NCERT Solutions for Class 9 Maths
NCERT solutions for class 9 maths comprise the analysis and explanation of all the problems covered by the NCERT textbook for 9th grade. The solutions are provided in a detailed manner to ensure that students can understand concepts as they go through them. These class 9 maths NCERT solutions are prepared by some of the most accomplished experts in the field of Mathematics.
NCERT solutions for class 9 maths are extremely helpful for students preparing for CBSE exams and state or national competitive tests such as Olympiads. The CBSE examinations follow the structure provided by the NCERT textbooks. Additionally, CBSE NCERT solutions for class 9 maths help in getting students ready to face class 10th and the corresponding board exams.
NCERT Solutions for Class 9 Maths PDF Download
Practicing these class 9 maths NCERT solutions can help students to increase their theoretical skills and practical knowledge. The solutions to all NCERT class 9 maths sums are available for free pdf download. You can find tips and tricks included with the solutions. The links to these pdf solutions are given below.
Class 9 Maths NCERT Solutions Chapter 1 to 15
As a student, you must indulge in constant revision and practice all the solutions. Thus, revisiting the above-mentioned links periodically is a must. Start by attempting the questions in the NCERT book and then have a look at these class 9 maths NCERT solutions. You can download the NCERT textbook given below and begin your learning journey.
NCERT Solutions for Class 9 Maths Chapter 1
NCERT solutions for class 9 maths chapter 1 number systems help students amass the tools required to identify rational and irrational numbers, locate real numbers on a number line, and convert a real number to its decimal form. The solutions to this chapter also give students a good understanding of simplifying expressions using certain identities and laws of exponents.
- √(ab) = √a √b
- √(a/b) = √a/√b
- (√a + √b)(√a - √b) = a - b
- (√a + √b)2 = a + b + 2√(ab)
|Class 9 Maths NCERT Solutions Chapter 1 Exercises|
Class 9 Maths Chapter 1 Real Numbers
Topics Covered: NCERT solutions for class 9 maths cover questions based on rationalizing the denominator, the rules of exponentiation, identities applied to positive real numbers, expanding a real number into its decimal form, and vice versa.
Total Questions: Chapter 1 has a total of 27 questions, out of which 5 are easy, 7 are moderate and 10 are difficult.
NCERT Solutions for Class 9 Maths Chapter 2
NCERT solutions for class 9 maths chapter 2 polynomials help students form the basics of polynomials that will be used in higher classes. Students learn about the types of polynomials in one variable, degree and zeros of a polynomial, the remainder theorem, factorization of polynomials, and the application of algebraic identities.
- (x + y)2 = x2 + 2xy + y2
- x2 - y2 = (x + y) (x - y)
- (x + a) (x + b) = x2 + (a + b)x + ab
- (x - y)2 = x2 - 2xy + y2
- (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
|Class 9 Maths NCERT Solutions Chapter 2 Exercises|
Class 9 Maths Chapter 2 Polynomials
Topics Covered: Factorization of different polynomials using the splitting method and factor theorem, finding the remainder of a polynomial, applying algebraic identities to simplify sums, and determining the zeros of a polynomial.
Total Questions: Class 9 maths chapter 2 polynomials have 33 questions. 10 are of a moderate level, 14 are easy and 9 are challenging sums.
NCERT Solutions for Class 9 Maths Chapter 3
NCERT solutions for class 9 maths chapter 3 coordinate geometry help students to get a grasp on how to describe the position of an object using coordinates and explanation of the cartesian system. It acts as a foundation for further chapters that require students to use their knowledge to solve geometrical problems.
|Class 9 Maths NCERT Solutions Chapter 3 Exercises|
Class 9 Maths Chapter 3 Coordinate Geometry
Topics Covered: Determining the x and y coordinates of a point in a cartesian plane, identifying quadrants, locating an object, and expressing it in the form of coordinates are the topics under class 9 maths NCERT solutions.
Total Questions: There are 6 questions in class 9 maths chapter 3 coordinate geometry. 3 are difficult, 2 are medium level and 1 is an easy question.
Class 9 Maths Chapter 3 Summary:
Class 9 maths NCERT solutions chapter 3 gives a brief overview of the definition of the origin, how to use the horizontal x-axis and vertical y-axis to pinpoint the position of an object. It also helps students determine what quadrant does a point lies in depending upon the signs of the coordinates.
NCERT Solutions for Class 9 Maths Chapter 4
NCERT solutions for class 9 maths chapter 4 linear equations in two variables introduce students to a linear equation in two variables by helping them understand the concepts behind it. The solutions include solving linear equations, graphical representation methods, and the depiction of parallel lines.
|Class 9 Maths NCERT Solutions Chapter 4 Exercises|
Linear equations are one of the most important chapters covered in class 9 maths NCERT solutions. The following are a few tips that can help you power through these solutions quickly and efficiently.
- Build an understanding of the concepts
- Revision of the topic at regular intervals
- Understanding the general form of a line
Class 9 Maths Chapter 4 Linear Equations in Two Variables
Topics Covered: Finding solutions to a linear equation by substituting values, representing a linear equation in two variables graphically, and how to write the equation of lines parallel to the x-axis.
Total Questions: Chapter 4 linear equations in two variables have a total of 16 questions. 8 require students to carefully think and attempt the problem, 5 are moderate and 4 are easy sums.
NCERT Solutions Class 9 Maths Chapter 5
NCERT solutions class 9 maths chapter 5 introduction to Euclid’s geometry gives students an insight into the definitions of points, lines, surfaces, and so on. Students also get an idea of the 5 postulates given by Euclid that are used to solve geometry problems.
|Class 9 Maths NCERT Solutions Chapter 5 Exercises|
Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry
Topics Covered: Euclid’s five postulates dealing with line segments, radii, perpendicular lines, and the variation of the fifth postulate are the topics covered under NCERT solutions class 9 maths.
Total Questions: Chapter 5 introduction to Euclid’s Geometry has 9 questions, 4 are simple, 3 are intermediate and 2 are tough sums.
The chapter deals with five postulates given by Euclid regarding a straight line, a terminating line, a circle, and its radius, as well as right angles. Students also learn about the two equivalent versions of Euclid’s fifth postulate that led to the discovery of other theorems.
Class 9 Maths NCERT Solutions Chapter 6
Class 9 maths NCERT solutions chapter 6 lines and angles give students an understanding of the basic definitions of various terms that are used in geometry. Students learn about the different types of angles, pairs of angles, transversals, parallel lines, angle sum property of triangles, and axioms associated with these concepts.
|Class 9 Maths NCERT Solutions Chapter 6 Exercises|
Class 9 Maths NCERT Solutions Chapter 6 Preparation Tips
There are a number of properties taught in this chapter that students must learn. The following points can help in streamlining this process.
- Practice several questions on properties of angles that get formed when a transversal intersects 2 or 3 parallel lines.
- Understand the reasoning behind theorems such as the angle sum property of a triangle.
- Make a note of the relationship between pairs of angles.
Class 9 Maths Chapter 6 Lines and Angles
Topics Covered: Adjacent angles, vertically opposite angles, linear pairs of angles, alternate angles, corresponding angles, parallel lines, and transversal are topics falling under Class 9 maths NCERT solutions. In addition to this, problems based on the angle sum property of a triangle are also included in these solutions.
Total Questions: There are a total of 18 questions in class 9 maths chapter 6 lines and angles. 10 are very easy property-based questions, 5 require some thinking on a student’s part and 3 are difficult sums.
NCERT Solutions for Class 9 Maths Chapter 7
NCERT solutions for class 9 maths chapter 7 triangles are a fantastic way for students to learn about the congruence of triangles using the SAS, SSS, ASA, RHS criteria. Several properties of triangles along with concepts on the inequality of triangles also fall under the scope of this chapter.
|Class 9 Maths NCERT Solutions Chapter 7 Exercises|
Class 9 Maths Chapter 7 Triangles
Topics Covered: Questions based on applying properties of triangles, proving the congruence of two triangles using different methods, and theorems on the inequality of triangles are the topics covered.
Total Questions: Class 9 maths chapter 7 has 31 questions. 10 are simple questions based on applying the properties of triangles, 9 are complex sums and 12 are moderate problems.
In this chapter, students learn about the SSS, SAS, ASA, and RHS congruence criteria. Students can get a good grasp on properties of triangles such as sides opposite to equal angles are equivalent, the sum of any two sides of a triangle is greater than the third side, and so on. Thus, after studying this chapter students can master the basic concepts of a triangle.
NCERT Solutions for Class 9 Maths Chapter 8
NCERT solutions for class 9 maths chapter 8 quadrilaterals comprise questions that teach students about the different types of quadrilaterals such as squares, rhombuses, rectangles, parallelograms, and their properties.
|Class 9 Maths NCERT Solutions Chapter 8 Exercises|
Class 9 Maths Chapter 8 Quadrilaterals
Topics Covered: Angle sum property of quadrilaterals, properties of a parallelogram, square, and rhombus that depend upon different factors such as length of diagonals, the measure of sides, etc. It also includes the mid-point theorem.
Total Questions: 19 sums - 8 are simple problems, 6 are of an intermediate level and 5 are challenging questions.
Class 9 Maths NCERT Solutions Chapter 9
Class 9 maths NCERT solutions chapter 9 areas of parallelograms and triangles include various theorems that help to simplify finding the area of complex parallelograms as well as triangles, properties to find the area of triangles divided by a median, and area of planar regions.
- Area of a triangle = ½ × base × height
- Area of a parallelogram = base × height
|Class 9 Maths NCERT Solutions Chapter 9 Exercises|
Class 9 Maths Chapter 9 Areas of Parallelograms and Triangles
Topics Covered: Topics covered incorporate questions on finding the area of two or more triangles and parallelograms with the same base between the same parallels, determining the area of triangles that are split by a median as well as the area of congruent figures.
Total Questions: Chapter 9 areas of parallelograms and triangles have a total of 31 questions. 8 questions belong to an optional exercise with a high difficulty level, 14 are intermediate and 9 are simple sums.
NCERT Solutions for Class 9 Maths Chapter 10
NCERT solutions for class 9 maths chapter 10 circles build a foundation for students by introducing them to the various terms related to circles. Students get the opportunity to build their knowledge base on properties of chords, the angle subtended by chord and arc, perpendicular to a point, the distance of a chord from the center as well as cyclic quadrilaterals.
|Class 9 Maths NCERT Solutions Chapter 10 Exercises|
NCERT solutions for class 9 maths chapter 10 are purely based on the properties of various elements of a circle. To memorize these theorems you can follow the topics listed below.
- Make a note of all the properties so that you can skim through them when required.
- Solve all NCERT textbook questions.
- Regularly revise the solutions so that you can recall them when necessary.
Class 9 Maths Chapter 10 Circles
Topics Covered: NCERT solutions class 9 maths encompass topics such as angles subtended by a chord at a point, perpendiculars dropped from the center to a chord, and how to calculate the distance of equal chords from the center. Other discussions include angles subtended by an arc of the circle, cyclic quadrilaterals, and their attributes.
Total Questions: There are a total of 35 questions in this chapter out of which 15 are relatively simpler, 10 are complex and 10 are of moderate complexity.
NCERT Solutions for Class 9 Maths Chapter 11
NCERT solutions for class 9 maths chapter 11 constructions will teach students how to precisely use geometrical instruments for basic constructions. Students learn to construct different angles, perpendicular bisectors, and triangles with a few given restraints.
|Class 9 Maths NCERT Solutions Chapter 11 Exercises|
Class 9 Maths Chapter 11 Constructions
Topics Covered: Constructing an angle from a ray, bisector of a given angle, and techniques to construct triangles when certain constraints such as the perimeter are mentioned, are the topics explained in this chapter.
Total Questions: Chapter 11 constructions have a total of 10 questions. 3 can be simply solved, 5 are moderate questions and 2 are difficult.
After studying NCERT solutions class 9 maths chapter 11 students will be adept at using a ruler and compass to bisect a given angle, construct angles of discrete values, draw the perpendicular bisector of a line segment, and building triangles.
NCERT Solutions for Class 9 Maths Chapter 12
NCERT solutions for class 9 maths chapter 12 heron’s formula is a formula given by the Hero of Alexandria to calculate the area of a triangle when the lengths of all three sides are known. In this chapter, students will learn more about the application of this formula to find the area of triangles and quadrilaterals.
- Semi Perimeter s = (a + b + c)/2
- Area of a Triangle = √[s (s - a) (s - b) (s - c)]
|Class 9 Maths NCERT Solutions Chapter 12 Exercises|
Class 9 Maths Chapter 12 Heron’s Formula
Topics Covered: NCERT Solutions class 9 maths chapter 12 cover the complete scope of Heron’s formula and how to apply it to questions for determining the area of a triangle. Additionally, this formula is also used for computing the area of a quadrilateral.
Total Questions: Chapter 12 heron’s formula consists of 15 well-curated questions out of which 5 are simple, 3 and medium level and 7 are difficult problems.
Class 9 Maths NCERT Solutions Chapter 13
Class 9 maths NCERT solutions chapter 13 surface areas and volumes introduce students to the concepts required to find the total surface areas, lateral or curved surface areas, and volumes of different shapes. The concepts and formulas in this chapter are very important as they form the basis of topics covered in higher education.
- TSA of a Cuboid = 2(lb + bh + hl)
- TSA of a Cube = 6a2
- TSA of a Cylinder = 2πr(r + h)
- TSA of a Cone = πr(l + r)
- TSA of a Sphere = 4πr2
- TSA of a Hemisphere = 3πr2
|Class 9 Maths NCERT Solutions Chapter 13 Exercises|
Class 9 Maths Chapter 13 Surface Areas and Volumes
Total Questions: Chapter 13 surface areas and volumes have 75 questions. 25 are easy formula-based sum, 31 are medium level and 10 are difficult problems that challenge a student’s thinking abilities.
Class 9 Maths NCERT Solutions Chapter 14
Class 9 maths NCERT solutions chapter 14 statistics provide an in-depth analysis of problems that are based on the collection and presentation of data for analytical purposes, graphical representation of data using different types of graphs, and measures of central tendency. This chapter enables students to develop their calculation speed and visualization skills.
|Class 9 Maths NCERT Solutions Chapter 14 Exercises|
Class 9 Maths Chapter 14 Statistics
Topics Covered: Representation of data in the form of frequency tables and drawing inferences from them, plotting data points using bar graphs, histograms, and frequency polygons, finding the mean, median, and mode of the given data.
Total Questions: Chapter 14 statistics have 26 problems that can be split into 8 long answer-type questions, 11 moderate, and 7 simple sums.
Once students finish practicing class 9 maths NCERT solutions chapter 14 they will have a good understanding of the purpose of well-organized data, the three measures of central tendency for ungrouped data namely mean, median, and mode. Students can effectively apply these formulas to all levels of problems after studying this topic.
NCERT Solutions Class 9 Maths Chapter 15
NCERT solutions class 9 maths chapter 15 probability help students deal with daily-life problems that require the use of probability concepts. This chapter helps students explore more about empirical probability, the definition of an event, and how to calculate the probability of different events.
|Class 9 Maths NCERT Solutions Chapter 15 Exercises|
☛ Chapter 15 Class 9 Maths Ex 15.1 - 13 Questions
Class 9 Maths Chapter 15 Study Tips:
NCERT solutions class 9 maths chapter 6 probability is a widely used topic that people require to perform their day-to-day operations. Given below are some study tips that can help in comprehending this chapter in a better manner.
- Apply the concept to simple daily life questions.
- Practice the solutions regularly to get the best results.
- Read the question carefully and break it down to get a better idea of what has to be done.
Class 9 Maths Chapter 15 Probability
Topics Covered: Finding the probability of the occurrence of heads or tails when a coin is flipped, tossing a die, and drawing conclusions from the result obtained are the topics covered in NCERT solutions class 9 maths.
Total Questions: Chapter 15 probability has 13 questions. 4 are short answer problems, 3 are difficult and 6 are moderate sums.
Class 9 Maths NCERT Solutions Important Formulas
Class 9 maths covers several topics such as polynomials, number systems, geometry, probability, etc. The formulas and concepts behind these topics are required to understand higher grade mathematics. Formulas are required to reduce the time taken to solve questions and increase accuracy. Given below is an overview of the important maths formulas encompassed in the NCERT solutions for class 9 maths.
- Area of a triangle = ½ * b × h
- Perimeter of a triangle = a + b + c
- Area of parallelogram = b × h
- Perimeter of a parallelogram = 2(a + b)
Surface Area and volume:
- TSA of a cuboid = 2(lb + bh +lh)
- Volume of a cuboid = l×b×h
- TSA of a cube = 6a2
- Volume of a cube = a3
- (a + b)2 = a2 + b2 + 2ab
- (a - b)2 = a2 + b2 - 2ab
- (a + b) (a - b) = a2 - b2
Importance of NCERT Solutions for Class 9 Maths
Class 9 Maths NCERT Solutions are extremely helpful for CBSE students as they give a means for efficient time management and improving auxiliary abilities. The following are the advantages of these solutions.
- Practical Applications: Many of the sums provided by the NCERT textbook are similar to real-life examples, especially when word problems are considered. By following the class 9 maths NCERT solutions students can come up with precise solutions that can also be applied to daily life problems.
- Practice: To solve the toughest sums within a matter of minutes you need due diligence and regular practice. These NCERT solutions for class 9 maths are very helpful for this purpose as they give students a detailed overview of how to solve different types of sums. They can also help a student to complete the syllabus well within the given time frame.
- Regular Revision: On examination days, students do not have the time to solve each and every question. Thus, these NCERT solutions class 9 maths help students with a quick revision of the entire syllabus. This also helps to free up time so that students can focus on ensuring that their concept formation is correct and can be accurately applied to sums.
FAQs on NCERT Solutions for Class 9 Maths
Why are NCERT Solutions for Class 9 Maths Important?
NCERT solutions for class 9 maths are vital for students as they provide reliable solutions that students can use to cross-check their answers. These class 9 maths NCERT solutions also guide students on how to appear for an examination so that students can write their answers without missing any steps and get the best possible score.
Where Can I Get Chapter-wise Solutions for NCERT Class 9 Maths?
The chapter-wise solutions for NCERT class 9 maths can be accessed by the links mentioned above. They are further segregated into exercise-wise pdfs that are available for free download. These pdfs are also compatible with mobile screens and are in a scrollable format for easy use.
How CBSE Students can utilize Class 9 Maths NCERT Solutions?
To effectively utilize class 9 maths NCERT solutions students need to ensure that they have an understanding of the concepts used to solve the problems. Thus, when they start consulting these solutions students will comprehend them in a better manner and will be able to retain what they learned more efficiently.
Do I Need to Make Notes while Referring to NCERT Solutions Class 9 Maths?
Notes act as a simplified guide for students when they are trying to build an understanding of the subject or if they are revising that question. It is a good way of retaining and recalling concepts when required. Thus, it is crucial that students make notes of the NCERT solutions class 9 maths to expedite their learning process.
What are the Best Ways to Learn Concepts Covered in NCERT Solutions Class 9th Maths?
The most impactful way to learn concepts covered in NCERT solutions class 9th maths is by constantly practicing these solutions. If a student finds some concept difficult and cannot proceed further while attempting a problem he needs to immediately clarify his doubts so that there is no gap in learning.
Can We Use Other Methods to Solve Problems Apart From Those Mentioned in NCERT Solutions for Class 9 Maths?
You may not want to use the method given in the NCERT solutions for class 9 maths however, the approach used and the answer obtained need to be universally recognized. Having said that you cannot use the tricks that are useful in solving competitive exam problems as these are not standardized techniques.
Why Choose Cuemath for Class 9 Maths NCERT Solutions?
The best website for class 9 maths NCERT solutions is Cuemath. These solutions are prepared by IITians and certified math experts. Thus, they are reliable, precise and give the best techniques to solving any problem. They also include some tips to enable students to solve sums with speed and accuracy.
How Many Chapters are there in NCERT Solutions for Class 9 Maths?
There are 15 chapters in class 9 maths NCERT solutions. The problems covered by these are fantastic to improve a student’s understanding of different concepts. The NCERT solutions give an in-depth analysis of all exercise problems.
Do I Need to Practice all Questions Provided in NCERT Solutions Class 9 Maths?
All questions that are included in the CBSE home examinations are based on the NCERT class 9 maths. Thus, in order to get good marks, it is necessary to be well-versed with all the questions covered in class 9 maths NCERT solutions that can only be achieved through regular practice and due diligence.
Are NCERT Solutions for Class 9 Maths Sufficient for CBSE Exams?
The NCERT solutions for class 9 maths are sufficient for all CBSE exams. As the school teachers set the paper they include questions from the NCERT textbook. The language or values given in the problem may differ however, the approach to solving it is more or less the same. As long as a student has a robust understanding of all chapters, he can successfully clear the examination.
What are the Important Topics Covered in NCERT Solutions for Class 9 Maths?
All the topics such as construction, circles, surface area, volume, statistics, etc. that are covered in the NCERT solutions for class 9 maths are required for class 10th. Thus, skipping a topic is not advisable. In addition to this, all topics within the scope of mathematics are interlinked making it necessary to give equal attention without leaving out anything.
What are the Important Formulas in NCERT Class 9 Maths Solutions?
The questions in NCERT solutions class 9 maths are mostly based on the application of theorems and axioms to prove a certain point. Thus, as there are a fewer number of formulas it is not only easy but necessary to memorize them all.
What are the Benefits of Class 9 Maths NCERT Solutions for CBSE Students?
The class 9 maths NCERT solutions help students to finish their syllabus on time giving them enough room for revision. They also provide a quick overview of all the problems that students can use as a reference while skimming through the chapter.
Why Should I Practice NCERT Solutions for Class 9 Maths Regularly?
The practice of NCERT solutions for class 9 maths will help a student achieve the goal of perfection. It helps students get comfortable with the covered concepts. Thus, during an examination students can keep their cool and breeze through the paper without getting flustered.
How is Class 9 Maths NCERT Solutions Promoting Problem Solving in Students?
NCERT solutions for class 9 maths start by explaining the questions and then breaking the problem down to elucidate it in a better way. It teaches students to think of the why and how behind a sum. These solutions encourage students to use a similar approach enabling them to harness their problem-solving skills and computational mindset.