assignment 3 class 9 math

NCERT Solutions for Class 9 Maths Chapter 3 Coordinate Geometry

Class 9 Maths Chapter 3 for CBSE Board Class 9 Maths Exercise 3.1 in English Class 9 Maths Exercise 3.2 in English

Class 9 Maths Chapter 3 for State Boards Class 9 Maths Chapter 3 Exercise 3.1 Class 9 Maths Chapter 3 Exercise 3.2 Class 9 Maths Chapter 3 Exercise 3.3

NCERT Solutions for Class 9 Maths Chapter 3 Coordinate Geometry in Hindi and English Medium modified and updated for new academic session 2024-25 based on latest Curriculum. All educational resources on Tiwari Academy are available for free. This is advantageous for students who not have access to paid educational materials or prefer to save on expenses. Question-answers and solutions are prepared according to new NCERT books for class 9 issued for 2024-25 exams.

Being an online platform, our materials are usually accessible 24/7, allowing students to study at their own pace and convenience. Our solutions can serve as a useful revision tool for students preparing for exams. They can quickly review key concepts and problem-solving techniques. Class 9 Maths Chapter 3 in Hindi Medium Class 9 Maths Exercise 3.1 in Hindi Class 9 Maths Exercise 3.2 in Hindi

Class 9 Maths Chapter 3 Study Material

• 9th Maths Study Material for 2024-25 – English Medium
• 9th Maths Study Material for 2024-25 – Hindi Medium
• Class 9 Mathematics Solutions Page
• Class 9 all Subjects Solutions

UP Board students can get the UP Board Solutions for Class 9 Maths Chapter 3 from here. They can download Exercise 3.1 and Exercise 3.2 in English Medium or Prashnavali 3.1 and Prashnavali 3.2 in Hindi Medium PDF format free. View all exercises in Video Format for the students of CBSE and UP Board following the new NCERT (https://ncert.nic.in/) Books 2024-25 for their final exams in 2024-25. All the questions are described using simple and easy steps so that students can understand all the sums of 9th Maths Chapter 3. If someone is facing any problem to use these solutions, please contact us for help. Download Class 9 Maths App for CBSE and UP Board for Offline use or Download Kaksha 9 Ganit App for offline use.

All the solutions are re-prepared removing the error are simplifying the steps of solving including the formula on each step for the simplification. NCERT Solutions 2024-25 are up to date according to the new CBSE curriculum for 2024-25.

Important Questions on 9th Maths Chapter 3

In which quadrant or on which axis do each of the points (– 2, 4), (3, – 1), (– 1, 0), (1, 2) and (– 3, – 5) lie.

(–2, 4): Second Quadrant (3, –1): Fourth Quadrant (–1, 0): x–axis (1, 2): First Quadrant (–3, –5): Third Quadrant

How can we locate the position of an object?

• To locate the position of an object or a point in a plane, we require two perpendicular lines. One of them is horizontal, and the other is vertical.

What is the coordinates of origin?

• The coordinates of the origin are (0, 0).

What is the coordinates of a point on x-axis or on y-axis?

• The coordinates of a point on the x-axis are of the form (x, 0) and that of the point on the y-axis are (0, y).

What are quadrants?

• The coordinate axes divide the plane into four parts called quadrants.

NCERT Solutions for Class 9 Maths Chapter 3 all exercises and Prashnavali are given to download free for new academic session 2024-25. You can view these solutions in Video Format also. Download Kaksha 9 Ganit App for offline use. These apps work well without internet also. Ask your doubts in Discussion Forum and share your knowledge with your friends and other users.

We are updating and revising our contents are per student’s suggestion, need or for the simplicity. For any inconvenience, you may call us, Whats App, Mail or Message us. Vedic Maths is helpful to make calculation easier and faster. NCERT Solutions 2024-25 for class 9 Science is also available to download as well as Online to study. We always response to all calls, if we are not picking your call, it means we are busy with another user, call us little bit later or after an hour. You may leave message us on Whats App. Follow us on Instagram for latest Updates.

I. In Cartesian system, there are two perpendicular straight lines XX’ (horizontal line) and YY’ (vertical line) which intersect at origin. II. The x and y-coordinates of a point is the perpendicular distance from the Y-axis and X-axis measured along the X and Y-axes respectively. III. The y-coordinate of any point X-axis is zero and x-coordinate of any point on Y-axis is zero. IV. The axes divides the plane into four quadrants . 1. In 1st quadrants x > 0, y > 0 i.e (+, +) 2. In 2nd quadrants x < 0, y > 0 i.e (-, +) 3. In 3rd quadrants x < 0, y < 0 i.e (-, -) 4. In 4th quadrants x > 0, y < 0 i.e (+, -)

What are the main points of Class 9 Maths Chapter 3 for exams?

The main points for the term exam preparation from chapter 3 class 9th Maths are:

• The plane is called the Cartesian, or coordinate plane and the lines are called the coordinate axes.
• The horizontal line is called the x -axis, and the vertical line is called the y – axis.
• The point of intersection of the axes is called the origin.
• The distance of a point from the y – axis is called its x-coordinate, or abscissa, and the distance of the point from the x-axis is called its y-coordinate, or ordinate.
• If the abscissa of a point is x and the ordinate is y, then (x, y) are called the coordinates of the point.
• The coordinates of a point are of the form (+ , +) in the first quadrant, (–, +) in the second quadrant, (–, –) in the third quadrant and (+, –) in the fourth quadrant, where + denotes a positive real number and – denotes a negative real number.
• If x ≠ y, then (x, y) ≠ (y, x), and (x, y) = (y, x), if x = y.

How many problems are there in chapter 3 of class 9th Maths for exams?

As per latest NCERT books for class 9 Maths, in chapter 3, there are two exercises. The first exercise (Ex 3.1) contains two questions, and the second exercise (Ex 3.2) contains two questions, and two examples (examples 1, 2). So, there are in all six problems (6 questions and four examples) in chapter 3 (Coordinate Geometry) of class 9th Maths.

Is it possible to solve chapter 3 Coordinate Geometry of class 9th Maths in one day?

No, it is not possible to solve chapter 3 (Coordinate Geometry) of class 9th Maths in one day. Students need a maximum of 3 days to solve chapter 3 (Coordinate Geometry) of class 9th Maths if they give at least 1-2 hours per day to this chapter. This time also depends on student’s speed, efficiency, capability, and many other factors.

Is chapter 3 of class 9th Maths complicated?

No, chapter 3 of class 9th Maths is not at all complicated. It is the easiest chapter of class 9th Maths. This chapter is the student’s favourite chapter. Students quickly solve all questions of this chapter without any difficulty because the questions of this chapter are nice and easy.

« Chapter 2. Polynomials

Chapter 4. linear equations in two variables ».

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• NCERT Solutions for Class 9 Maths Exercises
• NCERT Solutions

NCERT Solutions for Class 9 Maths Exercises - PDF Solutions

Class 9 Maths has a vivid CBSE syllabus covering excellent topics in Mathematics. The CBSE board has formulated the syllabus for the development of respective mathematical concepts and skills among the students. To prepare these chapters, refer to the NCERT Class 9 Maths All Exercises Solutions developed by the subject experts of Vedantu and learn how the fundamental questions can be solved accurately.

Learn the best approaches to solve such problems and score more in the exams.

Vedantu is a platform that provides free NCERT Solution and other study materials for students. Students can register and get access to the best and reliable source of study materials specially made by master teachers at Vedantu. You can download NCERT Solutions Class 9 Maths to help you to revise the complete Syllabus and score more marks in your examinations. Subjects like Science, Maths, English will become easy to study if you have access to NCERT Solution for Class 9 Science , Maths solutions, and solutions of other subjects that are available on Vedantu only.

NCERT Solutions for Class 9 Maths - Chapter-wise List

Given below are the chapter-wise NCERT Solutions for Class 9 Maths . These solutions are provided by the Maths experts at Vedantu in a detailed manner. Go through these chapter-wise solutions to be thoroughly familiar with the concepts.

NCERT Solutions for Class 9 Maths Chapter 1

NCERT Solutions for Class 9 Maths Chapter 2

Ncert solutions for class 9 maths chapter 3, ncert solutions for class 9 maths chapter 4, ncert solutions for class 9 maths chapter 5, ncert solutions for class 9 maths chapter 6, ncert solutions for class 9 maths chapter 7, ncert solutions for class 9 maths chapter 8, ncert solutions for class 9 maths chapter 9, ncert solutions for class 9 maths chapter 10, ncert solutions for class 9 maths chapter 11, ncert solutions for class 9 maths chapter 12, ncert solutions for class 9 maths chapter 13, ncert solutions for class 9 maths chapter 14, ncert solutions for class 9 maths chapter 15.

NCERT Solutions for Class 9 Maths All Exercises

What could be better if you get the simpler and readymade solutions of your Class 9 NCERT textbook exercises ? Moreover, all these are created by an experienced subject matter expert. Can you imagine how beneficial these could be? Sounds amazing, right! You must opt for these NCERT solutions for Class 9 Maths All Exercises right away. You can download NCERT solutions Class 9 Maths All Exercises from Vedantu official website (vedantu.com) by registering yourself with your phone number/ email address. You can also access the same study materials from Vedantu app as well. In that case, you need to download the Vedantu app first from Google play store.

You will get Class 9 Maths All Exercises NCERT solutions written in a simple and lucid manner. Solve all the problems/questions given in the exercises and evaluate your performance with these 100% accurate solutions so that you can learn how to write answers to the questions from any perspective during the final exam. Each of the answers in the Maths All Exercises NCERT solutions Class 9 is student-friendly and customized by the experts.

These NCERT solutions Class 9 Maths All Exercises contain step-by-step answers to all the questions from the exercise which are considered as an invaluable aid to the students when it comes to regular study, exam preparation or doing homework. Class 9 Maths All Exercises NCERT solutions will clear all your doubts in very less time. Hence your confidence will be boosted. By practising the questions from these books, you will be able to improve your time management skills. That will help you in solving all the questions in the stipulated time frame during the final exam.

NCERT Solutions for Class 9 Maths All Exercises are designed as per the latest CBSE curriculum and guidelines. Vedantu’s study materials are updated regularly as per the latest changes. Hence students of Class 9 who are referring to these materials will be able to score the best marks in the final exam. Once you start using these materials, the preparation for Class 9 exam and scoring the highest percentile amongst your peers will seem like a cakewalk at the end of the day.

NCERT Solutions for Class 9 Maths Chapter 1 Number System

6 exercises are included in the NCERT Solutions for Class 9 Maths Chapter 1 which are Exercise 1.1 to 1.6. Most of the questions are application based and a few are memory-based questions to test students’ application skills and cognizance.

NCERT Solutions for Class 9 Maths Chapter 2 Polynomials

Chapter 2 of NCERT Solutions for Class 9 Maths includes 5 exercises. The majority of questions are practical and application-based which allows you to improve the understanding of the chapter.

Class 9 Maths NCERT Solutions Chapter 3 Coordinate Geometry

With 3 exercises comprising both long and short questions, the exercises of NCERT Class 9 Solutions Maths for Chapter 3 are designed to test your application skills as well as cognizance.

NCERT Solutions for Class 9 Maths Chapter 4 Linear Equations in Two Variables

NCERT 9th class Maths book solutions include 4 exercises with 16 questions in total. This chapter includes application-based questions to test your understanding of the chapter.

NCERT Solutions for Class 9 Maths Chapter 5 Introduction to Euclid's Geometry

Chapter 5 of Class 9 Maths NCERT Solutions has 2 exercises. The questions are based on analysis to test your analytical skills and cognizance.

NCERT Solutions for Class 9 Maths Chapter 6 Lines and Angles

NCERT Solution Class 9 Maths of Chapter 6 includes 3 exercises.

NCERT Solutions for Class 9 Maths Chapter 7 Triangles

NCERT Solution Class 9 Maths of Chapter 7 includes 5 exercises . These questions are designed to test your cognizance.

NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

NCERT Solution Class 9 Maths of Chapter 8 includes 2 exercises. These questions are based on your application skills and analysis skills.

NCERT Solutions for Class 9 Maths Chapter 9 Areas of Parallelograms and Triangles

NCERT Solution Class 9 Maths of Chapter 9 includes 4 questions to test your application skills, numerical skills and memory skills.

NCERT Solutions for Class 9 Maths Chapter 10 Circles

9th class Maths book solutions Chapter 10 includes 6 exercises and solutions to all 35 questions given in your NCERT Textbook. These questions are based on your numerical abilities, application skills, and memory skills.

NCERT Solutions for Class 9 Maths Chapter 11 Constructions

Ncert solutions for class 9 maths chapter 12 heron’s formula.

NCERT Solution Class 9 Maths of Chapter 12 includes 2 exercises designed on your ability to decipher and apply the formula.

NCERT Solutions for Class 9 Maths Chapter 13 Surface Areas and Volumes

NCERT Solution Class 9 Maths of Chapter 13 includes 9 exercises and these questions will test your understanding ability and memory skills (required to retain the different formulae).

NCERT Solutions for Class 9 Maths Chapter 14 Statistics

NCERT Solution Class 9 Maths of Chapter 14 contains 4 exercises. The questions judge your comprehensive ability and analytical skills.

NCERT Solutions Class 9 Maths Chapter 15 Probability

NCERT Solution Class 9 Maths Chapter 15 comprises just a single exercise that will assess your comprehensive ability. To prepare for the exams, make sure to read the chapter thoroughly and practise all the exercise questions. This will help you to score better.

CBSE Class 9 Maths Chapter-Wise Marks Weightage 

Internal assessment for cbse class 9 maths, importance of cbse chapter wise list of class 9 maths all exercises.

The experts have formulated the exercise solutions following the latest CBSE syllabus of Class 9 Maths so that students can find them conveniently here. The 9th Maths All Exercises Book Answers for all the exercises have been simplified so that students can easily grab hold of the concepts and use them to develop their answering skills.

The Maths chapters included in this syllabus are all crucial for the conceptual development of the students. They will need these fundamental concepts and skills in the future to study advanced Mathematics in higher classes. Hence, the foundation needs to be strong enough to carry these skills.

This is where the solutions can act as the right guide for the students to develop such skills in a profound way.

Download NCERT Maths All Exercises Book Class 9 Solutions PDF

Perform the NCERT Maths All Exercises Book Class 9 Solutions PDF free download to complete your study material. Find out how the experts have skillfully solved all the questions utilising mathematical principles. Develop your skills accordingly and learn how to ace the Class 9 Maths exams .

Important Related Links for CBSE Class 9 Maths

Important Questions for CBSE Class 9 Math

RS Aggarwal Class 9 Solutions for Math book

NCERT Solutions for Class 9 Maths

CBSE Class 9 Maths Revision Notes

FAQs on NCERT Solutions for Class 9 Maths Exercises

1. How can I prepare all the Class 9 Maths chapters?

Complete preparing a chapter by solving its exercises. Compare your answers to the solutions to find the preparation gaps to fix them. Proceed to the next chapter. This is how you can complete the whole syllabus.

2. How many chapters are there in Class 9 Maths?

There are 15 chapters in the CBSE Class 9 Maths syllabus.

3. What is the best way to remember the Class 9 Maths formulas?

The best way to remember the Class 9 Maths formulas is to prepare and apply them. The more your practice the easier you can remember them all.

4. How can I revise the Class 9 Maths exercises before an exam?

Follow the solutions and revise the answers before an exam and reduce your preparation time.

5. Can I use the exercise solutions for doubt resolution?

The answers in the solutions can be used to resolve your doubts related to the exercise problems. All the answers are accurate and follow the latest CBSE guidelines .

NCERT Solutions for Class 9

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Assignments Class 9 Mathematics Pdf Download

Students can refer to Assignments for Class 9 Mathematics available for download in Pdf. We have given below links to subject-wise free printable Assignments for Mathematics Class 9 which you can download easily. All assignments have a collection of questions and answers designed for all topics given in your latest NCERT Books for Class 9 Mathematics for the current academic session. All Assignments for Mathematics Grade 9 have been designed by expert faculty members and have been designed based on the type of questions asked in standard 9 class tests and exams. All Free printable Assignments for NCERT CBSE Class 9, practice worksheets, and question banks have been designed to help you understand all concepts properly. Practicing questions given in CBSE NCERT printable assignments for Class 9 with solutions and answers will help you to further improve your understanding. Our faculty have used the latest syllabus for Class 9. You can click on the links below to download all Pdf assignments for class 9 for free. You can get the best collection of Kendriya Vidyalaya Class 9 Mathematics assignments and questions workbooks below.

Class 9 Mathematics Assignments Pdf Download

CBSE NCERT KVS Assignments for Mathematics Class 9 have been provided below covering all chapters given in your CBSE NCERT books. We have provided below a good collection of assignments in Pdf for Mathematics standard 9th covering Class 9 questions and answers for Mathematics. These practice test papers and workbooks with question banks for Class 9 Mathematics Pdf Download and free CBSE Assignments for Class 9 are really beneficial for you and will support in preparing for class tests and exams. Standard 9th students can download in Pdf by clicking on the links below.

Subjectwise Assignments for Class 9 Mathematics

More assignments for class 9 mathematics.

Benefits of Solving Class 9 Mathematics Assignments

• The best collection of Grade 9 assignments for Mathematics have been provided below which will help you in getting better marks in class tests and exams.
• The solved question for Class 9 Mathematics will help you to gain more confidence to attempt all types of problems in exams
• Latest NCERT Books for Class 9 Mathematics have been referred to for designing these assignments
• We have provided step by step solutions for all questions in the Class 9 assignments so that you can understand the solutions in detail.
• We have provided single click download links to all chapterwise worksheets and assignments in Pdf.
• Class 9 practice question banks will support to enhance subject knowledge and therefore help to get better marks in exam

FAQs by Mathematics Students in Class 9

At https://www.ncertbooksolutions.com is the best website that has the biggest collection of free printable assignments for Class 9 Mathematics.

We provide here Standard 9 subject-wise assignments which can be easily downloaded in Pdf format for free. Our teachers have provided these Grade 9 Mathematics test sheets for English given in your books.

You can click on the links above and get assignments for Mathematics in Grade 9, all chapters and topic-wise question banks with solutions have been provided here. You can click on the links to download in Pdf.

We have provided here subject-wise Grade 9 Mathematics question banks, revision notes and questions for all difficult topics, and other study material. You can download it all without any charge by clicking on the links provided above.

We have provided the best quality question bank for Class 9 for Mathematics available for Pdf Download. You can download them all and use them offline without the internet.

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Assignments For Class 9 Mathematics

Assignments for Class 9 Mathematics have been developed for Standard 9 students based on the latest syllabus and textbooks applicable in CBSE, NCERT and KVS schools. Parents and students can download the full collection of class assignments for class 9 Mathematics from our website as we have provided all topic wise assignments free in PDF format which can be downloaded easily. Students are recommended to do these assignments daily by taking printouts and going through the questions and answers for Grade 9 Mathematics. You should try to do these test assignments on a daily basis so that you are able to understand the concepts and details of each chapter in your Mathematics book and get good marks in class 9 exams.

Assignments with Answers for Class 9 Mathematics

Assignments for class 9 mathematics as per cbse and ncert pattern, more assignments for class 9 mathematics, assignments for class 9 mathematics as per cbse ncert pattern.

All students studying in Grade 9 Mathematics should download the assignments provided here and use them for their daily routine practice. This will help them to get better grades in Mathematics exam for standard 9. We have made sure that all topics given in your textbook for Mathematics which is suggested in Class 9 have been covered ad we have made assignments and test papers for all topics which your teacher has been teaching in your class. All chapter wise assignments have been made by our teachers after full research of each important topic in the textbooks so that you have enough questions and their solutions to help them practice so that they are able to get full practice and understanding of all important topics. Our teachers at https://www.assignmentsbag.com have made sure that all test papers have been designed as per CBSE, NCERT and KVS syllabus and examination pattern. These question banks have been recommended in various schools and have supported many students to practice and further enhance their scores in school and have also assisted them to appear in other school level tests and examinations. Its easy to take print of thee assignments as all are available in PDF format.

Some advantages of Free Assignments for Class 9 Mathematics

• Solving Assignments for Mathematics Class 9 helps to further enhance understanding of the topics given in your text book which will help you to get better marks
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• You will be able to understand the type of questions which are expected in your Mathematics class test
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• Parents will be able to take print out of the assignments and give to their child easily.

All free Printable practice assignments are in PDF single lick download format and have been prepared by Class 9 Mathematics teachers after full study of all topics which have been given in each chapter so that the students are able to take complete benefit from the worksheets. The Chapter wise question bank and revision assignments can be accessed free and anywhere. Go ahead and click on the links above to download free CBSE Class 9 Mathematics Assignments PDF.

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You can get free PDF downloadable assignments for Grade 9 Mathematics from our website which has been developed by teachers after doing extensive research in each topic.

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RS Aggarwal Solutions Class 9 Chapter 3 Factorisation of Polynomials

RS Aggarwal Solutions for Class 9 Maths Book Chapter 3 Factorisation of Polynomials are available here. Study path has prepared solutions of all the exercises of the chapter by our expert math teachers to help you to get good marks in exams. This lesson has a ton of questions that are very important from the examination point of view.

We at Study Path solved each questions step by step with detailed explanations. Students must practice from practice these problems to score high marks in Maths. Below we have listed the Class 9 RS Aggarwal Solutions Chapter 3 Factorisation of Polynomials Exercise 3A, Ex 3B, Ex 3C, Ex 3D, 3E, Ex 3F, Ex 3G and Multiple Choice Questions (MCQs).

Class 9 RS Aggarwal Solutions Chapter 3 Factorisation of Polynomials

Class 9 rs aggarwal solutions chapter 3 ex 3a.

Class 9 RS Aggarwal Solutions Chapter 3 Ex 3B

Class 9 RS Aggarwal Solutions Chapter 3 Ex 3C

Class 9 RS Aggarwal Solutions Chapter 3 Ex 3D

Class 9 RS Aggarwal Solutions Chapter 3 Ex 3E

Class 9 RS Aggarwal Solutions Chapter 3 Ex 3F

Class 9 RS Aggarwal Solutions Chapter 3 Ex 3G

Class 9 RS Aggarwal Solutions Chapter 3 MCQs

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CBSE NCERT Solutions

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Class 9 Mathematics Polynomials Assignments

We have provided below free printable Class 9 Mathematics Polynomials Assignments for Download in PDF. The Assignments have been designed based on the latest NCERT Book for Class 9 Mathematics Polynomials . These Assignments for Grade 9 Mathematics Polynomials cover all important topics which can come in your standard 9 tests and examinations. Free printable Assignments for CBSE Class 9 Mathematics Polynomials , school and class assignments, and practice test papers have been designed by our highly experienced class 9 faculty. You can free download CBSE NCERT printable Assignments for Mathematics Polynomials Class 9 with solutions and answers. All Assignments and test sheets have been prepared by expert teachers as per the latest Syllabus in Mathematics Polynomials Class 9. Students can click on the links below and download all Pdf Assignments for Mathematics Polynomials class 9 for free. All latest Kendriya Vidyalaya Class 9 Mathematics Polynomials Assignments with Answers and test papers are given below.

Mathematics Polynomials Class 9 Assignments Pdf Download

We have provided below the biggest collection of free CBSE NCERT KVS Assignments for Class 9 Mathematics Polynomials . Students and teachers can download and save all free Mathematics Polynomials assignments in Pdf for grade 9th. Our expert faculty have covered Class 9 important questions and answers for Mathematics Polynomials as per the latest syllabus for the current academic year. All test papers and question banks for Class 9 Mathematics Polynomials and CBSE Assignments for Mathematics Polynomials Class 9 will be really helpful for standard 9th students to prepare for the class tests and school examinations. Class 9th students can easily free download in Pdf all printable practice worksheets given below.

Topicwise Assignments for Class 9 Mathematics Polynomials Download in Pdf

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NCERT Solutions for Class 9 Maths Chapter 1 Number Systems

Ncert solutions class 9 maths chapter 1 – cbse free pdf download.

Download Exclusively Curated Chapter Notes for Class 9 Maths Chapter – 1 Number Systems

Download most important questions for class 9 maths chapter – 1 number systems.

In NCERT Solutions for Class 9 Maths Chapter 1 , students are introduced to several important topics that are considered to be very crucial for those who wish to pursue Mathematics as a subject in their higher classes. Based on these NCERT Solutions , students can practise and prepare for their upcoming CBSE exams, as well as equip themselves with the basics of Class 10. These Maths Solutions of NCERT Class 9 are helpful as they are prepared with respect to the latest update on the CBSE syllabus for 2023-24 and its guidelines.

• Chapter 1- Number Systems
• Chapter 2 Polynomials
• Chapter 3 Coordinate Geometry
• Chapter 4 Linear Equations in Two Variables
• Chapter 5 Introduction to Euclids Geometry
• Chapter 6 Lines and Angles
• Chapter 7 Triangles
• Chapter 8 Quadrilaterals
• Chapter 9 Areas of Parallelograms and Triangles
• Chapter 10 Circles
• Chapter 11 Constructions
• Chapter 12 Heron’s Formula
• Chapter 13 Surface Areas and Volumes
• Chapter 14 Statistics
• Chapter 15 Inroduction to Probability

NCERT Solutions for Class 9 Maths Chapter 1 – Number Systems

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Access Answers to NCERT Class 9 Maths Chapter 1 – Number Systems

Exercise 1.1 page: 5.

1. Is zero a rational number? Can you write it in the form p/q where p and q are integers and q ≠ 0?

We know that a number is said to be rational if it can be written in the form p/q , where p and q are integers and q ≠ 0.

Taking the case of ‘0’,

Zero can be written in the form 0/1, 0/2, 0/3 … as well as , 0/1, 0/2, 0/3 ..

Since it satisfies the necessary condition, we can conclude that 0 can be written in the p/q form, where q can either be positive or negative number.

Hence, 0 is a rational number.

2. Find six rational numbers between 3 and 4.

There are infinite rational numbers between 3 and 4.

As we have to find 6 rational numbers between 3 and 4, we will multiply both the numbers, 3 and 4, with 6+1 = 7 (or any number greater than 6)

i.e., 3 × (7/7) = 21/7

and, 4 × (7/7) = 28/7. The numbers in between 21/7 and 28/7 will be rational and will fall between 3 and 4.

Hence, 22/7, 23/7, 24/7, 25/7, 26/7, 27/7 are the 6 rational numbers between 3 and 4.

3. Find five rational numbers between 3/5 and 4/5.

There are infinite rational numbers between 3/5 and 4/5.

To find out 5 rational numbers between 3/5 and 4/5, we will multiply both the numbers 3/5 and 4/5

with 5+1=6 (or any number greater than 5)

i.e., (3/5) × (6/6) = 18/30

and, (4/5) × (6/6) = 24/30

The numbers in between18/30 and 24/30 will be rational and will fall between 3/5 and 4/5.

Hence,19/30, 20/30, 21/30, 22/30, 23/30 are the 5 rational numbers between 3/5 and 4/5

4. State whether the following statements are true or false. Give reasons for your answers.

(i) Every natural number is a whole number.

Natural numbers- Numbers starting from 1 to infinity (without fractions or decimals)

i.e., Natural numbers = 1,2,3,4…

Whole numbers – Numbers starting from 0 to infinity (without fractions or decimals)

i.e., Whole numbers = 0,1,2,3…

Or, we can say that whole numbers have all the elements of natural numbers and zero.

Every natural number is a whole number; however, every whole number is not a natural number.

(ii) Every integer is a whole number.

Integers- Integers are set of numbers that contain positive, negative and 0; excluding fractional and decimal numbers.

i.e., integers= {…-4,-3,-2,-1,0,1,2,3,4…}

Whole numbers- Numbers starting from 0 to infinity (without fractions or decimals)

i.e., Whole numbers= 0,1,2,3….

Hence, we can say that integers include whole numbers as well as negative numbers.

Every whole number is an integer; however, every integer is not a whole number.

(iii) Every rational number is a whole number.

Rational numbers- All numbers in the form p/q, where p and q are integers and q≠0.

i.e., Rational numbers = 0, 19/30 , 2, 9/-3, -12/7…

All whole numbers are rational, however, all rational numbers are not whole numbers.

Exercise 1.2 Page: 8

1. State whether the following statements are true or false. Justify your answers.

(i) Every irrational number is a real number.

Irrational Numbers – A number is said to be irrational, if it cannot be written in the p/q, where p and q are integers and q ≠ 0.

i.e., Irrational numbers = π, e, √3, 5+√2, 6.23146…. , 0.101001001000….

Real numbers – The collection of both rational and irrational numbers are known as real numbers.

i.e., Real numbers = √2, √5, , 0.102…

Every irrational number is a real number, however, every real number is not an irrational number.

(ii) Every point on the number line is of the form √m where m is a natural number.

The statement is false since as per the rule, a negative number cannot be expressed as square roots.

E.g., √9 =3 is a natural number.

But √2 = 1.414 is not a natural number.

Similarly, we know that there are negative numbers on the number line, but when we take the root of a negative number it becomes a complex number and not a natural number.

E.g., √-7 = 7i, where i = √-1

The statement that every point on the number line is of the form √m, where m is a natural number is false.

(iii) Every real number is an irrational number.

The statement is false. Real numbers include both irrational and rational numbers. Therefore, every real number cannot be an irrational number.

Every irrational number is a real number, however, every real number is not irrational.

2. Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.

No, the square roots of all positive integers are not irrational.

For example,

√4 = 2 is rational.

√9 = 3 is rational.

Hence, the square roots of positive integers 4 and 9 are not irrational. ( 2 and 3, respectively).

3. Show how √5 can be represented on the number line.

Step 1: Let line AB be of 2 unit on a number line.

Step 2: At B, draw a perpendicular line BC of length 1 unit.

Step 3: Join CA

Step 4: Now, ABC is a right angled triangle. Applying Pythagoras theorem,

AB 2 +BC 2 = CA 2

2 2 +1 2 = CA 2 = 5

⇒ CA = √5 . Thus, CA is a line of length √5 unit.

Step 4: Taking CA as a radius and A as a center draw an arc touching

the number line. The point at which number line get intersected by

arc is at √5 distance from 0 because it is a radius of the circle

whose center was A.

Thus, √5 is represented on the number line as shown in the figure.

4. Classroom activity (Constructing the ‘square root spiral’) : Take a large sheet of paper and construct the ‘square root spiral’ in the following fashion. Start with a point O and draw a line segment OP1 of unit length. Draw a line segment P1P2 perpendicular to OP 1 of unit length (see Fig. 1.9). Now draw a line segment P 2 P 3 perpendicular to OP 2 . Then draw a line segment P 3 P 4 perpendicular to OP 3 . Continuing in Fig. 1.9 :

Constructing this manner, you can get the line segment P n-1 Pn by square root spiral drawing a line segment of unit length perpendicular to OP n-1 . In this manner, you will have created the points P 2 , P 3 ,….,Pn,… ., and joined them to create a beautiful spiral depicting √2, √3, √4, …

Step 1: Mark a point O on the paper. Here, O will be the center of the square root spiral.

Step 2: From O, draw a straight line, OA, of 1cm horizontally.

Step 3: From A, draw a perpendicular line, AB, of 1 cm.

Step 4: Join OB. Here, OB will be of √2

Step 5: Now, from B, draw a perpendicular line of 1 cm and mark the end point C.

Step 6: Join OC. Here, OC will be of √3

Step 7: Repeat the steps to draw √4, √5, √6….

Exercise 1.3 Page: 14

1. Write the following in decimal form and say what kind of decimal expansion each has :

= 0.36 (Terminating)

= 4.125 (Terminating)

(vi) 329/400

= 0.8225 (Terminating)

2. You know that 1/7 = 0.142857. Can you predict what the decimal expansions of 2/7, 3/7, 4/7, 5/7, 6/7 are, without actually doing the long division? If so, how?

[Hint: Study the remainders while finding the value of 1/7 carefully.]

3. Express the following in the form p/q, where p and q are integers and q 0.

Assume that   x  = 0.666…

Then,10 x  = 6.666…

10 x  = 6 +  x

(ii) \(\begin{array}{l}0.4\overline{7}\end{array} \)

= (4/10)+(0.777/10)

Assume that  x  = 0.777…

Then, 10 x  = 7.777…

10 x  = 7 +  x

(4/10)+(0.777../10) = (4/10)+(7/90) ( x = 7/9 and x = 0.777…0.777…/10 = 7/(9×10) = 7/90 )

= (36/90)+(7/90) = 43/90

Assume that   x  = 0.001001…

Then, 1000 x  = 1.001001…

1000 x  = 1 +  x

4. Express 0.99999…. in the form p/q . Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

Assume that x  = 0.9999…..Eq (a)

Multiplying both sides by 10,

10 x  = 9.9999…. Eq. (b)

Eq.(b) – Eq.(a), we get

10 x  = 9.9999

– x  = -0.9999…

_____________

The difference between 1 and 0.999999 is 0.000001 which is negligible.

Hence, we can conclude that, 0.999 is too much near 1, therefore, 1 as the answer can be justified.

5. What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 1/17 ? Perform the division to check your answer.

Dividing 1 by 17:

There are 16 digits in the repeating block of the decimal expansion of 1/17.

6. Look at several examples of rational numbers in the form p/q (q ≠ 0), where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?

We observe that when q is 2, 4, 5, 8, 10… Then the decimal expansion is terminating. For example:

1/2 = 0. 5, denominator q = 2 1

7/8 = 0. 875, denominator q =2 3

4/5 = 0. 8, denominator q = 5 1

We can observe that the terminating decimal may be obtained in the situation where prime factorization of the denominator of the given fractions has the power of only 2 or only 5 or both.

7. Write three numbers whose decimal expansions are non-terminating non-recurring.

We know that all irrational numbers are non-terminating non-recurring. three numbers with decimal expansions that are non-terminating non-recurring are:

• √3 = 1.732050807568
• √26 =5.099019513592
• √101 = 10.04987562112

8. Find three different irrational numbers between the rational numbers 5/7 and 9/11.

Three different irrational numbers are:

• 0.73073007300073000073…
• 0.75075007300075000075…
• 0.76076007600076000076…

9.  Classify the following numbers as rational or irrational according to their type:

√23 = 4.79583152331…

Since the number is non-terminating and non-recurring therefore, it is an irrational number.

√225 = 15 = 15/1

Since the number can be represented in p/q form, it is a rational number.

(iii) 0.3796

Since the number,0.3796, is terminating, it is a rational number.

(iv) 7.478478

The number,7.478478, is non-terminating but recurring, it is a rational number.

(v) 1.101001000100001…

Since the number,1.101001000100001…, is non-terminating non-repeating (non-recurring), it is an irrational number.

Exercise 1.4 Page: 18

1. Visualise 3.765 on the number line, using successive magnification.

Exercise 1.5 Page: 24

1. Classify the following numbers as rational or irrational:

We know that, √5 = 2.2360679…

Here, 2.2360679…is non-terminating and non-recurring.

Now, substituting the value of √5 in 2 –√5, we get,

2-√5 = 2-2.2360679… = -0.2360679

Since the number, – 0.2360679…, is non-terminating non-recurring, 2 –√5 is an irrational number.

(ii) (3 +√23)- √23

(3 + √ 23) –√23 = 3+ √ 23–√23

Since the number 3/1 is in p/q form, ( 3 +√23)- √23 is rational.

(iii) 2√7/7√7

2√7/7√7 = ( 2/7)× (√7/√7)

We know that (√7/√7) = 1

Hence, ( 2/7)× (√7/√7) = (2/7)×1 = 2/7

Since the number, 2/7 is in p/q form, 2√7/7√7 is rational.

Multiplying and dividing numerator and denominator by √2 we get,

(1/√2) ×(√2/√2)= √2/2 ( since √2×√2 = 2)

We know that, √2 = 1.4142…

Then, √2/2 = 1.4142/2 = 0.7071..

Since the number , 0.7071..is non-terminating non-recurring, 1/√2 is an irrational number.

We know that, the value of = 3.1415

Hence, 2 = 2×3.1415.. = 6.2830…

Since the number, 6.2830…, is non-terminating non-recurring, 2 is an irrational number.

2. Simplify each of the following expressions:

(i) (3+√3)(2+√2)

(3+√3)(2+√2 )

Opening the brackets, we get, (3×2)+(3×√2)+(√3×2)+(√3×√2)

= 6+3√2+2√3+√6

(ii) (3+√3)(3-√3 )

(3+√3)(3-√3 ) = 3 2 -(√3) 2 = 9-3

(iii) (√5+√2) 2

(√5+√2) 2 = √5 2 +(2×√5×√2)+ √2 2

= 5+2×√10+2 = 7+2√10

(iv) (√5-√2)(√5+√2)

(√5-√2)(√5+√2) = (√5 2 -√2 2 ) = 5-2 = 3

3. Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter, (say d). That is, π =c/d. This seems to contradict the fact that π is irrational. How will you resolve this contradiction?

There is no contradiction. When we measure a value with a scale, we only obtain an approximate value. We never obtain an exact value. Therefore, we may not realize whether c or d is irrational. The value of π is almost equal to 22/7 or 3.142857…

4. Represent (√9.3) on the number line.

Step 1: Draw a 9.3 units long line segment, AB. Extend AB to C such that BC=1 unit.

Step 2: Now, AC = 10.3 units. Let the centre of AC be O.

Step 3: Draw a semi-circle of radius OC with centre O.

Step 4: Draw a BD perpendicular to AC at point B intersecting the semicircle at D. Join OD.

Step 5: OBD, obtained, is a right angled triangle.

Here, OD 10.3/2 (radius of semi-circle), OC = 10.3/2 , BC = 1

OB = OC – BC

⟹ (10.3/2)-1 = 8.3/2

Using Pythagoras theorem,

OD 2 =BD 2 +OB 2

⟹ (10.3/2) 2 = BD 2 +(8.3/2) 2

⟹ BD 2 = (10.3/2) 2 -(8.3/2) 2

⟹ (BD) 2 = (10.3/2)-(8.3/2)(10.3/2)+(8.3/2)

⟹ BD 2  = 9.3

⟹ BD =  √9.3

Thus, the length of BD is √9.3.

Step 6: Taking BD as radius and B as centre draw an arc which touches the line segment. The point where it touches the line segment is at a distance of √9.3 from O as shown in the figure.

5. Rationalize the denominators of the following:

Multiply and divide 1/√7 by √7

(1×√7)/(√7×√7) = √7/7

(ii) 1/(√7-√6)

Multiply and divide 1/(√7-√6) by (√7+√6)

= (√7+√6)/√7 2 -√6 2 [denominator is obtained by the property, (a+b)(a-b) = a 2 -b 2 ]

= (√7+√6)/(7-6)

= (√7+√6)/1

(iii) 1/(√5+√2)

Multiply and divide 1/(√5+√2) by (√5-√2)

= (√5-√2)/(√5 2 -√2 2 ) [denominator is obtained by the property, (a+b)(a-b) = a 2 -b 2 ]

= (√5-√2)/(5-2)

= (√5-√2)/3

(iv) 1/(√7-2)

Multiply and divide 1/(√7-2) by (√7+2)

1/(√7-2)×(√7+2)/(√7+2) = (√7+2)/(√7-2)(√7+2)

= (√7+2)/(√7 2 -2 2 ) [denominator is obtained by the property, (a+b)(a-b) = a 2 -b 2 ]

= (√7+2)/(7-4)

Exercise 1.6 Page: 26

64 1/2 = (8×8) 1/2

= 8 1 [⸪2×1/2 = 2/2 =1]

32 1/5 = (2 5 ) 1/5

= 2 1 [⸪5×1/5 = 1]

(iii)125 1/3

(125) 1/3 = (5×5×5) 1/3

= 5 1 (3×1/3 = 3/3 = 1)

9 3/2 = (3×3) 3/2

= (3 2 ) 3/2

= 3 3 [⸪2×3/2 = 3]

(ii) 32 2/5

32 2/5 = (2×2×2×2×2) 2/5

= (2 5 ) 2⁄5

= 2 2 [⸪5×2/5= 2]

(iii)16 3/4

16 3/4 = (2×2×2×2) 3/4

= (2 4 ) 3⁄4

= 2 3 [⸪4×3/4 = 3]

(iv) 125 -1/3

125 -1/3 = (5×5×5) -1/3

= (5 3 ) -1⁄3

= 5 -1 [⸪3×-1/3 = -1]

3. Simplify :

(i) 2 2/3 ×2 1/5

2 2/3 ×2 1/5 = 2 (2/3)+(1/5) [⸪Since, a m ×a n =a m+n ____ Laws of exponents]

= 2 13/15 [⸪2/3 + 1/5 = (2×5+3×1)/(3×5) = 13/15]

(ii) (1/3 3 ) 7

(1/3 3 ) 7 = (3 -3 ) 7 [⸪Since,(a m ) n = a m x n ____ Laws of exponents]

(iii) 11 1/2 /11 1/4

11 1/2 /11 1/4 = 11 (1/2)-(1/4)

= 11 1/4 [⸪(1/2) – (1/4) = (1×4-2×1)/(2×4) = 4-2)/8 = 2/8 = ¼ ]

(iv) 7 1/2 ×8 1/2

7 1/2 ×8 1/2 = (7×8) 1/2 [⸪Since, (a m ×b m = (a×b) m ____ Laws of exponents]

As the Number System is one of the important topics in Maths, it has a weightage of 8 marks in Class 9 Maths CBSE exams. On an average three questions are asked from this unit.

• One out of three questions in part A (1 marks).
• One out of three questions in part B (2 marks).
• One out of three questions in part C (3 marks).

This chapter talks about:

• Introduction of Number Systems
• Irrational Numbers
• Real Numbers and their Decimal Expansions
• Representing Real Numbers on the Number Line.
• Operations on Real Numbers
• Laws of Exponents for Real Numbers

List of Exercises in NCERT Solutions for Class 9 Maths Chapter 1:

Exercise 1.1 Solutions 4 Questions ( 2 long, 2 short)

Exercise 1.2 Solutions 4 Questions ( 3 long, 1 short)

Exercise 1.3 Solutions 9 Questions ( 9 long)

Exercise 1.4 Solutions 2 Questions ( 2 long)

Exercise 1.5 Solutions 5 Questions ( 4 long 1 short)

Exercise 1.6 Solutions 3 Questions ( 3 long)

NCERT Solutions for Class 9 Maths Chapter 1- Number Systems

NCERT Solutions for Class 9 Maths Chapter 1 Number System is the first chapter of Class 9 Maths. The Number System is discussed in detail in this chapter. The chapter discusses the Number Systems and their applications. The introduction of the chapter includes whole numbers, integers and rational numbers.

The chapter starts with the introduction of Number Systems in section 1.1, followed by two very important topics in sections 1.2 and 1.3

• Irrational Numbers – The numbers which can’t be written in the form of p/q.
• Real Numbers and their Decimal Expansions – Here, you study the decimal expansions of real numbers and see whether it can help in distinguishing between rational and irrational.

Next, it discusses the following topics.

• Representing Real Numbers on the Number Line – In this, the solutions for 2 problems in Exercise 1.4.
• Operations on Real Numbers – Here, you explore some of the operations like addition, subtraction, multiplication and division on irrational numbers.
• Laws of Exponents for Real Numbers – Use these laws of exponents to solve the questions.

Explore more about Number Systems and learn how to solve various kinds of problems only on  NCERT Solutions For Class 9 Maths . It is also one of the best academic resources to revise for your CBSE exams.

Key Advantages of NCERT Solutions for Class 9 Maths Chapter 1 – Number Systems

• These NCERT Solutions for Class 9 Maths help you solve and revise the whole CBSE syllabus of Class 9.
• After going through the step-wise solutions given by our subject expert teachers, you will be able to score more marks in the board exams.
• It follows NCERT guidelines.
• It contains all the important questions from the examination point of view.

The faculty have curated the solutions in a lucid manner to improve the problem-solving abilities of the students. For a more clear idea about Number Systems, students can refer to the study materials available at BYJU’S.

• RD Sharma Solutions for Class 9 Maths Number Systems

Disclaimer: 

Dropped Topics – 1.4 Representing real numbers on the number line.

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