Question

If $\varphi ϵR$ then $\Delta \left(\varphi \right)=\left|\begin{array}{ccc}\sin \varphi & \cos \varphi & \sin 2\varphi \\ \sin (\varphi +2\pi /3)& \cos (\varphi +2\pi /3)& \sin (2\varphi +4\pi /3)\\ \sin (\varphi -2\pi /3)& \cos (\varphi -2\pi /3)& \sin (2\varphi -2\pi /3)\end{array}\right|$

• A. is independent of $\varphi$
• B. has no local minimum
• C. is equal to a constant
• D. cannot be determined

Answer 1

Answer: (A), (B), (C)

The correct options are
A is independent of $\varphi$
B has no local minimum
C is equal to a constant

$\Delta \left(\varphi \right)=\left|\begin{array}{ccc}\sin \varphi & \cos \varphi & \sin 2\varphi \\ \sin (\varphi +2\pi /3)& \cos (\varphi +2\pi /3)& \sin (2\varphi +4\pi /3)\\ \sin (\varphi -2\pi /3)& \cos (\varphi -2\pi /3)& \sin (2\varphi -2\pi /3)\end{array}\right|$

Applying $R_2\to R_2+R_3$

$\Delta \left(\varphi \right)=\left|\begin{array}{ccc}\sin \varphi & \cos \varphi & \sin 2\varphi \\ 2\sin \left(\varphi \right)\cos 2\pi /3& 2\cos \left(\varphi \right)\cos 2\pi /3& 2\sin \left(2\varphi \right)\cos 4\pi /3\\ \sin (\varphi -2\pi /3)& \cos (\varphi -2\pi /3)& \sin (2\varphi -2\pi /3)\end{array}\right|$

$=\left|\begin{array}{ccc}\sin \varphi & \cos \varphi & \sin 2\varphi \\ 2\sin \left(\varphi \right)(-1/2)& 2\cos \left(\varphi \right)(-1/2)& 2\sin \left(2\varphi \right)(-1/2)\\ \sin (\varphi -2\pi /3)& \cos (\varphi -2\pi /3)& \sin (2\varphi -2\pi /3)\end{array}\right|$

$=-\left|\begin{array}{ccc}\sin \varphi & \cos \varphi & \sin 2\varphi \\ \sin \left(\varphi \right)& \cos \left(\varphi \right)& \sin \left(2\varphi \right)\\ \sin (\varphi -2\pi /3)& \cos (\varphi -2\pi /3)& \sin (2\varphi -2\pi /3)\end{array}\right|=0$