CBSE 2025 · Region 4 · Set 1 · Q26 · 3 marks
Prove that $\displaystyle \sqrt{2}$ is an irrational number.Let $\displaystyle x$ and $\displaystyle y$ be two distinct prime numbers and $\displaystyle p=x^{2} y^{3}, q=x y^{4}$, $\displaystyle r=x^{5} y^{2}$. Find the HCF and LCM of $\displaystyle p, q$ and $\displaystyle r$. Further check if $\displaystyle \operatorname{HCF}(p, q, r) \times \operatorname{LCM}(p, q, r)=p \times q \times r$ or not.
Prove that $\displaystyle \sqrt{2}$ is an irrational number.
Let $\displaystyle x$ and $\displaystyle y$ be two distinct prime numbers and $\displaystyle p=x^{2} y^{3}, q=x y^{4}$, $\displaystyle r=x^{5} y^{2}$. Find the HCF and LCM of $\displaystyle p, q$ and $\displaystyle r$. Further check if $\displaystyle \operatorname{HCF}(p, q, r) \times \operatorname{LCM}(p, q, r)=p \times q \times r$ or not.
Marking-scheme solution
Let \(\displaystyle \sqrt{2}\) be a rational number.
\(\displaystyle \therefore \sqrt{2}=\frac{\mathbf{p}}{\mathbf{q}}\), where \(\displaystyle q \neq 0\) and let \(\displaystyle p \& q\) be co-primes.
\(\displaystyle 2 q^{2}=p^{2} \Rightarrow p^{2}\) is divisible by \(\displaystyle 2 \Rightarrow p\) is divisible by $\displaystyle 2$ ----- (i)
\(\displaystyle \Rightarrow p=2 a\), where ' \(\displaystyle a\) ' is some integer
\(\displaystyle 4 a^{2}=2 q^{2} \Rightarrow q^{2}=2 a^{2} \Rightarrow q^{2}\) is divisible by \(\displaystyle 2 \Rightarrow q\) is divisible by $\displaystyle 2$ ----- (ii)
(i)
and (ii) leads to contradiction as ' \(\displaystyle p\) ' and ' \(\displaystyle q\) ' are co-primes.
\(\displaystyle \therefore \sqrt{2}\) is an irrational number.
\[\begin{aligned}
& p=x^{2} y^{3}, q=x y^{4}, r=x^{5} y^{2} \\
& \operatorname{HCF}(p, q, r)=x y^{2} \\
& \operatorname{LCM}(p, q, r)=x^{5} y^{4} \\
& \operatorname{HCF} \times \operatorname{LCM}=x^{6} y^{6} \\
& p \times q \times r=x^{8} y^{9} \\
& \Rightarrow \operatorname{HCF}(p, q, r) \times \operatorname{LCM}(p, q, r) \neq p \times q \times r
\end{aligned}
\]
Real NumbersProving a Number IrrationalApplyshort_answermedium
More from Real Numbers
- (1+√3)^2-(1-√3)^2 is:2025 · asked 3×
- If x is the LCM of 4,6, 8 and y is the LCM of 3,5, 7 and p is the LCM of x and y, then which of the following…2025 · asked 3×
- Prove that √3 is an irrational number. OR State true or false for each of the following statements and…2025 · asked 3×
- √0.4 is a/an2025 · asked 3×
- (√3+2)^2+(√3-2)^2 is a/an2025 · asked 3×
- Prove that √5 is an irrational number. OR Let p, q and r be three distinct prime numbers. Check whether p · q…2025 · asked 3×
- Find the smallest number which is divisible by both 644 and 462. OR Two numbers are in the ratio 4: 5 and…2025 · asked 3×
- Prove that √3 is an irrational number.2025 · asked 3×
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.