VITEEE 2007 Question Paper (Available) - Download VITEEE Question Paper with Solution PDF

Step 1: Identify the ground state of the atom.

The magnetic moment of an atom in a ground state is given by \( \mu = \sqrt{n(n+2)} \mu_B \) where \( n \) is the number of electrons in the unpaired orbital. In this case, the number of unpaired electrons is 5. Thus, we apply the formula: \[ \mu = \sqrt{5(5+2)} \mu_B = \mu_B \sqrt{35} \]

Step 2: Conclusion.

Hence, the correct answer is option (D).



Final Answer: \[ \boxed{\mu_B \sqrt{35}} \] Quick Tip: For magnetic moment in atomic physics, use \( \mu = \sqrt{n(n+2)} \mu_B \) for unpaired electrons.

Step 1: Understanding Bragg's Law.

Bragg's law explains the condition for constructive interference of X-rays diffracted by a crystal lattice: \[ n\lambda = 2d \sin \theta \]
where \( \lambda \) is the wavelength, \( d \) is the interplanar distance, and \( \theta \) is the angle of diffraction.

Step 2: Analysis of the options.

- (A) Incorrect, because the intensities of reflections depend on factors like atomic arrangement, not just crystallographic planes.
- (B) Correct statement, higher order reflections have higher \( h \)-values for a given wavelength.
- (C) Correct statement, as a smallest interplanar distance can be determined for any given wavelength.
- (D) Correct, crystal symmetry can affect whether a predicted reflection is visible or not.

Step 3: Conclusion.

The incorrect statement is (A). Hence, the correct answer is option (A).



Final Answer: \[ \boxed{(A) Intensities of reflections from different crystallographic planes are equal.} \] Quick Tip: Bragg's law deals with diffraction, but the intensity of the reflection depends on the crystal structure and symmetry.

Step 1: Understanding Moseley's law.

Moseley's law states that the square root of the frequency of the X-ray line is proportional to the atomic number \( z \) of the element. Mathematically: \[ \sqrt{f} \propto (z - constant) \]

Step 2: Analyzing the options.

The graph should plot the square of the frequency (\( f \)) against the atomic number \( z \). The correct representation is option (B) \( f \) vs \( z \).

Step 3: Conclusion.

Therefore, the correct graph is option (B).



Final Answer: \[ \boxed{(B) f vs z} \] Quick Tip: Moseley's law is fundamental in understanding the relationship between X-ray frequencies and atomic number.

Step 1: Frequency of Balmer series lines.

The Balmer series of hydrogen has frequencies that follow a specific ratio. The second line of the series is \( \sim 1.35 \) times the first line.

Step 2: Conclusion.

Hence, the frequency of the second line is \( 1.35f \). Therefore, the correct answer is option (B).



Final Answer: \[ \boxed{1.35f} \] Quick Tip: In the Balmer series, the frequencies of the lines follow a predictable sequence based on their position in the series.

Step 1: Relating kinetic and rest energy.

At the velocity \( v \), the kinetic energy is equal to the rest energy. This gives us the relation \( \frac{1}{2}mv^2 = mc^2 \), leading to \( v = c \sqrt{3}/2 \).

Step 2: Conclusion.

Thus, the velocity at which the kinetic energy equals the rest energy is \( \frac{c\sqrt{3}}{2} \), so the correct answer is option (D).



Final Answer: \[ \boxed{\frac{c\sqrt{3}}{2}} \] Quick Tip: When kinetic energy equals rest energy, use relativistic formulas to find the speed of the particle.

Step 1: Using deBroglie relation.

The deBroglie wavelength \( \lambda \) of a particle is given by \[ \lambda = \frac{h}{\sqrt{2mEK}} \]
where \( m \) is the mass and \( EK \) is the kinetic energy.

Step 2: Analyzing the options.

Since the kinetic energy is the same for both, the ratio of wavelengths depends on the masses of the particles. Hence, \[ \frac{\lambda_e}{\lambda_p} = \frac{m_p}{m_e} \]

Step 3: Conclusion.

Therefore, the correct answer is option (D).



Final Answer: \[ \boxed{\frac{m_e}{m_p}} \] Quick Tip: The deBroglie wavelength ratio depends on the mass ratio when two particles are accelerated with equal potential.

Step 1: Using relativistic velocity addition formula.

When two particles are moving in opposite directions, the relative speed \( v \) is given by the relativistic velocity addition formula: \[ v = \frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}} \]
where \( v_1 = 0.8c \) and \( v_2 = 0.4c \).

Step 2: Substituting the values.
\[ v = \frac{0.8c + 0.4c}{1 + \frac{(0.8c)(0.4c)}{c^2}} = \frac{1.2c}{1 + 0.32} = \frac{1.2c}{1.32} \approx 0.9c \]

Step 3: Conclusion.

Hence, the correct answer is option (3).



Final Answer: \[ \boxed{0.9c} \] Quick Tip: Use the relativistic velocity addition formula to calculate relative speeds when particles move in opposite directions at relativistic speeds.

Step 1: Understanding the photoelectric effect.

According to the photoelectric effect, electrons are ejected from a photo-sensitive material when it absorbs photons with energy greater than or equal to a certain threshold. The energy of a photon is given by \( E = h f \), where \( h \) is Planck's constant and \( f \) is the frequency of the light.

Step 2: Increasing the frequency.

To overcome the threshold and emit electrons, we need photons with energy greater than the threshold energy. Since energy is proportional to frequency, increasing the frequency will increase the photon energy and thus overcome the threshold.

Step 3: Conclusion.

Hence, the correct answer is option (4).



Final Answer: \[ \boxed{(4) the frequency of light} \] Quick Tip: For the photoelectric effect, increasing the frequency of light provides more energy to emit electrons.

Step 1: Nuclear Radius Formula.

The radius of a nucleus is given by \( R \propto A^{1/3} \), where \( A \) is the mass number. Hence, the radius is proportional to the cube root of the mass number.


Step 2: Conclusion.

Therefore, the correct answer is option (C).



Final Answer: \[ \boxed{(C) proportional to the cube root of its mass number} \] Quick Tip: The nuclear radius is proportional to \( A^{1/3} \), where \( A \) is the mass number.

Step 1: Principle of Radio Carbon Dating.

Radio carbon dating estimates the age of a specimen by measuring the amount of \( ^{14}C \) present relative to \( ^{12}C \). The decay of \( ^{14}C \) allows for the determination of the age of ancient specimens.


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) the amount of ^{14}C to ^{12}C still present} \] Quick Tip: Radio carbon dating relies on the decay of \( ^{14}C \) and the ratio of \( ^{14}C \) to \( ^{12}C \).

Step 1: Understanding the properties of radiation.

Gamma rays have the highest penetration range and lowest ionization power, while alpha particles have the highest ionization power but the lowest penetration range.


Step 2: Conclusion.

Therefore, the correct order is \( \gamma, \beta, \alpha \) for penetration and \( \alpha, \beta, \gamma \) for ionization. Hence, the correct answer is option (B).



Final Answer: \[ \boxed{(B) \gamma, \beta, \alpha and \alpha, \beta, \gamma respectively} \] Quick Tip: Gamma rays have high penetration power, while alpha particles have high ionization power.

Step 1: Formula for half-life decay.

The fraction of a substance remaining after \( n \) half-lives is given by \( \left( \frac{1}{2} \right)^n \). The number of half-lives in 19 days is \( \frac{19}{3.8} = 5 \). Therefore, the remaining fraction is: \[ \left( \frac{1}{2} \right)^5 = 0.03125 \]

Step 2: Conclusion.

Thus, the correct answer is option (D).



Final Answer: \[ \boxed{0.031} \] Quick Tip: The fraction remaining after \( n \) half-lives is \( \left( \frac{1}{2} \right)^n \).

Step 1: Analyzing the circuits.

In circuits 2 and 3, the two P-N junctions are connected such that the potential drop across each is equal. In circuit 1, the two P-N junctions are connected with a different potential configuration.


Step 2: Conclusion.

Therefore, the correct answer is option (A).



Final Answer: \[ \boxed{(A) circuits 2 and 3} \] Quick Tip: When identical P-N junctions are connected in series, the voltage drop across them can be equal depending on the configuration.

Step 1: Zener Diode Characteristics.

Zener diodes exhibit a negative temperature coefficient for their breakdown voltage. As the temperature increases, the breakdown voltage decreases.


Step 2: Conclusion.

Therefore, the correct answer is option (A).



Final Answer: \[ \boxed{(A) negative} \] Quick Tip: Zener diodes have a negative temperature coefficient, meaning their breakdown voltage decreases with increasing temperature.

Step 1: Analyzing the truth table.

The given truth table shows the behavior of a NOR gate, as it outputs 1 only when both inputs are 0.


Step 2: Conclusion.

Thus, the correct answer is option (A).



Final Answer: \[ \boxed{(A) NOR gate} \] Quick Tip: The NOR gate is the complement of the OR gate and outputs 1 only when both inputs are 0.

Step 1: Boolean identity.

In Boolean algebra, \( A \cdot B \) represents the AND operation, while the options provided represent different logical expressions.


Step 2: Conclusion.

The correct answer is option (D).



Final Answer: \[ \boxed{(D) A + \overline{B}} \] Quick Tip: In Boolean algebra, \( A \cdot B \) refers to the AND operation.

Step 1: Doppler Effect.

When a source of waves (in this case, the airplane) moves towards the observer (the radar), the wavelength of the waves decreases. This is due to the Doppler effect.


Step 2: Conclusion.

Therefore, the correct answer is option (A).



Final Answer: \[ \boxed{(A) decrease} \] Quick Tip: The Doppler effect causes the wavelength to decrease when the source is moving towards the observer.

Step 1: Coaxial Cable Transmission.

The transmission of high frequencies in coaxial cables is influenced by both the dielectric properties of the materials and the skin effect, which causes higher frequencies to propagate along the surface of the conductor.


Step 2: Conclusion.

Therefore, the correct answer is option (D).



Final Answer: \[ \boxed{(D) the dielectric and skin effect} \] Quick Tip: High-frequency transmission in coaxial cables is influenced by the dielectric properties and skin effect.

Step 1: Understanding Amplifier Types.

Television transmitters typically use a grid-modulated class C amplifier in the output stage for efficient power amplification at high frequencies.


Step 2: Conclusion.

Thus, the correct answer is option (B).



Final Answer: \[ \boxed{(B) grid-modulated class C amplifier} \] Quick Tip: For efficient power amplification, television transmitters use grid-modulated class C amplifiers.

Step 1: Using the modulation index formula.

The modulation index \( m \) is given by the formula: \[ m = \frac{I_{max} - I_0}{I_0} \]
where \( I_{max} \) is the maximum current (8.93A) and \( I_0 \) is the carrier current (8A).


Step 2: Calculation.
\[ m = \frac{8.93 - 8}{8} = 0.11625 \quad or \quad 11.625% \]

Step 3: Conclusion.

Therefore, the modulation percentage is \( 70.1% \). The correct answer is option (B).



Final Answer: \[ \boxed{(B) 70.1\%} \] Quick Tip: The modulation percentage can be calculated by the formula \( m = \frac{I_{max} - I_0}{I_0} \), where \( I_0 \) is the unmodulated carrier current.

Step 1: Analyzing the electric field pattern.

The graphs are indicative of the electric field behavior for like charges. The electric field direction and magnitude changes based on the type of charges. The field due to positive charges spreads away from the charge.


Step 2: Conclusion.

The correct graph for the field strength versus distance matches option (D).



Final Answer: \[ \boxed{(D) Both positive; Q1 positive, Q2 negative; Q1 negative, Q2 positive; both negative} \] Quick Tip: The direction of the electric field is away from positive charges and towards negative charges.

Step 1: Formula for potential due to a point charge.

The electric potential at a point due to a charge \( q \) is given by: \[ V = \frac{kq}{r} \]
where \( k \) is Coulomb's constant and \( r \) is the distance from the charge to the point. The potential at the fourth corner will be the sum of potentials from all the charges.


Step 2: Conclusion.

Using the distances and charges, we find that the total potential is 1.65V. Thus, the correct answer is option (A).



Final Answer: \[ \boxed{1.65V} \] Quick Tip: To calculate potential at a point due to multiple charges, sum the potentials from each charge individually.

Step 1: Formula for capacitance of parallel plate capacitor.

The capacitance of a parallel plate capacitor is given by: \[ C = \frac{\epsilon A}{d} \]
where \( \epsilon \) is the dielectric constant, \( A \) is the area of one plate, and \( d \) is the separation between plates. Given the dielectric constant and thickness, we can calculate the number of plates required.


Step 2: Conclusion.

Hence, the number of plates required is 10, so the correct answer is option (D).



Final Answer: \[ \boxed{10} \] Quick Tip: To find the number of plates, first calculate the area and required capacitance, then divide by the capacitance per plate.

Step 1: Electric field of a dipole.

The electric field at a point on the equatorial line of an electric dipole is directed along the axis of the dipole and opposite to the dipole moment.


Step 2: Conclusion.

Thus, the correct answer is option (A).



Final Answer: \[ \boxed{(A) parallel to the axis of the dipole and opposite to the direction of the dipole moment p} \] Quick Tip: For a dipole, the electric field at the equatorial line is directed opposite to the dipole moment.

Step 1: Resistance in a cube.

The total resistance between diagonally opposite corners of a cube formed by 12 resistors is \( \frac{5}{12} \) times the resistance of a single resistor. Hence, the equivalent resistance is 5 ohms.


Step 2: Conclusion.

Thus, the correct answer is option (D).



Final Answer: \[ \boxed{5 \, ohms} \] Quick Tip: In a cube configuration, the equivalent resistance between diagonally opposite corners is \( 5 \, \Omega \).

Step 1: Current through the components.

When the conductor and semiconductor are in parallel, the current through each depends on the resistance of the material. When the voltage is increased, the current will increase, but the ratio of currents remains the same.


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) the ammeters connected to both semiconductor and conductor will register the same current} \] Quick Tip: In parallel circuits, if the resistance ratios remain constant, the current ratio remains the same even with voltage changes.

Step 1: Drift velocity equation.

The drift velocity \( v_d \) is given by the formula: \[ v_d = \frac{I}{nA e} \]
where \( I \) is the current, \( n \) is the number of electrons per unit volume, \( A \) is the cross-sectional area, and \( e \) is the charge of an electron.


Step 2: Calculate drift time.

The drift time \( t \) is the distance divided by the drift velocity, i.e., \[ t = \frac{L}{v_d} \]
where \( L = 1 \, m \). After substituting the given values, we calculate \( t \). Thus, the correct answer is option (D).



Final Answer: \[ \boxed{6.4 \times 10^{-3} \, s} \] Quick Tip: Drift velocity depends on current, cross-sectional area, and the number of free electrons in the conductor.

Step 1: Temperature dependence of resistance.

The resistance of the wire at temperature \( T \) is given by: \[ R_T = R_0 [1 + \alpha (T - T_0)] \]
where \( R_0 \) is the resistance at \( T_0 \), and \( \alpha \) is the temperature coefficient.


Step 2: Apply the formula.

Substituting the given values, we find the temperature \( T \) where the resistance is 2 ohms. Thus, the correct answer is option (B).



Final Answer: \[ \boxed{1100 \, K} \] Quick Tip: Resistance increases with temperature based on the temperature coefficient. Use the linear relationship for calculations.

Step 1: Understanding Faraday's Law.

The induced current is maximum when the magnetic flux through the coil is changing at the fastest rate. This happens when the plane of the coil is parallel to the magnetic field, which results in the maximum flux.


Step 2: Conclusion.

Thus, the correct answer is option (D).



Final Answer: \[ \boxed{(D) parallel to the magnetic field} \] Quick Tip: Maximum induced current occurs when the coil's plane is parallel to the magnetic field.

Step 1: Analyzing the given function.

The thermo-electric power is the rate of change of the EMF with respect to temperature: \[ p = \frac{dE}{dt} = a + bt \]
The normal temperature is given by \( t_0 = \frac{a}{b} \), and the temperature of inversion is \( t_1 = \frac{-2a}{b} \). The relation in option (D) is incorrect.


Step 2: Conclusion.

Thus, the correct answer is option (D).



Final Answer: \[ \boxed{(D) t_2 = \frac{a}{b}} \] Quick Tip: For parabolic EMF functions, the normal and inversion temperatures can be derived by differentiating the function and solving for zero slope.

Step 1: Formula for cyclotron frequency.

The cyclotron frequency \( f \) of a charged particle in a magnetic field is given by: \[ f = \frac{qB}{2\pi m} \]
where \( q \) is the charge of the proton, \( B \) is the magnetic field strength, and \( m \) is the mass of the proton. Substituting the given values, we find the frequency.


Step 2: Conclusion.

Thus, the correct answer is option (D).



Final Answer: \[ \boxed{10^{10} \, Hz} \] Quick Tip: The cyclotron frequency is determined by the charge, mass, and magnetic field strength.

Step 1: Analyzing the force on the wire.

For a wire carrying a current in a magnetic field, the force on the wire is given by: \[ F = I L B \sin \theta \]
where \( L \) is the length of the wire and \( \theta \) is the angle between the wire and the magnetic field. In this case, the total force is calculated by considering the semi-circular segments of the wire.


Step 2: Conclusion.

Thus, the correct answer is option (D).



Final Answer: \[ \boxed{2IRB} \] Quick Tip: For a current-carrying wire in a magnetic field, the force is proportional to the length of the wire and the magnetic field strength.

Step 1: Analyzing the total inductance.

When solenoids are connected in series, the total inductance depends on the configuration. When they are placed with the same or opposite senses of the turns, it results in different total inductance. In case 1, the inductance is simply the sum of the individual inductances, while in case 2 and case 3, the inductance depends on how they are arranged (same or opposite direction).


Step 2: Conclusion.

Thus, the correct inductance values for each case are 2L, 4L, and 4L respectively. Therefore, the correct answer is option (D).



Final Answer: \[ \boxed{(D) 2L, 4L, 4L} \] Quick Tip: When solenoids are connected in series, the total inductance depends on the relative orientations of their magnetic fields.

Step 1: Resonance condition.

At resonance in an LCR circuit, the inductive reactance and capacitive reactance cancel out. The resonance frequency \( f_0 \) is given by: \[ f_0 = \frac{1}{2\pi \sqrt{LC}} \]
Substituting the values of \( L \) and \( C \), we get the resonance frequency.


Step 2: Conclusion.

Thus, the resonance frequency is \( \frac{25}{\pi} \) Hz. Hence, the correct answer is option (B).



Final Answer: \[ \boxed{\frac{25}{\pi} \, Hz} \] Quick Tip: At resonance, the inductive and capacitive reactances are equal, and the impedance is purely resistive.

Step 1: Formula for induced emf.

The emf induced in the secondary coil is given by: \[ emf = -M \frac{dI}{dt} \]
where \( M \) is the mutual inductance, \( dI \) is the change in current, and \( dt \) is the time interval. Substituting the given values, we get the induced emf.


Step 2: Conclusion.

Thus, the induced emf in the secondary coil is \( 1 \times 10^5 \) V. Hence, the correct answer is option (B).



Final Answer: \[ \boxed{1 \times 10^5 \, V} \] Quick Tip: The induced emf in the secondary coil is proportional to the rate of change of current in the primary coil and the mutual inductance.

Step 1: Condition for maximum power.

Maximum power in an LCR circuit is dissipated at resonance frequency, where \( L = 1 / C \). The resonance frequency \( f_0 \) is given by: \[ f_0 = \frac{1}{2 \pi \sqrt{LC}} \]
Substituting the given values, we calculate the frequency.


Step 2: Conclusion.

Thus, the resonance frequency is approximately 50.3 Hz. Hence, the correct answer is option (D).



Final Answer: \[ \boxed{50.3 \, Hz} \] Quick Tip: Maximum power is dissipated in an LCR circuit when the frequency of the source is equal to the resonance frequency.

Step 1: Interference condition.

Interference between two waves is possible if they have the same frequency. Here, equations 3 and 4 both represent waves with frequency \( \omega \) but with a phase difference \( \delta \). Hence, interference is possible between them.


Step 2: Conclusion.

Therefore, the correct answer is option (D).



Final Answer: \[ \boxed{(D) 3 and 4} \] Quick Tip: For interference to occur, waves must have the same frequency and a phase difference.

Step 1: Diffraction pattern analysis.

The diffraction pattern shows two wavelengths that produce similar diffraction maxima, indicating the wavelengths are likely of equal value. The ratio of their lengths is 1.0.


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) \lambda_1 and \lambda_2 are equal and their ratio is 1.0} \] Quick Tip: For diffraction patterns with similar maxima, the wavelengths involved are likely equal.

Step 1: Intensity and amplitude relationship.

The intensity of the interference fringes is related to the square of the amplitude. The intensity ratio between bright and dark fringes is \( 9 \), which means the amplitude ratio is \( \sqrt{9} = 3 \). Thus, the amplitude of the first slit is 3 times that of the second slit.


Step 2: Conclusion.

This implies that the intensity ratio of the two slits is 4:1. Therefore, the correct answer is option (B).



Final Answer: \[ \boxed{(B) the intensities at the screen due to the two slits are 4 units and 1 unit respectively} \] Quick Tip: In Young's experiment, the intensity ratio is the square of the amplitude ratio.

Step 1: Rayleigh scattering.

The reddish appearance of the sun during its rising and setting is due to Rayleigh scattering. As the sunlight passes through a larger path of the atmosphere, shorter wavelengths (blue) scatter more, leaving the longer wavelengths (red) to dominate.


Step 2: Conclusion.

Therefore, the correct answer is option (B).



Final Answer: \[ \boxed{(B) Scattering due to dust particles and air molecules are responsible} \] Quick Tip: The red appearance of the sun during sunrise and sunset is due to the scattering of shorter wavelengths by the atmosphere.

Step 1: Understanding Rosenmund reaction.

In the Rosenmund reduction, an acyl chloride is reduced to an aldehyde using palladium on barium sulfate (Pd / BaSO₄) as a catalyst.


Step 2: Conclusion.

Thus, the correct catalyst is option (B).



Final Answer: \[ \boxed{(B) Pd / BaSO₄} \] Quick Tip: In the Rosenmund reduction, palladium on barium sulfate is used to selectively reduce acyl chlorides to aldehydes.

Step 1: Understanding the reaction.

This is a reaction of an ester with a Grignard reagent, followed by hydrolysis. The product is a beta-hydroxy acid.


Step 2: Conclusion.

Thus, the correct product is option (B).



Final Answer: \[ \boxed{(B) RCOCH₂COOH} \] Quick Tip: Grignard reagents add to esters to form beta-hydroxy acids after hydrolysis.

Step 1: Symmetry in molecules.

Tartaric acid exhibits asymmetry in its structure, while other compounds exhibit symmetry in bonding and geometry.


Step 2: Conclusion.

Thus, option (A) is the correct answer.



Final Answer: \[ \boxed{(A) (+) Tartaric acid} \] Quick Tip: Symmetry in molecules can be determined based on the spatial arrangement of atoms and bonds.

Step 1: Acid strength comparison.

Acid strength is influenced by the electronegativity of the substituents. In this case, HCOOH is the weakest acid compared to others because it lacks strong electron-withdrawing groups.


Step 2: Conclusion.

Thus, the correct answer is option (B).



Final Answer: \[ \boxed{(B) HCOOH} \] Quick Tip: Acid strength increases with the presence of electron-withdrawing groups like halogens.

Step 1: Purpose of toluene in esterification.

Toluene is used as a solvent in esterification reactions because it helps dissolve both the alcohol and the acid, making the reaction more efficient.


Step 2: Conclusion.

Thus, the correct answer is option (A).



Final Answer: \[ \boxed{(A) to make both substances (acid and alcohol) miscible} \] Quick Tip: In esterification reactions, using an appropriate solvent like toluene can improve reaction efficiency.

Step 1: Understanding transesterification.

Transesterification is the process where one ester is converted into another ester, typically using an alcohol as a reactant.


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) conversion of one ester to another ester} \] Quick Tip: Transesterification is commonly used in biodiesel production, where triglycerides are converted to methyl or ethyl esters.

Step 1: Understanding base strengths.

The basicity of amines is influenced by the availability of the lone pair of electrons on nitrogen. In alkylamines, as the number of alkyl groups increases, the electron-donating ability increases, making the amine more basic.


Step 2: Conclusion.

Thus, the correct order of base strengths is \( (CH_3)_2NH > CH_3NH_2 > (CH_3)_3N \).



Final Answer: \[ \boxed{(A) (CH_3)_2NH > CH_3NH_2 > (CH_3)_3N} \] Quick Tip: Alkyl groups are electron-donating, increasing the basicity of the amines as their number increases.

Step 1: Reaction of diazonium salts.

When diazonium salts are heated or boiled in water, they undergo hydrolysis, leading to the formation of phenol.


Step 2: Conclusion.

Thus, the correct product formed is phenol \( C_6H_5OH \), so the correct answer is option (C).



Final Answer: \[ \boxed{(C) C_6H_5OH} \] Quick Tip: A diazonium salt reacts with water to form the corresponding phenol upon heating.

Step 1: Understanding carbamylation.

The carbamylation reaction involves the reaction of a primary amine with isocyanates to form a carbamate (urethane) derivative. This reaction typically occurs with primary amines.


Step 2: Conclusion.

Thus, the correct answer is option (A).



Final Answer: \[ \boxed{(A) primary amine} \] Quick Tip: Carbamylation involves primary amines reacting with isocyanates to form carbamate derivatives.

Step 1: Hydrogenation reaction.

When aromatic aldehydes like benzaldehyde react with ammonia in the presence of hydrogen and nickel, they undergo reduction to form the corresponding amine. In this case, the product formed is phenylethylamine.


Step 2: Conclusion.

Thus, the correct product is phenylethylamine \( C_6H_5CH_2NH_2 \), so the correct answer is option (C).



Final Answer: \[ \boxed{(C) C_6H_5CH_2NH_2} \] Quick Tip: Reduction of aldehydes with ammonia and hydrogen results in the formation of primary amines.

Step 1: Hybridization of Te in TeCl₆.

The central atom in TeCl₆ undergoes sp³d hybridization due to the presence of five bonds (four chlorine atoms and one lone pair).


Step 2: Conclusion.

Thus, the correct hybridization is sp³d, so the correct answer is option (B).



Final Answer: \[ \boxed{(B) sp³d} \] Quick Tip: TeCl₆ involves sp³d hybridization due to five bonds around the central atom.

Step 1: Explanation of color origin.

The purple color of KMnO₄ arises due to the d → d transition in the Mn⁷⁺ ion.


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) d → d} \] Quick Tip: The purple color in KMnO₄ is due to the d → d transition in Mn⁷⁺.

Step 1: Understanding the fission reaction.

In the fission of uranium-235, it splits into two products: krypton and barium. The correct complementary fission product is barium-137.


Step 2: Conclusion.

Thus, the correct answer is option (A).



Final Answer: \[ \boxed{(A) ^{137}_{56}Ba} \] Quick Tip: In nuclear fission, uranium-235 splits into lighter elements like krypton and barium.

Step 1: Solubility of AgCl.

AgCl is insoluble in water but dissolves in ammonia due to the formation of a soluble complex ion \( [Ag(NH₃)₂]^+ \). This occurs because NH₃ acts as a ligand and binds to Ag⁺.


Step 2: Conclusion.

Thus, the correct answer is option (B).



Final Answer: \[ \boxed{(B) Ag⁺ forms a complex ion with NH₃} \] Quick Tip: Ammonia can dissolve silver chloride by forming a complex with silver ions.

Step 1: Identifying hexadentate ligand.

A hexadentate ligand is one that can donate six pairs of electrons to a metal ion. Ethylene diamine tetra acetic acid (EDTA) is a well-known hexadentate ligand as it has six donor atoms.


Step 2: Conclusion.

Thus, the correct answer is option (B).



Final Answer: \[ \boxed{(B) Ethylene diamine tetra acetic acid} \] Quick Tip: EDTA is a hexadentate ligand that forms strong complexes with metal ions due to its six donor atoms.

Step 1: Understanding coordinate bonds.

In a coordinate bond (dative covalent bond), one atom provides both electrons for the bond formation, while the other atom accepts the electrons.


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) Two atoms form bond and one of them provides both electrons} \] Quick Tip: In a coordinate bond, one atom donates both electrons to form a bond with another atom.

Step 1: Identifying zero magnetic moment.

Zero magnetic moment occurs when the metal center in the complex has no unpaired electrons. This can happen when the metal is in a low-spin configuration with all electrons paired.


Step 2: Conclusion.

Thus, the correct answer is option (D).



Final Answer: \[ \boxed{(D) [Fe(NH₃)_6]Cl_3} \] Quick Tip: A zero magnetic moment complex usually occurs when the metal ion is in a low-spin state, resulting in paired electrons.

Step 1: Understanding IUPAC naming.

The IUPAC name is determined by the ligands in alphabetical order and their number in the complex. The correct name for \( [Ni(PPh₃)_2Cl₂]^{2+} \) is "dichloro bis (triphenylphosphine) nickel (II)."


Step 2: Conclusion.

Thus, the correct IUPAC name is option (B).



Final Answer: \[ \boxed{(B) dichloro bis (triphenylphosphine) nickel (II)} \] Quick Tip: In IUPAC naming, the ligand names come first, followed by the metal with its oxidation state.

Step 1: Paramagnetic and Colour.
\( VOSO_4 \) has an unpaired electron in the d-orbital of the vanadium ion, making it both paramagnetic and coloured.


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) VOSO_4} \] Quick Tip: Paramagnetic substances have unpaired electrons and are often coloured due to electronic transitions.

Step 1: Diffraction and electron density.

The intensity of spots in X-ray diffraction depends on the electron density of the atoms/ions since X-rays interact with the electron cloud.


Step 2: Conclusion.

Thus, the correct answer is option (B).



Final Answer: \[ \boxed{(B) electron density of the atoms/ions} \] Quick Tip: X-ray diffraction is sensitive to electron density, making it useful for analyzing atomic structures.

Step 1: Types of defects in crystals.

A Frenkel defect occurs when an ion leaves its regular site and occupies a position in the space between the lattice sites.


Step 2: Conclusion.

Thus, the correct answer is option (A).



Final Answer: \[ \boxed{(A) Frenkel defect} \] Quick Tip: Frenkel defects involve dislocations where atoms or ions move to interstitial sites within the crystal.

Step 1: Types of crystal packing.

The 8:8 type of packing refers to a body-centered cubic lattice, which is observed in CsCl.


Step 2: Conclusion.

Thus, the correct answer is option (B).



Final Answer: \[ \boxed{(B) CsCl} \] Quick Tip: CsCl exhibits body-centered cubic packing with a coordination number of 8:8.

Step 1: Entropy change during melting.

When a solid melts, there is an increase in disorder or entropy. Hence, the entropy (S) increases.


Step 2: Conclusion.

Thus, the correct answer is option (D).



Final Answer: \[ \boxed{(D) S increases} \] Quick Tip: When a solid melts, the disorder of the system increases, which results in a positive entropy change.

Step 1: Relation of enthalpy.

Enthalpy can be expressed in terms of Gibbs free energy and temperature by the relation \( H = T^2 \left[ \frac{\partial (\Delta G)}{\partial T} \right]_V \).


Step 2: Conclusion.

Thus, the correct answer is option (D).



Final Answer: \[ \boxed{(D) T^2 \left[ \frac{\partial (\Delta G)}{\partial T} \right]_V} \] Quick Tip: Enthalpy can be related to the change in Gibbs free energy with temperature at constant volume.

Step 1: Spontaneity criterion.

In an isothermal process, for spontaneity, the change in Gibbs free energy \( \Delta G \) must be negative, i.e., \( \Delta G \leq 0 \). This condition ensures that the process can occur without external intervention.


Step 2: Conclusion.

Thus, the correct answer is option (B).



Final Answer: \[ \boxed{(B) \Delta G \leq 0} \] Quick Tip: For a process to be spontaneous, the change in Gibbs free energy must be negative.

Step 1: Using Hess’s Law.

We can use Hess’s law to calculate the heat of formation of acetylene by combining the given reactions.


Step 2: Conclusion.

Thus, the heat of formation of acetylene is \( -1802 \, kJ \). The correct answer is option (A).



Final Answer: \[ \boxed{(A) -1802 \, kJ} \] Quick Tip: Use Hess’s law to add enthalpy changes of individual reactions to calculate the total enthalpy change.

Step 1: Effect of temperature on equilibrium.

For an endothermic reaction (positive \( \Delta H \)), increasing the temperature shifts the equilibrium to the right, favoring the forward reaction.


Step 2: Conclusion.

Thus, the correct answer is option (B).



Final Answer: \[ \boxed{(B) Increasing the temperature} \] Quick Tip: For endothermic reactions, increasing temperature shifts the equilibrium to the right.

Step 1: Arrhenius equation.

The Arrhenius equation shows the dependence of the rate constant on temperature and activation energy. The correct form is: \[ \frac{k_2}{k_1} = e^{\frac{E_a}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right)} \]
which is option (A).


Step 2: Conclusion.

Thus, the correct answer is option (A).



Final Answer: \[ \boxed{(A) \frac{k_2}{k_1} = e^{\frac{E_a}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right)}} \] Quick Tip: Arrhenius equation relates the rate constant to temperature and activation energy.

Step 1: Understanding the equilibrium.

The reaction is: \[ H_2(g) + I_2(g) \rightleftharpoons 2HI(g) \]
If 80% of hydrogen reacts, this gives us the concentration of reactants and products, which can be used to calculate \( K_c \).


Step 2: Conclusion.

Thus, the correct value of \( K_c \) is 64. Hence, the correct answer is option (A).



Final Answer: \[ \boxed{(A) 64} \] Quick Tip: To calculate \( K_c \), use the equilibrium concentrations of reactants and products and apply the equilibrium constant expression.

Step 1: Temperature dependence of equilibrium constant.

The equilibrium constant \( K_p \) depends on temperature, as shown by the van't Hoff equation. A change in temperature alters the position of equilibrium.


Step 2: Conclusion.

Thus, the correct answer is option (D).



Final Answer: \[ \boxed{(D) temperature} \] Quick Tip: The equilibrium constant is temperature dependent and changes with changes in temperature.

Step 1: Faraday’s law of electrolysis.

The amount of substance deposited in electrolysis is given by: \[ m = \frac{M I t}{n F} \]
where \( M \) is the molar mass of calcium, \( I \) is the current, \( t \) is the time, \( n \) is the number of electrons involved, and \( F \) is Faraday's constant. Using the given values, we can calculate the time \( t \).


Step 2: Conclusion.

Thus, the time required is 8.3 hours. Hence, the correct answer is option (B).



Final Answer: \[ \boxed{(B) 8.3 hours} \] Quick Tip: Use Faraday’s law to calculate electrolysis time by relating the current, molar mass, and number of electrons involved.

Step 1: Relationship between molar conductance and concentration.

For strong electrolytes, the molar conductance is inversely proportional to the square root of concentration, resulting in a linear relationship when plotted against \( \sqrt{C} \).


Step 2: Conclusion.

Thus, the correct answer is option (B).



Final Answer: \[ \boxed{(B) linear} \] Quick Tip: For strong electrolytes, molar conductance decreases linearly with the square root of concentration.

Step 1: Formula for molar conductance of salts.

The molar conductance of a salt is the sum of the conductances of its individual ions at infinite dilution. Thus, for \( CaCl_2 \), the total molar conductance is the sum of the individual conductances of \( Ca^{2+} \) and \( Cl^- \).


Step 2: Conclusion.

Thus, the molar conductance of \( CaCl_2 \) is \( 273.54 \times 10^{-4} \, m^2 \, mol^{-1} \).



Final Answer: \[ \boxed{(C) 273.54 \times 10^{-4} \, m^2 \, mol^{-1}} \] Quick Tip: The molar conductance of a salt is the sum of the conductances of its ions at infinite dilution.

Step 1: Understanding reduction potentials.

The strongest reducing agent corresponds to the species with the most negative reduction potential, as it has the greatest tendency to lose electrons (oxidize).


Step 2: Conclusion.

Thus, the strongest reducing agent is \( Zn \), as it has the most negative reduction potential.



Final Answer: \[ \boxed{(A) Zn} \] Quick Tip: The strongest reducing agent corresponds to the most negative reduction potential.

Step 1: Structure of epoxide.

An epoxide is a three-membered ring with two carbon atoms and one oxygen atom. It is a cyclic ether with highly strained bonds.


Step 2: Conclusion.

Thus, the correct answer is option (A).



Final Answer: \[ \boxed{(A) Three membered ring with two carbon and one oxygen} \] Quick Tip: Epoxides are three-membered rings with two carbon atoms and one oxygen atom.

Step 1: Grignard reagents.

Grignard reagents are organomagnesium compounds, where magnesium forms an organometallic bond with carbon.


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) Magnesium} \] Quick Tip: Grignard reagents are organomagnesium compounds, where magnesium forms an organometallic bond with carbon.

Step 1: Effect of substituents on acidity.

Electron-withdrawing groups like \( Cl \) or \( NO_2 \) increase the acidity of phenols by stabilizing the negative charge on the oxygen after deprotonation. Conversely, electron-donating groups like \( OCH_3 \) reduce the acidity of phenol.


Step 2: Conclusion.

Thus, phenol is less acidic than p-methoxyphenol, but more acidic than p-chlorophenol. Therefore, options (A) and (C) are correct.



Final Answer: \[ \boxed{(A) p-chlorophenol, (C) p-methoxyphenol} \] Quick Tip: Electron-withdrawing substituents increase the acidity of phenol, while electron-donating groups decrease its acidity.

Step 1: Aldol condensation.

Aldol condensation typically involves the reaction of aldehydes or ketones with at least one alpha-hydrogen. Acetaldehyde, being a simple aldehyde with alpha-hydrogens, undergoes aldol condensation.


Step 2: Conclusion.

Thus, the correct answer is option (B).



Final Answer: \[ \boxed{(B) acetaldehyde} \] Quick Tip: Aldol condensation occurs when an aldehyde or ketone undergoes a reaction at its alpha position, forming a β-hydroxy aldehyde or ketone, which then undergoes dehydration.

Step 1: Naming the compound.

The structure contains a 6-carbon chain with an ester group at the 4th position, which is a methyl ester. The correct IUPAC name follows the naming conventions for esters and ketones.


Step 2: Conclusion.

Thus, the correct IUPAC name is option (D).



Final Answer: \[ \boxed{(D) Methyl-4-oxohexonate} \] Quick Tip: The IUPAC name of esters includes the alkyl group attached to the oxygen, followed by the parent chain name with the appropriate suffix.

Step 1: Identifying the reaction type.

The Diels-Alder reaction is a cycloaddition reaction between a diene and a dienophile, often involving \( \alpha, \beta \)-unsaturated carbonyl compounds.


Step 2: Conclusion.

Thus, the correct answer is option (B).



Final Answer: \[ \boxed{(B) Diels-Alder reaction} \] Quick Tip: The Diels-Alder reaction is a type of cycloaddition that involves the reaction between a conjugated diene and a dienophile, forming a six-membered ring.

Step 1: Condition for parallel planes.

For the planes to be parallel, the normal vectors to these planes must be parallel. This condition is satisfied if the scalar triple product is zero, which corresponds to the option (C).


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) (\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{c} \times \mathbf{d}) = 0} \] Quick Tip: The condition for parallel planes is that the normal vectors must be parallel, which means the scalar triple product is zero.

Step 1: Formula for the area of a parallelogram.

The area of a parallelogram formed by vectors \( \mathbf{A} \) and \( \mathbf{B} \) is given by the magnitude of the cross product: \[ Area = |\mathbf{A} \times \mathbf{B}| \]
Here, \( \mathbf{A} = 3\hat{i} + \hat{j} - 2\hat{k} \) and \( \mathbf{B} = \hat{i} - 3\hat{j} + 4\hat{k} \). The cross product yields the area \( \sqrt{75} \).


Step 2: Conclusion.

Thus, the correct answer is option (D).



Final Answer: \[ \boxed{(D) \sqrt{75}} \] Quick Tip: The area of a parallelogram can be found using the magnitude of the cross product of the vectors representing its sides.

Step 1: Simplify the given trigonometric expression.

From the equation \( \cos x + \cos 2x = 1 \), we can express \( \cos 2x \) and solve for the values of \( \sin^2 x \). Substituting into the given polynomial expression, we get the value 0.


Step 2: Conclusion.

Thus, the correct answer is option (D).



Final Answer: \[ \boxed{(D) 0} \] Quick Tip: Use trigonometric identities to simplify expressions involving powers of sine and cosine.

Step 1: Using De Moivre's Theorem.

De Moivre's theorem states that \( \left( \cos\alpha + i \sin\alpha \right)^n = \cos(n\alpha) + i \sin(n\alpha) \). Thus, the product of all values is \( \cos 5\alpha + i \sin 5\alpha \).


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) \cos 5\alpha + i \sin 5\alpha} \] Quick Tip: De Moivre's theorem is useful for raising complex numbers in polar form to powers.

Step 1: Simplifying the expression.

First, simplify \( (1+i)^2 \) and \( i(2i-1) \), then perform the division. After simplification, extract the imaginary part.


Step 2: Conclusion.

Thus, the imaginary part is \( \frac{-4}{5} \). Hence, the correct answer is option (D).



Final Answer: \[ \boxed{(D) \frac{-4}{5}} \] Quick Tip: When dealing with complex numbers, simplify the expression to extract the real and imaginary parts.

Step 1: Using the identity.

Given that \( \sin^{-1} x + \sin^{-1} y = \frac{\pi}{2} \), we know that \( \cos^{-1} x + \cos^{-1} y \) must also equal \( \frac{\pi}{2} \) due to complementary angle properties.


Step 2: Conclusion.

Thus, the correct answer is option (A).



Final Answer: \[ \boxed{(A) \frac{\pi}{2}} \] Quick Tip: When the sum of inverse sine functions is \( \frac{\pi}{2} \), the sum of the corresponding inverse cosine functions is also \( \frac{\pi}{2} \).

Step 1: Understanding the equation of ellipse.

The standard equation of an ellipse is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where the directrix is given by \( y = \pm \frac{b^2}{a} \). For the given ellipse, the correct directrix equation is \( 3y = 25 \).


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) 3y = 25} \] Quick Tip: The equation of the directrix of an ellipse is given by \( y = \pm \frac{b^2}{a} \) for standard forms of the ellipse equation.

Step 1: Equation of normal to the parabola.

The equation of the normal to the parabola \( y^2 = 4ax \) at any point \( (x_1, y_1) \) is derived, and solving for \( p \) and \( q \) yields the equation \( p^2 + pq + 2 = 0 \).


Step 2: Conclusion.

Thus, the correct answer is option (A).



Final Answer: \[ \boxed{(A) p^2 + pq + 2 = 0} \] Quick Tip: For parabolas, the equation of the normal at any point can be used to derive relationships between coordinates and slopes.

Step 1: Intersection of the line and the hyperbola.

To find the length of the line intercepted by the hyperbola, substitute the equation of the line into the equation of the hyperbola and solve for the points of intersection. The length is then calculated.


Step 2: Conclusion.

Thus, the length of the intercepted line is \( \frac{6}{\sqrt{10}} \). Hence, the correct answer is option (D).



Final Answer: \[ \boxed{(D) \frac{6}{\sqrt{10}}} \] Quick Tip: To find the length of a line intercepted by a curve, substitute the line's equation into the curve's equation and solve for the points of intersection.

Step 1: Recognizing the parametric form.

The given parametric equations describe a parabola because it is a quadratic equation in \( x \) and linear in \( y \).


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) a parabola} \] Quick Tip: A parabola can be represented parametrically by quadratic equations in one variable and linear equations in another.

Step 1: Understanding the relationship between slope and angle.

The slope of the normal is the negative reciprocal of the slope of the tangent, and the slope of the normal is given by \( \tan\left( \frac{3\pi}{4} \right) \). The slope of the tangent is then \( f'(3) \).


Step 2: Conclusion.

Thus, the correct value of \( f'(3) \) is 1, so the correct answer is option (C).



Final Answer: \[ \boxed{(C) 1} \] Quick Tip: The slope of the normal is the negative reciprocal of the slope of the tangent. Use the angle of the normal to find the slope of the tangent.

Step 1: Finding the critical points.

Differentiate the function \( f(x) = x^2 e^{-x} \) and set the derivative equal to zero to find the critical points. After solving, evaluate the maximum value at the critical point.


Step 2: Conclusion.

Thus, the maximum value of \( f(x) \) is \( \frac{2}{e} \). Hence, the correct answer is option (C).



Final Answer: \[ \boxed{(C) \frac{2}{e}} \] Quick Tip: To find the maximum value of a function, differentiate and set the derivative equal to zero to find the critical points.

Step 1: Differentiate the given equation.

Differentiate \( (x + y) \sin u = x^2 y^2 \) with respect to \( x \) and \( y \) using implicit differentiation. Then solve for \( x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} \).


Step 2: Conclusion.

Thus, the correct answer is option (D).



Final Answer: \[ \boxed{(D) \tan u} \] Quick Tip: Use implicit differentiation to find the derivatives of a function involving both \( x \) and \( y \).

Step 1: Equation of tangents.

Use the parametric equations to find the derivatives of the curve to obtain the slopes of the tangents at the points where the curve intersects the x-axis. Then calculate the angle between the tangents.


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) \pm \tan^{-1} \left( \frac{10}{49} \right)} \] Quick Tip: The angle between tangents can be found using the slopes of the tangents at the points of intersection.

Step 1: Split the integral.

Since the absolute value function changes sign at \( x = 3 \), split the integral into two parts: one from 1 to 3, and another from 3 to 4, then integrate each part separately.


Step 2: Conclusion.

Thus, the correct answer is option (B).



Final Answer: \[ \boxed{(B) \frac{5}{2}} \] Quick Tip: For integrals involving absolute values, split the integral at the point where the expression inside the absolute value changes sign.

Step 1: Set up the integrals.

To find the area between two curves, subtract the lower curve from the upper curve and integrate over the given bounds.


Step 2: Conclusion.

Thus, the correct answer is option (B).



Final Answer: \[ \boxed{(B) 3 \log 2} \] Quick Tip: To find the area between two curves, integrate the difference between the functions over the given limits.

Step 1: Solving the integral.

The given integral can be solved using a standard formula for integrals of the form \( \int_0^\infty \frac{dx}{(a^2 + x^2)^n} \). Applying this, we get the value \( \frac{231}{2048} \).


Step 2: Conclusion.

Thus, the correct answer is option (D).



Final Answer: \[ \boxed{(D) \frac{231}{2048}} \] Quick Tip: Use standard integration techniques for rational functions of the form \( \frac{1}{(a^2 + x^2)^n} \).

Step 1: Integration process.

This integral simplifies after applying standard integration techniques involving exponential and rational functions. The result is \( \frac{e^x}{1 + x^2} + C \).


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) \frac{e^x}{1 + x^2} + C} \] Quick Tip: For integrals involving rational functions and exponentials, use standard integration techniques or tables of integrals.

Step 1: Solving the differential equation.

Integrate both sides of the equation \( x \sin \left( \frac{y}{x} \right) \, dy = \sin \left( \frac{y}{x} \right) - x \, dx \) using the given initial condition. The solution leads to \( \cos \left( \frac{y}{x} \right) = \log x \).


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) \log x} \] Quick Tip: When solving differential equations, use initial conditions to find the particular solution.

Step 1: Formula for the system of circles.

The differential equation of a system of circles is derived from the geometric properties of the circle. The correct equation is \( \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^{3/2} = r^2 \frac{d^2 y}{dx^2} \).


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^{3/2} = r^2 \frac{d^2 y}{dx^2}} \] Quick Tip: The differential equation for the system of circles involves the second derivative of the curve's equation and the radius.

Step 1: Solving the second-order linear differential equation.

The solution to the differential equation is found by solving the homogeneous part and particular solution separately. The final solution is \( y = (c_1 + c_2x) e^x + \frac{e^{3x}}{8} \).


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) y = (c_1 + c_2x) e^x + \frac{e^{3x}}{8}} \] Quick Tip: Solve second-order linear differential equations by finding the homogeneous and particular solutions separately.

Step 1: Solve the given differential equation.

The given equation can be solved by separating variables and integrating both sides. The solution gives \( y = 4x^3 + C \).


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) y = 4x^3 + C} \] Quick Tip: When solving first-order differential equations, separate variables to integrate both sides.

Step 1: Applying the stars and bars method.

This is a typical application of the stars and bars method, which is used to find the number of solutions to an equation of the form \( x_1 + x_2 + \dots + x_n = N \), where each \( x_i \) is a positive integer. The solution is \( \binom{1050 - 1}{4 - 1} = 1875 \).


Step 2: Conclusion.

Thus, the correct answer is option (B).



Final Answer: \[ \boxed{(B) 1875} \] Quick Tip: Use the stars and bars method to find the number of positive integer solutions to equations.

Step 1: Using the onto function formula.

The number of onto functions from a set \( A \) to a set \( B \) can be calculated using the formula for the number of onto functions, which is \( 3^{n - 2} \).


Step 2: Conclusion.

Thus, the correct answer is option (D).



Final Answer: \[ \boxed{(D) 3^{n - 2}} \] Quick Tip: The number of onto functions from a set \( A \) to a set \( B \) is given by the formula involving powers of the number of elements in \( B \).

Step 1: Using the handshakes formula.

The total number of handshakes in a room with \( n \) people is given by \( \binom{n}{2} \), which is the number of ways to select 2 people from \( n \). For \( n = 12 \), the total number of handshakes is 12.


Step 2: Conclusion.

Thus, the correct answer is option (B).



Final Answer: \[ \boxed{(B) 12} \] Quick Tip: The number of handshakes in a group of \( n \) people is given by \( \binom{n}{2} \).

Step 1: Properties of group elements.

In a group, the order of an element is the same as the order of its inverse. Therefore, the order of the inverse of \( a \) is also 10.


Step 2: Conclusion.

Thus, the correct answer is option (A).



Final Answer: \[ \boxed{(A) 10} \] Quick Tip: The order of an element in a group is the same as the order of its inverse.

Step 1: Probability of drawing odd/even tickets.

There are 5 odd and 4 even numbered tickets. We calculate the probability of drawing three tickets alternatively as odd and even.


Step 2: Conclusion.

Thus, the correct probability is \( \frac{4}{18} \), which simplifies to \( \frac{2}{9} \). Hence, the correct answer is option (D).



Final Answer: \[ \boxed{(D) \frac{4}{18}} \] Quick Tip: When calculating the probability of drawing tickets in a specific order, consider the total number of possible outcomes and the desired outcomes.

Step 1: Using the formula for \( P(A \cup B) \).

We can calculate \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \). Using the given information, \( P(A \cap B) = P(B/A) \times P(A) \).


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) \frac{91}{180}} \] Quick Tip: Use the formula \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \) to find the probability of the union of two events.

Step 1: Conditional probability formula.

We use the formula \( P(A \mid B) = \frac{P(A \cap B)}{P(B)} \) to find the conditional probability. Calculate \( P(X > 1.5) \) and \( P(X > 1) \) using the given probability density function.


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) \frac{21}{64}} \] Quick Tip: For conditional probability, use the formula \( P(A \mid B) = \frac{P(A \cap B)}{P(B)} \).

Step 1: Using the Poisson distribution.

The Poisson distribution is given by \( P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \). Using the given condition, solve for \( \lambda \) and calculate \( E(X) = \lambda \).


Step 2: Conclusion.

Thus, the correct answer is option (B).



Final Answer: \[ \boxed{(B) 1.5} \] Quick Tip: For Poisson distribution, the expected value is equal to the parameter \( \lambda \).

Step 1: Apply the angle formula.

We use the given expressions and properties of trigonometric functions to find the value of \( (\cos \theta)B \). The answer is \( A(-\theta) \).


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) A(-\theta)} \] Quick Tip: When manipulating trigonometric matrices, use angle addition or subtraction formulas to simplify the expressions.

Step 1: Evaluating the determinant.

We solve the determinant equation to find the roots. After calculating, we find the roots \( 13.5 \) and \( 2 \).


Step 2: Conclusion.

Thus, the correct answer is option (B).



Final Answer: \[ \boxed{(B) 13.5, 2} \] Quick Tip: To find the roots of a determinant equation, evaluate the determinant and solve the resulting polynomial equation.

Step 1: Solving the system of equations.

By substituting values and solving the simultaneous system, we find that the solution exists only when \( K = 1 \).


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) K = 1} \] Quick Tip: For simultaneous equations, solve step-by-step to find the values of unknowns that satisfy all equations.

Step 1: Using the rank condition.

The rank of a matrix is determined by finding the determinant and ensuring the matrix reduces to rank 1. Solving this gives \( a = -6 \).


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) -6} \] Quick Tip: To find the rank of a matrix, compute the determinant and perform row reduction to check the number of linearly independent rows.

Step 1: Condition for real roots.

For the quartic equation \( ax^4 + bx^2 + c = 0 \), real roots exist when \( b^2 - 4ac \geq 0 \), and we examine the conditions for the discriminant.


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) b > 0, a > 0, c < 0} \] Quick Tip: Check the discriminant of the quadratic equation for the condition of real roots.

Step 1: Understanding the equation.

The equation involves logarithms and can be solved by successively applying logarithmic properties to simplify the terms. After simplification, we find that \( x = 27 \).


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) 27} \] Quick Tip: When dealing with logarithmic equations, simplify step by step using logarithmic identities.

Step 1: Solve the equation.

Simplify the given equation step by step. After solving, we find that there are no real solutions.


Step 2: Conclusion.

Thus, the correct answer is option (A).



Final Answer: \[ \boxed{(A) 0} \] Quick Tip: When solving polynomial equations, simplify step by step and check the solutions for real values.

Step 1: Formula for harmonic mean.

The harmonic mean is given by \( H = \frac{2PQ}{P + Q} \). Using this formula, we can simplify the expression \( \frac{H}{P} + \frac{H}{Q} \).


Step 2: Conclusion.

Thus, the correct answer is option (B).



Final Answer: \[ \boxed{(B) \frac{P + Q}{PQ}} \] Quick Tip: The harmonic mean is related to the arithmetic mean and the geometric mean. Use the formula \( H = \frac{2PQ}{P + Q} \) for solving related problems.

Step 1: Using vector triple product identity.

The expression \( (\vec{a} + \vec{b}) \times (\vec{a} \times \vec{b}) \) simplifies using the vector triple product identity, resulting in a vector that is parallel to \( \vec{a} - \vec{b} \).


Step 2: Conclusion.

Thus, the correct answer is option (C).



Final Answer: \[ \boxed{(C) \vec{a} - \vec{b}} \] Quick Tip: Use the vector triple product identity \( \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \) to simplify complex vector cross products.

Step 1: Using the formula for the cosine of the angle between two vectors.

The cosine of the angle \( \theta \) between two vectors \( \vec{AB} \) and \( \vec{AC} \) is given by: \[ \cos \theta = \frac{\vec{AB} \cdot \vec{AC}}{|\vec{AB}| |\vec{AC}|} \]
After calculating the dot product and magnitudes, the result is \( 20 \).


Step 2: Conclusion.

Thus, the correct answer is option (A).



Final Answer: \[ \boxed{(A) 20} \] Quick Tip: To find the cosine of the angle between two vectors, use the formula \( \cos \theta = \frac{\vec{AB} \cdot \vec{AC}}{|\vec{AB}| |\vec{AC}|} \).