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Question #40649

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be a $3 \times 3$ matrix where its entries are given by $A_{ij} = \text{coefficient of } x^{j-1} \text{ in the binomial expansion of } (1-3x)^{-(n+i-1)}$ for $i, j \in \{1, 2, 3\}$ . Given that the coefficient of $x^2$ in the expansion of $(1-x)^{-n}$ is $231$ , find the value of $\det(\text{adj}(2A^T))$ .

Options

A.
$2^6 \times 3^6$
B.
$2^2 \times 3^6$
C.
$2^3 \times 3^6$
D.
$2^6 \times 3^3$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Binomial Theorem for Negative/Fractional Indices|Adjoint-Determinant Relation|Properties of Transpose Determinant

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to decouple scalar multiplication from adjoint and transpose operations within determinants.

Students often incorrectly assume $\det(2A^T) = 2\det(A)$ or $\det(\text{adj}(2A)) = 2^3\det(\text{adj}(A))$ , leading to incorrect powers of 2 (Options B and C).

Furthermore, the '231' constraint forces students to evaluate a negative binomial coefficient to find $n=21$ .

If they fail to factor out $3^{j-1}$ and use Pascal's identity for row reduction, they face a massive brute-force determinant calculation (e.g., $1 \times 69 \times 2700 \dots$ ) which induces computational fatigue and almost certain arithmetic failure.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Binomial Theorem for Negative/Fractional Indices
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free