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# NCERT Solutions Class 9 Maths PDF

Maths Class 9 is one of the most important subjects that students need to be proficient in. This class introduces them to the concept of Triangles and Polynomials. It also introduces them to Number Systems and many more. The exercise questions in this class are very difficult. Therefore, we recommend the students to follow the instructions given by reliable sources to get ready for the Maths Class 9 exam.

eSaral experts have prepared NCERT Class 9 Mathematics Solutions for students to complete the large syllabus in a hassle-free manner. The solutions will help students to understand the subject and solve the calculation problems easily and sequentially.

eSaral’s experts have come up with an effective solution to the problems that students face while learning Mathematics. All the theorems and concepts are explained in a simple manner so that students can solve difficult problems easily. These study materials serve as guides that help students to progress in class.

## Class 9 Maths NCERT Solutions

In Class 9 Mathematics, students need to be provided with the necessary guidance and support in order to achieve excellence in their studies. The chapters of the syllabus cover probability, circles, polynomials, statistics, triangles, heron's formula, surface areas and volumes etc. All these concepts have been explained in the solutions provided by NCERT Solutions. Through the use of these solutions, students are able to gain a comprehensive understanding of theorems, formulas, and calculations. Furthermore, they are able to complete the syllabus in a timely manner with the assistance of the experts of eSaral.

When it comes to Maths preparation, Class 9 students need to focus on all topics and chapters at the same time. With NCERT solutions for Class 9, students don’t need to go through the whole chapter as they can solve the exercise questions only and get a good understanding of the concepts. The solutions are given in a step by step format to make it easier for the students.

## NCERT Solutions for Class 9 Maths Chapter List

Chapter List of NCERT Solutions for Class 9 Maths is given below

## NCERT Solutions of Class 9 Book Chapter Brief

Students who are struggling to solve difficult Math problems can use these Mathematics Class 9 Solutions from NCERT for guidance and quick review. Completing these exercises in every chapter will help you in achieving good results.

### NCERT Solutions for Class 9 Maths Chapter 1 Number Systems

In this chapter, Rational and Irrational numbers will be discussed. They will also learn about the extended version of number lines and how to show numbers (integers and irrational numbers) on it. Students will learn about the representation of terminating and non-terminating repeating decimals (as well as the successive magnification method) and how to show square roots of numbers (2, 3 and others non-rational numbers) on the number lines. They will learn about the law of integral powers and the law of rational exponents (positive real bases) in the Number System.

Topics covered in Class 9 Maths Chapter 1 Number System

• Review of representation of natural numbers, integers, and rational numbers on the number line.

• Representation of terminating/non-terminating recurring decimals on the number line through successive magnification. Rational numbers as recurring/terminating decimals.

• Examples of nonrecurring/non-terminating decimals such as √2, √3, √5 etc.

• 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, every point on the number line represents a unique real number.

 NCERT Solutions for Class 9 Maths Chapter 1 - Number Systems Exercise 1.1 4 Questions Exercise 1.2 4 Questions Exercise 1.3 9 Questions Exercise 1.4 5 Questions Exercise 1.5 3 Questions

### NCERT Solutions for Class 9 Maths Chapter 2 Polynomials

This chapter provides an overview of the concept of polynomial, a type of algebras, and the terminology associated with it. Polynomials are expressions composed of variables and their associated operations, such as addition, subtractions, multiplications, and non-negligible integer exponents. The Remainder Theorem is discussed in this chapter, as well as the Factor Theorem, which is used to factorise the polynomial. Students will be presented with a variety of examples, as well as definitions of various terms, such as a polynomial, its degrees, its coefficients, its zeros, and its terms.

Topics covered in Class 9 Maths Chapter 2 Polynomials

• Definition of a polynomial in one variable, its coefficients, with examples and counter examples, its terms, zero polynomial

• Degree of a polynomial. Constant, linear, quadratic, cubic polynomials

• monomials, binomials, trinomials. Factors and multiples.

• Zeros/roots of a polynomial/equation.

• State and motivate the Remainder Theorem with examples and analogy to integers.

• Statement and proof of the Factor Theorem.

 NCERT Solutions for Class 9 Maths Chapter 2 - Polynomials Exercise 2.1 5 Questions Exercise 2.2 4 Questions Exercise 2.3 5 Questions Exercise 2.4 16 Questions

### NCERT Solutions for Class 9 Maths Chapter 3 Coordinate Geometry

The chapter of Coordinate Geometry covers the concepts of Cartesian plane, Xy–plane coordinates, terms, and notation related to the coordinate plane, such as X-axis, Y-axis, X-coordinate, Y-coordinate, Origin, Quadrants, and more. In this chapter, students will also learn about Abscissa and Ordinates of a Point, as well as Plotting and Naming a Point in Xy–plane. There are 2 exercises in this chapter that contain questions revolving around the topics mentioned in the chapter, helping the students get thorough with the concepts.

Topics covered in Class 9 Maths Chapter 3 Coordinate Geometry

• The Cartesian plane, coordinates of a point, names and terms associated with the coordinate plane, notations, plotting points in the plane, graph of linear equations as examples

• Focus on linear equations of the type ax + by + c = 0 by writing it as y =mx + c and linking with the chapter on linear equations in two variables.

 NCERT Solutions for Class 9 Maths Chapter 3 - Coordinate Geometry Exercise 3.1 2 Questions Exercise 3.2 2 Questions

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

In this chapter, students will not only be familiar with linear equations in a single variable, but they will also be introduced to linear equations in two variables (i.e., ax+by+c = 0). Additionally, students will be taught to plot linear equations on a graph of two variables. Four exercises are included in this chapter, which are related to solving linear equations, plotting linear equations on graphs, and other topics covered in this chapter.

Topics covered in Class 9 Maths Chapter 4 Linear Equations in Two Variables

• Introduction to the equation in two variables

• Prove 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 seem to lie on a line

• Examples, problems from real life, including problems on Ratio and Proportion and with algebraic and graphical solutions being done simultaneously.

 NCERT Solutions for Class 9 Maths Chapter 4 -Linear Equations in Two Variables Exercise 4.1 2 Questions Exercise 4.2 4 Questions

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

In this chapter, we'll be looking at Euclid's way of thinking about geometry and how it relates to modern geometry. We'll start off with an introduction to Euclid's Geometry, where we'll learn how to define common geometric shapes and concepts. Then, we'll move on to Axioms, Postulates and Theorems with these two exercises.

Topics covered in Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry

• History – Euclid and geometry in India

• Euclid’s method of formalising observed phenomenon into rigorous mathematics with definitions, common/obvious notions, axioms/postulates, and theorems

• The five postulates of Euclid

• Equivalent versions of the fifth postulate

• Showing the relationship between axiom and theorem

 NCERT Solutions for Class 9 Maths Chapter 5 - Introduction to Euclid’s Geometry Exercise 5.1 7 Questions

### NCERT Solutions for Class 9 Maths Chapter 6 Lines and Angles

This chapter focuses on the theorems that are included in the Lines and Angles topics. Students may often be required to demonstrate the statements made in the questions. The chapter includes 2 exercises that solve which students would be capable of fully comprehending the concepts discussed in the chapter. There are 4 axioms and 8 theorems in this chapter.

Topics covered in Class 9 Maths Chapter 6 Lines and Angles

• Introduction

• Basic Terms and Definition

• Intersecting Lines and Non-Intersecting Lines

• Pairs of Angles

• Parallel Lines and Transversal Line

• Lines Parallel to the Same Line

• Angle Sum Property of a Triangle

 NCERT Solutions for Class 9 Maths Chapter 6 - Lines and Angles Exercise 6.1 6 Questions Exercise 6.2 5 Questions

### NCERT Solutions for Class 9 Maths Chapter 7 Triangles

This chapter provides an in-depth exploration of the Congruence of Triangles, Congruence Rules, Certain Properties of Triangles, and Intervals in Triangles. It consists of five exercises in which students are required to demonstrate the properties they have learned in previous classes. Additionally, the chapter teaches students how to apply the different Congruence Rules when solving the exercises. Approximately eight Theorems are discussed in this chapter.

Topics covered in Class 9 Maths Chapter 7 Triangles

• Triangle congruence

• SAS, ASA, SSS and RHS Congruence Rule

• Triangle Properties

• Triangle Inequalities

 NCERT Solutions for Class 9 Maths Chapter 7 - Triangles Exercise 7.1 8 Questions Exercise 7.2 8 Questions Exercise 7.3 5 Questions

### NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

A quadrilateral is a number obtained by connecting four points in a certain order. In this chapter, the students are introduced to the topic of Quadrilaterals. There are two exercises in this chapter that only require one theorem to be proved. However, the total number of theorems in this chapter is nine. These theorems can be applied to solve application or conceptual level questions. In this chapter, students will learn the properties of a Quadrilateral, Types of Quadrilateral, Properties of a Pyramids, and Mid-Point Theorem.

Topics covered in Class 9 Maths Chapter 8 Quadrilaterals

• Angle sum property of a quadrilateral

• Properties of a parallelogram

• Conditions for a quadrilateral to be a parallelogram

• Mid-point theorem

 NCERT Solutions for Class 9 Maths Chapter 8 - Quadrilaterals Exercise 8.1 7 Questions Exercise 8.2 6 Questions

### NCERT Solutions for Class 9 Maths Chapter 9 Circles

A circle is defined as the sum of all points in the plane at a constant distance from a point on the plane. This chapter covers topics such as angle subtended by a chord at a point, equal chords and their distance from the centre, angle subtended by an arc of a circle, cyclic quadrilaterals, and other concepts related to circles. There are 12 theorems in this chapter and students will learn which theorems will help them understand the concepts better.

Topics covered in Class 9 Maths Chapter 9 Circles

• Introduction

• Circles and its related terms

• Angle subtended by a chord at a point

• Perpendicular from the centre of a chord

• Circle through three points

• Equal chords and their distances from the centre

• Angle subtended by an arc of a circle

• Cyclic quardrilateral

• Summary

 NCERT Solutions for Class 9 Maths Chapter 9 - Circles Exercise 9.1 2 Questions Exercise 9.2 6 Questions Exercise 9.3 12 Questions

### NCERT Solutions for Class 9 Maths Chapter 10 Heron’s Formula

In this chapter, we are going to talk about Heron's formula. Heron's formula can be used to determine the area of the triangle when all three sides of the triangle are given. In this formula, the angles and other distances of the triangle are not required. Heron's formula can not only be used to figure out the area of triangles, but also to figure out the areas of quadratic and other polygons if they are divided into triangles.There are two exercises in this chapter that help the students to understand how to solve the problems based on the Heron's formula.

Topics covered in Class 9 Maths Chapter 10 Heron’s Formula

• Area of a Triangle by Heron’s formula

 NCERT Solutions for Class 9 Maths Chapter 10 - Heron’s Formula Exercise 10.1 7 Questions

### NCERT Solutions for Class 9 Maths Chapter 11 Surface Areas and Volumes

In all our studies so far, we have focused on figures that are easy to draw on our notebook or blackboard. These figures are referred to as plane figures. In previous classes, we have learned about rectangles, squares, and circles, what they mean by their perimeter and area, and how to locate them. We have also learned about cubes, cylinders and other solids in earlier classes. Let us now see what would happen if we cut out a large number of planes of the same size and shape from cardboard sheets and stack them in a vertical stack. Through this process, we will be able to obtain some solid figures such as a cube, cylinder, etc. We will now learn about the surface areas and volume of cuboids in more detail and expand this study to other solids like cones and spheres.

Topics covered in Class 9 Maths Chapter 11 Surface Areas and Volumes

• Lateral and Total Surface Area of Cube and Cuboid

• Curved and Total Surface Area of a Cylinder

• Curved and Total Surface Area of right angular Cone

• Surface Area of a Sphere

• Curved and Total Surface Area of an Hemisphere

• Volume of Cube and Cuboid

• Volume of Cylinder

• Volume of Cone

• Volume of Sphere

• Volume of Hemisphere

 NCERT Solutions for Class 9 Maths Chapter 11 - Surface Areas and Volumes Exercise 11.1 8 Questions Exercise 11.2 9 Questions Exercise 11.3 9 Questions Exercise 11.4 10 Questions

### NCERT Solutions for Class 9 Maths Chapter 12 Statistics

We have already looked at how tables represent data. Now let’s look at another way of presenting data: graphical representation. As they say, “a picture is worth a thousand words”. Most of the time, the comparison between the individual items is best presented in the form of a graph. This way, the representation becomes easier to comprehend than the actual data.

Topics covered in Class 9 Maths Chapter 12 Statistics

• Bar Graphs

• Histograms of uniform width, and of varying widths

• Frequency polygons

 NCERT Solutions for Class 9 Maths Chapter 12 - Statistics Exercise 12.1 7 Questions

NCERT solutions for Class 9 maths by eSaral provide an analysis and explanation of every problem covered by NCERT textbook for Class 9. The solutions are given in detail so that students can understand the concepts as they are going through them. These Class 9 Maths NCERT solutions by eSaral are provided by Maths experts of eSaral.

NCERT Class 9 Maths Solutions are very useful for students who are preparing for CBSE examinations and state or national competitions like Olympiads. NCERT Solutions for class 9 Maths by eSaral help students in preparing for class 10th and corresponding board exams.