Consider two vectors $\overrightarrow{a}=3\widehat{i}-2\widehat{j}+4\widehat{k}$ and $\overrightarrow{b}=\hat{j}+2\hat{k}.$ If $\overrightarrow{c}$ is a unit vector and $k$ be the maximum value of $a·\left(\overrightarrow{b}\times \overrightarrow{c}\right)$, then the value of $k^2-50$ is equal to

$\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\widehat{i}& \widehat{j}& \widehat{k}\\ 3& -2& 4\\ 0& 1& 2\end{array}\right|=\widehat{i}\left(-8\right)-\widehat{j}\left(6\right)+\widehat{k}\left(3\right)$
$=-8\widehat{i}-6\widehat{j}+3\widehat{k}$
$\left[\begin{array}{ccc}\overrightarrow{a}& \overrightarrow{b}& \overrightarrow{c}\end{array}\right]=\left(\overrightarrow{a}\times \overrightarrow{b}\right)\cdot \overrightarrow{c}\le \left|\overrightarrow{a}\times \overrightarrow{b}\right|\left|\overrightarrow{c}\right|$
$\Rightarrow$ maximum value of $\left[\begin{array}{ccc}\overrightarrow{a}& \overrightarrow{b}& \overrightarrow{c}\end{array}\right]=\left|\overrightarrow{a}\times \overrightarrow{b}\right|\left|\overrightarrow{c}\right|$

$\Rightarrow k=\sqrt{64+36+9}.1=\sqrt{109}$
$\Rightarrow k^2-50=109-50=59$