Question

Let $f\left(x\right)=\left|\begin{array}{ccc}\cos (x+\alpha )+i\sin (x+\alpha )& \cos (x+\beta )+i\sin (x+\beta )& \cos (x+\gamma )+i\sin (x+\gamma )\\ \sin (x+\alpha )-i\cos (x+\alpha )& \sin (x+\beta )-i\cos (x+\beta )& \sin (x+\gamma )-i\cos (x+\gamma )\\ \sin (2\beta -2\gamma )& \sin (2\gamma -2\alpha )& \sin (2\alpha -2\beta )\end{array}\right|$.
Then the value of $f\left(\alpha \right)+f\left(\beta \right)-2f\left(\gamma \right)$ is

• A. dependent on $\alpha$
• B. dependent on $\gamma$
• C. Independent of all $\alpha ,\beta$ and $\gamma$
• D. dependent on $\beta$

Answer 1

The correct option is C Independent of all $\alpha ,\beta$ and $\gamma$

$f\left(x\right)=\left|\begin{array}{ccc}\cos (x+\alpha )+i\sin (x+\alpha )& \cos (x+\beta )+i\sin (x+\beta )& \cos (x+\gamma )+i\sin (x+\gamma )\\ \sin (x+\alpha )-i\cos (x+\alpha )& \sin (x+\beta )-i\cos (x+\beta )& \sin (x+\gamma )-i\cos (x+\gamma )\\ \sin (2\beta -2\gamma )& \sin (2\gamma -2\alpha )& \sin (2\alpha -2\beta )\end{array}\right|$

Multiplying second row by $i$
$=\frac{1}{i}\left|\begin{array}{ccc}\cos (x+\alpha )+i\sin (x+\alpha )& \cos (x+\beta )+i\sin (x+\beta )& \cos (x+\gamma )+i\sin (x+\gamma )\\ \cos (x+\alpha )+i\sin (x+\alpha )& \cos (x+\beta )+i\sin (x+\beta )& \cos (x+\gamma )+i\sin (x+\gamma )\\ \sin (2\beta -2\gamma )& \sin (2\gamma -2\alpha )& \sin (2\alpha -2\beta )\end{array}\right|$
$\because$ first and second row are same
$\therefore f\left(x\right)=0$
So $f\left(\alpha \right)+f\left(\beta \right)-2f\left(\gamma \right)=0$