(i) As per the question, it is given that,
$\left| \begin{matrix}
x & -7 \\
x & 5x+1 \\
\end{matrix} \right|$
$\left| A \right|=x\left( 5x+1 \right)-\left( -7 \right)x$
$\left| A \right|=5{{x}^{2}}+8x$
(ii) According to the question
$\left| \begin{matrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta \\
\end{matrix} \right|$
$\left| A \right|=\cos \theta \times \cos \theta -\left( -\sin \theta \right)\times \sin \theta $
$\left| A \right|={{\cos }^{2}}\theta +{{\sin }^{2}}\theta $
As we all know the trigonometric function ${{\cos }^{2}}\theta +{{\sin }^{2}}\theta =1$
$\left| A \right|=1$