Question

The parameter on which the value of the determinant $=\left|\begin{array}{ccc}1& a& a^2\\ \cos (p-d)x& \cos px& \cos (p+d)x\\ \sin (p-d)x& \sin px& \sin (p+d)x\end{array}\right|$ does not depend upon, is

• A. a
• B. p
• C. d
• D. x

Answer 1

Answer: (B)

$=\left|\begin{array}{ccc}1& a& a^2\\ \cos (p-d)x& \cos px& \cos (p+d)x\\ \sin (p-d)& \sin px& \sin (p+d)x\end{array}\right|$
Applying $C_1\to Cx+C_3$
$=\left|\begin{array}{ccc}1+a^2& a& a^2\\ \cos (p-d)x+\cos (p+d)x& \cos px& \cos (p+d)x\\ \sin (p-d)+\sin (p+d)x& \sin px& \sin (p+d)x\end{array}\right|$
$\Rightarrow \Delta =\left|\begin{array}{ccc}1+a^2& a& a^2\\ 2\cos px\cos dx& \cos px& \cos (p+d)x\\ 2\sin px\sin dx& \sin px& \sin (p+d)x\end{array}\right|$
Applying $C_1\to C_1-2\cos dx{C}_2$
$\Rightarrow \Delta =\left|\begin{array}{ccc}1+a^2-2a\cos dx& a& a^2\\ 0& \cos px& \cos (p+d)x\\ 0& \sin px& \sin (p+d)x\end{array}\right|$
$\Rightarrow \Delta =(1+a^2-2a\cos dx)[\sin (p+d)x\cos px-\sin px\cos (p+d)x]$
$\Rightarrow \Delta =(1+a^2-2a\cos dx)\sin dx$
which is independent of $p$.