NCERT Solutions for Class 10 Maths

NCERT Maths solution pdfs for Class 10 is a very reliable and accurate source of information. The study material contains an exhaustive explanation to all the questions. These answers are written clearly and step by step to maximise retention. While going through these solutions, students will find themselves understanding the concept better. Practising regularly is a prerequisite if students want to score well in mathematics. NCERT solutions Class 10th act as a perfect guide for you. You can consult the guide to brush up on your memory or help you if you get stuck somewhere.

The solutions provided by Vedantu are formulated and designed by professionals who have years of experience in the field of teaching. The answers are given in simple language for the students to understand the concept better. Vedantu aims at providing solutions for each question which is easy for students of different calibre to understand. All the syllabus and solutions are updated in accordance with the changes to the NCERT textbook or the syllabus guidelines given by the CBSE. Vedantu is the leading online platform providing answers for all the solutions for each and every subject and class. Helping students understand a subject and enjoy learning new things is what Vedantu strives for. Hence, the comfort and ease of the students are foremost when developing solutions for the questions. The solutions come in handy if the students get stuck or need to understand what they are doing wrong. It is a good way for the students to analyse their answers and performance and also get more practice by referring to the extra questions in the solutions pdf. 

NCERT Solutions for Class 10 Maths Chapter 1 - Real Numbers

There are 4 Exercises present in this chapter. Students will learn about real numbers and irrational numbers in the 1st chapter of Class 10 Maths. CBSE Class 10 Maths Chapter 1 explains Euclid’s Division Lemma, HCF, LCM, Divisibility of integers, rational number, irrational number and decimal expansion of rational numbers with the help of theorem.

Euclid’s Division Lemma states that “Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0≤r<b”. This gives the base for the division algorithm which is used to take out the HCF for two positive integers. The Fundamental Theorem of Arithmetic is also explained in this chapter. It explains how the LCM and HCF for two positive integers can be derived.

NCERT Solution for Class 10 Mathematics Chapter 2 - Polynomials

There are a total of 4 Exercises including an optional Exercise in the 2nd chapter of class 10 Maths. The first Exercise is about finding zeros of polynomials p(x). In the second Exercise, the students need to find a quadratic polynomial. The second Exercise is based on the relationship between 0’s and coefficients of a polynomial. The third Exercise has questions on the division of polynomials. The last Exercise covers questions from all the concepts of Class 10 Maths Chapter 2.

The students should also know the steps to derive the Division Algorithm for polynomials. It can be asked in the exam as it is an important concept in the chapter that has been explained.

NCERT Solutions for Class 10 Maths Chapter 3 - Pair of Linear Equations in Two Variables

The third chapter of class 10 maths is about the linear equations in two variables. It describes what a linear equation in two variables is and how to solve the problems of linear equations in two variables. In the first Exercise (Ex 3.1), representing a problem or situation algebraically and graphically is taught. The second Exercise, i.e, Ex 3.2, depicts the methods of solving linear equations by graphical method. The next four Exercises, Ex 3.3, 3.4, 3.5 and 3.6 explain the different types of methods of solving the pairs of linear equations. The methods explained here are algebraic method, cross-multiplication method, elimination method and substitution method. Exercise 3.7 is a mixed bag. It contains questions from all the Exercises.

NCERT Solutions for Class 10 Maths Chapter 4 - Quadratic Equations

Chapter 4 of Class 10 Maths NCERT book PDF solution is all about quadratic equations. This chapter explains what a quadratic equation is and the methods of solving quadratic equations. It also explains the factorization method of solving quadratic equations and the square method. The chapter gives the details of relationships between discriminants and the nature of roots. Also, the problems of real-life have been solved as examples in this chapter.

Important formulas for the students to keep in mind:

If b2 - 4ac > 0, then the answer obtained should be two distinct real roots. 

If b2 - 4ac = 0, then the roots of the equation ax2 + bx + c = 0 should be -b/2a

It can be said in this case then that the equation ax2 + bx + c = 0 has two equal roots. 

If b2 – 4ac < 0, then there is no real number whose square is b2 – 4ac. Hence, there are no real roots for the given quadratic equation in this case.

b2 – 4ac is called the discriminant of this quadratic equation since, b2 – 4ac determines whether the quadratic equation ax2 + bx + c = 0 has real roots or not

NCERT Solutions for Class 10 Mathematics Chapter 5 - Arithmetic Progressions

This chapter introduces a new topic to the students - Arithmetic Progression, commonly called AP. Students will learn what is AP, the derivation of the nth term, finding the sum of first n terms of the AP and solving the real-life problems using AP in this chapter. The first Exercise of the chapter teaches how to represent a problem or situation in the form of AP, how to find the first term and difference of the AP and to find whether the given series is AP or not. The next Exercise, Ex 5.2 explains how to find the nth term of the AP using the formula. The third Exercise describes how to find the sum of the first n terms of AP and contains questions related to the topic. The last and fourth exercises have questions related to the topics taught in the chapter.

Important formulas that come in handy while solving questions from the exercises in the textbook are:

1. Finding the nth term of a progression-

nth term= a + (n-1) d

2. Sum of the first n terms in sequence-

Sn= n/2 [2a + (n-1)d]

NCERT Solutions for Class 10 Maths Chapter 6 - Triangles

Chapter 6 of class 10 gives details about the triangles. The chapter gives details about the figures with the same shape but different sizes. It explains the similarity of the triangles, theorems associated with the similarity of triangles and the concept of congruent triangles. Further, theorems related to areas of triangles, the Pythagoras theorem and the converse of Pythagoras theorem is explained.

Important theorems that students should know thoroughly while preparing for the exam include:

Theorem 1: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

Theorem 2: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

Theorem 3: If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar.

Theorem 4: If in two triangles, sides of one triangle are proportional to (i.e., in the same ratio of ) the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar.

Theorem 5: If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.

Theorem 6: The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Theorem 7: If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.

Theorem 8: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Theorem 9: In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

All the exercises throughout the chapter test how well the students have understood these theorems and if they are able to apply them and get answers to the questions. If the students have enough practice of the questions from the exercise, then understanding the theorems will be relatively easy for them as well.

NCERT Solutions for Class 10 Maths Chapter 7 - Coordinate Geometry

Class 10 Maths Chapter 7 explains coordinate geometry, the maths of locating a given point with the help of an ordered pair of numbers. The coordinate or cartesian geometry helps to find the distance between two points whose coordinates are given. The concept of finding the area of a triangle formed by three given points. Also, the ways of finding the coordinates of the point which divides a line segment joining two given points in a given ratio are explained in the chapter. The students will learn distance formula, section formula and area of triangles in this chapter of NCERT Solutions.

NCERT Solutions for Class 10 Maths Chapter 8 - Introduction to Trigonometry

Chapter 8 of NCERT Solutions introduces trigonometry to the students. Trigonometry is the study of the ratio of right triangles with respect to the acute angles, which are known as trigonometric ratios of the angles. Also, the trigonometric ratios for angles 0∘ and 90∘. Students will also learn to calculate trigonometric ratios for the given angles and also a few trigonometric identities.

The chapter gives a basic introduction to trigonometry that comes in handy for the students in Class 11 and 12 as well. They are taught the introductory formulas of trigonometry so that they have a strong grip over the basics that will help them in higher classes and the future. In addition to the formulas, the relationship between the ratios and angles is also explained in the chapter. 

Trigonometric ratios that are important for complementary angles are:

sin (90° – A) = cos A,

cos (90° – A) = sin A,

tan (90° – A) = cot A,

cot (90° – A) = tan A,

sec (90° – A) = cosec A,

cosec (90° – A) = sec A

sin2 A + cos2 A = 1,

sec2 A – tan2 A = 1 for 0° ≤ A < 90°,

cosec2 A = 1 + cot2 A for 0° < A ≤ 90°

NCERT Solutions for Class 10 Maths Chapter 9 - Some Applications of Trigonometry

Chapter 9 is the continuation of chapter 8 where the students will learn about the application of trigonometry which they learnt in the previous chapter. This chapter helps to understand how trigonometry is applied to our everyday life to find out the height and distance of various objects. They will also learn how trigonometry is applied and used in navigation, construction and determining the position of any piece of land based on the total latitude and longitudes.

NCERT Solutions for Class 10 Mathematics Chapter 10 - Circles

Chapter 10 for class 10 maths, NCERT Solutions continues the details of Circles studies in previous classes. In this chapter, students will be introduced to the concept of tangents and the number of tangents from a point on a circle. 

Important Theorems –

Theorem 1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Theorem 2: The lengths of tangents drawn from an external point to a circle are equal.

NCERT Solutions for Class 10 Mathematics Chapter 11 - Constructions

Chapter 11 is one of the interesting chapters. This chapter is all about constructing various geometric figures. The students will learn to divide a line segment, and how to draw tangents to a circle. The methods of construction are well explained in the chapter to clear the concepts.

Methods and steps for the construction of the tangents to the circle are explained in the chapter with enough examples before each exercise. Examples before each exercise are given so that understanding the methods and theorems becomes easier for the students.

NCERT Solutions Class 10 Maths Chapter 12 - Areas Related to Circles

As the name suggests, the chapter is about the perimeter and area of a circle. The concepts of finding the area of a sector and segment of a circle are clarified here. This chapter further explains how to find the area of the figures that contains a circle or part of a circle.

Important formulas from within the chapter for the students to practice–

Circumference = 2πr

Area of the circle = πr2

Area of the sector of angle θ = (θ/360) × πr2

Length of an arc of a sector of angle θ = (θ/360) × 2 π r where r is the radius of the circle

NCERT Solutions for Class 10 Maths Chapter 13 - Surface Areas and Volumes

Chapter 13 of Class 10 Maths NCERT book PDF solution is about the surface areas and volumes of different objects and shapes. In the five Exercises of this chapter, find the surface area of any object which is made up of two or more different solid shapes. The solid shapes include cuboid, cone, sphere, hemisphere and cylinder. Then, there are questions on finding the volume of the objects that are again made up of two different solid shapes. Also, there are questions that ask about the conversion from one shape to another. This chapter further explains how to find the volume, curved surface area and total surface area of a frustum of a cone.

Important formulas used for calculating surface areas and volumes of different objects

TSA of new solid = CSA of one hemisphere + CSA of cylinder + CSA of other hemispheres

Diameter of sphere = 2r

The surface area of sphere = 4 π r2

The volume of Sphere = 4/3 π r3

The curved surface area of Cylinder = 2 πrh

Area of two circular bases = 2 πr2

The total surface area of Cylinder = Circumference of Cylinder + Curved surface area of Cylinder = 2 πrh + 2 πr2

Volume of Cylinder = π r2 h

Slant height of cone = l = √(r2 + h2)

Curved surface area of cone = πrl

Total surface area of cone = πr (l + r)

Volume of cone = ⅓ π r2 h

Perimeter of cuboid = 4(l + b +h)

Length of the longest diagonal of a cuboid = √(l2 + b2 + h2)

Total surface area of cuboid = 2(l×b + b×h + l×h)

Volume of Cuboid = l × b × h

NCERT Solutions for Class 10 Maths Chapter 14 - Statistics

This is another interesting chapter where the students learn about the numerical representation of data, grouped or ungrouped. They further learn about finding the mean, median and mode of the given data. In the next exercise, they will learn about the cumulative frequency distribution and draw cumulative frequency curves.

NCERT Solutions for Class 10 Maths Chapter 15 - Probability

Chapter 15 or the last chapter of NCERT Solutions for Class 10 Maths teaches probability. It elaborates and explains the difference between experimental probability and theoretical probability.  The chapter is full of examples to clarify the concept of probability to the learners.

Probability is also an extremely important chapter for the students to focus on as this chapter is further discussed and new topics are introduced in the classes ahead. It is important that the students get a strong grip on the basics of the chapter in class 10 so that keeping up with the formulas and new topics introduced in higher classes is easier. If the students understand well and practice enough it is also an easy chapter to score good marks in.

Finding accurate answers online is a difficult task but Vedantu’s NCERT class 10 Maths SolutionsPDF is a reliable source. Students can download the PDF easily and start going through the chapters. You will find clear categorisation and step by step solutions of all the questions with a clear conclusion. This will help you to understand how you can write answers in your exams to help you score well.

CBSE Class 10 Maths Exam Pattern for 2021-2022

The Central Board of Secondary Education conducts CBSE Class 10 Board exams every year. The exam is of 100 marks where 80 marks are for theory whereas 20 marks are given for internal assessment. To understand the marking scheme in a better way, you should understand the exam pattern clearly. To score well in the exam, you should know how much mark is awarded for each question and each step of the question. Every year, just before the board exams, the board releases sample papers and marking schemes so that all the doubts of the students are cleared. According to the revised pattern of CBSE, 25% of the theory questions are going to be objective. The number of questions is increased and the marks per question are reduced. This marking scheme will be helpful to the students as the maximum number of questions will become objective or very short answer type as the marks per question has been reduced.

CBSE focuses on the all-around development of Class 10 students to prepare them for their further education. To perform well in Class 10 then becomes fairly important. So students are advised to go through the NCERT solution for Class 10 PDF regularly. CBSE marking scheme is stepwise so you have scored accordingly. Therefore students should write each step clearly and give a proper conclusion at the end of the answer. The unit wise marks weightage of the mathematics paper is as follows:

The paper has four sections. There are a total of 30 questions for 80 marks.

Class 10 Maths Chapter-Wise Marks Weightage 

CBSE Class 10 Maths Question Paper Design 2021-22

Internal assessment includes:

Pen Paper Test - 10

Portfolio - 05

Lab Practical - 05

Total - 20 marks.
 

Benefits of NCERT Solution for Class 10 Maths

NCERT 10 Maths solutions PDF created at Vedantu is one of the most sought after study materials online. Vedantu’s online learning ecosystem has everything the students need. Students go through the PDF and gain an in-depth understanding of the topics which prepares them for their final exams and also for their further studies. Some key features of the NCERT Maths solution Class 10 PDF are:

• Step by step calculation of every question to improve understanding of the topic.

• An exhaustive explanation of the problems to encourage concept-based learning.

• Up to date with the improvements made in the syllabus.

• Categorised chapter wise to improve ease of use.

• Great for revision and brushing up before examinations.

• Additional questions to help you track your progress.

• Follow an exam pattern to help you score well in the examinations.

• Great for revision as the whole syllabus is consolidated in one solution PDF that makes it easier for the students to refer to before the exam.

• Solutions are provided in a simple easy to understand language. 

• Extra solved questions are also provided so that students can refer to them while practising extra questions

• Modified to any changes made in the syllabus or exam pattern by the CBSE board.

• They are available to download for free on the Vedantu website. The students can download them at any point they feel necessary and start focussing on their preparation with the help of the solutions pdf.

How will CBSE Class 10 Maths Solutions Help You To Score High Marks in Board Exams?

Maths is an important as well as difficult subject to crack. To score high marks in maths board exams, you need to have a clear picture of the concepts taught in grade 10. If you can grasp the concepts and topics of class 10 maths, you can definitely score very high marks in the board exams. But this is not easy. You need to do constant practise along with understanding the topics and the concepts behind them. To make the work smarter for you, NCERT Solutions are provided. Here, a detailed description of all the topics is provided by the highly qualified experts of Vedantu. They put all their experience in the pdf of NCERT Solutions for Class 10 Maths. So, if you want to score high marks in board exams do refer to our pdfs which will make your scoring high.

The solutions PDF can help you at any point for wherever you are getting stuck in the syllabus. It has extra practice questions for you to assess your performance once you have understood the concepts clearly. The concepts are explained in such an easy manner that the students can understand them quickly. Maths is a subject that requires thorough practice besides the understanding of the concept. The PDF gives easy to understand explanations and enough practise questions for the students to follow. It is their complete guide for how to prepare for exams well and score good marks in the board exams. The solutions pdf have enough extra questions for you to understand the kinds of questions to expect in the exam as well. The sample papers provided online can also be practised. This will give you a chance to assess your knowledge of all the topics and also how you would attempt the actual exam.

Why Refer NCERT Maths Class 10 Solutions PDF for Board Exams?

NCERT Maths Class 10 PDF is a free PDF available online for students to use while preparing for their Maths exam. The exam pattern focuses on application-based learning where the concepts taught in the textbooks are tested. So, students need to have everything on their tips. From formulas to theorems, from identities to diagrams, all these can look like an uphill task. In order to help you in your preparations, Vedantu’s NCERT solution for Class 10 is available. You can download the free PDF for free and start your preparations.

While preparing for your exams you will find the PDF rather helpful. It is categorised into chapters which makes it perfect to use as a reference study material. If you find yourself stuck anywhere you can go to the chapter directly and find the solution to the problem in no time. During exam preparations managing your time is very important. NCERT Solutions for Class 10 PDF is created to reduce the effort in finding accurate solutions to questions. Get your copy of the free PDF here.

The chapters are also focused upon depending on their marks weightage. The solutions are in an easy to understand language. The students after learning the concepts from the NCERT textbook can just revise and practice from the solutions pdf to score good marks. The solutions PDF can become their go-to book for revision for the exams. The solutions pdf is designed to make learning easier and more interesting for the students. The teachers writing the solutions have years of experience and understand how questions need to be answered and what the teachers correcting the answer sheets are looking for in a correct answer. 

The NCERT textbook is the main textbook prescribed by the CBSE board for class 10 Maths. NCERT books are usually recommended for all classes and subjects by the CBSE. The students and teachers can refer to extra books for practice but the textbook is from where the syllabus needs to be prepared. The NCERT book is the main book that the students need to prepare for the exam. All the questions need to be thoroughly practised and the concepts need to be understood. The concepts are explained in the form of examples before the exercises in the textbook. The board exam question papers are usually formed from the NCERT textbooks themselves. The questions asked may not be exactly the same but the questions follow a similar pattern. The NCERT textbook should be the main book of preparation for the students while preparing. They should have practised each question and example from the textbook. This book is the main insight as to what to expect in the question paper. The examples are a good way to understand how the answers need to be formulated and written in the exam.

The solutions PDF provided by Vedantu is in accordance with the NCERT book and is updated when there are changes in the textbook or by the CBSE. After the students have studied from the NCERT textbooks they can refer to the solutions pdf for extra help and practice. The solutions PDF also gives answers to the exercise questions if they are getting stuck anywhere. The NCERT textbook is the first book that the students need to start their preparation for the exams.

Important Tips for Scoring Well in the Board Exams

Class 10 is an important class in most students’ school lives and is the first class where the students have to appear for a board exam. The class may seem daunting but if the students have been thorough with their studies for the whole academic year then it will not be difficult for them to score good marks at all. To achieve a good score it is necessary that the students keep the following tips and suggestions in mind for the whole year and especially while preparing for the exams.

• Be regular- Being regular with what you are learning in school every day is very important. This basically means that you should go home and revise or practice what has been taught in school. Maths is a subject where you can only score good marks if you have enough practice. Make sure that you practice the topic or the questions that were taught in school every day. If you keep up with what is being taught every day in school you will be able to prepare for the exam better. You will not forget any topic and in the end, before the exam, the syllabus will not seem overwhelming as you will have a grip on it because of regular practice. 

• Practice as much as possible. In maths, it is important that you practice as many questions as possible. Firstly attempt all the examples and exercise questions in the NCERT book. You can assess your practice and answers with the help of the solutions PDF. Assess your answers and practice all the wrong questions till you understand them and have enough practice. You can refer to the solutions pdf and attempt the extra question available. Keep assessing your answers and reviewing the syllabus. 

• Make sure to clear all your concepts well in time. If you have any doubt you should ask your teacher to help you understand them better. Not having clarity of a topic is not ideal right before the exam. It is important that you understand the concepts well in time and just go through them before the exam. You can also refer to the easy to understand solutions and explanations given by Vedantu in the Solutions PDF for Class 10 Maths. 

• While starting to prepare for the exam, it is important that you are familiar with your syllabus and the marks weightage for each topic. Besides this, it is also important that you know the pattern of the paper. This information will help you prepare better for the exam. 

• Prepare a timetable. The most important thing while preparing for exams is to have a schedule. Having a schedule will help you keep a check on the amount of syllabus you are completing and it will also help you not skip any topics. Make a timetable once you know the pattern and mark the weightage of each topic. Give time to each chapter and topic depending on your strengths and weaknesses and also depending on their weightage in the exam. Set goals for each day and week as well. This will help you stay focused and motivated and get a much larger chunk of the syllabus covered. In the timetable, it is also important that you give yourself a break and put in time for yourself. Studying for long stretches can be exhausting and after a point, your brain does not retain much. It is important that you take time to refresh yourself and continue studying once you feel active again. Figure out the time when you feel the most refreshed and only start a new topic then.

• Know your strengths and weaknesses. Every student has some topics that they may know better than others. It is important that you give equal amounts of time to each and every topic. Practice well enough so that you are confident in each topic. Keep practising the topics that you may not be fully confident in. 

• Make notes. Maths requires a lot of written practice. You can only thoroughly prepare for this subject if you are writing and practising enough questions. It is important that you note down all the important theorems from the textbook and write all the formulas down as well. Writing all the formulas and theorems in one place will help you consolidate them for quick revision. Before the exam, you can just go through your list of formulas. Writing and learning formulas will also help you memorise them better and not make a mistake during exams.

• Practice sample papers. For board exams, it is recommended that you practice as many sample papers and question papers from the previous years. The pattern in board exams for questions and the kind of questions asked in the exam stay more or less the same. Practising samples and previous year’s papers will help you understand the kind of questions to expect in the exam. You can find sample pers on the Vedantu website as well. Sample papers also have solutions that you can refer to after attempting them. Practising sample papers will also help you understand how to answer the questions and in what manner. 

• Make sure to get enough rest before the exam. Sleep well and eat well the night before and get rid of all kinds of distractions. Any kind of distraction may lead to unnecessary confusion and might shake your confidence. Be confident and prepare well. Before the exam, it is ideal that you rest well and wake up refreshed for the exam. gather all the belongings that you would need for the exam, the night before. Do not leave all of this for the morning. 

• While answering sample papers make sure to attempt them like you would an actual exam. Time yourself and try to finish your exam well in time with extra minutes to revise your answers. Practising enough sample papers will help you manage your time well and get you prepared for the exam. This will assure that you do not leave any questions unattempted due to lack of time. Assess your answers and focus on the topics that may need more practice. Practice as many questions and papers as possible. 

• While answering your exam make sure to keep in mind how you present your answer sheet. Answer all the questions in the order that they are asked in the question paper. Highlight the final answer and the formula that is used in a particular answer. Presenting your paper well helps the person check the answer sheet. Make sure to not waste time on any questions. If you get stuck anywhere you can move forward and come back to the question later. Do not leave any questions unattempted. Even if you do not know how to solve the whole question it is important that you write down at least the formula or one or two steps that you might know. This will help you fetch at least one mark for the formulas or a step that you wrote. This will avoid you from getting a zero for a particular question.

Following the above-mentioned tips and being sincere in your preparation will help you score good marks in the exams. Having a strong base in Maths in Class 10 is definitely beneficial for classes ahead and in the future. This will help you have an edge over others even in competitive exams. Most competitive exams have maths sections and the questions generally asked pertain to what you may have learnt in school. Having a good grip over the formulas and the topics will definitely help you perform well in these exams as well.