Question

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