Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+5 \hat{\mathrm{j}}+\alpha \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\beta \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}$ be three vectors such that, $|\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}|=5 \sqrt{3}$ and $\overrightarrow{\mathrm{a}}$ is perpendicular to $\overrightarrow{\mathrm{b}}$. Then the greatest amongst the values of $|\vec{a}|^{2}$ is
since, $\vec{a} \cdot \vec{b}=0$
$1+15+\alpha \beta=0 \Rightarrow \alpha \beta=-16$
Also,
$|\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}|^{2}=75 \Rightarrow\left(10+\beta^{2}\right) 14-(5-3 \beta)^{2}=75$
$\Rightarrow 5 \beta^{2}+30 \beta+40=0$
$\Rightarrow \beta=-4,-2$
$\Rightarrow \alpha=4,8$
$\Rightarrow|\vec{a}|_{\max }^{2}=\left(26+\alpha^{2}\right)_{\max }=90$
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