✓ Board-verified
Mathematics · 2026 · 2 marks
CBSE 2026 · Region 5 · Set 2 · Q23
Prove that 4-2 √ 5 is an irrational number given that √ 5 is irrational.
Prove that $\displaystyle 4-2 \sqrt{5}$ is an irrational number given that $\displaystyle \sqrt{5}$ is irrational.
Marking-scheme solution
Sol.
Let \(\displaystyle 4-2 \sqrt{5}\) be a rational number.
\(\displaystyle \therefore 4-2 \sqrt{5}=\frac{\mathrm{a}}{\mathrm{b}}\) where a and b are integers and \(\displaystyle \mathrm{b} \neq 0\).
\(\displaystyle \sqrt{5}=\frac{4 \mathrm{~b}-\mathrm{a}}{2 \mathrm{~b}}\)
RHS is rational but LHS is an irrational which is a contradiction to our supposition. Hence \(\displaystyle 4-2 \sqrt{5}\) is an irrational number.
More from Real Numbers
- The quadratic equation x^2+x+1=0 has roots.2024
- Find the greatest 3-digit number which is divisible by 18, 24 and 36.2023
- Prove that √3 is an irrational number.2023
- The roots of the quadratic equation 4 x^2-5 x+4=0 are2024
- Three sets of Physics, Chemistry and Mathematics books have to be stacked in such a way that all the books are stored subject-wise and the height of…2025
- Prove that √5 is an irrational number.2023
- (1+√3)^2-(1-√3)^2 is:2025
- The probability for a randomly selected number out of 1,2,3,4, …, 25 to be a composite number is:2026
CBSE Class 10 Mathematics past-paper question from the 2026 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.