The value of x for which the angle between $\vec a = 2x^2\hat i + 4x\hat j +\hat k$ and $\vec b =7\hat i-2\hat j+x\hat k$ is obtuse and the angle between $\vec b$ and the z-axis is acute and less than $π/6$, are

none of these

The angle between $\vec a$ and $\vec b$ is obtuse.

$∴\vec a.\vec b<0$

$⇒14x^2 - 8x + x <0⇒ 7x (2x-1) <0⇒ 0 < x <1/2$   ...(i)

The angle between $\vec b$ and z-axis is acute and less than $π/6$.

$∴\frac{\vec b.\vec k}{|\vec b||\hat k|}>\cos\frac{π}{6}$   $[∵θ<\frac{π}{6}⇒\cos θ>\cos\frac{π}{6}]$

$⇒\frac{x}{\sqrt{x^2+53}}>\frac{\sqrt{3}}{2}$

$⇒4x^2 > 3x^2 + 159$

$⇒x^2 >159⇒x>\sqrt{159}$ or $x<-\sqrt{159}$   ...(ii)

Clearly, (i) and (ii) cannot hold together.