A binary operation ∗ with identity element zero is defined on the set R of real numbers by p∗q=p+q+3pq. For what value of p does the operation have no inverse?
Worked solution (try it first)
The identity is 0, so the inverse q of p satisfies p∗q=0, that is p+q+3pq=0.
Collect the q terms: q(1+3p)=−p.
Divide by 1+3p: q=1+3p−p.
This has no value when the bottom is zero: 1+3p=0, so 3p=−1 and p=−31, option B.