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

- Real Numbers Class 10 Ex 1.1
- Real Numbers Class 10 Ex 1.2
- Real Numbers Class 10 Ex 1.3
- Real Numbers Class 10 Ex 1.4

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

**Ex 1.3 Class 10 Maths Question 1.**Prove that âˆš5 is irrational.

**Solution:**

Let âˆš5 =Â Â be a rational number, where p and q are co-primes and q â‰ 0.

Then, âˆš5q = p => 5q

^{2}=p

^{2}â‡’Â p

^{2}Â â€“ Sq

^{2}Â Â Â â€¦ (i)

Since 5 divides p

^{2}, so it will divide p also.

Let p = 5r

Then p

^{2}Â â€“ 25rÂ

^{2}Â Â Â [Squaring both sides]

â‡’ 5q

^{2}Â = 25r

^{2}Â Â Â [From(i)]

â‡’ q

^{2}Â = 5r

^{2}Since 5 divides q

^{2}, so it will divide q also. Thus, 5 is a common factor of both p and q.

This contradicts our assumption that âˆš5 is rational.

Hence, âˆš5 is irrational. Hence, proved.

**Ex 1.3 Class 10 Maths Question 2.**

Show that 3 + âˆš5 is irrational.**Solution:**Let 3 + 2âˆš5 = be a rational number, where p and q are co-prime and q â‰ 0.

Then, 2âˆš5 = â€“ 3 =

â‡’ âˆš5 =

since is a rational number,

therefore, âˆš5 is a rational number. But, it is a contradiction.

Hence, 3 + âˆš5 is irrational. Hence, proved.

**Ex 1.3 Class 10 Maths Question 3.**Prove that the following are irrational.

**Solution:**

**(i)**Let = be a rational number,

where p and q are co-prime and q â‰ 0.

Then, âˆš2 =

Since is rational, therefore, âˆš2 is rational.

But, it is a contradiction that âˆš2 is rational, rather it is irrational.

Hence, is irrational.

Hence, proved.

**(ii)** Let 7âˆš5 = be a rational number, where p, q are co-primes and q â‰ 0.

Then, âˆš5 =

Since is rational therefore, âˆš5 is rational.

But, it is a contradiction that âˆš5 is rational rather it is irrational.

Hence, 7âˆš5 s is irrational.

Hence proved.

**(iii)** Let 6 + âˆš2 = be a rational number, where p, q are co-primes and q â‰ 0.

Then, âˆš2 = â€“ 6 =

Since is rational therefore, âˆš2 is rational.

But, it is a contradiction that âˆš2 is rational, rather it is irrational.

Hence, 6 + âˆš2 is irrational.

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