If vectors $ax \hat{i}+3 \hat{j}-5 \hat{k}$ and $x \hat{i}+2 \hat{j}+2 a x \hat{k}$ make an acute angle with each other, for all x ∈ R, then a belongs to the interval
Answer & explanation
Correct answer: option 3
Since vectors make an acute angle with each other so their dot product must be positive i.e. $a x^2-10 a x+6>0 ~\forall~ x \in R$
$\Rightarrow-ax^2+10 ax-6<0 ~\forall~ x \in R \Rightarrow-a<0$ and $100 a^2<24 a$
Hence (3) is correct answer.