If $${x^m}.{y^n} = {\left( {x + y} \right)^{m + n}},$$ then $${{{dy} \over {dx}}}$$ is

The set of points where $$f\left( x \right) = {x \over {1 + \left| x \right|}}$$ is differentiable is

Let $W$ denote the words in the English dictionary. Define the relation $R$ by

$R=\{(x, y) \in W \times W \mid$ the words $x$ and $y$ have at least one letter in common}. Then, $R$ is

A solid which is not transparent to visible light and whose conductivity increases with temperature is formed by