NCERT Solutions for Class 9 Maths - Free Latest PDF (2024-25)

Chapter 1 - Number System

NCERT Class 9 Maths Chapter 1 sets the foundation for the subject. Solutions of text exercises and end exercise of Number are explained elaborately, to help understand the basic concepts. The chapter discusses the topics of rational, irrational numbers and representing the same on the number line. Students are also taught how to represent square roots, as well as terminating and non-terminating decimals on the number lines.

1. Exercise 1.1: Deals with basic classification of numbers. You might be asked to identify different types of numbers (natural, whole, integers, rational, irrational) from a given set.

2. Exercise 1.2: This exercise could involve applying properties of operations (addition, subtraction, multiplication, division) to various number systems. You might be asked to check if a specific property holds true for a particular operation on a given set of numbers.

3. Exercise 1.3: Here, the focus might be on representing numbers on the number line. You could be solving problems that involve plotting numbers, finding the distance between two numbers, or comparing their relative positions.

4. Exercise 1.4: Here, you might be challenged with problems that involve comparing and ordering numbers of different types (e.g., rational vs. irrational).

5. Exercise 1.5: This exercise could be a mix of concepts covered earlier. You might be presented with a scenario that requires applying various aspects of number systems and their properties to solve it.

In Class 9 Maths Chapter 1 on Number System, we cover the following:

• Different Number Systems: natural, whole, integers, rational, irrational

• Properties of operations (addition, subtraction, multiplication, division) on these systems

• Representing numbers on the number line

• Relating terminating/non-terminating recurring decimals to rational numbers

• Understanding irrational numbers (e.g., √2, √3)

• Every real number has a unique position on the number line (and vice versa)

• Introduction to √x for positive real numbers

• Brief review of exponent laws (integral & rational powers)

• Brief introduction to rationalizing numbers

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

• Number System Important Questions

• Number System Important Formulas

• Number System Revision Notes

• Number System NCERT Exemplar Solutions

• Number System RD Sharma Solutions

• Number Systems RS Aggarwal Solutions

Chapter 2 - Polynomials

Chapter 2 of NCERT Solutions of Class 9 Maths is on Polynomials. Students are introduced to the concept of Polynomials, which are expressions consisting of variables and coefficients. Properties such as addition, subtraction, division, multiplication of polynomials are taught in this chapter along with their factorisation. The Factor Theorem and Remainder Theorem making use of these polynomials are also provided in this chapter.

1. Exercise 2.1: This exercise deals with identifying basic components of polynomials. You'll practice recognizing terms, coefficients, variables, and the degree of a polynomial expression.

2. Exercise 2.2: Here, the focus is on classifying polynomials based on their degree. You might be asked to identify polynomials as linear, quadratic, cubic, etc., based on the highest exponent of the variable.

3. Exercise 2.3: This exercise involves recognizing zeroes (or roots) of a polynomial. You might be presented with a polynomial equation and asked to find the values of the variable that make the equation true (where the expression evaluates to zero).

4. Exercise 2.4: This exercise covers on the Factor Theorem. This theorem relates the factors of a polynomial to its zeroes. You might be challenged with problems that involve finding factors based on zeroes or vice versa.

In Class 9 Maths Chapter 2 on Polynomials, we cover the following:

• Explanation of a polynomial with one variable, its coefficients, along with examples and non-examples, its terms, and the zero polynomial.

• Understanding the degree of a polynomial and types like constant, linear, quadratic, cubic polynomials; also monomials, binomials, and trinomials.

• Concepts of factors, multiples, and zeros/roots of a polynomial/equation.

• Introduction and rationale of the Remainder Theorem, with examples and comparisons to integers. Also, the statement and proof of the Factor Theorem.

• Techniques for factorization of expressions like ax2 + bx + c, a ≠ 0 where a, b, c are real numbers, and of cubic polynomials using the Factor Theorem.

• Definitions of the Remainder Theorem and the Factor Theorem.

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

• Polynomials Important Questions

• Polynomials Important Formulas

• Polynomials Revision Notes

• Polynomials NCERT Exemplar Solution

• Polynomials RS Aggarwal Solutions

Chapter 3 - Coordinate Geometry

Chapter 3 discusses the cartesian plane, coordinates of a point in the ‘xy’ plane, and several other significant terms associated with coordinate geometry. Students will learn how to plot points in the xy plane following the abscissa and ordinates, etc. With 3 exercises comprising both long and short questions, the exercises of NCERT Class 9 Solutions Maths for Chapter 3 are designed to test your application skills as well as cognizance.

1. Exercise 3.1: This exercise likely deals with the foundational concepts. You might be asked to explain how to describe the location of a point on a flat surface (like your desk) using coordinates. This involves understanding how perpendicular axes (X and Y) help pinpoint a position.

2. Exercise 3.2: Here, the focus is on translating coordinates into a visual representation. You'll probably practice plotting points on a graph based on their given coordinates (X and Y values).

In Class 9 Maths Chapter 3 on Coordinate Geometry, we cover the following:

• The Cartesian Plane: A plane with a horizontal axis (x-axis) and a vertical axis (y-axis) that intersect at the origin (0,0).

• Coordinates of a Point: Each point on the Cartesian plane is defined by an ordered pair (𝑥,𝑦), representing its position along the x-axis and y-axis.

• Names and Terms Associated with the Coordinate Plane: Terminology includes terms like origin, quadrants, axes, and coordinate points.

• Notations Used in Coordinate Geometry: Symbols and notations such as (x,y) for points, and 𝑂 for the origin.

• Plotting Points on the Plane: The process of marking points on the Cartesian plane based on their coordinates (𝑥,𝑦)

• Graph of Linear Equations with Examples: Representation of linear equations on the Cartesian plane, typically resulting in a straight line.

• Focusing on Linear Equations of the Form ax+by+c=0: Understanding and simplifying linear equations to the slope-intercept form 𝑦=𝑚𝑥+𝑐

• Linking with the Chapter on Linear Equations in Two Variables: Connecting concepts from coordinate geometry with those from the study of linear equations involving two variables.

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

• Coordinate Geometry Important Questions

• Coordinate Geometry Important Formulas

• Coordinate Geometry Revision Notes

• Coordinate Geometry RD Sharma Solutions

• Coordinate Geometry RS Aggarwal Solutions

Chapter 4 - Linear Equations in Two Variables

Chapter 4 also discusses the concept of linear equations having two variables. Students already equipped with the knowledge of coordinate geometry will learn how to plot the graph of a linear equation with two variables.

NCERT 9th class Maths book solutions include 4 exercises with 16 questions in total. This chapter includes application-based questions to test your understanding of the chapter.

1. Exercise 4.1: This exercise focuses on writing linear equations in two variables. You'll likely be given worded problems describing relationships between two variables and asked to translate them into mathematical equations.

2. Exercise 4.2: Here, you'll practice solving linear equations in two variables. The problems might involve finding the values of the variables (x and y) that satisfy the given equation.

In Class 9 Maths Chapter 4 on Linear Equations in Two Variables, we cover the following:

• Review of linear equations in one variable.

• Introduction to equations involving two variables.

• Demonstration that a linear equation in two variables has infinitely many solutions, represented as ordered pairs of real numbers.

• Plotting these solutions to show they form a straight line.

• Real-life examples and problems, including those on ratio and proportion, with both algebraic and graphical solutions presented simultaneously.

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

• Linear Equations in Two Variables Important Questions

• Linear Equations in Two Variables Important Formulas

• Linear Equations in Two Variables Revision Notes

• Linear Equations in Two Variables NCERT Exemplar Solutions

• Linear Equations in Two Variables RD Sharma Solutions

• Linear Equations in Two Variables RS Aggarwal Solutions

Chapter 5 - Introduction to Euclid's Geometry

The 5th Chapter, Introduction to Euclid's Geometry tries to establish a link between present-day geometry with that of Euclidean geometry. The axioms, theorems and postulates have been provided to provide students with an in-depth understanding. Chapter 5 of Class 9 Maths NCERT Solutions has 2 exercises. The questions are based on analysis to test your analytical skills.

1. Exercise 5.1: This exercise might involve basic definitions and properties of points and lines. You could be asked to identify points and lines in diagrams, understand how many points two lines can intersect, or differentiate between intersecting and parallel lines.

In Class 9 Maths Chapter 5 on Introduction to Euclid's Geometry, we cover the following:

• History – Euclid and the development of geometry in India.

• Euclid’s method of transforming observed phenomena into precise mathematics using definitions, common notions, axioms/postulates, and theorems.

• The five postulates of Euclid.

• Equivalent interpretations of the fifth postulate.

• Demonstrating the connection between axioms and theorems.

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

• Introduction to Euclids Geometry Important Questions

• Introduction to Euclids Geometry Important Formulas

• Introduction to Euclids Geometry Revision Notes

• Introduction to Euclids Geometry NCERT Exemplar Solutions

• Introduction to Euclids Geometry RD Sharma Solutions

• Introduction to Euclids Geometry RS Aggarwal Solutions

Chapter 6 - Lines and Angles

The 6th chapter in NCERT Solutions Class 9 Maths is on Lines and Angles. Students are mostly asked to prove the statements based on the axioms and theorems explained in the chapter. NCERT Solution Class 9 Maths of Chapter 6 includes 3 exercises.

1. Exercise 6.1: This exercise likely focuses on finding the measure of unknown angles based on given information. You might encounter problems where you need to use properties of angles (like a straight angle being 180 degrees) or identify special angle pairs (like linear pairs or verticals) to solve for missing angles.

In Class 9 Maths Chapter 6 on Lines and Angles, we cover the following:

• Sum of Adjacent Angles on a Line: If a ray stands on a line, the sum of the two adjacent angles formed is 180°, and vice versa.

• Vertically Opposite Angles: If two lines intersect, the vertically opposite angles are equal.

• Angles Formed by a Transversal with Parallel Lines: Results on corresponding angles, alternate angles, and interior angles when a transversal intersects two parallel lines.

• Parallel Lines: Lines parallel to a given line are also parallel.

• Sum of Angles in a Triangle: The sum of the angles in a triangle is 180°.

• Exterior Angle of a Triangle: If a side of a triangle is extended, the exterior angle formed is equal to the sum of the two opposite interior angles.

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

• Lines and Angles Important Questions

• Lines and Angles Important Formulas

• Lines and Angles Revision Notes

• Lines and Angles NCERT Exemplar Solutions

• Lines and Angles RD Sharma Solutions

• Lines and Angles RS Aggarwal Solutions

Chapter 7 - Triangles

Triangles is the 7th Chapter in NCERT Maths Class 9 Solutions. The properties of triangles such as inequalities, congruence, rules of congruence have been explained in this chapter. Students are also taught about the application of the congruence rules while solving the exercise questions. NCERT Solution Class 9 Maths of Chapter 7 includes 5 exercises. These questions are designed to test your skills.

1. Exercise 7.1: This exercise deals with proving triangles congruent (having exactly the same size and shape) based on given criteria. You'll likely use properties like side-side-side (SSS) or angle-side-angle (ASA) congruence rules.

2. Exercise 7.2: Here, the focus is on properties of angles in triangles. You might encounter problems that involve finding the relationship between interior angles (angles inside the triangle) and exterior angles (angle formed by one side of the triangle and the extension of the other side).

3. Exercise 7.3: This exercise introduces the concept of midpoints and perpendicular bisectors (a line segment that bisects a side at a 90-degree angle). You might be asked to prove that a segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

In Class 9 Maths Chapter 7 on Triangles, we cover the following:

• SAS Congruence: Two triangles are congruent if two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle.

• ASA Congruence: Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of the other triangle.

• SSS Congruence: Two triangles are congruent if all three sides of one triangle are equal to all three sides of the other triangle.

• Congruence in Right Triangles: Two right triangles are congruent if the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle.

• Angles Opposite Equal Sides: The angles opposite equal sides of a triangle are equal.

• Sides Opposite Equal Angles: The sides opposite equal angles of a triangle are equal.

• Triangle Inequalities: Triangle inequalities and the relationship between an angle and the side opposite it; inequalities within a triangle.

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

• Triangles Important Questions

• Triangles Important Formulas

• Triangles Revision Notes

• Triangles NCERT Exemplar Solutions

• Triangles RS Aggarwal Solutions

Chapter 8 - Quadrilaterals

In chapter 8 you will be introduced to Quadrilaterals and their various properties. A quadrilateral is a figure obtained by joining four distinct points on a plane. Students are provided in-depth knowledge about the topic in this chapter.

NCERT Solution Class 9 Maths of Chapter 8 includes 2 exercises. These questions are based on your application skills and analysis skills.

1. Exercise 8.1: This exercise involves using theorems related to quadrilaterals, especially parallelograms. You might encounter problems that ask you to prove properties of a quadrilateral based on given information about its sides or angles. For instance, you could be asked to show that a quadrilateral with opposite sides equal and parallel is a parallelogram.

2. Exercise 8.2: Here, the focus shifts towards applying the Mid-Point Theorem. This theorem states that the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and half its length. You might be solving problems that involve using this theorem to prove properties of quadrilaterals or find missing lengths and angles within them.

In Class 9 Maths Chapter 8 on Triangles, we cover the following:

• Diagonals of a Parallelogram: The diagonal of a parallelogram divides it into two congruent triangles.

• Equality of Opposite Sides in a Parallelogram: In a parallelogram, opposite sides are equal, and vice versa.

• Equality of Opposite Angles in a Parallelogram: In a parallelogram, opposite angles are equal, and conversely.

• Conditions for a Parallelogram: A quadrilateral is a parallelogram if a pair of its opposite sides are both parallel and equal.

• Bisection of Diagonals in a Parallelogram: In a parallelogram, the diagonals bisect each other, and conversely.

• Midpoint Theorem in Triangles: The line segment joining the midpoints of any two sides of a triangle is parallel to the third side, and its converse can be motivated.

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

• Quadrilaterals Important Questions

• Quadrilaterals Important Formulas

• Quadrilaterals Revision Notes

• Quadrilaterals NCERT Exemplar Solutions

• Quadrilaterals RD Sharma Solutions

• Quadrilaterals RS Aggarwal Solutions

Chapter 9 - Circles

In this chapter of Class 9 Maths, you will learn about circles. Definition of a circle, tangent, chord, arc, etc., are described in this chapter. Various other concepts such as the angle subtended by the arc of a circle, cyclic quadrilaterals, etc., are also explained. 9th class Maths book solutions Chapter 9 includes 6 exercises and solutions to all 35 questions given in your NCERT Textbook. These questions are based on your numerical abilities, application skills, and memory skills.

1. Exercise 9.1: Deals with theorems about equal chords subtending equal central angles and vice versa (converse). You'll practice identifying these relationships and their implications.

2. Exercise 9.2: This exercise involves problems applying these central angle and chord relationships to solve for missing measures (angles or chord lengths) based on the given information.

In Class 9 Maths Chapter 9 on Circles, we cover the following:

• Defining Circle-related Concepts: Understanding the terms radius, circumference, diameter, chord, arc, and subtended angle through examples.

• Equal Chords and Central Angles: Equal chords of a circle subtend equal angles at the centre, and vice versa.

• Perpendicular Bisector of a Chord: The perpendicular from the centre of a circle to a chord bisects the chord, and conversely, a line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.

• Uniqueness of Circles through Three Points: There is one and only one circle passing through three given non-collinear points.

• Equal Chords and Distance from Center: Equal chords of a circle (or of congruent circles) are equidistant from the centre(s), and vice versa.

• Angle Subtended by an Arc: 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.

• Angles in the Same Segment: Angles in the same segment of a circle are equal.

• Concyclic Points and Cyclic Quadrilaterals: If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the segment, the four points lie on a circle. The sum of either pair of opposite angles of a cyclic quadrilateral is 180°, and vice versa.

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

• Circles Important Questions

• Circles Important Formulas

• Circles Revision Notes

• Circles NCERT Exemplar Solutions

• Circles RD Sharma Solutions

• Circles RS Aggarwal Solutions

Chapter 10 - Heron’s Formula

Heron’s formula, the 10th chapter of Class 9 Maths, is used to calculate the area of a triangle when the lengths of its sides are given. Students are also taught how to calculate the area of quadrilaterals and other polygons by dividing them into triangles and applying Heron’s formula. NCERT Solution Class 9 Maths of Chapter 10 includes 2 exercises designed on your ability to decipher and apply the formula.

1. Exercise 10.1: This exercise involves practice problems where you'll be given the side lengths of a triangle and asked to find its area using Heron's Formula.

In Class 9 Maths Chapter 10 on Heron’s Formula, we cover the following:

• Area of a Triangle with Heron’s Formula: Finding the area of a triangle using Heron’s Formula, without going into the proof.

• Application in Quadrilaterals: Using Heron’s Formula to find the area of a quadrilateral.

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

• Heron’s Formula Important Questions

• Heron’s Formula Important Formulas

• Heron’s Formula Revision Notes

• Heron’s Formula RD Sharma Solutions

Chapter 11 - Surface Areas and Volumes

In this chapter, you will understand the significance of surface areas and volumes of cubes, cuboids, cones, etc. This chapter teaches students how to calculate the same. NCERT Solution Class 9 Maths of Chapter 11 includes 9 exercises and these questions will test your understanding ability and memory skills (required to retain the different formulae).

1. Exercise 11.1: Introduction to finding the curved surface area (CSA) of a cone. You'll be introduced to the relevant formula and practice using it on cones with different measurements.

2. Exercises 11.2 - 11.3: Further exploration of CSA. These exercises might involve variations like calculating CSA when the slant height or radius is missing, or solving problems with right cones vs. slanted cones.

In Class 9 Maths Chapter 11 on Surface Areas and Volumes, we cover the following:

• Introduction to Surface Areas and Volumes

• Surface Area of a Cuboid and a Cube:

• Surface Area of a Right Circular Cylinder

• Surface Area of a Right Circular Cone

• Surface Area of a Sphere

• Volume of a Cuboid

• Volume of a Cube

• Volume of a Right Circular Cylinder

• Volume of a Right Circular Cone

• Volume of a Sphere

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

• Surface Areas and Volumes Important Questions

• Surface Areas and Volumes Formulas

• Surface Areas and Volumes Revision Notes

• Surface Areas and Volumes NCERT Exemplar Solutions

Chapter 12 - Statistics

Statistics is the 12th chapter in Class 9 Maths. Students are introduced to the concept of statistics and its significance and application in different fields. Concepts of data, data distribution, and representation, different types of graphs, etc., are also explained. Students will mainly learn to calculate mean, median, and mode in this chapter. NCERT Solution Class 9 Maths of Chapter 12 contains 4 exercises. The questions judge your comprehensive ability and analytical skills.

1. Exercise 12.1: This exercise involves interpreting data presented in a textual format (like percentages) and converting it into a suitable format for creating a bar graph.

In Class 9 Maths Chapter 12 on Surface Areas and Volumes, we cover the following:

• Gathering data.

• Presenting data in various formats such as tables, ungrouped and grouped data.

• Creating bar graphs, histograms (with varying base lengths), and frequency polygons.

• Analysing data qualitatively to determine the appropriate method of presentation.

• Calculating mean, median, and mode for ungrouped data.

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

• Statistics Important Questions

• Statistics Formulas

• Statistics Revision Notes

CBSE Class 9 Maths Chapter-Wise Marks Weightage 

Internal Assessment for CBSE Class 9 Maths

Benefits of Vedantu’S NCERT Solutions for Class 9 Maths

NCERT or National Council of Educational Research and Training was created to standardize India’s education system. The NCERT solutions can help you get an in-depth and detailed understanding of the concepts and questions from the NCERT textbooks. Here are some more benefits of Vedantu’s NCERT Solutions for Class 9 Maths:

1. All the Concepts are Explained in Detail

In these solutions, you will find that all the concepts have been explained in easy language and in detail. It will help you get a better understanding of the concepts and score well in your exams.

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4. Get Your Doubts Cleared

If you are confused about any topic from your class 9 Maths syllabus, you can use these NCERT solutions for getting your doubts cleared. All the solutions are provided in an easy way so that you can have your concepts cleared.

5. Created by the Best Teachers

Vedantu’s NCERT Solutions for Class 9 Maths have been created by teachers who are subject matter experts. By studying through these high-quality study materials, you will be able to score more marks. The solutions have been designed after performing extensive research on the subject. Once you are done, you will be able to solve problems of varying difficulty levels.

6. Helps with Preparation For Exams

The NCERT Solutions contain a Chapter Wise List Of Class 9 Maths so that students can easily identify the important chapters that have significant weightage in the examinations. Thus, they can prepare for their exams well and score more marks by focusing on chapters that are crucial.

7. Makes Revision Easier

While revising the chapters, students don’t have to go through the entire book. The NCERT solutions have explained all the topics and concepts in a very easy manner. While solving the problems, students will learn about the theorems, formulas, and descriptions that are used. Thus, they will easily be able to complete the chapter without any interruptions.

Important Related Links for CBSE Class 9 Maths

• Important Questions for CBSE Class 9 Math

• RS Aggarwal Class 9 Solutions for Math book

• NCERT Class 9 Math Formulas

• NCERT Class 9 Maths Revision Notes

• NCERT Class 9 Maths Exemplar Solutions

• RD Sharma Solutions for NCERT Class 9

Conclusion

Vedantu's NCERT Solutions for Class 9 Maths provide comprehensive guidance and practice for each curriculum chapter. These solutions are essential for students aiming to understand concepts thoroughly and perform well in exams. Key areas to focus on include mastering formulas, practicing exercises, and understanding theorems and their applications. Additionally, topics such as Polynomials, Coordinate Geometry, Linear Equations, and Probability are crucial and often challenging, requiring extra attention and regular practice. Utilizing these solutions can significantly enhance a student's problem-solving skills and conceptual clarity.