let $\left|\begin{array}{ccc}{\tan }^{-1}\mathrm{x}& {\tan }^{-1}2\mathrm{x}& {\tan }^{-1}3\mathrm{x}\\ {\tan }^{-1}3\mathrm{x}& {\tan }^{-1}\mathrm{x}& {\tan }^{-1}2\mathrm{x}\\ {\tan }^{-1}2\mathrm{x}& {\tan }^{-1}3\mathrm{x}& {\tan }^{-1}\mathrm{x}\end{array}\right|=0$ then the number of values of x satisfying the equation is

$\left|\begin{array}{ccc}{\tan }^{-1}\mathrm{x}& {\tan }^{-1}2\mathrm{x}& {\tan }^{-1}3\mathrm{x}\\ {\tan }^{-1}3\mathrm{x}& {\tan }^{-1}\mathrm{x}& {\tan }^{-1}2\mathrm{x}\\ {\tan }^{-1}2\mathrm{x}& {\tan }^{-1}3\mathrm{x}& {\tan }^{-1}\mathrm{x}\end{array}\right|=0$

${\mathrm{R}}_2\leftrightarrow {\mathrm{R}}_3$

$\left|\begin{array}{ccc}{\tan }^{-1}\mathrm{x}& {\tan }^{-1}2\mathrm{x}& {\tan }^{-1}3\mathrm{x}\\ {\tan }^{-1}2\mathrm{x}& {\tan }^{-1}3\mathrm{x}& {\tan }^{-1}\mathrm{x}\\ {\tan }^{-1}3\mathrm{x}& {\tan }^{-1}\mathrm{x}& {\tan }^{-1}2\mathrm{x}\end{array}\right|=0$

let ${\tan }^{-1}\mathrm{x}=\mathrm{a},{\tan }^{-1}2\mathrm{x}=\mathrm{b}\cdot {\tan }^{-1}3\mathrm{x}=\mathrm{c}$

$\left|\begin{array}{l}\mathrm{a}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{b}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{c}\\ \mathrm{b}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{c}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{a}\\ \mathrm{c}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{a}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{b}\end{array}\right|=0\Rightarrow 3\mathrm{abc}-{\mathrm{a}}^3-{\mathrm{b}}^3-{\mathrm{c}}^3=0$

$\begin{array}{l}\Rightarrow {\mathrm{a}}^3+{\mathrm{b}}^3+{\mathrm{c}}^3=3\mathrm{abc}\\ \Rightarrow \mathrm{a}++\mathrm{b}+\mathrm{c}=0\\ \Rightarrow {\tan }^{-1}\mathrm{x}+{\tan }^{-1}2\mathrm{x}+{\tan }^{-1}3\mathrm{x}=0\end{array}$

$\therefore \mathrm{x}=0$ satisfies above equation

$\therefore$ no of values of x is 1