NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers all exercises in Hindi and English medium PDF format. The solution of Subject math is updated for the new academic year 2022-2023. 10th Mathematics NCERT Textbook solutions are useful for both UP Board (High School) Schools, and CBSE Board Schools.

Class 10 Maths Chapter 1 Solutions

Class 10 Maths Chapter 1 Main Topics

Class 10 Maths Chapter 1 Practice Tests

10th Maths Chapter 1 Question wise Solutions

Class 10 Maths Chapter 1 MCQ

Class 10 Maths Chapter 1 Assignments

## NCERT Solutions for Class 10 Maths Chapter 1

Get here topic wise NCERT Solutions for 10th Mathematics chapter 1 all exercises in Hindi and English medium. Students can take help of videos for solving questions of each exercise.

### 10th Maths Chapter 1 Exercise 1.1

### 10th Maths Chapter 1 Exercise 1.2

### 10th Maths Chapter 1 Exercise 1.3

### 10th Maths Chapter 1 Exercise 1.4

### 10th Maths Chapter 1 Exercise 1.1 Solutions

**Class 10 Maths Chapter 1 Exercise 1.1 Question 1**

*Use Euclid’s division algorithm to find the HCF of: (i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255.*

See the solutions given here and discuss part 1, part 2 and part 3 with us in detail for more clarification. After that use assignment and worksheet to get confident.

**Class 10 Maths Chapter 1 Exercise 1.1 Question 2**

*Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.*

To practice more, use assignment and discus the question with classmates. Later on see the solutions of this questions to verify answer.

**Class 10 Maths Chapter 1 Exercise 1.1 Question 3**

*An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?*

Question 3 of 10th Maths is one of the important questions in board exams. So, must do practice assignment and share the question using solutions given here.

**Class 10 Maths Chapter 1 Exercise 1.1 Question 4**

*Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.*

[Hint : Let x be any positive integer then it is of the form 3q, 3q + 1 or 3q + 2. Now square each of these and show that they can be rewritten in the form 3m or 3m + 1.]

The solution of question 4 is similar to question 2 and 5. Download assignment and share it with friends to discuss in better way.

**Class 10 Maths Chapter 1 Exercise 1.1 Question 5**

*Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.*

After doing question number 2 and 4, you will find question number 5 solution very easy. The practice questions based on this question will help you. If you have any doubt related to this question, please visit to Discussion forum.

#### 10th Maths Chapter 1 Exercise 1.2 Solutions

**Class 10 Maths Chapter 1 Exercise 1.2 Question 1**

*Express each number as a product of its prime factors: (i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429*

The step by step solutions of each part is given here with practice questions. You can discuss with us Part (i), Part (ii), Part (iii), Part (iv), and Part (v) to learn more about the solution.

**Class 10 Maths Chapter 1 Exercise 1.2 Question 2**

*Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54.*

Get the solution of question 2 here and discuss Part (i), Part (ii) and Part (iii) with us along with assignments and other doubts for more clarification.

**Class 10 Maths Chapter 1 Exercise 1.2 Question 3**

*Find the LCM and HCF of the following integers by applying the prime factorisation method. (i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25.*

Discuss with part one, two and three in detail and get worksheets to understand the solution in simple way.

**Class 10 Maths Chapter 1 Exercise 1.2 Question 4**

*Given that HCF (306, 657) = 9, find LCM (306, 657).*

The solution of this question is very simple and usually asked in MCQ type questions. For more revision, please write to us.

**Class 10 Maths Chapter 1 Exercise 1.2 Question 5**

*Check whether 6ⁿ can end with the digit 0 for any natural number n.*

The solution of question 5 is similar to the example given in textbook. Discuss with us and Practice here more questions based on question 5.

**Class 10 Maths Chapter 1 Exercise 1.2 Question 6**

*Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.*

Extra questions for practice based on the solution of question 6 are given here to download. Put your view to discuss with us.

**Class 10 Maths Chapter 1 Exercise 1.2 Question 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?*

Question number 7 is important for exams. Get here the solution and practice assignments with answers for revision.

#### 10th Maths Chapter 1 Exercise 1.3 Solutions

**Class 10 Maths Chapter 1 Exercise 1.3 Question 1**

*Prove that √5 is irrational.*

To understand the solution of question 1, student need to practice assignments and discuss the explanation in depth.

**Class 10 Maths Chapter 1 Exercise 1.3 Question 2**

*Prove that 3 + 2√5 is irrational.*

Get here the assignments and practice questions based on the solution of question 2. If still there is any doubt, please visit to discussion forum.

**Class 10 Maths Chapter 1 Exercise 1.3 Question 3**

*Prove that the following are irrationals: (i) 1/√2 (ii) 7√5 (iii) 6 + √2*

The solution of question 3 is similar to question 1 and 2. Get more practice to learn the concepts in better way. Discuss in detail here.

#### 10th Maths Chapter 1 Exercise 1.4 Solutions

**Class 10 Maths Chapter 1 Exercise 1.4 Question 1**

*Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion: (i) 13/3125 (ii) 17/8 (iii) 64/455 (iv) 15/1600 (v) 29/343 (vi) 23/(2³5²) (vii) 129/2²5⁷7⁵ (viii) 6/15 (ix) 35/50 (x) 77/210*

Question 1 solution is easy to understand. More revision questions and discussion makes this topic better than ever.

**Class 10 Maths Chapter 1 Exercise 1.4 Question 2**

*Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.*

To practice well question 2, there are assignments given along with solutions. Discuss here this question in detail with classmates.

**Class 10 Maths Chapter 1 Exercise 1.4 Question 3**

*The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form , p q what can you say about the prime factors of q? (i) 43.123456789 (ii) 0.120120012000120000… (iii) 43.123456789123456789…*

If you have any doubt related to the solution of this question, please discuss with us. Download the assignment to practice similar questions.

The complete chapter of 10th Maths chapter 1 is divided in four main parts. The explanation of each part with video explanation is given below.

Topics | Detail about the Topics |
---|---|

Topic: 1 | Euclid’s Division Lemma |

Topic: 2 | The Fundamental Theorem of Arithmetic |

Topic: 3 | Dealing with Irrational with Square Root |

Topic: 4 | Decimal Representation of Rational Numbers |

#### 10th Maths Chapter 1 Practice Test Papers

There are four exercises in chapter 1 covering all the four topics on Real Numbers. Students can practice well using the 6 question papers given below. It will help to prepare the chapter for CBSE exams also.

##### Class 10 Maths Chapter 1 Practice Tests with Answers

###### 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. Learn here how to make class 10 Maths chapter 1 easy to learn.

### How many exercises 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**:

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