Question

The number of solutions of the equation $x+2\tan x=\frac{\pi }{2}$ in the interval$[0,2\pi ]$ is

• A. $5$
• B. $2$
• C. $4$
• D. $3$

Answer 1

The correct option is D

$3$

Find the number of solutions for the given equation

Given, $x+2\tan x=\frac{\pi }{2}$

$$\begin{gathered} \Rightarrow 2\tan x=\frac{\pi }{2}–x \\ \Rightarrow \tan x=\frac{\pi }{4}–\frac{x}{2} \end{gathered}$$

plot the graph between $y=\tan x$and $y=\frac{\pi }{4}-\frac{x}{2}$

straight line cuts$y=\tan x,$ $3$ times in the interval $[0,2\pi ]$

Therefore, there are $3$ solutions of the given equation

Hence, option (D) is the correct answer.