If f ( x ) = | * 20 c ( x + ) & ( x + ) & ( x + ) ( x + ) & ( x + ) & ( x + ) ( + ) & ( + ) & ( + ) | and f(10) = 10 then f( ) is equal to

If f ( x ) = | * 20 c ( x + ) & ( x + ) & ( x + ) ( x + ) & ( x +…

MCQ

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

A
$0$
B
$\pi$
✓
$10$
D
None of these

✓

Answer

Correct option: C.

$10$

c
$f'\left( x \right) = \left| {\begin{array}{*{20}{c}}
{\cos \left( {x + \alpha } \right)}&{\cos \left( {x + \beta } \right)}&{\cos \left( {x + \gamma } \right)}\\
{\cos \left( {x + \alpha } \right)}&{\cos \left( {x + \beta } \right)}&{\cos \left( {x + \gamma } \right)}\\
{\sin \left( {\alpha + \beta } \right)}&{\sin \left( {\beta + \gamma } \right)}&{\sin \left( {\gamma + \alpha } \right)}
\end{array}} \right|$

$ + \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 $ $f\left( {10} \right) = 10\,\,\,\,\,\,\, \Rightarrow \boxed{f\left( x \right)10}$