### NCERT Solutions For Class 10 Maths Chapter – 1 Exercise – 1.3

- by Dev Bhardwaj

*NCERT Solutions For Class 10 Math Chapter – 1 Exercise – 1.3*

* *

**Q1. Prove that ****√****5 is irrational.**

**Solution:-**

** **

Let take √5 as irrational number.

If a and b are two co – prime and b is not equal to 0.

We can write√5 = a/b

b√5 = a

To remove root , squaring on both sides , we get

5b^{2} = a^{2}

Therefore , 5 divides a^{2} and according to theorem of rational number , for any prime number p which is divides a^{2} then it will divides a also.

That means 5 will divides a so we can write.

A = 5c

Putting value of a in equation (i) we get

5b^{2} = (5c)^{2}

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

Divide

B2/5 = c^{2}

Similarly , we get that b will divide by 5.

And we have already get that a is divisible by 5.

But a and b are co – prime number so it contradicts.

Hence √5 is not a rational number , it is irrational.

**Q2. Prove that 3 +2****√****5 is irrational.**

**Solution:-**

** **

Let take that 3 + 2√5 is a irrational number.

So we can write this number as

3 + 2√5 = a/b

Here a and b are two co – prime number and b is not equal to 0.

Subtract 3 both sides we get

2√5 = a/b – 3

2√5 = (a – 3b)/b

Now divide by 2, we get

√5 = (a – 3b)/2b

Here a and b are integers so (a – 3b)/2b is a rational number so √5 should be a rational number. But √5 is a irrational number so it contradicts.

Hence , 3 + 2√5 is a irrational number.

**Q3. Prove that the following are irrationals:**

**1/****√****2**

**Solution:-**

** **

Let take that 1/√2 is a rational number.

So we can write this number as

1/√2 = a/b

Here a and b are two co – prime number and b is not equal to 0.

Multiply by √2 both sides we get

1 = (a√2)/b

Now multiply by b.

B = a√2

Divide by a we get

B/a = √2

Here a and b are integer so b/a is a rational number so √2 should be a rational number But √2 is a irrational number so it contradicts.

Hence 1/√2 is a irrational number.

- Let take that 7√5 is rational number.

So we can write this number as

7/√5 = a/b

Here a and b are two co – prime number and b is not equal to 0.

Divide by 7 we get

√5 = a/(7b)

Here a and b are integer so a/7b is a rational numbers so √5 should be a rational number but √5 is irrational number so it contradicts.

Hence , 7√5 is a irrational number.

- Let take that 6 + √2 is a rational number.

So we can write this number as

6 + √2 = a/b

Here a and b are two co – prime number and b is not equal to 0.

Subtract 6 both sides we get

√2 = a/b – 6

√2 (a – 6b)/b

Here a and b are integer so ( a – 6b)/b is a rational number so √2 should be rational number.

But √2 is a irrational number so it contradicts.

Hence 6 + √2 is a irrational number.

NCERT Solutions For Class 10 Math Chapter – 1 Exercise – 1.3 Q1. Prove that √5 is irrational. Solution:- …

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