If θ ∈ [-(7π/6), 4π/3], then the number of solutions of √3 cosec^2 θ - 2(√3-1) cosec θ-4=0, is equal to

Question

If \(\theta \in\left[-\frac{7 \pi}{6}, \frac{4 \pi}{3}\right],\) then the number of solutions of \(\sqrt{3} \operatorname{cosec}^{2} \theta-2(\sqrt{3}-1) \operatorname{cosec} \theta-4=0,\) is equal to

(1) 6

(2) 8

(3) 10

(4) 7 

Answer 1

Correct option is: (1) 6 

\( \operatorname{cosec} \theta=\frac{2(\sqrt{3}-1) \pm \sqrt{4(3+1-2 \sqrt{3})+16 \sqrt{3}}}{2 \sqrt{3}}\)

\(=\frac{2(\sqrt{3-1}) \pm \sqrt{16+8 \sqrt{3}}}{2 \sqrt{3}} \)

\( =\frac{2(\sqrt{3}-1) \pm(2+2 \sqrt{3})}{2 \sqrt{3}}\)

\(\operatorname{cosec} \theta=2\ \text{or}\ \frac{-2}{\sqrt{3}}\)

\(\therefore\ \sin \theta=\frac{1}{2}\ \text{or}\ \frac{-\sqrt{3}}{2}\)

\(\therefore \sin \theta=\frac{1}{2}\) has 3 solutions & also \(\sin \theta=\frac{-\sqrt{3}}{2}\)

has 3 solutions in \(\left[\frac{-7 \pi}{6}, \frac{4 \pi}{3}\right]\)