Question
The parameter, on which the value of the determinant \[\left| \begin{array}{l}
\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,a\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{a^2}\\
\cos \left( {p - d} \right)x\,\,\,\,\,\,\,\,\,\,\cos px\,\,\,\,\,\,\,\,\,\,\,\,\,\cos \left( {p + d} \right)x\\
\sin \left( {p - d} \right)x\,\,\,\,\,\,\,\,\,\,\,\sin px\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sin \left( {p + d} \right)x
\end{array} \right|\]
does not depend upon is
A.
$$a$$
B.
$$p$$
C.
$$d$$
D.
$$x$$
Answer :
$$p$$
Solution :
\[{\rm{Let }}\,\Delta = \left| \begin{array}{l}
\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,a\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{a^2}\\
\cos \left( {p - d} \right)x\,\,\,\,\,\,\,\,\,\,\cos px\,\,\,\,\,\,\,\,\,\,\,\,\,\cos \left( {p + d} \right)x\\
\sin \left( {p - d} \right)x\,\,\,\,\,\,\,\,\,\,\,\sin px\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sin \left( {p + d} \right)x
\end{array} \right|\]
$${\text{Applying }}{C_1} \to {C_1} + {C_3}$$
\[ \Rightarrow \,\,\Delta = \left| \begin{array}{l}
\,\,\,\,\,\,\,\,\,1 + {a^2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,a\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{a^2}\\
2\cos px\cos dx\,\,\,\,\,\,\,\,\,\,\,\,\cos px\,\,\,\,\,\,\,\,\,\,\,\cos \left( {p + d} \right)x\\
2\sin px\cos dx\,\,\,\,\,\,\,\,\,\,\,\,\,\sin px\,\,\,\,\,\,\,\,\,\,\,\sin \left( {p + d} \right)x
\end{array} \right|\]
$${C_1} \to {C_1} - \left( {2\cos dx} \right){C_2}$$
\[\Delta = \left| \begin{array}{l}
1 + {a^2} - 2a\cos dx\,\,\,\,\,\,\,\,\,\,\,\,\,\,a\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{a^2}\\
\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\cos px\,\,\,\,\,\,\,\,\,\,\,\,\,\,\cos \left( {p + d} \right)x\\
\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sin px\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sin \left( {p + d} \right)x
\end{array} \right|\]
Expanding along $${C_1},$$ we get
$$\eqalign{
& \Delta = \left( {1 + {a^2} - 2a\cos dx} \right)\left[ {\sin \left( {p + d} \right)x\cos px - \sin px\cos \left( {p + d} \right)x} \right] \cr
& \Rightarrow \,\,\Delta = \left( {1 + {a^2} - 2a\cos dx} \right)\left[ {\sin \left\{ {\left( {p + d} \right)x - px} \right\}} \right] \cr
& \Rightarrow \,\,\Delta = \left( {1 + {a^2} - 2a\cos dx} \right)\left[ {\sin dx} \right] \cr} $$
which is independent of $$p.$$