Question

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

• A. a
• B. p
• C. d
• D. x

Answer 1

Answer: (B)

$=\begin{vmatrix}1 & a & a^2\\ \cos (p-d)x& \cos\ px & \cos(p+d)x\\ \sin (p-d)& \sin\ px& \sin(p+d)x\\\end{vmatrix}$
Applying $C_1 \rightarrow C x + C_3$
$=\begin{vmatrix}1+a^2 & a & a^2\\ \cos (p-d)x+\cos(p+d)x& \cos\ px & \cos(p+d)x\\ \sin (p-d)+ \sin(p+d)x& \sin\ px& \sin(p+d)x\\\end{vmatrix}$
$\Rightarrow \Delta=\begin{vmatrix}1+a^2 & a & a^2\\2 \cos\ px\ \cos\ dx& \cos\ px & \cos(p+d)x\\2 \sin\ px\ \sin\ dx& \sin\ px& \sin(p+d)x\\\end{vmatrix}$
Applying $C_1 \rightarrow C_1-2 \cos dx C_2$
$\Rightarrow \Delta=\begin{vmatrix}1+a^2-2a\ \cos\ dx & a & a^2\\0& \cos\ px & \cos(p+d)x\\0& \sin\ px& \sin(p+d)x\\\end{vmatrix}$
$\Rightarrow \Delta= (1 + a^2 - 2a \cos dx) [\sin (p + d) x \cos px - \sin p x \cos (p + d ) x]$
$\Rightarrow \Delta= (1 + a^2 - 2a \cos dx) \sin d x$
which is independent of $p$.