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Question #66491

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all points $(x, y) \in \mathbb{R}^2$ such that the matrix $A = \begin{pmatrix} \sin(\sin^{-1} x) & \cos(\sin^{-1} x) \\ \sin(\sin^{-1} y) & -\cos(\sin^{-1} y) \end{pmatrix}$ is an orthogonal matrix. Let $L$ be the total arc length of the locus $S$ , and let $k = \frac{L}{\pi}$ . If $M = \begin{pmatrix} k & 1 \\ -1 & k \end{pmatrix}$ , find the value of $\det(M^3 - 2M^2 + 2M)$ .

Options

A.
0
B.
5
C.
10
D.
25

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

Orthogonal Matrix Conditions|Inversion and Determinant Product|Variable Transformation Substitution

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 student's ability to handle domain restrictions during variable transformations in matrix equations.

When substituting $x = \sin\alpha$ and $y = \sin\beta$ , students often solve the orthogonality condition $\cos(\alpha+\beta)=0$ to get $\alpha+\beta = \pi/2$ and blindly apply algebraic simplification to get $x^2+y^2=1$ , assuming it represents the full circle (length $2\pi$ ).

They fail to realize that the principal domain $\alpha, \beta \in [-\pi/2, \pi/2]$ restricts the valid locus to only the first and third quadrants ( $xy \ge 0$ ).

This extraneous solution trap leads to an incorrect arc length, cascading into a wrong matrix determinant in the recursive step.

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