# NCERT Solutions for Class 10 Maths Chapter 1

NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers all exercises in Hindi and English medium PDF format updated for the new academic year 2021-2022. NCERT Textbook solutions are useful for both UP Board (High School) Schools, and CBSE Board Schools as Uttar Pradesh Board is now following NCERT Books for 2021-22. So, students can download UP Board Solutions for Class 10 Maths Chapter 1 from this page. Download Class 10 Maths App or Class 10 Ganit App or Offline Apps for other subjects for offline use in UP Board or CBSE or other boards based on NCERT Book’s Syllabus.

Visit Discussion Forum to share your knowledge and ask your doubts. In Chapter 1 of 10th Maths, we observe that Euclid’s division algorithm has to do with divisibility of integers. The result obtained from Euclid’s Lemma is quite easy to state and understand. In Class 10 Maths, there are many applications related to the divisibility properties of integers. Here, we will try a few of them to compute the HCF of two positive integers.

## NCERT Solutions for Class 10 Maths Chapter 1

 Class: 10 Maths (English and Hindi Medium) Chapter 1: Real Numbers

### 10th Maths Chapter 1 Solutions

Class 10 Maths Chapter 1 NCERT Solutions 2021-22 and study material related to 10th Maths chapter 1 is modified according to latest CBSE Syllabus 2021-2022. UP Board students also use this solution for their exams. The fundamental theorem of arithmetic and Euclid’s division lemma are the main topics of this chapter (Real Numbers). The Fundamental Theorem of Arithmetic and Euclid’s division lemma has many applications, both within mathematics and in other fields. NCERT Solutions of all other subjects are also available in PDF form.

• ### 10th Maths Chapter 1 Exercise 1.4

#### Changes in CBSE Syllabus 2021-2022 Class 10 Maths Chapter 1

Get the latest CBSE syllabus to observe the changes due to corona-virus with full detail for class 10 Maths Chapter 1. CBSE has issued a new syllabus for session 2021-2022 educational session. See all the changes made for Class 10 Maths Syllabus for 2021-2022.

#### Class 10 Maths Exercise 1.1 & 1.2 Solutions in Video

Class 10 Maths Exercise 1.1 Solutions in Video
Class 10 Maths Exercise 1.2 Solutions in Video

#### Class 10 Maths Exercise 1.3 & 1.4 Solutions in Video

Class 10 Maths Exercise 1.3 Solutions in Video
Class 10 Maths Exercise 1.4 Solutions in Video

#### Class 10 Maths Chaper 1 Exercise 1.1 and 1.2 Solution in Hindi

Class 10 Maths Chaper 1 Exercise 1.1 Solution in Hindi
Class 10 Maths Chaper 1 Exercise 1.2 Solution in Hindi

#### Class 10 Maths Chaper 1 Exercise 1.1 and 1.2 Solution in Hindi

Class 10 Maths Chaper 1 Exercise 1.3 Solution in Hindi
Class 10 Maths Chaper 1 Exercise 1.4 Solution in Hindi

## Difference between algorithm and lemma.

### Algorithm

An algorithm is a series of well defined steps which gives a procedure for solving a type of problem.

### Lemma

A lemma is a proven statement used for proving another statement.

### Class 10 Maths Chapter 1: Real Numbers

#### Class 10 Maths Exercise 1.1

In Exercise 1.1 of Class 10 Maths Chapter 1, there are only five questions. All the questions are based on Euclid’s Division Lemma and its applications. In Questions number 1 we have to find HCF applying Euclid’s Division Lemma. In Questions number 2, 4 and 5 also apply step by step Euclid’s Division Algorithms to prove the questions. In Questions 3, we have to find out the maximum number of columns in which they can march, that is, HCF of the two numbers.

#### Class 10 Maths Exercise 1.2

In Exercise 1.2 of Class 10 Maths, the Fundamental Theorem of Arithmetic is main to solve the questions. Question number 1, 2, 3 are based on the prime factorisation method of LCM and HCF. In Question 4, we can use direct formula a × b = LCM × HCF. Question number 6 is an important question as per exams point of view and based on Example Number 5. One can solve Question 6 by taking common or directly solving each number and showing that it has more than two factors. At last in Question 7, we have to find out a number which is divisible by both 12 and 18 that is LCM of the two numbers.

#### Class 10 Maths Exercise 1.3

In Exercise 1.2 of Class 10 Maths, only three questions are there. We know that the square root of a non-perfect square number is an irrational number. Here, we have to prove the same fact with a various set of numbers. In each of these questions, first of all, we assume that the given number is a rational number with numerator and denominator as coprime integers. Later on, we receive a false result due to the wrong assumption, so we conclude that the given number is irrational.

#### Class 10 Maths Exercise 1.4

Class 10 Maths Exercise 1.4 deals with that decimal representation of rational numbers. Questions of Exercise 1.4 are based on Theorem 1.5, Theorem 1.6 and Theorem 1.7 given in NCERT Book of Class 10 Maths Chapter 1. Using these theorems, we can find whether the given rational number is either terminating or non-terminating repeating or non-terminating non-repeating. Most often, one mark questions are asked in CBSE Board exams or school tests also from Exercise 1.4 of Class 10 Maths.

#### Euclid’s division algorithm

The step by step way to find the HCF of two positive integers, say c and d, with c > d.

• Step 1: Apply Euclid’s division lemma, to c and d. So, we find whole numbers, q and r such that c = dq + r, 0 £ r < d.
• Step 2: If r = 0, d is the HCF of c and d. If r ≠ 0, apply the division lemma to d and r.
• Step 3: Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

#### Test & Assignments

Download the practice test series with answers. Level 1 Test 1 contains basic questions for practicing the chapter Real Numbers. Most of the questions of Level 1 Test 2 are easy to understand and provides good practice. Answers of these test series will be available on website free to use.

##### Historical Facts on Real Numbers

The word algorithm come from the name of the name of the 9th century Persian mathematician al-Khwarizmi. The word ‘algebra’ is derived from a book, he wrote, called Hisab al-jabr w’al-muqabala.
An equivalent version of Fundamental theorem of Arithmetic was probably first recorded as Proposition 14 of Book IX in Euclid’s Elements, before it came to be known as the Fundamental Theorem of Arithmetic.
However, the first correct proof was given by Carl Friedrich Gauss in his al-Khwarizmi.
Carl Friedrich Gauss is often referred to as the ‘Prince of Mathematicians’ and is considered one of the three greatest mathematicians of all time, along with Archimedes and Newton. He has made fundamental contributions to both mathematics and science.

###### Prove that the square of an integer is of the form 9k or 3k + 1. ##### How many exercises, questions and examples are there in class 10 Maths chapter 1 Real numbers?

There are 4 exercises in class 10 chapter 1 (Real numbers):
In first exercise (Ex 1.1), there are 5 questions.
In second exercise (Ex 1.2), there are 7 questions.
In third exercise (Ex 1.3), there are 3 questions.
In fourth exercise (Ex 1.4), there are 3 questions.
So, there are in all 18 questions in class 10 chapter 1 (Real numbers).
There are in all 11 examples in class 10 chapter 1 (Real numbers).

• Examples 1,2,3,4 are based on Ex 1.1.
• Examples 5,6,7,8 are based on Ex 1.2.
• Examples 9, 10, 11 are based on Ex 1.3.
##### Are there any Theorems or Algorithm in 10th Maths chapter 1?

Yes, there are 7 Theorems and 1 Algorithm in class 10 chapter 1 (Real numbers).
First exercise (Ex 1.1) is based on Theorem 1.1 (Euclid’s Division Lemma) and Euclid’s Division algorithm.
Theorem 1.2 (Fundamental Theorem of Arithmetic) is used in second exercise (Ex 1.2).
Theorem 1.3 and Theorem 1.4 plays important role in third exercise (Ex 1.3).
Last exercise that is Ex 1.4 is based on Theorems 1.5, 1.6, 1.7.

##### What are the Important questions from Maths class 10 chapter 1 Real numbers?

Important questions from Maths class 10 chapter 1:

• In first exercise (Ex 1.1), all questions are important.
• In second exercise (Ex 1.2), Q2, 4, 5, 7 are important.
• In third exercise (Ex 1.3), all questions are important.
• In fourth exercise (Ex 1.4), Q1 and Q2 are important.
##### Which topics students should recall before starting class 10 Maths chapter 1?

Students should recall some topics of class 9 chapter 1 (Number systems).
Topics:

1. Types of numbers (Natural number, Whole number, Integers, Rational numbers, Irrational numbers and Real numbers).
2. Decimal expansion of Rational numbers (Terminating and Non-terminating recurring) and Irrational numbers (Non-terminating non-recurring).
3. HCF (Highest Common Factor) and LCM (Lowest/Least Common Multiple).
4. And some other topics like Meaning of Dividend, Divisor, Quotient and Remainder, Prime numbers, Composite numbers, Factors.                          