RD Chapter 1- Real Numbers Ex-1.1 |
RD Chapter 1- Real Numbers Ex-1.2 |
RD Chapter 1- Real Numbers Ex-1.3 |
RD Chapter 1- Real Numbers Ex-1.4 |
RD Chapter 1- Real Numbers Ex-1.6 |
RD Chapter 1- Real Numbers Ex-VSAQS |
RD Chapter 1- Real Numbers Ex-MCQS |

Show that the following numbers are irrational

(i) 1/√2

(ii) 7 √5

(iii) 6 + √2

(iv) 3 – √5

**Answer
1** :

But itcontradics that because √5 is irrational

3 – √5 is irrational

Prove that following numbers are irrationals :

(i) 2/√7

(ii) 3/2√5

(iii) 4 + √2

(iv) 5 √2

**Answer
2** :

5 √2 is an irrational number

**Answer
3** :

Let 2 – √3 is not an irrational number

√3 is a rational number

But it contradicts because √3 is an irrational number

2 – √3 is an irrational number

Hence proved.

**Answer
4** :
Let 3 + √2 is a rational number

and √2 is irrational

But our suppositon is wrong

3 + √2 is an irrational number

**Answer
5** :

Let 4 – 5 √2 is not are irrational number

and let 4 – 5 √2 is a rational number

and 4 – 5 √2 = a/b where a and b are positive prime integers

√2 is a rational number

But √2 is an irrational number

Our supposition is wrong

4 – 5 √2 is an irrational number

**Answer
6** :

Let 5 – 2 √3 is a rational number

Let 5 – 2 √3 = ab where a and b are positive integers

and √3 is a rational number

Our supposition is wrong

5 – 2 √3 is a rational number

**Answer
7** :

Let 2 √3 – 1 is not an irrational number

and let 2 √3 – 1 a ration number

and then 2 √3 – 1 = a/b where a, b positive prime integers

√3 is a rational number

But √3 is an irrational number

Our supposition is wrong

2 √3 – 1 is an irrational number

**Answer
8** :

Let 2 – 3 √5 is not an irrational number and let 2 – 3 √5 is a rational number

Let 2 – 3 √5 = a/b where a and b are positive prime integers

⟹2b−a3b=√5

√5 is a rational

But √5 is an irrational number

Our supposition is wrong

2 – 3 √5 is an irrational

**Answer
9** :

Let √5 + √3 is a rational number

and let √5 + √3 = a/b where a and b are co-primes

√3 is a rational number

But it contradics as √3 is irrational number

√5 + √3 is irrational

**Answer
10** :

Let us suppose that √2 + √3 is rational.

Let √2 + √3 = a, where a is rational.

Therefore, √2 = a – √3

Squaring on both sides, we get

which is a contradiction as the right hand side is a rational number while √3 is irrational.

Hence, √2 + √3 is irrational.

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