JEE Main 2021 (Online) 17th March Evening ShiftQuestion 1
The number of solutions of the equation $x+2\tan x=\frac{\pi}{2}$ in the interval $[0,2\pi]$ is :
Trigonometric Ratio & Identities
Solution
Concept. $\tan x=\frac{\pi/2-x}{2}$: the line meets each branch of $\tan x$ at most once, because $g(x)=x+2\tan x-\frac\pi2$ is increasing on every branch. Formulas used. (1) $g'(x)=1+2\sec^2x>0$Step 1 - $\left[0,\frac\pi2\right)$. $g(0)=-\frac\pi2<0$ and $g\to+\infty$: one root. Step 2 - $\left(\frac\pi2,\frac{3\pi}2\right)$. $g$ runs from $-\infty$ to $+\infty$: one root. Step 3 - $\left(\frac{3\pi}2,2\pi\right]$. $g\to-\infty$ at the left end and $g(2\pi)=\frac{3\pi}2>0$: one root. In the figure the straight line is $y=\frac{1}{2}\left(\frac\pi2-x\right)$:Step 4 - Total. $3$. Answer: (B)