JEE Main 2024 — 6 April, Shift 2 — Question 11

$\quad|(\vec{a} \times \vec{b} \times \vec{c})|=|\vec{a} \times \vec{b}||\vec{c}| \frac{\sqrt{3}}{2}$

$|\overrightarrow{\mathbf{c}}-\overrightarrow{\mathbf{a}}|=2 \sqrt{2}$

$|c|^{2}+|a|^{2}-2 \vec{c} \cdot \vec{a}=8$

$|z|^{2}+38-12|z|=8$

$|z|^{2}-12|z|+30=0$

$|z|=\frac{12 \pm \sqrt{144-120}}{2}$

$=\frac{12 \pm 2 \sqrt{6}}{2}$

$|z|=6+\sqrt{6}$

$\vec{a}\times \vec{b}=\left| \begin{matrix}\overset{\text{}}{\mathop{\ell }}\, & \overset{}{\mathop{j}}\, & \overset{}{\mathop{k}}\, \\6 & 1 & -1 \\1 & 1 & 0 \\\end{matrix} \right|$

$\hat{\ell}-\hat{\mathrm{j}}+5 \hat{\mathrm{k}}$

$|\vec{a} \times \vec{b}|=\sqrt{27}$

$|(\vec{a} \times b) \times z|=\sqrt{27}(6+\sqrt{6}) \frac{\sqrt{3}}{2}$

$\frac{9}{2}(6+\sqrt{6})$