The principal solutions, of the equation $\sqrt{3} \sec x+2=0$, are

The principal solutions, of the equation $\sqrt{3} \sec x+2=0$, are
  1. $\frac{2 \pi}{3}, \frac{4 \pi}{3}$
  2. $\frac{4 \pi}{3}, \frac{5 \pi}{3}$
  3. $\frac{5 \pi}{6}, \frac{7 \pi}{6}$
  4. $\frac{7 \pi}{6}, \frac{11 \pi}{6}$

Solution

$\begin{aligned} & \sqrt{3} \sec x+2=0 \\ & \Rightarrow \sec x=\frac{-2}{\sqrt{3}} \\ & \Rightarrow \cos x=\frac{-\sqrt{3}}{2} \\ & \Rightarrow \cos x=\cos \left(\pi-\frac{\pi}{6}\right)=\cos \frac{5 \pi}{6} \\ & \text { and } \cos x=\cos \left(\pi+\frac{\pi}{6}\right)=\cos \frac{7 \pi}{6} \end{aligned}$ $\therefore$ The principal solutions are $\frac{5 \pi}{6}$ and $\frac{7 \pi}{6}$

Asked in: MHT CET 2024 (02 May Shift 1)

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