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Question #73298

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{u}, \vec{v}, \vec{w}$ be three mutually orthogonal unit vectors in $\mathbb{R}^3$ . Let $A$ be the $3 \times 3$ matrix whose rows are $\vec{u}^T, \vec{v}^T, \vec{w}^T$ . Let $t = u_1 + v_2 + w_3$ and $\delta = \vec{u} \cdot (\vec{v} \times \vec{w})$ . If $[\delta t] = k$ for $k \in \{1, 2\}$ , where $[\cdot]$ denotes the greatest integer function, which of the following represents the exact range of $\text{tr}(A^2)$ ?

Options

A.
$[k^2 - 2k, k^2 - 1)$
B.
$[k^2 - 2k + 1, k^2)$
C.
$(k^2 - 2k, k^2 - 1]$
D.
$[k^2 - 1, k^2 + 2k)$

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

Direction Cosines constraint|Orthogonal Matrix Conditions|Greatest Integer Function Traps

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the spectral properties of orthogonal matrices combined with real analysis constraints.

The trap lies in treating the trace and determinant as independent continuous variables, or failing to recognize that the scalar triple product represents the determinant ( $\pm 1$ ).

Students often incorrectly expand $\text{tr}(A^2)$ without using the specific $1, e^{i\theta}, e^{-i\theta}$ eigenvalue structure, missing the algebraic identity $\text{tr}(A^2) = (\delta t)^2 - 2(\delta t)$ .

Furthermore, the GIF constraint forces an asymmetric domain $[k, k+1)$ ; ignoring the strict monotonicity of $x^2 - 2x$ on $[1, 3)$ leads to incorrect boundary inclusions.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Direction Cosines constraint
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free