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Question #40786

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all values of $\theta \in [0, 2\pi]$ for which the following three lines are concurrent: $L_1: x \cos \theta + y \sin \theta = 1$ $L_2: x \cos 2\theta + y \sin 2\theta = 2$ $L_3: x \cos 3\theta + y \sin 3\theta = 1$ Find the sum of all elements in $S$ .

Options

A.
$2\pi$
B.
$3\pi$
C.
$4\pi$
D.
$5\pi$

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

Normal Form Parameters|Cramer's Rule Constraints|Inconsistency Detection

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between a coefficient matrix having a determinant of zero and a system of linear equations having a unique point of concurrency.

Students often mistakenly assume $\Delta = 0$ is a sufficient condition for concurrency.

The trap is set by the roots $\theta = 0, \pi, 2\pi$ , which make the determinant zero but cause the lines to be parallel and distinct (an inconsistent system with no intersection).

Only by verifying the geometric reality of the lines for each root can the student discard the extraneous solutions (summing to $3\pi$ ) and find the true sum of $2\pi$ .

If they blindly sum all roots of the determinant equation, they will incorrectly choose $5\pi$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Normal Form Parameters
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free