If ƒ'(x) = tan^–1(secx + tanx), $$ - {\pi \over 2} < x < {\pi \over 2}$$,
and ƒ(0) = 0, then ƒ(1) is equal to :

If for some $$\alpha $$ and $$\beta $$ in R, the intersection of the following three places
x + 4y – 2z = 1
x + 7y – 5z = b
x + 5y + $$\alpha $$z = 5
is a line in R³, then $$\alpha $$ + $$\beta $$ is equal to :

If for x $$ \ge $$ 0, y = y(x) is the solution of the differential equation
(x + 1)dy = ((x + 1)² + y – 3)dx, y(2) = 0, then y(3) is equal to _______.

If e₁ and e₂ are the eccentricities of the ellipse, $${{{x^2}} \over {18}} + {{{y^2}} \over 4} = 1$$ and the hyperbola, $${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1$$ respectively and (e₁, e₂) is a point on the ellipse, 15x² + 3y² = k, then k is equal to :