case study of ch 1 class 9 maths

Class 9 Maths Case Study Questions of Chapter 1 Real Numbers

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Case study Questions in Class 9 Mathematics Chapter 1  are very important to solve for your exam. Class 9 Maths Chapter 1 Case Study Questions have been prepared for the latest exam pattern. You can check your knowledge by solving  Class 9 Maths Case Study Questions  Chapter 1 Real Numbers

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In CBSE Class 9 Maths Paper, Students will have to answer some questions based on Assertion and Reason. There will be a few questions based on case studies and passage-based as well. In that, a paragraph will be given, and then the MCQ questions based on it will be asked.

Real Numbers Case Study Questions With Answers

Here, we have provided case-based/passage-based questions for Class 9 Maths Chapter 1 Real Numbers

Case Study/Passage-Based Questions

Case Study 1: A Mathematics Exhibition is being conducted in your school and one of your friends is making a model of a factor tree. He has some difficulty and asks for your help in completing a quiz for the audience.

Observe the following factor tree and answer the following:

1. What will be the value of x?

Answer: b) 13915

2. What will be the value of y?

Answer: c) 11

3. What will be the value of z?

Answer: b) 23

4. According to the Fundamental Theorem of Arithmetic 13915 is a

a) Composite number

b) Prime number

c) Neither prime nor composite

d) Even number

Answer: a) Composite number

5. The prime factorization of 13915 is

a) 5 × 11 3  × 13 2

b) 5 × 11 3  × 23 2

c) 5 × 11 2  × 23

d) 5 × 11 2  × 13 2

Answer: c) 5 × 112 × 23

Case Study 2: Srikanth has made a project on real numbers, where he finely explained the applicability of exponential laws and divisibility conditions on real numbers. He also included some assessment questions at the end of his project as listed below. Answer them.

(i) For what value of n, 4 n  ends in 0?

(a) 10 (b) when n is even (c) when n is odd (d) no value of n

Answer: (d) no value of n3

(ii) If a is a positive rational number and n is a positive integer greater than 1, then for what value of n, an is a rational number?

(a) when n is any even integer (b) when n is any odd integer (c) for all n > 1 (d) only when n=0

Answer: (c) for all n > 1

(iii) If x and y are two odd positive integers, then which of the following is true?

(a) x 2 +y 2  is even (b) x 2 +y 2  is not divisible by 4 (c) x 2 +y 2   is odd (d) both (a) and (b)

Answer: (d) both (a) and (b)

(iv) The statement ‘One of every three consecutive positive integers is divisible by 3’ is

(a) always true (b) always false (c) sometimes true (d) None of these

Answer:(a) always true

(v) If n is any odd integer, then n 2 – 1 is divisible by

(a) 22 (b) 55 (c) 88 (d) 8

Answer: (d) 8

Hope the information shed above regarding Case Study and Passage Based Questions for Class 9 Mathematics Chapter 1 Real Numbers with Answers Pdf free download has been useful to an extent. If you have any other queries about CBSE Class 9 Maths Real Numbers Case Study and Passage Based Questions with Answers, feel free to comment below so that we can revert back to us at the earliest possible By Team Study Rate

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CBSE Class 9 Maths Case Study Questions PDF Download

Download Class 9 Maths Case Study Questions to prepare for the upcoming CBSE Class 9 Exams 2023-24. These Case Study and Passage Based questions are published by the experts of CBSE Experts for the students of CBSE Class 9 so that they can score 100% in Exams.

Case study questions play a pivotal role in enhancing students’ problem-solving skills. By presenting real-life scenarios, these questions encourage students to think beyond textbook formulas and apply mathematical concepts to practical situations. This approach not only strengthens their understanding of mathematical concepts but also develops their analytical thinking abilities.

Table of Contents

CBSE Class 9th MATHS: Chapterwise Case Study Questions

Inboard exams, students will find the questions based on assertion and reasoning. Also, there will be a few questions based on case studies. In that, a paragraph will be given, and then the MCQ questions based on it will be asked. For Class 9 Maths Case Study Questions, there would be 5 case-based sub-part questions, wherein a student has to attempt 4 sub-part questions.

Chapterwise Case Study Questions of Class 9 Maths

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

Checkout: Class 9 Science Case Study Questions

And for mathematical calculations, tap Math Calculators which are freely proposed to make use of by calculator-online.net

The above  Class 9 Maths Case Study Question s will help you to boost your scores as Case Study questions have been coming in your examinations. These CBSE Class 9 Maths Case Study Questions have been developed by experienced teachers of cbseexpert.com for the benefit of Class 10 students.

Class 9 Maths Syllabus 2023-24

UNIT I: NUMBER SYSTEMS

1. REAL NUMBERS (18 Periods)

1. Review of representation of natural numbers, integers, and rational numbers on the number line. Rational numbers as recurring/ terminating decimals. Operations on real numbers.

2. Examples of non-recurring/non-terminating decimals. Existence of non-rational numbers (irrational numbers) such as √2, √3 and their representation on the number line. Explaining that every real number is represented by a unique point on the number line and conversely, viz. every point on the number line represents a unique real number.

3. Definition of nth root of a real number.

4. Rationalization (with precise meaning) of real numbers of the type

(and their combinations) where x and y are natural number and a and b are integers.

5. Recall of laws of exponents with integral powers. Rational exponents with positive real bases (to be done by particular cases, allowing learner to arrive at the general laws.)

UNIT II: ALGEBRA

1. POLYNOMIALS (26 Periods)

Definition of a polynomial in one variable, with examples and counter examples. Coefficients of a polynomial, terms of a polynomial and zero polynomial. Degree of a polynomial. Constant, linear, quadratic and cubic polynomials. Monomials, binomials, trinomials. Factors and multiples. Zeros of a polynomial. Motivate and State the Remainder Theorem with examples. Statement and proof of the Factor Theorem. Factorization of ax2 + bx + c, a ≠ 0 where a, b and c are real numbers, and of cubic polynomials using the Factor Theorem. Recall of algebraic expressions and identities. Verification of identities:

RELATED STORIES

and their use in factorization of polynomials.

2. LINEAR EQUATIONS IN TWO VARIABLES (16 Periods)

Recall of linear equations in one variable. Introduction to the equation in two variables. Focus on linear equations of the type ax + by + c=0.Explain that a linear equation in two variables has infinitely many solutions and justify their being written as ordered pairs of real numbers, plotting them and showing that they lie on a line.

UNIT III: COORDINATE GEOMETRY COORDINATE GEOMETRY (7 Periods)

The Cartesian plane, coordinates of a point, names and terms associated with the coordinate plane, notations.

UNIT IV: GEOMETRY

1. INTRODUCTION TO EUCLID’S GEOMETRY (7 Periods)

History – Geometry in India and Euclid’s geometry. Euclid’s method of formalizing observed phenomenon into rigorous Mathematics with definitions, common/obvious notions, axioms/postulates and theorems. The five postulates of Euclid. Showing the relationship between axiom and theorem, for example: (Axiom)

1. Given two distinct points, there exists one and only one line through them. (Theorem)

2. (Prove) Two distinct lines cannot have more than one point in common.

2. LINES AND ANGLES (15 Periods)

1. (Motivate) If a ray stands on a line, then the sum of the two adjacent angles so formed is 180O and the converse.

2. (Prove) If two lines intersect, vertically opposite angles are equal.

3. (Motivate) Lines which are parallel to a given line are parallel.

3. TRIANGLES (22 Periods)

1. (Motivate) Two triangles are congruent if any two sides and the included angle of one triangle is equal to any two sides and the included angle of the other triangle (SAS Congruence).

2. (Prove) Two triangles are congruent if any two angles and the included side of one triangle is equal to any two angles and the included side of the other triangle (ASA Congruence).

3. (Motivate) Two triangles are congruent if the three sides of one triangle are equal to three sides of the other triangle (SSS Congruence).

4. (Motivate) Two right triangles are congruent if the hypotenuse and a side of one triangle are equal (respectively) to the hypotenuse and a side of the other triangle. (RHS Congruence)

5. (Prove) The angles opposite to equal sides of a triangle are equal.

6. (Motivate) The sides opposite to equal angles of a triangle are equal.

4. QUADRILATERALS (13 Periods)

1. (Prove) The diagonal divides a parallelogram into two congruent triangles.

2. (Motivate) In a parallelogram opposite sides are equal, and conversely.

3. (Motivate) In a parallelogram opposite angles are equal, and conversely.

4. (Motivate) A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and equal.

5. (Motivate) In a parallelogram, the diagonals bisect each other and conversely.

6. (Motivate) In a triangle, the line segment joining the mid points of any two sides is parallel to the third side and in half of it and (motivate) its converse.

5. CIRCLES (17 Periods)

1. (Prove) Equal chords of a circle subtend equal angles at the center and (motivate) its converse.

2. (Motivate) The perpendicular from the center of a circle to a chord bisects the chord and conversely, the line drawn through the center of a circle to bisect a chord is perpendicular to the chord.

3. (Motivate) Equal chords of a circle (or of congruent circles) are equidistant from the center (or their respective centers) and conversely.

4. (Prove) The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.

5. (Motivate) Angles in the same segment of a circle are equal.

6. (Motivate) If a line segment joining two points subtends equal angle at two other points lying on the same side of the line containing the segment, the four points lie on a circle.

7. (Motivate) The sum of either of the pair of the opposite angles of a cyclic quadrilateral is 180° and its converse.

UNIT V: MENSURATION 1.

1. AREAS (5 Periods)

Area of a triangle using Heron’s formula (without proof)

2. SURFACE AREAS AND VOLUMES (17 Periods)

Surface areas and volumes of spheres (including hemispheres) and right circular cones.

UNIT VI: STATISTICS & PROBABILITY

STATISTICS (15 Periods)

 Bar graphs, histograms (with varying base lengths), and frequency polygons.

To crack case study questions, Class 9 Mathematics students need to apply their mathematical knowledge to real-life situations. They should first read the question carefully and identify the key information. They should then identify the relevant mathematical concepts that can be applied to solve the question. Once they have done this, they can start solving the Class 9 Mathematics case study question.

Benefits of Practicing CBSE Class 9 Maths Case Study Questions

Regular practice of CBSE Class 9 Maths case study questions offers several benefits to students. Some of the key advantages include:

• Deeper Understanding : Case study questions foster a deeper understanding of mathematical concepts by connecting them to real-world scenarios. This improves retention and comprehension.
• Practical Application : Students learn to apply mathematical concepts to practical situations, preparing them for real-life problem-solving beyond the classroom.
• Critical Thinking : Case study questions require students to think critically, analyze data, and devise appropriate solutions. This nurtures their critical thinking abilities, which are valuable in various academic and professional domains.
• Exam Readiness : By practicing case study questions, students become familiar with the question format and gain confidence in their problem-solving abilities. This enhances their readiness for CBSE Class 9 Maths exams.
• Holistic Development: Solving case study questions cultivates not only mathematical skills but also essential life skills like analytical thinking, decision-making, and effective communication.

Tips to Solve CBSE Class 9 Maths Case Study Questions Effectively

Solving case study questions can be challenging, but with the right approach, you can excel. Here are some tips to enhance your problem-solving skills:

• Read the case study thoroughly and understand the problem statement before attempting to solve it.
• Identify the relevant data and extract the necessary information for your solution.
• Break down complex problems into smaller, manageable parts to simplify the solution process.
• Apply the appropriate mathematical concepts and formulas, ensuring a solid understanding of their principles.
• Clearly communicate your solution approach, including the steps followed, calculations made, and reasoning behind your choices.
• Practice regularly to familiarize yourself with different types of case study questions and enhance your problem-solving speed.Class 9 Maths Case Study Questions

Remember, solving case study questions is not just about finding the correct answer but also about demonstrating a logical and systematic approach. Now, let’s explore some resources that can aid your preparation for CBSE Class 9 Maths case study questions.

Q1. Are case study questions included in the Class 9 Maths Case Study Questions syllabus?

Yes, case study questions are an integral part of the CBSE Class 9 Maths syllabus. They are designed to enhance problem-solving skills and encourage the application of mathematical concepts to real-life scenarios.

Q2. How can solving case study questions benefit students ?

Solving case study questions enhances students’ problem-solving skills, analytical thinking, and decision-making abilities. It also bridges the gap between theoretical knowledge and practical application, making mathematics more relevant and engaging.

Q3. How do case study questions help in exam preparation?

Case study questions help in exam preparation by familiarizing students with the question format, improving analytical thinking skills, and developing a systematic approach to problem-solving. Regular practice of case study questions enhances exam readiness and boosts confidence in solving such questions.

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CBSE Class 9th Maths 2023 : 30 Most Important Case Study Questions with Answers; Download PDF

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CBSE Class 9 Maths exam 2022-23 will have a set of questions based on case studies in the form of MCQs. CBSE Class 9 Maths Question Bank on Case Studies given in this article can be very helpful in understanding the new format of questions.

Each question has five sub-questions, each followed by four options and one correct answer. Students can easily download these questions in PDF format and refer to them for exam preparation.

CBSE Class 9 All Students can also Download here Class 9 Other Study Materials in PDF Format.

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• CBSE Class 9th Maths 2023 : 30 Most Important Case Study Questions with Answers; Download PDF 10 February, 2023, 6:20 pm
• CBSE Class 9th Maths 2023 : Important Assertion Reason Question with Solution Download Pdf 9 February, 2023, 12:16 pm
• CBSE Class 9th Exam 2023 : Social Science Most Important Short Notes; Download PDF 16 January, 2023, 4:29 pm
• CBSE Class 9th Mathematics 2023 : Most Important Short Notes with Solutions 27 December, 2022, 6:05 pm
• CBSE Class 9th English 2023 : Chapter-wise Competency-Based Test Items with Answer; Download PDF 21 December, 2022, 5:16 pm
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CBSE Case Study Questions for Class 9 Maths - Pdf PDF Download

Cbse case study questions for class  9 maths.

CBSE Case Study Questions for Class 9 Maths are a type of assessment where students are given a real-world scenario or situation and they need to apply mathematical concepts to solve the problem. These types of questions help students to develop their problem-solving skills and apply their knowledge of mathematics to real-life situations.

Chapter Wise Case Based Questions for Class 9 Maths

The CBSE Class 9 Case Based Questions can be accessed from Chapetrwise Links provided below:

Chapter-wise case-based questions for Class 9 Maths are a set of questions based on specific chapters or topics covered in the maths textbook. These questions are designed to help students apply their understanding of mathematical concepts to real-world situations and events.

Chapter 1: Number System

• Case Based Questions: Number System

Chapter 2: Polynomial

• Case Based Questions: Polynomial

Chapter 3: Coordinate Geometry

• Case Based Questions: Coordinate Geometry

Chapter 4: Linear Equations

• Case Based Questions: Linear Equations - 1
• Case Based Questions: Linear Equations -2

Chapter 5: Introduction to Euclid’s Geometry

• Case Based Questions: Lines and Angles

Chapter 7: Triangles

• Case Based Questions: Triangles

Chapter 8: Quadrilaterals

• Case Based Questions: Quadrilaterals - 1
• Case Based Questions: Quadrilaterals - 2

Chapter 9: Areas of Parallelograms

• Case Based Questions: Circles

Chapter 11: Constructions

• Case Based Questions: Constructions

Chapter 12: Heron’s Formula

• Case Based Questions: Heron’s Formula

Chapter 13: Surface Areas and Volumes

• Case Based Questions: Surface Areas and Volumes

Chapter 14: Statistics

• Case Based Questions: Statistics

Chapter 15: Probability

• Case Based Questions: Probability

Weightage of Case Based Questions in Class 9 Maths

Why are Case Study Questions important in Maths Class  9?

• Enhance critical thinking:  Case study questions require students to analyze a real-life scenario and think critically to identify the problem and come up with possible solutions. This enhances their critical thinking and problem-solving skills.
• Apply theoretical concepts:  Case study questions allow students to apply theoretical concepts that they have learned in the classroom to real-life situations. This helps them to understand the practical application of the concepts and reinforces their learning.
• Develop decision-making skills:  Case study questions challenge students to make decisions based on the information provided in the scenario. This helps them to develop their decision-making skills and learn how to make informed decisions.
• Improve communication skills:  Case study questions often require students to present their findings and recommendations in written or oral form. This helps them to improve their communication skills and learn how to present their ideas effectively.
• Enhance teamwork skills:  Case study questions can also be done in groups, which helps students to develop teamwork skills and learn how to work collaboratively to solve problems.

In summary, case study questions are important in Class 9 because they enhance critical thinking, apply theoretical concepts, develop decision-making skills, improve communication skills, and enhance teamwork skills. They provide a practical and engaging way for students to learn and apply their knowledge and skills to real-life situations.

Class 9 Maths Curriculum at Glance

The Class 9 Maths curriculum in India covers a wide range of topics and concepts. Here is a brief overview of the Maths curriculum at a glance:

• Number Systems:  Students learn about the real number system, irrational numbers, rational numbers, decimal representation of rational numbers, and their properties.
• Algebra:  The Algebra section includes topics such as polynomials, linear equations in two variables, quadratic equations, and their solutions.
• Coordinate Geometry:  Students learn about the coordinate plane, distance formula, section formula, and slope of a line.
• Geometry:  This section includes topics such as Euclid’s geometry, lines and angles, triangles, and circles.
• Trigonometry: Students learn about trigonometric ratios, trigonometric identities, and their applications.
• Mensuration: This section includes topics such as area, volume, surface area, and their applications.
• Statistics and Probability:  Students learn about measures of central tendency, graphical representation of data, and probability.

The Class 9 Maths curriculum is designed to provide a strong foundation in mathematics and prepare students for higher education in the field. The curriculum is structured to develop critical thinking, problem-solving, and analytical skills, and to promote the application of mathematical concepts in real-life situations. The curriculum is also designed to help students prepare for competitive exams and develop a strong mathematical base for future academic and professional pursuits.

Students can also access Case Based Questions of all subjects of CBSE Class 9

• Case Based Questions for Class 9 Science
• Case Based Questions for Class 9 Social Science
• Case Based Questions for Class 9 English
• Case Based Questions for Class 9 Hindi
• Case Based Questions for Class 9 Sanskrit

Frequently Asked Questions (FAQs) on Case Based Questions for Class 9 Maths

What is case-based questions.

Case-Based Questions (CBQs) are open-ended problem solving tasks that require students to draw upon their knowledge of Maths concepts and processes to solve a novel problem. CBQs are often used as formative or summative assessments, as they can provide insights into how students reason through and apply mathematical principles in real-world problems.

What are case-based questions in Maths?

Case-based questions in Maths are problem-solving tasks that require students to apply their mathematical knowledge and skills to real-world situations or scenarios.

What are some common types of case-based questions in class 9 Maths?

Common types of case-based questions in class 9 Maths include word problems, real-world scenarios, and mathematical modeling tasks.

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CBSE Class 9 Maths Most Important Case Study Based Questions With Solution

Cbse class 9 mathematics case study questions.

In this post I have provided CBSE Class 9 Maths Case Study Based Questions With Solution. These questions are very important for those students who are preparing for their final class 9 maths exam.

All these questions provided in this article are with solution which will help students for solving the problems. Dear students need to practice all these questions carefully with the help of given solutions.

As you know CBSE Class 9 Maths exam will have a set of cased study based questions in the form of MCQs. CBSE Class 9 Maths Question Bank given in this article can be very helpful in understanding the new format of questions for new session.

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Case studies in class 9 mathematics.

The Central Board of Secondary Education (CBSE) has included case study based questions in the Class 9 Mathematics paper in current session. According to new pattern CBSE Class 9 Mathematics students will have to solve case based questions. This is a departure from the usual theoretical conceptual questions that are asked in Class 9 Maths exam in this year.

Each question provided in this post has five sub-questions, each followed by four options and one correct answer. All CBSE Class 9th Maths Students can easily download these questions in PDF form with the help of given download Links and refer for exam preparation.

There is many more free study materials are available at Maths And Physics With Pandey Sir website. For many more books and free study material all of you can visit at this website.

Given Below Are CBSE Class 9th Maths Case Based Questions With Their Respective Download Links.

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Chapter 1 Class 9 Number Systems

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Get solutions of all NCERT Questions of Chapter 1 Class 9 Number System free at teachoo. Answers to all NCERT Exercises and Examples are solved for your reference. Theory of concepts is also made for your easy understanding

In this chapter, we will learn

• Different Types of numbers like Natural Numbers, Whole numbers, Integers, Rational numbers
• How to find rational numbers between two rational numbers
• What is an irrational number
• Checking if number is irrational or not
• And how to draw an irrational number on the number line
• Then, we will study What a real number is
• And find Decimal expansions - Terminating, Non terminating - repeating, Non terminating Non repeating
• Converting non-terminating repeating numbers into p/q form
• Finding irrational numbers between two numbers
• Representing real numbers on the number line (we use magnification)
• We will learn how to add , subtract and multiply numbers with square root (like 5√2 + 3√3 - 8√2)
• We will learn some identities of numbers with square root (like (√a + √b) 2 )
• How to rationalize numbers
• We will also do questions on Law of Exponents (here, the exponents can also be in fractions)

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NCERT Solutions Class 9 Maths Chapter 1 Number Systems

NCERT solutions for class 9 maths chapter 1 number systems consists of an introduction about the number system and the different kinds of numbers in it. The number system has been classified into different types of numbers like natural numbers, whole numbers , integers, rational numbers, irrational numbers , etc. The NCERT solutions class 9 maths chapter 1 covers all the basics of the number system which will be helpful in forming the basic foundation of mathematics.

Class 9 maths chapter 1 number systems will help the students in differentiating between rational and irrational numbers, wherein irrational numbers cannot be expressed in the form of a ratio, and also about real numbers. Class 9 maths NCERT solutions chapter 1 number systems sample exercises can be downloaded from the links below and also you can find some of these in the exercises given below.

• NCERT Solutions Class 9 Maths Chapter 1 Ex 1.1
• NCERT Solutions Class 9 Maths Chapter 1 Ex 1.2
• NCERT Solutions Class 9 Maths Chapter 1 Ex 1.3
• NCERT Solutions Class 9 Maths Chapter 1 Ex 1.4
• NCERT Solutions Class 9 Maths Chapter 1 Ex 1.5
• NCERT Solutions Class 9 Maths Chapter 1 Ex 1.6

NCERT Solutions for Class 9 Maths Chapter 1 PDF

These NCERT solutions for class 9 maths involving the important concepts of real numbers , rational and irrational numbers, are available for free pdf download. The questions involving real numbers and their decimal form, the law of exponents are given below:

☛ Download Class 9 Maths NCERT Solutions Chapter 1 Number Systems

NCERT Class 9 Maths Chapter 1   Download PDF

NCERT Solutions for Class 9 Maths Chapter 1 Number Systems

It is advisable for the students to practice the questions in the above links as this will give them better clarity on the kind of numbers and their properties. An exercise-wise detailed analysis of NCERT Solutions Class 9 Maths Chapter 1 number systems is given below for reference.

• Class 9 Maths Chapter 1 Ex 1.1 - 4 Questions
• Class 9 Maths Chapter 1 Ex 1.2 - 4 Questions
• Class 9 Maths Chapter 1 Ex 1.3 - 9 Questions
• Class 9 Maths Chapter 1 Ex 1.4 - 2 Questions
• Class 9 Maths Chapter 1 Ex 1.5 - 5 Questions
• Class 9 Maths Chapter 1 Ex 1.6 - 11 Questions

☛ Download Class 9 Maths Chapter 1 NCERT Book

Topics Covered: The important topics focussed upon are irrational numbers, real numbers, and real numbers when expanded in the decimal form. The class 9 maths NCERT solutions chapter 1 covers the representation of real numbers on a number line, methods to perform operations on real numbers, and laws of exponents when dealing with real numbers.

Total Questions: Class 9 maths chapter 1 Number Systems consists of total 35 questions of which 30 are easy, 2 are moderate and 3 are long answer-type questions.

List of Formulas in NCERT Solutions Class 9 Maths Chapter 1

NCERT solutions class 9 maths chapter 1 covers important facts about the number systems which will help strengthen the math foundation. Like if a number ‘a’ is rational, and ‘b’ represents an irrational number, then ‘a+b’, and ‘a-b’ are irrational numbers, and ‘ab’ and ‘a/b’ are supposed to be irrational numbers, and ‘b’ is not equal to zero. For ‘a’ and ‘b’ positive real numbers the following formula or entities will be true:

• √ab = √a √b
• √(a/b) = √a / √b

Important Questions for Class 9 Maths NCERT Solutions Chapter 1

Video solutions for class 9 maths ncert chapter 1, faqs on ncert solutions class 9 maths chapter 1, do i need to practice all questions provided in ncert solutions class 9 maths number systems.

Practicing the NCERT solutions class 9 maths number systems and exercises on real numbers, rational numbers will help in exploring the number systems in a better way. The NCERT Solutions Class 9 Maths Number Systems will also provide a good insight into the solving of problems.

Why are Class 9 Maths NCERT Solutions Chapter 1 Important?

Since the number systems chapter deals with rational and irrational numbers, real numbers, and their expansion, their decimal form, also covering the law of exponents. Hence, this makes the NCERT solutions class 9 maths important for examinations.

What are the Important Formulas in NCERT Solutions Class 9 Maths Chapter 1?

There are several formulas or entities for positive real numbers which will be helpful in learning mathematics even for higher grades. Like if one wants to rationalize the denominator of 1/ ( √a + b ), then we can multiply and divide by its algebraic conjugate which is √a - b

How Many Questions are there in NCERT Solutions Class 9 Maths Chapter 1 Real Numbers?

The questions in the NCERT Solutions Class 9 Maths Chapter 1 are a great way for learning real numbers. There are around 35 questions dealing with number systems with 25 of them being simple and have straightforward logic, 6 of them are with medium complexity and 4 are elaborative questions.

What are the Important Topics Covered in NCERT Solutions Class 9 Maths Chapter 1?

The NCERT Solutions Class 9 Maths Chapter 1 deal with integers, real numbers, rational and irrational numbers. Apart from these the important topics covered are the real numbers, and what happens when they are expanded in decimal form, the law of exponents in the case of real numbers, how to differentiate between rational and irrational numbers etc.

How CBSE Students can utilize NCERT Solutions Class 9 Maths Chapter 1 effectively?

The students should first practice all the examples to understand the logic and problem solving technique and should try to solve all the exercise questions. The CBSE itself recommends the NCERT Solutions Class 9 Maths for the board exam studies.

NCERT Solutions for Class 9 Maths Chapter 1 Number Systems

NCERT Solutions for Class 9 Maths Chapter 1 Number Systems are provided here. Our NCERT Maths solutions contain all the questions of the NCERT textbook that are solved and explained beautifully. Here you will get complete NCERT Solutions for Class 9 Maths Chapter 1 all exercises Exercise in one place. These solutions are prepared by the subject experts and as per the latest NCERT syllabus and guidelines. CBSE Class 9 Students who wish to score good marks in the maths exam must practice these questions regularly.

Class 9 Maths Chapter 1 Number Systems NCERT Solutions

Below we have provided the solutions of each exercise of the chapter. Go through the links to access the solutions of exercises you want. You should also check out our NCERT Class 9 Solutions for other subjects to score good marks in the exams.

NCERT Solutions for Class 9 Maths Chapter 1 Exercise 1.1

NCERT Solutions for Class 9 Maths Chapter 1 Exercise 1.2

NCERT Solutions for Class 9 Maths Chapter 1 Exercise 1.3

NCERT Solutions for Class 9 Maths Chapter 1 Exercise 1.4

NCERT Solutions for Class 9 Maths Chapter 1 Exercise 1.5

NCERT Solutions for Class 9 Maths Chapter 1 Exercise 1.6

NCERT Solutions for Class 9 Maths Chapter 1 – Topic Discussion

Below we have listed the topics that have been discussed in this chapter. As Number System is one of the important topics in Maths, it has a weightage of 6 marks in class 9 Maths exams. 

• 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

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NCERT Solutions for Class 9 Mathematics Chapter 1- Number Systems

Home » NCERT Solutions » NCERT Solutions for Class 9 Mathematics Chapter 1- Number Systems

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Mathematics is a subject which requires a lot of practice. . The more you practice the better you become. . Therefore, you must practice to perfection. There are plenty of  examples to practice with in Extramarks NCERT Solutions for Class 9 Mathematics Chapter 1.

The Chapter -Number Systems of Class 9 Mathematics covers all the fundamentals of Mathematics and will help students understand the core concepts covered in higher classes. . As Mathematics is totally based on  numbers, this  chapter tells about all the different types of numbers and various applications of numbers in Mathematics. If you are looking for a thorough knowledge of the concepts of the Chapter, Extramarks is the right platform to get the right amount of practice and to develop your mathematical abilities and be confident at an early age. 

You can avail of NCERT Solutions for Class 9 Mathematics Chapter 1 on the Extramarks website and turn your child into a smart learner. Number systems- Chapter 1 of Class 9 Mathematics comprises all the  fundamental concepts.. Based on the CBSE NCERT latest 2021-2022 syllabus, we have provided points to ponder as well as detailed solutions for the better understanding of the subject. It also encourages students to be curious and look for answers themselves.  Students are recommended  to use the NCERT solutions Class 9 Mathematics to realise their  true potential and to enjoy the entire process of learning and stay ahead of the competition.    

Visit the Extramarks website to keep yourself updated about the CBSE syllabus, NCERT Solutions and exam patterns. You may  also search for NCERT Solutions Class 10 to step up  your preparation and stay ahead of others.

Key Topics Covered In NCERT Solution for Class 9 Mathematics Chapter 1

Number system is entirely the study of numeracy, and hence the students must understand the concepts and enjoy the learning experience.   It will directly connect to  chapters like Quadratic equations, Sets etc in higher classes. As a result, students aiming for good grades should be able to identify different types of numbers, know their representation and identities and should know how to rationalise them efficiently.

In Extramarks NCERT Solutions for Class 9 Mathematics Chapter 1, students can expect all topics to be covered and explained in detail. The  chapter includes sections like real numbers and their decimal expansion, representing real numbers on the number line, operation on real numbers etc. For complete study material for NCERT Solutions Class 9, NCERT Solutions Class 10, NCERT Solutions Class 11, and NCERT Solutions Class 12, visit the Extramarks website and app which is trusted by students across India and  their  numbers have been growing by leaps and bounds because of the unshakable trust and faith these schools have in us. 

The key topics covered in NCERT Solutions of Class 9 Mathematics Chapter 1:

NCERT Solutions for Class 9 Mathematics Chapter 1 requires students to apply and correlate whatever they have learnt in their previous classes. . Students can also access NCERT Solutions for Mathematics Class 8 and Class 7  to review the  concepts studied last year or earlier.

1.1 Introduction

This Chapter on Number Systems begins with the basic introduction of numbers and their applications in our daily lives. Further, it categorises  numbers as Natural numbers, Whole numbers, Integers, Rational numbers and Irrational numbers. The various examples provided in the chapter help recognise different numbers, which can help easily recall the concepts in prior  Classes.

1.2 Irrational numbers

This section deals entirely with what makes a number irrational and how one can distinguish between rational and irrational numbers. Students have to keep in mind specific points while deciding it is an irrational number which they will find in our NCERT Solutions for Class 9 Mathematics Chapter 1. Students will also read about the set of numbers called real numbers.

At the end of this section, students will get a proper understanding of irrational as well as real numbers. Also, they will be available to locate certain square roots of numbers on the number line. 

1.3 Real numbers and their decimal expansion

In this section, first, you will learn about  decimal expansions of real numbers. Then you would evaluate whether you can distinguish between rational and irrational numbers based on the decimal expansion. You come across different cases and will illustrate them on the basis of examples.

1.4 Representing Real numbers on the number line

As learnt in the previous t section about the decimal expansion of real numbers, we will use it for application on the number line. The decimal expansion helps represent real numbers and get good practice with examples.  

After going through this section, you would be able to locate points of the number line with ease, learn to visualize points on the number line in a systematic way, learn to round off to the nearest decimal and know that a unique point represents every real number.

1.5 Operation on Real numbers

In the earlier Classes, we have learnt that rational numbers follow commutative, associative and distributive properties for mathematical operations, i.e. when you add, subtract, multiply or divide a rational number, you get a rational number. Likewise, this holds true for irrational numbers also. . 

The set of rational and irrational numbers is called real numbers. Hence, this applies to real numbers too. 

After completing this section, you will be able to carry out operations on non-terminating and non-recurring decimal expansions with the help of illustrative examples. Refer to our NCERT Solutions for Class 9 Mathematics Chapter 1 to get access to more solved questions based on Operations on Real Numbers. 

1.6 Law of exponents for real numbers

You are already acquainted with exponents and laws of exponents from your earlier Classes. In this section, we will specifically learn about the laws of exponents on real numbers. The application of laws of exponents remains the same in the case of real numbers. You have to learn to convert the square root or the cube root of the number into exponential form.

NCERT Solutions for Class 9 Mathematics Chapter 1 Exercise &  Solutions

Find NCERT Solutions for Class 9 Mathematics Chapter 1  on the Extramarks website. From a detailed analysis of the Chapter to short notes, you can find everything to level up your preparation and gear up your performance in the exams. You will get access to  all the questions on Number Systems once you access the NCERT Solutions for Class 9 Mathematics on our website.

Click on the below links to view exercise specific questions and solutions for NCERT Solutions for Class 9 Mathematics Chapter 1:

•  Chapter 9: Exercise 1.1 Question and answers    
•  Chapter 9: Exercise 1.2 Question and answers
• Chapter 9: Exercise 1.3 Question and answers
• Chapter 9: Exercise 1.4 Question and answers
• Chapter 9: Exercise 1.5 Question and answers

Along with Class 9 Mathematics Solutions, students can explore NCERT Solutions on our Extramarks website for all primary and secondary  classes.

• NCERT Solutions Class 5 
• NCERT solutions Class 10
• NCERT solutions Class 11
• NCERT solutions Class 12

NCERT Exemplar for Class 9 Mathematics 

NCERT Exemplar Class 9 Mathematics is an excellent resource  for students preparing for their 9th standard exams. The book consists of a variety of questions of different levels  of difficulty. It  encourages students to develop more interest in  Mathematics and get more significant insights into the  chapter to become proficient in facing challenging questions in the exams.

 NCERT Exemplar helps students to develop confidence during their preparation as they have questions of basic level  as well as advanced level. It has proved to be quite  beneficial for students, especially for  those preparing for various competitive exams. It covers  the entire chapters in detail  , which makes it fruitful for all curriculum students.

After referring to the NCERT Solutions and NCERT Exemplar, the students are confident   to solve all the complicated and tricky questions. As a result, students can easily switch  to more advanced and higher-level conceptual questions. By studying from the Exemplar, you can prepare well for entrance exams like Olympiad, NTSE and KVPY.

Key Features of NCERT Solutions for Class 9 Mathematics Chapter 1

In order to obtain a good score in exams, revision of previous concepts is a must. Hence, NCERT Solutions for Class 9 Mathematics Chapter 1 offers a complete solution for all problems. The key features of NCERT solutions are:  : 

• Mathematics experienced faculty and subject experts have designed Extramarks NCERT Solutions for Class 9 Mathematics Chapter 1. It is a thoroughly researched material made in sync with CBSE examination guidelines.
• Students have a very clear understanding of the concepts and overcome  all their doubts with the help of Extramarks NCERT solutions.
• After completing the NCERT Solutions for Class 9 Mathematics Chapter 1 students will be able to solve all the basic and advanced level problems with better  accuracy.The systemic and well-laid out balanced study plan boosts their performance naturally and effortlessly. 
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Yes, 0 is a rational number. It can be represented as (0/1), (0/2), (0/3) etc.

Find six rational numbers between 3 and 4.

There are infinite rational numbers between 3 and 4. 3 and 4 can be represented as 24/8 and 32/8 respectively. The rational numbers between 3 and 4 are 25/8, 26/8, 27/8, 28/8, 29/8, 30/8.

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

Q.4 State whether the following statements are true or false. Give reasons for your answers. (i) Every natural number is a whole number. (ii) Every integer is a whole number. (iii) Every rational number is a whole number.

(i) True; since the collection of whole numbers contains all natural numbers. (ii) False; since integers may be negative but whole numbers are positive. For example: – 5 is an integer it is not a whole number. (iii) False; as rational number may be a fraction but whole number may not be a fraction. For example: 4/5 is a rational number and it is not a whole number.

State whether the following statements are true or false. Justify your answers. (i) Every irrational number is a real number. (ii) Every point on the number line is of the form

where m is a natural number. (iii) Every real number is an irrational number.

(i) True; because real number is a collection of rational and irrational number. (ii) False; as negative numbers cannot be represented as the square root of any other number. (iii) False; as real numbers include both rational and irrational numbers i.e., irrational number is a part of real number. Therefore, every real number cannot be an irrational number.

Q.6 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: The square roots of 4 and 9 are 2 and 3 respectively.

how 5 can be represented on the number line .

Write the following in decimal form and say what kind of

decimal expansion each has: i   36 100    ii   1 11    iii  4 1 8    iv   3 13    v   2 11    vi   329 400

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?

Express the following in the form

p q , where p and q are integers and q¹0. i  0   ii  0.4 7 ¯    iii  0. 001 ¯

Express 0.99999, in the form

p q . Are you surprised by your answer? With your teacher and classmates discussway the answer makes sense.

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 .

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 ?

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

Three numbers whose decimal expansions are non- terminating non-recurring are as follows:

0.030030012003000050004123000…

0.01200012500003500050010008879000102003…

1.5200050040060080010030010040038001…

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

Classify the following numbers as rational or irrational:

i   23    ii   225   iii  0.3796  iv  7.478478 . ..    v  1.101001000100001 . ..

Visualise 3.765 on the number line, using successive magnification.

3.765 can be visualised as in the following steps.

4 . 26 ¯ on the number line , upto 4 decimal places .

Classify the following numbers as rational or irrational:

i  2 − 5    ii   3+ 23 − 23    iii   2 7 7 7 iv   1 2   v  2π

each of the following expressions : ( i ) ( 3 + 3 ) ( 2 + 2 ) ( ii ) ( 3 + 3 ) ( 3 − 3 ) ( iii ) ( 5 + 2 ) 2 ( iv ) ( 5 − 2 ) ( 5 + 2 )

, 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. Remember that when you measure a length with a scale or any other device, you only get an approximate rational value. So, you may not realise that either c or d is irrational.

9 . 3 on the number line .

Mark a line segment AB = 9.3 on number line. Further, take BC of 1 unit. Draw a semi-circle on AC as diameter. Draw a perpendicular to line AC passing through point B. Let it intersect the semi circle at D. Taking B as centre and BD as radius, draw an arc intersecting number line at E. BE =

Rationalise

the denominators of the following : ( i ) 1 7 ( i i ) 1 7 − 6 ( i i i ) 1 5 + 2 ( i v ) 1 7 − 2

:   i 64 1 2       ii 32 1 5       iii 125 1 3

i 9 3 2    ii 32 2 5    iii 16 3 4     iv 125 − 1 3

:      ( i )   2 2 3 . 2 1 5       ( ii )   ( 1 3 3 ) 7       ( iii )   11 1 2 11 1 4         ( iv ) 7 1 2 . 8 1 2

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Faqs (frequently asked questions), 1. where should i search for the ncert solutions for class 9 mathematics chapter 1 online.

There are plenty of  online platforms that provide  study materials for Class 9 Mathematics. However students should rely on only those study solutions that are prepared by subject   experts and strictly follow the latest CBSE curriculum. 

Students can refer to Extramarks, one of the leading  e-learning platforms which  has made it possible for students to access NCERT Solutions for Class 9 Mathematics Chapter 1 as they are prepared by Mathematics subject matter experts with decades of experience. Along with Class 9th Solutions, one can find NCERT Solutions right from Class 1 to Class 12 on our website.  Extramarks has built its credibility and is trusted by students as well as private and government schools across India.

2. How to prepare for NCERT Class 9 Mathematics Chapter 1?

Students should start studying Class 9 Mathematics from NCERT textbook first. They should be attentive in their class lectures. Along with the NCERT textbook, students should solve questions from NCERT Exemplars to build a strong foundation. 

We highly recommend students  also register on reliable online learning platforms such as Extramarks which strictly follows NCERT books and provides solved exercises and practice questions to step up their learning experience and eliminate “mathematics phobia” among students. The additional support of online learning and classes  will  allow  students to clear  their doubts and strengthen their base. To get good grades in exams students  must refer to multiple study resources, practise a lot of questions  and  stick to a study schedule and follow it rigorously to come out with flying colours. 

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Extra questions for class 9 maths chapter 1, number systems | real numbers | rational & irrational numbers.

NCERT Class 9 Math / Number System Extra Questions

C hapter 1 of CBSE NCERT Class 9 Math covers number systems. Concepts covered in chapter 1 include rational numbers, irrational numbers, rationalizing irrational numbers by multiplying with their conjugates, decimal expansion of real numbers, operations on real numbers and laws of exponents or rules of indices. The extra questions given below include questions akin to HOTS (Higher Order Thinking Skills) questions and exemplar questions of NCERT.

Here is a quick recap of the key concepts that are covered in this chapter in the CBSE NCERT Class 9 Math text book.

What are rational numbers?

A number that can be written in the form \\frac{p}{q}\\) where p and q are integers and p ≠ 0 is a rational number.

Possibility 1 : If the decimal expansion of the number is terminating it is a rational number. Note: Integers are terminating decimals and are therefore, rational numbers.

Possibility 2 : If the decimal expansion of the number is non-terminating but is recurring , it is rational. Example \\frac{1}{3}\\) = 0.333.. is a non-terminating recurring decimal and is a rational number.

What are irrational numbers?

A number that CANNOT be written in the form \\frac{p}{q}\\) where p and q are integers and p ≠ 0 is an irrational number.

If the decimal expansion of the number is non-terminating AND non-recurring it is an irrational number. Example: \\sqrt{2}\\), π

How to Rationalize Irrational Numbers?

For an irrational number of the form a + √b, a - √b is its conjugate. And for an irrational number of the from a - √b, a + √b is its conjugate.

Important Laws of Exponents (Rules of Indices)

If a > 0 is a real number and m and n are rational numbers, the following laws of exponents hold good.

• a m × a n = a m + n Example .: 10 3 × 10 2 = 10 3 + 2 = 10 5
• (a m ) n = a mn Example : (10 3 ) 2 = 10 (3 \\times\\) 2) = 10 6
• \\frac{a^m}{a^n}\\) = a (m - n) Example : \\frac{10^3}{10^2}\\) = 10 (3 - 2) = 10
• a m b m = (ab) m Example : 2 2 × 5 2 = (2 × 5) 2 = 10 2

Extra Questions for Class 9 Maths - Number Systems

Prime Factorise & Rationalise Denominator: \\frac{14}{{\sqrt {108}} - {\sqrt {96}} + {\sqrt {192}} - {\sqrt {54}}}\\)

Rational numbers - Fractions: Find 5 rational numbers between \\frac{3}{4}) and \\frac{4}{5}).

Express as Fractions Express 1.363636... in the form \\frac{p}{q}), where p and q are integers and q ≠ 0.

Express in the form \\frac{p}{q}) Express 0.4323232… in the form \\frac{p}{q}), where p and q are integers and q ≠ 0.

Simplify the following (a) \({8 + \sqrt{5})}) \({8 - \sqrt{5})}) (b) \({10 + \sqrt{3})}) \({6 + \sqrt{2})}) (c) \{(\sqrt {3} + \sqrt {11})}^2) + \{(\sqrt {3} - \sqrt {11})}^2)

Rationalize the denominator: (a) \\frac{2}{\sqrt{3} - 1}) (b) \\frac{7}{\sqrt{12} - \sqrt{5}}) (c) \\frac{1}{8 + 3\sqrt{5}}) (d) \\frac{1}{4 + \sqrt{2} + \sqrt{5}})

Simplify and find the value of (a) \{(729)}^{\frac{1}{6}}) (b) \{(64)}^{\frac{2}{3}}) (c) \{(243)}^{\frac{6}{5}}) (d) \{(21)}^{\frac{3}{2}} \times {(21)}^{\frac{5}{2}}) (e) \\frac{{(81)}^{\frac{1}{3}}}{{(81)}^{\frac{1}{12}}})

Operation on real numbers & Algebraic identities If x = \\frac{3 - {\sqrt{13}}}{2}\\), what is the value of \x^2 + \frac{1}{x^2}\\)?

Rationalise & find value of cubic expression If x = \\frac{1}{8-\sqrt{60}}\\), what is the value of (x 3 - 5x 2 + 8x - 4) ?

Question 10

Rationalise the denominator \\frac{1}{9 + {\sqrt{5} + \sqrt{6}}}\\)

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• CBSE Class 9 Maths Worksheet Chapter 1 Number System

CBSE Class 9 Maths Worksheet Chapter 1 Number System - Download Free PDF with Solution

When it comes to Maths as a whole, not many people excel in this subject as it is a subject that solely relies on the logical reasoning functions of the brain. That is why the number system can be intimidating to most students. In Chapter 1, Number System for Class 9, students will learn the number system and their types and how to solve the equations. 

So, what is the number system, and what does the number system syllabus contain? A number system can be defined as an arithmetic system or practice of writing numbers to express them. It is the mathematical notation for continuously representing numbers of any given set by using a certain set of digits, symbols, or other characters. It offers a unique representation of every number. It signifies the arithmetic and algebraic structure of the given figures, permitting us to carry out mathematical calculations such as addition, subtraction, and division. 

All these figures carry their values, which can be determined by looking at the digit, the position in the number, and the base of the number. A number is a mathematical value used to count, measure, or label objects. Regarding the number system, these numbers are used as digits. 

With the help of worksheets such as the Number System Class 9 worksheet, Class 9 Maths Chapter 1 worksheet pdf, and worksheet for Class 9 Maths Chapter 1 with solutions and the operations on Real Numbers Class 9 worksheet, students will have a better understanding of what number systems are and how to solve them accurately.

Access Worksheet for Class 9 Maths Number System

1. It is impossible to represent a rational number in decimal form.

Terminating

Non- terminating

Repeating or Non- Terminating

Non-repeating or Non- terminating

2. Between two rational numbers

There is no rational number.

There is exactly one rational number.

There are infinitely many rational numbers.

There are only rational numbers and no irrational numbers.

3. The product of any two irrational numbers,

is always an irrational number.

is always a rational number.

is always an integer.

can be rational or irrational.

4. Which of the following is irrational?

$\sqrt{81}$

$\dfrac{\sqrt{12}}{\sqrt{3}}$

$\dfrac{\sqrt{4}}{9}$

5. What is the value is $\sqrt{4} \times \sqrt{81}$?

6. Fill in the blanks;

Any two integers are separated by a finite number of others …..

There are an ….. amount of rational numbers between 15 and 18.

X+Y is a rational number if x and y are both ……

Value of $\sqrt[3]{8}$ …….

7. Match the Column:

8. Using two irrational numbers as an example:

Product is an irrational number.

Difference is an irrational number.

Division is an irrational number.

9. Simplify; $(\sqrt{5}+\sqrt{6})(\sqrt{5}-\sqrt{6})$.

10. Simplify; $\sqrt[3]{1331}-\sqrt{100}+\sqrt{81}$.

11. Calculate the value of $\dfrac{11^{\dfrac{1}{2}}}{11^{\dfrac{1}{4}}}$.

12. Calculate the $\dfrac{x}{y}$ form of $0.777 . . . . .$, where $\mathbf{x}$ and $\mathbf{y}$ are integers and $\mathbf{y}$ does not equal to zero.

13. Find three rational number between $\dfrac{9}{11}$ and $\dfrac{5}{11}$.

14. The value of $\dfrac{\sqrt{8}+\sqrt{12}}{\sqrt{32}+\sqrt{48}}$.

15. The value of $a^b+b^a$, if $\mathbf{a}=2$ and $\mathbf{b}=3$

16. Simplify; $2^{\dfrac{2}{3}} \cdot 2^{\dfrac{1}{5}}$

17. Find the value of $\dfrac{1}{a^b+b^a}$, where $a=5, \mathbf{b}=2$

18. Arrange in ascending order $\sqrt[3]{2}, \sqrt{3}, \sqrt[6]{5} \text {. }$

19. Simplify $(4 \sqrt{5}+3 \sqrt{7})^2$

20. Find the value of a, If $\left(\dfrac{y}{x}\right)^{2a-8}=\left(\dfrac{x}{y}\right)^{a-1}$.

21. Rationalize the denominators of $\dfrac{1}{\sqrt{7}}$.

22. Recall, $\pi$ is defined as the ratio of circumference (say c) to its diameter (say d). That is $\pi=\dfrac{c}{d}$. This seems to contradict the fact that $\pi$ is irrational. How will you resolve this contradiction?

23. Express $0 . \overline{001}$ in the form of $\dfrac{p}{q}$, where $\mathrm{p}$ and $\mathrm{q}$ are integers and $\mathrm{q} \neq 0$.

24. Find five rational numbers between $\dfrac{3}{4}$ and $\dfrac{4}{5}$

25. Find six rational numbers between 3 and 4.

Answers to the Worksheet:

A rational number cannot have a non-terminating or non-repeating decimal form.

2. (c) 

Between two rational numbers, there are infinitely many rational numbers. 

E.g. $\dfrac{3}{5}$ and $\dfrac{4}{5}$ are two rational numbers, then $\dfrac{31}{50} \dfrac{32}{50} \dfrac{33}{50} \dfrac{34}{50} \dfrac{35}{50} \ldots$ are infinite rational number between them.

3. (d) 

The product of two irrational numbers can be rational or irrational depending on the two numbers.

For example, $\sqrt{3} \times \sqrt{3}$ is 3 which is a rational number whereas $\sqrt{2} \times \sqrt{4}$ is $\sqrt{8}$ which is an irrational number. As $\sqrt{3}, \sqrt{2}, \sqrt{4}$ are irrational.

Hence, option D is correct.

4. (a) $\sqrt{7}$ is an irrational number.

5. (b) 

$\sqrt{4} \times \sqrt{81}$ $= \sqrt{2^2} \times \sqrt{9^2}$ $= 2 \times 9$ = 18

6. Fill in the blanks.

Any two integers are separated by a finite number of other integers .

There are an endless amount of rational numbers between 15 and 18 .

$\mathrm{X}+\mathrm{Y}$ is a rational number if $\mathrm{x}$ and $\mathrm{y}$ are both rational numbers .

Value of $\sqrt[3]{8}$ is $\underline{2}$

7. Match The Column:

Explanation:

8. Given an example of two irrational numbers whose;

Product is an irrational number $\sqrt{6} \times \sqrt{3}=\sqrt{6 \times 3}=\sqrt{18}=3 \sqrt{2}$

Difference is a irrational number $\sqrt{6}-\sqrt{3}$ = $\sqrt{3}$

Division is an irrational number $\dfrac{\sqrt{6}}{\sqrt{3}}=\sqrt{\dfrac{6}{3}}=\sqrt{2}$

9. Simplify; $(\sqrt{5}+\sqrt{6})(\sqrt{5}-\sqrt{6})$ 

We know that, $(a+b)(a-b)=a^2-b^2$

= $\left((\sqrt{5})^2-(\sqrt{6})^2\right)$

10. $\sqrt[3]{1331}-\sqrt{100}+\sqrt{81}$

= $\sqrt[3]{11^3}-\sqrt{10^2}+\sqrt{9^2}$

= $11-10+9$

11. $\dfrac{11^{\dfrac{1}{2}}}{11^{\dfrac{1}{4}}}$

$\dfrac{11^{\dfrac{1}{2}}}{11^{\dfrac{1}{4}}}=11^{\dfrac{1}{2}-\dfrac{1}{4}}$

$=11^{\dfrac{2-1}{4}}$

$=11^{\dfrac{1}{4}}$

12. Let, 

$p= 0.777…$ ....   (1)

Multiply both side in above equation 10

Then, 

$10p= 7.777…$ ….(2)

Subtracting equation (1) from (2), we get;

$10p-p= 7.777… - 0.777…$

$p= \dfrac{7}{9}$

13. Three rational number between $\dfrac{9}{11}$ and $\dfrac{5}{11}$

Rational number of $\dfrac{9}{11}$ and $\dfrac{5}{11}$ is denominator same

$= \dfrac{9}{11}, \dfrac{8}{11}, \dfrac{7}{11}, \dfrac{6}{11}, \dfrac{5}{11}$

14. $\dfrac{\sqrt{8}+\sqrt{12}}{\sqrt{32}+\sqrt{48}}$

$= \dfrac{\sqrt{2^3}+\sqrt{4 \times 3}}{\sqrt{8 \times 4}+\sqrt{8 \times 6}}$

$= \dfrac{2 \sqrt{2}+2 \sqrt{3}}{4 \sqrt{2}+4 \sqrt{3}}$

$= \dfrac{2(\sqrt{2}+\sqrt{3})}{4(\sqrt{2}+\sqrt{3})}$

$= \dfrac{(\sqrt{2}+\sqrt{3})}{2(\sqrt{2}+\sqrt{3})}$

$= \dfrac{1}{2}$

15. If $a=2$ and $b=3$

The value of $a^b+b^a$

$= 2^3+3^2$

16. $2^{\dfrac{2}{3}} \cdot 2^{\dfrac{1}{5}}$

$2^{\dfrac{2}{3}} \cdot 2^{\dfrac{1}{5}}=2^{\dfrac{2}{3}+\dfrac{1}{5}} \quad \because a^p \cdot a^q=a^{p+q}$

$=2^{\dfrac{10+3}{15}}$

$=2^{\dfrac{13}{15}}$

17. Value of $\dfrac{1}{a^b+b^a}$, where $a=5, b=2$

$= \dfrac{1}{5^2+2^5}$

$= \dfrac{1}{25+32}$

$= \dfrac{1}{57}$

18. Here we have : $\sqrt[3]{2}, \sqrt{3}, \sqrt[5]{5}$

We can also write the expression in simpler form as follows:

$2^{\dfrac{1}{3}}, 3^{\dfrac{1}{2}}, 5^{\dfrac{1}{6}}$

Now we can see that in the denominators of the exponents we have: $3,2,6$

We will now take the LCM of $3,2,6$, which is 6 .

Now we will make all the denominators equal to 6 , so we have to multiply by the multiples in both numerator and denominator.

We can write the numbers as:

$2^{\dfrac{1}{3}} \times \dfrac{2}{2}=2^{\dfrac{2}{6}}$

For the second number we can write:

$3 \dfrac{1}{2} \times \dfrac{3}{3}=3 \dfrac{3}{6}$

Since in the third number we already have the desired denominator, so the third number is

$5^{\dfrac{1}{6}}$

Now we will again write the numbers in the root under, but we have to keep in mind that the numerator will turn as the exponential powers inside the root.

So we have the numbers as:

$\sqrt[6]{2^2}, \sqrt[5]{3^3}, \sqrt[5]{5}$

We will simplify the values inside the root, so we have:

$\sqrt[5]{4}, \sqrt[6]{27}, \sqrt[5]{5}$

From this we can write the smaller value in the front and then the larger value:

$\sqrt[5]{4}, \sqrt[6]{5}, \sqrt[5]{27}$

Hence the original numbers in ascending form are:

$\sqrt[3]{2}, \sqrt[6]{5}, \sqrt{3}$

19. $(4 \sqrt{5}+3 \sqrt{7})^2$

We know that,

$(a+b)^2=a^2+b^2+2 a b$

$=(4 \sqrt{5})^2+(3 \sqrt{7})^2+2 \times (4 \sqrt{5}) (3 \sqrt{7})$

$=80+63+24 \sqrt{5 \times 7}$

$=143+24 \sqrt{35}$

20. $\left(\dfrac{y}{x}\right)^{2 a-8}=\left(\dfrac{x}{y}\right)^{a-1}$

$\left(\dfrac{y}{x}\right)^{2 a-8}=\left(\dfrac{x}{y}\right)^{8-2 a}$   $ \because (x)^{-a}=\dfrac{1}{x^a}$

$\left(\dfrac{x}{y}\right)^{8-2 a}=\left(\dfrac{x}{y}\right)^{a-1}$

When the bases of both sides of an equation are the same, then their exponents are also equal.

$\Rightarrow 8-2 a=a-1$

$\Rightarrow 2 a+a=8+1$

$\Rightarrow 3 a=9$

$\Rightarrow a=\dfrac{9}{3}$

$\Rightarrow a=3$

21. $\dfrac{1}{\sqrt{7}}=\dfrac{1}{\sqrt{7}} \times \dfrac{\sqrt{7}}{\sqrt{7}}$

(Dividing and multiplying by $\sqrt{7}$ )

$=\dfrac{\sqrt{7}}{7}$

22. Writing $\pi$ as $\dfrac{22}{7}$ is only an approximate value and so we can't conclude that it is in the form of a rational. In fact, the value of $\pi$ is calculating as non-terminating, non-recurring decimal as $\pi=3.14159$ Whereas

If we calculate the value of $\dfrac{22}{7}$ it gives $3.142857$ and hence $\pi \neq \dfrac{22}{7}$

In conclusion $\pi$ is an irrational number.

23. Let $x=0.001001 \ldots \ldots$ (1)

Since 3 digits are repeated multiply both the sides of (1) by 1000

$1000 x=1.001001 \ldots$

$1000 x=1+0.001001 \ldots$

$1000 x=1+x$

$1000 x-x=1$

$x=\dfrac{1}{999}$

$\therefore 0 . \overline{001}=\dfrac{1}{999}$

24. Since we make the denominator the same first, then

$\dfrac{3}{4}=\dfrac{3 \times 5}{4 \times 5}=\dfrac{15}{20}$

$\dfrac{4}{5}=\dfrac{4 \times 4}{5 \times 4}=\dfrac{16}{20}$

Now we need to find 5 rational no.

$\dfrac{15}{20}  =\dfrac{15 \times 6}{20 \times 6}=\dfrac{90}{120}$

$\dfrac{16}{20}=\dfrac{16 \times 6}{20 \times 6}=\dfrac{96}{120}$

$\therefore$ Five rational numbers between $\dfrac{3}{4}$ and $\dfrac{4}{5}$ are $\dfrac{91}{120}, \dfrac{92}{120}, \dfrac{93}{120}, \dfrac{94}{120}$ and $\dfrac{95}{120}$

25. We can find any number of rational numbers between two rational numbers. First of all, we make the denominators same by multiplying or dividing the given rational numbers by a suitable number. If denominator is already same then depending on number of rational no. we need to find in question, we add one and multiply the result by numerator and denominator.

$3=\dfrac{3 \times 7}{7} \text { and } \quad 4=\dfrac{4 \times 7}{7}$

$3=\dfrac{21}{7} \quad \text { and } \quad 4=\dfrac{28}{7}$

We can choose 6 rational numbers as: $\dfrac{22}{7}, \dfrac{23}{7}, \dfrac{24}{7}, \dfrac{25}{7}, \dfrac{26}{7}$ and $\dfrac{27}{7}$

Benefits of Learning Number System in Class 9 Chapter 1 Maths Worksheet The Class 9 Maths Chapter 1 worksheet pdf contains more than enough material to help students better understand what number systems are and how to solve them. The worksheets come with extensive questions, attempting to clear any doubts the students might have about the number system and their types.

The Maths assignment for Class 9 Number System list of questions and answers provide thorough insights on the topic’s resources and offers easy tricks to identify quicker ways to solve the questions faster while also being more aware and making sure students don’t go wrong or commit any silly mistakes in their solutions.

All of these worksheets have been developed by the best mathematicians and experienced arithmetic representatives who are very aware of the needs and requirements of the students of Class 9.

Examples of Usage of Number System for Class 9

These are a few examples of Maths assignments for Class 9 Number System exercises’ examples :

Answer the following.

Find two irrational numbers and two rational numbers between 0.7 and 0.77.

Every integer is not a whole number. True or false?

Find at least 7 rational numbers between 2 and 9.

Write down 4567 in the decimal and binary number systems.

Is 0 a rational number? State your reasons based on your answer.

Interesting Facts About Number System for Class 9

There are nine types of number systems in mathematics. They are :

Natural numbers

Whole numbers

Rational numbers

Irrational numbers

Real numbers

Imaginary numbers

Prime and composite numbers

Natural numbers are the root forms of numbers between 0 to infinity. They are also named “positive numbers” or “counting numbers” and are represented by the symbol N. (1, 2, 3, 4, 5 and so on)

Whole numbers are natural numbers, with the only difference being the inclusion of 0. They are represented by the symbol W. (0, 1, 2, 3, 4, 5 and so on)

Integers contain whole numbers and the negative values of natural numbers and don’t include fractions, so their numbers can’t be written in the “a/b” format. It ranges from infinity at the negative end to infinity at the positive end, including 0 and is represented by Z. (...-3, -2, -1, 0, 1, 2, 3… and so on)

Fractions are numbers written in the “a/b” format, where “a” (numerator) is a whole number, and “b” (denominator) is a natural number. Hence, the denominator can never be 0. (2/4, 0/10, 5/7, etc.)

Rational numbers can be written in fractions where “a” and “b” are both integers and b ≠ 0. All fractions are rational numbers, but all rational numbers are not fractions.(-5/9, 3/9, -8/14, etc)

Irrational numbers are numbers that can’t be written in fractional forms. (√8, √.127, √3.209, etc)

Real numbers can be written in decimals, including whole numbers, integers, fractions, etc. All integers belong to real numbers, but not all real numbers belong to integers. (1.25, 0.467, 8.9, etc.)

Imaginary numbers are not real numbers, resulting in negative numbers when squared or put together. They are also named complex numbers and are represented by the symbol i. (√-3, √-16, √-1, etc.)

Numbers that don’t have other factors except 1 are called prime numbers, and the rest of the numbers - except 0 - are called composite numbers, as 0 is neither a prime nor a composite number. ( 2, 3, 5… are prime numbers whereas 4, 6, 8… are composite numbers)

Other definitions and elaborated explanations will be provided in the operations on real numbers class 9 worksheet pdf and more .

Important Topics for Class 9 Number System

The important topics students will have to learn in the number system syllabus for Class 9 are as follows :

What are number systems, and how to solve them?

What are the four types of number systems?

How to convert one number system to another number system.

Solving the problems and choosing the correct answer.

Various other exercises in the Number System Class 9 worksheet

What does the PDF Consist of?

Most schools have syllabuses that don’t include just spoon-feeding the information to the students.

It only means that the students must learn by themselves, with the teachers guiding and aiding them throughout their learning process.

With technology being part of most school curriculums, a huge part of their assignments, tests, and worksheets are online, favoured as pdfs.

Vedantu’s pdf format is highly sought-after as it is used for creating, editing, highlighting, saving, and sharing content.

The worksheet for Class 9 Maths Chapter 1 with Solutions pdfs is free for download at Vedantu’s website.

Rest assured, all the worksheets adhere to the CBSE guidelines' strict, updated, and revised rules.

Many other Number Systems Class 9 worksheet pdfs are present at Vedantu’s platform, created by their own arithmetic subject matter experts, ensuring that the students receive the best training and exercises needed to test their skills and excel in their examinations.

FAQs on CBSE Class 9 Maths Worksheet Chapter 1 Number System

1. Which number system is frequently used?

The decimal number system is the most widely used.

2. How are the values of various figures calculated?

All these figures carry their values, and these values can be determined by: 

looking at the digit 

the position in the number 

the base of the number. 

3. Can rational numbers be whole numbers?

No, since rational numbers may be fractional and whole numbers are not.

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Case study questions for class 9 maths chapter 9 areas of parallelograms and triangles, case study questions for class 9 maths chapter 6 lines and angles, case study questions for class 9 maths chapter 7 triangles, case study questions for class 9 maths chapter 5 introduction to euclid’s geometry, case study and passage based questions for class 9 maths chapter 14 statistics, case study questions for class 9 maths chapter 1 real numbers, case study questions for class 9 maths chapter 4 linear equations in two variables, case study questions for class 9 maths chapter 3 coordinate geometry, case study questions for class 9 maths chapter 15 probability, case study questions for class 9 maths chapter 13 surface area and volume, case study questions for class 9 maths chapter 10 circles, case study questions for class 9 maths chapter 9 quadrilaterals, case study questions for class 9 maths chapter 2 polynomials.

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• Exercise 1.1

NCERT Solutions for class 9 Maths Chapter 1 - Number Systems Exercise 1.1

NCERT Solutions Class 9 Maths Chapter 1 Number Systems Exercise 1.1 are provided here. Our subject experts have prepared the NCERT Maths solutions for Class 9 chapter-wise so that it helps students to solve problems easily while using it as a reference. They also focus on creating solutions for these exercises in such a way that it is easy to understand for the students.

The first exercise in Number Systems Exercise 1.1 is the introduction. They provide a detailed and stepwise explanation of each answer to the questions given in the exercises in the NCERT textbook for Class 9. The NCERT Solutions are always prepared by following NCERT guidelines so that it covers the whole syllabus accordingly. These are very helpful in scoring well in CBSE examinations.

NCERT Solutions for Class 9 Maths Chapter 1- Number Systems Exercise 1.1

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

Exercise 1.6 Solutions 3 Questions (3 long)

Access Answers to Maths NCERT Class 9 Chapter 1 – Number Systems Exercise 1.1

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 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 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…}

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.

NCERT Solutions for Class 9 Maths Chapter 1 Number Systems Exercise 1.1 is the first exercise of Chapter 1 of Class 9 Maths. This exercise explains how to find rational numbers between two given numbers.

Key Features of NCERT Solutions for Class 9 Maths Chapter 1 – Number Systems Exercise 1.1

• These NCERT Solutions help you solve and revise all questions of Exercise 1.1.
• After going through the stepwise solutions given by our subject expert teachers, you will be able to score more marks.
• It follows NCERT guidelines which help in preparing the students accordingly.
• It contains all the important questions from the examination point of view.

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