Question

If $\Delta =\left|\begin{array}{ccc}\sin \alpha & \cos \alpha & \sin (\alpha +\delta )\\ \sin \beta & \cos \beta & \sin (\beta +\delta )\\ \sin \gamma & \cos \gamma & \sin (\gamma +\delta )\end{array}\right|$ then $\Delta$ is independent of

• A. $\alpha$
• B. $\beta$
• C. $\gamma$
• D. $\delta$

Answer 1

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

The correct options are
A $\alpha$
B $\beta$
C $\gamma$
D $\delta$

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

Applying $C_3\to C_3-{\cos \delta C}_1-\sin \delta C_2$

$\Delta =\left|\begin{array}{ccc}\sin \alpha & \cos \alpha & 0\\ \sin \beta & \cos \beta & 0\\ \sin \gamma & \cos \gamma & 0\end{array}\right|=0$