Maharashtra Board 2024 Class 12 Physics (54-J-837) Question Paper (Available) :Download Solution PDF with Answer Key

Step 1: The moment of inertia of a solid disc rotating about its central axis is given by:

I = (1/2) MR²

Step 2: Comparing with the given options, the correct choice is option (B).

Step 1: Surface tension is defined as force per unit length:

T = Force / Length

Since force has a dimensional formula:

[F] = M L T⁻²

Dividing by length (L) gives:

[T] = M L⁻¹ T⁻²

Step 2: Comparing with the given options, the correct choice is option (D).

Step 1: A stationary wave is formed by the superposition of two identical waves moving in opposite directions.

Step 2: The phase difference (Δφ) between a node (zero displacement) and an adjacent antinode (maximum displacement) is:

Δφ = π/2 rad

This means that a particle at an antinode is a quarter cycle ahead of a particle at a node.

Step 1: Electric potential (V) at a point is defined as the work done per unit charge to bring a positive test charge from infinity to that point.

V = W/q

Step 2: This potential depends on the electric field but not on the path taken, as electrostatic forces are conservative.

Step 1: A moving coil galvanometer is a sensitive instrument that detects small currents.

Step 2: To convert it into an ammeter (which measures large currents), a shunt resistance (Rₛ) is connected in parallel:

Rₛ = G / ((I/Ig) -1)

where G is the galvanometer resistance, I is the total current, and Ig is the full-scale deflection current of the galvanometer.

Step 1: According to Einstein's photoelectric equation:

KE = hf - φ

Step 2: If f is doubled, then:

KE' = h(2f) - φ = 2hf - φ

which is more than twice the initial kinetic energy.

Step 1: In a cyclic process, the system returns to its initial state, meaning the change in internal energy is zero:

ΔU = 0

Step 2: From the First Law of Thermodynamics:

Q = ΔU + W

Since ΔU = 0, we get:

Q = W

Step 1: According to Faraday’s law of electromagnetic induction, the induced electromotive force (e.m.f.) in a coil is given by:

ε = -L (dI/dt)

Step 2: Given L = 20H, dI = (50A - 10A) = 40A, dt = 0.1s:

ε = - (20) × (40 / 0.1) = 8000V

Step 1: The velocity in simple harmonic motion is given by:

v = ω √(A² - x²)

where A = 10 cm, x = 6 cm, and ω = 2π/T = π rad/s.

Step 2: Substituting values:

v = π √(10² - 6²) = π √(100 - 36) = π √64 = 8π

Step 1: In Young’s double-slit experiment, the distance of the nth bright fringe from the central fringe is given by:

xₙ = n λ D / d

Since the setup remains unchanged, the ratio of fringe positions is:

x₃₀ / x₂₀ = 30 / 20 = 3/2

Step 2: Given x₂₀ = 1.2 cm,

x₃₀ = (3/2) × 1.2 = 1.8 cm

Centripetal force is the force that acts on an object moving in a circular path and is directed towards the center of the circle. It is given by:

F = mv²/r

Detergents reduce the surface tension of water, allowing it to spread and wet clothes more effectively. They also form micelles that trap grease and dirt, allowing them to be rinsed away.

An ideal voltmeter has infinite resistance so that it does not draw any current from the circuit and provides an accurate voltage measurement.

The torque (τ) on a current-carrying coil in a magnetic field (B) is:

τ = M × B

The binding energy of a hydrogen atom is the energy required to remove the electron from the nucleus. It is given by the formula:

E_b = -13.6 eV

In thermodynamics, the surroundings refer to everything outside the system under study. It includes all external objects and the environment that can exchange energy and matter with the system.

The maximum kinetic energy of a photoelectron is equal to the energy provided by the photon minus the work function of the material. The kinetic energy is given by:

K.E. = eV = 1.5 eV

For capacitors in series, the resultant capacitance (C_eq) is given by:

1/C_eq = 1/C₁ + 1/C₂

Substituting C₁ = 5μF and C₂ = 10μF:

1/C_eq = 1/5 + 1/10 = 3/10

C_eq = 10/3 = 3.33 μF

The internal energy of a thermodynamic system is the total energy contained within the system, including both kinetic and potential energy. When a gas is heated, its temperature increases, which results in an increase in the average kinetic energy of the molecules. The increase in kinetic energy leads to an increase in the internal energy, as the internal energy of an ideal gas is directly proportional to its temperature.

ΔU = nC_V ΔT

where ΔU is the change in internal energy, n is the number of moles, C_V is the molar specific heat at constant volume, and ΔT is the change in temperature.

According to Huygens' principle, every point on a wavefront can be considered as a source of secondary spherical wavelets. As these wavelets propagate outward, they form a new wavefront. For a spherical wavefront, the wavelets are generated from a point source, and as they propagate, they form spherical surfaces around the source. The radius of each spherical wavefront increases with time, and the new wavefront is tangential to the wavelets.

Magnetization is the vector quantity that represents the magnetic moment per unit volume of a material. It is denoted by M, and it gives the measure of the strength and direction of the magnetic field inside the material.

The SI unit of magnetization is A/m (Ampere per meter), and its dimensions are:

[M] = [I L⁻¹]

where I is current and L is length.

For simple harmonic motion (SHM), the restoring force is proportional to the displacement x from the equilibrium position:

F = -kx

By Newton's second law, F = ma, where a is the acceleration. Therefore, we have:

ma = -kx

Since acceleration a = d²x/dt², the equation becomes:

m d²x/dt² = -kx

This is the differential equation of SHM, which can be written as:

d²x/dt² + (k/m) x = 0

To convert a galvanometer to a voltmeter, we need to add a resistance in series with the galvanometer. The total resistance R is given by:

R = V/I

where V = 10V is the range of the voltmeter and I = 20μA = 20 × 10⁻⁶ A is the full-scale deflection current. Substituting these values:

R = 10 / (20 × 10⁻⁶) = 500,000 Ω

Since the galvanometer already has a resistance of 30Ω, the resistance to be added is:

R_added = 500,000 - 30 = 499,970 Ω

Biot-Savart law gives the magnetic field B due to a small current element. It states that the magnetic field at a point due to a current element is directly proportional to the current and the length of the current element and inversely proportional to the square of the distance between the point and the element. The formula is:

B = (μ₀/4π) × (I dℓ × r̂) / r²

where μ₀ is the permeability of free space, I is the current, dℓ is the current element, r̂ is the unit vector from the current element to the observation point, and r is the distance between the current element and the point.

A Light Emitting Diode (LED) is a semiconductor device that emits light when current flows through it in the forward direction. The LED works on the principle of electroluminescence, where electrons recombine with holes, releasing energy in the form of photons (light).

The circuit symbol for an LED is:

The e.m.f. induced between the tips of the wings is given by the formula:

ε = B × v × L

where B = 6 × 10⁻⁵ T is the magnetic field, v = 500 m/s is the speed, and L = 60 m is the wingspan of the aircraft. Substituting the values:

ε = (6 × 10⁻⁵) × (500) × (60) = 1.8 V

The maximum speed of a vehicle moving along a horizontal circular track occurs when the centripetal force is equal to the frictional force. The centripetal force F_c is given by:

F_c = mv²/r

where m is the mass of the vehicle, v is the velocity, and r is the radius of the track. The frictional force F_f is given by:

F_f = μmg

where μ is the coefficient of friction and g is the acceleration due to gravity. Equating the two forces:

mv²/r = μmg

Solving for v, we get the maximum speed:

v_max = √(μgr)

The viscosity η of a fluid is given by the formula:

η = (F × d) / (A × v)

where F = 0.5 N is the force, d = 0.5 × 10⁻³ m is the thickness of the fluid layer, A = 10⁻² m² is the area, and v = 3 × 10⁻² m/s is the velocity.

Substituting the values:

η = (0.5 × 0.5 × 10⁻³) / (10⁻² × 3 × 10⁻²) = 0.833 N·s/m²

The wavelength λ of a sound wave is related to the frequency f and the velocity of sound v by the equation:

λ = v/f

For the two tuning forks, the wavelengths are:

λ₁ = 340 / 320 = 1.0625 m,

λ₂ = 340 / 340 = 1.0 m

The difference in wavelength is:

Δλ = λ₁ - λ₂ = 1.0625 - 1 = 0.0625 m

The angular momentum L of an electron in a hydrogen atom is quantized and given by:

L = nℏ

where n is the principal quantum number and ℏ is the reduced Planck's constant. For the third orbit, n = 3, and for the first orbit, n = 1. The change in angular momentum is:

ΔL = L₁ - L₃ = 1ℏ - 3ℏ = -2ℏ

Since ℏ = 1.055 × 10⁻³⁴ Js, the change in angular momentum is:

ΔL = -2 × 1.055 × 10⁻³⁴ = -2.11 × 10⁻³⁴ Js

Thus, the change in angular momentum is 2.11 × 10⁻³⁴ Js.

The magnetic field at the centre of a circular coil is given by:

B = (μ₀ N I) / (2R)

where N = 200 is the number of turns, I = 0.5 A is the current, R = 10 cm = 0.1 m is the radius, and μ₀ = 4π × 10⁻⁷ T·m/A is the permeability of free space.

Substituting the values:

B = (4π × 10⁻⁷ × 200 × 0.5) / (2 × 0.1) = 6.28 × 10⁻⁵ T

The photoelectric effect is the phenomenon in which electrons are emitted from a material (typically a metal) when it is exposed to light of sufficient frequency. This effect demonstrates the particle nature of light, where photons transfer energy to electrons.

Experimental Set-Up:

• A light source (e.g., UV light) of known frequency.
• A photosensitive material (metal plate) connected to an electrometer.
• A variable voltage source to control the stopping potential.
• A collector to collect the emitted photoelectrons.

The light is incident on the metal surface, and the emitted electrons are measured by the electrometer.

For a transistor, the current gain α_DC and β_DC are defined as:

α_DC = I_C / I_E,   β_DC = I_C / I_B

where I_C is the collector current, I_E is the emitter current, and I_B is the base current.

The relation between α_DC and β_DC is given by:

β_DC = α_DC / (1 - α_DC)

Surface energy is the energy required to increase the surface area of a liquid by unit area. It is associated with the force that acts at the surface of the liquid and resists its expansion.

The surface tension of a liquid is the force per unit length acting along the surface, and it is related to surface energy.

The relation between surface energy (E) and surface tension (T) is given by:

E = T × L

where L is the length of the line along the surface.

An isothermal process is a process in which the temperature of the system remains constant (ΔT = 0). In this process, the internal energy of an ideal gas does not change, and the work done is related to the pressure and volume changes.

The work done by a gas in an isothermal process is given by:

W = nRT ln(V_f / V_i)

where n is the number of moles, R is the universal gas constant, T is the temperature, V_i and V_f are the initial and final volumes.

A stationary wave is formed when two progressive waves of the same frequency and amplitude, traveling in opposite directions, superpose on a stretched string. The equation of the two waves can be written as:

y₁(x,t) = A sin(kx - ωt)

y₂(x,t) = A sin(kx + ωt)

where:

• A is the amplitude,
• k is the wave number (k = 2π/λ, where λ is the wavelength),
• ω is the angular frequency,
• x is the position along the string,
• t is time.

The resultant displacement y(x,t) of the string is the sum of these two waves:

y(x,t) = y₁(x,t) + y₂(x,t)

Substituting the expressions for y₁(x,t) and y₂(x,t):

y(x,t) = A sin(kx - ωt) + A sin(kx + ωt)

Using the trigonometric identity for the sum of sines, we get:

y(x,t) = 2A sin(kx) cos(ωt)

Identifying Nodes and Antinodes:

• Nodes occur where sin(kx) = 0, which happens at x_n = (nπ/k), where n is an integer.
• Antinodes occur where sin(kx) = ±1, leading to maximum displacement.

Distance Between Two Successive Nodes or Antinodes:

The distance between two successive nodes is:

Δx = x_n+1 - x_n = (n+1)π/k - nπ/k = π/k

Since the wavelength λ = 2π/k, we get:

Δx = λ/2

Similarly, the distance between two successive antinodes is also λ/2.

An LCR circuit consists of a resistor (R), an inductor (L), and a capacitor (C) connected in series to an AC power supply. The impedance (Z) of a series LCR circuit is defined as the opposition to the flow of alternating current, and it is a complex quantity due to the phase difference between the voltage and the current.

The total impedance Z of a series LCR circuit is given by:

Z = √(R² + (X_L - X_C)²)

where:

• R is the resistance,
• X_L = ωL is the inductive reactance,
• X_C = 1/(ωC) is the capacitive reactance.

The phase difference (ϕ) between the current and the voltage is given by:

tan(ϕ) = (X_L - X_C) / R

Phasor Diagram:

The wavelength (λ) of a spectral line in the hydrogen atom can be calculated using the Rydberg formula:

1/λ = R_H (1/n₁² - 1/n₂²)

For the first line in the Balmer series (n₁ = 2, n₂ = 3):

λ₁ = 656 nm

For the second line in the Balmer series (n₁ = 2, n₂ = 4):

λ₂ = 486 nm

The percentage increase in the magnetic field due to the presence of lithium is given by:

Percentage increase = χ × 100

Substituting the value of χ:

Percentage increase = 2.1 × 10⁻⁵ × 100 = 0.0021%

The angle of banking (θ) is given by:

tan(θ) = v² / (r g)

Substituting v = 25 m/s, r = 200 m, and g = 9.8 m/s²:

tan(θ) = 0.318

θ = tan⁻¹(0.318) = 17.7°

Using the first law of thermodynamics:

dQ = dU + p dV

Applying the ideal gas law:

C_p - C_v = R/J

(i) The peak value of e.m.f. is the amplitude:

Peak value = 8 V

(ii) The angular frequency (ω) is related to the frequency (f) by:

ω = 2πf

Solving for f:

f = 100 Hz

(iii) The instantaneous value of e.m.f. at t = 10 ms is:

e = 8 sin(6.284)

Since sin(6.284) ≈ 0:

e = 0 V

A transformer is an electrical device used to change the voltage and current in an alternating current (AC) circuit. It works on the principle of electromagnetic induction.

Construction:

• A primary coil (input coil) connected to an AC source.
• A secondary coil (output coil) connected to the load.
• A magnetic core, usually made of iron, that is common to both coils.

Working:

When an alternating current flows through the primary coil, it generates a time-varying magnetic field. This varying magnetic flux is linked to the secondary coil, inducing an emf in it (according to Faraday's Law of Induction).

Equation for Transformer:

The relation between the voltages and the number of turns in the primary and secondary coils is given by:

V_s / V_p = N_s / N_p

where:

• V_s is the secondary voltage,
• V_p is the primary voltage,
• N_s is the number of turns in the secondary coil,
• N_p is the number of turns in the primary coil.

In the double-slit experiment, the interference pattern is produced when light from two slits interferes. The fringe width (β) is the distance between two consecutive maxima (or minima) on the screen.

The path difference for two waves reaching a point on the screen is given by:

Δ Path = d sin θ

For constructive interference (bright fringes), the path difference is an integer multiple of the wavelength (λ):

d sin θ = mλ

For small angles θ, sin θ ≈ tan θ, and the position of the maxima is given by:

y_m = m (λL / d)

The fringe width (β) is the distance between two successive maxima, which is:

β = λL / d

Ferry's perfectly black body:

Comparison of RMS Speed:

The root mean square (rms) speed (v_rms) of a gas molecule is given by:

v_rms = √(3kT / m)

For hydrogen at 500 K and oxygen at 400 K:

v_{rms, H₂} / v_{rms, O₂} = √(500 × 32) / (400 × 2) = √(20) ≈ 4.472

Thus, the rms speed of hydrogen molecules is approximately 4.472 times the rms speed of oxygen molecules.

Energy Stored in a Charged Capacitor:

The energy (U) stored in a charged capacitor is given by:

U = ½ C V²

where:

• C is the capacitance of the capacitor,
• V is the voltage across the capacitor.

Electric Field Calculation:

The electric field (E) at a distance (r) from a charged sphere is given by:

E = (1 / 4πε₀) × (Q / r²)

Given:

• Q = 2 × 10⁻⁶ C
• r = 0.20 m

Substituting values:

E = (9 × 10⁹ × 2 × 10⁻⁶) / (0.2)²

E = 4.5 × 10⁵ N/C

Thus, the electric field at a distance of 20 cm from the sphere is 4.5 × 10⁵ N/C.