NCERT Solutions For Class 9 Maths
NCERT Solutions for class 9 Maths is one of most popular study material of Physics Wallah. These solutions are prepared by senior teachers and well know mathematician of NCERT Solutions for Class 9 Maths chapter-13 Surface Areas and Volumes is prepared by academic team of Physics Wallah. We have prepared NCERT Solutions for all exercise of chapter-13. Maths is among the most crucial subjects for students in class 9.
We at Physics Wallah have come up with the most advanced NCERT solutions for class 9 maths study material. Students can find a detailed solution for their class 9 maths. NCERT Solutions of NCERT Solutions for Class 9 Maths chapter-13 Surface Areas and Volumes is prepared by academic team of Physics Wallah. We have prepared NCERT Solutions for all exercise of chapter-13 is recommended by most of the maths teacher for reference.
Since maths is very challenging to many students we have made efforts to make it simple. The very reason of negligence among students is removed through continuous efforts. The students can easily get acquainted with the subject matter through NCERT solutions for class 9 maths as it is very concise.
NCERT Solutions for Class 9 Maths Chapters (Updated 2022-23)
NCERT Solutions for Class 9 Maths Free PDF Download
Class 9th Maths give you strong foundation for upcoming mathematics class and NCERT solutions of class 9 maths will help you to solve and understand difficult numericals. It has been observed that transitioin of maths in 9th from 8th is bit challenging. Maths in class 9th is quite complex as compared to class 8th. The chapters covered in Class 9 Maths are
Chapter 1 Number System
The Pythagoreans in Greece, followers of the famous mathematician and philosopher Pythagoras, were the first to discover the numbers which were not rationals, around 400 BC. These numbers are called irrational numbers (irrationals), because they cannot be written in the form of a ratio of integers.
There are many myths surrounding the discovery of irrational numbers by the Pythagorean, Hippacus of Croton. In all the myths, Hippacus has an unfortunate end, either for discovering that root 2 is irrational or for disclosing the secret about root 2 to people outside the secret Pythagorean section.
Syllabus covered in this chapter are Review of representation of natural numbers, integers rational numbers on the number line. Representation of terminating/non-terminating recurring decimals on the number line through successive magnification. Rational numbers as recurring/ terminating decimals.
Examples of non-recurring/non-terminating decimals such as root 2,3,5 etc.Existence of non-rational numbers (irrational numbers) such as root 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, every point on the number line represents a unique real number.
Existence of root X for a given positive real number x (visual proof to be emphasized). Definition of nth root of a real number. 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).
Also, access the helpful resources for Class 9 Maths Chapter 1 Number Systems at Physics Wallah
Important Questions for class 9 maths Chapter 1 Number Systems |
RD Sharma Solutions for Class 9 Chapter 1 Number Systems |
Maths revision notes for class 9 maths Chapter 1 Number system |
Chapter 2 Polynomials
The concept of polynomial functions goes way back to perhaps Babylonians times, since for example as simple a need of computing the area of a square y = x^{2 }is a polynomial, and is needed in buildings and survey, fundamental to core civilization.
The Pythagorean theorem x^{2} + y^{2} = z^{2} is also a polynomial equation, and much basic number theory have been expressed algorithmetically in Greek or pre-Greek era.The modern concept of polynomial as a function of integer powers and their symbolic manipulation is developed in 1600s and 1700s. Finding solutions of polynomials as ready-made formulas is a spectacular chapter in the history of mathematics.
You will learn Definition of a polynomial in one variable, its coefficients, with examples and counter-examples, its terms, zero polynomial. Degree of a polynomial. Constant, linear, quadratic, cubic polynomials: monomials, binomials,trinomials. Factors and multiples. Zeros/roots of a polynomial/equation.
State and motivate the Remainder Theorem with examples and analogy to integers. Statement and proof of the Factor Theorem. Factorization of ax^{2}+bx+c not equal to zero where a, b, c are real numbers, and of cubic polynomials using the Factor Theorem.
Also, access the helpful resources for Class 9 Maths Chapter 2 Polynomials at Physics Wallah
Chapter 3 Coordinate Geometry
René Déscartes, the great French mathematician of the seventeenth century, liked to lie in bed and think! One day, when resting in bed, he solved the problem of describing the position of a point in a plane. His method was a development of the older idea of latitude and longitude. In honour of Déscartes, the system used for describing the position of a point in a plane is also known as the Cartesian system.
Sub topics covered in this chapter are The Cartesian plane, coordinates of a point, names and terms associated with the coordinate plane, notations, plotting points in the plane, graph of linear equations as examples; focus on linear equations of the type ax + by + c = 0 by writing it as y = mx + c and linking with the chapter on linear equations in two variables.
Also, access the helpful resources for Class 9 Maths Chapter 3 Coordinate Geometry at Physics Wallah
Chapter 4 Linear Equations in Two Variables
An equation is simply the mathematical way to describe a relationship between two variables. The variables may be physical quantities, perhaps temperature and position for instance, in which case the equation tells us how one quantity depends on the other, so how the temperature varies with position.
The simplest kind of relationship that two such variables can have is a linear relationship. This means that to find one quantity from the other you multiply the first by some number, then add another number to the result. Put mathematically, if we call the quantities x and y, then they are related by the equation? y = mx + c, where we can choose any values for m and c.
This is a linear equation. Fortunately, in real physical problems, quantities often are related linearly, so this equation is very commonly used.This chapter covers- Recall of linear equations in one variable. Introduction to the equation in two variables.
Prove 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 seem to lie on a line. Examples, problems from real life, including problems on Ratio and Proportion and with algebraic and graphical solutions being done simultaneously.
Also, access the helpful resources for Class 9 Maths Chapter 4 Linear Equations in Two Variables at Physics Wallah
Chapter 5 Introduction to Euclids Geometry
There are three basic terms in geometry, namely "Point", "Line" and "Plane". It is not possible to define these three terms precisely. So, these are taken as undefined terms.Axioms or postulates are the basic facts which are taken for granted without proof.
Theorems are statements which are proved through logical reasoning based on previously proved results and some axioms.
Following are the incidence axioms :
- Axiom 1 : A line contains infinitely many points.
- Axiom 2 : Through a given point pass infinitely many lines.
- Axiom 3 : Given two points A and B, there is one and only one line that contains both both the points.
- Two distinct lines cannot have more than one point in common.
- Two lines are intersecting if they have a common point. The common point is called the point of intersection.
- Two lines are parallel if they do not have a common point i.e. they do not intersect.
- If A and B are two points on a line, then the part of the line with end points at A and B is called the line segment AB. The distance between A and B is called the length of line segment AB.
Also, access the helpful resources for Class 9 Maths Chapter 5 Introduction to Euclids Geometry at Physics Wallah
Chapter 6 Lines and Angles
This chapter explains very important theorems which are used in Lines and Angles mentioned below. There are four axioms and eight theorems which are explained detail in the chapter.
- If a ray stands on a line, then the sum of the two adjacent angles is 180° and the converse.
- (Prove that if two lines intersect, the vertically opposite angles are equal.
- When a transversal intersects two parallel lines, it results in a corresponding, alternate, and interior angles.
- Lines, which are parallel to a given line, are parallel.
- Prove that the sum of the angles of a triangle is 180°.
- If a side of a triangle is formed, then the exterior angle so produced is equal to the sum of the given two interior opposite angles.
Also, access the helpful resources for Class 9 Maths Chapter 6 Lines and Angles at Physics Wallah
Chapter 7 Triangles
Similarity is some degree of symmetry in either analogy and resemblance between two or more concepts or objects. In philosophy, similarity is defined as sharing properties or characteristic traits Looking around you will see many objects which are of the same shape but of same or different sizes.
For example, photographs of different sizes developed from the same negative are of same shape but different sizes, the miniature model of a building and the building itself are of same shape but different sizes. All those objects which have the same shape but different sizes are called similar object.
Definitions, examples, counter examples of similar triangles.
- If a line is drawn parallel to one side of triangle to intersect the other two sides in distinct points the other two sides are divided in the same ratio.
- If a line divides two sides of a triangle in the same ratio, the line is parallel to the third side.
- If in two triangles, the corresponding angles are equal, their corresponding sides are proportional and the triangles are similar.
- If the corresponding sides of two triangles are proportional, their corresponding angles are equal and the two triangles are similar.
- If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the two triangles are similar.
- If a perpendicular is drawn from the vertex of the right angle of right to the whole triangle and too each other.
- The ratio of the areas of two similar triangles is equal to the ratio of the squares on their corresponding sides.
- In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
- In a triangle, if the square on one side is equal to sum of the squares on the other two sides, the angles opposite to the first side is a right triangle.
Also, access the helpful resources for Class 9 Maths Chapter 7 Triangles at Physics Wallah
Chapter 8 Quadrilaterals
A quadrilateral, sometimes also known as a tetragon or quadrangle is a four-sided polygon. If not explicitly stated, all four polygon vertices are generally taken to lie in a plane. (If the points do not lie in a plane, the quadrilateral is called a skew quadrilateral.)
There are three topological types of quadrilaterals convex quadrilaterals concave quadrilaterals and crossed quadrilaterals. A quadrilateral with two sides parallel is called a trapezoid, whereas a quadrilateral with opposite pairs of sides parallel is called a parallelogram.
A special type of quadrilateral is the cyclic quadrilateral, for which a circle can be circumscribed so that it touches each polygon vertex. Another special type is a tangential quadrilateral, for which a circle can be inscribed so it is tangent to each edge. A quadrilateral that is both cyclic and tangential is called a bicentric quadrilateral.
(Prove) The diagonal divides a parallelogram into two congruent triangles. (Motivate) In a parallelogram opposite sides are equal and conversely. (Motivate) In a parallelogram opposite angles are equal and conversely. (Motivate) A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and equal. (Motivate) In a parallelogram, the diagonals bisect each other and conversely. (Motivate) In a triangle, the line segment joining the mid points of any two sides is parallel to the third side and (motivate) its converse.
Also, access the helpful resources for Class 9 Maths Chapter 8 Quadrilaterals at Physics Wallah
Chapter 9 Areas of Parallelograms and Triangles
You cannot gain insight by the approach into the conditions, under which two triangles or parallelograms have the same area and how to convert a figure of a definite shape into a figure of another shape, that is, how you can draw figures with the same area. Since the square and the rectangle are special forms of parallelograms, you will suspect that you should start from the last.
First of all, obviously, equally oriented, congruent parallelograms have the same area, because they can be made to cover each other. An examination of two symmetric parallelograms, which have the same area, because they will cover each other on reflection, is a step in the right direction. The area of a parallelogram is obtained by sub-division into two congruent triangles. The area of a trapezoid also involves its being cut up by a diagonal into two triangles.
Review concept of area, recall area of a rectangle. Prove Parallelograms on the same base and between the same parallels have the same area. Motivate Triangles on the same base and between the same parallels are equal in area and its converse.
Also, access the helpful resources for Class 9 Maths Chapter 9 Areas of Parallelograms and Triangles at Physics Wallah
Chapter 10 Circles
The circle has been known since before the beginning of recorded history. It is the basis for the wheel which, with related inventions such as gears, makes much of modern civilization possible. In mathematics, the study of the circle has helped inspire the development of geometry and calculus.
Circles are simple closed curves which divide the plane into an interior and an exterior. The circumference of a circle is the perimeter of the circle, and the interior of the circle is called a disk. An arc is any connected part of a circle. A circle of infinite radius is considered to be a straight line. A circle with zero radius is considered as a point.
Through examples, arrive at definitions of circle related concepts, radius, circumference, diameter, chord, arc, subtended angle.
- Equal chords of a circle subtend equal angles at the center and (motivate) its converse.
- 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.
- There is one and only one circle passing through three given non-collinear points.
- Equal chords of a circle (or of congruent circles) are equidistant from the center(s) and conversely.
- 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.
- Angles in the same segment of a circle are equal.
- 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.
- The sum of the either pairs of the opposite angles of a cyclic quadrilateral is 180º and its converse.
Also, access the helpful resources for Class 9 Maths Chapter 10 Circles at Physics Wallah
Chapter 11 Constructions
Geometry (Greek geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers.Classic geometry was focused in compass and straightedge constructions.
As they are the composition of five elemental constructions over a set of elements, as an algebra over an axiomatic system, the barrier between algebra and geometry began to fade out.In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, many modern branches of the field are barely recognizable as the descendants of early geometry.
- Division of a line segment in a given ratio (internally).
- Tangent to a circle from a point outside it.
- Construction of a triangle similar to a given triangle.
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Chapter 12 Heron’s Formula
The chapter discusses Heron’s formula, when the length of all three sides is given it is used to calculate the area of a triangle. Heron’s formula can be used in two thinks one to find the area of triangles and second to find the areas of quadrilaterals and other polygons by dividing them into triangles.
Also, access the helpful resources for Class 9 Maths Chapter 12 Heron’s Formula at Physics Wallah
Important Questions for class 9 maths Chapter 12 Heron’s Formula |
RD Sharma Solutions for Class 9 Chapter 12 Heron’s Formula |
Maths revision notes for class 9 maths Chapter 12 Heron’s Formula |
Chapter 13 Surface Areas and Volumes
In chapter-13, students will learn to find the surface areas and volumes of cuboids and cylinders in detail and extend the study to other solids such as cones and spheres. This chapter is just an expanded version of the measurement chapter, where students learned about surface areas and volumes in earlier classes. There are eight exercises in this chapter, which contain problems based on the surface areas and volumes of various solids such as cubes, cuboids, spheres, cylinders, cones, and hemispheres.
Also, access the helpful resources for Class 9 Maths Chapter 13 Surface Areas and Volumes at Physics Wallah
Chapter 14 Statistics
We have studied in previous class about classification of given data into ungrouped as well as grouped frequency distributions. We have also learnt to represent the data with the help of various graphs like bar graphs, histogarms and frequency polygons. Now, we will study about certain numerical representatives of the ungrouped data, mean, median and mode.
In this chapter we will study mean, median and mode from ungrouped data to that of grouped data. Besides this, we will know the concept of cumulative frequency, the cumulative frequency distribution and also how to draw cumulative frequency curves.
- Sub topics
- Introduction
- Direct Method
- Assumed Mean Method
- Step-Deviation Method
- Mode of Grouped Data
- Median of Grouped Data
- Median of Grouped Frequency Distribution
- Graphical Representation of Cumulative Frequency Distribution
Also, access the helpful resources for Class 9 Maths Chapter 14 Statistics at Physics Wallah
Chapter 15 Probability
In investigating weighing, and sorting and finding an efficient code, you may note a common thread. In each case, and in many others that we will encounter later, we start with a (usually large) number of possible situations, and are interested in determining the true situation from among them. Thus we have devised schemes for doing so in the contexts mentioned.
Probability theory, which by the way was developed initially in the study of gambling schemes, is intended to provide a framework for modeling problems of this kind, with the aim of providing information about what we can “expect” to happen in each context.
This is done by defining a Sample Space whose points or elements are the possible situations, each one of which is given a weight or probability according to the proportion of the time we expect it to be the true situation.Then we determine what to “expect” as the answer to some question, by representing the answer as a function defined on the points of the sample space, and averaging that function over them.
In most contexts, including gambling and all that we have considered so far, the potential situations can be represented by strings of binary bits (or ternary symbols, in the case of weighing). In the following discussion we assume that our sample space consists of points each of which is such a bit string.
Also, access the helpful resources for Class 9 Maths Chapter 15 Probability at Physics Wallah
CBSE Class 9 Maths Unit-wise weightage for 2022-23
Unit | Unit Name | Marks |
I | Number systems | 10 |
II | Algebra | 20 |
III | Coordinate Geometry | 04 |
IV | Geometry | 27 |
V | Mensuration | 13 |
VI | Statistics and Probability | 06 |
Total | 80 | |
Internal Assessment | 20 | |
Total | 100 |
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- Solving Maths numerical is best habit to improve your skill in maths. This habit will help students to not only score great marks in their examination but will help in score good marks in competitive exams.
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All students are expected is to go through this study material by heart. The very objective of the team at Physics Wallah is to ensure that students get acquainted with the best in the class study material. The students can get themselves the expert guidance with the Physics Wallah.
Tips to use NCERT Solutions for Class 9 Maths
On must Use of NCERT Maths and NCERT solutions for class 9 with proper strategies, Physics Wallah team prepared few steps to have better understanding on class 9 maths one can us these step to have more conceptual clearly.
- Step-1 Read the theory part given in each chapter of NCERT book don’t jump to next topics without proper completion of first follow the chapter sequence as given in NCERT book don’t mix and match the chapter. Once you completely understand the chapter by reading and understand the theory part than go to next step.
- Step-2 Start solving the solved question by yourself without the help of solution. Always try to solve questions which are difficult more than two to three time before taking help for the solutions.
- Step-3 Start solving the exercise given in the NCERT book. Try to solve all question don’t take help of NCERT solutions for class 9 maths. When you can’t solve the questions by attempting multiple time now you can take help from Physics Wallah NCERT solutions for class 9 maths.
- Step-4 This step is important to have more depth in the subjects at this step you must understand the chapter completely now just login to Physics Wallah and solved the question given in the website and go for chapter wise online test. If you wants to extent your learning read the extra theory part of chapter mentioned in Physics Wallah.
Frequently Asked Question (FAQs)
Q1. How to score Good marks in class 9 Maths?
Ans. To score good marks in class 9 Maths one must follow right strategies from day one of his/her class 9. Start your preparation with NCERT text book read theory carefully and try to prepare your notes of each and every chapter with mentioning all important formulas required in the chapter.
Move to solve the exercise and try to solve all questions given in NCERT exercise with the help of NCERT solutions for class 9 Maths. While preparing note make sure you have added all important points of the chapter. Do as many as question you can do use NCERT Exemplar to solve more questions of class 9 Maths.
Q2. How Many chapters are there in NCERT Class 9 Maths?
Ans. NCERT Class 9 Maths is very well-designed syllabus having 15 chapters all total and all chapters are very important for upcoming class. NCERT Class 9 Maths is divided into 7 units such as Number system, Algebra, Coordinate Geometry, Geometry, Mensuration, Statistics & Probability.
Q3. What are the important chapters in NCERT class 9 Maths?
Ans. Based on weightage of allocated marks mensuration is highest weightage unit and if you want a good foundation in Class 9 Maths for class 10 and 11 you need to focus on few chapters like Algebra, Coordinate Geometry, Geometry, Probability.
Q4. Do we need to solve additional MCQ based Questions in class 9 Maths?
Ans. Objective questions are best way to check your concepts and become a good habit to handle MCQ-based questions. Solving MCQ based question help you to identify the error in the concept and you will get a direction to modify your mistakes. Do solve MCQ based questions uploaded by Physics Wallah.
Q5. How to use NCERT solutions for class 9 Maths ?
Ans. The best way to use NCERT solutions for class 9 Maths is use it as a reference for those questions which you can’t able to solve after taking multiple attempts. Solving class 9 maths exercise need time so try to solve the question by yourself. Never depend on NCERT solutions for class 9 Maths.
Q6. Do we need to solve the Sample papers along with NCERT class 9 Maths?
Ans. Sample papers for Class 9 Maths are very important for the final revision. One must do practice form sample papers. To help you out our academic team uploaded several sample papers for class 9 maths having all types of questions like subjective and MCQ based questions.
Q7. If we are preparing for JEE and other competitive exam do, we need additional resources?
Ans. Now a day’s foundation course is doing very well preparation for JEE or Olympiad exam start form very beginning like from class 8. If you are preparing for foundation course than you need some additional resource apart form NCERT class 9 Maths. For such students Physics Wallah prepared separate section for class 9 Math Notes do read all theory and solve additional question given in this section.
Recent Concepts
- chapter-1 Number Systems
- chapter-2 Polynomials
- chapter-3 Coordinate Geometry
- chapter-4 Linear Equations in Two Variables
- chapter-5 Introduction to Euclid 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 Herons Formula
- chapter 13-Surface Areas And Volumes
- chapter-14 Statistics
- chapter-15 Probability