An operation * is defined on the set, R, of real numbers by (p * q = p + q + 2pq). If the identity element is 0, find the value of p for which the operation has no inverse.
-
A.
(frac{-1}{2}) -
B.
(0) -
C.
(frac{2}{3}) -
D.
(2)
Correct Answer: Option A
Explanation
Given the formula for p * q as: (p + q + 2pq) and its identity element is 0, such that if, say, t is the inverse of p, then
(p * t = 0), then (p + t + 2pt = 0 therefore p + (1 + 2p)t = 0)
(t = frac{-1}{1 + 2p}) is the formula for the inverse of p and is undefined on R when
(1 + 2p) = 0) i.e when (2p = -1; p = frac{-1}{2}).