Total number of solutions of the equation $3\mathrm{x}+2\tan \mathrm{x}=\frac{5\mathrm{\pi }}{2}\text{ in }\mathrm{x}\in [0,2\mathrm{\pi }]$ is equal to

Solution:

We have, $3\mathrm{x}+2\tan \mathrm{x}=\frac{5\mathrm{\pi }}{2}$

$\Rightarrow \tan \mathrm{x}=\frac{5\mathrm{\pi }}{4}-\frac{3\mathrm{x}}{2}$

Let   $\frac{5\mathrm{\pi }}{4}-\frac{3\mathrm{x}}{2}=\mathrm{y}$

and  $\mathrm{y}=\tan \mathrm{x}$

$\therefore$$\text{ Graph of }\mathrm{y}=\frac{5\mathrm{\pi }}{4}-\frac{3\mathrm{x}}{2}\text{ and }\mathrm{y}=\tan \mathrm{x}\text{ meet exactly three }$times in $[0,2\mathrm{\pi }]$.Thus,

   the number of solutions of given equation is 3.