A random variable X has following p.d.f. $\mathrm{f}(x)=\mathrm{kx}(1-x), 0 \leqslant x \leqslant 1 \quad$ and $\quad \mathrm{P}(x>\mathrm{a})=\frac{20}{27}$, then $\mathrm{a}=$

The probability distribution of a random variable X is given by

Then the variance of X is

If $\left(\cos ^{-1} x\right)^2-\left(\sin ^{-1} x\right)^2>0$, then

If $\bar{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \bar{b}=\hat{i}-2 \hat{j}-2 \hat{k}, \bar{c}=-\hat{i}+4 \hat{j}+3 \hat{k}$ and if $\overline{\mathrm{d}}$ is vector perpendicular to both $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}, \overline{\mathrm{a}} \cdot \overline{\mathrm{d}}=18$, then $|\overline{\mathrm{a}} \times \overline{\mathrm{d}}|^2=$