MHT CET 2024 15 May Shift 2 Question Paper (Available): Download PCM Question Paper with Answers PDF

We simplify the given Boolean expression step by step:

1. The expression is: (p ∧ (~q)) ∨ ((~p) ∧ q) ∨ ((~p) ∧ (~q))

2. Group the terms: (p ∧ (~q)) ∨ [(~p ∧ q) ∨ (~p ∧ ~q)]

3. Simplify (~p ∧ q) ∨ (~p ∧ ~q) using the distributive law: (~p ∧ (q ∨ ~q)) = ~p

4. Substitute: (p ∧ (~q)) ∨ ~p

5. Rearrange: ~p ∨ (p ∧ ~q)

6. Apply the absorption law: ~p ∨ ~q

Thus, the simplified expression is (~p) ∨ (~q).

We are given B as the adjoint of A and |A| = 4.

1. The relationship between the adjoint and determinant is A · adj(A) = |A| · I.

2. Given B = adj(A), its structure ensures that the determinant condition holds for |A| = 4.

3. By symmetry and determinant validation, the value of a must be 1 for consistency.

Thus, a = 1.

To find the inverse of a 3×3 matrix, compute the determinant and the adjoint.

1. Calculate det(A): Use cofactor expansion.

2. Compute adj(A): Find cofactors and take the transpose.

3. Compute A⁻¹: Use the formula A⁻¹ = adj(A) / det(A).

Verification shows that A⁻¹ = [[1/2, -1/2, 1/2], [-4, 3, -1], [5/2, -3/2, 1/2]].

Let θ = cot⁻¹(x). Then, cot(θ) = x.

1. Represent θ in a right triangle: cot(θ) = adjacent / opposite = x / 1.

2. Hypotenuse = √(x² + 1).

3. sin(θ) = opposite / hypotenuse = 1 / √(x² + 1).

Thus, sin(cot⁻¹(x)) = 1 / √(1 + x²).

We are given:

dy/dx = -5sin(x) - 3cos(x).

Step 1: Compute the second derivative.

Differentiate dy/dx with respect to x:

d²y/dx² = -5cos(x) + 3sin(x).

Step 2: Add y to the second derivative.

y = 5cos(x) - 3sin(x). Add y to d²y/dx²:

d²y/dx² + y = (-5cos(x) + 3sin(x)) + (5cos(x) - 3sin(x)).

Step 3: Simplify the expression.

Combine terms:

d²y/dx² + y = 0.

Let x be one part and y be the other part.

Step 1: Write the condition.

x + y = 20, so y = 20 - x.

We aim to maximize f(x) = (20 - x)x³ = 20x³ - x⁴.

Step 2: Differentiate to find critical points.

f'(x) = 60x² - 4x³. Solve f'(x) = 0:

4x²(15 - x) = 0, so x = 0 or x = 15.

Step 3: Evaluate the second derivative.

f''(x) = 120x - 12x². At x = 15:

f''(15) = -900 < 0, so f(x) is maximum at x = 15.

Step 4: Compute the product.

y = 20 - x = 5. Product = 15 × 5 = 75.

Step 1: Rewrite the integral:

I = ∫ sec²/³(x) csc⁴/³(x) dx from π/6 to π/3.

Step 2: Use substitution t = tan(x). Adjust limits:

x = π/6 → t = 1/√3; x = π/3 → t = √3.

Step 3: Simplify using substitution.

I = ∫ t^(2/3)/(1 + t²)^(2/3) dt. Further simplify and solve:

I = 3^(7/6) - 3^(5/6).

Step 1: Recall the definition of the cumulative distribution function (CDF). The CDF F(X) gives the probability that the random variable X takes a value less than or equal to x. Mathematically:

F(x) = P[X ≤ x].

Step 2: Compute P[X = 4].

The probability of X taking an exact value is the difference between the CDF values at consecutive integers:

P[X = 4] = F(4) - F(3).

From the given data:

F(4) = 0.62, F(3) = 0.48.

P[X = 4] = 0.62 - 0.48 = 0.14.

Step 3: Compute P[X = 5].

P[X = 5] = F(5) - F(4).

From the given data:

F(5) = 0.85, F(4) = 0.62.

P[X = 5] = 0.85 - 0.62 = 0.23.

Step 4: Combine the probabilities.

P[X = 4] + P[X = 5] = 0.14 + 0.23 = 0.37.

Step 1: Parametric equations of the first line:

(x - k)/2 = (y + 1)/3 = (z - 1)/4 = t.

Parametric equations: x = 2t + k, y = 3t - 1, z = 4t + 1.

Step 2: Parametric equations of the second line:

(x - 3)/1 = (y - 9/2)/2 = z = s.

Parametric equations: x = s + 3, y = 2s + 9/2, z = s.

Step 3: Solve for intersection.

Equate the parametric equations for x, y, z:

2t + k = s + 3, 3t - 1 = 2s + 9/2, 4t + 1 = s.

From z equation: s = 4t + 1.

Substitute s into the other equations and solve:

k = 1.

A tertiary alkane is characterized by a central carbon atom that is directly bonded to three other carbon atoms. In 2,2-Dimethylpropane, the central carbon atom is bonded to three methyl groups, fulfilling this criterion and making it a tertiary alkane. The remaining options do not feature a carbon atom bonded to three other carbons, and therefore, they are not tertiary alkanes.

A primary amine is characterized by the functional group -NH₂, where one hydrogen atom of ammonia is replaced by an alkyl or aryl group. Among the given options, Ethylamine has the structure CH₃CH₂NH₂, which includes the -NH₂ group, identifying it as a primary amine. In contrast, Diethylamine and Triethylamine are secondary and tertiary amines, respectively, and Aniline features an aromatic -NH₂ group, making it distinct from a primary amine.

A chiral carbon is defined as a carbon atom bonded to four different groups. Analyzing the structure of 2-chloro-3,4-dimethylhexane:

• The second carbon (C-2) is chiral because it is attached to four distinct groups: a chlorine atom, a methyl group, an ethyl group, and a hydrogen atom.
• The third carbon (C-3) is also chiral since it is connected to four different groups: a methyl group, a 2-chloropropyl group, a hydrogen atom, and another unique substituent.

The remaining carbon atoms in the molecule do not meet the criteria for chirality. Therefore, the compound contains two chiral carbons.

Using the Henderson-Hasselbalch equation for a basic buffer:

pOH = pKb - log([base]/[salt]), and pH = 14 - pOH.

Given: pKb = 4.75, [base] = 0.1 M, [salt] = 0.05 M.

Step 1: Substitute into the equation:

pOH = 4.75 - log(0.1/0.05).

Step 2: Simplify:

log(0.1/0.05) = log(2) ≈ 0.301.

pOH = 4.75 - 0.301 = 4.449.

Step 3: Calculate pH:

pH = 14 - 4.449 = 9.551 ≈ 9.55.

The volume of a unit cell is determined by the edge length a, and for a cubic unit cell, it is given by:

Volume = a³.

In the case of a body-centered cubic (BCC) structure, the relationship between the edge length a and the atomic radius r is:

a = 4r / √3.

Substitute this value of a into the volume formula:

Volume = (4r / √3)³.

Simplify the expression:

Volume = (4r)³ / (√3)³ = 64r³ / 3√3.

Thus, the total volume of the body-centered cubic unit cell is:

64r³ / 3√3.

To name the compound systematically using IUPAC rules:

1. Identify the longest carbon chain that includes the functional group (-OH). In this case, the chain has 5 carbon atoms, making it a "pentane" derivative.

2. Number the chain starting from the end nearest to the -OH group to ensure it gets the lowest possible number. The hydroxyl group is on carbon 1.

3. Identify and number substituents. Two methyl groups are attached to carbons 2 and 5 of the main chain.

4. Combine the name, listing substituents alphabetically with their positions and the suffix "-ol" to indicate the alcohol group.

The correct IUPAC name is: 2,5-dimethylpentanol.