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Where to find NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Exercise 1.3 to complete homework.

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Providing you CBSE NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.3 PDF will be helpful in knowing the concepts insights of the chapter. Class 10 Maths NCERT Solutions is helpful in building fundamentals.

Book NameClass 10 Mathematics NCERT Textbook
ChapterChapter 1 Real Numbers
ExerciseEx 1.3

1. Prove that √5 is irrational.

Solution

Let √5 be a rational number.

∴ We have to find two integers a and b (where, b ≠ 0 and a and b are coprime) such that

a/b = √5

⇒ a = √5.b      ... (1)

Squaring both sides, we have

a2 = 5b2

∴ 5 divides a2

⇒ 5 divides a   ...(2)

[∵ a prime number ‘p’ divides a2 then ‘p’ divides ‘a’, where ‘a’ is positive integer.]

∴ a  =  5c,  where c is an integer.

∴ Putting a = 5c in (1), we have

5c = √5. b    

or (5c)2 =  5b2

⇒ 25c2 = 5b2

⇒ 5c2 = b2

⇒ 5 divides b2

⇒ 5 divides b         ...(3)

From (2) and (3)

a and b have at least 5 as a common factor.

i.e., a and b are not coprime.

∴ Our supposition that √5 is rational is wrong.

Hence, √5 is irrational.

2. Prove that 3+2√5 is irrational.

Solution

Let 3+2√5 is rational.

∴ We can find two coprime integers ‘a’ and ‘b’ such that

[3+2√5] =  a/b,  where  b ≠ 0

⇒ (1) is a rational

⇒ √5 is a rational

But this contradicts the fact that √5 is irrational.

∴ Our supposition is wrong.

3+2√5  is an irrational.

3. Prove that the following are irrationals:

(i) 1/√2      

(ii) 7√5  

(iii) 6+√2

Solution

(i) We have

since, the division of two integers is rational.

∴ 2a/b is a rational.

From (1), √2 is a rational number which contradicts the fact that √2 is irrational.

∴ Our assumption is wrong.

Thus, 1√2 is irrational.

(ii) Let us suppose that 7√5 is rational.

Let there be two coprime integers ‘a’ and ‘b’.

such that 7√5 =  a/b , where b ≠ 0

Now, = 7√5 = a/b

⇒ √5 is a rational

This contradicts the fact that √5 is irrational.

∴ We conclude that 7√5 is irrational.

(iii) Let us suppose that 6+√2 is rational.

∴ We can find two coprime integers ‘a’ and ‘b’ (b ≠ 0), such that

6+√2  = a/b

[∵ subtraction of integers is also an integer]

[∵ Division of two integers is a rational number]

⇒ a-6b/b is a rational number.

From (1), √2 is a rational number, which contradicts the fact that √2 is an irrational number.

∴ Our supposition is wrong.

⇒ 6+ √2 is an irrational number.

These Class 10 Maths Chapter 1 Real numbers NCERT Solutions will be useful in prepare own answers by taking help.

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