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STD 12 - 3 and 4 . determinant and metrices — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 3 and 4 . determinant and metrices4 Marks
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 $ $f\left( {10} \right) = 10\,\,\,\,\,\,\, \Rightarrow \boxed{f\left( x \right)10}$

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