Let $$\theta$$ be the angle between the planes $$P_{1}: \vec{r} \cdot(\hat{i}+\hat{j}+2 \hat{k})=9$$ and $$P_{2}: \vec{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})=15$$. Let $$\mathrm{L}$$ be the line that meets $$P_{2}$$ at the point $$(4,-2,5)$$ and makes an angle $$\theta$$ with the normal of $$P_{2}$$. If $$\alpha$$ is the angle between $$\mathrm{L}$$ and $$P_{2}$$, then $$\left(\tan ^{2} \theta\right)\left(\cot ^{2} \alpha\right)$$ is equal to ____________.

If the variance of the frequency distribution

is 3, then $$\alpha$$ is equal to _____________.

The initial speed of a projectile fired from ground is $$\mathrm{u}$$. At the highest point during its motion, the speed of projectile is $$\frac{\sqrt{3}}{2} u$$. The time of flight of the projectile is :

A bar magnet with a magnetic moment $$5.0 \mathrm{Am}^{2}$$ is placed in parallel position relative to a magnetic field of $$0.4 \mathrm{~T}$$. The amount of required work done in turning the magnet from parallel to antiparallel position relative to the field direction is _____________.