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Question #41265

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a} = \left(\sqrt{x^2} - 5\right)\hat{i} + x\hat{j} + \hat{k}$ , $\vec{b} = x\hat{i} + \left(\sqrt{x^2} - 5\right)\hat{j} + \hat{k}$ , and $\vec{c} = 2\hat{i} + \hat{j} + \hat{k}$ . Let $S$ be the set of all real values of $x$ for which the vectors $\vec{a}, \vec{b}$ , and $\vec{c}$ are coplanar. Find the value of $4 \sum_{x \in S} x^2$ .

Options

A.
89
B.
64
C.
114
D.
45

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

Linear Dependency of Rows|Scalar Triple Product Volume Logic|Unit Vector Normalization|Square Root and Modulus Logic

Why This Question Is Tricky

Common Mistakes

This question tests the evaluation of the scalar triple product for coplanarity while embedding a critical algebraic trap: the simplification of $\sqrt{x^2}$ .

Students frequently and incorrectly assume $\sqrt{x^2} = x$ instead of the rigorous $|x|$ .

If this mistake is made, the determinant equation collapses to a linear equation yielding only the positive root ( $x=4$ ), causing them to miss the degenerate case for the negative root ( $x=-5/2$ ) where $\vec{a}$ and $\vec{b}$ become identical vectors.

Falling for this trap leads directly to the incorrect option 64.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Linear Dependency of Rows
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free