NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2

Irrational numbers are numbers that cannot be written in the form p/q, where p and q are integers and q ≠ 0. NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2 connect Real Numbers with proof by contradiction, irrational square roots and the Fundamental Theorem of Arithmetic.

Chapter 1 Real Numbers first explains prime factorisation and then uses it to prove why numbers such as √2, √3 and √5 are irrational. Exercise 1.2 focuses on short proof-based questions where students prove root expressions are irrational. NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2 cover all three questions in textbook order, including prove root 5 is irrational, prove 3 + 2 root 5 is irrational, and prove irrational numbers Class 10 questions involving 1/√2, 7√5 and 6 + √2. The textbook uses the Fundamental Theorem of Arithmetic Class 10 to support these irrationality proofs.

Key Takeaways

• Irrational Number: It cannot be written as p/q, where p and q are integers and q ≠ 0.
• Proof Method: Most questions use proof by contradiction.
• Prime Root Result: If p is prime, then √p is irrational.
• Rational + Irrational: The sum of a rational number and an irrational number is irrational.

NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2 Structure 2026

NCERT Class 10 Maths Chapter 1 Exercise 1.2 Solutions

Exercise 1.2 is based on irrational numbers Class 10. The main idea is to assume that the given expression is rational and then show that this assumption makes a known irrational number rational, which is impossible.

Class 10 Real Numbers Solutions for Irrational Numbers

The textbook explains that a number is irrational if it cannot be written as p/q, where p and q are integers and q ≠ 0. It also proves √2 and √3 irrational before Exercise 1.2.

Q1. Prove that √5 is irrational.

To prove that √5 is irrational, assume the opposite.

Assume:

√5 is rational.

Then √5 can be written as:

√5 = a/b

where a and b are coprime integers and b ≠ 0.

Now square both sides:

5 = a²/b²

So:

a² = 5b²

This means:

5 divides a²

Since 5 is prime, if 5 divides a², then 5 divides a.

So, let:

a = 5c

for some integer c.

Substitute a = 5c in a² = 5b²:

(5c)² = 5b²

25c² = 5b²

b² = 5c²

This means:

5 divides b²

So:

5 divides b

Now, 5 divides both a and b.

This contradicts the fact that a and b are coprime.

Therefore, the assumption that √5 is rational is false.

Answer:

√5 is irrational.

NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2: Prove 3 + 2 Root 5 Is Irrational

This question uses the result from Q1. Since √5 is irrational, any expression that would make √5 rational leads to a contradiction.

Q2. Prove that 3 + 2√5 is irrational.

To prove that 3 + 2√5 is irrational, assume the opposite.

Assume:

3 + 2√5 is rational.

Let:

3 + 2√5 = r

where r is rational.

Now subtract 3 from both sides:

2√5 = r - 3

Divide both sides by 2:

√5 = (r - 3)/2

Since r is rational, r - 3 is rational.

So:

(r - 3)/2 is rational.

Therefore:

√5 is rational.

This contradicts the fact that √5 is irrational.

Therefore, the assumption that 3 + 2√5 is rational is false.

Answer:

3 + 2√5 is irrational.

NCERT Class 10 Maths Chapter 1 Exercise 1.2 Solutions for Related Irrational Expressions

Question 3 asks students to prove three expressions irrational. Each part uses a known irrational number and shows that assuming the expression rational gives a contradiction.

Q3. Prove that the following are irrationals.

Q3(i). 1/√2

To prove that 1/√2 is irrational, assume the opposite.

Assume:

1/√2 is rational.

Let:

1/√2 = r

where r is rational and r ≠ 0.

Now take reciprocal on both sides:

√2 = 1/r

Since r is rational and non-zero:

1/r is rational.

Therefore:

√2 is rational.

This contradicts the known result that √2 is irrational.

Therefore, the assumption that 1/√2 is rational is false.

Answer:

1/√2 is irrational.

Q3(ii). 7√5

To prove that 7√5 is irrational, assume the opposite.

Assume:

7√5 is rational.

Let:

7√5 = r

where r is rational.

Divide both sides by 7:

√5 = r/7

Since r is rational:

r/7 is rational.

Therefore:

√5 is rational.

This contradicts the result that √5 is irrational.

Therefore, the assumption that 7√5 is rational is false.

Answer:

7√5 is irrational.

Q3(iii). 6 + √2

To prove that 6 + √2 is irrational, assume the opposite.

Assume:

6 + √2 is rational.

Let:

6 + √2 = r

where r is rational.

Subtract 6 from both sides:

√2 = r - 6

Since r is rational:

r - 6 is rational.

Therefore:

√2 is rational.

This contradicts the known result that √2 is irrational.

Therefore, the assumption that 6 + √2 is rational is false.

Answer:

6 + √2 is irrational.

Real Numbers Class 10: Concepts Used in Exercise 1.2

Exercise 1.2 uses the definition of irrational numbers and earlier results from the chapter. The textbook proves √2 and √3 irrational using contradiction and prime divisibility, then asks students to apply the same logic to new expressions.

Irrational Numbers Class 10

An irrational number cannot be expressed as a ratio of two integers.

Copy-friendly definition:

A number is irrational if it cannot be written as p/q, where p and q are integers and q ≠ 0.

Examples:

√2

√3

√5

Fundamental Theorem of Arithmetic Class 10

The Fundamental Theorem of Arithmetic says that every composite number can be expressed as a product of primes in a unique way.

This helps prove results like:

√2 is irrational

√3 is irrational

√5 is irrational

Prove Root 5 Is Irrational

The proof of √5 follows the same pattern as the textbook proof for √2.

Copy-friendly proof flow:

Assume √5 = a/b

Then:

a² = 5b²

So:

5 divides a²

Therefore:

5 divides a

Let:

a = 5c

Then:

b² = 5c²

So:

5 divides b

This contradicts that a and b are coprime.

Therefore:

√5 is irrational.

Prove 3 + 2 Root 5 Is Irrational

Use the result that √5 is irrational.

Copy-friendly proof flow:

Assume 3 + 2√5 is rational.

Let:

3 + 2√5 = r

Then:

√5 = (r - 3)/2

This makes √5 rational, which is impossible.

Therefore:

3 + 2√5 is irrational.

Class 10 Maths Real Numbers Solutions: Proof Pattern for Exercise 1.2

All questions in Exercise 1.2 can be solved using one common proof pattern.

Step 1: Assume the Given Number Is Rational

Start by assuming the opposite of what you need to prove.

Example:

Assume 6 + √2 is rational.

Step 2: Rearrange to Isolate the Irrational Part

Move rational terms to the other side.

Example:

6 + √2 = r

√2 = r - 6

Step 3: Use Rational Number Properties

The difference, product or quotient of rational numbers is rational, as long as division by zero is avoided.

So:

r - 6 is rational.

Step 4: Reach a Contradiction

If the rearrangement says √2 or √5 is rational, it contradicts the known result.

Conclusion:

The original expression is irrational.

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