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Question #67166

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \sqrt{x-2} + \frac{8}{\sqrt{x-2}}$ for $x > 2$ . Consider the matrix $M(x) = \begin{pmatrix} f(x)-7 & -4 & -1 \\ -4 & f(x)-1 & 2 \\ -1 & 2 & f(x)+5 \end{pmatrix}$ Let $S$ be the sum of all real values of $x$ for which $M(x)$ is a singular matrix. Find the value of $S$ .

Options

A.
69
B.
93
C.
75
D.
24

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

Determinant Scaling Property|Quadratic Roots in Sequences|Extraneous Solution Rejection

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the evaluation of determinants involving functional entries and the critical rejection of extraneous solutions.

The determinant simplifies to a cubic polynomial in $f(x)$ .

When solving for $x$ , the quadratic equations in $\sqrt{x-2}$ yield both positive and negative roots.

The trap lies in blindly squaring the negative roots (e.g., $\sqrt{x-2} = -2 \implies x = 6$ ), failing to recognize that the principal square root function is strictly non-negative.

Students who do not reject these extraneous roots will incorrectly add them to the final sum, arriving at the trap answer 93.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Determinant Scaling Property
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free