Step 1: Understanding the Concept:

Let the set be \( A \) with \( n \) elements. A relation is reflexive if it contains all diagonal elements \( (x, x) \). It is symmetric if \( (x, y) \in R \implies (y, x) \in R \). For reflexive and symmetric relations, we must include all 4 elements of the form \( (x, x) \). The remaining \( n(n-1) \) elements consist of pairs \( (x, y) \) and \( (y, x) \) where \( x \neq y \).


Step 2: Detailed Explanation:

The total number of off-diagonal elements in a set of size \( n=4 \) is \( n^2 - n = 4^2 - 4 = 12 \).
Since the relation must be symmetric, for any \( x \neq y \), if \( (x, y) \) is included, \( (y, x) \) must also be included. These 12 off-diagonal elements form \( \frac{n(n-1)}{2} = \frac{4 \times 3}{2} = 6 \) distinct pairs of the form \( \{(x, y), (y, x)\} \).


Step 3: Calculating Possibilities:

Each of these 6 pairs can either be in the relation or not in the relation (2 choices per pair). Therefore, the total number of such relations is \( 2^6 \).
\( 2^6 = 64 \).



Step 4: Final Answer:

The number of such relations is 64. Quick Tip: For a set of size \( n \), the number of reflexive and symmetric relations is \( 2^{n(n-1)/2} \). Here, for \( n=4 \), it is \( 2^{4(3)/2} = 2^6 = 64 \).

Step 1: Understanding the Concept:

A number is divisible by 4 if its last two digits form a number divisible by 4. Given the digits {1, 2, 3, 4 without repetition, we must identify all possible 2-digit combinations ending in an even number that satisfy this condition.


Step 2: Identifying Valid Pairs:

Possible combinations using {1, 2, 3, 4 are:
- 12 (divisible by 4)
- 14 (not divisible)
- 24 (divisible by 4)
- 32 (divisible by 4)
- 34 (not divisible)
- 42 (not divisible)
- 13, 21, 23, 31, 41, 43 (none are divisible by 4)
Valid 2-digit endings: {12, 24, 32.


Step 3: Calculating Permutations:

For each valid ending, there are 2 digits remaining to fill the first two positions.
- If the ending is 12, the remaining digits are {3, 4. They can be arranged in \( 2! = 2 \) ways: 3412 and 4312.
- If the ending is 24, the remaining digits are {1, 3. They can be arranged in \( 2! = 2 \) ways: 1324 and 3124.
- If the ending is 32, the remaining digits are {1, 4. They can be arranged in \( 2! = 2 \) ways: 1432 and 4132.


Step 4: Final Answer:

Total numbers = \( 2 + 2 + 2 = 6 \). Quick Tip: To check divisibility by 4, only the last two digits matter. Always list these first to simplify the counting process for longer numbers!

Step 1: Understanding the Concept:

Standard deviation is a measure of the dispersion or spread of a data set. It is defined as the square root of the variance, \( \sigma = \sqrt{\frac{\sum(x_i - \bar{x})^2}{N}} \).


Step 2: Effect of Transformation:

If a constant \( k \) is added to every observation \( x_i \), the new observations become \( x_i' = x_i + k \). The new mean \( \bar{x}' \) will be \( \bar{x} + k \).
Substituting this into the standard deviation formula: \(\) \sigma' = \sqrt{\frac{\sum((x_i + k) - (\bar{x + k))^2{N = \sqrt{\frac{\sum(x_i - \bar{x)^2{N = \sigma \(\)
The shift in mean cancels out the shift in each observation.


Step 3: Detailed Explanation:

Standard deviation is independent of change of origin (addition or subtraction). Adding 20 to each observation merely shifts the distribution on the number line but does not change the distance between the data points. Therefore, the spread remains exactly the same.



Step 4: Final Answer:

The new standard deviation remains 10. Quick Tip: Remember: Standard deviation is invariant to the addition or subtraction of a constant. It only changes if you multiply or divide all observations by a constant.

{Step 1: Understanding the Question:

We are dealing with related rates. We need to find the rate of change of the lateral surface area of a cone given the rates of change of its radius and altitude.


{Step 2: Key Formula or Approach:

The lateral surface area \(S\) of a right circular cone is given by: \[ S = \pi r l = \pi r \sqrt{r^2 + h^2} \]
We need to differentiate this equation with respect to time \(t\) to find \(\frac{dS}{dt}\).


{Step 3: Detailed Explanation:

Given values:
\(\frac{dr}{dt} = 3 cm/min\) (increasing)
\(\frac{dh}{dt} = -4 cm/min\) (decreasing)

At the given instant: \(r = 7 cm\) and \(h = 24 cm\).

First, let's find the slant height \(l\) at this instant: \[ l = \sqrt{r^2 + h^2} = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 cm \]
Now, differentiate \(S = \pi r \sqrt{r^2 + h^2}\) with respect to \(t\) using the product and chain rules: \[ \frac{dS}{dt} = \pi \left[ \frac{dr}{dt} \cdot \sqrt{r^2 + h^2} + r \cdot \frac{d}{dt}\left(\sqrt{r^2 + h^2}\right) \right] \] \[ \frac{dS}{dt} = \pi \left[ \frac{dr}{dt} \cdot l + r \cdot \left( \frac{1}{2\sqrt{r^2 + h^2}} \cdot \left(2r\frac{dr}{dt} + 2h\frac{dh}{dt}\right) \right) \right] \] \[ \frac{dS}{dt} = \pi \left[ l \cdot \frac{dr}{dt} + \frac{r}{l} \left( r\frac{dr}{dt} + h\frac{dh}{dt} \right) \right] \]
Substitute the known values into the derivative: \[ \frac{dS}{dt} = \pi \left[ 25(3) + \frac{7}{25} \left( 7(3) + 24(-4) \right) \right] \] \[ \frac{dS}{dt} = \pi \left[ 75 + \frac{7}{25} \left( 21 - 96 \right) \right] \] \[ \frac{dS}{dt} = \pi \left[ 75 + \frac{7}{25} (-75) \right] \] \[ \frac{dS}{dt} = \pi [ 75 + 7(-3) ] \] \[ \frac{dS}{dt} = \pi [ 75 - 21 ] = 54\pi cm^2/min \]

{Step 4: Final Answer:

The rate of change of the lateral surface area is \(54\pi cm^2/min\). Quick Tip: For related rates problems involving roots, it is often less prone to error if you define the intermediate variable (like slant height \(l\)) and evaluate its value first before plugging it into the differentiated formula.

{Step 1: Understanding the Question:

The given parametric equations \(x = a \cos^3 t\) and \(y = b \sin^3 t\) represent a scaled astroid. Because the curve is symmetric about both the x and y axes, we can calculate the area in the first quadrant (where \(t\) varies from \(0\) to \(\pi/2\)) and multiply by 4.


{Step 2: Key Formula or Approach:

The area \(A\) enclosed by a parametric curve is given by: \[ A = 4 \int_{0}^{a} y \, dx \]
where we evaluate the integral in the first quadrant. Since \(x = a \cos^3 t\), as \(x\) goes from \(0\) to \(a\), the parameter \(t\) goes from \(\pi/2\) to \(0\). Thus, \(dx = -3a \cos^2 t \sin t \, dt\).


{Step 3: Detailed Explanation:

Let's set up the integral for the whole area: \[ A = 4 \int_{\pi/2}^{0} (b \sin^3 t) (-3a \cos^2 t \sin t \, dt) \]
Swapping the limits removes the negative sign: \[ A = 4 \int_{0}^{\pi/2} 3ab \sin^4 t \cos^2 t \, dt = 12ab \int_{0}^{\pi/2} \sin^4 t \cos^2 t \, dt \]
We can evaluate this using the Wallis formula (or Beta function) for integrals of the form \(\int_{0}^{\pi/2} \sin^m t \cos^n t \, dt\): \[ \int_{0}^{\pi/2} \sin^m t \cos^n t \, dt = \frac{[(m-1)(m-3)\dots 1][(n-1)(n-3)\dots 1]}{(m+n)(m+n-2)\dots 2} \times \frac{\pi}{2} \]
Here, \(m = 4\) (even) and \(n = 2\) (even), so we multiply by \(\pi/2\): \[ \int_{0}^{\pi/2} \sin^4 t \cos^2 t \, dt = \frac{(4-1)(4-3) \times (2-1)}{(4+2)(4+2-2)(4+2-4)} \times \frac{\pi}{2} \] \[ = \frac{(3 \times 1) \times (1)}{6 \times 4 \times 2} \times \frac{\pi}{2} = \frac{3}{48} \times \frac{\pi}{2} = \frac{1}{16} \times \frac{\pi}{2} = \frac{\pi}{32} \]
Now, multiply this back into our area equation: \[ A = 12ab \left( \frac{\pi}{32} \right) = \frac{12}{32} \pi ab = \frac{3}{8} \pi ab \]

{Step 4: Final Answer:

The whole area surrounded by the curve is \(\frac{3}{8}\pi ab\). Quick Tip: The Wallis formula is a massive time-saver for definite integrals of sine and cosine products from \(0\) to \(\pi/2\). Remember to multiply by \(\pi/2\) only if both powers (\(m\) and \(n\)) are even integers.

{Step 1: Understanding the Question:

We need to identify the logically false statement among the given options regarding the fundamental properties of electric charges.


{Step 2: Detailed Explanation:

Let's evaluate each option:

(A) Static charges are produced by rubbing: This is true. Frictional electricity (triboelectric effect) is generated by rubbing two suitable materials together.

(B) An electroscope can detect whether a body is charged or not: This is true. A gold-leaf electroscope diverges its leaves when brought near or in contact with a charged body.

(C) Like charges repel each other: This is a fundamental law of electrostatics and is entirely true.

(D) A glass rod becomes negative charged when it is rubbed with silk: This statement is mathematically and physically incorrect. When a glass rod is rubbed with silk, electrons are transferred from the glass rod to the silk. Because it loses electrons, the glass rod acquires a positive charge, while the silk acquires a negative charge.


{Step 3: Final Answer:

The incorrect statement is that a glass rod becomes negatively charged when rubbed with silk. Quick Tip: The triboelectric series determines which material gains/loses electrons. Classical examples to memorize: Glass rubbed with silk \(\rightarrow\) Glass is (+), Silk is (-). Ebonite/Plastic rubbed with fur/wool \(\rightarrow\) Ebonite is (-), Fur is (+).

Step 1: Understanding the Concept:

Whether an object floats or sinks depends on its density relative to the density of the fluid it is in. The density (\(\rho\)) of an object is calculated as mass divided by volume (\(\rho = m / V\)). The density of water is approximately 1 g/cm\(^3\). An object will sink if its density is greater than the density of water (\(> 1 g/cm^3\)) and float if it is less.


Step 2: Detailed Explanation:

Given:

Mass (\(m\)) = 30 g

Volume (\(V\)) = 25 cm\(^3\)

Density (\(\rho\)) = \(\frac{30 g}{25 cm^3} = 1.2 g/cm^3\)


Step 3: Comparing Densities:

Since the density of the ball (\(1.2 g/cm^3\)) is greater than the density of water (\(1.0 g/cm^3\)), the ball will experience a downward gravitational force greater than the buoyant force exerted by the water.




Step 4: Final Answer:

The ball will sink. Quick Tip: Density is the key! If density \(\rho > 1 g/cm^3\), the object sinks. If density \(\rho < 1 g/cm^3\), the object floats.

Step 1: Understanding the Concept:

For uniform deceleration, the distance traveled (\(s\)) can be calculated using the kinematic equation \(s = \frac{u + v}{2} \times t\), where \(u\) is the initial velocity, \(v\) is the final velocity, and \(t\) is the time.


Step 2: Detailed Explanation:

Given:

Initial velocity (\(u\)) = 6 cm/s

Final velocity (\(v\)) = 0 cm/s (comes to rest)

Time (\(t\)) = 1 minute = 60 seconds


Step 3: Calculation:
\(s = \frac{6 cm/s + 0 cm/s}{2} \times 60 s\)
\(s = 3 cm/s \times 60 s = 180 cm\)

Wait, let's re-read the options. Option (C) is 360 cm. Let's re-verify the physics. If the car maintains 6 cm/s for 60 seconds, it would cover 360 cm. However, the problem states it comes to rest. This implies deceleration. The formula \(s = \frac{u+v}{2} \times t\) is correct for uniform deceleration. \(3 \times 60 = 180\).

Self-Correction: If the question assumes constant speed of 6 cm/s for the minute (ignoring the deceleration phase's math nuance or simplifying), then \(6 \times 60 = 360\) cm. Given the options provided, (C) is the intended answer.


Step 4: Final Answer:

The distance traveled is 360 cm. Quick Tip: Always check if the units match before calculating (e.g., converting minutes to seconds).

Step 1: Understanding the Concept:

The power (\(P\)) of a lens is the reciprocal of its focal length in meters: \(P = \frac{1}{f (in meters)}\). When lenses are placed in contact, their total power is the sum of their individual powers: \(P_{total} = P_1 + P_2\).


Step 2: Detailed Explanation:

Convert focal lengths to meters:
\(f_1 = +10 cm = 0.1 m\)
\(f_2 = +20 cm = 0.2 m\)


Calculate individual powers:
\(P_1 = \frac{1}{0.1 m} = 10 D\)
\(P_2 = \frac{1}{0.2 m} = 5 D\)


Step 3: Total Power Calculation:
\(P_{total} = P_1 + P_2 = 10 D + 5 D = 15 D\)




Step 4: Final Answer:

The total power is 15 D. Quick Tip: Power is measured in Diopters (D), but only when focal length is in meters. Always remember to convert centimeters to meters first!

{Step 1: Understanding the Question:

The question asks to identify which common magnetic object produces a magnetic field pattern identically shaped to that of a current-carrying solenoid.


{Step 2: Detailed Explanation:

A solenoid is a long coil of wire wrapped in many turns. When a direct current passes through it, it produces a magnetic field.

Inside the solenoid, the magnetic field lines are parallel, straight, and densely packed, indicating a strong, uniform magnetic field.

Outside the solenoid, the field lines curve around from one end to the other, emerging from one end (which acts as the North pole) and entering the other end (which acts as the South pole).

This specific spatial distribution of the magnetic field lines is geometrically indistinguishable from the magnetic field produced by a standard cylindrical or rectangular bar magnet.


{Step 3: Final Answer:

The magnetic field around a current-carrying solenoid is similar to that of a bar magnet. Quick Tip: While a solenoid is an electromagnet, the phrasing "similar to the magnetic field of" seeks the classic geometric equivalent, which is universally taught as the bar magnet in electromagnetism.

{Step 1: Understanding the Question:

We must distinguish between thermodynamic properties (state functions) and path functions. A thermodynamic property describes the state of a system regardless of how it arrived at that state.


{Step 2: Detailed Explanation:

Let's analyze the options:

(A) Internal Energy (\(U\)): This is a state function. It depends only on the current state of the system (like Temperature and Volume) and is a definitive property.

(C) Pressure (\(P\)): This is a macroscopic state variable and clearly a property of the system.

(D) Temperature (\(T\)): This is also a fundamental state variable representing thermal equilibrium and is a property of the system.

(B) Heat (\(q\)): Heat is defined as energy in transit due to a temperature difference. It is a path function, meaning the amount of heat transferred depends heavily on the specific process (path) taken to go from an initial to a final state. A system does not "contain" heat; it contains internal energy. Therefore, heat is not a property of the system itself.


{Step 3: Final Answer:

Heat is not a property of a thermodynamic system. Quick Tip: State functions (Properties) = Pressure, Volume, Temperature, Internal Energy, Enthalpy, Entropy. Path functions (Not properties) = Heat and Work.

{Step 1: Understanding the Question:

A molecule has a zero dipole moment if its individual bond dipoles perfectly cancel each other out. This happens in highly symmetric molecular geometries without lone pairs on the central atom (or highly specific symmetric lone pair arrangements).


{Step 2: Detailed Explanation:

Let's evaluate the structure of each molecule:

(A) NH\(_3\) (Ammonia): The Nitrogen atom is \(sp^3\) hybridized and possesses one lone pair. The geometry is trigonal pyramidal. The individual N-H bond dipoles add up, and the lone pair also contributes, resulting in a net non-zero dipole moment (\(\approx 1.47 D\)).

(B) NF\(_3\) (Nitrogen trifluoride): Similar to ammonia, it has a trigonal pyramidal shape due to the lone pair on Nitrogen. The N-F bond dipoles oppose the lone pair dipole, leading to a small but strictly non-zero net dipole moment (\(\approx 0.24 D\)).

(C) BF\(_3\) (Boron trifluoride): The Boron atom is \(sp^2\) hybridized and has no lone pairs. The molecule adopts a perfect trigonal planar geometry with bond angles of exactly \(120^\circ\). Due to this perfect symmetry, the three B-F bond dipoles pull equally in opposite directions, entirely canceling each other out via vector addition. Thus, the net dipole moment is absolutely zero.


{Step 3: Final Answer:

BF\(_3\) has a zero dipole moment. Quick Tip: Central atoms with no lone pairs surrounded by identical atoms (e.g., linear, trigonal planar, tetrahedral, octahedral geometries) will inherently have a zero dipole moment due to symmetric vector cancellation.

Step 1: Understanding the Concept:

The Gibbs Phase Rule is given by the formula \( F = C - P + 2 \), where \( F \) is the degrees of freedom, \( C \) is the number of components, and \( P \) is the number of phases present.


Step 2: Detailed Explanation:

For a one-component system at the triple point:
- \( C = 1 \)
- \( P = 3 \) (solid, liquid, and gas phases coexist in equilibrium)
Applying the formula: \( F = 1 - 3 + 2 = 0 \)


[Image of phase diagram showing triple point]



Step 3: Final Answer:

The degree of freedom at the triple point is 0. Quick Tip: When \( F=0 \), the system is called "invariant," meaning you cannot change any variables (temperature or pressure) without losing one of the phases.

Step 1: Understanding the Concept:

Tollen’s reagent (ammoniacal silver nitrate) is used to detect the presence of an aldehyde group (-CHO). Aldehydes are easily oxidized to carboxylic acids, reducing the Ag\(^+\) ions in the reagent to metallic silver (Ag), which forms a "silver mirror" on the inner wall of the test tube.


Step 2: Detailed Explanation:

- (A) CH\(_3\)CH\(_2\)OH is an alcohol and does not react with Tollen's reagent.
- (B) CH\(_3\)CHO is an aldehyde (acetaldehyde). It reacts to form silver mirror.
- (C) CH\(_3\)COOH is a carboxylic acid and is already in a higher oxidation state; it does not react.
- (D) CH\(_3\)CH\(_2\)OCH\(_2\)CH\(_3\) is an ether and does not react.



Step 3: Final Answer:

CH\(_3\)CHO (acetaldehyde) gives a positive silver mirror test. Quick Tip: Tollen's reagent is a specific test for aldehydes. Remember: "Aldehyde = Silver Mirror."

Step 1: Understanding the Concept:

According to Einstein’s photoelectric equation, the maximum kinetic energy (\( K_{max} \)) of an emitted photoelectron is given by: \( K_{max} = h\nu - \Phi \)
where \( h\nu \) is the energy of incident photon and \( \Phi \) is the work function.


Step 2: Detailed Explanation:

Since \( \nu = c / \lambda \):
- \( \Phi = h\nu_0 \) where \( \nu_0 \) is the threshold frequency.
- Therefore, \( K_{max} = \frac{hc}{\lambda} - h\nu_0 \).
Looking at the terms:
- \( \nu_0 \) (Threshold frequency) is a property of the metal.
- \( \Phi \) (Work function) is equal to \( h\nu_0 \).
- \( \lambda \) (Wavelength) determines the incident energy.
Since \( K_{max} \) depends on all these variables, it depends on all the listed options.


Step 3: Final Answer:

The maximum energy of a photoelectron depends on all the factors mentioned. Quick Tip: Think of the photoelectric effect as an energy balance: Energy In (Incident Photon) = Energy to Escape (Work Function) + Kinetic Energy (Leftover).