Question

The values of $x$ for which the angle between the vectros $2 x^{2} \hat{i}+4 x \hat{j}+\hat{k}$ and $7 \hat{i}-2 \hat{j}+x \hat{k}$ are obtuse and the angle between the z-axis $7 \hat{i}-2 \hat{j}+x \hat{k}$ is acute and less than $\frac{\pi}{6}$ is given by

• A. $0 < x < \frac{1}{2}$
• B. $x > \frac{1}{2}$ or $x < 0$
• C. $\frac{1}{2} < x < 15$
• D. there is no such value for $x$

Answer 1

Answer: (D)

Let $\vec{ a }=2 x ^{2} \hat{ i }+4 x \hat{ j }+\hat{ k }$ and $\vec{ b }=7 \hat{ i }-2 \hat{ j }+ x \hat{ k }$.
The angle between between $\vec{a}$ and $\vec{b}$ is obtuse
$\Rightarrow \vec{a} \cdot \vec{b}<0 $
$\Rightarrow 14 x^{2}-8 x+x<0$
$\Rightarrow 14 x^{2}-7 x<0$
$\Rightarrow 2 x^{2}-x < 0 $
$\Rightarrow x(2 x-1)< 0$
$\Rightarrow x \in\left(0, \frac{1}{2}\right)\,\,\, ...(i)$
Also it is given $\vec{b} . \hat{k}=x$ and $\frac{\vec{b} \hat{k}}{|\vec{b}|}< \left(\cos \frac{\pi}{6}=\frac{\sqrt{3}}{2}\right)$
$\Rightarrow 2 x>\sqrt{3} \sqrt{53+x^{2}}$
$\Rightarrow x^{2}>159\,\,\, ...(ii)$
There is no common value for Eqs (i) and (ii)