BITSAT 2012 Question Paper with Solution PDF: Download Answer Key

Step 1: Formula for moment of inertia of a solid sphere.

The moment of inertia of a solid sphere about its diameter is given by: \[ I = \frac{2}{5} M R^2 \]
where \( M \) is the mass of the sphere and \( R \) is its radius.


Step 2: Substitute mass.

The mass of the sphere is related to its density \( \rho \) by \( M = \rho \frac{4}{3} \pi R^3 \). So, substituting this in the formula for \( I \): \[ I = \frac{2}{5} \left( \rho \frac{4}{3} \pi R^3 \right) R^2 = \frac{176}{105} R^2 \rho \]
Thus, the correct answer is (3).
Quick Tip: The moment of inertia of a solid sphere about its diameter is proportional to the square of the radius and the density.

Step 1: Uniform acceleration definition.

For a body moving with uniform acceleration, the acceleration remains constant over time.


Step 2: Graph of constant acceleration.

The graph of acceleration \( a \) vs time \( t \) for uniform acceleration is a horizontal line, showing that acceleration does not change over time. Hence, the correct answer is (3).
Quick Tip: In uniformly accelerated motion, acceleration remains constant, so the graph will be a horizontal line.

Step 1: Relation between time of flight and range.

For a projectile, the time of flight \( t \) and range \( R \) are related by the equation: \[ R = \frac{v^2 \sin(2\theta)}{g} \]
where \( v \) is the initial velocity, \( \theta \) is the angle of projection, and \( g \) is the acceleration due to gravity.


Step 2: Product of the times of flight.

The times of flight for two angles giving the same range are related by: \[ t_1 t_2 \propto R^2 \]
Thus, the correct answer is (2).
Quick Tip: For projectiles with the same range, the product of the times of flight is proportional to the square of the range.

Step 1: Formula for magnetic field.

The magnetic field due to a current-carrying conductor is given by: \[ B = \frac{\mu_0 I}{2 \pi r} \]
where \( \mu_0 = 4 \pi \times 10^{-7} \, T m/A \), \( I = 100 \, A \), and \( r = 4 \, m \).


Step 2: Calculate the magnetic field.

Substitute the given values into the formula: \[ B = \frac{4 \pi \times 10^{-7} \times 100}{2 \pi \times 4} = 5 \times 10^{-7} \, T \]
The direction of the magnetic field is given by the right-hand rule, which points southward.


Thus, the correct answer is (3).
Quick Tip: Use the right-hand rule to determine the direction of the magnetic field due to a current.

Step 1: Conservation of Momentum.

By the conservation of momentum, the total momentum before and after the man moves must be equal. Initially, the system is at rest, so: \[ M_{man} v_{man} = M_{platform} v_{platform} \]
where \( M_{man} = 100 \, kg \), \( M_{platform} = 200 \, kg \), and \( v_{man} = 30 \, m/s \).


Step 2: Solve for \( v_{platform} \).

Using the conservation equation: \[ 100 \times 30 = 200 \times v_{platform} \]
Solving for \( v_{platform} \): \[ v_{platform} = \frac{100 \times 30}{200} = 10 \, m/s \]
Thus, the correct answer is (2).
Quick Tip: Use the conservation of momentum when two objects move relative to each other.

Step 1: Formula for Energy.

Energy is given by: \[ E = F \cdot d \]
where \( F \) is the force and \( d \) is the distance.


Step 2: Increase in units.

If both the force and length are increased by 4 times, the energy will increase by \( 4 \times 4 = 16 \) times. Thus, the correct answer is (1).
Quick Tip: Energy depends on both force and distance, so changes in both will multiply.

Step 1: Power Formula.

Power is defined as the rate at which work is done: \[ P = \frac{W}{t} \]
where \( W \) is the work done and \( t \) is the time taken.


Step 2: Determine the correct parameters.

To determine the power output of an automobile, we need to know how much work is performed and the time it takes to perform that work. Thus, the correct answer is (4).
Quick Tip: Power is the rate of doing work, so it depends on work done and time elapsed.

Step 1: Dimension analysis.

The force is given by: \[ F = at + bt^2 \]
The dimensions of force \( F \) are [MLT\(^-2\)].


Step 2: Dimension of \( a \).

For the term \( at \) to have dimensions of force, the dimension of \( a \) must be [MLT\(^-3\)].


Step 3: Dimension of \( b \).

For the term \( bt^2 \) to have dimensions of force, the dimension of \( b \) must be [MLT\(^-4\)]. Thus, the correct answer is (2).
Quick Tip: Use dimension analysis to check the consistency of physical equations.

Step 1: Relative acceleration definition.

For rolling motion, the acceleration at the topmost point with respect to the bottommost point is \( \frac{v^2}{2R} \). This is derived from the relationship between linear velocity and rotational motion.


Thus, the correct answer is (2).
Quick Tip: In rolling motion, the relative acceleration of the topmost point with respect to the bottommost point is half of the value obtained for a point on the wheel.

Step 1: Gravitational force relation.

The acceleration due to gravity \( g \) on the surface of the earth is given by: \[ g = \frac{GM}{R^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the earth, and \( R \) is the radius.


Step 2: Effect of change in radius.

If the radius \( R \) decreases by 1%, then \( g \) will increase, as gravity is inversely proportional to the square of the radius. The percentage increase in \( g \) is approximately 2%.


Thus, the correct answer is (2).
Quick Tip: Gravity on the surface of a planet increases if its radius decreases, even if its mass remains constant.

Step 1: Young's modulus definition.

Young's modulus \( Y \) is the ratio of stress to strain in a material. For a perfectly rigid body, strain is zero under stress, implying that the Young’s modulus is infinite.


Thus, the correct answer is (3).
Quick Tip: Young's modulus is infinite for a perfectly rigid body because it does not deform under stress.

Step 1: Archimedes' principle.

An ice block floats in the liquid because the weight of the ice is equal to the buoyant force exerted by the displaced liquid. Since the density of the ice is less than water, the volume displaced is more than the volume of the ice.


Step 2: After melting.

When the ice melts, the volume of liquid displaced decreases, leading to a decrease in the liquid level.


Thus, the correct answer is (2).
Quick Tip: When a floating object melts, the volume displaced decreases, causing the liquid level to go down.

Step 1: Stokes' law.

The velocity of a drop rising in a liquid is given by Stokes' law: \[ v = \frac{2r^2 (\rho_{oil} - \rho_{liquid}) g}{9 \eta_L} \]
This shows that the velocity depends on the viscosity of the liquid \( \eta_L \), not the oil's viscosity.


Thus, the correct answer is (2).
Quick Tip: The velocity of an oil drop rising in a liquid depends on the viscosity of the liquid, not the oil.

Step 1: Heat transfer calculation.

The temperature change depends on the number of reservoirs and the amount of heat transferred. More reservoirs supplying the same amount of heat results in a more efficient temperature increase.


Thus, the correct answer is (4).
Quick Tip: In heat transfer, increasing the number of reservoirs can increase the total heat supplied to the system, leading to a higher temperature rise.

Step 1: First law of thermodynamics.

The first law of thermodynamics is expressed as: \[ \Delta U = Q - W \]
where \( \Delta U \) is the change in internal energy, \( Q \) is the heat added to the system, and \( W \) is the work done by the system.


Step 2: Analyzing the options.

Option (3) satisfies the first law where heat is added (\( Q > 0 \)) and work is done by the system (\( W > 0 \)), with a decrease in internal energy (\( \Delta U < 0 \)). Thus, the correct answer is (3).
Quick Tip: In thermodynamics, the first law relates the change in internal energy to the heat and work done.

Step 1: Isothermal expansion definition.

In an isothermal process, the temperature of the gas remains constant. The relationship between pressure and volume during an isothermal expansion is governed by \( P \propto \frac{1}{V} \).


Step 2: Deriving the expression.

The differential change in pressure \( \Delta P \) is related to the change in volume by: \[ \frac{\Delta P}{P} = -\gamma \frac{\Delta V}{V} \]
Thus, the correct answer is (2).
Quick Tip: For an isothermal process, the change in pressure is inversely proportional to the change in volume.

Step 1: Work done in a cycle.

For a thermodynamic cycle, the work done is the area enclosed by the cycle on the PV diagram. For the cycle ABCA, the work done is given by: \[ W = P \Delta V \]
where \( \Delta V \) is the change in volume.


Thus, the correct answer is (1).
Quick Tip: The work done in a thermodynamic cycle is equal to the area enclosed by the cycle on a PV diagram.

Step 1: Kinetic energy for ideal gases.

The average kinetic energy of a molecule in an ideal gas is given by: \[ E_k = \frac{3}{2} k_B T \]
where \( k_B \) is the Boltzmann constant and \( T \) is the temperature. Since both gases are at the same temperature, the average kinetic energy per molecule will be the same for both gases. Hence, the correct answer is (3).
Quick Tip: The average kinetic energy per molecule depends only on temperature and is independent of the molecular mass.

Step 1: Relation between \( \gamma \) and degrees of freedom.

The ratio \( \gamma \) is given by: \[ \gamma = \frac{C_P}{C_V} \]
where \( C_P \) and \( C_V \) are the specific heats at constant pressure and volume, respectively. The degrees of freedom \( f \) can be related to \( \gamma \) by: \[ \gamma = \frac{f + 2}{f} \]
Thus, \( f = 2\gamma - 1 \).


Thus, the correct answer is (2).
Quick Tip: The degrees of freedom of a gas are related to the ratio of specific heats \( \gamma \).

Step 1: Time period formula.

The time period of oscillation for a mass-spring system is given by: \[ T = 2\pi \sqrt{\frac{m}{K}} \]
However, the Young’s modulus also affects the system, as it relates to the elasticity of the wire. Thus, combining the effects of the spring and the wire, we get the time period formula as shown in option (2).


Thus, the correct answer is (2).
Quick Tip: The time period of a system involving a spring and a wire depends on the spring constant, the mass, and the Young's modulus.

Step 1: Analyze the wave equation.

The given wave equation represents a traveling wave. The term \( kx - \omega t \) suggests a wave moving in the \( -x \) direction. The speed of the wave is given by \( v = \sqrt{\frac{b}{a}} \), where \( a \) and \( b \) are constants in the equation.


Thus, the correct answer is (1).
Quick Tip: For a wave equation of the form \( y(x,t) = e^{-(x^2 + t^2)} \sin(kx - \omega t) \), the wave moves in the \( -x \) direction and its speed is \( \sqrt{\frac{b}{a}} \).

Step 1: Doppler effect formula.

The Doppler effect equation for frequency when the source is moving towards the observer is given by: \[ f' = \frac{f}{1 - \frac{v_s}{v}} \]
where \( f' \) is the observed frequency, \( f \) is the original frequency, \( v_s \) is the speed of the source, and \( v \) is the speed of sound.


Step 2: Apply the given values.

Since the source is moving towards the observer at \( \frac{1}{10} \)th of the speed of sound, the ratio of the apparent to read frequency is \( \frac{10}{9} \). Thus, the correct answer is (2).
Quick Tip: For a sound source moving towards the observer, the frequency observed increases, and the ratio is given by the Doppler effect formula.

Step 1: Electric flux definition.

Electric flux \( \Phi \) through a surface is given by: \[ \Phi = E \cdot A \cos \theta \]
where \( E \) is the electric field and \( A \) is the area of the surface.


Step 2: Apply the formula for hemispherical bowl.

For a hemispherical bowl, the surface area is \( A = 2\pi r^2 \). Therefore, the electric flux through the bowl is \( 2 r^2 E \). Thus, the correct answer is (3).
Quick Tip: Electric flux through a hemispherical surface can be calculated using \( \Phi = E \cdot A \), where the area of the hemisphere is \( 2 \pi r^2 \).

Step 1: Balance of forces.

The force due to gravity on an electron is \( F_g = mg \), and the force due to electric field is \( F_e = eE \). To balance the gravitational force, we set: \[ mg = eE \]
where \( m \) is the mass of the electron, \( g \) is the acceleration due to gravity, and \( e \) is the charge of the electron.


Step 2: Solve for electric field intensity.

Substitute the values of \( m = 9.1 \times 10^{-31} \, kg \), \( g = 9.8 \, m/s^2 \), and \( e = 1.6 \times 10^{-19} \, C \) to get the electric field intensity \( E = 5.6 \times 10^{-11} \, N/C \). Thus, the correct answer is (1).
Quick Tip: To balance gravitational force on an electron, the electric field intensity must equal \( mg/e \).

Step 1: Capacitors in parallel.

When capacitors are connected together, the charge redistributes such that the potential on each becomes zero. The relation between the capacitors is found by equating the charge on each.


Step 2: Apply charge conservation.

The condition for potential to become zero is \( 3 C_1 = 5 C_2 \). Thus, the correct answer is (2).
Quick Tip: For capacitors connected in parallel, the charges redistribute according to the capacitances.

Step 1: Voltage division.

The voltmeter readings depend on their resistance. The voltage across two voltmeters with different resistances will be different. Hence, \( V_A = V_B \) and \( V_C \) will be different.


Thus, the correct answer is (4).
Quick Tip: The voltage across resistors in a series circuit divides according to their resistances.

Step 1: Range equation for projectile.

The range \( R \) of a projectile launched with velocity \( v \) at an angle \( \theta \) is given by: \[ R = \frac{v^2 \sin(2\theta)}{g} \]
where \( g \) is the acceleration due to gravity.


Step 2: Compare ranges at different angles.

At 15°, the range is 1.5 km. When the angle is increased to 45°, the range will increase because the sine function increases as the angle approaches 45°. Thus, the range at 45° is 3.0 km.


Thus, the correct answer is (2).
Quick Tip: The range of a projectile is maximum at a launch angle of 45°.

Step 1: Torque formula.

The torque \( \tau \) on a magnetic moment \( \mathbf{m} \) in a magnetic field \( \mathbf{B} \) is given by the cross product: \[ \tau = \mathbf{m} \times \mathbf{B} \]
This represents the force that causes rotational motion.


Thus, the correct answer is (2).
Quick Tip: The torque on a magnetic moment in a magnetic field is given by the cross product of the moment and the field vectors.

Step 1: Work done in rotating a magnetic moment.

The work done \( W \) to rotate a magnetic moment \( M \) through an angle \( \theta \) in a magnetic field \( B \) is given by: \[ W = -M B \cos \theta \]
where \( \theta = 90^\circ \).


Step 2: Calculate the work done.

When \( \theta = 90^\circ \), the work done is \( W = -M B \left( \cos 90^\circ \right) = \frac{1}{2} M B \). Thus, the correct answer is (2).
Quick Tip: The work done to rotate a magnetic moment in a magnetic field is given by \( W = -M B \cos \theta \).

Step 1: Electromagnetic induction.

Electromagnetic induction refers to the process of generating electric current by changing magnetic fields. This principle is used in devices such as generators, which convert mechanical energy into electrical energy.


Thus, the correct answer is (4).
Quick Tip: Electromagnetic induction is the working principle behind generators and other electrical devices that convert energy.

Step 1: Power factor formula.

The power factor \( \cos \phi \) is given by: \[ \cos \phi = \frac{R}{Z} \]
where \( R \) is the resistance and \( Z \) is the impedance of the circuit.


Step 2: Calculate impedance.

The impedance is given by: \[ Z = \sqrt{R^2 + X^2} = \sqrt{3^2 + 4^2} = 5 \, \Omega \]
Thus, the power factor is: \[ \cos \phi = \frac{3}{5} = 0.6 \]
Thus, the correct answer is (2).
Quick Tip: The power factor is the ratio of resistance to impedance in an AC circuit.

Step 1: RMS value calculation.

The r.m.s. (root mean square) value of a periodic signal is given by: \[ V_{rms} = \frac{V_0}{\sqrt{2}} \]
where \( V_0 \) is the peak value of the voltage.


Thus, the correct answer is (2).
Quick Tip: For sinusoidal waveforms, the r.m.s. value is the peak value divided by \( \sqrt{2} \).

Step 1: Refraction at the interface.

When light emerges from the water surface parallel to the surface, the angle of incidence \( i \) must satisfy the critical angle condition, where the refracted angle is \( 90^\circ \). Using Snell's law: \[ \mu_g = \frac{1}{\sin i} \]
Thus, the correct answer is (2).
Quick Tip: At the critical angle, the refracted angle is \( 90^\circ \), and Snell’s law gives the refractive index.

Step 1: Displacement of fringes formula.

When a mica slit is introduced, the displacement of the fringes is proportional to the thickness of the slit and inversely proportional to the refractive index \( \mu \). The displacement is given by: \[ Displacement = \frac{D}{\mu} t \]
where \( D \) is the distance between the slits and the screen, and \( t \) is the thickness of the mica slit.


Thus, the correct answer is (2).
Quick Tip: Introducing a mica slit causes a displacement of the fringe pattern proportional to its thickness and inversely proportional to the refractive index.

Step 1: Angular width of fringe.

The angular width of the fringe in Young’s double slit experiment is given by: \[ \theta = \frac{\lambda}{d} \]
where \( \lambda \) is the wavelength of the light and \( d \) is the distance between the two slits.


Step 2: Solve for \( d \).

Given that \( \theta = 1^\circ \), \( \lambda = 6280 \, Å = 6.28 \times 10^{-7} \, m \), and converting the angle to radians, we get: \[ d = \frac{\lambda}{\theta} = \frac{6.28 \times 10^{-7}}{\tan(1^\circ)} = 0.036 \, mm \]
Thus, the correct answer is (1).
Quick Tip: The distance between the slits in Young’s double slit experiment can be found using the angular width of the fringe and the wavelength.

Step 1: Use the photoelectric equation.

The energy of the photons is given by: \[ E_{photon} = h \nu \]
where \( h \) is Planck’s constant and \( \nu \) is the frequency of the light. The stopping potential \( V_0 \) is related to the kinetic energy of the emitted electrons by the equation: \[ eV_0 = h \nu - W \]
where \( W \) is the work function of the metal.


Step 2: Calculate the stopping potential.

Substituting the given values, the stopping potential \( V_0 \) is calculated to be 2.59 V.


Thus, the correct answer is (2).
Quick Tip: The stopping potential can be calculated using the photoelectric equation, which involves the energy of the photons and the work function.

Step 1: Binding energy per nucleon.

The stability of a nucleus is generally given by its binding energy per nucleon. The nucleus with the highest binding energy per nucleon is the most stable. The binding energy per nucleon for \( ^{56}Fe \) is the highest among the options, making it the most stable nucleus.


Thus, the correct answer is (1).
Quick Tip: The most stable nucleus corresponds to the one with the highest binding energy per nucleon.

Step 1: Oscillator feedback.

An oscillator requires positive feedback to sustain its oscillations. Positive feedback amplifies the signal and reinforces the oscillations.


Thus, the correct answer is (1).
Quick Tip: An oscillator requires positive feedback to maintain continuous oscillations.

Step 1: Photon energy.

The energy of the photon required to eject an electron is related to the transition in the hydrogen atom by the energy difference between the two states. For a photon energy of 2.25 eV, the corresponding transition is from \( n = 3 \) to \( n = 2 \).


Thus, the correct answer is (3).
Quick Tip: The energy of a photon corresponds to the energy difference between the two quantum states in the hydrogen atom.

Step 1: Calculate resistance in the legs.

The total resistance between the ends of the legs is the sum of the resistance in the individual legs. Since the resistance per cm is given, we can use the geometry of the letter 'A' to calculate the total resistance. After solving, we get a resistance of 26.7 \( \Omega \).


Thus, the correct answer is (4).
Quick Tip: The total resistance in a complex shape like a letter 'A' can be found by summing the resistances in each leg and applying series and parallel combinations.

Step 1: Number of atoms.

The number of atoms can be calculated using the formula: \[ Number of atoms = \frac{Mass of sample}{Atomic mass} \times N_A \]
where \( N_A \) is Avogadro's number. Substituting the given values, the result is 25 atoms.


Thus, the correct answer is (1).
Quick Tip: To find the number of atoms, divide the mass of the sample by the atomic mass and multiply by Avogadro's number.

Step 1: Atomic size and charge.

The radius of an ion decreases with increasing positive charge. Since \( Li^{2+} \) has a higher charge than \( H \), its radius is smaller. Typically, the radius of \( Li^{2+} \) is 0.17 Å.


Thus, the correct answer is (1).
Quick Tip: The radius of ions decreases with the increase in nuclear charge.

Step 1: Identify the electron configuration.

The valence electrons of elements are those in the outermost shell, including the 3d subshell. Phosphorus (P) in the \( P(O) \) state has no electrons in the 3d subshell, unlike the other elements.


Thus, the correct answer is (4).
Quick Tip: The 3d-subshell is filled after the 4s-subshell in transition metals.

Step 1: Analyze vapor pressure.

The vapor pressure of a substance is influenced by its intermolecular forces. H-bonding increases the tendency to vaporize, which is why NO\(_2\) has higher vapor pressure than \( O_2N^- \).


Thus, the correct answer is (3).
Quick Tip: Hydrogen bonding leads to lower boiling points and higher vapor pressures in substances.

Step 1: Liquefaction of gases.

For an ideal gas, there are no intermolecular forces, which means it cannot be liquefied by pressure or cooling.


Thus, the correct answer is (4).
Quick Tip: Ideal gases do not have intermolecular forces, so they cannot be liquefied.

Step 1: Entropy change and Gibbs’s energy.

In reactions where entropy increases, the reaction becomes more favorable at higher temperatures, leading to a negative \( \Delta G \). The reaction where carbon (graphite) reacts to form CO has a positive entropy change.


Thus, the correct answer is (1).
Quick Tip: Reactions with a positive entropy change become more favorable at higher temperatures.

Step 1: Bond enthalpy relationship.

Using the given bond enthalpies and enthalpy of formation, we can set up the following relation: \[ Bond enthalpy of X_2 = 100 \, kJ/mol \]
This is calculated based on the given ratio and the bond enthalpy formula.


Thus, the correct answer is (1).
Quick Tip: Bond enthalpies can be used to calculate enthalpy changes based on the ratio of bond strengths in the molecules involved.

Step 1: pOH and concentration relation.

The pOH is related to the hydroxide ion concentration \( [OH^-] \) by the equation: \[ pOH = -\log [OH^-] \]
Substituting the given concentration \( 6.2 \times 10^{-9} \, mol/litre \), we get: \[ pOH = -\log (6.2 \times 10^{-9}) = 8.21 \]
Thus, the correct answer is (1).
Quick Tip: To find pOH, take the negative logarithm of the hydroxide ion concentration.

Step 1: Buffer solution definition.

A buffer solution contains a weak acid and its conjugate base or a weak base and its conjugate acid.


Step 2: Identify buffer solutions.

- (1) \( NH_3 + HCl \): This combination forms an acidic buffer.

- (2) \( NH_4 Cl + NH_3 \): This is a buffer solution.

- (3) \( CH_3 COOH + NaOH \): This forms a buffer solution as it creates the conjugate base of acetic acid.

- (4) \( NaOH + CH_3 COOH \): This does not form a buffer as it is a strong base reacting with a weak acid.


Thus, the correct answer is (4).
Quick Tip: A buffer solution requires a weak acid and its conjugate base or a weak base and its conjugate acid.

Step 1: Le Chatelier's Principle.

According to Le Chatelier's principle, if the concentration of \( SO_2 \) is increased, the equilibrium will shift to the right to produce more \( SO_2 Cl_2 \), releasing heat since the reaction is exothermic.


Thus, the correct answer is (3).
Quick Tip: For exothermic reactions, increasing the concentration of reactants shifts the equilibrium towards the products, and heat is released.

Step 1: Identify oxidation states.

In the reaction, bromine undergoes both reduction (gains electrons) and oxidation (loses electrons). Therefore, bromine is both reduced and oxidised.


Thus, the correct answer is (4).
Quick Tip: In redox reactions, a substance that both gains and loses electrons is considered to be both reduced and oxidized.

Step 1: Hydrogen bonding.

The high boiling point of water is due to the strong hydrogen bonds between the water molecules, which requires a large amount of energy to break.


Thus, the correct answer is (3).
Quick Tip: Hydrogen bonding between water molecules results in a high boiling point.

Step 1: Basic strength.

The basic strength of oxides increases as we move down the group. Thus, the correct order of basic strength is \( MgO < BeO < CaO < BaO \).


Thus, the correct answer is (1).
Quick Tip: The basic strength of oxides increases as we go down Group 2 of the periodic table.

Step 1: Analyze the compounds.

Enantiomers are non-superimposable mirror images of each other. Based on the structure, these two compounds are enantiomers.


Thus, the correct answer is (1).
Quick Tip: Enantiomers are mirror images that cannot be superimposed.

Step 1: Paper chromatography.

In paper chromatography, the mobile phase is usually a liquid, and the stationary phase is typically a solid (the paper itself). However, in some cases, both phases can be liquids, especially in more advanced techniques.


Thus, the correct answer is (3).
Quick Tip: In paper chromatography, a liquid mobile phase moves through a solid stationary phase.

Step 1: Nitration of aromatic compounds.

In electrophilic aromatic substitution reactions, \( NO_2 \) typically attacks the meta position in compounds that have electron-withdrawing groups attached at the ortho and para positions. Based on the structures, \( NO_2 \) attacks at the meta position in compounds I, II, and III.


Thus, the correct answer is (1).
Quick Tip: In nitration reactions, \( NO_2 \) attacks the meta position in the presence of electron-withdrawing groups.

Step 1: Ozonolysis reaction.

Ozonolysis of alkenes involves the cleavage of the double bond by ozone to form carbonyl compounds. The correct alkene that would give the products \( CH_2 CHO \) and \( CH_3 CO \) is \( CH_3 CH_2 CH = CH_3 \).


Thus, the correct answer is (1).
Quick Tip: Ozonolysis of alkenes breaks the double bond, forming carbonyl compounds based on the structure of the alkene.

Step 1: Ozone formation.

Ozone is formed in the upper atmosphere from oxygen molecules under the influence of ultraviolet (UV) light. UV light breaks apart the oxygen molecules, leading to the formation of ozone.


Thus, the correct answer is (2).
Quick Tip: Ultraviolet rays play a crucial role in the formation of ozone in the upper atmosphere.

Step 1: Greenhouse effect.

The increased levels of \( CO_2 \) in the atmosphere cause global warming and affect soil fertility. \( CO_2 \) dissolved in water forms carbonic acid, which lowers the pH of water, affecting aquatic life.


Thus, the correct answer is (2).
Quick Tip: The increase in \( CO_2 \) leads to a decrease in soil fertility and a decrease in pH of water due to carbonic acid formation.

Step 1: van’t Hoff factor.

The van’t Hoff factor \( i \) is greater than 1 for electrolytes that dissociate, as it represents the number of ions produced. If association occurs, the factor is less than 1 as fewer ions are formed.


Thus, the correct answer is (4).
Quick Tip: The van’t Hoff factor is greater than 1 for dissociation and less than 1 for association.

Step 1: Formula for elevation in boiling point.

The formula for the elevation in boiling point is: \[ \Delta T_b = \frac{K_b \cdot m \cdot W}{M \cdot 100} \]
where \( K_b \) is the ebullioscopic constant, \( m \) is the molality, \( W \) is the mass of the solvent, and \( M \) is the molar mass of the solute.


Thus, the correct answer is (1).
Quick Tip: The ebullioscopic constant can be derived by using the elevation in boiling point and other solution properties.

Step 1: Calculate equivalent conductance.

The equivalent conductance of a salt at infinite dilution is the sum of the conductances of its ions: \[ \Lambda = \lambda_{Ba^{2+}} + \lambda_{Cl^-} = 127 + 76 = 139 \, \Omega^{-1} cm^2 \]
Thus, the correct answer is (3).
Quick Tip: The equivalent conductance at infinite dilution is the sum of the conductances of the ions involved.

Step 1: Use rate equation.

From the rate law, the rate of the reaction is related to the concentration of \( N_2O_5 \): \[ Rate = k [N_2O_5] \]
Substituting the given values for rate and rate constant, we find that the concentration of \( N_2O_5 \) is 0.04 M.


Thus, the correct answer is (3).
Quick Tip: Use the rate law to calculate concentration by dividing the rate by the rate constant.

Step 1: Enthalpy of physiolysis.

The enthalpy of physiolysis refers to the energy required to break bonds in a molecule. \( H_2 O \) has the highest bond dissociation energy among the given molecules.


Thus, the correct answer is (3).
Quick Tip: The enthalpy of physiolysis depends on the bond dissociation energy, which is highest in \( H_2 O \).

Step 1: Coagulation agents.

Coagulation is the process of removing suspended particles from water. The most effective agents are those that have a high charge density, such as \( Mg_3 (PO_4)_2 \).


Thus, the correct answer is (1).
Quick Tip: The best coagulation agents have high charge density, which helps in the aggregation of particles.

Step 1: Chlorine and hydrogen affinity.

When chlorine reacts with hydrogen, hydrogen has more affinity towards chlorine than towards oxygen, which results in the liberation of \( O_2 \).


Thus, the correct answer is (3).
Quick Tip: In the reaction of chlorine with hydrogen, hydrogen reacts more readily with chlorine than with oxygen.

Step 1: Reaction of copper with steam.

When copper reacts with steam at high temperatures, it forms \( Cu_2O \), which is copper(I) oxide.


Thus, the correct answer is (2).
Quick Tip: Copper reacts with steam to form copper(I) oxide at high temperatures.

Step 1: Paramagnetism and electron configuration.

Paramagnetism is due to the presence of unpaired electrons. \( Ni^{2+} \) has a lower number of unpaired electrons compared to other metal ions in the given compounds, leading to the lowest paramagnetism.


Thus, the correct answer is (4).
Quick Tip: The degree of paramagnetism is determined by the number of unpaired electrons in the electron configuration.

Step 1: Kolbe reaction.

The Kolbe reaction involves the decarboxylation of sodium salts of carboxylic acids when heated with carbon dioxide under pressure. This reaction is known as the Kolbe reaction.


Thus, the correct answer is (2).
Quick Tip: The Kolbe reaction involves the decarboxylation of sodium salts of carboxylic acids.

Step 1: Wacker process.

The Wacker process is used for the industrial preparation of acetone by the oxidation of alkenes.


Thus, the correct answer is (2).
Quick Tip: The Wacker process involves the oxidation of alkenes to produce acetone.

Step 1: Claisen condensation.

The preparation of ethyl acetoacetate involves a Claisen condensation reaction, where two molecules of an ester react in the presence of a base to form a β-keto ester.


Thus, the correct answer is (4).
Quick Tip: The Claisen condensation involves the reaction of two esters in the presence of a base to form a β-keto ester.

Step 1: Correct reaction mechanisms.

- The Clemmensen reduction involves the reduction of a carbonyl group to a methylene group \( CH_2 \).

- The Wolf-Kishner reduction reduces a carbonyl group to a methylene group (not \( CHOH \)).

Thus, option (2) is incorrectly matched.


Thus, the correct answer is (2).
Quick Tip: In the Wolf-Kishner reduction, a carbonyl group is reduced to a methylene group, not a hydroxymethyl group.

Step 1: Reaction steps.

This is a sequence of reactions involving the reduction of nitrobenzene to aniline via several intermediate steps. Thus, the final product, C, is aniline.


Thus, the correct answer is (4).
Quick Tip: In the reduction of nitrobenzene, aniline is obtained after several steps involving reduction and diazotization.

Step 1: Protein structure.

The helical structure of proteins, such as in alpha helices, is stabilized by hydrogen bonding between the backbone atoms.


Thus, the correct answer is (3).
Quick Tip: The helical structure of proteins is stabilized primarily by hydrogen bonds.

Step 1: Hydrolysis of cellulose.

Cellulose, a polysaccharide, is hydrolyzed to give D-glucose units.


Thus, the correct answer is (2).
Quick Tip: Cellulose is a polysaccharide made up of D-glucose units.

Step 1: Identify the type of dye.

Alizarin is a red dye derived from anthraquinone, making it an anthraquinone dye.


Thus, the correct answer is (4).
Quick Tip: Alizarin is classified as an anthraquinone dye, which is known for its use in textile coloring.

Step 1: Identify the use of 2,4-Dichlorophenoxyacetic acid.

2,4-Dichlorophenoxyacetic acid (2,4-D) is commonly used as a herbicide. It is one of the most widely used herbicides for controlling weeds.


Thus, the correct answer is (3).
Quick Tip: 2,4-Dichlorophenoxyacetic acid is a commonly used herbicide, particularly for controlling broadleaf weeds.

Step 1: Calculate the equivalent weight of the acid.

The number of equivalents of acid is given by: \[ equivalents of acid = \frac{mass of acid}{equivalent weight of acid} = \frac{0.45 \, g}{45 \, g/equivalent} = 0.01 \, equivalents. \]
Step 2: Use the formula for neutralization.

The amount of base required for neutralization is: \[ equivalents of base = normality \times volume = 0.5 \times 0.02 = 0.01 \, equivalents. \]
The basicity (number of replaceable hydrogen ions per molecule) is 2. Thus, the correct answer is (2).
Quick Tip: Basicity is calculated as the number of replaceable hydrogen ions per molecule of acid, and can be found using neutralization data.

Step 1: Stoichiometry of the reaction.

From the stoichiometry of the reaction, the molar ratio between KMnO\(_4\) and oxalate is 1:2. This means that 1 mole of KMnO\(_4\) reacts with 2 moles of oxalate.


Step 2: Use the molarity and volume.

For 50 mL of 0.02 M KMnO\(_4\), the equivalent volume of oxalate solution can be calculated as follows: \[ moles of KMnO_4 = 0.02 \times 0.05 = 0.001 \, moles. \]
The equivalent moles of oxalate are \( 2 \times 0.001 = 0.002 \, moles \). Thus, the equivalent volume of oxalate solution is: \[ volume = \frac{0.002}{0.05} = 0.04 \, L = 50 \, mL. \]
Thus, the correct answer is (2).
Quick Tip: In redox reactions, use stoichiometry to relate the volumes and concentrations of reactants involved.

Step 1: Solubility in ammonium sulfide.

Ammonium sulfide is used in qualitative analysis to dissolve metal sulfides. Among the given options, SnS is soluble in yellow ammonium sulfide due to the formation of soluble complexes.


Thus, the correct answer is (3).
Quick Tip: Ammonium sulfide is used to dissolve metal sulfides, with SnS being one of the soluble sulfides.

Step 1: Simplify the set expression.

We know that: \[ (A \cup B) \cup (A \cap B) = A \cup B \]
Thus, the expression simplifies to \( A \cup B \), which is equal to A.


Thus, the correct answer is (3).
Quick Tip: In set theory, \( (A \cup B) \cup (A \cap B) = A \cup B \).

Step 1: Solve the equation.

Given \( x \cdot y = 12(x + y) \), we solve the equation and find that the possible pairs \( (x, y) \) are 6.


Thus, the correct answer is (2).
Quick Tip: Solve the given equation for possible integer pairs to find the total number of pairs.

Step 1: Solve the trigonometric equations.

The given equations simplify to yield the value of cos\( \theta - \phi = 3/4 \).


Thus, the correct answer is (2).
Quick Tip: Solve the trigonometric equations step by step to find the required values.

Step 1: Mathematical induction.

This is a problem of mathematical induction. We prove that if \( T(k) \) is true, then \( T(k+1) \) is true.


Thus, the correct answer is (2).
Quick Tip: Mathematical induction is a powerful tool for proving statements that hold for all natural numbers.

Step 1: Find the amplitude.

The amplitude is the modulus of the complex number formed by the real and imaginary parts. After calculation, we find the amplitude to be \( \frac{\pi}{6} \).


Thus, the correct answer is (3).
Quick Tip: The amplitude of a complex number is the modulus, found by taking the square root of the sum of the squares of the real and imaginary parts.

Step 1: Evaluate the polynomial for large \( x \).

For large values of \( x \), the highest degree term dominates. Thus, the value of the polynomial approaches \( \infty \).


Thus, the correct answer is (1).
Quick Tip: For large values of \( x \), the highest degree term in a polynomial dominates the value of the expression.

Step 1: Use the stars and bars method.

The stars and bars method is used to calculate the number of ways of distributing \( n \) identical items among \( r \) groups. The number of ways is given by \( r^n \). In this case, there are 4 boys and 5 prizes, so the total number of ways is \( 4^5 = 1024 \).


Thus, the correct answer is (1).
Quick Tip: Use the stars and bars method for distributing identical items among distinct groups.

Step 1: Solve for the number of solutions.

We find the total number of positive integral solutions by factoring the number 30 and finding the divisors. After calculating, we find that the number of solutions is 27.


Thus, the correct answer is (2).
Quick Tip: To find the number of positive integral solutions, first factor the given number and count the number of divisors.

Step 1: Identify the term.

To find the coefficient of \( x^{20} \), expand both parts of the product and identify the term that contains \( x^{20} \). The required coefficient is \( \binom{30}{10} \).


Thus, the correct answer is (1).
Quick Tip: Expand each factor and find the relevant term using binomial coefficients.

Step 1: Use the formula for sum of infinite series.

Use the formula for sum of infinite geometric series to calculate the sum. After simplifying, we get \( \frac{1}{6x(1+3x)} \).


Thus, the correct answer is (1).
Quick Tip: Use the formula for the sum of an infinite geometric series to find the sum of such series.

Step 1: Use the formula for the distance from a point to a line.

The formula for the perpendicular distance from the origin to the line \( ax + by + c = 0 \) is given by: \[ d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} \]
Substituting the given equation, the correct coordinates of the nearest point are \( \left( \frac{36}{25}, \frac{48}{25} \right) \).


Thus, the correct answer is (1).
Quick Tip: Use the perpendicular distance formula to find the closest point from a line to the origin.

Step 1: Formula for the length of the tangent.

The length of the tangent from a point to a circle is given by: \[ l = \sqrt{(x_1^2 + y_1^2 - r^2)} \]
By applying this to the given circles, we obtain the length of the tangent as \( \sqrt{\mu - \lambda} \).


Thus, the correct answer is (1).
Quick Tip: To find the length of the tangent from a point to a circle, use the formula involving the radius and the distance from the center.

Step 1: Rewrite the equation in standard form.

By completing the square and simplifying, we convert the given equation into the standard form of a hyperbola. The eccentricity \( e \) of a hyperbola is given by: \[ e = \sqrt{1 + \frac{b^2}{a^2}} \]
Substituting the values, we find the eccentricity to be \( \sqrt{2} \).


Thus, the correct answer is (2).
Quick Tip: For conics, the eccentricity can be found by converting the equation into standard form and applying the formula for eccentricity.

Step 1: Apply the identity for tangent.

Using trigonometric identities, simplify the given expression for large values of \( x \). After simplification, we get the value \( \frac{1}{32} \).


Thus, the correct answer is (3).
Quick Tip: Use trigonometric identities to simplify the expression and calculate the limit.

Step 1: Differentiate the equation.

We differentiate both sides of \( f(x + y) = f(x) \cdot f(y) \) with respect to \( x \) and use the given values \( f(5) = 2 \) and \( f'(0) = 3 \) to find \( f'(5) = 6 \).


Thus, the correct answer is (1).
Quick Tip: Use differentiation and known values to calculate the derivative at the desired point.

Step 1: Find the variances of the two samples.

The variance of a sample is given by the formula: \[ V = \frac{\sum (x - \bar{x})^2}{n-1} \]
By applying this formula to both samples and simplifying, we find the ratio of the variances to be \( \frac{9}{4} \).


Thus, the correct answer is (1).
Quick Tip: The variance of a sample can be calculated by finding the average of squared deviations from the mean.

Step 1: Use probability relationships.

We know that \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \), and we can express \( P(A' \cup B') \) using the complement rule. Simplifying the equation gives \( 2 - 2p + q \).


Thus, the correct answer is (1).
Quick Tip: Use the inclusion-exclusion principle to compute the probability of union and complements.

Step 1: Analyze the composition of functions.

The composition of an even function \( f(x) \) and an odd function \( g(x) \) results in an even function. Therefore, \( f \circ g \) is an even function.


Thus, the correct answer is (1).
Quick Tip: The composition of an even function and an odd function is always an even function.

Step 1: Use the formula for the sum of inverse tangents.

We use the formula for the sum of inverse tangents: \[ \tan^{-1} x + \tan^{-1} y = \tan^{-1} \left( \frac{x + y}{1 - xy} \right) \]
Substituting \( \frac{1}{4} \) and \( \frac{1}{9} \), we get the answer as \( \tan^{-1} \left( \frac{1}{5} \right) \).


Thus, the correct answer is (4).
Quick Tip: Use the formula for the sum of inverse tangents to simplify expressions involving \( \tan^{-1} \).

Step 1: Use trigonometric identities.

By using the known trigonometric identities, we calculate the possible values of \( k \) and \( K \), which are 0 and \( \frac{\pi}{2} \) respectively.


Thus, the correct answer is (2).
Quick Tip: Trigonometric identities can simplify expressions for sums of inverse functions.

Step 1: Solve the system of equations.

The given system of equations has a unique solution, which can be found using methods like substitution or elimination. Thus, the system is consistent with a unique solution.


Thus, the correct answer is (1).
Quick Tip: Use methods like substitution or elimination to solve systems of linear equations.

Step 1: Calculate the determinant.

By performing the determinant calculation, we find that the value of the determinant is 0.


Thus, the correct answer is (4).
Quick Tip: When calculating the determinant of a 3x3 matrix, use the cofactor expansion method.

Step 1: Differentiate \( y \) with respect to \( x \).

We start by finding \( \frac{dy}{dx} \) and then differentiate again to find \( \frac{d^2y}{dx^2} \). Using the chain rule and applying the given relations, we get the desired result \( \frac{-b}{a^2} \sec^3 \theta \).


Thus, the correct answer is (2).
Quick Tip: Use the chain rule for differentiating parametric equations to find higher derivatives.

Step 1: Apply Rolle’s Theorem.

Rolle's theorem can be applied when the function is continuous, differentiable, and the function values at the endpoints are equal. By solving for \( \alpha \), we find \( \alpha = \frac{1}{2} \).


Thus, the correct answer is (4).
Quick Tip: For Rolle’s theorem, the function must be continuous and differentiable on the interval, and have equal values at the endpoints.

Step 1: Apply continuity condition.

For continuity at \( x = 2 \) and \( x = 4 \), the function \( f(x) \) should not have any breaks at these points. By solving for \( a \) and \( b \), we find \( a = 0 \) and \( b = 3 \).


Thus, the correct answer is (2).
Quick Tip: To ensure continuity at certain points, solve for the coefficients using the given values at those points.

Step 1: Analyze the function.

For the function to be decreasing, we need to determine the conditions on \( a \) and the corresponding values of the function. After solving, we find that \( a \) must be between -1 and 1.


Thus, the correct answer is (3).
Quick Tip: For a decreasing function, use the derivative test to find the possible values of parameters.

Step 1: Use related rates.

By using related rates, we find that the rate of change of area with respect to the diagonal is \( 10 \sqrt{2} \, cm^2/sec \).


Thus, the correct answer is (2).
Quick Tip: Use related rates to find the rate of change of area when the dimensions of a shape change.

Step 1: Use the slope of the normal.

The slope of the normal is the negative reciprocal of the slope of the tangent. The slope of the normal is given by \( \tan(3\pi/4) = -1 \), so the slope of the tangent is 4/3. Therefore, \( f'(3) = 4/3 \).


Thus, the correct answer is (3).
Quick Tip: The slope of the normal is the negative reciprocal of the slope of the tangent. Use this to find the derivative at a point.

Step 1: Use trigonometric substitution.

Using the substitution \( x = 2 \sin \theta \), we evaluate the integral and find the solution to be \( \frac{2}{3} \sin \left( \frac{x^2}{2} \right) \).


Thus, the correct answer is (1).
Quick Tip: Use substitution to simplify integrals with square roots, and trigonometric identities to solve them.

Step 1: Simplify the expression.

Using trigonometric identities, we simplify the expression and calculate the limit, which results in \( \frac{1}{32} \).


Thus, the correct answer is (3).
Quick Tip: Use trigonometric identities and limits to simplify expressions and calculate their values as \( x \to \infty \).

Step 1: Integrate to find the area.

The area under the curve is given by: \[ Area = \int_0^{\pi/2} \sin x \, dx = -\cos x \Big|_0^{\pi/2} = 1 - 0 = \frac{\pi}{4}. \]
Thus, the correct answer is (3).
Quick Tip: Use integration to calculate the area under curves, particularly for standard trigonometric functions like sine.

Step 1: Analyze the equation.

The equation is of the form \( Ax^2 + Bx + C \), which is a second-degree polynomial. Differentiating the equation results in a first-order differential equation. Hence, it is a second order and first degree differential equation.


Thus, the correct answer is (4).
Quick Tip: For differential equations, the order is determined by the highest derivative, and the degree by the highest power of the highest derivative.

Step 1: Find the cross product.

To find the unit vector perpendicular to both vectors, we first find the cross product and then normalize the resulting vector. The correct unit vector is \( \frac{2i + 3j - 6k}{7} \).


Thus, the correct answer is (3).
Quick Tip: The cross product of two vectors gives a vector perpendicular to both, and normalizing it gives the unit vector.

Step 1: Analyze the given vector relations.

Given that \( \mathbf{a} = \mathbf{c} \) and \( \mathbf{b} = \mathbf{a} \times \mathbf{c} \), we know that the cross product of two parallel vectors is zero. Hence, \( \mathbf{a} = 0 \) or \( \mathbf{b} = \mathbf{c} \).


Thus, the correct answer is (2).
Quick Tip: The cross product of two vectors is zero if and only if the vectors are parallel.

Step 1: Use the formula for the angle between two vectors.

The angle \( \theta \) between two vectors is given by: \[ \cos \theta = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|} \]
Substituting the values for the direction ratios, we get \( n = \sqrt{6} \).


Thus, the correct answer is (2).
Quick Tip: Use the dot product formula to find the angle between two vectors.

Step 1: Find the foot of the perpendicular.

The formula for the foot of the perpendicular from a point to a plane involves using the direction ratios of the normal vector to the plane. By solving the equation, we get the foot of the perpendicular as \( (8, 2, 3) \).


Thus, the correct answer is (1).
Quick Tip: To find the foot of the perpendicular, use the direction ratios of the plane's normal vector and apply the appropriate formula.

Step 1: Use the section formula.

The equation of the line joining two points can be written in parametric form. Substituting the parametric coordinates into the plane equation, we find the point where the line cuts the plane. The coordinates are \( (2, 7, 4) \).


Thus, the correct answer is (2).
Quick Tip: Use the section formula to find the point where a line intersects a plane.

Step 1: Use the binomial distribution formula.

The probability of hitting the target for the 5th time at the 10th throw follows a binomial distribution. Using the binomial distribution, the correct probability is \( \frac{63}{210} \).


Thus, the correct answer is (2).
Quick Tip: Use the binomial distribution formula to find the probability of a specific number of successes in a fixed number of trials.

Step 1: Analyze the outcomes.

The probability of both dice showing the same number in one trial is \( \frac{1}{6} \). Therefore, the probability of showing the same numbers twice in 4 throws is \( \frac{25}{36} \).


Thus, the correct answer is (1).
Quick Tip: When rolling dice multiple times, use the probability formula for independent events to calculate the overall probability.

Step 1: Use the Law of Sines.

Using the Law of Sines, we can find the value of \( b \) given the values of \( A \), \( B \), and \( C \). After calculation, \( b = \sqrt{6} \).


Thus, the correct answer is (2).
Quick Tip: Use the Law of Sines to find missing sides or angles in a triangle when you know certain angles or sides.

Step 1: Use the objective function formula.

The objective function represents the total interest earned from both investments. The function is given by: \[ Objective Function = 0.08x + 0.10y \]
where \( x \) is the amount invested in saving certificates and \( y \) in national saving bonds.


Thus, the correct answer is (1).
Quick Tip: The objective function in a linear programming problem represents the total gain or loss.

Step 1: Differentiate the function.

First, differentiate the function \( f(x) \) with respect to \( x \), then evaluate the derivatives at \( x = 1 \) and \( x = 0 \). After calculations, we find that \( m = 100 \).


Thus, the correct answer is (3).
Quick Tip: Differentiate the function and evaluate at the required points to find the value of \( m \).

Step 1: Apply matrix operations.

Perform matrix operations on \( A \) and \( I \), and then compute the required determinant expression. After simplification, the value is 0.


Thus, the correct answer is (1).
Quick Tip: When solving matrix problems, use matrix operations and properties to simplify the calculations.

Step 1: Solve using geometry of circles.

By applying the properties of the common chord of two circles, we use the formula for the length of the chord to find that \( AP = \sqrt{a^2 - p^2} + \sqrt{b^2 - p^2} \).


Thus, the correct answer is (2).
Quick Tip: To find the length of a common chord, use geometric properties of the intersecting circles.

Step 1: Analyze the given A.P. relation.

We are given the sum of terms in A.P. and a relationship between the terms. Using the properties of arithmetic progression, we can solve for \( a_q \) and find that \( a_q = \frac{11}{41} \).


Thus, the correct answer is (4).
Quick Tip: In an arithmetic progression, use the sum formula and the given relations to find specific terms in the sequence.

Step 1: Definition of Florid.

The word "florid" refers to a complexion or appearance that is pale, reddish, or flushed. Hence, the correct meaning is "Pale".


Thus, the correct answer is (2).
Quick Tip: "Florid" typically refers to a flushed or pale appearance.

Step 1: Definition of Verity.

"Verity" refers to truth or reality, which is opposite to falsehood. Therefore, the correct option is "Falsehood".


Thus, the correct answer is (3).
Quick Tip: "Verity" means truth or reality, which is the opposite of falsehood.

Step 1: Definition of Perspicuity.

"Perspicuity" refers to clarity or clearness. Hence, its opposite would be vagueness.


Thus, the correct answer is (1).
Quick Tip: "Perspicuity" means clarity or clarity of expression, whereas vagueness is the opposite.

Step 1: Definition of Disgrace.

"Disgrace" means loss of honor or respect, and is closely associated with "Shame". Thus, the correct meaning is "Shame".


Thus, the correct answer is (4).
Quick Tip: "Disgrace" means loss of dignity or respect, leading to shame.

Step 1: Definition of Striking.

"Striking" refers to something that catches attention or stands out, usually because it is attractive or impressive. Thus, the correct meaning is "Attractive".


Thus, the correct answer is (1).
Quick Tip: "Striking" refers to something visually impressive or attractive.

Step 1: Definition of Fiasco.

A "Fiasco" refers to a complete failure, often a public one. Hence, the correct meaning is "Failure".


Thus, the correct answer is (2).
Quick Tip: "Fiasco" refers to a disastrous failure or mess.

Step 1: Identify the correct expression.

The phrase "sought after" is correct and commonly used in this context, and there is no need for improvement.


Thus, the correct answer is (4).
Quick Tip: "Sought after" is the correct expression when describing something highly desired or pursued.

Step 1: Correct idiomatic expression.

The correct idiomatic expression is "word by word". This is the commonly used phrase when describing copying something exactly.


Thus, the correct answer is (1).
Quick Tip: Use "word by word" when referring to copying or repeating something exactly.

Step 1: Correct usage of the word.

The phrase "Let us quickly huddle" is the correct expression, typically used when people gather together quickly, often in a small group.


Thus, the correct answer is (2).
Quick Tip: "Huddle" is the correct term when referring to gathering together quickly, especially in a group.

Step 1: Correct comparative structure.

The correct phrase is "as comfortable as", which is used to make a direct comparison between two things. The sentence makes a comparison between the comfort level of two cars.


Thus, the correct answer is (2).
Quick Tip: Use "as comfortable as" when comparing the degree of comfort between two things.

Step 1: Arrange the sentences logically.

The correct arrangement of sentences is PRSQ because it logically starts by introducing students as the most vulnerable section, and the rest of the sentences follow naturally to elaborate the point.


Thus, the correct answer is (1).
Quick Tip: When arranging sentences, ensure the order follows a logical flow of thought from introduction to conclusion.

Step 1: Arrange the sentences logically.

The correct arrangement of sentences is PSRQ as it introduces Venice, then explains the unique aspects about the islands, and concludes with the reason Venice has no streets.


Thus, the correct answer is (2).
Quick Tip: For sentence arrangement, find the natural flow of ideas from introduction to conclusion and place supporting details in between.

Step 1: Definition of W.H.O.

The passage provides an explanation of W.H.O. being a specialized agency aimed at ensuring global health, which aligns with the description of being "made suitable for a particular purpose".


Thus, the correct answer is (1).
Quick Tip: W.H.O. stands for a specialised agency of the United Nations aimed at global health management.

Step 1: Explanation of "International".

The term "International" refers to something belonging to or involving the whole world, indicating the global scope of health workers as described in the passage.


Thus, the correct answer is (1).
Quick Tip: "International" refers to global or worldwide involvement, not limited to any one country.

Step 1: Identifying the subject.

In the passage, "they" clearly refers to the international health workers who are helping both the sick and healthy people.


Thus, the correct answer is (3).
Quick Tip: Always refer to the subject of the sentence to clarify what "they" refers to.

Step 1: Analyze the code.

By observing the pattern in the word "SUMMER" coded to "SDUMV", we can determine that each letter is substituted in a particular sequence. Using this pattern, "WINTER" is coded as "SDMUW".


Thus, the correct answer is (1).
Quick Tip: When deciphering coded words, identify the pattern of letter shifts or replacements.

The given puzzle involves multiplying the numbers in a pattern. The first column is multiplied by the second column, and the third column is the sum of those values. The correct result is 888.


Thus, the correct answer is (1).
Quick Tip: Look for patterns in the arrangement of numbers when solving puzzles of this type.

61 days from Monday is equivalent to \( 61 \div 7 = 8 \) weeks and 5 days. Counting 5 days ahead from Monday gives us Saturday.


Thus, the correct answer is (2).
Quick Tip: Use division by 7 to find the remainder when calculating days of the week after a certain number of days.

Let the total number of persons be \( N \). Using the given positions of Rahul and Nitesh, we can set up an equation to find \( N \). After finding \( N \), we can determine Nitesh's new position from the right. The new position is 31.


Thus, the correct answer is (4).
Quick Tip: When solving position problems, use the total number of persons in the row and apply algebraic equations to find the unknown.

By observing the pattern of the series, we can see that 414 does not follow the established rule, while the others do. Therefore, 414 is the wrong number.


Thus, the correct answer is (2).
Quick Tip: When solving number series, identify the pattern and spot the number that does not fit.

By substituting the correct mathematical operations from the choices, the equation becomes true. The correct sequence is \( = \times + = \).


Thus, the correct answer is (2).
Quick Tip: When solving equations with missing operations, test each option by substituting different operations.

The diagram that shows carrot as a vegetable and food is the correct representation.


Thus, the correct answer is (1).
Quick Tip: Look for diagrams that match the given categories and choose the one that correctly represents them.

The correct interpretation of the statement leads to the conclusion that no women are members of the Tennis club.


Thus, the correct answer is (4).
Quick Tip: Pay attention to logical phrasing and interpret the statements carefully.

By visualizing the mirror image, the correct answer is (d), which is the exact mirror image of the given figure.


Thus, the correct answer is (4).
Quick Tip: To solve mirror image problems, visualize how the figure would appear in a mirror.

By visualizing the mirror image, the correct answer is (c), which represents the exact mirror image of the given figure.


Thus, the correct answer is (3).
Quick Tip: To solve mirror image problems, carefully observe how the figure will be reflected.