The polymer obtained from the following reaction is :

Let $$f\left( x \right) = \left\{ {\matrix{ { - 1} & { - 2 \le x < 0} \cr {{x^2} - 1,} & {0 \le x \le 2} \cr } } \right.$$ and

$$g(x) = \left| {f\left( x \right)} \right| + f\left( {\left| x \right|} \right).$$

Then, in the interval (–2, 2), g is :

The value of r for which ²⁰C_r ²⁰C₀ + ²⁰C_r$$-$$1 ²⁰C₁ + ²⁰C_r$$-$$2 ²⁰C₂ + . . . . .+ ²⁰C₀ ²⁰C_r  is maximum, is

The value of the integral $$\int\limits_{ - 2}^2 {{{{{\sin }^2}x} \over { \left[ {{x \over \pi }} \right] + {1 \over 2}}}} \,dx$$ (where [x] denotes the greatest integer less than or equal to x) is