Maharashtra BoardEnglish MediumSTD 10MathsReal Numbers5 Marks
Question
Prove that $\big(5-2\sqrt3\big)$ is an irrational number.
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Answer
Let $\text{x}=5-2\sqrt3$ be a rational number.
$\text{x}=5-2\sqrt3$
$\Rightarrow\text{x}^2=\big(5-2\sqrt3\big)^2$
$\Rightarrow\text{x}^2=(5)^2+\big(2\sqrt3\big)^2-2(5)\big(2\sqrt3\big)$
$\Rightarrow\text{x}^2=25+12-20\sqrt3$
$\Rightarrow\text{x}^2-37=-20\sqrt3$
$\Rightarrow\frac{37-\text{x}^2}{20}=\sqrt3$
Since $x$ is a rational number, $x^2$ is also a rational number.
$\Rightarrow 37 − x^2$ is a rational number
$\Rightarrow\frac{37-\text{x}^2}{20}$ is a rational number
$\Rightarrow\sqrt3$ is a rational number
But $\sqrt3$ is an irrational number, which is a contradiction.
Hence, our assumption is wrong.
Thus, $\big(5-2\sqrt3\big)$ is an irrational number.
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