maths

Question
If x and y are irrational, which of the following expression must be irrational?
A. 2x
B. x + 2y
C. 2xy
D. All of the above.

Answer
A.

Explanation
Counterexample for B.
Let x = 2√3, y = -√3, both x and y are irrational,
then x + 2y = 2√3 + 2(-√3) = 0 is rational.

Counterexample for C.
Let x = y = √2, both x and y are irrational,
then 2xy = 2√2√2 = 4 is rational.

Then, clearly D is also wrong.

By elimination, B, C, D are wrong, the answer is A.

With more explanation:
Suppose 2x is rational, then 2x = p/q where p and q are integers.
Then, x = p/(2q).
Here p and 2q are both integers, so x should be rational.
We have "2x is rational" ⇒ "x is rational".
Therefore, by proof of contrapositive,
we have "x is irrational" ⇒ "2x is irrational".

Therefore, A is correct (that is, must be irrational).

must be irrational: b, c ,d
-------------------------------------------------a is correct -------------------------------------

i think that the answer may be D

i think is D

maybe