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If $f\left( x \right) = \left| {\begin{array}{*{20}{c}}
{\sin \left( {x + \alpha } \right)}&{\sin \left( {x + \beta } \right)}&{\sin \left( {x + \gamma } \right)} \\
{\cos \left( {x + \alpha } \right)}&{\cos \left( {x + \beta } \right)}&{\cos \left( {x + \gamma } \right)} \\
{\sin \left( {\alpha + \beta } \right)}&{\sin \left( {\beta + \gamma } \right)}&{\sin \left( {\gamma + \alpha } \right)}
\end{array}} \right|$ and $f(10) = 10$ then $f(\pi)$ is equal to
$0$
$\pi$
$10$
None of these
Solution
$f'\left( x \right) = \left| {\begin{array}{*{20}{c}}
{\cos \left( {x + \alpha } \right)}&{\cos \left( {x + \beta } \right)}&{\cos \left( {x + \gamma } \right)}\\
{\cos \left( {x + \alpha } \right)}&{\cos \left( {x + \beta } \right)}&{\cos \left( {x + \gamma } \right)}\\
{\sin \left( {\alpha + \beta } \right)}&{\sin \left( {\beta + \gamma } \right)}&{\sin \left( {\gamma + \alpha } \right)}
\end{array}} \right|$
$ + \left( { – 1} \right)\left| {\begin{array}{*{20}{c}}
{\sin \left( {x + \alpha } \right)}&{\sin \left( {x + \beta } \right)}&{\sin \left( {x + \gamma } \right)}\\
{\sin \left( {x + \alpha } \right)}&{\sin \left( {x + \beta } \right)}&{\sin \left( {x + \gamma } \right)}\\
{\sin \left( {\alpha + \beta } \right)}&{\sin \left( {\beta + \gamma } \right)}&{\sin \left( {\gamma + \alpha } \right)}
\end{array}} \right|$
$ + \left| {\begin{array}{*{20}{c}}
{\sin \left( {x + \alpha } \right)}&{\sin \left( {x + \alpha } \right)}&{\sin \left( {x + \gamma } \right)}\\
{\cos \left( {x + \alpha } \right)}&{\cos \left( {x + \alpha } \right)}&{\cos \left( {x + \gamma } \right)}\\
0&0&0
\end{array}} \right|$
$ = 0 – 0 + 0 = 0$
Henc, $f(x)$ is a constant $f'n;$
$\because $ $f\left( {10} \right) = 10\,\,\,\,\,\,\, \Rightarrow \boxed{f\left( x \right)10}$
