CBSE 2025 · Region 5 · Set 1 · Q26 · 3 marks
Prove that $\displaystyle \sqrt{3}$ is an irrational number.State true or false for each of the following statements and justify in each case :(i)$\displaystyle 2 \times 3 \times 5 \times 7+7$ is a composite number.(ii)$\displaystyle 2$ × $\displaystyle 3$ × $\displaystyle 5$ × $\displaystyle 7$ + $\displaystyle 1$ is a composite number.
Prove that $\displaystyle \sqrt{3}$ is an irrational number.
State true or false for each of the following statements and justify in each case :
(i)
$\displaystyle 2 \times 3 \times 5 \times 7+7$ is a composite number.
(ii)
$\displaystyle 2$ × $\displaystyle 3$ × $\displaystyle 5$ × $\displaystyle 7$ + $\displaystyle 1$ is a composite number.
Marking-scheme solution
Let \(\displaystyle \sqrt{3}\) be a rational number.
\[\begin{aligned}
& \therefore \sqrt{3}=\frac{\mathrm{p}}{\mathrm{q}}, \text { where } \mathrm{q} \neq 0 \text { and let } \mathrm{p} \text { \& } \mathrm{q} \text { be coprimes. } \\
& \Rightarrow 3 \mathrm{q}^{2}=\mathrm{p}^{2} \\
& \Rightarrow \mathrm{p}^{2} \text { is divisible by } 3 . \\
& \Rightarrow \mathrm{p} \text { is divisible by } 3 . \quad \text { ----- (1) }
\end{aligned}
\]
Let \(\displaystyle \mathrm{p}=3 \mathrm{a}\), where 'a' is some integer
\[\begin{aligned}
& \therefore 9 \mathrm{a}^{2}=3 \mathrm{q}^{2} \\
& \Rightarrow \mathrm{q}^{2}=3 \mathrm{a}^{2} \\
& \Rightarrow \mathrm{q}^{2} \text { is divisible by } 3 \\
& \Rightarrow \mathrm{q} \text { is divisible by } 3 \quad \text { ----- (2) }
\end{aligned}
\]
∴ $\displaystyle 3$ divides both p & q.
(1)
and ($\displaystyle 2$) leads to contradiction as p and q are coprimes.
Hence, \(\displaystyle \sqrt{3}\) is an irrational number.
(b)
True,
\[\because 2 \times 3 \times 5 \times 7+7=7 \times(2 \times 3 \times 5+1) \text { has more than two factors. }
\]
(ii)
False,
\[\because 2 \times 3 \times 5 \times 7+1=211 \text { has only two factors. }
\]
Real NumbersProving a Number IrrationalUnderstandshort_answermedium
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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.