Question

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

• A. a
• B. p
• C. d
• D. x

Answer 1

The correct option is B p

Expanding along first row, we have $1\left[cospxsin\right(p+d)x-cos(p+d\left)xsinpx\right]$
$-a\left[cos\right(p-d\left)xsin\right(p+d)x-cos(p+d\left)xsin\right(p-d\left)x\right]$
$+a^2\left[cos\right(p-d)xsinpx-cospxsin(p-d\left)x\right]$
$=sindx-asin2dx+a^{2}sindx$
which is independent of p.