QJEEM24Ma9fdd: JEE Main 2024 Physics - Class 11 JEEM24Ma9fdd Solution: The set of all , for which the vectors and are inclined at an…

Given $\vec{a}=\alpha t \hat{i}+6 \hat{j}-3 \hat{k}$

and $\vec{b}=t \hat{i}-2 \hat{j}-2 \alpha t \hat{k}$

angle between $\vec{a}$ and $\vec{b}$ is given by

$\cos \theta=\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$

We have, \cos \theta < 0(\because angle between $\vec{a}$ and $\vec{b}$ is obtuse)

$\begin{aligned}& \Rightarrow \quad \vec{a} \cdot \vec{b}<0 \\& \Rightarrow \alpha t^2-12+6 \alpha t<0 \forall t \in \mathbb{R}\end{aligned}$

If $\alpha=0$ , then -12<0 (condition holds)

If \alpha \neq 0 \Rightarrow \alpha<0\quad \text{.... (i)}

And maximum value of \alpha t^2+6 \alpha t-12<0

$\begin{aligned}& \Rightarrow \frac{-D}{4 a}<0 \text { (where } D \text { is discriminant and } a=\alpha) \\& \Rightarrow \frac{36 \alpha^2+48 \alpha}{4 \alpha}>0 \\& \Rightarrow \alpha>\frac{-4}{3} \\& \therefore \alpha \in\left(\frac{-4}{3}, 0\right]\end{aligned}$