Short Intro
In this post, students can find complete step-by-step solutions for Class 10 Maths Chapter 1 Real Numbers Exercise 1.2. All questions are solved using simple explanations based on the latest NCERT syllabus and CBSE pattern to help students prepare effectively for board exams.
Exercise 1.2 focuses on irrational numbers and proofs using contradiction method.
Quick Information Box
| Item | Details |
|---|---|
| Board | CBSE |
| Class | 10 |
| Subject | Maths |
| Chapter | Real Numbers |
| Exercise | 1.2 |
| Topic | Irrational Numbers |
Concepts Used (Topics Covered)
- Irrational Numbers
- Proof by Contradiction
- Rational and Irrational Numbers
- Properties of Square Roots
- Fundamental Theorem of Arithmetic
The exercise is based on irrationality proofs discussed in Chapter 1.
Important Formulas
Rational Number Form
Irrational Number Concept
Questions & Step-by-step Solutions
Question 1
Prove that √5 is irrational.
Solution
Assume, to the contrary, that:
where:
- a and b are coprime integers
- b ≠ 0
Squaring both sides:
5 = a²/b²
a² = 5b²
Thus, 5 divides a².
By the theorem:
If p divides a², then p divides a.
Therefore, 5 divides a.
Let:
a = 5c
Substituting:
25c² = 5b²
5c² = b²
Thus, 5 divides b² and hence 5 divides b.
Therefore, a and b have a common factor 5.
But this contradicts the assumption that a and b are coprime.
Hence,
Final Answer:
√5 is irrational.
Question 2
Prove that 3 + 2√5 is irrational.
Solution
Assume, to the contrary, that:
3 + 2√5 is rational
Subtracting 3 from both sides:
2√5 is rational
Dividing by 2:
\sqrt{5}=\frac{\text{rational}}{2}
This implies √5 is rational.
But from Question 1, we know:
√5 is irrational.
This contradiction arises because our assumption was wrong.
Hence,
Final Answer:
3 + 2√5 is irrational.
Question 3
Prove that the following are irrational numbers.
(i) 1/√2
Solution
Assume:
\frac{1}{\sqrt{2}}
is rational.
Then:
1/√2 = a/b
Rearranging:
√2 = b/a
Thus √2 becomes rational.
But √2 is irrational.
Contradiction.
Hence,
1/√2 is irrational.
(ii) 7√5
Solution
Assume:
7√5 is rational.
Dividing by 7:
√5 becomes rational.
But √5 is irrational.
Contradiction.
Hence,
7√5 is irrational.
(iii) 6 + √2
Solution
Assume:
6 + √2 is rational.
Subtracting 6:
√2 becomes rational.
But √2 is irrational.
Contradiction.
Hence,
6 + √2 is irrational.
Common Mistakes
- Forgetting the contradiction step
- Not assuming numbers are coprime
- Incorrect rearrangement of equations
- Confusing rational and irrational numbers
Exam Tips
- Always start proofs with “Assume, to the contrary…”
- Write contradiction clearly
- Mention theorem properly
- Practice square root irrationality proofs regularly
Practice MCQs
MCQ 1
√7 is:
A. Rational
B. Irrational
C. Integer
D. Whole number
Answer:
B. Irrational
MCQ 2
Which of the following is irrational?
A. 3/5
B. 0.25
C. √11
D. 7
Answer:
C. √11
MCQ 3
The sum of a rational and irrational number is:
A. Rational
B. Irrational
C. Integer
D. Natural
Answer:
B. Irrational
MCQ 4
Which theorem is used in irrationality proofs?
A. Pythagoras Theorem
B. Fundamental Theorem of Arithmetic
C. Euclid’s Theorem
D. Binomial Theorem
Answer:
B. Fundamental Theorem of Arithmetic
11. FAQ Section
What is an irrational number?
A number that cannot be expressed in the form:
p/q
where q ≠ 0.
Is √5 rational?
No, √5 is irrational.
Which method is used in Exercise 1.2?
Proof by contradiction method.
Is Exercise 1.2 important for board exams?
Yes, irrationality proofs are frequently asked in CBSE board exams.
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