Question

The arbitrary constant on which the value of the determinant
$\begin{vmatrix}1 & \alpha & \alpha^{2} \\ \cos (p-d) a & \cos p a & \cos (p-d) a \\ \sin (p-d) a & \sin p a & \sin (p-d) a\end{vmatrix}$
does not depend, is

• A. $ \alpha $
• B. $p$
• C. $d$
• D. $a$

Answer 1

Answer: (B)

Let $\Delta=\begin{vmatrix}1 & \alpha & \alpha^{2} \\ \cos (p-d) a & \cos p a & \cos (p-d) a \\ \sin (p-d) a & \sin p a & \sin (p-d) a\end{vmatrix}$
Applying $C_{3} \rightarrow C_{3}-C_{1}$, we get
$\Rightarrow \Delta=\begin{vmatrix}1 & \alpha & \alpha^{2}-1 \\ \cos (p-d) a & \cos p a & 0 \\ \sin (p-d) a & \sin p a & 0\end{vmatrix}$
$=\left(\alpha^{2}-1\right)\{-\cos p a \sin (p-d) a+\sin p a \cos (p-d) a\}$
$=\left(\alpha^{2}-1\right) \sin \{-p(p-d) a+p a\}$
$\Rightarrow \Delta=\left(\alpha^{2}-1\right) \sin d a$
which is independent of $p .$