If , then the number of solutions of 2 2 cos 2 + ( 2 – 6 ) cos – 3 = 0 , is equal to : [2025]

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Solution

Q.
If $θ \in [- 2 π, 2 π]$ , then the number of solutions of $2 \sqrt{2} \cos^{2} θ + (2 - \sqrt{6}) \cos θ - \sqrt{3} = 0$ , is equal to : [2025]

1 8

2 12

3 10

4 6

Ans.
(1)

We have, $2 \sqrt{2} \cos^{2} θ + 2 \cos θ - \sqrt{6} \cos θ - \sqrt{3} = 0$

$\Rightarrow (2 \cos θ - \sqrt{3}) (\sqrt{2} \cos θ + 1) = 0 \Rightarrow \cos θ = [\frac{\sqrt{3}}{2}, \frac{- 1}{\sqrt{2}}]$

$\Rightarrow \frac{- 11 π}{6}, \frac{- π}{6}, \frac{π}{6}, \frac{11 π}{6}, \frac{5 π}{4}, \frac{3 π}{4}, \frac{- 5 π}{4}, \frac{- 3 π}{4}$

Hence, number of solutions = 8.

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