JEE Main 2024 — 6 April, Shift 2 — Question 27

$\quad \overline{\mathrm{a}}_{1}=\lambda \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$

$\overline{\mathrm{a}}_{2}=-2 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$

$\overrightarrow{\mathrm{p}}-=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$

$\vec{q}-=-3 \hat{i}+2 \hat{j}+4 \hat{k}$

$(\lambda+2) \hat{\mathrm{i}}+7 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}=\overline{\mathrm{a}}_{1}-\overline{\mathrm{a}}_{2}$

$\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{q}}-=-6 \hat{\mathrm{i}}-15 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$

$\frac{44}{\sqrt{30}}=\frac{|-6 \lambda-12-105-9|}{\sqrt{(-6)^{2}+(-15)^{2}+3^{2}}}$

$\frac{44}{\sqrt{30}}=\frac{|6 \lambda+126|}{3 \sqrt{30}}$

$132=|6 \lambda+126|$

$\lambda=1, \lambda=-43$

$|\lambda|=43$