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 on

• A. $a$
• B. $p$
• C. $d$
• D. $x$

Answer 1

The correct option is B $p$

Let$\Delta =\left|\begin{array}{ccc}1& a& a^2\\ \cos (p-d)x& \cos px& \cos (p+d)x\\ \sin (p-d)x& \sin px& \sin (p+d)x\end{array}\right|$
Applying $C_1\to C_1+C_3$
​​$\Rightarrow \Delta =\left|\begin{array}{ccc}1+a^2& a& a^2\\ \cos (p-d)x+\cos (p+d)x& \cos px& \cos (p+d)x\\ \sin (p-d)x+\sin (p+d)x& \sin px& \sin (p+d)x\end{array}\right|\Rightarrow \Delta =\left|\begin{array}{ccc}1+a^2& a& a^2\\ 2\cos px\cos dx& \cos px& \cos (p+d)x\\ 2\sin px\cos dx& \sin px& \sin (p+d)x\end{array}\right|$

Applying $C_1\to C_1-2\cos dx{C}_2$

$\Rightarrow \Delta =\left|\begin{array}{ccc}1+a^2-2a\cos dx& a& a^2\\ 0& \cos px& \cos (p+d)x\\ 0& \sin px& \sin (p+d)x\end{array}\right|$

$\Rightarrow \Delta =(1+a^2-2a\cos dx)\left[\sin \right(p+d)x\cos px-\sin px\cos (p+d)x$

$\Rightarrow \Delta =(1+a^2-2a\cos dx)\sin dx$

Which is independent of $p$