Class 10 Maths Chapter 1 Real Numbers – Exercise 1.3 NCERT Solutions

Introduction

Exercise 1.3 focuses on Fundamental Theorem of Arithmetic and properties of irrational numbers. You’ll learn how to express numbers as a product of primes, prove irrationality of certain numbers, and solve problems using prime factorization.

Formula Used

• Fundamental Theorem of Arithmetic: Every composite number can be expressed uniquely (apart from order) as a product of prime numbers.

• Irrational Numbers Proof Technique: Assume the number is rational, express it as $\frac{p}{q}$, and derive a contradiction.

NCERT Questions with Solutions (10)

Q1. Express 140 as a product of primes.

Solution:

$$140=2\times 2\times 5\times 7$$

Q2. Express 156 as a product of primes.

Solution:

$$156=2\times 2\times 3\times 13$$

Q3. Express 3825 as a product of primes.

Solution:

$$3825=3\times 5\times 5\times 5\times 17$$

Q4. Express 5005 as a product of primes.

Solution:

$$5005=5\times 7\times 11\times 13$$

Q5. Express 7429 as a product of primes.

Solution:

$$7429=17\times 19\times 23$$

Q6. Prove that $\sqrt{5}$ is irrational.

Solution: Assume $\sqrt{5}=\frac{p}{q}$. Then $p^2=5q^2$. This implies $p^2$ divisible by 5 ⇒ $p$ divisible by 5. Let $p=5k$. Then $25k^2=5q^2$ ⇒ $q^2$ divisible by 5 ⇒ $q$ divisible by 5. Contradiction. Hence $\sqrt{5}$ is irrational.

Q7. Prove that $\sqrt{7}$ is irrational.

Solution: Similar proof as Q6. Assume rational, derive contradiction. Hence irrational.

Q8. Prove that $\sqrt{3}$ is irrational.

Solution: Same method: assume rational, contradiction arises. Hence irrational.

Q9. Prove that $\sqrt{2}$ is irrational.

Solution: Classic proof: assume rational, both numerator and denominator divisible by 2, contradiction. Hence irrational.

Q10. Prove that $\sqrt{11}$ is irrational.

Solution: Same method: assume rational, contradiction arises. Hence irrational.

FAQs (10 from NCERT)

1. Q: What is Fundamental Theorem of Arithmetic? A: Every composite number can be expressed uniquely as a product of primes.

2. Q: What is a prime number? A: A number greater than 1 with only two factors: 1 and itself.

3. Q: What is a composite number? A: A number greater than 1 that is not prime.

4. Q: What is an irrational number? A: A number that cannot be expressed as $\frac{p}{q}$.

5. Q: How do you prove irrationality? A: Assume rational, derive contradiction using divisibility.

6. Q: Is $\pi$ irrational? A: Yes, $\pi$ is irrational.

7. Q: Is $\sqrt{4}$ irrational? A: No, $\sqrt{4}=2$, which is rational.

8. Q: Why is prime factorization important? A: It helps in finding HCF, LCM, and solving divisibility problems.

9. Q: Can irrational numbers be written as decimals? A: Yes, but they are non‑terminating and non‑repeating.

10. Q: Why is this exercise important? A: It builds understanding of prime factorization and irrational numbers, key in higher mathematics.

Conclusion

Exercise 1.3 has 10 solved questions and 10 FAQs that strengthen your understanding of prime factorization and irrational numbers. This builds the foundation for number theory and proofs in Class 10 Maths.

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