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Question #41518

Difficulty: 8/10 | 🟢 Free

The Question

Let $P$ and $Q$ be $2 \times 2$ real matrices such that $P^2 = Q^2 = I$ and $PQ + QP = \mathbf{0}$ , where $I$ is the identity matrix and $\mathbf{0}$ is the zero matrix. Let $A(x) = P \sin^2 x + Q \cos^2 x$ and $B(x) = P \cos^2 x + Q \sin^2 x$ . Define the function $f(x) = \det \left( \frac{d}{dx} (A(x) B(x)) \right)$ . Find the value of $\frac{4}{\pi} \int_0^{\pi/2} f(x) \, dx$ .

Options

A.
1
B.
3
C.
5
D.
8

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

Wallis' Formula|Differentiation of Determinants|Matrix Multiplication Non-Commutativity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle the differentiation of matrix products and determinants involving non-commutative matrices.

The primary trap lies in the product rule for matrices: students often incorrectly assume commutativity and write $\frac{d}{dx}(AB) = A'B + B'A$ instead of $A'B + AB'$ .

If they make this error, the $PQ$ terms cancel out, leading to an incorrect determinant of $\sin^2 4x$ and a final answer of 1.

Furthermore, it elegantly bridges abstract linear algebra (anti-commuting involutory matrices) with calculus (Wallis' integral properties), a hallmark of high-tier JEE Advanced problems.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Wallis' Formula
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free