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Question #75897

Difficulty: 8/10 | 🟢 Free

The Question

Let $P$ be a $3 \times 3$ matrix with $\det(P) = 2024$ . Let $Q = \begin{pmatrix} \sin x & \cos x & \sin(x - \lambda x^\circ) \\ \sin y & \cos y & \sin(y - \lambda y^\circ) \\ \sin z & \cos z & \sin(z - \lambda z^\circ) \end{pmatrix}$ where $x, y, z \in \mathbb{R}$ are measured in radians and $x^\circ$ denotes the equivalent angle in degrees. Let $M = P Q^T$ . If the row vectors of $M$ are coplanar for all $x, y, z \in \mathbb{R}$ , let $S$ be the sum of all possible strictly positive values of $\lambda$ . Find $S$ .

Options

A.
$\frac{540}{\pi}$
B.
$\frac{360}{\pi}$
C.
$\frac{180}{\pi}$
D.
$\frac{720}{\pi}$

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

Coplanarity Condition|Determinant Product Rule|Transpose of Product|Radian-Degree Conversion Identity

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the integration of the Determinant Product Rule with vector coplanarity, while embedding a rigorous Radian-Degree Conversion trap.

Students must recognize that $\det(PQ^T) = \det(P)\det(Q) = 0$ .

The primary trap lies in the term $\sin(x - \lambda x^\circ)$ .

Students often incorrectly treat $x^\circ$ as a constant angle shift, assuming the third column is trivially a linear combination of the first two via standard trigonometric expansion.

However, $x^\circ$ is a variable dependent on $x$ ( $x^\circ = \frac{\pi}{180}x$ ).

Recognizing this unit mismatch forces the student to solve the functional identity $\sin(kx) = A\sin x + B\cos x$ , rigorously restricting $k$ to $\{-1, 0, 1\}$ and yielding the correct values for $\lambda$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Coplanarity Condition
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free