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

If $$\int x^5 e^{-4 x^3} \mathrm{~d} x=\frac{1}{48} \mathrm{e}^{-4 x^3} \mathrm{f}(x)+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration, then $$\mathrm{f}(x)$$ is given by

The solution of the differential equation $$\mathrm{e}^{-x}(y+1) \mathrm{d} y+\left(\cos ^2 x-\sin 2 x\right) y \mathrm{~d} x=0$$ at $$x=0$$, $$y=1$$ is