## NCERT Solutions for Class 9 Maths (updated 2021-22 )

**NCERT’s** Class 9 Mathematical Solutions incorporates solutions to all questions provided in the NCERT Class 9 textbook. Students can download a PDF of the chapter-specific solutions to these issues, from the links provided at the bottom of this page. These NCERT Class 9 solutions cover all topics included in the NCERT textbook-like Number System, Coordinate Geometry, Polynomials, Euclid’s Geometry, Quadrilaterals, Triangles, Circles, Constructions, Surface Areas and Volumes, Statistics, Probability, etc.

**Chapter 1 Number System**

**Chapter 2 Polynomials**

**Chapter 3 Coordinate Geometry**

**Chapter 4 Linear Equations in Two Variables**

**Chapter 5 Introduction to Euclids Geometry**

**Chapter 6 Lines and Angles**

**Chapter 7 Triangles**

## Chapter 8 Quadrilaterals

## Chapter 9 Areas of Parallelograms and Triangles

## Chapter 10 Circles

## Chapter 11 Constructions

## Chapter 12 Heron’s Formula

**Chapter 13 Surface Areas and Volumes**

**Chapter 14 Statistics**

**Chapter 15 Probability**

### NCERT Solutions Class 9 Maths Chapter 1 Number System – Term I

This chapter discusses different topics, including rational numbers and irrational numbers. Students will also be learning the extended version of the number line and how to represent numbers (integers, rational and irrational) on it. A total of 6 exercises are present in this chapter that contains the problems based on all the topics asked in the chapter. This chapter also teaches students the representation of terminating/non-terminating recurring decimals (and successive magnification method) as well as the presentation of square roots of 2, 3 and other non-rational numbers on the number line. The chapter also deals with the laws of integral powers and rational exponents with positive real bases in Number System.

*Topics Covered in Class 9 Maths Chapter 1 Number System for First Term:*

*Topics Covered in Class 9 Maths Chapter 1 Number System for First Term:*

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

- Examples of non-recurring/non-terminating decimals. Existence of non-rational numbers (irrational numbers) such as, √2, √3 and their representation on the number
- Rationalization (with precise meaning) of real numbers of the type \frac{1}{a+b\sqrt{x}}\: and\: \frac{1}{\sqrt{x}+\sqrt{\sqrt{y}}}
*a*+*b**x*1*a**n**d**x*+*y*1 (and their combinations) where x and y are natural number and a and b are integers. - 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.)

**Important Formulas – **

**Operations on Real Numbers**\\\sqrt{ab}=\sqrt{a}\sqrt{b} \\ \\\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}*a**b*=*a**b**b**a*=*b**a*

(√a + √b) (√a – √b) = a – b

(a + √b) (a – √b) = a^{2} – b

(√a + √b) (√c + √d) = √ac + √ad + √bc + √bd

(√a + √b)^{2} = a + 2√ab + b

**Laws of Exponents for Real Numbers**

a^{m} . a^{n} = a^{m + n}

(a^{m})^{n} = a^{mn}

a^{m}/a^{n} = a^{m – n}, m > n

a^{m}b^{m} = (ab)^{m}

### NCERT Solutions Class 9 Maths Chapter 2 Polynomials – Term II

This chapter discusses a particular type of algebraic expression called polynomial and terminology related to it. Polynomial is an expression that consists of variables and coefficients, involving the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The chapter also deals with Remainder Theorem and the Factor Theorem with the uses of these theorems in the factorisation of polynomials. Students will be taught several examples as well as the definition of different terms like polynomial, degrees, coefficient, zeros and terms of a polynomial. A total of 5 exercises are present in this chapter which includes problems related to all the topics mentioned in the chapter.

*Topics Covered in Class 9 Maths Chapter 2 Polynomials for Second Term:*

*Topics Covered in Class 9 Maths Chapter 2 Polynomials for Second Term:*

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. Factorization of ax^{2} + 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

(x + y + z)^{2} = x^{2} + y^{2} + z^{2} + 2xy + 2yz + 2zx

(x ± y)^{3} = x^{3} ± y^{3} ± 3xy (x ± y)

x^{3} ± y^{3} = (x ± y) (x^{2} ± xy + y^{2})

and their use in factorization of polynomials.

**Important Formulas –**

(x + y)^{2} = x^{2} + 2xy + y^{2}

(x – y)^{2} = x^{2} – 2xy + y^{2}

x^{2} – y^{2} = (x + y) (x – y)

(x + y + z)^{2} = x^{2} + y^{2} + z^{2} + 2xy + 2yz + 2zx

(x + y)^{3} = x^{3} + y^{3} + 3xy(x + y)

(x – y)^{3} = x^{3} – y^{3} – 3xy(x – y)

x^{3} + y^{3} + z^{3} – 3xyz = (x + y + z) (x^{2} + y^{2} + z^{2} – xy – yz – zx)

Dividend = (Divisor × Quotient) + Remainder

**Remainder theorem**

Let p(x) be any polynomial of degree greater than or equal to one and let a be any real number. If p(x) is divided by the linear polynomial x – a, then the remainder is p(a).

**Factor theorem**

If p(x) is a polynomial of degree n > 1 and a is any real number, then (i) x – a is a factor of p(x), if p(a) = 0, and (ii) p(a) = 0, if x – a is a factor of p(x).

### NCERT Solutions Class 9 Maths Chapter 3 Coordinate Geometry – Term I

The chapter Coordinate Geometry includes the concepts of the cartesian plane, coordinates of a point in xy – plane, terms, notations associated with the coordinate plane, including the x-axis, y-axis, x- coordinate, y-coordinate, origin, quadrants and more. Students, in this chapter, will also be studying the concepts of Abscissa and ordinates of a point as well as plotting and naming a point in xy – plane. There are 3 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 for First Term:*

*Topics Covered in Class 9 Maths Chapter 3 Coordinate Geometry for First Term:*

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

**Important Points – **

- To locate the position of an object or a point in a plane, we require two perpendicular lines. One of them is horizontal, and the other is vertical.
- The plane is called the Cartesian, or coordinate plane and the lines are called the coordinate axes.
- The horizontal line is called the x -axis, and the vertical line is called the y – axis.
- The coordinate axes divide the plane into four parts called quadrants.
- The point of intersection of the axes is called the origin.
- The distance of a point from the y – axis is called its x-coordinate, or abscissa, and the distance of the point from the x-axis is called its y-coordinate, or ordinate.

### NCERT Solutions Class 9 Maths Chapter 4 Linear Equations in Two Variables – Term I

Along with recalling the knowledge of linear equations in one variable, this chapter will introduce the students to the linear equation in two variables, i.e., ax + by + c = 0. Students will also learn to plot the graph of a linear equation in two variables. There are 4 exercises in this chapter that consist of questions related to finding the solutions of a linear equation, plotting a linear equation on the graph and other topics discussed in the chapter.

*Topics Covered in Class 9 Maths Chapter 4 Linear Equations in Two Variables for First Term:*

*Topics Covered in Class 9 Maths Chapter 4 Linear Equations in Two Variables for First Term:*

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. Graph of linear equations in two variables. Examples, problems from real life with algebraic and graphical solutions being done simultaneously.

**Important Points –**

- An equation of the form ax + by + c = 0, where a, b and c are real numbers, such that a and b are not both zero, is called a linear equation in two variables.
- A linear equation in two variables has infinitely many solutions.
- The graph of every linear equation in two variables is a straight line.
- x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis.
- The graph of x = a is a straight line parallel to the y-axis.
- The graph of y = a is a straight line parallel to the x-axis.
- An equation of the type y = mx represents a line passing through the origin.