The values of `x` for which the angle between ` vec a=2x^2\ hat i+4x hat j+ hat k ,\ 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 `pi/6` are `x >1/2or\ x<0` b. `0

Explanation

To solve the problem, we need to find the values of \( x \) for which the angle between the vectors \( \vec{a} \) and \( \vec{b} \) is obtuse and the angle between \( \vec{b} \) and the z-axis is acute and less than \( \frac{\pi}{6} \).

### Step 1: Define the vectors
Given:
\[ \vec{a} = 2x^2 \hat{i} + 4x \hat{j} + \hat{k} \]
\[
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