Step 1: Understanding the Concept:

The inquiry centers on identifying which of Maxwell's fundamental equations of electromagnetism logically precludes the existence of magnetic monopoles. A magnetic monopole would be an isolated magnetic pole, either a north or a south, analogous to an isolated electric charge.


Step 2: Detailed Explanation:

An examination of the provided options, each representing one of Maxwell's equations, is necessary:


\( \nabla \cdot \vec{B} = 0 \): This equation is known as Gauss's law for magnetism. It mathematically asserts that the divergence of the magnetic field \( \vec{B} \) is invariably zero. Physically, this signifies that there are no points in space that act as sources or sinks for magnetic field lines. Consequently, magnetic field lines must form continuous, closed loops, never originating from or terminating at a single point. A hypothetical magnetic monopole would function as a source (a north pole) or a sink (a south pole), which would necessitate a non-zero divergence. Thus, the equation \( \nabla \cdot \vec{B} = 0 \) serves as the formal declaration of the non-existence of magnetic monopoles.
\( \nabla \cdot \vec{D} = \rho \): This is Gauss's law for electricity, where \( \vec{D} \) is the electric displacement field and \( \rho \) represents the density of free electric charges. This law explains that electric field lines begin on positive charges and end on negative charges, thereby describing the existence of electric monopoles (charges), not addressing magnetic ones.
\( \nabla \cdot \vec{E} = 0 \): This is a specialized version of Gauss's law for electricity, applicable only to regions of space where there is no net electric charge.
\( \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} \): This equation is Faraday's law of induction. It articulates how a magnetic field that changes over time induces a circulating electric field. It is not concerned with the existence or non-existence of magnetic monopoles.


Step 3: Final Answer:

The equation \( \nabla \cdot \vec{B} = 0 \) is the one that directly establishes the absence of magnetic monopoles, as it requires the total magnetic flux passing through any closed surface to be zero.
Quick Tip: Remember the physical meaning of divergence: it measures the "outflow" of a vector field from a point. Zero divergence for the magnetic field means no point sources or sinks, hence no monopoles. In contrast, the divergence of the electric field is proportional to charge density, indicating that charges are the sources/sinks of the electric field.

Step 1: Understanding the Concept:

The magnetic energy within a system is stored in the magnetic field that the system generates. Our task is to determine the magnetic field present in the space between the two cylinders of the coaxial cable. Subsequently, we must integrate the density of this magnetic energy throughout the entire volume of that region to find the total stored energy.


Step 2: Key Formula or Approach:

The total magnetic energy, denoted as \( U_m \), contained within a specific volume \( V \) is calculated by integrating the magnetic energy density, \( u_m \), over that volume:
\[ U_m = \int_V u_m dV \]
The energy density itself is defined by the strength of the magnetic field \( B \):
\[ u_m = \frac{B^2}{2\mu_0} \]
To determine the magnetic field \( B \), we will employ Ampere's law, which is suitable for this symmetric current distribution.


Step 3: Detailed Explanation:

1. Determine the magnetic field (B):

We construct a circular Amperian loop with a radius \( r \) that is positioned between the inner and outer cylinders, i.e., \( r_1 < r < r_2 \). The total current enclosed by this loop is simply 'I'. Applying Ampere's Law:
\[ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} \]
For our circular loop, this simplifies to: \[ B \cdot (2\pi r) = \mu_0 I \]
Solving for \( B \) gives: \[ B = \frac{\mu_0 I}{2\pi r} \]
It is important to note that the magnetic field is non-existent for \( r < r_1 \) (inside the inner cylinder) and for \( r > r_2 \) (outside the outer cylinder) because the net enclosed current is zero in those regions.


2. Compute the magnetic energy density (\( u_m \)):

Substituting the expression for B into the energy density formula:
\[ u_m = \frac{B^2}{2\mu_0} = \frac{1}{2\mu_0} \left( \frac{\mu_0 I}{2\pi r} \right)^2 = \frac{\mu_0^2 I^2}{2\mu_0 (4\pi^2 r^2)} = \frac{\mu_0 I^2}{8\pi^2 r^2} \]

3. Integrate to obtain the total magnetic energy (\( U_m \)):

The next step is to integrate this energy density over the volume spanning the region between the cylinders for a given length 'L'. In cylindrical coordinates, a differential volume element \( dV \) can be represented as a thin cylindrical shell of radius \( r \), thickness \( dr \), and length \( L \), so \( dV = (2\pi r) L dr \).
\[ U_m = \int_{r_1}^{r_2} u_m dV = \int_{r_1}^{r_2} \left( \frac{\mu_0 I^2}{8\pi^2 r^2} \right) (2\pi r L dr) \]
Simplifying the expression before integrating: \[ U_m = \frac{\mu_0 I^2 L}{4\pi} \int_{r_1}^{r_2} \frac{1}{r} dr \]
Performing the integration: \[ U_m = \frac{\mu_0 I^2 L}{4\pi} \left[\ln(r)\right]_{r_1}^{r_2} \] \[ U_m = \frac{\mu_0 I^2 L}{4\pi} (\ln(r_2) - \ln(r_1)) \]

Step 4: Final Answer:

By applying the logarithmic identity \( \ln(a) - \ln(b) = \ln(a/b) \), the final expression becomes:
\[ U_m = \frac{\mu_0 I^2 L}{4\pi} \ln\left(\frac{r_2}{r_1}\right) \]
This result corresponds to option (C).
Quick Tip: For problems involving energy in fields (electric or magnetic), the standard procedure is: 1. Find the field (\(\vec{E}\) or \(\vec{B}\)) using Gauss's Law or Ampere's Law. 2. Calculate the energy density (\(u_E = \frac{1}{2}\epsilon_0 E^2\) or \(u_m = \frac{1}{2\mu_0} B^2\)). 3. Integrate the energy density over the relevant volume.

Step 1: Understanding the Concept:

Within magnetic materials, the alignment of countless microscopic magnetic dipoles, which arise from the intrinsic spin and orbital motion of electrons, produces a large-scale magnetic effect known as magnetization (\( \vec{M} \)). This collective magnetization can be effectively modeled as equivalent macroscopic currents. These include a bound volume current (\( \vec{J}_b \)) flowing inside the material and a bound surface current (\( \vec{K}_b \)) on its boundary. They are termed "bound" because they originate from the movement of charges that are confined to their respective atoms, in contrast to free charges that constitute a conduction current.


Step 2: Detailed Explanation:

Let us assess the validity of each provided statement:


(A) which is associated with magnetization of the material. This statement is accurate. Bound currents are a direct manifestation of a material's magnetization \( \vec{M} \). They are a theoretical construct used to simplify the calculation of the magnetic field generated by the magnetized material.

(B) which involves spin and orbital motion of electrons. This is also accurate. The microscopic source of magnetization \( \vec{M} \) lies in the quantum mechanical properties of electrons—specifically, their spin magnetic dipole moment and orbital magnetic dipole moment. The organized alignment of these atomic-level dipoles creates the macroscopic magnetization, which is then described by the bound currents.

(C) which is the result of linear motion of charge when electric polarization changes. This statement is incorrect. A time-varying electric polarization \( \vec{P} \) induces a different type of current known as the polarization current density, defined as \( \vec{J}_P = \frac{\partial \vec{P}}{\partial t} \). This is fundamentally an electric phenomenon, distinct from the bound currents associated with magnetism.

(D) which is given by \( \nabla \times \vec{M} \) (where \( \vec{M} \) is magnetization). This is a correct mathematical definition. This expression defines the bound volume current density \( \vec{J}_b \). A spatially varying magnetization (specifically, one with a non-zero curl) within the material results in a net effective flow of current. The corresponding bound surface current is defined by \( \vec{K}_b = \vec{M} \times \hat{n} \), with \( \hat{n} \) being the unit vector normal to the surface.


Step 3: Final Answer:

Statements (A), (B), and (D) provide correct descriptions and definitions concerning bound magnetic currents. Statement (C), however, pertains to polarization current, an unrelated concept. Thus, the correct selection is the combination of (A), (B), and (D).
Quick Tip: Remember the key relationships for bound currents: Volume bound current: \( \vec{J}_b = \nabla \times \vec{M} \) Surface bound current: \( \vec{K}_b = \vec{M} \times \hat{n} \) Distinguish these from electric polarization concepts: Volume bound charge: \( \rho_b = -\nabla \cdot \vec{P} \) Surface bound charge: \( \sigma_b = \vec{P} \cdot \hat{n} \)

Step 1: Understanding the Concept:

This question asks to correctly pair fundamental terms and principles from electromagnetism (List-I) with their corresponding mathematical expressions (List-II). The goal is to accurately identify the formula for each concept.


Step 2: Detailed Explanation:


(A) Displacement current (\(J_d\)): The displacement current density, \( \vec{J}_d \), was introduced by Maxwell to generalize Ampere's law. It is defined by the expression \( \vec{J}_d = \epsilon_0 \frac{\partial \vec{E}}{\partial t} \). This crucial term reveals that a magnetic field can be generated not only by moving charges but also by an electric field that changes with time. This correctly matches with (IV).

(B) Poynting vector (\(\vec{S}\)): The Poynting vector describes the magnitude and direction of the flow of energy in an electromagnetic field. It represents the energy flux density, or the power transferred per unit area. Its definition is \( \vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B}) \). This correctly matches with (III).

(C) Energy stored in electric field (\(\vec{E}\)): An electric field contains stored energy. The density of this energy (energy per unit volume) is given by \( u_E = \frac{1}{2}\epsilon_0 E^2 \). To find the total energy \( U_E \) within a given volume \( \tau \), one must integrate this density function over that volume, leading to the expression \( U_E = \int u_E d\tau = \int \frac{1}{2}\epsilon_0 E^2 d\tau = \frac{\epsilon_0}{2} \int E^2 d\tau \). This correctly matches with (I).

(D) Gauss's Law: As one of the four pillars of Maxwell's equations, Gauss's law establishes the relationship between an electric field and the electric charges that create it. The law states that the divergence of the electric field is proportional to the local charge density \( \rho \). In its differential form, it is written as \( \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} \). This correctly matches with (II).


Step 3: Final Answer:

Following the detailed analysis, the correct associations are as follows:

(A) is paired with (IV)
(B) is paired with (III)
(C) is paired with (I)
(D) is paired with (II)

This combination corresponds to option (B).
Quick Tip: For matching questions, it's often efficient to identify the one or two pairings you are most confident about first. For example, Gauss's Law (\( \nabla \cdot \vec{E} = \rho/\epsilon_0 \)) is a very standard definition. Finding this match (D -> II) can help eliminate incorrect options quickly.

Step 1: Understanding the Concept:

The thermal voltage, symbolized as \( V_T \), is a fundamental parameter in the study of semiconductor devices. It quantifies the amount of thermal energy carried per unit of elementary charge at a given temperature and is directly proportional to the absolute temperature of the system.


Step 2: Key Formula or Approach:

The mathematical definition for the thermal voltage is:
\[ V_T = \frac{kT}{q} \]
where the variables are:
\( k \): Boltzmann's constant, which is \( 1.38 \times 10^{-23} \) J/K
\( T \): The absolute temperature measured in Kelvin (K)
\( q \): The magnitude of the charge of a single electron, which is \( 1.602 \times 10^{-19} \) C


Step 3: Detailed Explanation:

1. Convert the Given Temperature to the Absolute Scale (Kelvin):

The problem provides the temperature as 47°C. The first essential step is to convert this to Kelvin by adding 273.15.
\[ T(in Kelvin) = 47 + 273.15 = 320.15 \, K \]

2. Substitute the Known Values into the Thermal Voltage Formula:

Now, we insert the values for \(k\), \(T\), and \(q\) into the equation.
\[ V_T = \frac{(1.38 \times 10^{-23} \, J/K) \times (320.15 \, K)}{1.602 \times 10^{-19} \, C} \]
First, calculate the numerator: \[ V_T = \frac{4.41807 \times 10^{-21}}{1.602 \times 10^{-19}} \, V \]
Then, perform the division: \[ V_T \approx 0.02758 \, V \]

3. Express the Result in Millivolts (mV):

The calculated voltage is in Volts. To convert it to millivolts, we multiply the value by 1000.
\[ V_T \approx 0.02758 \times 1000 \, mV \] \[ V_T \approx 27.58 \, mV \]

Step 4: Final Answer:

The computed thermal voltage is approximately 27.6 mV, which aligns with option (A).
Quick Tip: A useful shortcut for competitive exams is to remember the thermal voltage at room temperature (\(T \approx 300 \, K\) or \(27^\circ C\)). At 300 K, \(V_T \approx 25.85 \, mV \approx 26 \, mV\). You can use this as a benchmark to quickly estimate the answer for other temperatures. Since 47°C is slightly warmer than room temperature, the thermal voltage should be slightly higher than 26 mV.

Step 1: Understanding the Concept:

The Peak Inverse Voltage (PIV) represents the maximum possible reverse-bias voltage that a diode in a rectifier circuit will experience. It is a critical parameter because if this voltage is exceeded, the diode may suffer from reverse breakdown and be permanently damaged. In a half-wave rectifier, this maximum stress occurs during the negative half-cycle of the AC input, when the diode is blocking current flow. The voltage across the diode at this point is the peak voltage of the AC signal supplied by the transformer's secondary coil.


Step 2: Key Formula or Approach:

The solution involves a three-step process:
1. Determine the RMS voltage on the secondary side of the transformer (\(V_{s,rms}\)).
2. Convert this RMS voltage to the peak voltage on the secondary side (\(V_{s,peak}\)).
3. Recognize that for a half-wave rectifier, the PIV is equal to this peak secondary voltage.


The necessary equations are:

Transformer voltage relation: \( \frac{V_s}{V_p} = \frac{N_s}{N_p} \), which gives \( V_{s,rms} = V_{p,rms} \times \frac{N_s}{N_p} \)

Peak vs. RMS voltage for a sine wave: \( V_{peak} = V_{rms} \times \sqrt{2} \)


Step 3: Detailed Explanation:

1. Calculate the secondary RMS voltage (\(V_{s,rms}\)):

We are given the primary RMS voltage, \(V_{p,rms} = 220\) V.
The turns ratio is specified as 20:1, meaning the primary-to-secondary turns ratio is \( \frac{N_p}{N_s} = \frac{20}{1} \). This is a step-down transformer. \[ V_{s,rms} = V_{p,rms} \times \frac{N_s}{N_p} = 220 \, V \times \frac{1}{20} = 11 \, V \]

2. Calculate the secondary peak voltage (\(V_{s,peak}\)):

The relationship between the peak and RMS values for a sinusoidal AC voltage allows us to find the maximum instantaneous voltage.
\[ V_{s,peak} = V_{s,rms} \times \sqrt{2} \]
Using the approximation \( \sqrt{2} \approx 1.414 \): \[ V_{s,peak} = 11 \, V \times 1.414 \approx 15.554 \, V \]

3. Identify the Peak Inverse Voltage (PIV):

For a half-wave rectifier circuit, during the non-conducting half-cycle, the diode acts like an open switch. The entire secondary voltage appears across its terminals, reaching a maximum value at the peak of the sine wave.
\[ PIV = V_{s,peak} \approx 15.554 \, V \]

Step 4: Final Answer:

The maximum reverse voltage the diode must withstand is 15.554 V. This aligns with option (D).
Quick Tip: Always be careful whether the given AC voltage is an RMS value or a peak value. In standard practice (like mains supply), the value given (e.g., 220V, 110V) is always the RMS value unless specified otherwise. Remember PIV values for standard rectifiers: Half-wave: PIV = \(V_{s,peak}\) Full-wave (center-tapped): PIV = \(2V_{s,peak}\) (where \(V_{s,peak}\) is across half the secondary) Full-wave (bridge): PIV = \(V_{s,peak}\)

Step 1: Understanding the Concept:

This question requires identifying the defining characteristic, primary function, or a key associated property for several common semiconductor devices.


Step 2: Detailed Explanation:

We will examine each device from List-I and determine its most appropriate match from List-II.


(A) Zener diode: This diode is engineered to operate reliably in its reverse breakdown region. A key feature is that once the Zener voltage is reached, the voltage across the diode remains almost constant even as the reverse current changes significantly. This stability makes it an excellent component for a voltage regulator circuit. Therefore, (A) correctly pairs with (II).

(B) Tunnel diode: Constructed from very heavily doped semiconductor material, the tunnel diode's operation is governed by a quantum mechanical phenomenon called tunneling. Its most distinctive feature is a segment in its forward I-V characteristic where an increase in voltage leads to a decrease in current. This is termed a negative resistance region and is exploited in the construction of high-frequency electronic oscillators. Thus, (B) matches with (I).

(C) Rectifier: The fundamental purpose of a rectifier circuit is to convert alternating current (AC) into direct current (DC). However, the raw output from a basic rectifier is not a constant DC level. Instead, it is a pulsating d.c. waveform that still contains significant ripple from the original AC signal. This output must be filtered to become smooth DC. So, (C) is correctly associated with (III).

(D) Light emitting diode (LED): An LED is a p-n junction diode that emits photons (light) when it is forward-biased, a process known as electroluminescence. The specific compound semiconductor material used in its fabrication determines the wavelength, and therefore the color, of the emitted light. Gallium Arsenide Phosphide (GaAsP) is a ternary semiconductor compound commonly used to produce LEDs in the red to yellow part of the spectrum. Hence, (D) matches with (IV).


Step 3: Final Answer:

The correct pairings are determined to be:

(A) with (II)
(B) with (I)
(C) with (III)
(D) with (IV)

This set of matches corresponds to option (A).
Quick Tip: Associate each electronic component with its key unique feature: Zener Diode \(\rightarrow\) Reverse Breakdown, Voltage Regulation Tunnel Diode \(\rightarrow\) Heavy Doping, Negative Resistance Rectifier \(\rightarrow\) AC to DC conversion, Pulsating DC Output LED \(\rightarrow\) Light Emission, specific semiconductor materials for colors (e.g., GaAsP, GaN)

Step 1: Understanding the Concept:

In an ideal Bipolar Junction Transistor (BJT), no current would flow if there is no base current. However, in reality, a small leakage current exists, primarily due to thermally generated minority carriers. This leakage current is specified differently depending on the transistor configuration.

\( I_{CEO} \): This represents the Collector-to-Emitter leakage current when the base terminal is left open (unconnected). It is relevant to the Common Emitter (CE) configuration.
\( I_{CBO} \): This represents the Collector-to-Base leakage current when the emitter terminal is open. It is relevant to the Common Base (CB) configuration.

A fundamental relationship exists between these two parameters, linked by the transistor's current gain.


Step 2: Key Formula or Approach:

The connection between the common emitter leakage current (\( I_{CEO} \)) and the common base leakage current (\( I_{CBO} \)) is established by the following equation:
\[ I_{CEO} = (\beta + 1) I_{CBO} \]
Here, \(\beta\) (beta) is the DC current gain in the common emitter configuration. The problem provides \( I_{CEO} \) and \(\beta\), and our goal is to determine \( I_{CBO} \).


Step 3: Detailed Explanation:

1. List the provided values:

The leakage current in the common emitter connection is \( I_{CEO} = 300 \, \mu A \).

The common emitter current gain is \( \beta = 130 \).


2. Isolate the desired variable, \( I_{CBO} \), by rearranging the formula:
\[ I_{CBO} = \frac{I_{CEO}}{\beta + 1} \]

3. Substitute the numerical values and compute the result:
\[ I_{CBO} = \frac{300 \, \mu A}{130 + 1} \] \[ I_{CBO} = \frac{300}{131} \, \mu A \] \[ I_{CBO} \approx 2.290 \, \mu A \]

Step 4: Final Answer:

The calculated value for the leakage current in the common base arrangement is approximately 2.29 \(\mu\)A. This value is closest to the 2.28 \(\mu\)A given in option (A).
Quick Tip: Remember that the leakage current in the common emitter configuration (\(I_{CEO}\)) is always significantly larger than in the common base configuration (\(I_{CBO}\)) by a factor of (\(\beta+1\)). This is because the small \(I_{CBO}\) is amplified by the transistor action in the CE setup.

Step 1: Understanding the Concept:

The decibel (dB) is a logarithmic unit that expresses the ratio between two values of a physical quantity, most commonly power or intensity. The use of a logarithmic scale is advantageous for representing very wide-ranging quantities on a more compressed scale and aligns well with the logarithmic nature of human perception for senses like hearing and sight.


Step 2: Key Formula or Approach:

The equation to convert a power gain, which is expressed as a ratio \( A_p = P_{out}/P_{in} \), into its decibel equivalent is:
\[ G_{dB} = 10 \log_{10}(A_p) \]
It is important to note that for quantities like voltage gain (\( A_v \)) or current gain (\( A_i \)), the formula uses a factor of 20, i.e., \( 20 \log_{10}(A_v) \), due to power being proportional to the square of voltage or current.


Step 3: Detailed Explanation:

1. State the given information:

We are provided with a power gain ratio, \( A_p = 100 \).


2. Apply the decibel conversion formula for power:

Substitute the given power gain into the equation.
\[ G_{dB} = 10 \log_{10}(100) \]

3. Compute the value of the logarithm:

The base-10 logarithm of 100 is the power to which 10 must be raised to get 100. Since \( 10^2 = 100 \), we have \( \log_{10}(100) = 2 \).


4. Perform the final calculation:
\[ G_{dB} = 10 \times 2 = 20 \, dB \]

Step 4: Final Answer:

A power gain of 100 corresponds to a level of 20 dB.
Quick Tip: Memorize some common logarithmic values for quick calculations: \(\log_{10}(1) = 0\) \(\implies\) 0 dB gain (no change) \(\log_{10}(2) \approx 0.3\) \(\implies\) 3 dB gain (power doubles) \(\log_{10}(10) = 1\) \(\implies\) 10 dB gain (power increases by 10x) \(\log_{10}(100) = 2\) \(\implies\) 20 dB gain (power increases by 100x) For every factor of 10 increase in power gain, you add 10 dB.

Step 1: Understanding the Concept:

This problem deals with the concept of relative velocity within the framework of classical mechanics, specifically using the Galilean transformation for velocities. We are provided with the particle's motion in a stationary frame (S) and the motion of a second, moving frame (S') relative to S. Our objective is to determine how the particle's velocity appears to an observer situated in the moving frame S'.


Step 2: Key Formula or Approach:

The Galilean velocity transformation provides the relationship between the particle's velocity in frame S (\(\vec{v}_{p,S}\)), its velocity in frame S' (\(\vec{v}_{p,S'}\)), and the velocity of frame S' relative to frame S (\(\vec{v}_{S',S}\)):
\[ \vec{v}_{p,S} = \vec{v}_{p,S'} + \vec{v}_{S',S} \]
To find the particle's velocity in the S' frame, we need to rearrange this equation:
\[ \vec{v}_{p,S'} = \vec{v}_{p,S} - \vec{v}_{S',S} \]

Step 3: Detailed Explanation:

1. Determine the particle's velocity in the stationary frame S (\(\vec{v}_{p,S}\)):

The position of the particle in frame S is given as a function of time:
\[ \vec{r}_{p,S}(t) = (5t + 3)\hat{i} + (6t)\hat{j} + 5\hat{k} \]
Velocity is the time derivative of position. We differentiate this vector with respect to \( t \):
\[ \vec{v}_{p,S} = \frac{d\vec{r}_{p,S}}{dt} = \frac{d}{dt}\left[(5t + 3)\hat{i} + (6t)\hat{j} + 5\hat{k}\right] \] \[ \vec{v}_{p,S} = 5\hat{i} + 6\hat{j} + 0\hat{k} = (5\hat{i} + 6\hat{j}) \, m/s \]

2. State the velocity of the moving frame S' relative to S (\(\vec{v}_{S',S}\)):

This velocity is directly given in the problem statement:
\[ \vec{v}_{S',S} = (3\hat{i} + 4\hat{j}) \, m/s \]

3. Compute the particle's velocity in the moving frame S' (\(\vec{v}_{p,S'}\)):

We now apply the rearranged transformation formula by subtracting the frame's velocity from the particle's absolute velocity:
\[ \vec{v}_{p,S'} = \vec{v}_{p,S} - \vec{v}_{S',S} \] \[ \vec{v}_{p,S'} = (5\hat{i} + 6\hat{j}) - (3\hat{i} + 4\hat{j}) \]
Performing the vector subtraction component by component: \[ \vec{v}_{p,S'} = (5-3)\hat{i} + (6-4)\hat{j} \] \[ \vec{v}_{p,S'} = (2\hat{i} + 2\hat{j}) \, m/s \]

Step 4: Final Answer:

The velocity of the particle as measured from within frame S' is \( (2\hat{i} + 2\hat{j}) \) m/s. This result matches option (A).
Quick Tip: Remember the subscription notation for relative velocity: \( \vec{v}_{A,B} \) means "velocity of A with respect to B". The transformation rule can be written as \( \vec{v}_{A,C} = \vec{v}_{A,B} + \vec{v}_{B,C} \). In this problem, let A = particle, B = frame S', C = frame S. We want \( \vec{v}_{p,S'} \). We know \( \vec{v}_{p,S} \) and \( \vec{v}_{S',S} \). The relation is \( \vec{v}_{p,S} = \vec{v}_{p,S'} + \vec{v}_{S',S} \), which gives the required formula.

Step 1: Understanding the Concept:

The problem statement presents a contradiction in terms. A "perfectly elastic collision" is defined by the conservation of both momentum and kinetic energy. In contrast, a collision where the "particles stick together" is the definition of a "perfectly inelastic collision," in which kinetic energy is maximally lost (converted to other forms like heat and sound), although momentum is still conserved. Given that the question explicitly asks for the final velocity under the condition of the particles coalescing, we must proceed by analyzing it as a perfectly inelastic collision. The fundamental law that governs this scenario is the conservation of linear momentum.


Step 2: Key Formula or Approach:

The principle of conservation of linear momentum dictates that for an isolated system, the total momentum before an interaction (collision) is identical to the total momentum after the interaction.
\[ \vec{P}_{initial} = \vec{P}_{final} \]
Mathematically, for two particles, this is expressed as: \[ m_1 \vec{v}_1 + m_2 \vec{v}_2 = (m_1 + m_2) \vec{v}_f \]
Here, \( \vec{v}_f \) represents the final common velocity of the combined mass after they stick together.


Step 3: Detailed Explanation:

Let's denote the initial velocities of the particles with masses \(m_1\) and \(m_2\) as \( v_1 \) and \( v_2 \), respectively.
The total initial momentum of the system is the vector sum of the individual momenta: \( P_{initial} = m_1 v_1 + m_2 v_2 \).

Following the collision, the two particles merge into a single entity with a combined mass of \( (m_1 + m_2) \). This new entity moves with a single final velocity, \( v_f \).

The total final momentum of the system is therefore: \( P_{final} = (m_1 + m_2) v_f \).

Equating the initial and final momenta based on the conservation principle:
\[ m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f \]
To find the final velocity \( v_f \), we simply rearrange the equation by dividing by the total mass:
\[ v_f = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} \]

Step 4: Final Answer:

The final velocity of the combined mass after the particles stick together is determined by the formula for the velocity of the system's center of mass, which corresponds to option (B).
Quick Tip: Be aware of contradictory statements in questions. "Sticking together" is the defining characteristic of a perfectly inelastic collision. In such collisions, linear momentum is always conserved, but kinetic energy is lost (converted to heat, sound, etc.). The final velocity is the velocity of the center of mass of the system.

Step 1: Understanding the Concept:

This question assesses comprehension of the fundamental definitions and distinguishing characteristics of conservative and non-conservative forces, which are central concepts in the study of work and energy in physics.


Step 2: Detailed Explanation:

We will evaluate the correctness of each statement individually:


(A) For conservative and non-conservative forces conservation of energy holds good. This statement is correct. The Law of Conservation of Energy is a universal principle stating that the total energy of an isolated system remains constant. When only conservative forces act, the *mechanical energy* (sum of kinetic and potential energy) is conserved. When non-conservative forces (like friction) are present, mechanical energy is converted into other forms, such as thermal energy, but the *total* energy of the system is still conserved.

(B) Friction is a conservative force. This statement is incorrect. Friction is the archetypal example of a non-conservative force. The work done by friction is dependent on the total distance traveled (the path) and results in the dissipation of mechanical energy from the system, typically as heat.

(C) The work done by a conservative force in a closed path is zero. This statement is correct. This is a defining property of a conservative force. If an object is moved along any closed trajectory that returns it to its starting point, the net work performed by a conservative force (such as gravity or the electrostatic force) throughout this journey is exactly zero.

(D) Friction is a non-conservative force. This statement is correct. As detailed in the explanation for (B), friction is a dissipative force. The work it does depends on the path taken, not just the start and end points, which is the hallmark of a non-conservative force.


Step 3: Final Answer:

The statements (A), (C), and (D) are factually correct, while statement (B) is false. Consequently, the option that includes the combination of (A), (C), and (D) is the right answer.
Quick Tip: Key distinctions to remember: \textbf{Conservative Force}: Work is path-independent; work in a closed loop is zero; mechanical energy is conserved. Examples: Gravity, electrostatic force, ideal spring force. \textbf{Non-Conservative Force}: Work is path-dependent; work in a closed loop is not zero; mechanical energy is dissipated. Examples: Friction, air resistance, viscous drag. The Law of Conservation of *Total* Energy always holds.

Step 1: Understanding the Concept:

The problem explores the relationship between the phase difference (\(\Delta\phi\)) of two points along a traveling wave and their spatial separation, known as the path difference (\(\Delta x\)). This relationship is fundamentally linked to the wavelength (\(\lambda\)) of the wave, which serves as the spatial period. The first task is to determine the wavelength using the provided wave speed and frequency.


Step 2: Key Formula or Approach:

The solution requires two key relationships:
1. The fundamental wave equation to find the wavelength (\( \lambda \)): \( \lambda = \frac{v}{f} \), where \( v \) is the wave's velocity and \( f \) is its frequency.
2. The formula connecting phase difference (\( \Delta \phi \)) to path difference (\( \Delta x \)): \( \Delta \phi = \frac{2\pi}{\lambda} \Delta x \). This shows that a phase difference of \(2\pi\) corresponds to a path difference of one full wavelength.


Step 3: Detailed Explanation:

1. Determine the wavelength (\( \lambda \)) of the wave:

We are given \( f = 500 \) Hz and \( v = 1000 \) m/s. \[ \lambda = \frac{v}{f} = \frac{1000 \, m/s}{500 \, Hz} = 2.0 \, m \]

2. Calculate the path difference (\( \Delta x \)) corresponding to the phase difference:

The specified phase difference is \( \Delta \phi = \frac{\pi}{3} \).
We rearrange the relationship formula to solve for the path difference, \( \Delta x \): \( \Delta x = \Delta \phi \cdot \frac{\lambda}{2\pi} \). \[ \Delta x = \left(\frac{\pi}{3}\right) \times \frac{2.0 \, m}{2\pi} = \frac{1}{3} \, m \approx 0.333 \, m \]
This calculated value of 0.333 m is not among the given options. This discrepancy strongly suggests a typographical error in the problem statement. Let's investigate if a more common phase difference would yield one of the answers. A phase difference of \( \frac{\pi}{2} \) (a quarter cycle) is frequently used in such problems. Let's recalculate with this assumption.

Assuming \( \Delta \phi = \frac{\pi}{2} \): \[ \Delta x = \left(\frac{\pi}{2}\right) \times \frac{2.0 \, m}{2\pi} = \frac{2.0}{4} \, m = 0.5 \, m \]
This result perfectly matches option (D). It is therefore highly probable that the intended phase difference in the question was \( \frac{\pi}{2} \) rather than \( \frac{\pi}{3} \).


Step 4: Final Answer:

Based on the calculation that assumes a likely typo in the question (phase difference should be \( \frac{\pi}{2} \)), the distance between the two points is 0.50 m.
Quick Tip: When your calculated answer doesn't match any options, double-check your calculations. If they are correct, consider the possibility of a typo in the question. You can work backwards from the answers to see which simple change in the input data would lead to one of the options. For waves, phase differences of \( \pi/2 \), \( \pi \), and \( 2\pi \) are very common.

Step 1: Understanding the Concept:

This problem integrates two distinct physical principles: first, the phenomenon of beats, which arises from the superposition of two waves with nearly identical frequencies, and second, the physics of a sonometer, where the fundamental frequency of a vibrating string is inversely proportional to its length (assuming tension and mass per unit length are constant).


Step 2: Key Formula or Approach:

1. **Beat Frequency:** The number of beats heard per second is equal to the absolute difference between the frequencies of the two sources, \( f_1 \) and \( f_2 \).
\[ f_{beat} = |f_1 - f_2| \]
2. **Sonometer Law:** The fundamental frequency (\(f\)) of a string is given by \( f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \). When the tension (\(T\)) and linear mass density (\(\mu\)) are held constant, frequency is inversely proportional to the length (\(L\)).
\[ f \propto \frac{1}{L} \quad or \quad \frac{f_1}{f_2} = \frac{L_2}{L_1} \]

Step 3: Detailed Explanation:

Let the frequencies of the two tuning forks be designated as \( f_1 \) and \( f_2 \).
From the problem, we know the beat frequency is 6 Hz, so: \[ |f_1 - f_2| = 6 \, Hz \]
Let \( f_1 \) be the frequency corresponding to the sonometer wire of length \( L_1 = 1.5 \) m.
Let \( f_2 \) be the frequency corresponding to the sonometer wire of length \( L_2 = 2.0 \) m.
According to the sonometer principle (\(f \propto 1/L\)), the shorter wire will produce a higher frequency. Therefore, we can conclude that \( f_1 > f_2 \).
This allows us to remove the absolute value from the beat equation: \[ f_1 - f_2 = 6 \quad (Equation 1) \]
Next, we use the ratio relationship from the sonometer law: \[ \frac{f_1}{f_2} = \frac{L_2}{L_1} = \frac{2.0 \, m}{1.5 \, m} = \frac{4}{3} \]
From this, we can express \( f_1 \) in terms of \( f_2 \): \[ f_1 = \frac{4}{3} f_2 \quad (Equation 2) \]
We now have a system of two linear equations with two unknowns. We can solve this by substituting Equation 2 into Equation 1: \[ \left(\frac{4}{3} f_2\right) - f_2 = 6 \]
Factoring out \( f_2 \): \[ \left(\frac{4}{3} - 1\right) f_2 = 6 \] \[ \frac{1}{3} f_2 = 6 \]
Solving for \( f_2 \): \[ f_2 = 18 \, Hz \]
Finally, we substitute this value back into Equation 1 to find \( f_1 \): \[ f_1 = f_2 + 6 = 18 + 6 = 24 \, Hz \]

Step 4: Final Answer:

The determined frequencies of the two tuning forks are 24 Hz and 18 Hz. This result corresponds to option (C).
Quick Tip: For sonometer problems, the key relationship to remember is \( f \propto 1/L \). This means a shorter wire produces a higher pitch (higher frequency), and a longer wire produces a lower pitch. This helps in correctly setting up the beat frequency equation (\(f_{higher} - f_{lower}\)).

Step 1: Understanding the Concept:

This question requires establishing the direct correspondence between the fundamental quantities used to describe motion along a line (translational motion) and those used to describe motion around an axis (rotational motion). For nearly every concept in linear mechanics, there exists a parallel or analogous concept in rotational mechanics.


Step 2: Detailed Explanation:

We will analyze each quantity from List-I and identify its direct counterpart in List-II based on their physical roles.


(A) Force (\(\vec{F}\)): In translational motion, a force is an influence that causes a change in an object's linear velocity, resulting in linear acceleration, as described by Newton's second law, \(\vec{F} = m\vec{a}\). The rotational equivalent is Torque (\(\vec{\tau}\)), which is a rotational force that causes a change in an object's angular velocity, resulting in angular acceleration. The rotational form of Newton's second law is \(\vec{\tau} = I\vec{\alpha}\). Therefore, (A) is analogous to (I).

(B) Distance covered (s): This quantity measures the linear displacement, or the change in position of an object along a path. The rotational counterpart is the angular displacement, which is the Angle described (\(\theta\)), measuring the change in angular position of a rotating body. Therefore, (B) is analogous to (II).

(C) Mass (m): Mass serves as the measure of a body's linear inertia, which is its intrinsic resistance to being linearly accelerated. The rotational analogue is the Moment of inertia (I), which quantifies a body's resistance to being angularly accelerated. It depends not only on the mass but also on how that mass is distributed relative to the axis of rotation. Therefore, (C) is analogous to (III).

(D) Linear velocity (\(\vec{v}\)): This vector quantity represents the rate at which an object's position changes over time (\(v = ds/dt\)). The rotational parallel is Angular velocity (\(\vec{\omega}\)), which represents the rate at which the angular position changes over time (\(\omega = d\theta/dt\)). Therefore, (D) is analogous to (IV).


Step 3: Final Answer:

Based on this analysis of the physical roles of the quantities, the correct pairings are (A) with (I), (B) with (II), (C) with (III), and (D) with (IV). This set of matches corresponds directly to option (A).
Quick Tip: Creating a simple two-column table of linear and rotational analogues can be very helpful for studying this topic. Position (x) \(\leftrightarrow\) Angle (\(\theta\)) Velocity (v) \(\leftrightarrow\) Angular Velocity (\(\omega\)) Acceleration (a) \(\leftrightarrow\) Angular Acceleration (\(\alpha\)) Mass (m) \(\leftrightarrow\) Moment of Inertia (I) Force (F) \(\leftrightarrow\) Torque (\(\tau\)) Momentum (p=mv) \(\leftrightarrow\) Angular Momentum (L=I\(\omega\))

Step 1: Understanding the Concept:

This question focuses on the mathematical relationships that connect the key quantities of rotational dynamics: rotational kinetic energy (E), angular momentum (J), moment of inertia (I), and angular velocity (\(\omega\)). The task is to examine the given equations and identify the one that does not represent a valid physical relationship.


Step 2: Key Formula or Approach:

We start with the two most fundamental definitions in rotational motion:
1. Rotational Kinetic Energy: The energy of a body due to its rotation is defined as \( E = \frac{1}{2} I \omega^2 \).
2. Angular Momentum: For a rigid body, the angular momentum is defined as the product of its moment of inertia and angular velocity, \( J = I \omega \).
By algebraically manipulating these two core equations, we can verify the correctness of the other options.


Step 3: Detailed Explanation:

Let us systematically evaluate each of the given options:


(A) \( E = \frac{1}{2} I \omega^2 \): This equation is the standard definition of the kinetic energy of a rotating rigid body. It is the rotational analog of the linear kinetic energy formula \( K = \frac{1}{2} m v^2 \). This statement is correct.

(B) \( J = I \omega \): This equation is the standard definition of the angular momentum of a rigid body rotating about a principal axis. It is the rotational analog of the linear momentum formula \( p = mv \). This statement is correct.

(C) \( E = \frac{J^2}{2I} \): This relationship can be derived from the fundamental definitions. From the angular momentum equation, \( J = I \omega \), we can express the angular velocity as \( \omega = \frac{J}{I} \). Substituting this expression for \( \omega \) into the kinetic energy formula gives:

\[ E = \frac{1}{2} I \omega^2 = \frac{1}{2} I \left(\frac{J}{I}\right)^2 = \frac{1}{2} I \frac{J^2}{I^2} = \frac{J^2}{2I} \]
This shows the direct relationship between energy and the square of angular momentum. This statement is correct.

(D) \( E = JI \): This equation does not represent a valid physical law and cannot be derived from the fundamental principles. A dimensional analysis also shows it is incorrect: The units of E are Joules (\(kg \cdot m^2/s^2\)), while the units of JI are (\(kg \cdot m^2/s\)) \(\cdot\) (\(kg \cdot m^2\)), which is inconsistent. This statement is incorrect.


Step 4: Final Answer:

The equation that presents an incorrect relationship between the given quantities is E = JI.
Quick Tip: Remember the analogy between linear and rotational motion. Kinetic Energy \( K = \frac{1}{2} m v^2 \) is analogous to \( E = \frac{1}{2} I \omega^2 \). Linear momentum \( p = mv \) is analogous to \( J = I \omega \). The relationship \( K = \frac{p^2}{2m} \) is analogous to \( E = \frac{J^2}{2I} \). This can help you quickly recall or verify these formulas.

Step 1: Understanding the Concept:

This question requires an evaluation of several statements concerning the behavior of a series LCR (Inductor-Capacitor-Resistor) circuit. The key concepts involved are the quality factor (Q-factor), the role of resistance in damping, the nature of capacitor discharge, and the meaning of resonance.


Step 2: Detailed Explanation:

Let us carefully assess each statement's validity:


(A) In purely inductive circuit (R = 0), Quality factor is infinite. The quality factor, or Q-factor, for a series LCR circuit is mathematically defined as \( Q = \frac{\omega_0 L}{R} \), where \( \omega_0 \) is the natural resonant frequency. This formula shows that Q is inversely proportional to the resistance R. In an idealized circuit where the resistance is zero (\(R=0\)), representing a lossless system, the Q-factor would mathematically approach infinity (\( Q \to \infty \) as \( R \to 0 \)). Thus, this statement is true.

(B) Resistance 'R' is alone responsible for damping of oscillations. In the context of a simple LCR circuit, the inductor and capacitor are ideal energy storage elements, trading energy between the magnetic field and the electric field. The resistor, however, is an energy dissipative element, converting electrical energy into heat through Joule heating. This continuous energy loss is what causes the amplitude of the oscillations to decrease over time, a phenomenon known as damping. Therefore, this statement is true.

(C) Discharge of capacitor is not oscillatory in character. This statement is false. When a charged capacitor is connected to an inductor (forming an LC or LCR circuit), the energy does not simply drain away. Instead, it is transferred to the inductor's magnetic field, then back to the capacitor's electric field, and so on. This back-and-forth transfer of energy between the capacitor and inductor is the very definition of an electrical oscillation. The presence of resistance only serves to damp these oscillations, but their fundamental character is oscillatory (unless the circuit is critically damped or overdamped, in which case it is a limiting non-oscillatory case).

(D) Q-factor is measure of sharpness of resonance in case of a driven oscillator. This is one of the most important physical interpretations of the Q-factor. In a driven LCR circuit, resonance occurs when the driving frequency matches the natural frequency. A high Q-factor signifies a circuit with low damping, which responds very strongly only to frequencies extremely close to resonance, resulting in a sharp and narrow resonance peak on a graph of amplitude versus frequency. Conversely, a low Q-factor indicates a broad, flat resonance curve. Therefore, this statement is true.


Step 3: Final Answer:

Statements (A), (B), and (D) accurately describe the properties of an LCR circuit, whereas statement (C) is incorrect. The correct choice is therefore the one that combines (A), (B), and (D).
Quick Tip: Remember the physical meaning of the Q-factor. It's a measure of the "quality" of an oscillator. \( Q = 2\pi \times \frac{Energy stored per cycle}{Energy dissipated per cycle} \). For R=0, energy dissipated is zero, so Q is infinite. High Q means low damping and sharp resonance.

Step 1: Understanding the Concept:

The kinetic energy of a satellite in a stable circular orbit is determined by its mass, the mass of the central body it is orbiting (in this case, Earth), and its distance from the center of that body (the orbital radius). The orbital radius is not the same as the altitude; it is the sum of the Earth's radius and the satellite's altitude above the surface.


Step 2: Key Formula or Approach:

For a satellite of mass \(m\) moving in a circular orbit of radius \(r\) around the Earth (mass \(M_E\)), the gravitational force of attraction provides the required centripetal force to maintain the orbit:
\[ F_{gravity} = F_{centripetal} \implies \frac{G M_E m}{r^2} = \frac{m v^2}{r} \]
We can find an expression for kinetic energy, \( KE = \frac{1}{2} m v^2 \), from this equilibrium condition. First, solve for \(mv^2\):
\[ m v^2 = \frac{G M_E m}{r} \]
Then, substitute this into the kinetic energy formula: \[ KE = \frac{1}{2} m v^2 = \frac{G M_E m}{2r} \]
This crucial result shows that for a given satellite mass and central body, the kinetic energy is inversely proportional to the orbital radius \(r\).


Step 3: Detailed Explanation:

1. Calculate the orbital radius for each satellite:

The orbital radius \(r\) is measured from the center of the Earth. The formula is \( r = R_{earth} + altitude \).
For satellite A, the altitude is \(h_A = 2R\). Its orbital radius is \(r_A = R + 2R = 3R\).

For satellite B, the altitude is \(h_B = 5R\). Its orbital radius is \(r_B = R + 5R = 6R\).


2. Formulate the ratio of the kinetic energies:

The mass of both satellites is given as 'M'.
The kinetic energy of satellite A is: \[ KE_A = \frac{G M_E M}{2 r_A} = \frac{G M_E M}{2(3R)} \]
The kinetic energy of satellite B is: \[ KE_B = \frac{G M_E M}{2 r_B} = \frac{G M_E M}{2(6R)} \]
To find the ratio \( \frac{KE_A}{KE_B} \), we divide the two expressions. The constant term \( \frac{G M_E M}{2} \) cancels out. \[ \frac{KE_A}{KE_B} = \frac{1/r_A}{1/r_B} = \frac{r_B}{r_A} = \frac{6R}{3R} = \frac{2}{1} \]

Step 4: Final Answer:

The ratio of the kinetic energy of satellite A to that of satellite B is 2 to 1.
Quick Tip: For a satellite in a circular orbit, remember these energy relationships: Kinetic Energy \(KE = \frac{GMm}{2r}\) (Always positive) Potential Energy \(PE = -\frac{GMm}{r}\) (Always negative) Total Energy \(E = KE + PE = -\frac{GMm}{2r}\) Notice that \(KE = -E\) and \(KE = -\frac{1}{2}PE\). This can save time in problems involving ratios of different types of energy.

Step 1: Understanding the Concept:

Newton's rings are an interference pattern created by the reflection of light between two surfaces: a spherical surface (plano-convex lens) and an adjacent flat surface (glass plate). The interference occurs in the thin film of air between these surfaces. A critical factor in this setup is the phase change that light undergoes upon reflection at the boundary of a medium with a higher refractive index.


Step 2: Key Formula or Approach:

The total optical path difference between the two interfering rays (one reflecting from the top of the air film, the other from the bottom) determines whether interference is constructive or destructive.
1. **Reflection and Phase Change:** The light ray reflecting from the top surface of the flat glass plate (the bottom boundary of the air film) travels from a rarer medium (air, n≈1) to a denser medium (glass, n≈1.5). This reflection introduces an additional phase shift of \(\pi\) radians, which is equivalent to adding a path difference of \(\lambda/2\). The ray reflecting from the bottom surface of the lens (top boundary of the air film) does not undergo a phase change, as it reflects from a denser-to-rarer medium interface.
2. **Geometrical Path Difference:** A ray traveling through the air film of thickness 't', reflecting off the bottom surface, and returning, travels an extra geometrical distance of \(2t\) (for near-normal viewing).
3. **Total Path Difference:** The total optical path difference (\(\Delta\)) is the sum of the geometrical path difference and the path difference due to phase change: \( \Delta = 2t + \frac{\lambda}{2} \).
4. **Condition for Constructive Interference (Bright Fringe):** For a bright fringe to appear, the two waves must be in phase, meaning their total path difference must be an integer multiple of the wavelength: \( \Delta = m\lambda \), where \(m = 1, 2, 3, ...\).


Step 3: Detailed Explanation:

We now set the total path difference equal to the condition for a bright fringe (constructive interference):
\[ Total path difference = Integer multiple of wavelength \] \[ 2t + \frac{\lambda}{2} = m\lambda \]
To find the condition on the film thickness \(t\), we rearrange the equation to solve for \(2t\):
\[ 2t = m\lambda - \frac{\lambda}{2} \]
Factoring out \(\lambda\), we get: \[ 2t = \left(m - \frac{1}{2}\right)\lambda \]
To express this with a single fraction, we find a common denominator: \[ 2t = \frac{(2m - 1)\lambda}{2} \]
This expression is the condition for the thickness \(t\) that will produce the \(m^{th}\) bright ring. This result matches option (C). Note that option (A), \( 2t = m\lambda + \frac{\lambda}{2} \), would be \( 2t = (m+1/2)\lambda \), which is also a condition for constructive interference but uses a different indexing for \(m\). Option (C) is the conventional form.


Step 4: Final Answer:

The condition on the film thickness \(t\) for the formation of a bright ring is \( 2t = \frac{(2m-1)\lambda}{2} \).
Quick Tip: For interference in thin films, always check for phase changes on reflection. A reflection from a denser medium adds \(\lambda/2\) to the path difference. This is why in Newton's rings (and soap bubbles), the conditions for bright and dark fringes are "swapped" compared to what you might expect from the geometrical path difference alone. The center of Newton's rings (where t=0) is dark because the \(\lambda/2\) phase shift causes destructive interference.

Step 1: Understanding the Concept:

The acronym LASER stands for Light Amplification by Stimulated Emission of Radiation. The physics of this process imbues the resulting light with a set of unique and defining properties that distinguish it sharply from incoherent light sources like the sun or an incandescent bulb. The question asks to identify which of the given options is contrary to these established properties.


Step 2: Detailed Explanation:

Let's review the principal characteristics of laser light and evaluate each option:


(A) Highly coherent: This is a fundamental property. Coherence means that all the light waves in the laser beam are in phase with one another, both in space (spatial coherence) and in time (temporal coherence). This lock-step propagation is what allows for a stable interference pattern and is a direct result of the stimulated emission process. So, (A) is a key characteristic.
(B) Highly penetrating: This term is not a standard descriptor for laser light. Penetration depth depends heavily on the laser's wavelength and the material it interacts with. While a high-power laser can cut or drill materials, this is due to intense localized heating, not because the light itself is inherently "penetrating" in the way X-rays or gamma rays are. So, this is not a fundamental characteristic.
(C) Highly intense: Because all the light energy is concentrated into a very narrow, low-divergence beam, the power per unit area (intensity) of a laser can be extraordinarily high, orders of magnitude greater than that of conventional sources. So, (C) is a key characteristic.
(D) Highly divergent: This statement is the direct opposite of a primary laser characteristic. Laser light is renowned for its high degree of collimation, or directionality. This means the beam spreads out, or diverges, very little as it travels over long distances. A typical laser has a very small angle of divergence. Therefore, stating it is "highly divergent" is incorrect.


Step 3: Final Answer:

When comparing the options, "Highly divergent" is a direct and unambiguous contradiction of one of the most well-known properties of a laser (its low divergence or high directionality). While "Highly penetrating" is also not a standard property, "Highly divergent" is the clear antithesis of a fundamental characteristic. Therefore, it is the most definitively incorrect statement and the correct answer.
Quick Tip: Remember the four main properties of laser light: 1. \textbf{Monochromaticity:} Very narrow range of wavelengths (single color). 2. \textbf{Coherence:} All waves are in phase. 3. \textbf{Directionality:} Travels in a straight, narrow beam (low divergence). 4. \textbf{High Intensity:} Concentrated power. "Highly divergent" is the opposite of Directionality.

Step 1: Understanding the Concept:

The operation of a laser is not a single event but a carefully orchestrated sequence of physical processes that must occur in a specific order within the active laser medium. This question asks for the correct chronological flow of these essential steps.


Step 2: Detailed Explanation:

Let's break down the process of generating a laser beam into its logical, sequential stages:

1. (A) Pumping: The entire process must be initiated by supplying energy from an external source to the atoms or molecules of the lasing medium. This is known as pumping. The pump energy excites the atoms, raising them from their ground state to a higher energy level. This is the prerequisite for all subsequent steps.

2. (B) Population Inversion: For a laser to work, the pumping process must be efficient enough to create a highly unnatural state called a population inversion. Under normal thermal equilibrium, lower energy states are always more populated than higher energy states. Population inversion is the condition where there are more atoms in a specific excited (metastable) state than in a lower energy state to which they can transition. This is the necessary condition for light amplification to be possible.

3. (C) Stimulated Emission: Once a population inversion exists, the process of amplification can begin. It is typically triggered by a spontaneously emitted photon. When this photon of the correct energy passes near an excited atom, it stimulates that atom to drop to its lower energy state, releasing a second photon. Crucially, this new photon is an exact clone of the first: it has the same frequency, phase, direction, and polarization.

4. (D) Light Amplification: The stimulated emission of one photon creates two, these two create four, and so on. This cascade effect, where photons stimulate the emission of more identical photons, is a chain reaction that rapidly increases the number of coherent photons. This process, often enhanced by using an optical cavity (mirrors) to pass the light back and forth through the inverted medium, results in a massive amplification of the initial light. This is the essence of the LASER acronym: Light Amplification by Stimulated Emission of Radiation.


Step 3: Final Answer:

The logical and causal sequence of these events is: Pumping provides the energy to create Population Inversion, which then allows Stimulated Emission to occur, leading to a chain reaction of Light Amplification. Therefore, the correct chronological order is (A), (B), (C), (D).
Quick Tip: Think of it like setting up a chain of dominoes. - \textbf{Pumping} is lifting the dominoes and setting them up. - \textbf{Population Inversion} is the state when all the dominoes are standing up, ready to fall. - \textbf{Stimulated Emission} is the first domino being tipped over, which then hits the next one. - \textbf{Light Amplification} is the entire chain reaction of dominoes falling.

Step 1: Understanding the Concept:

This question requires matching various optical phenomena and components (List-I) with their corresponding physical experiments, devices, or defining concepts (List-II). It tests knowledge across interference, diffraction, and polarization.


Step 2: Detailed Explanation:


(A) Circular Fringes: The defining characteristic of the Newton's Ring experiment is the formation of a concentric series of bright and dark circular interference fringes. This pattern arises because the thickness of the air film trapped between the plano-convex lens and the flat glass plate is constant along any circle centered on the point of contact. These circles of equal thickness are called fringes of equal thickness. Thus, (A) matches with (II).

(B) Straight parallel and equidistant interference pattern: When a thin film of air is created in the shape of a wedge (for instance, by placing a thin foil between two glass slides at one end), the loci of equal film thickness are straight lines that are parallel to the apex of the wedge. The resulting interference pattern is therefore a set of straight, parallel, and equally spaced fringes. Thus, (B) matches with (III).

(C) Polarizer: A polarizer is any optical device that can convert a beam of unpolarized light into a beam of polarized light. One of the classic and historically important devices used for this purpose is the Nicol prism. It is constructed from a calcite crystal that is cut and cemented in a specific way to separate the ordinary and extraordinary rays by total internal reflection, transmitting only one of them. Thus, (C) matches with (I).

(D) E-ray and O-ray travel with same speed: In an anisotropic, birefringent crystal like calcite, an incident light ray splits into two components: the ordinary ray (O-ray), which obeys Snell's law, and the extraordinary ray (E-ray), which does not. These two rays have different polarizations and, in general, travel at different speeds. However, there exists a specific direction or axis within the crystal along which this distinction vanishes, and both rays propagate with the same velocity. This special direction is known as the Optic axis of the crystal. Thus, (D) matches with (IV).


Step 3: Final Answer:

The correctly established pairings are: (A) with (II), (B) with (III), (C) with (I), and (D) with (IV). This combination corresponds exactly to option (A).
Quick Tip: Associate fringe shapes with the geometry of the film: - \textbf{Circular symmetry} (Newton's Rings) \(\rightarrow\) \textbf{Circular fringes}. - \textbf{Linear wedge} \(\rightarrow\) \textbf{Straight, parallel fringes}. Also, remember key optical components: Nicol Prism = Polarizer, and the definition of the Optic Axis in birefringent crystals.

Step 1: Understanding the Concept:

The phenomenon described is polarization by reflection, which is governed by Brewster's Law. This law states that for a specific angle of incidence, known as the polarizing angle or Brewster's angle (\(\theta_p\)), the reflected light from a dielectric surface will be perfectly plane-polarized. The question relates this physical law to a geometrical scenario, asking for the sun's angle relative to the horizon, which is the reflecting surface. This angle is complementary to the angle of incidence.


Step 2: Key Formula or Approach:

1. **Brewster's Law:** This law provides the mathematical relationship between the polarizing angle (\(\theta_p\)) and the refractive index (\(n\)) of the material: \( n = \tan(\theta_p) \).
2. **Geometric Relationship:** The angle of incidence (\(\theta_p\)) is always measured with respect to the normal (a line perpendicular to the reflecting surface). The sun's angle with the horizon (\(\alpha\)) is measured with respect to the surface itself. A diagram shows that these two angles add up to 90 degrees: \( \theta_p + \alpha = 90^\circ \).


Step 3: Detailed Explanation:

1. Apply Brewster's Law to find the Polarizing Angle (\(\theta_p\)):

We are given the refractive index of the material, \( n = 1.732 \).

It is useful to recognize that this value is a very close approximation of the square root of 3 (\( \sqrt{3} \approx 1.73205 \)).

Substituting this into Brewster's Law:
\[ \tan(\theta_p) = n = \sqrt{3} \]
From trigonometry, we know that the angle whose tangent is \(\sqrt{3}\) is \(60^\circ\).
\[ \theta_p = 60^\circ \]
This means the sunlight must be striking the material at an angle of 60° to the normal for the reflected light to be fully polarized.

2. Determine the Sun's Angle with the Horizon (\(\alpha\)):

The angle of incidence, \(\theta_p = 60^\circ\), is the angle between the incoming sunlight and the vertical normal line. The question asks for the angle between the sunlight and the horizontal surface (the horizon). This angle, \(\alpha\), is the complement of \(\theta_p\).

Using the geometric relationship: \( \alpha = 90^\circ - \theta_p \).
\[ \alpha = 90^\circ - 60^\circ = 30^\circ \]

Step 4: Final Answer:

At the instant of complete polarization, the angle between the sun and the horizon is 30°.
Quick Tip: Always be careful about which angle is being asked for in problems involving reflection and refraction. The laws use the angle with the normal, but questions might ask for the angle with the surface (glancing/grazing angle) or the angle of deviation. Drawing a simple diagram of the incident ray, the surface, and the normal can prevent confusion.

Step 1: Understanding the Concept:

The intensity distribution in an interference pattern is a function of the phase difference (\(\delta\)) between the interfering light waves. A maximum intensity corresponds to perfect constructive interference, where the crests of both waves align. We need to determine the value of the phase difference \(\delta\) for which the given mathematical expression for intensity reaches its maximum possible value.


Step 2: Key Formula or Approach:

The intensity \(I\) of the interference pattern is provided as:
\[ I = 4A^2 \cos^2\left(\frac{\delta}{2}\right) \]
The goal is to maximize this function. Since \(4A^2\) is a constant amplitude factor, we must find the maximum value of the trigonometric term, \( \cos^2(\delta/2) \).


Step 3: Detailed Explanation:

The function \( \cos^2(x) \) represents the square of the cosine function. The cosine function itself, \( \cos(x) \), oscillates between -1 and +1. Squaring this function means its value will oscillate between 0 and 1.

The maximum value of \( \cos^2(\delta/2) \) is therefore 1.

So, for maximum intensity, we must have:
\[ \cos^2\left(\frac{\delta}{2}\right) = 1 \]
Taking the square root of both sides gives: \[ \cos\left(\frac{\delta}{2}\right) = \pm 1 \]
The cosine function equals +1 or -1 whenever its argument is an integer multiple of \(\pi\). Let this integer be \(n\). \[ \frac{\delta}{2} = n\pi, \quad where n is an integer (n = 0, \pm 1, \pm 2, \ldots) \]
To find the condition for the phase difference \(\delta\), we multiply both sides by 2:
\[ \delta = 2n\pi \]
This result signifies that the phase difference \(\delta\) must be an even integer multiple of \(\pi\). This is equivalent to saying that \(\delta\) must be an integral multiple of \(2\pi\). This is the general condition for constructive interference.


Step 4: Final Answer:

The intensity reaches its maximum value when the phase difference \(\delta\) is an integral multiple of \(2\pi\).
Quick Tip: For interference, remember the conditions for intensity maxima and minima. \textbf{Maximum Intensity (Constructive Interference):} Phase difference \( \delta = 2n\pi \), Path difference \( \Delta x = n\lambda \). \textbf{Minimum Intensity (Destructive Interference):} Phase difference \( \delta = (2n+1)\pi \), Path difference \( \Delta x = (n+1/2)\lambda \).

Step 1: Understanding the Concept:

The pattern observed from a double slit is not just pure interference. It is a combination of the interference pattern caused by the two slits acting as coherent sources and the diffraction pattern produced by the light passing through each individual slit. A "missing order" occurs when the condition for an interference maximum (a bright fringe) coincides with the condition for a diffraction minimum (a dark band) at the exact same angular position. The diffraction minimum effectively cancels out the light, causing the interference maximum to be absent.


Step 2: Key Formula or Approach:

Let \(a\) represent the width of a single slit and \(d\) represent the center-to-center separation between the two slits.
The angular position (\(\theta\)) of the \(n^{th}\) order interference maximum is given by the condition for constructive interference: \[ d \sin\theta = n\lambda, \quad where n = 0, 1, 2, \ldots \]
The angular position (\(\theta\)) of the \(m^{th}\) order diffraction minimum is given by the condition for destructive interference from a single slit: \[ a \sin\theta = m\lambda, \quad where m = 1, 2, 3, \ldots (note: m is a non-zero integer) \]
An interference order \(n\) will be missing if both equations are satisfied for the same angle \(\theta\). We can find the condition for this by isolating \( \sin\theta \) in both equations and setting them equal, or more simply, by dividing the first equation by the second:
\[ \frac{d \sin\theta}{a \sin\theta} = \frac{n\lambda}{m\lambda} \implies \frac{d}{a} = \frac{n}{m} \]

Step 3: Detailed Explanation:

The problem provides the following values:
Slit width, \( a = 0.12 \) mm.

Slit separation, \( d = 0.6 \) mm.

First, we compute the critical ratio of these two dimensions:
\[ \frac{d}{a} = \frac{0.6 \, mm}{0.12 \, mm} = 5 \]
The condition for missing orders therefore simplifies to:
\[ \frac{n}{m} = 5 \implies n = 5m \]
We can now find the specific interference orders (\(n\)) that are missing by substituting the allowed integer values for the diffraction minimum order (\( m = 1, 2, 3, \ldots \)):


When \(m=1\) (the first diffraction minimum), the missing interference order is \(n = 5(1) = 5\).
When \(m=2\) (the second diffraction minimum), the missing interference order is \(n = 5(2) = 10\).
When \(m=3\) (the third diffraction minimum), the missing interference order is \(n = 5(3) = 15\).

So, the sequence of missing orders is 5, 10, 15, 20, and so on. None of the provided options match this result. This strongly indicates a typographical error in either the given values or the options. Let's re-evaluate based on the options. Option (A) shows multiples of 6. Let's see what ratio \(d/a\) would produce this. If \(n=6m\), then \(d/a\) must be 6.
Let's assume the slit separation was intended to be \(d = 0.72\) mm. Then the ratio would be: \[ \frac{d}{a} = \frac{0.72 \, mm}{0.12 \, mm} = 6 \]
Under this assumption, the condition for missing orders becomes: \[ n = 6m \]
The missing orders would then be:


For \(m=1\), \(n = 6\).
For \(m=2\), \(n = 12\).
For \(m=3\), \(n = 18\).
For \(m=4\), \(n = 24\).

This sequence (6, 12, 18, 24) is a perfect match for option (A). Given this match, it is extremely likely that the intended value for the slit separation was 0.72 mm.


Step 4: Final Answer:

Assuming a typographical error in the problem where the slit separation \(d\) should have been 0.72 mm instead of 0.6 mm, the missing interference orders are 6, 12, 18, and 24.
Quick Tip: For missing order problems, the key is the ratio \(d/a\). If \(d/a = k\), then the missing interference orders are \(n = k, 2k, 3k, \ldots\). Always calculate this ratio first. If your answer doesn't match the options, re-check for possible typos in the given values that would lead to one of the options.

Step 1: Understanding the Concept:

A zone plate is a unique optical device that focuses light, not through refraction as a conventional lens does, but through the principles of diffraction and interference. It consists of a set of concentric transparent and opaque rings whose radii are precisely controlled. This structure alters the phase and amplitude of transmitted light in such a way as to produce a focusing effect.


Step 2: Detailed Explanation:

Let's analyze the optical properties of a zone plate in relation to the given statements:


Multiple Foci: The design of a zone plate causes light from different zones to interfere constructively at several points along its central axis. This results in not just one focal point, but a series of focal points. There is a principal focus which is the most intense, but weaker, secondary foci also exist at distances \( f/3, f/5, f/7, \ldots \) from the plate. Therefore, statement (A) "has only one focus" and statement (D) "has only two foci" are both incorrect.

Focusing Action (Converging): The primary function of a zone plate is to bring parallel light to a focus. This action of converging light to form a real image is precisely the function of a convex lens. Therefore, statement (B) "can not act as convex lens" is incorrect.

Converging and Diverging Action: The mathematics of diffraction by a zone plate shows that in addition to producing a series of real foci (like a convex lens), it also produces a corresponding series of virtual foci on the opposite side of the plate (the side from which the light originates). Light appears to diverge from these virtual focal points. This diverging action is characteristic of a concave lens. Because a zone plate produces both real (converging) and virtual (diverging) foci, it is accurately described as acting simultaneously like a convex lens and a concave lens. Therefore, statement (C) is the most complete and correct description.


Step 3: Final Answer:

Among the given choices, the most accurate description of a zone plate's properties is that it simultaneously exhibits the converging behavior of a convex lens (by forming real foci) and the diverging behavior of a concave lens (by forming virtual foci).
Quick Tip: The key difference between a lens and a zone plate is how they work: Lens \(\rightarrow\) Refraction, Zone Plate \(\rightarrow\) Diffraction. This fundamental difference leads to the zone plate having multiple foci (both real and virtual), whereas a simple lens has one primary focal point.

Step 1: Understanding the Concept:

This problem involves calculating the work done during an adiabatic compression of a gas. In an adiabatic process, there is no heat exchange between the system (the gas) and its surroundings (\(Q=0\)). According to the first law of thermodynamics (\(\Delta U = Q - W\)), the work done *on* the gas (\(-W\)) is equal to the increase in its internal energy (\(\Delta U\)). We can calculate this work directly using the initial and final pressure and volume of the gas.


Step 2: Key Formula or Approach:

The work done *by* a gas during a polytropic process from state 1 to state 2 is given by \( W = \frac{P_1 V_1 - P_2 V_2}{\gamma - 1} \).
The work done *on* the gas during compression is the negative of this:
\[ W_{on} = -W = \frac{P_2 V_2 - P_1 V_1}{\gamma - 1} \]
To relate the initial and final states, we use the equation for an adiabatic process:
\[ P_1 V_1^\gamma = P_2 V_2^\gamma \]

Step 3: Detailed Explanation:

1. Establish the Initial Conditions (State 1) in CGS units:

The gas is initially at Normal Temperature and Pressure (NTP).
Mass, \(m = 1\) gram.
Density, \(\rho = 0.0001465\) g/cm³.
The initial volume is calculated from mass and density: \( V_1 = \frac{m}{\rho} = \frac{1 \, g}{0.0001465 \, g/cm^3} \approx 6825.94 \, cm^3 \).

Normal pressure is 1 atm. For CGS calculations leading to ergs, we need pressure in dynes/cm². \(1 \, atm \approx 1.013 \times 10^6 \) dyn/cm². To simplify and match the likely intent for such problems, we will approximate NTP pressure as \( P_1 = 10^6 \) dyn/cm² (1 bar).


2. Establish the Final Conditions (State 2):

The gas is compressed to one-fourth of its initial volume.
\( V_2 = \frac{V_1}{4} \).

We find the final pressure, \(P_2\), using the adiabatic relation:
\[ P_2 = P_1 \left(\frac{V_1}{V_2}\right)^\gamma = P_1 (4)^{\gamma} = P_1 (4)^{1.5} \] \[ P_2 = P_1 (4^{3/2}) = P_1 (\sqrt{4}^3) = P_1 (2^3) = 8P_1 \]
So, \( P_2 = 8 \times 10^6 \, dyn/cm^2 \).

3. Calculate the Work Done on the Gas:

Using the formula for work done in an adiabatic process: \[ W_{on} = \frac{P_2 V_2 - P_1 V_1}{\gamma - 1} \]
It is often easier to substitute the relationships before the numbers: substitute \(P_2 = 8P_1\) and \(V_2 = V_1/4\):
\[ W_{on} = \frac{(8P_1) (\frac{V_1}{4}) - P_1 V_1}{1.5 - 1} = \frac{2P_1 V_1 - P_1 V_1}{0.5} = \frac{P_1 V_1}{0.5} = 2 P_1 V_1 \]
This gives a simple expression for the work done. Now, we insert the numerical values for the initial state: \[ W_{on} = 2 \times (10^6 \, dyn/cm^2) \times (6825.94 \, cm^3) \] \[ W_{on} = 13651880000 \, ergs \approx 1.365 \times 10^{10} \, ergs \]
The unit 'erg' is the CGS unit of energy, equivalent to a dyne-centimeter.

Step 4: Final Answer:

The work done to adiabatically compress the air is approximately \( 1.365 \times 10^{10} \) ergs.
Quick Tip: In thermodynamics problems, pay close attention to units. The CGS unit of energy is the 'erg' (\(1 \, erg = 1 \, dyn \cdot cm\)). Ensure your pressure is in dyn/cm² and volume is in cm³ to get the work in ergs. Also, be aware of standard approximations for NTP/STP; sometimes 1 atm is taken as \(10^5\) Pa (SI) or \(10^6\) dyn/cm² (CGS) for simplicity.

Step 1: Understanding the Concept:

The change in entropy, a measure of a system's thermal disorder, is defined for a reversible process by the differential relation \( dS = \frac{dQ_{rev}}{T} \). Our goal is to derive a specific formula for the total entropy change, \( \Delta S \), when a perfect gas undergoes a process at constant volume (an isochoric process).


Step 2: Key Formula or Approach:

The derivation begins with the first law of thermodynamics, which relates heat (\(dQ\)), internal energy (\(dU\)), and work (\(dW\)): \( dQ = dU + dW \). For a perfect gas, the change in internal energy is given by \( dU = nC_V dT \), and the work done by the gas is \( dW = P dV \). Combining these with the definition of entropy for a reversible process, we get:
\[ TdS = nC_V dT + P dV \]
This general equation can be rearranged to express the differential change in entropy:
\[ dS = nC_V \frac{dT}{T} + \frac{P}{T} dV \]
We will now apply the specific constraint of an isochoric process to simplify this general expression.


Step 3: Detailed Explanation:

An isochoric process is, by definition, a process that occurs at a constant volume. This implies that the final volume is equal to the initial volume (\( V_2 = V_1 \)), and therefore, any infinitesimal change in volume is zero (\( dV = 0 \)).

Applying this condition to the general expression for \(dS\):
\[ dS = nC_V \frac{dT}{T} + \frac{P}{T} (0) = nC_V \frac{dT}{T} \]
To find the total change in entropy, \( \Delta S \), as the system moves from an initial state (1) to a final state (2), we integrate this expression with respect to temperature:
\[ \Delta S = S_2 - S_1 = \int_{T_1}^{T_2} nC_V \frac{dT}{T} = nC_V \left[ \ln(T) \right]_{T_1}^{T_2} = nC_V \ln\left(\frac{T_2}{T_1}\right) \]
The result is in terms of temperature, but the options are in terms of pressure or volume. For a perfect gas undergoing a constant-volume process, the relationship between pressure and temperature is described by Gay-Lussac's Law:
\[ \frac{P_1}{T_1} = \frac{P_2}{T_2}, \quad which implies \quad \frac{T_2}{T_1} = \frac{P_2}{P_1} \]
We can substitute this pressure ratio into our expression for entropy change. Assuming the \(C_V\) in the options represents the total heat capacity for the amount of gas present (i.e., \(nC_V\)):
\[ \Delta S = C_V \ln\left(\frac{P_2}{P_1}\right) \]

Step 4: Final Answer:

The change in entropy for a perfect gas in a constant-volume process is correctly expressed as \( C_V \log_e \frac{P_2}{P_1} \).
Quick Tip: You can derive the entropy change formulas for all basic processes (isothermal, isobaric, isochoric) from the general expression \( \Delta S = C_V \ln(T_2/T_1) + R \ln(V_2/V_1) \) for 1 mole of an ideal gas. \textbf{Isochoric (V=const):} The second term is zero, \( \Delta S = C_V \ln(T_2/T_1) = C_V \ln(P_2/P_1) \). \textbf{Isobaric (P=const):} \( \Delta S = C_P \ln(T_2/T_1) = C_P \ln(V_2/V_1) \). \textbf{Isothermal (T=const):} The first term is zero, \( \Delta S = R \ln(V_2/V_1) \).

Step 1: Understanding the Concept:

The coefficient of volume expansion measures the fractional change in a substance's volume per unit change in temperature. This question asks for the ratio of this coefficient under two different thermodynamic conditions: adiabatic (constant entropy) and isobaric (constant pressure).


Step 2: Key Formula or Approach:

The general definition for the coefficient of volume expansion is \( \frac{1}{V}\left(\frac{\partial V}{\partial T}\right) \). We must evaluate this for both isobaric and adiabatic conditions for a perfect gas.
The isobaric coefficient of volume expansion is formally defined as:
\[ \alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_P \]
The adiabatic coefficient of volume expansion is formally defined as:
\[ \beta = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_S \]
Our task is to derive expressions for \( \alpha \) and \( \beta \) and then compute their ratio, \( \beta / \alpha \).


Step 3: Detailed Explanation:

1. Derivation of the Isobaric Coefficient (\(\alpha\)):

We begin with the ideal gas law for one mole: \( PV = RT \). To find the partial derivative with respect to T at constant P, we rearrange it as \( V = \frac{RT}{P} \).

Now, we differentiate V with respect to T while treating P as a constant: \[ \left( \frac{\partial V}{\partial T} \right)_P = \frac{R}{P} \]
From the ideal gas law, we know that \( \frac{R}{P} = \frac{V}{T} \). Substituting this back into the definition of \( \alpha \): \[ \alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_P = \frac{1}{V} \left( \frac{V}{T} \right) = \frac{1}{T} \]

2. Derivation of the Adiabatic Coefficient (\(\beta\)):

For a reversible adiabatic process, the relationship between temperature and volume for a perfect gas is \( TV^{\gamma-1} = K \), where K is a constant. To find the derivative, we can differentiate this equation implicitly with respect to T:
\[ \frac{d}{dT}(TV^{\gamma-1}) = \frac{d}{dT}(K) \] \[ 1 \cdot V^{\gamma-1} + T \cdot (\gamma-1)V^{\gamma-2} \frac{dV}{dT} = 0 \]
Now, we solve for the derivative \( \frac{dV}{dT} \), which under these adiabatic conditions is \( \left( \frac{\partial V}{\partial T} \right)_S \): \[ T(\gamma-1)V^{\gamma-2}\left( \frac{\partial V}{\partial T} \right)_S = -V^{\gamma-1} \] \[ \left( \frac{\partial V}{\partial T} \right)_S = -\frac{V^{\gamma-1}}{T(\gamma-1)V^{\gamma-2}} = -\frac{V}{T(\gamma-1)} \]
Substituting this into the definition of \( \beta \): \[ \beta = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_S = \frac{1}{V} \left( -\frac{V}{T(\gamma-1)} \right) = -\frac{1}{T(\gamma-1)} \]

3. Calculation of the Ratio \( \frac{\beta}{\alpha} \):

We now divide the expression for the adiabatic coefficient by the expression for the isobaric coefficient: \[ \frac{\beta}{\alpha} = \frac{-\frac{1}{T(\gamma-1)}}{\frac{1}{T}} = -\frac{1}{\gamma-1} \]
This result can be algebraically manipulated by multiplying the numerator and denominator by -1: \[ \frac{-1}{\gamma-1} = \frac{1}{-(\gamma-1)} = \frac{1}{1 - \gamma} \]

Step 4: Final Answer:

The ratio of the adiabatic coefficient of expansion to the isobaric coefficient of expansion is \( \frac{1}{1-\gamma} \).
Quick Tip: This result is related to the ratio of adiabatic compressibility (\(\kappa_S\)) to isothermal compressibility (\(\kappa_T\)), which is a more common relation: \( \frac{\kappa_S}{\kappa_T} = \frac{1}{\gamma} \). While the question asks about coefficients of expansion, the method involves similar derivations from the gas laws for different processes.

Step 1: Understanding the Concept:

The TdS equations are powerful thermodynamic relations that express the change in entropy, \(dS\), in terms of changes in more easily measurable state variables (like T, P, V) and material properties (like heat capacities). The "first" TdS equation is derived by considering entropy \(S\) as a state function of temperature \(T\) and volume \(V\).


Step 2: Key Formula or Approach:

Since entropy is a state function, we can express it as \( S = S(T, V) \). According to the rules of multivariable calculus, the total differential \(dS\) can be written as the sum of its partial derivatives:
\[ dS = \left(\frac{\partial S}{\partial T}\right)_V dT + \left(\frac{\partial S}{\partial V}\right)_T dV \]
The derivation proceeds by finding thermodynamic expressions for the two partial derivative terms in this equation.


Step 3: Detailed Explanation:

1. Evaluating the first term, \( (\partial S / \partial T)_V \):

We use the definition of the heat capacity at constant volume, \( C_V \). From the second law, \(dQ_{rev} = TdS\). At constant volume, \(dQ_V = C_V dT\). Combining these gives \(TdS_V = C_V dT\). Rearranging this gives the partial derivative: \[ \left(\frac{\partial S}{\partial T}\right)_V = \frac{C_V}{T} \]

2. Evaluating the second term, \( (\partial S / \partial V)_T \):

This term cannot be evaluated from basic definitions alone; it requires the use of a Maxwell relation. The Maxwell relations are derived from the fact that the order of differentiation does not matter for second derivatives of the thermodynamic potentials. The specific relation needed here is derived from the Helmholtz free energy (\(F = U - TS\)) and states:
\[ \left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V \]
This relation elegantly connects the change in entropy with volume to the change in pressure with temperature.

3. Assembling the equation for dS:

Now we substitute the expressions we found in steps 1 and 2 back into the total differential equation for \(dS\):
\[ dS = \left(\frac{C_V}{T}\right) dT + \left(\frac{\partial P}{\partial T}\right)_V dV \]

4. Deriving the final TdS equation:

To obtain the standard form of the "TdS" equation, we multiply the entire expression by the temperature T:
\[ TdS = T \left(\frac{C_V}{T}\right) dT + T \left(\frac{\partial P}{\partial T}\right)_V dV \] \[ TdS = C_V dT + T \left(\frac{\partial P}{\partial T}\right)_V dV \]

Step 4: Final Answer:

The correct mathematical form of the first TdS equation is \( TdS = C_V dT + T \left(\frac{\partial P}{\partial T}\right)_V dV \), which corresponds to option B.
Quick Tip: There are two main TdS equations. Memorizing them is useful: \textbf{1st TdS Equation (S in terms of T, V):} \( TdS = C_V dT + T \left(\frac{\partial P}{\partial T}\right)_V dV \) \textbf{2nd TdS Equation (S in terms of T, P):} \( TdS = C_P dT - T \left(\frac{\partial V}{\partial T}\right)_P dP \) Remembering that the first involves \(C_V\) and \(dV\), while the second involves \(C_P\) and \(dP\), helps to distinguish them.

Step 1: Understanding the Concept:

This question requires identifying the correct mathematical expressions for several key concepts, equations, and thermodynamic potentials used in the study of thermodynamics.


Step 2: Detailed Explanation:


(A) Clausius-Clapeyron equation: This important equation describes the relationship between pressure and temperature along a phase coexistence line (e.g., liquid-vapor or solid-liquid). It states that the slope of this line on a P-T diagram, \( \frac{dP}{dT} \), is equal to the latent heat of the transformation (L) divided by the product of the absolute temperature (T) and the change in specific volume (\(V_2 - V_1\)) during the phase change. This matches the expression \( \frac{dP}{dT} = \frac{L}{T (V_2-V_1)} \). Therefore, (A) matches with (IV).

(B) Gibbs Function (G): The Gibbs free energy, often called the Gibbs function, is a thermodynamic potential that is particularly useful for processes occurring at constant pressure and temperature. It is defined as \( G = H - TS \), where H is enthalpy, T is temperature, and S is entropy. Since enthalpy itself is defined as \( H = U + PV \) (where U is internal energy), we can expand the definition of G to \( G = (U + PV) - TS \). This matches the expression \( U-TS + PV \). Therefore, (B) matches with (III).

(C) Enthalpy (H): Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy (U), which is the energy required to create the system, and the amount of energy required to make room for it by displacing its environment (PV). It is defined as the sum \( H = U + PV \). Therefore, (C) matches with (II).

(D) Adiabatic change in Perfect Gas: An adiabatic process is one in which no heat is exchanged with the surroundings. For a perfect gas undergoing a reversible adiabatic expansion or compression, the pressure (P) and volume (V) are related by the equation \( PV^\gamma = constant \), where \( \gamma \) is the adiabatic index or heat capacity ratio (\(C_p/C_v\)). Therefore, (D) matches with (I).


Step 3: Final Answer:

Based on the definitions and equations above, the correct pairings are:

(A) - (IV)
(B) - (III)
(C) - (II)
(D) - (I)

This combination corresponds to option (A).
Quick Tip: Remember the four main thermodynamic potentials: Internal Energy: U Enthalpy: H = U + PV Helmholtz Free Energy: F = U - TS Gibbs Free Energy: G = H - TS = U + PV - TS Knowing these definitions by heart is crucial for thermodynamics questions.

Step 1: Understanding the Concept:

The Rayleigh-Jeans law represents an attempt to describe the spectrum of thermal radiation emitted by a black body using the principles of classical statistical mechanics and electromagnetism. The question asks us to identify the correct statements describing the law's performance in predicting experimental observations.


Step 2: Detailed Explanation:

We will evaluate each statement based on the historical and physical context of the law.


(A) agrees well with experimental results at low frequencies. This statement is true. In the limit of low frequencies, the predictions of the Rayleigh-Jeans law align almost perfectly with the experimentally measured black-body radiation curve. This was its main success.

(B) agrees well with experimental results at longer wavelengths. This statement is also true. The relationship between frequency (\(f\)) and wavelength (\(\lambda\)) is \(c = f\lambda\), where \(c\) is the speed of light. This inverse relationship means that the low-frequency region corresponds to the long-wavelength region. Thus, this statement is physically equivalent to statement (A).

(C) shows ultra-violet catastrophe at higher frequencies. This statement is true and describes the law's famous failure. The formula predicts that the energy radiated by the black body should increase in proportion to the square of the frequency (\( E \propto f^2 \)). This implies that as the frequency increases towards the ultraviolet part of the spectrum and beyond, the radiated energy should approach infinity. This theoretical prediction of infinite energy at high frequencies starkly contradicted experimental data, which showed that the energy peaked and then decreased to zero. This dramatic failure of classical physics is known as the "ultraviolet catastrophe."

(D) agrees well with experimental results at higher frequencies. This statement is false. As explained above, the high-frequency (short-wavelength) region is precisely where the Rayleigh-Jeans law fails catastrophically. The disagreement between the law's prediction and reality in this region was a major impetus for the development of quantum mechanics.


Step 3: Final Answer:

The accurate descriptions of the Rayleigh-Jeans law are given in statements (A), (B), and (C). Statement (D) is incorrect. Therefore, the correct option is the one that includes (A), (B), and (C).
Quick Tip: Remember the key features of the black-body radiation laws: \textbf{Rayleigh-Jeans Law:} Classical, works for low frequencies/long wavelengths, fails at high frequencies (ultraviolet catastrophe). \textbf{Wien's Law:} Early quantum theory, works for high frequencies/short wavelengths, fails at low frequencies. \textbf{Planck's Law:} Full quantum theory, works perfectly across all frequencies.

Step 1: Understanding the Concept:

The problem requires calculating the total amount of thermal energy radiated by a hot object (a filament) over a specific time period. This is governed by the Stefan-Boltzmann law, which quantifies the rate of energy radiation (power) as a function of the object's temperature, surface area, and emissivity.


Step 2: Key Formula or Approach:

First, we calculate the power (P), which is the energy radiated per second, using the Stefan-Boltzmann law for a real (non-blackbody) object:
\[ P = e \sigma A T^4 \]
Here, \(e\) is the emissivity, \(\sigma\) is the Stefan-Boltzmann constant, A is the surface area, and T is the absolute temperature in Kelvin.
Second, we find the total energy (E) radiated over a time interval (t) by multiplying the power by the time:
\[ E = P \times t \]

Step 3: Detailed Explanation:

1. List all given quantities with their units:

Emissivity, \( e = 0.85 \) (dimensionless).

Stefan's constant, \( \sigma = 5.7 \times 10^{-8} \, J \cdot m^{-2} \cdot s^{-1} \cdot K^{-4} \).

Surface area, \( A = 4 \times 10^{-5} \, m^2 \).

Temperature, \( T = 2000 \, K \).

Time interval, \( t = 1 \, minute = 60 \, seconds \).


2. First, compute the power (P) radiated by the filament:
\[ P = (0.85) \times (5.7 \times 10^{-8}) \times (4 \times 10^{-5}) \times (2000)^4 \]
Calculate the temperature term first: \( (2000)^4 = (2 \times 10^3)^4 = 16 \times 10^{12} \, K^4 \).
Now, multiply all the numerical parts and the powers of ten separately: \[ P = (0.85 \times 5.7 \times 4 \times 16) \times (10^{-8} \times 10^{-5} \times 10^{12}) \] \[ P = 310.08 \times 10^{-13 + 12} = 310.08 \times 10^{-1} = 31.008 \, J/s or Watts \]

3. Next, calculate the total energy (E) radiated in one minute:
\[ E = P \times t = 31.008 \, J/s \times 60 \, s \approx 1860.5 \, J \]
This calculated value (1860.5 J) does not match any of the options. This often happens in exam questions due to typos in the provided constants or options. Let's examine the options for clues. Option (D) is 1641.6 J and option (B) is 27.36 J. Notice that \( 27.36 \times 60 = 1641.6 \). This strongly suggests that the intended power was 27.36 W (option B) and the intended total energy was 1641.6 J (option D). The question asks for energy per minute, making option (D) the target answer. The discrepancy likely arises from using a slightly different value for \(\sigma\) (e.g., \(5.67 \times 10^{-8}\) might be intended, or there is another typo). Assuming the intended answer is among the options, we conclude:
Intended Power \( P = 27.36 \, W \).
Intended Energy \( E = P \times t = 27.36 \, J/s \times 60 \, s = 1641.6 \, J \).


Step 4: Final Answer:

Based on the high probability of a typo in the provided constant and the relationship between the options, the intended total energy radiated in one minute is 1641.6 J.
Quick Tip: In exam questions with numerical calculations, if your result doesn't match any option, first double-check your own math. If it's correct, look for relationships between the options. Here, one option is 60 times another, a strong hint that they represent power (W or J/s) and energy per minute (J), respectively. This can guide you to the intended answer even if the problem data is slightly off.

Step 1: Understanding the Concept:

This problem deals with the Compton effect, a cornerstone of quantum physics. It describes the scattering of high-energy photons (like X-rays) by free or loosely bound electrons. A key outcome of this interaction is that the scattered photon has less energy, and therefore a longer wavelength, than the incident photon. The change in wavelength is dependent on the angle at which the photon is scattered.


Step 2: Key Formula or Approach:

The Compton shift formula quantifies the increase in wavelength (\( \Delta\lambda \)) as a function of the scattering angle (\(\theta\)):
\[ \Delta\lambda = \lambda' - \lambda = \lambda_c (1 - \cos\theta) \]
where:
- \( \lambda' \) is the wavelength of the scattered photon.
- \( \lambda \) is the wavelength of the incident photon.
- \( \lambda_c \) is the Compton wavelength of the electron, a constant value given by \(h/(m_e c)\).
- \( \theta \) is the angle between the direction of the incident and scattered photon.


Step 3: Detailed Explanation:

1. Identify and list the given values:

Incident wavelength, \( \lambda = 15 \) pm.

Compton wavelength, \( \lambda_c = 2.426 \) pm.

Scattering angle, \( \theta = 60^\circ \).


2. First, calculate the increase in wavelength (\( \Delta\lambda \)):

We need the value of \( \cos(60^\circ) \), which from trigonometry is exactly 0.5.

Substitute the known values into the Compton shift formula:
\[ \Delta\lambda = 2.426 \, pm \times (1 - \cos(60^\circ)) \] \[ \Delta\lambda = 2.426 \, pm \times (1 - 0.5) \] \[ \Delta\lambda = 2.426 \, pm \times 0.5 = 1.213 \, pm \]
This value represents how much longer the wavelength becomes after scattering.

3. Finally, calculate the new wavelength of the scattered photon (\( \lambda' \)):

The scattered wavelength is the sum of the initial wavelength and the Compton shift.
\[ \lambda' = \lambda + \Delta\lambda \] \[ \lambda' = 15 \, pm + 1.213 \, pm = 16.213 \, pm \]

Step 4: Final Answer:

After scattering through an angle of 60°, the new wavelength of the X-rays is 16.213 pm.
Quick Tip: Remember that in Compton scattering, the scattered photon always has a longer wavelength (and therefore lower energy) than the incident photon, except for forward scattering (\(\theta=0\)), where the wavelength does not change. This helps you eliminate any options that suggest a decrease in wavelength.

Step 1: Understanding the Concept:

The photoelectric effect provided critical evidence for the quantization of light into discrete packets of energy called photons. The question asks to evaluate several statements about this phenomenon based on the photon model proposed by Einstein.


Step 2: Detailed Explanation:

Let's analyze each statement in the context of Einstein's photoelectric equation, \( KE_{max} = hf - \phi \), where \(KE_{max}\) is the maximum kinetic energy of an emitted electron, \(h\) is Planck's constant, \(f\) is the light frequency, and \(\phi\) is the work function of the material.

(A) There is no time interval... This is a correct experimental observation. In the photon model, the effect is instantaneous because the energy of a single photon is transferred to a single electron in a discrete event. If the photon's energy (\(hf\)) is sufficient to overcome the work function, the electron is ejected immediately. This contrasts with the classical wave model, which predicted a measurable time delay for an electron to absorb enough energy.

(B) Higher the frequency of light, more is the kinetic energy... This is a correct statement and a direct consequence of the photoelectric equation. The equation shows that \(KE_{max}\) increases linearly with the frequency \(f\), as long as \(f\) is above the threshold frequency (\(f > \phi/h\)). A higher frequency photon carries more energy, and this excess energy (after subtracting the work function) becomes the kinetic energy of the photoelectron.

(C) A bright light yields more photo-electrons... This is also correct. The intensity (brightness) of a light beam is proportional to the number of photons it delivers per second. A brighter light means more photons are striking the surface per second. Since each photon can eject at most one electron, a higher flux of photons results in a higher rate of electron emission, which is observed as a larger photoelectric current.

(D) Blue light emits slower electrons than red light. This statement is incorrect. In the visible spectrum, blue light has a higher frequency than red light (\(f_{blue} > f_{red}\)). According to the photoelectric equation, the higher frequency of blue light will impart a *greater* maximum kinetic energy to the photoelectrons compared to red light (assuming both can cause emission). Greater kinetic energy means the electrons will have higher maximum speeds. Therefore, blue light emits faster electrons than red light.


Step 3: Final Answer:

The correct statements describing the photoelectric effect are (A), (B), and (C). Statement (D) is incorrect. Therefore, the correct option is (A).
Quick Tip: To remember the photoelectric effect laws, think in terms of photons: \textbf{Frequency (Energy of one photon):} Determines the kinetic energy of the ejected electron (\(KE = hf - \phi\)). Higher frequency \(\rightarrow\) higher KE. \textbf{Intensity (Number of photons):} Determines the number of ejected electrons. Higher intensity \(\rightarrow\) more electrons (more current).

Step 1: Understanding the Concept:

The problem asks to identify the mathematical relationship between the four roots of a given quartic equation. A powerful tool for this is to assume a certain relationship (like an Arithmetic Progression) and then use Vieta's formulas, which connect the polynomial's coefficients to the sums and products of its roots, to verify the assumption.


Step 2: Key Formula or Approach:

For a quartic polynomial \( x^4 + Bx^3 + Cx^2 + Dx + E = 0 \), Vieta's formulas state:
- Sum of roots: \( \sum r_i = -B \)
- Product of roots: \( r_1 r_2 r_3 r_4 = E \)
If the roots are in an Arithmetic Progression (AP), we can represent them symmetrically as \( a-3d, a-d, a+d, a+3d \), where 'a' is the arithmetic mean and '2d' is the common difference between consecutive terms in the inner pair.


Step 3: Detailed Explanation:

The given polynomial is \( x^4 - 20x^3 + 140x^2 - 400x + 384 = 0 \).
1. Assume the roots are in Arithmetic Progression (AP):

Let the four roots be \( (a-3d), (a-d), (a+d), (a+3d) \).
2. Use the Sum of Roots to find 'a':

According to Vieta's formulas, the sum of the roots is equal to the negative of the coefficient of the \(x^3\) term. \[ (a-3d) + (a-d) + (a+d) + (a+3d) = -(-20) = 20 \]
The 'd' terms cancel out, simplifying the equation: \[ 4a = 20 \implies a = 5 \]
This tells us the arithmetic mean of the roots is 5. The roots are of the form \( 5-3d, 5-d, 5+d, 5+3d \).
3. Use the Product of Roots to find 'd':

The product of the roots is equal to the constant term. \[ (a-3d)(a+d)(a-d)(a+3d) = 384 \]
Substitute \(a=5\): \[ (5-3d)(5+3d)(5-d)(5+d) = 384 \]
Using the difference of squares formula, \( (x-y)(x+y) = x^2 - y^2 \): \[ (25 - 9d^2)(25 - d^2) = 384 \]
To solve this equation, let \( y = d^2 \). \[ (25 - 9y)(25 - y) = 384 \]
Expand the left side: \[ 625 - 25y - 225y + 9y^2 = 384 \] \[ 9y^2 - 250y + 241 = 0 \]
This is a quadratic equation for \(y\). We can test simple integer values. Let's test if \(y=1\) is a solution: \[ 9(1)^2 - 250(1) + 241 = 9 - 250 + 241 = 0 \]
The equation holds true, so \( y = 1 \) is a root. This means \( d^2 = 1 \), which gives \( d = \pm 1 \).
4. Determine the roots:

Choosing \( d = 1 \) (choosing \( d = -1 \) would give the same set of roots in reverse order), the four roots are:
- \( r_1 = a-3d = 5-3(1) = 2 \)
- \( r_2 = a-d = 5-1 = 4 \)
- \( r_3 = a+d = 5+1 = 6 \)
- \( r_4 = a+3d = 5+3(1) = 8 \)
The roots are 2, 4, 6, and 8. This is clearly a set of numbers in Arithmetic Progression with a common difference of 2. Our assumption was correct.


Step 4: Final Answer:

The roots of the given equation are in an Arithmetic Progression.
Quick Tip: For questions asking about the nature of roots (AP, GP, etc.), assuming the pattern and using the sum and product of roots from Vieta's formulas is usually the fastest method. For a quartic equation in AP, the sum of roots immediately gives you the mean value 'a'.

Step 1: Understanding the Concept:

The problem deals with the \(n^{th}\) roots of unity, which are the complex number solutions to the equation \( x^n - 1 = 0 \). We are asked to find the value of a specific product involving all of these roots except for the root \(x=1\).


Step 2: Key Formula or Approach:

A fundamental theorem of algebra states that a polynomial can be factored in terms of its roots. The polynomial \( P(x) = x^n - 1 \) has the roots \(1, \alpha_1, \alpha_2, \ldots, \alpha_{n-1}\). Therefore, we can write it in factored form:
\[ x^n - 1 = (x-1)(x-\alpha_1)(x-\alpha_2)\ldots(x-\alpha_{n-1}) \]
This identity is true for all complex numbers \(x\). We can manipulate this identity to isolate the product we need to evaluate.


Step 3: Detailed Explanation:

From the factored form of the polynomial, we can divide both sides by the term \( (x-1) \) (for \(x \neq 1\)): \[ \frac{x^n - 1}{x - 1} = (x-\alpha_1)(x-\alpha_2)\ldots(x-\alpha_{n-1}) \]
The left side is the formula for the sum of a finite geometric series: \[ x^{n-1} + x^{n-2} + \ldots + x + 1 = (x-\alpha_1)(x-\alpha_2)\ldots(x-\alpha_{n-1}) \]
This creates a new polynomial identity. The expression we need to evaluate is \( (1-\alpha_1)(1-\alpha_2)\ldots(1-\alpha_{n-1}) \). We can see that this is exactly the value of the polynomial on the right-hand side when we substitute \( x = 1 \).
Let's perform this substitution: \[ (1-\alpha_1)(1-\alpha_2)\ldots(1-\alpha_{n-1}) = 1^{n-1} + 1^{n-2} + \ldots + 1^1 + 1^0 \]
The right-hand side is a sum of powers of 1. Each term is simply 1. We need to count the number of terms in the sum. The powers range from \(n-1\) down to 0, so there are \((n-1) - 0 + 1 = n\) terms in total. \[ (1-\alpha_1)(1-\alpha_2)\ldots(1-\alpha_{n-1}) = \underbrace{1 + 1 + \ldots + 1 + 1}_{n times} \]
Therefore, the value of the sum is \(n\).

Step 4: Final Answer:

The value of the product \( (1-\alpha_1)(1-\alpha_2)(1-\alpha_3)\ldots(1-\alpha_{n-1}) \) is equal to \( n \).
Quick Tip: This is a standard result related to roots of unity. The polynomial \( \Phi_n(x) = \frac{x^n-1}{x-1} \) is called the nth cyclotomic polynomial, and its roots are the primitive nth roots of unity (for prime n). The value of this polynomial at \(x=1\) gives the desired product, which is always \(n\).

Step 1: Understanding the Concept:

The rank of a matrix corresponds to the maximum number of linearly independent rows or columns. For a square \(n \times n\) matrix, the rank is less than \(n\) if and only if the rows (and columns) are linearly dependent. This condition of linear dependence is equivalent to the matrix being singular, which means its determinant is zero.


Step 2: Key Formula or Approach:

The given matrix is 3x3. The problem states that its rank is 2. This implies two things:
1. The rank is less than 3, so the determinant of the 3x3 matrix must be zero.
2. The rank is at least 2, meaning there must be at least one 2x2 submatrix (a minor) that has a non-zero determinant.
Our strategy will be to first confirm the rank is at least 2, and then set the determinant of the full 3x3 matrix to zero to solve for the unknown value \( \lambda \).


Step 3: Detailed Explanation:

Let the given matrix be denoted by A:
\[ A = \begin{pmatrix} 1 & 2 & 3
4 & 5 & 6
0 & 7 & \lambda \end{pmatrix} \]
1. Confirm that the rank is at least 2:

We can check the determinant of any 2x2 submatrix. Let's choose the top-left minor: \[ \det \begin{pmatrix} 1 & 2
4 & 5 \end{pmatrix} = (1)(5) - (2)(4) = 5 - 8 = -3 \]
Since this determinant is -3 (which is non-zero), the first two rows and columns are linearly independent. This confirms that the rank of matrix A is at least 2.

2. Impose the condition that the rank is exactly 2:

For the rank to be exactly 2 (and not 3), the entire 3x3 matrix must be singular. This requires its determinant to be zero. We calculate the determinant of A, for instance by expanding along the first column:
\[ \det(A) = 1 \cdot \det\begin{pmatrix} 5 & 6
7 & \lambda \end{pmatrix} - 4 \cdot \det\begin{pmatrix} 2 & 3
7 & \lambda \end{pmatrix} + 0 \cdot \det\begin{pmatrix} 2 & 3
5 & 6 \end{pmatrix} = 0 \]
Now, we compute the 2x2 determinants: \[ 1 \cdot (5\lambda - (6)(7)) - 4 \cdot (2\lambda - (3)(7)) + 0 = 0 \] \[ (5\lambda - 42) - 4(2\lambda - 21) = 0 \]
Distribute the terms: \[ 5\lambda - 42 - 8\lambda + 84 = 0 \]
Combine like terms: \[ (5\lambda - 8\lambda) + (-42 + 84) = 0 \] \[ -3\lambda + 42 = 0 \]
Solve for \( \lambda \): \[ 3\lambda = 42 \] \[ \lambda = \frac{42}{3} = 14 \]

Step 4: Final Answer:

For the rank of the matrix to be 2, the value of \( \lambda \) must be 14.
Quick Tip: For a 3x3 matrix, the condition "rank = 2" is almost always equivalent to "determinant = 0". It's a quick and reliable way to solve such problems in an exam setting.

Step 1: Understanding the Concept:

The fundamental principle we will employ is the Cayley-Hamilton theorem. This powerful theorem asserts that every square matrix is a "root" of its own characteristic polynomial. Therefore, our primary objective is to formulate the characteristic equation specifically for the provided matrix A.


Step 2: Key Formula or Approach:

The characteristic equation of a matrix A is formally given by the condition \( \det(A - \lambda I) = 0 \), where \( \lambda \) represents an eigenvalue and I is the identity matrix of the same dimension. For a 3x3 matrix, this relationship can be expressed through a highly efficient shortcut formula:
\[ \lambda^3 - (tr(A))\lambda^2 + (sum of cofactors of diagonal elements)\lambda - \det(A) = 0 \]
We will systematically compute each coefficient in this formula to construct the final equation.


Step 3: Detailed Explanation:

Given the matrix \( A = \begin{pmatrix} 2 & 0 & -1
5 & 1 & 0
0 & 1 & 3 \end{pmatrix} \).

1. Calculate the Trace of A:

The trace, denoted as tr(A), is the sum of the elements on the main diagonal.
\[ tr(A) = 2 + 1 + 3 = 6 \]

2. Calculate the Sum of the Cofactors of the Diagonal Elements:

The cofactor of an element \(a_{ij}\) is calculated by taking the determinant of the submatrix formed by removing the i-th row and j-th column, multiplied by \((-1)^{i+j}\).
For \(a_{11}\) (2), the cofactor is \( M_{11} = \det\begin{pmatrix} 1 & 0
1 & 3 \end{pmatrix} = (1)(3) - (0)(1) = 3 \).

For \(a_{22}\) (1), the cofactor is \( M_{22} = \det\begin{pmatrix} 2 & -1
0 & 3 \end{pmatrix} = (2)(3) - (-1)(0) = 6 \).

For \(a_{33}\) (3), the cofactor is \( M_{33} = \det\begin{pmatrix} 2 & 0
5 & 1 \end{pmatrix} = (2)(1) - (0)(5) = 2 \).

The sum of these cofactors is \( 3 + 6 + 2 = 11 \).


3. Calculate the Determinant of A:

We compute the determinant by expanding along the first row:
\[ \det(A) = 2 \begin{vmatrix} 1 & 0
1 & 3 \end{vmatrix} - 0 \begin{vmatrix} 5 & 0
0 & 3 \end{vmatrix} + (-1) \begin{vmatrix} 5 & 1
0 & 1 \end{vmatrix} \] \[ \det(A) = 2(3 - 0) - 0 + (-1)(5 - 0) = 6 - 5 = 1 \]

4. Form the Characteristic Equation:

Substituting the calculated values (trace=6, sum of cofactors=11, determinant=1) into the formula:
\[ \lambda^3 - (6)\lambda^2 + (11)\lambda - (1) = 0 \] \[ \lambda^3 - 6\lambda^2 + 11\lambda - 1 = 0 \]

5. Apply the Cayley-Hamilton Theorem:

The theorem allows us to substitute the matrix A for \( \lambda \) and \( I \) (the identity matrix) for the constant term's multiplier.
\[ A^3 - 6A^2 + 11A - 1 \cdot I = 0 \]

Step 4: Final Answer:

Thus, the specific matrix equation that A must satisfy is \( A^3 - 6A^2 + 11A - I = 0 \).
Quick Tip: Using the formula \( \lambda^3 - (tr(A))\lambda^2 + (sum of diagonal cofactors)\lambda - \det(A) = 0 \) is a much faster way to find the characteristic equation for a 3x3 matrix than expanding \( \det(A - \lambda I) \) directly, reducing the chance of algebraic errors.

Step 1: Understanding the Concept:

A system of linear equations is deemed inconsistent when it possesses no possible solution. This situation arises during the process of Gaussian elimination when we encounter a logically impossible statement, such as \(0x + 0y + 0z = k\), where \(k\) is a non-zero number. From a matrix perspective, inconsistency means the rank of the coefficient matrix A is strictly less than the rank of the augmented matrix [A|B].


Step 2: Key Formula or Approach:

Our method involves representing the system of equations as an augmented matrix. We will then apply a sequence of elementary row operations to transform this matrix into its row echelon form, which will make the conditions for inconsistency clear. The initial augmented matrix is:
\[ \left[ \begin{array}{ccc|c} 1 & 2 & 1 & 6
1 & 4 & 3 & 10
1 & 4 & \lambda & \mu \end{array} \right] \]

Step 3: Detailed Explanation:

We proceed with row reduction to simplify the matrix.

First, to create zeros in the first column below the leading 1, we perform the operations:

1. \( R_2 \rightarrow R_2 - R_1 \)

2. \( R_3 \rightarrow R_3 - R_1 \)

This yields the new matrix: \[ \left[ \begin{array}{ccc|c} 1 & 2 & 1 & 6
0 & 2 & 2 & 4
0 & 2 & \lambda-1 & \mu-6 \end{array} \right] \]
Next, to create a zero in the second column of the third row, we apply \( R_3 \rightarrow R_3 - R_2 \):
\[ \left[ \begin{array}{ccc|c} 1 & 2 & 1 & 6
0 & 2 & 2 & 4
0 & 0 & (\lambda-1)-2 & (\mu-6)-4 \end{array} \right] \]
Simplifying the last row gives the final echelon form: \[ \left[ \begin{array}{ccc|c} 1 & 2 & 1 & 6
0 & 2 & 2 & 4
0 & 0 & \lambda-3 & \mu-10 \end{array} \right] \]
The final row of this matrix corresponds to the equation \( 0x + 0y + (\lambda-3)z = \mu-10 \).

For this system to be inconsistent, this final equation must represent a contradiction. A contradiction occurs when the coefficient of the variable is zero, but the constant on the other side is non-zero.

The coefficient of z becomes zero if:
\[ \lambda - 3 = 0 \implies \lambda = 3 \]
The constant on the right side is non-zero if:
\[ \mu - 10 \neq 0 \implies \mu \neq 10 \]

Step 4: Final Answer:

For the system to have no solution, the conditions must be \( \lambda = 3 \) and simultaneously \( \mu \neq 10 \). This directly corresponds to option (C).
Quick Tip: For a system of linear equations to be inconsistent, you need a row in the echelon form of the augmented matrix that looks like \([0 \ 0 \ \ldots \ 0 \ | \ k]\) where \(k \neq 0\). This single condition encapsulates the requirement for no solution.

Step 1: Understanding the Concept:

This problem asks us to systematically assess the validity of five different statements involving matrix operations, specifically matrix multiplication and the calculation of matrix powers. We will analyze each claim one by one to determine its correctness.


Step 2: Detailed Explanation:


Statement (A): Evaluate the claim \( B^n = \begin{pmatrix} 1 & nq
0 & 1 \end{pmatrix} \).
Let's check the base case for n=2: \( B^2 = \begin{pmatrix} 1 & q
0 & 1 \end{pmatrix} \begin{pmatrix} 1 & q
0 & 1 \end{pmatrix} = \begin{pmatrix} 1\cdot1+q\cdot0 & 1\cdot q+q\cdot1
0\cdot1+1\cdot0 & 0\cdot q+1\cdot1 \end{pmatrix} = \begin{pmatrix} 1 & 2q
0 & 1 \end{pmatrix} \). The pattern holds for n=2. This suggests the formula is correct, which can be formally proven using mathematical induction. This statement is correct.
Statement (B): Evaluate the claim \( A^n = \begin{pmatrix} p^n & q\frac{p^n-1}{p-1}
0 & 1 \end{pmatrix} \).
For n=2, we calculate: \( A^2 = \begin{pmatrix} p & q
0 & 1 \end{pmatrix} \begin{pmatrix} p & q
0 & 1 \end{pmatrix} = \begin{pmatrix} p^2 & pq+q
0 & 1 \end{pmatrix} = \begin{pmatrix} p^2 & q(p+1)
0 & 1 \end{pmatrix} \).
Now we check the given formula for n=2: The top-right element is \( q\frac{p^2-1}{p-1} = q\frac{(p-1)(p+1)}{p-1} = q(p+1) \). Since the calculated value matches the formula's result, the pattern is consistent. The top-right element is the sum of a geometric series \( q(1+p+p^2+...+p^{n-1}) \). This statement is correct.
Statement (C): Evaluate the claim \( AB = \begin{pmatrix} p & pq+q
0 & 1 \end{pmatrix} \).
We perform the matrix multiplication directly:
\( AB = \begin{pmatrix} p & q
0 & 1 \end{pmatrix} \begin{pmatrix} 1 & q
0 & 1 \end{pmatrix} = \begin{pmatrix} p(1)+q(0) & p(q)+q(1)
0(1)+1(0) & 0(q)+1(1) \end{pmatrix} = \begin{pmatrix} p & pq+q
0 & 1 \end{pmatrix} \).
The calculation confirms the statement. This statement is correct.
Statement (D): Evaluate the claim \( B^{n-1} = \begin{pmatrix} 1 & (n+1)q
0 & 1 \end{pmatrix} \).
Based on our confirmed result from statement (A), we know that \( B^k = \begin{pmatrix} 1 & kq
0 & 1 \end{pmatrix} \). By substituting \(k = n-1\), we get \( B^{n-1} = \begin{pmatrix} 1 & (n-1)q
0 & 1 \end{pmatrix} \). This contradicts the given statement. This statement is therefore incorrect.
Statement (E): Evaluate the claim \( AB^n = \begin{pmatrix} p & (np+1)q
0 & 1 \end{pmatrix} \).
Using the verified formula for \(B^n\) from statement (A):
\( AB^n = \begin{pmatrix} p & q
0 & 1 \end{pmatrix} \begin{pmatrix} 1 & nq
0 & 1 \end{pmatrix} = \begin{pmatrix} p(1)+q(0) & p(nq)+q(1)
0(1)+1(0) & 0(nq)+1(1) \end{pmatrix} = \begin{pmatrix} p & npq+q
0 & 1 \end{pmatrix} \).
Factoring q from the top-right element gives \( \begin{pmatrix} p & (np+1)q
0 & 1 \end{pmatrix} \).
The calculation verifies the statement. This statement is correct.


Step 3: Final Answer:

Upon review, statements (A), (B), (C), and (E) have been verified as correct, whereas statement (D) is incorrect. Consequently, the correct choice is the option that includes only (A), (B), (C), and (E).
Quick Tip: When dealing with matrix powers, especially for 2x2 matrices, look for simple patterns. Proving by induction is the formal method, but for multiple-choice questions, testing for n=2 or n=3 is often sufficient to verify or disprove a given formula.

Step 1: Understanding the Concept:

A well-known condition for three distinct complex numbers \(z_1, z_2, z_3\) to represent the vertices of an equilateral triangle is the algebraic relationship \( z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1 \). For this specific problem, the vertices are specified as \(z_1\), \(z_2\), and the origin (0), so we can simplify the general condition by setting \(z_3 = 0\).


Step 2: Key Formula or Approach:

An alternative and more intuitive method relies on the geometric interpretation of complex multiplication as rotation. If the points 0, \(z_1\), and \(z_2\) form an equilateral triangle, it implies that the vector representing \(z_2\) can be obtained by rotating the vector for \(z_1\) around the origin by an angle of \(60^\circ\) (\(\pi/3\) radians), either counter-clockwise or clockwise. This geometric fact is expressed algebraically as:
\[ z_2 = z_1 \cdot e^{\pm i\pi/3} \]

Step 3: Detailed Explanation:

Starting from the rotation formula, we can isolate the ratio of the complex numbers:
\[ \frac{z_2}{z_1} = e^{\pm i\pi/3} \]
Using Euler's formula, we can expand the exponential term: \[ \frac{z_2}{z_1} = \cos(\pm\pi/3) + i\sin(\pm\pi/3) = \cos(\pi/3) \pm i\sin(\pi/3) = \frac{1}{2} \pm i\frac{\sqrt{3}}{2} \]
Let's define a new variable \( \omega = \frac{z_2}{z_1} \). This variable has two possible values, which are the roots of a quadratic equation. We can find this equation by considering the sum and product of these roots.
Sum of the roots: \( \left(\frac{1}{2} + i\frac{\sqrt{3}}{2}\right) + \left(\frac{1}{2} - i\frac{\sqrt{3}}{2}\right) = 1 \).

Product of the roots: \( \left(\frac{1}{2} + i\frac{\sqrt{3}}{2}\right)\left(\frac{1}{2} - i\frac{\sqrt{3}}{2}\right) = \left(\frac{1}{2}\right)^2 - \left(i\frac{\sqrt{3}}{2}\right)^2 = \frac{1}{4} - \left(-\frac{3}{4}\right) = 1 \).

A quadratic equation with these roots is \( \omega^2 - (sum of roots)\omega + (product of roots) = 0 \).
\[ \omega^2 - 1\omega + 1 = 0 \]
Now, we substitute the definition of \( \omega \) back into this equation:
\[ \left(\frac{z_2}{z_1}\right)^2 - \left(\frac{z_2}{z_1}\right) + 1 = 0 \]
To eliminate the fraction, we multiply the entire equation by \(z_1^2\) (we can assume \(z_1 \neq 0\), otherwise the points are not distinct vertices):
\[ z_1^2 \cdot \left(\frac{z_2^2}{z_1^2}\right) - z_1^2 \cdot \left(\frac{z_2}{z_1}\right) + z_1^2 \cdot 1 = 0 \] \[ z_2^2 - z_1 z_2 + z_1^2 = 0 \]
Rearranging the terms to match the standard form gives the final condition:
\[ z_1^2 + z_2^2 - z_1 z_2 = 0 \]

Step 4: Final Answer:

The necessary and sufficient condition for the origin, \(z_1\), and \(z_2\) to be the vertices of an equilateral triangle is \( z_1^2 + z_2^2 - z_1 z_2 = 0 \).
Quick Tip: Remember the general condition for three points \(z_1, z_2, z_3\) forming an equilateral triangle: \(z_1^2 + z_2^2 + z_3^2 = z_1z_2 + z_2z_3 + z_3z_1\). If one of the points is the origin, you can simply set it to zero in this formula to get the required condition quickly.

Step 1: Understanding the Concept:

This problem requires an examination of the core properties of the exponential function when its argument is a complex number, \(e^z\). We will analyze each statement based on the definition \(e^z = e^{x+iy} = e^x(\cos y + i\sin y)\).


Step 2: Detailed Explanation:


(A) \(e^z\) is never zero: Let the complex number be \(z = x+iy\). The exponential function is \(e^z = e^{x+iy} = e^x e^{iy}\). The magnitude (or modulus) of this complex number is \(|e^z| = |e^x| |e^{iy}|\). Since \(x\) is real, \(e^x\) is always a positive real number. The magnitude of \(e^{iy}\) is \(|\cos y + i\sin y| = \sqrt{\cos^2 y + \sin^2 y} = 1\). Thus, \(|e^z| = e^x\). As \(e^x\) is never zero for any finite real \(x\), the magnitude of \(e^z\) is never zero, which means \(e^z\) itself can never be zero. This statement is True.
(B) \(|e^{ix}|=1\) if x is real: This is a direct application of Euler's formula, which states \(e^{ix} = \cos(x) + i\sin(x)\). The magnitude of this complex number is defined as \(|e^{ix}| = \sqrt{(real part)^2 + (imaginary part)^2} = \sqrt{\cos^2(x) + \sin^2(x)}\). Due to the fundamental trigonometric identity, this simplifies to \(\sqrt{1} = 1\). This statement is True.
(C) \(e^z = 1\) if z is an integral multiple of \(2\pi i\): Let \(z = 2n\pi i\), where \(n\) is any integer. We evaluate \(e^z = e^{2n\pi i}\). Using Euler's formula, this becomes \(\cos(2n\pi) + i\sin(2n\pi)\). For any integer \(n\), the cosine of an even multiple of \(\pi\) is 1, and the sine is 0. So, \(e^z = 1 + i(0) = 1\). This statement is True.
(D) \(e^{z_1} = e^{z_2}\) ...: The equality \(e^{z_1} = e^{z_2}\) implies \(e^{z_1}/e^{z_2} = 1\), which means \(e^{z_1 - z_2} = 1\). From the property verified in statement (C), this is true if and only if the exponent \(z_1 - z_2\) is an integer multiple of \(2\pi i\). That is, \(z_1 - z_2 = 2n\pi i\) for some integer \(n\). The statement given includes an extraneous factor of \(1/\sqrt{3}\), making it incorrect. This statement is False.
(E) \(|e^z| > e^z\) for \(z \neq 0\): This statement involves an inequality between a real number (\(|e^z|\)) and a complex number (\(e^z\)). The greater-than/less-than relations are not defined for complex numbers unless they are purely real. For example, if \(z=2\) (a real number), then \(|e^2| = e^2\), and the statement \(e^2 > e^2\) is false. Therefore, the statement is not universally true. This statement is False.


Step 3: Final Answer:

Based on the analysis, statements (A), (B), and (C) are correct. Therefore, the option that correctly groups these true statements is (C).
Quick Tip: The complex exponential function \(e^z\) has a period of \(2\pi i\). This is a fundamental property. It means \(e^{z + 2n\pi i} = e^z\) for any integer \(n\). This fact is the basis for both statements (C) and (D).

Step 1: Understanding the Concept:

This problem requires solving four distinct complex number equations. Each solution set must then be correctly matched to its corresponding description or list of roots from List-II. We will solve each equation methodically.


Step 2: Detailed Explanation:


(A) \((z + 1)^{2n} + (z - 1)^{2n} = 0\): We rearrange the equation to \( (z+1)^{2n} = -(z-1)^{2n} \), which gives \( \left(\frac{z+1}{z-1}\right)^{2n} = -1 \). The complex number -1 can be written in polar form as \( e^{i(\pi + 2k\pi)} = e^{i(2k+1)\pi} \). Taking the \(2n\)-th root, we get \(\frac{z+1}{z-1} = e^{i\frac{(2k+1)\pi}{2n}}\). Let \(w = \frac{z+1}{z-1}\). Note that \(|w|=1\), so \(w\) lies on the unit circle. Solving for z, we find \(z(w-1) = w+1\), so \(z = \frac{w+1}{w-1}\). If \(w=e^{i\theta}\), then \(z = \frac{e^{i\theta}+1}{e^{i\theta}-1} = \frac{e^{i\theta/2}(e^{i\theta/2}+e^{-i\theta/2})}{e^{i\theta/2}(e^{i\theta/2}-e^{-i\theta/2})} = \frac{2\cos(\theta/2)}{2i\sin(\theta/2)} = -i\cot(\theta/2)\). As \(\cot(\theta/2)\) is a real number, the roots for z are all purely imaginary. (A) matches (II).
(B) \( z^5 + z^4 + z^3 + z^2 + z + 1 = 0 \): This equation represents a finite geometric series. We can use the sum formula: \(S_n = \frac{a(r^n-1)}{r-1}\). Here, the sum is \(\frac{1(z^6-1)}{z-1}=0\). This implies that \(z^6-1=0\), with the condition that \(z \neq 1\). The equation \(z^6=1\) asks for the six 6th roots of unity, which are \(e^{i2\pi k/6}\) for k=0,1,2,3,4,5. Since we must exclude \(z=1\) (the k=0 case), the roots are for k=1,2,3,4,5. These are \(e^{i\pi/3}, e^{i2\pi/3}, e^{i\pi}, e^{i4\pi/3}, e^{i5\pi/3}\). Converting to Cartesian coordinates gives \(\frac{1}{2}+i\frac{\sqrt{3}}{2}, -\frac{1}{2}+i\frac{\sqrt{3}}{2}, -1, -\frac{1}{2}-i\frac{\sqrt{3}}{2}, \frac{1}{2}-i\frac{\sqrt{3}}{2}\). This collection of roots is described by \(-1, \pm\frac{1}{2} \pm i\frac{\sqrt{3}}{2}\). (B) matches (III).
(C) \( (z-1)^5 + z^5 = 0 \): This can be written as \( z^5 = -(z-1)^5 \), or \( (\frac{z}{z-1})^5 = -1 \). Taking the 5th root, \( \frac{z}{z-1} = e^{i\frac{(2k+1)\pi}{5}} \). Let \(w_k = e^{i\frac{(2k+1)\pi}{5}}\). Then \( z = w_k(z-1) \implies z = w_k z - w_k \implies z(1-w_k)=-w_k \implies z = \frac{-w_k}{1-w_k} = \frac{w_k}{w_k-1}\). An alternative manipulation gives \(\frac{z-1}{z} = 1 - \frac{1}{z} = e^{i\frac{(2k+1)\pi}{5}}\), so \(\frac{1}{z} = 1 - e^{i\theta_k}\) where \(\theta_k = \frac{(2k+1)\pi}{5}\). Then \(z = \frac{1}{1-e^{i\theta_k}} = \frac{1}{1-(\cos\theta_k+i\sin\theta_k)} = \frac{1}{(1-\cos\theta_k)-i\sin\theta_k} = \frac{1}{2\sin^2(\theta_k/2)-i2\sin(\theta_k/2)\cos(\theta_k/2)}\). After simplification, this becomes \(z = \frac{1}{2}[1+i\cot(\frac{\theta_k}{2})] = \frac{1}{2}[1+i\cot(\frac{(2k+1)\pi}{10})]\). For k=0,1,2,3,4, the values in the cotangent argument correspond to p=1,3,5,7,9. (C) matches (IV).
(D) \( z^5 + 1 = 0 \): This equation is equivalent to \(z^5 = -1\). In polar form, \(-1 = e^{i(\pi+2k\pi)}\). Therefore, the roots are \(z_k = (e^{i(2k+1)\pi})^{1/5} = e^{i(2k+1)\pi/5}\) for k=0,1,2,3,4.
For k=0: \(e^{i\pi/5} = \cos(\pi/5)+i\sin(\pi/5)\).
For k=1: \(e^{i3\pi/5} = \cos(3\pi/5)+i\sin(3\pi/5)\).
For k=2: \(e^{i5\pi/5} = e^{i\pi} = -1\).
For k=3: \(e^{i7\pi/5} = \cos(7\pi/5)+i\sin(7\pi/5) = \cos(3\pi/5)-i\sin(3\pi/5)\) (conjugate of k=1 root).
For k=4: \(e^{i9\pi/5} = \cos(9\pi/5)+i\sin(9\pi/5) = \cos(\pi/5)-i\sin(\pi/5)\) (conjugate of k=0 root).
The full set of roots is \(-1, \cos(\pi/5)\pm i\sin(\pi/5), \cos(3\pi/5)\pm i\sin(3\pi/5)\). (D) matches (I).


Step 3: Final Answer:

The derived pairings are: (A) with (II), (B) with (III), (C) with (IV), and (D) with (I). This unique combination corresponds to option (A).
Quick Tip: Problems involving sums or differences of powers of complex binomials, like \((z+a)^n \pm (z-a)^n = 0\), are often simplified by rearranging to \((\frac{z+a}{z-a})^n = \mp 1\) and then solving for z.

Step 1: Understanding the Concept:

The problem tests the conditions under which the principal value identity for the sum of two inverse tangent functions holds. The standard formula \( \tan^{-1}x + \tan^{-1}y = \tan^{-1}\left(\frac{x+y}{1-xy}\right) \) is not universally true; its validity is restricted by the product of the arguments \(x\) and \(y\).


Step 2: Key Formula or Approach:

The identity for the sum of inverse tangent functions is conditional: \[ \tan^{-1}x + \tan^{-1}y = \begin{cases} \tan^{-1}\left(\frac{x+y}{1-xy}\right) & if xy < 1
\pi + \tan^{-1}\left(\frac{x+y}{1-xy}\right) & if xy > 1 and x, y > 0
-\pi + \tan^{-1}\left(\frac{x+y}{1-xy}\right) & if xy > 1 and x, y < 0 \end{cases} \]
The equation presented in the question corresponds to the first case, so we must determine the values of \(n\) for which this specific condition is met.


Step 3: Detailed Explanation:

In the context of the given equation, we identify \(x = 3\) and \(y = n\).
The equation is presented in the form of the principal value identity. For this identity to be the correct one to use, the condition associated with it must be satisfied.
The required condition is: \[ xy < 1 \]
Substituting our specific values for \(x\) and \(y\): \[ (3)(n) < 1 \]
To solve for \(n\), we divide both sides of the inequality by 3: \[ n < \frac{1}{3} \]

Step 4: Final Answer:

Therefore, the provided equation is valid precisely for the values of \(n\) that satisfy the inequality \(n < \frac{1}{3}\).
Quick Tip: Always remember to check the condition \(xy < 1\) when applying the formula \(\tan^{-1}x + \tan^{-1}y = \tan^{-1}\left(\frac{x+y}{1-xy}\right)\). This is a common pitfall in trigonometry problems. The range of the principal value of \(\tan^{-1}\) is \( (-\pi/2, \pi/2) \), and this condition ensures the sum of the angles on the left also falls within a range that can be represented by a single \(\tan^{-1}\) function without adding or subtracting \(\pi\).

Step 1: Understanding the Concept:

Asymptotes are straight lines that a curve continuously approaches but never touches as the curve extends to infinity. For this algebraic curve, we are looking for horizontal and vertical asymptotes. Horizontal asymptotes describe the behavior of y as x approaches \(\pm\infty\), while vertical asymptotes describe the values of x where y approaches \(\pm\infty\).


Step 2: Key Formula or Approach:

To find the asymptotes, it's beneficial to first simplify the given equation.
The initial equation is: \( (x^2 - a^2)(y^2 - b^2) = a^2b^2 \)

Let's expand the left side of the equation: \[ x^2y^2 - b^2x^2 - a^2y^2 + a^2b^2 = a^2b^2 \]
Subtracting \(a^2b^2\) from both sides leaves a simplified form: \[ x^2y^2 - b^2x^2 - a^2y^2 = 0 \]
From this form, we can isolate \(y^2\) to find horizontal asymptotes and isolate \(x^2\) to find vertical asymptotes.


Step 3: Detailed Explanation:

1. Finding Horizontal Asymptotes (as \(x \to \pm\infty\)):

We rearrange the simplified equation to express \(y^2\) as a function of \(x\). \[ y^2(x^2 - a^2) = b^2x^2 \] \[ y^2 = \frac{b^2x^2}{x^2 - a^2} \]
To determine the behavior of y at infinity, we take the limit of \(y^2\) as \(x \to \infty\). We can divide the numerator and denominator by the highest power of x, which is \(x^2\): \[ \lim_{x\to\infty} y^2 = \lim_{x\to\infty} \frac{b^2x^2/x^2}{(x^2 - a^2)/x^2} = \lim_{x\to\infty} \frac{b^2}{1 - a^2/x^2} \]
As \(x \to \infty\), the term \(a^2/x^2\) approaches 0. \[ \lim_{x\to\infty} y^2 = \frac{b^2}{1-0} = b^2 \]
This tells us that as x becomes infinitely large, \(y^2\) approaches \(b^2\). This implies that y approaches \(\pm\sqrt{b^2}\), so \(y = \pm b\).
These are the two horizontal asymptotes: \(y = b\) and \(y = -b\).


2. Finding Vertical Asymptotes (where \(y \to \pm\infty\)):

Vertical asymptotes occur at x-values for which the function is undefined, typically because of division by zero. Looking at our expression for \(y^2\), this happens when the denominator is zero.
Set the denominator to zero: \[ x^2 - a^2 = 0 \]
Solving for x gives: \[ x^2 = a^2 \implies x = \pm a \]
These are the two vertical asymptotes: \(x = a\) and \(x = -a\).


Step 4: Final Answer:

The complete set of asymptotes for the curve consists of two horizontal lines, \(y = \pm b\), and two vertical lines, \(x = \pm a\).
Quick Tip: For algebraic curves, an easy way to find horizontal and vertical asymptotes is to set the coefficients of the highest power of x and y to zero, respectively, after clearing fractions. In \(x^2y^2 - b^2x^2 - a^2y^2 = 0\), the coefficient of \(y^2\) is \(x^2-a^2\). Setting this to zero gives the vertical asymptotes \(x=\pm a\). The coefficient of \(x^2\) is \(y^2-b^2\). Setting this to zero gives the horizontal asymptotes \(y=\pm b\).

Step 1: Understanding the Concept:

The task is to compute the surface area of a solid formed by rotating a parametrically defined curve around the y-axis. The formula for this involves integrating the circumference of the circle traced by a point \((x,y)\) on the curve, multiplied by the differential arc length of the curve.


Step 2: Key Formula or Approach:

When a parametric curve defined by \(x(t)\) and \(y(t)\) for \(a \le t \le b\) is revolved around the y-axis, the resulting surface area \(S\) is given by the integral: \[ S = \int_{a}^{b} 2\pi x(t) \, ds \]
The term \(ds\) represents the differential arc length and is calculated using the formula: \[ ds = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \]
Our process will be to first find the derivatives, then compute \(ds\), and finally set up and solve the integral for \(S\).


Step 3: Detailed Explanation:

1. Find the Derivatives of x(t) and y(t):

Using the product rule for differentiation: \( x = e^t \cos t \implies \frac{dx}{dt} = (e^t)'\cos t + e^t(\cos t)' = e^t\cos t - e^t\sin t = e^t(\cos t - \sin t) \)
\( y = e^t \sin t \implies \frac{dy}{dt} = (e^t)'\sin t + e^t(\sin t)' = e^t\sin t + e^t\cos t = e^t(\sin t + \cos t) \)


2. Compute the Arc Length Element \(ds\):

First, we find the sum of the squares of the derivatives: \[ \left(\frac{dx}{dt}\right)^2 = [e^t(\cos t - \sin t)]^2 = e^{2t}(\cos^2 t - 2\sin t\cos t + \sin^2 t) = e^{2t}(1 - \sin(2t)) \] \[ \left(\frac{dy}{dt}\right)^2 = [e^t(\sin t + \cos t)]^2 = e^{2t}(\sin^2 t + 2\sin t\cos t + \cos^2 t) = e^{2t}(1 + \sin(2t)) \]
Adding these two expressions together: \[ \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 = e^{2t}(1 - \sin(2t) + 1 + \sin(2t)) = e^{2t}(2) = 2e^{2t} \]
Now, we find \(ds\) by taking the square root: \[ ds = \sqrt{2e^{2t}} \, dt = \sqrt{2} e^t \, dt \]

3. Construct and Solve the Surface Area Integral:

We substitute \(x(t)\) and \(ds\) into the surface area formula, with limits from \(0\) to \(\pi/2\): \[ S = \int_{0}^{\pi/2} 2\pi (e^t \cos t) (\sqrt{2} e^t \, dt) = 2\sqrt{2}\pi \int_{0}^{\pi/2} e^{2t} \cos t \, dt \]
This integral requires integration by parts, or we can use the standard formula: \( \int e^{at}\cos(bt) dt = \frac{e^{at}}{a^2+b^2}(a\cos(bt) + b\sin(bt)) \).
With a=2 and b=1, we get: \[ \int e^{2t} \cos t \, dt = \frac{e^{2t}}{2^2+1^2}(2\cos t + 1\sin t) = \frac{e^{2t}}{5}(2\cos t + \sin t) \]
Now we evaluate this antiderivative at the given limits: \[ \left[ \frac{e^{2t}}{5}(2\cos t + \sin t) \right]_{0}^{\pi/2} = \left(\frac{e^{2(\pi/2)}}{5}(2\cos(\pi/2) + \sin(\pi/2))\right) - \left(\frac{e^{2(0)}}{5}(2\cos(0) + \sin(0))\right) \] \[ = \left(\frac{e^{\pi}}{5}(2(0) + 1)\right) - \left(\frac{1}{5}(2(1) + 0)\right) = \frac{e^{\pi}}{5} - \frac{2}{5} = \frac{e^\pi - 2}{5} \]

4. Calculate the Final Surface Area:

Finally, we multiply this result by the constant factor from the integral: \[ S = 2\sqrt{2}\pi \left( \frac{e^\pi - 2}{5} \right) = \frac{2\sqrt{2}\pi(e^\pi - 2)}{5} \]

Step 4: Final Answer:

The total surface area of the generated solid is \( \frac{2\sqrt{2}}{5}\pi(e^\pi - 2) \) square units.
Quick Tip: The curve \(x=ae^{kt}\cos(kt), y=ae^{kt}\sin(kt)\) is a logarithmic spiral. Calculating \(ds\) for such curves often results in a neat simplification. The integral of the form \(\int e^{ax}\cos(bx)dx\) is very common in physics and engineering problems, and memorizing the formula can save a lot of time compared to performing integration by parts twice.

Step 1: Understanding the Concept:

The radius of curvature, denoted by \(\rho\), at a specific point on a curve provides the radius of the "osculating circle," which is the circle that best fits the curve's bend at that point. Our goal is to compute this radius for the function \(y=e^x\) at the point (0, 1) and then use the result to find the values of \(\alpha\) and \(\beta\).


Step 2: Key Formula or Approach:

For a function given explicitly as \(y=f(x)\), the radius of curvature is calculated using the standard formula involving its first and second derivatives: \[ \rho = \frac{\left[1 + (y')^2\right]^{3/2}}{|y''|} \]
Here, \(y' = \frac{dy}{dx}\) and \(y'' = \frac{d^2y}{dx^2}\). We need to evaluate these derivatives at the given point.


Step 3: Detailed Explanation:

1. Find the First and Second Derivatives of the Function:

The function is \(y = e^x\).
The first derivative is: \[ y' = \frac{d}{dx}(e^x) = e^x \]
The second derivative is: \[ y'' = \frac{d}{dx}(e^x) = e^x \]

2. Evaluate the Derivatives at the Specified Point (0, 1):

The x-coordinate of the point is \(x=0\). We substitute this value into our derivative expressions. \[ y'(0) = e^0 = 1 \] \[ y''(0) = e^0 = 1 \]

3. Calculate the Radius of Curvature \(\rho\):

Now we plug these values, \(y'(0) = 1\) and \(y''(0) = 1\), into the radius of curvature formula: \[ \rho = \frac{[1 + (1)^2]^{3/2}}{|1|} = \frac{[1 + 1]^{3/2}}{1} = 2^{3/2} \]
To simplify \(2^{3/2}\), we can write it as \( (2^3)^{1/2} = \sqrt{8} \). Factoring out the largest perfect square from 8 gives \( \sqrt{4 \times 2} = 2\sqrt{2} \).
So, \( \rho = 2\sqrt{2} \).

4. Determine the values of \(\alpha\) and \(\beta\):

The problem states that the radius of curvature has the form \(\alpha\sqrt{\beta}\). By comparing our result \( \rho = 2\sqrt{2} \) to this form, we can directly identify the constants: \[ \alpha = 2, \quad \beta = 2 \]

5. Calculate the final expression \(\alpha^2 + \beta\):

We substitute the determined values of \(\alpha\) and \(\beta\): \[ \alpha^2 + \beta = (2)^2 + 2 = 4 + 2 = 6 \]

Step 4: Final Answer:

The final computed value of the expression \(\alpha^2 + \beta\) is 6.
Quick Tip: The radius of curvature formula is a standard application of differentiation. Remember to evaluate the derivatives at the specified point *before* plugging them into the formula. The expression \(2^{3/2}\) can be tricky; remember that \(x^{a/b} = (\sqrt[b]{x})^a\), so \(2^{3/2} = (\sqrt{2})^3 = 2\sqrt{2}\).

Step 1: Understanding the Concept:

The goal is to derive a general recurrence relation for the higher-order derivatives of the function \(y = \tan^{-1}x\). The standard technique for finding such a relation involves establishing a simple differential equation and then applying Leibniz's theorem to differentiate it multiple times.


Step 2: Key Formula or Approach:

Our strategy consists of two main parts:
1. Differentiate \(y = \tan^{-1}x\) one or two times to establish a simple differential equation without fractions.
2. Apply Leibniz's theorem for the \(n\)-th derivative of a product, which states: \( D^n(uv) = D^n(u)v + \binom{n}{1}D^{n-1}(u)D(v) + \binom{n}{2}D^{n-2}(u)D^2(v) + \ldots \)
where \(D^k(f)\) is the \(k\)-th derivative of \(f\).


Step 3: Detailed Explanation:

Given the function \( y = \tan^{-1}x \).
First, we compute the first derivative, which we denote as \(y_1\): \[ y_1 = \frac{dy}{dx} = \frac{1}{1+x^2} \]
To avoid using the quotient rule repeatedly, we rearrange this into a product: \[ (1+x^2)y_1 = 1 \]
Now, differentiate this equation with respect to x using the product rule. Let \(u = (1+x^2)\) and \(v = y_1\). \[ u'v + uv' = 0 \implies (2x)y_1 + (1+x^2)y_2 = 0 \]
This gives us the second-order differential equation: \( (1+x^2)y_2 + 2xy_1 = 0 \).
We will now differentiate this entire equation \(n\) times using Leibniz's theorem. Let's apply it to each term separately.
For the first term, \(D^n[(1+x^2)y_2]\):
Let \(u=y_2\) and \(v=1+x^2\). \(v'=2x\), \(v''=2\), \(v'''=0\). \[ D^n(vy_2) = v y_{n+2} + n v' y_{n+1} + \frac{n(n-1)}{2!} v'' y_n + \ldots \] \[ = (1+x^2)y_{n+2} + n(2x)y_{n+1} + \frac{n(n-1)}{2}(2)y_n \] \[ = (1+x^2)y_{n+2} + 2nxy_{n+1} + n(n-1)y_n \]
For the second term, \(D^n[2xy_1]\):
Let \(u=y_1\) and \(v=2x\). \(v'=2\), \(v''=0\). \[ D^n(vy_1) = v y_{n+1} + n v' y_n + \ldots \] \[ = (2x)y_{n+1} + n(2)y_n = 2xy_{n+1} + 2ny_n \]
Now, we add the results of these two differentiations and set the sum to zero: \[ [(1+x^2)y_{n+2} + 2nxy_{n+1} + n(n-1)y_n] + [2xy_{n+1} + 2ny_n] = 0 \]
Finally, we group the terms by the order of the derivative: \[ (1+x^2)y_{n+2} + (2nx + 2x)y_{n+1} + (n(n-1) + 2n)y_n = 0 \]
Simplifying the coefficients gives: \[ (1+x^2)y_{n+2} + (2n+2)xy_{n+1} + (n^2-n+2n)y_n = 0 \] \[ (1+x^2)y_{n+2} + (2n+2)xy_{n+1} + (n^2+n)y_n = 0 \] \[ (1+x^2)y_{n+2} + (2n+2)xy_{n+1} + n(n+1)y_n = 0 \]

Step 4: Final Answer:

Translating the subscript notation back to the differential operator notation, we arrive at the equation presented in option (A).
Quick Tip: For problems involving finding the \(n^{th}\) order differential equation, the standard procedure is to differentiate once or twice until you can form an equation that is a polynomial in x multiplying derivatives of y. Then, apply Leibniz's theorem. The derivatives of the polynomial part (like \(1+x^2\) or \(2x\)) will terminate after a few terms, making the application straightforward.

Step 1: Understanding the Concept:

Lagrange's Mean Value Theorem (MVT) provides a fundamental link between the derivative of a function and its average rate of change over an interval. The theorem is only guaranteed to apply if two specific preconditions are met: the function \(f\) must be continuous over the closed interval \([a, b]\) and differentiable over the open interval \((a, b)\). We must scrutinize each function and interval to see if both of these conditions hold.


Step 2: Detailed Explanation:


(A) f(x) = |x| in [-1, 2]: The absolute value function is continuous everywhere, so it is continuous on [-1, 2]. However, the function has a sharp corner at x=0, which means it is not differentiable at this point. Since \(0\) is within the open interval \((-1, 2)\), the differentiability condition is violated. Therefore, the MVT does not necessarily apply. Statement (A) is false.
(B) f(x) = cosx in [0, \(\pi\)/2]: The cosine function is known to be continuous and differentiable for all real numbers. As a result, it is guaranteed to be continuous on the closed interval [0, \(\pi\)/2] and differentiable on the open interval (0, \(\pi\)/2). Since both preconditions are satisfied, the MVT applies. Statement (B) is true.
(C) f(x) = \( \frac{1}{x} \) in [-1, 2]: The function \(f(x) = 1/x\) has a discontinuity at x=0, as the function is undefined there. Because the point of discontinuity \(x=0\) falls within the interval [-1, 2], the function is not continuous over the entire closed interval. The first condition for the MVT fails. Statement (C) is false.
(D) f(x) = x(x-1)(x-2) in [0, 1/2]: This function is a polynomial (\(f(x) = x^3 - 3x^2 + 2x\)). All polynomial functions are continuous and differentiable over the entire real number line. Consequently, they satisfy the conditions for any sub-interval. Both continuity on [0, 1/2] and differentiability on (0, 1/2) are assured. The MVT applies. Statement (D) is true.
(E) f(x) = \(x^{1/3}\) in [-1, 1]: The cube root function is continuous for all real numbers, so it is continuous on [-1, 1]. To check for differentiability, we find its derivative: \(f'(x) = \frac{1}{3}x^{-2/3} = \frac{1}{3(x^2)^{1/3}}\). At x=0, the denominator becomes zero, so the derivative is undefined. This indicates a vertical tangent line at the origin. Since \(x=0\) is within the open interval \((-1, 1)\), the function is not differentiable over the required interval. The MVT is not guaranteed to apply. Statement (E) is false.


Step 3: Final Answer:

After evaluating all five cases, we conclude that the conditions for Lagrange's Mean Value Theorem are only met for statements (B) and (D). Thus, the correct option is (C).
Quick Tip: To quickly check for the applicability of Mean Value Theorems (Rolle's or Lagrange's), look for "problem spots" in the given interval: Points where the function is undefined (e.g., division by zero). Points where the function might not be differentiable (e.g., sharp corners like in |x|, or vertical tangents like in \(x^{1/3}\)). If any such point lies within the interval, the theorem's conditions are likely violated.

Step 1: Understanding the Concept:

This question assesses knowledge of symmetry tests for curves expressed in polar coordinates, \(r = f(\theta)\). We must apply the standard tests for three types of symmetry:

Symmetry about the initial line (x-axis, \(\theta=0\)): The equation should remain unchanged if we replace the point \((r, \theta)\) with \((r, -\theta)\).
Symmetry about the line \(\theta=\pi/2\) (y-axis): The equation should remain unchanged if we replace the point \((r, \theta)\) with \((r, \pi-\theta)\).
Symmetry about the pole (origin): The equation should remain unchanged if we replace the point \((r, \theta)\) with \((-r, \theta)\) or with \((r, \theta+\pi)\).


Step 2: Detailed Explanation:


(A) r = a(1 + cos\(\theta\)): To test for symmetry about the initial line, we replace \(\theta\) with \(-\theta\). The equation becomes \(r = a(1 + \cos(-\theta))\). Since the cosine function is an even function, \(\cos(-\theta) = \cos(\theta)\). The equation becomes \(r = a(1 + \cos\theta)\), which is identical to the original. Thus, the curve is symmetrical about the initial line. Statement (A) is true.
(B) r = 2(1 - 2 sin\(\theta\)): We test for symmetry about the initial line by replacing \(\theta\) with \(-\theta\). The equation becomes \(r = 2(1 - 2\sin(-\theta))\). Since the sine function is an odd function, \(\sin(-\theta) = -\sin(\theta)\). The equation transforms to \(r = 2(1 - 2(-\sin\theta)) = 2(1+2\sin\theta)\), which is not the same as the original. Statement (B) is false.
(C) r = a(1 + sin\(\theta\)): To test for symmetry about the line \(\theta=\pi/2\), we replace \(\theta\) with \(\pi-\theta\). The equation becomes \(r = a(1 + \sin(\pi-\theta))\). Using the trigonometric identity \(\sin(\pi-\theta) = \sin(\theta)\), the equation remains \(r = a(1 + \sin\theta)\). It is unchanged, so the curve is symmetrical about the line \(\theta=\pi/2\). Statement (C) is true.
(D) r = a sin(3\(\theta\)): We test for symmetry about the initial line by replacing \(\theta\) with \(-\theta\). The equation becomes \(r = a\sin(3(-\theta)) = a\sin(-3\theta)\). Since sine is an odd function, this is \(r = -a\sin(3\theta)\), which is different from the original equation. Statement (D) is false.
(E) r² = a²cos(2\(\theta\)): To test for symmetry about the pole, we can replace \(r\) with \(-r\). The equation becomes \((-r)^2 = a^2\cos(2\theta)\), which simplifies to \(r^2 = a^2\cos(2\theta)\). Since the equation is identical to the original, the curve is symmetrical about the pole. Statement (E) is true.


Step 3: Final Answer:

The statements that hold true are (A), (C), and (E). This selection corresponds to option (D).
Quick Tip: Quick rules for polar symmetry: If the equation only contains \(\cos\theta\), it's symmetric about the initial line (x-axis). If the equation only contains \(\sin\theta\), it's symmetric about \(\theta=\pi/2\) (y-axis). If \(r\) appears as \(r^2\) or \(\theta\) appears as a multiple within a function (like \(\cos(n\theta)\)), it often has pole symmetry. These are useful shortcuts, but always apply the formal tests (\(\theta \to -\theta\), etc.) if unsure.

Step 1: Understanding the Concept:

This problem requires us to evaluate four distinct limits. Each limit presents an indeterminate form, necessitating techniques such as using known standard limits, algebraic manipulation, or applying L'Hôpital's Rule to find a determinate value.


Step 2: Detailed Explanation:


(A) \( \lim_{x\to 0} \frac{\ln(1+x)}{\sin x} \): As \(x \to 0\), both \(\ln(1+x) \to \ln(1) = 0\) and \(\sin x \to 0\). This is the indeterminate form \( \frac{0}{0} \). We can cleverly rewrite the expression using the standard limits \( \lim_{u\to 0} \frac{\ln(1+u)}{u} = 1 \) and \( \lim_{u\to 0} \frac{\sin u}{u} = 1 \).
\[ \lim_{x\to 0} \frac{\ln(1+x)}{\sin x} = \lim_{x\to 0} \left(\frac{\ln(1+x)}{x} \cdot \frac{x}{\sin x}\right) = \left(\lim_{x\to 0} \frac{\ln(1+x)}{x}\right) \cdot \left(\lim_{x\to 0} \frac{1}{\sin x / x}\right) = (1) \cdot \left(\frac{1}{1}\right) = 1 \]
So, (A) matches (I).
(B) \( \lim_{x\to \infty} 2x \tan(1/x) \): As \( x \to \infty \), \(2x \to \infty\) and \(1/x \to 0\), so \(\tan(1/x) \to 0\). This is the indeterminate form \( \infty \cdot 0 \). To resolve this, we use a substitution. Let \( u = 1/x \). As \( x \to \infty \), it follows that \( u \to 0^+ \). The expression becomes:
\[ \lim_{u\to 0^+} 2\left(\frac{1}{u}\right) \tan(u) = 2 \lim_{u\to 0^+} \frac{\tan u}{u} \]
This is a well-known standard limit whose value is 1.
\[ 2 \times 1 = 2 \]
So, (B) matches (III).
(C) \( \lim_{x\to \infty} \frac{x^2}{e^x} \): As \(x \to \infty\), both \(x^2 \to \infty\) and \(e^x \to \infty\). This is the indeterminate form \( \frac{\infty}{\infty} \). We can apply L'Hôpital's Rule, which allows us to differentiate the numerator and denominator.
\[ \lim_{x\to \infty} \frac{\frac{d}{dx}(x^2)}{\frac{d}{dx}(e^x)} = \lim_{x\to \infty} \frac{2x}{e^x} \]
This is still of the form \( \frac{\infty}{\infty} \), so we apply the rule again:
\[ \lim_{x\to \infty} \frac{\frac{d}{dx}(2x)}{\frac{d}{dx}(e^x)} = \lim_{x\to \infty} \frac{2}{e^x} \]
As \( x \to \infty \), \(e^x \to \infty\), so the limit is \( \frac{2}{\infty} = 0 \).
So, (C) matches (II).
(D) \( \lim_{x\to 1} x^{1/(x-1)} \): As \(x \to 1\), the base \(x \to 1\) and the exponent \(1/(x-1) \to \infty\). This is the indeterminate form \( 1^\infty \). To evaluate this, we let \( L \) be the limit and take the natural logarithm of both sides.
\[ \ln L = \ln\left(\lim_{x\to 1} x^{1/(x-1)}\right) = \lim_{x\to 1} \ln\left(x^{1/(x-1)}\right) = \lim_{x\to 1} \frac{1}{x-1}\ln x = \lim_{x\to 1} \frac{\ln x}{x-1} \]
This is now a \( \frac{0}{0} \) form, suitable for L'Hôpital's Rule.
\[ \ln L = \lim_{x\to 1} \frac{\frac{d}{dx}(\ln x)}{\frac{d}{dx}(x-1)} = \lim_{x\to 1} \frac{1/x}{1} = \frac{1/1}{1} = 1 \]
We found that \( \ln L = 1 \). To find \(L\), we exponentiate both sides: \( L = e^1 = e \).
So, (D) matches (IV).


Step 3: Final Answer:

The final pairings are (A)-(I), (B)-(III), (C)-(II), and (D)-(IV). This combination corresponds exactly to option (B).
Quick Tip: For limits, quickly classify the indeterminate form: \(0/0, \infty/\infty, 0 \cdot \infty, \infty - \infty, 1^\infty, 0^0, \infty^0\). For \(0/0\) or \(\infty/\infty\), use L'Hôpital's Rule or standard limits/series expansions. For \(1^\infty, 0^0, \infty^0\), take the logarithm first to turn it into a \(0 \cdot \infty\) form, then rearrange for L'Hôpital's Rule.

Step 1: Understanding the Concept:

The concavity of a function's graph is determined by the sign of its second derivative, \(f''(x)\). The rule is as follows:

If \(f''(x) > 0\) over an interval, the function's graph is concave upward (also called convex) on that interval.
If \(f''(x) < 0\) over an interval, the function's graph is concave downward on that interval.

Our procedure for each function will be to compute the second derivative and then analyze its sign to determine the regions of concavity.


Step 2: Detailed Explanation:


(A) \(f(x) = e^{-x^2}\):
First derivative: \(f'(x) = e^{-x^2} \cdot (-2x) = -2xe^{-x^2}\).
Second derivative (using product rule): \(f''(x) = (-2)e^{-x^2} + (-2x)(e^{-x^2} \cdot -2x) = -2e^{-x^2} + 4x^2e^{-x^2} = 2e^{-x^2}(2x^2-1)\).
To find where it's concave downward, we set \(f''(x) < 0\). Since \(2e^{-x^2}\) is always positive, we only need to solve \(2x^2-1 < 0 \implies x^2 < 1/2 \implies -1/\sqrt{2} < x < 1/\sqrt{2}\).
The function is concave downward in the interval \( (-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}) \). (A) matches (III).
(B) \(f(x) = (1+x^2)e^{-x}\):
First derivative: \(f'(x) = (2x)e^{-x} + (1+x^2)(-e^{-x}) = e^{-x}(2x - 1 - x^2) = -e^{-x}(x-1)^2\).
Second derivative: \(f''(x) = -[(-e^{-x})(x-1)^2 + e^{-x}(2(x-1))] = -e^{-x}(x-1)[-(x-1)+2] = -e^{-x}(x-1)(-x+3) = e^{-x}(x-1)(x-3)\).
To find where it's concave upward, we set \(f''(x) > 0\). Since \(e^{-x}\) is always positive, we solve \((x-1)(x-3) > 0\). This inequality holds when \(x < 1\) or \(x > 3\).
The function is concave upward in \((-\infty, 1)\). (B) matches (II).
(C) \(f(x) = 3x^4+4x^3-6x^2+12x+12\):
First derivative: \(f'(x) = 12x^3+12x^2-12x+12\).
Second derivative: \(f''(x) = 36x^2+24x-12 = 12(3x^2+2x-1) = 12(3x-1)(x+1)\).
To find where it's concave downward, we set \(f''(x) < 0\), which requires \((3x-1)(x+1) < 0\). This is true for \(-1 < x < 1/3\).
List-II's option (I) states "Concave downward in \((-\infty, -1)\)", which is incorrect. However, by eliminating the other options, we deduce this must be the intended match. (C) matches (I).
(D) \(f(x) = (x+1)^{1/3}\):
First derivative: \(f'(x) = \frac{1}{3}(x+1)^{-2/3}\).
Second derivative: \(f''(x) = \frac{1}{3}(-\frac{2}{3})(x+1)^{-5/3} = -\frac{2}{9(x+1)^{5/3}}\).
To find where it's concave upward, we set \(f''(x) > 0\). This requires \(-\frac{2}{9(x+1)^{5/3}} > 0\), which simplifies to \((x+1)^{5/3} < 0\). This is true when \(x+1 < 0 \implies x < -1\).
The function is concave upward in \((-\infty, -1)\). (D) matches (IV).


Step 3: Final Answer:

The correct pairings are (A)-(III), (B)-(II), and (D)-(IV). By process of elimination, (C) must be paired with (I), despite the interval in (I) being incorrect for the function in (C). This combination corresponds to option (B).
Quick Tip: When doing concavity problems, find the second derivative \(f''(x)\) and then find its roots. These roots are the potential inflection points. Test the sign of \(f''(x)\) in the intervals between these roots to determine the concavity. A simple sign chart is very effective.

Step 1: Understanding the Concept:

The general second-degree equation can represent any of the standard conic sections (ellipse, parabola, hyperbola), a circle, or it can degenerate into a simpler form. One such degenerate case is a pair of straight lines. There is a specific algebraic condition involving the coefficients of the equation that determines whether this degeneration occurs.


Step 2: Key Formula or Approach:

The condition for the general second-degree equation to represent a pair of straight lines is determined by evaluating the determinant of a specific 3x3 symmetric matrix formed from the coefficients. This determinant is often denoted by \( \Delta \).
The matrix is constructed as follows: \[ \Delta = \begin{vmatrix} a & h & g
h & b & f
g & f & c \end{vmatrix} \]
The equation represents a pair of straight lines if and only if this determinant is equal to zero (\( \Delta = 0 \)).


Step 3: Detailed Explanation:

Our task is to explicitly calculate the determinant of the matrix \( \Delta \). We can expand the determinant along the first row: \[ \det(\Delta) = a \begin{vmatrix} b & f
f & c \end{vmatrix} - h \begin{vmatrix} h & f
g & c \end{vmatrix} + g \begin{vmatrix} h & b
g & f \end{vmatrix} \] \[ = a(bc - f^2) - h(hc - fg) + g(hf - bg) \]
Distributing the terms, we get: \[ = abc - af^2 - h^2c + fgh + fgh - bg^2 \]
Combining the like terms, we arrive at the standard expression: \[ = abc + 2fgh - af^2 - bg^2 - ch^2 \]
The condition for the equation to represent a pair of straight lines is that this entire expression must be equal to zero. \[ abc + 2fgh - af^2 - bg^2 - ch^2 = 0 \]

Step 4: Final Answer:

The required necessary and sufficient condition is precisely \( abc + 2fgh - af^2 - bg^2 - ch^2 = 0 \), which matches option (C).
Quick Tip: A useful mnemonic to remember the expression for the determinant is "A Hot Girl Had Beautiful Face, Go For Chocolates": \(a(bc - f^2)\) \(-h(hc - fg)\) \(+g(hf - bg)\) Another way is to remember the phrase: "All handsome guys having beautiful faces go for coffee", which helps to set up the matrix \(\begin{pmatrix} a & h & g
h & b & f
g & f & c \end{pmatrix}\).

Step 1: Understanding the Concept:

The geometric condition for a plane to be tangent to a sphere is that the shortest distance from the center of the sphere to the plane must be exactly equal to the sphere's radius. Our strategy is to find the sphere's center and radius, calculate the distance from the center to the given plane, and then set this distance equal to the radius to solve for the unknown parameter \(\lambda\).


Step 2: Key Formula or Approach:

1. **Sphere's Center and Radius:** For an equation \(x^2+y^2+z^2+2ux+2vy+2wz+d=0\), the center is \((-u,-v,-w)\) and radius is \(R=\sqrt{u^2+v^2+w^2-d}\).
2. **Distance from a Point to a Plane:** The perpendicular distance \(d\) from a point \((x_0, y_0, z_0)\) to a plane \(Ax + By + Cz + D = 0\) is given by the formula \(d = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}}\).
3. **Tangency Condition:** Set \(d = R\).


Step 3: Detailed Explanation:

1. Find the Center and Radius of the Sphere:

The given equation is \(x^2 + y^2 + z^2 - 2x - 2y - 2z - 6 = 0\).
To find the center and radius, we complete the square for the x, y, and z terms: \[ (x^2 - 2x) + (y^2 - 2y) + (z^2 - 2z) = 6 \] \[ (x^2 - 2x + 1) + (y^2 - 2y + 1) + (z^2 - 2z + 1) = 6 + 1 + 1 + 1 \] \[ (x-1)^2 + (y-1)^2 + (z-1)^2 = 9 \]
From this standard form, we can identify the center of the sphere as \(C = (1, 1, 1)\) and the radius as \(R = \sqrt{9} = 3\).


2. Find the Distance from the Center to the Plane:

The equation of the plane is \(x + y + z = \sqrt{3}\lambda\), which we rewrite as \(x + y + z - \sqrt{3}\lambda = 0\).
The center of the sphere is \((x_0, y_0, z_0) = (1, 1, 1)\).
Using the distance formula: \[ d = \frac{|1(1) + 1(1) + 1(1) - \sqrt{3}\lambda|}{\sqrt{1^2 + 1^2 + 1^2}} = \frac{|3 - \sqrt{3}\lambda|}{\sqrt{3}} \]

3. Apply the Tangency Condition and Solve for \(\lambda\):

We set the distance \(d\) equal to the radius \(R\): \[ \frac{|3 - \sqrt{3}\lambda|}{\sqrt{3}} = 3 \]
Multiply both sides by \(\sqrt{3}\): \[ |3 - \sqrt{3}\lambda| = 3\sqrt{3} \]
This absolute value equation leads to two separate linear equations:
Case 1: \( 3 - \sqrt{3}\lambda = 3\sqrt{3} \) \[ 3 - 3\sqrt{3} = \sqrt{3}\lambda \] \[ \lambda = \frac{3 - 3\sqrt{3}}{\sqrt{3}} = \frac{3}{\sqrt{3}} - 3 = \sqrt{3} - 3 \]
Case 2: \( 3 - \sqrt{3}\lambda = -3\sqrt{3} \) \[ 3 + 3\sqrt{3} = \sqrt{3}\lambda \] \[ \lambda = \frac{3 + 3\sqrt{3}}{\sqrt{3}} = \frac{3}{\sqrt{3}} + 3 = \sqrt{3} + 3 \]
Combining these two solutions, we get \( \lambda = \sqrt{3} \pm 3 \).


Step 4: Final Answer:

The values of \(\lambda\) for which the plane touches the sphere are \( \sqrt{3} + 3 \) and \( \sqrt{3} - 3 \).
Quick Tip: To find the center and radius of a sphere from the general equation \(x^2+y^2+z^2+2ux+2vy+2wz+d=0\), the center is \((-u, -v, -w)\) and the radius is \(R = \sqrt{u^2+v^2+w^2-d}\). This is faster than completing the square. In our case, \(2u=-2 \implies u=-1\), \(2v=-2 \implies v=-1\), \(2w=-2 \implies w=-1\), and \(d=-6\). Center is (1,1,1), Radius is \(\sqrt{(-1)^2+(-1)^2+(-1)^2 - (-6)} = \sqrt{1+1+1+6} = \sqrt{9} = 3\).

Step 1: Understanding the Concept:

The problem asks for the equation of a cone whose vertex is at the origin and whose base is the circle formed by the intersection of a sphere and a plane. The method to solve this is called homogenization. We use the linear equation of the plane to make the quadratic equation of the sphere homogeneous (i.e., ensure every term has the same degree), resulting in the equation of the cone.


Step 2: Key Formula or Approach:

Given the sphere \(S \equiv x^2 + y^2 + z^2 + x - 2z + 3y - 4 = 0\) and the plane \(P \equiv x - y + z = 2\).
1. Rewrite the plane equation to express the constant '1': \( 1 = \frac{x-y+z}{2} \).
2. Substitute this expression into the sphere's equation to elevate the degree of the lower-degree terms. The linear terms (degree 1) are multiplied by this factor, and the constant term (degree 0) is multiplied by the square of this factor.
The resulting homogeneous equation will be: \[ (x^2 + y^2 + z^2) + (x + 3y - 2z)\left(\frac{x-y+z}{2}\right) - 4\left(\frac{x-y+z}{2}\right)^2 = 0 \]

Step 3: Detailed Explanation:

Let's expand the expression derived from homogenization. It is important to note that the provided question likely contains typos, as a direct calculation does not yield any of the options. However, we will proceed with the calculation to demonstrate the correct method.
The equation to simplify is: \[ (x^2 + y^2 + z^2) + \frac{1}{2}(x + 3y - 2z)(x-y+z) - 4\left(\frac{(x-y+z)^2}{4}\right) = 0 \] \[ (x^2 + y^2 + z^2) + \frac{1}{2}(x^2 - xy + xz + 3xy - 3y^2 + 3yz - 2xz + 2yz - 2z^2) - (x^2+y^2+z^2-2xy+2xz-2yz) = 0 \]
Multiply the entire equation by 2 to eliminate the fraction: \[ 2(x^2 + y^2 + z^2) + (x^2 + 2xy - xz - 3y^2 + 5yz - 2z^2) - 2(x^2+y^2+z^2-2xy+2xz-2yz) = 0 \]
Now, group the terms by variable:

\(x^2\) term: \( 2x^2 + x^2 - 2x^2 = x^2 \)
\(y^2\) term: \( 2y^2 - 3y^2 - 2y^2 = -3y^2 \)
\(z^2\) term: \( 2z^2 - 2z^2 - 2z^2 = -2z^2 \)
\(xy\) term: \( 2xy + 4xy = 6xy \)
\(yz\) term: \( 5yz + 4yz = 9yz \)
\(xz\) term: \( -xz - 4xz = -5xz \)

The resulting equation is \( x^2 - 3y^2 - 2z^2 + 6xy + 9yz - 5xz = 0 \). As this result does not align with any of the options, it confirms the presence of errors in the problem statement. The correct procedure was followed, but the initial data is inconsistent with the provided answers.


Step 4: Final Answer:

The problem as stated cannot be solved to match any of the given options due to apparent typos in the initial equations. The method of homogenization is the correct approach, but the flawed data prevents a valid derivation.
Quick Tip: The process of finding the equation of a cone with a vertex at the origin and a guiding curve defined by \(S=0\) and \(P=0\) is called homogenization. Always write the plane equation as \(P/k=1\) and substitute it into the non-homogeneous terms of the surface equation \(S=0\).

Step 1: Understanding the Concept:

This problem requires us to verify five distinct statements related to the geometry of planes and vectors in three-dimensional space. We will use standard formulas to test the validity of each claim.


Step 2: Detailed Explanation:


(A): To find the equation of a plane given a point \((x_0, y_0, z_0)\) and a normal vector with direction ratios (a, b, c), we use the formula \(a(x-x_0) + b(y-y_0) + c(z-z_0) = 0\).
With point (1, -1, 2) and normal (2, 3, 2), we have:
\(2(x-1) + 3(y-(-1)) + 2(z-2) = 0 \implies 2x - 2 + 3y + 3 + 2z - 4 = 0 \implies 2x + 3y + 2z - 3 = 0\).
This simplifies to \(2x + 3y + 2z = 3\). Statement (A) is true.
(B): The angle between two planes is the angle between their normal vectors. The normal vectors are \(\vec{n_1} = \langle 2, -1, 1 \rangle\) and \(\vec{n_2} = \langle 1, 1, 2 \rangle\). The cosine of the angle \(\theta\) between them is given by the dot product formula:
\(\cos\theta = \frac{|\vec{n_1} \cdot \vec{n_2}|}{|\vec{n_1}||\vec{n_2}|} = \frac{|(2)(1) + (-1)(1) + (1)(2)|}{\sqrt{2^2+(-1)^2+1^2}\sqrt{1^2+1^2+2^2}} = \frac{|2-1+2|}{\sqrt{6}\sqrt{6}} = \frac{3}{6} = \frac{1}{2}\).
An angle with a cosine of 1/2 is \(\theta = \pi/3\). Statement (B) is true.
(C): The normal vector to the plane is \(\vec{n} = \langle 4, 8, 1 \rangle\). The direction vector for the z-axis is \(\vec{k} = \langle 0, 0, 1 \rangle\). The angle \(\alpha\) between these vectors is found using the dot product:
\(\cos\alpha = \frac{|\vec{n} \cdot \vec{k}|}{|\vec{n}||\vec{k}|} = \frac{|(4)(0) + (8)(0) + (1)(1)|}{\sqrt{4^2+8^2+1^2}\sqrt{0^2+0^2+1^2}} = \frac{1}{\sqrt{16+64+1}} = \frac{1}{\sqrt{81}} = \frac{1}{9}\).
The angle is \(\alpha = \cos^{-1}(1/9)\), not \(\sin^{-1}(1/9)\). Statement (C) is false.
(D): A vector normal to the plane is the vector connecting the two points: \(\vec{n} = \langle 2-3, -1-4, 5-(-1) \rangle = \langle -1, -5, 6 \rangle\). We can use the parallel vector \(\langle 1, 5, -6 \rangle\) for simplicity. The plane passes through (3, -3, 1).
The equation is: \(1(x-3) + 5(y-(-3)) - 6(z-1) = 0 \implies x-3+5y+15-6z+6=0 \implies x+5y-6z = -18\).
The provided equation is \(x + 5y + 6z = -18\), which has a sign error in the z-term. Statement (D) is false.
(E): The coefficients of x, y, and z in the plane's equation, \(2x - y + 2z = 5\), give the components of a normal vector: \(\vec{n} = \langle 2, -1, 2 \rangle\), which is \(2\vec{i} - \vec{j} + 2\vec{k}\). Any non-zero scalar multiple of this vector is also a normal vector. The vector \( \frac{1}{3}(2\vec{i} - \vec{j} + 2\vec{k}) \) is simply \(\vec{n}\) scaled by 1/3, making it a valid (unit) normal vector. Statement (E) is true.


Step 3: Final Answer:

The statements that have been verified as true are (A), (B), and (E). This combination matches option (D).
Quick Tip: The coefficients of x, y, and z in the equation of a plane \(ax+by+cz=d\) directly give the direction ratios (a, b, c) of the normal vector. This is the starting point for most problems involving angles with planes.

Step 1: Understanding the Concept:

To classify the conic section represented by the general second-degree equation \(ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0\), we use two main discriminants. First, we check if the conic is degenerate (a pair of straight lines) using \( \Delta = abc + 2fgh - af^2 - bg^2 - ch^2 \). If \( \Delta = 0 \), it's a pair of lines. If \( \Delta \neq 0 \), we classify the non-degenerate conic using the sign of \( h^2 - ab \).

\( h^2 - ab < 0 \): Ellipse
\( h^2 - ab = 0 \): Parabola
\( h^2 - ab > 0 \): Hyperbola


Step 2: Detailed Explanation:

For each equation, we identify the coefficients a, h, and b (recalling that the coefficient of xy is 2h).

(A) \(9x^2 - 12xy + 4y^2 - 74x - 98y + 324 = 0\):
Here, \(a=9\), \(2h=-12 \implies h=-6\), and \(b=4\).
We calculate the discriminant: \(h^2 - ab = (-6)^2 - (9)(4) = 36 - 36 = 0\).
A value of 0 indicates the conic is a Parabola. (A) matches (IV).
(B) \(12x^2 + 7xy - 12y^2 + 10x + 55y - 125 = 0\):
Here, \(a=12\), \(2h=7 \implies h=7/2\), and \(b=-12\).
We calculate the discriminant: \(h^2 - ab = (7/2)^2 - (12)(-12) = 49/4 + 144\).
Since this value is clearly positive, the conic is a Hyperbola. (B) matches (I).
(C) \(x^2 + 3xy + 2y^2 + x + y = 0\):
Here, \(a=1, 2h=3 \implies h=3/2, b=2, 2g=1 \implies g=1/2, 2f=1 \implies f=1/2, c=0\).
Since "pair of straight lines" is an option, we should first check the determinant \( \Delta \).
\( \Delta = abc + 2fgh - af^2 - bg^2 - ch^2 \)
\( \Delta = (1)(2)(0) + 2(1/2)(1/2)(3/2) - (1)(1/2)^2 - (2)(1/2)^2 - (0)(3/2)^2 \)
\( \Delta = 0 + 3/4 - 1/4 - 2/4 = 0 \).
Since \( \Delta = 0 \), the equation represents a pair of straight lines. (C) matches (II).
(D) \(5x^2 + y^2 - 30x + 1 = 0\):
Here, \(a=5\), \(h=0\) (no xy term), and \(b=1\).
We calculate the discriminant: \(h^2 - ab = 0^2 - (5)(1) = -5\).
Since this value is negative, the conic is an Ellipse. (D) matches (III).


Step 3: Final Answer:

Compiling our results, we have the matches: (A)-(IV), (B)-(I), (C)-(II), and (D)-(III). This sequence corresponds to option (B).
Quick Tip: The sign of \(h^2 - ab\) is the quickest way to classify a conic section when \(\Delta \neq 0\). \(h^2-ab > 0\): Hyperbola (think \(y^2-x^2=1\)) \(h^2-ab < 0\): Ellipse (think \(x^2+y^2=1\)) \(h^2-ab = 0\): Parabola (think \(y=x^2\)) Always check for degeneracy (\(\Delta=0\)) first if the option of "pair of straight lines" is available.

Step 1: Understanding the Concept:

This problem involves calculating the eccentricity and identifying the asymptotes for various hyperbolas. For a standard hyperbola, the eccentricity \(e\) is given by \(e = \sqrt{1 + (semi-conjugate axis)^2 / (semi-transverse axis)^2}\). The asymptotes pass through the center and are determined by the ratio of the semi-axes. We will analyze each case and match it to the correct property.


Step 2: Detailed Explanation:


(A) \( \frac{y^2}{36} - \frac{x^2}{16} = 1 \):
This is a vertical hyperbola (opens up/down) with its transverse axis along the y-axis. Here, \(a^2 = 36\) and \(b^2 = 16\).
The eccentricity is \(e = \sqrt{1 + \frac{b^2}{a^2}} = \sqrt{1 + \frac{16}{36}} = \sqrt{\frac{36+16}{36}} = \sqrt{\frac{52}{36}} = \frac{\sqrt{4 \cdot 13}}{6} = \frac{2\sqrt{13}}{6} = \frac{\sqrt{13}}{3}\).
Thus, (A) matches (III).
(C) \( 7x^2 - y^2 = 224 \):
To get the standard form, we divide by 224: \(\frac{7x^2}{224} - \frac{y^2}{224} = 1 \implies \frac{x^2}{32} - \frac{y^2}{224} = 1\).
This is a horizontal hyperbola with \(a^2=32\) and \(b^2=224\).
The eccentricity is \(e = \sqrt{1 + \frac{b^2}{a^2}} = \sqrt{1 + \frac{224}{32}} = \sqrt{1 + 7} = \sqrt{8} = 2\sqrt{2}\).
Thus, (C) matches (I).
(D) \( \frac{x^2}{16} - \frac{y^2}{20} = \frac{1}{9} \):
First, we multiply by 9 to set the right side to 1: \(\frac{9x^2}{16} - \frac{9y^2}{20} = 1\).
This can be rewritten as \(\frac{x^2}{16/9} - \frac{y^2}{20/9} = 1\).
This is a horizontal hyperbola with \(a^2=16/9\) and \(b^2=20/9\).
The eccentricity is \(e = \sqrt{1 + \frac{b^2}{a^2}} = \sqrt{1 + \frac{20/9}{16/9}} = \sqrt{1 + \frac{20}{16}} = \sqrt{1+\frac{5}{4}} = \sqrt{\frac{9}{4}} = \frac{3}{2}\).
Thus, (D) matches (II).
(B) \( 7x^2 + 12xy - 2y^2 - 2x + 4y - 7 = 0 \):
This is a rotated hyperbola. Finding its properties is complex. However, since we have matched every other option, we can deduce by elimination that (B) must match with the remaining property.
Thus, (B) matches (IV).


Step 3: Final Answer:

The correctly determined pairings are (A)-(III), (B)-(IV), (C)-(I), and (D)-(II). This combination corresponds to option (C).
Quick Tip: For standard hyperbolas, quickly identify if it's horizontal (\(x^2\) term positive) or vertical (\(y^2\) term positive). This determines which value is \(a^2\) (the denominator of the positive term) and which is \(b^2\). The formula for eccentricity \(e^2 = 1+b^2/a^2\) is the same, but the roles of the denominators switch.

Step 1: Understanding the Concept:

This problem deals with vertical motion under the influence of constant gravitational acceleration. When an object is thrown upwards, it reaches a maximum height where its velocity momentarily becomes zero. We can relate the initial velocity, final velocity, acceleration, and displacement (height) using a standard equation of motion.


Step 2: Key Formula or Approach:

The kinematic equation that connects final velocity (\(v\)), initial velocity (\(u\)), acceleration (\(a\)), and displacement (\(s\)) is: \[ v^2 = u^2 + 2as \]
For our scenario:

The final velocity at the maximum height is \(v = 0\).
The acceleration due to gravity acts downwards, so \(a = -g\).
The displacement is the maximum height, \(s = H\).

Substituting these into the equation gives \( 0^2 = u^2 + 2(-g)H \).
We can rearrange this to solve for the maximum height \(H\): \[ 2gH = u^2 \implies H = \frac{u^2}{2g} \]
This result shows a crucial relationship: the maximum height attained is directly proportional to the square of the initial velocity (\(H \propto u^2\)), since \(2g\) is a constant.


Step 3: Detailed Explanation:

Let the initial velocities of the two stones be \(u_1\) and \(u_2\), and their corresponding maximum heights be \(H_1\) and \(H_2\).
We are given the ratio of their initial velocities: \[ \frac{u_1}{u_2} = \frac{2}{5} \]
Using the proportionality relationship \(H \propto u^2\), we can set up a ratio for their heights: \[ \frac{H_1}{H_2} = \frac{k \cdot u_1^2}{k \cdot u_2^2} \quad (where k = \frac{1}{2g}) \] \[ \frac{H_1}{H_2} = \frac{u_1^2}{u_2^2} = \left(\frac{u_1}{u_2}\right)^2 \]
Now we can substitute the given velocity ratio into this equation: \[ \frac{H_1}{H_2} = \left(\frac{2}{5}\right)^2 = \frac{2^2}{5^2} = \frac{4}{25} \]

Step 4: Final Answer:

The ratio of the maximum heights, \(H_1:H_2\), is 4:25.
Quick Tip: For projectile motion problems, remember the key relationships: Maximum height \(H \propto u^2\) (for a given angle) Time of flight \(T \propto u\) Range \(R \propto u^2\) These proportionality relations allow you to solve ratio problems very quickly without calculating the actual values.

Step 1: Understanding the Concept:

When three forces acting on a single point are in equilibrium, their vector sum is zero. This implies that if we represent the forces as vectors and arrange them head-to-tail, they will form a closed triangle. The lengths of the sides of this triangle will be equal to the magnitudes of the forces. We can then use the Law of Cosines on this triangle to find the angles.


Step 2: Key Formula or Approach:

Let the magnitudes of the three forces be \(a=5\) N, \(b=4\) N, and \(c=8\) N. These form the sides of a force triangle. The question asks for the angle between the two smaller forces (4 N and 5 N). In the force triangle, this corresponds to the internal angle opposite the side representing the largest force (8 N).
The Law of Cosines states: \( c^2 = a^2 + b^2 - 2ab\cos\theta \), where \(\theta\) is the angle opposite side \(c\).


Step 3: Detailed Explanation:

Let the two smaller forces be represented by sides of length \(a=5\) and \(b=4\). The third force is represented by the side of length \(c=8\). We want to find the angle \(\theta\) between the sides \(a\) and \(b\), which is the angle opposite side \(c\).
Applying the Law of Cosines: \[ c^2 = a^2 + b^2 - 2ab\cos\theta \]
Substitute the given force magnitudes: \[ 8^2 = 5^2 + 4^2 - 2(5)(4)\cos\theta \] \[ 64 = 25 + 16 - 40\cos\theta \]
Combine the constant terms on the right side: \[ 64 = 41 - 40\cos\theta \]
Isolate the term with \(\cos\theta\): \[ 64 - 41 = -40\cos\theta \] \[ 23 = -40\cos\theta \]
Finally, solve for \(\cos\theta\): \[ \cos\theta = -\frac{23}{40} \]
This result matches option (B). It is important to note a discrepancy in the provided problem, as the official "Correct Answer" is listed as (A), while the calculation correctly yields (B). Based on the standard application of physics principles, the derived answer is \(\cos^{-1}(-\frac{23}{40})\).


Step 4: Final Answer:

Following the direct calculation based on the Law of Cosines applied to the force triangle, the angle between the two smaller forces is \( \cos^{-1}\left(-\frac{23}{40}\right) \).
Quick Tip: For three forces in equilibrium, the vector sum is zero, and they form a closed triangle. You can apply either the Law of Sines or the Law of Cosines to this triangle. The angle *between* two force vectors (when placed tail-to-tail) is supplementary to the internal angle of the triangle at their vertex. Be careful which angle the question asks for.

Step 1: Understanding the Concept:

The principle of equilibrium, as described by Newton's First Law, states that an object will remain at rest or in uniform motion unless acted upon by a net external force. For a point to be in equilibrium, the vector sum of all forces acting on it must be the zero vector. Here, we analyze the specific case of just two forces.


Step 2: Key Formula or Approach:

Let the two forces be represented by the vectors \( \vec{F}_1 \) and \( \vec{F}_2 \). The mathematical condition for equilibrium is: \[ \vec{F}_{net} = \vec{F}_1 + \vec{F}_2 = \vec{0} \]
We will analyze the implications of this single vector equation.


Step 3: Detailed Explanation:

The equilibrium equation \( \vec{F}_1 + \vec{F}_2 = \vec{0} \) can be rearranged to \( \vec{F}_1 = -\vec{F}_2 \). This simple vector relationship contains three distinct and necessary conditions:

Condition on Magnitude: The magnitude of a vector is its length, which is always non-negative. If we take the magnitude of both sides of the equation, we get \( |\vec{F}_1| = |-\vec{F}_2| \). The negative sign does not affect the magnitude, so \( |\vec{F}_1| = |\vec{F}_2| \). This means the forces must be equal in magnitude. Therefore, statement (A) is true, and statement (E) is false.
Condition on Direction: The negative sign in \( \vec{F}_1 = -\vec{F}_2 \) explicitly means that the direction of vector \( \vec{F}_1 \) is exactly the opposite of the direction of vector \( \vec{F}_2 \). Therefore, statement (C) is true, and statement (B) is false.
Condition on Line of Action: For two vectors to be perfect opposites, they must not only point in opposite directions but also lie along the same line. If they were parallel but on different lines, they would create a couple and induce rotation, violating equilibrium. Thus, they must act along the same straight line (be collinear). Therefore, statement (D) is true.


Step 4: Final Answer:

For two forces to ensure equilibrium, they must satisfy all three conditions: be equal in magnitude, act in opposite directions, and share the same line of action. Thus, statements (A), (C), and (D) are the correct conditions.
Quick Tip: Think of a tug-of-war. For the rope to stay still (in equilibrium), the two teams must pull with the same force (equal magnitude) in perfectly opposite directions along the line of the rope. All three conditions (equal magnitude, opposite direction, same line of action) are necessary.

Step 1: Understanding the Concept:

The magnitude of the resultant vector of two forces \(P\) and \(Q\) acting at a common point with an angle \(\alpha\) between them is found using the Law of Cosines for vector addition. This problem requires us to apply this law to four specific scenarios.


Step 2: Key Formula or Approach:

The general formula for the magnitude of the resultant force \(R\) is given by: \[ R^2 = P^2 + Q^2 + 2PQ\cos\alpha \quad \implies \quad R = \sqrt{P^2 + Q^2 + 2PQ\cos\alpha} \]
We will systematically evaluate this formula for each of the conditions given in List-I.


Step 3: Detailed Explanation:


(A) P and Q are perpendicular: Perpendicular forces have an angle of \(\alpha = 90^\circ\) between them. The cosine of this angle is \(\cos(90^\circ) = 0\).
Substituting into the general formula:
\[ R = \sqrt{P^2 + Q^2 + 2PQ(0)} = \sqrt{P^2 + Q^2} \]
This is essentially the Pythagorean theorem for vectors. (A) matches with (IV).
(B) P and Q are equal forces (\(Q=P\)): We substitute \(Q=P\) into the formula:
\[ R = \sqrt{P^2 + P^2 + 2P(P)\cos\alpha} = \sqrt{2P^2 + 2P^2\cos\alpha} = \sqrt{2P^2(1 + \cos\alpha)} \]
Using the half-angle trigonometric identity \(1 + \cos\alpha = 2\cos^2(\alpha/2)\), we get:
\[ R = \sqrt{2P^2 \cdot 2\cos^2(\alpha/2)} = \sqrt{4P^2\cos^2(\alpha/2)} = 2P\cos(\alpha/2) \]
(Assuming the angle gives a positive cosine). (B) matches with (III).
(C) P and Q act in the same direction: This corresponds to an angle of \(\alpha = 0^\circ\). The cosine is \(\cos(0^\circ) = 1\).
\[ R = \sqrt{P^2 + Q^2 + 2PQ(1)} = \sqrt{(P+Q)^2} = P+Q \]
The resultant is the simple arithmetic sum of the magnitudes. (C) matches with (II).
(D) P and Q act in the opposite direction: This corresponds to an angle of \(\alpha = 180^\circ\). The cosine is \(\cos(180^\circ) = -1\).
\[ R = \sqrt{P^2 + Q^2 + 2PQ(-1)} = \sqrt{P^2 - 2PQ + Q^2} = \sqrt{(P-Q)^2} = |P-Q| \]
The resultant is the absolute difference of the magnitudes. (D) matches with (I).


Step 4: Final Answer:

The correct matches are (A)-(IV), (B)-(III), (C)-(II), and (D)-(I), which corresponds to option (D).
Quick Tip: Memorize the general formula \(R^2 = P^2 + Q^2 + 2PQ\cos\alpha\). All other cases (perpendicular, parallel, anti-parallel, equal forces) are just specializations of this one rule. This is much more efficient than memorizing four separate formulas.

Step 1: Understanding the Concept:

The problem requires evaluating a definite integral of the dot product of a vector function \(\vec{A}(t)\) with its own time derivative. The key to solving this efficiently is to recognize that the integrand is related to the derivative of the vector's squared magnitude.


Step 2: Key Formula or Approach:

We start by considering the derivative of the squared magnitude of \(\vec{A}\), which is \(A^2 = \vec{A} \cdot \vec{A}\). Applying the product rule for differentiation to this dot product: \[ \frac{d}{dt}(A^2) = \frac{d}{dt}(\vec{A} \cdot \vec{A}) = \frac{d\vec{A}}{dt} \cdot \vec{A} + \vec{A} \cdot \frac{d\vec{A}}{dt} \]
Since the dot product is commutative (\(\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}\)), the two terms on the right are identical. \[ \frac{d(A^2)}{dt} = 2 \left(\vec{A} \cdot \frac{d\vec{A}}{dt}\right) \]
From this identity, we can express the integrand of our problem as: \[ \vec{A} \cdot \frac{d\vec{A}}{dt} = \frac{1}{2} \frac{d(A^2)}{dt} \]

Step 3: Detailed Explanation:

By substituting this identity into the original integral, we get a much simpler form: \[ \int_2^3 \vec{A} \cdot \frac{d\vec{A}}{dt} dt = \int_2^3 \frac{1}{2} \frac{d(A^2)}{dt} dt \]
According to the Fundamental Theorem of Calculus, integrating a derivative with respect to its variable simply yields the original function evaluated at the limits of integration. \[ \int_2^3 \frac{1}{2} \frac{d(A^2)}{dt} dt = \frac{1}{2} \left[ A^2 \right]_2^3 = \frac{1}{2} \left( |\vec{A}(3)|^2 - |\vec{A}(2)|^2 \right) \]
Now, we just need to compute the squared magnitudes of the vector at \(t=2\) and \(t=3\).
For \(t=2\), \( \vec{A}(2) = 2\hat{i} - \hat{j} + 2\hat{k} \): \[ |\vec{A}(2)|^2 = (2)^2 + (-1)^2 + (2)^2 = 4 + 1 + 4 = 9 \]
For \(t=3\), \( \vec{A}(3) = 4\hat{i} - 2\hat{j} + 3\hat{k} \): \[ |\vec{A}(3)|^2 = (4)^2 + (-2)^2 + (3)^2 = 16 + 4 + 9 = 29 \]
Finally, we substitute these values back into our expression: \[ Value = \frac{1}{2} (29 - 9) = \frac{1}{2} (20) = 10 \]

Step 4: Final Answer:

The value of the definite integral is exactly 10.
Quick Tip: The identity \( \vec{A} \cdot \frac{d\vec{A}}{dt} = \frac{1}{2} \frac{d}{dt}(A^2) = A \frac{dA}{dt} \) is extremely useful in vector calculus and mechanics (where it relates to the rate of change of kinetic energy). Recognizing this identity immediately converts a complex integral into a simple evaluation at the endpoints.

Step 1: Understanding the Concept:

This question asks for the correct mathematical statement of Green's theorem. This fundamental theorem of vector calculus establishes a relationship between a line integral around a simple, closed, counterclockwise-oriented curve \(C\) and a double integral over the plane region \(R\) that is enclosed by \(C\).


Step 2: Key Formula or Approach:

Green's theorem is typically expressed in the context of a 2D vector field \( \vec{F}(x,y) = M(x,y)\hat{i} + N(x,y)\hat{j} \).
The theorem relates the circulation of the vector field around the boundary curve \(C\) to the integral of the "microscopic circulation" (the curl) over the interior region \(R\).
The circulation is given by the line integral: \( \oint_C \vec{F} \cdot d\vec{r} = \oint_C Mdx + Ndy \).
The microscopic circulation is given by the z-component of the curl of \( \vec{F} \): \( (curl \vec{F})_z = \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \).
The theorem equates these two concepts: \[ \oint_C Mdx + Ndy = \iint_R \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) dx dy \]

Step 3: Detailed Explanation:

We must carefully compare the standard formula with the four options provided.

Option (A) presents \( \frac{\partial M}{\partial x} - \frac{\partial N}{\partial y} \), which has incorrect partial derivatives and the wrong sign.
Option (B) presents \( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \). This precisely matches the correct formulation of Green's theorem for circulation.
Option (C) presents \( \frac{\partial M}{\partial y} - \frac{\partial N}{\partial x} \), which is the negative of the correct expression.
Option (D) presents \( \frac{\partial M}{\partial x} + \frac{\partial N}{\partial y} \), which is the divergence of the vector field, not the curl. This expression appears in the flux-divergence form of Green's theorem, not the circulation-curl form asked here.


Step 4: Final Answer:

The correct and standard statement of Green's (circulation) theorem is the one provided in option (B).
Quick Tip: A simple way to remember the order of terms in Green's theorem is to think of the vector field \( \vec{F} = (M, N) \). The integrand is \( \frac{\partial(second component)}{\partial(first variable)} - \frac{\partial(first component)}{\partial(second variable)} \), which is \( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \). This is the 2D "curl".

Step 1: Understanding the Concept:

We are tasked with finding the area of a specific portion of a plane. This portion of the plane is situated directly above a defined region in the xy-plane. The region in the xy-plane is its projection. We can calculate the surface area by integrating a specific factor over this projection region.


Step 2: Key Formula or Approach:

The area of a surface \(z = f(x, y)\) that lies above a region R in the xy-plane is given by the double integral: \[ S = \iint_R \sqrt{1 + \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2} \, dA \]
For a plane, the term inside the square root is constant, which simplifies the calculation to \(S = (constant factor) \times (Area of R)\).


Step 3: Detailed Explanation:

1. Determine the Surface and its Derivatives:

The equation of the plane is \(x + 2y + 2z = 12\). We first solve for \(z\) to express it as a function \(z(x, y)\): \[ 2z = 12 - x - 2y \implies z = 6 - \frac{1}{2}x - y \]
Next, we find the partial derivatives of \(z\) with respect to \(x\) and \(y\): \[ \frac{\partial z}{\partial x} = -\frac{1}{2} \] \[ \frac{\partial z}{\partial y} = -1 \]
2. Calculate the Integrand:

Now we compute the square root term for the surface area formula: \[ \sqrt{1 + \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2} = \sqrt{1 + \left(-\frac{1}{2}\right)^2 + (-1)^2} = \sqrt{1 + \frac{1}{4} + 1} = \sqrt{\frac{9}{4}} = \frac{3}{2} \]
3. Determine the Region of Integration R:

The plane is "cut off" by the boundaries \(x=0\), \(y=0\), and the cylinder \(x^2+y^2=16\). This describes the projection of our surface area onto the xy-plane. The region R is the part of the disk of radius 4 (\(r=\sqrt{16}\)) that lies in the first quadrant.
The area of a full circle is \( \pi r^2 = \pi (4)^2 = 16\pi \).
The area of the region R, being one quadrant, is one-fourth of the full circle's area: \[ Area of R = \frac{1}{4} (16\pi) = 4\pi \]
4. Calculate the Surface Area:

Since the integrand is a constant (\(3/2\)), the surface area integral simplifies: \[ S = \iint_R \frac{3}{2} \, dA = \frac{3}{2} \times (Area of R) = \frac{3}{2} \times (4\pi) = 6\pi \]

Step 4: Final Answer:

The area of the portion of the plane cut off by the given boundaries is \(6\pi\).
Quick Tip: The formula \( S = \frac{A_{proj}}{|\cos\theta|} \), where \(A_{proj}\) is the area of the projection and \(\theta\) is the angle between the normal to the surface and the normal to the projection plane, is very powerful for finding the area of a planar region. Here, \(\cos\theta = \frac{\vec{n} \cdot \hat{k}}{|\vec{n}||\hat{k}|} = \frac{2}{3}\), so \(S = \frac{4\pi}{2/3} = 6\pi\).

Step 1: Understanding the Concept:

The problem asks for the evaluation of a surface integral of a vector field over a closed surface \(S\). This integral represents the net flux of the vector field \(\vec{F}\) emerging from the volume enclosed by the surface. The surface described is a closed cylinder, so the Divergence Theorem (Gauss's theorem) provides the most direct method of solution by converting the surface integral into a volume integral.


Step 2: Key Formula or Approach:

The Divergence Theorem states that the total flux through a closed surface \(S\) is equal to the volume integral of the divergence of the vector field over the enclosed volume \(V\): \[ \oint_S \vec{F} \cdot d\vec{s} = \iiint_V (\nabla \cdot \vec{F}) \, dV \]
Our procedure will be:
1. Calculate the divergence of \( \vec{F} \).
2. Set up and evaluate the triple integral of the divergence over the volume of the cylinder.


Step 3: Detailed Explanation:

1. Calculate the Divergence of \(\vec{F}\):

The divergence of a vector field \( \vec{F} = F_x\hat{i} + F_y\hat{j} + F_z\hat{k} \) is \( \nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \).
For our field, \( \vec{F} = 4x\hat{i} - 2y^2\hat{j} + z^2\hat{k} \): \[ \nabla \cdot \vec{F} = \frac{\partial}{\partial x}(4x) + \frac{\partial}{\partial y}(-2y^2) + \frac{\partial}{\partial z}(z^2) = 4 - 4y + 2z \]

2. Set up and Evaluate the Volume Integral:

The volume \(V\) is a cylinder defined by \(x^2+y^2 \le 4\) and \(0 \le z \le 3\). It is most convenient to use cylindrical coordinates for integration over this volume.
The transformations are: \( x = r\cos\theta, y = r\sin\theta, z = z \), and the volume element is \( dV = r \, dz \, dr \, d\theta \).
The limits of integration are: \( 0 \le r \le 2 \), \( 0 \le \theta \le 2\pi \), and \( 0 \le z \le 3 \).
The integral becomes: \[ \iiint_V (4 - 4y + 2z) \, dV = \int_0^{2\pi} \int_0^2 \int_0^3 (4 - 4r\sin\theta + 2z) \, r \, dz \, dr \, d\theta \]
We can evaluate this term by term: \[ \int_0^{2\pi} \int_0^2 \int_0^3 4r \, dz dr d\theta - \int_0^{2\pi} \int_0^2 \int_0^3 4r^2\sin\theta \, dz dr d\theta + \int_0^{2\pi} \int_0^2 \int_0^3 2zr \, dz dr d\theta \]
Term 1: \( \int_0^{2\pi} d\theta \cdot \int_0^2 r dr \cdot \int_0^3 4 dz = [2\pi] \cdot [\frac{r^2}{2}]_0^2 \cdot [4z]_0^3 = (2\pi)(2)(12) = 48\pi \).
Term 2: This integral involves \( \int_0^{2\pi} \sin\theta \, d\theta \). Since the integral of sine over a full period is zero (\([-\cos\theta]_0^{2\pi} = -1 - (-1) = 0\)), this entire term evaluates to 0.
Term 3: \( \int_0^{2\pi} d\theta \cdot \int_0^2 r dr \cdot \int_0^3 2z dz = [2\pi] \cdot [\frac{r^2}{2}]_0^2 \cdot [z^2]_0^3 = (2\pi)(2)(9) = 36\pi \).
Summing the results: \[ Total Flux = 48\pi - 0 + 36\pi = 84\pi \]

Step 4: Final Answer:

By applying the Divergence Theorem, the value of the surface integral is found to be \(84\pi\).
Quick Tip: When asked to evaluate a surface integral over a simple closed surface (like a sphere, cylinder, or cube), always check if the Divergence Theorem can be applied first. It often simplifies the problem from multiple surface integrals to a single, often easier, volume integral. Also, look for symmetries that might make parts of the integral zero, like integrating \(y\) or \(sin\theta\) over a symmetric domain.

Step 1: Understanding the Concept:

This problem is a multi-step vector calculus task. It requires finding the directional derivative of a scalar field, \(\phi\), at a specific point. The scalar field itself is derived from another function, \(f\), by applying the Laplacian operator (\(\phi = \nabla \cdot (\nabla f) = \nabla^2 f\)). The direction for the derivative is given by the normal vector to a specified surface at that same point.


Step 2: Key Formula or Approach:

The process can be broken down into five distinct calculations:
1. Compute the scalar field \(\phi\) by finding the Laplacian of \(f\): \( \phi = \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} \).
2. Find the gradient of this new field, \( \nabla \phi \), and evaluate it at the given point (1, -2, 1).
3. Find the direction vector by calculating the gradient of the function defining the surface, \( S(x,y,z) = xy^2z - 3x - z^2 = 0 \). Evaluate \( \nabla S \) at (1, -2, 1).
4. Normalize the direction vector from step 3 to get a unit vector, \( \hat{u} \).
5. Compute the directional derivative using the dot product: \( D_{\hat{u}}\phi = (\nabla \phi) \cdot \hat{u} \) at the point.


Step 3: Detailed Explanation:

1. Calculate \( \phi = \nabla^2 f \):

Given \( f = 2x^3y^2z^4 \), we find its second partial derivatives: \( \frac{\partial f}{\partial x} = 6x^2y^2z^4 \implies \frac{\partial^2 f}{\partial x^2} = 12xy^2z^4 \) \( \frac{\partial f}{\partial y} = 4x^3yz^4 \implies \frac{\partial^2 f}{\partial y^2} = 4x^3z^4 \) \( \frac{\partial f}{\partial z} = 8x^3y^2z^3 \implies \frac{\partial^2 f}{\partial z^2} = 24x^3y^2z^2 \)
Summing them gives the Laplacian: \( \phi(x,y,z) = 12xy^2z^4 + 4x^3z^4 + 24x^3y^2z^2 \).


2. Find the Gradient of \( \phi \) at (1, -2, 1):

We need the partial derivatives of \(\phi\): \( \frac{\partial \phi}{\partial x} = 12y^2z^4 + 12x^2z^4 + 72x^2y^2z^2 \). At (1,-2,1): \( 12(-2)^2(1)^4 + 12(1)^2(1)^4 + 72(1)^2(-2)^2(1)^2 = 48 + 12 + 288 = 348 \). \( \frac{\partial \phi}{\partial y} = 24xyz^4 + 48x^3yz^2 \). At (1,-2,1): \( 24(1)(-2)(1)^4 + 48(1)^3(-2)(1)^2 = -48 - 96 = -144 \). \( \frac{\partial \phi}{\partial z} = 48xy^2z^3 + 16x^3z^3 + 48x^3y^2z \). At (1,-2,1): \( 48(1)(-2)^2(1)^3 + 16(1)^3(1)^3 + 48(1)^3(-2)^2(1) = 192 + 16 + 192 = 400 \).
So, at (1, -2, 1), the gradient is \( \nabla \phi = 348\hat{i} - 144\hat{j} + 400\hat{k} \).


3. Find the Normal Vector to the Surface:

The surface is given by \( xy^2z - 3x - z^2 = 0 \). Let \( S(x,y,z) \) be this function. The normal vector is \( \nabla S \). \( \nabla S = \frac{\partial S}{\partial x}\hat{i} + \frac{\partial S}{\partial y}\hat{j} + \frac{\partial S}{\partial z}\hat{k} = (y^2z - 3)\hat{i} + (2xyz)\hat{j} + (xy^2 - 2z)\hat{k} \).
Evaluate at (1, -2, 1): \( \nabla S = ((-2)^2(1) - 3)\hat{i} + (2(1)(-2)(1))\hat{j} + ((1)(-2)^2 - 2(1))\hat{k} = (4 - 3)\hat{i} - 4\hat{j} + (4 - 2)\hat{k} = \hat{i} - 4\hat{j} + 2\hat{k} \).


4. Find the Unit Direction Vector \( \hat{u} \):

The magnitude of the normal vector is \( |\nabla S| = \sqrt{1^2 + (-4)^2 + 2^2} = \sqrt{1 + 16 + 4} = \sqrt{21} \).
The unit vector is \( \hat{u} = \frac{\nabla S}{|\nabla S|} = \frac{1}{\sqrt{21}}(\hat{i} - 4\hat{j} + 2\hat{k}) \).


5. Calculate the Final Directional Derivative:
\( D_{\hat{u}}\phi = \nabla \phi \cdot \hat{u} = (348\hat{i} - 144\hat{j} + 400\hat{k}) \cdot \frac{1}{\sqrt{21}}(\hat{i} - 4\hat{j} + 2\hat{k}) \) \( = \frac{1}{\sqrt{21}} [(348)(1) + (-144)(-4) + (400)(2)] \) \( = \frac{1}{\sqrt{21}} [348 + 576 + 800] = \frac{1724}{\sqrt{21}} \).

Step 4: Final Answer:

The directional derivative at the specified point and direction is \( \frac{1724}{\sqrt{21}} \).
Quick Tip: Break down complex vector calculus problems into smaller, manageable steps: 1. Identify the function you're differentiating. 2. Identify the direction vector. 3. Compute the gradient of the function. 4. Normalize the direction vector. 5. Take the dot product. Be methodical with partial derivatives to avoid errors.

Step 1: Understanding the Concept:

This question requires the identification of correct vector calculus identities involving the gradient (grad, \( \nabla \)), divergence (div, \( \nabla \cdot \)), and curl (\(\nabla \times \)) operators. We will verify each statement by expanding the definitions of these operators.


Step 2: Detailed Explanation:


(A) div (grad f) = \( \nabla^2 f \):
First, we find the gradient of the scalar function f: \( grad f = \nabla f = \frac{\partial f}{\partial x}\hat{i} + \frac{\partial f}{\partial y}\hat{j} + \frac{\partial f}{\partial z}\hat{k} \).
Next, we take the divergence of this resulting vector field: \( div (grad f) = \nabla \cdot (\nabla f) = \frac{\partial}{\partial x}(\frac{\partial f}{\partial x}) + \frac{\partial}{\partial y}(\frac{\partial f}{\partial y}) + \frac{\partial}{\partial z}(\frac{\partial f}{\partial z}) = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} \).
This expression is the definition of the Laplacian operator, \( \nabla^2 f \). Thus, statement (A) is correct.
(B) curl curl \( \vec{F} \) = grad curl \( \vec{F} \) - \( \nabla^2 \vec{F} \):
This refers to the "vector triple product" identity for the del operator. The correct identity is \( \nabla \times (\nabla \times \vec{F}) = \nabla(\nabla \cdot \vec{F}) - (\nabla \cdot \nabla)\vec{F} \).
In words, this is \( curl(curl \vec{F}) = grad(div \vec{F}) - \nabla^2 \vec{F} \).
The statement provides "grad curl \( \vec{F} \)", which is incorrect as the identity involves the gradient of the divergence. Therefore, statement (B) is incorrect.
(C) div curl \( \vec{F} \) = \( \vec{0} \):
The divergence of the curl of any sufficiently smooth vector field is always the scalar zero: \( \nabla \cdot (\nabla \times \vec{F}) = 0 \). The statement claims the result is the zero vector (\( \vec{0} \)), whereas the divergence of a vector field is a scalar. This is a notational error, making the statement technically incorrect as written.
(D) curl grad f = \( \vec{0} \):
The curl of the gradient of any sufficiently smooth scalar function is always the zero vector: \( \nabla \times (\nabla f) = \vec{0} \). This is a fundamental identity that reflects the conservative nature of gradient fields. Statement (D) is correct.
(E) div (\(f\vec{F}\)) = f div \( \vec{F} \) + (grad f) \( \times \vec{F} \):
This is a product rule for divergence. The correct identity is \( \nabla \cdot (f\vec{F}) = (\nabla f) \cdot \vec{F} + f(\nabla \cdot \vec{F}) \), which is \( div(f\vec{F}) = (grad f) \cdot \vec{F} + f div \vec{F} \).
The statement incorrectly uses a cross product (\(\times\)) instead of the correct dot product (\(\cdot\)). Therefore, statement (E) is incorrect.


Step 3: Final Answer:

The only statements that are definitively correct in both form and notation are (A) and (D). This combination corresponds to option (B).
Quick Tip: Memorize the two fundamental "zero" identities: The curl of a gradient is always the zero vector: \( \nabla \times (\nabla f) = \vec{0} \). The divergence of a curl is always zero: \( \nabla \cdot (\nabla \times \vec{F}) = 0 \). Also, remember the "vector BAC-CAB rule" analogue: \( \nabla \times (\nabla \times \vec{F}) = \nabla(\nabla \cdot \vec{F}) - (\nabla \cdot \nabla)\vec{F} \).

Step 1: Understanding the Concept:

A set of functions is defined as linearly dependent if at least one function in the set can be expressed as a linear combination of the others. Algebraically, a set of functions \( f_1(x), f_2(x), \ldots, f_n(x) \) is linearly dependent if there exists a set of constants \( c_1, c_2, \ldots, c_n \), not all of which are zero, such that the equation \( c_1 f_1(x) + c_2 f_2(x) + \ldots + c_n f_n(x) = 0 \) holds true for all values of x. If the only solution is \( c_1 = c_2 = \ldots = c_n = 0 \), the functions are linearly independent. We need to find the statement that makes a false claim about dependency.


Step 2: Detailed Explanation:


(A) \( f_1=x^2-1, f_2=3x^2, f_3=2-5x^2 \): We check if there are non-zero constants \(c_1, c_2, c_3\) such that \( c_1(x^2-1) + c_2(3x^2) + c_3(2-5x^2) = 0 \).
Combining terms by powers of x gives: \( (c_1 + 3c_2 - 5c_3)x^2 + (-c_1 + 2c_3) = 0 \).
For this polynomial to be zero for all x, its coefficients must be zero:
1) \( c_1 + 3c_2 - 5c_3 = 0 \)
2) \( -c_1 + 2c_3 = 0 \implies c_1 = 2c_3 \).
Substituting (2) into (1): \( (2c_3) + 3c_2 - 5c_3 = 0 \implies 3c_2 - 3c_3 = 0 \implies c_2 = c_3 \).
We can choose a non-zero value, for instance, \( c_3 = 1 \). This gives \( c_2 = 1 \) and \( c_1 = 2 \). Since a non-trivial solution (2, 1, 1) exists, the functions are linearly dependent. The statement is correct.
(B) \( x, x^2, x^3 \): The equation is \( c_1x + c_2x^2 + c_3x^3 = 0 \). A non-zero polynomial of degree 3 can have at most 3 roots. For this equation to be true for all x (infinitely many roots), it must be the zero polynomial, meaning all coefficients are zero: \(c_1=c_2=c_3=0\). This is the trivial solution, so the functions are linearly independent. The statement is correct.
(C) 1, sinx, cosx: The equation is \( c_1(1) + c_2\sin x + c_3\cos x = 0 \). To determine the constants, we can evaluate the equation at different values of x:
At \(x=0\): \(c_1 + c_2(0) + c_3(1) = 0 \implies c_1+c_3=0\).
At \(x=\pi\): \(c_1 + c_2(0) + c_3(-1) = 0 \implies c_1-c_3=0\).
Adding these two results gives \(2c_1=0 \implies c_1=0\), which in turn means \(c_3=0\).
Now substitute these into the original equation: \( c_2\sin x = 0 \). Since this must hold for all x, and \(\sin x\) is not always zero, we must have \(c_2=0\).
The only solution is \(c_1=c_2=c_3=0\). Therefore, the functions are linearly independent. The statement claims they are dependent, so the statement is not correct.
(D) x and \( \frac{1}{x} \): The equation is \( c_1x + c_2\frac{1}{x} = 0 \). Multiplying by x (for \(x \neq 0\)) gives \( c_1x^2 + c_2 = 0 \). For this polynomial to be zero for all x, both coefficients must be zero: \(c_1=0\) and \(c_2=0\). The functions are linearly independent. The statement is correct.


Step 3: Final Answer:

The only incorrect statement among the options is (C).
Quick Tip: To quickly test linear independence, consider if one function can be written as a linear combination of the others. In (A), you can see that \(2(x^2-1) + 1(3x^2) = 5x^2-2\), which is \(-1 \times (2-5x^2)\). So \(2f_1 + f_2 + f_3 = 0\). For (C), it's impossible to write, for example, \(\sin x\) as \(c_1(1) + c_2(\cos x)\) for all x.

Step 1: Understanding the Concept:

Finding the Laplace transform of \( \cos\sqrt{t} \) is not straightforward using a standard table of transforms. The presence of the square root inside the cosine function suggests that a different technique is required. The most effective method is to represent the function as a power series (its Taylor series) and then apply the Laplace transform to each term of the series.


Step 2: Key Formula or Approach:

The solution relies on two key mathematical facts:
1. The Maclaurin series (Taylor series centered at 0) for the cosine function is: \( \cos(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \ldots \).
2. The Laplace transform of a power of \(t\) is given by: \( \mathcal{L}\{t^k\} = \frac{k!}{s^{k+1}} \), where k is a non-negative integer.


Step 3: Detailed Explanation:

1. Express \( \cos\sqrt{t} \) as a Power Series:

We begin by taking the known series for \(\cos(x)\) and substituting \( x = \sqrt{t} \). \[ \cos\sqrt{t} = \sum_{n=0}^\infty \frac{(-1)^n (\sqrt{t})^{2n}}{(2n)!} \]
Simplifying the term \((\sqrt{t})^{2n} = (t^{1/2})^{2n} = t^n\), the series becomes: \[ \cos\sqrt{t} = \sum_{n=0}^\infty \frac{(-1)^n t^n}{(2n)!} = \frac{t^0}{0!} - \frac{t^1}{2!} + \frac{t^2}{4!} - \frac{t^3}{6!} + \ldots \]

2. Apply the Laplace Transform to the Series:

We apply the Laplace transform operator \( \mathcal{L} \) to the entire series. Assuming that we can interchange the operations of summation and transformation (which is valid for this series): \[ \mathcal{L}\{\cos\sqrt{t}\} = \mathcal{L}\left\{ \sum_{n=0}^\infty \frac{(-1)^n t^n}{(2n)!} \right\} = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!} \mathcal{L}\{t^n\} \]
Now, we use the formula for the Laplace transform of \(t^n\), which is \( \mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}} \).
Substituting this into our series gives the final result: \[ \mathcal{L}\{\cos\sqrt{t}\} = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!} \cdot \frac{n!}{s^{n+1}} = \sum_{n=0}^\infty \frac{(-1)^n n!}{(2n)! s^{n+1}} \]

Step 4: Final Answer:

The resulting series expression for the Laplace transform is \( \sum_{n=0}^\infty \frac{(-1)^n n!}{(2n)! s^{n+1}} \), which corresponds exactly to option (B).
Quick Tip: When faced with a Laplace transform of a function for which you don't have a standard formula (especially involving compositions like \( \cos\sqrt{t} \) or \( \frac{\sin t}{t} \)), the Taylor series expansion is a powerful technique. Expand the function, then transform term-by-term using the basic formula \( \mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}} \).

Step 1: Understanding the Concept:

To find the general solution of a linear, non-homogeneous, second-order differential equation with constant coefficients, we follow a two-part process. First, we find the complementary function (\(y_c\)), which is the general solution to the associated homogeneous equation. Second, we find a particular integral (\(y_p\)), which is any specific solution to the full non-homogeneous equation. The general solution is then their sum, \(y = y_c + y_p\).


Step 2: Key Formula or Approach:

1. **Find \(y_c\):** Solve the homogeneous equation \(y'' + 9y = 0\) by finding the roots of its auxiliary (characteristic) equation, \(m^2 + 9 = 0\).
2. **Find \(y_p\):** The form of the right-hand side, \(\cos(3x)\), suggests using the method of undetermined coefficients. However, we must first check if this term is already part of \(y_c\). If it is, this signifies a case of resonance, and our guess for \(y_p\) must be modified. Alternatively, the operator method provides a direct way to handle resonance cases.


Step 3: Detailed Explanation:

1. Finding the Complementary Function (\(y_c\)):

We start with the homogeneous equation: \(y'' + 9y = 0\).
The auxiliary equation is \(m^2 + 9 = 0\), which gives \(m^2 = -9\), so the roots are complex: \(m = \pm \sqrt{-9} = \pm 3i\).
For complex roots of the form \(m = \alpha \pm i\beta\), the solution is \(y = e^{\alpha x}(C_1\cos(\beta x) + C_2\sin(\beta x))\). Here, \(\alpha=0\) and \(\beta=3\), so the complementary function is: \[y_c = C_1\cos(3x) + C_2\sin(3x)\]

2. Finding the Particular Integral (\(y_p\)):

The forcing function on the right-hand side is \(\cos(3x)\). We notice that this term is already present in our complementary function \(y_c\). This is a special case known as resonance. When this occurs, the standard method of undetermined coefficients fails, and a modified approach is needed. Using the operator method is efficient here. Let \(D = d/dx\).
The particular integral is \( y_p = \frac{1}{D^2+9} \cos(3x) \).
Normally, for \(\cos(ax)\), we substitute \(D^2 = -a^2\). Here, \(a=3\), so we would substitute \(D^2 = -3^2 = -9\). However, this makes the denominator zero, confirming resonance. For this specific case, we use the standard resonance formula: \[ \frac{1}{D^2+a^2} \cos(ax) = \frac{x}{2a} \sin(ax) \]
Applying this formula with \(a=3\): \[ y_p = \frac{x}{2(3)} \sin(3x) = \frac{x}{6}\sin(3x) \]

3. Forming the General Solution:

The complete general solution is the sum of the complementary function and the particular integral: \[ y(x) = y_c + y_p = C_1\cos(3x) + C_2\sin(3x) + \frac{x}{6}\sin(3x) \]

Step 4: Final Answer:

The general solution to the differential equation is \( y(x) = C_1\cos(3x) + C_2\sin(3x) + \frac{x}{6}\sin(3x) \), which corresponds to option (B).
Quick Tip: When solving \( (D^2+a^2)y = \cos(ax) \) or \( (D^2+a^2)y = \sin(ax) \), always check if the roots of the auxiliary equation (\(\pm ai\)) match the frequency of the forcing term. If they do, you are in a resonance case, and the particular solution will involve multiplication by x. Remember the formulas: \( \frac{1}{D^2+a^2}\cos(ax) = \frac{x}{2a}\sin(ax) \) \( \frac{1}{D^2+a^2}\sin(ax) = -\frac{x}{2a}\cos(ax) \)

Step 1: Understanding the Concept:

The task is to convert a differential equation with given boundary conditions (a Boundary Value Problem or BVP) into an equivalent integral equation. This is a common procedure in the study of differential equations and mathematical physics, often accomplished by using a Green's function, which acts as the kernel of the integral equation. The provided kernel \(k(x,t)\) is the Green's function for this specific problem.


Step 2: Key Formula or Approach:

We start by rewriting the BVP as \( y''(x) = f(x) \), where we temporarily set the forcing function \(f(x) = -\lambda y(x)\). The goal is to solve for \(y(x)\) in terms of an integral involving \(f(x)\). We can achieve this by integrating the equation twice and applying the boundary conditions.
1. Integrate \(y''(x) = f(x)\) from 0 to x.
2. Integrate the result again from 0 to x.
3. Use the boundary conditions \(y(0)=0\) and \(y(1)=0\) to solve for the constants of integration.
4. Substitute \(f(t) = -\lambda y(t)\) back into the resulting integral form.


Step 3: Detailed Explanation:

Starting with \( y''(t) = f(t) \), we integrate with respect to \(t\) from 0 to \(x\): \[ \int_0^x y''(t) dt = \int_0^x f(t) dt \implies y'(x) - y'(0) = \int_0^x f(t) dt \]
Let \(C_1 = y'(0)\). Then \( y'(x) = \int_0^x f(t) dt + C_1 \).
Integrate again with respect to \(x\) from 0 to \(x\): \[ \int_0^x y'(s) ds = \int_0^x \left(\int_0^s f(t) dt\right) ds + \int_0^x C_1 ds \] \[ y(x) - y(0) = \int_0^x (x-t)f(t) dt + C_1x \]
Using the first boundary condition, \(y(0)=0\), this becomes: \[ y(x) = \int_0^x (x-t)f(t) dt + C_1x \]
Now, apply the second boundary condition, \(y(1)=0\): \[ 0 = y(1) = \int_0^1 (1-t)f(t) dt + C_1 \implies C_1 = - \int_0^1 (1-t)f(t) dt \]
Substitute this expression for \(C_1\) back into the equation for \(y(x)\): \[ y(x) = \int_0^x (x-t)f(t) dt - x \int_0^1 (1-t)f(t) dt \]
We can combine these into a single integral by splitting the second integral: \[ y(x) = \int_0^x (x-t)f(t) dt - x \int_0^x (1-t)f(t) dt - x \int_x^1 (1-t)f(t) dt \] \[ y(x) = \int_0^x [(x-t) - x(1-t)] f(t) dt - \int_x^1 x(1-t)f(t) dt \] \[ y(x) = \int_0^x [t(x-1)] f(t) dt + \int_x^1 [x(1-t)]f(t) dt \]
This can be written as \( y(x) = \int_0^1 G(x,t) f(t) dt \), where \( G(x,t) = \begin{cases} t(x-1) & t < x
x(1-t) & t > x \end{cases} \).
Finally, we replace the placeholder \(f(t)\) with its original expression, \(f(t) = -\lambda y(t)\): \[ y(x) = \int_0^1 G(x,t) (-\lambda y(t)) dt = \lambda \int_0^1 [-G(x,t)] y(t) dt \]
Observing that \( -G(x,t) = \begin{cases} t(1-x) & t < x
x(t-1) & t > x \end{cases} \), we see that \(k(x,t)=-G(x,t)\) is precisely the kernel given in the problem.
Thus, the integral equation is: \[ y(x) = \lambda \int_0^1 k(x,t) y(t) dt \]

Step 4: Final Answer:

The boundary value problem is equivalent to the Fredholm integral equation \( y(x) = \lambda \int_0^1 k(x,t) y(t) dt \), which is option (A).
Quick Tip: The conversion of a Sturm-Liouville boundary value problem \( (p(x)y')' + q(x)y + \lambda r(x)y = 0 \) with homogeneous boundary conditions into an integral equation \( y(x) = \lambda \int_a^b G(x,t) r(t) y(t) dt \) is a standard procedure. The function \(G(x,t)\) is the Green's function for the operator \(L = (py')'+qy\). For the simple case \(y''+\lambda y=0\) with \(y(0)=y(1)=0\), the resulting integral equation is \(y(x) = \lambda \int_0^1 k(x,t) y(t) dt\), where \(k(x,t)\) is the specific kernel given.

Step 1: Understanding the Concept:

To find the family of curves that are orthogonal (perpendicular at every intersection) to a given family of curves, we follow a three-step process. First, we find the differential equation that describes the original family by differentiating and eliminating the parameter \(c\). Second, we find the differential equation for the orthogonal family by replacing the slope \( \frac{dy}{dx} \) with its negative reciprocal, \( -\frac{dx}{dy} \). Third, we solve this new differential equation.


Step 2: Detailed Explanation:


(A) Family: \(xy = c\) (Rectangular Hyperbolas)
1. Differentiate w.r.t. x: \(1 \cdot y + x \cdot \frac{dy}{dx} = 0 \implies \frac{dy}{dx} = -\frac{y}{x}\).
2. Replace slope for orthogonality: \( \frac{dy}{dx} \to -\frac{dx}{dy} \). The new DE is \( -\frac{dx}{dy} = -\frac{y}{x} \implies y \, dy = x \, dx \).
3. Solve by integration: \( \int y \, dy = \int x \, dx \implies \frac{y^2}{2} = \frac{x^2}{2} + C' \implies y^2 - x^2 = 2C' \). This is a family of hyperbolas of the form \(y^2 - x^2 = K\). This matches (III).

(B) Family: \(e^x + e^{-y} = c\)
1. Differentiate w.r.t. x: \(e^x + e^{-y}(-\frac{dy}{dx}) = 0 \implies \frac{dy}{dx} = \frac{e^x}{e^{-y}} = e^{x+y}\).
2. Replace slope: \( \frac{dy}{dx} \to -\frac{dx}{dy} \). The new DE is \( -\frac{dx}{dy} = e^{x+y} \implies \frac{dy}{dx} = -e^{-x-y} = -e^{-x}e^{-y} \).
3. Solve by separating variables: \( e^y \, dy = -e^{-x} \, dx \). Integrate both sides: \( \int e^y \, dy = \int -e^{-x} \, dx \implies e^y = e^{-x} + K \). This is the family \( e^y - e^{-x} = K \). This matches (IV).

(C) Family: \(y^2 = cx\) (Parabolas opening sideways)
1. Differentiate: \(2y \frac{dy}{dx} = c\). Eliminate \(c\) using \(c=y^2/x\): \(2y \frac{dy}{dx} = \frac{y^2}{x} \implies \frac{dy}{dx} = \frac{y}{2x}\).
2. Replace slope: \( -\frac{dx}{dy} = \frac{y}{2x} \implies -2x \, dx = y \, dy \).
3. Solve by integration: \( \int -2x \, dx = \int y \, dy \implies -x^2 = \frac{y^2}{2} + C' \implies x^2 + \frac{y^2}{2} = K \). This is a family of ellipses. This matches (I).

(D) Family: \(x^2 - y^2 = cx\)
1. Differentiate: \(2x - 2y\frac{dy}{dx} = c\). Eliminate \(c\) using \(c=(x^2-y^2)/x\): \(2x - 2y\frac{dy}{dx} = \frac{x^2-y^2}{x} \).
\( 2x^2 - 2xy\frac{dy}{dx} = x^2 - y^2 \implies 2xy\frac{dy}{dx} = x^2+y^2 \implies \frac{dy}{dx} = \frac{x^2+y^2}{2xy} \).
2. Replace slope: \( -\frac{dx}{dy} = \frac{x^2+y^2}{2xy} \implies \frac{dy}{dx} = -\frac{2xy}{x^2+y^2} \).
3. This is a homogeneous DE. Let \(y=vx\), so \(\frac{dy}{dx} = v + x\frac{dv}{dx}\).
\( v + x\frac{dv}{dx} = -\frac{2x(vx)}{x^2+(vx)^2} = -\frac{2v}{1+v^2} \).
\( x\frac{dv}{dx} = -\frac{2v}{1+v^2} - v = \frac{-2v - v(1+v^2)}{1+v^2} = \frac{-3v-v^3}{1+v^2} \).
Separate variables: \( \frac{1+v^2}{v(3+v^2)}dv = -\frac{dx}{x} \). Integrating (via partial fractions) gives \( \frac{1}{3}\ln|v| + \frac{1}{3}\ln|v^2+3| = -\ln|x| + C' \).
This simplifies to \( v(v^2+3) = K/x^3 \). Substituting \(v=y/x\) back gives \( \frac{y}{x}(\frac{y^2}{x^2}+3) = \frac{K}{x^3} \), which simplifies to \( y(y^2+3x^2) = K \). This matches (II).


Step 3: Final Answer:

The correct set of matches is (A)-(III), (B)-(IV), (C)-(I), and (D)-(II), which corresponds to option (A).
Quick Tip: The procedure for finding orthogonal trajectories is systematic: 1. Find the differential equation of the family (eliminate the constant). 2. Replace \(y'\) with \(-1/y'\). 3. Solve the new differential equation. For homogeneous equations like in part (D), the substitution \(y=vx\) is the standard method.

Step 1: Understanding the Concept:

We are asked to find a particular integral (\(y_p\)) for a second-order linear non-homogeneous differential equation. The left side is \(y'' + 2y' + y\), which corresponds to the operator \( (D+1)^2 \). The right-hand side is of the form \(e^{ax}V(x)\), which makes the operator shift theorem an ideal method for finding the particular integral.


Step 2: Key Formula or Approach:

The particular integral \(y_p\) can be found using the inverse operator: \[ y_p = \frac{1}{D^2+2D+1} \left(e^{-x}\log x\right) = \frac{1}{(D+1)^2} \left(e^{-x}\log x\right) \]
We apply the operator shift theorem, which states \( \frac{1}{f(D)} e^{ax}V(x) = e^{ax} \frac{1}{f(D+a)}V(x) \).
In our case, \(a=-1\) and \(V(x)=\log x\).


Step 3: Detailed Explanation:

Applying the shift theorem to our problem: \[ y_p = e^{-x} \frac{1}{((D-1)+1)^2} \log x = e^{-x} \frac{1}{D^2} \log x \]
The operator \( \frac{1}{D} \) signifies integration. Thus, the operator \( \frac{1}{D^2} \) means we must integrate the function \(\log x\) twice consecutively.
First Integration: We use integration by parts for \( \int \log x \, dx \), with \(u=\log x\) and \(dv=dx\). This gives \(du = \frac{1}{x}dx\) and \(v=x\). \[ \frac{1}{D}(\log x) = \int 1 \cdot \log x \, dx = x\log x - \int x \cdot \frac{1}{x} dx = x\log x - \int 1 \, dx = x\log x - x \]
Second Integration: Now we integrate the result. \[ \frac{1}{D^2}(\log x) = \int (x\log x - x) \, dx = \int x\log x \, dx - \int x \, dx \]
The second term is simple: \( \int x \, dx = \frac{x^2}{2} \).
For the first term, \( \int x\log x \, dx \), we use integration by parts again with \(u=\log x\) and \(dv=x dx\), so \(v=\frac{x^2}{2}\). \[ \int x\log x \, dx = \frac{x^2}{2}\log x - \int \frac{x^2}{2} \cdot \frac{1}{x} dx = \frac{x^2}{2}\log x - \int \frac{x}{2} dx = \frac{x^2}{2}\log x - \frac{x^2}{4} \]
Combining these results for the second integration: \[ \frac{1}{D^2}(\log x) = \left(\frac{x^2}{2}\log x - \frac{x^2}{4}\right) - \frac{x^2}{2} = \frac{x^2}{2}\log x - \frac{3x^2}{4} \]
Finally, we re-introduce the exponential term to get the full particular integral: \[ y_p = e^{-x} \left( \frac{x^2}{2}\log x - \frac{3x^2}{4} \right) \]
This derived solution is correct, but the answer options are in a different format. We must verify if one of the options simplifies to our result. Let's check option (B): \[ y_p = \frac{x^2e^{-x}}{2}\left(\frac{1}{2}-\log x\right) + x^2e^{-x}(\log x - 1) \] \[ = e^{-x} \left[ \frac{x^2}{4} - \frac{x^2}{2}\log x + x^2\log x - x^2 \right] \] \[ = e^{-x} \left[ (\frac{x^2}{4} - x^2) + (-\frac{x^2}{2}\log x + x^2\log x) \right] \] \[ = e^{-x} \left[ -\frac{3x^2}{4} + \frac{x^2}{2}\log x \right] = e^{-x} \left( \frac{x^2}{2}\log x - \frac{3x^2}{4} \right) \]
The simplified form of option (B) matches our derived solution.

Step 4: Final Answer:

The particular integral of the differential equation is correctly represented by option (B).
Quick Tip: The operator shift theorem is very powerful for particular integrals where the forcing function is of the form \(e^{ax}V(x)\). It reduces the problem to finding the particular integral for the function \(V(x)\) with a simpler operator. For repeated roots in the complementary function, like \((D+a)^2 y = ...\), the method of variation of parameters is also a reliable, albeit sometimes longer, alternative.