If $f: Z \rightarrow N$ is defined by

$$ f(n)=\left\{\begin{array}{cll} 2 n, & \text { if } & n>0 \\ 1, & \text { if } & n=0, \text { then } f \text { is } \\ -2 n-1, & \text { if } & n<0 \end{array}\right. $$

If $\frac{x^5-5}{x^3+x^2}=f(x)+\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x+1}$, then the larger value of $K$ for which $f(K)+A+B+C=1$, is

Given that for any $n \in \mathbf{N}$ there exist an odd integer $q$ and a non-negative integer $r$ such that, $n$ can be written uniquely as $n=q \times 2^r$.

Let $f: \mathbf{N} \rightarrow \mathbf{N} \times \mathbf{N}$ be function defined by $f(n)=\left(r+1, \frac{q+1}{2}\right)$. Then,

2. If $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined by $f(x)=x+2|x+1|+2|x-1|$, then the element in the co-domain, which has unique pre image in the domain is