NCERT Solutions Class 9 Maths: Chapter-Wise Details

NCERT Solutions Class 9 Maths are collated by subject-matter experts at SimplyAcad to help students solve various mathematical problems of different chapters. Students will be extremely benefitted with the provided solutions as they guide them through all the questions asked in the prescribed exercises of NCERT Maths textbook. Students must use the solutions to do regular practice and revisions. This will ensure them to score maximum marks in their paper. 

 

NCERT Solutions Class 9 Maths is available below, students can access the solutions of each chapter by clicking on the specific chapter. The solutions of all chapters including Number System, Coordinate Geometry, Euclid’s Geometry, Polynomials, Triangles, Circles, Quadrilaterals, Constructions, etc. are explained in detail through stepwise manner. 

NCERT Solutions Class 9 Maths Chapters

 

Chapter Wise ncert class 9 maths solutions

Chapter 1 Number System

Chapter 2 Polynomials

Chapter 3 Coordinate Geometry

Chapter 4 Linear Equations in Two Variables

Chapter 5 Introduction to Euclid’s 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

Chapter 1 of ncert class 9 maths: Number System deals with various concepts of the topic including rational numbers and irrational numbers, representing numbers such as integers, rational and irrational on the number line,etc. There are a total of 6 exercises in the chapter covering all the sections discussed to help students strengthen their understanding.

Overview of the exercises of Chapter 1: Number System

Overview of the exercises of Chapter – Number Systems Chapter 1

Exercise 1.1: The first exercise deals with helping students distinguish between natural, integer, and whole numbers, and represent the correct format of real numbers.

Exercise 1.2: The second exercise focuses on forming the basics of irrational numbers.

Exercise 1.3: The third exercise deals with the concept of rational and irrational numbers.

Exercise 1.4: The fourth exercise focuses on explaining the process of successive magnification.

Exercise 1.5: The fifth exercise deals with addition and subtraction of a rational and irrational number resulting in an irrational number.

Exercise 1.6: The final exercise of the number systems focuses on the application of the law of exponents for real numbers.

Important Formulas:

Operations on Real Numbers

√(a/b) = √a/√b

√ab = √a √b

(√a + √b) (√a – √b) = a – b

(a + √b) (a – √b) = a2 – b

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

(√a + √b)2 = a + 2√ab + b

Laws of Exponents for Real Numbers

am . an = am + n

(am)n = amn

am/an = am – n, m > n

ambm = (ab)m

 

Class 9 Maths NCERT Solutions Chapter 1 Exercises

Exercise 1.1 – 4 Questions (4 short answers) Exercise 1.2 – 4 Questions (4 short answers) Exercise 1.3 – 9 Questions (8 short answers, 1 long answer) Exercise 1.4 – 2 Questions (2 long answers) Exercise 1.5 – 5 Questions (4 short answers, 1 long answer) Exercise 1.6 – 3 Questions (3 short answers)

NCERT Solutions Class 9 Maths Chapter 2 Polynomials

Chapter 2 of ncert class 9 maths: Polynomials deals with a certain form of algebraic expression called polynomials and its related functions. It is basically an expression comprising variables and coefficients involving various operations such as addition, subtraction, multiplication, etc. Remainder theorem is also a prominent section included in the chapter for factorisation of polynomials. There are a total of 5 exercises included in the chapter.

Overview of the exercise of Chapter 2 Polynomials 

Exercise 2.1: The first exercise focuses on explaining the basic concept of a polynomial and its components.

Exercise 2.2: The second exercise deals with the concept of zero of the polynomial. 

Exercise 2.3: The third exercise questions on the remainder theorem.

Exercise 2.4: The fourth exercise deals with the concepts of polynomial factorization.

Exercise 2.5: The final exercise is based on algebraic entities.

Important Formulas:

(x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx

(x ± y)3 = x3 ± y3 ± 3xy (x ± y)

x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – 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).

 

Class 9 Maths NCERT Solutions Chapter 2 Exercises

Exercise 2.1 – 5 Questions (5 short answers) Exercise 2.2 – 4 Questions (4 short answers) Exercise 2.3 – 3 Questions (3 short answers) Exercise 2.4 – 5 Questions (4 short answers, 1 long answer) Exercise 2.5 – 16 Questions (9 short answers, 5 long answers, 2 very long answers)

NCERT Solutions Class 9 Maths Chapter 3 Coordinate Geometry

Chapter 3 of ncert class 9 maths, Coordinate Geometry focuses on the concepts of cartesian plane, coordinates of a point in xy–plane, followed by terms, and notations related with the coordinate plane, comprising the origin, quadrants,  x-axis, y-axis, x- coordinate, and y-coordinate. There are a total of two exercises in the chapter, go through the provided outline of the exercises to get a brief idea about them.

Overview of the Exercises of Chapter 3: Coordinate Geometry

Exercise 3.1: The first exercise deals with introducing the basic concepts of coordinate geometry including Cartesian plane, quadrants, coordinates of points,  distance formula, and the section formula.

Exercise 3.2: The second exercise revolves around the different forms of equations of a straight line. Finding the slope and intercept of a line is also focused.

 

Class 9 Maths NCERT Solutions Chapter 3 Exercises

Exercise 3.1 – 2 Questions (1 short answer, 1 long answer) Exercise 3.2 – 2 Questions (2 short answers)

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

Chapter 4 of the ncert class 9 maths, Linear Equations in Two Variables advances the topic to the next level where students will be taught to solve linear equations two variables, and plotting the graph of the same. There are a total of 4 exercises prescribed in the chapter for students to understand the concepts better. Here is an overview of the exercises that students must complete to score well in the exams. 

Overview of the Exercises Of Chapter 4 Linear Equations in Two Variables

 

Exercise 4.1: The first exercise deals with the standard form of linear equation having two variables. The questions are based on representing a statement, expressing the linear equation in two variables form, and then identifying a,b,c in a linear equation.

Exercise 4.2: The second exercise is the sub part of the first and mainly focuses on the above mentioned two variable linear equations. Identifying the number of solutions of the equation, finding the solution and the unknown term is the basis of the questions asked here.

Exercise 4.3: The third exercise deals with drawing graphs of the given data. This section also includes a few word problems to be solved. 

Exercise 4.4 : The final exercise includes geometrical representations of the given set of data.

 

Class 9 Maths NCERT Solutions Chapter 4 Exercises

Exercise 4.1 – 2 Questions (2 short answers) Exercise 4.2 – 4 Questions (3 short answers, 1 long answer ) Exercise 4.3 – 8 Questions (4 short answers, 4 long answers) Exercise 4.4 – 2 Questions (2 long answers)

 

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

Chapter 5 of the ncert class 9 maths, Introduction to Euclid’s Geometry allows students to understand the approach of euclid to geometry and how to link it with the present day geometry. There are a total of two exercises included in the chapter, go through the overview to learn more  about them.

Overview of the Exercises of Chapter 5: Introduction to Euclid’s Geometry

Exercise 5.1: The first exercise includes true and false questions to check the basics of students, followed by drawing figures to explain and prove equations on a line, and proof related to the midpoint segment.

Exercise 5.2 : The second exercise of the chapter includes a total of two questions of Euclid fifth postulate. 

Class 9 Maths NCERT Solutions Chapter 5 Exercises

Exercise 5.1 – 7 Questions (4 short answers, 3 long answers) Exercise 5.2 – 2 Questions (1 short answer, 1 long answer)

NCERT Solutions Class 9 Maths Chapter 6 Lines and Angles

Chapter 6 of the ncert solutions class 9 maths: Lines and Angles is a crucial section of the syllabus prescribed to the students. Students must go through the important theorem discussed to score better. 

Class 9 Maths NCERT Solutions Chapter 6 Exercises

Exercise 6.1 – 6 Questions (5 short answers, 1 long answer) Exercise 6.2 – 6 Questions (3 short answers, 3 long answers) Exercise 6.3 – 6 Questions (5 short answers, 1 long answer)

 

NCERT Solutions Class 9 Maths Chapter 7 Triangles

Chapter 7 of the ncert class 9 maths, Triangles deals with its different types, properties and congruence of triangles. There are a total of three exercises present in the chapter to help students learn various concepts, theorems and axioms related to triangles. Check out the given outline of the exercises to get a brief idea about them.

Overview of the Exercises of Chapter 7 Triangles 

Exercise 7.1: The first deals with the basic properties of a triangle, including interior and exterior angles and inequalities.

Exercise 7.2: The second exercise deals with criteria for congruence of triangles, including SSS, SAS, ASA, RHS, and AAS.

Exercise 7.3: The third exercise focuses on the properties of isosceles triangles as well as equilateral triangles.

 

Class 9 Maths NCERT Solutions Chapter 7 Exercises

Exercise 7.1 – 8 Questions (6 short answers, 2 long answers) Exercise 7.2 – 8 Questions (6 short answers, 2 long answers) Exercise 7.3 – 5 Questions (3 short answers, 2 long answers)

NCERT Solutions Class 9 Maths Chapter 8 Quadrilaterals

Chapter 8 of the ncert class 9 maths: Quadrilateral deals with various problems related to a four side enclosed figure. Some of the important sections the chapter contains are the angle sum property of a quadrilateral, different types of quadrilaterals, the properties of a parallelogram, and the mid-point theorem. There are a total of 2 exercises in the chapter including theorems to prove and solve the questions asked. 

Overview of the Exercises of Chapter 8 Quadrilaterals

 

Exercise 8.1: The first exercise includes concepts of quadrilaterals, students are asked to identify the properties and its different types. 

Exercise 8.2: The second exercise deals with application of the properties of the quadrilaterals, students are asked to prove the properties and apply them to solve problems.

 

Class 9 Maths NCERT Solutions Chapter 8 Exercises

Exercise 8.1 – 12 Questions (2 short answers, 6 long answers, 4 very long answers) Exercise 8.2 – 7 Questions (2 short answers, 2 long answers, 3 very long answers)

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

Chapter 9 of ncert class 9 maths, Areas of Parallelograms and Triangles focuses on helping students grasp the formulas to calculate areas of different figures by finding the relationship between different sides, geometric facts, and theories. There are a total of 4 exercises included  in the chapter.

Overview of the Exercises of Chapter 9 Areas of Parallelograms and Triangles

 

Exercise 9.1: The first exercise focuses on defining a parallelogram and its related properties, students are asked questions testing their basics, such as identifying and distinguishing the different types.

Exercise 9.2: The second exercise goes around to establish the properties and types of triangles according to their sides and angles. Heron’s formula and Pythagorean theorem are also focused here.

Exercise 9.3: The third exercise focuses on calculating areas of the different types of quadrilateral such as parallelograms, rectangles, squares, and rhombuses.

Exercise 9.4: The final exercises contain questions in word problem format based on real life situations. Students are asked to apply the concepts and formulas discussed in the above exercise to solve the problems.

Class 9 Maths NCERT Solutions Chapter 9 Exercises

Exercise 9.1 – 1 Question (1 short answer) Exercise 9.2 – 6 Questions (5 short answers, 1 long answer) Exercise 9.3 – 16 Questions (12 short answers, 4 long answers) Exercise 9.4 – 8 Questions (4 short answers, 1 long answer, 3 very long answers)

NCERT Solutions Class 9 Maths Chapter 10 Circles

Chapter 10 of the ncert class 9 maths, Circles discussed various concepts such as equal chords and their distances from the centre, angle subtended by a chord at a point, the angle subtended by an arc of a circle, cyclic quadrilaterals,etc. There are a total of 6 exercises covered in the chapter, you can go through the given summary of all the mentioned exercises to get a brief idea about them.

Overview of the Exercises of Chapter 9 Circles

Exercise 10.1: The first chapter focuses on the understanding of the basics, including definition and properties of a circle. Students are asked to find the radius of the circle here. 

Exercise 10.2: The second exercise deals with tangents and its properties of a circle. Students are asked to calculate the length of the tangents, the angle between the radius and tangent, and the distance of point from the centre of the circle.

Exercise 10.3: The third exercise deals with secants of a circle.Students are asked to find the length, angle between the secant and the tangent, and the intersecting point of two secants. 

Exercise 10.4: The fourth exercise focuses on the chords of a circle. Students are asked to find the length of the chord, the angle between two chords, and its perpendicular bisector.

Exercise 10.5: The fifth exercise focuses on the practical applications of circles. Students are asked to find the radius of a wheel and the length of the belt, followed by calculating the distance between the two parallel tracks, and the different number of revolutions completed by the wheel.

Exercise 10.6: The final exercise is a compilation of the above ones. It covers all the concepts learnt in the chapter and its applications to solve the questions. These problems consist of calculating the areas of the shaded portion, calculating the length of the tangents, etc.

 

Class 9 Maths NCERT Solutions Chapter 10 Exercises

Exercise 10.1 – 2 Questions (2 short answers) Exercise 10.2 – 2 Questions (2 long answers) Exercise 10.3 – 3 Questions (3 long answers) Exercise 10.4 – 6 Questions (6 long answers) Exercise 10.5 – 12 Questions (12 long answers) Exercise 10.6 – 10 Questions (10 long answers)

 

NCERT Solutions Class 9 Maths Chapter 11 Constructions

Chapter 11 of ncert class 9 maths: Constructions deals with the basics of the topic and students learn to construct different types of triangles using several theorems discussed in the chapter. There are a total of two exercises presented to boost their understanding and help them to solve various types of questions with distinct parameters.

Overview of the Exercises of Chapter 11 Constructions

Exercise 11.1: The first exercise deals with constructing geometric figures, including angles, triangles, parallelograms and performing angle bisector through ruler and a compass.  

Exercise 11.2: The second exercise deals with application of geometry to solve real life situation questions including constructing a quadrilateral, triangle, hexagon, triangle with certain sides given or using a median to calculate the length.

 

Class 9 Maths NCERT Solutions Chapter 11 Exercises

Exercise 11.1 – 5 Questions (2 short answers, 2 long answers, 1 very long answer) Exercise 11.2 – 5 Questions (5 very long answers)

 

NCERT Solutions Class 9 Maths Chapter 12 Heron’s Formula

Chapter 12 of the ncert class 9 maths, Heron’s Formula teaches students to use the formula to calculate the area of the triangles when the lengths of all the sides are already mentioned. There are two important exercises included in the chapter to help students understand the usage of formulas in the right way.

Overview of the Exercises of Chapter 10 Heron’s Formula

Exercise 12.1: The first exercise of the chapter focuses on the basics concepts of Heron’s Formula. Students are asked to find the area, missing sides and height of a triangle, when certain specifics are given.

Exercise 12.2: The second exercise consists of a total of 9 questions and is the continuation of the first exercise. 

Important Formulas:

Area of a triangle = 1/2 × base × height

Area of a triangle by Heron’s Formula = 

Where a, b and c are the sides of a triangle

s is the semi-perimeter, i.e. half of the perimeter of a triangle = (a + b + c)/2

 

Class 9 Maths NCERT Solutions Chapter 12 Exercises

Exercise 12.1 – 6 Questions (2 short answers, 2 long answers, 2 very long answers) Exercise 12.2 – 9 Questions (4 long answers, 5 very long answers)

NCERT Solutions Class 9 Maths Chapter 13 Surface Areas and Volumes

Chapter 13 of the ncert class 9 maths, Surface Areas and Volumes teaches students to calculate the surface areas and volumes of cylinders and cuboids, and other solid objects including cones, spheres, hemisphere etc. The chapter is an advanced form of the Mensuration that was already covered in details in the previous maths classes. There are a total of 8 exercises included in the chapter for students to complete and practise. This will allow them to master the topic and perform well in exams. 

Overview of the Exercises of Chapter 11 Surface Areas and Volumes

Exercise 13.:1 The first exercise focuses on  explaining the geometry of cuboids through simple cardboard based questions.

Exercise 13.2: The second exercise of the chapter focuses on the details of a right circular cylinder.

Exercise 13.3: The third exercise focuses on introducing a three-dimensional image of the right circular cone.

Exercise 13.4: The fourth exercise is built on the concept of a sphere’s surface area. 

Exercise 13.5: The fifth exercise of the chapter focuses on the concept of volume, that the space occupied by an object is measurable, it’s the volume of the object. 

Exercise 13.6: The sixth exercise consists of the more defined questions on volume including finding the volume of the cylinder.

Exercise 13.7: The seventh exercise focuses on how the volume of a right circular cone is equal to one-third the volume of the cylinder.

Exercise 13.8: The final exercise of the chapter deals with the volume of a sphere. 

Important Formulas:

• Surface Area of a Cuboid = 2(lb + bh + hl)

where l, b and h are, respectively, the three edges of the cuboid

• Surface Area of a Cube = 6a2

where a is the edge of the cube.

• Curved Surface Area of a Cylinder = 2πrh

where r is the radius of the base of the cylinder and h is the height of the cylinder

• Total Surface Area of a Cylinder = 2πr(r + h)

where h is the height of the cylinder and r is its radius

• Curved Surface Area of a Cone = 1/2 × l × 2πr = πrl

where r is its base radius, and l is its slant height

• Total Surface Area of a Cone = πrl + πr2 = πr(l + r)
• Surface Area of a Sphere = 4 π r2

where r is the radius of the sphere.

• Curved Surface Area of a Hemisphere = 2πr2

where r is the radius of the sphere of which the hemisphere is a part.

• Total Surface Area of a Hemisphere = 3πr2
• Volume of a Cuboid = base area × height = length × breadth × height
• Volume of a Cube = edge × edge × edge = a3
• Volume of a Cylinder = πr2h

where r is the base radius, and h is the height of the cylinder.

• Volume of a Cone = 1/3 πr2h

where r is the base radius, and h is the height of the cone.

• Volume of a Sphere = 4/3 πr3

where r is the radius of the sphere.

• Volume of a Hemisphere = 2/3 πr3

where r is the radius of the hemisphere.

Class 9 Maths NCERT Solutions Chapter 13 Exercises

Exercise 13.1 – 8 questions Exercise 13.2 – 11 questions Exercise 13.3 – 8 questions Exercise 13.4 – 9 questions Exercise 13.5 – 9 questions Exercise 13.6 – 8 questions Exercise 13.7 – 9 questions Exercise 13.8 – 10 questions

NCERT Solutions Class 9 Maths Chapter 14 Statistics

Chapter 14 of ncert class 9 maths, Statistics focuses on collecting data from different aspects of life and extracting meaningful information from them. The chapters include graphical representations of data accumulated through graphs such as bar graphs, histograms, frequency polygons, etc. Mean, median and mode of the raw data is also discussed in the other sections of the chapter. There are a total of 4 exercises in the chapter to help students learn the topics better and score well in their exams. 

Important Formulas:

Statistics deals with the collection, presentation, and analysis of data, as well as the drawing of meaningful conclusions from the data.

Mean – It is found by adding all the values of the observations and dividing it by the total number of observations. It is denoted by

𝑥―

So,

For an ungrouped frequency distribution, it is

 

Median – It is the value of the middle-most observation (s).

If n is an odd number, the median = value of the [(n + 1)/2]th observation

If n is an even number, median = Mean of the values of the (n/2)th and [(n/2) + 1]th observations.

Mode – The mode is the most frequently occurring observation.

Class 9 Maths NCERT Solutions Chapter 14 Exercises

Exercise 14.1 – 2 questions (2 short answers)  Exercise 14.2 – 9 questions (9 long answers)  Exercise 14.3 – 9 questions (6 short answers, 3 long answers)  Exercise 14.4 – 6 questions (2 short answers, 4 long answers)

 

NCERT Solutions Class 9 Maths Chapter 15 Probability

Chapter 15 of ncert class 9 maths deals with collecting various results of any experiment termed as events, and the possibility of an event is called Probability. There is a single exercise based on real-life events to help students learn the concept of possibilities and chances through the chapter.

Overview of the Exercises of Chapter 15 Probability

Exercise 15.1: The exercise focuses on testing the theoretical approach to probability and related calculations. 

 

Class 9 Maths NCERT Solutions Chapter 15 Exercises

Exercise 15.1 – 10 Questions ( 4 short answers, 3 long answers, 3 very long answers)