MCQ
If a is rational and $\sqrt{\text{b}}$ is irrational ,than $a +\sqrt{\text{b}}$ is :
- Aan integer
- Ba natural number
- Ca rational number
- ✓an irrational number
✓
Answer
Correct option: D.
an irrational number
let a be rational and $\sqrt{\text{b}}$ is irrational.
If possible let $\text{a}+\sqrt{\text{b}}$ be rational .
then $\text{a}+\sqrt{\text{b}}$ is rational and a is rational.
$\Rightarrow\Big[\Big(\text{a}+\sqrt{\text{b}}\Big)-\text{a}\Big]$ is rational $[$Difference of two rationals is rationals$]$
$\Rightarrow\sqrt{\text{b}}$ is rational.
This contradicts the fact that $\sqrt{\text{b}}$ is irrational.
The contradication arises by assuming that $\text{a}+\sqrt{\text{b}}$ is rational.
Therefore, $\text{a}+\sqrt{\text{b}}$ is irrational .
If possible let $\text{a}+\sqrt{\text{b}}$ be rational .
then $\text{a}+\sqrt{\text{b}}$ is rational and a is rational.
$\Rightarrow\Big[\Big(\text{a}+\sqrt{\text{b}}\Big)-\text{a}\Big]$ is rational $[$Difference of two rationals is rationals$]$
$\Rightarrow\sqrt{\text{b}}$ is rational.
This contradicts the fact that $\sqrt{\text{b}}$ is irrational.
The contradication arises by assuming that $\text{a}+\sqrt{\text{b}}$ is rational.
Therefore, $\text{a}+\sqrt{\text{b}}$ is irrational .
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