Chapter 1 · Real Numbers

NCERT Class 10 Maths Exercise 1.2

All 3 questions of NCERT Class 10 Maths Exercise 1.2 (Real Numbers), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   Prove ​that $\sqrt{5}$ is ​irrational.
   Show answer

   The assumption that $\sqrt{5}=\frac{p}{q}$ (in lowest terms) forces $5$ to divide both $p$ and $q$, contradicting that they are coprime. Hence $\sqrt{5}$ is irrational.

   Step-by-step solution →
2. Q2
   Prove ​that $3+2\sqrt{5}$ ​is irrational.
   Show answer

   If $3+2\sqrt{5}$ were rational, then $\sqrt{5}=\frac{p-3q}{2q}$ would also be rational, contradicting the known irrationality of $\sqrt{5}$. Hence $3+2\sqrt{5}$ is irrational.

   Step-by-step solution →
3. Q3
   Prove ​that the ​following ​are ​irrationals:

   1. (i)$\frac{1}{\sqrt{2}}$
   2. (ii)$7\sqrt{5}$ ​
   3. (iii)$6+\sqrt{2}$
   Show answer

   Each assumption that the number is rational leads to $\sqrt{2}$ or $\sqrt{5}$ being rational, which is false. Hence all three are irrational.

   Step-by-step solution →