Question

If $f\left(x\right)=\left|\begin{array}{ccc}\cos (x+\alpha )& \cos (x+\beta )& \cos (x+\gamma )\\ \sin (x+\alpha )& \sin (x+\beta )& \sin (x+\gamma )\\ \sin (\beta -\gamma )& \sin (\gamma -\alpha )& \sin (\alpha -\beta )\end{array}\right|$ where $\alpha ,\beta ,\gamma \in R$ and $f\left(0\right)=-2,$ then $\sum _{r=1}^{30}\left|f\right(r\left)\right|$ equals

• A. $2$
• B. $30$
• C. $60$
• D. $120$

Answer 1

The correct option is C $60$

$f\left(x\right)=\left|\begin{array}{ccc}\cos (x+\alpha )& \cos (x+\beta )& \cos (x+\gamma )\\ \sin (x+\alpha )& \sin (x+\beta )& \sin (x+\gamma )\\ \sin (\beta -\gamma )& \sin (\gamma -\alpha )& \sin (\alpha -\beta )\end{array}\right|$

$f^{\prime }\left(x\right)=\left|\begin{array}{ccc}-\sin (x+\alpha )& -\sin (x+\beta )& -\sin (x+\gamma )\\ \sin (x+\alpha )& \sin (x+\beta )& \sin (x+\gamma )\\ \sin (\beta -\gamma )& \sin (\gamma -\alpha )& \sin (\alpha -\beta )\end{array}\right|$

$+\left|\begin{array}{ccc}\cos (x+\alpha )& \cos (x+\beta )& \cos (x+\gamma )\\ \cos (x+\alpha )& \cos (x+\beta )& \cos (x+\gamma )\\ \sin (\beta -\gamma )& \sin (\gamma -\alpha )& \sin (\alpha -\beta )\end{array}\right|$

$+\left|\begin{array}{ccc}\cos (x+\alpha )& \cos (x+\beta )& \cos (x+\gamma )\\ \sin (x+\alpha )& \sin (x+\beta )& \sin (x+\gamma )\\ 0& 0& 0\end{array}\right|$

$f^{\prime }\left(x\right)=0+0+0=0$
$\Rightarrow f\left(x\right)$ is constant.
Since $f\left(0\right)=-2,$
$\Rightarrow f\left(x\right)=-2\forall x\in R$
$\therefore \left|f\right(1\left)\right|+\left|f\right(2\left)\right|+\cdots +\left|f\right(30\left)\right|=2\times 30=60$