Let a, b $$\in$$ R, b $$\in$$ 0, Define a function

$$f(x) = \left\{ {\matrix{ {a\sin {\pi \over 2}(x - 1),} & {for\,x \le 0} \cr {{{\tan 2x - \sin 2x} \over {b{x^3}}},} & {for\,x > 0} \cr } } \right.$$.

If f is continuous at x = 0, then 10 $$-$$ ab is equal to ________________.

Let $$f:[0,3] \to R$$ be defined by $$f(x) = \min \{ x - [x],1 + [x] - x\} $$ where [x] is the greatest integer less than or equal to x. Let P denote the set containing all x $$\in$$ [0, 3] where f i discontinuous, and Q denote the set containing all x $$\in$$ (0, 3) where f is not differentiable. Then the sum of number of elements in P and Q is equal to ______________.

Consider the function


where P(x) is a polynomial such that P'' (x) is always a constant and P(3) = 9. If f(x) is continuous at x = 2, then P(5) is equal to _____________.

Let f : R $$\to$$ R be a function defined as $$f(x) = \left\{ {\matrix{ {3\left( {1 - {{|x|} \over 2}} \right)} & {if} & {|x|\, \le 2} \cr 0 & {if} & {|x|\, > 2} \cr } } \right.$$

Let g : R $$\to$$ R be given by $$g(x) = f(x + 2) - f(x - 2)$$. If n and m denote the number of points in R where g is not continuous and not differentiable, respectively, then n + m is equal to ______________.