Q. The arbitrary constant on which the value of the determinant $\left| \begin{matrix} 1 & \alpha & \alpha^{2} \\ \cos{(p - d)a} & \cos{pa} & \cos{(p - d)a} \\ \sin{(p - d)a} & \sin{pa} & \sin{(p - d)a} \end{matrix} \right|$ does not depend, is
A) $\alpha$
B) $p$
C) $d$
D) $a$
Explanation
(b)
Applying $C_{3} \rightarrow C_{3} - C_{1}$ ,we get
$\mathrm{\Delta} = \left| \begin{matrix}
1 & \alpha & \alpha^{2} - 1 \\
\cos{(p - d)a} & \cos{pa} & 0 \\
\sin{(p - d)a} & \sin{pa} & 0
\end{matrix} \right|$
$= \left( \alpha^{2} - 1 \right)\{ - \cos{pa}\ \sin{(p - d)a} + \sin{pa}\cos{(p - d)a}\}\$
$= \left( \alpha^{2} - 1 \right)\sin{\{ - (p - d)a + pa\}}$
$\Rightarrow \ \mathrm{\Delta} = \left( \alpha^{2} - 1 \right)\sin{da}$
Which is independent of $p$.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.