Question

The number of solutions of the equation $ 2x + 3\tan x = \pi $, $ x \in [-2\pi, 2\pi] - \left\{ \pm \frac{\pi}{2}, \pm \frac{3\pi}{2} \right\} $ is

• A. 6
• B. 5
• C. 4
• D. 3

Hint:

To find the number of solutions of an equation involving trigonometric and linear functions, sketch the graphs of both functions and count the number of intersection points.

Answer 1

The Correct Option is B

Given equation is \( 2x + 3\tan x = \pi \)
Rearranging the terms, we get \( \tan x = \frac{\pi - 2x}{3} \) 
Let \( f(x) = \tan x \) and \( g(x) = \frac{\pi - 2x}{3} \) 
We need to find the number of intersection points of these two functions in the given interval \( [-2\pi, 2\pi] \). 
\( f(x) = \tan x \) has vertical asymptotes at \( x = \pm \frac{\pi}{2} \) and \( x = \pm \frac{3\pi}{2} \). \( g(x) = \frac{\pi - 2x}{3} \) is a straight line with slope \( -\frac{2}{3} \) and y-intercept \( \frac{\pi}{3} \). 
We can analyze the intersection points graphically or by analyzing intervals. In the interval \( [-2\pi, 2\pi] \), we have the following intervals to consider: \( [-2\pi, -\frac{3\pi}{2}) \), \( (-\frac{3\pi}{2}, -\frac{\pi}{2}) \), \( (-\frac{\pi}{2}, \frac{\pi}{2}) \), \( (\frac{\pi}{2}, \frac{3\pi}{2}) \), \( (\frac{3\pi}{2}, 2\pi] \). 
By sketching the graphs or analyzing the behavior of the functions in each interval, we can observe that there are 5 intersection points.
  
Therefore, the number of solutions is 5.