Examine whether the operation * defines on R by a*b=ab+1 is (i) a binary or not. (ii) if a binary operation, is it associative or not?

Given operation a*b=ab+1,
If any operation is a binary operation then it must follow the closure property.
Let $\mathrm{a}\in \mathrm{R}, \mathrm{b}\in \mathrm{R}$
then ${\mathrm{a}}^{\ast }\mathrm{b}\in \mathrm{R}$
also, $\mathrm{ab}+1\in \mathrm{R}$

i.e. ${\mathrm{a}}^{\ast }\mathrm{b}\in \mathrm{R}$
so * on R satisfies the closure property
Now if this binary operation satisfies associative law then
$\begin{array}{rcl}\left(\mathrm{a}*\mathrm{b}\right)& =& *\mathrm{c}=\mathrm{a}*\left(\mathrm{b}*\mathrm{c}\right)\\ \left(\mathrm{a}*\mathrm{b}\right)*\mathrm{c}& =& \left(\mathrm{db}+1\right)*\mathrm{c}\\ & =& \left(\mathrm{ab}+1\right)\mathrm{c}+1\\ & =& \mathrm{abc}+\mathrm{c}+1\\ & & \\ \mathrm{a}*\left(\mathrm{b}*\mathrm{c}\right)& =& \mathrm{a}*\left(\mathrm{bc}+1\right)\\ & =& \mathrm{a}\left(\mathrm{bc}+1\right)+1\\ & =& \mathrm{abc}+\mathrm{a}+1\\ \therefore \left(\mathrm{a}*\mathrm{b}\right)*\mathrm{c}& \ne & \mathrm{a}*\left(\mathrm{b}*\mathrm{c}\right)\end{array}$
i.e., * operation does not follow associative law.
Therefore, (i)  the operation * defines on R by a*b=ab+1 is a binary, but 
                   (ii) it is not associative.