$$\int_\limits{\frac{\pi}{6}}^{\frac{\pi}{2}} \frac{\operatorname{cosec} x \cdot \cot x}{1+\operatorname{cosec}^2 x} d x=$$

A spherical raindrop evaporates at a rate proportional to its surface area. If its radius originally is 3 mm. and 1 hour later has been reduced to 2 mm, then the expression of radius r of the raindrop at any time t is (where 0 $$\le$$ t < 3)

The differential equation of all parabolas having vertex at the origin and axis along positive Y-axis is

If the vectors $$2 \hat{i}-\hat{j}-\hat{k} ; \hat{i}+2 \hat{j}-3 \hat{k}$$ and $$3 \hat{i}+\lambda \hat{j}+5 \hat{k}$$ are coplanar, then the value of $$\lambda$$ is