Trigonometrical Equations 3 Question 2

2. The number of solutions of the pair of equations $2 \sin ^{2} \theta-\cos 2 \theta=0$ and $2 \cos ^{2} \theta-3 \sin \theta=0$ in the interval $[0,2 \pi]$ is

(a) 0

(b) 1

(c) 2

(d) 4

(2007, 3M)

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Answer:

Correct Answer: 2. (c)

Solution:

  1. $2 \sin ^{2} \theta-\cos 2 \theta=0$

$\Rightarrow \quad \sin ^{2} \theta=\frac{1}{4}$

Also, $\quad 2 \cos ^{2} \theta=3 \sin \theta$

$\therefore \sin \theta=\frac{1}{2}$

$[\because \sin \theta+2 \neq 0]$

$\Rightarrow$ Two solutions exist in the interval $[0,2 \pi]$.



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