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NCERT Solutions for Class 9 Maths – FREE PDF Updated for 2025-26 Session

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NCERT Maths Class 9 Solutions - All Chapters Question Answers - FREE PDF Download

NCERT Solutions for Class 9 Maths provide complete, chapter-wise answers to all questions given in the NCERT Maths Class 9 textbook. These detailed solutions are designed to help students understand concepts clearly, practise a wide variety of problems, and prepare effectively for school exams as well as future board-level studies.

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The Class 9 Maths Syllabus plays a crucial role in improving mathematical concepts. Topics such as Number Systems, Polynomials, Coordinate Geometry, Euclid’s Geometry, Lines and Angles, Triangles, Quadrilaterals, Circles, Surface Areas and Volumes, Statistics, and Probability are explained in a step-by-step manner, making learning more structured and stress-free.


All NCERT Maths Class 9 Solutions are prepared by subject experts and strictly follow the latest NCERT guidelines. With FREE PDFs, important questions, and exemplar-style practice, students can prepare better for Class 10 and competitive exams.


NCERT Solutions for Class 9 Maths - Chapter-wise List

Given below are the chapter-wise NCERT Solutions for Class 9 Maths. Go through these  9th maths solutions chapter-wise to be thoroughly familiar with the concepts.



Note: The chapters on Areas of Parallelograms and Triangles, Constructions and  Probability have been excluded from the Class 9 Maths textbook for the 2025-26 academic year


Also Check for: The Concepts of NCERT Class 9 Maths

Competitive Exams after 12th Science
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NCERT Solutions for Class 9 Maths Chapter Overview

Students who find certain Class 9 Maths problems challenging can refer to the NCERT Solutions for Class 9 Maths for clear explanations and quick revision. Practising chapter-wise exercises regularly helps improve accuracy, strengthen concepts, and build confidence for exams.


NCERT Solutions Class 9 Maths Chapter 1 Number System

Chapter 1: Number System introduces students to important concepts such as rational and irrational numbers. The chapter explains the extended number line and shows how to represent integers, rational numbers, and irrational numbers on it.


This chapter consists of six exercises covering:


  • Representation of terminating and non-terminating recurring decimals

  • Use of the successive magnification method

  • Representation of $\sqrt{2}$, $\sqrt{3}$, and other irrational numbers on the number line

  • Laws of integral powers

  • Rational exponents with positive real bases


These topics form the foundation of NCERT Maths Class 9 and are essential for understanding higher-level concepts in later chapters and Class 10.


Topics Covered in Class 9 Maths Chapter 1 Number System

  • Representation of natural numbers, integers, and rational numbers on the number line

  • Terminating and non-terminating recurring decimals using successive magnification

  • Irrational numbers $\left ( \sqrt{2},\sqrt{3},\sqrt{5} \right )$ and their representation on the number line

  • Every real number corresponds to a unique point on the number line

  • Existence of $\sqrt{x}$ and definition of the nth root of a real number

  • Laws of integral and rational exponents

  • Rationalisation of expressions involving surds


Important Formulas Number System

  • $\sqrt{\left ( a/b \right )}=\sqrt{a}/\sqrt{b}$

  • $\sqrt{ab}=\sqrt{a}\sqrt{b}$

  • $\left ( \sqrt{a}+\sqrt{b} \right )\left ( \sqrt{a}-\sqrt{b} \right )=a-b$

  • aᵐ · aⁿ = aᵐ⁺ⁿ

  • (aᵐ)ⁿ = aᵐⁿ


Class 9 Maths NCERT Solutions Chapter 1 Exercises

Exercise 1.1 - 4 Questions & Solutions (4 short Answers)

Exercise 1.2 - 4 Questions & Solutions (4 short Answers)

Exercise 1.3 - 9 Questions & Solutions (8 short Answers, 1 long Answer)

Exercise 1.4 - 5 Questions & Solutions (4 short Answers, 1 long Answer)

Exercise 1.5 - 3 Questions & Solutions (3 short Answers)


Along with this, students can also download additional study materials provided by Vedantu, for Chapter 1 of CBSE Class 9 Maths Solutions–



NCERT Solutions for Class 9 Maths Chapter 2 Polynomials

Class 9 Maths Chapter 2 Polynomials introduces students to algebraic expressions known as polynomials and the key terminology associated with them. A polynomial is an expression formed using variables and coefficients with operations such as addition, subtraction, multiplication, and non-negative integer powers of variables.


The chapter explains important concepts like degree of a polynomial, coefficients, terms, and zeroes, along with different types of polynomials. Students also study the Remainder Theorem and Factor Theorem, which are widely used in the factorisation of polynomials. The chapter contains five exercises covering all major concepts through step-by-step problems.


Topics Covered in Class 9 Maths Chapter 2 Polynomials

  • Definition of a polynomial in one variable

  • Terms, coefficients, zero polynomial

  • Degree of a polynomial

  • Constant, linear, quadratic, and cubic polynomials

  • Monomials, binomials, trinomials

  • Factors and multiples of polynomials

  • Zeroes (roots) of a polynomial

  • Remainder Theorem – statement and applications

  • Factor Theorem – statement and proof

  • Factorisation of quadratic and cubic polynomials

  • Use of algebraic identities in factorisation


Important Algebraic Identities for Chapter 2 Polynomials

  • (x + y)² = x² + 2xy + y²

  • (x − y)² = x² − 2xy + y²

  • x² − y² = (x + y)(x − y)

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

  • (x + y)³ = x³ + y³ + 3xy(x + y)

  • (x − y)³ = x³ − y³ − 3xy(x − y)

  • x³ + y³ + z³ − 3xyz = (x + y + z)(x² + y² + z² − xy − yz − zx)


Important Theorems for Polynomials

Division Algorithm

Dividend = (Divisor × Quotient) + Remainder


Remainder Theorem

If a polynomial p(x) is divided by (x − a), the remainder is p(a).


Factor Theorem

For a polynomial p(x):


  • x − a is a factor of p(x) if p(a) = 0

  • If x − a is a factor of p(x), then p(a) = 0


Class 9 Maths NCERT Solutions Chapter 2 Exercises

Exercise 2.1 - 5 Questions & Solutions (5 Short Answers)

Exercise 2.2 - 4 Questions & Solutions (4 Short Answers)

Exercise 2.3 - 3 Questions & Solutions (3 Short Answers)

Exercise 2.4 - 5 Questions & Solutions (4 Short Answers, 1 Long Answer)


Along with this, students can also download additional study materials provided by Vedantu, for Chapter 2 of CBSE Class 9 Maths Solutions–



NCERT Solutions for Class 9 Maths Chapter 3 Coordinate Geometry

Class 9 Maths Chapter 3 Coordinate Geometry introduces students to the basics of locating points on a plane using coordinates. The chapter explains the Cartesian plane, coordinates of a point, and important terms related to the coordinate system.


Students learn how to identify the x-axis, y-axis, origin, and quadrants, along with understanding the abscissa and ordinate of a point. The chapter also focuses on plotting and naming points on the xy-plane. There are three exercises in this chapter that help students practise and strengthen these foundational concepts.


Topics Covered in Class 9 Maths Chapter 3 Coordinate Geometry

  • The Cartesian (coordinate) plane

  • Coordinates of a point in the xy-plane

  • Terms and notations related to the coordinate plane

  • Plotting points on the plane

  • Graphs of linear equations as examples

  • Linear equations of the form ax + by + c = 0, written as y = mx + c, linked to linear equations in two variables


Important Points for Coordinate Geometry

  • To locate a point in a plane, two perpendicular lines are required

  • These lines form the Cartesian or coordinate plane

  • The horizontal line is the x-axis and the vertical line is the y-axis

  • The coordinate axes divide the plane into four quadrants

  • The point where the axes intersect is called the origin

  • The distance of a point from the y-axis is its x-coordinate (abscissa)

  • The distance of a point from the x-axis is its y-coordinate (ordinate)


Class 9 Maths NCERT Solutions Chapter 3 Exercises

Exercise 3.1 - 2 Questions & Solutions (1 Short Answer, 1 Long Answer)

Exercise 3.2 - 2 Questions & Solutions (2 Short Answers)


Along with this, students can also download additional study materials provided by Vedantu, for Chapter 3 of CBSE Class 9 Maths Solutions



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

Class 9 Maths Chapter 4 builds on the understanding of linear equations in one variable and introduces linear equations in two variables of the form ax + by + c = 0. In this chapter, students learn how to find solutions of such equations and represent them graphically on the coordinate plane.


Students also understand how linear equations in two variables relate to real life situations. The chapter contains four exercises that include problems on finding solutions, plotting graphs, and solving word problems using both algebraic and graphical methods.


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

  • Revision of linear equations in one variable

  • Introduction to linear equations in two variables

  • Infinite number of solutions of a linear equation in two variables

  • Representation of solutions as ordered pairs of real numbers

  • Graphical representation of linear equations

  • Real life problems including ratio and proportion

  • Simultaneous use of algebraic and graphical methods


Important Points Linear Equations in Two Variables

  • An equation of the form ax + by + c = 0, where a, b, and c are real numbers and 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 a linear equation in two variables is always a straight line

  • x = 0 represents the y axis and y = 0 represents 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 form y = mx represents a straight line passing through the origin


Class 9 Maths NCERT Solutions Chapter 4 Exercises

Exercise 4.1 - 2 Questions & Solutions (2 Short Answers)

Exercise 4.2 - 4 Questions & Solutions (3 Short Answers, 1 Long Answer)


Along with this, students can also download additional study materials provided by Vedantu, for Chapter 4 of CBSE Class 9 Maths Solutions–



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

Class 9 Maths Chapter 5 introduces students to Euclid’s Geometry and explains how geometry was developed as a logical system. This chapter connects Euclid’s classical approach to geometry with modern day geometry, helping students understand how mathematical reasoning is built using definitions, axioms, postulates, and theorems.


Students learn how basic geometrical ideas are formalised and how logical statements are used to prove results. The chapter contains two exercises that focus on understanding axioms, postulates, and simple proofs.


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

  • History of geometry including Euclid and geometry in India

  • Euclid’s method of converting observed facts into mathematics

  • Definitions, axioms or common notions, postulates, and theorems

  • Euclid’s five postulates and equivalent forms of the fifth postulate

  • Relationship between axioms and theorems

  • Statement and proof based on:

    • Given two distinct points, there exists one and only one line through them

    • Two distinct lines cannot have more than one point in common


Important Axioms of Euclid for Chapter 5 Introduction to Euclid’s Geometry

  • Things equal to the same thing are equal to one another

  • If equals are added to equals, the wholes are equal

  • If equals are subtracted from equals, the remainders are equal

  • Things that coincide with one another are equal

  • The whole is greater than the part

  • Things which are double of the same things are equal

  • Things which are halves of the same thing are equal


Euclid’s Five Postulates 

  • Postulate 1
    A straight line can be drawn joining any two points

  • Postulate 2
    A line segment can be extended indefinitely

  • Postulate 3
    A circle can be drawn with any centre and any radius

  • Postulate 4
    All right angles are equal

  • Postulate 5
    If a straight line intersects two straight lines such that the interior angles on the same side add up to less than two right angles, then the two lines meet on that side when extended


Class 9 Maths NCERT Solutions Chapter 5 Exercises

Exercise 5.1 - 7 Questions & Solutions (4 Short Answers, 3 Long Answers)


Along with this, students can also download additional study materials provided by Vedantu, for Chapter 5 of CBSE Class 9 Maths Solutions–



NCERT Solutions for Class 9 Maths Chapter 6 Lines and Angles

Class 9 Maths Chapter 6 focuses on important axioms and theorems related to lines and angles. This chapter mainly deals with logical reasoning and proof based questions, which are commonly asked in school exams.


The chapter contains three exercises that help students understand angle relationships formed by intersecting lines and transversals. In total, students study four axioms and eight theorems, which form the base for higher level geometry topics.


Topics Covered in Class 9 Maths Chapter 6 Lines and Angles

  • If a ray stands on a line, the sum of the two adjacent angles is 180° and its converse

  • Vertically opposite angles formed when two lines intersect

  • Corresponding angles, alternate interior angles, and interior angles when a transversal intersects parallel lines

  • Lines parallel to the same line are parallel to each other

  • Sum of the interior angles of a triangle is 180°

  • Exterior angle of a triangle equals the sum of its two interior opposite angles


Important Points Lines and Angles

  • If a ray stands on a line, the adjacent angles formed add up to 180°, and vice versa. This is known as the linear pair axiom

  • When two lines intersect, vertically opposite angles are equal

  • If a transversal intersects two parallel lines, then

    • Corresponding angles are equal

    • Alternate interior angles are equal

    • Interior angles on the same side of the transversal are supplementary

  • If a transversal intersects two lines such that

    • One pair of corresponding angles is equal, or

    • One pair of alternate interior angles is equal, or

    • Interior angles on the same side are supplementary

  • then the two lines are parallel


Class 9 Maths NCERT Solutions Chapter 6 Exercises

Exercise 6.1 - 6 Questions & Solutions (5 Short Answers, 1 Long Answer)


Along with this, students can also download additional study materials provided by Vedantu, for Chapter 6 of CBSE Class 9 Maths Solutions



NCERT Solutions for Class 9 Maths Chapter 7 Triangles

Class 9 Maths Chapter 7 deals with the congruence of triangles, important congruence rules, properties of triangles, and inequalities in triangles. This chapter is heavily proof based and helps students develop logical reasoning and step by step mathematical thinking.


The chapter contains five exercises that include both to prove questions and application based problems. Students also revise and formally prove properties of triangles learned in earlier classes. Around eight key theorems are covered in this chapter, making it an important part of the Class 9 Maths syllabus.


Topics Covered in Class 9 Maths Chapter 7 Triangles

  • Congruence of triangles

  • SAS congruence rule

  • ASA congruence rule

  • SSS congruence rule

  • RHS congruence rule for right triangles

  • Angles opposite equal sides of a triangle

  • Sides opposite equal angles of a triangle

  • Triangle inequalities and relation between angles and opposite sides


Important Axioms and Theorems Triangles

  • Axiom 7.1 SAS Congruence Rule
    Two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding two sides and included angle of another triangle

  • Theorem 7.1 ASA Congruence Rule
    Two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding two angles and included side of another triangle

  • Theorem 7.2
    Angles opposite equal sides of an isosceles triangle are equal

  • Theorem 7.3
    Sides opposite equal angles of a triangle are equal

  • Theorem 7.4 SSS Congruence Rule
    If three sides of one triangle are equal to the three sides of another triangle, the triangles are congruent

  • Theorem 7.5 RHS Congruence Rule
    In two right triangles, if the hypotenuse and one corresponding side are equal, the triangles are congruent

  • Theorem 7.6
    If two sides of a triangle are unequal, the angle opposite the longer side is greater

  • Theorem 7.7
    In a triangle, the side opposite the greater angle is longer

  • Theorem 7.8
    The sum of any two sides of a triangle is greater than the third side


Class 9 Maths NCERT Solutions Chapter 7 Exercises

Exercise 7.1 - 8 Questions & Solutions (6 Short Answers, 2 Long Answers)

Exercise 7.2 - 8 Questions & Solutions (6 Short Answers, 2 Long Answers)

Exercise 7.3 - 5 Questions & Solutions (3 Short Answers, 2 Long Answers)


Along with this, students can also download additional study materials provided by Vedantu, for Chapter 7 of CBSE Class 9 Maths Solutions–



NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals

Class 9 Maths Chapter 8 introduces students to quadrilaterals, which are figures formed by joining four points in order. This chapter explains different types of quadrilaterals, with a major focus on parallelograms and their properties.


The chapter contains two exercises, mainly based on proof questions. Although only a few theorems are directly proved, students learn to apply multiple theorems to solve conceptual and application based problems. Important ideas such as the angle sum property of a quadrilateral, properties of parallelograms, and the mid point theorem are covered in detail.


Topics Covered in Class 9 Maths Chapter 8 Quadrilaterals

  • Diagonal of a parallelogram divides it into two congruent triangles

  • Properties of parallelograms including opposite sides and angles

  • Conditions for a quadrilateral to be a parallelogram

  • Diagonals of a parallelogram and their properties

  • Mid point theorem and its converse

  • Angle sum property of a quadrilateral


Important Theorems for Chapter 8 Quadrilaterals

  • A quadrilateral has four sides, four angles, and four vertices

  • Angle Sum Property
    The sum of the interior angles of a quadrilateral is 360°

  • Theorem 8.1
    A diagonal of a parallelogram divides it into two congruent triangles

  • Theorem 8.2
    In a parallelogram, opposite sides are equal

  • Theorem 8.3
    If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram

  • Theorem 8.4
    In a parallelogram, opposite angles are equal

  • Theorem 8.5
    If each pair of opposite angles of a quadrilateral is equal, then it is a parallelogram

  • Theorem 8.6
    The diagonals of a parallelogram bisect each other

  • Theorem 8.7
    If the diagonals of a quadrilateral bisect each other, then it is a parallelogram

  • Theorem 8.8
    A quadrilateral is a parallelogram if one pair of opposite sides is equal and parallel

  • Theorem 8.9
    The line segment joining the mid points of two sides of a triangle is parallel to the third side

  • Theorem 8.10
    A line drawn through the mid point of one side of a triangle parallel to another side bisects the third side


Class 9 Maths NCERT Solutions Chapter 8 Exercises

Exercise 8.1 - 12 Questions & Solutions (2 Short Answers, 6 Long Answers, 4 Very Long Answers)

Exercise 8.2 - 7 Questions & Solutions (2 Short Answers, 3 Long Answers, 3 Very Long Answers)


Along with this, students can also download additional study materials provided by Vedantu, for Chapter 8 of CBSE Class 9 Maths Solutions–



NCERT Solutions for Class 9 Maths Chapter 9 Circles

Class 9 Maths Chapter 9 introduces students to the basic concepts of circles and their important properties. A circle is defined as the set of all points in a plane that are at a fixed distance from a fixed point called the centre.


In this chapter, students study key ideas such as angles subtended by chords and arcs, properties of equal chords, cyclic quadrilaterals, and relationships between angles and the centre of a circle. The chapter includes six exercises and covers around twelve theorems, most of which are proof based and concept driven.


Topics Covered in Class 9 Maths Chapter 9 Circles

  • Basic terms related to circles such as radius, diameter, chord, arc, circumference, and subtended angle

  • Equal chords of a circle and the angles subtended by them at the centre

  • Relationship between chords and their distance from the centre

  • Construction of a circle through three non collinear points

  • Angle subtended by an arc at the centre and at a point on the circle

  • Angles in the same segment of a circle

  • Cyclic quadrilaterals and their properties


Important Theorems for Chapter 9 Circles

  • Theorem 9.1
    Equal chords of a circle subtend equal angles at the centre

  • Theorem 9.2
    If angles subtended by two chords at the centre are equal, then the chords are equal

  • Theorem 9.3
    The perpendicular drawn from the centre of a circle to a chord bisects the chord

  • Theorem 9.4
    A line drawn through the centre to bisect a chord is perpendicular to the chord

  • Theorem 9.5
    There is one and only one circle passing through three given non collinear points

  • Theorem 9.6
    Equal chords of a circle or congruent circles are equidistant from the centre

  • Theorem 9.7
    Chords equidistant from the centre of a circle are equal in length

  • Theorem 9.8
    The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle

  • Theorem 9.9
    Angles in the same segment of a circle are equal

  • Theorem 9.10
    If a line segment subtends equal angles at two points on the same side, the four points lie on a circle

  • Theorem 9.11
    The sum of either pair of opposite angles of a cyclic quadrilateral is 180 degrees

  • Theorem 9.12
    If the sum of a pair of opposite angles of a quadrilateral is 180 degrees, then the quadrilateral is cyclic


Class 9 Maths NCERT Solutions Chapter 9 Exercises

Exercise 9.1 - 2 Questions & Solutions (2 Short Answers)

Exercise 9.2 - 2 Questions & Solutions (2 Long Answers)

Exercise 9.3 - 3 Questions & Solutions (3 Long Answers)


Along with this, students can also download additional study materials provided by Vedantu, for Chapter 9 of CBSE Class 9 Maths Solutions



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

Class 9 Maths Chapter 10 introduces Heron’s Formula, a useful method to find the area of a triangle when the lengths of all three sides are known. Using this formula, students can calculate the area without finding angles or heights, making it especially helpful for solving application-based problems.


The chapter also explains how Heron’s Formula can be applied to find the area of quadrilaterals and other polygons by dividing them into triangles. There are two exercises in this chapter that help students practise and understand the correct use of the formula.


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

  • Area of a triangle using Heron’s Formula without proof

  • Application of Heron’s Formula to find the area of a quadrilateral

  • Conversion of complex shapes into triangles for area calculation


Important Formulas Heron’s Formula

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

  • Area of a triangle using Heron’s Formula
    Area = √[s(s − a)(s − b)(s − c)]
    where
    a, b, c are the sides of the triangle
    s is the semi perimeter = (a + b + c) / 2


Class 9 Maths NCERT Solutions Chapter 10 Exercises

Exercise 10.1 - 6 Questions & Solutions (2 Short Answers, 2 Long Answer, 2 Very Long Answers)


Along with this, students can also download additional study materials provided by Vedantu, for Chapter 10 of CBSE Class 9 Maths Solutions



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

Class 9 Maths Chapter 11 focuses on calculating the surface areas and volumes of solid figures. In this chapter, students learn in detail about cuboids and cylinders and then extend these concepts to other solids such as cones, spheres, and hemispheres.


This chapter builds on the ideas of mensuration studied in earlier classes and helps students apply formulas to solve practical problems. There are eight exercises in this chapter, covering a wide range of questions based on the surface areas and volumes of different three-dimensional solids.


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

  • Surface areas and volumes of cubes and cuboids

  • Surface areas and volumes of right circular cylinders

  • Surface areas and volumes of cones

  • Surface areas and volumes of spheres and hemispheres


Important Formulas Surface Areas and Volumes

Cuboid

  • Total surface area = 2(lb + bh + hl)

  • Volume = l × b × h


Cube

  • Total surface area = 6a²

  • Volume =


Cylinder

  • Curved surface area = 2πrh

  • Total surface area = 2πr(r + h)

  • Volume = πr²h


Cone

  • Curved surface area = πrl

  • Total surface area = πr(l + r)

  • Volume = 1/3 πr²h


Sphere

  • Surface area = 4πr²

  • Volume = 4/3 πr³


Hemisphere

  • Curved surface area = 2πr²

  • Total surface area = 3πr²

  • Volume = 2/3 πr³


where r is the radius, h is the height, l is the slant height, and a is the edge of the cube.


Class 9 Maths NCERT Solutions Chapter 11 Exercises

Exercise 11.1 - 8 Questions & Solutions 

Exercise 11.2 - 11 Questions & Solutions

Exercise 11.3 - 8 Questions & Solutions


Along with this, students can also download additional study materials provided by Vedantu, for Chapter 10 of CBSE Class 9 Maths Solutions



NCERT Solutions for Class 9 Maths Chapter 14 Statistics

Class 9 Maths Chapter 14 introduces students to Statistics, a branch of Mathematics used to collect, organise, present, and analyse data meaningfully. This chapter helps students understand how numerical data related to daily life is represented and interpreted.


Students learn different methods of data presentation, including tables and graphical forms such as bar graphs, histograms, and frequency polygons. The chapter also explains important measures of central tendency namely mean, median, and mode for ungrouped data. There are four exercises in this chapter that cover all these concepts through practical problems.


Topics Covered in Class 9 Maths Chapter 14 Statistics

  • Introduction to Statistics

  • Collection of data

  • Presentation of data in:

    • Tabular form

    • Ungrouped and grouped data

  • Graphical representation of data:

    • Bar graphs

    • Histograms including varying base lengths

    • Frequency polygons

  • Choosing the correct method of data representation

  • Mean, median, and mode of ungrouped data


Important Concepts and Formulas Statistics

Mean

The mean is the average of all observations.

Mean = Sum of all observations ÷ Number of observations

For ungrouped data:
Mean = (x₁ + x₂ + x₃ + … + xₙ) ÷ n


Median

The median is the middle value of the arranged data.


  • If n is odd,
    Median = Value of the (n + 1) ÷ 2 th observation

  • If n is even,
    Median = Mean of the (n ÷ 2) th and [(n ÷ 2) + 1] th observations


Mode

The mode is the most frequently occurring value in the data.


Class 9 Maths NCERT Solutions Chapter 12 Exercises

Exercise 12.1 - 2 Questions & Solutions (2 Short Answers)


Along with this, students can also download additional study materials provided by Vedantu, for Chapter 10 of CBSE Class 9 Maths Solutions



CBSE Class 9 Maths Chapter-Wise Marks Weightage 

S. No

Chapter Name

Marks

1

Number Systems

10

2

Algebra

20

3

Coordinate Geometry

4

4

Geometry

27

5

Mensuration

13

6

Statistics & Probability

6


Total

80


Internal Assessment

20


Total

100


Internal Assessment for CBSE Class 9 Maths

Internal Assessment

Pen Paper Test and Multiple Assessment

10 Marks

Portfolio 

5 Marks

Lab Practical

5 Marks

Total

20 Marks


Why Vedantu’s NCERT Solutions for Class 9 Maths are Helpful for Students?

  • Class 9 Maths NCERT solutions are explained in simple, step-by-step format, making it easier for students to understand concepts clearly.

  • With NCERT solutions for Class 9 Maths, students can quickly clear doubts and revise topics without confusion.

  • Chapter-wise and exercise-wise PDFs of Class 9th Maths NCERT solutions help students practise systematically and stay organised.

  • Detailed explanations improve logical thinking and strengthen problem-solving skills required for exams.

  • Problems are solved using the most efficient and exam-oriented methods, aligned with the CBSE syllabus.

  • Clear diagrams and illustrations help students visualise questions and understand solutions better.


Best Way to Learn Maths Class 9 with NCERT Solutions and Concept Clarity

After going through the Class 9 NCERT Maths solutions for Chapter 1 Number System, students often look for a clearer and more engaging way to understand concepts in depth. To strengthen fundamentals and practise effectively, combining class 9th maths NCERT solutions with concept-based learning is highly beneficial.


Vedantu helps students learn Maths Class 9 through structured explanations, visual learning, and exam-focused practice that aligns with the NCERT Class 9 Maths syllabus. This approach makes it easier to understand concepts, revise formulas, and apply methods correctly in exams.


How Vedantu Supports Class 9 Maths Learning?

Students preparing with NCERT Class 9 Maths solutions on Vedantu get access to:


  • Concept-wise explanations linked with Class 9 NCERT Maths solutions

  • Engaging video lessons that simplify complex topics

  • Practice questions based on Maths Class 9 NCERT patterns

  • Important questions and formula revision support

  • Chapter-wise practice tests to track progress

  • Anytime access to learning resources for flexible study


Important Related Links for CBSE Class 9 Maths

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FAQs on NCERT Solutions for Class 9 Maths – FREE PDF Updated for 2025-26 Session

1. How can I learn the Class 9 Maths chapters in a fast and efficient way?

The most effective way to learn Class 9 Maths is by practising questions directly from the NCERT Maths Class 9 textbook along with NCERT Solutions for Class 9 Maths. The textbook covers all exam-relevant questions, while the solutions explain each step clearly, helping students understand the correct approach. Regular practice of chapters such as Class 9 Maths Chapter 1, Class 9 Maths Chapter 2, and Class 9 Maths Chapter 9 also improves logical thinking and problem-solving skills, which are essential for scoring well in exams.

2. Can I get solved answers for the questions from the NCERT Class 9 Maths chapters?

Yes, students can access Class 9 Maths NCERT solutions that provide fully solved answers for every question from the Class 9 Maths book. These NCERT Class 9 Maths solutions are prepared by subject experts and strictly follow the latest CBSE syllabus and exam pattern. Chapter-wise and exercise-wise solutions are available, making it easy to revise topics like Class 9 Maths triangles and other important chapters using a structured format.

3. Should I daily solve the questions from the NCERT Solutions for Class 9 Maths?

Yes, daily practice using NCERT Solutions Class 9 Maths is highly recommended. Maths is a practice-based subject, and regular problem-solving helps students build confidence, accuracy, and speed. Solving Class 9 Maths question answer sets every day also strengthens fundamental concepts of Maths Class 9, which are crucial for higher classes and board-level preparation.

4. How can I download the NCERT Solutions for Class 9 Maths at Vedantu?

Students looking for NCERT solutions for Class 9 Maths can easily access chapter-wise PDFs that are useful for revision and exam preparation. These resources include solutions aligned with the Class 9 Maths textbook PDF and cover all important chapters. Exercise-wise links help students practise systematically and revise concepts from the Class 9 Maths NCERT Solutions at their own pace.

5. What is the best way to prepare for Class 9 Maths?

The best way to prepare for Class 9 Maths is to study directly from the NCERT Maths Class 9 textbook and practise regularly using NCERT Solutions for Class 9 Maths. This ensures full syllabus coverage and helps students understand the correct problem-solving approach.

6. Are NCERT Solutions for Class 9 Maths enough for exams?

Yes, Class 9 Maths NCERT Solutions are sufficient for school exams when practised thoroughly. Since questions are based on the Class 9 Maths book, solving all textbook exercises and examples gives students strong exam readiness.

7. Where can I find reliable NCERT Maths Class 9 solutions?

Students can find NCERT Class 9 Maths solutions online from Vedantu in a chapter-wise format. These solutions explain every step clearly and are aligned with the latest CBSE syllabus, making them ideal for daily practice and revision.

8. Is the Class 9 Maths textbook PDF useful for revision?

Yes, the Class 9 Maths textbook PDF is very useful for quick revision. When combined with NCERT solutions for Class 9 Maths, it helps students revise formulas, concepts, and questions efficiently before exams.

9. What does NCERT Solutions Class 9 Maths include?

NCERT Solutions Class 9 Maths include:

  • Step-by-step answers for all textbook questions

  • Chapter-wise explanations

  • Important formulas and methods

  • Practice support for every Maths Class 9 chapter

10. Which chapters are most important in Maths Class 9?

Some important chapters in Maths Class 9 include:

  • Class 9 Maths Chapter 1 Number System

  • Class 9 Maths Chapter 2 Polynomials

  • Class 9 Maths Chapter 9 Circles

  • Class 9 Maths Triangles

Practising these chapters using Class 9 NCERT Maths solutions is essential for strong fundamentals.

11. Are Class 9th Maths NCERT solutions helpful for weak students?

Yes, Class 9th Maths NCERT solutions are especially helpful for students who find Maths difficult. The step-by-step explanations make concepts easier to understand and improve confidence in solving problems.

12. How often should students practise NCERT Class 9 Maths solutions?

Students should practise NCERT Class 9 Maths solutions daily or after completing each topic. Regular practice improves accuracy, speed, and understanding of Class 9 Maths question answer patterns.

13. Are NCERT Class 9 Maths solutions useful for Class 10 preparation?

Yes, NCERT Maths Class 9 concepts form the base for Class 10. Using NCERT solutions for Class 9 Maths helps students prepare smoothly for higher-level chapters and board exam topics.

14. Do NCERT Maths Class 9 solutions follow the latest CBSE syllabus?

Yes, NCERT Maths Class 9 solutions strictly follow the latest CBSE syllabus and NCERT guidelines, making them safe and reliable for exam preparation.

15. Is it necessary to solve all questions from the Class 9 Maths NCERT book?

Yes, solving all questions from the Class 9 Maths NCERT book is important, as school exam papers are based directly on textbook exercises and examples.