# Class10 NCRT Real Numbers Exercise – 1.3 pdf || UP Board

# __Real Numbers__

__Exercise – 1.3__

__Real Numbers__

__Exercise – 1.3__

**1. ****Prove that**** **√5** ****is irrational.****
**

**Solution****:**

Let√5 is a rational
number.

So 5 will divide b^{2}, Therefore 5 will divide b .…………(iii)

Now, Equations (i) and (ii) show that a and b have a common factor of 5 which is
contrary to our assumption (since a and b are co-prime numbers i.e. a and b
have no common factor other than 1.)

So this contradiction proves that contrary to our imagination, √5 is an irrational number.

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

**Solution****:**

Let **3+2**√5 is rational

However,
we know that √5 is an irrational
number, So this contradiction proves that contrary to our imagination, 3+2√5** **is an irrational number.

**3. ****Prove that the following are irrationals:****
**

**(ii) ****7**√5

**Solution****:**

Let **7**√5 is rational

However,
we know that √5 is an irrational
number, So this contradiction proves that contrary to our imagination, 7√5** **is an irrational number.

**(iii) ****6+****√2**

**Solution****:**

Let **6+****√2** is rational

However,
we know that **√2** is an irrational
number, So this contradiction proves that contrary to our imagination, **6+****√2**** **is an irrational number.

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