If the projection of $\bar{a}$ on $\bar{b}+\bar{c}$ is twice the projection of $\bar{b}+\bar{c}$ on $\bar{a}$ also if $|\bar{b}|=2 \sqrt{2},|\bar{c}|=4$ and the angle between $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ is $\frac{\pi}{4}$ then $|\overline{\mathrm{a}}|=$

The equation of the plane passing through the point of intersection of the planes $2 x-y+z-3=0$ and $4 x-3 y+5 z+9=0$ and parallel to the line $\frac{x+1}{2}=\frac{y+3}{4}=\frac{z-3}{5}$ is $\alpha x+\beta y+\gamma z+d=0$ Then $\alpha+\beta+\gamma+d=$

If the points $\mathrm{A}(1,1,2), \mathrm{B}(2,1, \mathrm{p}), \mathrm{C}(1,0,3)$ and $D(2,2,0)$ are coplanar then the value of $p$ is

$$ \int \frac{\mathrm{e}^{\tan ^{-1} 2 x}}{1+4 x^2}= $$