Question

Match List-I with List-II:

• A. (A)-(II), (B)-(I), (C)-(IV), (D)-(III)
• B. (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
• C. (A)-(IV), (B)-(I), (C)-(III), (D)-(II)
• D. (A)-(I), (B)-(III), (C)-(II), (D)-(IV)

Hint:

Determinants of matrices with identical rows or columns are always zero.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Evaluate each \( 2 \times 2 \) determinant \( \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc \).

Step 2: Detailed Explanation:
(A) \( (6 \times 3) - (2 \times 4) = 18 - 8 = 10 \). Matches (III).
(B) Rows are identical, so determinant = 0. Matches (I).
(C) \( 1 - (\log_b a)^2 \). If \( a=b \), this is \( 1-1 = 0 \). If \( \log_b a = 0 \), this is \( 1 \). Correction: Using log identity properties, this simplifies based on value.
(D) Expansion: \( (x-1)(x^2+x+1) - x^3(x^2+1) = (x^3-1) - x^5 - x^3 = -x^5 - 1 \). At \( x=0 \), this is -1. Matches (II).

Step 3: Final Answer:
Matching: (A)-(III), (B)-(I), (C)-(IV), (D)-(II).