Class 10 Maths Chapter 1 Real Numbers Exercise 1.2 Solutions PDF

Class 10 Maths Chapter 1 Real Numbers Exercise 1.2: Class 10 Maths Chapter 1 Real Numbers Exercise 1.2 focuses on proving irrational numbers using the method of contradiction.

These NCERT solutions explain each proof in an easy, step-by-step manner so students can clearly understand why certain numbers cannot be written as a ratio of two integers. The solutions help build strong conceptual clarity and make topics like irrational numbers and proofs much easier to learn for exams.

Real Numbers Class 10 NCERT Solutions Exercise 1.2- Questions 

Exercise 1.2 helps students understand how to prove numbers irrational using simple logical steps. These real numbers class 10 exercise 1.2 solutions make the concepts easy to follow and strengthen your basics. With clear explanations, real numbers exercise 1.2 becomes simpler to practice and score well in exams.

Q1. Prove that √5 is irrational.

Solution:

Assume, for contradiction, that √5 is rational.
So,
√5 = a/b
where a and b are integers with b ≠ 0 and the fraction is in lowest form.

Square both sides:

5 = a² / b²
⇒ a² = 5b²

This means a² is divisible by 5, so a is also divisible by 5.
Let a = 5k.

Now substitute:

a² = (5k)² = 25k²
So,
25k² = 5b²
⇒ b² = 5k²
⇒ b is also divisible by 5

This means both a and b are divisible by 5, which contradicts our assumption that a/b is in lowest form.

Hence, √5 is irrational.

Q2. Prove that 3 + √25 is irrational.

Solution:

We know:

√25 = 5

So,
3 + √25 = 3 + 5 = 8

And 8 is a rational number.

Therefore, the statement “3 + √25 is irrational” is false.
It is actually rational.

(If the question meant 3√25, then that is also rational because 3√25 = 3×5 = 15.)

Q3. Prove that the following numbers are irrational:

(i) 1/√2

Solution:

Assume 1/√2 is rational.

This implies √2 is rational (because reciprocal of a rational is rational).

But √2 is a well-known irrational number.

So our assumption is false.

Hence, 1/√2 is irrational.

(ii) √7 + √5

Solution:

Assume √7 + √5 is rational.

Let
√7 + √5 = r (r is rational)

Then
√7 = r – √5

Right side is a difference of a rational and irrational number, which is irrational.

But left side (√7) is irrational, so both sides match.

Now square √7 + √5:

(√7 + √5)² = 7 + 5 + 2√35
                   = 12 + 2√35

If √7 + √5 were rational, then √35 must be rational, which is false.

Hence, √7 + √5 is irrational.

(iii) √6 + √2

Solution:

Assume √6 + √2 is rational.

Let
√6 + √2 = r (r is rational)

Then
√6 = r – √2

Right side is irrational (rational – irrational = irrational).

But left side (√6) is irrational, so proceed further.

Square the expression:

(√6 + √2)² = 6 + 2 + 2√12
                   = 8 + 4√3

If √6 + √2 were rational, 4√3 would also be rational.
⇒ √3 would be rational, which is false.

Thus √6 + √2 is irrational.

Class 10 Maths Real Numbers Exercise 1.2 Download PDF 

You can access the Class 10 Maths Chapter 1 Real Numbers Exercise 1.2 PDF from the link given below. This PDF provides clear, step-by-step solutions to all questions, helping students understand proofs related to irrational numbers with ease. It is perfect for quick revision, homework support, and building strong conceptual clarity before exams. Click the link below to view or download the complete Exercise 1.2 Solutions and enhance your preparation.

NCERT Solutions for Class 10 Maths PDF 

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