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Question #60064

Difficulty: 5/10 | 🟢 Free

The Question

Let $A(\theta) = \begin{pmatrix} \sin\theta & -1 & \cos\theta \\ 1 & \sin\theta & -1 \\ -\cos\theta & 1 & \sin\theta \end{pmatrix}$ . If $S$ is the set of all values of $\theta \in [0, 2\pi]$ for which $A(\theta)$ is invertible, find the range of $|A^{-1}(\theta)|$ for $\theta \in S$ .

Options

A.
$\left[-\frac{1}{3}, \frac{1}{3}\right]$
B.
$\left(-\infty, -\frac{1}{3}\right] \cup \left[\frac{1}{3}, \infty\right)$
C.
$\left[-3, 3\right]$
D.
$\left(-\infty, -3\right] \cup \left[3, \infty\right)$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Range and Coefficient of Range|Inversion and Determinant Product|Minors and Cofactors Sign Convention

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply the minors and cofactors sign convention during determinant expansion and subsequently analyze the range of the inverse determinant.

The primary trap lies in the inversion of the range: students often incorrectly assume that if $f(x) \in [-a, a]$ , then $1/f(x) \in [-1/a, 1/a]$ .

However, because the determinant's range crosses zero, the reciprocal function becomes unbounded, splitting into $(-\infty, -1/a] \cup [1/a, \infty)$ .

Failing to account for the asymptote at zero leads directly to the trap option.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Range and Coefficient of Range
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free