"On the set of ℝ suppose operation * is defined by a * b = 1 + ab. Is * a binary operation?"
This question is confusing. I get to see addition right there, yet the variables a and b are multiplied. I don't know what operation the * stands for.
your confusion comes from seeing * and thinking ordinary multiplication.
In this problem it's not.
think of * in this case as *(a,b) = 1 + ab where a and b are mutliplied using ordinary real multiplication
for example 2*5 = 1 + (2)(5) = 11
* takes two real arguments and returns a real result so yes it is a binary operator on the real numbers