If $$\hat{\mathrm{a}}$$ and $$\hat{\mathrm{b}}$$ are unit vectors and $$\overline{\mathrm{c}}=\hat{\mathrm{b}}-(\hat{\mathrm{a}} \times \overline{\mathrm{c}})$$, then minimum value of $$[\hat{a} \hat{b} \bar{c}]$$ is

Angles of a triangle are in the ratio $$4: 1: 1$$. Then the ratio of its greatest side to its perimeter is

If a continuous random variable $$\mathrm{X}$$ has probability density function $$\mathrm{f}(x)$$ given by

$$f(x)=\left\{\begin{array}{cl} a x & , \text { if } 0 \leq x<1 \\ a & , \text { if } 1 \leq x<2 \\ 3 a-a x & , \text { if } 2 \leq x \leq 3 \\ 0 & , \text { otherwise } \end{array}\right.$$,

then a has the value

The value of $$\begin{aligned} \cos \left(18^{\circ}-\mathrm{A}\right) \cdot \cos ( & \left.18^{\circ}+\mathrm{A}\right) \\ & -\cos \left(72^{\circ}-\mathrm{A}\right) \cos \left(72^{\circ}+\mathrm{A}\right) \text { is }\end{aligned}$$