VITEEE 2021 Question Paper (Available) - Download VITEEE Question Paper with Solution PDF and Answer Key

Step 1: Gravitational Force at Zero.

At the point where the gravitational force between the moon and the earth is zero, the forces due to both bodies must be equal in magnitude and opposite in direction.


Step 2: Gravitational force equations.

Let the distance from the centre of the earth be \( r \). Then, using the inverse square law of gravitation, we can express the force due to the earth and moon at distance \( r \) and \( D - r \).


Step 3: Solving the equation.

After solving for \( r \), we find that the distance at which the gravitational force is zero is \( \frac{4D}{3} \).
Quick Tip: For the gravitational force to be zero, the magnitudes of the forces from both bodies must be equal.

Step 1: Understand the relationship.

The increase in length of the wire is proportional to the force applied and inversely proportional to the product of the material's Young's modulus and the cross-sectional area.


Step 2: Apply the formula.

From the formula \( \Delta L = \frac{F L}{A Y} \), where \( F \) is the force, \( L \) is the length, \( A \) is the cross-sectional area, and \( Y \) is the Young's modulus.


Step 3: Compare the two wires.

Since the lengths are in the ratio \( 1:2 \) and the diameters in the ratio \( 2:1 \), the increase in length will be in the ratio \( 1:4 \).
Quick Tip: The increase in length is inversely proportional to the cross-sectional area, so smaller cross-sectional area results in a larger increase in length.

Step 1: Analyzing the equation.

The equation given is \( x = at + bt^2 \). The distance \( x \) is in kilometers, and time \( t \) is in seconds.


Step 2: Determine the unit of \( b \).

In the term \( bt^2 \), \( t^2 \) has units of seconds squared, and for the equation to be dimensionally consistent, \( b \) must have units of km/s\(^2\).


Step 3: Conclusion.

The unit of \( b \) is km/s\(^2\).
Quick Tip: Always check the units on both sides of the equation to ensure consistency.

Step 1: Understanding the problem.

In the case of two soap bubbles, the film separating the bubbles will have a radius equal to the sum of the radii of the two bubbles because of the pressure difference between the inside and outside of the bubbles.


Step 2: Applying the principle.

The pressure difference between the two soap films results in the separation distance being the sum of their radii. Thus, the radius of the film separating the bubbles is \( r_2 + r_1 \).
Quick Tip: The film separating two soap bubbles has a radius equal to the sum of the radii of the bubbles due to the pressure difference.

Step 1: Understanding the force.

The force acting on a charged particle moving in a magnetic field is given by the equation \( F = qvB \sin \theta \), where \( \theta \) is the angle between the velocity and magnetic field. For \( \theta = 90^\circ \), this simplifies to \( F = qvB \).


Step 2: Conclusion.

The force on the charge is \( qvB \).
Quick Tip: The force on a moving charge in a magnetic field is given by \( F = qvB \sin \theta \).

Step 1: Understanding the center of mass.

The center of mass of a system is determined by the weighted average of the positions of the masses.


Step 2: Calculation for two spheres.

In this case, the COM of the two spheres lies at the point of contact because their centers are aligned, and their masses are in the ratio 1:2.
Quick Tip: The center of mass of two spheres in contact lies at the point of contact if their masses are in a simple ratio.

Step 1: Understanding friction types.

Static friction is the force that resists the initiation of sliding motion between two surfaces, while kinetic friction is the force that opposes the motion once the surfaces are sliding.


Step 2: Comparing coefficients.

The coefficient of static friction is generally higher than the coefficient of kinetic friction because it requires more force to initiate motion than to keep an object in motion.


Step 3: Conclusion.

Thus, the correct answer is that the coefficient of static friction is more than the coefficient of kinetic friction.
Quick Tip: Static friction is always greater than kinetic friction for the same surfaces and materials.

Step 1: Use the equation of motion.

The distance travelled in the \(n^{th}\) second is given by the equation: \[ S_n = u + \frac{a(2n-1)}{2} \]
where \( u \) is the initial velocity, \( a \) is the acceleration, and \( n \) is the second.


Step 2: Apply the values.

For the third second, we have \( u = 0 \), \( a = 3 \, m/s^2 \), and \( n = 3 \). Substituting these values, we get: \[ S_3 = 0 + \frac{3(2(3)-1)}{2} = 20 \, m. \] Quick Tip: Use the equation of motion to calculate the distance travelled in a particular second when acceleration is constant.

Step 1: Energy of the photon.

The energy of the photon can be calculated using the equation: \[ E = \phi + K.E. \]
where \( \phi \) is the work function and \( K.E. \) is the kinetic energy of the emitted electron.


Step 2: Calculate kinetic energy.

The kinetic energy is given by: \[ K.E. = \frac{1}{2} m v^2 \]
where \( m = 9.1 \times 10^{-31} \, kg \) and \( v = 1 \times 10^6 \, m/s \). Substituting these values, we get: \[ K.E. = 4.55 \times 10^{-19} \, J = 2.8 \, eV. \]
Thus, the total energy of the photon is: \[ E = 1 \, eV + 2 \, eV = 3 \, eV. \] Quick Tip: To find the energy of the photon, use the work function and the kinetic energy of the emitted photoelectron.

Step 1: Use the formula for efficiency.

The efficiency of a heat engine is given by: \[ \eta = \frac{W}{Q_H} \]
where \( W \) is the work done, and \( Q_H \) is the heat absorbed.


Step 2: Apply the efficiency.

Given that the efficiency \( \eta = 25% = 0.25 \), and the work done \( W = Q_H - Q_C \), where \( Q_C \) is the heat rejected. The efficiency can be used to find the heat absorbed.
Quick Tip: For heat engines, efficiency is the ratio of work done to the heat absorbed.

Step 1: Electric potential due to point charges.

The electric potential at a point due to a point charge is given by: \[ V = \frac{kq}{r} \]
where \( k \) is Coulomb's constant, \( q \) is the charge, and \( r \) is the distance from the charge.


Step 2: Calculate the potential at the midpoint.

At the midpoint between the charges \( +q \) and \( -q \), the potentials due to both charges cancel each other out, resulting in a net potential of zero.
Quick Tip: When two equal but opposite charges are placed symmetrically, the electric potential at the midpoint will be zero.

Step 1: Formula for horizontal range.

The horizontal range \( R \) of a projectile is given by the formula: \[ R = \frac{v^2 \sin 2\theta}{g} \]
where \( v \) is the velocity, \( \theta \) is the angle of projection, and \( g \) is the acceleration due to gravity.


Step 2: Comparing ranges.

Since the velocities and the acceleration due to gravity are the same for both cases, the horizontal ranges will be the same because: \[ \sin 2 \times 30^\circ = \sin 60^\circ \quad and \quad \sin 2 \times 60^\circ = \sin 120^\circ \]
Thus, the ratio of the horizontal ranges is \( 1 : 1 \).
Quick Tip: For two projectiles with the same velocity, the horizontal ranges are equal if the angles of projection sum to \( 90^\circ \).

Step 1: Use Coulomb's Law.

The force between two charges is given by Coulomb's law: \[ F = \frac{k q_1 q_2}{r^2} \]
where \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between them.


Step 2: Apply the new charges.

When \( -5 \, \mu C \) is added to each charge, the new charges become \( -2 \, \mu C \) and \( +3 \, \mu C \). Substituting these into Coulomb’s law, the new force is: \[ F_{new} = \frac{k (-2 \, \mu C)(3 \, \mu C)}{r^2} = 10 \, N \] Quick Tip: When charges of opposite signs are added to each other, the force can change its direction and magnitude based on the new charge configuration.

Step 1: Rolling motion equation.

For a rolling sphere, the acceleration \( a \) is given by: \[ a = \frac{g \sin \theta}{1 + k^2/r^2} \]
where \( k \) is the radius of gyration and \( r \) is the radius of the sphere. For a solid sphere, \( k^2/r^2 = 2/5 \).


Step 2: Substitute values.

Substituting \( k^2/r^2 = 2/5 \) into the equation, we get the acceleration as: \[ a = \frac{5g}{7} \sin \theta \] Quick Tip: For a rolling body, the acceleration is affected by the moment of inertia. A solid sphere has \( k^2/r^2 = 2/5 \).

Step 1: Understanding the deviation.

In a prism, the deviation of light depends on the angle of the prism and the refractive index of the material.


Step 2: Adding additional prisms.

When additional prisms are placed in sequence, the total deviation will increase if the angle of incidence remains the same, leading to a greater deviation.


Step 3: Conclusion.

Thus, the ray will now suffer a greater deviation.
Quick Tip: The total deviation increases with the addition of more prisms, as it alters the path of the light further.

Step 1: Apply Ohm’s Law.

From Ohm’s Law, the current \( I \) is given by: \[ I = \frac{V}{R} \]
where \( V \) is the voltage and \( R \) is the resistance.


Step 2: Substitute the values.

Given \( V = 36 \, V \) and \( R = 120 \, \Omega \), the current is: \[ I = \frac{36}{120} = 0.3 \, A \] Quick Tip: Always use Ohm’s Law to calculate current when voltage and resistance are known.

Step 1: Use the equation for root mean square velocity.

The root mean square velocity \( v_{rms} \) is related to temperature by the equation: \[ v_{rms} = \sqrt{\frac{3kT}{m}} \]
where \( k \) is Boltzmann's constant, \( T \) is temperature, and \( m \) is the mass of the molecule.


Step 2: Use the temperature ratio.

The velocity ratio for two gases is given by: \[ \frac{v_{rms2}}{v_{rms1}} = \sqrt{\frac{T_2}{T_1}} \]
Substituting the values for hydrogen and oxygen, we get: \[ v_{rms2} = \sqrt{\frac{1200}{300}} \times 1930 = 586 \, m/s \] Quick Tip: The velocity of molecules is proportional to the square root of the temperature.

Step 1: Use Faraday's Law of Induction.

According to Faraday’s Law, the induced emf is given by: \[ \mathcal{E} = N \frac{d\Phi}{dt} \]
where \( N \) is the number of turns, \( \Phi \) is the magnetic flux, and \( \frac{d\Phi}{dt} \) is the rate of change of magnetic flux.


Step 2: Calculate the rate of change of flux.

Substituting the given values: \[ 10 = 5 \times \frac{d\Phi}{dt} \quad \Rightarrow \quad \frac{d\Phi}{dt} = 2 \, Wb/s \] Quick Tip: The induced emf is proportional to the rate of change of magnetic flux through the coil.

Step 1: Formula for capacitance.

The capacitance \( C \) of a parallel plate capacitor is given by: \[ C = \frac{\varepsilon A}{d} \]
where \( \varepsilon \) is the dielectric constant, \( A \) is the area of the plates, and \( d \) is the distance between them.


Step 2: Effects of halving the distance and introducing the dielectric.

If the distance is reduced by half and the dielectric constant \( k \) is introduced, the capacitance becomes: \[ C' = k \times \frac{\varepsilon A}{d/2} = 2k \times C \]
Substituting \( k = 6 \), we get: \[ C' = 72 \, \mu F \] Quick Tip: Capacitance increases when the distance between plates is reduced or when a dielectric is introduced.

Step 1: Relationship between potential energy and displacement.

The potential energy \( PE \) in SHM is proportional to the square of the displacement, given by: \[ PE = \frac{1}{2} k x^2 \]
where \( x \) is the displacement and \( k \) is the spring constant.


Step 2: Graph of potential energy.

Since the displacement follows a cosine function, the potential energy will vary as the square of the cosine function, which matches graph II.
Quick Tip: In SHM, potential energy varies as the square of displacement, and it has a periodic behavior.

Step 1: Use the decay formula.

The decay of a radioactive substance follows the exponential decay formula: \[ N(t) = N_0 e^{-\lambda t} \]
where \( N_0 \) is the initial quantity, \( \lambda \) is the decay constant, and \( t \) is time.


Step 2: Calculate the decay fraction.

The fraction of A that decays in 3 days can be calculated using the half-life formula \( \lambda = \frac{\ln 2}{t_{1/2}} \). For A, with a half-life of 5 days, we find: \[ Fraction decayed = 1 - e^{-\lambda t} = 1 - e^{-\frac{3}{5}} \approx 0.2 \] Quick Tip: The fraction decayed in a given time can be calculated using the exponential decay formula with the decay constant derived from the half-life.

Step 1: Use the wave speed formula.

The speed \( v \) of a wave on a string is given by: \[ v = \sqrt{\frac{T}{\mu}} \]
where \( T \) is the tension and \( \mu \) is the linear mass density (\( \mu = \frac{m}{L} \)).


Step 2: Apply the values.

Given \( T = 50 \, N \), \( m = 0.035 \, kg \), and \( L = 3 \, m \), we find: \[ \mu = \frac{0.035}{3} = 0.01167 \, kg/m \]
Substituting into the formula for \( v \), we get: \[ v = \sqrt{\frac{50}{0.01167}} = 15.4 \, m/s \] Quick Tip: The speed of a wave on a string depends on the tension and the linear mass density of the string.

Step 1: Identify the logic gate.

The given circuit is a combination of transistors that forms an AND gate. The output is high only when both inputs are high.


Step 2: Conclusion.

Thus, the given circuit represents an AND gate.
Quick Tip: An AND gate outputs high only when both inputs are high.

Step 1: Formula for kinetic energy.

The kinetic energy \( K.E. \) is given by: \[ K.E. = \frac{1}{2} m v^2 \]
where \( m = 10 \, kg \) and \( v = 5 \, m/s \).


Step 2: Calculate kinetic energy.

Substituting the values, we get: \[ K.E. = \frac{1}{2} \times 10 \times 5^2 = 250 \, J \] Quick Tip: Kinetic energy is proportional to the square of the velocity, so doubling the velocity increases the kinetic energy by a factor of four.

Step 1: Use the Doppler effect formula.

When the source of sound is approaching the observer, the apparent frequency \( f' \) is given by the Doppler effect equation: \[ f' = f \left( \frac{v + v_s}{v} \right) \]
where \( f \) is the frequency of the source, \( v \) is the speed of sound, and \( v_s \) is the velocity of the source.


Step 2: Calculate the wavelength change.

The change in wavelength \( \Delta \lambda \) can be found using the relation \( \Delta \lambda = \frac{v}{f} - \lambda_0 \), where \( \lambda_0 \) is the initial wavelength.


Step 3: Apply the values.

Given \( f = 170 \, Hz \), \( v = 340 \, m/s \), and \( v_s = 17 \, m/s \), the apparent wavelength change is: \[ \Delta \lambda = \frac{340}{170} \times \left( \frac{340 + 17}{340} \right) \approx 0.2 \, m \] Quick Tip: The apparent wavelength change for a moving source can be calculated using the Doppler effect equation.

Step 1: Analyze the reactions.

We are given two reactions. To calculate the total number of atoms, we need to balance the elements in the reactants and products.


Step 2: Calculate the total number of atoms.

In one molecule of each compound (A), (B), and (C), we calculate the sum of the atoms involved. The total comes out to be 18.
Quick Tip: To calculate the total number of atoms, carefully balance the chemical equation and count the atoms on both sides of the reaction.

Step 1: Hydrolysis of Xenon Hexafluoride.

On partial hydrolysis, xenon hexafluoride reacts with water to form various compounds. The products formed are Xenon oxyfluoride and xenon dioxide fluoride.


Step 2: Oxidation states.

The oxidation state of Xenon in XeOF\(_4\) is +6, and in XeO\(_2\)F\(_2\), it is also +6.
Quick Tip: The oxidation states of elements in oxyfluorides can be determined by balancing the charges of the components.

Step 1: Use the formula for density.

The density \( \rho \) of a cubic unit cell is given by: \[ \rho = \frac{Z M}{N_A V_{cell}} \]
where \( Z \) is the number of atoms per unit cell, \( M \) is the molar mass, \( N_A \) is Avogadro's number, and \( V_{cell} \) is the volume of the unit cell.


Step 2: Calculate the radius.

Using the given values for \( M \), \( \rho \), and the unit cell edge length, we find the radius of the metal atom to be 217 pm.
Quick Tip: The radius of the metal atom can be determined from the density and molar mass by using the relationship between these quantities.

Step 1: Analyzing the statements.

- Statement I is false because for zero-order reactions, the rate is independent of the concentration of the reactant.
- Statement II is true because the rate constant is equal to collision frequency when \( E_a = 0 \).
- Statement III is false because the rate constant is zero when \( E_a = \infty \).
- Statement IV is true because \( \ln k \) vs \( T \) forms a straight line in the Arrhenius equation.
- Statement V is true because \( 1/T \) vs \( \ln k \) is a straight line in the Arrhenius equation.


Step 2: Conclusion.

Thus, the correct statements are II and V.
Quick Tip: The Arrhenius equation relates the rate constant and temperature, where the logarithm of the rate constant is a linear function of \( \frac{1}{T} \).

Step 1: Use Faraday’s Law of Electrolysis.

The amount of electricity required is given by: \[ Q = \frac{m M}{z F} \]
where \( m \) is the mass of the substance deposited, \( M \) is the molar mass of copper, \( z \) is the valency of copper, and \( F \) is Faraday's constant.


Step 2: Apply the values.

For copper, \( M = 63.5 \, g/mol \), \( z = 2 \), and \( F = 96500 \, C/mol \). Substituting the values, we get: \[ Q = \frac{0.634 \times 63.5}{2 \times 96500} \approx 1930 \, C \] Quick Tip: Use Faraday's law to calculate the amount of electricity required for electrolysis by considering the mass of the substance, its molar mass, and valency.

Step 1: Understanding the skew conformation.

The skew conformation of ethane has the hydrogen atoms on the two carbons in a staggered arrangement. This minimizes repulsion and gives the dihedral angle of 149°.


Step 2: Conclusion.

Therefore, the dihedral angle between \( H' - C - C - H'' \) is 149°.
Quick Tip: The staggered conformation of ethane minimizes steric repulsion and results in a dihedral angle of approximately 149°.

Step 1: Understand the reagents.

- NaBH\(_4\) reduces the aldehyde group to a primary alcohol.
- Pb\(_3\) is involved in the formation of a reaction intermediate.
- Mg/ether helps in Grignard reagent formation.
- CO\(_2\)/H\(_2\)O introduces a carboxyl group, forming a carboxylic acid.


Step 2: Final product.

The final product after these reactions is COOH (carboxylic acid).
Quick Tip: Grignard reagents react with CO\(_2\) to form carboxylic acids, and NaBH\(_4\) reduces aldehydes to alcohols.

Step 1: Stepwise analysis.

- The reaction with P\(_1,2\) converts ethanol to an aldehyde (propanal).
- The reaction with Mg/ether forms a Grignard reagent.
- The reaction with formaldehyde results in a secondary alcohol (butanol).
- The final reaction with water gives n-propyl alcohol.


Step 2: Conclusion.

Thus, compound D is n-propyl alcohol.
Quick Tip: Grignard reagents react with formaldehyde to form alcohols.

Step 1: Reduction of nitrobenzene.

Aniline can be obtained by reducing nitrobenzene with zinc in hydrochloric acid.


Step 2: Final product.

Thus, the correct reaction is \( C_6H_5NO_2 + ZnHCl \), which produces aniline as the main product.
Quick Tip: Reduction of nitrobenzene with zinc in hydrochloric acid gives aniline.

The secondary structure of a protein refers to the regular folding of continuous portions of the polypeptide chain, such as alpha-helices and beta-pleated sheets.


Step 2: Conclusion.

Thus, the correct answer is (4) regular folding patterns of continuous portions of the polypeptide chain.
Quick Tip: The secondary structure of proteins includes alpha helices and beta sheets, formed by hydrogen bonds between backbone atoms.

The values of e/m for different particles (e = electron, p = proton, n = neutron, \( \alpha \) = alpha particle) follow this increasing order: \[ Neutron (n) < Alpha particle (\alpha) < Proton (p) < Electron (e) \] Quick Tip: The electron has the largest e/m ratio among the fundamental particles.

Both \( Ni^{2+} \) and \( Ti^{3+} \) are coloured ions in aqueous solutions, whereas \( Sc^{3+} \) and \( Cu^{2+} \) are colourless.


Step 2: Conclusion.

Thus, the correct answer is (4) \( Ni^{2+}, Ti^{3+} \).
Quick Tip: Transition metal ions with partially filled d-orbitals typically form coloured solutions due to electronic transitions.

For square-planar coordination complexes, the number of possible isomers depends on the ligands and their positions. For the given complex, we find that there are 12 possible isomers.


Step 2: Conclusion.

Thus, the correct answer is 12.
Quick Tip: Isomerism in coordination compounds is determined by the positions of ligands and their ability to form different spatial arrangements.

A large equilibrium constant does suggest that the reaction favors the formation of products. However, catalysts are needed to increase the rate of achieving equilibrium, not to shift the equilibrium. Thus, option (1) is incorrect.


Step 2: Conclusion.

Thus, the correct answer is (1).
Quick Tip: A catalyst speeds up a reaction by lowering the activation energy but does not affect the equilibrium constant.

For a reaction with half-life inversely proportional to the square of the initial concentration, the reaction follows second-order kinetics. This can be derived from the integrated rate law for a second-order reaction.


Step 2: Conclusion.

Thus, the correct answer is (3) second order.
Quick Tip: For second-order reactions, the half-life is inversely proportional to the initial concentration.

The standard e.m.f of the cell is given by: \[ E^\circ_{cell} = E^\circ_{cathode} - E^\circ_{anode} = 1.33 \, V - 0.77 \, V = 0.56 \, V \]
Current flows from anode to cathode, and the anode is where oxidation occurs, so electrode A (Cr) will act as the anode, and electrode B (Fe) will act as the cathode. Therefore, the given statements are true, making option (4) correct.
Quick Tip: In a galvanic cell, current flows from the anode to the cathode in the external circuit. The standard e.m.f is the difference between the reduction potentials of the two electrodes.

Keto-enol tautomerism is a type of isomerism where a compound exists in two forms: a keto form and an enol form. Both compounds 1 and 2 exhibit this type of tautomerism.


Step 2: Conclusion.

Thus, the correct answer is (4) Both (1) and (2).
Quick Tip: In keto-enol tautomerism, the equilibrium between the keto form and enol form depends on factors like temperature and solvent.

- The reaction of ethylbenzene with \( KMnO_4 \) results in oxidation, forming a carboxylic acid group.
- The subsequent reaction with \( FeCl_3 \) leads to substitution at the benzylic position.
- Finally, the hydration of the product gives the final product as \( Br \), i.e., a bromo-substituted product.


Step 2: Conclusion.

Thus, the correct product is (4) Br.
Quick Tip: The reaction of ethylbenzene with oxidizing agents such as \( KMnO_4 \) leads to carboxylic acids, and subsequent reactions can lead to substitution.

Step 1: Hydrolysis of nitrile.

The hydrolysis of a nitrile (CH\(_3\)CH\(_2\)CN) leads to the formation of an amide (CH\(_3\)CH\(_2\)CONH\(_2\)).


Step 2: Reaction with ammonia.

When the amide reacts with ammonia, it undergoes reduction to form an amine.


Step 3: Conclusion.

The final product is the primary amine \( CH_3CH_2NH_2 \).
Quick Tip: Amides can be converted to amines by reduction with ammonia.

Step 1: Reaction of acetic acid with thionyl chloride.

Acetic acid reacts with thionyl chloride (SOCl\(_2\)) to form acetyl chloride.


Step 2: Friedel-Crafts acylation.

Acetyl chloride reacts with benzene in the presence of AlCl\(_3\) to form acetophenone.


Step 3: Hydrolysis.

The next reaction involves hydrolysis of the nitrile group to form a carboxylic acid, yielding the final product as \( C_6H_5COOH \) (benzoic acid).
Quick Tip: Friedel-Crafts acylation is used to add an acyl group (RCO-) to an aromatic ring.

Fructose has multiple chiral centers, leading to the formation of 16 optical isomers. This can be calculated by the formula \( 2^n \), where \( n \) is the number of chiral centers. Fructose has 4 chiral centers, so the number of optical isomers is \( 2^4 = 16 \).


Step 2: Conclusion.

Thus, the correct answer is 16.
Quick Tip: The number of optical isomers is \( 2^n \), where \( n \) is the number of chiral centers in the molecule.

The uncertainty principle \( \Delta x \, \Delta p \geq \frac{h}{4\pi} \) relates the uncertainty in position and momentum. Since the mass of the helium atom is larger than that of the electron, the uncertainty in its momentum is smaller. Therefore, the minimum uncertainty in the momentum of the helium atom is \( 5.0 \times 10^{-26} \, kg \, ms^{-1} \).


Step 2: Conclusion.

Thus, the correct answer is \( 5.0 \times 10^{-26} \, kg \, ms^{-1} \).
Quick Tip: The uncertainty in momentum decreases with the mass of the particle. The helium atom's larger mass results in a smaller uncertainty compared to the electron.

To calculate the value of \( \log_{10} K \), we use the equation: \[ \log_{10} K = \frac{-\Delta G^\circ}{2.303RT} \]
Using the relation \( \Delta G^\circ = \Delta H^\circ - T \Delta S^\circ \), we can substitute the values and simplify the expression, resulting in \( \log_{10} K = 10 \).
Quick Tip: To find \( \log_{10} K \), use the relationship between Gibbs free energy, enthalpy, and entropy, along with the gas constant \( R \).

The heat of formation of CO can be derived from Hess's Law, by adding and subtracting the enthalpy changes of the reactions. The result is \( R - S \).


Step 2: Conclusion.

Thus, the correct answer is \( R - S \).
Quick Tip: Hess's law allows us to calculate the heat of formation by adding or subtracting known enthalpy changes.

Markovnikov's rule states that in the addition of HX to an alkene, the hydrogen atom adds to the carbon with the greater number of hydrogen atoms. The exception is the compound \( CH_3CH_2C = CH_2 \), which does not follow this rule.


Step 2: Conclusion.

Thus, the correct answer is \( CH_3CH_2C = CH_2 \).
Quick Tip: Markovnikov's law is followed in most alkene additions, but exceptions occur due to factors like stability of carbocation intermediates.

According to Rolle’s Theorem, there exists at least one point \( c \) in \( (a, b) \) such that \( f'(c) = 0 \), where \( f(a) = f(b) \). For the given function, we find that \( c = \frac{3\pi}{4} \) satisfies the condition.
Quick Tip: Rolle's Theorem guarantees the existence of at least one point where the derivative is zero if the function is continuous and differentiable on the interval, and \( f(a) = f(b) \).

Solving the system of equations, we find that there exists a unique solution for \( x \) and \( y \), making the system consistent with a unique solution.


Step 2: Conclusion.

Thus, the correct answer is (1).
Quick Tip: A system of linear equations is consistent and has a unique solution if the determinant of the coefficient matrix is non-zero.

To calculate the shortest distance between the two skew lines, we use the formula for the distance between two skew lines. After solving, we find the shortest distance to be 2 units.


Step 2: Conclusion.

Thus, the correct answer is 2.
Quick Tip: The shortest distance between two skew lines can be found using the vector cross-product formula.

To find the point \( Q \), we need to find the equation of the tangent to the curve at \( P(1, 1) \). Using the point-slope form of the line and substituting the values of the curve, we find that the coordinates of \( Q \) are \( \left( \frac{9}{4}, \frac{3}{8} \right) \).


Step 2: Conclusion.

Thus, the correct answer is \( \left( \frac{9}{4}, \frac{3}{8} \right) \).
Quick Tip: To find the second intersection point of a tangent with a curve, use the point-slope form of the equation and solve for the coordinates.

At \( x = 0 \), the function behaves discontinuously due to the powers of \( x^2 \). Therefore, the function is not continuous and is not differentiable.
Quick Tip: Examine the behavior of the terms in an infinite series at specific values of \( x \) to determine if the series is continuous or differentiable.

The equation of a circle is \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center and \( r \) is the radius. Comparing the given equation with this form, we get \( r = \sqrt{36} = 6 \).
Quick Tip: The radius of a circle can be obtained by taking the square root of the constant term on the right-hand side of the equation.

Using the cross-product formula, \[ \mathbf{a} \times \mathbf{c} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k}
2 & -2 & 1
-1 & 0 & 2 \end{vmatrix} \]
we get the result \( 2\mathbf{i} - 2\mathbf{j} + \sqrt{5} \mathbf{k} \).
Quick Tip: To calculate the cross product, use the determinant method with unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) in the first row.

The diagonals of a square are perpendicular and bisect each other. Using the point \( (-4, 5) \) and the given diagonal equation, we can derive the equation of the second diagonal by using the condition for perpendicularity.


Step 2: Conclusion.

Thus, the correct answer is \( x + 7y = 31 \).
Quick Tip: The diagonals of a square are perpendicular and bisect each other. This helps in finding the equation of the second diagonal.

\( P = Q \) can be written as \( p \sim q \), as this denotes equality in a specific context, such as in relation to logic or set theory.
Quick Tip: In set theory or logic, the symbol \( \sim \) denotes equivalence or similarity between two objects.

The integral can be simplified and \( g(x) \) turns out to be the inverse sine function \( \sin^{-1} x \).
Quick Tip: The inverse sine function can appear when integrating functions involving square roots and cubes.

The set of all prime numbers is infinite, as there is no largest prime number.


Step 2: Conclusion.

Thus, the correct answer is (d) The set of all primes.
Quick Tip: The set of prime numbers is infinite, and it is one of the fundamental infinite sets in number theory.

The domain of the function is determined by the values of \( x \) for which both square roots are defined. After solving, the valid range for \( x \) is \( [-2, 1] \).


Step 2: Conclusion.

Thus, the correct answer is \( [-2, 1] \).
Quick Tip: The domain of a function with square roots is restricted by the values that make the expression inside the square roots non-negative.

The area under the curve \( y = \log x \) can be calculated by the integral: \[ Area = \int_1^e \log x \, dx = x \log x - x \Big|_1^e = 1 \]
Thus, the area is 1.
Quick Tip: The area under \( y = \log x \) from \( x = 1 \) to \( x = e \) is straightforward to compute using integration.

The angle of intersection between the two curves can be determined using the formula for the angle between two curves: \[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]
where \( m_1 \) and \( m_2 \) are the slopes of the tangents to the curves at the point of intersection. After calculation, the angle \( \theta = \frac{\pi}{2} \).
Quick Tip: To find the angle of intersection between curves, use the formula involving their slopes at the point of intersection.

The angle between the tangent and the Y-axis is given by the inverse tangent of the slope of the tangent. The slope of the tangent at \( (2, -3) \) is calculated by differentiating the equation \( y = x^2 + 4x - 17 \) and finding the slope at the point \( x = 2 \). After calculation, the angle is \( \frac{\pi}{2} - \tan^{-1} 9 \).
Quick Tip: The angle between the Y-axis and a tangent line can be found using the slope of the tangent at a given point.

To find \( (1 + i)^4 \), first express \( (1 + i) \) in polar form and then use De Moivre's Theorem to calculate the power. The result is 16.
Quick Tip: Use De Moivre's Theorem to raise complex numbers to a power. Convert the number to polar form before applying the theorem.

The relation \( R \) is defined by the condition \( |x^2 - y^2| < 16 \). We need to check the pairs that satisfy this condition. After checking, we find that none of the given options are correct.
Quick Tip: When working with relations, check each pair of elements to see if they satisfy the given condition.

This is a standard integral. Using substitution and simplification, we get the solution as \( -e^{-x} \).
Quick Tip: For integrals of the form \( \frac{dx}{(e^x + e^{-x})^2} \), simplify using trigonometric identities or substitution.

Using the addition formula for inverse tangents, we find the sum of the angles is \( \pi \).
Quick Tip: Use the addition formula for inverse tangents: \[ \tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1} \left( \frac{a + b}{1 - ab} \right) \]

The bacteria count follows exponential growth, and the formula for exponential growth is \( N = N_0 e^{kt} \), where \( N_0 \) is the initial count, \( N \) is the final count, \( k \) is the rate constant, and \( t \) is time. After solving for the time, we find \( t = \frac{\log 2}{\log 11} \).
Quick Tip: Exponential growth can be solved using the equation \( N = N_0 e^{kt} \), and logarithms are useful for finding the rate constant.

The angle between two lines is given by the formula: \[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]
where \( m_1 \) and \( m_2 \) are the slopes of the two lines. After substituting the values for \( m_1 \) and \( m_2 \), we find the angle \( \theta = 60^\circ \).
Quick Tip: To find the angle between two lines, use the formula involving the slopes of the lines.

Using the formula for the angle between a line and a plane, we can find the value of \( n \) by substituting the given values. After solving, we find \( n = \frac{5}{3} \).
Quick Tip: The angle between a line and a plane can be found using the direction ratios of the line and the normal to the plane.

First, find the intersection point of the line and the plane. Then, calculate the distance from the given point to the intersection using the distance formula. The distance is 13.
Quick Tip: The distance from a point to a line and plane can be found using the appropriate vector formulas for distance.

By solving the integral using the substitution method and simplifying the expression, we obtain the result \( \sqrt{3} \).
Quick Tip: Use substitution to simplify integrals involving trigonometric functions like \( \sec^2(x) \).

To solve the inequality, first break it into cases based on the absolute value. After solving, we get \( x \in (-5, -2) \cup (-1, \infty) \).
Quick Tip: When solving inequalities involving absolute values, split the inequality into cases based on the sign of the expression inside the absolute value.

The paragraph states that members can qualify for alternative credit for drills by proving previous experience in actual hazmat emergency response, meaning prior real emergency experience qualifies them to attend fewer drills.
Quick Tip: F.A.S.T. allows for alternative credit based on previous real emergency experience.

The main subject of the passage is about the requirements for completing F.A.S.T. membership, including training and certification requirements.
Quick Tip: The passage primarily discusses the requirements and conditions for maintaining F.A.S.T. membership.

The passage states that applicants must be available for training three days each month.
Quick Tip: F.A.S.T. requires its members to commit to three days of training per month.

Jatin moves in a series of right and left turns. After calculating the total displacement, we find that his distance from the starting point is 10 m.
Quick Tip: When solving problems involving displacement, break the motion into components and calculate the resultant displacement.

From the given statements, we can infer that employed members of Mohan's family are honest based on the given relationships between employment and honesty.
Quick Tip: In logical problems, use the process of elimination and logical deduction to identify the correct inference.