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 $p$

$\left|\begin{array}{ccc}1& a& a^2\\ \cos (p-d)x& \cos px& \cos (p+d)x\\ \sin (p-d)x& \sin px& \sin (p+d)x\end{array}\right|$
$C_3\to C_3+C_1$
$=\left|\begin{array}{ccc}1& a& a^2+1\\ \cos (p-d)x& \cos px& cos(p+d)x+cos(p-d)x\\ \sin (p-d)x& \sin px& sin(p+d)x+sin(p-d)x\end{array}\right|$
$=\left|\begin{array}{ccc}1& a& a^2+1\\ \cos (p-d)x& \cos px& 2\cos px\cos dx\\ \sin (p-d)x& \sin px& 2\sin px\cos dx\end{array}\right|$
$=\sin 2px(\cos dx-\cos dx)-2a\sin dx\cos dx+{(a}^2+1)\sin dx$
$=-2a\sin dx\cos dx+{(a}^2+1)\sin dx$
Hence, the value of determinant is independent of p