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Question #41285

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = [a_{ij}]$ be a $4 \times 4$ matrix where its entries are given by the binomial sum: $a_{ij} = \frac{1}{10} \sum_{k=0}^2 (-1)^k \binom{2}{k} (i-j+1-k)^5$ for $1 \le i, j \le 4$ . Let matrix $B$ be obtained from $A$ by applying the following sequence of elementary operations in order: $R_1 \leftrightarrow R_3$ , $C_2 \leftrightarrow C_4$ , and $R_2 \leftrightarrow R_4$ . Find the value of $\left[ \frac{|B|}{1000} \right]$ , where $[ \cdot ]$ denotes the greatest integer function.

Options

A.
-21
B.
-20
C.
20
D.
21

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Skew-Symmetric Determinant|Properties of Binomial Coefficients|Integral Part and Fractional Part|Row/Column Exchange Parity

Why This Question Is Tricky

Common Mistakes

This question tests the evaluation of finite differences disguised as binomial sums to reveal a skew-symmetric matrix.

The core JEE Advanced trap lies in the sequence of elementary operations and the greatest integer function.

Students often correctly compute the determinant of the even-order skew-symmetric matrix as a perfect square (20736).

However, they frequently miss the parity shift caused by the odd number of row/column swaps (3 swaps), leading to a sign error.

Furthermore, the greatest integer function acts as a secondary trap: even if they get the negative sign, they might incorrectly evaluate the greatest integer of -20.736 as -20 instead of -21.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Skew-Symmetric Determinant
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free