Question

Let $\alpha ,\beta ,\gamma ,a,b,c,xϵR$ and let
$\Delta =\left|\begin{array}{ccc}\sin (x+\alpha )& \cos (x+\alpha )& a+x\sin \alpha \\ \sin (x+\beta )& \cos (x+\beta )& b+x\sin \beta \\ \sin (x+\gamma )& \cos (x+\gamma )& c+x\sin \gamma \end{array}\right|$
then $\Delta$ is

• A. independent of $x$
• B. dependent of $\alpha ,\beta ,\gamma$
• C. dependent on $a,b,c$
• D. $a$ constant

Answer 1

Answer: (A), (D)

The correct options are
A independent of $x$
D $a$ constant

$\Delta =\left|\begin{array}{ccc}\sin (x+\alpha )& \cos (x+\alpha )& a+x\sin \alpha \\ \sin (x+\beta )& \cos (x+\beta )& b+x\sin \beta \\ \sin (x+\gamma )& \cos (x+\gamma )& c+x\sin \gamma \end{array}\right|=\left|\begin{array}{ccc}\sin x\cos \alpha +\cos x\sin \alpha & \cos x\cos \alpha -\sin x\sin \alpha & a+x\sin \alpha \\ \sin x\cos \beta +\cos x\sin \beta & \cos x\cos \beta -\sin x\sin \beta & b+x\sin \beta \\ \sin x\cos \gamma +\cos x\sin \gamma & \cos x\cos \gamma -\sin x\sin \gamma & c+x\sin \gamma \end{array}\right|=\left|\begin{array}{ccc}\sin x\cos \alpha & \cos x\cos \alpha & a\\ \sin x\cos \beta & \cos x\cos \beta & b\\ \sin x\cos \gamma & \cos x\cos \gamma & c\end{array}\right|+\left|\begin{array}{ccc}\cos x\sin \alpha & -\sin x\sin \alpha & x\sin \alpha \\ \cos x\sin \beta & -\sin x\sin \beta & x\sin \beta \\ \cos x\sin \gamma & -\sin x\sin \gamma & x\sin \gamma \end{array}\right|=\sin x\cos x\left|\begin{array}{ccc}\cos \alpha & \cos \alpha & a\\ \cos \beta & \cos \beta & b\\ \cos \gamma & \cos \gamma & c\end{array}\right|+\left(\cos x\right)(-\sin x)\left(x\right)\left|\begin{array}{ccc}\sin \alpha & \sin \alpha & \sin \alpha \\ \sin \beta & \sin \beta & \sin \beta \\ \sin \gamma & \sin \gamma & \sin \gamma \end{array}\right|=0$