If $$\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$$ are unit vectors and $$\theta$$ is angle between $$\overline{\mathrm{a}}$$ and $$\bar{c}$$ and $$\bar{a}+2 \bar{b}+2 \bar{c}=\overline{0}$$, then $$|\bar{a} \times \bar{c}|=$$

The integral $$\int_\limits{\frac{\pi}{6}}^{\frac{\pi}{3}} \sec ^{\frac{2}{3}} x \operatorname{cosec}^{\frac{4}{3}} x d x$$ is equal to

The principal solutions of the equation $$\sec x+\tan x=2 \cos x$$ are

If $$\bar{a}, \bar{b}, \bar{c}$$ are three vectors with magnitudes $$\sqrt{3}$$, 1, 2 respectively, such that $$\bar{a} \times(\bar{a} \times \bar{c})+3 \bar{b}=\overline{0}$$, if $$\theta$$ is the angle between $$\bar{a}$$ and $$\bar{c}$$, then $$\sec ^2 \theta$$ is