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

No such value for x

$14x^2-8x+x<0$

$14x^2-7x<0$

$7x(2x-1)<0$  $⇒x∈(0,\frac{1}{2})$

$7\hat i-2\hat j+x\hat k$

$\hat k∈(0,\frac{\pi}{6});cosθ∈(\frac{\sqrt{3}}{2},1)$

$\frac{\sqrt{3}}{2}<\frac{x}{\sqrt{53+x^2}}<1⇒\frac{3}{4}<\frac{x^2}{53+x^2}<1⇒4x^2>159+3x^2⇒x^2>159$