The value of $$\alpha$$, so that the volume of parallelopiped formed by $$\hat{i}+\alpha \hat{j}+\hat{k}, \hat{j}+\alpha \hat{k}$$ and $$\alpha \hat{\mathrm{i}}+\hat{\mathrm{k}}$$ becomes minimum, is

In a certain culture of bacteria, the rate of increase is proportional to the number of bacteria present at that instant. It is found that there are 10,000 bacteria at the end of 3 hours and 40,000 bacteria at the end of 5 hours, then the number of bacteria present in the beginning are

If $$x, y, z$$ are in A.P. and $$\tan ^{-1} x, \tan ^{-1} y$$ and $$\tan ^{-1} z$$ are also in A.P., then

Mean and variance of six observations are 8 and 16 respectively. If each observation is multiplied by 3, then new variance of the resulting observations is