# NCERT Solutions for class 10 Maths chapter 1 pdf

(Last Updated On: September 7, 2023)

NCERT Solutions for class 10 Maths chapter 1 Real Numbers Questionnaire 1.1 is solved with practice and chapter wise questions and answers for class 10 students along with chapter wise review which is useful for UPSC, SSC, UPPSC, BPSC and others. Helping students preparing for competitive exams and entrance exams.

## NCERT Solutions for class 10 Maths chapter 1 pdf

Class 10 CBSE Board Chapter 1. Real Numbers Important Key Points for Quick Revision for Board Exams, UPSSSC , SSC CGL ,SSC MTS and UPSC ,BPSC ,UPPSC Exam Preparation. – 1. Real Numbers – Exercise 1.1: NCERT Book Solutions for Class 10th. 1. All Solutions for Real Numbers and Addition or Addition Solved Questions: Exercise 1.1 Maths Class 10th: Hindi Medium NCERT Book Solutions. Class 10 Chapter 1. Real Numbers Important Key Points for Quick Revision for the preparation of Board Exams, SSC & UPSC Exams.

### All Question Answers of Class 10 Exercise 1.1 in Hindi, English Chapter 1. real numbers

1.1 Maths Class 10th: Hindi, English Medium NCERT Book Solutions

Class 10 Chapter 1. Real Numbers Important Key Points for Quick Revision for the preparation of Board Exams, SSC & UPSC Exams. – 1. Real Numbers – Exercise 1.1: NCERT Book Solutions for Class 10th. 1. All Solutions for Real Numbers and Addition or Addition Solved Questions: Exercise 1.1 Maths Class 10th: Hindi Medium NCERT Book Solutions.

Class 10 1. Real Numbers Exercise 1.1: NCERT Book Solutions

### NCERT Books Subject wise for Class 10th Hindi Medium and English Medium

1. Real numbers
Class 10 Chapter 1. Real Numbers Important Key Points for Quick Revision for Board Exams, SSC, BPSC , RPSC ,UKPSC and UPSC Exam Preparation.

Topics and Sub Topics in Class 10 Maths Chapter 1 Real Numbers:

 Section Name Topic Name 1 Real Numbers 1.1 Introduction 1.2 Euclid’s Division Lemma 1.3 The Fundamental Theorem of Arithmetic 1.4 Revisiting Irrational Numbers 1.5 Revisiting Rational Numbers and Their Decimal Expansions 1.6 Summary

## NCERT Solutions For Class 10 Maths Chapter 1 Real Numbers Ex 1.1

NCERT Solutions for Class 10 Maths for CBSE Board Chapter 6 Triangles Example 1.1 are part of the NCERT Solutions for Class 10 Maths. Here we have given NCERT CBSE Pattern Solutions for Class 10 Maths Chapter 6 Triangles Exercise 1.1

 Board CBSE Textbook NCERT Class Class 10 Subject Maths Chapter Chapter 1 Chapter Name Real Numbers Exercise Ex 1.1 Number of Questions Solved 5 Category NCERT Solutions

#### प्रश्नावली 1.1

अभ्यास 1.1

Q1. Find the HCF using Euclid’s division algorithm.

(i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255

Solution:

(1) 135 and 225

a = 225, b = 135 {The largest number is considered as a and the smallest number is considered as b}

using Euclid’s division algorithm

a = bq + r (then)

225 = 135 × 1 + 90

135 = 90 × 1 + 45

90 = 45 × 2 + 0 {When we get r=0 we stop solving further}

b = 45 {then the value of b is HCF;}

HCF = 45

(ii) 196 and 38220 Solution

a = 38220, b = 196 {The largest number is considered as a and the smallest number is considered as b}

using Euclid’s division algorithm

a = bq + r (then)

38220= 196 ×195 + 0 {When we get r=0 we stop solving further}

b = 196{Then the value of b is HCF;}

HCF = 196:

(iii) 867 and 255 Solution

a = 867, b = 255 {The largest number is considered as a and the smallest number is considered as b}

using Euclid’s division algorithm

a = bq + r (then)

38220= 196 ×195 + 0 {When we get r=0 we stop solving further}

b = 196 {then the value of b is HCF;}

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.

Solution:

Show: a = 6q + 1, 6q+3 or 6q+5

Let a be any positive odd integer; where b = 6,

When we divide a by 6 we get remainders 0, 1, 2, 3, 4 and 5 respectively;

where 0 ≤ r < b

Here a is an odd number, so the remainder is also an odd number.

remainder will be 1 or 3 or 5

Using Euclid division algorithm we get;

a = 6q + 1, 6q+3 or 6q+5

Q3. In a parade, an army contingent of 616 members is to march behind an army band of 32 members. Both the groups are to march in equal number of columns. What is the maximum number of columns in which they can march?

Solution:

Maximum number of columns = HCF(616, 32)

a = 616, b = 32 {let the largest number be a and the smallest number be b}

using Euclid’s division algorithm

a = bq + r (then)

616 = 32 ×19 + 8 {when we get r=0 we stop solving further}

32 = 8 × 4 + 0

b = 8 {value of b is HCF}

HCF = 8

Hence maximum number of columns = 8

Question 4. Using Euclid’s division lemma, show that the square of any positive integer is of the form 3m or 3m + 1 for some integer m.

Solution:

Show that : a2 = 3m or 3m + 1

a = bq + r

Let a be any positive integer where b = 3 and r = 0, 1, 2 because 0 ≤ r < 3

Then a = 3q + r for some integer q ≥ 0

Therefore, a = 3q + 0 or 3q + 1 or 3q + 2

Now we get;

⇒ a2 = (3q + 0)2 or (3q + 1)2 or (3q +2)2

⇒ a2 = 9q2 or 9q2 + 6q + 1 or 9q2 + 12q + 4

⇒ a2 = 9q2 or 9q2 + 6q + 1 or 9q2 + 12q + 3 + 1

⇒ a2 = 3(3q2) or 3(3q2 + 2q) + 1 or 3(3q2 + 4q + 1) + 1

If m = (3q2) or (3q2 + 2q)  or (3q2 + 4q + 1) हो तो

We get That;

a2 = 3m or 3m + 1 or 3m + 1

Q5. Using Euclid’s division lemma, show that the cube of a positive integer is of the form 9m, 9m + 1 or 9m + 8.

Solution:

Let a be any positive integer;

Using Euclid’s division lemma;

a = bq + r where; 0 ≤ r < b

By keeping b = 9

a = 9q + r where; 0 ≤ r < 9

When r = 0;

a = 9q + 0 = 9q

a3  = (9q)3 = 9(81q3) or 9m where m = 81q3

When r = 1

a = 9q + 1

a3 = (9q + 1)3 = 9(81q3 + 27q2 + 3q) + 1

= 9m + 1  where m = 81q3 + 27q2 + 3q

when r = 2

a = 9q + 2

a3  = (9q + 2)3 = 9(81q3 + 54q2 + 12q) + 8

= 9m + 2  where m = 81q3 + 54q2 + 12q

Thus, the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.

### NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers

This chapter helps the students to understand the fundamental theorem of arithmetic which has many real life and scientific applications. Real numbers are also used in other related fields which are explained in this chapter. So for a strong foundation in maths that will aid in further higher education as well, our NCERT Solutions for Class 10 Maths will surely help. Rational and irrational numbers are the main types of real numbers. Theorems will explain these with suitable examples and applications.

#### Let us discuss the sub-topics in detail –

1.1: Introduction:

In this chapter, students will explore the world of real numbers and their respective applications. This chapter covers some very important properties of positive numbers.

1.2: Euclid’s Division Lemma:

In this lesson, students will learn a technique for computing the highest common factor (HCF) of two given positive integers by Euclid’s algorithm.

1.3: Fundamental Theorem of Arithmetic:

Students will learn that every composite number can be uniquely expressed as the product of prime numbers. This property is a fundamental theorem of arithmetic.

1.4: Irrational numbers revisited:

This chapter redefines irrational numbers. Some relevant examples will help in understanding the concept easily. A method of contradiction will help to prove them.

1.5: Revisiting the rational numbers and their decimal expansions:

Students will revisit the concept of rational numbers using fraction expressions as well as decimal expansion. This is because the decimal expansion of every rational number is either terminating or repeatedly terminating.

NCERT Solutions for Class 10 Maths