(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 ${{\cos }^{2}}\theta +{{\sin }^{2}}\theta =1$

$\left| A \right|=1$