JEE Main 2026 — 22 January, Evening Shift — Question 21

$\frac{\overrightarrow{\mathrm{a}} \cdot \hat{\mathrm{k}}}{|\overrightarrow{\mathrm{a}}|}=\cos \theta \Rightarrow \frac{\lambda}{\sqrt{3+\lambda^{2}}}=\cos \theta$

$\Rightarrow 0<\frac{\lambda}{\sqrt{3+\lambda^{2}}}<\frac{\sqrt{3}}{2}$

$\Rightarrow \lambda>0 \& 4 \lambda^{2}<9+3 \lambda^{2} \Rightarrow \lambda^{2}<9$ $\begin{gathered} \Rightarrow \lambda \in(0,3) \end{gathered}$

$\Rightarrow \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}<0 \Rightarrow-\sqrt{2} \lambda^{2}-4 \sqrt{2}+4 \sqrt{2} \lambda<0$

$\Rightarrow \lambda^{2}-4 \lambda+4>0 \Rightarrow(\lambda-2)^{2}>0$

$\begin{gathered} \Rightarrow \lambda \neq 2 \end{gathered}$ from (1) & (2) $\lambda \in(0,3)-\{2\}$

$\therefore \alpha=0, \beta=3, \gamma=2$

$\Rightarrow \alpha+\beta+\gamma=5$