NCERT Solutions for Class 10 Mathematics Chapter 1 Real Numbers Ex 1.3 are part of NCERT Solutions for Class 10 Maths. Here we have given NCERT Solutions for Class 10 Mathematics Chapter 1 Real Numbers Ex 1.3.

Board |
CBSE |

Textbook |
NCERT |

Class |
Class 10 |

Subject |
Maths |

Chapter |
Chapter 1 |

Chapter Name |
Real Numbers |

Exercise |
Ex 1.3 |

Number of Questions Solved |
3 |

Category |
NCERT Solutions |

### NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Ex 1.3

**Question 1.**

**Prove that √5 is irrational.**

**Solutions:**

Let us assume that is rational.

∴ There exists co-prime integers a and b (b ≠ 0) such that

√5 = \(\frac { a }{ b }\) ⇒ √5b= 0

Squaring on both sides, we get

5b^{2}= a^{2}…… (i)

⇒ 5 divides a^{2} ⇒ 5 divides a

So, we can write a = 5c for some integer c.

From (i) and (ii)

5b^{2} = 25c^{2}

⇒ b^{2} = 5c^{2}

⇒ 5 divides b^{2}

⇒ 5 divides b

∴ 5 is a common factor of a and b.

But this contradicts the fact that a and b are co-primes.

This contradiction has arisen because of our incorrect assumption that √5 is rational.

Hence, √5 is irrational.

**Question 2.**

**Prove that 3 + 2√5 is irrational.**

**Solutions:**

Let us assume that 3 + 2√5 is rational.

∴ There exists co-prime integers a and b(b ≠ 0) such that

But this contradicts the fact that √5 is irrational.

This contradiction has arisen because of our incorrect assumption that 3 + 2√5 is rational. Hence, we conclude that 3 + 2√5 is irrational.

**Question 3.**

**Prove that the following are irrationals.**

**Solutions:**

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