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Question #41274

Difficulty: 8/10 | 🟢 Free

The Question

Consider the following system of linear equations for $\theta \in (0, \frac{\pi}{2}) \setminus \{\frac{\pi}{4}\}$ : $x \cos \theta + y \sin \theta + z = \cos^2 \theta$ $x \sin \theta + y \cos \theta + z = \sin^2 \theta + \sqrt{1 - \sin 2\theta}$ $x + y + 2z = 1$ Let $(x(\theta), y(\theta), z(\theta))$ be the unique solution to this system. If $f(\theta) = x(\theta) - y(\theta)$ , and $L_1 = \lim_{\theta \to \frac{\pi}{4}^-} f(\theta)$ , $L_2 = \lim_{\theta \to \frac{\pi}{4}^+} f(\theta)$ , then the value of $L_1^2 + L_2^2 + L_1 L_2$ is:

Options

A.
5
B.
7
C.
9
D.
3

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

Elementary Row Operation Invariance|Cramer's Rule Constraints|The Quadrant Adjustment Logic|Rational Function Asymptotes

Why This Question Is Tricky

Common Mistakes

This question tests the interaction between Cramer's Rule, Elementary Row Operations, and limits near a singular state.

Students often attempt to compute the full determinants $\Delta, \Delta_x, \Delta_y$ which is computationally heavy, missing the efficiency of the row operation $R_1 - R_2$ .

The core trap lies in the 'Quadrant Adjustment' of the radical $\sqrt{1-\sin 2\theta} = |\cos\theta - \sin\theta|$ .

As the system approaches the singularity at $\theta = \pi/4$ , the absolute value induces a sign flip in the Cramer ratio depending on the direction of approach.

Failing to account for this non-differentiable behavior near the asymptote leads to incorrect identical limits for $L_1$ and $L_2$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Elementary Row Operation Invariance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free