NCERT Solutions for Class 9 Maths

Mathematics NCERT Grade 9, Chapter 1: Number Systems- Basics of number line and representation on number line is already discussed in earlier classes. Rational numbers and Irrational numbers are defined in this chapter and students can practice the questions based on them from exercise 1.1.
One can learn the representation of the irrational number on the number line in section 1.2.
A short exercise 1.2 is given which also has a small activity on square root spiral. Moving forward students will learn about real numbers and their decimal expansions.
In the next section, the topic of representing real numbers on the number line is explained through successive magnification. Exercise 1.4 contains only two questions. After that, operations on real numbers is discussed including rationalising the denominator. Section 1.6 explains laws of exponents for real numbers. Later in section 1.7 summary of the chapter is given.  

Mathematics NCERT Grade 9, Chapter 2: Polynomials: Polynomials are a particular type of algebraic expression. In this chapter, students will also study the remainder theorem and factor theorem. Some more algebraic identities will be discussed in this chapter. Their use in factorisation and in evaluating some given expressions will also be discussed. 
Starting with Polynomials in One Variable, the chapter will move further to the topic zeroes of a polynomial. Different types of polynomials and their degree are explained in this section. Exercise 2.2 is based on the same concept.
The next section explains the topic remainder theorem. In this section, the solved example is given in which each step of the solution is explained in detail.
Section 2.5 covers the topic, factorisation of polynomials including splitting middle term and factor theorem. Section 2.6 is based on the topic, algebraic identities. 8 identities are discussed in this section. In the end, summary of the chapter is given. 

Mathematics NCERT Grade 9, Chapter 3: Coordinate Geometry: How to locate a point on the number line is already discussed. In this chapter we will study how to locate a point on the coordinate plane. This is explained with the help of a small activity followed by exercise 3.1.  The next section is about the Cartesian system and related aspects like xy - plane, quadrants, origin, abscissa, ordinate. 2 solved examples are given in this section. Exercise 3.2 is a short exercise containing only 2 questions. 
After that topic- Plotting a point in the plane if its coordinates are given is discussed. It is explained by two solved examples. 
Activity is also given in this section to make the topic more interesting. The last exercise is 3.3 comprising of 2 questions.
At the end the chapter is summarised using 11 points.

Mathematics NCERT Grade 9, Chapter 4 Linear Equations In Two Variables: Linear equation in one variable was already explained in earlier classes. Linear equations in one variable have a unique solution. In this chapter, students will learn about linear equations in two variables. Exercise 4.1 is a short exercise containing 2 questions.
The next part of the chapter deals with solution of linear equation. A linear equation in two variables has infinitely many solutions. Exercise 4.2 contains 4 questions based on the same.
Moving ahead students will find a detailed description about the topic- Graph of a linear equation in two variables. A degree one, polynomial equation will be referred to as linear equation as it will form a straight line. Solved examples are given to make the topic clear to students followed by the exercise 4.3.
Last section deals with the topic- Equations of lines parallel to the x - axis and y - axis. Exercise 4.4 is a short exercise and contains only two questions. After that conclusion of the chapter is provided. 

Mathematics NCERT Grade 9, Chapter 5: Introduction To Euclid's Geometry - The word geometry comes from the Greek word 'geo' meaning the 'earth' and 'metrein' meaning 'to measure'. A detailed description of geometry is given in the introduction part. In this chapter Euclid's Definitions, Axioms and Postulates are given. This is followed by solved examples and theorem that two distinct lines cannot have more than one point in common. Exercise 5.1 contains 7 questions out of which 1 question is of true/ false.
Section 5.3 is about the topic - Equivalent Versions of Euclid's Fifth Postulate. There are several equivalent versions, but in this class we will study only two versions. One of them is Playfair's Axiom and the other one is that two distinct intersecting lines cannot be parallel to same line. Exercise 5.2 is a short exercise that contains 2 questions. 
For final conclusion the summary of the chapter is given.  

Mathematics NCERT Grade 9, Chapter 6: Lines and Angles: In this chapter students will study the properties of the angle formed when two lines intersect each other and properties of the angle formed when a line intersects two or more parallel lines at distinct points. The chapter starts from basic level, the first topic of the chapter being Basic Terms and Definitions where angles like straight, reflex, right, acute, obtuse, supplementary and complementary angles are discussed.
Section 6.3 is about the topic- Intersecting Lines and Non-Intersecting Lines.
After that, the topic- Pair of Angles is discussed. 2 axioms, 1 theorem, and some solved examples are also given in this section. Exercise 6.1 contains various questions based on all the concepts discussed in the previous section.
The next section is about Parallel Lines and a Transversal. Axioms and theorems on this concept are explained.
Moving on, students will find a detailed description on Lines Parallel to the Same Line. Exercise 6.2 contains 6 questions based on the same.
Another important topic discussed in this chapter is Angle Sum Property. This topic is supported by exercise 6.3.
In the end, a summary of the chapter is provided.

Mathematics NCERT Grade 9, Chapter 7: Triangles- Students have already studied about triangles and their various properties. A closed figure formed by three intersecting lines is called a triangle. In this chapter, students will study in detail about the congruence of triangles, rules of congruence, some more properties of triangles and inequalities in a triangle.
The first section lays emphasis on the Congruence of Triangles. After that, students will learn about the Criteria for the Congruence of Triangles.
SAS(Side-Angle-Side), ASA(Angle-Side-Angle), AAS(Angle-Angle-Side), SSS(Side-Side-Side), RHS(Right angle-Hypotenuse-Side) congruence rule are some of the congruence criteria.
Later on, Some properties of a triangle are discussed involving isosceles triangles.
Exercise 7.4 is based on the topic inequalities in a triangle. Ample number of questions are given as solved examples and in exercises for the assessment of students. Three theorems are explained through two activities in this particular section.
For the final conclusion summary of the chapter is discussed. 

Mathematics NCERT Grade 9, Chapter 8: Quadrilaterals- Ever wondered marking four points in some specific pattern forms what? In this chapter curiosity of students about a four-point closed figure will be quenched. A quadrilateral has the following properties:
1. Four sides 
2. Four angles
3. Four vertices

Mathematics NCERT Grade 9, Chapter 9: Areas of Parallelograms and Triangles- The chapter starts with a detailed introduction followed by the brief discription on the topic Figures on the Same Base and Between the same parallels. Exercise 9.1 contains only one question. The next part is about Parallelograms on the Same Base and between the same Parallels where one studies that the two parallelograms have the same area. A theorem is stated and proof of the same is also given. After this problems are given in exercise 9.2 for practice. 
Moving on the next topic which is Triangles on the same base and between the same Parallels relating the areas of the two. Exercise 9.3 is based on the same concept. Exercise 9.4 is an optional exercise containing 8 questions which is not given from the examination point of view. 
In the end summary of the chapter is provided. 

Mathematics NCERT Grade 9, Chapter 10: Circles- We all see many objects in our daily life which are round in shape.
In this chapter, students will study about circles, other related terms and some properties of a circle.
Firstly, Circles and Its Related Terms like minor and major arc, minor and major segment, minor and major sector of a circle are discussed. Exercise 10.1 contains two short questions.
The next section is about Angle Subtended by a Chord at a point. 2 theorems are discussed in this section. Exercise 10.2 contains only two questions.
After this topic- Perpendiculars Drawn from the Centre to a Chord is discussed. This is followed by a description of the topic - Circle through Three Points or circumcircle. Moving on, 3 questions are given in exercise 10.3. Section 10.6 is about Equal Chords and Their Distance from the Centre. Some solved examples are given for better understanding. Before moving further to the next topic students must solve the questions of exercise 10.4.
Angle Subtended by an Arc of a Circle and Cyclic Quadrilaterals is explained later. The last exercise is 10.5 after that an optional exercise is given.
The summary of the chapter-Circles is provided in the end. 

Mathematics NCERT Grade 9, Chapter 11: Constructions- This chapter starts with a brief introduction about the geometrical instruments and their usage.
After that, the first section is given which discusses Basic Constructions. 3 constructions are explained stepwise.
1. To construct the bisector of a given angle.
2. To construct the perpendicular bisector of a given line segment. 
3. To construct an angle of 60$°$ at the initial point of a given ray.
This is followed by the topic Some Constructions of Triangles. This includes construction of triangles when its base, base angle and sum/difference of other two sides are given. Also, construction of triangle when its perimeter is given.
And the last exercise i.e 11.2 is given. It is not a very long exercise and contains only 5 questions but this chapter needs a lot of practice so that you can attempt all questions of construction in minimum possible time with complete accuracy.
In the end summary of the chapter is provided.  

Mathematics NCERT Grade 9, Chapter 12: Heron's Formula- Students have already studied figures of different shapes such as squares, rectangles, triangles, and quadrilaterals and also calculated the perimeters and areas of figures. 
Students are already familiar with the formula of the area of the triangle using the height and base length. We can apply the previously learned formula to the right-angled triangles only. But to calculate the area of the equilateral triangle, isosceles triangle, and scalar triangle, we need Heron's formula. 
Starting with the topic- Area of a triangle- by Heron's formula. This formula uses all the sides of the triangle and semi-perimeter. 3 solved examples are given to make the application of formula clear to students. Exercise 12.1 contains 6 questions, some are in the form of word problems in which students have to find the area of the triangle formed in different situations.
After that application of Heron's Formula in Finding Area of Quadrilaterals is given. Next exercise 12.2 is based on the same.
To end the chapter, a summary of the chapter- Heron's Formula is given. 

Mathematics NCERT Grade 9, Chapter 13: Surface Areas And Volumes- In this chapter, we will learn about how to find the total surface areas, curved surface areas and volumes of cuboids, cone, sphere, hemisphere and cylinders in detail. The first section is about the surface area of a cuboid and a cube. Exercise 13.1 is about the same. In a similar way, surface area of different solid shapes are discussed and solved examples with corresponding exercise questions are given.
Later, in this chapter volume will be discussed.
Firstly, the volume of the cuboid is discussed, then the volume of the cylinder is explained followed by the volume of a right circular cone.
Lastly, we will discuss the volume of a sphere and hemisphere. Again, the concept of volume can be understood through different questions given in the NCERT.
After that, an optional exercise is given which is not from the examination point of view.
In the end summary of the chapter is discussed.  

Mathematics NCERT Grade 9, Chapter 14: Statistics- Numbers, facts, figures, tables, graphs, these are all common in our daily lives. Extraction of meaningful information is studied in a branch of mathematics called statistics. The first section is about the Collection of Data. This particular section is explained with the help of an activity.
Moving to the next topic- Presentation of Data. Here students will come across terms like raw data, range, frequency, ungrouped frequency distribution table, tally marks​, class limits etc. In this topic, students will find solved examples to make the concept more comprehensible.
After that, the Graphical Representation of Data is explained. They include:
a. Bar graphs
b. Histograms of uniform width, and of varying widths
c. Frequency of polygons
The measure of central tendency is the last topic of this chapter. It includes measures like mean, median and mode. The chapter is filled with solved examples and exercise questions so that the chapter is easy to understand and grasp.
In the end summary of the chapter is provided. 

Mathematics NCERT Grade 9, Chapter 15: Probability- We all feel that there are some probabilities that we face in our daily lives such as it will probably rain today, most probably she will go today, etc. 
The very first section given after a short introduction is Probability- an Experimental Approach. Here we study about the experimental or empirical probability and the various other terms like trial, event, outcomes of an experiment etc.

• An event for an experiment is the collection of some outcomes of the experiment.