Explanation:

The correct answer is: B

In a mass spectrometer, the ion moves in a circular path due to the magnetic force. The magnetic force \( q v B \) provides the centripetal force \( \frac{m v^2}{r} \). Setting them equal:

\[ \begin{aligned} q v B &= \frac{m v^2}{r} \\[1em] q B &= \frac{m v}{r} \quad (\text{divide by } v) \\[1em] v &= \frac{q B r}{m} \end{aligned} \]

This matches option B, \( \frac{B q r}{m} \).

Explanation:

The correct answer is: A

The wavelength \( \lambda \) is calculated using \( v = f \lambda \), where \( v = 331 \, \text{m} \, \text{s}^{-1} \) (speed) and \( f = 512 \, \text{s}^{-1} \) (frequency in Hz). Solving for \( \lambda \):

\[ \begin{aligned} \lambda &= \frac{v}{f} \\[1em] &= \frac{331}{512} \\[1em] &\approx 0.646 \, \text{m} \end{aligned} \]

Units check: \( \frac{\text{m} \, \text{s}^{-1}}{\text{s}^{-1}} = \text{m} \), which is correct for wavelength. Thus, the wavelength is 0.646 m.

Explanation:

The correct answer is: A

In the Rutherford scattering experiment, the large deflections of alpha particles indicated that the atom’s positive charge is concentrated in a tiny nucleus.

Most particles passed through, showing atoms are mostly empty space, while the few significant deflections confirmed the nucleus is positively charged and occupies a small volume, repelling the positively charged alpha particles.

Explanation:

The correct answer is: C

The work done \( W \) is calculated using the formula \( W = Q \cdot V \), where \( Q = 10 \, \text{C} \) (charge) and \( V = 220 \, \text{V} \) (potential difference):

\[ \begin{aligned} W &= Q \cdot V \\[1em] &= 10 \cdot 220 \\[1em] &= 2200 \, \text{J} \end{aligned} \]

Thus, the work done is 2200 J.

Explanation:

The correct answer is: A

The magnetic field strength of a solenoid is given by \( B = \mu n I \), where \( \mu \) is the permeability of the core.

Option A (brass rod) is correct because brass is non-magnetic (\( \mu \approx \mu_0 \)), so it does not increase the field strength compared to an air core.

The other options are incorrect because they involve ferromagnetic materials with high permeability (\( \mu \gg \mu_0 \)): B (soft iron), C (steel), and D (nickel) all enhance the magnetic field significantly due to their ability to be magnetized.

Explanation:

The correct answer is: C

A solid dielectric enhances a capacitor by increasing its capacitance and maximum potential difference, and by maintaining plate separation.

Option C is correct as the exception because a dielectric should be an insulator; increasing electrical conduction would allow current to flow between plates, reducing the capacitor’s charge storage ability.

Options A, B, and D are incorrect as exceptions because they are true advantages: A increases the breakdown voltage, B boosts capacitance via the dielectric constant, and D ensures fixed plate separation.

Explanation:

The correct answer is: B

Option B is correct because ferromagnetic materials have a high magnetic susceptibility (\( \chi \gg 1 \)), allowing them to be strongly magnetized in an external magnetic field.

Options A, C, and D are incorrect: A is wrong because ferromagnets have high relative permeability, not low; C is wrong because they attract, not repel, other magnetic materials; and D is wrong because they do not attract non-magnetic materials.

Explanation:

The correct answer is: A

Option A is correct because both balls experience the same gravitational acceleration (\( g \approx 9.8 \, \text{m s}^{-2} \)) downward, regardless of their initial velocities or directions of motion.

Options B, C, and D are incorrect: B is wrong because the accelerations are in the same direction (downward), not opposite; C and D are wrong because the accelerations are equal, not different.

Explanation:

The correct answer is: C

The escape velocity is the minimum speed required for an object to escape a planet’s gravitational pull, reaching infinity with zero kinetic energy. Using conservation of energy, the kinetic energy at the surface, \( \frac{1}{2} m v^2 \), equals the gravitational potential energy, \( \frac{G M m}{R} \). Setting them equal:

\[ \begin{aligned} \frac{1}{2} m v^2 &= \frac{G M m}{R} \\[1em] \frac{1}{2} v^2 &= \frac{G M}{R} \quad (\text{divide by } m) \\[1em] v^2 &= \frac{2 G M}{R} \\[1em] v &= \left( \frac{2 G M}{R} \right)^{1/2} \end{aligned} \]

This matches option C, \( \left( \frac{2 G M}{R} \right)^{1/2} \).

Explanation:

The correct answer is: D

The specific latent heat of vaporization is the amount of thermal energy required to change 1 kg of a substance from liquid (molten) to vapor phase at constant temperature and pressure.

Here, the value \( 8.7 \times 10^5 \, \mathrm{J \, kg^{-1}} \) means:

\[ \text{Energy required} = 8.7 \times 10^5 \, \mathrm{J} \] to change \[ 1\, \mathrm{kg} \] of molten lead to vapour at constant temperature and pressure.

It does not apply to the solid phase directly, nor does it depend on temperature change during the phase transition.

Therefore, the correct answer is D.

Explanation:

The correct answer is: B

The forces opposing the motion are the component of the weight down the incline and the frictional force.

Given:

Mass, \( m = 100\, \mathrm{kg} \)
Incline angle, \( \theta = 30^\circ \)
Coefficient of friction, \( \mu = 0.25 \)
Gravity, \( g = 10\, \mathrm{m/s}^2 \)

Weight component down the incline:

\[ W_{\parallel} = mg \sin \theta \] \[ = 100 \times 10 \times \sin 30^\circ \] \[ = 1000 \times 0.5 \] \[ = 500\, \mathrm{N} \]

Normal force:

\[ N = mg \cos \theta \] \[ = 100 \times 10 \times \cos 30^\circ \] \[ = 1000 \times 0.866 \] \[ = 866\, \mathrm{N} \]

Frictional force:

\[ f = \mu N \] \[ = 0.25 \times 866 \] \[ = 216.5\, \mathrm{N} \]

Total opposing force is sum of weight component and friction:

\[ F_{\text{opposing}} = W_{\parallel} + f \] \[ = 500 + 216.5 \] \[ = \boxed{716.5\, \mathrm{N}} \]

Thus, the correct answer is B. 716.5 N.

Explanation:

The correct answer is: D

Both objects start from rest, with object A having acceleration \( a \) and object B having acceleration \( 2a \). Assuming they accelerate for the same time \( t \), the distance traveled is given by:

\[ s = \frac{1}{2} a t^2 \]

For object A:

\[ s_A = \frac{1}{2} a t^2 \]

For object B:

\[ \begin{aligned} s_B &= \frac{1}{2} (2a) t^2 \\ &= a t^2 \end{aligned} \]

The ratio of distances is:

\[ \begin{aligned} \frac{s_B}{s_A} &= \frac{a t^2}{\frac{1}{2} a t^2} \\ &= 2 \end{aligned} \]

Thus, B travels twice as far as A, matching option D.

Explanation:

The correct answer is: D

The hot metal at the center radiates heat uniformly. Convection causes hot air to rise to the top of the box, making the top face (E) the hottest. The bottom face (G) is the coolest due to sinking cool air, while the side faces (F and H) are intermediate. Thus, thermometer E registers the highest temperature.

Explanation:

The correct answer is: A

Topic: Evaporation and Molecular Energy

When a liquid is left in an open dish, evaporation occurs. Evaporation happens because the most energetic molecules at the liquid's surface gain enough energy to overcome intermolecular forces and escape into the air.

This results in a gradual loss of liquid over time.

Therefore, the correct explanation is that most energetic molecules leave the surface and escape into the air, which corresponds to option A.

Explanation:

The correct answer is: C

Topic: Visible Light Spectrum and Wavelengths

In the visible light spectrum, violet has the shortest wavelength, while red has the longest wavelength.

Therefore, the colors with the shortest and longest wavelengths respectively are violet and red, which matches option C.

Explanation:

The correct answer is: A

Topic: Mirror Formula and Optics

The standard mirror and lens formulas include:

\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]

\[ m = \frac{v}{u} \quad \text{and} \quad f = \frac{1}{D} \quad \ \]

However, the formula \[ N = \frac{360}{\theta} = -1 \] is not related to mirror equations and does not make physical or mathematical sense in this context.

Hence, option A is not a valid mirror formula.

Explanation:

The correct answer is: C

Topic: Atomic Energy Levels

An excited state occurs when an atom absorbs energy, causing one or more electrons to move from a lower energy level (ground state) to a higher one.

This state is unstable, and the atom may release energy to return to its ground state.

Thus, the correct option is C: electrons have moved to a higher energy level.

Explanation:

The correct answer is: D

Topic: Magnetism – Testing for Magnetic Properties

While attraction is a necessary condition, it is not sufficient because magnetic materials (like soft iron) can be attracted to magnets without being magnets themselves.

The conclusive test for magnetism is directional behavior: a freely suspended magnet will always align itself along the magnetic North-South direction of the Earth.

Therefore, the steel bar settling in the magnetic N-S direction confirms it has become magnetized.

Explanation:

The correct answer is: C

Topic: Electrostatics – Charge Detection

The gold leaf electroscope is an instrument used to detect the presence and relative magnitude of electric charge on a body.

When a charged object is brought near the metal cap of the electroscope, it induces a charge that causes the gold leaves to diverge.

The degree of divergence gives a visual indication of the magnitude of the charge — the greater the divergence, the larger the charge.

Therefore, the gold leaf electroscope is the correct device for comparing the relative magnitudes of charges.

Explanation:

The correct answer is: C

Topic: Capacitors and Electric Circuits

A parallel plate capacitor has two conductive plates separated by a dielectric (insulating) material.

Direct current (DC) requires a continuous conducting path, but the dielectric is non-conducting, so no actual current can pass through the capacitor.

Instead, a displacement current is established in AC circuits, but for DC, after the plates are charged, current ceases to flow.

Therefore, the correct explanation is that the dielectric between the plates is non-conducting.

Explanation:

The correct answer is: B

Topic: Electromagnetic Induction

Electromagnetic induction is the process of generating an electric current in a conductor by changing the magnetic field around it.

This phenomenon was discovered by Michael Faraday and forms the basis for the operation of generators and transformers.

It occurs when there is relative motion between a conductor and a magnetic field, or when the magnetic field itself changes with time.

Hence, the correct option is B: electromagnetic induction.

Explanation:

The correct answer is: B

Topic: Thermal Expansion of Solids

Overhead electrical wires are made of metals, which expand when heated by the sun during the day and contract when cooled at night.

This phenomenon is known as linear expansion.

To prevent breakage during cold conditions, wires are intentionally hung with some slack.

As temperature increases, the wires expand and appear to sag more due to increased length.

Thus, the sagging of the wires is due to linear expansion.

Explanation:

The correct answer is: B

Topic: Electrical Circuits – EMF and Internal Resistance

Using Ohm’s law and the formula for EMF:

\[ \text{EMF} = I(R + r) \]

Where:

• \( \text{EMF} = 2 \ \text{V} \)
• \( I = 0.4 \ \text{A} \)
• \( r = 0.5 \ \Omega \) (internal resistance)
• \( R = \text{resistance of P} \)

Substitute into the equation:

\[ 2 = 0.4(R + 0.5) \]

Divide both sides by 0.4:

\[ \frac{2}{0.4} = R + 0.5 \]

\[ 5 = R + 0.5 \]

\[ R = 5 - 0.5 = 4.0 \ \Omega \]

Therefore, the resistance of P is 4.0 Ω.

Explanation:

The correct answer is: B

Topic: Heat and Temperature

Heat flows from a body at a higher temperature to one at a lower temperature when they are in thermal contact. The property that dictates this flow is the temperature, not heat capacity, mass, or power.

Temperature is a measure of the average kinetic energy of particles in a substance, and it directly governs the direction of thermal energy transfer.

Explanation:

The correct answer is: A

Topic: RLC Series Circuits and Resonance

At resonance in an RLC series circuit, the inductive reactance \( X_L \) is equal to the capacitive reactance \( X_C \).

Thus, the voltage across the inductor \( V_L \) and the voltage across the capacitor \( V_C \) are equal in magnitude but opposite in direction:

\[ V_L = -V_C \]

Given:

\[ V_L = 40 \text{ V} \Rightarrow V_C = 40 \text{ V} \]

So, the voltage across the capacitor is 40 V.

Explanation:

The correct answer is: A

Topic: Electric Motors and Commutators

In a direct current (DC) motor, the commutator serves the crucial function of reversing the direction of the current in the armature coil every half rotation.

This reversal ensures that the torque acting on the coil always acts in the same rotational direction, enabling continuous rotation.

Without the commutator, the coil would experience a torque reversal every half turn, causing it to oscillate rather than rotate continuously.

Explanation:

The correct answer is: D

Topic: Conservation of Momentum and Kinetic Energy

Given:

\( m_1 = 0.05\, \text{kg}, \quad v_1 = 200\, \text{m s}^{-1} \)
\( m_2 = 0.95\, \text{kg}, \quad v_2 = 0\, \text{m s}^{-1} \)

Assuming the bullet embeds in the block (perfectly inelastic collision), they move together at speed \( v \).

By conservation of momentum:

\[ m_1 v_1 + m_2 v_2 = (m_1 + m_2) v \]

Substituting values:

\[ \begin{aligned} (0.05)(200) + (0.95)(0) &= (0.05 + 0.95) v \\ 10 &= 1.0 \times v \\ v &= 10\, \text{m s}^{-1} \end{aligned} \]

Kinetic energy of block after impact:

\[ \begin{aligned} KE &= \frac{1}{2} m_2 v^2 \\ &= \frac{1}{2} \times 0.95 \times (10)^2 \\ &= 0.475 \times 100 \\ &= 47.5\, \text{J} \approx 50\, \text{J} \end{aligned} \]

Therefore, the kinetic energy is approximately 50 J.

Explanation:

The correct answer is: A

Topic: Nuclear Physics

The force that holds protons and neutrons (nucleons) together in the nucleus of an atom is known as the nuclear force. This force is very strong but acts over a very short range, overcoming the electrostatic repulsion between positively charged protons.

Explanation:

The correct answer is: C

Topic: Nuclear Reactor Control

Control rods absorb neutrons to regulate the rate of the fission reaction.

This affects how much energy is generated but not the radiation emitted.

Radiation mostly comes from fission fragments and radioactive decay, which control rods do not influence.

Explanation:

The correct answer is: B

Topic: Electrical Power and Safety

Step 1: Convert the power rating from kilowatts to watts.

\[ P = 1.06\,\text{kW} = 1060\,\text{W} \]

Step 2: Use the formula \( I = \frac{P}{V} \) to calculate the current.

\[ I = \frac{1060}{220} \approx 4.82\,\text{A} \]

Step 3: Choose a fuse slightly above the calculated current.

So, the appropriate fuse rating is 5 A.

Explanation:

The correct answer is: A

Topic: Reflection by Curved Mirrors

Dentists use concave mirrors because they form magnified and upright images.

This helps them clearly see the details of the teeth being examined.

Concave mirrors also focus light to illuminate the area effectively.

Explanation:

The correct answer is: A

The diagram shows a standing wave with 5 loops, indicating the fifth harmonic (\( n = 5 \)). For a wire fixed at both ends, \( \lambda = \frac{2L}{n} \), where \( L = 1.25 \, \text{m} \):

\[ \begin{aligned} \lambda &= \frac{2 \times 1.25}{5} \\[1em] &= 0.50 \, \text{m} \end{aligned} \]

Thus, the wavelength is 0.50 m.

Explanation:

The correct answer is: B

Using the wavelength \( \lambda = 0.50 \, \text{m} \) from Question 39 and the wave speed \( v = 120 \, \text{m/s} \), the frequency is:

\[ \begin{aligned} f &= \frac{v}{\lambda} \\[1em] &= \frac{120}{0.50} \\[1em] &= 240 \, \text{Hz} \end{aligned} \]

Thus, the frequency of the waves is 240 Hz.

Explanation:

The correct answer is: A

Topic: Electric Fields

The electric field between charged parallel plates is approximately uniform, but not everywhere.

Near the edges of the plates, fringing causes the field to become non-uniform.

Therefore, stating that the field is uniform everywhere is incorrect.

The correct answer is: C

Topic: Relative Velocity

The velocity of \( P \) relative to \( Q \), denoted \( v_{PQ} \), is given by:

\[ v_{PQ} = v_P - v_Q \]

Substituting the given velocities:

\[ v_{PQ} = 30\ \mathrm{m\,s^{-1}} - 20\ \mathrm{m\,s^{-1}} \]

Calculating the difference:

\[ v_{PQ} = 10\ \mathrm{m\,s^{-1}} \]

This means car \( P \) is moving at \( 10\ \mathrm{m\,s^{-1}} \) faster than car \( Q \) in the same direction.

The correct answer is: C

Topic: Electromagnetic Spectrum – Infrared Waves

The wavelength of infrared radiation lies within the range:

\[ 10^{-3}\ \mathrm{m} \quad \text{to} \quad 10^{-6}\ \mathrm{m} \]

This means the wavelength is longer than \(10^{-6}\ \mathrm{m}\) and shorter than \(10^{-3}\ \mathrm{m}\).

Therefore, the correct statement is that the wavelength is longer than \(10^{-6}\ \mathrm{m}\).

The correct answer is: D

Explanation:

Topic: Atomic Energy Levels and Photon Emission

When an electron jumps from a higher energy level \( n=4 \) to a lower energy level \( n=2 \),

a photon is emitted because the electron loses energy.

This photon corresponds to the energy difference between the two orbits.

The correct answer is: D

Explanation:

Topic: Newton’s Second Law and Impulse-Momentum Theorem

Impulse is given by \( F \times t \), which is also equal to the change in momentum \( \Delta p \).

Since both bodies experience the same force for the same time, they receive the same impulse.

Therefore, both acquire the same momentum regardless of their masses.

The correct answer is: B

Topic: AC Circuits and Electromagnetic Devices

Explanation:

A step-up transformer operates based on electromagnetic induction, which requires a changing magnetic field.

Such a changing field is produced only by alternating current (AC), not direct current (DC).

Therefore, step-up transformers are exclusively used with AC.

Explanation:

The correct answer is: B

Topic: Equations of Motion

The equation for distance travelled under uniform acceleration is:

\[ S = ut + \frac{1}{2} a t^2 \]

Given initial velocity \( u = 20 \, \text{m/s} \) and time \( t = 15 \, \text{s} \), substitute:

\[ S = 20 \times 15 + \frac{1}{2} a \times 15^2 \]

Calculate the terms:

\[ S = 300 + \frac{1}{2} a \times 225 \]

Simplify the expression:

\[ S = 300 + \frac{225a}{2} \]

This matches option B.

Explanation:

The correct answer is: B

Topic: Projectile Motion

For a projectile, the maximum range \( R \) and maximum height \( H \) are given by:

\[ R = \frac{u^2 \sin(2\theta)}{g}, \quad H = \frac{u^2 \sin^2\theta}{2g} \]

Taking the ratio:

\[ \frac{R}{H} = \frac{ \frac{u^2 \sin(2\theta)}{g} }{ \frac{u^2 \sin^2\theta}{2g} } = \frac{2 \sin(2\theta)}{\sin^2\theta} \]

Given \( \frac{R}{H} = 4 \), substitute and simplify:

\[ \frac{2 \sin(2\theta)}{\sin^2\theta} = 4 \Rightarrow \frac{ \sin(2\theta) }{ \sin^2\theta } = 2 \]

Using the identity \( \sin(2\theta) = 2\sin\theta\cos\theta \):

\[ \frac{2 \sin\theta \cos\theta}{\sin^2\theta} = 2 \]

\[ \Rightarrow \frac{2 \cos\theta}{\sin\theta} = 2 \]

\[ \Rightarrow \cot\theta = 1 \]

Therefore, \( \theta = 45^\circ \).

The correct answer is: D

Topic: Newton's Third Law of Motion

Newton’s third law states that for every action, there is an equal and opposite reaction.

So, the force exerted on the bus by the towing vehicle is equal in magnitude and opposite in direction to the force exerted by the bus on the towing vehicle (statement II is correct).

However, only statement II is fully correct in describing observable forces, since the others misinterpret interactions between the road and the vehicles.

Explanation:

The correct answer is: C

The distance between X and Y is the area under the velocity-time graph from \( t = 4 \, \text{s} \) to \( t = 8 \, \text{s} \). The velocity is constant at 40 m/s, so the area is a rectangle:

\[ \begin{aligned} \text{Area} &= \text{velocity} \times \text{time} \\[1em] &= 40 \, \text{m/s} \times (8 - 4) \, \text{s} \\[1em] &= 40 \times 4 \\[1em] &= 160 \, \text{m} \end{aligned} \]

Thus, the distance is 160.0 m.