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

Let $$\alpha $$ and $$\beta $$ be the roots of the equation x² + x + 1 = 0. Then for y $$ \ne $$ 0 in R,
$$$\left| {\matrix{ {y + 1} & \alpha & \beta \cr \alpha & {y + \beta } & 1 \cr \beta & 1 & {y + \alpha } \cr } } \right|$$$ is equal to