If $$a > 0$$ and $$z=\frac{(1+i)^2}{a+i},(i=\sqrt{-1})$$ has magnitude $$\frac{2}{\sqrt{5}}$$, then $$\bar{z}$$ is equal to

The integral $$\int \frac{\sin ^2 x \cos ^2 x}{\left(\sin ^5 x+\cos ^3 x \sin ^2 x+\sin ^3 x \cos ^2 x+\cos ^5 x\right)^2} \mathrm{~d} x$$ is equal to

The equation of the plane through $$(-1,1,2)$$ whose normal makes equal acute angles with co-ordinate axes is

If $$\mathrm{T}_{\mathrm{n}}$$ denotes the number of triangles which can be formed using the vertices of regular polygon of $$\mathrm{n}$$ sides and $$T_{n+1}-T_n=21$$, then $$\mathrm{n}=$$