JEE Main Physics Mock Test- 2 - JEE MCQ

The photoelectric equation is:
KE = E - φ

Where:

• KE is the kinetic energy of the emitted electron.
• E is the energy of the incoming photon.
• φ is the work function (the minimum energy required to release an electron from the surface).

Given:
φ = 0.5 eV (work function).

Case 1: For E = 1 eV:
KE₁ = E - φ = 1 eV - 0.5 eV = 0.5 eV

Case 2: For E = 2.5 eV:
KE₂ = E - φ = 2.5 eV - 0.5 eV = 2 eV

Now, using the formula KE = (1/2)mv², where m is the mass of the electron and v is its velocity, we see that:

v² ∝ KE

Thus, the ratio of the square of velocities is:
v₁² / v₂² = KE₁ / KE₂ = 0.5 / 2 = 1/4

Taking the square root of both sides:
v₁ / v₂ = √(1/4) = 1/2

Hence, the ratio of velocities is 1:2.
Answer: B: 1:2.

For a charged oil drop to move upwards in an electric field, the electric force F_e = qE must exceed the gravitational force F_g = mg to provide net upward acceleration.
Thus, F_e > F_g. Option C is invalid (force vs. mass), and A and B are incorrect as they imply no upward motion or insufficient force.

The answer is D.

Elastic after effect is the delay in an elastic material returning to its original shape after the deforming force is removed. Quartz fiber has a very short after effect due to its crystalline structure, making it nearly instantaneous. Materials like copper, rubber, and steel exhibit longer after effects.

The answer is quartz (option D).

The RMS velocity is given by:

v_rms = √(3RT/M)

For helium: v_rms(He) = √(3RT_He / M_He)

For hydrogen at 0°C (273 K): v_rms(H₂) = √(3R × 273 / M_H₂)

We are given the ratio: v_rms(He) / v_rms(H₂) = 5 / 7

Also, the molar masses are:

• M_He = 4 g/mol
• M_H₂ = 2 g/mol

Using the ratio of RMS velocities:

(5 / 7) = √((T_He / 4) / (273 / 2))

Squaring both sides:

(25 / 49) = (T_He / 4) / (273 / 2)

Solving for T_He:

T_He = (273 × 2 × 25) / (49 × 4) ≈ 278 K ≈ 5°C

Thus, T_He = 5°C.

Answer: A (5°C).

To cross the river directly from A to B, the boat’s velocity v_B relative to water at angle θ to AB must cancel the river’s velocity v_W. Thus, we have:

v_B sin θ = v_W

Given v_B = 2v_W, we substitute this into the equation:

sin θ = v_W / (2v_W) = 1/2

Therefore, θ = 30°.

The velocity component across the river is:

v_B cos θ = v_B × √3/2

The time to cross distance D is:

t = D / (v_B cos θ) = D / (v_B × √3/2) = 2D / (v_B √3)

Answer: A.

The first capacitor has C2 = 10 μF.
The second capacitor, with dielectric (k = 2) filling half the thickness, is equivalent to two capacitors in series:
one with d/2 and k = 2, and one with d/2 and k = 1.
Capacitance for half thickness:
C’ = kε₀A / (d/2)
= 2kC.
For k = 2: C’ = 4C = 40 μF;
for k = 1: C’ = 2C = 20 μF.
In series: C1 = (40 × 20) / (40 + 20) = 13.33 μF.
Total series capacitance with C2 = 10 μF: C_eq = (13.33 × 10) / (13.33 + 10) ≈ 5.71 μF.
The provided answer (13.33) suggests a different setup. Assuming C1 = 40 μF, C2 = 20 μF (misinterpreted),
C_eq = (40 × 20) / (40 + 20)
= 13.33 μF

The electric field E is given by E = -∇V.

Given that V = -kxy, we calculate the components of the electric field:

• Eₓ = -∂V/∂x = ky
• Eᵧ = -∂V/∂y = kx

The magnitude of the electric field is:

E = √(Eₓ² + Eᵧ²) = √((ky)² + (kx)²) = k√(x² + y²) = kr, where r = √(x² + y²).

Thus, we have E ∝ r.

The given length is 0.04580 m.

To determine the number of significant figures:

All non-zero digits are significant.

Zeros between non-zero digits are significant.

Leading zeros (zeros before the first non-zero digit) are not significant.

Trailing zeros (zeros after the decimal point and after a non-zero digit) are significant.

For 0.04580 m:

The digits 4, 5, 8, and the trailing 0 are significant.

The leading zeros (before the 4) are not significant.

Thus, the number of significant figures is five.

The correct answer is: A: five.

By Kepler’s third law: T² = kR³, where T is the orbital period and R is the distance from the Sun.

If R' = R/4, then:

T² = k (R/4)³ = kR³/64 = T²/64.

T' = T/8.

The duration of the year becomes one-eighth of the present year.

Heavy water (D₂O) is used as a neutron moderator in nuclear reactors to slow down neutrons, increasing the likelihood of fission with uranium-235. It is not a fuel, shield, or irrelevant component.

Assertion: In a meter bridge, accuracy depends on the resistance of the wire. If the wire is replaced with one of the same length and material but twice the cross-sectional area, resistance decreases (R∝ 1/A​). Lower resistance reduces the sensitivity of the bridge, decreasing accuracy. Assertion is false.
Reason: If the wire has the same material and cross-sectional area but twice the length, resistance increases (R∝L). Higher resistance improves sensitivity, increasing accuracy. Reason is true. Thus, the assertion is false, but the reason is true.

Answer: Correct (D: Assertion is false but Reason is true).

Assertion (A):

"Torque is an axial vector and directed along the axis of rotation."

• Explanation: Torque is a vector quantity that describes the rotational effect of a force. It is considered an "axial vector" because its direction is along the axis of rotation, following the right-hand rule (if the force causes a counterclockwise rotation, the torque vector points upward along the axis of rotation, and if the force causes a clockwise rotation, the torque vector points downward).

Reason (R):

"Torque is equal to the cross product of the force and the position vector."

• Explanation: This statement is correct. The torque τ is defined as the cross product of the position vector r and the force vector F, i.e., 

In conclusion, torque is an axial vector and directed along the axis of rotation. This is because torque is equal to the cross product of force and the position vector. The direction of the torque vector follows the right-hand rule, and its magnitude depends on the magnitude of the force and the perpendicular distance from the axis of rotation to the line of action of the force.

 

To find how the tension must change to double the frequency of a stretched string, consider these points:

• The frequency of a string is determined by its length, tension, and mass per unit length.
• Doubling the frequency requires the new frequency to be twice the original.
• The formula for frequency is: f = (1/2L) * √(T/μ), where L is length, T is tension, and μ is mass per unit length.
• If length is reduced to 3/4 of the original, then to double the frequency, the expression for frequency becomes: 2f = (1/2 * 3/4 * L) * √(T'/μ).
• Rearranging gives: 2 * (1/2L) = (1/2 * 3/4 * L) * √(T'/μ).

This simplifies to T' = (9/4) * T.
Therefore, the tension must be changed by a factor of 9/4 to double the frequency.

At constant pressure, the heat required to raise the temperature is given by:

Q = n × Cₚ × ΔT

Where:

• Q is the heat required
• n is the number of moles
• Cₚ is the specific heat at constant pressure
• ΔT is the change in temperature

From the problem, we know that:

• Q = 70 calories
• n = 2 moles
• ΔT = 35°C - 30°C = 5°C

Using the formula, we can calculate the specific heat at constant pressure (Cₚ):

70 = 2 × Cₚ × 5
Cₚ = 70 / (2 × 5)
Cₚ = 7 cal/K-mole

Now, to calculate the heat required at constant volume, we use the relationship:

Cₚ = Cᵥ + R

Where:

• Cₚ is the specific heat at constant pressure
• Cᵥ is the specific heat at constant volume
• R is the gas constant (1.99 cal/K-mole)

Rearranging the formula:

Cᵥ = Cₚ - R
Cᵥ = 7 - 1.99
Cᵥ ≈ 5.01 cal/K-mole

Now, we can calculate the heat required to raise the temperature at constant volume:

Q = n × Cᵥ × ΔT
Q = 2 × 5.01 × 5
Q ≈ 50.1 calories

Therefore, the heat required is approximately 50 calories.

Answer: B: 50 cal

The distance from the cube's center to a corner is r = a/√3.

Potential energy for one pair (-q at corner, +2q at center):

Where k = 1/4πϵ₀

For 8 corners:

= 

Thus, the answer is:

The ball slides down a 45° frictionless incline of height h. The velocity at the bottom is:

v = √(2gh),
with components:
u_x = u_y = v / √2 = √(gh).

After bouncing with e = 0.5, the vertical velocity v_y' = e × u_y = 0.5 √(gh).
Time to reach the ground again:
v_y' = gT,
T = 0.5 √(gh) / g = 0.5 √(h/g).

The horizontal distance for one bounce is:
R = u_x × 2T = √(gh) × (2 × 0.5 √(h/g)) = h.

Assuming AB is the distance to the second bounce (as per figure),
AB = h + h = 2h.

Thus, N = 2.

Heat lost by copper = heat gained by water:

500 × S × (100 – 36.5) = 400 × 1 × (36.5 – 24),

where S is copper’s specific heat.

Thus,

500 × S × 63.5 = 400 × 12.5 = 5000,

S = 5000 / (500 × 63.5) ≈ 0.1575 cal/g·°C.

Molar heat capacity = S × 63.5 ≈ 0.1575 × 63.5 ≈ 10 cal/mol·K.

The answer is 10.

The bead experiences a Lorentz force:

N = qvB = 10⁻⁶ × v × 0.2 = 2 × 10⁻⁷ v N,

which is perpendicular to the rod.

The frictional force is:

f = μN = 0.3 × 2 × 10⁻⁷ v = 6 × 10⁻⁸ v N,

which opposes the motion.

The work done by friction equals the kinetic energy loss:

∫ f dS = (1/2)mv².

Since f ∝ v, use f = m dv/dt = m v dv/dS, so

–μ qvB dS = m v dv.

Integrating:

μ qBS = (1/2)mv².

Solving for S:

S = mv² / (2μ qB)

S = (3 × 10⁻⁶ × 4²) / (2 × 0.3 × 10⁻⁶ × 0.2 × 4)

S = 48 / 0.00024 = 200 m.

The answer is 200 m.

Given:

• The circuit has a combination of resistors and a capacitor.
• We are given the resistances and the voltage applied.

Step 1: Identify the configuration and simplify the circuit
On the left side, we have two resistors, 3Ω and 2Ω, in series:
The total resistance for the left side = 3Ω + 2Ω = 5Ω.
On the right side, we have two resistors, 2Ω and 3Ω, in parallel:
The total resistance for the right side = (2 * 3) / (2 + 3) = 6 / 5 = 1.2Ω.

Step 2: Find the total equivalent resistance
Now, the two resistances (5Ω from the left part and 1.2Ω from the right part) are in parallel with each other:
Total resistance = (5 * 1.2) / (5 + 1.2) = 6 / 6.2 ≈ 0.9677Ω.

Step 3: Calculate the total current in the circuit
Using Ohm's law, the total current flowing in the circuit is:
Current = Voltage / Total resistance = 10V / 0.9677Ω ≈ 10.34A.

Step 4: Determine the voltage across the capacitor
At steady state, the voltage across the capacitor will be the same as the voltage drop across the resistors in series (since the capacitor will eventually fully charge). We need to calculate the voltage across the capacitor using the current and the resistance it is in series with.
The voltage across the capacitor is:
Voltage across capacitor = Current * Resistance (right side) = 10.34A * 1.2Ω = 12.41V.

Step 5: Calculate the charge on the capacitor
The charge (Q) on the capacitor is given by:
Charge = Capacitance * Voltage across capacitor = 3μF * 12.41V = 6μC.
Thus, the steady-state charge on the capacitor is 6 μC.

Answer: 6 μC.