Let x and y be rational and irrational numbers , respectively. Is x+y necessarily an irrational number ?

Options

A. True
B. False
C. Can not be determined
D. None of these

Answer

A

Explanation

Yes ,(x+y) is necessarily an irrational number .
e.g… Let `" " x=2, Y=sqrt3`
Then, ` " " x+y =2+sqrt3`
if possible, let x+y =2 `+ sqrt3` be a rational number.
Consider , ` " " a=2,+sqrt3`
On Squaring both sides, we get
`a^(2)=(2+sqrt3)^(2) " " [ "unsing identity"(a+b)^(2)=a^(2) + b^(2) + 2ab]`
` implies " " a^(2) = 2^(2) + (sqrt3)^(2))+2(2)(sqrt(3))`
`a^(2)=4 +3+4sqrt3 implies (a^(2)-7)/4=sqrt(3)`
So, a is rational `Rightarrow (a^(2) -7)/4` is rational ` implies sqrt3` is rational.
But, this contradicits the fact that `sqrt3` is an irrational number. thus, our assumption is wrong.
Hence, x+ y is an irraional number.