To find the velocity, we differentiate the displacement equation with respect to time.

The velocity (v) is given by:

v = dx/dt = d/dt [10sin(2t - π/6)]

Using the chain rule:

v = 10 × 2cos(2t - π/6)

v = 20cos(2t - π/6)

When the displacement is 6 m, we substitute this into the displacement equation:

6 = 10sin(2t - π/6)

sin(2t - π/6) = 6/10 = 0.6

2t - π/6 = sin⁻¹(0.6)

2t - π/6 = 0.6435 rad

2t = 0.6435 + π/6

2t = 0.6435 + 0.5236 = 1.1671 rad

t = 1.1671 / 2 = 0.5836 s

Now, substitute t = 0.5836 s into the velocity equation:

v = 20cos(2 × 0.5836 - π/6)

v = 20cos(1.1671 - 0.5236)

v = 20cos(0.6435) = 20 × 0.8 = 16 m/s

The velocity of the particle when its displacement is 6 m is 16 m/s.

Calculatipon:

The total energy of a particle executing Simple Harmonic Motion (S.H.M.) is given by the equation:

E = K + U

Where E is the total energy, K is the kinetic energy, and U is the potential energy.

For a particle in SHM, the kinetic energy is:

K = (1/2)mv²

And the potential energy is:

U = (1/2)kx²

The total energy (E) remains constant throughout the motion, and it is the sum of the kinetic and potential energies:

E = (1/2)kA²

Where A is the amplitude of the motion, and k is the spring constant.

Thus, the total energy is directly proportional to the square of the amplitude of the motion.

Correct Answer: Option 4 - square of amplitude of motion

Calculation:

For a pipe of length L open at both ends, the fundamental frequency (f) in air is given by the equation:

f = v/λ = (v / 2L)

⇒ λ =2L

Now, when the pipe is dipped vertically in water and half of it is submerged, the length of the air column is halved. 

⇒ L' = L/2 

⇒ λ' =4L' = 2L 

Hence, f' = v / λ' = v / 2L = f 

Correct Answer: Option 2 - f

Calculation:

For two capacitors in series, the total potential difference across both capacitors is equal to the battery voltage. The formula for the electrostatic energy stored in a capacitor is given by:

U = (1/2) × C × V²

Where:

• U is the energy stored in the capacitor,
• C is the capacitance,
• V is the voltage across the capacitor.

The capacitance of a parallel plate capacitor is given by:

C = (ε₀ × A) / d

Where:

• ε₀ is the permittivity of free space (for air, it is 1),
• A is the area of the plates,
• d is the separation between the plates.

For capacitor X, the dielectric constant is 1 (for air), and for capacitor Y, the dielectric constant is 2. The capacitance of Y will therefore be twice the capacitance of X. Since the capacitors are in series, the total capacitance is reduced, and the voltage across each capacitor is inversely proportional to their capacitance.

The energy stored in each capacitor depends on its capacitance, and since Y has twice the capacitance of X, the energy stored in X will be half of that stored in Y.

Correct Answer: Option 3 - 2 : 1

Calculation:

The potential at a point outside a charged spherical shell is given by the formula:

V = k × Q / r

Where:

• V is the potential at a distance r from the center of the sphere,
• k is the electrostatic constant (8.99 × 10⁹ N·m²/C²),
• Q is the charge on the sphere,
• r is the distance from the center of the sphere.

For a hollow sphere, the potential outside the sphere is the same as if all the charge were concentrated at the center. Since the potential at the surface of the sphere (at r = 5 cm) is 10 V, the potential inside the sphere and at any point inside (such as 2 cm from the center) remains the same.

Correct Answer: Option 2 - 10V

Calculation:

The electrostatic force between two charges is given by Coulomb’s law:

F_e = (k × |q₁ × q₂|) / r²

Where:

• k is Coulomb’s constant (9 × 10⁹ N·m²/C²),
• q₁ and q₂ are the charges on the particles,
• r is the distance between the particles.

The gravitational force between two masses is given by Newton's law of gravitation:

F_g = (G × m₁ × m₂) / r²

Where:

• G is the gravitational constant (6.67 × 10⁻¹¹ N·m²/kg²),
• m₁ and m₂ are the masses of the particles,
• r is the distance between the particles.

The forces cancel when:

F_e = F_g

Solving the equation, we find that the charge difference Δe (the difference between the proton and electron charge) is of the order of 10^-20 C.

Correct Answer: Option 4 - 10^-20 C

Calcultion:

The I-V characteristics shown in the figure represent a non-linear relationship between current (I) and voltage (V). This type of graph is typical of non-ohmic conductors, where the resistance varies with the applied voltage.

In contrast to ohmic conductors, where the I-V graph is a straight line (indicating constant resistance), non-ohmic conductors exhibit a curve, meaning their resistance changes with voltage.

Correct Answer: Option 2 - Non-ohmic conductors