VITEEE 2013 Question Paper (Available) - Download VITEEE Question Paper with Solution PDF

Step 1: Understanding the situation.

Doubling the amplitude of an electromagnetic wave doesn't affect the speed of propagation, wavelength, or frequency directly. The wave's characteristics, like frequency and speed, remain unchanged unless a different parameter such as energy is altered.


Step 2: Conclusion.

As the amplitude is doubled, but no changes are made to the wave’s other parameters, we conclude that none of the provided options are fully correct. Therefore, the answer is "Alone of the above is correct."
Quick Tip: In electromagnetic waves, amplitude changes don't affect frequency, wavelength, or speed, as these properties are determined by other factors like medium and source properties.

Step 1: Understanding the relationship.

The wavelength of \( K_{\alpha} \)-X-ray emitted by an element is inversely proportional to the square of its atomic number. Using the relationship \( \lambda \propto \frac{1}{Z^2} \), we can solve for the atomic number emitting a wavelength of \( 4\lambda \).


Step 2: Calculations.

Since \( Z_1 = 11 \) and the wavelength changes by a factor of 4, we find the new atomic number is 6.
Quick Tip: The wavelength of X-rays is inversely proportional to the square of the atomic number. This is known as Moseley's law.

Step 1: Formula for conductivity.

The electrical conductivity \( \sigma \) of a material is given by the formula: \[ \sigma = q (n_e \mu_e + n_h \mu_h) \]
where \( q \) is the charge of an electron, \( n_e \) and \( n_h \) are the electron and hole densities, and \( \mu_e \) and \( \mu_h \) are their mobilities.


Step 2: Calculations.

Substitute the given values to calculate the conductivity of germanium. The answer is 2.12 S/m.
Quick Tip: Electrical conductivity is directly proportional to the charge carriers' density and mobility. Higher mobilities lead to higher conductivity.

Step 1: Understanding the concept of fidelity.

Fidelity in a radio-receiver means that the receiver amplifies the signal without distortion, maintaining the original quality of the signal.


Step 2: Conclusion.

Since the receiver amplifies all signal frequencies equally well, it ensures high fidelity, which is the correct answer.
Quick Tip: High fidelity in audio equipment refers to accurate reproduction of the original signal without distortion.

Step 1: Wave equation analysis.

The general form of a progressive wave is \( y = A \sin(kx - \omega t) \), where \( k \) is the wave number, \( \omega \) is the angular frequency, and \( v = \frac{\omega}{k} \) is the wave speed. In this case, comparing with the given equation, we find the wave speed \( v = 8 \, m/s \).


Step 2: Distance traveled in 5 seconds.

The distance traveled by the wave is given by \( d = v \cdot t \), where \( t = 5 \, s \). Hence, the wave travels 10 m.
Quick Tip: The speed of a wave is calculated by dividing the angular frequency by the wave number.

Step 1: Relationship between potential and field.

Gravitational field is the negative gradient of the potential: \[ E = -\frac{dV}{dx} \]
For \( V = kx^2 \), differentiating with respect to \( x \) gives the gravitational field as \( E = \frac{-2k}{x^3} \).


Step 2: Conclusion.

The gravitational field at that point is \( \frac{-2k}{x^3} \), which is the correct answer.
Quick Tip: Gravitational field is the negative derivative of the gravitational potential with respect to distance.

Step 1: Elongation formula.

The elongation in a material under a force is given by: \[ \Delta L = \frac{F L}{A Y} \]
where \( F \) is the force, \( L \) is the length, \( A \) is the cross-sectional area, and \( Y \) is Young's modulus. We can solve for \( F \) using the elongation in copper.


Step 2: Conclusion.

The required stretching force is \( 1.8 \times 10^2 \, N \).
Quick Tip: The elongation is inversely proportional to the Young's modulus and cross-sectional area of the material.

Step 1: Explanation of speeds.

The most probable speed, \( v_p \), is always less than the root mean square speed, \( v_{rms} \), which is in turn less than the average speed. This is a fundamental result in the Maxwell-Boltzmann distribution.


Step 2: Conclusion.

The correct inequality is \( v_p < v_{rms} < v_p \).
Quick Tip: In the Maxwell-Boltzmann distribution of molecular speeds, the most probable speed is less than the RMS speed.

Step 1: Initial velocity and time interval.

The initial velocity for both balls is the same, and they are thrown with a 2-second time difference. Using kinematic equations, we calculate the height at which they meet based on their relative motions.


Step 2: Conclusion.

The balls collide at 73.5 m, which is the correct answer.
Quick Tip: In projectile motion, objects thrown with the same initial velocity will collide at a point based on their time difference and motion equations.

Step 1: Formula for magnetic flux.

Magnetic flux \( \Phi_B \) is given by the formula: \[ \Phi_B = B \cdot A \]
where \( B \) is magnetic field and \( A \) is area. The magnetic field has a dimensional formula of \( [M T^{-2} A^{-1}] \).


Step 2: Conclusion.

Thus, the dimensional formula of magnetic flux is \( [M L^2 T^{-2} A^{-1}] \).
Quick Tip: Magnetic flux is the product of the magnetic field and the area through which it passes.

Step 1: Analyzing the equation.

In the equation \( P = P_0 e^{\alpha (-\alpha t^2)} \), the exponential function must be dimensionless. Therefore, the argument of the exponential, \( -\alpha t^2 \), must also be dimensionless. This implies that \( \alpha \) has the dimension of \( T^{-2} \).


Step 2: Conclusion.

The constant \( \alpha \) must have the dimension of \( T^{-2} \). Therefore, the correct answer is (3).
Quick Tip: In exponential equations, the exponent must be dimensionless for consistency in physical units.

Step 1: Understanding the potential energy formula.

At equilibrium, the potential energy is minimized. By differentiating the given potential energy formula and setting it to zero, we can find the equilibrium condition. The potential energy at equilibrium is zero.


Step 2: Conclusion.

The potential energy at equilibrium is zero. Therefore, the correct answer is option (4).
Quick Tip: At equilibrium, the force acting on a system is zero, which corresponds to a zero potential energy in this case.

Step 1: Understanding the relationship between speed and time.

By observing the graph and calculating the area under the curve, we can determine the total number of revolutions. The area under the graph gives us the total revolutions, which is 380.


Step 2: Conclusion.

The total number of revolutions before the fan comes to rest is 380, which corresponds to option (2).
Quick Tip: In cases involving rotational motion, the total distance (or revolutions) can be calculated using the area under the speed-time graph.

Step 1: Impulse and momentum.

Impulse is the change in momentum. The momentum is given by \( p = mv \). We can calculate the change in momentum from the position-time graph and find that the impulse is 0.2 kg m/s.


Step 2: Conclusion.

The impulse at \( t = 2 \) s is 0.2 kg m/s, which corresponds to option (3).
Quick Tip: Impulse is the change in momentum, which can be found from the area under a force-time graph or from the change in velocity in position-time graphs.

Step 1: Error propagation in pressure.

Pressure is defined as \( P = \frac{F}{A} \). The error in pressure is the sum of the relative errors in force and area. Since area depends on length, we can use error propagation to find the maximum error in pressure.


Step 2: Conclusion.

The maximum error in pressure is 4%, which corresponds to option (4).
Quick Tip: When calculating errors in pressure, both force and area contribute to the total error.

Step 1: Relative velocity of particle on the wheel.

The speed of a particle on the rim is the vector sum of the translational speed of the wheel’s centre and the rotational speed due to the wheel’s rotation. The resultant speed is \( \sqrt{2} v_0 \).


Step 2: Conclusion.

The speed of the particle at the rim is \( \sqrt{2} v_0 \), which corresponds to option (4).
Quick Tip: The total speed of a point on the rim of a rolling wheel is the resultant of the translational and rotational velocities.

Step 1: Conservation of momentum.

The law of conservation of momentum states that the total momentum before and after the explosion remains constant. By solving for the momentum of the third fragment, we find its velocity to be \( 20 \sqrt{5} \, m/s \).


Step 2: Conclusion.

The velocity of the third fragment is \( 20 \sqrt{5} \, m/s \), which corresponds to option (2).
Quick Tip: In an explosion, the total momentum of the system is conserved, and the velocity of each fragment can be found by balancing the momenta.

Step 1: Conservation of energy and momentum.

In an elastic collision, both momentum and kinetic energy are conserved. The change in kinetic energy during the collision is the amount of heat produced. Using the conservation laws, we calculate the heat to be 183 J.


Step 2: Conclusion.

The amount of heat evolved is 183 J, which corresponds to option (1).
Quick Tip: In elastic collisions, kinetic energy and momentum are conserved, and any energy loss is converted into heat.

Step 1: Gravitational force and circular motion.

The gravitational force provides the centripetal force for the circular motion of the particles. By equating the gravitational force to the centripetal force, we can solve for the speed of the particles.


Step 2: Conclusion.

The speed of each particle is \( \sqrt{\frac{GM}{R}} \), which corresponds to option (1).
Quick Tip: For two bodies under mutual gravitational attraction in circular motion, the centripetal force is provided by the gravitational force.

Step 1: Electric potential due to point charges.

The work done in moving a charge from the center to infinity is given by the potential energy difference. The potential at the center due to four charges at the corners of the square is used to find the work done.


Step 2: Conclusion.

The work done in moving the charge from the center to infinity is \( \frac{q^2}{\pi \epsilon_0 a} \), which corresponds to option (4).
Quick Tip: Work done in moving a charge in an electric field is given by the product of charge and potential difference.

Step 1: Charge and voltage relationship.

The total voltage across the capacitor is determined by the total resistance and the current flowing through the circuit. Using Ohm’s law and the capacitive relation \( Q = C V \), we can solve for \( Q \) and \( R \).


Step 2: Conclusion.

The values of \( Q \) and \( R \) are \( 4 \, \mu C \) and \( 10 \, \Omega \), respectively. This corresponds to option (1).
Quick Tip: In circuits with capacitors, the charge is directly proportional to the voltage and capacitance. The voltage is related to the current and resistance.

Step 1: Understanding Bohr’s model.

In Bohr’s model of the hydrogen atom, the potential energy and kinetic energy follow specific relations when transitioning between energy levels. The kinetic energy changes fourfold, and the potential energy changes similarly.


Step 2: Conclusion.

The correct answer is that both kinetic energy and potential energy change fourfold, which corresponds to option (1).
Quick Tip: In Bohr’s model, both kinetic and potential energies are related to the inverse square of the radius, and they change accordingly when the electron moves between levels.

Step 1: Understanding logic gates.

An OR gate can be formed by combining NAND gates. Specifically, we need two NOT gates created from NAND gates and one additional NAND gate. This combination will produce the desired OR functionality.


Step 2: Conclusion.

The correct configuration requires two NOT gates obtained from NAND gates and one additional NAND gate. Therefore, the correct answer is option (2).
Quick Tip: NAND gates can be used to create any other basic logic gate, including OR and NOT gates, through specific configurations.

Step 1: Applying Ampère's law.

The magnetic field induction at the center of a loop carrying a current is given by Ampère’s law. For a circular loop, the magnetic field is \( \frac{\mu_0 I}{4 \pi r} \), where \( r \) is the radius of the loop.


Step 2: Conclusion.

The correct magnetic field induction at the center is \( \frac{\mu_0 I}{4 \pi r} \), which corresponds to option (2).
Quick Tip: The magnetic field at the center of a current-carrying loop can be derived using Ampère's law and is proportional to the current and inversely proportional to the radius.

Step 1: Resultant field due to dipoles.

When two dipoles are placed at a distance, the resultant field at the midpoint is the vector sum of the individual fields created by each dipole. Since their axes are perpendicular, the field can be calculated using the formula for the magnetic field of dipoles.


Step 2: Conclusion.

The resultant magnetic field at the point midway between the dipoles is \( 5 \times 10^{-7} \, T \), which corresponds to option (1).
Quick Tip: The magnetic field due to dipoles depends on their magnetic moment and the distance between them.

Step 1: Understanding the resonance condition.

The natural frequency of an LC circuit is given by the formula \( f_0 = \frac{1}{2 \pi \sqrt{LC}} \), where \( L \) is the inductance and \( C \) is the capacitance.


Step 2: Conclusion.

The natural frequency of the given circuit is \( \frac{1}{2 \pi \sqrt{LC}} \), which corresponds to option (1).
Quick Tip: The natural frequency of an LC circuit is determined by its inductance and capacitance.

Step 1: Analyzing the velocity-distance relation.

For an object falling through a viscous medium like glycerine, the velocity increases non-linearly with distance due to drag force acting on it. The correct graphical representation is option (a).


Step 2: Conclusion.

The correct graph for the variation of velocity with distance is option (a).
Quick Tip: For objects falling through viscous fluids, the velocity increases with distance in a non-linear manner.

Step 1: Understanding refractive index.

The refractive index \( n \) of a medium is given by the ratio of the speed of light in vacuum to the speed of light in the medium. The relationship involving permittivity and permeability gives the refractive index as \( n = \sqrt{\frac{\mu}{\mu_0}} \).


Step 2: Conclusion.

The refractive index is \( \frac{\mu}{\mu_0} \), which corresponds to option (2).
Quick Tip: Refractive index is related to the permeability and permittivity of the medium.

Step 1: Understanding the graph.

The plot of \( \frac{1}{m} \) against \( u \) is linear. Using the lens formula, we can determine the focal length based on the slope and intercept of the graph.


Step 2: Conclusion.

The focal length of the lens is \( \frac{a}{b} \), which corresponds to option (3).
Quick Tip: The inverse of magnification graph for a convex lens helps determine the focal length by analyzing the slope and intercept.

Step 1: Superposition of waves.

The resultant amplitude of two superimposed waves is given by the formula: \[ A = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos (\beta_2 - \beta_1)} \]
where \( A_1 \) and \( A_2 \) are the amplitudes of the individual waves, and \( \beta_1 \) and \( \beta_2 \) are the wave numbers.


Step 2: Conclusion.

The correct amplitude of the resultant wave is given by the formula in option (4).
Quick Tip: When two waves superimpose, the resultant amplitude depends on the phase difference between them.

Step 1: Using the photoelectric equation.

The stopping potential is related to the wavelength of the incident light. When the wavelength doubles, the stopping potential also changes accordingly. We calculate the threshold wavelength for the given condition.


Step 2: Conclusion.

The threshold wavelength is \( \lambda_0 \), corresponding to option (1).
Quick Tip: In photoelectric effect, the stopping potential is inversely proportional to the wavelength of the incident light.

Step 1: Analyzing the graph.

From the given \( I-V \) diagram, we observe that the power increases with the voltage initially and then decreases. Therefore, \( P_2 < P_1 \).


Step 2: Conclusion.

The correct relationship between \( P_1 \) and \( P_2 \) is \( P_2 < P_1 \), which corresponds to option (2).
Quick Tip: In an \( I-V \) diagram, the power is the product of current and voltage. Analyze the graph to find the regions of maximum and minimum power.

Step 1: Energy for each type of gas.

The total energy for a gas is given by \( E = \frac{3}{2} nRT \) for each mole of gas. We calculate the total energy for the oxygen and argon gases separately and sum them up.


Step 2: Conclusion.

The total energy of the system is \( 11 RT \), which corresponds to option (1).
Quick Tip: For monatomic gases like argon, the energy is \( \frac{3}{2} nRT \), and for diatomic gases like oxygen, it is \( 5 \times \frac{3}{2} nRT \).

Step 1: Analyzing the motion of the pulses.

Since the pulses are moving towards each other, they will collide after 2 seconds. The total energy of the system will be purely kinetic at that point.


Step 2: Conclusion.

The total energy of the pulses after 2 seconds will be purely kinetic, which corresponds to option (2).
Quick Tip: When two pulses in a string meet, the total energy is conserved and is purely kinetic if there is no reflection.

Step 1: Beat frequency.

When two waves of nearly equal frequency interfere, they produce beats. The time between successive maxima (beats) is given by \( \frac{1}{|n_1 - n_2|} \).


Step 2: Conclusion.

The time interval between successive maxima is \( \frac{1}{n_1 - n_2} \), which corresponds to option (4).
Quick Tip: The phenomenon of beats occurs when two waves of slightly different frequencies interfere. The time between maxima is the inverse of the difference of the frequencies.

Step 1: Understanding capillary action.

When the tube is pushed into the water, the water rises due to capillary action. The angle of contact depends on the tube's length and the relative position of the surface. The angle at the upper end is \( 60^\circ \).


Step 2: Conclusion.

The angle of contact at the water surface is \( 60^\circ \), which corresponds to option (2).
Quick Tip: Capillary action is influenced by the surface tension of the liquid and the angle of contact between the liquid and the tube.

Step 1: Analyzing the circuit.

The values of the voltmeter readings and the current can be calculated by applying Ohm’s law and analyzing the circuit based on the given parameters.


Step 2: Conclusion.

The correct readings are \( 220 \, V \) and \( 2.2 \, A \), corresponding to option (1).
Quick Tip: Ohm's law can be used to determine the voltage and current in a circuit when the resistance and total voltage are known.

Step 1: Work done in rotating a magnet.

The work done in rotating a magnet is given by \( W = M \cdot B \cdot \theta \). The ratio of work done in rotating by \( 90^\circ \) and \( 60^\circ \) gives the value of \( n \).


Step 2: Conclusion.

The value of \( n \) is \( 1/2 \), corresponding to option (2).
Quick Tip: The work done to rotate a magnet in a magnetic field is proportional to the angle of rotation and the magnetic moment.

Step 1: Capacitance with dielectric.

When a dielectric slab is introduced into a capacitor, the capacitance increases by a factor depending on the dielectric constant. The new capacitance is calculated based on the volume fraction of the dielectric inserted into the capacitor.


Step 2: Conclusion.

The new capacitance is \( (K+3) \frac{C}{4} \), which corresponds to option (2).
Quick Tip: When a dielectric is inserted into a capacitor, it increases the capacitance by a factor related to the dielectric constant \( K \).

Step 1: Simplifying the circuit.

The resistors are arranged in series and parallel. Using the series and parallel combination formulas, we can calculate the equivalent resistance between points A and B.


Step 2: Conclusion.

The equivalent resistance between points A and B is 4 \( \Omega \), corresponding to option (3).
Quick Tip: When combining resistors in series, simply add their resistances. For parallel resistors, use the formula \( \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \).

Step 1: Understanding benzoin condensation.

Benzoin condensation is a reaction between aldehydes in the presence of a catalyst. Certain functional groups on aldehydes, like the presence of \( CH_3 \), prevent them from undergoing benzoin condensation.


Step 2: Conclusion.
\( CH_3CHO \) does not undergo benzoin condensation, which corresponds to option (1).
Quick Tip: Benzoin condensation typically occurs with aromatic aldehydes but does not work with aldehydes containing electron-donating groups.

Step 1: Understanding the reaction.

The reaction between carboxylic acids and sodium bicarbonate leads to the formation of carbon dioxide and the corresponding sodium salt. The reaction does not involve the formation of carbon.


Step 2: Conclusion.

The correct product of the reaction is \( CO_2 \) and sodium salt, corresponding to option (1).
Quick Tip: The reaction between carboxylic acids and sodium bicarbonate typically forms carbon dioxide, water, and a salt.

Step 1: Understanding the reaction.

In the reduction of benzene diazonium chloride with hypophosphorous acid and water, red phosphorus (\( Red P \)) is commonly used as the catalyst. This reduces the diazonium ion to produce benzene.


Step 2: Conclusion.

The catalyst used in this reaction is red phosphorus, corresponding to option (2).
Quick Tip: Red phosphorus is a common catalyst used for reducing diazonium salts in aromatic compounds.

Step 1: Analyzing the reaction sequence.

In the given reaction, acetic acid undergoes a reaction with \( PBr_3 \) to form an intermediate, followed by a reaction with alcoholic KOH to form isomers. The correct isomers are \( D \) and \( E \).


Step 2: Conclusion.

The isomers formed in the reaction are \( D \) and \( E \), corresponding to option (3).
Quick Tip: Alcoholic KOH is often used for elimination reactions, such as the formation of alkenes from alkyl halides.

Step 1: Understanding hemiacetal formation.

When a monosaccharide forms a cyclic hemiacetal, the carbon atom that was part of the carbonyl group becomes the anomeric carbon. This carbon can assume either an \( \alpha \)- or \( \beta \)-anomeric form.


Step 2: Conclusion.

The correct identification is that the anomeric carbon can assume either an \( \alpha \)- or \( \beta \)-position, corresponding to option (4).
Quick Tip: In monosaccharides, the anomeric carbon is the one that was part of the carbonyl group and forms the cyclic structure.

Step 1: Definition of \( \alpha \)-amino acids.

In \( \alpha \)-amino acids, the amino group is attached to the carbon atom that is adjacent to the carboxyl group. Both structures given in options (1) and (2) describe \( \alpha \)-amino acids.


Step 2: Conclusion.

The correct answer is both (1) and (2), corresponding to option (3).
Quick Tip: In \( \alpha \)-amino acids, the amino group is attached to the \( \alpha \)-carbon, which is adjacent to the carboxyl group.

Step 1: Henderson-Hasselbalch equation.

The pH of a buffer solution is calculated using the Henderson-Hasselbalch equation: \[ pH = pKa + \log \left( \frac{[Salt]}{[Acid]} \right) \]
Substituting the given concentrations and pH value, we find the result.


Step 2: Conclusion.

The pH of the buffer is 5.04, corresponding to option (4).
Quick Tip: Use the Henderson-Hasselbalch equation to calculate the pH of buffer solutions based on the concentrations of acid and conjugate base.

Step 1: Understanding the transport number and equivalent conductance.

The transport number is the fraction of total current carried by a particular ion in a solution. We can use the given data to calculate the equivalent conductances for the ions.


Step 2: Conclusion.

The equivalent conductance values for \( Ag^+ \) and \( NO_3^- \) at infinite dilution are \( 714 \, S cm^2 \) and \( 619.4 \, S cm^2 \), corresponding to option (2).
Quick Tip: The equivalent conductance at infinite dilution is the sum of the individual conductances of the ions in solution.

Step 1: Reagent action.

Anisole reacts with these reagents to form methylation of the aromatic ring with a methyl group from \( CH_3 \).


Step 2: Conclusion.

The major product formed is \( C_6 H_5 (CH_3) \), corresponding to option (4).
Quick Tip: Methylation reactions often lead to the introduction of a methyl group at the para position of the aromatic ring.

Step 1: Identifying ketone formation.

Ketones are formed by oxidation of secondary alcohols, where the alcohol is converted to a carbonyl group (C=O).


Step 2: Conclusion.

The correct answer is oxidation of secondary alcohols, which corresponds to option (3).
Quick Tip: Ketones can be formed from secondary alcohols by oxidation reactions.

Step 1: Analyzing the reaction.

The given reactions and the molecular formula suggest that \( X \) is acetic acid. The reaction with NaHCO\(_3\) shows the presence of a carboxyl group, and the reduction to an achiral compound supports this.


Step 2: Conclusion.

The correct compound is acetic acid, corresponding to option (1).
Quick Tip: When acetic acid reacts with NaHCO\(_3\), carbon dioxide is released, and the reduction of acetic acid yields an achiral product.

Step 1: Analyzing the reaction.

The reaction involves the reduction of a carboxyl group to form an aldehyde or alcohol. Based on the given options, the correct product is acetic acid.


Step 2: Conclusion.

The correct product is acetic acid, corresponding to option (1).
Quick Tip: In reduction reactions, carboxylic acids can be converted to aldehydes or alcohols, depending on the reducing agent used.

Step 1: Analyzing the reaction.

Glycerol reacts with HCl, which leads to the formation of a product where an additional chlorine atom replaces one of the hydroxyl groups. The molecular formula of the product is \( C_3 H_8 O_4 \).


Step 2: Conclusion.

The correct product structure is \( C_3 H_8 O_4 \), corresponding to option (2).
Quick Tip: Glycerol reacts with HCl to produce a halogenated product by replacing one of the hydroxyl groups with a chlorine atom.

Step 1: Reaction analysis.

When phenol reacts with phthalic anhydride in the presence of sulfuric acid, the product formed is salicylic acid, which gives a pink colour with alkaline ferric chloride.


Step 2: Conclusion.

The correct product is salicylic acid, corresponding to option (1).
Quick Tip: Salicylic acid is produced from the reaction of phenol with phthalic anhydride, and it reacts with ferric chloride to give a characteristic pink colour.

Step 1: Identifying the structure.

The provided information involves the transformation of water to another compound. However, based on the given context, none of the provided options corresponds to the transformation described.


Step 2: Conclusion.

The correct answer is "None of these" because no option matches the reaction.
Quick Tip: Always verify the reaction conditions and expected products to match the correct transformation.

Step 1: Reaction mechanism.

The van-Arkel method involves the reduction of metal halides using hydrogen and halogens. In this case, B is obtained from its halide by the reaction where 3I is involved in the process.


Step 2: Conclusion.

The correct reaction is \( B_2 \) + 3I, corresponding to option (1).
Quick Tip: The van-Arkel method is used for the purification of metals and involves the reduction of metal halides.

Step 1: Analyzing the reaction.

When \( NH_4 Cl \) is heated, it dissociates into ammonia \( NH_3 \) and HCl. The ammonia diffuses faster than HCl, which changes the litmus paper.


Step 2: Conclusion.

The correct answer is that the ammonia diffuses faster than HCl, corresponding to option (3).
Quick Tip: Ammonia diffuses faster than HCl due to its lower molar mass, which is why it affects the litmus paper first.

Step 1: Identifying peroxy acids.

Peroxy acids are compounds where the oxygen atom is attached to two other oxygen atoms, and they can be formed from sulfuric acid. \( H_2 SO_5 \) is the peroxy acid.


Step 2: Conclusion.

The correct peroxy acid is \( H_2 SO_5 \), corresponding to option (1).
Quick Tip: Peroxy acids have an O-O bond, which is typical of compounds with peroxy groups.

Step 1: Volume and density relationship.

The volume occupied by argon can be calculated from the density, and the volume of a single argon atom can be calculated assuming it is a sphere. By comparing these volumes, we find that 38% of the solid argon is empty space.


Step 2: Conclusion.

The percentage of empty space in solid argon is 38%, which corresponds to option (2).
Quick Tip: To calculate the empty space in a solid, compare the total volume of atoms to the volume occupied by the atoms themselves.

Step 1: Applying the adiabatic condition.

For an adiabatic expansion of gases, the relation \( P_1 V_1^\gamma = P_2 V_2^\gamma \) can be used. Using the given temperature and volume, we can find the final volume.


Step 2: Conclusion.

The final volume of the gas is 80L, which corresponds to option (4).
Quick Tip: In adiabatic processes, the relationship between temperature and volume is given by \( T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1} \).

Step 1: Using the equation for hydrolysis.

For first order reactions, the rate constant \( k \) is related to the change in volume of NaOH required to neutralise the acid. If the ester is 50% neutralised, then the final volume \( V_\infty \) is equal to \( V_t \), the volume at time \( t \).


Step 2: Conclusion.

The correct relationship is \( V_\infty = V_t \), corresponding to option (3).
Quick Tip: For first order reactions, the volume of NaOH required to neutralise the acid can be used to determine the rate of reaction.

Step 1: Understanding the process.

The near UV photon is absorbed and then emitted as two lower energy photons. The wavelength of the second photon is calculated using energy conservation and the properties of photon emission.


Step 2: Conclusion.

The wavelength of the second photon is 496nm, which corresponds to option (2).
Quick Tip: When photons are absorbed and re-emitted as lower energy photons, the sum of their energies is equal to the energy of the absorbed photon.

Step 1: Color of ions in solution.

Transition metal ions such as \( Fe^{3+} \) often exhibit color in solution due to d-d transitions, whereas ions like \( Ni^{2+} \) and \( Al^{3+} \) do not show such color.


Step 2: Conclusion.

The \( Fe^{3+} \) ion is expected to be colored in solution, corresponding to option (1).
Quick Tip: Transition metal ions with unfilled d-orbitals can show color due to d-d transitions when placed in a solution.

Step 1: Understanding the statement.

Alkane nitriles contain a nitrile group, which has both electrophilic and nucleophilic centers, making it a versatile functional group for various reactions.


Step 2: Conclusion.

The correct statement is that alkane nitriles have electrophilic and nucleophilic centers, corresponding to option (2).
Quick Tip: Nitrile groups are versatile and can participate in both nucleophilic and electrophilic reactions.

Step 1: Identifying the reaction.

The reaction mechanism leads to the formation of a chloro nitrobenzene derivative at the meta position due to the positions of the substituents in the starting material.


Step 2: Conclusion.

The product is m-chloro nitrobenzene, corresponding to option (3).
Quick Tip: In electrophilic aromatic substitution, the nitro group directs new substituents to the meta position on the benzene ring.

Step 1: Analyzing the reaction.

The reaction of the given compound with the reactants forms benzoic acid and HBr as products.


Step 2: Conclusion.

The end product is \( C_6 H_5 COOH + HBr \), corresponding to option (1).
Quick Tip: The reaction of a benzene derivative with a halogen in the presence of an electrophilic catalyst often leads to substitution reactions.

Step 1: Identifying isomers.

The compounds are described as geometrical isomers, and the given options are evaluated based on cis-trans isomerism. Option (1) provides the correct sequence of cis and trans forms.


Step 2: Conclusion.

The correct answer is \( cis \), \( cis \), \( trans \), corresponding to option (1).
Quick Tip: Geometrical isomerism is observed when two groups or atoms can occupy different positions around a double bond or a ring structure.

Step 1: Understanding ester basicity.

In esters, the carbonyl oxygen is more basic than the carboxyl oxygen due to its greater ability to donate electron density.


Step 2: Conclusion.

The carbonyl oxygen is more basic, corresponding to option (1).
Quick Tip: The carbonyl oxygen in esters is more basic than the carboxyl oxygen, as it is less electron-withdrawing.

Step 1: Understanding the Claisen condensation mechanism.

In a Claisen condensation, a strong base removes a proton from the \( \alpha \)-carbon of an ester. The resulting carbanion acts as a nucleophile and forms a new \( C-C \) bond with another ester molecule.


Step 2: Conclusion.

All the statements provided are correct, corresponding to option (4).
Quick Tip: Claisen condensation is a key reaction for the formation of carbon-carbon bonds in organic synthesis.

Step 1: Reaction with Grignard reagent.

The reaction of ethyl magnesium iodide (a Grignard reagent) with a carbonyl compound like an aldehyde or ketone forms an alcohol. The fact that \( B \) does not react with PCC or PDC suggests that \( B \) is an alcohol, specifically a secondary alcohol.


Step 2: Conclusion.

The correct compound \( A \) is acetic acid, corresponding to option (1).
Quick Tip: Grignard reagents react with carbonyl compounds to form alcohols, and PCC or PDC can oxidize primary alcohols to aldehydes but not secondary alcohols.

Step 1: Grignard reagent reaction.

Grignard reagents react with carbonyl compounds to form an alcohol. In this case, the reaction with the ketone forms a secondary alcohol.


Step 2: Conclusion.

The product is a secondary alcohol, \( CH_3CH_2CH_2C(OH)CH_3 \), corresponding to option (3).
Quick Tip: Grignard reagents add to carbonyl compounds, forming alcohols, and are particularly useful in forming carbon-carbon bonds.

Step 1: Using the Nernst equation.

The Nernst equation can be used to calculate the cell potential by relating the individual electrode potentials. We can find the value of \( E^\circ_{Cu^{2+}/Cu} \) using the given values.


Step 2: Conclusion.

The value of \( E^\circ_{Cu^{2+}/Cu} \) is 1.61V, corresponding to option (2).
Quick Tip: The standard electrode potentials can be used to calculate the cell potential by using the formula \( E^\circ_{cell} = E^\circ_{cathode} - E^\circ_{anode} \).

Step 1: Calculating moles of Cl\(_2\) produced.

Using Faraday's law and the molar volume of gas at STP, the volume of \( Cl_2 \) gas produced can be calculated by considering the number of moles of electrons involved.


Step 2: Conclusion.

The volume of Cl\(_2\) gas produced is 30.0L, corresponding to option (1).
Quick Tip: Faraday's law can be used to relate the amount of substance produced in an electrolysis reaction to the current and time.

Step 1: Analyzing the reaction.

This reaction involves a conditional situation based on structure and variable outcomes in the reaction system. Conditions for a reaction are discussed.


Step 2: Conclusion.

Option (2) describes the correct outcome, i.e., the structure is based on specific conditions.
Quick Tip: When analyzing reactions, understand the role of conditions and catalysts in directing product formation.

Step 1: Applying the entropy change equation.

The entropy change of a system is related to the energy produced in the system. By using the given values for entropy and potential change, the required value is calculated.


Step 2: Conclusion.

The entropy change value is \( 2 \times 10^4 \, V \, K^{-1} \), corresponding to option (1).
Quick Tip: Entropy change is related to the amount of disorder in a system and can be calculated using thermodynamic equations.

Step 1: Understanding the transitions.

The wavelength of the transition is determined by the change in energy levels. For He\(^+\), the wavelength of the transition from \( n = 4 \) to \( n = 2 \) will match the transition from \( n = 2 \) to \( n = 1 \) in the hydrogen spectrum.


Step 2: Conclusion.

The correct transition is from \( n = 2 \) to \( n = 1 \), corresponding to option (3).
Quick Tip: The wavelengths of transitions in hydrogen-like atoms are related to the change in energy levels and can be calculated using the Rydberg formula.

Step 1: Understanding degeneracy.

The degeneracy of an energy level in the hydrogen atom is given by \( 2n^2 \), where \( n \) is the principal quantum number. For the given energy, \( n = 3 \), so the degeneracy is \( 9 \).


Step 2: Conclusion.

The degeneracy of the level is 9, corresponding to option (2).
Quick Tip: The degeneracy of energy levels in the hydrogen atom is proportional to \( n^2 \), where \( n \) is the principal quantum number.

Step 1: Identifying the correct pairs.

- Dry ice (solid CO\(_2\)) is used as a refrigerant, matching with 4.
- Semiconductor is used in electronic devices, matching with 2.
- Solder is used for joining circuits, matching with 3.
- TEL (Tetraethyl lead) is used as an anti-knocking compound, matching with 1.


Step 2: Conclusion.

The correct match is IV III II I, corresponding to option (4).
Quick Tip: Dry ice, semiconductors, solder, and TEL are all important substances used in various industrial and technological applications.

Step 1: Understanding tetradentate ligands.

A tetradentate ligand is one that can form four bonds with a metal ion. Both \( NH_2 \) and \( N \) can act as tetradentate ligands in certain coordination complexes.


Step 2: Conclusion.

Both \( NH_2 \) and \( N \) are tetradentate ligands, corresponding to option (3).
Quick Tip: Tetradentate ligands are capable of forming four bonds with a metal ion in a coordination complex.

Step 1: Understanding Effective Atomic Number (EAN).

The EAN is calculated by adding the number of electrons from the metal ion and the ligands. For this complex, the number of electrons is 22.


Step 2: Conclusion.

The EAN of the complex is 22, corresponding to option (2).
Quick Tip: The Effective Atomic Number (EAN) is a useful concept in coordination chemistry to determine the stability of a complex.

Step 1: Identifying the relation.

The relation is defined as \( y = |x| \), so each element of the set \( A \) corresponds to the absolute value of \( x \). The correct set corresponds to option (1).


Step 2: Conclusion.

The correct answer is option (1).
Quick Tip: In relations involving absolute values, both positive and negative values of \( x \) yield the same \( y \) value.

Step 1: Analyzing the differential equation.

The given equation is separable and can be solved by integrating. The solution to the differential equation yields the form \( f(x) = y + C + C \).


Step 2: Conclusion.

The correct solution is \( f(x) = y + C + C \), corresponding to option (2).
Quick Tip: To solve differential equations, consider using separation of variables and integration techniques.

Step 1: Analyzing the determinant.

Since the rows and columns of the determinant are linear, we know that the determinant of the matrix will be zero when \( a \), \( b \), and \( c \) are in arithmetic progression.


Step 2: Conclusion.

The determinant equals zero, corresponding to option (1).
Quick Tip: For matrices where rows or columns are in arithmetic progression, the determinant is often zero.

Step 1: Using probability relations.

The odds against \( A \) give \( P(A) = \frac{1}{3} \), and the odds in favor of \( A \cup B \) give \( P(A \cup B) = \frac{3}{4} \). By using these values, we can solve for the range of \( P(B) \).


Step 2: Conclusion.

The probability \( P(B) \) lies between \( \frac{5}{12} \) and \( \frac{3}{4} \), corresponding to option (2).
Quick Tip: To find the probability of events in combination, use the relationship \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \).

Step 1: Simplifying the expression.

The expression involves inverse trigonometric identities. Using standard trigonometric simplifications, we find that the value of the expression simplifies to \( \tan^{-1} x \).


Step 2: Conclusion.

The correct value is \( \tan^{-1} x \), corresponding to option (1).
Quick Tip: Inverse trigonometric identities can be simplified by using basic algebraic operations and known formulas.

Step 1: Logical expression equivalency.

Using logical equivalencies, \( \neg (p \iff q) \) is equivalent to \( (p \vee q) \land (\neg p \vee \neg q) \).


Step 2: Conclusion.

The correct equivalence is \( (p \vee q) \land (\neg p \vee \neg q) \), corresponding to option (2).
Quick Tip: The logical equivalence of negating a biconditional is the disjunction of the negations of the components.

Step 1: Evaluating the logical expression.

Since \( p = F \) and \( q = T \), \( p \vee q \) is T. Therefore, \( \neg(p \vee q) \) is F.


Step 2: Conclusion.

The truth value is F, corresponding to option (2).
Quick Tip: When evaluating logical expressions, first evaluate the truth value of the compound expression, then apply negations.

Step 1: Understanding the surface area change.

The surface area \( A \) of a sphere is given by \( A = 4\pi r^2 \). The rate of change of surface area is proportional to \( r^2 \). Since the radius is increasing at a rate of \( 2 \, cm/s \), the rate of change of surface area is proportional to \( \frac{1}{r^2} \).


Step 2: Conclusion.

The rate of change of the surface area is proportional to \( \frac{1}{r^2} \), corresponding to option (2).
Quick Tip: The rate of change of surface area of a sphere can be found by differentiating the surface area formula with respect to time.

Step 1: Verifying equivalence properties.

The relation satisfies the reflexive, symmetric, and transitive properties. Therefore, it is an equivalence relation.


Step 2: Conclusion.

The correct answer is that the relation is an equivalence relation, corresponding to option (4).
Quick Tip: An equivalence relation must be reflexive, symmetric, and transitive.

Step 1: Understanding the condition.

The argument of a complex number is given by the angle it makes with the real axis. The given condition indicates that the complex number lies on a circle.


Step 2: Conclusion.

The correct answer is that the complex number lies on a circle, corresponding to option (3).
Quick Tip: The argument of a complex number is the angle it forms with the positive real axis in the complex plane.

Step 1: Applying the AM-GM inequality.

By the Arithmetic Mean-Geometric Mean (AM-GM) inequality, we know that for positive real numbers \( a_1, a_2, a_3 \), the arithmetic mean is always greater than or equal to the geometric mean. This gives the relation \( a_1^2 + a_2^2 + a_3^2 \geq 3a_1a_2a_3 \).


Step 2: Conclusion.

The correct statement is option (2).
Quick Tip: The AM-GM inequality states that for positive real numbers, the arithmetic mean is greater than or equal to the geometric mean.

Step 1: Solving the quadratic equation.

The given quadratic equation is \( x^2 + 2x - 5 = 0 \). Using the quadratic formula, we find the values of \( x \) as \( x = -2, 2, 4 \).


Step 2: Conclusion.

The correct values of \( x \) are \( -2, 2, 4 \), corresponding to option (2).
Quick Tip: To solve quadratic equations, use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Step 1: Analyzing the problem.

The centres of the circles lie on the given circle \( x^2 + y^2 = 25 \). The radius of each circle is 3, so the locus of any point in the set lies on this circle.


Step 2: Conclusion.

The locus of the points is \( x^2 + y^2 = 25 \), corresponding to option (1).
Quick Tip: The locus of points forming the centre of a set of circles can be determined by the equation of the circle they lie on.

Step 1: Analyzing the geometry.

The relationship between the height of the tower, the angles of elevation, and the distance from the point of observation can be used to derive the tangent of \( \theta \). Using trigonometry, \( \tan \theta = \frac{1}{2} \).


Step 2: Conclusion.

The correct value of \( \tan \theta \) is \( \frac{1}{2} \), corresponding to option (3).
Quick Tip: Use trigonometric identities and geometry to solve problems involving angles of elevation and depression.

Step 1: Using the given information.

We know that the sum of the roots \( \theta + \gamma = \alpha \) and the product of the roots \( \theta \gamma = \beta \). Using the identity \( \theta^2 + \gamma^2 = (\theta + \gamma)^2 - 2\theta \gamma \), we can substitute the values for \( \theta + \gamma \) and \( \theta \gamma \).


Step 2: Conclusion.

The value of \( \theta^2 + \gamma^2 \) is \( \alpha^2 - 4\beta \), corresponding to option (3).
Quick Tip: To find \( \theta^2 + \gamma^2 \), use the identity \( \theta^2 + \gamma^2 = (\theta + \gamma)^2 - 2\theta \gamma \).

Step 1: Analyzing the intersection.

The angle of intersection between two circles is given by the formula involving the slopes of the tangents to the circles at the points of intersection. After finding the slopes, we can compute the angle using the tangent inverse.


Step 2: Conclusion.

The angle of intersection is \( \tan^{-1} \left( \frac{9}{8} \right) \), corresponding to option (1).
Quick Tip: The angle of intersection of two curves can be calculated using the formula \( \theta = \tan^{-1} \left( \frac{m_1 - m_2}{1 + m_1 m_2} \right) \), where \( m_1 \) and \( m_2 \) are the slopes of the tangents.

Step 1: Understanding the series.

The given series is a binomial expansion series that can be simplified using the general formula for a geometric series. After evaluating, the sum is found to be \( 10^6 \).


Step 2: Conclusion.

The correct expansion value is \( 10^6 \), corresponding to option (3).
Quick Tip: To expand binomial series, first express the series in a standard form and then apply geometric series summation techniques.

Step 1: Decomposing the vector.

To find the component of the vector along \( \hat{i} + \hat{k} \), use the projection formula. The resulting vector will be \( 3\hat{i} + 2\hat{k} \).


Step 2: Conclusion.

The correct value of \( \mathbf{v} \) is \( 3\hat{i} + 2\hat{k} \), corresponding to option (1).
Quick Tip: To decompose a vector into components, use the projection formula to find the vector parallel to a given direction.

Step 1: Understanding collinearity.

When three points are collinear, the rank of the matrix formed by these points will always be 2, as they lie on a straight line.


Step 2: Conclusion.

The correct rank is 2, corresponding to option (2).
Quick Tip: When points are collinear, the matrix formed by the coordinates of these points will have a rank of 2.

Step 1: Evaluating the determinant.

To evaluate this determinant, apply cofactor expansion. The value of the determinant simplifies to \( \alpha^2 - \beta^2 \).


Step 2: Conclusion.

The correct answer is \( \alpha^2 - \beta^2 \), corresponding to option (2).
Quick Tip: To solve matrix determinants, use cofactor expansion or row/column operations.

Step 1: Solving for \( K \).

We solve the equation for \( K \) by equating it with the general form of a sinusoidal equation. The number of possible integral values of \( K \) is 8.


Step 2: Conclusion.

The correct number of integral values of \( K \) is 8, corresponding to option (2).
Quick Tip: For trigonometric equations, find the range of the expression and solve for the variable.

Step 1: Rotating the line.

The new equation after rotating the line can be obtained using the rotation formula for coordinates. After applying the transformation, the equation becomes \( \sqrt{3}x - y - 2\sqrt{5} = 0 \).


Step 2: Conclusion.

The new equation of the line is \( \sqrt{3}x - y - 2\sqrt{5} = 0 \), corresponding to option (1).
Quick Tip: To rotate a line, use the rotation transformation for coordinates: \( x' = x \cos \theta - y \sin \theta \), \( y' = x \sin \theta + y \cos \theta \).

Step 1: Finding the point of contact.

The equation of the line and the curve are used to find the point of contact by substituting the values and solving the system of equations.


Step 2: Conclusion.

The point of contact is \( (4, -6) \), corresponding to option (1).
Quick Tip: To find the point of contact between a line and a curve, solve the system of equations formed by the line equation and the curve equation.

Step 1: Using Pick's Theorem.

Pick's Theorem gives a way to calculate the number of lattice points inside a polygon. The formula is \( A = I + \frac{B}{2} - 1 \), where \( A \) is the area, \( I \) is the number of interior lattice points, and \( B \) is the number of boundary lattice points.


Step 2: Conclusion.

The number of interior lattice points is 105, corresponding to option (3).
Quick Tip: Pick's Theorem relates the area of a polygon to the number of lattice points inside and on the boundary of the polygon.

Step 1: Simplifying the expression.

The integral is solved by breaking down the expression and using integration by parts. After solving, we find that the result is \( e^x + C \).


Step 2: Conclusion.

The integral evaluates to \( e^x + C \), corresponding to option (3).
Quick Tip: To solve integrals involving exponential functions, consider using integration by parts or simplifying the expression.

Step 1: Understanding the function.

The function \( f(x) = x - \lfloor x \rfloor \) is the fractional part of \( x \), which is periodic with a period of 1. The integral over one period results in 0.


Step 2: Conclusion.

The value of the integral is 0, corresponding to option (1).
Quick Tip: The fractional part function is periodic, and its integral over one period is 0.

Step 1: Simplifying the integral.

This integral is solved by substitution and breaking it down into simpler components. We use standard integration techniques to solve the integral.


Step 2: Conclusion.

The value of the integral is \( 4 \log 4 \), corresponding to option (3).
Quick Tip: When integrating complex rational functions, use substitution and simplify the integrand before attempting to solve.

Step 1: Using geometry of the ellipse.

The area of \( \triangle OAB \) can be calculated using the lengths of the major and minor axes and applying the appropriate geometric formulas for triangles.


Step 2: Conclusion.

The area of \( \triangle OAB \) is 48 sq units, corresponding to option (1).
Quick Tip: For tangents to ellipses, use the equation of the tangent and geometry of the ellipse to find the area of the triangle formed by the tangent and axes.

Step 1: Understanding the locus.

The locus of midpoints of tangents intercepted between the axes of an ellipse forms an ellipse with a scaled equation. This scaled ellipse has a factor of 2 in the equation.


Step 2: Conclusion.

The correct equation for the locus is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 2 \), corresponding to option (1).
Quick Tip: The locus of midpoints of tangents to an ellipse is another ellipse with a scaled equation.

Step 1: Using properties of hyperbolas.

For a double ordinate of a hyperbola and an equilateral triangle formed with the center of the hyperbola, the eccentricity satisfies the condition \( 1 < e < \frac{2}{\sqrt{3}} \).


Step 2: Conclusion.

The correct range for the eccentricity is \( 1 < e < \frac{2}{\sqrt{3}} \), corresponding to option (1).
Quick Tip: For hyperbolas with double ordinates, use the properties of the eccentricity and the geometry of the curve to solve for the required values.

Step 1: Understanding the problem.

The number of triangles that can be formed is given by selecting 3 points from the 3 sides. The calculation uses combinatorics and the number of ways to select points on each side.


Step 2: Conclusion.

The total number of triangles that can be formed is 205, corresponding to option (1).
Quick Tip: To find the number of triangles from given points, use combinations to select points from each side.

Step 1: Identifying the expansion.

The expansion of \( a + bx \) will result in terms of the form \( ax^r \). The coefficient of \( x^r \) is found by using the general form of the expansion and applying the binomial theorem.


Step 2: Conclusion.

The coefficient of \( x^r \) is \( (1 - r) a - br \), corresponding to option (3).
Quick Tip: Use the binomial expansion to find coefficients of terms in algebraic expressions.

Step 1: Using logarithmic properties.

The sum of logarithms can be expressed as the logarithm of the product. Therefore, \( \sum_{i=1}^{1999} \log x_i \) is equal to \( \log(1999!) \).


Step 2: Conclusion.

The correct answer is \( \log 1999! \), corresponding to option (4).
Quick Tip: The sum of logarithms can be simplified to the logarithm of the product of the terms.

Step 1: Understanding the geometry.

The intercepts of the plane with the axes form a relationship based on the geometry of the situation. By applying properties of planes and intercepts, we find the sum of the reciprocals equals \( \frac{3a}{2} \).


Step 2: Conclusion.

The sum of the reciprocals is \( \frac{3a}{2} \), corresponding to option (1).
Quick Tip: To find the sum of reciprocals of intercepts, use the properties of planes and geometry.

Step 1: Understanding the problem.

We are given the curves \( y = x - x^2 \) and \( y = mx \), and we need to find the value of \( m \) such that the area enclosed by the curves is 5. This can be done by setting up the area integral and solving for \( m \).


Step 2: Conclusion.

The value of \( m \) is \( -2 \), corresponding to option (2).
Quick Tip: To find the area between curves, integrate the difference of the functions over the range of intersection.

Step 1: Analyzing the derivative.

Given that \( f'(x) = 6 \), the solution function that satisfies the derivative and the given values is \( e^{3x} \). The function \( f(x) = e^{3x} \) satisfies \( f(1) = 3 \) and \( f'(1) = 6 \).


Step 2: Conclusion.

The correct function is \( e^{3x} \), corresponding to option (3).
Quick Tip: To find a function given its derivative and specific values, integrate the derivative and use initial conditions.

Step 1: Applying continuity conditions.

For the function to be continuous at \( x = 0 \), the values of \( a \) and \( b \) must satisfy the condition \( \lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) \). Using this condition, we solve for \( a = 2 \) and \( b = e^3 \).


Step 2: Conclusion.

The values of \( a \) and \( b \) are \( a = 2 \) and \( b = e^3 \), corresponding to option (2).
Quick Tip: For continuity at a point, the left-hand and right-hand limits must be equal at that point.

Step 1: Determining the domain.

The domain of the function is determined by the constraints on the logarithmic and square root terms. The function is valid when \( 1 - x > 0 \) and \( x + 2 \geq 0 \). Solving these inequalities gives \( [-3, 2] \).


Step 2: Conclusion.

The domain of the function is \( [-3, 2] \), corresponding to option (2).
Quick Tip: To find the domain of a function, check the constraints on logarithms and square roots to ensure the arguments are within the allowed range.

Step 1: Solving the differential equation.

The given differential equation is separable and can be solved by separation of variables. After solving, we find that the solution is \( x \cdot e^{2y} = e^y + K \).


Step 2: Conclusion.

The correct solution is \( x \cdot e^{2y} = e^y + K \), corresponding to option (2).
Quick Tip: To solve separable differential equations, separate variables and integrate each side independently.

Step 1: Analyzing the gradient.

The given expression for the gradient of the tangent involves both \( x \) and \( y \). Using the relationship \( \frac{dy}{dx} = \frac{1}{x} \), we integrate the equation to find the equation of the curve.


Step 2: Conclusion.

The equation of the curve is \( y = \cot(\log x) \), corresponding to option (1).
Quick Tip: To solve gradient-based problems, differentiate and integrate to find the equation of the curve.