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 Name | Class 10 Mathematics NCERT Textbook |
Chapter | Chapter 1 Real Numbers |
Exercise | Ex 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
a^{2} = 5b^{2}
∴ 5 divides a^{2}
⇒ 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} = 5b^{2}
⇒ 25c^{2} = 5b^{2}
⇒ 5c^{2} = b^{2}
⇒ 5 divides b^{2}
⇒ 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.