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Question #41161

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The Question

Let $A$ be a $3 \times 3$ matrix defined as: $A = \begin{pmatrix} \ln(\tan \frac{\pi}{4}) & \cos^{-1}(\cos 5) & \tan^{-1}(\tan 4) \\ \sin^{-1}(\sin 5) & \sin^{-1}(\cos \frac{\pi}{2}) & \sec^{-1}(\sec 9) \\ \sin^{-1}(\sin 4) & \csc^{-1}(\csc 9) - \pi & \cos^{-1}(\sin \frac{\pi}{2}) \end{pmatrix}$ Let $S$ be the set of all column vectors $B \in \mathbb{R}^3$ such that the matrix equation $AX = B$ has at least one solution for $X \in \mathbb{R}^3$ . Which of the following statements is true?

Options

A.
$S = \mathbb{R}^3$
B.
$S \text{ is a 2-dimensional subspace of } \mathbb{R}^3$
C.
$S \text{ is a 1-dimensional subspace of } \mathbb{R}^3$
D.
$S = \{ \mathbf{0} \}$

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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

The Quadrant Adjustment Logic|Skew-Symmetric Determinant|Inversion and Determinant Product

Why This Question Is Tricky

Common Mistakes

This question tests the 'Quadrant Adjustment Logic' for inverse trigonometric functions and the properties of skew-symmetric matrices.

The primary trap is that students might attempt to directly expand the determinant or incorrectly evaluate the inverse trig functions (e.g., assuming $\cos^{-1}(\cos 5) = 5$ ), which obscures the matrix's true structure.

By correctly applying domain restrictions, the matrix reveals itself as a $3 \times 3$ skew-symmetric matrix, which inherently has a determinant of zero.

Consequently, the rank is exactly 2, meaning the column space (the set of all valid $B$ vectors for $AX=B$ ) forms a 2-dimensional subspace, bypassing the need for any heavy algebraic computation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	The Quadrant Adjustment Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free