std 9 maths assignment solutions 2023

NCERT Solutions for Class 9 Maths

NCERT Solutions for Class 9 Maths in Hindi and English Medium updated for 2024-25 session. We have carefully done Class 9 Mathematics Textbook Solutions to meet the needs of CBSE Grade 9 students.

Class 9 Maths Syllabus for 2024-25 Study Plan to Perform Well in Class 9 Exams Chapter 1. Number Systems 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. Circles Chapter 10. Heron’s Formula Chapter 11. Surface Areas and Volumes Chapter 12. Statistics For other State Boards Chapter 13. Surface Areas and Volumes Chapter 14. Statistics Chapter 15. Probability Tiwari Academy’s NCERT Solutions for Class 9 Maths are designed to enhance conceptual understanding and improve problem-solving skills, making them an invaluable resource for students.

NCERT Solutions of Class 9 Maths

The class 9 mathematics solutions and answers have been suitably modified and revised taking into account the guidelines of CBSE and State board 9th Maths curriculum 2024-25 . Your valuable feedback and Review would help us to improve further our answers. Class 9 Solution Apps Download Class 9 Math solutions are available in multiple formats such as printed books, online resources and mobile apps. Platforms like Tiwari Academy take their social responsibility seriously by providing free access to NCERT solutions, ensuring that quality education is accessible to everyone. These free resources help students learn and succeed without financial barriers.

CBSE Syllabus for Class 9 Maths Revised for 2024-25

• CBSE Syllabus for Class 9 Maths
• Class 9 All Subjects NCERT Solutions

NCERT solutions for Class 9 Mathematics are incredibly useful for students because they are created by subject experts based on the CBSE curriculum. If you’re searching for the revised Class 9 Maths UP Board NCERT Solutions for the 2024-25 exams, Tiwari Academy has you covered. They offer updated solutions for all chapters, making it easier for students to understand key concepts and score higher in exams. These solutions not only strengthen mathematical understanding but also help students excel in their studies.

NCERT Solution Apps for 9th Class You can get to know about various concepts while solving Class 9 Maths NCERT Solutions , which will be useful in your higher classes. Get Class 9th Maths question answers for the new Academic year based on latest NCERT book syllabus.

Our Educational App offers notifications and reminders for important dates, exams and study schedules, helping students stay organised and focused. The app also allows students to track their scores, monitor exercise completion and identify areas for improvement, making it a valuable tool for staying on track with their studies.

How to Score well in class 9 Maths Exams For a Class 9 student aiming to excel in Maths, a structured approach to learning and practice is the key. Start by thoroughly understanding the current syllabus and the fundamental concepts it covers.

Physical exercise, hobbies, or short walks can refresh your mind and improve focus while studying. Staying positive and believing in yourself helps you grow. Regular quizzes or timed math practice show weak areas and build confidence.

Solutions for NCERT Class 9 Mathematics All Chapters

These Chapter Wise Class 9 Mathematics Solutions App is well designed so that class 9 students can easily understand each sum’s solutions. The Class 9th NCERT Math textbook has a total of 12 chapters which are divided into six units. Solving Class 9th Maths solution in each chapter will ensure positive results. Chapters covered in the NCERT 9th Math textbook are Number System, Polynomials, Introduction to Euclids Geometry, Quadrilaterals, Surface Areas, Heron’s Formula, and Volumes, Statistics etc. 

NCERT Class 9 Maths Chapters Solutions

Students who are facing any trouble in solving tough Class 9th Maths problems can get best help from these CBSE Maths of Class 9 Solutions of NCERT (https://ncert.nic.in/) textbooks. The syllabus contains 12 chapters with important topics. Here all solutions build your fundamental very clear and efficiency by solving theses exercise-wise solutions.

NCERT Solutions for Class 9 Maths Chapter 1 – Number System

NCERT Solutions for Class 9 Maths Chapter 1 introduces key topics like rational numbers and irrational numbers. You will learn how to represent integers, rational numbers and irrational numbers on a number line. This chapter provides a comprehensive understanding of the number line, helping students grasp these fundamental concepts effectively.

Class 9 Maths Chapter 1 for CBSE Board Class 9 Maths Exercise 1.1 in English Class 9 Maths Exercise 1.2 in English Class 9 Maths Exercise 1.3 in English Class 9 Maths Exercise 1.4 in English Class 9 Maths Exercise 1.5 in English

Class 9 Maths Chapter 1 Solution for State Boards Class 9 Maths Chapter 1 Exercise 1.1 Class 9 Maths Chapter 1 Exercise 1.2 Class 9 Maths Chapter 1 Exercise 1.3 Class 9 Maths Chapter 1 Exercise 1.4 Class 9 Maths Chapter 1 Exercise 1.5 Class 9 Maths Chapter 1 Exercise 1.6

Class 9 Maths Chapter 1 in Hindi Medium Class 9 Maths Exercise 1.1 in Hindi Class 9 Maths Exercise 1.2 in Hindi Class 9 Maths Exercise 1.3 in Hindi Class 9 Maths Exercise 1.4 in Hindi Class 9 Maths Exercise 1.5 in Hindi

There are five exercises in this NCERT Solutions for Class 9 Maths Chapter 1 Number System, where you can also find all word problems based on 9th Class Maths NCERT Book. You can learn from these 9th Class Maths Book Solution, the representation of terminating and non-terminating recurring decimals, presentation of square roots of 2, 3, other non-rational numbers on the number line.

Topics to Study in Class 9 Maths Chapter 1 Number Systems Review of representation of natural numbers, integers, and rational numbers on the number line. Rational numbers as recurring/ terminating decimals. Operations on real numbers. 2. Examples of non-recurring/non-terminating decimals. Existence of non-rational numbers (irrational numbers) such as √2, √3 and their representation on the number line. Explaining that every real number is represented by a unique point on the number line and conversely, viz. every point on the number line represents a unique real number. 3. Definition of nth root of a real number. 4. Rationalization (with precise meaning) of real numbers of the type 1/(a + b√x) and 1/(√x + √y) (and their combinations) where x and y are natural number and a and b are integers. 5. 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.)

NCERT Solutions for Class 9 Maths Chapter 2 – Polynomials

Get NCERT Solutions for Class 9 Maths Chapter 2 here with free of cost, and all solutions are solved by the professional Maths teacher. Class 9 Maths Chapter 2 for CBSE Board Class 9 Maths Exercise 2.1 in English Class 9 Maths Exercise 2.2 in English Class 9 Maths Exercise 2.3 in English Class 9 Maths Exercise 2.4 in English

Class 9 Maths Chapter 2 Solution for State Boards Class 9 Maths Chapter 2 Exercise 2.1 Class 9 Maths Chapter 2 Exercise 2.2 Class 9 Maths Chapter 2 Exercise 2.3 Class 9 Maths Chapter 2 Exercise 2.4 Class 9 Maths Chapter 2 Exercise 2.5

Class 9 Maths Chapter 2 in Hindi Medium Class 9 Maths Exercise 2.1 in Hindi Class 9 Maths Exercise 2.2 in Hindi Class 9 Maths Exercise 2.3 in Hindi Class 9 Maths Exercise 2.4 in Hindi

This chapter Polynomial consists of variables and coefficients, its operations of addition, subtraction, multiplication and also non-negative integer exponents of variables.

Topics of Class 9 Maths Chapter 2 Polynomials 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. Motivate and State the Remainder Theorem with examples. Statement and proof of the Factor Theorem. Factorization of ax² + 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 = x² + y² + z² + 2xy + 2yz + 2zx (x + y)³ = x³ + y³ + 3xy(x + y) (x – y)³ = x³ – y³ – 3xy(x – y) x³ + y³ = (x + y)(x² – xy + y²) x³ – y³ = (x – y)(x² + xy + y²) x³ + y³ + z³ – 3xyz = (x + y)(x² + y² + z² – xy – yz – 2zx) and their use in factorization of polynomials.

NCERT Solutions for Class 9 Maths Chapter 3 – Coordinate Geometry

Students can get NCERT exercises solutions for Class 9 Maths Chapter 3 with free of cost. Chapter 3 introduces the concepts of the Cartesian plane, including coordinates of a point, terms, coordinate plane notations, x-axis, y-axis, x-coordinate, y-coordinate, quadrants, origin and more. It provides a thorough understanding of these fundamental terms, helping students navigate and plot points on the coordinate plane effectively.

Class 9 Maths Chapter 3 for CBSE Board Class 9 Maths Exercise 3.1 in English Class 9 Maths Exercise 3.2 in English

Class 9 Maths Chapter 3 for State Boards Class 9 Maths Chapter 3 Exercise 3.1 Class 9 Maths Chapter 3 Exercise 3.2 Class 9 Maths Chapter 3 Exercise 3.3

Class 9 Maths Chapter 3 in Hindi Medium Class 9 Maths Exercise 3.1 in Hindi Class 9 Maths Exercise 3.2 in Hindi

In NCERT Solutions for Class 9 Maths Chapter 3, Coordinate Geometry, students learn about the coordinate plane and the mutually perpendicular lines, known as axes. They can download the NCERT Solutions for Class 9 Maths PDF to study offline anytime, making it convenient for learning on the go. Get Chapter Number System Class 9 Maths Solutions in Hindi medium and English medium.

Class 9 Maths Chapter 3 Coordinate Geometry Main Points The Cartesian plane, coordinates of a point, names and terms associated with the coordinate plane, notations.

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

NCERT Solutions for Class 9 Maths Chapter 4 4 covers linear equations in one variable and introduces linear equations in two variables, like ax + by + c = 0. In this chapter, students will also learn how to plot the graph of a linear equation in two variables, gaining a solid understanding of these key concepts.

Class 9 Maths Chapter 4 for CBSE Board Class 9 Maths Exercise 4.1 in English Class 9 Maths Exercise 4.2 in English

Class 9 Maths Chapter 4 for State Boards Class 9 Maths Chapter 4 Exercise 4.1 Class 9 Maths Chapter 4 Exercise 4.2 Class 9 Maths Chapter 4 Exercise 4.3 Class 9 Maths Chapter 4 Exercise 4.4

Class 9 Maths Chapter 4 in Hindi Medium Class 9 Maths Exercise 4.1 in Hindi Class 9 Maths Exercise 4.2 in Hindi

NCERT Solutions for Class 9 Maths Chapter 9, Linear Equations in Two Variables, includes just two exercises. Students can download the free NCERT Solutions for Class 9 Maths PDF to study anytime, anywhere. These solutions help students clear all concepts for each math problem. Download unlimited free solutions for Chapter 9 Linear Equations in Two Variables in both Hindi Medium and English Medium.

Topics to be covered in Class 9 Maths Chapter 4 Linear Equations in Two Variables 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.

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

Get NCERT Solutions for Class 9 Maths Chapter 5 for all exercises. This chapter covers Euclid’s approach to geometry, which forms the foundation of present-day geometry. In the NCERT exercises for Class 9 Maths, Introduction to Euclid’s Geometry, students will learn about defining common geometrical shapes and terms, providing a solid understanding of the basics of geometry.

Class 9 Maths Chapter 5 for CBSE Board Class 9 Maths Exercise 5.1 in English

Class 9 Maths Chapter 5 for State Boards Class 9 Maths Chapter 5 Exercise 5.1 Class 9 Maths Chapter 5 Exercise 5.2

Class 9 Maths Chapter 5 in Hindi Medium Class 9 Maths Exercise 5.1 in Hindi

This 9th Class Maths Book Solution includes only one exercise that provides a deeper understanding of the relationship between axioms, postulates, and theorems. Download Chapter Euclid’s Geometry Class 9 Maths Solutions in both Hindi Medium and English Medium for free to enhance your learning experience.

Syllabus of Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry History – Geometry in India and Euclid’s geometry. Euclid’s method of formalizing observed phenomenon into rigorous Mathematics with definitions, common/obvious notions, axioms/postulates and theorems. The five postulates of Euclid. Showing the relationship between axiom and theorem, for example: (Axiom) 1. Given two distinct points, there exists one and only one line through them. (Theorem) 2. (Prove) Two distinct lines cannot have more than one point in common.

NCERT Solutions for Class 9 Maths Chapter 6 – Lines and Angles

NCERT Solutions for Class 9 Maths Chapter 6 revolves around the theorems present in the topics of Lines and Angles.

Class 9 Maths Chapter 6 Solutions for CBSE Board Class 9 Maths Exercise 6.1 in English Class 9 Maths Exercise 6.2 in English

Class 9 Maths Chapter 6 Solutions for State Boards Class 9 Maths Chapter 6 Exercise 6.1 Class 9 Maths Chapter 6 Exercise 6.2 Class 9 Maths Chapter 6 Exercise 6.3

Class 9 Maths Chapter 6 in Hindi Medium Class 9 Maths Exercise 6.1 in Hindi Class 9 Maths Exercise 6.2 in Hindi

The 9th Maths Book Solutions provided here are carefully reviewed for accuracy. Get free NCERT Solutions for Class 9 Maths in PDF format for offline study. Practice these solutions thoroughly to score your best in the final exams. Download Chapter Lines and Angles Class 9 Maths Solutions in both Hindi and English medium.

Class 9 Maths Chapter 6 Lines and Angles Topics 1. (Motivate) If a ray stands on a line, then the sum of the two adjacent angles so formed is 180° and the converse. 2. (Prove) If two lines intersect, vertically opposite angles are equal. 3. (Motivate) Lines which are parallel to a given line are parallel.

NCERT Solutions for Class 9 Maths Chapter 7 – Triangles

Get NCERT Solutions for Class 9 Maths Chapter 7 , and students will study detail knowledge about the triangle, congruence of triangles, rules of congruence, and its properties of triangles and many more.

Class 9 Maths Chapter 7 Solutions for CBSE Board Class 9 Maths Exercise 7.1 in English Class 9 Maths Exercise 7.2 in English Class 9 Maths Exercise 7.3 in English

Class 9 Maths Chapter 7 Solutions for State Boards Class 9 Maths Chapter 7 Exercise 7.1 Class 9 Maths Chapter 7 Exercise 7.2 Class 9 Maths Chapter 7 Exercise 7.3 Class 9 Maths Chapter 7 Exercise 7.4 Class 9 Maths Chapter 7 Exercise 7.5

Class 9 Maths Chapter 7 in Hindi Medium Class 9 Maths Exercise 7.1 in Hindi Class 9 Maths Exercise 7.2 in Hindi Class 9 Maths Exercise 7.3 in Hindi

In this 9th Maths Book Solution, students will prove the properties of triangles learned in previous classes. While solving problems, they will apply various congruence rules. This chapter includes eight theorems, and there are three exercises covered in the NCERT Solutions for Class 9 Maths Chapter 7, Triangles. Download Chapter Triangles Class 9 Maths Solutions in both Hindi Medium and English Medium for a comprehensive understanding.

Class 9 Maths Chapter 7 Triangles Main Points to Study 1. (Motivate) Two triangles are congruent if any two sides and the included angle of one triangle is equal to any two sides and the included angle of the other triangle (SAS Congruence). 2. (Prove) Two triangles are congruent if any two angles and the included side of one triangle is equal to any two angles and the included side of the other triangle (ASA Congruence). 3. (Motivate) Two triangles are congruent if the three sides of one triangle are equal to three sides of the other triangle (SSS Congruence). 4. (Motivate) Two right triangles are congruent if the hypotenuse and a side of one triangle are equal (respectively) to the hypotenuse and a side of the other triangle. (RHS Congruence) 5. (Prove) The angles opposite to equal sides of a triangle are equal. 6. (Motivate) The sides opposite to equal angles of a triangle are equal.

NCERT Solutions for Class 9 Maths Chapter 8 – Quadrilaterals

Find NCERT Solutions for Class 9 Maths Chapter 8 , which covers the topic Quadrilaterals. The joining four points in order is called a quadrilateral.

Class 9 Maths Chapter 8 for CBSE and State Boards Class 9 Maths Exercise 8.1 in English Class 9 Maths Exercise 8.2 in English

Class 9 Maths Chapter 8 in Hindi Medium Class 9 Maths Exercise 8.1 in Hindi Class 9 Maths Exercise 8.2 in Hindi

Class 9 Maths Chapter 8 includes two exercises and focuses on proving one theorem. These solutions cover important concepts such as the angle sum property of a quadrilateral, properties of a parallelogram, and different types of quadrilaterals. Download Chapter Quadrilaterals Class 9 Maths Solutions in both Hindi Medium and English Medium for free.

Class 9 Maths Chapter 8 Quadrilaterals Focus Points 1. (Prove) The diagonal divides a parallelogram into two congruent triangles. 2. (Motivate) In a parallelogram opposite sides are equal, and conversely. 3. (Motivate) In a parallelogram opposite angles are equal, and conversely. 4. (Motivate) A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and equal. 5. (Motivate) In a parallelogram, the diagonals bisect each other and conversely. 6. (Motivate) In a triangle, the line segment joining the mid points of any two sides is parallel to the third side and in half of it and (motivate) its converse.

NCERT Solutions for Class 9 Maths Chapter 9 – Circles

Get well-explained NCERT Solutions for Class 9 Maths Chapter Circles for free, with no need to log in or register. Access solutions for all exercises at no cost, making learning easy and convenient. In this NCERT Solutions for Class 9 Maths Chapter 9 Circle, learn that circle is a collection of all the points in a plane at a fixed distance.

Class 9 Maths Chapter 9 for CBSE Board Class 9 Maths Exercise 9.1 in English Class 9 Maths Exercise 9.2 in English Class 9 Maths Exercise 9.3 in English

Class 9 Maths Chapter 9 Solutions for State Boards Class 9 Maths Chapter 9 Exercise 9.1 Class 9 Maths Chapter 9 Exercise 9.2 Class 9 Maths Chapter 9 Exercise 9.3 Class 9 Maths Chapter 9 Exercise 9.4

Class 9 Maths Chapter 9 in Hindi Medium Class 9 Maths Exercise 9.1 in Hindi Class 9 Maths Exercise 9.2 in Hindi Class 9 Maths Exercise 9.3 in Hindi

Learn about circles, including Chord at a Point and Equal Chords, with twelve theorems in total. Download NCERT Solutions for Class 9 Maths for free, with no registration needed. Get Chapter 9 Circles Solutions in both Hindi and English Medium at no cost.

Class 9 Maths Chapter 9 Circles Curriculum for 2024-25 1.(Prove) Equal chords of a circle subtend equal angles at the center and (motivate) its converse. 2.(Motivate) The perpendicular from the center of a circle to a chord bisects the chord and conversely, the line drawn through the center of a circle to bisect a chord is perpendicular to the chord. 3. (Motivate) Equal chords of a circle (or of congruent circles) are equidistant from the center (or their respective centers) and conversely. 4. (Prove) The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle. 5. (Motivate) Angles in the same segment of a circle are equal. 6. (Motivate) If a line segment joining two points subtends equal angle at two other points lying on the same side of the line containing the segment, the four points lie on a circle. 7. (Motivate) The sum of either of the pair of the opposite angles of a cyclic quadrilateral is 180° and its converse.

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

Students get NCERT Solutions for Class 9 Maths Chapter 10 , which covers Heron’s Formula, which helps calculate the area of a triangle when the lengths of all three sides are given. With Heron’s Formula, you can find the area of a triangle without needing to calculate the heights or distances directly.

Class 9 Maths Chapter 10 for CBSE Board Class 9 Maths Exercise 10.1 in English

Class 9 Maths Chapter 10 for State Boards Class 9 Maths Chapter 10 Exercise 10.1 Class 9 Maths Chapter 10 Exercise 10.2 Class 9 Maths Chapter 10 Exercise 10.3 Class 9 Maths Chapter 10 Exercise 10.4 Class 9 Maths Chapter 10 Exercise 10.5 Class 9 Maths Chapter 10 Exercise 10.6

Class 9 Maths Chapter 10 in Hindi Medium Class 9 Maths Exercise 10.1 in Hindi

Heron’s Formula is used to find the areas of quadrilaterals, triangles, and other polygons. There is only one exercise in NCERT Solutions for Class 9 Maths Chapter 10, Heron’s Formula. You can download the NCERT Solutions for Class 9 Maths Chapter 12 PDF for free and study anytime, offline, on your desktop or mobile.

Class 9 Maths Chapter 10 Heron’s Formula Main Points To Study Area of a triangle using Heron’s formula (without proof)

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

NCERT Solutions for Class 9 Maths Chapter 11 , learn to find the surface areas and volumes of cuboids and cylinders in detail. In NCERT Solutions for Class 9 Maths Chapter 11, Surface Areas and Volumes, you will also explore the surface areas of other solids, such as cones and spheres.

Class 9 Maths Chapter 11 for CBSE Board Class 9 Maths Exercise 11.1 in English Class 9 Maths Exercise 11.2 in English Class 9 Maths Exercise 11.3 in English Class 9 Maths Exercise 11.4 in English

Class 9 Maths Chapter 11 Solutions for State Boards Class 9 Maths Chapter 11 Exercise 11.1 Class 9 Maths Chapter 11 Exercise 11.2

Class 9 Maths Chapter 11 in Hindi Medium Class 9 Maths Exercise 11.1 in Hindi Class 9 Maths Exercise 11.2 in Hindi Class 9 Maths Exercise 11.3 in Hindi Class 9 Maths Exercise 11.4 in Hindi

In this chapter, students will deepen their understanding of surface areas and volumes, building on concepts from earlier classes. Chapter 11 includes four exercises covering the surface areas and volumes of various solids such as cubes, cuboids, hemispheres, spheres, cylinders and cones. Download the NCERT Solutions for Class 9 Maths PDF for free without any login or registration required. Get Chapter Surface Areas and Volumes  Class 9 Maths Solutions in Hindi medium and English medium.

The Points to be covered in Class 9 Maths Chapter 11 Surface Areas and Volumes Surface areas and volumes of spheres (including hemispheres) and right circular cones.

NCERT Solutions for Class 9 Maths Chapter 12 – Statistics

These best easy NCERT Solutions for Class 9 Maths Chapter 12 , deals with Statistics which the extraction of meaningful information is studied is known as Statistics.

Class 9 Maths Chapter 12 for CBSE Board Class 9 Maths Exercise 12.1 in English

Class 9 Maths Chapter 12 Solutions for State Boards Class 9 Maths Chapter 12 Exercise 12.1 Class 9 Maths Chapter 12 Exercise 12.2

Class 9 Maths Chapter 12 in Hindi Medium Class 9 Maths Exercise 12.1 in Hindi

Statistics plays an important role in daily life, influencing various aspects like people’s lives and state functions. NCERT Solutions for Class 9 Maths Chapter 12, Statistics, teaches how to present data in different ways, including frequency distribution. You’ll also learn how to organize data in tabular and row form, group it in regular intervals and calculate the mean, mode and median. Get Chapter Statistics Class 9 Maths Solutions in Hindi medium and English medium.

Class 9 Maths Chapter 12 Statistics Main Points Bar graphs, histograms (with varying base lengths), and frequency polygons.

NCERT Solutions for Class 9 Maths Chapter 13 – Surface Areas and Volumes

Tiwari Academy offers NCERT solutions on Surface Areas and Volumes, based on the state board syllabus for the 2024-25 session, in both Hindi and English mediums. Master geometry with Tiwari Academy’s expert NCERT Solutions for Class 9 Maths Exercise 13.8, specifically designed for students of the 2024-2025 session, available in both languages. Class 9 Maths Chapter 13 for State Boards Class 9 Maths Chapter 13 Exercise 13.1 Class 9 Maths Chapter 13 Exercise 13.2 Class 9 Maths Chapter 13 Exercise 13.3 Class 9 Maths Chapter 13 Exercise 13.4 Class 9 Maths Chapter 13 Exercise 13.5 Class 9 Maths Chapter 13 Exercise 13.6 Class 9 Maths Chapter 13 Exercise 13.7 Class 9 Maths Chapter 13 Exercise 13.8 Class 9 Maths Chapter 13 Exercise 13.9

NCERT Solutions for Class 9 Maths Chapter 14 Statistics

Access Class 9 Maths Chapter 14 Statistics solutions in both Hindi and English at Tiwari Academy. Perfect for CBSE students, our resources are comprehensive, user-friendly, and designed to help you excel in your studies. Class 9 Maths Chapter 1 Solution for State Boards Class 9 Maths Chapter 14 Exercise 14.1 Class 9 Maths Chapter 14 Exercise 14.2 Class 9 Maths Chapter 14 Exercise 14.3 Class 9 Maths Chapter 14 Exercise 14.4

NCERT Solutions for Class 9 Maths Chapter 15 Probability

Master Class 9 Maths Chapter 15, Probability, with our solutions in English and Hindi. Clear and concise, these expert resources help you excel in CBSE exams and understand Probability easily. Class 9 Maths Chapter 15 Solutions for State boards Class 9 Maths Chapter 15 Exercise 15.1

Importance of NCERT Solutions for Class 9 Maths

NCERT, or the National Council of Educational Research and Training, creates standard textbooks and supports education in India. NCERT Class 9 Mathematics Solugions give clear, detailed answers to textbook questions, helping students understand concepts better. These solutions improve learning and boost exam performance with many added benefits.

Explain Each and Every Concept in Detail Class 9 Maths Question Answers explain every concept in detail using simple language, making it easier for students to understand. These solutions are essential for a better grasp of the subject and help in scoring good marks in Class 9 exams.

Most Accurate Solutions All the NCERT Solutions for Class 9 Maths for all chapters are provided by the many years experienced teachers. All solutions are well-reviewed before provided. Students who have misguided by the wrong study materials before can study with these most accurate NCERT Solutions for Class 9 Maths.

Student can learn Offline NCERT Solutions for Class 9 Maths are available as downloadable PDFs for all chapters, allowing you to study offline. You can access these solutions anytime and anywhere, making learning more convenient and flexible.

Clear all Doubts Class 9 Students who have confusion with many topics in class 9 Maths, they can learn with these NCERT Solutions for Class 9 Maths for all chapters. These solutions are given in easy method, and it clear your all doubts and concepts clear.

Created the Best Subject Experts in Field Class 9 Maths solutions are prepared by expert teachers and subject specialists to help you score higher. Each solution is carefully crafted after thorough research, allowing you to tackle problems of any difficulty.

How are CBSE 9 Maths Solutions of NCERT Helpful for Final Exams?

CBSE 9 Maths Solutions of NCERT is sufficient to score more in 9th boards exams. With NCERT Solutions for Class 9 Maths , students will get complete guidance on every topics and sub-topics related to the respective chapter.

NCERT Solutions for Class 9 Maths simplify complex concepts, helping students understand topics thoroughly, improve problem-solving skills, and boost confidence which is good for anyone aiming to excel in their exams. To maximise success in CBSE Class 9 Maths, it is essential to cover all exercises and thoroughly learn every theorem in the syllabus. Skipping any topic could impact your overall understanding and performance in exams. The NCERT solutions provide detailed explanations for each exercise, ensuring you master every concept. By staying consistent and covering all topics in CBSE 9th Maths syllabus , students can build a strong foundation and perform well in both school assessments and final exams.

A panel of experts at Tiwari Academy recently conducted a two-day workshop to thoroughly review and enhance the content quality of their educational materials. During this workshop, they meticulously analysed both the English and Hindi versions of the content, providing valuable suggestions for improvements. Based on these expert recommendations, necessary amendments were made to ensure that the solutions meet high educational standards and offer clear, concise explanations for students. Following these discussions, the final draft of the solutions was prepared, adhering to the new syllabus for 2024-25 . We offer well-structured materials with a classical approach to geometry and constructions, making key concepts easier to grasp. For UP Board High School students, comprehensive solutions, sample papers and notes are available for the 2024-25 syllabus in Hindi and English. These resources support exam success and concept mastery.

How to practice Class 9 Maths using NCERT Solutions?

For the academic session 2024-25, NCERT has significantly reduced the syllabus for Class 9 Maths in the latest textbooks. As a result, the NCERT Solutions for Class 9 Maths have also been updated and revised accordingly. These solutions are essential for students aiming to excel in their CBSE Class 9 Mathematics exams. Follow the steps below to learn how you can effectively use the updated Class 9 Maths solutions to improve your exam performance.

Step 1: Use NCERT Solutions for Comprehensive Coverage of class 9 Maths textbook.

Step 2: verify your answers and learn about the exam pattern., step 3: revise the syllabus and use class 9 maths ncert solution as a revision aid., step 4: make effective self study in 9th maths using ncert solutions., step 5: confined to new rationalised ncert textbooks and solve it using ncert solutions..

What are the most important chapters in Class 9 Maths for the final exams?

The key chapters for Class 9 Maths exams are Number Systems, Polynomials, and Quadrilaterals, as they build a foundation for future classes. Coordinate Geometry and Statistics are also important for skills like plotting and data analysis. Focusing on Triangles and Circles is crucial too, as they carry high marks. Understanding these chapters well can help you score better, as they include both objective and subjective questions and strengthen Maths basics for higher classes.

How many chapters are there in NCERT Class 9 Maths?

According to new syllabus, there are a total of 12 chapters present in the NCERT Class 9 Maths. 

• 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- Circles, 
• Chapter 10- Heron’s Formula, 
• Chapter 11- Surface Areas and Volumes, 
• Chapter 12- Statistics, 

These are the chapters in Class 9 of NCERT Maths. And you need to cover all the chapters to get full marks in your final exam.

How can I score full marks in Class 9 Maths?

To score full marks in Class 9 Maths, regular practice and a clear understanding of concepts are crucial. Begin by solving all NCERT textbook questions as they are designed to cover the entire syllabus comprehensively. Also, make sure to understand and practice each theorem, as many problems are based on applying these mathematical principles. In addition, revising formulas regularly, solving sample papers and taking mock tests can help boost your confidence and time management skills during the exam. It’s important to identify weak areas and work on them through additional exercises or reference materials. With consistent effort and the right strategy, scoring full marks is achievable.

Where can I find free NCERT Solutions for Class 9 Maths?

Free NCERT Solutions for Class 9 Maths are widely available on various educational websites like Tiwari Academy, etc. These platforms provide chapter-wise solutions in a clear, step-by-step format. Additionally, you can download the official Tiwari Academy app, which offers solutions and textbooks for free. These resources not only help in solving textbook problems but also provide practice questions and mock tests. For students looking to excel in their exams, having access to these solutions ensures a thorough understanding of each topic and helps in revising important questions quickly.

How many exercises are there in the NCERT Class 9 Maths book?

The NCERT Class 9 Maths book contains a total of 15 chapters, each consisting of multiple exercises. The number of exercises varies across chapters, but on average, each chapter has around 3-4 exercises, amounting to over 80 exercises in the entire book. Some chapters like Triangles and Quadrilaterals have more exercises due to the complexity of the topics. These exercises cover various types of questions, from easy to difficult, ensuring students develop a comprehensive understanding of each topic. By solving all these exercises, students can build a strong foundation and effectively prepare for their exams.

How can I get more than 80% marks in Class 9 Maths Exams?

Most of the chapters in Class 9 Maths are easy to understand. If we try to learn by doing, the mathematics become simple and interesting subject otherwise it would be a burden for us. Get concentrated on your NCERT Textbook 2024-25 and start practicing with understanding to score well with less efforts. How to practice questions: Read the preface of the chapter and see the pattern of each example given before the exercises. Now taking the hints from these solved example, try to solve the questions of exercises by own. Don’t take help from teachers or online website when your are solving the questions. After trying many time if still not getting the correct answers, then consult your respective teacher or visit to Tiwari Academy to take help from PDF Solutions or videos solutions.

What is the importance of the theorems in Class 9 Maths?

Theorems in Class 9 Maths play a crucial role as they lay the foundation for understanding geometry and problem-solving techniques. Important theorems like Pythagoras’ theorem, properties of triangles and circles not only appear in exams but also form the base for higher-level Maths. Understanding the proof of these theorems helps students apply them effectively to solve complex questions. Learning these theorems thoroughly enhances logical thinking and analytical skills, which are essential not only for Maths but also for subjects like Physics. These theorems appear frequently in exams, so knowing them well can contribute to scoring full marks in related questions.

Is class 9 maths difficult?

In real Maths of class, 9th and 10th are both very easy. But practically you can say class 9th mathematics is tough for you because you introduced some new topics and topics in the advanced level study here. In previous classes like 8th, 7th and 6th, the math you study is very basic and somehow simple. So your mindset is for that simple Maths method with very fewer logics and formulas. Some student class 9 get confusion with the syllabus.

What is the best way to practice Class 9 Maths?

The best way to practice Class 9 Maths is through consistent problem-solving from the NCERT textbook and additional reference materials. Start by solving all the textbook exercises and then move on to sample papers and previous years’ question papers. This helps in familiarising yourself with the exam pattern and types of questions that are commonly asked. Regular revision of key concepts and formulas ensures that you retain important information. Online platforms offering practice tests can also be beneficial for testing your speed and accuracy. Time management and regular revision are key to mastering the subject.

Is Class 9 Maths easy?

Class 9 Maths can feel challenging for some students due to the introduction of more complex concepts like algebraic expressions, geometry and coordinate geometry. However, with consistent practice and a clear understanding of concepts, it becomes easy. The difficulty level depends largely on how well students grasp the foundational topics and apply them to solve problems. Students who regularly practice the NCERT exercises and seek help for difficult areas often find that the subject becomes easier over time. Proper guidance and use of supplementary resources like NCERT Solutions can make the subject less daunting.

How can I perform best in Terminal exams of Class 9 Maths?

There are so many chapters in class 9 Maths which are easy and interesting to work with. When you are in energetic mood, start doing the chapter which you considered as difficult one but if you are tired and still want to practice mathematics, start with an easier chapter. Studying in this manner make the mathematics interesting and easier.

Are the NCERT solutions enough for scoring well in Class 9 Maths?

NCERT Solutions are a great resource and can be sufficient for scoring well in Class 9 Maths if studied thoroughly. The solutions cover all the essential concepts and types of questions that could appear in the exam. They are structured in a step-by-step manner to help students grasp the logic behind each problem. However, for students aiming for perfection, practicing additional questions from reference books or mock tests might provide the extra edge. Solving a variety of problems ensures familiarity with different question formats, boosting overall performance in exams.

How can I improve my problem-solving skills in Class 9 Maths?

Improving problem-solving skills in Class 9 Maths requires regular practice and a strategic approach to tackling different types of questions. Start by understanding the concepts behind each topic and then apply them to solve NCERT textbook problems. After mastering the basics, move on to solving sample papers and taking timed mock tests to improve speed and accuracy. Analysing mistakes and reviewing difficult problems is equally important for improvement. Revising key formulas and practicing mental Maths can also enhance problem-solving efficiency. Engaging in group discussions and seeking help from teachers or peers can further improve understanding and performance.

Can I download Class 9 Maths solutions for offline study?

Yes, you can easily download Class 9 Maths solutions for offline study from various websites like Tiwari Academy and other similar websites or the official NCERT website. These platforms offer downloadable PDFs of chapter-wise solutions, which can be saved for offline access. Tiwari Academy Offline mobile apps also provide NCERT solutions, allowing you to study without the need for an internet connection. Offline access to solutions is helpful for revising anytime and anywhere, making it convenient for students who may not always have internet access. Having these solutions on hand ensures you can practice regularly and clarify doubts as they arise.

Shikhar Tiwari

Having graduated from Electronics and Communication Engineering from AKTU – Noida, India, in 2021, working for Tiwari Academy as a content writer and reviewer. My main focus is to provide an easy to understand methods in all subjects specially mathematics and making study material with step by step explanation.

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• Chapter 10: Circles

NCERT Solutions for Class 9 Maths Chapter 10 Circles

* According to the CBSE Syllabus 2023-24, this chapter has been renumbered as Chapter 9.

NCERT Solutions for Class 9 Maths Chapter 10 Circles are provided here in PDF format, which can be downloaded for free. The NCERT Solutions for the chapter Circles are included as per the latest update of the CBSE curriculum (2023-24) and have been designed by our expert teachers.

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Download most important questions for class 9 maths chapter – 10 circles.

All the solved questions of Chapter 10 Circles are with respect to the CBSE syllabus and guidelines to help students solve each exercise question present in the book and prepare for the exam. These serve as reference tools for the students to do homework and also support them in scoring good marks. Students can also get the solutions for Class 9th Maths all chapters exercise-wise and practise well for the exams.

• 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 Introduction to Probability

NCERT Solutions for Class 9 Maths Chapter 10 – Circles

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List of Exercises in Class 9 Maths Chapter 10 Exercise 10.1 Solutions 2 Questions (2 Short) Exercise 10.2 Solutions 2 Questions (2 long) Exercise 10.3 Solutions 3 Questions (3 long) Exercise 10.4 Solutions 6 Questions (6 long) Exercise 10.5 Solutions 12 Questions (12 long) Exercise 10.6 Solutions 10 Questions (10 long)

Access Answers of Maths NCERT Class 9 Chapter 10 Circles

Exercise: 10.1 (Page No: 171)

1. Fill in the blanks.

(i) The centre of a circle lies in ____________ of the circle. (exterior/ interior)

(ii) A point whose distance from the centre of a circle is greater than its radius lies in __________ of the circle. (exterior/ interior)

(iii) The longest chord of a circle is a _____________ of the circle.

(iv) An arc is a ___________ when its ends are the ends of a diameter.

(v) Segment of a circle is the region between an arc and _____________ of the circle.

(vi) A circle divides the plane, on which it lies, in _____________ parts.

(i) The centre of a circle lies in interior of the circle.

(ii) A point, whose distance from the centre of a circle is greater than its radius lies in exterior of the circle.

(iii) The longest chord of a circle is a diameter of the circle.

(iv) An arc is a semicircle when its ends are the ends of a diameter.

(v) Segment of a circle is the region between an arc and chord of the circle.

(vi) A circle divides the plane, on which it lies, in 3 (three) parts.

2. Write True or False. Give reasons for your solutions.

(i) Line segment joining the centre to any point on the circle is a radius of the circle.

(ii) A circle has only a finite number of equal chords.

(iii) If a circle is divided into three equal arcs, each is a major arc.

(iv) A chord of a circle, which is twice as long as its radius, is the diameter of the circle.

(v) Sector is the region between the chord and its corresponding arc.

(vi) A circle is a plane figure.

(i) True. Any line segment drawn from the centre of the circle to any point on it is the radius of the circle and will be of equal length.

(ii) False. There can be infinite numbers of equal chords in a circle.

(iii) False. For unequal arcs, there can be major and minor arcs. So, equal arcs on a circle cannot be said to be major arcs or minor arcs.

(iv) True. Any chord whose length is twice as long as the radius of the circle always passes through the centre of the circle, and thus, it is known as the diameter of the circle.

(v) False. A sector is a region of a circle between the arc and the two radii of the circle.

(vi) True. A circle is a 2d figure, and it can be drawn on a plane.

Exercise: 10.2 (Page No: 173)

1. Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.

To recall, a circle is a collection of points whose every point is equidistant from its centre. So, two circles can be congruent only when the distance of every point of both circles is equal from the centre.

For the second part of the question, it is given that AB = CD, i.e., two equal chords.

Now, it is to be proven that angle AOB is equal to angle COD.

Consider the triangles ΔAOB and ΔCOD.

OA = OC and OB = OD (Since they are the radii of the circle.)

AB = CD (As given in the question.)

So, by SSS congruency, ΔAOB ≅  ΔCOD

∴ By CPCT, we have,

∠ AOB = ∠ COD (Hence, proved).

2. Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

Consider the following diagram.

Here, it is given that ∠ AOB = ∠ COD, i.e., they are equal angles.

Now, we will have to prove that the line segments AB and CD are equal, i.e., AB = CD.

In triangles AOB and COD,

∠ AOB = ∠ COD (As given in the question.)

OA = OC and OB = OD (These are the radii of the circle.)

So, by SAS congruency, ΔAOB ≅  ΔCOD

∴ By the rule of CPCT, we have,

AB = CD (Hence, proved.)

Exercise: 10.3 (Page No: 176)

1. Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?

In these two circles, no point is common.

Here, only one point, ‘P’, is common.

Even here, P is the common point.

Here, two points are common, which are P and Q.

No point is common in the above circle.

2. Suppose you are given a circle. Give a construction to find its centre.

The construction steps to find the centre of the circle is:

Step I: Draw a circle first.

Step II: Draw 2 chords, AB and CD, in the circle.

Step III: Draw the perpendicular bisectors of AB and CD.

Step IV: Connect the two perpendicular bisectors at a point. This intersection point of the two perpendicular bisectors is the centre of the circle.

3. If two circles intersect at two points, prove that their centres lie on the perpendicular bisector of the common chord.

It is given that two circles intersect each other at P and Q.

OO’ is perpendicular bisector of PQ.

(i) PR = RQ

(ii) ∠PRO = ∠PRO’ = ∠QRO = ∠QRO’ = 90 0

In triangles ΔPOO’ and ΔQOO’,

OP = OQ and O’P = O’Q (Since they are also the radii.)

OO’ = OO’ (It is the common side.)

So, it can be said that ΔPOO’ ≅  ΔQOO’ (SSS Congruence rule)

∴ ∠ POO’ = ∠ QOO’ (c.p.c.t)— (i)

Even triangles ΔPOR and ΔQOR are similar by SAS congruency.

OP = OQ (Radii)

∠ POR = ∠ QOR (As ∠ POO’ = ∠ QOO’)

OR = OR (Common arm)

So, ΔOPO’ ≅  ΔOQO’ (SAS Congruence rule)

∴ PR = QR and ∠ PRO = ∠ QRO (c.p.c.t) …. (ii)

As PQ is a line

∠ PRO + ∠ QRO = 180°

∠ PRO + ∠ PRO = 180° (Using (ii))

2 ∠ PRO = 180°

∠ PRO = 90°

So ∠ QRO = ∠ PRO = 90°

∠ PRO’ = ∠QRO = 90° and ∠QRO’ = ∠PRO = 90° (Vertically opposite angles)

∠ PRO = ∠ QRO = ∠ PRO’ = ∠QRO’ = 90°

So, OO’ is the perpendicular bisector of PQ.

Exercise: 10.4 (Page No: 179)

1. Two circles of radii 5 cm and 3 cm intersect at two points, and the distance between their centres is 4 cm. Find the length of the common chord.

The perpendicular bisector of the common chord passes through the centres of both circles.

As the circles intersect at two points, we can construct the above figure.

Consider AB as the common chord and O and O’ as the centres of the circles.

As the radius of the bigger circle is more than the distance between the two centres, we know that the centre of the smaller circle lies inside the bigger circle.

The perpendicular bisector of AB is OO’.

OA = OB = 3 cm

As O is the midpoint of AB

AB = 3 cm + 3 cm = 6 cm

The length of the common chord is 6 cm.

It is clear that the common chord is the diameter of the smaller circle.

2. If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.

Let AB and CD be two equal cords (i.e., AB = CD). In the above question, it is given that AB and CD intersect at a point, say, E.

It is now to be proven that the line segments AE = DE and CE = BE

Construction Steps

Step 1: From the centre of the circle, draw a perpendicular to AB, i.e., OM ⊥ AB.

Step 2: Similarly, draw ON ⊥ CD.

Step 3: Join OE.

Now, the diagram is as follows:

From the diagram, it is seen that OM bisects AB, and so OM ⊥ AB

Similarly, ON bisects CD, and so ON ⊥ CD.

It is known that AB = CD. So,

AM = ND — (i)

and MB = CN — (ii)

Now, triangles ΔOME and ΔONE are similar by RHS congruency, since

∠ OME = ∠ ONE (They are perpendiculars.)

OE = OE (It is the common side.)

OM = ON (AB and CD are equal, and so they are equidistant from the centre.)

∴ ΔOME ≅  ΔONE

ME = EN (by CPCT) — (iii)

Now, from equations (i) and (ii), we get

AM+ME = ND+EN

So, AE = ED

Now from equations (ii) and (iii), we get

MB-ME = CN-EN

So, EB = CE (Hence, proved)

3. If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords.

From the question, we know the following:

(i) AB and CD are 2 chords which are intersecting at point E.

(ii) PQ is the diameter of the circle.

(iii) AB = CD.

Now, we will have to prove that ∠ BEQ = ∠ CEQ

For this, the following construction has to be done.

Construction:

Draw two perpendiculars are drawn as OM ⊥ AB and ON ⊥ D. Now, join OE. The constructed diagram will look as follows:

Now, consider the triangles ΔOEM and ΔOEN.

So, by RHS congruency criterion, ΔOEM ≅ ΔOEN.

Hence, by the CPCT rule, ∠ MEO = ∠ NEO

∴ ∠ BEQ = ∠ CEQ (Hence, proved)

4. If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that AB = CD (see Fig. 10.25).

The given image is as follows:

First, draw a line segment from O to AD, such that OM ⊥ AD.

So, now OM is bisecting AD since OM ⊥ AD.

Therefore, AM = MD — (i)

Also, since OM ⊥ BC, OM bisects BC.

Therefore, BM = MC — (ii)

From equation (i) and equation (ii),

AM-BM = MD-MC

5. Three girls, Reshma, Salma and Mandip, are playing a game by standing on a circle of radius 5m drawn in a park. Reshma throws a ball to Salma, Salma to Mandip, and Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is 6m each, what is the distance between Reshma and Mandip?

Let the positions of Reshma, Salma and Mandip be represented as A, B and C, respectively.

From the question, we know that AB = BC = 6cm

So, the radius of the circle, i.e., OA = 5cm

Now, draw a perpendicular BM ⊥ AC.

Since AB = BC, ABC can be considered an isosceles triangle. M is the mid-point of AC. BM is the perpendicular bisector of AC, and thus it passes through the centre of the circle.

let AM = y and

So, BM will be = (5-x).

By applying the Pythagorean theorem in ΔOAM, we get

OA 2 = OM 2 +AM 2

⇒ 5 2 = x 2 +y 2 — (i)

Again, by applying the Pythagorean theorem in ΔAMB,

AB 2 = BM 2 +AM 2

⇒ 6 2 = (5-x) 2 +y 2 — (ii)

Subtracting equation (i) from equation (ii), we get

36-25 = (5-x) 2 +y 2  -x 2 -y 2

Now, solving this equation, we get the value of x as

Substituting the value of x in equation (i), we get

y 2 +(49/25) = 25

⇒ y 2 = 25 – (49/25)

Solving it, we get the value of y as

= 2×(24/5) m

So, the distance between Reshma and Mandip is 9.6 m.

6. A circular park of radius 20m is situated in a colony. Three boys, Ankur, Syed and David, are sitting at equal distances on its boundary, each having a toy telephone in his hands to talk to each other. Find the length of the string of each phone.

First, draw a diagram according to the given statements. The diagram will look as follows:

Here, the positions of Ankur, Syed and David are represented as A, B and C, respectively. Since they are sitting at equal distances, the triangle ABC will form an equilateral triangle.

AD ⊥ BC is drawn. Now, AD is the median of ΔABC, and it passes through the centre O.

Also, O is the centroid of the ΔABC. OA is the radius of the triangle.

OA = 2/3 AD

Let the side of a triangle a metres, then BD = a/2 m.

Applying Pythagoras’ theorem in ΔABD,

AB 2 = BD 2 +AD 2

⇒ AD 2 = AB 2 -BD 2

⇒ AD 2 = a 2 -(a/2) 2

⇒ AD 2 = 3a 2 /4

⇒ AD = √3a/2

20 m = 2/3 × √3a/2

So, the length of the string of the toy is 20√3 m.

Exercise: 10.5 (Page No: 184)

1. In Fig. 10.36, A, B and C are three points on a circle with centre O, such that ∠ BOC = 30° and ∠ AOB = 60°. If D is a point on the circle other than the arc ABC, find ∠ ADC.

It is given that,

∠ AOC = ∠ AOB+ ∠ BOC

So, ∠ AOC = 60°+30°

∴ ∠ AOC = 90°

It is known that an angle which is subtended by an arc at the centre of the circle is double the angle subtended by that arc at any point on the remaining part of the circle.

∠ ADC = (½) ∠ AOC

= (½)× 90° = 45°

2. A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

Here, the chord AB is equal to the radius of the circle. In the above diagram, OA and OB are the two radii of the circle.

Now, consider the ΔOAB. Here,

AB = OA = OB = radius of the circle

So, it can be said that ΔOAB has all equal sides, and thus, it is an equilateral triangle.

∴ ∠ AOC = 60°

And, ∠ ACB = ½ ∠ AOB

So, ∠ ACB = ½ × 60° = 30°

Now, since ACBD is a cyclic quadrilateral,

∠ ADB + ∠ ACB = 180° (They are the opposite angles of a cyclic quadrilateral)

So, ∠ ADB = 180°-30° = 150°

So, the angle subtended by the chord at a point on the minor arc and also at a point on the major arc is 150° and 30°, respectively.

3. In Fig. 10.37, ∠ PQR = 100°, where P, Q and R are points on a circle with centre O. Find ∠ OPR.

Since the angle which is subtended by an arc at the centre of the circle is double the angle subtended by that arc at any point on the remaining part of the circle.

So, the reflex ∠ POR = 2× ∠ PQR

We know the values of angle PQR as 100°.

So, ∠ POR = 2×100° = 200°

∴ ∠ POR = 360°-200° = 160°

Now, in ΔOPR,

OP and OR are the radii of the circle.

So, OP = OR

Also, ∠ OPR = ∠ ORP

Now, we know the sum of the angles in a triangle is equal to 180 degrees.

∠ POR+ ∠ OPR+ ∠ ORP = 180°

∠ OPR+ ∠ OPR = 180°-160°

As ∠ OPR = ∠ ORP

2 ∠ OPR = 20°

Thus, ∠ OPR = 10°

4. In Fig. 10.38, ∠ ABC = 69°, ∠ ACB = 31°, find ∠ BDC.

We know that angles in the segment of the circle are equal, so,

∠BAC = ∠BDC

Now. in the ΔABC, the sum of all the interior angles will be 180°.

So, ∠ABC+∠BAC+∠ACB = 180°

Now, by putting the values,

∠BAC = 180°-69°-31°

So, ∠BAC = 80°

∴ ∠BDC = 80°

5. In Fig. 10.39, A, B, C and D are four points on a circle. AC and BD intersect at a point E, such that ∠ BEC = 130° and ∠ ECD = 20°. Find BAC.

We know that the angles in the segment of the circle are equal.

∠ BAC = ∠ CDE

Now, by using the exterior angles property of the triangle,

In ΔCDE, we get

∠ CEB = ∠ CDE+∠ DCE

We know that ∠ DCE is equal to 20°.

So, ∠ CDE = 110°

∠ BAC and ∠ CDE are equal

∴ ∠ BAC = 110°

6. ABCD is a cyclic quadrilateral whose diagonals intersect at point E. If ∠ DBC = 70°, ∠ BAC is 30°, find ∠ BCD. Further, if AB = BC, find ∠ ECD.

Consider the chord CD.

We know that angles in the same segment are equal.

So, ∠ CBD = ∠ CAD

∴ ∠ CAD = 70°

Now, ∠ BAD will be equal to the sum of angles BAC and CAD.

So, ∠ BAD = ∠ BAC+∠ CAD

∴ ∠ BAD = 100°

We know that the opposite angles of a cyclic quadrilateral sum up to 180 degrees.

∠ BCD+∠ BAD = 180°

It is known that ∠ BAD = 100°

So, ∠ BCD = 80°

Now, consider the ΔABC.

Here, it is given that AB = BC

Also, ∠ BCA = ∠ CAB (They are the angles opposite to equal sides of a triangle)

∠ BCA = 30°

also, ∠ BCD = 80°

∠ BCA +∠ ACD = 80°

Thus, ∠ ACD = 50° and ∠ ECD = 50°

7. If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

Draw a cyclic quadrilateral ABCD inside a circle with centre O, such that its diagonal AC and BD are two diameters of the circle.

We know that the angles in the semi-circle are equal.

So, ∠ ABC = ∠ BCD = ∠ CDA = ∠ DAB = 90°

So, as each internal angle is 90°, it can be said that the quadrilateral ABCD is a rectangle.

8. If the non-parallel sides of a trapezium are equal, prove that it is cyclic.

9. Two circles intersect at two points, B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q, respectively (see Fig. 10.40). Prove that ∠ ACP = ∠ QCD.

Join the chords AP and DQ.

For chord AP, we know that angles in the same segment are equal.

So, ∠ PBA = ∠ ACP — (i)

Similarly, for chord DQ,

∠ DBQ = ∠ QCD — (ii)

It is known that ABD and PBQ are two line segments which are intersecting at B.

At B, the vertically opposite angles will be equal.

∴ ∠ PBA = ∠ DBQ — (iii)

From equation (i), equation (ii) and equation (iii), we get

∠ ACP = ∠ QCD

10. If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lies on the third side.

First, draw a triangle ABC and then two circles having diameters of AB and AC, respectively.

We will have to now prove that D lies on BC and BDC is a straight line.

We know that angles in the semi-circle are equal.

So, ∠ ADB = ∠ ADC = 90°

Hence, ∠ ADB+∠ ADC = 180°

∴ ∠ BDC is a straight line.

So, it can be said that D lies on the line BC.

11. ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠ CAD = ∠CBD.

We know that AC is the common hypotenuse and ∠ B = ∠ D = 90°.

Now, it has to be proven that ∠ CAD = ∠ CBD

Since ∠ ABC and ∠ ADC are 90°, it can be said that they lie in a semi-circle.

So, triangles ABC and ADC are in the semi-circle, and the points A, B, C and D are concyclic.

Hence, CD is the chord of the circle with centre O.

We know that the angles which are in the same segment of the circle are equal.

∴ ∠ CAD = ∠ CBD

12. Prove that a cyclic parallelogram is a rectangle.

It is given that ABCD is a cyclic parallelogram, and we will have to prove that ABCD is a rectangle.

Thus, ABCD is a rectangle.

Exercise: 10.6 (Page No: 186)

1.  Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection.

In ΔPOO’ and ΔQOO’

OP = OQ          (Radius of circle 1)

O’P = O’Q        (Radius of circle 2)

OO’ = OO’        (Common arm)

So, by SSS congruency, ΔPOO’ ≅ ΔQOO’

Thus, ∠OPO’ = ∠OQO’ (proved).

2. Two chords AB and CD of lengths 5 cm and 11 cm, respectively, of a circle, are parallel to each other and are on opposite sides of its centre. If the distance between AB and CD is 6, find the radius of the circle.

Here, OM ⊥ AB and ON ⊥ CD are drawn, and OB and OD are joined.

We know that AB bisects BM as the perpendicular from the centre bisects the chord.

Since AB = 5 so,

BM = AB/2 = 5/2

Similarly, ND = CD/2 = 11/2

Now, let ON be x.

So, OM = 6−x.

Consider ΔMOB,

OB 2 = OM 2 +MB 2

Consider ΔNOD,

OD 2 = ON 2 + ND 2

We know, OB = OD (radii)

From equation 1 and equation 2, we get

Now, from equation (2), we have,

OD 2 = 1 2 +(121/4)

Or OD = (5/2)×√5 cm

3. The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at a distance 4 cm from the centre, what is the distance of the other chord from the centre?

Here, AB and CD are 2 parallel chords. Now, join OB and OD.

Distance of smaller chord AB from the centre of the circle = 4 cm

So, OM = 4 cm

MB = AB/2 = 3 cm

Consider ΔOMB.

Or, OB = 5cm

Now, consider ΔOND.

OB = OD = 5 (Since they are the radii.)

ND = CD/2 = 4 cm

Now, OD 2 = ON 2 +ND 2

Or, ON = 3 cm

4. Let the vertex of an angle ABC be located outside a circle, and let the sides of the angle intersect equal chords AD and CE with the circle. Prove that ∠ABC is equal to half the difference of the angles subtended by the chords AC and DE at the centre.

Consider the diagram.

Here AD = CE

We know any exterior angle of a triangle is equal to the sum of interior opposite angles.

∠DAE = ∠ABC+∠AEC (in ΔBAE) ——————-(i)

DE subtends ∠DOE at the centre and ∠DAE in the remaining part of the circle.

∠DAE = (½)∠DOE ——————-(ii)

Similarly, ∠AEC = (½)∠AOC  ——————-(iii)

Now, from equations (i), (ii), and (iii), we get

(½)∠DOE = ∠ABC+(½)∠AOC

Or, ∠ABC = (½)[∠DOE-∠AOC]  (Hence, proved)

5. Prove that the circle drawn with any side of a rhombus as diameter passes through the point of intersection of its diagonals.

To prove: A circle drawn with Q as the centre will pass through A, B and O (i.e., QA = QB = QO).

Since all sides of a rhombus are equal,

Now, multiply (½) on both sides.

(½)AB = (½)DC

So, AQ = DP

Since Q is the midpoint of AB,

Again, as PQ is drawn parallel to AD,

Now, as AQ = BQ and RA = QO, we get

QA = QB = QO (Hence, proved)

6. ABCD is a parallelogram. The circle through A, B and C intersect CD (produced if necessary) at E. Prove that AE = AD.

Here, ABCE is a cyclic quadrilateral. In a cyclic quadrilateral, the sum of the opposite angles is 180°.

So, ∠AEC+∠CBA = 180°

As ∠AEC and ∠AED are linear pairs,

∠AEC+∠AED = 180°

Or, ∠AED = ∠CBA … (1)

We know in a parallelogram, opposite angles are equal.

So, ∠ADE = ∠CBA … (2)

Now, from equations (1) and (2), we get

∠AED = ∠ADE

Now, AD and AE are angles opposite to equal sides of a triangle.

∴ AD = AE (proved)

7. AC and BD are chords of a circle which bisect each other. Prove that (i) AC and BD are diameters; (ii) ABCD is a rectangle.

Here, chords AB and CD intersect each other at O.

Consider ΔAOB and ΔCOD.

∠AOB = ∠COD (They are vertically opposite angles.)

OB = OD (Given in the question.)

OA = OC (Given in the question.)

So, by SAS congruency, ΔAOB ≅ ΔCOD

Also, AB = CD (By CPCT)

Similarly, ΔAOD ≅ ΔCOB

Or, AD = CB (By CPCT)

In quadrilateral ACBD, opposite sides are equal.

So, ACBD is a parallelogram.

We know that opposite angles of a parallelogram are equal.

So, ∠A = ∠C

Also, as ABCD is a cyclic quadrilateral,

∠A+∠C = 180°

⇒∠A+∠A = 180°

Or, ∠A = 90°

As ACBD is a parallelogram and one of its interior angles is 90°, so, it is a rectangle.

∠A is the angle subtended by chord BD. And as ∠A = 90°, therefore, BD should be the diameter of the circle. Similarly, AC is the diameter of the circle.

8. Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F, respectively. Prove that the angles of the triangle DEF are 90°–(½)A, 90°–(½)B and 90°–(½)C.

Here, ABC is inscribed in a circle with centre O, and the bisectors of ∠A, ∠B and ∠C intersect the circumcircle at D, E and F, respectively.

Now, join DE, EF and FD.

As angles in the same segment are equal, so,

∠EDA = ∠FCA ————-(i)

∠FDA = ∠EBA ————-(i)

By adding equations (i) and (ii), we get

∠FDA+∠EDA = ∠FCA+∠EBA

Or, ∠FDE = ∠FCA+∠EBA = (½)∠C+(½)∠B

We know, ∠A +∠B+∠C = 180°

In a similar way,

∠FED = [90° -(∠B/2)] °

∠EFD = [90° -(∠C/2)] °

9. Two congruent circles intersect each other at points A and B. Through A, any line segment PAQ is drawn so that P, Q lie on the two circles. Prove that BP = BQ.

The diagram will be

Here, ∠APB = ∠AQB (as AB is the common chord in both the congruent circles.)

Now, consider ΔBPQ.

∠APB = ∠AQB

So, the angles are opposite to equal sides of a triangle.

10. In any triangle ABC, if the angle bisector of ∠A and perpendicular bisector of BC intersect, prove that they intersect on the circumcircle of the triangle ABC.

Consider this diagram.

Here, join BE and CE.

Now, since AE is the bisector of ∠BAC,

∠BAE = ∠CAE

∴arc BE = arc EC

This implies chord BE = chord EC

Now, consider triangles ΔBDE and ΔCDE.

DE = DE     (It is the common side)

BD = CD     (It is given in the question)

BE = CE      (Already proved)

So, by SSS congruency, ΔBDE ≅ ΔCDE.

Thus, ∴∠BDE = ∠CDE

We know, ∠BDE = ∠CDE = 180°

Or, ∠BDE = ∠CDE = 90°

∴ DE ⊥ BC (Hence, proved).

Chapter 10, Circles, of Grade 9, is one of the most important chapters, whose concepts will also be used in Class 10. The weightage of this chapter in the final exam is around 15 marks. Therefore, students are advised to read the chapter carefully and practise each and every question included in the textbook with the help of NCERT Solutions ,  along with examples, to have good practice.

Topics covered in Chapter 10 Circles, are

• Circles and the related terms
• Angle Subtended by a Chord at a Point
• Perpendicular from the Centre to a Chord
• Circle through Three Points
• Equal Chords and Their Distances from the Centre
• Angle Subtended by an Arc of a Circle
• Cyclic Quadrilaterals

NCERT Solutions for Class 9 Maths Chapter 10 – Circles are made available for students looking to solve all the problems of Ex-10.1. The methods by which problems have been solved, in a broad way, so that students find it easy to understand the fundamentals of circles. Some of the important points of this chapter are

• A circle is a simple closed geometrical shape equidistant from a central point. It is an important shape in the field of geometry.
• Every circle has its centre.
• The straight line from the centre to the circumference of a circle is called the radius of the circle.
• The length of the line through the centre that touches two points on the edge of the circle is called a diameter.
• The total distance around the circle is called the Circumference.
• The area of the circle can be calculated by applying the formula: A = π r 2 , where A is the Area, r is the radius, and the value of π is 3.14.

Key Features of NCERT Solutions for Class 9 Maths Chapter 10 – Circles

• The solutions for the chapter Circles work as a reference for the students.
• Students will be able to resolve all the problems of this chapter in a faster way.
• It is good learning material for exam preparation and to do the revision for Class 9 Maths Chapter 10.
• The questions of Circles are solved by our subject experts.
• The NCERT Solutions are given as per the latest update on the CBSE syllabus and guidelines.

Disclaimer:

Dropped Topics –  10.1 Introduction, 10.2 Circles and its related terms: Review and Circle through three points.

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NCERT Solutions For Class 9 Maths

NCERT Solutions for Class 9 Maths All Chapters includes solutions to all the questions given in the NCERT textbook for Class 9th. You can download PDF of each chapter for free through our app. All these solutions are designed and prepared by our subject experts. These solutions are updated as per the latest NCERT books and CBSE guidelines.

NCERT Solutions For Class 9 Maths All Chapters

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 for cbse class 9 maths chapter-wise discussion.

To ensure that it is easily understood by the students, our teachers designed NCERT Class 9 Maths solutions in a well-structured manner. Regular practice of these solutions improves your marks in the exam. Download today the NCERT Class 9 Maths solutions as free PDF through our app to enhance your marks.

In previous classes, you have learned about number lines. This chapter extends the topic and explains the number systems in detail. The chapter has a total of six exercises that explains topics like real numbers, decimal numbers. Along with this, you will also know about the laws of exponents and integral power.

• Exercise 1.1
• Exercise 1.2
• Exercise 1.3
• Exercise 1.4
• Exercise 1.5
• Exercise 1.6

This chapter helps you to understand the algebraic expressions called polynomials. It has five exercises, and each exercise has plenty of problems to solve. Topics like zeroes of a polynomial, remainder theorem, factorisation of polynomials, and algebraic identities are explained in this chapter.

• Exercise 2.1
• Exercise 2.2
• Exercise 2.3
• Exercise 2.4
• Exercise 2.5

Chapter 3 has a total of exercises on it. In this chapter, you will learn about the cartesian plane, coordinate, and various terms associated with the cartesian system. Further, you will learn how to plot coordinates in the XY plane.

• Exercise 3.1
• Exercise 3.2
• Exercise 3.3

Linear equation in two variables is a new topic introduced in the NCERT class 9 maths textbook. With a total of four exercises, the chapter includes a lot of problems related to linear equations. In this chapter, you will learn about solutions and graphs of linear equations, etc.

• Exercise 4.1
• Exercise 4.2
• Exercise 4.3
• Exercise 4.4

The chapter called Euclid’s geometry starts with an introduction that includes short information about our Indian geometry. The chapter has two exercises on it. In this chapter, you will know about the relationship between axiom, postulates, and theorems.

• Exercise 5.1
• Exercise 5.2

As the name suggests, the chapter deals with lines and angles. This chapter explains the intersecting and non-intersecting lines, pairs of angles, parallel lines, angle sum property of a triangle, etc. The chapter contains three exercises that include a lot of problems related to the topics mentioned above.

• Exercise 6.1
• Exercise 6.2
• Exercise 6.3

You have already read about triangles in your previous classes. In this chapter, you will learn about the congruence of triangles, criteria for congruence of triangles and some properties of a triangle. In the end, the chapter explains the inequalities in a triangle.

• Exercise 7.1
• Exercise 7.2
• Exercise 7.3
• Exercise 7.4
• Exercise 7.5

A closed figure obtained by joining four sides is known as a quadrilateral. This chapter contains only two exercises. However, there are nine theorems and many problems related to quadrilaterals that you need to solve.

• Exercise 8.1
• Exercise 8.2

As the name suggests, this chapter explains the areas related to parallelogram and triangles and their combinations. Further, the chapter includes a few theorems along with their proofs that are essential for solving the exercise problems.

• Exercise 9.1
• Exercise 9.2
• Exercise 9.3
• Exercise 9.4

In this chapter, you will learn about some important topics like angle subtended by a chord at a point, circle through three points, Equal Chords and their Distances from the Centre, Angle Subtended by an Arc of a Circle and Cyclic Quadrilaterals.

• Exercise 10.1
• Exercise 10.2
• Exercise 10.3
• Exercise 10.4
• Exercise 10.5
• Exercise 10.6

In this chapter, you will learn how to construct basic figures using a compass and ruler. The chapter has only two exercises. The first exercise deals with the construction of a certain angle and the bisector of a given angle while the second exercise deals with the constructions of triangles for given parameters.

• Exercise 11.1
• Exercise 11.2

There are two exercises in this chapter. In the first exercise, you will learn how to calculate the area of triangles by Heron’s formula while in the second you will learn applications Heron’s formula in finding the areas of quadrilaterals.

• Exercise 12.1
• Exercise 12.2

This chapter is the lengthiest chapter of your NCERT class 9 maths textbook. There are a total of nine exercises in this chapter. Here you will learn how to calculate the areas and volumes of a cube, cuboid, right circular cone, right circular cylinder, sphere and their combinations.

• Exercise 13.1
• Exercise 13.2
• Exercise 13.3
• Exercise 13.4
• Exercise 13.5
• Exercise 13.6
• Exercise 13.7
• Exercise 13.8
• Exercise 13.9

Statistics is defined as the collection of data on different aspects of the life of people. In this chapter, you will learn how to present and interpret a group of data. Further, you will learn how to represent a collection of data using different graphs such as Bar graphs, Histograms, Frequency polygons etc.

• Exercise 14.1
• Exercise 14.2
• Exercise 14.3
• Exercise 14.4

The chances of occurrence of an event are known as probability. The value of probability lies between 0 and 1. In this chapter, you will learn to calculate the chance of occurrence of a particular outcome in an experiment. This chapter has only one exercise, and the problems in this exercise are related to real-life situations.

Here in this article, we have discussed each chapter of NCERT class 9 maths textbook and provided their solutions. If you find any error in our solutions, please let us know about it by commenting below. Also, share your experience with our site and suggest how can we improve our contents in the comment section. We always love to hear from you.

Frequently Asked Questions on NCERT Solution for Class 9 Maths

How many chapters are present in ncert class 9 maths.

A total of 15 chapters are present in the NCERT Class 9 Maths. 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 are the chapters in Class 9 of NCERT Maths.

Is it important to practice NCERT Maths?

NCERT solutions provide answers to each and every textbook question, which is a good way to get acquainted with the chapter. Since Maths is an application-based subject, it cannot be memorised. Therefore, it is essential that you practise Maths every day, especially the difficult chapters.

Will CBSE Class 9 Maths help me in other exams?

The topics and concepts that you will study now will come in handy in other competitive exams such as JEE, IAS and NDA.

Key advantages of NCERT Solutions for Class 9 Maths

• NCERT Solutions for Class 9 Maths helps you solve and revise the whole syllabus of class 9.
• After going through the step-wise solutions given by our subject expert teachers, you will be able to score more marks.
• It follows NCERT guidelines.
• It contains all the important questions from the examination point of view.
• It helps in scoring well in Maths in board exams.

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NCERT Solutions For class 9 Maths

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NCERT Solutions for Class 9 Maths - Free PDF Download

Class 9 Maths is an important subject to master for students. In this class, they are introduced to concepts of Triangles, Polynomials, Number Systems, and much more. Also, the exercise questions are challenging, and thus we suggest the students to take guidance from reliable sources to prepare for the Class 9 Maths exam. In order to complete the vast syllabus in an easy and efficient way, Vedantu experts have compiled NCERT Class 9 Math Solution for students. These study materials act as guides that can help them progress in class. Students should download class 9th maths NCERT solutions PDF, for their preparation.

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 2024-25 academic year

Also Check for: The Concepts of NCERT Class 9 Maths

Glance on NCERT Solutions Class 9 Maths

NCERT Solutions for Class 9 Maths for all the chapters and exercises from Chapters 1 to 12 are provided.

Practising the textbook questions using these solutions can help students analyse their level of preparation and understanding of concepts.

Covering chapters like Number Systems, Algebraic Expressions, Linear Equations in Two Variables, Geometry, and Statistics.

The link also provides details about the exam pattern, marks weightage, and question paper design for CBSE Class 9 Maths.

This article also provides resources such as NCERT notes, important questions, exemplar solutions, RD Sharma, and RS Aggarwal solutions PDF for further reference.

NCERT Solutions for Class 9 Maths Chapter Details, Formulas and Exercises PDF

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.

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.

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.

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.

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

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.

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.

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.

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).

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 ax 2 + 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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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.

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.

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).

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.

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

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.

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.

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.

2. Accurate Solutions

The NCERT solutions for class 9 maths PDF by Vedantu have been created by experienced teachers. The solutions are well-reviewed so that students don’t get misguided by inaccurate study materials.

3. Learn Online and Offline

Vedantu’s NCERT Solutions for Class 9 Maths are available in PDF format online. You can download these PDFs and learn offline as well. This means you can study anytime and anywhere. It is also possible to share the links of the NCERT Maths class 9 solutions PDF download with your friends.

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

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.

FAQs on NCERT Solutions For class 9 Maths

1. How to study NCERT Solutions for Class 9 Maths in an easy way?

Vedantu offers a free PDF for NCERT Solutions for Maths Class 9. In addition to being totally free, the NCERT Maths class 9 solutions PDF download can be accessed anytime as per your convenience and from anywhere and what’s more important is that these are designed by experienced teachers and thus, are error-free and reliable.

2. How to download NCERT Solutions for Class 9 Maths PDFs?

The latest edition of  Class 9 NCERT Solutions for Maths are available at Vedantu from where they can be downloaded by clicking on the links provided. Based on the CBSE syllabus , these solutions will enhance your ability for exam preparation and the option for convenient download makes things more hassle-free. You can now prepare for Class 9 Maths as per the latest syllabus and get all the necessary study material at your fingertips.

3. Which are the important chapters in CBSE Class 9 Maths?

The Class 9 Maths important chapters include:

Polynomials

Surface Areas and Volume

Areas of Parallelograms and Triangles

Quadrilaterals

4. Which chapters and exercises are deleted from CBSE Class 9th Maths CBSE?

The chapters deleted from NCERT Class 9 CBSE Maths syllabus are Introduction to Euclid's Geometry and Areas of Parallelograms and Triangles. In addition to these, parts of various chapters are deleted as well.

5. Are these NCERT Solutions enough to score better in the Class 9 Maths Board Exam?

NCERT Solutions for Class 9 Maths help students to find answers to the textbook exercise problems and clear all their doubts regarding the questions given in the NCERT book. The solutions are provided by expert teachers who are well versed in the curriculum. By referring to these, students can learn how to write answers in the exam that would fetch more marks. Students should, however, first make a habit of understanding the chapters and then refer to the NCERT Solutions if they are having trouble in solving the questions given in the textbook. Students are also encouraged to practise questions from previous years and solve sample papers to improve their scores in the exam.

6. Why is it important to refer to NCERT Solutions for Class 9 Maths?

Studying from the NCERT Solutions for Class 9 Maths is very important as they are a reliable way for you to cross-check your answers. You can use these as a guide on how your answers should appear in an examination. It will help you ensure that you don’t miss any steps and score the best possible marks.

7. Should I make notes while going through the NCERT Solutions for Class 9 Maths?

When you are creating notes, you are creating a simple guide that will help you build a thorough understanding of the subject as well as when you want to revise it. It helps you retain and recall concepts when needed. So, if you want to expedite your learning process, you need to make notes as well.

8. What is the best way to learn the concepts that are covered in the NCERT Solutions for Class 9 Maths?

The best way for you to learn the concepts that are covered in the solutions is through regular practice. If you are facing some difficulty with a concept and are unable to proceed further, you need to clarify your doubts immediately. This will ensure that there are no gaps in your learning.

9. Can you use any other method for solving questions apart from the one provided in the NCERT Solutions for Class 9 Maths pdf?

Yes, you can use some other method for solving problems as long as it is universally recognized. You cannot use any tricks to solve exam problems. You have to use standardized techniques.

10. Why should I refer to Vedantu’s NCERT Solutions Class 9 Maths?

Vedantu’s NCERT Solutions Class 9 Maths contains answers to all the questions in the NCERT Class 9 Maths book. The solutions are present in a simple manner so that students can understand easily. These solutions are framed by subject-matter experts and the solutions are updated according to the latest curriculum of the CBSE. Hence, these are very reliable. It will help students to clear their doubts and strengthen their concepts.

11. Should I practice NCERT Solutions for Class 9 Maths regularly?

Yes, you should practice NCERT Solutions for Class 9 Maths regularly. Solving questions daily will improve your speed and accuracy. Your fundamental concepts will be cleared, and you will understand chapters of higher classes easily. If you practice daily, you will gain sufficient confidence to attempt new questions, and you can even play around with the concepts, which will enhance your analytical skills.

12. What is the best reference book for Class 9 Maths?

Vedantu’s NCERT Solutions for Class 9 Maths is the best reference book for Class 9 Maths. It has detailed and comprehensive solutions to all questions in your NCERT Maths book . All the topics are covered in this book, and it focuses on strengthening your concepts. This is prepared by top faculties and is in accordance with the latest CBSE curriculum , and is very reliable.

13. Which is the hardest chapter in Maths Class 9?

The hardest chapter in Class 9 Maths often varies for different students based on their individual strengths and weaknesses. However, many students find "Chapter 8: Quadrilaterals" and "Chapter 10: Circles" challenging due to the abstract concepts and theorems involved. Additionally, "Chapter 13: Surface Areas and Volumes" can also be tough due to the complex calculations and spatial visualization required.

14. Which chapter is most important in Class 9 Maths?

All chapters in Class 9 Maths are important as they lay the foundation for higher classes. However, some particularly crucial chapters include "Chapter 1: Number Systems," "Chapter 6: Lines and Angles," "Chapter 8: Quadrilaterals," and "Chapter 14: Statistics." These chapters introduce fundamental concepts that are extensively used in higher-level mathematics.

15. Is Class 9 Maths easy?

Class 9 Maths can be easy for some students and challenging for others. It largely depends on a student's grasp of the basics, their interest in the subject, and the amount of practice they put in. Consistent study, understanding the core concepts, and regular practice can make Class 9 Maths manageable and even enjoyable.

16. Is Class 9 Maths important?

Yes, Class 9 Maths is very important. It builds the foundation for Class 10 and higher secondary level mathematics. The concepts learned in Class 9 are crucial for understanding more advanced topics in future classes. A strong understanding of Class 9 Maths also aids in various competitive exams and entrance tests later on.

By focusing on understanding the concepts thoroughly and practising regularly, students can excel in Class 9 Maths and find it to be a rewarding subject.

NCERT SOLUTIONS FOR CLASS 9

Cbse class 9 study materials.

Class 9 Maths NCERT Solutions, Notes, Study Material

Top courses for class 9, mathematics (maths) class 9 cbse exam pattern 2024-2025.

Mathematics (Maths) Class 9 Syllabus 2024-2025 PDF Download

Detailed Syllabus for Class 9 Mathematics:

Number System:

• Real numbers and their decimal expansions
• Operations on real numbers
• Laws of exponents for real numbers
• Rational and irrational numbers

Polynomials:

• Definition of polynomials
• Degree of a polynomial
• Addition, subtraction and multiplication of polynomials
• Division of polynomials

Coordinate Geometry:

• Cartesian plane
• Coordinates of a point
• Distance formula
• Section formula
• Area of a triangle

Linear Equations in Two Variables:

• Definition of linear equations in two variables
• Methods of solving linear equations
• Graphical representation of linear equations
• Equations of lines parallel and perpendicular to the x-axis and y-axis

Introduction to Euclid's Geometry:

• Axioms and postulates
• Geometry of lines and angles
• Geometry of triangles and quadrilaterals

Lines and Angles:

• Basic concepts of lines and angles
• Types of angles
• Properties of parallel lines
• Angle sum property of a triangle
• Basic concepts of triangles
• Types of triangles
• Properties of triangles
• Construction of triangles

Quadrilaterals:

• Basic concepts of quadrilaterals
• Types of quadrilaterals
• Properties of quadrilaterals
• Construction of quadrilaterals
• Basic concepts of circles
• Chords and tangents
• Angles in a circle
• Cyclic quadrilaterals

Heron's Formula:

• Area of a triangle using Heron's formula
• Applications of Heron's formula

Surface Area and Volumes:

• Surface area and volume of a cuboid, cube, cylinder, cone and sphere
• Applications of surface area and volume

Statistics:

• Collection and presentation of data
• Mean, median and mode
• Range and quartiles

Area of Parallelograms and Triangles (Old Syllabus):

• Basic concepts of parallelograms and triangles
• Area of a parallelogram and a triangle

Construction (Old Syllabus):

• Basic constructions using ruler and compass
• Construction of triangles, quadrilaterals and circles

Probability (Old Syllabus):

• Basic concepts of probability
• Experimental probability and theoretical probability
• Addition and multiplication rules of probability

Additional Resources:

• CBSE Sample Question Papers
• RD Sharma Solutions
• RS Aggarwal Solutions
• NCERT Textbooks and Solutions
• Short and Long Question Answers
• NCERT Exemplar

This course is helpful for the following exams: Class 9 , Grade 9

How to Prepare Mathematics (Maths) Class 9?

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