Let u → , v → and w → be vectors in three-dimensional space, where u → and v →…

Volume of the parallelopiped is $\left[\overrightarrow{u}\overrightarrow{v}\overrightarrow{w}\right]=\sqrt{2}$

$\Rightarrow {\left[\overrightarrow{u}\overrightarrow{v}\overrightarrow{w}\right]}^2=2$

$\Rightarrow \left|\begin{array}{ccc}\overrightarrow{u}·\overrightarrow{u}& \overrightarrow{u}·v& \overrightarrow{u}·\overrightarrow{w}\\ \overrightarrow{v}·\overrightarrow{u}& \overrightarrow{v}·\overrightarrow{v}& \overrightarrow{v}·\overrightarrow{w}\\ \overrightarrow{w}·\overrightarrow{u}& \overrightarrow{w}·\overrightarrow{v}& \overrightarrow{w}·\overrightarrow{w}\end{array}\right|=2$

Using $\overrightarrow{u}·\overrightarrow{w}=1, \overrightarrow{v}·\overrightarrow{w}=1, \overrightarrow{w}·\overrightarrow{w}=4$, we get

$\left|\begin{array}{ccc}\overrightarrow{u}·\overrightarrow{u}& \overrightarrow{u}·v& \overrightarrow{u}·\overrightarrow{w}\\ \overrightarrow{v}·\overrightarrow{u}& \overrightarrow{v}·\overrightarrow{v}& \overrightarrow{v}·\overrightarrow{w}\\ \overrightarrow{w}·\overrightarrow{u}& \overrightarrow{w}·\overrightarrow{v}& \overrightarrow{w}·\overrightarrow{w}\end{array}\right|=\left|\begin{array}{ccc}1& \overrightarrow{u}·\overrightarrow{v}& 1\\ \overrightarrow{v}·\overrightarrow{u}& 1& 1\\ 1& 1& 4\end{array}\right|=2$

$\Rightarrow 3-\left(\overrightarrow{u}·\overrightarrow{v}\right)\left(4\overrightarrow{u}·\overrightarrow{v}-1\right)+\overrightarrow{v}·\overrightarrow{u}-1=2$

$\Rightarrow \left(\overrightarrow{u}·\overrightarrow{v}\right)(1-4\overrightarrow{u}·\overrightarrow{v}+1)=0$

$\Rightarrow \overrightarrow{u}·\overrightarrow{v}=\frac{1}{2} (\because \overrightarrow{u}·\overrightarrow{v}\ne 0)$

Also, $\left|\overrightarrow{u}\right|=1, \left|\overrightarrow{v}\right|=1$

So, $|3\overrightarrow{u}+5\overrightarrow{v}|=\sqrt{{\left(|3\overrightarrow{u}+5\overrightarrow{v}|\right)}^2}$

$=\sqrt{9{\left|\overrightarrow{u}\right|}^2+25{\left|\overrightarrow{v}\right|}^2+30\overrightarrow{u}·\overrightarrow{v}}$

$=7$