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. `altxlt1//2`
B. `1//2ltxlt15`
C. `xlt1//2orxlt0`
D. none of these

Answer

b

Explanation

The angle between `veca and vecb` is obtuse , therefore,
`veca.vecb lt 0`
`Rightaarrow 14x^(2) - 8x + x lt 0`
`or 7x ( 2x - 1) lt 0`
`or 0 lt x lt 1//2`
The angle between `vecb` and the z-axis is acute and less than `pi//6` therefore,
`(vecb.veck)/(|vecb||veck|)gtcospi//6 ( thetalt pi//6 Rightarrowcos thetagtcos pi//6)`
`Rightarrow x/(sqrt(x^(2)+53) gt sqrt3/2`
`or 4x^(2) gt 3x^(2) + 159`
`or x^(2) gt 159`
`Rightarrow x gt sqrt159 or x gt - sqrt159`
clearly, (i) and (ii) cannot hold together.