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Question #66448

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x)$ be a $3 \times 3$ symmetric orthogonal matrix with continuous entries for all $x \in \mathbb{R}$ . The trace of $A(x)$ is $1$ and its first column is given by $C_1(x) = \begin{pmatrix} \cos 2x \\ -2\sin^2 x \cos x \\ -2\sin x \cos^2 x \end{pmatrix}$ If $B = \int_{-\pi/2}^{\pi/2} A(x) \, dx$ , then the value of $\frac{9}{\pi} \det(B)$ is

Options

A.
-4
B.
4
C.
-2
D.
0

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Orthogonal Matrix Conditions|Symmetry and Periodicity|Even and Odd Function Symmetry

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 properties of symmetric orthogonal matrices and the exploitation of parity in definite integrals.

The primary trap lies in the implicit definition of the matrix A(x).

Students who attempt to solve for the 9 entries of A(x) using brute-force algebraic equations (A^T A = I) will be overwhelmed.

The elegant path requires recognizing that a symmetric orthogonal matrix with a trace of 1 must be a Householder reflection of the form I - 2uu^T.

Furthermore, a secondary trap awaits during integration: students who fail to identify that the off-diagonal elements involving products of sine and cosine are odd functions will waste significant time integrating them, missing the fact that the integrated matrix B is sparse.

Finally, recognizing that B_11 = 0 completely bypasses the need to integrate A_22, rewarding structural analysis over mechanical calculation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Orthogonal Matrix Conditions
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free