MHT CET 2024 3 May Shift 1 Question Paper (Available): Download PCM Question Paper with Answers PDF

Step 1: Analyze the hcp structure.

In a Hexagonal Close-Packed (hcp) structure, the coordination number is 12 because each atom is surrounded by 12 nearest neighbors:

• 6 atoms in the same plane as the central atom,
• 3 atoms above,
• 3 atoms below.

This maximizes the packing efficiency and provides the characteristic coordination number of 12.

Step 1: Recall the number of atoms in an FCC unit cell.

In a face-centered cubic (FCC) structure, there are 4 atoms per unit cell.

Step 2: Calculate the density.

Density is defined as the mass per unit volume:

Density = (Mass of unit cell) / (Volume of unit cell).

The mass of the unit cell is 4M, where M is the atomic mass. The volume of the unit cell is a^3, where a is the edge length.

Thus, the density is:

Density = 4M / a^3.

Step 1: Recall the Gibbs free energy equation.

The change in Gibbs free energy (ΔG) is calculated using the formula:

ΔG = ΔH - TΔS,

where:

• ΔG is the change in Gibbs free energy,
• ΔH is the change in enthalpy,
• ΔS is the change in entropy,
• T is the temperature in Kelvin.

Step 2: For spontaneity, ΔG must be negative.

The process will be spontaneous if the enthalpy decreases (exothermic reaction) and/or the entropy increases.

Step 1: Understand essential amino acids.

Essential amino acids cannot be synthesized by the body and must be obtained from the diet. Leucine is one such amino acid.

Step 2: Differentiate from non-essential amino acids.

Non-essential amino acids like alanine, glycine, and glutamine can be synthesized by the human body.

Step 1: Define reducing sugars.

Reducing sugars contain an aldehyde or free ketone group in their open-chain form, which enables them to donate electrons and reduce other compounds.

Step 2: Identify the feature in glucose.

In its open-chain form, glucose contains an aldehyde group, making it a reducing sugar. In its cyclic form, glucose forms a hemiacetal, but the aldehyde group is exposed in the open-chain form.

Step 1: Understand the definition of a homopolymer.

A homopolymer consists of only one type of monomer. Among the options:

• Polystyrene is a homopolymer formed from styrene monomers.
• Nylon is a copolymer made from two different monomers.
• Bakelite is a thermosetting polymer formed from phenol and formaldehyde.
• PVC is a homopolymer made from vinyl chloride monomers.

Thus, the correct answer is Polystyrene.

Step 1: Recall the Gattermann-Koch reaction.

The Gattermann-Koch reaction involves the reaction of benzene with carbon monoxide and hydrochloric acid in the presence of a Lewis acid catalyst (e.g., AlCl3).

This reaction adds an aldehyde group (-CHO) to the benzene ring, forming benzaldehyde.

Other Methods:

• Friedel-Crafts acylation forms ketones, not aldehydes.
• Reduction of benzoic acid produces benzyl alcohol, not benzaldehyde.
• Ozonolysis of styrene produces aldehydes or ketones but not specifically benzaldehyde.

Step 1: Understand zero-order reactions.

In a zero-order reaction, the rate of reaction does not depend on the concentration of reactants. The rate law is given by:

Rate = k,

where k is the rate constant.

Step 2: Analyze the options:

• Option (1): Represents a first-order reaction where Rate = k[A].
• Option (2): Represents a second-order reaction where Rate = k[A]^2.
• Option (3): Correctly represents a zero-order reaction where Rate = k.
• Option (4): Represents a general reaction where the order is n.

Step 1: Understand the Rosenmund reduction.

This reaction involves the reduction of an acyl chloride to an aldehyde using hydrogen gas in the presence of a palladium catalyst poisoned by sulfur (e.g., palladium on barium sulfate).

The poisoning of the catalyst prevents the aldehyde from further reducing to an alcohol, making the reaction selective.

Step 1: Analyze solubility factors.

Alcohols are soluble in water due to the presence of the hydroxyl (-OH) group, which forms hydrogen bonds with water molecules.

Other compounds:

• Nonpolar hydrocarbons are generally insoluble in water due to lack of hydrogen bonding.
• Ionic compounds vary in solubility based on ion size and charge.
• Gaseous compounds can have varying solubility, but most are not highly soluble in water.

Step 1: Understand molar conductivity.

Molar conductivity is defined as the conductivity of an electrolyte solution divided by the molar concentration of the electrolyte.

Factors affecting molar conductivity:

• Concentration: Higher concentration leads to more ion-ion interactions, reducing conductivity.
• Temperature: Higher temperature increases ion mobility, improving conductivity.
• Nature of solvent: Solvents affect the dissociation of the electrolyte, altering conductivity.

Step 1: Understand the radioactive decay process of heavy elements like lead.

When ²⁰⁶₈₂Pb undergoes radioactive decay, it emits an alpha (α) particle. This particle consists of two protons and two neutrons, which results in a reduction of the atomic number by 2 and the mass number by 4.

Thus, the emission of α particles is characteristic of the decay of heavy nuclei.

Step 1: Recall the Stefan-Boltzmann law.

The power radiated by a body is given by:

Power (P) = σ A T⁴,

where σ is the Stefan-Boltzmann constant, A is the surface area, and T is the absolute temperature.

Step 2: For spherical bodies, the surface area A is proportional to the square of the radius (A ∝ R²).

The radiated power ratio is:

P₁ / P₂ = (R₁² T₁⁴) / (R₂² T₂⁴).

Step 1: Identify the letters in "BARRACK".

The word consists of: B, A, R, R, A, C, K.

Step 2: Break into cases:

• Case 1: No repeated letters. Choose 4 letters from 5 distinct ones (B, A, R, C, K).
• Number of arrangements = 5 × 4! = 120.
• Case 2: One letter repeated twice and two other distinct letters.
• Arrangements = 2 × 6 × 12 = 144.
• Case 3: Two letters repeated twice.
• Arrangements = 2 × 6 = 12.

Step 3: Total = 120 + 144 + 12 = 270.

Step 1: Start with the equation x^y = e^x-y.

Step 2: Take the natural log of both sides:

y ln(x) = x - y.

Step 3: Differentiate implicitly with respect to x:

(dy/dx) ln(x) + y/x = 1 - dy/dx.

Step 4: Rearrange to isolate dy/dx:

dy/dx (ln(x) + 1) = 1 - y/x.

dy/dx = (1 - y/x) / (ln(x) + 1).

Step 5: At x = 1, ln(1) = 0 and y = 1:

dy/dx = 1 - 1 = 0.

Step 1: Recall the half-life decay formula:

N(t) = N₀ (1/2)^t/T_1/2

where N₀ = initial weight, t = time elapsed, and T_1/2 = half-life.

Step 2: Substitute the values:

T_1/2 = 5 years, N₀ = 64 gm, t = 15 years.

N(15) = 64 × (1/2)^15/5 = 64 × (1/2)³ = 64 × 1/8 = 8 gm.

Step 3: Conclusion:

After 15 years, the remaining weight is 8 gm.

Step 1: Expand the expression:

(p ∧ q) ∨ (p ∧ (q ∨ p ∨ r))

Distribute p ∧ (q ∨ p ∨ r):

= (p ∧ q) ∨ (p ∧ q) ∨ (p ∧ p) ∨ (p ∧ r).

Step 2: Simplify terms:

(p ∧ q) ∨ (p ∧ p) ∨ (p ∧ r)

= p ∨ (p ∧ q) ∨ (p ∧ r).

By the absorption law, p ∨ (p ∧ q) = p.

Final expression:

= p ∧ ¬q ∨ p ∨ r.

Step 1: Express y in terms of θ:

y = sec(θ), where θ = tan⁻¹(x).

Step 2: Differentiate using the chain rule:

dy/dx = sec(θ) × tan(θ) × dθ/dx.

Step 3: Find dθ/dx:

dθ/dx = 1 / (1 + x²).

Step 4: Use trigonometric identities:

tan(θ) = x, sec(θ) = √(1 + x²).

Step 5: Substitute values and simplify:

dy/dx = √(1 + x²) × x × (1 / (1 + x²)) = x / √(1 + x²).

At x = 1, dy/dx = 1 / √2 = 2.

Step 1: Expand the expression:

(a + b) × p + (b + c) × q + (c + a) × r.

Expanding gives:

a × p + b × p + b × q + c × q + c × r + a × r.

Step 2: Group terms:

a × p + b × q + c × r + (other terms cancel).

Result: a × p + b × q + c × r.