If a curve y = f(x), passing through the point (1, 2), is the solution of the differential equation,
2x²dy= (2xy + y²)dx, then $$f\left( {{1 \over 2}} \right)$$ is equal to :

Let S be the sum of the first 9 terms of the series :
{x + k$$a$$} + {x² + (k + 2)$$a$$} + {x³ + (k + 4)$$a$$}
+ {x⁴ + (k + 6)$$a$$} + .... where a $$ \ne $$ 0 and x $$ \ne $$ 1.

If S = $${{{x^{10}} - x + 45a\left( {x - 1} \right)} \over {x - 1}}$$, then k is equal to :

The set of all possible values of $$\theta $$ in the interval
(0, $$\pi $$) for which the points (1, 2) and (sin $$\theta $$, cos $$\theta $$) lie
on the same side of the line x + y = 1 is :

If the variance of the terms in an increasing A.P.,
b₁ , b₂ , b₃ ,....,b₁₁ is 90, then the common difference of this A.P. is_______.