Step 1: Understanding the Question:

We are given the displacement and time interval for a moving body and asked to calculate its velocity. The key part of the question is to express the final answer with the correct number of significant figures.


Step 2: Key Formula or Approach:

The formula for velocity is:
\[ Velocity = \frac{Displacement}{Time} \]
The rule for significant figures in division states that the result should have the same number of significant figures as the quantity with the least number of significant figures.


Step 3: Detailed Explanation:

Given values:

Displacement = 151.25 cm

Time = 2.25 s


First, let's determine the number of significant figures in each given value:

- In 151.25, there are 5 significant figures.

- In 2.25, there are 3 significant figures.


The minimum number of significant figures is 3. Therefore, our final answer must be rounded to 3 significant figures.


Now, we calculate the velocity:
\[ Velocity = \frac{151.25 cm}{2.25 s} \approx 67.2222... cm s^{-1} \]

Step 4: Final Answer:

We need to round the calculated velocity (67.2222...) to 3 significant figures.

The first three significant figures are 6, 7, and 2. The fourth digit is 2, which is less than 5, so we round down.

So, the velocity in the correct number of significant figures is 67.2 cm s\(^{-1}\).

This corresponds to option (E).
Quick Tip: In multiplication and division, the final answer should have the same number of significant figures as the input value with the fewest significant figures.
For addition and subtraction, the answer should have the same number of decimal places as the input value with the fewest decimal places.
Always check the rules for significant figures in measurement-based calculations.

Step 1: Understanding the Question:

The question asks for the dimensional formula of torque.


Step 2: Key Formula or Approach:

Torque (\(\tau\)) is defined as the product of force (F) and the perpendicular distance (r) from the axis of rotation to the line of action of the force.
\[ \tau = Force \times Perpendicular Distance \]
We need to find the dimensions of force and distance and then multiply them.


Step 3: Detailed Explanation:

First, let's find the dimensions of Force (F).

Force = Mass (m) \(\times\) Acceleration (a)

The dimension of Mass is \([M]\).

The dimension of Acceleration (change in velocity/time) is \([LT^{-2}]\).

So, the dimension of Force is:
\[ [F] = [M] \times [LT^{-2}] = [MLT^{-2}] \]

Next, the dimension of Perpendicular Distance (r) is simply length \([L]\).


Now, we can find the dimensions of Torque (\(\tau\)):
\[ [\tau] = [F] \times [r] \] \[ [\tau] = [MLT^{-2}] \times [L] \] \[ [\tau] = [ML^{1+1}T^{-2}] = [ML^2T^{-2}] \]

Step 4: Final Answer:

The dimensional formula for torque is \([ML^2T^{-2}]\).

This corresponds to option (D).
Quick Tip: The dimensions of torque are the same as the dimensions of work or energy.
Work = Force \(\times\) Displacement, which also gives \([MLT^{-2}] \times [L] = [ML^2T^{-2}]\).
Remembering this equivalence can save time in exams.

Step 1: Understanding the Question:

The question gives the velocity of a particle, \(\vec{v} = 6\hat{i} - 4\hat{j}\), and asks for its velocity when it reaches the x-axis "again". The wording suggests a reflection from the x-axis.


Step 2: Key Formula or Approach:

In an elastic collision with a horizontal surface (the x-axis):

1. The horizontal component of velocity (\(v_x\)) remains unchanged.

2. The vertical component of velocity (\(v_y\)) reverses its direction but keeps the same magnitude.


Step 3: Detailed Explanation:

Let the velocity of the particle just before hitting the x-axis be \(\vec{v}_{before} = 6\hat{i} - 4\hat{j}\).

The components are:

- Horizontal component: \(v_x = 6\)

- Vertical component: \(v_y = -4\)


During the elastic reflection from the x-axis:

- The new horizontal component, \(v_x'\), will be the same as the old one: \(v_x' = v_x = 6\).

- The new vertical component, \(v_y'\), will be the negative of the old one: \(v_y' = -v_y = -(-4) = +4\).


So, the velocity of the particle immediately after reflecting from the x-axis is:
\[ \vec{v}_{after} = v_x'\hat{i} + v_y'\hat{j} = 6\hat{i} + 4\hat{j} \]

This interpretation is the only one that leads to one of the given options.


Step 4: Final Answer:

The velocity of the body on reaching the x-axis again (after reflection) is \(6\hat{i} + 4\hat{j}\).

This corresponds to option (E).
Quick Tip: In projectile motion problems, always analyze the horizontal and vertical components of motion separately.
If a question's phrasing seems odd, consider alternative physical scenarios like reflection or collision, especially if a direct application of formulas doesn't match the options.
Here, "reaching the x-axis again" is best interpreted as colliding and bouncing off it.

Step 1: Understanding the Question:

We are given information about the displacement-time (x-t) graph of a moving body. We need to determine the nature of its motion.


Step 2: Key Formula or Approach:

The key concept is the physical meaning of the slope of a displacement-time graph.
\[ Velocity (v) = \frac{dx}{dt} = Slope of the x-t graph \]
The slope of a straight line is constant.


Step 3: Detailed Explanation:

The problem states that the x-t graph is a straight line. A straight line has a constant slope.

Since the slope of the x-t graph represents the velocity of the body, a constant slope implies that the velocity of the body is constant.

Motion with constant velocity is known as uniform velocity.


The angle given (\(45^{\circ}\)) allows us to calculate the value of this uniform velocity:
\[ Slope = \tan(\theta) \] \[ v = \tan(45^{\circ}) = 1 \]
Since the velocity is a constant value of 1 (in appropriate units), the body is moving with uniform velocity. Acceleration, which is the rate of change of velocity, is zero.


Step 4: Final Answer:

Because the displacement-time graph is a straight line, its slope is constant. This means the velocity is constant. Therefore, the body is moving with uniform velocity.

This corresponds to option (A).
Quick Tip: Remember the relationship between motion graphs:
- Slope of Displacement-Time graph = Velocity
- Slope of Velocity-Time graph = Acceleration
- Area under Velocity-Time graph = Displacement
- Area under Acceleration-Time graph = Change in Velocity

Step 1: Understanding the Question:

A ball is projected vertically upwards with a given initial speed. We need to find the maximum height it reaches, using the given value for the acceleration due to gravity.


Step 2: Key Formula or Approach:

We can use the third equation of motion for an object moving under constant acceleration:
\[ v^2 = u^2 + 2as \]
Here: \(v\) is the final velocity, \(u\) is the initial velocity, \(a\) is the acceleration, and \(s\) is the displacement (\(h_{max}\)).


Step 3: Detailed Explanation:

Let's list the known values:

- Initial speed, \(u = 6.0 m s^{-1}\).

- At the maximum height, the final speed is momentarily zero, so \(v = 0\).

- The acceleration is due to gravity, acting downwards. So, \(a = -g = -10 m s^{-2}\).

- The displacement is the maximum height, \(s = h_{max}\).


Now, substitute these values into the equation of motion:
\[ (0)^2 = (6.0)^2 + 2(-10)h_{max} \] \[ 0 = 36 - 20h_{max} \]
Rearrange the equation to solve for \(h_{max}\):
\[ 20h_{max} = 36 \] \[ h_{max} = \frac{36}{20} \] \[ h_{max} = 1.8 m \]

Step 4: Final Answer:

The maximum height reached by the ball is 1.8 m.

This corresponds to option (D).
Quick Tip: For vertical motion problems, you can also use the principle of conservation of energy.
The initial kinetic energy is converted into potential energy at the maximum height.
\[ \frac{1}{2}mu^2 = mgh_{max} \] \[ h_{max} = \frac{u^2}{2g} = \frac{6^2}{2 \times 10} = \frac{36}{20} = 1.8 m \] This method is often faster.

Step 1: Understanding the Question:

We need to identify the statement that is physically incorrect among the given five options.


Step 3: Detailed Explanation:

Let's analyze each statement:

(A) Forces in nature always occur between pair of bodies: This is a statement of Newton's Third Law. Forces are interactions between two objects. This is correct.


(B) Action and reaction forces are simultaneous forces: According to Newton's Third Law, the action and reaction forces arise at the exact same instant. This is correct.


(C) Coefficient of static friction is greater than the coefficient of kinetic friction: It generally takes more force to start an object moving than to keep it moving. Thus, \(\mu_s > \mu_k\). This is correct.


(D) Force is always in the direction of motion: This statement is incorrect. According to Newton's Second Law (\(\vec{F} = m\vec{a}\)), the net force is in the direction of acceleration, not necessarily its velocity. For example, in circular motion, velocity is tangential while the centripetal force is radial.


(E) Centripetal force acts towards the centre of a circle: This is the definition of centripetal force, which is necessary for circular motion. This is correct.


Step 4: Final Answer:

The statement that is incorrect is (D) because force determines the direction of acceleration, not necessarily the direction of motion.
Quick Tip: Be very careful with the words "always" and "never" in physics statements.
The statement "Force is always in the direction of motion" is a common misconception.
Remember that force is a vector that causes a change in momentum (acceleration).

Step 1: Understanding the Question:

A bullet with a given mass and initial velocity penetrates a certain distance into a tree and stops. We need to calculate the average resistive force.


Step 2: Key Formula or Approach:

The Work-Energy Theorem states that the net work done on an object equals the change in its kinetic energy.
\[ W_{net} = \Delta K = K_f - K_i \]
The work done by the resistive force F over a distance s is \(W = -F \cdot s\).


Step 3: Detailed Explanation:

First, convert all quantities to SI units.

- Mass, \(m = 10 g = 0.01\) kg.

- Initial velocity, \(u = 250\) m/s.

- Final velocity, \(v = 0\).

- Distance, \(s = 5 cm = 0.05\) m.


Calculate the change in kinetic energy.

- Initial Kinetic Energy, \(K_i = \frac{1}{2}mu^2 = \frac{1}{2}(0.01)(250)^2 = 312.5 J\).

- Final Kinetic Energy, \(K_f = 0\) J.


Apply the Work-Energy Theorem:
\[ W = K_f - K_i \] \[ -F \cdot s = 0 - 312.5 \] \[ -F \times (0.05) = -312.5 \] \[ F = \frac{312.5}{0.05} = 6250 N \]

Step 4: Final Answer:

The magnitude of the force exerted by the tree limb is 6250 N.

This corresponds to option (E).
Quick Tip: Alternatively, use Newton's second law and equations of motion.
1. Find acceleration using \(v^2 = u^2 + 2as\): \(0^2 = 250^2 + 2a(0.05) \implies a = -625000 m/s^2\).
2. Find force using \(F = ma\): \(F = (0.01) \times (-625000) = -6250\) N. The magnitude is 6250 N.
Choose the method you find quicker.

Step 1: Understanding the Question:

A block in a lift accelerating downwards has an apparent weight of \(\frac{Mg}{4}\). We need to find the downward acceleration.


Step 2: Key Formula or Approach:

Apply Newton's Second Law (\(F_{net} = Ma\)) to the block. The force exerted by the block on the floor is equal in magnitude to the normal reaction force N.


Step 3: Detailed Explanation:

Let 'a' be the downward acceleration of the lift.

The forces on the block are:

1. Gravitational force, \(W = Mg\), downwards.

2. Normal reaction force, \(N\), upwards.


We are given \(N = \frac{Mg}{4}\).


Applying Newton's Second Law in the downward direction:
\[ F_{net} = Ma \] \[ Mg - N = Ma \]
Substitute the value of N:
\[ Mg - \frac{Mg}{4} = Ma \] \[ \frac{3Mg}{4} = Ma \]
Cancel M from both sides:
\[ a = \frac{3g}{4} \]

Step 4: Final Answer:

The downward acceleration of the lift must be \(\frac{3g}{4}\).

This corresponds to option (E).
Quick Tip: Remember the general formulas for apparent weight in a lift:
- Accelerating upwards: \(N = M(g+a)\).
- Accelerating downwards: \(N = M(g-a)\).
Here, \(\frac{Mg}{4} = M(g-a)\), which quickly gives \(\frac{g}{4} = g-a\), and thus \(a = \frac{3g}{4}\).

Step 1: Understanding the Question:

We are given the mass, amplitude, and time period of a particle in SHM. We need to calculate its total mechanical energy.


Step 2: Key Formula or Approach:

The total mechanical energy (E) in SHM is given by:
\[ E = \frac{1}{2}m\omega^2A^2 \]
where \(m\) is mass, \(\omega\) is angular frequency, and \(A\) is amplitude. Angular frequency \(\omega = \frac{2\pi}{T}\).


Step 3: Detailed Explanation:

Convert all quantities to SI units.

- Mass, \(m = 40 g = 0.04\) kg.

- Amplitude, \(A = 2.0 cm = 0.02\) m.

- Time period, \(T = \frac{\pi}{20}\) s.


Calculate the angular frequency \(\omega\):
\[ \omega = \frac{2\pi}{T} = \frac{2\pi}{(\pi/20)} = 40 rad/s \]

Calculate the total energy:
\[ E = \frac{1}{2} \times (0.04) \times (40)^2 \times (0.02)^2 \] \[ E = (0.02) \times (1600) \times (0.0004) \] \[ E = 0.0128 J \]

Convert Joules to millijoules (1 J = 1000 mJ):
\[ E = 0.0128 \times 1000 mJ = 12.8 mJ \]

Step 4: Final Answer:

The total mechanical energy of the system is 12.8 mJ.

This corresponds to option (C).
Quick Tip: Always be careful with units. Convert all given values to standard SI units (kg, m, s) at the beginning of the calculation to avoid errors. Then, convert the final answer to the unit required by the options.

Step 1: Understanding the Question:

Given the percentage increase in kinetic energy, find the percentage increase in linear momentum.


Step 2: Key Formula or Approach:

The relationship between kinetic energy (K) and linear momentum (p) is:
\[ K = \frac{p^2}{2m} \implies p = \sqrt{2mK} \]
This shows that momentum is directly proportional to the square root of the kinetic energy (\(p \propto \sqrt{K}\)).


Step 3: Detailed Explanation:

Let the initial kinetic energy be \(K_1\) and initial momentum be \(p_1\).

A 21% increase in K means the final kinetic energy \(K_2 = K_1 + 0.21K_1 = 1.21K_1\).

Let the final momentum be \(p_2\).

Using the proportionality:
\[ \frac{p_2}{p_1} = \sqrt{\frac{K_2}{K_1}} = \sqrt{\frac{1.21K_1}{K_1}} = \sqrt{1.21} = 1.1 \]
So, \(p_2 = 1.1 p_1\).


The percentage increase in momentum is:
\[ % Increase = \frac{p_2 - p_1}{p_1} \times 100% = \frac{1.1p_1 - p_1}{p_1} \times 100% \] \[ % Increase = 0.1 \times 100% = 10% \]

Step 4: Final Answer:

The percentage increase in the linear momentum is 10%.

This corresponds to option (A).
Quick Tip: For small percentage changes (\(x\)), if \(y \propto x^n\), then the percentage change in \(y\) is approximately \(n\) times the percentage change in \(x\). Here the change is not small, so the approximation doesn't work well. Using ratios is the exact method. A 21% increase means the new value is 1.21 times the old. The new momentum is \(\sqrt{1.21} = 1.1\) times the old, which is a 10% increase.

Step 1: Understanding the Question:

A ball is thrown up, and due to air resistance, it reaches a specific height. We need to find the work done by air resistance.


Step 2: Key Formula or Approach:

Use the Work-Energy Theorem for non-conservative systems: The work done by non-conservative forces equals the change in total mechanical energy.
\[ W_{resistance} = \Delta E_{mech} = (K_f + U_f) - (K_i + U_i) \]

Step 3: Detailed Explanation:

Convert mass to SI units: \(m = 50 g = 0.05\) kg.

Assume \(g \approx 10 m/s^2\). Let the launch point be the reference height (\(h=0\)).


Initial State (at launch):

- \(K_i = \frac{1}{2}mu^2 = \frac{1}{2}(0.05)(25)^2 = 15.625\) J.

- \(U_i = mgh_i = 0\).

- \(E_i = 15.625\) J.


Final State (at max height):

- \(K_f = 0\) (since \(v=0\)).

- \(U_f = mgh_f = (0.05)(10)(25) = 12.5\) J.

- \(E_f = 12.5\) J.


Now, calculate the work done by resistance:
\[ W_{resistance} = E_f - E_i = 12.5 J - 15.625 J = -3.125 J \]
The work done by the resistance force is -3.125 J. The options are all positive, implying they might ask for the magnitude of energy loss, which is 3.125 J.


Step 4: Final Answer:

The calculated work is -3.125 J. None of the options match this value. The closest option (E) is off by a factor of 10. Due to this discrepancy, the question was cancelled.
Quick Tip: When a question in an exam leads to an answer that doesn't match any options, re-check your calculations. If you're confident, it's likely a flawed question. Such questions are often cancelled, and marks are awarded to all students.

Step 1: Understanding the Question:

Find the radius of gyration (K) for a uniform circular disc rotating about its diameter.


Step 2: Key Formula or Approach:

The radius of gyration is defined by \(I = MK^2\). We first need the moment of inertia (I) of the disc about its diameter. We can find this using the Perpendicular Axis Theorem.


Step 3: Detailed Explanation:

The Perpendicular Axis Theorem states \(I_z = I_x + I_y\).

For a disc, the moment of inertia about an axis perpendicular to its plane and through its center is \(I_z = \frac{1}{2}MR^2\).

Let x and y axes be two perpendicular diameters. By symmetry, \(I_x = I_y = I_{dia}\).

Applying the theorem:
\[ I_z = I_{dia} + I_{dia} = 2I_{dia} \] \[ \frac{1}{2}MR^2 = 2I_{dia} \implies I_{dia} = \frac{1}{4}MR^2 \]
Now, set this equal to \(MK^2\):
\[ MK^2 = \frac{1}{4}MR^2 \] \[ K^2 = \frac{R^2}{4} \implies K = \frac{R}{2} \]

Step 4: Final Answer:

The radius of gyration of the circular disc about its diameter is \(\frac{R}{2}\).

This corresponds to option (B).
Quick Tip: Memorize the moments of inertia for common shapes. For a disc:
- About axis perpendicular to plane: \(\frac{1}{2}MR^2\)
- About a diameter: \(\frac{1}{4}MR^2\)
The Perpendicular Axis Theorem is crucial for planar objects.

Step 1: Understanding the Question:

We need to find the angular velocity (\(\omega\)) of the second hand of an analog clock.


Step 2: Key Formula or Approach:

For uniform circular motion, angular velocity is calculated as:
\[ \omega = \frac{Angular Displacement}{Time} = \frac{2\pi}{T} \]
where T is the time period for one complete revolution.


Step 3: Detailed Explanation:

The second hand completes one full circle (\(2\pi\) radians) in 60 seconds.

So, the time period T = 60 s.


Calculate the angular velocity:
\[ \omega = \frac{2\pi rad}{60 s} = \frac{\pi}{30} rad s^{-1} \]

Step 4: Final Answer:

The angular velocity of the second hand is \(\frac{\pi}{30}\) rad s\(^{-1}\).

This corresponds to option (E).
Quick Tip: It's useful to also know the angular velocities for the minute and hour hands:
- \textbf{Minute Hand:} T = 3600 s. \(\omega_{min} = \frac{2\pi}{3600} = \frac{\pi}{1800}\) rad/s.
- \textbf{Hour Hand:} T = 12 hours = 43200 s. \(\omega_{hour} = \frac{2\pi}{43200} = \frac{\pi}{21600}\) rad/s.

Step 1: Understanding the Question:

We are given the relative \(g\) and \(R\) for a planet compared to Earth and asked for the ratio of their escape velocities.


Step 2: Key Formula or Approach:

The escape velocity (\(v_e\)) is given by \(v_e = \sqrt{2gR}\). This formula is most direct as the problem provides \(g\) and \(R\).


Step 3: Detailed Explanation:

Let subscript 'p' be for the planet and 'e' for Earth.

Given: \(g_p = 2.5 g_e\) and \(R_p = 10 R_e\).


The ratio of escape velocities is:
\[ \frac{v_p}{v_e} = \frac{\sqrt{2g_pR_p}}{\sqrt{2g_eR_e}} = \sqrt{\left(\frac{g_p}{g_e}\right) \left(\frac{R_p}{R_e}\right)} \]
Substitute the given relations:
\[ \frac{v_p}{v_e} = \sqrt{(2.5) \times (10)} = \sqrt{25} = 5 \]
The ratio is \(\frac{5}{1}\) or 5:1.


Step 4: Final Answer:

The ratio of the escape velocity on the planet to that on Earth is 5:1.

This corresponds to option (E).
Quick Tip: When solving ratio problems, write the formula, form the ratio, and then substitute the given proportionalities. This is faster and less error-prone than calculating individual values.

Step 1: Understanding the Question:

The question asks for the relationship between the orbital period (T) of a planet and the semi-major axis (a) of its orbit.


Step 2: Key Formula or Approach:

This is a direct question about Kepler's Third Law of Planetary Motion.


Step 3: Detailed Explanation:

Kepler's Third Law states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

Mathematically, this is expressed as:
\[ T^2 \propto a^3 \]

Step 4: Final Answer:

The correct relationship is \(T^2 \propto a^3\).

This corresponds to option (C).
Quick Tip: Remember all three of Kepler's Laws:
1. \textbf{Law of Orbits:} Planets move in elliptical orbits with the Sun at one focus.
2. \textbf{Law of Areas:} The line joining a planet and the Sun sweeps out equal areas in equal intervals of time.
3. \textbf{Law of Periods:} \(T^2 \propto a^3\).

Step 1: Understanding the Question:

The question asks which principle arises from applying mass conservation to incompressible liquid flow.


Step 3: Detailed Explanation:

The law of conservation of mass states that the mass flow rate is constant in a pipe. Mass flow rate is \(\rho A v\), where \(\rho\) is density, A is cross-sectional area, and v is velocity.
\[ \rho_1 A_1 v_1 = \rho_2 A_2 v_2 \]
For an incompressible liquid, density \(\rho\) is constant. Thus, \(\rho_1 = \rho_2\). The equation simplifies to:
\[ A_1 v_1 = A_2 v_2 \quad or \quad Av = constant \]
This is known as the equation of continuity.


The other laws are based on different principles:

- Bernoulli's law: Conservation of energy.

- Stoke's law: Viscous drag force.

- Toricelli's law: Special case of Bernoulli's law.

- Pascal's law: Pressure in static fluids.


Step 4: Final Answer:

The conservation of mass for an incompressible fluid flow leads to the equation of continuity.

This corresponds to option (A).
Quick Tip: Associate fundamental conservation laws with the key equations of fluid dynamics:
- \textbf{Conservation of Mass} \(\implies\) \textbf{Equation of Continuity}.
- \textbf{Conservation of Energy} \(\implies\) \textbf{Bernoulli's Principle}.

Step 1: Understanding the Question:

This is a definition-based question about the transition from streamlined (laminar) to turbulent flow.


Step 3: Detailed Explanation:

Fluid flow can be streamlined (orderly) or turbulent (chaotic). The transition between these two regimes occurs as the fluid velocity increases.

- Critical Velocity is the specific term for the maximum velocity a fluid can have while its flow remains streamlined. Above this velocity, the flow tends to become turbulent.

- Terminal velocity is the constant speed of a falling object when air resistance equals gravity.

The other terms are not standard definitions for this transition point.


Step 4: Final Answer:

The maximum velocity for which fluid flow remains streamlined is called the critical velocity.

This corresponds to option (B).
Quick Tip: The transition between laminar and turbulent flow is characterized by the Reynolds number (\(R_e\)). The critical velocity is the speed corresponding to the critical Reynolds number where the transition begins.

Step 1: Understanding the Question:

Given the coefficient of linear expansion (\(\alpha\)), find the coefficient of volume expansion (\(\gamma\)).


Step 2: Key Formula or Approach:

For isotropic materials, the coefficient of volume expansion (\(\gamma\)) is approximately three times the coefficient of linear expansion (\(\alpha\)).
\[ \gamma \approx 3\alpha \]

Step 3: Detailed Explanation:

Given \(\alpha = 2.5 \times 10^{-5} K^{-1}\).
\[ \gamma = 3 \times (2.5 \times 10^{-5} K^{-1}) \] \[ \gamma = 7.5 \times 10^{-5} K^{-1} \]

Step 4: Final Answer:

The coefficient of volume expansion for aluminum is \(7.5 \times 10^{-5} K^{-1}\).

This corresponds to option (C).
Quick Tip: Remember the ratio for thermal expansion coefficients: \(\alpha : \beta : \gamma \approx 1 : 2 : 3\), where \(\beta\) is the coefficient of area expansion. This is a very useful shortcut.

Step 1: Understanding the Question:

Calculate the efficiency of a Carnot engine operating between the freezing and boiling points of water.


Step 2: Key Formula or Approach:

The efficiency (\(\eta\)) of a Carnot engine is given by:
\[ \eta = 1 - \frac{T_C}{T_H} \]
where \(T_C\) and \(T_H\) are the absolute temperatures (in Kelvin) of the cold and hot reservoirs.


Step 3: Detailed Explanation:

First, convert the temperatures to Kelvin.

- Cold reservoir (ice point): \(T_C = 0^{\circ}C = 273.15\) K.

- Hot reservoir (steam point): \(T_H = 100^{\circ}C = 373.15\) K.


Now, calculate the efficiency:
\[ \eta = 1 - \frac{273.15}{373.15} = \frac{373.15 - 273.15}{373.15} = \frac{100}{373.15} \approx 0.268 \]
As a percentage, \(\eta% \approx 26.8%\).


Step 4: Final Answer:

The calculated efficiency is approximately 26.8%, which is closest to 27%.

This corresponds to option (D).
Quick Tip: Always convert temperatures to Kelvin when using thermodynamic formulas like Carnot efficiency. The Carnot cycle gives the maximum possible efficiency for a heat engine operating between two temperatures.

Step 1: Understanding the Question:

Identify the thermodynamic processes X and Y from their curves on a PV diagram.


Step 2: Key Formula or Approach:

On a PV diagram, the slope of an expansion/compression curve helps identify the process. The magnitude of the slope is given by \(|dP/dV|\).

- Isothermal Process (\(PV = const\)): Slope is \(-\frac{P}{V}\).

- Adiabatic Process (\(PV^\gamma = const\)): Slope is \(-\gamma\frac{P}{V}\).

Since \(\gamma > 1\), the adiabatic curve is always steeper than the isothermal curve at any given point.


Step 3: Detailed Explanation:

In the given diagram, both curves represent expansion. Curve X is clearly steeper than curve Y.

- Steeper curve (X) \(\implies\) Adiabatic process.

- Less steep curve (Y) \(\implies\) Isothermal process.

So, the processes are Adiabatic (X) and Isothermal (Y), which corresponds to option (E).


Reason for Cancellation:

This question was officially cancelled. A probable reason is the flawed diagram, which shows the processes originating from the origin (P=0, V=0). This is not a physically realistic state for a gas expansion, making the diagram fundamentally incorrect and confusing.


Step 4: Final Answer:

Based on the relative slopes, the answer should be (E) Adiabatic and isothermal. However, the question was cancelled by the examining authority due to the flawed diagram.
Quick Tip: Remember: \textbf{Adiabatic curves are steeper than isothermal curves} on a PV diagram. In an adiabatic expansion, the gas does work and its temperature drops, causing pressure to fall more rapidly than in an isothermal expansion where temperature is kept constant.

Step 1: Understanding the Question:

We have two metallic rods connected in series. They have the same length and cross-sectional area but different thermal conductivities. Their ends are at different temperatures. We need to find the temperature at the junction between the two rods in the steady state.


Step 2: Key Formula or Approach:

In the steady state, the rate of heat flow (heat current, H) through the first rod is equal to the rate of heat flow through the second rod. The formula for the rate of heat flow is:
\[ H = \frac{KA(T_{hot} - T_{cold})}{l} \]
where K is the thermal conductivity, A is the area, l is the length, and \(T_{hot} - T_{cold}\) is the temperature difference across the rod.


Step 3: Detailed Explanation:

Let T be the temperature at the junction of the two rods.

Since T\(_{1} >\) T\(_{2}\), heat will flow from the end at T\(_{1}\) to the end at T\(_{2}\).


For the first rod (conductivity K):

The ends are at temperatures T\(_{1}\) and T. The rate of heat flow is:
\[ H_1 = \frac{K A (T_1 - T)}{l} \]

For the second rod (conductivity K/2):

The ends are at temperatures T and T\(_{2}\). The rate of heat flow is:
\[ H_2 = \frac{(K/2) A (T - T_2)}{l} \]

In the steady state, the rate of heat flow must be the same through both rods:
\[ H_1 = H_2 \] \[ \frac{K A (T_1 - T)}{l} = \frac{(K/2) A (T - T_2)}{l} \]
We can cancel the common terms \( \frac{A}{l} \) from both sides:
\[ K(T_1 - T) = \frac{K}{2}(T - T_2) \]
Cancel K from both sides:
\[ T_1 - T = \frac{1}{2}(T - T_2) \]
Multiply by 2 to clear the fraction:
\[ 2(T_1 - T) = T - T_2 \] \[ 2T_1 - 2T = T - T_2 \]
Rearrange the equation to solve for T:
\[ 2T_1 + T_2 = T + 2T \] \[ 2T_1 + T_2 = 3T \] \[ T = \frac{2T_1 + T_2}{3} \]

Step 4: Final Answer:

The temperature at the joining point in the steady state is \(\frac{2T_1+T_2}{3}\).

This corresponds to option (C).
Quick Tip: This problem is analogous to two resistors in series in an electrical circuit. Thermal resistance is given by \(R_{th} = \frac{l}{KA}\). Here, \(R_1 = \frac{l}{KA}\) and \(R_2 = \frac{l}{(K/2)A} = 2R_1\). The junction temperature is like the potential at the junction of two resistors, which can be found using a potential divider concept.

Step 1: Understanding the Question:

The question asks for the average energy associated with a single translational degree of freedom and a single rotational degree of freedom, based on the equipartition of energy theorem.


Step 2: Key Formula or Approach:

The Law of Equipartition of Energy states that for any dynamical system in thermal equilibrium, the total energy is shared equally among all its degrees of freedom. The average energy associated with each degree of freedom that is quadratic in position or momentum is \(\frac{1}{2}k_BT\), where \(k_B\) is the Boltzmann constant and T is the absolute temperature.


Step 3: Detailed Explanation:

- Translational Degree of Freedom: The kinetic energy of a particle moving in three dimensions can be written as \(K_{trans} = \frac{1}{2}mv_x^2 + \frac{1}{2}mv_y^2 + \frac{1}{2}mv_z^2\). Each term (\(\frac{1}{2}mv^2\)) is quadratic in a velocity (momentum) component. According to the equipartition theorem, the average energy contributed by each of these terms, and thus each translational degree of freedom, is \(\frac{1}{2}k_BT\).

- Rotational Degree of Freedom: The rotational kinetic energy can be written in terms of angular velocity components, e.g., \(K_{rot} = \frac{1}{2}I_x\omega_x^2 + \frac{1}{2}I_y\omega_y^2 + \dots\). Each of these terms is also quadratic. Therefore, the average energy contributed by each rotational degree of freedom is also \(\frac{1}{2}k_BT\).


Step 4: Final Answer:

The energy contributed by each translational degree of freedom is \(\frac{1}{2}k_BT\), and the energy contributed by each rotational degree of freedom is also \(\frac{1}{2}k_BT\).

This corresponds to option (A). (Note: Option D is identical to A).
Quick Tip: Remember that the equipartition theorem applies to each *quadratic* degree of freedom. This includes translational, rotational, and even vibrational modes (which contribute \(k_BT\) each, \(\frac{1}{2}k_BT\) for kinetic and \(\frac{1}{2}k_BT\) for potential energy). The key value to remember is \(\frac{1}{2}k_BT\) per quadratic term.

Step 1: Understanding the Question:

We are given that the total kinetic energy of a certain number of moles of a diatomic gas is equal to the total kinetic energy of 'n' moles of a monoatomic gas at the same temperature. We need to find 'n'.


Step 2: Key Formula or Approach:

The total internal kinetic energy of \(N\) moles of an ideal gas is given by:
\[ E = N \times \left(\frac{f}{2}\right)RT \]
where \(f\) is the number of degrees of freedom, R is the universal gas constant, and T is the absolute temperature.


Step 3: Detailed Explanation:

First, determine the degrees of freedom for each type of gas (assuming they are at a temperature where vibrational modes are not active).

- For a monoatomic gas (like He, Ar), there are only 3 translational degrees of freedom. So, \(f_{mono} = 3\).

- For a diatomic gas (like O\(_{2}\), N\(_{2}\)), there are 3 translational and 2 rotational degrees of freedom. So, \(f_{di} = 3 + 2 = 5\).


Now, write the expressions for the kinetic energy for both gases.

- Kinetic energy of 3 moles of the diatomic gas:
\[ E_{di} = 3 \times \left(\frac{f_{di}}{2}\right)RT = 3 \times \left(\frac{5}{2}\right)RT = \frac{15}{2}RT \]
- Kinetic energy of n moles of the monoatomic gas:
\[ E_{mono} = n \times \left(\frac{f_{mono}}{2}\right)RT = n \times \left(\frac{3}{2}\right)RT \]

We are given that the kinetic energies are the same:
\[ E_{di} = E_{mono} \] \[ \frac{15}{2}RT = n \times \frac{3}{2}RT \]
We can cancel the common term \(\frac{RT}{2}\) from both sides:
\[ 15 = 3n \]
Solving for n:
\[ n = \frac{15}{3} = 5 \]

Step 4: Final Answer:

The value of n is 5.

This corresponds to option (D).
Quick Tip: Memorize the degrees of freedom for common gas types at normal temperatures:
- Monoatomic: \(f=3\)
- Diatomic: \(f=5\)
- Polyatomic (non-linear): \(f=6\)
These values are frequently used in problems related to kinetic theory and thermodynamics.

Step 1: Understanding the Question:

We have a string fixed at both ends, vibrating in its fundamental mode. We need to find the angular wave number (k) of this standing wave.


Step 2: Key Formula or Approach:

1. For a string fixed at both ends, the fundamental mode of vibration (the first harmonic) has a wavelength (\(\lambda\)) that is twice the length of the string.
\[ \lambda = 2L \]
2. The angular wave number (k) is related to the wavelength (\(\lambda\)) by the formula:
\[ k = \frac{2\pi}{\lambda} \]

Step 3: Detailed Explanation:

In the fundamental mode, the string forms a single loop, with nodes at the two fixed ends and an antinode in the middle. The length of the string corresponds to half a wavelength.
\[ L = \frac{\lambda}{2} \implies \lambda = 2L \]
Now, we can calculate the angular wave number (k) using its definition:
\[ k = \frac{2\pi}{\lambda} \]
Substitute the value of \(\lambda\) we found:
\[ k = \frac{2\pi}{2L} \] \[ k = \frac{\pi}{L} \]

Step 4: Final Answer:

The angular wave number of the standing wave is \(\frac{\pi}{L}\).

This corresponds to option (D).
Quick Tip: Remember the general conditions for standing waves on a string of length L fixed at both ends:
- Allowed wavelengths: \(\lambda_n = \frac{2L}{n}\) for \(n=1, 2, 3, \dots\)
- Allowed frequencies: \(f_n = \frac{v}{\lambda_n} = n \frac{v}{2L}\)
- Angular wave numbers: \(k_n = \frac{2\pi}{\lambda_n} = \frac{n\pi}{L}\)
The fundamental mode corresponds to \(n=1\).

Step 1: Understanding the Question:

We need to find the ratio of the frequency of the third harmonic of a closed organ pipe to that of an open organ pipe, given that both pipes have the same length.


Step 2: Key Formula or Approach:

We need the formulas for the harmonic frequencies for both types of pipes. Let v be the speed of sound and L be the length of the pipe.

- Closed Organ Pipe (one end closed, one open): Only odd harmonics are present. The frequency of the n-th harmonic is given by \(f_n = n \frac{v}{4L}\), where n = 1, 3, 5, ...

- Open Organ Pipe (both ends open): All harmonics are present. The frequency of the n-th harmonic is given by \(f'_n = n \frac{v}{2L}\), where n = 1, 2, 3, ...


Step 3: Detailed Explanation:

First, find the frequency of the third harmonic for the closed pipe.

For the third harmonic of a closed pipe, n = 3.
\[ f_{3, closed} = 3 \frac{v}{4L} \]

Next, find the frequency of the third harmonic for the open pipe.

For the third harmonic of an open pipe, n = 3.
\[ f_{3, open} = 3 \frac{v}{2L} \]

Now, find the ratio of these two frequencies:
\[ \frac{f_{3, closed}}{f_{3, open}} = \frac{3 \frac{v}{4L}}{3 \frac{v}{2L}} \]
We can cancel the common terms \(3v\) and \(L\):
\[ \frac{f_{3, closed}}{f_{3, open}} = \frac{1/4}{1/2} = \frac{1}{4} \times \frac{2}{1} = \frac{2}{4} = \frac{1}{2} \]
So, the ratio is 1:2.


Step 4: Final Answer:

The ratio between the frequencies of the third harmonics in the closed organ pipe and open organ pipe is 1:2.

This corresponds to option (B).
Quick Tip: A common point of confusion is the term "overtone".
- For an open pipe: 1st overtone = 2nd harmonic, 2nd overtone = 3rd harmonic.
- For a closed pipe: 1st overtone = 3rd harmonic, 2nd overtone = 5th harmonic.
The question asks for the "third harmonic", which is straightforward (n=3 for both). Be careful if the question uses the term "overtone".

Step 1: Understanding the Question:

A sound source (tuning fork) with a fixed frequency produces sound waves in two different media (air and water). We are given the velocities of sound in these media and need to find the ratio of the wavelengths.


Step 2: Key Formula or Approach:

The key principle is that the frequency of a wave is determined by its source and does not change when the wave enters a new medium.

The relationship between wave velocity (v), frequency (f), and wavelength (\(\lambda\)) is:
\[ v = f \lambda \implies \lambda = \frac{v}{f} \]

Step 3: Detailed Explanation:

Let the quantities in air be denoted by subscript 'a' and in water by 'w'.

We are given:

- Frequency of the source, \(f = 300\) Hz. This frequency is the same in both air and water.

- Velocity of sound in air, \(v_a = 350 m/s\).

- Velocity of sound in water, \(v_w = 1500 m/s\).


Calculate the wavelength in air:
\[ \lambda_a = \frac{v_a}{f} = \frac{350}{300} m \]

Calculate the wavelength in water:
\[ \lambda_w = \frac{v_w}{f} = \frac{1500}{300} m \]

Now, find the required ratio of the wavelength in air to that in water:
\[ \frac{\lambda_a}{\lambda_w} = \frac{350/300}{1500/300} \]
The frequency (300) cancels out:
\[ \frac{\lambda_a}{\lambda_w} = \frac{350}{1500} \]
Simplify the fraction:
\[ \frac{\lambda_a}{\lambda_w} = \frac{35}{150} = \frac{7 \times 5}{30 \times 5} = \frac{7}{30} \]
So, the ratio is 7:30.


Step 4: Final Answer:

The ratio of the wavelength in air to that in water is 7:30.

This corresponds to option (D).
Quick Tip: Remember this crucial concept: When a wave passes from one medium to another, its frequency remains constant, while its velocity and wavelength change. The ratio of wavelengths in two media is equal to the ratio of the wave velocities in those media: \(\frac{\lambda_1}{\lambda_2} = \frac{v_1}{v_2}\).

Step 1: Understanding the Question:

We need to calculate the ratio of the electrostatic force in two different scenarios: (1) between an electron and a proton, and (2) between a proton and an alpha particle. The distance of separation is the same in both cases.


Step 2: Key Formula or Approach:

We will use Coulomb's Law for the magnitude of the electrostatic force between two point charges \(q_1\) and \(q_2\):
\[ F = k \frac{|q_1 q_2|}{r^2} \]
where \(k\) is Coulomb's constant and \(r\) is the distance between the charges.


Step 3: Detailed Explanation:

First, let's define the charges of the particles in terms of the elementary charge, \(e\).

- Charge of an electron, \(q_e = -e\). Magnitude is \(|q_e| = e\).

- Charge of a proton, \(q_p = +e\). Magnitude is \(|q_p| = e\).

- An alpha particle is a helium nucleus, consisting of 2 protons and 2 neutrons. Its charge is \(q_\alpha = +2e\). Magnitude is \(|q_\alpha| = 2e\).


Case 1: Force between an electron and a proton (\(F_1\))

The charges are \(q_e\) and \(q_p\).
\[ F_1 = k \frac{|(-e)(+e)|}{r^2} = k \frac{e^2}{r^2} \]

Case 2: Force between a proton and an alpha particle (\(F_2\))

The charges are \(q_p\) and \(q_\alpha\).
\[ F_2 = k \frac{|(+e)(+2e)|}{r^2} = k \frac{2e^2}{r^2} \]

Now, find the ratio \(\frac{F_1}{F_2}\):
\[ \frac{F_1}{F_2} = \frac{k \frac{e^2}{r^2}}{k \frac{2e^2}{r^2}} \]
Cancel the common terms \(k\), \(e^2\), and \(r^2\):
\[ \frac{F_1}{F_2} = \frac{1}{2} \]
The ratio is 1:2.


Step 4: Final Answer:

The required ratio of the forces is 1:2.

This corresponds to option (E).
Quick Tip: For ratio problems involving Coulomb's law, notice that if the distance 'r' is the same, the force is directly proportional to the product of the magnitudes of the charges: \(F \propto |q_1 q_2|\). You can quickly find the ratio by just comparing the product of charges: \(|q_e q_p| : |q_p q_\alpha| \implies e^2 : 2e^2 \implies 1:2\).

Step 1: Understanding the Question:

The question asks for the relationship between the electric field (E), linear charge density (\(\lambda\)), and radial distance (r) for an infinitely long charged wire.


Step 2: Key Formula or Approach:

The electric field of an infinitely long straight wire is a standard result derived from Gauss's Law. The formula for the magnitude of the electric field is:
\[ E = \frac{\lambda}{2\pi\epsilon_0 r} \]
where \(\epsilon_0\) is the permittivity of free space.


Step 3: Detailed Explanation:

From the formula \(E = \frac{\lambda}{2\pi\epsilon_0 r}\), we can analyze the proportionality.

The term \(2\pi\epsilon_0\) is a constant.

Therefore, the electric field E is directly proportional to the linear charge density \(\lambda\) and inversely proportional to the radial distance r.

This can be written as:
\[ E \propto \frac{\lambda}{r} \]
The options seem to show the combined proportionality. The electric field is proportional to the term \(\frac{\lambda}{r}\).


Step 4: Final Answer:

The electric field is proportional to \(\frac{\lambda}{r}\).

This corresponds to option (B).
Quick Tip: It's essential to memorize the electric field formulas for standard charge distributions:
- Point Charge: \(E \propto \frac{1}{r^2}\)
- Infinite Line Charge: \(E \propto \frac{1}{r}\)
- Infinite Sheet of Charge: \(E\) is constant (independent of r).
These are fundamental results from Gauss's Law and are frequently tested.

Step 1: Understanding the Question:

Two conducting spheres of different radii, initially with the same charge Q, are connected by a conducting wire. We need to compare their final charges.


Step 2: Key Formula or Approach:

When two conductors are connected by a wire, they form a single equipotential system. This means that charge will flow between them until they reach the same electric potential.

1. Principle of Electrostatic Equilibrium: Potential is constant everywhere on the surface of the connected conductors. So, \(V_A = V_B\).

2. Potential of a Sphere: The electric potential on the surface of a sphere with charge q and radius R is \(V = \frac{1}{4\pi\epsilon_0} \frac{q}{R}\).

3. Conservation of Charge: The total charge of the system is conserved. The initial total charge is \(Q + Q = 2Q\). The final total charge is \(Q_A + Q_B = 2Q\).


Step 3: Detailed Explanation:

Let the final charges on sphere A and sphere B be \(Q_A\) and \(Q_B\), respectively.

After connecting, their potentials become equal:
\[ V_A = V_B \] \[ \frac{1}{4\pi\epsilon_0} \frac{Q_A}{R_A} = \frac{1}{4\pi\epsilon_0} \frac{Q_B}{R_B} \]
Cancel the constant term:
\[ \frac{Q_A}{R_A} = \frac{Q_B}{R_B} \]
This gives the ratio of the final charges:
\[ \frac{Q_A}{Q_B} = \frac{R_A}{R_B} \]
We are given in the question that \(R_A > R_B\) (since \(R_B < R_A\)).

Since the ratio \(\frac{R_A}{R_B} > 1\), it follows that the ratio \(\frac{Q_A}{Q_B} > 1\).

This implies that \(Q_A > Q_B\).

The charge distributes itself in direct proportion to the radius, so the larger sphere holds more charge.


Step 4: Final Answer:

Since the radius of sphere A is greater than the radius of sphere B, the final charge on A will be greater than the final charge on B.

This corresponds to option (E).
Quick Tip: For connected spherical conductors, remember that charge distributes in proportion to the radius (\(Q \propto R\)), while the surface charge density (\(\sigma = Q/A = Q/(4\pi R^2)\)) distributes in inverse proportion to the radius (\(\sigma \propto 1/R\)). This means charge accumulates more densely on sharper points (smaller radius of curvature).

Step 1: Understanding the Question:

We need to calculate the electrical conductivity of an intrinsic silicon wafer given the intrinsic carrier concentration, and the mobilities of electrons and holes.


Step 2: Key Formula or Approach:

The conductivity (\(\sigma\)) of an intrinsic semiconductor is given by the formula:
\[ \sigma = n_i e (\mu_e + \mu_h) \]
where:

- \(n_i\) is the intrinsic carrier concentration (number of electron-hole pairs per unit volume).

- \(e\) is the magnitude of the elementary charge (\(1.6 \times 10^{-19}\) C).

- \(\mu_e\) is the mobility of electrons.

- \(\mu_h\) is the mobility of holes.

We must be careful with units. All given units are in terms of cm, so the result will be in \((\Omega \cdot cm)^{-1}\) or mho cm\(^{-1}\).


Step 3: Detailed Explanation:

Let's list the given values:

- \(n_i = 1.1 \times 10^{10} cm^{-3}\)

- \(\mu_e = 1500 cm^2 V^{-1}s^{-1}\)

- \(\mu_h = 500 cm^2 V^{-1}s^{-1}\)

- \(e = 1.6 \times 10^{-19} C\)


Substitute these values into the conductivity formula:
\[ \sigma = (1.1 \times 10^{10} cm^{-3}) \times (1.6 \times 10^{-19} C) \times (1500 + 500) cm^2 V^{-1}s^{-1} \] \[ \sigma = (1.1 \times 1.6 \times 10^{-9}) \times (2000) \quad [units: C cm^{-1} V^{-1}s^{-1} = (\Omega \cdot cm)^{-1}] \] \[ \sigma = (1.76 \times 10^{-9}) \times (2 \times 10^3) \] \[ \sigma = 3.52 \times 10^{-6} \, (\Omega \cdot cm)^{-1} \]
The unit \((\Omega)^{-1}\) is also called a mho. So, \(\sigma = 3.52 \times 10^{-6} mho cm^{-1}\).


The question asks for the answer in micro-mho per cm (\(\mumho cm^{-1}\)).

Since \(1 \, \mumho = 10^{-6} mho\), we can convert the units:
\[ \sigma = 3.52 \times 10^{-6} \frac{mho}{cm} \times \left( \frac{1 \, \mumho}{10^{-6} mho} \right) = 3.52 \, \mumho cm^{-1} \]

Step 4: Final Answer:

The conductivity of the Si wafer is nearly 3.52 \(\mu\)mho cm\(^{-1}\).

This corresponds to option (C).
Quick Tip: In semiconductor physics, pay extremely close attention to units. Problems are often given in cgs-based units (like cm\(^3\), cm\(^2\)/Vs) rather than SI units. Performing calculations consistently in the given units is often easier than converting everything to SI and then converting back. The unit `mho` is an older name for siemens (S), the SI unit of conductance.

Step 1: Understanding the Question:

This is a definition-based question asking for the name of a specific physical quantity.


Step 2: Key Formula or Approach:

The quantity in question is defined as \(\frac{Magnitude of drift velocity}{Electric field}\). We need to identify this from the standard definitions in electricity and magnetism.


Step 3: Detailed Explanation:

Let's analyze the definition. The drift velocity (\(v_d\)) of charge carriers (like electrons) in a conductor is the average velocity they attain due to an applied electric field (E). The relationship between them is:
\[ v_d = \mu E \]
where the constant of proportionality, \(\mu\), is called the mobility of the charge carrier.

Rearranging this formula gives the definition of mobility:
\[ \mu = \frac{v_d}{E} \]
So, mobility is the magnitude of the drift velocity per unit electric field. It is a measure of how easily a charge carrier can move through a material under the influence of an electric field.


Let's review the other options:

- Displacement current: A concept introduced by Maxwell, related to a changing electric field (\(I_D = \epsilon_0 \frac{d\Phi_E}{dt}\)).

- Electric resistance: The opposition to the flow of electric current (\(R = V/I\)).

- Electrical conductivity: The inverse of resistivity, a measure of a material's ability to conduct electric current (\(\sigma = 1/\rho\)).

- Relaxation time: The average time between collisions for a charge carrier in a material.


Step 4: Final Answer:

The magnitude of drift velocity per unit electric field is the definition of mobility.

This corresponds to option (B).
Quick Tip: It is helpful to remember the micro- and macroscopic relations in current electricity.
- Microscopic: \(v_d = \mu E\), and the current density \(J = nq v_d\). This leads to \(J = (nq\mu)E\).
- Macroscopic: Ohm's law \(V=IR\). The microscopic form is \(E = \rho J\) or \(J = \sigma E\).
Comparing \(J = \sigma E\) and \(J = (nq\mu)E\), you can see the relation between conductivity and mobility: \(\sigma = nq\mu\).

Step 1: Understanding the Question:

The question asks for the y-intercept of a graph for a simple DC circuit. However, the description of the axes is confusing: "terminal voltage V with load resistance R along y and x-axis, respectively". A graph of V vs R is not a straight line. The most common graphical representation of a cell's properties is the graph of terminal voltage (V) versus current (I). The y-intercept in that standard graph has a clear physical meaning. Given the options, it is highly probable that the question is implicitly referring to the standard V-I graph, or asking for the value of V when the condition for a y-intercept (x-variable=0) is met in a physically meaningful way.


Step 2: Key Formula or Approach:

The relationship between the terminal voltage (V) across a cell, its electromotive force (emf, \(\varepsilon\)), internal resistance (r), and the current (I) it delivers is:
\[ V = \varepsilon - Ir \]
This equation is in the form of a straight line, \(y = c + mx\), if we plot V (on the y-axis) against I (on the x-axis).


Step 3: Detailed Explanation:

Let's assume the standard V-I graph. The equation is:
\[ V = (-r)I + \varepsilon \]
Comparing this to the equation of a straight line, \(y = mx + c\):

- The y-variable is the terminal voltage, V.

- The x-variable is the current, I.

- The slope, m, is the negative of the internal resistance, -r.

- The y-intercept, c, is the value of y (V) when x (I) is zero.


The y-intercept occurs when I = 0. Setting I = 0 in the equation:
\[ V = \varepsilon - (0)r = \varepsilon \]
Physically, a current of zero (I=0) means the circuit is open (the load resistance R is infinite). In an open circuit, no current flows, and the terminal voltage is equal to the cell's emf. This is the maximum possible terminal voltage.


Even if we consider a V vs. R graph, the concept of an intercept is often tied to a limiting value. As \(R \to \infty\), the current \(I = \frac{\varepsilon}{R+r} \to 0\), and the terminal voltage \(V = IR = \frac{\varepsilon R}{R+r} \to \varepsilon\). This limiting value is the emf. Thus, interpreting the "intercept" as the asymptote gives \(\varepsilon\).


Step 4: Final Answer:

The y-intercept on a standard V vs. I graph for a cell is its emf, \(\varepsilon\). This is the most logical interpretation of the question.

This corresponds to option (A).
Quick Tip: For a real battery (with internal resistance), the terminal voltage V is equal to the emf \(\varepsilon\) only when no current is flowing (open circuit). When the battery supplies current, the terminal voltage is always less than the emf (\(V = \varepsilon - Ir\)). When the battery is being charged, the terminal voltage is greater than the emf (\(V = \varepsilon + Ir\)).

Step 1: Understanding the Question:

We are looking for a condition on electric (E) and magnetic (B) fields under which a charged particle will *always* continue to move in the same direction. "Moving in the same direction" implies that its velocity vector does not change direction. The most straightforward interpretation is that the velocity vector \(\vec{v}\) is constant, which means the net force on the particle is zero.


Step 2: Key Formula or Approach:

The net force on a charged particle in electric and magnetic fields is given by the Lorentz force law:
\[ \vec{F} = q(\vec{E} + \vec{v} \times \vec{B}) \]
For the velocity to remain constant, the net force \(\vec{F}\) must be zero.


Step 3: Detailed Explanation:

Let's analyze the options:

- (A) E = 0, B = 0: The Lorentz force equation becomes \(\vec{F} = q(0 + \vec{v} \times 0) = 0\). With zero force, there is no acceleration, and the particle continues to move with constant velocity (i.e., in the same direction with the same speed). This condition is always true.


- (B) E \(\neq\) 0, B \(\neq\) 0: The force is \(\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})\). It is possible for this force to be zero if \(\vec{E} = -(\vec{v} \times \vec{B})\). This requires specific conditions on the fields and the velocity (e.g., in a velocity selector). However, it is not a general condition that guarantees the particle will continue in the same direction for any initial velocity.


- (C) E = 0, B \(\neq\) 0: The force is \(\vec{F} = q(\vec{v} \times \vec{B})\). The direction of this force is perpendicular to \(\vec{v}\), which will change the direction of motion unless \(\vec{v}\) is parallel to \(\vec{B}\). If \(\vec{v} || \vec{B}\), then \(\vec{v} \times \vec{B} = 0\) and \(\vec{F} = 0\). So, it's possible but not guaranteed.


- (D) E \(\neq\) 0, B = 0: The force is \(\vec{F} = q\vec{E}\). This force will cause the particle to accelerate, changing its velocity. The direction of motion will only remain the same if the initial velocity \(\vec{v}\) is parallel to \(\vec{E}\).


The question asks for a condition where the particle *will* continue to move in the same direction, which implies a general condition. Option (A) is the only one that guarantees zero force and hence constant velocity, regardless of the particle's initial velocity.


Step 4: Final Answer:

In a region with no electric or magnetic fields, a charged particle will experience no force and continue to move with constant velocity.

This corresponds to option (A).
Quick Tip: When analyzing particle motion in fields, always start with the Lorentz Force law. Remember that a magnetic field can only change the direction of a charged particle's velocity (it does no work), while an electric field can change both its magnitude (speed) and direction. Constant velocity implies zero net force.

Step 1: Understanding the Question:

A proton and an alpha particle enter a uniform magnetic field at a right angle and move in circles of the same radius. We need to find the ratio of their velocities.


Step 2: Key Formula or Approach:

When a charged particle of charge q and mass m moves with velocity v perpendicular to a magnetic field B, the magnetic force provides the necessary centripetal force for circular motion.
\[ F_{magnetic} = F_{centripetal} \] \[ qvB = \frac{mv^2}{r} \]
From this, we can derive an expression for the radius of the circular path:
\[ r = \frac{mv}{qB} \]

Step 3: Detailed Explanation:

Let's denote quantities for the alpha particle with subscript \(\alpha\) and for the proton with subscript p.

We need the properties of these particles:

- Proton (p): Mass \(m_p\), Charge \(q_p = e\)

- Alpha particle (\(\alpha\)): Mass \(m_\alpha \approx 4m_p\), Charge \(q_\alpha = 2e\)


We are given that their radii are the same: \(r_\alpha = r_p\).

Using the formula \(r = \frac{mv}{qB}\):
\[ \frac{m_\alpha v_\alpha}{q_\alpha B} = \frac{m_p v_p}{q_p B} \]
The magnetic field B is the same for both, so it cancels out:
\[ \frac{m_\alpha v_\alpha}{q_\alpha} = \frac{m_p v_p}{q_p} \]
We want to find the ratio of their velocities, \(\frac{v_\alpha}{v_p}\) or \(\frac{v_p}{v_\alpha}\). The options are given without specifying the order, but "respective velocities" usually implies proton:alpha or alpha:proton. Let's find \(\frac{v_p}{v_\alpha}\) first.
\[ \frac{v_p}{v_\alpha} = \frac{m_\alpha}{m_p} \frac{q_p}{q_\alpha} \]
Substitute the particle properties:
\[ \frac{v_p}{v_\alpha} = \left(\frac{4m_p}{m_p}\right) \left(\frac{e}{2e}\right) = (4) \left(\frac{1}{2}\right) = 2 \]
So, the ratio of the proton's velocity to the alpha particle's velocity is 2:1. The question asks for the ratio of their respective velocities, implying \(\frac{v_\alpha}{v_p}\). \[ \frac{v_\alpha}{v_p} = \frac{1}{2} \]
This corresponds to a ratio of 1:2.


Step 4: Final Answer:

The ratio of the velocity of the alpha particle to the velocity of the proton is 1:2.

This corresponds to option (D). (Note: The PDF indicates this question was cancelled, but the physical derivation is sound and leads to option D).
Quick Tip: The radius of the circular path of a charged particle in a magnetic field is given by \(r = \frac{p}{qB}\), where \(p=mv\) is the momentum. If the radii are the same, it means their "magnetic rigidity" (\(p/q\)) is the same. This is a useful shortcut: if \(r_\alpha = r_p\), then \(\frac{p_\alpha}{q_\alpha} = \frac{p_p}{q_p}\).

Step 1: Understanding the Question:

We are given the voltage rating of a power supply and need to find the amplitude (or peak value) of this voltage.


Step 2: Key Formula or Approach:

Standard AC voltage values given for power supplies (like 200 V, 120 V, 240 V) are the Root Mean Square (RMS) values. The RMS voltage (\(V_{rms}\)) is related to the amplitude or peak voltage (\(V_0\)) by the formula:
\[ V_{rms} = \frac{V_0}{\sqrt{2}} \]
The information about current (3A) and frequency (50 Hz) is extra information not needed to solve for the voltage amplitude.


Step 3: Detailed Explanation:

We are given the RMS voltage:
\[ V_{rms} = 200 V \]
We need to find the amplitude, \(V_0\). Rearranging the formula:
\[ V_0 = V_{rms} \times \sqrt{2} \]
Substitute the given value:
\[ V_0 = 200 \times \sqrt{2} \]
Using the approximation \(\sqrt{2} \approx 1.414\):
\[ V_0 \approx 200 \times 1.414 = 282.8 V \]

Step 4: Final Answer:

The amplitude of the supply voltage is approximately 283 V.

This corresponds to option (C).
Quick Tip: Always assume that stated AC voltages and currents are RMS values unless explicitly stated otherwise (e.g., "peak" or "amplitude"). To convert from RMS to peak, multiply by \(\sqrt{2}\). To convert from peak to RMS, divide by \(\sqrt{2}\).

Step 1: Understanding the Question:

We are given the mathematical equation for the magnetic field component of a plane electromagnetic wave. We need to find the wavelength of this wave.


Step 2: Key Formula or Approach:

The standard equation for a plane wave traveling along the x-axis is:
\[ y(x, t) = A \sin(\omega t + kx + \phi) \]
In this equation:

- \(\omega\) is the angular frequency.

- \(k\) is the angular wave number.

The angular wave number \(k\) is related to the wavelength \(\lambda\) by the formula:
\[ k = \frac{2\pi}{\lambda} \]

Step 3: Detailed Explanation:

The given equation is:
\[ B_y = (8 \times 10^{-6}) \sin [2 \times 10^{-11} t + 200 \pi x] \]
By comparing this with the standard wave equation, we can identify the angular wave number \(k\) as the coefficient of x:
\[ k = 200\pi \]
Assuming that x is in standard SI units (meters), the unit of k is rad/m.


Now, we use the relationship between \(k\) and \(\lambda\) to find the wavelength:
\[ k = \frac{2\pi}{\lambda} \] \[ 200\pi = \frac{2\pi}{\lambda} \]
We can cancel \(2\pi\) from both sides:
\[ 100 = \frac{1}{\lambda} \] \[ \lambda = \frac{1}{100} meters \]
The question asks for the wavelength in centimeters (cm). We know that 1 meter = 100 cm.
\[ \lambda = \frac{1}{100} m \times \frac{100 cm}{1 m} = 1 cm \]

Step 4: Final Answer:

The wavelength of the electromagnetic wave is 1 cm.

This corresponds to option (A).
Quick Tip: In a wave equation of the form \(A \sin(\omega t \pm kx)\):
- The coefficient of time (t) is the angular frequency, \(\omega = 2\pi f\).
- The coefficient of position (x) is the angular wave number, \(k = 2\pi/\lambda\).
- The speed of the wave is \(v = \frac{\omega}{k}\).
Quickly identifying these terms is key to solving such problems.

Step 1: Understanding the Question:

The question states that a certain path length in air is "equal" to a path length x in another medium. In optics, this "equality" usually refers to the optical path length being the same. The optical path length represents the equivalent distance the light would have traveled in a vacuum in the same amount of time.


Step 2: Key Formula or Approach:

The optical path length (OPL) is defined as the product of the geometric path length (d) and the refractive index (n) of the medium.
\[ OPL = n \times d \]
The principle behind this is that light travels slower in a denser medium, so it covers less geometric distance in the same amount of time.


Step 3: Detailed Explanation:

Let's calculate the optical path length for both cases and set them equal.

The refractive index of air is taken as \(n_{air} \approx 1\).


Case 1: Path in Air

- Geometric path length, \(d_{air} = 1\) m.

- Refractive index, \(n_{air} = 1\).

- Optical path length, \(OPL_{air} = n_{air} \times d_{air} = 1 \times 1 = 1\) m.


Case 2: Path in Medium

- Geometric path length, \(d_{medium} = x\) m.

- Refractive index, \(n_{medium} = 1.5\).

- Optical path length, \(OPL_{medium} = n_{medium} \times d_{medium} = 1.5 \times x\).


We are given that these path lengths are equal, which means their optical path lengths are equal:
\[ OPL_{air} = OPL_{medium} \] \[ 1 = 1.5 \times x \]
Now, solve for x:
\[ x = \frac{1}{1.5} \]
Since \(1.5 = \frac{3}{2}\), we have:
\[ x = \frac{1}{3/2} = \frac{2}{3} m \]

Step 4: Final Answer:

The value of x is \(\frac{2}{3}\) metre.

This corresponds to option (D).
Quick Tip: Think of optical path length as a measure of the number of wavelengths that fit into a geometric path. Since wavelength shortens in a denser medium (\(\lambda' = \lambda/n\)), the geometric path must also be shorter to contain the same number of wavelengths. This gives the relation \(d_1 n_1 = d_2 n_2\).

Disclaimer: This question, as stated, contains physical inconsistencies when analyzed with standard prism optics. The derivation below makes a non-standard assumption to arrive at the given correct answer.


Step 1: Understanding the Question:

Light enters a prism at face PQ with an angle of incidence \(\alpha\). It then strikes face PR and undergoes total internal reflection (TIR). We are given the condition for TIR in terms of a minimum value of \(\alpha\), and we need to find the prism angle \(\theta\).


Step 2: Key Formulas and Standard Analysis:

1. Snell's Law at face PQ: \(n_1 \sin\alpha = n_2 \sin r_1\). Here, \(1 \cdot \sin\alpha = \mu \sin r_1\).

2. Prism Geometry: For the prism angle \(\theta\), the relation between the angle of refraction at the first face (\(r_1\)) and the angle of incidence at the second face (\(r_2\)) is \(\theta = r_1 + r_2\).

3. Condition for TIR at face PR: The angle of incidence \(r_2\) must be greater than or equal to the critical angle C, where \(\sin C = \frac{1}{\mu}\).


Step 3: Standard Derivation (which shows the question's flaw):

- First, find the critical angle C: \(\sin C = \frac{1}{\sqrt{2}} \implies C = 45^{\circ}\).

- The question states TIR occurs when \(\alpha\) has a minimum value of 45\(^{\circ}\). As \(\alpha\) increases, \(r_1\) increases, and thus \(r_2 = \theta - r_1\) decreases. This means TIR (which requires a large \(r_2\)) occurs for *small* values of \(\alpha\), not large ones. The question's premise is reversed. It should likely state a *maximum* value of \(\alpha\) is 45\(^{\circ}\) for TIR to occur.

- Assuming it's a typo and the boundary condition is \(\alpha = 45^{\circ}\) causing \(r_2 = C = 45^{\circ}\):

- From Snell's Law: \(1 \cdot \sin(45^{\circ}) = \sqrt{2} \sin r_1 \implies \frac{1}{\sqrt{2}} = \sqrt{2} \sin r_1 \implies \sin r_1 = \frac{1}{2} \implies r_1 = 30^{\circ}\).

- Using the standard prism formula: \(\theta = r_1 + r_2 = 30^{\circ} + 45^{\circ} = 75^{\circ}\).

- This result (75\(^{\circ}\)) is not among the options. The question or the provided answer key is incorrect.


Step 4: Non-Standard Derivation to Match the Answer Key:

To arrive at the answer \(\theta = 15^{\circ}\), one must assume a non-standard geometric relation inside the prism. Using the values calculated at the boundary condition (\(r_1 = 30^{\circ}\) and \(r_2 = 45^{\circ}\)), if we assume the prism angle is given by the difference between these angles, we get: \[ \theta = r_2 - r_1 \] \[ \theta = 45^{\circ} - 30^{\circ} = 15^{\circ} \]
This geometric relationship is not standard for a prism where light enters one face and exits another adjacent face but is the only apparent way to reconcile the given numbers with the provided answer. This implies a significant error in the problem's formulation.


Final Answer:

Based on a non-standard interpretation forced by the inconsistency between the problem statement and the given answer, the angle \(\theta\) is 15\(^{\circ}\).

This corresponds to option (A).
Quick Tip: When encountering a physics problem where your standard derivation leads to a result not in the options, double-check your formulas and understanding. If the discrepancy persists, the question may be flawed. In an exam, you might have to guess or try to find an alternative (even if non-standard) path to one of the answers, but be aware of the inconsistency.

Step 1: Understanding the Question:

We need to find the de Broglie wavelength of a particle with mass m moving at a specific fraction of the speed of light.


Step 2: Key Formula or Approach:

The de Broglie wavelength (\(\lambda\)) of a particle is given by the equation:
\[ \lambda = \frac{h}{p} \]
where h is Planck's constant and p is the momentum of the particle. The momentum is given by \(p = mv\), where m is the mass and v is the velocity.


Step 3: Detailed Explanation:

We are given the following information:

- Mass of the particle = m

- Velocity of the particle, \(v = \frac{1}{10}c\)


First, calculate the momentum (p) of the particle:
\[ p = mv = m \left(\frac{c}{10}\right) = \frac{mc}{10} \]
Now, substitute this momentum into the de Broglie wavelength formula:
\[ \lambda = \frac{h}{p} = \frac{h}{\frac{mc}{10}} \]
Simplifying the expression gives:
\[ \lambda = h \times \frac{10}{mc} = \frac{10h}{mc} \]

Step 4: Final Answer:

The de Broglie wavelength associated with the particle is \(\frac{10h}{mc}\).

This corresponds to option (C).
Quick Tip: The de Broglie wavelength equation, \(\lambda = h/p\), is fundamental to wave-particle duality. Remember that 'p' is the relativistic momentum for particles moving at speeds close to c, but for speeds much less than c (like 0.1c), using the classical momentum \(p=mv\) is usually a valid approximation.

Step 1: Understanding the Question:

The question asks for the relationship between the mean life (\(\tau\)) and the half-life (\(t_{1/2}\)) of a radioactive substance.


Step 2: Key Formula or Approach:

Both mean life and half-life are related to the radioactive decay constant, \(\lambda\).

1. The half-life, \(t_{1/2}\), is the time it takes for half of the radioactive nuclei to decay. Its formula is:
\[ t_{1/2} = \frac{\ln 2}{\lambda} \]
2. The mean life (or average lifetime), \(\tau\), is the average lifetime of all the nuclei in a sample. Its formula is:
\[ \tau = \frac{1}{\lambda} \]

Step 3: Detailed Explanation:

We have two equations:

(i) \(t_{1/2} = \frac{\ln 2}{\lambda}\)

(ii) \(\tau = \frac{1}{\lambda}\)

From equation (ii), we can express the decay constant as \(\lambda = \frac{1}{\tau}\).

Now, substitute this expression for \(\lambda\) into equation (i):
\[ t_{1/2} = \frac{\ln 2}{(1/\tau)} \]
Simplifying this gives the relationship:
\[ t_{1/2} = \tau \ln 2 \]

Step 4: Final Answer:

The relationship between half-life and mean life is \(t_{1/2} = \tau \ln 2\).

This corresponds to option (B).
Quick Tip: Since \(\ln 2 \approx 0.693\), the relationship is often written as \(t_{1/2} = 0.693 \tau\). This shows that the half-life is always shorter than the mean life for a given radioactive substance. Remembering this fact can help you eliminate incorrect options in a multiple-choice question.

Step 1: Understanding the Question:

The question asks for the reason why the binding energy per nucleon (BE/A) is nearly constant for nuclei with mass numbers (A) ranging from about 30 to 170.


Step 2: Detailed Explanation:

The nuclear force is a very strong, short-range force. This means a nucleon (proton or neutron) inside a nucleus only interacts with its immediate neighbours. It does not interact with all the other nucleons in the nucleus. This property is known as the saturation of the nuclear force. As more nucleons are added to a medium-sized nucleus, each new nucleon is only bound to its few neighbours, contributing a roughly constant amount to the total binding energy. Therefore, the average binding energy per nucleon remains relatively constant. Both "short range nature" and "saturation" are correct concepts, but the short-range nature is the fundamental cause of the saturation effect. Given the options, "short range nature of nuclear force" is the most fundamental reason.


Step 3: Final Answer:

The constancy of the binding energy per nucleon for medium-sized nuclei is a consequence of the short-range nature of the nuclear force, which leads to its saturation property.

This corresponds to option (A).
Quick Tip: The binding energy per nucleon curve is a key concept in nuclear physics. Remember its main features: it peaks around A=56 (Iron), indicating maximum stability. The relative flatness in the middle is due to the saturation/short-range nature of the nuclear force. This explains why fusion of light nuclei and fission of heavy nuclei release energy.

Step 1: Understanding the Question:

The question asks for the mathematical expression for the fraction of radioactive atoms that have not yet decayed after a time interval t.


Step 2: Key Formula or Approach:

The law of radioactive decay states that the number of undecayed nuclei, N(t), at any time t is given by:
\[ N(t) = N_0 e^{-\lambda t} \]
where \(N_0\) is the initial number of nuclei at t=0, and \(\lambda\) is the radioactive decay constant.


Step 3: Detailed Explanation:

We want to find the fraction of atoms left undecayed. This is the ratio of the number of atoms at time t to the initial number of atoms.
\[ Fraction left = \frac{N(t)}{N_0} \]
Substituting the formula for N(t):
\[ Fraction left = \frac{N_0 e^{-\lambda t}}{N_0} = e^{-\lambda t} \]

Step 4: Final Answer:

The fraction of the number of atoms left undecayed after time t is \(e^{-\lambda t}\).

This corresponds to option (B).
Quick Tip: Remember the key equations for radioactive decay:
- Number of nuclei remaining: \(N(t) = N_0 e^{-\lambda t}\)
- Activity of the sample: \(A(t) = A_0 e^{-\lambda t}\)
The fraction remaining is always the exponential term \(e^{-\lambda t}\).

Step 1: Understanding the Question:

The question shows a nuclear reaction representing electron emission (beta-minus decay) and asks to identify the particle 'q' that is also emitted.


Step 2: Detailed Explanation:

The process shown is beta-minus (\(\beta^-\)) decay. In this decay, a neutron inside the nucleus transforms into a proton, an electron, and an electron antineutrino.
\[ n \to p + e^{-} + \bar{\nu}_e \]
The proton remains in the nucleus, increasing the atomic number Z by one (\(Z \to Z+1\)) while the mass number A remains the same. The electron (beta particle) and the electron antineutrino are ejected from the nucleus. The emission of the antineutrino is necessary to conserve energy, momentum, and lepton number. Therefore, the particle 'q' is an antineutrino.


Step 3: Final Answer:

In electron emission (beta-minus decay), an electron antineutrino is emitted along with the electron.

This corresponds to option (C).
Quick Tip: Remember the pairings in beta decay:
- \(\beta^-\) decay (electron emission): An \textbf{antineutrino} is emitted.
- \(\beta^+\) decay (positron emission): A \textbf{neutrino} is emitted.
An easy way to remember is that the "anti" particle (antineutrino) is paired with the "normal" particle (electron).

Step 1: Understanding the Question:

The question asks to identify the type of current that flows from the p-side to the n-side in a pn junction.


Step 2: Detailed Explanation:

There are two main types of current in a pn junction:

1. Diffusion Current: This is caused by the movement of majority charge carriers (holes from the p-side and electrons from the n-side) across the junction due to a concentration gradient. The conventional current direction for this process is from p to n. This current is significant under forward bias.

2. Drift Current: This is caused by the movement of minority charge carriers being swept across the depletion region by the built-in electric field. The direction of this current is from n to p.


The question specifies the current flowing from the p to n side. This is the direction of the diffusion current of majority carriers. While the phrase "irrespective of biasing" is slightly misleading (as diffusion current is heavily dependent on bias), the directionality (p to n) is the defining characteristic of diffusion current. The answer key confirms this interpretation.


Step 3: Final Answer:

The current due to the flow of majority carriers from the p-side to the n-side is the diffusion current.

This corresponds to option (B).
Quick Tip: Remember the directions in a PN junction:
- \textbf{Diffusion Current}: Majority carriers, P \(\to\) N. Dominant in forward bias.
- \textbf{Drift Current}: Minority carriers, N \(\to\) P. Dominant in reverse bias.

Step 1: Understanding the Question:

The question asks for the value of the energy band gap (\(E_g\)) for the semiconductor Germanium (Ge).


Step 2: Detailed Explanation:

The "forbidden band" or energy band gap is the energy difference between the top of the valence band and the bottom of the conduction band in a solid. An electron must gain at least this amount of energy to become a free electron and contribute to conduction. This is a fundamental property of a semiconductor material. For Germanium (Ge) at room temperature (around 300 K), the energy band gap is approximately 0.72 electron-volts (eV).


Step 3: Final Answer:

The energy band gap for Germanium is about 0.72 eV.

This corresponds to option (A).
Quick Tip: It is very useful to memorize the approximate band gaps for the two most common semiconductors at room temperature:
- \textbf{Silicon (Si):} \(E_g \approx 1.1\) eV
- \textbf{Germanium (Ge):} \(E_g \approx 0.7\) eV
These values are frequently required in problems on semiconductor devices.

Step 1: Understanding the Question:

We need to calculate the molarity of a sodium hydroxide (NaOH) solution.


Step 2: Key Formula or Approach:

Molarity (M) is defined as the number of moles of solute per liter of solution.
\[ Molarity = \frac{Moles of solute}{Volume of solution (L)} \] \[ Moles = \frac{Mass of solute}{Molar mass of solute} \]

Step 3: Detailed Explanation:

1. Calculate moles of NaOH:

Moles = \(\frac{6 g}{40 g/mol} = 0.15 mol\)

2. Convert volume to Liters:

Volume = 600 mL = 0.600 L

3. Calculate Molarity:

Molarity = \(\frac{0.15 mol}{0.600 L} = 0.25 mol/L\) or 0.25 M.


Step 4: Final Answer:

The molarity of the solution is 0.25 M.

This corresponds to option (C).
Quick Tip: Always ensure your units are correct before the final calculation for molarity: mass should be used to find moles, and the volume of the solution must be in liters.

Step 1: Understanding the Question:

We need to find the volume of pure ethanol required to make a specific aqueous solution. This involves a multi-step calculation starting from the desired solution's properties.


Step 2: Detailed Explanation:

1. Find moles of ethanol needed:

Moles = Molarity \(\times\) Volume (L) = 0.25 mol/L \(\times\) 3 L = 0.75 mol.

2. Find mass of ethanol needed:

Mass = Moles \(\times\) Molar Mass = 0.75 mol \(\times\) 60 g/mol = 45 g.

3. Use density to find volume. The density is 0.36 kg/L. Let's convert the mass to kg to match the units.

Mass = 45 g = 0.045 kg.

Volume = \(\frac{Mass}{Density} = \frac{0.045 kg}{0.36 kg/L} = 0.125 L\).

4. Convert final volume to mL:

Volume = 0.125 L \(\times\) 1000 mL/L = 125 mL.


Step 3: Final Answer:

The volume of ethanol required is 125 mL.

This corresponds to option (A).
Quick Tip: For solution preparation problems, work backwards from the final solution to the pure substance:
Desired Solution (Volume, Molarity) \(\to\) Moles Needed \(\to\) Mass Needed \(\to\) Volume of Pure Substance (using density). Pay close attention to unit conversions (g to kg, L to mL).

Step 1: Understanding the Question:

The question asks to identify which statement is incorrect *about* Bohr's model. This means we are looking for a statement that falsely describes the model's postulates, successes, or failures.


Step 2: Detailed Explanation:

- (A) and (B) and (E) are all correct statements describing the limitations or failures of Bohr's model. It couldn't explain the fine structure, the Zeeman effect (splitting in magnetic fields), or the Stark effect (splitting in electric fields). So, these statements about the model are true.

- (C) is a correct statement about a key postulate of Bohr's model (\(L = n\frac{h}{2\pi}\)). This statement is true.

- (D) Bohr's model is a model for a single-electron atom. It provides no mechanism or explanation for how atoms interact to form chemical bonds and molecules. Therefore, to list "The ability of atoms to form molecule" as a feature or aspect of the Bohr model is incorrect. The model simply does not address this.


Step 3: Final Answer:

The statement that is incorrect in the context of describing Bohr's model is (D), as chemical bonding is a concept entirely outside the scope of what the model explained or postulated.

This corresponds to option (D).
Quick Tip: Bohr's model's successes were explaining the hydrogen spectrum and quantizing energy levels. Its failures include applicability only to H-like atoms and its inability to explain fine structure, Zeeman/Stark effects, and chemical bonding.

Step 1: Understanding the Question:

We need to arrange the elements Beryllium (Be), Carbon (C), Nitrogen (N), and Oxygen (O) in order of decreasing first ionization enthalpy (IE1).


Step 2: Key Concepts and Explanation:

1. General Trend: First ionization enthalpy generally increases across a period (from left to right) due to increasing effective nuclear charge. The expected order would be Be \(<\) C \(<\) N \(<\) O.

2. Exceptions to the Trend:

- Be vs. C: Carbon is to the right of Beryllium, so IE1(C) \(>\) IE1(Be). This is consistent with the general trend.

- N vs. O: Nitrogen has a half-filled p-orbital electron configuration (\(2p^3\)), which is particularly stable. Oxygen has a \(2p^4\) configuration. Removing an electron from the paired orbital in oxygen is easier due to inter-electronic repulsion, so IE1(N) \(>\) IE1(O). This is a key exception.

3. Combining the Trends: We know IE1(N) > IE1(O). Also, IE1(O) is generally higher than IE1(C) (follows the trend). And IE1(C) is higher than IE1(Be). So the final decreasing order is N \(>\) O \(>\) C \(>\) Be.


Step 3: Final Answer:

The correct decreasing order of first ionization enthalpy is N \(>\) O \(>\) C \(>\) Be.

This corresponds to option (A).
Quick Tip: Remember the two key exceptions for ionization energy across Period 2:
- IE1 of Be (Group 2) \(>\) IE1 of B (Group 13) due to a stable, full 2s orbital.
- IE1 of N (Group 15) \(>\) IE1 of O (Group 16) due to a stable, half-full 2p orbital.

Step 1: Understanding the Question:

We need to determine the hybridization of the central chromium atom in the coordination complex \([CrF_6]^{3-}\).


Step 2: Detailed Explanation:

1. Find the oxidation state of Cr: Let the oxidation state be x. The charge on each fluoride ion (F\(^-\)) is -1.

x + 6(-1) = -3 \(\implies\) x - 6 = -3 \(\implies\) x = +3. So, we have Cr(III).

2. Write the electron configuration:

- Neutral Cr (Z=24): [Ar] \(3d^5 4s^1\)

- Cr\(^{3+}\) ion: [Ar] \(3d^3\)

3. Consider the ligand and geometry: There are six F\(^-\) ligands, which means the complex has an octahedral geometry. F\(^-\) is generally a weak-field ligand.

4. Determine hybridization: For an octahedral complex, the hybridization can be either \(sp^3d^2\) (outer orbital) or \(d^2sp^3\) (inner orbital). To form six bonds, Cr\(^{3+}\) needs six empty orbitals. The electron configuration of Cr\(^{3+}\) is \(3d^3\), meaning three of the five 3d orbitals are singly occupied. This leaves two 3d orbitals empty. The atom can use these two empty 3d orbitals, the one 4s orbital, and the three 4p orbitals to form six hybrid orbitals.

\[ Orbitals used: \underbrace{d_1, d_2}_{3d}, \underbrace{s}_{4s}, \underbrace{p_1, p_2, p_3}_{4p} \implies d^2sp^3 \]
This is an inner orbital complex.


Step 3: Final Answer:

The hybridization of Cr in \([CrF_6]^{3-}\) is d\(^2\)sp\(^3\).

This corresponds to option (A).
Quick Tip: For octahedral complexes with \(d^1\), \(d^2\), or \(d^3\) electron configurations on the central metal ion, the hybridization will always be \(d^2sp^3\) (inner orbital), regardless of whether the ligand is strong-field or weak-field, because there are vacant inner d-orbitals readily available.

Step 1: Understanding the Question:

We need to determine the molecular orbital (MO) configuration for the valence electrons of a dicarbon molecule (C\(_2\)).


Step 2: Detailed Explanation:

1. A carbon atom (C) has an atomic number of 6, so its electron configuration is \(1s^2 2s^2 2p^2\).

2. The C\(_2\) molecule has a total of 12 electrons. The valence electrons are those in the n=2 shell, so there are \(4+4=8\) valence electrons to place in the molecular orbitals.

3. For diatomic molecules up to N\(_2\), the energy order of the valence molecular orbitals is: \(\sigma_{2s} < \sigma^*_{2s} < \pi_{2p} < \sigma_{2p}\).

4. We fill these orbitals with the 8 valence electrons according to the Aufbau principle:

- Two electrons in \(\sigma_{2s}\): \((\sigma_{2s})^2\)

- Two electrons in \(\sigma^*_{2s}\): \((\sigma_{2s})^2(\sigma^*_{2s})^2\)

- The remaining four electrons fill the degenerate \(\pi_{2p}\) orbitals: \((\sigma_{2s})^2(\sigma^*_{2s})^2(\pi_{2p})^4\)


Step 3: Final Answer:

The correct valence electron MO configuration for C\(_2\) is \((\sigma_{2s})^2(\sigma^*_{2s})^2(\pi_{2p})^4\).

This corresponds to option (B).
Quick Tip: The energy ordering of \(\sigma_{2p}\) and \(\pi_{2p}\) orbitals changes after N\(_2\). For O\(_2\), F\(_2\), and Ne\(_2\), the \(\sigma_{2p}\) orbital is lower in energy than the \(\pi_{2p}\) orbitals. Memorizing the two different filling orders is crucial for MO theory questions.

Step 1: Understanding the Question:

This is a factual question asking to identify the anode material in a mercury cell.


Step 2: Detailed Explanation:

In a mercury cell, the electrochemical reactions involve the following components:

- Anode (Oxidation): A zinc-mercury amalgam (Zn(Hg)). Zinc is oxidized.

- Cathode (Reduction): A paste of mercuric oxide (HgO) and carbon. Mercuric oxide is reduced.

- Electrolyte: A paste of potassium hydroxide (KOH) and zinc oxide (ZnO).

The anode is where oxidation occurs, which in this cell is the zinc-mercury amalgam.


Step 3: Final Answer:

The anode in a mercury cell is a Zinc-Mercury amalgam.

This corresponds to option (E).
Quick Tip: For common electrochemical cells (like Leclanché cell, lead-acid battery, mercury cell, fuel cell), it's important to memorize the materials used for the anode, cathode, and electrolyte, as these are frequently asked in exams.

Step 1: Understanding the Question:

We need to find the thermodynamic conditions (\(\Delta H\) and \(\Delta S\)) for a reaction that is spontaneous only at high temperatures.


Step 2: Key Formula and Explanation:

The spontaneity of a reaction is determined by the Gibbs free energy change: \(\Delta G = \Delta H - T\Delta S\). A reaction is spontaneous when \(\Delta G < 0\).

For a reaction to become spontaneous as temperature (T) increases, the term \(-T\Delta S\) must be negative and become large enough at high T to overcome a positive \(\Delta H\).

- This requires \(\Delta S\) to be positive (\(\Delta S > 0\)).

- It also requires \(\Delta H\) to be positive (\(\Delta H > 0\), an endothermic reaction).

Under these conditions (\(\Delta H > 0, \Delta S > 0\)), at low T, \(\Delta G\) will be positive (non-spontaneous), but at high T, the \(-T\Delta S\) term will dominate, making \(\Delta G\) negative (spontaneous). The condition \(\Delta G < 0\) is the result, not the cause.


Step 3: Final Answer:

The conditions are a positive enthalpy change (\(\Delta H > 0\)) and a positive entropy change (\(\Delta S > 0\)). The overall result is a negative Gibbs free energy (\(\Delta G < 0\)) at high T.

This corresponds to option (C).
Quick Tip: Remember the four spontaneity cases based on the signs of \(\Delta H\) and \(\Delta S\):
- (\(\Delta H < 0, \Delta S > 0\)): Spontaneous at all T.
- (\(\Delta H > 0, \Delta S < 0\)): Non-spontaneous at all T.
- (\(\Delta H < 0, \Delta S < 0\)): Spontaneous at low T.
- (\(\Delta H > 0, \Delta S > 0\)): Spontaneous at high T.

Step 1: Understanding the Question:

We need to calculate the change in internal energy (\(\Delta U\)) of a system using the First Law of Thermodynamics.


Step 2: Key Formula and Explanation:

The First Law of Thermodynamics is given by \(\Delta U = q + w\).

- \(q\) is heat. It is positive if heat is absorbed by the system.

- \(w\) is work. It is positive if work is done on the system, and negative if work is done by the system.

Given:

- Heat is absorbed: \(q = +600\) J.

- Work is done by the system: \(w = -375\) J.

Calculation:
\[ \Delta U = (+600 J) + (-375 J) = 225 J \]

Step 3: Final Answer:

The change in internal energy of the process is 225 J.

This corresponds to option (E).
Quick Tip: Pay close attention to the sign conventions in thermodynamics. Heat absorbed `by` the system is positive. Work done `by` the system is negative. Work done `on` the system is positive.

Step 1: Understanding the Question:

Given the equilibrium constant (K\(_c\)) and the equilibrium concentration of the reactant, we need to find the equilibrium concentration of the product.


Step 2: Key Formula and Explanation:

For the reaction \(2NO_{2(g)} \rightleftharpoons N_2O_{4(g)}\), the equilibrium constant expression is:
\[ K_c = \frac{[N_2O_4]}{[NO_2]^2} \]
We need to solve for \([N_2O_4]\):
\[ [N_2O_4] = K_c \times [NO_2]^2 \]
Given values:

- \(K_c = 2 \times 10^{-40}\) M\(^{-1}\) (Note: mol\(^{-1}\) dm\(^3\) is the same as L/mol or M\(^{-1}\))

- \([NO_2] = 2 \times 10^{-2}\) M

Calculation:
\[ [N_2O_4] = (2 \times 10^{-40}) \times (2 \times 10^{-2})^2 \] \[ [N_2O_4] = (2 \times 10^{-40}) \times (4 \times 10^{-4}) \] \[ [N_2O_4] = 8 \times 10^{-44} M \]

Step 3: Final Answer:

The equilibrium concentration of N\(_2\)O\(_4\) is \(8 \times 10^{-44}\) M.

This corresponds to option (C).
Quick Tip: Always write down the equilibrium expression carefully, paying attention to the stoichiometric coefficients which become exponents in the expression. A common mistake is forgetting to square the concentration of NO\(_2\).

Step 1: Understanding the Question:

We need to calculate the amount of charge in Faradays (F) required for the electrolysis of Al\(_2\)O\(_3\) to produce a specific mass of aluminum.


Step 2: Detailed Explanation:

1. Half-reaction for Al deposition: In molten Al\(_2\)O\(_3\), aluminum exists as Al\(^{3+}\). The reduction half-reaction is:

\[ Al^{3+} + 3e^- \to Al_{(s)} \]
This equation shows that 3 moles of electrons are required to produce 1 mole of aluminum.

2. Relate moles of electrons to charge: 1 mole of electrons corresponds to 1 Faraday (F) of charge. So, 3 F are needed for 1 mole of Al.

3. Calculate moles of Al to be produced:

Molar mass of Al = 27 g/mol.

Moles of Al = \(\frac{Mass}{Molar Mass} = \frac{18 g}{27 g/mol} = \frac{2}{3}\) mol.

4. Calculate required charge:

Charge = (Moles of Al) \(\times\) (Charge per mole of Al)

Charge = \(\frac{2}{3} mol \times 3 F/mol = 2 F\).


Step 3: Final Answer:

The quantity of electricity required is 2 Faradays.

This corresponds to option (A).
Quick Tip: Remember Faraday's laws of electrolysis. The key is to write the correct half-reaction to find the number of moles of electrons ('n-factor') required per mole of substance produced. Then use the formula: Moles produced = (Charge in F) / n.

Step 1: Understanding the Question:

We need to calculate the average oxidation state of a sulfur atom in the tetrathionate ion, S\(_4\)O\(_6^{2-}\).


Step 2: Detailed Explanation:

1. Let the oxidation state of sulfur (S) be \(x\).

2. The oxidation state of oxygen (O) is typically -2.

3. The overall charge of the ion is -2.

4. Set up the equation by summing the oxidation states of all atoms and equating it to the ion's charge:

\[ 4(x) + 6(-2) = -2 \]
\[ 4x - 12 = -2 \]
\[ 4x = 10 \]
\[ x = \frac{10}{4} = +2.5 \]
This is the average oxidation state. The individual sulfur atoms in the S\(_4\)O\(_6^{2-}\) structure actually have different oxidation states (two are +5 and two are 0).


Step 3: Final Answer:

The average oxidation state of sulphur in the tetrathionate ion is +2.5.

This corresponds to option (B).
Quick Tip: For polyatomic ions with multiple atoms of the same element, this algebraic method gives the *average* oxidation state. The actual oxidation states can differ and can be determined by drawing the Lewis structure. However, for most exam questions, the average value is what is required.

Step 1: Understanding the Question:

We need to calculate the mass percentage of a solution.


Step 2: Key Formula and Explanation:

Mass percentage of a component is given by:
\[ Mass % = \frac{Mass of component (solute)}{Total mass of solution} \times 100% \]
Given:

- Mass of solute (glucose) = 6 g.

- Mass of solvent (acetonitrile) = 294 g.

Calculation:

1. Total mass of solution = Mass of solute + Mass of solvent = 6 g + 294 g = 300 g.

2. Mass % of glucose = \(\frac{6 g}{300 g} \times 100% = \frac{1}{50} \times 100% = 2%\).


Step 3: Final Answer:

The mass percentage of glucose in the solution is 2%.

This corresponds to option (E).
Quick Tip: A common mistake in mass percentage calculations is to divide the solute mass by the solvent mass instead of the total solution mass. Always add the mass of the solute and solvent first to get the denominator.

Step 1: Understanding the Question:

We need to calculate the time required for a first-order reaction to proceed from an initial amount to a final amount, given the rate constant.


Step 2: Key Formula and Explanation:

The integrated rate law for a first-order reaction is:
\[ t = \frac{2.303}{k} \log_{10}\left(\frac{A_0}{A_t}\right) \]
where \(A_0\) is the initial amount and \(A_t\) is the amount at time t.

Given:

- \(k = 4.606 \times 10^{-3}\) s\(^{-1}\)

- \(A_0 = 20\) g

- \(A_t = 2\) g

Calculation:
\[ t = \frac{2.303}{4.606 \times 10^{-3}} \log_{10}\left(\frac{20}{2}\right) \]
Notice that \(4.606 = 2 \times 2.303\).
\[ t = \frac{2.303}{2 \times 2.303 \times 10^{-3}} \log_{10}(10) \] \[ t = \frac{1}{2 \times 10^{-3}} \times 1 = \frac{1000}{2} = 500 s \]

Step 3: Final Answer:

The time taken is 500 s.

This corresponds to option (B).
Quick Tip: Look for convenient numbers in kinetics problems. Here, 4.606 is exactly twice 2.303. Recognizing this relationship (\(2.303 \times 2 = 4.606\)) simplifies the calculation significantly.

Step 1: Understanding the Question:

We need to find the total order of a reaction from its given rate law.


Step 2: Key Formula and Explanation:

The total (or overall) order of a reaction is the sum of the exponents of the concentration terms in the experimentally determined rate law.

Given rate law: rate = \(k[A]^1 [B]^{3/2}\)

- Order with respect to A = 1

- Order with respect to B = 3/2 = 1.5

Total Order = \(1 + 1.5 = 2.5\)


Step 3: Final Answer:

The total order of the reaction is 2.5.

This corresponds to option (B).
Quick Tip: The order of a reaction is an experimental quantity and cannot be determined from the stoichiometry of the balanced chemical equation, unless the reaction is an elementary step. Reaction orders can be integers, fractions, or even zero.

Step 1: Understanding the Question:

This is a factual question asking to identify a pair of liquids that form an azeotropic mixture.


Step 2: Detailed Explanation:

An azeotrope is a liquid mixture that has a constant boiling point and a vapor with the same composition as the liquid. This occurs in non-ideal solutions showing significant deviation from Raoult's law.

- Nitric acid-water: This is a classic textbook example of a maximum boiling azeotrope (negative deviation from Raoult's Law).

- Other options like ethanol-acetone, chloroform-acetone, and CS\(_2\)-acetone also form azeotropes. However, Nitric acid-water is a very common and expected example in this context. The question is slightly flawed for having multiple correct options, but B is a very standard answer.


Step 3: Final Answer:

The mixture of Nitric acid and water forms a well-known azeotrope.

This corresponds to option (B).
Quick Tip: Remember common examples of azeotropes. Ethanol-water (95% ethanol) is a famous minimum boiling azeotrope. Nitric acid-water (68% nitric acid) is a famous maximum boiling azeotrope. Knowing these helps in questions about distillation and non-ideal solutions.

Step 1: Understanding the Question:

We need to identify the cobalt-containing coordination compound that functions as Vitamin B\(_{12}\), the factor that prevents pernicious anaemia.


Step 2: Detailed Explanation:

Pernicious anaemia is a condition caused by the inability to absorb Vitamin B\(_{12}\). Vitamin B\(_{12}\) is a complex biomolecule and a coordination compound. Its chemical name is cyanocobalamin. It contains a cobalt(III) ion at its center, coordinated to a corrin ring. The other options are incorrect: carboxypeptidase is a zinc-containing enzyme, and haemoglobin/myoglobin are iron-containing proteins.


Step 3: Final Answer:

The antipernicious anaemia factor is cyanocobalamin (Vitamin B\(_{12}\)).

This corresponds to option (A).
Quick Tip: Associate key metals with their biological roles:
- \textbf{Fe:} Haemoglobin, Myoglobin
- \textbf{Co:} Vitamin B\(_{12}\) (Cyanocobalamin)
- \textbf{Mg:} Chlorophyll
- \textbf{Zn:} Carboxypeptidase, Carbonic anhydrase
- \textbf{Pt:} Cisplatin (anti-cancer drug)

Step 1: Understanding the Question:

We need to determine the electronic transition (d-d transition) that occurs when the complex [Ti(H\(_2\)O)\(_6\)]\(^{3+}\) absorbs light.


Step 2: Detailed Explanation:

1. Oxidation state of Ti: H\(_2\)O is a neutral ligand, so the charge on the complex is the charge on the metal ion. Ti is in the +3 oxidation state.

2. Electron configuration of Ti\(^{3+}\): Neutral Ti (Z=22) is [Ar] \(3d^2 4s^2\). Ti\(^{3+}\) is [Ar] \(3d^1\).

3. Splitting in Octahedral Field: In an octahedral complex, the five d-orbitals split into a lower energy \(t_{2g}\) set and a higher energy \(e_g\) set.

4. Ground State: The single 3d electron will occupy an orbital in the lower-energy \(t_{2g}\) set. So, the ground state configuration is \(t_{2g}^1 e_g^0\).

5. Transition (Absorption): When the complex absorbs light of the correct energy, this electron is promoted to the higher-energy \(e_g\) set. The excited state configuration is \(t_{2g}^0 e_g^1\).

6. The transition is therefore from the ground state to the excited state: \(t_{2g}^1 e_g^0 \to t_{2g}^0 e_g^1\).


Step 3: Final Answer:

The d-d transition is \(t_{2g}^1 e_g^0 \to t_{2g}^0 e_g^1\).

This corresponds to option (B).
Quick Tip: A d-d transition involves the absorption of energy (light) to promote an electron from a lower-energy d-orbital to a higher-energy d-orbital. The process always goes from a more stable (ground) state to a less stable (excited) state. The color of transition metal complexes is due to these d-d transitions.

Step 1: Understanding the Question:

This is a direct question asking to arrange a given set of ligands according to increasing field strength, which is the order of the spectrochemical series.


Step 2: Detailed Explanation:

The spectrochemical series is an empirically determined list of ligands ordered by their ability to cause d-orbital splitting (\(\Delta_o\)).

- Ligands that cause small splitting are called weak-field ligands.

- Ligands that cause large splitting are called strong-field ligands.

The general trend is: Halide donors < O-donors < N-donors < C-donors.

From the given ligands:

- I\(^-\) and Cl\(^-\) are halide donors (weak field). Generally, I\(^-\) is the weakest common ligand.

- H\(_2\)O is an O-donor (intermediate field).

- CO (carbonyl) is a C-donor (very strong field).

Arranging them in increasing order of field strength gives: I\(^-\) < Cl\(^-\) < H\(_2\)O < CO.


Step 3: Final Answer:

The correct increasing order of field strength is I\(^-\) < Cl\(^-\) < H\(_2\)O < CO.

This corresponds to option (E).
Quick Tip: A useful mnemonic to remember the general trend of the spectrochemical series: "I Bring Some Cloudy Water, Nitrates, Ammonia, Ethylenediamine, and No Cyanide or Carbonyls" (I\(^-\), Br\(^-\), SCN\(^-\), Cl\(^-\), H\(_2\)O, NO\(^-_3\), NH\(_3\), en, NO, CN\(^-\), CO). The order follows this sentence.

Step 1: Understanding the Question:

We need to identify a suitable catalyst for the reaction between iodide ions and peroxodisulfate ions.


Step 2: Detailed Explanation:

This reaction, known as the persulfate-iodide clock reaction, is slow because it involves the reaction between two negatively charged ions (I\(^-\) and S\(_2\)O\(_8^{2-}\)), leading to strong electrostatic repulsion. A catalyst is needed to provide an alternative pathway with a lower activation energy. Transition metal ions like Fe\(^{3+}\) or Fe\(^{2+}\) are effective catalysts.

The mechanism with Fe\(^{3+}\) is:

1. \(2Fe^{3+} + 2I^- \to 2Fe^{2+} + I_2\)

2. \(2Fe^{2+} + S_2O_8^{2-} \to 2Fe^{3+} + 2SO_4^{2-}\)

The Fe\(^{3+}\) is consumed in the first step and regenerated in the second, fulfilling its role as a catalyst. Iron(II) can also initiate the catalysis. Given the options, Iron(III) is a correct choice.


Step 3: Final Answer:

The reaction is catalyzed by Iron(III) ions.

This corresponds to option (C).
Quick Tip: Many redox reactions, especially between ions of the same charge, are catalyzed by transition metal ions that have multiple stable oxidation states. This allows the catalyst to be oxidized and then reduced (or vice versa), providing an alternative reaction pathway.

Step 1: Understanding the Question:

We need to identify the chemical compound used in chelation therapy for lead poisoning.


Step 2: Detailed Explanation:

Lead poisoning is treated by administering a chelating agent. A chelating agent is a polydentate ligand that can bind tightly to a central metal ion (in this case, Pb\(^{2+}\)) to form a stable, water-soluble complex called a chelate. This complex can then be safely excreted from the body.

EDTA (Ethylenediaminetetraacetic acid) is a powerful hexadentate chelating agent. Its anion, EDTA\(^{4-}\), forms very stable complexes with many metal ions, including Pb\(^{2+}\), and is used medically for this purpose (often as its calcium disodium salt to prevent depleting the body's calcium). The other options are also ligands but are not typically used for treating lead poisoning.


Step 3: Final Answer:

EDTA is used in the treatment of lead poisoning.

This corresponds to option (A).
Quick Tip: The effectiveness of a chelating agent in therapy depends on the stability of the complex it forms with the toxic metal (the formation constant, K\(_f\), should be very high) and its ability to be safely administered and excreted. EDTA is a prime example of a therapeutically useful chelating agent.

Step 1: Understanding the Question:

We need to arrange three carboxylic acids in order of increasing acidity.


Step 2: Key Concepts and Explanation:

The acidity of carboxylic acids is influenced by the stability of the conjugate base (carboxylate anion). Electron-donating groups (+I effect) attached to the carboxyl group destabilize the anion by increasing the negative charge density, thereby decreasing the acid strength. Alkyl groups are electron-donating.

- (i) (CH\(_3\))\(_3\)C-COOH: The tert-butyl group has the strongest +I effect due to three methyl groups.

- (ii) (CH\(_3\))\(_2\)CH-COOH: The isopropyl group has a moderate +I effect.

- (iii) CH\(_3\)CH\(_2\)COOH: The ethyl group has the weakest +I effect among the three.

A stronger +I effect leads to a weaker acid. Therefore, the order of acidity is: (i) \(<\) (ii) \(<\) (iii).


Step 3: Final Answer:

The correct increasing order of acid strength is (i) \(<\) (ii) \(<\) (iii).

This corresponds to option (E).
Quick Tip: Remember the electronic effects on acidity:
- \textbf{Electron-Donating Groups (+I, +R)} decrease acidity.
- \textbf{Electron-Withdrawing Groups (-I, -R)} increase acidity.
The more alkyl branching near the -COOH group, the stronger the +I effect and the weaker the acid.

Step 1: Understanding the Question:

We need to arrange three given carbocations in order of decreasing stability.


Step 2: Key Concepts and Explanation:

Carbocation stability is primarily determined by hyperconjugation and the inductive effect. The more alkyl groups attached to the positively charged carbon, the more stable the carbocation. The general order is: Tertiary (3\(^\circ\)) \(>\) Secondary (2\(^\circ\)) \(>\) Primary (1\(^\circ\)) \(>\) Methyl.

- (i) (CH\(_3\))\(_3\)C\(^+\): This is a tertiary carbocation. It is stabilized by the +I effect of three methyl groups and 9 \(\alpha\)-hydrogens for hyperconjugation. It is very stable.

- (ii) (CH\(_3\))\(_2\)C-CH\(_2\)\(^{+}\): This structure is likely a typo and should be (CH\(_3\))\(_2\)CH-CH\(_2\)\(^{+}\) (isobutyl cation), which is a primary carbocation. It is the least stable. Even if interpreted as the neopentyl cation, it's still primary.

- \textbf(iii) CH\(_3\)CH\(^{+}\)-CH\(_2\)CH\(_3\): This is a secondary carbocation. It is stabilized by two alkyl groups and has 5 \(\alpha\)-hydrogens (3 on the left methyl, 2 on the right methylene).

The stability order is Tertiary \(>\) Secondary \(>\) Primary. Therefore, the decreasing order of stability is (i) \(>\) (iii) \(>\) (ii).


Step 3: Final Answer:

The correct decreasing order of stability is (i) \(>\) (iii) \(>\) (ii).

This corresponds to option (D).
Quick Tip: To determine carbocation stability, first classify it as primary (1\(^\circ\)), secondary (2\(^\circ\)), or tertiary (3\(^\circ\)). Then, among carbocations of the same class, the one with more possibilities for hyperconjugation (more \(\alpha\)-hydrogens) or resonance will be more stable.

Step 1: Understanding the Question:

We need to find the number of unpaired electrons in the octahedral complex [CoF\(_6\)]\(^{3-}\).


Step 2: Detailed Explanation:

1. Oxidation state of Co: Since F has a charge of -1, the oxidation state of Co is +3.

2. Electron configuration of Co\(^{3+}\): Neutral Co (Z=27) is [Ar] \(3d^7 4s^2\). Co\(^{3+}\) is [Ar] \(3d^6\).

3. Ligand Field Strength: F\(^-\) is a weak-field ligand. This means it causes a small d-orbital splitting (\(\Delta_o\)) and the pairing energy is high.

4. Electron Filling: For a \(d^6\) ion in a weak octahedral field (high-spin case), electrons will occupy the orbitals singly before pairing up, following Hund's rule. The configuration will be \(t_{2g}^4 e_g^2\).

- \(t_{2g}\) orbitals: [\(\uparrow\downarrow\)] [\(\uparrow\)] [\(\uparrow\)]

- \(e_g\) orbitals: [\(\uparrow\)] [\(\uparrow\)]

5. Unpaired Electrons: Counting the singly occupied orbitals, we find there are 4 unpaired electrons.


Step 3: Final Answer:

There are four unpaired electrons in [CoF\(_6\)]\(^{3-}\).

This corresponds to option (B).
Quick Tip: For octahedral complexes, the key is to identify the metal's d-electron count and whether the ligand is strong-field or weak-field. Weak-field ligands lead to high-spin complexes (maximum unpaired electrons). Strong-field ligands lead to low-spin complexes (minimum unpaired electrons).

Step 1: Understanding the Question:

We need to identify an alkene that produces pentan-3-one and methanal upon ozonolysis.


Step 2: Reconstructing the Alkene:

Ozonolysis cleaves a C=C double bond and replaces it with two C=O double bonds. To find the original alkene, we can reverse this process.

1. Write the structures of the products:

- Pentan-3-one: CH\(_3\)-CH\(_2\)-C(=O)-CH\(_2\)-CH\(_3\)

- Methanal (formaldehyde): H-C(=O)-H, which can be written as CH\(_2\)=O

2. Reconstruct the double bond: Remove the two oxygen atoms from the carbonyl groups and form a double bond between the two carbonyl carbons.

\[ CH_3-CH_2-C(CH_2-CH_3)= + =CH_2 \to CH_3-CH_2-C(CH_2-CH_3)=CH_2 \]
3. Name the resulting alkene:

The longest carbon chain containing the double bond is a butane chain (4 carbons). The double bond starts at carbon 1 (\(-but-1-ene\)). There is an ethyl group (\(-CH_2-CH_3\)) on carbon 2.

The IUPAC name is 2-ethylbut-1-ene.


Step 3: Final Answer:

The alkene is 2-ethylbut-1-ene.

This corresponds to option (C).
Quick Tip: Working backwards from ozonolysis products is a common and effective strategy. Simply take the two carbonyl compounds, align them so the C=O groups face each other, erase the oxygens, and connect the carbons with a double bond to find the parent alkene.

Step 1: Identify the starting material (X).

The molecular formula is C\(_4\)H\(_9\)Br. The problem states it's a tertiary (3\(^\circ\)) alkyl halide. The only possible structure is tert-butyl bromide, (CH\(_3\))\(_3\)C-Br.


Step 2: Reaction of X with alc. KOH (to find Y).

The reaction of an alkyl halide with alcoholic KOH is an elimination reaction (dehydrohalogenation). This removes HBr from the molecule to form an alkene.

(CH\(_3\))\(_3\)C-Br \(\xrightarrow{alc. KOH}\) CH\(_2\)=C(CH\(_3\))\(_2\) + KBr + H\(_2\)O

The product (Y) is 2-methylpropene.


Step 3: Reaction of Y with HBr in presence of peroxide (to find Z).

This is the addition of HBr to an alkene in the presence of peroxide, which proceeds via the anti-Markovnikov rule (free radical mechanism). The bromine atom adds to the carbon atom of the double bond that has more hydrogen atoms.

CH\(_2\)=C(CH\(_3\))\(_2\) + HBr \(\xrightarrow{Peroxide}\) Br-CH\(_2\)-CH(CH\(_3\))\(_2\)

The product (Z) is 1-bromo-2-methylpropane.


Step 4: Final Answer.

Compound Y is 2-methylpropene and compound Z is 1-bromo-2-methylpropane.

This corresponds to option (B).
Quick Tip: Remember the key regioselectivity rules:
- \textbf{Markovnikov's Rule:} In addition of HX to an unsymmetrical alkene, the negative part (X\(^-\)) goes to the carbon with fewer hydrogens. (No peroxide).
- \textbf{Anti-Markovnikov's Rule:} In addition of HBr in the presence of peroxide, Br goes to the carbon with more hydrogens.
- \textbf{Zaitsev's Rule (Elimination):} The more substituted (more stable) alkene is the major product.

Step 1: Identify the ether and the reaction type.

The compound is an unsymmetrical ether: 2-methylbutyl ethyl ether. The reaction is the cleavage of an ether using hydroiodic acid (HI).


Step 2: Determine the reaction mechanism.

The reaction of HI with ethers containing primary or secondary alkyl groups proceeds via an S\(_N\)2 mechanism. The iodide ion (I\(^-\)) acts as a nucleophile and attacks the less sterically hindered alkyl group attached to the oxygen atom.


Step 3: Compare the alkyl groups.

The two alkyl groups are:

- Ethyl group: -CH\(_2\)-CH\(_3\) (primary)

- 2-methylbutyl group: -CH\(_2\)-CH(CH\(_3\))CH\(_2\)CH\(_3\) (primary)

Comparing the two, the ethyl group is smaller and less sterically hindered.


Step 4: Predict the products.

The nucleophile I\(^-\) will attack the carbon of the less hindered ethyl group. This breaks the C-O bond, forming iodoethane. The other part becomes the alcohol.

Products: CH\(_3\)CH\(_2\)-I (iodoethane) and CH\(_3\)CH\(_2\)CH(CH\(_3\))CH\(_2\)OH (2-methylbutan-1-ol).


Step 5: Final Answer.

The major products are 2-methylbutan-1-ol and iodoethane.

This corresponds to option (A).
Quick Tip: For ether cleavage with HX (like HI or HBr):
- If both alkyl groups are 1\(^\circ\) or 2\(^\circ\), the mechanism is S\(_N\)2 and the halide attacks the less hindered group.
- If one group is 3\(^\circ\), the mechanism is S\(_N\)1 and the halide attacks the 3\(^\circ\) group because it forms a stable carbocation.

Step 1: Analyze the transformation.

The reactant is but-2-ene (CH\(_3\)-CH=CH-CH\(_3\)). The product is ethanal (CH\(_3\)CHO). The reaction involves breaking the C=C double bond of the alkene and forming two molecules of an aldehyde.


Step 2: Identify the correct reagent.

This type of reaction is called oxidative cleavage. Reductive ozonolysis is the specific reaction that achieves this.

- Ozonolysis: The alkene reacts with ozone (O\(_3\)) to form an ozonide, which is then cleaved.

- Reductive Workup: Using a mild reducing agent like zinc dust and water (Zn/H\(_2\)O) cleaves the ozonide to produce aldehydes and/or ketones.

Applying this to but-2-ene:

CH\(_3\)-CH=CH-CH\(_3\) \(\xrightarrow{1. O_3 \quad 2. Zn/H_2O}\) 2 CH\(_3\)-CHO (ethanal)


Step 3: Final Answer.

The reagent for this conversion is ozone followed by a reductive workup with zinc dust and water.

This corresponds to option (D).
Quick Tip: Ozonolysis is a powerful tool in organic synthesis to cleave double or triple bonds. Remember the two types of workup:
- \textbf{Reductive workup (Zn/H\(_2\)O or (CH\(_3\))\(_2\)S):} Gives aldehydes and ketones.
- \textbf{Oxidative workup (H\(_2\)O\(_2\)):} Oxidizes any resulting aldehydes to carboxylic acids.

Step 1: Understanding the Question.

This is a factual question from the application of amines, specifically their role as pheromones or attractants.


Step 2: Identifying the Compound.

N,N-Dimethylmethanamine is the IUPAC name for trimethylamine, ((CH\(_3\))\(_3\)N). This simple tertiary amine has a strong, fishy odor and is known to function as a kairomone (a chemical emitted by one species that benefits another) or attractant for certain insects, such as the apple maggot fly. While many insect pheromones are more complex molecules, simple amines can also serve this function.


Step 3: Final Answer.

N,N-Dimethylmethanamine (trimethylamine) is known to act as an insect attractant.

This corresponds to option (B).
Quick Tip: Amines are known for their characteristic odors. Lower aliphatic amines smell fishy, while aromatic amines are often toxic and have unpleasant odors. These properties are sometimes exploited by other organisms.

Step 1: Understanding the Question.

This is a factual question asking for the constituent monosaccharide units of the disaccharide lactose.


Step 2: Recalling the Structure of Lactose.

Lactose, also known as milk sugar, is a disaccharide. Upon hydrolysis, it yields two smaller sugar units. It is composed of one molecule of D-galactose and one molecule of D-glucose. Specifically, the linkage is a \(\beta\)-1,4 glycosidic bond between C1 of \(\beta\)-D-galactose and C4 of \(\beta\)-D-glucose. Therefore, its constituent units are \(\beta\)-D-galactose and \(\beta\)-D-glucose.


Step 3: Final Answer.

Lactose is composed of \(\beta\)-D-galactose and \(\beta\)-D-glucose.

This corresponds to option (C).
Quick Tip: Memorize the composition of the three common disaccharides:
- \textbf{Sucrose} = \(\alpha\)-D-glucose + \(\beta\)-D-fructose
- \textbf{Lactose} = \(\beta\)-D-galactose + \(\beta\)-D-glucose
- \textbf{Maltose} = \(\alpha\)-D-glucose + \(\alpha\)-D-glucose

Step 1: Understanding the Question.

We are given the number of elements in a set A, the union of A and B, and the intersection of A and B. We need to find the number of elements in set B.


Step 2: Key Formula or Approach.

We use the Principle of Inclusion-Exclusion for two sets:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
where n(S) represents the number of elements in set S.


Step 3: Detailed Explanation.

We are given:

- \(n(A) = 20\)

- \(n(A \cup B) = 32\)

- \(n(A \cap B) = 10\)

Substitute these values into the formula:
\[ 32 = 20 + n(B) - 10 \] \[ 32 = 10 + n(B) \]
Solve for n(B):
\[ n(B) = 32 - 10 = 22 \]

Step 4: Final Answer.

The number of elements in the set B is 22.

This corresponds to option (A).
Quick Tip: The inclusion-exclusion principle is a fundamental tool in set theory and combinatorics. Visualizing the problem with a Venn diagram can also be very helpful. The union is the total area, from which you can find the size of one circle if you know the other circle and the overlap.

Step 1: Analyze the relation.

First, we simplify the condition \(x^2 - 4xy + 3y^2 = 0\). This is a quadratic in x. We can factor it:
\(x^2 - xy - 3xy + 3y^2 = 0\)
\(x(x-y) - 3y(x-y) = 0\)
\((x-y)(x-3y) = 0\)

So, (x,y) \(\in\) R means either \(x = y\) or \(x = 3y\).


Step 2: Check the properties.

- Reflexive: Is (x,x) \(\in\) R for all x \(\in\) N? The condition is \(x = x\) or \(x = 3x\). Since \(x=x\) is always true, the relation is reflexive.

- Symmetric: If (x,y) \(\in\) R, is (y,x) \(\in\) R? Let's take a counterexample. Let y=1. Then for x=3, we have \(x=3y\), so (3,1) \(\in\) R. Now we check if (1,3) \(\in\) R. The condition would be \(1=3\) or \(1=3(3)\), both of which are false. So the relation is not symmetric.

- Transitive: If (x,y) \(\in\) R and (y,z) \(\in\) R, is (x,z) \(\in\) R? Let's take a counterexample. Let z=1. Then for y=3, we have \(y=3z\), so (3,1) \(\in\) R. For x=9, we have \(x=3y\), so (9,3) \(\in\) R. Now we check if (9,1) \(\in\) R. The condition would be \(9=1\) or \(9=3(1)\), both of which are false. So the relation is not transitive.


Step 3: Final Answer.

The relation is only reflexive.

This corresponds to option (A).
Quick Tip: To disprove properties like symmetry and transitivity, finding a single counterexample is sufficient. To prove them, you must show they hold true for all possible cases.

Step 1: Understanding the Question.

We have a piecewise function and need to find the positive input value 'x' that gives an output of 2.


Step 2: Test each piece of the function.

We set each definition of \(f(x)\) equal to 2 and check if the resulting x-value falls within the corresponding domain for that piece.

- Case 1: \(x < 0\)

\(x^2 = 2 \implies x = \pm\sqrt{2}\). The value \(x = -\sqrt{2}\) is in the domain \(x<0\), so it is a valid solution. However, the question asks for the *positive* value.

- Case 2: \(0 \le x \le 2\)

\(5x - 3 = 2 \implies 5x = 5 \implies x = 1\). The value \(x=1\) is within the domain \([0, 2]\). This is a positive solution.

- Case 3: \(x > 2\)

\(x^2 + 1 = 2 \implies x^2 = 1 \implies x = \pm 1\). Neither \(x=1\) nor \(x=-1\) is within the domain \(x>2\). So there are no solutions from this piece.


Step 3: Final Answer.

The only positive value of x for which f(x) = 2 is x = 1.

This corresponds to option (D).
Quick Tip: When solving equations involving piecewise functions, it is crucial to check that your solution for 'x' lies within the valid domain for the piece of the function you used. Solutions outside the domain must be discarded.

Step 1: Analyze the function.

The function \(f(x) = -8(x+5)^2\) describes a parabola. Since the coefficient of the squared term is negative (-8), the parabola opens downwards. The vertex of the parabola is at the point where \(x+5=0\), i.e., at \(x=-5\). The value of the function at the vertex is \(f(-5) = -8(0)^2 = 0\).


Step 2: Determine the range.

Since the parabola opens downwards from its vertex at (-5, 0), the maximum value of the function is 0. The set of all possible output values (the range) is therefore \((-\infty, 0]\).


Step 3: Relate one-to-one property, codomain, and range.

For the function to be one-to-one, its domain X must be restricted to one side of the vertex, for example, \(X = (-\infty, -5]\) or \(X = [-5, \infty)\). The question asks for the codomain Y. While the codomain can be any set containing the range, in the context of making a function one-to-one and onto, the codomain is typically chosen to be equal to the range. Assuming the function is also meant to be onto, the codomain Y must be the range of the function on the restricted domain. The range is \((-\infty, 0]\).


Step 4: Final Answer.

The range of the function is \((-\infty, 0]\). This is the only suitable choice for the codomain Y among the options.

This corresponds to option (A).
Quick Tip: The range of a quadratic function \(f(x) = a(x-h)^2 + k\) is \([k, \infty)\) if \(a>0\) (opens up) and \((-\infty, k]\) if \(a<0\) (opens down). The vertex is at \((h, k)\). Identifying these properties quickly helps determine the range.

Step 1: Use the property \(|w|^2 = w\bar{w}\).

It is easier to work with the squares of the moduli. Squaring both sides of the given equation:
\[ |z+16|^2 = 16|z+1|^2 \]
Now, apply the property \(|w|^2 = w\bar{w}\):
\[ (z+16)(\overline{z+16}) = 16(z+1)(\overline{z+1}) \] \[ (z+16)(\bar{z}+16) = 16(z+1)(\bar{z}+1) \]

Step 2: Expand and simplify.
\[ z\bar{z} + 16z + 16\bar{z} + 256 = 16(z\bar{z} + z + \bar{z} + 1) \] \[ |z|^2 + 16(z+\bar{z}) + 256 = 16|z|^2 + 16(z+\bar{z}) + 16 \]
The term \(16(z+\bar{z})\) cancels from both sides.
\[ |z|^2 + 256 = 16|z|^2 + 16 \]

Step 3: Solve for \(|z|\).
\[ 256 - 16 = 16|z|^2 - |z|^2 \] \[ 240 = 15|z|^2 \] \[ |z|^2 = \frac{240}{15} = 16 \] \[ |z| = \sqrt{16} = 4 \] (Since modulus must be non-negative)

Step 4: Final Answer.

The modulus of the complex number z is 4.

This corresponds to option (B).
Quick Tip: The equation \(|z-a| = k|z-b|\) (where k \(\neq\) 1) represents a circle in the complex plane (a Circle of Apollonius). Squaring both sides and using \(|w|^2 = w\bar{w}\) is the standard algebraic method to find its properties.

Step 1: Form a quadratic equation from the given value of z.

We are given \(2z = 7 + i\sqrt{3}\). To eliminate the imaginary part, isolate it and square both sides.
\[ 2z - 7 = i\sqrt{3} \]
Squaring both sides:
\[ (2z - 7)^2 = (i\sqrt{3})^2 \] \[ 4z^2 - 28z + 49 = i^2 \cdot 3 = -3 \] \[ 4z^2 - 28z + 52 = 0 \]

Step 2: Simplify the equation.

Divide the entire equation by 4:
\[ z^2 - 7z + 13 = 0 \]

Step 3: Use the result to find the value of the given expression.

We have found that \(z^2 - 7z = -13\). We need to find the value of the expression \(z^2 - 7z + 4\).

Substitute the value of \(z^2 - 7z\):
\[ (z^2 - 7z) + 4 = (-13) + 4 = -9 \]

Step 4: Final Answer.

The value of the expression is -9.

This corresponds to option (C).
Quick Tip: When a complex number is given and you need to evaluate a polynomial expression involving it, it's often much faster to first create a quadratic equation that the complex number satisfies. This allows you to simplify the polynomial expression without direct substitution.

Step 1: Simplify the base of the expression.

We first simplify the fraction \(\frac{1-i}{1+i}\) by multiplying the numerator and denominator by the conjugate of the denominator.
\[ \frac{1-i}{1+i} = \frac{(1-i)(1-i)}{(1+i)(1-i)} = \frac{1^2 - 2i + i^2}{1^2 - i^2} = \frac{1 - 2i - 1}{1 - (-1)} = \frac{-2i}{2} = -i \]

Step 2: Evaluate the power.

Now we need to calculate \((-i)^{10}\).
\[ (-i)^{10} = ((-1) \cdot i)^{10} = (-1)^{10} \cdot i^{10} = (1) \cdot i^{10} \]
To evaluate \(i^{10}\), we use the cyclic property of powers of i (\(i^2 = -1, i^4 = 1\)).
\[ i^{10} = i^{4 \times 2 + 2} = (i^4)^2 \cdot i^2 = (1)^2 \cdot (-1) = -1 \]

Step 3: Determine a and b.

So, \((\frac{1-i}{1+i})^{10} = -1\).

We are given that this is equal to \(a+ib\).
\[ a + ib = -1 + 0i \]
Comparing the real and imaginary parts, we get \(a = -1\) and \(b = 0\).


Step 4: Final Answer.

The values of a and b are -1 and 0, respectively.

This corresponds to option (C).
Quick Tip: It is very helpful to memorize the results of \(\frac{1-i}{1+i} = -i\) and \(\frac{1+i}{1-i} = i\). These fractions appear frequently in problems involving complex numbers and knowing them saves calculation time.

Step 1: Use the property \(|z_1|=1\).

The condition \(|z_1| = 1\) implies \(|z_1|^2 = 1\), which means \(z_1\bar{z_1} = 1\). From this, we can write \(\bar{z_1} = \frac{1}{z_1}\).


Step 2: Substitute and simplify the expression.

Let's look at the denominator of the fraction inside the modulus: \(1 - \bar{z_1}z_2\).

Substitute \(\bar{z_1} = \frac{1}{z_1}\):
\[ 1 - \bar{z_1}z_2 = 1 - \frac{1}{z_1}z_2 = \frac{z_1 - z_2}{z_1} \]
Now substitute this back into the original expression:
\[ |\frac{z_1-z_2}{1-\bar{z_1}z_2}| = |\frac{z_1-z_2}{(z_1 - z_2)/z_1}| \]
Assuming \(z_1 \neq z_2\), we can cancel the term \((z_1 - z_2)\):
\[ |\frac{1}{1/z_1}| = |z_1| \]

Step 3: Final Answer.

We are given that \(|z_1| = 1\). Therefore, the value of the expression is 1.

This corresponds to option (D).
Quick Tip: This is a standard identity in complex numbers. Any expression of the form \(|\frac{z-w}{1-\bar{w}z}|\) equals 1 if either \(|z|=1\) or \(|w|=1\) (and the denominator is non-zero). Recognizing this property can lead to an instant solution.

Step 1: Represent the terms of the G.P.

For problems involving products of terms in a Geometric Progression (G.P.), it is often convenient to represent three consecutive terms as \(\frac{a}{r}, a, ar\). Here, 'a' is the middle term (the second term) and 'r' is the common ratio.


Step 2: Use the given information.

We are given that the second term is 4. In our chosen representation, this means \(a = 4\).


Step 3: Calculate the product.

The product of the first three terms is:
\[ (\frac{a}{r}) \times (a) \times (ar) = a^3 \]
Substitute the value of \(a = 4\):
\[ Product = 4^3 = 64 \]

Step 4: Final Answer.

The product of the first three terms is 64.

This corresponds to option (D).
Quick Tip: Choosing the terms of a G.P. as \(\frac{a}{r}, a, ar\) for 3 terms, or \(\frac{a}{r^3}, \frac{a}{r}, ar, ar^3\) for 4 terms, simplifies calculations involving products because the common ratio 'r' cancels out.

Step 1: Find the first term 'a' of the G.P.

Let the first three terms be \(a, ar, ar^2\).

Product = \(a \cdot ar \cdot ar^2 = a^3 r^3 = (ar)^3\).

Given Product = 64, so \((ar)^3 = 64 \implies ar = 4\).
Given common ratio \(r = 1/2\).
\(a \cdot (\frac{1}{2}) = 4 \implies a = 8\).

*(Based on the question's text, the first term is 8. The sum S\(_10\) would be 1023/64. As this is not an option, we assume the question intended for the first term to be 4, which would yield a product of 8, not 64).*

Assuming a typo in the question and a = 4:


Step 2: Calculate the sum of the first 10 terms.

The formula for the sum of the first n terms of a G.P. is \(S_n = \frac{a(1-r^n)}{1-r}\).

Here, \(a=4\), \(r = 1/2\), and \(n = 10\).
\[ S_{10} = \frac{4(1 - (\frac{1}{2})^{10})}{1 - \frac{1}{2}} = \frac{4(1 - \frac{1}{1024})}{\frac{1}{2}} \] \[ S_{10} = 8 \left( \frac{1024 - 1}{1024} \right) = 8 \left( \frac{1023}{1024} \right) \] \[ S_{10} = \frac{1023}{128} \]

Step 3: Final Answer.

Assuming a typo in the problem's data, the sum of the first 10 terms is \(\frac{1023}{128}\).

This corresponds to option (A).
Quick Tip: If your derived answer is not among the options in a multiple-choice test, re-read the question for possible misinterpretations. If none are found, consider if a simple typo (e.g., product is 8 instead of 64) would lead to one of the given answers. This can be a useful strategy in time-constrained exams where questions might contain errors.

Step 1: Define the terms of both G.P.s

The first G.P. is \(a, b, c, d\). So, we can write \(b=ar\), \(c=ar^2\), and \(d=ar^3\).

The second G.P. has terms \(T_1 = \frac{1}{a^3+b^3}\) and \(T_2 = \frac{1}{b^3+c^3}\).


Step 2: Find the common ratio of the second G.P.

The common ratio (let's call it R) of the second G.P. is given by the ratio of any two consecutive terms, for example, \(R = \frac{T_2}{T_1}\).
\[ R = \frac{\frac{1}{b^3+c^3}}{\frac{1}{a^3+b^3}} = \frac{a^3+b^3}{b^3+c^3} \]

Step 3: Substitute the terms of the first G.P. into the expression for R.

Substitute \(b=ar\) and \(c=ar^2\):
\[ R = \frac{a^3 + (ar)^3}{(ar)^3 + (ar^2)^3} = \frac{a^3 + a^3r^3}{a^3r^3 + a^3r^6} \]
Factor out the common terms from the numerator and the denominator:
\[ R = \frac{a^3(1+r^3)}{a^3r^3(1+r^3)} \]
Cancel the common factor \(a^3(1+r^3)\):
\[ R = \frac{1}{r^3} \]

Step 4: Final Answer.

The common ratio of the second G.P. is \(\frac{1}{r^3}\).

This corresponds to option (E).
Quick Tip: When dealing with problems involving terms of a G.P., immediately express all terms using the first term 'a' and the common ratio 'r'. This simplifies the expressions and often leads to cancellations that reveal the solution.

Step 1: Use calculus to find the minimum value.

To find the minimum value of a function, we find its derivative, set it to zero to find critical points, and then check the second derivative.

The function is \(f(x) = 7x^4 + 28x + 31\).


Step 2: Find the first derivative and critical points.
\[ f'(x) = \frac{d}{dx}(7x^4 + 28x + 31) = 28x^3 + 28 \]
Set the derivative to zero:
\[ 28x^3 + 28 = 0 \implies x^3 + 1 = 0 \implies x^3 = -1 \implies x = -1 \]
There is one critical point at \(x = -1\).


Step 3: Check the second derivative.
\[ f''(x) = \frac{d}{dx}(28x^3 + 28) = 84x^2 \]
At \(x = -1\), \(f''(-1) = 84(-1)^2 = 84\). Since \(f''(-1) > 0\), the function has a local minimum at \(x = -1\).


Step 4: Calculate the minimum value.

Substitute \(x=-1\) back into the original function:
\[ f(-1) = 7(-1)^4 + 28(-1) + 31 = 7(1) - 28 + 31 = 7 + 3 = 10 \]

Step 5: Final Answer.

The minimum value of the function is 10.

This corresponds to option (B).
Quick Tip: To find the minimum or maximum of a differentiable function, the standard procedure is to find the derivative, set it to zero to locate critical points, and use the second derivative test to classify them (positive for minimum, negative for maximum).

Step 1: Recall the Binomial Theorem.

The binomial theorem states that for any integer n \(\ge\) 0:
\[ (x+y)^n = \binom{n}{0}x^n y^0 + \binom{n}{1}x^{n-1}y^1 + \dots + \binom{n}{n}x^0 y^n \]
A useful identity derived from this is by setting \(x=1\) and \(y=1\):
\[ (1+1)^n = 2^n = \binom{n}{0} + \binom{n}{1} + \dots + \binom{n}{n} \]

Step 2: Apply the identity to the given problem.

For \(n=10\), the identity is:
\[ \binom{10}{0} + \binom{10}{1} + \binom{10}{2} + \dots + \binom{10}{10} = 2^{10} \]
The sum we need to find is \(S = \binom{10}{1} + \binom{10}{2} + \dots + \binom{10}{10}\).

This is the full sum minus the first term, \(\binom{10}{0}\).


Step 3: Calculate the final value.
\[ S = 2^{10} - \binom{10}{0} \]
We know that \(2^{10} = 1024\) and \(\binom{n}{0} = 1\).
\[ S = 1024 - 1 = 1023 \]

Step 4: Final Answer.

The value of the sum is 1023. This matches option (A).
Quick Tip: The sum of all binomial coefficients for a given n is \(2^n\). The sum of coefficients from \(\binom{n}{1}\) to \(\binom{n}{n}\) is \(2^n - 1\). This is a very common identity in combinatorics.

Assuming the intended expression is \((\frac{1}{\sqrt{x}} - x)^6\)


Step 1: Write the general term of the expansion.

The general term, \(T_{r+1}\), in the expansion of \((a+b)^n\) is \(T_{r+1} = \binom{n}{r} a^{n-r} b^r\).

For this problem, \(n=6\), \(a = x^{-1/2}\), and \(b = -x\).
\[ T_{r+1} = \binom{6}{r} (x^{-1/2})^{6-r} (-x)^r \]

Step 2: Simplify the expression for the general term.
\[ T_{r+1} = \binom{6}{r} (-1)^r x^{\frac{-(6-r)}{2}} x^r \] \[ T_{r+1} = \binom{6}{r} (-1)^r x^{-3 + \frac{r}{2} + r} = \binom{6}{r} (-1)^r x^{-3 + \frac{3r}{2}} \]

Step 3: Find the value of 'r' for the \(x^3\) term.

We set the exponent of x equal to 3:
\[ -3 + \frac{3r}{2} = 3 \] \[ \frac{3r}{2} = 6 \] \[ 3r = 12 \implies r = 4 \]
Since r=4 is an integer between 0 and 6, this is a valid term.


Step 4: Calculate the coefficient.

The coefficient is \(\binom{6}{r} (-1)^r\) with \(r=4\).

Coefficient = \(\binom{6}{4} (-1)^4 = \binom{6}{2} \cdot 1 = \frac{6 \times 5}{2 \times 1} = 15\)


Step 5: Final Answer.

Under the assumption of a typo in the question, the coefficient of \(x^3\) is 15.

This corresponds to option (B).
Quick Tip: When finding the coefficient of a specific power of x in a binomial expansion, first write the general term \(T_{r+1}\). Then, collect all the powers of x and set the resulting exponent equal to the desired power to solve for 'r'. If 'r' is not a non-negative integer, the term does not exist.

Step 1: Recall the relationship between permutations and combinations.

The number of permutations (\(^n P_r\)) and combinations (\(^n C_r\)) are related by the formula:
\[ ^n P_r = r! \times ^n C_r \]

Step 2: Substitute the given values into the formula.

We are given \(^n P_r = 480\) and \(^n C_r = 20\).
\[ 480 = r! \times 20 \]

Step 3: Solve for r!.
\[ r! = \frac{480}{20} = 24 \]

Step 4: Determine the value of r.

We need to find the integer 'r' whose factorial is 24. We can calculate the factorials of small integers:

- \(1! = 1\)

- \(2! = 2 \times 1 = 2\)

- \(3! = 3 \times 2 \times 1 = 6\)

- \(4! = 4 \times 3 \times 2 \times 1 = 24\)

Thus, we find that \(r = 4\).


Step 5: Final Answer.

The value of r is 4.

This corresponds to option (C).
Quick Tip: The relationship \(^n P_r = r! \times ^n C_r\) is fundamental and provides a quick way to solve problems where both permutation and combination values are given. Memorizing the first few factorial values (1! to 6!) is also very useful for speed.

Step 1: Write the general term.

The general term \(T_{r+1}\) in the expansion of \((a+b)^n\) is \(\binom{n}{r}a^{n-r}b^r\).

Here, \(n=10\), \(a=x^2\), \(b=x^{-1}\).
\[ T_{r+1} = \binom{10}{r}(x^3)^{10-r}(x^{-2})^r = \binom{10}{r}x^{30-3r}x^{-2r} = \binom{10}{r}x^{30-5r} \]

Step 2: Find 'r' for the constant term.

The constant term is the term where the power of x is 0.

Set exponent to 0: \(30-5r = 0 \implies 5r = 30 \implies r=6\).


Step 3: Calculate the coefficient.

The constant term is the coefficient when \(r=6\).

Coefficient = \(\binom{10}{6} = \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 10 \times 3 \times 7 = 210\).


Step 4: Final Answer.

The constant term in the expansion is 210.

This corresponds to option (A).
Quick Tip: To find the constant term (or term independent of x) in a binomial expansion, write the general term \(T_{r+1}\), simplify the powers of x, and set the final exponent of x to zero to solve for r. The coefficient is the numerical part of that term.

Step 1: Perform the scalar multiplication and matrix addition.

First, multiply the first matrix by 2:
\[ 2\begin{bmatrix} 3 & 4
5 & x \end{bmatrix} = \begin{bmatrix} 6 & 8
10 & 2x \end{bmatrix} \]
Now add this to the second matrix:
\[ \begin{bmatrix} 6 & 8
10 & 2x \end{bmatrix} + \begin{bmatrix} 1 & y
0 & 1 \end{bmatrix} = \begin{bmatrix} 6+1 & 8+y
10+0 & 2x+1 \end{bmatrix} = \begin{bmatrix} 7 & 8+y
10 & 2x+1 \end{bmatrix} \]

Step 2: Equate the resulting matrix to the given matrix.
\[ \begin{bmatrix} 7 & 8+y
10 & 2x+1 \end{bmatrix} = \begin{bmatrix} 7 & 0
10 & 5 \end{bmatrix} \]

Step 3: Solve for x and y by comparing corresponding elements.

From the top-right element: \(8 + y = 0 \implies y = -8\).

From the bottom-right element: \(2x + 1 = 5 \implies 2x = 4 \implies x = 2\).


Step 4: Calculate x - y.
\[ x - y = 2 - (-8) = 2 + 8 = 10 \]

Step 5: Final Answer.

The value of x - y is 10.

This corresponds to option (D).
Quick Tip: When solving matrix equations, perform the operations (scalar multiplication, addition) on one side first. Then, equate the corresponding elements of the matrices on both sides of the equation to form a system of linear equations to solve for the unknown variables.

Step 1: Recall the property relating a matrix, its adjoint, and its determinant.

For any n \(\times\) n matrix A, the determinant of its adjoint is given by:
\[ |adj(A)| = |A|^{n-1} \]
In this problem, B = adj(A) and the matrix is \(3 \times 3\), so n = 3.

Therefore, \(|B| = |A|^{3-1} = |A|^2\).


Step 2: Substitute the given value of \(|A|\).

We are given \(|A| = 4\).
\[ |B| = 4^2 = 16 \]

Step 3: Calculate the determinant of B and solve for \(\alpha\).
\[ |B| = \begin{vmatrix} 1 & \alpha & 3
1 & 3 & 3
2 & 4 & 4 \end{vmatrix} \]
Expand along the first row:
\[ |B| = 1(3 \cdot 4 - 3 \cdot 4) - \alpha(1 \cdot 4 - 3 \cdot 2) + 3(1 \cdot 4 - 3 \cdot 2) \] \[ |B| = 1(12 - 12) - \alpha(4 - 6) + 3(4 - 6) \] \[ |B| = 1(0) - \alpha(-2) + 3(-2) \] \[ |B| = 2\alpha - 6 \]

Step 4: Equate the two expressions for \(|B|\).
\[ 2\alpha - 6 = 16 \] \[ 2\alpha = 22 \] \[ \alpha = 11 \]

Step 5: Final Answer.

The value of \(\alpha\) is 11.

This corresponds to option (D).
Quick Tip: The property \(|adj(A)| = |A|^{n-1}\) is extremely useful and frequently tested. Memorizing it can save a lot of time compared to trying to find matrix A first.

Step 1: Recall the condition for collinearity of three points.

Three points \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) are collinear if the slope of the line joining the first two points is equal to the slope of the line joining the second and third points.
\[ \frac{y_2 - y_1}{x_2 - x_1} = \frac{y_3 - y_2}{x_3 - x_2} \]
Alternatively, the area of the triangle formed by the three points is zero.


Step 2: Apply the slope condition.

Let the points be A(2, -3), B(\(\lambda\), -1), and C(0, 4).

Slope of AB = \(\frac{-1 - (-3)}{\lambda - 2} = \frac{2}{\lambda - 2}\)

Slope of BC = \(\frac{4 - (-1)}{0 - \lambda} = \frac{5}{-\lambda}\)


Step 3: Equate the slopes and solve for \(\lambda\).
\[ \frac{2}{\lambda - 2} = \frac{5}{-\lambda} \]
Cross-multiply:
\[ 2(-\lambda) = 5(\lambda - 2) \] \[ -2\lambda = 5\lambda - 10 \] \[ 10 = 5\lambda + 2\lambda \] \[ 10 = 7\lambda \] \[ \lambda = \frac{10}{7} \]

Step 4: Final Answer.

The value of \(\lambda\) is \(\frac{10}{7}\).

This corresponds to option (E).
Quick Tip: Using the slope condition for collinearity is often the fastest method. Another common method is to set the determinant for the area of the triangle to zero: \(\frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1
x_2 & y_2 & 1
x_3 & y_3 & 1 \end{vmatrix} = 0\). Choose whichever method you are more comfortable with.

Step 1: Solve the compound inequality.

We have \(-5 \le \frac{2-3x}{4} \le 9\). We need to isolate x.


Step 2: Multiply by 4.

Multiply all parts of the inequality by 4:
\[ 4(-5) \le 2-3x \le 4(9) \] \[ -20 \le 2-3x \le 36 \]

Step 3: Subtract 2.

Subtract 2 from all parts:
\[ -20 - 2 \le -3x \le 36 - 2 \] \[ -22 \le -3x \le 34 \]

Step 4: Divide by -3 and reverse inequality signs.

Divide all parts by -3. Remember to reverse the direction of the inequality signs when multiplying or dividing by a negative number.
\[ \frac{-22}{-3} \ge x \ge \frac{34}{-3} \] \[ \frac{22}{3} \ge x \ge -\frac{34}{3} \]

Step 5: Write the solution set.

Rewriting the inequality in the standard order (smallest to largest):
\[ -\frac{34}{3} \le x \le \frac{22}{3} \]
In interval notation, this is \([-\frac{34}{3}, \frac{22}{3}]\).

This matches option (C). The question was likely cancelled due to the typo in option (A) making it ambiguous.
Quick Tip: The most important rule to remember when solving inequalities is to flip the inequality sign (\(<\) to \(>\) and vice versa) whenever you multiply or divide the entire inequality by a negative number.

Step 1: Expand the determinant.

We expand the determinant along the first row:
\[ x \begin{vmatrix} 1 & x
-1 & 3 \end{vmatrix} - 1 \begin{vmatrix} 4 & x
1 & 3 \end{vmatrix} + 2 \begin{vmatrix} 4 & 1
1 & -1 \end{vmatrix} = -10 \] \[ x(1 \cdot 3 - x \cdot (-1)) - 1(4 \cdot 3 - x \cdot 1) + 2(4 \cdot (-1) - 1 \cdot 1) = -10 \] \[ x(3 + x) - (12 - x) + 2(-4 - 1) = -10 \]

Step 2: Simplify the expression to form a quadratic equation.
\[ 3x + x^2 - 12 + x + 2(-5) = -10 \] \[ x^2 + 4x - 12 - 10 = -10 \] \[ x^2 + 4x - 22 = -10 \] \[ x^2 + 4x - 12 = 0 \]

Step 3: Solve the quadratic equation.

We can solve this by factoring. We need two numbers that multiply to -12 and add to 4. These numbers are 6 and -2.
\[ (x+6)(x-2) = 0 \]
This gives two possible values for x:
\[ x+6=0 \implies x = -6 \] \[ x-2=0 \implies x = 2 \]

Step 4: Final Answer.

The values of x are 2 and -6.

This corresponds to option (E).
Quick Tip: Be careful with signs when calculating determinants. Double-check the cofactor signs (+, -, + for the first row) and the calculations within the 2x2 determinants.

Step 1: Recall the formula for the determinant using cofactors.

The determinant of a matrix can be calculated by expanding along any row or column. The formula for expanding along the first row (i=1) is:
\[ |A| = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} \]
where \(C_{ij}\) is the cofactor of the element \(a_{ij}\).


Step 2: Relate cofactors to minors.

The cofactor \(C_{ij}\) is related to the minor \(M_{ij}\) by the formula:
\[ C_{ij} = (-1)^{i+j}M_{ij} \]

Step 3: Calculate the required cofactors.

- \(C_{11} = (-1)^{1+1}M_{11} = (1)(-40) = -40\)

- \(C_{12} = (-1)^{1+2}M_{12} = (-1)(-10) = 10\)

- \(C_{13} = (-1)^{1+3}M_{13} = (1)(35) = 35\)


Step 4: Calculate the determinant.

Now substitute the elements \(a_{1j}\) and cofactors \(C_{1j}\) into the determinant formula:
\[ |A| = (1)(-40) + (3)(10) + (-2)(35) \] \[ |A| = -40 + 30 - 70 \] \[ |A| = -10 - 70 = -80 \]

Step 5: Final Answer.

The value of the determinant \(|A|\) is -80.

This corresponds to option (B).
Quick Tip: Remember the "checkerboard" pattern of signs for converting minors to cofactors: \(\begin{pmatrix} + & - & +
- & + & -
+ & - & + \end{pmatrix}\). A common error is to forget to apply these signs and use the minors directly in the determinant expansion.

Step 1: Simplify the trigonometric expression.

Use the Pythagorean identity \(\tan^2\theta + 1 = \sec^2\theta\), which means \(\sec^2\theta - 1 = \tan^2\theta\).
\[ \frac{\sec^2 15^\circ - 1}{\sec^2 15^\circ} = \frac{\tan^2 15^\circ}{\sec^2 15^\circ} \]
Now, express tan and sec in terms of sin and cos:
\[ \frac{(\sin 15^\circ / \cos 15^\circ)^2}{(1 / \cos 15^\circ)^2} = \frac{\sin^2 15^\circ / \cos^2 15^\circ}{1 / \cos^2 15^\circ} = \sin^2 15^\circ \]

Step 2: Find the value of \(\sin^2 15^\circ\).

Use the half-angle identity for cosine: \(\cos(2\theta) = 1 - 2\sin^2\theta\), which gives \(\sin^2\theta = \frac{1-\cos(2\theta)}{2}\).

Let \(\theta = 15^\circ\), so \(2\theta = 30^\circ\).
\[ \sin^2 15^\circ = \frac{1 - \cos 30^\circ}{2} \]
We know \(\cos 30^\circ = \frac{\sqrt{3}}{2}\).
\[ \sin^2 15^\circ = \frac{1 - \frac{\sqrt{3}}{2}}{2} = \frac{\frac{2-\sqrt{3}}{2}}{2} = \frac{2-\sqrt{3}}{4} \]

Step 3: Final Answer.

The value of the expression is \(\frac{2-\sqrt{3}}{4}\).

This corresponds to option (A). Quick Tip: Simplifying a complex trigonometric expression using fundamental identities (Pythagorean, quotient, reciprocal) is often the first step. For evaluating expressions with angles like 15\(^\circ\) or 22.5\(^\circ\), the half-angle identities are indispensable.

Step 1: Use trigonometric identities to relate the angles.

Notice the relationship between the two angles: \(\frac{3\pi}{8} + \frac{7\pi}{8} \neq \pi\). Let's try another relation.
Notice that \(\frac{7\pi}{8} = \pi - \frac{\pi}{8}\). This is not immediately useful.

Let's try \(\frac{7\pi}{8} = \frac{4\pi+3\pi}{8} = \frac{\pi}{2} + \frac{3\pi}{8}\). This seems promising.

We use the identity \(\sin(\frac{\pi}{2} + \theta) = \cos(\theta)\).

So, \(\sin(\frac{7\pi}{8}) = \sin(\frac{\pi}{2} + \frac{3\pi}{8})\). This is incorrect. The angle sum is \(\frac{4\pi+3\pi}{8}\), which is wrong.

Let's use \(\frac{7\pi}{8} = \pi - \frac{\pi}{8}\). This leads to \(\sin(\frac{7\pi}{8}) = \sin(\pi - \frac{\pi}{8}) = \sin(\frac{\pi}{8})\).

This gives \(\sin^2(\frac{3\pi}{8}) + \sin^2(\frac{\pi}{8})\).
Now, notice that \(\frac{3\pi}{8} = \frac{\pi}{2} - \frac{\pi}{8}\).

So, \(\sin(\frac{3\pi}{8}) = \sin(\frac{\pi}{2} - \frac{\pi}{8}) = \cos(\frac{\pi}{8})\).


Step 2: Substitute and simplify.

The original expression becomes:
\[ \cos^2(\frac{\pi}{8}) + \sin^2(\frac{\pi}{8}) \]
Using the fundamental Pythagorean identity \(\cos^2\theta + \sin^2\theta = 1\), the value of the expression is 1.


Step 3: Final Answer.

The value of the expression is 1.

This corresponds to option (B).
Quick Tip: When you see a sum of \(\sin^2\) terms, always look for angle relationships like \(\theta_1 + \theta_2 = 90^\circ (\pi/2)\) or \(\theta_1 + \theta_2 = 180^\circ (\pi)\), as they often allow you to use identities like \(\sin(\pi/2 - \theta) = \cos\theta\) to simplify the expression to \(\sin^2\theta + \cos^2\theta = 1\).

Step 1: Simplify the algebraic expression.

Let's find a common denominator for the two terms:
\[ \sqrt{\frac{a+b}{a-b}} + \sqrt{\frac{a-b}{a+b}} = \frac{(\sqrt{a+b})(\sqrt{a+b}) + (\sqrt{a-b})(\sqrt{a-b})}{(\sqrt{a-b})(\sqrt{a+b})} \] \[ = \frac{(a+b) + (a-b)}{\sqrt{(a-b)(a+b)}} = \frac{2a}{\sqrt{a^2-b^2}} \]

Step 2: Use the given trigonometric relation.

We are given \(\sin\theta = \frac{b}{a}\). This implies \(b = a\sin\theta\).

Substitute this into the simplified expression:
\[ \frac{2a}{\sqrt{a^2 - (a\sin\theta)^2}} = \frac{2a}{\sqrt{a^2 - a^2\sin^2\theta}} \] \[ = \frac{2a}{\sqrt{a^2(1-\sin^2\theta)}} = \frac{2a}{\sqrt{a^2\cos^2\theta}} \]
Assuming a and \(\cos\theta\) are positive (as is typical for such problems):
\[ = \frac{2a}{a\cos\theta} = \frac{2}{\cos\theta} \]

Step 3: Final Answer.

The value of the expression is \(\frac{2}{\cos\theta}\).

This corresponds to option (A).
Quick Tip: When an algebraic expression is given along with a trigonometric substitution, it's often best to simplify the algebra first before making the substitution. This usually makes the problem much cleaner.

Step 1: Simplify the function using a double-angle identity.

Recall the double-angle identity for sine: \(\sin(2\theta) = 2\sin\theta\cos\theta\).

Let \(\theta = 4x\). Then the given function is:
\[ f(x) = 2 \sin(4x) \cos(4x) = \sin(2 \cdot 4x) = \sin(8x) \]

Step 2: Determine the period of the simplified function.

The period of a standard sine function \(\sin(x)\) is \(2\pi\).

For a function of the form \(f(x) = A\sin(Bx + C) + D\), the period T is given by:
\[ T = \frac{2\pi}{|B|} \]
In our function, \(f(x) = \sin(8x)\), the value of B is 8.
\[ T = \frac{2\pi}{8} = \frac{\pi}{4} \]

Step 3: Final Answer.

The period of the function is \(\frac{\pi}{4}\).

This corresponds to option (D).
Quick Tip: Always try to simplify trigonometric functions using identities before finding properties like period or amplitude. Recognizing patterns like \(2\sin\theta\cos\theta\) is key to quick simplification.

Step 1: Find the domain of the numerator.

The domain of the inverse sine function, \(\sin^{-1}(u)\), is defined for \(-1 \le u \le 1\).

Here, \(u = x-3\). So, we must have:
\[ -1 \le x-3 \le 1 \]
Add 3 to all parts:
\[ -1+3 \le x \le 1+3 \] \[ 2 \le x \le 4 \]
So, the domain from the numerator is \([2, 4]\).


Step 2: Find the domain of the denominator.

The expression under the square root, \(9-x^2\), must be strictly greater than zero because it's in the denominator (we cannot divide by zero).
\[ 9-x^2 > 0 \] \[ 9 > x^2 \implies x^2 < 9 \]
Taking the square root of both sides gives:
\[ |x| < 3 \implies -3 < x < 3 \]
So, the domain from the denominator is \((-3, 3)\).


Step 3: Find the intersection of the two domains.

The overall domain of the function is the intersection of the domains from the numerator and the denominator.

We need the values of x that satisfy both \([2, 4]\) AND \((-3, 3)\).

The intersection of these two intervals is \([2, 3)\).


Step 4: Final Answer.

The domain of the function is \([2, 3)\).

This corresponds to option (C).
Quick Tip: To find the domain of a complex function, find the domain of each individual part (numerator, denominator, functions like log, sqrt, arcsin, etc.) and then find the intersection of all these individual domains.

Step 1: Use the AM-GM inequality.

For any two non-negative numbers a and b, the Arithmetic Mean (AM) is always greater than or equal to the Geometric Mean (GM).
\[ \frac{a+b}{2} \ge \sqrt{ab} \]
In the interval \(x \in (0, \frac{\pi}{2})\), both \(\tan x\) and \(\cot x\) are positive. Therefore, \(\tan^2 x\) and \(\cot^2 x\) are also positive. We can apply the AM-GM inequality with \(a = \tan^2 x\) and \(b = \cot^2 x\).


Step 2: Apply the inequality to the given expression.
\[ \frac{\tan^2 x + \cot^2 x}{2} \ge \sqrt{\tan^2 x \cdot \cot^2 x} \]
Since \(\cot x = \frac{1}{\tan x}\), their product is 1.
\[ \frac{\tan^2 x + \cot^2 x}{2} \ge \sqrt{1} \] \[ \frac{\alpha}{2} \ge 1 \] \[ \alpha \ge 2 \]

Step 3: Determine the interval.

The minimum value of \(\alpha\) is 2. The value is achieved when \(\tan^2 x = \cot^2 x\), which means \(\tan x = 1\) or \(x = \pi/4\). For all other values in the domain, \(\alpha > 2\).

Therefore, \(\alpha\) lies in the interval \([2, \infty)\).


Step 4: Final Answer.

The interval for \(\alpha\) is \([2, \infty)\).

This corresponds to option (E).
Quick Tip: The expression \(f(x) = y + \frac{1}{y}\) where y is positive always has a minimum value of 2, which occurs at y=1. Recognizing expressions like \(\tan^2 x + \cot^2 x\) as this form is a useful shortcut that avoids the formal AM-GM inequality.

Step 1: Use the sum identity for inverse tangent.

The formula for the sum of two inverse tangent functions is:
\[ \tan^{-1}(x) + \tan^{-1}(y) = \tan^{-1}\left(\frac{x+y}{1-xy}\right) \quad (if xy < 1) \]
Here, \(x = 3/4\) and \(y = 2/3\). Let's check the condition: \(xy = (3/4)(2/3) = 6/12 = 1/2 < 1\). So the formula is applicable.


Step 2: Apply the formula.
\[ \tan^{-1}(\frac{3}{4}) + \tan^{-1}(\frac{2}{3}) = \tan^{-1}\left(\frac{\frac{3}{4}+\frac{2}{3}}{1-\frac{3}{4}\cdot\frac{2}{3}}\right) \] \[ = \tan^{-1}\left(\frac{\frac{9+8}{12}}{1-\frac{6}{12}}\right) = \tan^{-1}\left(\frac{\frac{17}{12}}{\frac{6}{12}}\right) = \tan^{-1}\left(\frac{17}{6}\right) \]

Step 3: Evaluate the final expression.

The original expression is:
\[ \tan\left[\tan^{-1}\left(\frac{17}{6}\right)\right] \]
Since \(\tan(\tan^{-1}(z)) = z\), the value is simply \(\frac{17}{6}\).


Step 4: Final Answer.

The value of the expression is \(\frac{17}{6}\).

This corresponds to option (A).
Quick Tip: Memorize the sum and difference formulas for inverse trigonometric functions, especially for \(\tan^{-1}\). They are very frequently used to simplify expressions inside other trigonometric functions. Always check the condition (e.g., \(xy<1\)) before applying the principal value formula.

Step 1: Recall the identity for expressions of the form \(a\sin\theta + b\cos\theta\).

For any expressions \(a\sin\theta + b\cos\theta\) and \(b\sin\theta - a\cos\theta\), we have the identity:
\[ (a\sin\theta + b\cos\theta)^2 + (b\sin\theta - a\cos\theta)^2 = a^2 + b^2 \]
This can be proven by expanding the squares.


Step 2: Apply the identity to the given problem.

Here, \(a=3\) and \(b=5\).

We are given \(3\sin\theta + 5\cos\theta = 5\).

Let the expression we want to find be \(X = 5\sin\theta - 3\cos\theta\).

Substituting into the identity:
\[ (3\sin\theta + 5\cos\theta)^2 + (5\sin\theta - 3\cos\theta)^2 = 3^2 + 5^2 \] \[ (5)^2 + X^2 = 9 + 25 \] \[ 25 + X^2 = 34 \]

Step 3: Solve for X.
\[ X^2 = 34 - 25 = 9 \] \[ X = \pm\sqrt{9} = \pm 3 \]
The options include the positive value, 3.


Step 4: Final Answer.

The value of the expression is 3.

This corresponds to option (C).
Quick Tip: The identity \((a\sin x + b\cos x)^2 + (b\sin x - a\cos x)^2 = a^2 + b^2\) is a very powerful tool for this specific type of problem and should be memorized. It allows for a solution without having to find the value of \(\theta\).

Step 1: Understand the expression.

We need to find the cosine of an angle whose cotangent is \(\frac{7}{24}\). Let \(\theta = \cot^{-1}(\frac{7}{24})\). This means \(\cot\theta = \frac{7}{24}\).


Step 2: Construct a right-angled triangle.

We can visualize this relationship using a right-angled triangle. Since \(\cot\theta = \frac{Adjacent}{Opposite}\), we can set:

- Adjacent side = 7

- Opposite side = 24

Now, we find the hypotenuse using the Pythagorean theorem:

- Hypotenuse = \(\sqrt{Adjacent^2 + Opposite^2} = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25\).


Step 3: Find the value of \(\cos\theta\).

Now that we have all three sides of the triangle, we can find \(\cos\theta\).
\[ \cos\theta = \frac{Adjacent}{Hypotenuse} = \frac{7}{25} \]
Since \(\theta = \cot^{-1}(\frac{7}{24})\), our original expression is simply \(\cos\theta\).


Step 4: Final Answer.

The value of \(\cos(\cot^{-1}(\frac{7}{24}))\) is \(\frac{7}{25}\).

This corresponds to option (D).
Quick Tip: For problems of the form trig(inverse\_trig(value)), the right-angled triangle method is the most intuitive and reliable approach. Define the angle, draw the triangle based on the inner inverse function, find the missing side, and then calculate the value of the outer trigonometric function.

Step 1: Use the half-angle formula for \(\tan^2(\frac{\theta}{2})\).

The identity is: \(\tan^2(\frac{\theta}{2}) = \frac{1 - \cos\theta}{1 + \cos\theta}\).


Step 2: Substitute the given expression for \(\cos\theta\).
\[ \tan^2(\frac{\theta}{2}) = \frac{1 - (\frac{2\cos\alpha + 1}{2 + \cos\alpha})}{1 + (\frac{2\cos\alpha + 1}{2 + \cos\alpha})} \]
Simplify the numerator and the denominator:

Numerator: \(1 - \frac{2\cos\alpha + 1}{2 + \cos\alpha} = \frac{(2+\cos\alpha) - (2\cos\alpha + 1)}{2 + \cos\alpha} = \frac{1 - \cos\alpha}{2 + \cos\alpha}\)

Denominator: \(1 + \frac{2\cos\alpha + 1}{2 + \cos\alpha} = \frac{(2+\cos\alpha) + (2\cos\alpha + 1)}{2 + \cos\alpha} = \frac{3 + 3\cos\alpha}{2 + \cos\alpha}\)


Step 3: Combine the simplified numerator and denominator.
\[ \tan^2(\frac{\theta}{2}) = \frac{\frac{1 - \cos\alpha}{2 + \cos\alpha}}{\frac{3 + 3\cos\alpha}{2 + \cos\alpha}} = \frac{1 - \cos\alpha}{3(1 + \cos\alpha)} \]

Step 4: Use the half-angle formula for \(\tan^2(\frac{\alpha}{2})\).

We know that \(\tan^2(\frac{\alpha}{2}) = \frac{1 - \cos\alpha}{1 + \cos\alpha}\). We can see this pattern in our result.
\[ \tan^2(\frac{\theta}{2}) = \frac{1}{3} \left( \frac{1 - \cos\alpha}{1 + \cos\alpha} \right) = \frac{1}{3} \tan^2(\frac{\alpha}{2}) \]

Step 5: Final Answer.

The expression is equal to \(\frac{1}{3}\tan^2(\frac{\alpha}{2})\).

This corresponds to option (A).
Quick Tip: This problem is a test of algebraic manipulation combined with knowledge of half-angle identities. The key is to apply the identity for \(\tan^2(\theta/2)\) and then simplify patiently until you can re-apply the same identity for the angle \(\alpha\).

Step 1: Recall the property of direction cosines.

If a vector makes angles \(\alpha, \beta, \gamma\) with the x, y, and z axes (i.e., with \(\hat{i}, \hat{j}, \hat{k}\)), then the direction cosines \(l=\cos\alpha, m=\cos\beta, n=\cos\gamma\) must satisfy the relation:
\[ \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1 \]

Step 2: Substitute the given angles.

We are given \(\alpha = \frac{\pi}{3}\) and \(\beta = \frac{\pi}{4}\).
\[ \cos^2(\frac{\pi}{3}) + \cos^2(\frac{\pi}{4}) + \cos^2\gamma = 1 \]
We know \(\cos(\frac{\pi}{3}) = \frac{1}{2}\) and \(\cos(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}\).
\[ (\frac{1}{2})^2 + (\frac{1}{\sqrt{2}})^2 + \cos^2\gamma = 1 \] \[ \frac{1}{4} + \frac{1}{2} + \cos^2\gamma = 1 \] \[ \frac{3}{4} + \cos^2\gamma = 1 \]

Step 3: Solve for \(\cos\gamma\).
\[ \cos^2\gamma = 1 - \frac{3}{4} = \frac{1}{4} \] \[ \cos\gamma = \pm\sqrt{\frac{1}{4}} = \pm\frac{1}{2} \]

Step 4: Use the given interval for \(\gamma\) to find the correct angle.

We are given that \(\gamma \in (\frac{\pi}{2}, \pi)\). This is the second quadrant. In the second quadrant, the cosine function is negative. Therefore, we must choose the negative value:
\[ \cos\gamma = -\frac{1}{2} \]
The angle \(\gamma\) in the second quadrant for which \(\cos\gamma = -1/2\) is \(\gamma = \frac{2\pi}{3}\).


Step 5: Final Answer.

The angle \(\gamma\) is \(\frac{2\pi}{3}\).

This corresponds to option (E).
Quick Tip: The identity \(\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1\) is fundamental for direction cosines and is frequently tested. Always pay attention to any constraints given on the angles (like the quadrant), as this is crucial for choosing the correct sign when taking a square root.

Step 1: Use the given vector sum.

We are given \(\vec{u}+\vec{v}+\vec{w}=\vec{0}\).


Step 2: Take the dot product of the sum with itself.

If the vector sum is the zero vector, its magnitude squared is zero. The magnitude squared of a vector is its dot product with itself.
\[ |\vec{u}+\vec{v}+\vec{w}|^2 = (\vec{u}+\vec{v}+\vec{w}) \cdot (\vec{u}+\vec{v}+\vec{w}) = 0 \]

Step 3: Expand the dot product.
\[ \vec{u}\cdot\vec{u} + \vec{u}\cdot\vec{v} + \vec{u}\cdot\vec{w} + \vec{v}\cdot\vec{u} + \vec{v}\cdot\vec{v} + \vec{v}\cdot\vec{w} + \vec{w}\cdot\vec{u} + \vec{w}\cdot\vec{v} + \vec{w}\cdot\vec{w} = 0 \]
Using \(\vec{a}\cdot\vec{a} = |\vec{a}|^2\) and the commutative property of the dot product (\(\vec{a}\cdot\vec{b} = \vec{b}\cdot\vec{a}\)), we can simplify this:
\[ |\vec{u}|^2 + |\vec{v}|^2 + |\vec{w}|^2 + 2(\vec{u}\cdot\vec{v} + \vec{v}\cdot\vec{w} + \vec{w}\cdot\vec{u}) = 0 \]

Step 4: Substitute the given magnitudes and solve.

We are given \(|\vec{u}|=3, |\vec{v}|=4, |\vec{w}|=5\).
\[ 3^2 + 4^2 + 5^2 + 2(\vec{u}\cdot\vec{v} + \vec{v}\cdot\vec{w} + \vec{w}\cdot\vec{u}) = 0 \] \[ 9 + 16 + 25 + 2(\vec{u}\cdot\vec{v} + \vec{v}\cdot\vec{w} + \vec{w}\cdot\vec{u}) = 0 \] \[ 50 + 2(\vec{u}\cdot\vec{v} + \vec{v}\cdot\vec{w} + \vec{w}\cdot\vec{u}) = 0 \] \[ 2(\vec{u}\cdot\vec{v} + \vec{v}\cdot\vec{w} + \vec{w}\cdot\vec{u}) = -50 \] \[ \vec{u}\cdot\vec{v} + \vec{v}\cdot\vec{w} + \vec{w}\cdot\vec{u} = -\frac{50}{2} = -25 \]

Step 5: Final Answer.

The value of the expression is -25.

This corresponds to option (B).
Quick Tip: This is a standard problem type. Whenever you are given a sum of vectors equal to zero and their magnitudes, and asked to find a combination of their dot products, the key is to square the magnitude of the vector sum: \(|\vec{a}+\vec{b}+\vec{c}|^2=0\). This immediately relates the magnitudes to the dot products.

Step 1: Calculate the sum of vectors \(\vec{a}\) and \(\vec{c}\).
\[ \vec{a} = \hat{i} - \hat{j} \] \[ \vec{c} = \hat{k} - \hat{i} \] \[ \vec{a} + \vec{c} = (\hat{i} - \hat{j}) + (\hat{k} - \hat{i}) = (1-1)\hat{i} - \hat{j} + \hat{k} = 0\hat{i} - \hat{j} + \hat{k} \]

Step 2: Calculate the dot product of \(\vec{b}\) with \((\vec{a}+\vec{c})\).
\[ \vec{b} = \hat{j} - \hat{k} = 0\hat{i} + \hat{j} - \hat{k} \] \[ \vec{a}+\vec{c} = -\hat{j} + \hat{k} = 0\hat{i} - \hat{j} + \hat{k} \]
The dot product is the sum of the products of the corresponding components:
\[ \vec{b} \cdot (\vec{a}+\vec{c}) = (0)(0) + (1)(-1) + (-1)(1) \] \[ = 0 - 1 - 1 = -2 \]

Step 3: Final Answer.

The value of the expression is -2.

This corresponds to option (E).
Quick Tip: You could also use the distributive property of the dot product: \(\vec{b}\cdot(\vec{a}+\vec{c}) = \vec{b}\cdot\vec{a} + \vec{b}\cdot\vec{c}\).
\(\vec{b}\cdot\vec{a} = (\hat{j}-\hat{k})\cdot(\hat{i}-\hat{j}) = -1\)
\(\vec{b}\cdot\vec{c} = (\hat{j}-\hat{k})\cdot(\hat{k}-\hat{i}) = -1\)
Sum = \(-1 + (-1) = -2\). Choose whichever method seems faster to you.

Step 1: Translate the perpendicularity conditions into dot products.

- \(\vec{a} \perp (\vec{b}+\vec{c}) \implies \vec{a}\cdot(\vec{b}+\vec{c}) = 0 \implies \vec{a}\cdot\vec{b} + \vec{a}\cdot\vec{c} = 0\)

- \(\vec{b} \perp (\vec{c}+\vec{a}) \implies \vec{b}\cdot(\vec{c}+\vec{a}) = 0 \implies \vec{b}\cdot\vec{c} + \vec{b}\cdot\vec{a} = 0\)

- \(\vec{c} \perp (\vec{a}+\vec{b}) \implies \vec{c}\cdot(\vec{a}+\vec{b}) = 0 \implies \vec{c}\cdot\vec{a} + \vec{c}\cdot\vec{b} = 0\)


Step 2: Find the sum of the dot products.

Adding all three equations from Step 1:
\((\vec{a}\cdot\vec{b} + \vec{a}\cdot\vec{c}) + (\vec{b}\cdot\vec{c} + \vec{b}\cdot\vec{a}) + (\vec{c}\cdot\vec{a} + \vec{c}\cdot\vec{b}) = 0\)
\[ 2(\vec{a}\cdot\vec{b} + \vec{b}\cdot\vec{c} + \vec{c}\cdot\vec{a}) = 0 \] \[ \vec{a}\cdot\vec{b} + \vec{b}\cdot\vec{c} + \vec{c}\cdot\vec{a} = 0 \]

Step 3: Calculate the required magnitude.

We want to find \(|\vec{a}+\vec{b}+\vec{c}|\). It is easiest to find its square first.
\[ |\vec{a}+\vec{b}+\vec{c}|^2 = (\vec{a}+\vec{b}+\vec{c})\cdot(\vec{a}+\vec{b}+\vec{c}) \] \[ = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(\vec{a}\cdot\vec{b} + \vec{b}\cdot\vec{c} + \vec{c}\cdot\vec{a}) \]

Step 4: Substitute the known values.

We are given \(|\vec{a}|=4, |\vec{b}|=4, |\vec{c}|=2\), and we found the sum of dot products is 0.
\[ |\vec{a}+\vec{b}+\vec{c}|^2 = 4^2 + 4^2 + 2^2 + 2(0) \] \[ |\vec{a}+\vec{b}+\vec{c}|^2 = 16 + 16 + 4 + 0 = 36 \]
Taking the square root:
\[ |\vec{a}+\vec{b}+\vec{c}| = \sqrt{36} = 6 \]

Step 5: Final Answer.

The value of \(|\vec{a}+\vec{b}+\vec{c}|\) is 6.

This corresponds to option (B).
Quick Tip: The identity \(|\vec{a}+\vec{b}+\vec{c}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(\vec{a}\cdot\vec{b} + \vec{b}\cdot\vec{c} + \vec{c}\cdot\vec{a})\) is extremely useful and is the vector equivalent of the algebraic identity \((a+b+c)^2\).

Step 1: Use the condition for perpendicular vectors.

Two vectors are perpendicular if their dot product is zero. \(\vec{a} \cdot \vec{b} = 0\).


Step 2: Calculate the dot product.
\[ \vec{a} \cdot \vec{b} = (\cos\alpha)(\sin\alpha) + (\sin\alpha)(-\cos\alpha) + (\sin\frac{\alpha}{2})(\cos\frac{\alpha}{2}) = 0 \]
The first two terms cancel each other out:
\[ \cos\alpha\sin\alpha - \sin\alpha\cos\alpha + \sin\frac{\alpha}{2}\cos\frac{\alpha}{2} = 0 \] \[ \sin\frac{\alpha}{2}\cos\frac{\alpha}{2} = 0 \]

Step 3: Solve the trigonometric equation.

Use the double-angle identity \(\sin(2\theta) = 2\sin\theta\cos\theta\). Let \(\theta = \frac{\alpha}{2}\).

The equation becomes:
\[ \frac{1}{2} \sin(2 \cdot \frac{\alpha}{2}) = 0 \] \[ \frac{1}{2} \sin\alpha = 0 \] \[ \sin\alpha = 0 \]

Step 4: Find the values of \(\alpha\).

The general solution for \(\sin\alpha = 0\) is \(\alpha = n\pi\), where n is an integer.

Looking at the options, the values given are 0 and \(\pi\).

- If n=0, \(\alpha = 0\).

- If n=1, \(\alpha = \pi\).

Both of these values are listed in option (C).


Step 5: Final Answer.

The values of \(\alpha\) are 0 and \(\pi\).

This corresponds to option (C).
Quick Tip: When calculating the dot product, look for terms that might cancel out. Also, be ready to use trigonometric identities, especially double-angle identities like \(2\sin\theta\cos\theta = \sin(2\theta)\), to simplify the resulting equation.

Step 1: Find the center of the circle.

The equation of a circle is given by \(x^2+y^2+2gx+2fy+c=0\), where the center is \((-g, -f)\).

Comparing this with the given equation \(x^2+y^2-4x-6y+11=0\):

- \(2g = -4 \implies g = -2\)

- \(2f = -6 \implies f = -3\)

The center of the circle is \((-g, -f) = (2, 3)\).


Step 2: Use the midpoint formula.

The center of the circle is the midpoint of any diameter. Let the given end of the diameter be \(A = (3,4)\) and the other end be \(B = (x,y)\). The center is \(C = (2,3)\).

Using the midpoint formula:
\[ C = \left(\frac{x_A+x_B}{2}, \frac{y_A+y_B}{2}\right) \] \[ (2,3) = \left(\frac{3+x}{2}, \frac{4+y}{2}\right) \]

Step 3: Solve for the coordinates of the other end.

Equate the x and y coordinates separately:

- For the x-coordinate: \(2 = \frac{3+x}{2} \implies 4 = 3+x \implies x=1\)

- For the y-coordinate: \(3 = \frac{4+y}{2} \implies 6 = 4+y \implies y=2\)

The coordinates of the other end of the diameter are (1,2).


Step 4: Final Answer.

The coordinate of the other end of the diameter is (1,2).

This corresponds to option (C).
Quick Tip: To quickly find the center of a circle from the equation \(x^2+y^2+Dx+Ey+F=0\), the center is simply \( (-D/2, -E/2) \). The center is the midpoint of the diameter, which is the key property used to solve this problem.

Step 1: Determine the orientation and vertex of the parabola.

- The focus is at (0, -3) on the y-axis.

- The directrix is the horizontal line y = 3.

Since the focus is below the directrix, the parabola opens downwards. The axis of symmetry is the y-axis (x=0).

The vertex of the parabola is the midpoint between the focus and the directrix.

Vertex = \(\left(\frac{0+0}{2}, \frac{-3+3}{2}\right) = (0,0)\).


Step 2: Find the value of 'a'.

'a' is the distance from the vertex to the focus.
\(a = distance between (0,0) and (0,-3) = 3\).


Step 3: Use the standard equation for a downward-opening parabola.

The standard equation for a parabola with vertex at the origin that opens downwards is:
\[ x^2 = -4ay \]
Substitute the value \(a=3\):
\[ x^2 = -4(3)y \] \[ x^2 = -12y \]

Step 4: Final Answer.

The equation of the parabola is \(x^2 = -12y\).

This corresponds to option (B).
Quick Tip: Quickly determine the parabola's orientation:
- Focus \((0, a)\), Directrix \(y = -a\) \(\implies\) Opens up, \(x^2 = 4ay\).
- Focus \((0, -a)\), Directrix \(y = a\) \(\implies\) Opens down, \(x^2 = -4ay\).
- Focus \((a, 0)\), Directrix \(x = -a\) \(\implies\) Opens right, \(y^2 = 4ax\).
- Focus \((-a, 0)\), Directrix \(x = a\) \(\implies\) Opens left, \(y^2 = -4ax\).

Step 1: Extract information from the given data.

- The foci are at \((\pm c, 0)\), so we have a horizontal ellipse with \(c=2\).

- The eccentricity is \(e = \frac{1}{3}\).


Step 2: Find the semi-major axis 'a'.

The relationship between eccentricity, c, and a is \(e = \frac{c}{a}\).
\[ \frac{1}{3} = \frac{2}{a} \implies a = 6 \]

Step 3: Find the semi-minor axis 'b'.

For an ellipse, the relationship between a, b, and c is \(c^2 = a^2 - b^2\).
\[ b^2 = a^2 - c^2 \] \[ b^2 = 6^2 - 2^2 = 36 - 4 = 32 \] \[ b = \sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2} \]

Step 4: Calculate the length of the minor axis.

The length of the minor axis is \(2b\).

Length = \(2 \times 4\sqrt{2} = 8\sqrt{2}\).


Step 5: Final Answer.

The length of the minor axis is \(8\sqrt{2}\).

This corresponds to option (E).
Quick Tip: For ellipses, remember the key relationships:
- Eccentricity: \(e = c/a\)
- Focal relationship: \(c^2 = a^2 - b^2\)
- Length of major axis = \(2a\)
- Length of minor axis = \(2b\)
Always start by identifying 'a', 'b', and 'c' from the given information.

Step 1: Find the slope of the given line.

The equation of the given line is \(x+y+1=0\). We can write this in slope-intercept form (\(y=mx+c\)):
\[ y = -x - 1 \]
The slope of this line, \(m_1\), is -1.


Step 2: Find the slope of the perpendicular line.

If two lines are perpendicular, the product of their slopes is -1. Let the slope of the required line be \(m_2\).
\[ m_1 \times m_2 = -1 \] \[ (-1) \times m_2 = -1 \implies m_2 = 1 \]

Step 3: Use the point-slope form to find the equation of the required line.

The required line has a slope of 1 and passes through the point (1,2). The point-slope form is \(y - y_1 = m(x - x_1)\).
\[ y - 2 = 1(x - 1) \] \[ y - 2 = x - 1 \] \[ y - x - 1 = 0 \]

Step 4: Final Answer.

The equation of the line is \(y-x-1=0\).

This corresponds to option (C).
Quick Tip: A quick way to find the equation of a line perpendicular to \(Ax+By+C=0\) is to write \(Bx-Ay+K=0\). For the line \(x+y+1=0\), the perpendicular line will be \(x-y+K=0\). Substitute the point (1,2) to find K: \(1-2+K=0 \implies K=1\). The equation is \(x-y+1=0\), which is equivalent to \(y-x-1=0\).

Step 1: Use the condition that the first line passes through (13, 32).

Substitute \(x=13\) and \(y=32\) into the equation \(\frac{x}{5} + \frac{y}{b} = 1\).
\[ \frac{13}{5} + \frac{32}{b} = 1 \] \[ \frac{32}{b} = 1 - \frac{13}{5} = \frac{5-13}{5} = -\frac{8}{5} \] \[ b = \frac{32 \times 5}{-8} = -4 \times 5 = -20 \]

Step 2: Use the condition that the lines are parallel.

Two lines are parallel if their slopes are equal. We find the slopes of both lines by rewriting them in the form \(y=mx+c\).

- First line: \(\frac{y}{b} = 1 - \frac{x}{5} \implies y = b(1 - \frac{x}{5}) = -\frac{b}{5}x + b\). The slope is \(m_1 = -\frac{b}{5}\).

- Second line: \(\frac{y}{3} = 1 - \frac{x}{c} \implies y = 3(1 - \frac{x}{c}) = -\frac{3}{c}x + 3\). The slope is \(m_2 = -\frac{3}{c}\).

Since the lines are parallel, \(m_1 = m_2\).
\[ -\frac{b}{5} = -\frac{3}{c} \implies \frac{b}{5} = \frac{3}{c} \]

Step 3: Solve for c.

Substitute the value of \(b = -20\) into the slope equation:
\[ \frac{-20}{5} = \frac{3}{c} \] \[ -4 = \frac{3}{c} \implies c = -\frac{3}{4} \]

Step 4: Final Answer.

The values are \(b=-20\) and \(c=-\frac{3}{4}\).

This corresponds to option (A).
Quick Tip: The slope of a line in intercept form \(\frac{x}{p} + \frac{y}{q} = 1\) can be quickly found as \(m = -\frac{q}{p}\). Using this shortcut, the slopes are \(m_1 = -b/5\) and \(m_2 = -3/c\). Equating them gives \(b/5 = 3/c\).

Step 1: Use the reflection principle.

Let the initial point be A(1,2) and the final point be B(5,6). The reflection occurs at point P on the x-axis. Let the coordinates of P be \((x, 0)\). The principle of reflection states that the path from A to P to B is equivalent to the straight-line path from the reflection of A across the x-axis, let's call it A', to B.

The reflection of A(1,2) across the x-axis is A'(1,-2).


Step 2: Find the equation of the line passing through A' and B.

The points are A'(1,-2) and B(5,6).

The slope of the line A'B is \(m = \frac{6 - (-2)}{5 - 1} = \frac{8}{4} = 2\).

Using the point-slope form with point A'(1,-2):
\[ y - (-2) = 2(x - 1) \] \[ y + 2 = 2x - 2 \] \[ y = 2x - 4 \]

Step 3: Find the point P.

The point P is the intersection of this line with the x-axis. On the x-axis, the y-coordinate is 0.

Substitute \(y=0\) into the line equation:
\[ 0 = 2x - 4 \] \[ 2x = 4 \] \[ x = 2 \]

Final Answer.

The abscissa (x-coordinate) of the point P is 2.

This corresponds to option (C).
Quick Tip: The reflection principle is very powerful. To find the path of reflection from a line, reflect one of the points across the line and draw a straight line between the reflected point and the other original point. The intersection with the reflection line is the point of reflection.

Step 1: Use the condition that the point lies on the line.

If a point lies on a line, its coordinates must satisfy the equation of the line. We substitute the coordinates of the point (-1, 3, 2) for x, y, and z in the line's equation.
\[ \frac{-1-a}{1} = \frac{3-b}{2} = \frac{2-3}{-1} \]

Step 2: Solve the equations.

First, evaluate the constant part of the equation:
\[ \frac{2-3}{-1} = \frac{-1}{-1} = 1 \]
Now, we can set each of the other parts equal to 1.

- For 'a':

\[ \frac{-1-a}{1} = 1 \implies -1-a = 1 \implies -a = 2 \implies a = -2 \]
- For 'b':

\[ \frac{3-b}{2} = 1 \implies 3-b = 2 \implies -b = -1 \implies b = 1 \]

Step 3: Final Answer.

The values are \(a=-2\) and \(b=1\).

This corresponds to option (D).
Quick Tip: The symmetric form of a line equation \(\frac{x-x_1}{l} = \frac{y-y_1}{m} = \frac{z-z_1}{n}\) tells you that \((x_1, y_1, z_1)\) is a point on the line and \(\langle l, m, n \rangle\) is the direction vector. The question here is essentially asking for the point \((a,b,3)\) that defines the line, given another point on it.

Step 1: Identify the direction vectors of the two lines.

The direction vector of a line in the form \(\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}\) is \(\vec{d} = \langle a, b, c \rangle\).

- For the first line, the direction vector is \(\vec{d_1} = \langle -2, 1, 3 \rangle\).

- For the second line, the direction vector is \(\vec{d_2} = \langle \mu, \mu+1, \mu+2 \rangle\).


Step 2: Use the condition for perpendicular lines.

Two lines are perpendicular if their direction vectors are orthogonal, which means their dot product is zero.
\[ \vec{d_1} \cdot \vec{d_2} = 0 \]

Step 3: Calculate the dot product and solve for \(\mu\).
\[ (-2)(\mu) + (1)(\mu+1) + (3)(\mu+2) = 0 \] \[ -2\mu + \mu + 1 + 3\mu + 6 = 0 \]
Combine the terms with \(\mu\) and the constant terms:
\[ (-2+1+3)\mu + (1+6) = 0 \] \[ 2\mu + 7 = 0 \] \[ 2\mu = -7 \] \[ \mu = -\frac{7}{2} \]

Step 4: Final Answer.

The value of \(\mu\) is \(-\frac{7}{2}\).

This corresponds to option (D).
Quick Tip: For lines in 3D:
- \textbf{Perpendicular Condition:} Dot product of direction vectors is zero (\(a_1a_2 + b_1b_2 + c_1c_2 = 0\)).
- \textbf{Parallel Condition:} Direction vectors are proportional (\(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\)).
Memorize these two conditions as they are fundamental to problems involving the orientation of lines in space.

Step 1: Identify the direction vectors of the two lines.

- The direction vector of the first line is \(\vec{d_1} = \langle 2, 3, -1 \rangle\).

- The direction vector of the second line is \(\vec{d_2} = \langle a, b, -1 \rangle\).


Step 2: Use the condition for parallel lines.

Two lines are parallel if their direction vectors are proportional. This means \(\vec{d_1} = k \vec{d_2}\) for some scalar k, or equivalently, the ratio of their corresponding components is constant.
\[ \frac{2}{a} = \frac{3}{b} = \frac{-1}{-1} \]
From the last part of the equality, we find the constant of proportionality:
\[ \frac{-1}{-1} = 1 \]

Step 3: Solve for a and b.

- Equating the first and last parts: \(\frac{2}{a} = 1 \implies a = 2\).

- Equating the second and last parts: \(\frac{3}{b} = 1 \implies b = 3\).


Step 4: Calculate \(a^2 + b^2\).
\[ a^2 + b^2 = 2^2 + 3^2 = 4 + 9 = 13 \]

Step 5: Final Answer.

The value of \(a^2 + b^2\) is 13.

This corresponds to option (B).
Quick Tip: For lines in 3D:
- \textbf{Parallel Condition:} Direction vectors are proportional (\(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\)).
- \textbf{Perpendicular Condition:} Dot product of direction vectors is zero (\(a_1a_2 + b_1b_2 + c_1c_2 = 0\)).

Step 1: Identify the direction vectors of the two lines.

- The direction vector of the first line is \(\vec{d_1} = \langle 1, 1, 1 \rangle\).

- The direction vector of the second line is \(\vec{d_2} = \langle 0, 1, -1 \rangle\).


Step 2: Use the dot product formula for the angle between two vectors.

The angle \(\theta\) between two vectors is given by \(\cos\theta = \frac{\vec{d_1} \cdot \vec{d_2}}{|\vec{d_1}| |\vec{d_2}|}\).


Step 3: Calculate the dot product and magnitudes.

- \(\vec{d_1} \cdot \vec{d_2} = (1)(0) + (1)(1) + (1)(-1) = 0 + 1 - 1 = 0\).

- Since the dot product is zero, the vectors are orthogonal (perpendicular). The angle between them is \(\frac{\pi}{2}\) radians or 90 degrees. We don't need to calculate the magnitudes.


Step 4: Final Answer.

The angle between the lines is \(\frac{\pi}{2}\).

This corresponds to option (A).
Quick Tip: When finding the angle between two vectors (or lines), always calculate the dot product first. If it's zero, you immediately know the vectors are perpendicular and the angle is \(\pi/2\), saving you the effort of calculating the magnitudes.

Step 1: Find the total number of ways to choose 3 numbers.

The total number of ways to choose 3 distinct numbers from the first 50 natural numbers is \(\binom{50}{3}\).
\[ \binom{50}{3} = \frac{50 \times 49 \times 48}{3 \times 2 \times 1} = 50 \times 49 \times 8 = 19600 \]

Step 2: Find the number of favorable outcomes.

We want to choose 3 numbers that are divisible by both 2 and 3. A number divisible by both 2 and 3 is divisible by their LCM, which is 6.

Let's find how many numbers from 1 to 50 are divisible by 6.
\[ \lfloor \frac{50}{6} \rfloor = 8 \]
The numbers are 6, 12, 18, 24, 30, 36, 42, 48. There are 8 such numbers.

The number of ways to choose 3 numbers from these 8 numbers is \(\binom{8}{3}\).
\[ \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]

Step 3: Calculate the probability.
\[ P(event) = \frac{Number of favorable outcomes}{Total number of outcomes} = \frac{56}{19600} \]
Simplify the fraction:
\[ \frac{56}{19600} = \frac{28}{9800} = \frac{14}{4900} = \frac{7}{2450} = \frac{1}{350} \]

Step 4: Final Answer.

The probability is \(\frac{1}{350}\).

This corresponds to option (E).
Quick Tip: In probability problems involving "divisible by a and b", it's equivalent to being "divisible by LCM(a, b)". This simplifies the counting of favorable outcomes.

Step 1: Use the property of exhaustive events.

If events are mutually exclusive and exhaustive, the sum of their probabilities must be equal to 1.
\[ \frac{1+3p}{4} + \frac{1-p}{3} + \frac{1-3p}{2} = 1 \]

Step 2: Solve the equation for p.

Find a common denominator, which is 12.
\[ \frac{3(1+3p)}{12} + \frac{4(1-p)}{12} + \frac{6(1-3p)}{12} = 1 \] \[ 3(1+3p) + 4(1-p) + 6(1-3p) = 12 \] \[ 3 + 9p + 4 - 4p + 6 - 18p = 12 \]
Combine the terms:
\[ (3+4+6) + (9p - 4p - 18p) = 12 \] \[ 13 - 13p = 12 \] \[ 1 = 13p \implies p = \frac{1}{13} \]

Step 3: Check the validity of the probabilities.

Since each term is a probability, its value must be between 0 and 1. Let's check for \(p = 1/13\).

- \(P_1 = \frac{1+3(1/13)}{4} = \frac{16/13}{4} = \frac{4}{13}\) (Valid)

- \(P_2 = \frac{1-(1/13)}{3} = \frac{12/13}{3} = \frac{4}{13}\) (Valid)

- \(P_3 = \frac{1-3(1/13)}{2} = \frac{10/13}{2} = \frac{5}{13}\) (Valid)

The values are valid.


Step 4: Final Answer.

The value of p is \(\frac{1}{13}\).

This corresponds to option (D).
Quick Tip: The key properties for this type of problem are:
1. For any event A, \(0 \le P(A) \le 1\).
2. For mutually exclusive and exhaustive events \(A_1, A_2, \dots, A_n\), the sum of their probabilities is \(\sum P(A_i) = 1\).
Always use the sum property to solve for the variable, and then (if time permits) check that each individual probability is valid.

Step 1: Calculate the mean of the data.

The numbers are 3, 10, 10, 4, 7, 10, 5. There are n=7 numbers.

Sum = \(3 + 10 + 10 + 4 + 7 + 10 + 5 = 49\).

Mean (\(\bar{x}\)) = \(\frac{Sum}{n} = \frac{49}{7} = 7\).


Step 2: Calculate the absolute deviations from the mean.

We find the absolute difference between each data point and the mean (\(|x_i - \bar{x}|\)).

- \(|3 - 7| = 4\)

- \(|10 - 7| = 3\)

- \(|10 - 7| = 3\)

- \(|4 - 7| = 3\)

- \(|7 - 7| = 0\)

- \(|10 - 7| = 3\)

- \(|5 - 7| = 2\)


Step 3: Calculate the mean deviation.

The mean deviation is the average of these absolute deviations.

Sum of deviations = \(4 + 3 + 3 + 3 + 0 + 3 + 2 = 18\).

Mean Deviation = \(\frac{\sum |x_i - \bar{x}|}{n} = \frac{18}{7}\).


Step 4: Convert the fraction to a decimal.
\[ \frac{18}{7} \approx 2.5714... \]
This is approximately 2.57.


Step 5: Final Answer.

The mean deviation is approximately 2.57.

This corresponds to option (C).
Quick Tip: The steps for calculating mean deviation are:
1. Find the mean of the data set.
2. Find the absolute difference of each data point from the mean.
3. Find the average of these differences.
Be careful not to mix this up with standard deviation, which involves squaring the differences.

Step 1: Find the derivative of g(x).

The function is \(g(x) = -(25-x^2)^{1/2}\). We use the chain rule to differentiate.

Let \(u = 25-x^2\), so \(g(x) = -u^{1/2}\).
\[ \frac{dg}{dx} = \frac{dg}{du} \cdot \frac{du}{dx} \] \[ g'(x) = - \left( \frac{1}{2}u^{-1/2} \right) \cdot (-2x) \] \[ g'(x) = - \frac{1}{2\sqrt{u}} \cdot (-2x) = \frac{x}{\sqrt{u}} \]
Substitute back \(u = 25-x^2\):
\[ g'(x) = \frac{x}{\sqrt{25-x^2}} \]

Step 2: Evaluate the derivative at x=1.

Substitute \(x=1\) into the expression for \(g'(x)\).
\[ g'(1) = \frac{1}{\sqrt{25-1^2}} = \frac{1}{\sqrt{24}} \]

Step 3: Final Answer.

The value of \(g'(1)\) is \(\frac{1}{\sqrt{24}}\).

This corresponds to option (D).
Quick Tip: The derivative of \(\sqrt{f(x)}\) is a common pattern: \(\frac{d}{dx}\sqrt{f(x)} = \frac{f'(x)}{2\sqrt{f(x)}}\). Memorizing this can speed up chain rule calculations involving square roots. In this problem, \(f(x) = 25-x^2\), so the derivative is \(\frac{-2x}{2\sqrt{25-x^2}}\). Don't forget the minus sign from the original function!

Step 1: Identify the indeterminate form.

As \(x \to 0\), the numerator becomes \(\sin 0 + \sin 0 = 0\) and the denominator becomes \(\sin 0 + \sin 0 = 0\). This is the indeterminate form \(\frac{0}{0}\).


Step 2: Apply the standard limit \(\lim_{u \to 0} \frac{\sin u}{u} = 1\).

Divide both the numerator and the denominator by x:
\[ \lim_{x \to 0} \frac{\frac{\sin 2x + \sin 5x}{x}}{\frac{\sin 4x + \sin 6x}{x}} = \frac{\lim_{x \to 0} (\frac{\sin 2x}{x} + \frac{\sin 5x}{x})}{\lim_{x \to 0} (\frac{\sin 4x}{x} + \frac{\sin 6x}{x})} \]
Now, use the property \(\lim_{x \to 0} \frac{\sin ax}{x} = a\).
\[ = \frac{2 + 5}{4 + 6} = \frac{7}{10} \]

Alternative Step 2: L'Hôpital's Rule.

Since it is a \(\frac{0}{0}\) form, we can apply L'Hôpital's Rule by differentiating the numerator and the denominator separately.
\[ \lim_{x \to 0} \frac{\frac{d}{dx}(\sin 2x + \sin 5x)}{\frac{d}{dx}(\sin 4x + \sin 6x)} = \lim_{x \to 0} \frac{2\cos 2x + 5\cos 5x}{4\cos 4x + 6\cos 6x} \]
Now, substitute \(x=0\). Since \(\cos 0 = 1\):
\[ = \frac{2(1) + 5(1)}{4(1) + 6(1)} = \frac{2+5}{4+6} = \frac{7}{10} \]

Step 3: Final Answer.

The value of the limit is \(\frac{7}{10}\).

This corresponds to option (D).
Quick Tip: For limits involving sums of sine functions of the form \(\lim_{x \to 0} \frac{\sin(ax) + \sin(bx)}{\sin(cx) + \sin(dx)}\), the answer is simply the ratio of the sums of the coefficients: \(\frac{a+b}{c+d}\). This is a quick shortcut based on the standard limit or L'Hôpital's Rule.

Step 1: Recall the condition for continuity.

For a function to be continuous at a point \(x=a\), the left-hand limit (LHL), the right-hand limit (RHL), and the function's value at that point must all be equal.
\[ \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a) \]

Step 2: Apply the condition at \(x = \frac{\pi}{2}\).

- Left-Hand Limit (LHL) and Function Value: For \(x \le \frac{\pi}{2}\), \(f(x) = mx+1\).

\[ \lim_{x \to (\pi/2)^-} f(x) = f(\frac{\pi}{2}) = m(\frac{\pi}{2}) + 1 \]
- Right-Hand Limit (RHL): For \(x > \frac{\pi}{2}\), \(f(x) = \sin x + n\).

\[ \lim_{x \to (\pi/2)^+} f(x) = \sin(\frac{\pi}{2}) + n = 1 + n \]

Step 3: Equate the limits to find the relationship between m and n.

For continuity, LHL = RHL.
\[ m(\frac{\pi}{2}) + 1 = 1 + n \]
Subtract 1 from both sides:
\[ \frac{m\pi}{2} = n \]

Step 4: Final Answer.

The relationship between m and n is \(n = \frac{m\pi}{2}\).

This corresponds to option (C).
Quick Tip: For piecewise functions, the test for continuity at a boundary point always involves setting the left-hand limit equal to the right-hand limit. Simply substitute the boundary point into the expressions on both sides of the boundary and set them equal.

Step 1: Understand the function.

The function \(f(x) = x - [x]\) is the fractional part function. This function is known to be discontinuous at every integer value.


Step 2: Identify the integers within the given domain.

The domain is the open interval \((-1, 2)\). The integers within this interval are 0 and 1.


Step 3: Analyze continuity at the integer points.

- At \(x=0\):

- LHL: \(\lim_{x \to 0^-} (x - [x]) = 0 - (-1) = 1\).

- RHL: \(\lim_{x \to 0^+} (x - [x]) = 0 - (0) = 0\).

Since LHL \(\neq\) RHL, the function is discontinuous at \(x=0\).

- At \(x=1\):

- LHL: \(\lim_{x \to 1^-} (x - [x]) = 1 - (0) = 1\).

- RHL: \(\lim_{x \to 1^+} (x - [x]) = 1 - (1) = 0\).

Since LHL \(\neq\) RHL, the function is discontinuous at \(x=1\).

The function is continuous at all other non-integer points in the interval.


Step 4: Final Answer.

The function is not continuous at the integer points x=0 and x=1. There are 2 such points in the given interval.

This corresponds to option (B).
Quick Tip: The greatest integer function \([x]\) is discontinuous at all integers. Consequently, the fractional part function \(x - [x]\) is also discontinuous at all integers. To solve this type of problem, simply count the number of integers within the specified domain.

Step 1: Simplify the absolute value function in the given interval.

The function is \(f(x) = |\cos x - \sin x|\). We need to determine the sign of the expression inside the absolute value in the interval \(x \in (\frac{\pi}{4}, \frac{\pi}{2})\).

In this interval (between 45\(^\circ\) and 90\(^\circ\)), the value of \(\sin x\) is greater than the value of \(\cos x\).

Therefore, \(\cos x - \sin x\) is negative.

So, for this interval, we can write the function as:
\[ f(x) = -(\cos x - \sin x) = \sin x - \cos x \]

Step 2: Find the derivative of the simplified function.
\[ f'(x) = \frac{d}{dx}(\sin x - \cos x) = \cos x - (-\sin x) = \cos x + \sin x \]

Step 3: Evaluate the derivative at \(x = \frac{\pi}{3}\).
\[ f'(\frac{\pi}{3}) = \cos(\frac{\pi}{3}) + \sin(\frac{\pi}{3}) \]
We know the values: \(\cos(\frac{\pi}{3}) = \frac{1}{2}\) and \(\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}\).
\[ f'(\frac{\pi}{3}) = \frac{1}{2} + \frac{\sqrt{3}}{2} = \frac{1+\sqrt{3}}{2} \]

Step 4: Final Answer.

The value of \(f'(\frac{\pi}{3})\) is \(\frac{\sqrt{3}+1}{2}\).

This corresponds to option (C).
Quick Tip: When differentiating a function with an absolute value, the first step is always to remove the absolute value sign by considering the sign of the expression inside it over the relevant domain. This often simplifies the function into a standard form that is easy to differentiate.

Step 1: Simplify the function.

We can simplify \(f(x) = \sin^{-1}(\cos x)\) using the identity \(\cos x = \sin(\frac{\pi}{2} - x)\).
\[ f(x) = \sin^{-1}(\sin(\frac{\pi}{2} - x)) \]
For the principal value range of \(\sin^{-1}\), we must ensure that the argument of sin is in \([-\pi/2, \pi/2]\). For \(x = \pi/4\), the argument is \(\pi/2 - \pi/4 = \pi/4\), which is in the correct range. So, in the neighborhood of \(x=\pi/4\), we can simplify the function to:
\[ y = f(x) = \frac{\pi}{2} - x \]

Step 2: Find the first and second derivatives.
\[ \frac{dy}{dx} = \frac{d}{dx}(\frac{\pi}{2} - x) = -1 \] \[ \frac{d^2y}{dx^2} = \frac{d}{dx}(-1) = 0 \]

Step 3: Evaluate the second derivative at \(x=\frac{\pi}{4}\).

The second derivative is a constant, 0. Its value at \(x=\frac{\pi}{4}\) is 0.


Step 4: Final Answer.

The value of the second derivative at \(x=\frac{\pi}{4}\) is 0.

This corresponds to option (E).
Quick Tip: Before differentiating compositions of trigonometric and inverse trigonometric functions, always try to simplify them using identities like \(\sin(\pi/2 - x) = \cos x\). This can often turn a complicated chain rule problem into a very simple differentiation. Be mindful of the principal value ranges.

Step 1: Simplify the expression inside the \(\tan^{-1}\) function.

Divide both the numerator and the denominator by \(\cos x\):
\[ \frac{\cos x - \sin x}{\cos x + \sin x} = \frac{\frac{\cos x}{\cos x} - \frac{\sin x}{\cos x}}{\frac{\cos x}{\cos x} + \frac{\sin x}{\cos x}} = \frac{1 - \tan x}{1 + \tan x} \]

Step 2: Use a tangent identity.

Recall the tangent subtraction formula: \(\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}\).

We can write \(1 = \tan(\frac{\pi}{4})\). So the expression becomes:
\[ \frac{\tan(\frac{\pi}{4}) - \tan x}{1 + \tan(\frac{\pi}{4})\tan x} = \tan(\frac{\pi}{4} - x) \]

Step 3: Simplify the function y.
\[ y = \tan^{-1}\left[\tan(\frac{\pi}{4} - x)\right] \]
For the given domain \(-\frac{\pi}{2} < x < \frac{\pi}{2}\), the argument \(\frac{\pi}{4} - x\) lies within the principal value range of \(\tan^{-1}\). So we can simplify:
\[ y = \frac{\pi}{4} - x \]

Step 4: Differentiate with respect to x.
\[ \frac{dy}{dx} = \frac{d}{dx}(\frac{\pi}{4} - x) = -1 \]

Step 5: Final Answer.

The value of \(\frac{dy}{dx}\) is -1.

This corresponds to option (D).
Quick Tip: Expressions of the form \(\frac{\cos x \pm \sin x}{\cos x \mp \sin x}\) can almost always be simplified into \(\tan(\frac{\pi}{4} \pm x)\) by dividing by \(\cos x\). Recognizing this pattern is a key shortcut for problems involving inverse trigonometric functions.

Step 1: Find the derivative using the quotient rule.

The quotient rule is \(\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}\).

Here, \(u = x^2\) and \(v = x-1\). So, \(u' = 2x\) and \(v' = 1\).
\[ \frac{dy}{dx} = \frac{(2x)(x-1) - (x^2)(1)}{(x-1)^2} \] \[ \frac{dy}{dx} = \frac{2x^2 - 2x - x^2}{(x-1)^2} = \frac{x^2 - 2x}{(x-1)^2} \]

Step 2: Evaluate the derivative at \(x=-1\).

Substitute \(x=-1\) into the derivative expression:
\[ \frac{dy}{dx}\bigg|_{x=-1} = \frac{(-1)^2 - 2(-1)}{(-1-1)^2} = \frac{1 + 2}{(-2)^2} = \frac{3}{4} \]

Step 3: Final Answer.

The value of the derivative at \(x=-1\) is \(\frac{3}{4}\).

This corresponds to option (E).
Quick Tip: When applying the quotient rule, be careful with the order of terms in the numerator (it's \(u'v - uv'\), not the other way around) and the signs, especially when substituting negative values.

Step 1: Find the derivative of the function.

A function is decreasing where its first derivative is negative (\(f'(x) < 0\)).
\[ f'(x) = \frac{d}{dx}(2x^3 + 9x^2 + 12x - 1) = 6x^2 + 18x + 12 \]

Step 2: Find the intervals where the derivative is negative.

We need to solve the inequality \(6x^2 + 18x + 12 < 0\).

First, find the roots of the corresponding equation \(6x^2 + 18x + 12 = 0\).

Divide by 6: \(x^2 + 3x + 2 = 0\).

Factor the quadratic: \((x+2)(x+1) = 0\).

The roots are \(x=-2\) and \(x=-1\).


Step 3: Determine the sign of the derivative in the intervals defined by the roots.

The roots -2 and -1 divide the number line into three intervals: \((-\infty, -2)\), \((-2, -1)\), and \((-1, \infty)\). We can test a point in each interval to find the sign of \(f'(x) = 6(x+2)(x+1)\).

- For \(x < -2\) (e.g., x=-3): \(f'(-3) = 6(-1)(-2) = 12 > 0\) (increasing).

- For \(-2 < x < -1\) (e.g., x=-1.5): \(f'(-1.5) = 6(0.5)(-0.5) = -1.5 < 0\) (decreasing).

- For \(x > -1\) (e.g., x=0): \(f'(0) = 6(2)(1) = 12 > 0\) (increasing).

The function is decreasing in the interval \((-2, -1)\).


Step 4: Final Answer.

The function is decreasing in the interval \((-2, -1)\).

This corresponds to option (B).
Quick Tip: To find intervals of increase/decrease:
1. Find the derivative \(f'(x)\).
2. Find the critical points by setting \(f'(x)=0\).
3. Use the critical points to divide the number line into intervals.
4. Test a point in each interval to determine the sign of \(f'(x)\). \(f'>0\) means increasing, \(f'<0\) means decreasing.

Step 1: Analyze the function.

The function is \(y = 12 - |x-12|\). To maximize y, we need to minimize the term that is being subtracted, which is \(|x-12|\).


Step 2: Minimize the absolute value term.

The absolute value function \(|u|\) is always non-negative (\(\ge 0\)). Its minimum value is 0.

The term \(|x-12|\) has a minimum value of 0, which occurs when \(x-12=0\), i.e., at \(x=12\).

At \(x=12\), the value of the function would be \(y = 12 - 0 = 12\).


Step 3: Consider the given range for x.

The given range is \(-11 \le x \le 11\). The point \(x=12\) where the absolute maximum occurs is outside this range.

Since the maximum of the function is at \(x=12\), and the function \(|x-12|\) increases as we move away from \(x=12\), the value of y will decrease as we move away from \(x=12\).

Therefore, within the interval \([-11, 11]\), the maximum value must occur at the endpoint that is closest to \(x=12\). That endpoint is \(x=11\).


Step 4: Calculate the value at the endpoint.

Evaluate the function at \(x=11\):
\[ y = 12 - |11-12| = 12 - |-1| = 12 - 1 = 11 \]

Step 5: Final Answer.

The maximum value of the function in the given range is 11.

This corresponds to option (B).
Quick Tip: To find the maximum/minimum of a function on a closed interval, you must check the function's value at all critical points within the interval and also at the endpoints of the interval. The largest of these values is the absolute maximum.

Step 1: Identify the indeterminate form.

Substituting \(x=10\) gives \(\frac{10-10}{\sqrt{16}-4} = \frac{0}{0}\), which is an indeterminate form.


Step 2: Rationalize the denominator.

Multiply the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{x+6}+4\).
\[ \lim_{x \to 10} \frac{(x-10)(\sqrt{x+6}+4)}{(\sqrt{x+6}-4)(\sqrt{x+6}+4)} \] \[ = \lim_{x \to 10} \frac{(x-10)(\sqrt{x+6}+4)}{(x+6) - 16} \] \[ = \lim_{x \to 10} \frac{(x-10)(\sqrt{x+6}+4)}{x-10} \]
Cancel the \((x-10)\) term:
\[ = \lim_{x \to 10} (\sqrt{x+6}+4) \]

Step 3: Evaluate the limit.

Substitute \(x=10\) into the simplified expression:
\[ \sqrt{10+6}+4 = \sqrt{16}+4 = 4+4 = 8 \]

Step 4: Final Answer.

The value of the limit is 8.

This corresponds to option (B).
Quick Tip: For limits of the form \(\frac{0}{0}\) involving square roots, multiplying by the conjugate is the standard technique. Alternatively, L'Hôpital's Rule can be used, but differentiation can sometimes be more complex than the algebraic method.

Step 1: Manipulate the integrand.

Rewrite \(e^{-x}\) as \(\frac{1}{e^x}\) and simplify the expression.
\[ \frac{1}{1+e^{-x}} = \frac{1}{1+\frac{1}{e^x}} = \frac{1}{\frac{e^x+1}{e^x}} = \frac{e^x}{e^x+1} \]
So the integral becomes:
\[ \int \frac{e^x}{e^x+1} dx \]

Step 2: Use substitution to solve the integral.

Let \(u = e^x+1\). Then the derivative is \(du = e^x dx\).

The integral transforms perfectly into:
\[ \int \frac{1}{u} du \]
The integral of \(\frac{1}{u}\) is \(\ln|u| + C\).


Step 3: Substitute back for x.
\[ \ln|e^x+1| + C \]
Since \(e^x\) is always positive, \(e^x+1\) is also always positive, so we can drop the absolute value bars. The options use 'log' which usually means natural logarithm in this context.
\[ \log(1+e^x) + C \]

Step 4: Final Answer.

The integral is \(\log|1+e^x| + C\).

This corresponds to option (B).
Quick Tip: When you see an integral with \(e^x\) and \(e^{-x}\), a useful trick is often to multiply the numerator and denominator by \(e^x\). This converts all terms to powers of \(e^x\), which is often easier to handle with substitution.

Step 1: Use Integration by Parts.

The formula for integration by parts is \(\int u dv = uv - \int v du\).

We need to choose u and dv. Using the LIATE rule (Logarithmic, Inverse, Algebraic, Trigonometric, Exponential), we choose the algebraic part as u.

- Let \(u = x\) \(\implies\) \(du = dx\)

- Let \(dv = \cos x dx\) \(\implies\) \(v = \int \cos x dx = \sin x\)


Step 2: Apply the formula.
\[ \int x \cos x dx = (x)(\sin x) - \int (\sin x)(dx) \] \[ = x \sin x - (-\cos x) + C \] \[ = x \sin x + \cos x + C \]

Step 3: Final Answer.

The integral is \(x \sin x + \cos x + C\).

This corresponds to option (D).
Quick Tip: The LIATE rule is a helpful mnemonic for choosing 'u' in integration by parts. It prioritizes which function type to choose as 'u' to make the new integral \(\int v du\) simpler than the original.

Note: The question asks for an indefinite integral but the options provided are definite values, which is a contradiction. The question is flawed and was likely cancelled. We will solve the indefinite integral.


Step 1: Choose the method of integration.

The integrand is \(xe^{x^2}\). Notice that the derivative of the exponent \(x^2\) is \(2x\), and we have a factor of \(x\) in the integrand. This suggests using the method of substitution.


Step 2: Perform the substitution.

Let \(u = x^2\).

Then, \(du = 2x dx\), which means \(x dx = \frac{1}{2} du\).

The integral becomes:
\[ \int e^{x^2} (x dx) = \int e^u (\frac{1}{2} du) = \frac{1}{2} \int e^u du \]

Step 3: Evaluate the simplified integral.
\[ \frac{1}{2} \int e^u du = \frac{1}{2} e^u + C \]

Step 4: Substitute back for x.
\[ \frac{1}{2} e^{x^2} + C \]

Step 5: Final Answer.

The indefinite integral is \(\frac{1}{2}e^{x^2} + C\). Since this does not match any of the definite value options, the question is invalid.
Quick Tip: When integrating, always look for a function and its derivative (or a multiple of its derivative) within the integrand. This is the classic sign to use u-substitution, which is often the simplest method.

Note: The question as written is solvable and leads to option D. The reason for cancellation is unknown. We will proceed with the correct solution.



Step 1: Manipulate the integral to match the standard form.

The standard integral form for inverse sine is \(\int \frac{du}{\sqrt{a^2-u^2}} = \sin^{-1}(\frac{u}{a})+C\).

Let's rewrite our integral to match this form.
\[ \int \frac{dx}{\sqrt{16-9x^2}} = \int \frac{dx}{\sqrt{4^2-(3x)^2}} \]

Step 2: Use substitution.

Let \(u = 3x\). Then \(du = 3 dx\), which means \(dx = \frac{1}{3} du\).

The integral becomes:
\[ \int \frac{\frac{1}{3} du}{\sqrt{4^2-u^2}} = \frac{1}{3} \int \frac{du}{\sqrt{4^2-u^2}} \]

Step 3: Apply the standard integral formula.

Here, \(a=4\).
\[ \frac{1}{3} \sin^{-1}(\frac{u}{4}) + C \]

Step 4: Substitute back for x.
\[ \frac{1}{3} \sin^{-1}(\frac{3x}{4}) + C \]

Step 5: Compare with the given form to find A and B.

We are given the form \(A\sin^{-1}(Bx)+C\).

By comparison:

- \(A = \frac{1}{3}\)

- \(B = \frac{3}{4}\)


Step 6: Calculate A + B.
\[ A + B = \frac{1}{3} + \frac{3}{4} = \frac{4+9}{12} = \frac{13}{12} \]
Since this result is not among the options, there is a clear error in the question or the options. If the question was \(\int \frac{dx}{\sqrt{16-x^2}}\) then A=1, B=1/4, A+B=5/4. If the question was \(\int \frac{3dx}{\sqrt{1-9x^2}}\), then A=1, B=3, A+B=4. The question is flawed. However, if we assume B should be multiplied with x inside the sin\textsuperscript{-1 function, so the form is \(A \sin^{-1}(\frac{x}{B})\), then A=1/3 and B=4/3, and A+B = 5/3, which is still not in the options. Due to these inconsistencies, the question was cancelled.
Quick Tip: To solve integrals of the form \(\int \frac{dx}{\sqrt{a^2-b^2x^2}}\), always factor out the \(b^2\) to get it into the standard arcsin form. The result is always \(\frac{1}{b}\sin^{-1}(\frac{bx}{a})+C\). Knowing this pattern saves time.

Step 1: Manipulate the integrand.

Factor out \(x^4\) from the term in the parenthesis.
\[ \int \frac{dx}{x^2 [x^4(1+\frac{1}{x^4})]^{3/4}} = \int \frac{dx}{x^2 (x^4)^{3/4} (1+x^{-4})^{3/4}} \] \[ = \int \frac{dx}{x^2 \cdot x^3 (1+x^{-4})^{3/4}} = \int \frac{dx}{x^5 (1+x^{-4})^{3/4}} = \int x^{-5} (1+x^{-4})^{-3/4} dx \]

Step 2: Use substitution.

Let \(u = 1+x^{-4}\).

Then \(du = -4x^{-5} dx\), which means \(x^{-5} dx = -\frac{1}{4} du\).

The integral becomes:
\[ \int u^{-3/4} (-\frac{1}{4} du) = -\frac{1}{4} \int u^{-3/4} du \]

Step 3: Evaluate the integral.

Using the power rule for integration, \(\int u^n du = \frac{u^{n+1}}{n+1}\).
\[ -\frac{1}{4} \left[ \frac{u^{-3/4 + 1}}{-3/4 + 1} \right] + C = -\frac{1}{4} \left[ \frac{u^{1/4}}{1/4} \right] + C = -u^{1/4} + C \]

Step 4: Substitute back for x.
\[ -(1+x^{-4})^{1/4} + C = -\left(1+\frac{1}{x^4}\right)^{1/4} + C = -\left(\frac{x^4+1}{x^4}\right)^{1/4} + C \]

Step 5: Final Answer.

The result of the integration is \(-\left(\frac{x^4+1}{x^4}\right)^{1/4} + C\).

This corresponds to option (C).
Quick Tip: For integrals of the form \(\int \frac{dx}{x^m(ax^n+b)^p}\), a standard trick is to factor out the highest power of x from the parenthesis. This often sets up a simple u-substitution where u is the term inside the parenthesis.

Step 1: Simplify the integrand using logarithm properties.

Use the property \(n\log a = \log a^n\) and the identity \(e^{\log u} = u\).

- \(e^{6\log x} = e^{\log x^6} = x^6\)

- \(e^{5\log x} = x^5\)

- \(e^{4\log x} = x^4\)

- \(e^{3\log x} = x^3\)

The integral becomes:
\[ \int \frac{x^6 - x^5}{x^4 - x^3} dx \]

Step 2: Simplify the rational expression.

Factor out the common terms from the numerator and the denominator.
\[ \int \frac{x^5(x-1)}{x^3(x-1)} dx \]
Assuming \(x \neq 1\), we can cancel the \((x-1)\) term.
\[ \int \frac{x^5}{x^3} dx = \int x^2 dx \]

Step 3: Evaluate the integral.

Using the power rule for integration:
\[ \int x^2 dx = \frac{x^3}{3} + C \]

Step 4: Final Answer.

The integral is \(\frac{x^3}{3} + C\).

This corresponds to option (D).
Quick Tip: Always simplify the integrand as much as possible before attempting to integrate. In this case, applying the properties of logarithms (\(e^{\ln u} = u\)) is the key first step that transforms a complex-looking integral into a simple power rule problem.

Step 1: Simplify the integrand and apply a property of definite integrals.

First, simplify the term inside the logarithm:
\[ \log\left(\frac{1}{x}-1\right) = \log\left(\frac{1-x}{x}\right) \]
Let \(I = \int_0^1 \log\left(\frac{1-x}{x}\right) dx\).

We use the property of definite integrals: \(\int_0^a f(x) dx = \int_0^a f(a-x) dx\).

Here, \(a=1\).
\[ I = \int_0^1 \log\left(\frac{1-(1-x)}{1-x}\right) dx = \int_0^1 \log\left(\frac{x}{1-x}\right) dx \]

Step 2: Combine the two forms of the integral.

We have two expressions for I:

1) \(I = \int_0^1 \log\left(\frac{1-x}{x}\right) dx\)

2) \(I = \int_0^1 \log\left(\frac{x}{1-x}\right) dx\)

Add these two equations:
\[ 2I = \int_0^1 \log\left(\frac{1-x}{x}\right) dx + \int_0^1 \log\left(\frac{x}{1-x}\right) dx \] \[ 2I = \int_0^1 \left[ \log\left(\frac{1-x}{x}\right) + \log\left(\frac{x}{1-x}\right) \right] dx \]
Using the property \(\log A + \log B = \log(AB)\):
\[ 2I = \int_0^1 \log\left( \frac{1-x}{x} \cdot \frac{x}{1-x} \right) dx = \int_0^1 \log(1) dx \]

Step 3: Evaluate the final integral.

Since \(\log(1) = 0\):
\[ 2I = \int_0^1 0 dx = 0 \] \[ I = 0 \]

Step 4: Final Answer.

The value of the definite integral is 0.

This corresponds to option (E).
Quick Tip: The definite integral property \(\int_0^a f(x) dx = \int_0^a f(a-x) dx\) is extremely useful for integrals over the interval \([0, a]\). When you apply it and the integrand transforms into a form that is somehow opposite to the original (like here, where the fraction in the log inverted), adding the original and transformed integrals often leads to a simple result.

Step 1: Analyze the integrand and the limits of integration.

The integral is over a symmetric interval \([-\pi/2, \pi/2]\). This suggests checking if the integrand is an even or odd function.

Let \(f(x) = \sin^7 x \cos^2 x\).


Step 2: Test for even or odd function.

We check the value of \(f(-x)\).
\[ f(-x) = \sin^7(-x) \cos^2(-x) \]
Using the properties \(\sin(-x) = -\sin x\) and \(\cos(-x) = \cos x\):
\[ f(-x) = (-\sin x)^7 (\cos x)^2 = -\sin^7 x \cos^2 x = -f(x) \]
Since \(f(-x) = -f(x)\), the integrand is an odd function.


Step 3: Apply the property of definite integrals for odd functions.

For any odd function \(f(x)\), the definite integral over a symmetric interval \([-a, a]\) is always zero.
\[ \int_{-a}^{a} f(x) dx = 0 \quad if f(x) is odd \]
Therefore,
\[ \int_{-\pi/2}^{\pi/2} \sin^7 x \cos^2 x dx = 0 \]

Step 4: Final Answer.

The value of the definite integral is 0.

This corresponds to option (E).
Quick Tip: Whenever you see a definite integral with symmetric limits (like \([-a, a]\)), immediately check if the integrand is an even function (\(f(-x)=f(x)\)) or an odd function (\(f(-x)=-f(x)\)).
- If odd, the integral is 0.
- If even, the integral is \(2\int_0^a f(x) dx\).
This can save a lot of calculation time.

Step 1: Find the points of intersection of the two curves.

Set the two equations equal to each other:
\[ x^2 = 2x \] \[ x^2 - 2x = 0 \] \[ x(x-2) = 0 \]
The points of intersection are at \(x=0\) and \(x=2\).

When \(x=0, y=0\). When \(x=2, y=4\). The points are (0,0) and (2,4).


Step 2: Set up the definite integral for the area.

The area between two curves \(y=f(x)\) and \(y=g(x)\) from \(x=a\) to \(x=b\) is given by \(\int_a^b |f(x)-g(x)| dx\).

In the interval \([0, 2]\), we need to determine which function is on top. Let's test a point, e.g., \(x=1\).

- For \(y=2x\), at \(x=1, y=2\).

- For \(y=x^2\), at \(x=1, y=1\).

So, the line \(y=2x\) is above the parabola \(y=x^2\) in this interval.

The area is:
\[ A = \int_0^2 (2x - x^2) dx \]

Step 3: Evaluate the integral.
\[ A = \left[ 2\frac{x^2}{2} - \frac{x^3}{3} \right]_0^2 = \left[ x^2 - \frac{x^3}{3} \right]_0^2 \] \[ A = \left( 2^2 - \frac{2^3}{3} \right) - \left( 0^2 - \frac{0^3}{3} \right) \] \[ A = \left( 4 - \frac{8}{3} \right) - 0 = \frac{12-8}{3} = \frac{4}{3} \]

Step 4: Final Answer.

The area bounded by the curves is \(\frac{4}{3}\) square units.

This corresponds to option (C).
Quick Tip: To find the area between curves:
1. Find the intersection points to determine the limits of integration.
2. Determine which curve is the "upper" function and which is the "lower" function over the interval.
3. Integrate the difference (Upper - Lower) between the limits.

Step 1: Set up the integral for the area.

The circle is \(x^2+y^2=25\), which has a radius \(R=5\). We need the area of the segment cut by the line \(x=3\).

The equation for the upper semi-circle is \(y = \sqrt{25-x^2}\).

The area of the smaller segment to the right of \(x=3\) is given by the integral from \(x=3\) to \(x=5\). Due to symmetry, we can calculate the area in the first quadrant and multiply by 2.
\[ A = 2 \int_3^5 \sqrt{25-x^2} dx \]

Step 2: Use the standard integral formula.

The standard integral is \(\int \sqrt{a^2-x^2} dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}(\frac{x}{a})\).

Here, \(a=5\).
\[ A = 2 \left[ \frac{x}{2}\sqrt{25-x^2} + \frac{25}{2}\sin^{-1}(\frac{x}{5}) \right]_3^5 \]

Step 3: Evaluate the definite integral.
\[ A = 2 \left[ \left(\frac{5}{2}\sqrt{25-25} + \frac{25}{2}\sin^{-1}(\frac{5}{5})\right) - \left(\frac{3}{2}\sqrt{25-9} + \frac{25}{2}\sin^{-1}(\frac{3}{5})\right) \right] \] \[ A = 2 \left[ \left(0 + \frac{25}{2}\sin^{-1}(1)\right) - \left(\frac{3}{2}\sqrt{16} + \frac{25}{2}\sin^{-1}(\frac{3}{5})\right) \right] \] \[ A = 2 \left[ \frac{25}{2}(\frac{\pi}{2}) - \frac{3}{2}(4) - \frac{25}{2}\sin^{-1}(\frac{3}{5}) \right] \] \[ A = 25(\frac{\pi}{2}) - 12 - 25\sin^{-1}(\frac{3}{5}) \]

Step 4: Convert to \(\cos^{-1}\) form.

Using the identity \(\sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2}\), we have \(\sin^{-1}(\frac{3}{5}) = \frac{\pi}{2} - \cos^{-1}(\frac{3}{5})\).
\[ A = 25(\frac{\pi}{2}) - 12 - 25(\frac{\pi}{2} - \cos^{-1}(\frac{3}{5})) \] \[ A = 25\frac{\pi}{2} - 12 - 25\frac{\pi}{2} + 25\cos^{-1}(\frac{3}{5}) \] \[ A = 25\cos^{-1}(\frac{3}{5}) - 12 \]

Step 5: Final Answer.

The area of the smaller segment is \(25\cos^{-1}(\frac{3}{5}) - 12\).

This corresponds to option (C).
Quick Tip: The integral \(\int \sqrt{a^2-x^2} dx\) is one of the standard integral forms that should be memorized. When the answer involves inverse trig functions, be prepared to use identities like \(\sin^{-1}(x) + \cos^{-1}(x) = \pi/2\) to match the format of the options.

Step 1: Solve the differential equation.

The equation is \(y\frac{dy}{dx} + x = A\). This is a separable differential equation.
\[ y\frac{dy}{dx} = A - x \] \[ y \, dy = (A-x) \, dx \]
Integrate both sides:
\[ \int y \, dy = \int (A-x) \, dx \] \[ \frac{y^2}{2} = Ax - \frac{x^2}{2} + C_1 \]
Multiply by 2 and rearrange:
\[ y^2 = 2Ax - x^2 + 2C_1 \] \[ x^2 - 2Ax + y^2 = C_2 \quad (where C_2 = 2C_1) \]

Step 3: Final Answer.

This corresponds to option (A).
Quick Tip: To identify the family of curves represented by a differential equation, the best method is to solve the DE. This usually involves techniques like separation of variables, integrating factors (for linear DEs), or recognizing exact equations. The resulting equation, with its arbitrary constant, defines the family of curves.

Step 1: Identify the type of differential equation.

The equation \(\frac{dy}{dx} + y = 5\) can be solved using two methods: as a linear first-order DE or by separation of variables. Separation of variables is more direct here.


Step 2: Solve by separation of variables.

Rearrange the equation to separate the y and x terms.
\[ \frac{dy}{dx} = 5 - y \] \[ \frac{dy}{5-y} = dx \]
Integrate both sides:
\[ \int \frac{dy}{5-y} = \int dx \]
The left integral is of the form \(\int \frac{1}{u} du\) with a substitution. Let \(u = 5-y\), then \(du = -dy\).
\[ \int \frac{-du}{u} = \int dx \] \[ -\ln|u| = x + C \]
Substitute back for u:
\[ -\ln|5-y| = x + C \]
In this context, \(\ln\) and \(\log\) are used interchangeably for the natural logarithm.
\[ -\log|5-y| = x+C \]

Step 3: Final Answer.

The general solution is \(-\log|5-y| = x+C\).

This corresponds to option (A).
Quick Tip: When integrating expressions like \(\int \frac{dx}{a-bx}\), be careful with the chain rule. The result is \(-\frac{1}{b}\ln|a-bx| + C\). The minus sign is often forgotten.

Step 1: Understand the definition of Order and Degree.

- Order: The order of a differential equation is the order of the highest derivative appearing in it.

- Degree: The degree of a differential equation is the power of the highest-order derivative, provided the equation is a polynomial in its derivatives.


Step 2: Analyze the given differential equation.

The equation is \((y'')^2 + (\sin y')^4 + y = 0\).

The highest derivative is \(y''\) (the second derivative), so the order of the DE is 2.

To find the degree, we check if the equation is a polynomial in its derivatives (\(y', y'', \dots\)). The term \((\sin y')^4\) involves the sine function of a derivative (\(y'\)). Because of this transcendental function of a derivative, the equation is not a polynomial in its derivatives.


Step 3: Conclude the degree.

Since the differential equation cannot be expressed as a polynomial in \(y', y'', \dots\), its degree is not defined.


Step 4: Final Answer.

The degree of the differential equation is not defined.

This corresponds to option (E).
Quick Tip: For the degree of a differential equation to be defined, the equation must be a polynomial in all its derivatives. Look out for terms like \(\sin(y')\), \(e^{y''}\), \(\log(y')\), etc. The presence of any such term makes the degree undefined.

Step 1: Identify the feasible region.

The constraints are:

- \(x \ge 0\) and \(y \ge 0\) (first quadrant)

- \(x \le 3\) (region to the left of the vertical line x=3)

- \(y \le 2\) (region below the horizontal line y=2)

The feasible region is a rectangle in the first quadrant with vertices at (0,0), (3,0), (3,2), and (0,2).


Step 2: Evaluate the objective function at the corner points.

The fundamental theorem of linear programming states that the optimal solution (maximum or minimum) must occur at one of the vertices (corner points) of the feasible region.

The objective function is \(z = 11x + 7y\).

- At (0,0): \(z = 11(0) + 7(0) = 0\)

- At (3,0): \(z = 11(3) + 7(0) = 33\)

- At (0,2): \(z = 11(0) + 7(2) = 14\)

- At (3,2): \(z = 11(3) + 7(2) = 33 + 14 = 47\)


Step 3: Find the maximum value.

Comparing the values of z at the vertices, the maximum value is 47, which occurs at the point (3,2).


Step 4: Final Answer.

The optimal solution (the point that maximizes z) is (3,2).

This corresponds to option (A).
Quick Tip: For a linear programming problem:
1. Graph the inequalities to find the feasible region.
2. Identify the coordinates of all corner points (vertices) of the feasible region.
3. Substitute the coordinates of each corner point into the objective function.
4. The largest value is the maximum, and the smallest is the minimum.