# NCERT Solutions for Class 9 Maths

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NCERT - Mathematics

NCERT Solutions for Class 9 Maths are the best resource for Maths exam preparation. The Maths subject of CBSE Class 9th is significantly complex as compared to the previous classes. Neglecting NCERT textbook is the major mistake most of the students often do while preparing for the Class 9th Maths subject.

It is advised to prepare for the exam from the NCERT Maths book of Class 9, before referring any other book. Rather, you can refer to the Class 9 Maths NCERT Solutions below. At Goprep, our experienced subject experts have prepared accurate and easy solutions to all the chapters of the NCERT textbook.

## NCERT Class 9 Maths Solutions - All Chapters

The solutions given are based on the latest syllabus approved by the Central Board of Secondary Education (CBSE). To perform better in the exam, it is important to make your basics strong. Our experts have tried to make the solutions as easy as possible. You will not find any concept difficult to understand. Also, we have provided solutions to all the questions of each and every chapter of CBSE Class 9 Maths.

## NCERT Solutions for Class 9 Maths (Chapter-wise description)

### Chapter 1: Number System

Introduction: Before you start-off this chapter, you must have a basic understanding of rational numbers. Rational numbers can be written in the form of p/q, where p and q are integers and q ≠ 0. In NCERT Class 9 Maths textbook, the first new topics that you will learn is an irrational number. Unlike rational numbers, irrational numbers cannot be written in the form of p/q.

As you proceed, you will learn that the decimal expansion of a rational number is either terminating or non-terminating recurring. Amid fresh topics, you must recall real numbers that comprise natural, whole, integers, rational and irrational numbers. In the last exercise, you will study the concept of rationalization and solve related questions using the laws of exponents.

Topics

• Rational Numbers
• Irrational Numbers
• Real Numbers and their Decimal Expansions
• Representing Real Numbers on the Number Line
• Operations on Real Numbers
• Laws of Exponents for Real Numbers

Important Formulas of "Number System"

1. For positive real numbers a and b, you can apply one of the following identities:

• √ab = √a √b
• √a/b = √a/ √b
• (√a + √b) (√a  - √b) = a - b
• (√a + √b) (√a  - √b) = a2 - b
• (√a + √b)2 = a + 2 √ab + b
2. To rationalize (1/ √a + b), multiply (√a - b/ √a - b), where a and b are integers.

### Chapter 2: Polynomials

Introduction: NCERT Class 9 Maths textbook questions will require you to have prior knowledge of factorization and algebraic identities. In class 8, you studied three identities and their use in factorization. In this class, you will come across two important theorems- Remainder Theorem and the Factor Theorem.

A polynomial with one term is called a monomial whereas polynomials with two and three terms are called binomial and trinomial respectively. Based on the degree, a polynomial can be classified as linear (having degree 1), quadratic (having degree 2) and cubic (having degree 3).

Topics

• Polynomials in One Variable
• Zeros of a Polynomial
• Remainder Theorem
• Factorization of Polynomials
• Algebraic Identities

Important Formulas/ Identities of "Polynomials

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

2. (x + y)3 = x3 + y3 + 3xy (x + y)

3. (x - y)3 = x3 - y3 - 3xy (x - y)

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

### Chapter 3: Coordinate Geometry

Introduction: The concept of coordinate geometry revolves around locating a point on a number line, which you have already learnt in previous classes. You can locate the position of a point in a plane in a cartesian or coordinate plane. To form a cartesian plane, draw two perpendicular lines- horizontal and vertical.

These coordinate axes divide the plane into four parts called quadrants. The meeting point of the horizontal and vertical line is called the origin. Next, you will learn a few new terminologies such as abscissa and ordinate. The abscissa is the distance of a point from the y-axis, also called its x-coordinate. The ordinate is the distance of a point from the x-axis, also called its y-coordinate.

In this chapter, you will find a total of 27 questions in the back exercises from the following topics.

Topics

• Cartesian System
• Plotting a Point in the Plane if its Coordinates are Given

### Chapter 4: Linear Equations in Two Variables

Introduction: In the previous edition of NCERT textbook, you have learnt linear equations in one variable. You studied that such equations have a unique solution. Besides, you are already familiar with how to represent the solution on a number line. In this chapter, you will revisit the concept of linear equations in one variable and extend your knowledge to that of two variables.

This chapter consists of a total of 33 questions based on the following concepts.

Topics

• Linear Equations
• Solution of a Linear Equation
• Graph of a Linear Equation in Two Variables
• Equations of Line Parallel to the x-axis and y-axis

### Chapter 5: Introduction to Euclid's Geometry

Introduction: Since the ancient civilization, people have deployed the knowledge of geometry in carrying out various practical problems. Originally, Euclid defined a point, a line, and a plane, but other mathematicians did not agree with his findings. Euclid’s axioms and postulates are mere assumptions and are not proved.

You can now draw differences between theorems and axioms. Theorems are statements that hold valid explanation, i.e. they have been proved using definitions. On the other hand, axioms were previously proved statements with deductive reasoning.

Topics

• Euclid’s Definitions, Axioms and Postulates
• Equivalent Versions of Euclid’s Fifth Postulate

### Chapter 6: Lines and Angles

Introduction: In the previous chapter, you have studied that a line has an indefinite length from both its ends. While a line segment can be formed by joining two points. To form a ray, you need to keep one end fixed and extend the other end up to infinity. When a pair of rays meet at a certain point, the region covered between them is called an angle.

In NCERT Class 9 Maths textbook, you will learn the properties of the angles formed when two lines intersect each other. Further, you will come across properties of the angles when a line intersects two or more parallel lines at distinct points.

It is advised to revise a few properties of lines and angles from NCERT Maths textbooks of previous classes before you solve questions of this chapter. The properties that you are already familiar with include linear pair axiom, vertically opposite angles, angle sum property etc.

Topics

• Line, Line Segment & Ray
• Intersecting lines and Non-intersecting lines
• Pair of Angles
• Parallel lines and a Transversal
• Lines Parallel to the Same Line
• Angle Sum Property of a Triangle

Important Properties of "Lines and Angles

1. Linear Pair Axiom: Two angles are said to form a linear pair, when a ray intersects a line. The sum of the two adjacent angles, thus formed is 180°.

2. Angle Sum Property of a Triangle: The sum of interior angles of a triangle is 180°.

3. Exterior Angle Property: The exterior angle of a triangle is equal to the measure of two opposite interior angles.

4. Vertically Opposite Angles: When two lines intersect each other, then the angles so formed are vertically opposite angles.

### Chapter 7: Triangles

Introduction: A triangle is a polygon of three sides and three angles. In class 7, you have already learnt the properties of the triangle and proved them congruent. You are already familiar with the types of triangles which include acute, equilateral, isosceles, obtuse, scalene and right-angled.

Two triangles are said to be congruent if they have similar sides and angles. According to Side Angle Side congruence rule, two triangles are said to be congruent when their two adjacent sides and an angle are equal. In the case of Angle Side Angle rule, their two angles and a side are equal. This way, you can prove two triangles congruent. Other rules of congruency include SSS, AAS and RHS.

Topics

• Congruence of Triangles
• Criteria for Congruence of Triangles
• Some Properties of a Triangle
• Some More Criteria for Congruence of Triangles
• Inequalities in a Triangle

Important Rules of Congruency of "Triangles

1. SAS (Side Angle Side): When two triangles have two equal adjacent sides and an angle. They can be proved congruent by SAS.

2. AAS (Angle Side Angle): When two triangles have two equal angles and an equal side, then the congruence rule applied is AAS.

3. SSS (Side Side Side): If all three sides of a triangle are equal, the two triangles can be proved equal by SSS.

4. RHS (Right Hand Side Rule): When two right-angled triangles have a hypotenuse of the same length and an equal side then both triangles are said to be congruent by Right Hand Side Rule.

Introduction: In earlier classes, you already learnt that quadrilateral is a four-sided shape having 360° measure of the sum of its interior angles. Besides, you also read properties of common quadrilaterals including parallelogram, kite, rhombus, square and rectangle. In this chapter, you will revisit all these properties once again.

Further, you will be introduced to three new theorems, which you will have to apply while solving exercise questions. This chapter contains 31 questions that will require a strong understanding of the following concepts.

Topics

• Angle Sum Property of a Quadrilateral
• Properties of a Parallelogram
• Condition for a Quadrilateral to be a Parallelogram
• The Mid-point Theorem

### Chapter 9: Areas of Parallelograms and Triangles

Introduction: In NCERT Maths Book of Class 7 and 8, you learnt to calculate the area of parallelogram and triangle. This year, you will develop an in-depth understanding of various new properties of a triangle and a parallelogram. Let us discuss the first theorem of triangles. It states that when two triangles are on the same base and their opposite vertex is on the parallel line, then their area must be equal.

You will also study some new terminologies including a median of a triangle, the centroid etc. What if a parallelogram and a triangle shares the same base and between the same parallel? The triangle becomes half of the area of the parallelogram.

Once you complete studying the following concepts and terminologies, you can go through NCERT Solutions of Class 9 Maths chapter-8 for cross-checking your answers.

Topics

• Figures on the Same Base and Between the Same Parallels
• Parallelograms on the same Base and Between the same Parallels
• Triangles on the same Base and between the same Parallels

Important Formulas/ Theorems of "Areas of Parallelograms and Triangles

1. Area of Parallelogram = Base x Height

2. Area of Triangle = ½ x Base x Height

3. Triangles on the same base and between the same parallels

Ar (ABC) = Ar (DBC)

4. A Parallelogram and a Triangle on the same base and between the same parallel

Ar (△ABC) = ½  Ar (ABCE)

### Chapter 10: Circles

Introduction: In the previous two editions of the NCERT Maths textbook, you learnt to calculate the area and circumference of a circle. You are also well-versed with the terms related to circles such as arc, circumference, chord and diameter. This chapter covers 12 theorems based on a circle and its properties.

To start from the top, the first theorem states that two equal chords of a circle subtend equal angles at the centre. The second theorem explains that the perpendicular drawn from the centre of a circle to a chord bisects the chord.

As stated above, there are twelve theorems in total that will come to play when you solve NCERT textbook questions. You must be thorough with the following concepts before you start attempting the questions.

Topics

• Circles and Its 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

### Chapter 11: Constructions

Introduction: In NCERT Maths textbook of 6th standard, you have learnt how to construct angles of 30°, 45°, 60°, 90° and 120°. Also, you studied the procedure of constructing a circle and the perpendicular bisector of a line segment. Moving further, you will acquire knowledge of bisecting a given angle, drawing the perpendicular bisector of a given line segment etc.

Having mastered the basic constructions, you will be able to construct a triangle based on three different situations given below.

Topics

• Bisecting a given angle
• To draw the perpendicular bisector of a given line segment
• Construction of 60° angle
• Construction of a triangle given its base, a base angle and the sum of the other two sides
• Construction of a triangle in which base, a base angle and the difference of the other two sides are given
• Construction of a triangle given its perimeter and its two base angles

### Chapter 12: Heron's Formula

Introduction: In earlier classes, the NCERT textbooks covered the formula of area and perimeter of different shapes. Among those formulas, the area of triangle will be majorly applicable in this chapter. You must also be well-versed with the properties of a right-angled triangle and Pythagoras theorem by now.

In this chapter, you will understand the famous formula for the area of triangle derived by Heron. With the use of this formula, you will be able to find the area of the quadrilateral. Check out the list of topics given in this chapter below.

Topics

• Area of a Triangle by Heron’s Formula
• Application of Heron’s Formula in Finding Areas of Quadrilaterals

Important Formulas of "Heron's Formula"

1. Heron’s Area of triangle = √s (s - a) (s - b) (s - c), where a, b and c are the sides of the triangle

2. Heron’s Perimeter of triangle (s) = (a + b + c)/ 2

### Chapter 13: Surface Areas and Volumes

Introduction: In the previous lessons of surface areas and volumes, you have learnt to find the surface areas and volumes of cuboids, cubes and cylinders. This year, you will get to extend your knowledge regarding some other solids such as cones and spheres.

There is no point in simply mugging up the formulas of this chapter. Instead, you should focus on deriving the TSA, CSA and volume for each shape. We have listed down the topics and formulas of this chapter below.

Topics

• 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 Cylinder
• Volume of a Right Circular Cone
• Volume of a Sphere
Important Formulas of "Surface Areas and Volumes"

1. Total Surface Area of a Cuboid =  2 (lb + bh + hl)

2. Total Surface Area of a Cube = 6a2

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

4. Total Surface Area of a Right Circular Cone = πr (l + r)

5. Total Surface Area of a Sphere = 4πr2

6. Total Surface Area of a Hemisphere = 3πr2

7. Curved Surface Area of a Cylinder = 2πrh

8. Curved Surface Area of a Cone = πrl

9. Curved Surface Area of a Hemisphere = 2πr2

10. Volume of a Cuboid = l x b x h

11. Volume of a Cube = a3

12. Volume of a Cylinder = πr2h

13. Volume of a Cone = ⅓ πr2h

14. Volume of a Sphere of radius r = 4/3 πr3

15. Volume of a Hemisphere = ⅔ πr3

## Chapter 14: Statistics

Introduction: Statistics is a branch of mathematics in which data is extracted on different aspects of the life of people. It deals with analysis, collection, organization and interpretation of data. In the previous edition of NCERT Maths textbook, you have learnt to present data graphically in the form of bar graphs, histograms and frequency polygons.

Further, we will discuss the three measures of central tendency for ungrouped data which are mean, median and mode.

Topics

• Collection of Data
• Presentation of Data
• Graphical Representation of Data
• Measures of Central Tendency

Important Formulas of "Statistics"

1. Mean

For grouped distribution (x) = Σni = 1 xi / n

For ungrouped frequency distribution (x) = Σni = 1 (fixi)/  Σni = 1 fi

2. Median

If n is an odd number, the middle-most observation s = {(n + 1)/2}th

If n is an even number, the middle-most observation s = (n/2)th and {(n/2) + 1}th

### Chapter 15: Probability

Introduction: The term ‘Probability’ means the chances of an event to occur. In other words, when an event is uncertain to take place, we call it a probability. For a better understanding of this chapter, you can perform experiments like the tossing of coins, throwing of dice etc. The main motive of this chapter is to help us measure the chance of occurrence of a particular outcome in an experiment.

Topics

• Probability
• Empirical Probability

Important Formulae of "Probability "

Empirical Probability of an event E is given by

P(E) = Number of trials in which an event has happened/ Total number of trials

## CBSE Class 9 Maths Syllabus & Marks Distribution (Unit wise)

Based on the syllabus of CBSE Class 9 Maths, the question paper includes questions from the following units. Below is the list of units along with their allocation of marks.

 Name of Unit Allocation of Marks 1 Number Systems 08 2 Algebra 17 3 Coordinate Geometry 04 4 Geometry 28 6 Mensuration 13 7 Statistics & Probability 10 Total 80 Marks

## CBSE Class 9 Maths Exam Pattern 2020

Once you learn about the approaching exams, you should always check the latest CBSE Syllabus and CBSE Exam Pattern for Class 9 Maths and other subjects. Here we have provided the latest exam pattern of CBSE Class 9 Maths that will help you understand the design of question paper and marking scheme.

 Section No. of Questions Weightage of Each Question Weightage of Each Section A 20 1 20 B 6 2 12 C 8 3 24 D 6 4 24 Total 40 Questions 80 Marks

## Best Maths Books for Class 9

Many students aim to score good marks in Class 9 Maths by solving questions from multiple Maths reference books at once, but it is surely not a good practice. They should refer to only 1-2 reference books at once apart from NCERT Class 9 Maths textbook. Below are some of the best books for CBSE Class 9 Maths.

 List of Best Books for Class 9 Maths Chapter-wise Solutions RD Sharma Class 9 Book Click Here RS Aggarwal Class 9 Book Click Here

## Q. Is it true that the level of difficulty is more in Class 9 in comparison to Class 10?

To understand the difficulty level, let us pick major pointers to differentiate between Class 9 and 10 based on its difficulty level.

 Class 9 Class 10 1 You are introduced to new concepts in Maths such as coordinate geometry, advanced algebra, etc. 1 Most chapters covered in Class 10 Maths NCERT book include formulas and derivations that you have already done in Class 9. 2 Science gets segregated into three different sections. Physics and Chemistry include topics with an in-depth explanation. 2 You develop a habit of doing a detailed study after you clear Class 9. You no longer find it exhausting to write derivations. 3 In the case of theoretical subjects like English, Social Science, and Hindi, evaluation of mark sheets becomes strict. 3 Having learnt the stringent marking structure in Class 9, you get familiarized and avoid losing marks unnecessarily.

## Benefits of using NCERT Solution for Class 9 Maths by Goprep

There are several advantages of adding NCERT Maths solution to your preparation plan. We have listed a few of them for you.

• The solutions are based on the latest syllabus approved by CBSE
• You will get all the solutions to the questions of each chapter of NCERT textbook
• The solutions are free from the errors
• The methodology of each solution is simple and effective
• Not only solutions but our experts have tried to explain the concepts/formulas as well
• The solutions are free of cost and you can easily access them
• Our experts have good knowledge of the subject and own huge experience
• You can easily prepare for Class 9th Maths exam from the Goprep NCERT Solution

The best way to make your basics of Class 9th Maths strong is by using NCERT Solution for Class 9 Maths. The methodology used to explain even the most complex topics are much easier. If you find yourself stuck in the middle of any question of NCERT maths textbook, you can refer to the Goprep NCERT Solutions for Class 9. The solutions are prepared chapter-wise. If you prepare for the maths exam from the NCERT solutions, you will get the sure shot success.

• What are NCERT Class 9 Maths Solutions?

NCERT Solutions are the descriptive and structured answers to the questions asked in the CBSE textbook of Class 9th Maths subject. The solutions given are based on the latest syllabus approved by Central Board of Secondary Education. It covers all the chapters.

• How to score good marks in CBSE Class 9 Maths?

Follow these simple steps to score high marks in Class 9th Maths.

• Create a schedule of studying Maths for 1 hour everyday.
• Pay attention to school teachers and coaching tutors when they are explaining.
• Approach each question in step-by-step manner and raise a query to your teacher to resolve it immediately.
• Do your homework everyday and try solving extra questions from reference books that are related to same concept.
• Do not let go even a single day without practicing questions.
• Prepare a small pocket-size notebook and jot down important formulas and graphs so that you can study it while travelling in bus or train.
• Practice sample papers to get an idea about the difficulty level of questions, the type of questions asked, exam pattern, and marking scheme.
• How can I prepare for CBSE Class 9th Maths exam?

To get good marks in the Class 9th Maths subject, you should definitely prepare Mathematics questions from NCERT Solutions for Class 9. To get good marks in the exam, it is important to understand the official syllabus. The Goprep Maths solution covers all the chapters from the official syllabus. The solutions are prepared and made as simple as we can. If you aim to score better, you should refer to the Class 9th Maths Solutions first before you refer to any other reference book of Maths.

• Do the NCERT Solutions cover all the chapters?

Yes, it covers all the 15 chapters, namely; Number Systems, Polynomials, Coordinate Geometry, Linear Equations in Two Variables, Introduction to Euclid's Geometry, Lines and Angles, Triangles, Quadrilaterals, Areas of Parallelograms and Triangles, Circles, Constructions, Heron's Formula, Surface Areas and Volumes, Statistics and Probability.

• Is it easy to score good marks in Class 9?

The level of difficulty increases drastically when you are promoted to Class 9. Maths and Science are two subjects that share some common topics such as algebra, square root, exponents, etc. If you are good at Maths, you will naturally do well in Science.

Next comes language subjects like English and Hindi, where students lose marks mainly in the grammar and writing section. If you master grammar and improve your writing skills, it will become a cakewalk for you to score over 90% marks in both the language subjects.

Social Science is a theoretical subject in which your performance wholly depends on how much you can learn and remember. To make things a bit easier, you can easily memorize important dates and events when you have them written in a pocket-size notebook.

• Where can I find the best CBSE Class 9 Maths Solutions?

If you are looking out for the best and the most effective solutions, you can refer to Goprep Class 9th Maths NCERT Solutions. The solutions are prepared by skilled and experienced subject experts. The solutions are prepared on the basis of the most recent syllabus approved by CBSE. You will find the concepts and explanations easy and useful.

Not just the solutions, our experts have also explained the difficult concepts and listed important topics as well. The solutions given are error-free.