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Question #40983

Difficulty: 8/10 | 🟢 Free

The Question

Let $a_1, a_2, a_3$ be the roots of the equation $x^3 - 3x^2 + 4x - 5 = 0$ . Let $B$ be a $3 \times 3$ matrix defined as: $B = \begin{pmatrix} \frac{a_2+a_1}{2} & 0 & \frac{a_2+a_3}{2} \\ 0 & \frac{a_1+a_2}{2} & \frac{a_1+a_3}{2} \\ \frac{a_3+a_1}{2} & \frac{a_3+a_2}{2} & 0 \end{pmatrix}$ Find the value of $|B \cdot \text{adj}(-2B)|$ .

Options

A.
-343
B.
343
C.
-2744
D.
2744

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

Properties of Arithmetic Mean|Adjoint-Determinant Relation|Row/Column Exchange Parity

Why This Question Is Tricky

Common Mistakes

This problem tests the interaction between the multilinearity of determinants and row exchange parity.

Students often overlook that matrix $B$ is a row-swapped version of the standard symmetric arithmetic mean matrix $M$ , missing the crucial negative sign ( $|B| = -|M|$ ).

Furthermore, they frequently mishandle the scalar multiplication in the adjoint relation, incorrectly evaluating $\text{adj}(-2B)$ as $-2\text{adj}(B)$ or $-8\text{adj}(B)$ , or failing to cube the scalar when taking the determinant of $4|B|I_3$ , leading to sign errors or incorrect magnitudes.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Properties of Arithmetic Mean
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free