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Mathematics · 2023 · 3 marks
CBSE 2023 · Region 6 · Set 1 · Q27
Find the HCF and LCM of $\displaystyle 26$, $\displaystyle 65$ and $\displaystyle 117$, using prime factorisation.Prove that $\displaystyle \sqrt{2}$ is an irrational number.
Find the HCF and LCM of $\displaystyle 26$, $\displaystyle 65$ and $\displaystyle 117$, using prime factorisation.
Prove that $\displaystyle \sqrt{2}$ is an irrational number.
Marking-scheme solution
\[\left.\begin{array}{l}
26=13 \times 2 \\
65=13 \times 5 \\
117=13 \times 3 \times 3
\end{array}\right\}
\]
\[\therefore \mathrm{HCF}=13
\]
\[\mathrm{LCM}=13 \times 2 \times 3 \times 5 \times 3=1170
\]
Let \(\displaystyle \sqrt{2}\) be a rational number.
\[\therefore \sqrt{2}=\frac{\mathbf{p}}{\mathbf{q}} \text {, where } \mathrm{q} \neq 0 \text { and let } \mathrm{p} \text { \& } \mathrm{q} \text { be co-primes. }
\]
\[2 q^{2}=p^{2} \Rightarrow p^{2} \text { is divisible by } 2 \Rightarrow p \text { is divisible by } 2
\]
\[\Rightarrow p=2 a \text {, where 'a' is some integer }
\]
\[4 a^{2}=2 q^{2} \Rightarrow q^{2}=2 a^{2} \Rightarrow q^{2} \text { is divisible by } 2 \Rightarrow q \text { is divisible by } 2
\]
\[\Rightarrow q=2 b \text {, where 'b' is some integer }
\]
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CBSE Class 10 Mathematics past-paper question from the 2023board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.