The value of x for which the angle between `veca=2x^(2)hati+4xhatj + hatkand vecb=7hati-2hatj+hatk` is obtuse and the angle between `vecb` and the z-axis is acute and less then `pi//6`

Options

A. ` a lt x lt 1//2`
B. ` 1//2 lt x lt 15`
C. ` x gt 1//2 or x lt 0`
D. none of these

Answer

D

Explanation

The angle between `veca and vecb` is obtuse.
`veca.vecb lt 0`
` Rightarrow 14x^(2) - 8x +x gt 0 Rightarrow 7x(2x -1) gt 0 Rightarrow 0 lt x lt 1//2 …(1)`
The angle between `vecb` and z-axis is acute and less thann `pi//6` .
` ( vecb.veck)/(|vecb||veck|) gt cos "" pi/6 " " [ therefore, theta , pi/6 Rightarrow cos theta gt cos "" pi/6]`
` x/ (sqrt(x^(2) +53)) gt sqrt3/2`
` Rightarrow 4x^(2) gt 3x^(2) + 159`
` Rightarrow x^(2) gt 159 Rightarrow gt sqrt(159) or x gt - sqrt159 `
Clearly, (i) and (ii) cannot hold togerther.