Question:
Which of the following is a correct statement?
(a) Sum of two irrational numbers is always irrational
(b) Sum of a rational and irrational number is always an irrational number
(c) Square of an irrational number is always a rational number
(d) Sum of two rational numbers can never be an integer
Solution:
The sum of irrational number and rational number is always irrational number.
Let a be a rational number and b be an irrational number.
Then,
$(a+b)^{2}=a^{2}+b^{2}+2 a b$
$=\left(a^{2}+b^{2}\right)+2 a b$
As $2 a b$ is irrational therefore $(a+b)^{2}$ is irrational.
Hence $(a+b)$ is irrational.
Therefore answer is $b$.
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