The solution of the differential equation

$$x{{dy} \over {dx}} + 2y$$ = x² (x $$ \ne $$ 0) with y(1) = 1, is :

The integral $$\int {{\rm{se}}{{\rm{c}}^{{\rm{2/ 3}}}}\,{\rm{x }}\,{\rm{cose}}{{\rm{c}}^{{\rm{4 / 3}}}}{\rm{x \,dx}}} $$ is equal to (Hence C is a constant of integration)

Let the sum of the first n terms of a non-constant A.P., a₁, a₂, a₃, ..... be $$50n + {{n(n - 7)} \over 2}A$$, where A is a constant. If d is the common difference of this A.P., then the ordered pair (d, a₅₀) is equal to

The area (in sq. units) of the region

A = {(x, y) : x² $$ \le $$ y $$ \le $$ x + 2} is