If x is a rational number and y is an irrational number, then-

A. Both x + y and xy are necessarily rational

B. Both x + y and xy are necessarily irrational

C. xy is necessarily irrational, but x + y can be either rational or irrational

D. x + y is necessarily irrational, but xy can be either rational or irrational

Answer: Option D

Solution(By Examveda Team)

(a) Let x = 0 and y = $$\sqrt 2 $$
Then, x is rational and y is irrational.
∴ x + y = 0 + $$\sqrt 2 $$  = $$\sqrt 2 $$ , which is irrational
Thus, x + y is not rational
(b) Let x = 0 and y = $$\sqrt 2 $$
Then, x is rational and y is irrational
∴ xy = 0 × $$\sqrt 2 $$  = 0, which is rational
Hence, xy is not irrational
(c) As shown in (B), xy is not necessarily irrational
(d) x + y is necessary irrational. But xy can be either rational or irrational.
Hence, (D) is true.