RD Sharma Class 10 Solutions Chapter 1 - Real Numbers

RD Sharma Class 10 Solutions Chapter 1: Real Numbers is a key chapter in the CBSE Class 10 Maths syllabus, focusing on understanding real numbers and their properties. Topics like rational and irrational numbers, Euclid's division algorithm, and the number system are introduced, forming the foundation for algebra and calculus.

RD Sharma Class 10 Solutions provide step-by-step solutions, helping students better understand problem-solving methods. Students can also access the RD Sharma Class 10 Book PDF Free Download with Solutions online for easy reference. Additionally, RD Sharma Class 10 Chapter 1 PDF Questions offer ample practice material to strengthen your grasp on the concepts.

Whether preparing for exams or building a solid foundation, RD Sharma’s solutions are a reliable resource, guiding students through real numbers while enhancing their problem-solving skills. Use these materials to unlock your potential and excel in mathematics.

RD Sharma Class 10 Chapter 1 PDF with Solutions

RD Sharma Class 10 Chapter 1 PDF with Solutions is a great resource for students looking to understand real numbers and related concepts. This chapter covers everything from rational and irrational numbers to Euclid’s division algorithm, offering clear, step-by-step solutions to help you master the topic. It also includes helpful RD Sharma Class 10 Chapter 1 PDF Questions for practice, allowing you to test your understanding and improve problem-solving skills.

By downloading the RD Sharma Class 10 Chapter 1 Questions PDF, you get access to extra questions and examples that will make learning easier and more engaging. This PDF is a valuable tool for both exam preparation and building a solid foundation in mathematics.

RD Sharma Class 10 Chapter 1 – Real Numbers Important Questions with Solutions

1. If a and b are odd positive integers and a > b, prove that one of (a+b)/2 or (a−b)/2 is odd and other is even.

Solution: Let a = 2m + 1 and b = 2n + 1. Then a + b = 2(m + n + 1), so (a+b)/2 = m + n + 1 (an integer). And a − b = 2(m − n), hence (a−b)/2 = m − n (also an integer). From here, one will be odd and the other even.

2. Prove the product of 3 consecutive positive integers is divisible by 6.

Solution: Let numbers be n−1, n, n+1. Their product = (n−1)n(n+1). Among three consecutive numbers, one is divisible by 2, and one by 3. So, divisible by 6.

3. Show any positive odd integer is of form 4q+1 or 4q+3.

Solution: Any number can be written as 4q, 4q+1, 4q+2, or 4q+3. The odd ones are 4q+1 or 4q+3.

4. Prove n³ − n is divisible by 6.

Solution: n³ − n = n(n+1)(n−1), product of 3 consecutive numbers, divisible by 6.

5. Find HCF of 336 and 54 using Euclid’s algorithm.

Solution: 336 ÷ 54 = 6 rem 12 → 54 ÷ 12 = 4 rem 6 → 12 ÷ 6 = 2 rem 0. So HCF = 6.

6. If a = 6q+5 (positive integer), prove it's of form 3k+2.

Solution: a = 6q+5 = 3(2q+1)+2, so it's of the form 3k+2.

7. Prove square of any integer is of form 3m or 3m+1.

Solution: Take a = 3k, 3k+1, 3k+2. Squares are: (3k)² = 9k² = 3(3k²), (3k+1)² = ..., (3k+2)² = 9k² +12k + 4 = 3(3k² + 4k + 1) + 1. Never gives remainder 2.

8. Prove that square of any integer is of form 4q or 4q+1.

Solution: a = 2k or 2k+1 → Squares are (2k)² = 4k² = 4q, and (2k+1)² = 4k(k+1)+1 = 4q+1.

9. Prove square of integer is of form 5q, 5q+1, or 5q+4.

Solution: Try a = 5k, 5k+1, 5k+2, etc. Only possible forms of squares mod 5 are 0,1,4 → 5q, 5q+1, 5q+4.

10. Prove square of odd int is of form 8q+1.

Solution: Let n = 2k+1 ⇒ n² = 4k(k+1)+1. Since 4k(k+1) divisible by 8 ⇒ n² = 8q+1.

11. Convert 2.333... into a rational number.

Solution: x = 2.333... ⇒ 10x = 23.333... ⇒ 10x − x = 21 ⇒ x = 21/9 = 7/3

12. Prove √3 is irrational.

Solution: Assume √3 = p/q. Then 3q² = p² ⇒ p divisible by 3 ⇒ p = 3k ⇒ q also divisible by 3 ⇒ contradiction.

13. Find LCM of 120 and 144 using prime factorization.

Solution: 120 = 2³ × 3 × 5; 144 = 2⁴ × 3²; LCM = 2⁴ × 3² × 5 = 720

14. Show any odd number is of form 6q+1, 6q+3, or 6q+5.

Solution: Dividing number by 6 leaves remainder 1, 3 or 5 for odd numbers.

15. Classify √16, √20, 0.1010010001... as rational or irrational.

Solution: √16 = 4 → rational; √20 = 2√5 → irrational; 0.101001... → irrational (non-terminating, non-repeating).

16. Determine if 7/12 has a terminating decimal.

Solution: 7/12 = 7/(2² × 3) ⇒ contains prime factor other than 2 or 5 → non-terminating repeating decimal.

17. Verify: LCM × HCF = Product of numbers (e.g., 26 and 91).

Solution: HCF = 13, LCM = 182 ⇒ 13 × 182 = 2366 = 26 × 91 

18. Simplify surd √72.

Solution: √72 = √(36 × 2) = 6√2

19. Prove 5 + √2 is irrational.

Solution: Rational + irrational = irrational ⇒ 5 + √2 is irrational.

20. Rationalize: 1 / (√3 − 1)

Solution: Multiply numerator and denominator by (√3 + 1): (1 × (√3 + 1)) / ((√3 − 1)(√3 + 1)) = (√3 + 1) / (3 − 1) = (√3 + 1) / 2

RD Sharma Class 10 Chapter 1 PDF Overview

RD Sharma Class 10 Chapter 1 PDF help students to learn maths easy and fast. This chapter called "Real Numbers" is start of the book and cover important topics for exams.

Main topics in Chapter 1:

• Types of Numbers: real, rational, irrational, whole, natural, integers
• Euclid’s Division Lemma (how to divide numbers)
• Fundamental Theorem of Arithmetic (about prime factors)
• LCM and HCF questions and solutions
• Terminating and non-terminating decimals

PDF contains step by step solutions, hints, and simple explaination for all questions. It is made for CBSE syllabus and good for practice and revision at home. With this PDF, you can check answers anytime if you get stuck. Sometimes students think chapter is tough, but with solutions in PDF, become not so hard.

Why use RD Sharma Class 10 Chapter 1 PDF?

• All questions solved in easy way
• Good for exam preparation and homework
• Formula and key points explained with steps
• Can download and study offline without internet
• Free to access and help full even if you make mistake

Advantages of Solving RD Sharma Class 10 Chapter 1: Real Numbers

1. Strengthens Key Concepts: Solving RD Sharma Chapter 1 helps students grasp important topics like rational and irrational numbers, HCF, LCM, and Euclid's Division Lemma. These are the building blocks for more advanced maths topics later.
2. Improves Problem-Solving Skills: With a wide range of practice problems, including extra questions from RD Sharma, students get to work on their problem-solving and analytical skills, which are essential for tackling difficult questions.
3. Board Exam Preparation: The practice questions in the RD Sharma Class 10 Chapter 1 are aligned with the latest CBSE syllabus, helping students get ready for their Class 10 Maths board exams.
4. Boosts Speed and Accuracy: Regular practice with RD Sharma Class 10 Chapter 1 PDF Questions improves both speed and accuracy, preparing students to solve problems faster and more efficiently during exams.
5. Comprehensive Solutions: Each exercise comes with detailed solutions, helping students understand every step and resolve any confusion they may have while solving problems.
6. Prepares for Competitive Exams: A solid understanding of real numbers is also crucial for exams like JEE Mains and NEET. Working through this chapter lays the groundwork for these competitive exams too.