If $\overline{\mathrm{a}}=2\overline{\mathrm{i}}+\overline{\mathrm{j}}-3\overline{\mathrm{k}},\overline{\mathrm{b}}=\overline{\mathrm{i}}-2\overline{\mathrm{j}}+\overline{\mathrm{k}},\overline{\mathrm{c}}=-\overline{\mathrm{i}}+\overline{\mathrm{j}}-4\overline{\mathrm{k}},\overline{\mathrm{d}}=\overline{\mathrm{i}}+\overline{\mathrm{j}}+\mathrm{p}\overline{\mathrm{k}},$ where p is a positive real number and $\left|\left(\overline{\mathrm{a}}\times \overline{\mathrm{b}}\right)\times \left(\overline{\mathrm{c}}\times \overline{\mathrm{d}}\right)\right|=5\sqrt{114},$  then a value of p is

$$\begin{gathered} \left(\overline{\mathrm{a}}\times \overline{\mathrm{b}}\right)\times (\overline{\mathrm{c}}\times \overline{\mathrm{d}})=\left[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{d}}\right] \overline{\mathrm{c}}-\left[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}\right] \overline{\mathrm{d}} \\ \left[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{d}}\right]=\left|\begin{array}{ccc}2& 1& -3\\ 1& -2& 1\\ 1& 1& \mathrm{p}\end{array}\right|=1(-5),-1\left(5\right)+\mathrm{p}(-5) \\ =-10-5\mathrm{p} \\ \left[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}\right]=\left|\begin{array}{ccc}2& 1& -3\\ 1& -2& 1\\ -1& 1& -4\end{array}\right|=2\left(7\right)-(-3)-3(-1) \\ =14+3+3=20 \\ \left(\overline{\mathrm{a}}\times \overline{\mathrm{b}}\right)\times (\overline{\mathrm{c}}\times \overline{\mathrm{d}}) =(-10-5\mathrm{p}) (-\mathrm{i}+\mathrm{j}-4\mathrm{k}) -20(\mathrm{i}+\mathrm{j}+\mathrm{pk}) \\ =\mathrm{i}(5\mathrm{p}-10)+\mathrm{j}(-30-5\mathrm{p})+\mathrm{k}\left(40\right) \\ \left|\left(\overline{\mathrm{a}}\times \overline{\mathrm{b}}\right)\times (\overline{\mathrm{c}}\times \overline{\mathrm{d}})\right|=\sqrt{{(5\mathrm{p}-10)}^2+{(5\mathrm{p}+30)}^2+\left(40\right)}=5\sqrt{114} \\ \Rightarrow \mathrm{p}=2 \end{gathered}$$