MCQ
If a is rational and $\sqrt{b}$ is irrational, then $a+\sqrt{b}$ is:
- ✓an irrational number
- Ban integer
- Ca natural number
- Da rational number
✓
Answer
Correct option: A.
an irrational number
(A) an irrational number
Explanation: Let a be rational and $\sqrt{b}$ is irrational.
If possible let $a+\sqrt{b}$ be rational.
Then $a+\sqrt{b}$ is rational and a is rational.
$\Rightarrow[(a+\sqrt{b})-a]$ is rational [Difference of two rationals is rational]
$\Rightarrow \sqrt{b}$ is rational.
The contradiction arises by assuming that $a+\sqrt{b}$ is rational.
Therefore, $a+\sqrt{b}$ is irrational.
Explanation: Let a be rational and $\sqrt{b}$ is irrational.
If possible let $a+\sqrt{b}$ be rational.
Then $a+\sqrt{b}$ is rational and a is rational.
$\Rightarrow[(a+\sqrt{b})-a]$ is rational [Difference of two rationals is rational]
$\Rightarrow \sqrt{b}$ is rational.
The contradiction arises by assuming that $a+\sqrt{b}$ is rational.
Therefore, $a+\sqrt{b}$ is irrational.
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