If the volume of a parallelopiped, whose coterminus edges are given by the vectors
$\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{n} \hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-\mathrm{n} \hat{\mathrm{k}}$
and $\overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}+n \hat{\mathrm{j}}+3 \hat{\mathrm{k}} \quad(\mathrm{n} \geq 0)$, is $158 \mathrm{cu}$. units, then :
Correct Option: , 3
$v=[\vec{a} \vec{b} \vec{c}]$
$158=\left|\begin{array}{ccc}1 & 1 & n \\ 2 & 4 & -n \\ 1 & n & 3\end{array}\right|, n \geq 0$
$158=1\left(12+n^{2}\right)-(6+n)+n(2 n-4)$
$158=n^{2}+12-6-n+2 n^{2}-4 n$
$3 n^{2}-5 n-152=0$
$\mathrm{n}=8,-\frac{38}{6}$ (rejected)
$\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=1+\mathrm{n}+3 \mathrm{n}=1+4 \mathrm{n}=33$
$\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}=2+4 \mathrm{n}-3 \mathrm{n}=2+\mathrm{n}=10$
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