Step 1: Understanding the Concept:

When a body tied to a spring moves in a horizontal circular path, the spring undergoes an extension. This extension creates a restoring force (spring force) that acts as the necessary centripetal force required for the circular motion.


Step 2: Key Formula or Approach:

1. Spring Force: \( F_s = kx \), where \( k \) is the spring constant and \( x \) is the extension.

2. Centripetal Force: \( F_c = m \omega^2 r \).

3. Radius of circular path: \( r = L + x \), where \( L \) is the natural length of the spring.


Step 3: Detailed Explanation:

At equilibrium in the rotating frame, the spring force balances the centripetal requirements:
\[ kx = m \omega^2 (L + x) \]

Given:
\( m = 200 g = 0.2 kg \)
\( k = 12.5 N/m \)
\( \omega = 5 rad/s \)

Substituting these values:
\[ 12.5x = 0.2 \times (5)^2 \times (L + x) \]
\[ 12.5x = 0.2 \times 25 \times (L + x) \]
\[ 12.5x = 5(L + x) \]
\[ 12.5x = 5L + 5x \]
\[ 12.5x - 5x = 5L \]
\[ 7.5x = 5L \]

To find the ratio of extension (\( x \)) to natural length (\( L \)):
\[ \frac{x}{L} = \frac{5}{7.5} \]
\[ \frac{x}{L} = \frac{50}{75} = \frac{2}{3} \]


Step 4: Final Answer:

The ratio of extension in the spring to its natural length is 2:3.
Quick Tip: In circular motion involving springs, always remember that the radius of the circle is the "stretched length", which is \( (Natural Length + Extension) \).

Step 1: Understanding the Concept:

Different communication systems utilize specific frequency bands of the electromagnetic spectrum as allocated by regulatory bodies for efficient transmission and to avoid interference.


Step 2: Detailed Explanation:

- AM Broadcast (Amplitude Modulation): Operates in the Medium Frequency (MF) range. The standard band is approximately 540 to 1600 kHz. Thus, A matches with II.

- FM Broadcast (Frequency Modulation): Operates in the Very High Frequency (VHF) range, commonly known to be 88 to 108 MHz. Thus, B matches with I.

- Television: Television signals use a wider range including VHF and UHF (Ultra High Frequency), spanning approximately 54 MHz to 890 MHz. Thus, C matches with IV.

- Satellite Communication: Requires very high frequencies to penetrate the ionosphere, typically in the Gigahertz (GHz) range. The 3.7 to 4.2 GHz range is common for C-band satellite communication. Thus, D matches with III.


Step 3: Final Answer:

The correct matching is A-II, B-I, C-IV, D-III.
Quick Tip: Remembering the standard FM radio band (88-108 MHz) usually allows you to eliminate most options in these matching questions immediately.

Step 1: Understanding the Concept:

According to the principle of homogeneity of dimensions, the dimensions of the physical quantities on both sides of an equation must be identical.


Step 2: Key Formula or Approach:

We determine the dimensions of each quantity:

- Frequency (\( \nu \)): \( [T^{-1}] = [M^0 L^0 T^{-1}] \)

- Radius (\( r \)): \( [L] \)

- Density (\( \rho \)): \( [Mass/Volume] = [M L^{-3}] \)

- Surface Tension (\( s \)): \( [Force/Length] = [M L T^{-2} \cdot L^{-1}] = [M T^{-2}] \)


Step 3: Detailed Explanation:

Substitute the dimensions into the given expression \( \nu = r^a \rho^b s^c \):
\[ [M^0 L^0 T^{-1}] = [L]^a [M L^{-3}]^b [M T^{-2}]^c \]
\[ [M^0 L^0 T^{-1}] = M^{b+c} L^{a-3b} T^{-2c} \]

Equating the powers of M, L, and T:

1. For \( T \): \( -2c = -1 \implies c = \frac{1}{2} \)

2. For \( M \): \( b + c = 0 \implies b = -c = -\frac{1}{2} \)

3. For \( L \): \( a - 3b = 0 \implies a = 3b = 3 \left( -\frac{1}{2} \right) = -\frac{3}{2} \)


Step 4: Final Answer:

The values are \( a = -\frac{3}{2} \), \( b = -\frac{1}{2} \), and \( c = \frac{1}{2} \).
Quick Tip: Surface tension is energy per unit area as well as force per unit length. Its dimensions \( [MT^{-2}] \) are very common in liquid properties questions.

Step 1: Understanding the Concept:

Electric potential is a scalar quantity. The total potential at a point due to multiple charge distributions is the algebraic sum of the individual potentials.


Step 2: Key Formula or Approach:

The potential \( dV \) due to a small charge element \( dq \) at a distance \( R \) is:
\[ dV = \frac{1}{4 \pi \epsilon_0} \frac{dq}{R} \]

For a continuous arc, \( V = \frac{1}{4 \pi \epsilon_0 R} \int dq = \frac{Q_{total}}{4 \pi \epsilon_0 R} \).


Step 3: Detailed Explanation:

For a half ring with radius \( R \) and linear charge density \( \lambda \), the total charge is:
\[ Q = \lambda \times (circumference of semi-circle) = \lambda (\pi R) \]

The potential at the center due to one half ring is:
\[ V = \frac{\lambda \pi R}{4 \pi \epsilon_0 R} = \frac{\lambda}{4 \epsilon_0} \]

Notice that the potential is independent of the radius \( R \) because as the radius increases, the amount of charge also increases proportionally.

Total potential at the common center:
\[ V_{total} = V_1 + V_2 \]
\[ V_{total} = \frac{\lambda}{4 \epsilon_0} + \frac{\lambda}{4 \epsilon_0} = \frac{2 \lambda}{4 \epsilon_0} = \frac{\lambda}{2 \epsilon_0} \]


Step 4: Final Answer:

The net electric potential at the center is \( \frac{\lambda}{2 \epsilon_0} \).
Quick Tip: Potential at the center of any arc with uniform \( \lambda \) is \( V = \frac{\lambda \theta}{4 \pi \epsilon_0} \) where \( \theta \) is the angle in radians. For a semi-circle, \( \theta = \pi \), so \( V = \frac{\lambda}{4 \epsilon_0} \).

Step 1: Understanding the Concept:

According to Kepler's Third Law of planetary motion, the square of the time period of revolution (\( T \)) of a planet is proportional to the cube of its mean distance (\( r \)) from the Sun.


Step 2: Key Formula or Approach:
\[ T^2 \propto r^3 \implies \frac{T_1^2}{T_2^2} = \frac{r_1^3}{r_2^3} \]


Step 3: Detailed Explanation:

Let \( T_1 = 1 year \) (Earth's period) and \( r_1 = 1.5 \times 10^8 km \) (Earth's distance).

Given \( T_2 = 2.83 years \). We need to find \( r_2 \).

Note that \( (2.83)^2 \approx 8 \) (since \( 2.83 \approx \sqrt{8} \approx 2\sqrt{2} \)).

Using the formula:
\[ \frac{(1)^2}{(2.83)^2} = \frac{(1.5 \times 10^8)^3}{r_2^3} \]
\[ \frac{1}{8} = \left( \frac{1.5 \times 10^8}{r_2} \right)^3 \]

Taking the cube root on both sides:
\[ \sqrt[3]{\frac{1}{8}} = \frac{1.5 \times 10^8}{r_2} \]
\[ \frac{1}{2} = \frac{1.5 \times 10^8}{r_2} \]
\[ r_2 = 2 \times 1.5 \times 10^8 = 3.0 \times 10^8 km \]


Step 4: Final Answer:

The distance of the imaginary planet from the Sun is \( 3 \times 10^8 \) km.
Quick Tip: In gravitation problems, look for numerical clues: \( 2.83 \approx 2\sqrt{2} = \sqrt{8} \). Squaring it gives 8, which is a perfect cube (\( 2^3 \)), making calculations easy.

Step 1: Understanding the Concept:

The ratio of specific heats \( \gamma \) is related to the degrees of freedom (\( f \)) of a gas molecule by the formula \( \gamma = 1 + \frac{2}{f} \).


Step 2: Key Formula or Approach:

1. For a monoatomic gas, \( f = 3 \) (3 translational).

2. For a diatomic gas (rigid rotator), \( f = 5 \) (3 translational + 2 rotational).


Step 3: Detailed Explanation:

Calculate \( \gamma_1 \) for monoatomic gas:
\[ \gamma_1 = 1 + \frac{2}{3} = \frac{5}{3} \]

Calculate \( \gamma_2 \) for diatomic rigid rotator:
\[ \gamma_2 = 1 + \frac{2}{5} = \frac{7}{5} \]

Calculate the ratio \( \frac{\gamma_1}{\gamma_2} \):
\[ \frac{\gamma_1}{\gamma_2} = \frac{5/3}{7/5} = \frac{5}{3} \times \frac{5}{7} = \frac{25}{21} \]


Step 4: Final Answer:

The ratio \( \frac{\gamma_1}{\gamma_2} \) is 25/21.
Quick Tip: Always clarify if the diatomic molecule is "rigid" or "vibrating". A non-rigid rotator has \( f=7 \) due to 2 additional vibrational degrees of freedom at high temperatures.

Step 1: Understanding the Concept:

The time period (\( T \)) of a simple pendulum is the time it takes to complete one full oscillation. If \( T \) increases, the clock takes more time for each tick, meaning it runs slow. If \( T \) decreases, it runs fast.


Step 2: Key Formula or Approach:
\[ T = 2 \pi \sqrt{\frac{L}{g}} \implies T \propto \frac{1}{\sqrt{g}} \]


Step 3: Detailed Explanation:

- Evaluating Reason R: Acceleration due to gravity (\( g \)) decreases with increase in altitude. Mount Everest is at a high altitude, so \( g_{Everest} < g_{surface} \). Reason R is correct.

- Evaluating Assertion A: Since \( g \) decreases on Mount Everest, the time period \( T \) of the pendulum clock increases (\( T \uparrow \)).

- An increased time period means the clock is ticking more slowly than a standard clock. Thus, the clock becomes slow, not fast. Assertion A is incorrect.


Step 4: Final Answer:

Assertion A is false, but Reason R is true.
Quick Tip: "Time Period Increases" = "Clock Slows Down" = "Losing Time".
"Time Period Decreases" = "Clock Speeds Up" = "Gaining Time".

Step 1: Understanding the Concept:

The value of acceleration due to gravity (\( g \)) is maximum at the Earth's surface and varies as one moves away from the surface (either into space or towards the core).


Step 2: Key Formula or Approach:

1. At height \( h \) (\( h \ll R \)): \( g_h = g \left( 1 - \frac{2h}{R} \right) \).

2. At depth \( d \): \( g_d = g \left( 1 - \frac{d}{R} \right) \).


Step 3: Detailed Explanation:

- Statement I: Moving 'up' increases the distance from the Earth's center, decreasing \( g \). Moving 'down' reduces the effective mass of the Earth attracting the object, also decreasing \( g \). Hence, \( g \) decreases in both cases. Statement I is correct.

- Statement II: For the same distance \( x \) (where \( h=d=x \)):

Decrease with height \( \Delta g_h = g \left( \frac{2x}{R} \right) \).

Decrease with depth \( \Delta g_d = g \left( \frac{x}{R} \right) \).

The decrease at height \( h \) is twice the decrease at depth \( d \) (for small distances). Therefore, the values of gravity are not the same. Statement II is incorrect.


Step 4: Final Answer:

Statement I is correct, and Statement II is incorrect.
Quick Tip: Gravity decreases twice as fast as you go up compared to as you go down (near the surface). To have equal \( g \), you would need to go to a depth \( d = 2h \).

Step 1: Understanding the Concept:

When a conductor moves across a magnetic field, the free electrons in it experience a Lorentz force, leading to a potential difference between its ends. This is called motional electromotive force (emf).


Step 2: Key Formula or Approach:

The induced emf \( d\epsilon \) for a small length element \( dr \) moving with velocity \( v \) perpendicular to magnetic field \( B \) is \( d\epsilon = B v \, dr \).


Step 3: Detailed Explanation:

For a rod of length \( L \) rotating with angular velocity \( \omega \), the linear velocity of a point at a distance \( r \) from the axis of rotation is \( v = r \omega \).

The total induced emf \( \epsilon \) is the integral over the entire length of the rod:
\[ \epsilon = \int_{0}^{L} B v \, dr \]
\[ \epsilon = \int_{0}^{L} B (r \omega) \, dr \]
\[ \epsilon = B \omega \int_{0}^{L} r \, dr \]
\[ \epsilon = B \omega \left[ \frac{r^2}{2} \right]_{0}^{L} \]
\[ \epsilon = \frac{1}{2} B L^2 \omega \]


Step 4: Final Answer:

The induced emf across the ends of the rod is \( \frac{1}{2} B L^2 \omega \).
Quick Tip: This formula can also be visualized as \( \epsilon = B \times (Area swept per second) \). The area of the circle is \( \pi L^2 \). In one period \( T \), it sweeps this area.
Area swept per sec = \( \frac{\pi L^2}{T} = \frac{\pi L^2}{2\pi/\omega} = \frac{1}{2} L^2 \omega \). Thus \( \epsilon = B \frac{1}{2} L^2 \omega \).

Step 1: Understanding the Concept:

Two vectors are said to be perpendicular (orthogonal) if their scalar (dot) product is equal to zero.


Step 2: Key Formula or Approach:

If \( \vec{A} \perp \vec{B} \), then \( \vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z = 0 \).


Step 3: Detailed Explanation:

Given:
\( \vec{p} = 1 \hat{i} + 2m \hat{j} + m \hat{k} \)
\( \vec{Q} = 4 \hat{i} - 2 \hat{j} + m \hat{k} \)

Since they are perpendicular:
\[ \vec{p} \cdot \vec{Q} = (1)(4) + (2m)(-2) + (m)(m) = 0 \]
\[ 4 - 4m + m^2 = 0 \]

This is a quadratic equation which can be rewritten as:
\[ m^2 - 4m + 4 = 0 \]
\[ (m - 2)^2 = 0 \]

Taking the square root on both sides:
\[ m - 2 = 0 \implies m = 2 \]


Step 4: Final Answer:

The value of \( m \) is 2.
Quick Tip: The dot product is the fastest way to check for perpendicularity. If vectors were parallel, you would check if their components are proportional: \( \frac{p_x}{Q_x} = \frac{p_y}{Q_y} = \frac{p_z}{Q_z} \).

Step 1: Understanding the Concept:

In a plane electromagnetic wave traveling in vacuum, the amplitudes of the electric field (\( E_0 \)) and magnetic field (\( B_0 \)) are related to each other via the speed of light (\( c \)).


Step 2: Key Formula or Approach:

1. Relationship between amplitudes: \( \frac{E_0}{B_0} = c \).

2. Speed of wave: \( c = \frac{\omega}{k} \).


Step 3: Detailed Explanation:

From the standard properties of EM waves:
\[ \frac{E_0}{B_0} = c \]

Substituting the value of \( c = \frac{\omega}{k} \):
\[ \frac{E_0}{B_0} = \frac{\omega}{k} \]

Cross-multiplying gives:
\[ k E_0 = \omega B_0 \]


Step 4: Final Answer:

The correct relation is \( k E_0 = \omega B_0 \).
Quick Tip: The phase of the wave is \( (kz - \omega t) \). Dimensional check: \( [k] = [L^{-1}] \) and \( [\omega] = [T^{-1}] \). Thus \( \omega/k \) has dimensions of speed \( [LT^{-1}] \), which confirms \( E/B = \omega/k \).

Step 1: Understanding the Concept:

The given circuit is a transistor-based or simple switch-based resistor logic circuit. The output \( Y \) is taken from the collector/top node.


Step 2: Detailed Explanation:

- The output \( Y \) is connected to a +5V supply through a resistor.

- There are two parallel switches (or transistors) \( A_1 \) and \( B_1 \) connected between the output node and the ground (0V).

- If either switch \( A_1 \) or switch \( B_1 \) is closed (Input = 1), a path is created to ground. The potential at \( Y \) drops to 0V (Output = 0).

- If both switches \( A_1 \) and \( B_1 \) are open (Inputs = 0, 0), no current flows to ground. The potential at \( Y \) remains at +5V (Output = 1).


Truth Table:


\begin{tabular{|c|c|c|
\hline
A & B & Y
\hline
0 & 0 & 1
\hline
0 & 1 & 0
\hline
1 & 0 & 0
\hline
1 & 1 & 0
\hline
\end{tabular

The resulting logic \( Y = \overline{A + B} \) corresponds exactly to a NOR gate.


Step 3: Final Answer:

The equivalent logic gate is NOR.
Quick Tip: When switches are in \textbf{parallel} pulling the output to ground, it's a \textbf{NOR} configuration. When switches are in \textbf{series} pulling the output to ground, it's a \textbf{NAND} configuration.

Step 1: Understanding the Concept:

The de-Broglie wavelength of a particle is inversely proportional to its momentum. When particles have the same kinetic energy, their wavelengths depend on their masses.


Step 2: Key Formula or Approach:

The relation between de-Broglie wavelength (\(\lambda\)) and kinetic energy (\(K\)) is:
\[ \lambda = \frac{h}{p} = \frac{h}{\sqrt{2mK}} \]


Step 3: Detailed Explanation:

Given that kinetic energy \(K\) is constant for all three particles:
\[ \lambda \propto \frac{1}{\sqrt{m}} \]

We know the relative masses of the particles are:

1. Mass of electron (\(m_e\)) \(\approx 9.1 \times 10^{-31}\) kg

2. Mass of proton (\(m_p\)) \(\approx 1.67 \times 10^{-27}\) kg

3. Mass of \(\alpha\)-particle (\(m_{\alpha}\)) \(\approx 4 \times m_p \approx 6.64 \times 10^{-27}\) kg

Comparing the masses:
\[ m_{\alpha} > m_p > m_e \]

Since wavelength is inversely proportional to the square root of mass:
\[ \lambda_{\alpha} < \lambda_p < \lambda_e \]


Step 4: Final Answer:

The correct order of wavelengths is \(\lambda_{\alpha} < \lambda_p < \lambda_e\).
Quick Tip: For the same kinetic energy, the lighter the particle, the longer its de-Broglie wavelength. Since electrons are the lightest subatomic particles among the options, they will always have the maximum wavelength.

Step 1: Understanding the Concept:

Lenses are made of dispersive materials where the refractive index varies with the wavelength of light. This causes different colors to focus at different points.


Step 2: Key Formula or Approach:

According to Lens Maker's Formula:
\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]

Cauchy's equation states: \(\mu \approx A + \frac{B}{\lambda^2}\).


Step 3: Detailed Explanation:

Since refractive index (\(\mu\)) depends on wavelength (\(\lambda\)), and focal length (\(f\)) depends on \(\mu\), different colors (wavelengths) will have different focal lengths.

Violet light has a higher refractive index than red light (\(\mu_V > \mu_R\)), thus violet light converges closer to the lens than red light (\(f_V < f_R\)).

The failure of a lens to focus all colors at the same point is known as chromatic aberration.


Step 4: Final Answer:

The phenomenon is called Chromatic aberration.
Quick Tip: Remember: "Chrome" means color. Aberration caused by colors is Chromatic Aberration. Aberration caused by the shape (geometry) of the lens is Spherical Aberration.

Step 1: Understanding the Concept:

A long solenoid produces a uniform magnetic field inside its core that is proportional to the number of turns per unit length and the current flowing through it.


Step 2: Key Formula or Approach:

Magnetic field inside a long solenoid is given by:
\[ B = \mu_0 n I \]

where \(n\) is the number of turns per unit meter.


Step 3: Detailed Explanation:

Given:

Number of turns per cm = 70.
\(n = 70 turns/cm = 70 \times 10^2 turns/m = 7000 turns/m\).

Current \(I = 2.0\) A.

Permeability \(\mu_0 = 4\pi \times 10^{-7} TmA^{-1}\).

Calculating \(B\):
\[ B = (4\pi \times 10^{-7}) \times 7000 \times 2 \]
\[ B = 8\pi \times 7 \times 10^{-4} \]
\[ B = 56\pi \times 10^{-4} \]

Using \(\pi \approx 3.142\):
\[ B \approx 56 \times 3.142 \times 10^{-4} \]
\[ B \approx 175.952 \times 10^{-4} \approx 176 \times 10^{-4} T \]


Step 4: Final Answer:

The magnetic field produced inside the solenoid is \(176 \times 10^{-4}\) T.
Quick Tip: Be very careful with units of '\(n\)'. Always convert 'turns per cm' to 'turns per meter' by multiplying by 100 before substituting into the standard formula.

Step 1: Understanding the Concept:

In a velocity-time (\(v-t\)) graph:

- Displacement is the algebraic sum of the areas (positive above the axis, negative below).

- Distance is the sum of the magnitudes of the areas (all treated as positive).


Step 2: Key Formula or Approach:

Displacement \(= \int v \, dt = \sum Area_i\) (with signs).

Distance \(= \int |v| \, dt = \sum |Area_i|\).


Step 3: Detailed Explanation:

From the graph, we divide the motion into segments:

1. \(t = 0\) to \(2s\): \(v = 8 m/s\). Area \(A_1 = 2 \times 8 = 16 m\).

2. \(t = 2\) to \(4s\): \(v = -4 m/s\). Area \(A_2 = 2 \times (-4) = -8 m\).

3. \(t = 4\) to \(6s\): \(v = 4 m/s\). Area \(A_3 = 2 \times 4 = 8 m\).

4. \(t = 6\) to \(8s\): \(v = 0 m/s\). Area \(A_4 = 0 m\).

5. \(t = 8\) to \(10s\): Looking at the graph, \(v = -2 m/s\). Area \(A_5 = 2 \times (-2) = -4 m\).

Total Displacement:
\[ S = 16 - 8 + 8 + 0 - 4 = 12 m \]

Total Distance:
\[ D = 16 + 8 + 8 + 0 + 4 = 36 m \]

The ratio of displacement to distance is:
\[ Ratio = \frac{12}{36} = \frac{1}{3} \]


Step 4: Final Answer:

The ratio is 1:3.
Quick Tip: Always visually double-check the height of segments on the y-axis. Segment (8-10) is half the height of segment (2-4) in the negative region, indicating a velocity of -2 m/s.

Step 1: Understanding the Concept:

When an electron transitions between energy levels in a hydrogen atom, it emits a photon with energy equal to the difference between the initial and final levels.


Step 2: Key Formula or Approach:

Energy levels in H-atom: \(E_n = -\frac{13.6}{n^2} eV\).

Energy emitted: \(\Delta E = E_4 - E_1\).

Wavelength: \(\lambda = \frac{hc}{\Delta E}\).


Step 3: Detailed Explanation:

Given \(h = 4 \times 10^{-15} eVs\) and \(c = 3 \times 10^8 m/s\).

Calculate energy difference:
\[ \Delta E = -13.6 \left( \frac{1}{4^2} - \frac{1}{1^2} \right) = 13.6 \left( 1 - \frac{1}{16} \right) \]
\[ \Delta E = 13.6 \times \frac{15}{16} = 12.75 eV \]

Calculate wavelength:
\[ \lambda = \frac{(4 \times 10^{-15} eVs) \times (3 \times 10^8 m/s)}{12.75 eV} \]
\[ \lambda = \frac{12 \times 10^{-7}}{12.75} m \]
\[ \lambda \approx 0.9411 \times 10^{-7} m = 94.11 nm \]


Step 4: Final Answer:

The corresponding wavelength is 94.1 nm.
Quick Tip: Using the simplified value \(hc \approx 1242 eV\cdotnm\) (if standard \(h\) is used) or calculating \(hc\) directly from given values helps in avoiding unit conversion errors.

Step 1: Understanding the Concept:

Elasticity in physics refers to the property of a material to return to its original shape. A "more elastic" material requires more force to produce a given strain (high Young's modulus).


Step 2: Detailed Explanation:

Assertion A: Steel is indeed the primary material for skyscrapers and bridges due to its structural integrity.

Reason R: In materials science, steel has a higher Young's modulus than materials like copper or aluminum, meaning it resists deformation under heavy loads. A high elastic limit means it can sustain significant stress without undergoing permanent plastic deformation.

The high elasticity and elastic limit are precisely why engineers choose steel for construction, as it ensures the structure remains safe and stable under load. Thus, R is the correct explanation for A.


Step 3: Final Answer:

Both A and R are correct and R is the correct explanation of A.
Quick Tip: Common misconception: People think rubber is more elastic than steel. In Physics, steel is more elastic because it resists deformation more strongly (higher Stress/Strain ratio).

Step 1: Understanding the Concept:

For an ideal gas undergoing an isothermal process, the product of pressure (\(P\)) and volume (\(V\)) is constant according to Boyle's Law.


Step 2: Key Formula or Approach:

Ideal gas equation: \(PV = nRT\).

For a fixed amount of gas, at a higher temperature, the \(PV\) product is larger.


Step 3: Detailed Explanation:

The graph of \(P\) vs \(V\) for an isothermal process is a rectangular hyperbola.

As temperature increases, the curve moves further away from the origin because for any given volume \(V\), the pressure \(P = \frac{nRT}{V}\) will be higher if \(T\) is higher.

Given \(T_3 > T_2 > T_1\):

- The outermost curve (highest \(P\) for same \(V\)) must correspond to \(T_3\).

- The innermost curve must correspond to \(T_1\).

Looking at the diagrams, Option 1 shows the curves correctly labeled with \(T_3\) as the outermost and \(T_1\) as the innermost curve.


Step 4: Final Answer:

The correct representation is given in Option 1.
Quick Tip: Draw a vertical line (constant Volume) across the curves. The point where the line hits the highest pressure corresponds to the highest temperature isotherm.

Step 1: Understanding the Concept:

A real voltmeter has a finite resistance. When connected in parallel to a resistor, it changes the equivalent resistance of that part of the circuit, affecting the voltage distribution.


Step 2: Key Formula or Approach:

Parallel resistance: \(R_p = \frac{R_1 R_v}{R_1 + R_v}\).

Voltage Divider Rule: \(V_{out} = V_{total} \left( \frac{R_{parallel}}{R_{parallel} + R_{other}} \right)\).


Step 3: Detailed Explanation:

Circuit components: \(V = 90\) V, \(R_1 = 100\Omega\), \(R_2 = 100\Omega\), Voltmeter resistance \(R_v = 400\Omega\).

The voltmeter is connected across one resistor (say \(R_1\)).

Equivalent resistance of the voltmeter and \(R_1\):
\[ R_p = \frac{100 \times 400}{100 + 400} = \frac{40000}{500} = 80\Omega \]

Total resistance of the circuit now:
\[ R_{total} = R_p + R_2 = 80 + 100 = 180\Omega \]

Current from the cell:
\[ I = \frac{V}{R_{total}} = \frac{90}{180} = 0.5 A \]

The voltmeter reading is the voltage across the parallel combination:
\[ V_{reading} = I \times R_p = 0.5 \times 80 = 40 V \]


Step 4: Final Answer:

The reading of the voltmeter will be 40 V.
Quick Tip: If the voltmeter were ideal (infinite resistance), it would read \(45\) V. Since a real voltmeter draws some current, it always reads slightly lower than the ideal value for this configuration.

Step 1: Understanding the Concept:

The magnetic force on a current-carrying straight wire segment is determined by the length of the wire, the current, the magnetic field, and the angle between the wire and the field.


Step 2: Key Formula or Approach:

Force magnitude: \(F = I L B \sin\theta\).


Step 3: Detailed Explanation:

Given a right triangle with sides \(a=5, b=12, c=13\). Since \(5^2 + 12^2 = 13^2\), the angle opposite the 13 cm side is \(90^\circ\).

The magnetic field \(\vec{B}\) is parallel to the 13 cm hypotenuse.

We need the force on the 5 cm side. Let the angle between the 5 cm side and the 13 cm side be \(\theta\).

From the triangle properties:
\[ \sin\theta = \frac{opposite side}{hypotenuse} = \frac{12}{13} \]

Now, calculate the force on the 5 cm side (\(L = 0.05\) m):
\[ F = I L B \sin\theta \]
\[ F = 2 \times 0.05 \times 0.75 \times \frac{12}{13} \]
\[ F = 0.1 \times 0.75 \times \frac{12}{13} = 0.075 \times \frac{12}{13} \]
\[ F = \frac{0.9}{13} = \frac{9}{130} N \]

Comparing with the given form \(\frac{x}{130}\), we get \(x = 9\).


Step 4: Final Answer:

The value of \(x\) is 9.
Quick Tip: For right triangles, if the field is parallel to the hypotenuse, the force on one leg involves the sine of the angle which is just the ratio of the other leg to the hypotenuse.

Step 1: Understanding the Concept:

When a body falls through a viscous fluid and reaches terminal velocity (constant velocity), the net force on it is zero. This means the viscous force plus buoyant force balances the weight.


Step 2: Key Formula or Approach:

At terminal velocity: \(F_{viscous} = W - F_b = V\rho_s g - V\rho_l g = Vg(\rho_s - \rho_l)\).


Step 3: Detailed Explanation:

Given:

Radius \(r = 1 mm = 10^{-3} m\).

Density of sphere \(\rho_s = 10.5 g/cc = 10500 kg/m^3\).

Density of liquid \(\rho_l = 1.5 g/cc = 1500 kg/m^3\).

Viscous force \(F\):
\[ F = \frac{4}{3} \pi r^3 g (\rho_s - \rho_l) \]
\[ F = \frac{4}{3} \times \frac{22}{7} \times (10^{-3})^3 \times 9.8 \times (10500 - 1500) \]
\[ F = \frac{4}{3} \times \frac{22}{7} \times 10^{-9} \times 9.8 \times 9000 \]
\[ F = \frac{4}{3} \times 22 \times 10^{-9} \times 1.4 \times 9000 \] (since \(9.8/7 = 1.4\))
\[ F = 4 \times 22 \times 1.4 \times 10^{-9} \times 3000 \]
\[ F = 88 \times 4.2 \times 10^{-6} = 369.6 \times 10^{-6} N \]

We need the form \(3696 \times 10^{-x}\):
\[ 369.6 \times 10^{-6} = 3696 \times 10^{-7} \]

Thus, \(x = 7\).


Step 4: Final Answer:

The value of \(x\) is 7.
Quick Tip: Avoid calculating terminal velocity first unless asked. Use the equilibrium condition (Weight = Buoyancy + Viscous Force) to find the force directly from the volumes and densities.

Step 1: Understanding the Concept:

The focal length of a lens depends on the relative refractive index of the lens material with respect to the surrounding medium.


Step 2: Key Formula or Approach:

Lens Maker's Formula: \(\frac{1}{f} = \left( \frac{\mu_{lens}}{\mu_{med}} - 1 \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)\).


Step 3: Detailed Explanation:

In air (\(\mu_{med} = 1\)):
\[ \frac{1}{18} = (1.5 - 1)K = 0.5K \implies K = \frac{1}{9} \]

In water (\(\mu_{med} = 4/3\)):
\[ \frac{1}{f_w} = \left( \frac{1.5}{4/3} - 1 \right)K = \left( \frac{4.5}{4} - 1 \right)K = \left( \frac{9}{8} - 1 \right)K \]
\[ \frac{1}{f_w} = \frac{1}{8} K = \frac{1}{8} \times \frac{1}{9} = \frac{1}{72} \]

So, \(f_w = 72\) cm.

The change in focal length \(\Delta f\):
\[ \Delta f = f_w - f_{air} = 72 - 18 = 54 cm \]


Step 4: Final Answer:

The change in focal length is 54 cm.
Quick Tip: A useful shortcut: For a glass lens (\(\mu=1.5\)) immersed in water (\(\mu=1.33\)), the focal length always becomes approximately 4 times its focal length in air (\(f_w \approx 4f_a\)).

Step 1: Understanding the Concept:

Power is defined as the dot product of the force acting on a body and its instantaneous velocity.


Step 2: Key Formula or Approach:
\(P = \vec{F} \cdot \vec{v}\).
\(\vec{a} = \frac{\vec{F}}{m}\).
\(\vec{v} = \int \vec{a} \, dt\).


Step 3: Detailed Explanation:

Given mass \(m = 1\) kg and \(\vec{F} = (t\hat{i} + 3t^2\hat{j})\).

Acceleration:
\[ \vec{a} = \frac{\vec{F}}{m} = (t\hat{i} + 3t^2\hat{j}) m/s^2 \]

Velocity (assuming starting from rest):
\[ \vec{v} = \int \vec{a} \, dt = \int (t\hat{i} + 3t^2\hat{j}) \, dt = \frac{t^2}{2}\hat{i} + t^3\hat{j} \]

Now, find the power at \(t = 2s\):

At \(t = 2s\), Force \(\vec{F} = 2\hat{i} + 3(2^2)\hat{j} = 2\hat{i} + 12\hat{j}\).

At \(t = 2s\), Velocity \(\vec{v} = \frac{2^2}{2}\hat{i} + 2^3\hat{j} = 2\hat{i} + 8\hat{j}\).

Power \(P = \vec{F} \cdot \vec{v}\):
\[ P = (2 \times 2) + (12 \times 8) = 4 + 96 = 100 W \]


Step 4: Final Answer:

The power developed at \(t=2s\) is 100 W.
Quick Tip: For variable force, remember that power is also \(P = \frac{d}{dt}(K.E.)\). You can find kinetic energy \(K = \frac{1}{2}mv^2\) as a function of time and then differentiate it to get power.

Step 1: Understanding the Concept:

The total energy released in a nuclear fission process is the product of the energy released per single fission event and the total number of nuclei undergoing fission.


Step 2: Key Formula or Approach:

1. Number of moles (\(n\)) = \(\frac{Given mass (m)}{Atomic mass (M)}\).

2. Number of nuclei (\(N\)) = \(n \times N_A\).

3. Total energy (\(E_{total}\)) = \(N \times E_{fission}\).


Step 3: Detailed Explanation:

Given:

Mass of \(^{240}X\), \(m = 120\) g.

Atomic mass of \(X\), \(M = 240\) g/mol.

Energy per fission, \(E_{fission} = 200\) MeV.

Avogadro number, \(N_A = 6 \times 10^{23}\) atoms/mol.


Calculate the number of moles:
\[ n = \frac{120}{240} = 0.5 mol \]

Calculate the number of nuclei:
\[ N = 0.5 \times 6 \times 10^{23} = 3 \times 10^{23} nuclei \]

Calculate the total energy released:
\[ E_{total} = (3 \times 10^{23}) \times 200 MeV \]
\[ E_{total} = 600 \times 10^{23} MeV = 6 \times 10^{25} MeV \]


Step 4: Final Answer:

Comparing this with the form \(x \times 10^{25}\) MeV, we find the value of \(x\) is 6.
Quick Tip: In nuclear physics problems, always ensure that mass is divided by atomic mass to get moles. Remember \(1 mole = 6 \times 10^{23}\) particles.

Step 1: Understanding the Concept:

The capacitance of a parallel plate capacitor depends on the area of plates, the distance between them, and the dielectric constant of the material filling the gap.


Step 2: Key Formula or Approach:

Capacitance with air: \(C_0 = \frac{\epsilon_0 A}{d}\).

Capacitance with dielectric \(K\) and new distance \(d'\): \(C = \frac{K \epsilon_0 A}{d'}\).


Step 3: Detailed Explanation:

Given:

Initial capacitance, \(C_0 = 15\) pF.

New distance, \(d' = 2d\).

Dielectric constant, \(K = 3.5\).


New capacitance \(C\):
\[ C = \frac{K \epsilon_0 A}{d'} = \frac{3.5 \epsilon_0 A}{2d} \]

Substituting \(\frac{\epsilon_0 A}{d} = C_0 = 15\):
\[ C = \frac{3.5}{2} \times 15 = 1.75 \times 15 = 26.25 pF \]

The question states the new capacitance is \(\frac{x}{4}\) pF:
\[ \frac{x}{4} = 26.25 \]
\[ x = 26.25 \times 4 = 105 \]


Step 4: Final Answer:

The value of \(x\) is 105.
Quick Tip: Capacitance increases with the dielectric constant (\(K\)) and decreases as the separation distance (\(d\)) increases. \(C \propto \frac{K}{d}\).

Step 1: Understanding the Concept:

The time period of a mass-spring system in Simple Harmonic Motion (SHM) is directly proportional to the square root of the attached mass.


Step 2: Key Formula or Approach:

Time period, \(T = 2\pi \sqrt{\frac{m}{k}}\), where \(k\) is the spring constant.


Step 3: Detailed Explanation:

Case 1: \(T_1 = 1\) s for mass \(m\).
\[ 1 = 2\pi \sqrt{\frac{m}{k}} \quad ---(i) \]

Case 2: Mass becomes \((m+3)\), and the time period increases by 1 s, so \(T_2 = 1 + 1 = 2\) s.
\[ 2 = 2\pi \sqrt{\frac{m+3}{k}} \quad ---(ii) \]

Dividing equation (ii) by equation (i):
\[ \frac{2}{1} = \frac{2\pi \sqrt{\frac{m+3}{k}}}{2\pi \sqrt{\frac{m}{k}}} = \sqrt{\frac{m+3}{m}} \]

Squaring both sides:
\[ 4 = \frac{m+3}{m} \]
\[ 4m = m+3 \]
\[ 3m = 3 \implies m = 1 kg \]


Step 4: Final Answer:

The initial mass \(m\) is 1 kg.
Quick Tip: Since \(T \propto \sqrt{m}\), doubling the time period requires the mass to be quadrupled. \(4m = m + 3\) leads directly to the answer.

Step 1: Understanding the Concept:

The moment of inertia of a solid cylinder about its geometric axis depends on its mass and the square of its radius.


Step 2: Key Formula or Approach:

Moment of Inertia of a solid cylinder, \(I = \frac{1}{2} M R^2\).

Mass, \(M = Density (\rho) \times Volume (V) = \rho \pi R^2 L\).


Step 3: Detailed Explanation:

Let \(\rho\) be the density of the cylinder material.

For the original cylinder:
\(M_1 = \rho \pi R^2 L\)
\(I_1 = \frac{1}{2} M_1 R^2 = \frac{1}{2} (\rho \pi R^2 L) R^2 = \frac{1}{2} \rho \pi R^4 L\).


For the carved-out cylinder:

Radius, \(R' = R/2\)

Length, \(L' = L/2\)

Mass, \(M_2 = \rho \pi (R/2)^2 (L/2) = \rho \pi \frac{R^2}{4} \frac{L}{2} = \frac{1}{8} (\rho \pi R^2 L) = \frac{M_1}{8}\).

Moment of Inertia \(I_2\):
\(I_2 = \frac{1}{2} M_2 (R')^2 = \frac{1}{2} (\frac{M_1}{8}) (\frac{R}{2})^2\)
\(I_2 = \frac{1}{2} \cdot \frac{M_1}{8} \cdot \frac{R^2}{4} = \frac{1}{32} (\frac{1}{2} M_1 R^2) = \frac{I_1}{32}\).


Calculating the ratio:
\[ \frac{I_1}{I_2} = 32 \]


Step 4: Final Answer:

The ratio \(\frac{I_1}{I_2}\) is 32.
Quick Tip: For same density, \(I \propto R^4 L\). Here \(R\) is halved (\(\frac{1}{16}\) factor) and \(L\) is halved (\(\frac{1}{2}\) factor). Total factor is \(\frac{1}{16 \times 2} = \frac{1}{32}\).

Step 1: Understanding the Concept:

When a wire is stretched, its volume remains constant. An increase in length is accompanied by a decrease in cross-sectional area.


Step 2: Key Formula or Approach:

Resistance, \(R = \rho \frac{l}{A}\).

Volume, \(V = A \times l = constant\).

Therefore, \(A = \frac{V}{l}\), which gives \(R = \rho \frac{l^2}{V} \implies R \propto l^2\).


Step 3: Detailed Explanation:

Initial length = \(l_1\).

Final length, \(l_2 = l_1 + 20% of l_1 = 1.2 l_1\).

Since \(R \propto l^2\):
\[ \frac{R_2}{R_1} = \left( \frac{l_2}{l_1} \right)^2 = (1.2)^2 = 1.44 \]

Percentage increase in resistance:
\[ % Increase = \frac{R_2 - R_1}{R_1} \times 100 = (1.44 - 1) \times 100 = 44% \]


Step 4: Final Answer:

The percentage increase in resistance is 44%.
Quick Tip: For a small percentage increase (\(<5%\)), use \(\Delta R/R \approx 2 \Delta l/l\). For larger changes, always use the square factor: \((1 + n/100)^2 - 1\).

Step 1: Understanding the Concept:

"Long after the switch is closed" refers to the steady-state condition. In steady state, inductors act as ideal wires (zero resistance) because the current is no longer changing (\(\frac{di}{dt} = 0\)).


Step 2: Key Formula or Approach:

1. In steady state, \(V_L = L \frac{di}{dt} = 0\). Replace inductors with short circuits.

2. Calculate the equivalent resistance (\(R_{eq}\)) of the remaining network.

3. Use Ohm's Law: \(I = \frac{V}{R_{eq}}\).


Step 3: Detailed Explanation:

In the circuit diagram:

- There are three parallel branches across the 12 V battery.

- Branch 1 contains an inductor \(L\) and a resistor \(R\). In steady state, it is just \(R = 12 \Omega\).

- Branch 2 contains a resistor \(R = 12 \Omega\).

- Branch 3 contains an inductor \(L\) and a resistor \(R\). In steady state, it is just \(R = 12 \Omega\).

The equivalent resistance of three \(12 \Omega\) resistors in parallel:
\[ \frac{1}{R_{eq}} = \frac{1}{12} + \frac{1}{12} + \frac{1}{12} = \frac{3}{12} = \frac{1}{4} \implies R_{eq} = 4 \Omega \]

Current through the battery:
\[ I = \frac{V}{R_{eq}} = \frac{12}{4} = 3 A \]


Step 4: Final Answer:

The current is 3 A.
Quick Tip: Always remember: At \(t=0\), inductors are open circuits; at \(t \to \infty\), they are short circuits. Capacitors are exactly the opposite!