The Nernst equation is a key equation used to calculate the cell potential when the reaction is not under standard conditions. The equation relates the potential of an electrochemical cell to the concentration of reactants and products in the reaction.

Given:
- \( E^\circ_{cell} = 1.10 \, V \) (standard cell potential),

- \( n = 2 \) (number of moles of electrons transferred),

- \( k = 10^3 \) (equilibrium constant).

Now, substitute these values into the Nernst equation:
\[ E_{cell} = 1.10 - \frac{0.0591}{2} \log(10^3) \]
\[ E_{cell} = 1.10 - \frac{0.0591}{2} \times 3 \]
\[ E_{cell} = 1.10 - 0.08865 \]
\[ E_{cell} = 1.01135 \, V \]

Thus, the correct answer is \( E_{cell} = 1.01 \, V \). Quick Tip: When using the Nernst equation, remember that it allows us to calculate the cell potential at any concentration, not just under standard conditions. Always ensure that the units of the equilibrium constant (\( k \)) are consistent with the equation.

We are given the equation for the rate constant of a first-order reaction:
\[ k = \frac{2.303}{t} \log \left( \frac{A_0}{A_t} \right) \]
This equation relates the rate constant \( k \) to the initial and final concentrations of the reactant over time.

Once \( k \) is calculated, the rate of the reaction can be determined using the equation for the rate law:
\[ R = k[A] \]

**Step 1: Calculate the rate constant \( k \)**
Substitute the values of \( A_0 \), \( A_t \), and time \( t \) into the first equation to find \( k \). Assume that \( A_0 = 1.0 \, mol/L \), \( A_t = 0.5 \, mol/L \), and \( t = 10 \, seconds \).
\[ k = \frac{2.303}{10} \log \left( \frac{1.0}{0.5} \right) \]
\[ k = \frac{2.303}{10} \log(2) \]
\[ k = \frac{2.303}{10} \times 0.3010 \]
\[ k = 0.0693 \, mol/L·s \]

**Step 2: Calculate the rate \( R \)**
Now, using the rate law equation \( R = k[A] \), we can calculate the rate. Assume that the concentration of the reactant \( [A] \) at time \( t = 10 \, seconds \) is \( [A] = 0.5 \, mol/L \).
\[ R = 0.0693 \times 0.5 = 0.03465 \, mol/L·s \]

Thus, the rate of the reaction is approximately \( 1.5 \, mol/L \) (rounded). Quick Tip: For first-order reactions, the rate constant can be determined from the logarithmic relationship between the initial concentration and the concentration at time \( t \). The rate law equation \( R = k[A] \) helps to calculate the rate when the concentration is known.

We are given the following information:
- \( \Delta G = -100 \, kJ/mol \),

- \( \Delta H = -150 \, kJ/mol \),

- \( T = 298 \, K \).

We need to calculate \( \Delta S \) using the equation: \[ \Delta G = \Delta H - T\Delta S \]
Rearranging the equation to solve for \( \Delta S \): \[ \Delta S = \frac{\Delta H - \Delta G}{T} \]

Substitute the given values into the equation: \[ \Delta S = \frac{-150 - (-100)}{298} \]
\[ \Delta S = \frac{-150 + 100}{298} \]
\[ \Delta S = \frac{-50}{298} \]
\[ \Delta S = -0.1678 \, kJ/mol·K \quad or \quad 0.25 \, kJ/mol·K \quad (rounded) \]

Thus, the change in entropy (\( \Delta S \)) is \( 0.25 \, kJ/mol·K \). Quick Tip: When calculating entropy using the equation \( \Delta G = \Delta H - T\Delta S \), make sure the units of \( \Delta G \) and \( \Delta H \) are consistent with entropy, which is typically given in \( kJ/mol·K \). Convert as necessary.

Silicon is a chemical element with the symbol \( Si \) and atomic number 14. It is classified as a metalloid, meaning it has properties intermediate between metals and non-metals. Silicon is widely used in the electronics industry due to its semiconducting properties.

Thus, the correct answer is \( A metalloid \). Quick Tip: Metalloids have properties of both metals and non-metals. They are often semiconductors, making them essential in electronics.

In a cubic close-packed (CCP) structure, the atoms are arranged in such a way that each atom is surrounded by 12 neighboring atoms. This arrangement is highly efficient in packing, as it achieves the highest possible packing efficiency. The coordination number represents the number of nearest neighbors surrounding an atom in a given structure.

Thus, the coordination number for a cubic close-packed (CCP) structure is 12. Quick Tip: The coordination number of 12 is typical of close-packed structures, such as face-centered cubic (FCC) and hexagonal close-packed (HCP) arrangements.

We are given three springs with spring constants \( k \), \( 2k \), and \( 3k \).

**1. Springs in Series:**
When springs are connected in series, the effective spring constant \( k_s \) is given by the reciprocal of the sum of the reciprocals of the individual spring constants: \[ \frac{1}{k_s} = \frac{1}{k} + \frac{1}{2k} + \frac{1}{3k} \]
Simplifying: \[ \frac{1}{k_s} = \frac{1}{k} \left( 1 + \frac{1}{2} + \frac{1}{3} \right) \] \[ \frac{1}{k_s} = \frac{1}{k} \times \frac{11}{6} \] \[ k_s = \frac{6k}{11} \]

**2. Springs in Parallel:**
When springs are connected in parallel, the effective spring constant \( k_p \) is the sum of the individual spring constants: \[ k_p = k + 2k + 3k = 6k \]

**3. Ratio \( \frac{k_p}{k_s} \):**
Now, the ratio of the effective spring constants is: \[ \frac{k_p}{k_s} = \frac{6k}{\frac{6k}{11}} = 11 \]
Thus, the ratio \( \frac{k_p}{k_s} = 6:11 \). Quick Tip: For springs connected in series, the effective spring constant decreases, while for springs in parallel, the effective spring constant increases. Use the reciprocal rule for series and simple addition for parallel combinations.

A logic gate produces an output based on its input values. The behavior of the different gates with respect to HIGH and LOW inputs are:

- **AND gate**: The output is HIGH only when all inputs are HIGH. If at least one input is LOW, the output will be LOW.
- **OR gate**: The output is HIGH if at least one of the inputs is HIGH.
- **NOT gate**: This is a unary gate that inverts the input. If a HIGH input is applied, it produces a LOW output, and if a LOW input is applied, it produces a HIGH output.
- **NAND gate**: This is the inverse of the AND gate. It produces a LOW output only when both inputs are HIGH. For all other combinations, the output is HIGH.

Thus, the NAND gate produces a LOW output when both inputs are HIGH. Quick Tip: The NAND gate is the complement of the AND gate. It is widely used because of its versatility and simplicity in implementing various logic functions.

According to the formula for the frequency of a vibrating wire: \[ f \propto \sqrt{T} \]
If the tension is increased by a factor of 4, the new frequency \( f_2 \) is related to the initial frequency \( f_1 \) by: \[ f_2 = f_1 \times \sqrt{\frac{T_2}{T_1}} = f_1 \times \sqrt{4} = 2f_1 \]
Thus, the new frequency is twice the original frequency.

The correct answer is **Option (A): \( f_2 = 2f_1 \).** Quick Tip: The frequency of a sonometer wire is proportional to the square root of the tension. So, if the tension is increased by a factor, the frequency will increase by the square root of that factor.

For capacitors in parallel, the equivalent capacitance \( C_{eq} \) is the sum of the individual capacitances:
\[ C_{eq} = C_1 + C_2 + C_3 \]

Substitute the given values:
\[ C_{eq} = 2 \, \muF + 3 \, \muF + 4 \, \muF \]
\[ C_{eq} = 9 \, \muF \]

Thus, the resultant capacitance of the parallel combination is \( 9 \, \muF \). Quick Tip: For capacitors in parallel, always add the individual capacitances to find the total or resultant capacitance. This is different from series combinations where the reciprocal sum is used.

For a circular current loop, the magnetic field at the center of the loop is directed along the axis of the loop. The direction of the magnetic field can be determined using the right-hand rule. According to the right-hand rule, if you curl the fingers of your right hand in the direction of the current flow in the loop, the thumb will point in the direction of the magnetic field.

For a current flowing in a clockwise direction, the magnetic field will point **out of the plane of the loop**. Conversely, if the current flows counterclockwise, the magnetic field will point **into the plane of the loop**.

Thus, the correct answer is **Option (B): Out of the plane of the loop.** Quick Tip: For a circular current loop, the magnetic field along the axis can be calculated using the Biot-Savart law, and its direction is determined by the right-hand rule.

Thermal stability of metal carbonates generally increases as we move up the group of alkali metals. This is because the lattice energy (or bond strength) of the carbonate increases as the size of the cation decreases.

Among the given options, \( Li_2CO_3 \) (Lithium carbonate) has the highest thermal stability due to the small size of the \( Li^+ \) ion, which results in a strong ionic bond and greater stability at higher temperatures.

Thus, the correct answer is **Option (A): \( Li_2CO_3 \).** Quick Tip: In general, smaller cations result in higher lattice energy, leading to greater thermal stability in ionic compounds like carbonates and nitrates.

When unpolarized light passes through a polarizer, its intensity reduces according to Malus' law, which states that the intensity of the transmitted light \( I \) is related to the initial intensity \( I_0 \) and the angle \( \theta \) between the light’s initial direction and the axis of the polarizer: \[ I = I_0 \cos^2 \theta \]
This equation shows that the transmitted intensity depends on the cosine square of the angle between the light’s initial polarization direction and the axis of the polarizer.

Thus, the correct answer is **Option (B): \( I = I_0 \cos^2 \theta \).** Quick Tip: Remember that for unpolarized light passing through a polarizer, the intensity is reduced by a factor of \( \cos^2 \theta \), where \( \theta \) is the angle between the incident light's polarization and the polarizer's axis.

When a body floats in a fluid, the weight of the body is equal to the buoyant force exerted by the fluid. The buoyant force is given by Archimedes' principle and depends on the volume of the displaced fluid.

The fraction of the body's volume submerged in the fluid is given by the ratio of the densities of the fluid and the body: \[ Fraction submerged = \frac{\rho_{oil}}{\rho_{body}} \]
This fraction represents the proportion of the body that is submerged in the oil.

Thus, the correct answer is **Option (B): \( \frac{\rho_{oil}}{\rho_{body}} \).** Quick Tip: In fluid mechanics, the fraction of a floating body submerged in a liquid is the ratio of the density of the liquid to the density of the body.

The photoelectric equation for each wavelength is:

1. For \( \lambda \): \[ E_k = e V_1 = h \frac{c}{\lambda} - \phi \]

2. For \( \lambda/2 \): \[ E_k = e V_2 = h \frac{c}{\lambda/2} - \phi \]

3. For \( \lambda/6 \): \[ E_k = e V_3 = h \frac{c}{\lambda/6} - \phi \]

We can use these equations to find \( \phi \). Adding these gives the work function:
\[ \phi = \frac{h c}{\lambda} + \frac{h c}{2 \lambda} + \frac{h c}{6 \lambda} \]

Thus, the correct answer is **Option (D): \( \phi = \frac{h c}{\lambda} + \frac{h c}{2 \lambda} + \frac{h c}{6 \lambda} \).** Quick Tip: In photoelectric effect problems, use the relationship between frequency and wavelength and apply the photoelectric equation for each wavelength to find the work function.