CBSE BoardEnglish MediumSTD 10MathsReal Numbers5 Marks
Question
Prove that $\big(5-2\sqrt3\big)$ is an irrational number.
✓
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
$\Rightarrow \frac{37-x^2}{20}$ is a rational number
$\Rightarrow \sqrt{3}$ is a rational number
But $\sqrt{3}$ is an irrational number, which is a contradiction.
Hence, our assumption is wrong.
Thus, $(5-2 \sqrt{3})$ is an irrational number.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.