CBSE 2025 · Region 1 · Set 1 · Q29 · 3 marks
Prove that $\displaystyle \frac{1}{\sqrt{5}}$ is an irrational number.
Marking-scheme solution
Let \(\displaystyle \frac{1}{\sqrt{5}}\) be a rational number.
\(\displaystyle \therefore \frac{1}{\sqrt{5}}=\frac{\mathrm{p}}{\mathrm{q}}\), where \(\displaystyle \mathrm{q} \neq 0\) and let p & q be the co-primes.
\(\displaystyle 5 \mathrm{p}^{2}=\mathrm{q}^{2} \Rightarrow \mathrm{q}^{2}\) is divisible by 5.
⇒ q is divisible by 5.
let \(\displaystyle \mathrm{q}=5 \mathrm{a}\), where 'a' is some integer.
\(\displaystyle 25 \mathrm{a}^{2}=5 \mathrm{p}^{2} \Rightarrow \mathrm{p}^{2}=5 \mathrm{a}^{2} \Rightarrow \mathrm{p}^{2}\) is divisible by 5.
⇒ p is divisible by $\displaystyle 5$
(i)
and (ii) leads to contradiction as p and q are coprimes.
\(\displaystyle \therefore \frac{1}{\sqrt{5}}\) is an irrational number
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CBSE Class 10 Mathematics past-paper question from the 2025board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.