JEE Main 2026 — 22 January, Evening Shift — Question 8

$\vec{a}=2 \hat{i}-\hat{j}+\hat{k}$

$\overrightarrow{\mathrm{b}}=\lambda \hat{\mathrm{j}}+2 \hat{\mathrm{k}} ; \lambda \in \mathrm{Z}$

$\overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=(-2-\lambda) \hat{\mathrm{i}}-4 \hat{\mathrm{j}}+2 \lambda \hat{\mathrm{k}}$

$|\overrightarrow{\mathrm{c}}|=\sqrt{53}$ $\Rightarrow 5 \lambda^{2}+4 \lambda-33=0$

$\lambda=2.2$ or -3 $\Rightarrow \lambda=-3$

$\overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}-4 \hat{\mathrm{j}}-6 \hat{\mathrm{k}}$

let $\vec{d}=y \hat{j}+z \hat{k}$

$|\overrightarrow{\mathrm{d}}|=2$

$\Rightarrow \mathrm{y}^{2}+\mathrm{z}^{2}=4$

$(\vec{c} \cdot \vec{d})^{2}=(4 y+6 z)^{2} \leq\left(\sqrt{4^{2}+6^{2}} \times \sqrt{y^{2}+z^{2}}\right)^{2} \leq 208$