Number of solutions of 3 cos 2 + 8 cos + 3 3 = 0 , [ - 3 , ? 2 ] is: [2026]

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Solution

Q.
Number of solutions of $\sqrt{3} \cos 2 θ + 8 \cos θ + 3 \sqrt{3} = 0,$ $θ \in [- 3 π, 2 π]$ is: [2026]

1 5

2 0

3 3

4 4

Ans.
(1)

$\sqrt{3} (2 \cos^{2} θ - 1) + 8 \cos θ + 3 \sqrt{3} = 0$

$2 \sqrt{3} \cos^{2} θ + 8 \cos θ + 2 \sqrt{3} = 0$

$(\sqrt{3} \cos θ + 1) (\cos θ + \sqrt{3}) = 0$

$\cos θ = - \frac{1}{\sqrt{3}}$

$as - \sqrt{3} (reject)$

$∴$ $θ will have 5 values in [- 3 π, 2 π]$

$Ans. = 5 \Rightarrow option (1)$

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