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Question #40567

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x) = \begin{pmatrix} x-17 & 8 & 9 \\ 7 & x-16 & 9 \\ 7 & 8 & x-15 \end{pmatrix}$ . Let $n$ be the non-zero repeated root of the equation $|A(x) \cdot A^T(x)| = 0$ . If $n$ represents the number of ways to arrange $k$ distinct objects in a circle, find the value of $k^4 - 15$ .

Options

A.
610
B.
241
C.
1281
D.
256

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Circular Permutations Constraint|Properties of Transpose Determinant|Factor Theorem in Determinants

Why This Question Is Tricky

Common Mistakes

This question tests the integration of matrix determinant properties, the Factor Theorem, and circular permutations.

The core trap lies in the circular permutation constraint: students frequently miscalculate the number of circular arrangements of $k$ objects as $k!$ instead of $(k-1)!$ .

When they correctly identify the determinant's repeated root as $n=24$ using the transpose property $|A \cdot A^T| = |A|^2$ and the Factor Theorem (noting $x=24$ makes all rows identical), the trap springs.

Equating $k! = 24$ yields $k=4$ , leading directly to the distractor option 241 ( $4^4 - 15$ ).

The correct path requires $(k-1)! = 24 \implies k=5$ , yielding 610.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Circular Permutations Constraint
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free