Question

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

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

Answer 1

Answer: (B)

Let $\Delta =\left|\begin{array}{ccc}1& \alpha & {\alpha }^2\\ \cos (p-d)a& \cos pa& \cos (p-d)a\\ \sin (p-d)a& \sin pa& \sin (p-d)a\end{array}\right|$
Applying $C_3\to C_3-C_1$, we get
$\Rightarrow \Delta =\left|\begin{array}{ccc}1& \alpha & {\alpha }^2-1\\ \cos (p-d)a& \cos pa& 0\\ \sin (p-d)a& \sin pa& 0\end{array}\right|$
$=\left({\alpha }^2-1\right)\{-\cos pa\sin (p-d)a+\sin pa\cos (p-d)a\}$
$=\left({\alpha }^2-1\right)\sin \{-p(p-d)a+pa\}$
$\Rightarrow \Delta =\left({\alpha }^2-1\right)\sin da$
which is independent of $p.$