For a binomial variate $$\mathrm{X}$$ with $$\mathrm{n}=6$$ if $$P(X=4)=\frac{135}{2^{12}}$$, then its variance is

If $$\overline{\mathrm{a}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}, \overline{\mathrm{b}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$$ and $$\overline{\mathrm{c}}=\hat{\mathrm{i}}+3 \hat{\mathrm{j}}$$ are such that $$(\bar{a}+\lambda \bar{b})$$ is perpendicular to $$\bar{c}$$, then the value of $$\lambda$$ is

The number of integral values of $$\mathrm{p}$$ in the domain $$[-5,5]$$, such that the equation $$2 x^2+4 x y-p y^2+4 x+q y+1=0$$ represents pair of lines, are

An open metallic tank is to be constructed, with a square base and vertical sides, having volume 500 cubic meter. Then the dimensions of the tank, for minimum area of the sheet metal used in its construction, are