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Question #40504

Difficulty: 8/10 | 🟢 Free

The Question

The number of values of $\theta \in [0, 2\pi]$ for which the quadratic equations $x^2 - 2|\sin \theta| x - 1 = 0$ and $x^2 - 2\sin \theta x - \cos 2\theta = 0$ have a common real root is:

Options

A.
3
B.
4
C.
5
D.
7

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

Condition for Common Roots|Symmetry and Periodicity|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the 'Modulus Sign Dependency' trap within the context of common roots.

Students commonly ignore the piecewise definition of the modulus function, assuming $|\sin \theta| = \sin \theta$ universally.

If they do this, they only solve the case where the equations have identical coefficients, yielding 3 solutions ( $\theta = 0, \pi, 2\pi$ ).

By enforcing the modulus, a second valid case emerges where $\sin \theta < 0$ .

Here, the equations are distinct but still intersect at a common root, revealing 2 additional hidden solutions.

Furthermore, students who find the second case might forget to check the domain constraint $\sin \theta < 0$ , incorrectly adding extraneous solutions.

This forces a transition from blind algebraic manipulation to rigorous, domain-aware analytical checking.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Condition for Common Roots
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free