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Question #40349

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \begin{vmatrix} x^2 + \sqrt{x^2} - 2 & x^2 - \sqrt{x^2} & 1 \\ x^3 + x\sqrt{x^2} - 2x & x^3 - x\sqrt{x^2} & 2 \\ -3 & -4 & 1 \end{vmatrix}$ .

If $S$ is the sum of the squares of all distinct real roots of the equation $f(x) = 0$ , find the value of $S$ .

Options

A.
6
B.
69
C.
5
D.
134

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

Polynomial Function Parity|Factor Theorem in Determinants|Elementary Row Operation Invariance|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the student's vigilance with the modulus function disguised as $\sqrt{x^2}$ within a determinant.

Students frequently make the false simplification $\sqrt{x^2} = x$ .

If this mistake is made, the row operation $R_2 \to R_2 - x R_1$ still perfectly zeroes out the first two elements, but yields the incorrect polynomial $-(x-2)(x^2+7x-8)=0$ .

This leads to an extraneous root $x=-8$ and misses the root $x=-1$ (resulting in the trap answer 69).

The problem exploits the invariance of row operations, which creates a false sense of security and masks the domain-specific behavior of the modulus function hidden within the polynomial coefficients.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Polynomial Function Parity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free