Let $$a = \min \{ {x^2} + 2x + 3:x \in R\} $$ and $$b = \mathop {\lim }\limits_{\theta \to 0} {{1 - \cos \theta } \over {{\theta ^2}}}$$. Then $$\sum\limits_{r = 0}^n {{a^r}{b^{n - r}}} $$ is

A particle starts at the origin and moves 1 unit horizontally to the right and reaches P₁, then it moves $${1 \over 2}$$ unit vertically up and reaches P₂, then it moves $${1 \over 4}$$ unit horizontally to right and reaches P₃, then it moves $${1 \over 8}$$ unit vertically down and reaches P₄, then it moves $${1 \over 16}$$ unit horizontally to right and reaches P₅ and so on. Let P_n = (x_n, y_n) and $$\mathop {\lim }\limits_{n \to \infty } {x_n} = \alpha $$ and $$\mathop {\lim }\limits_{n \to \infty } {y_n} = \beta $$. Then, ($$\alpha$$, $$\beta$$) is

The value of $$\mathop {\lim }\limits_{x \to {0^ + }} {x \over p}\left[ {{q \over x}} \right]$$ is

Let f : [a, b] $$ \to $$ R be differentiable on [a, b] and k $$ \in $$ R. Let f(a) = 0 = f(b). Also let J(x) = f'(x) + kf(x). Then