NCERT Solutions For Class-10 Maths Exercise 1.3

1. Prove that √5 is irrational.

To prove that √5 is irrational, we will use a proof by contradiction.
Let us assume, for the sake of contradiction, that √5 is a rational number.
That is, √5 = x/y, where x and y are co-prime integers (i.e., they have no common factors other than 1).
simplifying we get:
y√5 = x
Now, let’s square both sides of the equation:
(y√5)² = x²
Now, square both sides: (y√5)² = x²
This simplifies to: 5y² = x² …………… (1)
From equation (1), we find that x² is divisible by 5, so x must also be divisible by 5. Let’s write x as x = 5k, where k is an integer.
Substituting this into equation (1): 5y² = (5k)²
Further simplifying: y² = 5k²
Now, y² is also divisible by 5, meaning y must be divisible by 5 as well.
However, this contradicts our initial assumption that x and y are co-prime, having no common factors other than 1.
Since our assumption that √5 is rational leads to a contradiction, we conclude that √5 is indeed irrational.
Therefore, √5 is an irrational number.

2. Prove that 3 + 2√5 + is irrational.

3. Prove that the following are irrational.

(i) 1/√2

(ii) 7√5

(iii) 6 + √2