AP EAMCET · Maths · Trigonometric Equations

Statement-I : In the interval \([0,2 \pi]\), the number of common solutions of the equations \(2 \sin ^2 \theta-\cos 2 \theta=0\) and \(2 \cos ^2 \theta-3 \sin \theta=0\) is two.
Statement-II : The number of solutions of \(2 \cos ^2 \theta-3 \sin \theta=0\) in \([0, \pi]\) is two.

A Statement-I and Statement-II are both true
B Statement-I is true, Statement-II is false
C Statement-I is false, Statement-II is true
D Statement-I and Statement-II are both false

Verified Solution

Answer & Solution

Correct Answer

(A) Statement-I and Statement-II are both true

Step-by-step Solution

Detailed explanation

Statement-I: First equation: \(2 \sin^2 \theta - \cos 2\theta = 0\) \(2 \sin^2 \theta - (1 - 2 \sin^2 \theta) = 0 \implies 4 \sin^2 \theta - 1 = 0 \implies \sin \theta = \pm \frac{1}{2}\) Solutions in \( [0, 2\pi] \):…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Statement-I : In the interval \([0,2 \pi]\), the number of common solutions of the equations \(2 \sin ^2 \theta-\cos 2 \theta=0\) and \(2 \cos ^2 \theta-3 \sin \theta=0\) is two.Statement-II : The number of solutions of \(2 \cos ^2 \theta-3 \sin \theta=0\) in \([0, \pi]\) is two.

Step-by-step Solution

Statement-I : In the interval \([0,2 \pi]\), the number of common solutions of the equations \(2 \sin ^2 \theta-\cos 2 \theta=0\) and \(2 \cos ^2 \theta-3 \sin \theta=0\) is two.
Statement-II : The number of solutions of \(2 \cos ^2 \theta-3 \sin \theta=0\) in \([0, \pi]\) is two.