Class 10 maths chapter 1 Real Numbers deals with the foundation of numeric system. Students preparing for their Class 10 exams will be able to clear all their concepts on real numbers class 10 at the root level. Expert faculty of Toppr produced these solutions to assist students with their first term exam preparations. It covers all of the major concepts in detail, allowing students to understand the ideas better.
NCERT solutions for class 10 maths chapter 1 concentrates on the essential concepts such as Euclid's division lemma, Prime Numbers, Composite Numbers, Fundamental Theorem of Arithmetic, HCF and LCM by Prime Factorization Method, and Irrational Numbers, etc.
Maths class 10 chapter 1 Questions and Answers are very useful for getting good grades in tests and properly preparing you with all of the important concepts. These NCERT Solutions are valuable tools that can assist you not only in covering the full syllabus but also in providing in-depth analysis of the subjects. The Class 10 Maths NCERT Solutions Chapter 1 are available in pdf format below, and some of them are also included in the exercises.
Let x be any positive integer. Then, it is of the form 3q or, 3q + 1 or, 3q + 2.
So, we have the following cases :
Case I : When x = 3q.
then, x3 = (3q)3 = 27q3 = 9 (3q3) = 9m, where m = 3q3.
Case II : When x = 3q + 1
then, x3 = (3q + 1)3
= 27q3 + 27q2 + 9q + 1
= 9 q (3q2 + 3q + 1) + 1
= 9m + 1, where m = q (3q2 + 3q + 1)
Case III. When x = 3q + 2
then, x3 = (3q + 2)3
= 27 q3 + 54q2 + 36q + 8
= 9q (3q2 + 6q + 4) + 8
= 9 m + 8, where m = q (3q2 + 6q + 4)
Hence, x3 is either of the form 9 m or 9 m + 1 or, 9 m + 8.
Both rational and irrational numbers are included in real numbers. A rational number is a number that can be stated as a fraction of two integers, with the denominator not equal to zero. Irrational numbers are those that cannot be stated as fractions. Irrational numbers have decimal expansions that don't end and don't become periodic. Integers, decimals, and fractions are examples of rational numbers, whereas irrational numbers include root overs, pi (22/7), and so on. In a nutshell, real numbers are all numbers other than imaginary numbers.
Euclid's division algorithm is a method for calculating the Highest Common Factor (HCF) of two positive numbers. Euclid's Division Lemma states that:
Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.
Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
Using this theorem, we can express every natural number as a prime number multiplication, for example, 35 = 7 × 5, 253 = 11 x 23, etc.
Let x = p/q be a rational number, such that the prime factorisation of q is of the form 2n5m, where n, m are non-negative integers. Then x has a decimal expansion which terminates.
Let x = p/q be a rational number, such that the prime factorisation of q is not of the form 2n5m, where n, m are non-negative integers. Then x has a decimal expansion which is non-terminating repeating (recurring).
Q1. Explain the Graphical Method of Solutions of a Pair of Linear Equations.
Answer: Two lines are represented by the graph of a pair of linear equations with two variables.
Q2. Are Real Numbers NCERT Solutions for Class 10 Maths Chapter 1 important for exams?
Answer: Yes, Chapter 1 of NCERT Class 10 Maths Solutions is significant for the exam. Your NCERT Class 10 Maths book contains critical questions. This chapter contains both short-answer and long-answer questions. This is a foundational chapter, and the topics presented will be used in subsequent chapters. This chapter will help you improve your problem-solving abilities.
Q3. What are the Key Topics in NCERT Solutions Class 10 Maths Chapter 1?
Answer: Euclid's Division Lemma and the Fundamental Theorem of Arithmetic are two major subtopics taught in NCERT Solutions Class 10 Maths Chapter 1. Students can utilize Euclid's Division Lemma to compute the HCF of two positive integers, and the Fundamental Theorem of Arithmetic to find the HCF and LCM of two positive integers. Students can practice questions relating to both areas to have a thorough comprehension of the material.
Q4. How many exercises are in NCERT Solutions for Class 10 Maths Chapter 1?
Answer: Real Numbers is Chapter 1 of the Class 10 NCERT Maths textbook. This chapter contains a total of four exercises. Every workout has a unique PDF. The NCERT Solutions Class 10 Maths Chapter 1 Real numbers have a total of 18 well-researched problems. The 18 questions are divided into three categories: long replies, middle-level answers, and easy answers. These solutions will enable you to complete the NCERT syllabus and be prepared for the Class 10 Maths exam.