A function $$f:\,\,{N^ + } \to {N^ + },$$ defined on the set of positive integers $${N^ + },$$ satisfies the following properties: $$$\eqalign{ & f\left( n \right) = f\left( {n/2} \right)\,\,\,\,if\,\,\,\,n\,\,\,\,is\,\,\,\,even \cr & f\left( n \right) = f\left( {n + 5} \right)\,\,\,\,if\,\,\,\,n\,\,\,\,is\,\,\,\,odd \cr} $$$

Let $$R = \left\{ i \right.|\exists j:f\left( j \right) = \left. i \right\}$$ be the set of distinct values that $$f$$ takes. The maximum possible size of $$R$$ is _____________________.

$$\mathop {\lim }\limits_{x \to 4} {{\sin \left( {x - 4} \right)} \over {x - 4}} = \_\_\_\_\_\_\_.$$

Consider the following experiment.
Step1: Flip a fair coin twice.
Step2: If the outcomes are (TAILS, HEADS) then output $$Y$$ and stop.
Step3: If the outcomes are either (HEADS, HEADS) or (HEADS, TAILS), then output $$N$$ and stop.
Step4: If the outcomes are (TAILS, TAILS), then go to Step 1.

The probability that the output of the experiment is $$Y$$ is (up to two decimal places) _____________.

A probability density function on the interval $$\left[ {a,1} \right]$$ is given by $$1/{x^2}$$ and outside this interval the value of the function is zero. The value of $$a$$ is _________.