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Question #65489

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be a $3 \times 3$ real matrix such that $A^3 = O$ and $A^2 \neq O$ , where $O$ is the null matrix. Let $P = \cos(\tan^{-1}(A))$ and $Q = \sec(\sin^{-1}(A))$ . Which of the following statements is INCORRECT?

Options

A.
$PQ = I$
B.
$P^2 + Q^{-2} = 2I - 2A^2$
C.
$\det(P+Q) = 8$
D.
$\text{Tr}(Q - P) > 0$

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

Composition Inverse Identity Traps|Composition Non-Commutativity|Polynomial Function Parity

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the evaluation of matrix-valued inverse trigonometric compositions using Taylor series truncation for nilpotent matrices.

The primary trap lies in Option D: students frequently confuse $\text{Tr}(A^2)$ with $\text{Tr}(AA^T)$ .

While $\text{Tr}(AA^T) > 0$ for any non-zero real matrix, a nilpotent matrix has all zero eigenvalues, making $\text{Tr}(A^2) = 0$ regardless of $A^2 \neq O$ .

Additionally, the identity trap is deployed by making students verify $PQ=I$ algebraically rather than assuming functional inverses imply matrix inverses.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Composition Inverse Identity Traps
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free