class 10 maths chapter 1 exercise 1.1 solutions

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Chapter 1 Math Exercises

Chapter 1

EXERCISE 1.1

1. Express each number as a product of its prime factors:

(i) 140

Solution:

  • Divide 140 by the smallest prime number repeatedly:
  • \( 140 \div 2 = 70 \)
  • \( 70 \div 2 = 35 \)
  • \( 35 \div 5 = 7 \)
  • \( 7 \div 7 = 1 \)
  • Prime factors: \( 2 \times 2 \times 5 \times 7 = 2^2 \times 5 \times 7 \)

(ii) 156

Solution:

  • Divide 156 by the smallest prime number:
  • \( 156 \div 2 = 78 \)
  • \( 78 \div 2 = 39 \)
  • \( 39 \div 3 = 13 \)
  • \( 13 \div 13 = 1 \)
  • Prime factors: \( 2 \times 2 \times 3 \times 13 = 2^2 \times 3 \times 13 \)

(iii) 3825

Solution:

  • Divide 3825:
  • \( 3825 \div 5 = 765 \)
  • \( 765 \div 5 = 153 \)
  • \( 153 \div 3 = 51 \)
  • \( 51 \div 3 = 17 \)
  • \( 17 \div 17 = 1 \)
  • Prime factors: \( 3 \times 3 \times 5 \times 5 \times 17 = 3^2 \times 5^2 \times 17 \)

(iv) 5005

Solution:

  • Divide 5005:
  • \( \begin{aligned} & 5005 \div 5 = 1001 \\ & 1001 \div 7 = 143 \\ & 143 \div 11 = 13 \\ & 13 \div 13 = 1 \end{aligned} \)
  • Prime factors: \( 5 \times 7 \times 11 \times 13 = \mathbf{5} \times \mathbf{7} \times \mathbf{11} \times \mathbf{13} \)

(v) 7429

Solution:

  • Divide 7429:
  • \( \begin{aligned} & 7429 \div 17 = 437 \\ & 437 \div 19 = 23 \\ & 23 \div 23 = 1 \end{aligned} \)
  • Prime factors: \( 17 \times 19 \times 23 = \mathbf{17} \times \mathbf{19} \times \mathbf{23} \)

2. Find the LCM and HCF of the following pairs of integers and verify that \(\mathrm{LCM} \times \mathrm{HCF} = \) product of the two numbers:

(i) 26 and 91

Solution:

  • Prime factors:
  • \( \begin{aligned} & 26 = 2 \times 13 \\ & 91 = 7 \times 13 \end{aligned} \)
  • HCF: Common factor \( = 13 \). So, \(\mathrm{HCF} = \mathbf{13}\).
  • LCM: Take the highest power of each prime: \( 2^1 \times 7^1 \times 13^1 = 2 \times 7 \times 13 = 182 \).
  • Verification: \(\mathrm{LCM} \times \mathrm{HCF} = 182 \times 13 = 2366\). Product of numbers \( = 26 \times 91 = 2366 \). Verified.

(ii) 510 and 92

Solution:

  • Prime factors:
  • \( \begin{aligned} & 510 = 2 \times 3 \times 5 \times 17 \\ & 92 = 2^2 \times 23 \end{aligned} \)
  • HCF: Common factor \( = 2 \). So, \(\mathrm{HCF} = \mathbf{2}\).
  • LCM: Highest powers: \( 2^2 \times 3^1 \times 5^1 \times 17^1 \times 23^1 = 4 \times 3 \times 5 \times 17 \times 23 = 23460 \).
  • Verification: \(\mathrm{LCM} \times \mathrm{HCF} = 23460 \times 2 = 46920\). Product \( = 510 \times 92 = 46920 \). Verified.

(iii) 336 and 54

Solution:

  • Prime factors:
  • \( \begin{aligned} & 336 = 2^4 \times 3 \times 7 \\ & 54 = 2 \times 3^3 \end{aligned} \)
  • HCF: Common factors \( = 2^1 \times 3^1 = 6 \). So, \(\mathrm{HCF} = \mathbf{6}\).
  • LCM: Highest powers: \( 2^4 \times 3^3 \times 7^1 = 16 \times 27 \times 7 = \mathbf{3024} \).
  • Verification: \(\mathrm{LCM} \times \mathrm{HCF} = 3024 \times 6 = 18144\). Product \( = 336 \times 54 = 18144 \). Verified.

3. Find the LCM and HCF of the following integers by applying the prime factorisation method:

(i) 12, 15, and 21

Solution:

  • Prime factors:
  • \( \begin{aligned} & 12 = 2^2 \times 3 \\ & 15 = 3 \times 5 \\ & 21 = 3 \times 7 \end{aligned} \)
  • HCF: Common factor \( = 3 \). So, \(\mathrm{HCF} = \mathbf{3}\).
  • LCM: Highest powers: \( 2^2 \times 3^1 \times 5^1 \times 7^1 = 4 \times 3 \times 5 \times 7 = \mathbf{420} \).

(ii) 17, 23, and 29

Solution:

  • Prime factors:
  • \( \begin{aligned} & 17 = 17 \\ & 23 = 23 \\ & 29 = 29 \end{aligned} \)
  • HCF: No common factors. So, \(\mathrm{HCF} = \mathbf{1}\).
  • LCM: \( 17 \times 23 \times 29 = \mathbf{11339} \).

(iii) 8, 9, and 25

Solution:

  • Prime factors:
  • \( \begin{aligned} & 8 = 2^3 \\ & 9 = 3^2 \\ & 25 = 5^2 \end{aligned} \)
  • HCF: No common factors. So, \(\mathrm{HCF} = \mathbf{1}\).
  • LCM: \( 2^3 \times 3^2 \times 5^2 = 8 \times 9 \times 25 = 1800 \).

4. Given that \(\operatorname{HCF}(306, 657) = 9\), find \(\operatorname{LCM}(306, 657)\).

Solution:

  • Use the formula: \(\mathrm{LCM} \times \mathrm{HCF} = \) Product of the numbers.
  • \(\mathrm{HCF} = 9\), Numbers \( = 306 \) and \( 657 \).
  • Product \( = 306 \times 657 = 201042 \).
  • \(\mathrm{LCM} = \) Product \( \div \mathrm{HCF} = 201042 \div 9 = 22338 \).

5. Check whether \(6^{\mathrm{n}}\) can end with the digit 0 for any natural number \(n\).

Solution:

  • A number ends with 0 if it is divisible by 10, i.e., it has at least one 2 and one 5 in its prime factors.
  • \( 6 = 2 \times 3 \), so \( 6^{\mathrm{n}} = (2 \times 3)^{\mathrm{n}} = 2^{\mathrm{n}} \times 3^{\mathrm{n}} \).
  • \( 6^{\mathrm{n}} \) has no 5 in its prime factors, only 2 and 3.
  • Since 5 is missing, \( 6^{\mathrm{n}} \) cannot be divisible by 10.
  • Therefore, \( 6^{\mathrm{n}} \) cannot end with the digit 0 for any natural number \( n \).

6. Explain why \(7 \times 11 \times 13 + 13\) and \(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5\) are composite numbers.

Solution:

  • For \( 7 \times 11 \times 13 + 13 \):
  • Rewrite: \( 13 \times (7 \times 11 + 1) = 13 \times (77 + 1) = 13 \times 78 \).
  • \( 78 = 2 \times 3 \times 13 \), so the number \( = 13 \times 2 \times 3 \times 13 = 2 \times 3 \times 13^2 \).
  • It has multiple prime factors \( (2, 3, 13) \), so it is composite.
  • For \( 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 \):
  • Let \(\mathrm{P} = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 7! = 5040\).
  • Number \( = 5040 + 5 = 5045 \).
  • Factorize 5045: \( 5045 \div 5 = 1009 \). Check if 1009 is prime (not divisible by primes \( < \sqrt{1009} \approx 31.7 \), i.e., \( 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 \)).
  • 1009 is prime (no divisors found).
  • So, \( 5045 = 5 \times 1009 \), which has two prime factors, making it composite.

7. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

Solution:

  • They meet again when the time is a common multiple of their round times (18 and 12 minutes).
  • Find LCM of 18 and 12:
  • \( 18 = 2 \times 3^2, 12 = 2^2 \times 3 \).
  • LCM \( = 2^2 \times 3^2 = 4 \times 9 = 36 \).
  • They meet after 36 minutes.

EXERCISE 1.2

1. Prove that \(\sqrt{5}\) is irrational.

Solution:

  • Assume \(\sqrt{5}\) is rational, i.e., \(\sqrt{5} = \mathrm{a} / \mathrm{b}\), where \(\mathrm{a}\) and \(\mathrm{b}\) are coprime integers, \(\mathrm{b} \neq 0\).
  • Square both sides: \( 5 = \mathrm{a}^2 / \mathrm{b}^2 \).
  • So, \(\mathrm{a}^2 = 5 \mathrm{b}^2\).
  • \(\mathrm{a}^2\) is divisible by 5, so \(\mathrm{a}\) must be divisible by 5 (since 5 is prime). Let \(\mathrm{a} = 5 \mathrm{m}\).
  • Substitute: \( (5 \mathrm{m})^2 = 5 \mathrm{b}^2 \rightarrow 25 \mathrm{m}^2 = 5 \mathrm{b}^2 \rightarrow \mathrm{b}^2 = 5 \mathrm{m}^2 \).
  • \(\mathrm{b}^2\) is divisible by 5, so \(\mathrm{b}\) is divisible by 5.
  • Both \(\mathrm{a}\) and \(\mathrm{b}\) are divisible by 5, contradicting the assumption that \(\mathrm{a}\) and \(\mathrm{b}\) are coprime.
  • Therefore, \(\sqrt{5}\) is irrational.

2. Prove that \(3 + 2 \sqrt{5}\) is irrational.

Solution:

  • Assume \( 3 + 2 \sqrt{5} \) is rational, i.e., \( 3 + 2 \sqrt{5} = \mathrm{a} / \mathrm{b} \), where \(\mathrm{a}\) and \(\mathrm{b}\) are coprime, \(\mathrm{b} \neq 0\).
  • Rearrange: \( 2 \sqrt{5} = \mathrm{a} / \mathrm{b} - 3 = (\mathrm{a} - 3 \mathrm{b}) / \mathrm{b} \).
  • So, \(\sqrt{5} = (\mathrm{a} - 3 \mathrm{b}) / (2 \mathrm{b})\).
  • Since \(\mathrm{a}, \mathrm{b}, 3, 2\) are integers, \( (\mathrm{a} - 3 \mathrm{b}) / (2 \mathrm{b}) \) is rational.
  • But \(\sqrt{5}\) is irrational (from Q1), so this is a contradiction.
  • Therefore, \( 3 + 2 \sqrt{5} \) is irrational.

3. Prove that the following are irrational:

(i) \( 1 / \sqrt{2} \)

Solution:

  • Assume \( 1 / \sqrt{2} \) is rational, i.e., \( 1 / \sqrt{2} = \mathrm{a} / \mathrm{b} \), where \(\mathrm{a}\) and \(\mathrm{b}\) are coprime, \(\mathrm{b} \neq 0\).
  • Rewrite: \(\sqrt{2} = \mathrm{b} / \mathrm{a}\).
  • Since \(\mathrm{a}\) and \(\mathrm{b}\) are integers, \(\mathrm{b} / \mathrm{a}\) is rational.
  • But \(\sqrt{2}\) is irrational (similar proof to \(\sqrt{5}\)), so this is a contradiction.
  • Therefore, \( 1 / \sqrt{2} \) is irrational.

(ii) \( 7 \sqrt{5} \)

Solution:

  • Assume \( 7 \sqrt{5} \) is rational, i.e., \( 7 \sqrt{5} = \mathrm{a} / \mathrm{b} \), where \(\mathrm{a}\) and \(\mathrm{b}\) are coprime, \(\mathrm{b} \neq 0\).
  • Rewrite: \(\sqrt{5} = \mathrm{a} / (7 \mathrm{b})\).
  • \(\mathrm{a} / (7 \mathrm{b})\) is rational (since \(\mathrm{a}, \mathrm{b}, 7\) are integers).
  • But \(\sqrt{5}\) is irrational, so this is a contradiction.
  • Therefore, \( 7 \sqrt{5} \) is irrational.

(iii) \( 6 + \sqrt{2} \)

Solution:

  • Assume \( 6 + \sqrt{2} \) is rational, i.e., \( 6 + \sqrt{2} = \mathrm{a} / \mathrm{b} \), where \(\mathrm{a}\) and \(\mathrm{b}\) are coprime, \(\mathrm{b} \neq 0\).
  • Rearrange: \(\sqrt{2} = \mathrm{a} / \mathrm{b} - 6 = (\mathrm{a} - 6 \mathrm{b}) / \mathrm{b}\).
  • \( (\mathrm{a} - 6 \mathrm{b}) / \mathrm{b} \) is rational (since \(\mathrm{a}, \mathrm{b}, 6\) are integers).
  • But \(\sqrt{2}\) is irrational, so this is a contradiction.
  • Therefore, \( 6 + \sqrt{2} \) is irrational.

Class 10 Maths Chapter 1 Solutions: Your Guide to Success

Welcome, Class 10 students! If you’re diving into the NCERT Class 10 Maths book, Chapter 1, titled “Real Numbers,” is where your journey begins. For board exams, mastering class 10 Maths Chapter 1 solutions is key to understanding the CBSE syllabus and acing your exam preparation. This chapter lays the foundation for the Class 10 Maths syllabus, and with the right approach, you can tackle it easily. In this article, we’ll explore class 10 Maths Chapter 1 solutions in simple language, break down the concepts, share tips on how students can approach it, and explain how www.growupncert.com can help you succeed in your board exams. Let’s get started!

Why Class 10 Maths Chapter 1 Solutions Matter

The NCERT Class 10 Maths book, part of the CBSE syllabus, is your go-to resource for board exams. Chapter 1, “Real Numbers,” introduces essential ideas that connect to other chapters and real-life applications. Understanding class 10 Maths Chapter 1 solutions helps you solve problems confidently, score well in exams, and build a strong base for topics like polynomials and geometry. The solutions for Class 10 Maths chapters, especially Chapter 1, guide you through exercises step by step. With support from websites like www.growupncert.com, you’ll be ready for exam preparation. Let’s dive into the chapter!

Understanding Class 10 Maths Chapter 1 : Real Numbers

What Is Chapter 1 About?

Real Numbers is the first chapter in the Class 10 Maths syllabus. It covers all types of numbers you’ll use in Maths—integers, fractions, decimals, rational, and irrational numbers. The NCERT book explains these in a simple way, with examples and exercises to practice. Class 10 Maths Chapter 1 solutions show you how to solve these problems correctly, making it a key part of your board exam prep.

Class 10 Maths Chapter 1 Solutions
Key Concepts in Real Numbers

Here’s a simple breakdown of the main topics:

  • Euclid’s Division Lemma: This is a rule to divide numbers. For any two positive integers a and b, you can write a = bq + r, where q is the quotient and r is the remainder (0 ≤ r < b). For example, if a = 17 and b = 5, then 17 = 5 × 3 + 2.
  • Fundamental Theorem of Arithmetic: Every number greater than 1 is either prime or can be written as a unique product of prime numbers. For example, 12 = 2 × 2 × 3.
  • Rational and Irrational Numbers: Rational numbers are like 1/2, 3, or 0.5 (can be written as fractions). Irrational numbers, like √2 or π, can’t be fractions and have non-repeating decimals.
  • Properties of Real Numbers: Real numbers include both rational and irrational numbers. You’ll learn to prove numbers like √3 are irrational.

Why It’s Important for Board Exams

Chapter 1 is a building block for the CBSE syllabus. It carries 6-8 marks in board exams, often with questions on division lemma, prime factorization, or proofs. Mastering class 10 Maths Chapter 1 solutions helps you solve these quickly and prepares you for related topics in later chapters, like Polynomials or Quadratic Equations.

NCERT Exercises in Chapter 1

The NCERT book has four exercises in Chapter 1:

  • Exercise 1.1: Focuses on Euclid’s Division Lemma. Example: Use it to find the quotient and remainder when 25 is divided by 4.
  • Exercise 1.2: Covers the Fundamental Theorem of Arithmetic. You’ll practice prime factorization, like breaking 48 into 2 × 2 × 2 × 2 × 3.
  • Exercise 1.3: Deals with rational and irrational numbers. You might prove √5 is irrational.
  • Exercise 1.4: Revisits concepts with mixed problems, combining division and factorization. Solutions for Class 10 Maths Chapter 1 guide you through each step, making exam preparation easier.

Sample Class 10 Maths Chapter 1 Solutions

Let’s look at a few sample problems and their solutions to help you understand.

Exercise 1.1, Question 1

Problem: Use Euclid’s Division Lemma to show that any positive integer is of the form 3q, 3q + 1, or 3q + 2. Solution:

  • Take a positive integer a and divide it by 3.
  • By Euclid’s Lemma, a = 3q + r, where r is the remainder and 0 ≤ r < 3.
  • So, r can be 0, 1, or 2.
  • If r = 0, a = 3q (e.g., 9 = 3 × 3).
  • If r = 1, a = 3q + 1 (e.g., 10 = 3 × 3 + 1).
  • If r = 2, a = 3q + 2 (e.g., 11 = 3 × 3 + 2).
  • Thus, any positive integer fits one of these forms.

Exercise 1.2, Question 2

Problem: Find the HCF of 56 and 88 by prime factorization. Solution:

  • Prime factorization of 56 = 2 × 2 × 2 × 7.
  • Prime factorization of 88 = 2 × 2 × 2 × 11.
  • Common factors are 2 × 2 × 2.
  • HCF = 2 × 2 × 2 = 8.
  • So, the HCF of 56 and 88 is 8.

Exercise 1.3, Question 3

Problem: Prove that √3 is irrational. Solution:

  • Assume √3 is rational, so √3 = p/q, where p and q are integers, q ≠ 0, and p and q have no common factors (simplest form).
  • Square both sides: 3 = p²/q².
  • Multiply by q²: 3q² = p².
  • This means p² is divisible by 3, so p must be divisible by 3 (e.g., p = 3m).
  • Substitute: 3q² = (3m)² = 9m².
  • Divide by 3: q² = 3m².
  • Now q² is divisible by 3, so q is divisible by 3.
  • If p and q are both divisible by 3, they have a common factor, contradicting our assumption.
  • Thus, √3 cannot be rational and is irrational.

These class 10 Maths Chapter 1 solutions show step-by-step thinking, perfect for board exam practice.

How Students Can Approach Class 10 Maths Chapter 1 Solutions

Mastering class 10 Maths Chapter 1 solutions is easy with the right plan. Here are practical tips for exam preparation within the CBSE syllabus.

1. Understand the CBSE Syllabus

  • What to Do: Check the Class 10 Maths syllabus on the CBSE website or with your teacher. Chapter 1 is worth 6-8 marks in board exams.
  • How It Helps: Focuses your study on key topics like Euclid’s Lemma and irrational numbers.
  • Tip: Note the weightage and prioritize accordingly.

2. Read the Chapter Carefully

  • What to Do: Start with the NCERT Class 10 Maths book. Read the explanation of real numbers, rational, and irrational numbers.
  • How It Helps: Builds a clear base for solving problems.
  • Tip: Focus on examples before jumping to exercises.

3. Learn Key Concepts

  • What to Do: Study Euclid’s Division Lemma, the Fundamental Theorem, and proofs of irrationality.
  • How It Helps: Makes class 10 Maths Chapter 1 solutions easier to follow.
  • Tip: Write down key points: e.g., a = bq + r for division.

4. Solve NCERT Exercises

  • What to Do: Work through Exercises 1.1 to 1.4. Start with simple problems, like division, then try proofs.
  • How It Helps: Practice builds confidence for board exams.
  • Tip: Don’t skip—solve all questions in order.

5. Practice Step-by-Step

  • What to Do: Break problems into steps. For example, in Exercise 1.2, factorize a number, find common factors, then calculate HCF.
  • How It Helps: Reduces mistakes and clarifies thinking.
  • Tip: Write each step neatly for the board exam.

6. Memorize Key Methods

  • What to Do: Learn the process for Euclid’s Lemma, prime factorization, and proving irrationality.
  • How It Helps: Speeds up solving in exams.
  • Tip: Make a small note card: e.g., “HCF = product of common prime factors.”

7. Use Previous Years’ Question Papers

  • What to Do: Solve past CBSE board exam papers related to Chapter 1.
  • How It Helps: Shows question types (e.g., HCF, proofs) and helps with time management.
  • Tip: Practice 5-10 questions under exam conditions.

8. Check Solutions for Class 10 Maths Chapter 1

  • What to Do: After solving, compare with class 10 Maths Chapter 1 solutions from NCERT or other resources.
  • How It Helps: Corrects errors and teaches better methods.
  • Tip: Note where you went wrong and retry.

9. Revise Regularly

  • What to Do: Review Chapter 1 weekly. Re-solve tough problems like proving √2 is irrational.
  • How It Helps: Keeps concepts fresh for the board exam.
  • Tip: Spend 30 minutes weekly on Real Numbers.

10. Clear Doubts Quickly

  • What to Do: Ask your teacher or friends if a concept like irrational numbers is tricky.
  • How It Helps: Strengthens your base for exam preparation.
  • Tip: Use online resources or study groups for extra help.

11. Practice Time Management

  • What to Do: Solve 5-10 problems in 20-30 minutes daily to build speed.
  • How It Helps: Prepares you for the 3-hour board exam.
  • Tip: Focus on quick, accurate solving for short questions.

12. Stay Positive

  • What to Do: Believe in yourself! Practice daily and stay calm.
  • How It Helps: Reduces Maths fear and boosts confidence.
  • Tip: Take breaks after solving exercises—relax and recharge!

How www.growupncert.com Helps Students

The website www.growupncert.com is a fantastic tool for Class 10 students. Here’s how it supports your exam preparation for class 10 Maths Chapter 1 solutions and the CBSE syllabus:

  • Class 10 Maths Chapter 1 Solutions: Find step-by-step answers to NCERT exercises (1.1 to 1.4). Stuck on proving √3 is irrational? The site explains it simply.
  • Solutions for Class 10 Maths Chapters: Beyond Chapter 1, get solutions for all chapters to strengthen your Class 10 Maths syllabus prep.
  • Previous Years’ Question Papers for Class 10: Download past CBSE board exam papers. Practice Real Numbers questions to know what to expect.
  • CBSE Syllabus Alignment: The site follows the Class 10 Maths syllabus, ensuring you focus on board exam topics.
  • Practice Tests: Take mock tests for Chapter 1 to check your skills and speed, perfect for exam preparation.
  • Simple Explanations: Concepts like Euclid’s Lemma or prime factorization are broken down in easy language.
  • Revision Notes: Get short notes on key points, like properties of real numbers, for quick review.
  • Doubt Support: Some sections may offer tips or ways to ask questions, helping with tricky problems.
  • How to Use It: Visit growupncert.com, find class 10 Maths Chapter 1 solutions, practice with past papers, and review notes. Combine this with your NCERT book for success.

Time Management for Board Exam Prep

  • Daily Study: Spend 30-60 minutes on Chapter 1. Solve 5-10 problems from class 10 Maths Chapter 1 solutions.
  • Weekly Goals: Master Real Numbers in 1-2 weeks, then move to other chapters.
  • Before Exam: Revise Chapter 1 in 2-3 days. Focus on HCF, proofs, and division.
  • Practice: Solve past papers in 3 hours to mimic the board exam.

Common Challenges and Solutions

  • Challenge: Confusion with Euclid’s Lemma.
  • Challenge: Trouble proving irrationality.
    • Solution: Follow step-by-step class 10 Maths Chapter 1 solutions and retry.
  • Challenge: Silly mistakes.
    • Solution: Write steps clearly and verify with NCERT or the website.
  • Challenge: Exam time pressure.
    • Solution: Practice timed questions from past papers.

Benefits of Mastering Class 10 Maths Chapter 1 Solutions

  • Board Exam Success: Chapter 1 questions (6-8 marks) are common in CBSE exams. Solutions help you nail them.
  • Strong Foundation: Prepares you for later chapters and future studies.
  • Confidence Boost: Practice and resources like growupncert.com make Maths easy.

Final Thoughts : Class 10 Maths Chapter 1 Solutions

Class 10 Maths Chapter 1 solutions are your key to mastering Real Numbers in the CBSE syllabus. This chapter, part of the Class 10 Maths syllabus, builds your base with concepts like Euclid’s Lemma, prime factorization, and irrational numbers. For exam preparation, read the NCERT book, solve exercises, practice step by step, and use previous years’ question papers for Class 10. The website www.growupncert.com helps with clear solutions, practice tests, and notes. Study daily, revise regularly, and stay positive. With this approach, you’ll ace Chapter 1 and shine in your board exams. Start today—you’ve got this!