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## Unit 1: Numbers and operations

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- NCERT Solutions For Class 8
- NCERT Solutions

## NCERT Solutions for Class 8 - Free PDF Download

Till Class 8th , most of the basics are covered for the various subjects, and the students are provided with a brief idea about all the concepts and the subjects. To understand every subject briefly and be the topper of the class, it is necessary to practice all the subjects well, including the questions available in the exercises. Our expert mentors and tutors with immense knowledge of the subjects precisely design all the NCERT Solutions for Class 8 to help the students pass with outstanding results. From English and Hindi to Maths, Science and Social Studies, we provide the best Class 8 NCERT Solutions , keeping the evaluation schemes of the schools in mind. Students can have all the discretional accurate answers with examples stated (whenever necessary), for all the subjects.

Students of Class 8 need all the support and guidance that they can get so that they can perform well in their examinations. These NCERT solutions provided by the experts at Vedantu will definitely help them get the best results for sure. These solutions are provided subject-wise and chapter-wise to ensure that students are not having any trouble while going through the solutions. They can easily download the solutions for any specific subject or chapter. Download NCERT Solutions Class 8 and start the preparation for your exams. You can also download NCERT Solutions for Class 8 Maths and NCERT Solution for Class 8 Science to help you to revise the complete syllabus and score more marks in your examinations.

## Detailed Overview of NCERT Solutions for Class 8

Subject wise ncert solutions for class 8.

Mentioned below in the table is the link to subject-wise NCERT solutions for Class 8. The NCERT solutions for Maths, English, Hindi, Science and Social Science are included in the table.

## NCERT Solutions for Class 8

## NCERT Solutions for Class 8 in Hindi Medium

Ncert solutions for class 8 maths, science, english, hindi & social science - free pdf download.

NCERT solutions are highly important for the students, as from there, they can understand all the answers in detail, without much effort and money expenditures. Students can also download the NCERT solutions Class 8 free pdf of each subject through the link provided below. This pdf includes chapter-wise solutions with accurate descriptions. The solution materials are specifically designed to clear all the doubts of the students. In the PDF, apart from the NCERT Class 8 solutions , we also provide additional questions with the solution to ensure that the students do not leave anything behind while preparing for the examinations.

NCERT 8th solutions is easily available and accessible in Pdf format. The students can download the material anytime and access them on any device, whenever they wish to, without any hassles. Additionally, as the questions are stated too along with the solutions, the students do not even need to carry their books everywhere, and they can directly learn everything with perfect practice, using the Pdfs of the subject materials.

## Important NCERT Class 8 Solutions For Mathematics are given below in detail

For Mathematics, we provide the complete NCERT Solutions of Class 8, including a detailed explanation of all the theorems, their real-life applications and examples, and step-wise explanations to help the students understand them well. Further, all the numerical are well explained, and we even provide multiple solutions whenever required. Some of the important chapters and their topics which are discussed in the NCERT solutions are as follows:

Rational Numbers – This is the first chapter and you have to learn a total of 4 exercises in it. The first exercise in this chapter is the properties of rational numbers . In this exercise, you have to learn all the rational number properties, such as closure, distributive, commutative, associative, etc. Practice a few examples for each and understand the concept properly. Solve the exercise questions and you can use the NCERT solutions for Class 8 as reference. In the next exercise, you will learn how to represent rational numbers on a number line, and learn the ways for both proper and improper fractions.

Practice the steps and solve exercise examples to understand better. The last exercise is about the rational number between two rational numbers, in this exercise you have to learn how to find a rational number between two rational numbers. Practice the formulas and conditions. Solve the exercise examples and questions. You can use the NCERT Maths Solutions as a reference and for additional help.

We Cover All the Exercises in Chapter Given Below:-

Practical Geometry – This is the 4 th chapter and it is an important one. The chapter is short and quick to learn but it requires practice. There are 5 exercises in it. In the first exercise, you will be revising the basic examples and concepts. The second exercise is constructing a quadrilateral. In this exercise, you have to learn the construction of quadrilaterals when their sides, diagonals, and angles are given, learn all the steps for each set of given information such as 4 sides and 1 diagonal, 3 sides and 2 angles, etc. Practice the exercise examples and be thorough with the concepts. The last exercise is on special cases, in this exercise you have to learn about the rhombus and how to construct one. Learn and practice all the steps. You can use the NCERT solutions for extra help.

Data Handling – In this chapter there are 3 exercises. The first exercise is looking for information. Learn this exercise properly and solve the exercise questions and examples. The second exercise is organizing data, in this exercise you have to learn how to group and organize data. Learn the process and go through the exercise examples. Use the NCERT solutions for the exercise questions. Learn about tally charts properly. The 4 th exercise is a circle graph or pie chart.

This exercise is very important and you have to learn it properly, for additional help you can always use the NCERT solutions. Learn the steps for creating a pie chart and also learn the formulas. Solve the exercise examples and the questions. The last exercise in this exercise is chance and probability, you will learn about chance and probability of events, you will also learn the probability formula, etc. Practice all the examples before solving the exercise questions. Use the NCERT solutions as they will help you understand the topics and concepts better.

Mensuration – This chapter has 4 exercises and it is one of the salient chapters. The first important exercise is the area of a trapezium, you have to learn the definition and basic concept of a trapezium. Next, you have to learn the properties of a trapezium and the formula of area and perimeter of a trapezium. Practice examples and questions related to it. You also have to learn the derivation of area. Finally, you have to learn the applications of a trapezium. Be thorough with all the topics and solve the exercise questions.

The next exercise is the area of general quadrilaterals. In this exercise, you have to learn the properties of a quadrilateral and the area of the quadrilateral. Solve examples and exercise questions. Another important exercise in this chapter is the area of the polygon. Learn the definition, formula, and the formula of area. Practice examples and understand the concept properly. The 7 th exercise in this chapter is the surface area of a cube, cuboid, and cylinder. In this exercise, you have to learn the formulas and practise examples to understand the concept properly. Learn the total surface area formula of a cube, cuboid, and cylinder and solve as many questions as possible related to them. The next exercise is the volume of cube, cuboid, and cylinder, in this you have to learn how to solve questions on volume.

Practice examples and exercise questions using the Class 8 NCERT solutions for better understanding. The last exercise of this chapter is volume and capacity, in this exercise you have to learn about capacity and its formula. Solve questions related to it and be thorough. You also have to learn the difference between volume and capacity. Solve the exercise questions and practice additional questions from the NCERT solutions.

Another basic but important chapter is the introduction to graphs. In the first exercise, you have to learn about the types of graphs, how to plot graphs, uses, examples, etc. practice all the types of line graphs and go through a few examples for each. The next exercise is the linear graph, you have to learn the linear graph equation and solve questions from the exercise. You must also be thorough with the exercise examples. The last exercise is the application of a linear graph, this exercise is fairly easy and you can finish it quickly. Solve questions based on linear graph application and practice examples before moving to exercise questions. While solving the exercise questions use the NCERT solutions for help, also the topics are excellently explained and you will find it easier to learn.

## NCERT Class 8 Solutions For Science

We provide NCERT solutions Class 8 for Science, including all the chapters under the classification of Physics, Chemistry, and Biology. With the help of these solutions, the students can learn the subject well and clear all the basics. Science is already an important subject, and Class 8 has many vital chapters that everyone must know well. To ensure an easy understanding, we provide the best NCERT Solutions for Class 8 Science.

Students can access NCERT Solutions of Class 8 Physics, Chemistry, and Biology pdf through the link below

## NCERT Solutions of Class 8 Physics

In the Physics syllabus, there are 7 chapters. The NCERT solutions have covered all the topics and also provide a better explanation. Some of the important chapters and topics in the class 8 physics solutions are as follows:

Force and Pressure – In this chapter, there are 10 questions and all of them are important. Learn about the types of force, contact force, non-contact force, etc. You also have to learn what push is and what pull is, their difference, etc. Solve the questions, and be thorough. Learn about pressure, its definition, etc. You also have to learn about atmospheric pressure and pressure exerted by liquids and gases. Solve all the exercise questions and understand the concepts properly.

Friction – In this chapter there are 10 questions. Learn about friction, factors affecting, fluid friction, its uses, etc. Practice and understand the topics. Be very thorough while studying and use the NCERT solutions for the questions, this will help you to prepare better.

Sound – In this chapter, there are 13 questions. In this, you have to learn how sound is produced by a vibrating body. Be thorough while reading and understand the concept. In the next, you will learn how sound is produced by humans. You also have to focus on the topics like amplitude, time period, and frequency. Learn about noise and noise pollution also. Practice all the questions and revise daily.

Some Natural Phenomena – In this chapter, you will learn about natural phenomena like lightning, transfer of charge, charging by rubbing, etc. You also have to learn about lighting safety and earthquakes. Solve the exercise questions and study the topics carefully. Maintain separate notes for revision and practice. The story of lighting is very important. You must study it thoroughly. Use the NCERT solutions for extra support and help.

Starts and The Solar System – This is the last chapter in your physics syllabus. In this chapter, you will learn about the moon, the stars, constellations, the solar system, etc. Learn all the topics thoroughly, and practice the questions.

NCERT Solutions of Class 8 Chemistry – In the chemistry solutions there are 5 chapters. Some of the important chapters and their topics are as follows:

Synthetic Fibres and Plastics – There are 15 questions in this chapter. Learn about synthetic fibres, their types, characteristics, etc. You also have to learn about plastics, their advantages, and their disadvantages.

Coal and Petroleum – In this chapter, there are 9 questions . In this chapter, you will learn about coal, petroleum, natural gas, etc. You also have to learn their use, advantages, and disadvantages. Be thorough while solving the questions and use the NCERT solutions as a reference.

Pollution in Air and Water – This is the last chapter and there are 11 questions . You have to learn about air pollution, the greenhouse effect, etc. study the cases, and be thorough with the chapter. You can also highlight the salient points. Next, you have to learn about water pollution, how it gets polluted, precautions and ways to minimize them, etc. Practice and revise the chapter properly. This chapter looks lengthy but it is very easy and you can score well in it. NCERT solutions provide a great explanation for all the above topics thus, you can prepare from it.

## NCERT Class 8 Solutions For Social Studies

Social Studies itself is a vast subject, and it has three sub-parts, including History, Civics, and Geography. We provide NCERT solution Class 8 for Social Studies to help the students understand all the chapters well, from all three subdivisions of the subject. The list of chapter in Social Studies for Class 8 include:

## NCERT Solutions for Class 8 English

Given below are the chapter-wise NCERT Solutions for Class 8 English. These solutions are provided by the English experts at Vedantu in a detailed manner. Go through these chapter-wise solutions to be thoroughly familiar with the concepts.

## NCERT Solutions for Class 8 Hindi

Given below are the chapter-wise NCERT Solutions for Class 8 Hindi. These solutions are provided by the Hindi experts at Vedantu in a detailed manner. Go through these chapter-wise solutions to be thoroughly familiar with the concepts.

## NCERT Class 8 Subjects - Marks Weightage

Apart from having the best Class 8th NCERT solutions, it is also necessary to know the marks Weightage well, as it helps a lot in the preparations. Every subject carries a total of 100 marks in Class 8, and in that 20 marks are for internal assessment and 80 for the final examinations.

## Significance NCERT Solutions for Class 8 All Subjects

Students of Class 8 need to complete their syllabus properly if they want to stay ahead in class and perform well in their exams. They can manage to cover the syllabus if they follow the classes regularly and study the chapters. However, in order to get some detailed insights about each subject and specific chapters, students need something more than classes. So, on that note, the Class 8 All Subject Question Answer PDF along with NCERT solutions will help the students completely dominate their syllabus and score good marks in the examination. Students can easily find the solutions to any question from the NCERT books of Class 8 . Not to mention that they can also get detailed solutions for other additional important topics that might be asked in the exams.

Regular practice of the NCERT solutions will enable students to understand the exam pattern of NCERT . Vedantu experts have taken their time to formulate these solutions according to the CBSE guidelines. Thus students can definitely get the guidance and support that they need to improve their answering skills and expand their knowledge.

## Benefits of Downloading NCERT Solutions Class 8 Chapter Wise List

The experts are well-versed in every subject and have designed the solutions in such a way that students will get correct explanations of the different questions. This way, they will be able to better understand the concepts and subjects taught in class 8.

The solutions are specifically designed to make sure that the students do not miss out on anything, and get the best preparation for the examinations.

The solutions provided are systematically designed, and well supported with examples and in-depth descriptions.

Along with the solutions, the important questions are also marked, and the students are also provided real-life applications of the topics.

The NCERT solutions also carry the explanations of the lessons, ensuring that the students understand every concept briefly.

The students will be able to analyse the important questions as well as NCERT solutions to figure out the probable questions that might be asked in the examinations. So, this way they will be able to prepare themselves better for the final exams and score more.

Even if you miss out on some of your classes, you can just download NCERT solutions from Vedantu for Class 8 and study from there. These PDF files contain notes, solutions, important questions, and other study materials that will help you learn better.

Download Free NCERT Class 8 Solutions from Vedantu right now and learn about the important topics covered in Class 8 NCERT Solutions . Students will be able to boost their exam preparation strategy by downloading and studying from these important solutions. They can find out the right way to answer questions so as to score more marks in the exam. Not to mention that students will also be able to finish their revision early with the important solutions and notes and thus enhance their exam performance.

## Important Related Links for NCERT Class 8

CBSE Class 8 Revision Notes

NCERT Books for Class 8

Important Questions for CBSE Class 8

NCERT Exemplar Solutions for Class 8

CBSE Class 8 Syllabus

CBSE Class 8 Maths Formulas

## FAQs on NCERT Solutions For Class 8

1. Why is NCERT solutions Class 8 Pdf important?

Having NCERT solution of Class 8 in pdf format is very useful for the students to learn well and get the best preparation. The pdf format of the solution ensures that they can be easily downloaded anytime and anywhere, and can be easily accessed on any device. These pdf formats of the solutions also contain questions and problem statements. It ensures that the students do not need to carry the entire books everywhere. Students can secure good marks referring to them.

2. What are the advantages of having the Class 8 NCERT solution?

There are many advantages of having an NCERT solution for class 8, like:

These solutions are designed by the experts, making sure that the students get a proper understanding of them.

Having the NCERT solutions ensures that the students will not face any hurdles while preparing.

Having the NCERT solutions also makes sure that the students will get the best preparation materials and will not lag anywhere.

These solution materials also contain the explanation of the chapters, making it easier for the students to understand every concept.

In the solution pdfs, the important questions are also marked, and examples of every solution are also stated.

3. What should I do to get a good percentage in Class 8 exams?

If you are a Class 8 student and wish to get a good percentage in exams then follow the tips given below.

You must start preparing for the exams at an early stage right from the starting of your academic session. Doing so will enable you to complete your course on time so that you can invest your extra time in revision.

Try to attend your school daily.

Revise all subjects and their respective chapters at your home.

Practice the textbook questions along with the NCERT Solutions of Class 8 to grasp the concepts.

4. How can I make a study plan for Class 8?

The steps for making a good study plan for Class 8 are-

First, you should begin with knowing the faults in your current timetable. You have to rearrange your timetable and do those activities first that are important. For instance, preparing for class tests, home assignments, etc.

Know when you like to study. Whether you like studying early in the morning or at night. Just know the time and start studying.

Be sure to take small breaks every 1-2 hours of studying. You can take a 5 to 10 minutes break and then resume.

Try to give at least one hour to every subject for revising concepts. However, you can increase or decrease the time duration as per your preference and preparation.

5. How can I download the NCERT Solutions of Class 8 for different subjects?

Ans: Here are the steps to download the NCERT Solutions of Class 8 for different subjects.

There will be a download link for each subject and chapter in this page.

Click on the link.

You will see a download PDF option on that page.

Click on it to download offline for free.

6. Are subjects of Class 8 easy or difficult?

Well, the answer to this question depends on the calibre of the student. Some students may find this class easy to clear as they are good at academics. At the same time, there are some students too, who will find this class tough as they are not good in their studies. Such students are recommended to begin with their studies at an early phase so that they can complete their syllabus on time along with understanding the concepts thoroughly. Moreover, they can refer to the NCERT Solutions for all subjects and the chapters for better comprehension. All the study material is available on the Vedantu app and the website.

7. Why is Vedantu considered the best online learning platform to study for Class 8?

Vedantu is considered the best online learning platform to study Class 8 due to the following reasons-

It provides different types of study materials including the NCERT Solutions of the chapters, revision notes, important questions, sample question papers, etc., so that the students can learn and understand the concepts clearly.

These study materials are prepared by professionals who have years of experience in the field of teaching.

The content in these study materials is easy to comprehend.

They offer the facility of video tutorials for efficient learning.

## CBSE Class 8 Study Materials

- NCERT Solutions
- NCERT Class 8

## NCERT Solutions for Class 8

Ncert solutions for class 8 – download pdfs.

NCERT Solutions for Class 8 is a very crucial study material as it is responsible for laying a strong foundation for your higher classes. However, there is no need to get too stressed or overwhelmed. The best study material any Class 8 student can use is NCERT Books . The textbook explains all the concepts that have to be learned by a Class 8 student neatly and efficiently.

Class 8 students should have a firm grip on the answers to the questions given in the NCERT textbook. The NCERT Solutions for Class 8 Maths and Science are given here for you to refer to. These solutions provided at BYJU’S are incredibly accurate and offer detailed solutions for every question present in each chapter of Maths and Science. These comprehensive explanations, along with the answers, are designed by expert subject teachers after extensive research to produce an authentic and appropriate source of NCERT Solutions for you. The solutions for Class 8 NCERT Science and Maths will help you clear your doubts instantly and understand the concepts quickly. These NCERT Solutions, if studied meticulously, can help you get rid of all your doubts.

Maths and science are subjects of extreme importance in the curriculum of a student. Hence, emphasizing practising the questions and solutions to understand any concepts in the given chapters is of utmost importance to secure good marks and advance in your desired career. For more practice, you can check out NCERT Books for Class 8 and practise the questions and answers based on the CBSE Syllabus for Class 8 .

NCERT Class 8 Solutions comprises chapter-wise solutions, providing you with the key to unlocking your problem-solving skills. Choosing the right learning strategy will make a profound impact on your academic career. These solutions will help you solve the questions as well as cross-verify your answers simultaneously. The prime motto of providing the perfect NCERT Solutions is to help you understand the topics and concepts comprehensively. This understanding helps you in building a strong foundation for higher grades.

## NCERT Solutions for Class 8 Maths

BYJU’S provides NCERT Solutions for Class 8 Maths Chapters 1 to 16, prepared by our expert faculty, to help you in your exam preparations. Download NCERT Class 8 Maths Solutions of all chapter-wise PDFs to find a better approach to solving the questions in the book. These solutions will also help you have a deeper understanding of concepts covered in NCERT Class 8 Maths Textbook . Practising the textbook questions will help you analyse your level of preparation and knowledge of concepts in the book.

The chapter-wise answers to all the questions present in the 16 chapters of NCERT Class 8 Maths, written in a precise and lucid manner, are provided here. You can refer to the PDFs of Class 8 Maths NCERT Solutions as your additional references and study materials to help you learn the essence faster. Practising textbook exercises will surely help you to prepare efficiently for your examination.

## NCERT Solutions for Class 8 Science

BYJU’S brings you Science NCERT Class 8 Solutions framed by our subject experts to facilitate a practical and smooth understanding of the concepts related to physics, chemistry and biology. These CBSE 8 Science NCERT Solutions can be accessed anytime and anywhere, at your convenience, to understand the concepts in a better way. These Class 8 Science NCERT Solutions are explained beautifully using a step-by-step approach for all the questions given in the textbooks. It acts as an essential tool for you in your exam preparation.

We have also provided NCERT Solutions Class 8 Science PDFs in an easily accessible format to help you prepare adequately. You can download and practise these NCERT Science Solutions for Class 8 to get a thorough knowledge of the concepts covered in the NCERT Books for Class 8 Science .

## NCERT Solutions for Class 8 Social Science

BYJU’S brings you Social Science NCERT Class 8 Solutions framed by our subject experts to facilitate a practical and smooth understanding of the concepts related to History, Geography and Civics. These CBSE 8 Social Science NCERT Solutions can be accessed anytime and anywhere, at your convenience, to understand the concepts in a better way. These Class 8 Social Science NCERT Solutions are elaborately explained for all the questions given in the textbooks. It acts as an essential tool for you in your exam preparation. Also, access the NCERT Solutions of Class 8 History, Geography and Political Science (Civics) from the links provided below:

We have also provided NCERT Solutions Class 8 Social Science PDFs in an easily accessible format to help you prepare adequately. You can download and practise these NCERT Social Science Solutions for Class 8 to get a thorough knowledge of the concepts covered in the NCERT Books for Class 8 Social Science .

## NCERT Solutions for Class 8 English

BYJU’S brings you English NCERT Class 8 Solutions designed by our subject experts to facilitate a practical and smooth understanding of the concepts which help prepare you for the board exams. These CBSE 8 English NCERT Solutions can be accessed anytime and anywhere, at your convenience, to understand the concepts in a better way. These NCERT Solutions for Class 8 English are explained perfectly for all the questions given in the textbooks. It acts as an essential tool for you in your exam preparation. The solutions are prepared according to the latest syllabus and guidelines as it contains answers to all the questions present in the textbook, along with sample papers and practice papers.

## How and When to Solve NCERT Solutions

It is often seen that the students directly jump to solving the questions present in the NCERT textbook. But is that the right way to learn? No, it is definitely not how you should learn. Here you go with the proper method of learning NCERT Solutions.

- For theoretical subjects, like Science, the first thing you have to do is to read the NCERT textbooks thoroughly. However, just reading it without giving any attention to it will do no good to you. You should make it a habit to read the textbook by understanding the concepts. You should read the book at least three times before moving on to solving the questions present in the NCERT book.
- In the case of numerical subjects, like Maths, you should give attention to the teachers in the class to understand the concepts. Once you get familiarised with the concept, the next thing you should do is understand the method of solving a question. To understand the problem-solving method, you should first try solving the solved examples given in the NCERT textbook. If you think that you are thorough with the examples, try solving them again without looking at the solutions given in the textbook. Compare your answers with the answers given in the textbook.
- If you follow the steps mentioned above, by the time you are done with it, you would be having a strong knowledge of the concepts as well as the problem-solving method of the chapter you have just learnt. Now, you can go ahead and try solving all the questions in the textbook. Make sure that you try solving them yourself without referring to any solutions.
- Once you are done with solving the questions in the exercises or at the end of the chapter, take out the NCERT Solutions provided at BYJU’S, and compare your answers with it.
- Give yourself a pat when you find that you have answered the questions right. But, on the other hand, if you have given incorrect answers, note down the questions and understand the right way to solve them by referring to the solutions. Practise it again and again till you are thorough with the solutions, and once you feel you are proficient in them, try solving the questions again and check the answers. Repeat the step until you find yourself giving the right answers for all the questions given in the textbook.

## Exam Preparation With NCERT Solutions

NCERT books & NCERT Solutions for Class 8 are more than enough to score high marks in the exam if you focus and read them with full concentration. The chances of out-of-syllabus questions being asked in the Class 8 exams are very less. Scoring high marks in the Class 8 examination is very easy. To score decent marks in the Class 8 examination, you don’t need to have an extraordinary brain. On the other hand, all you should have is a firm grip on the concepts, a logical approach and rational thinking. To score high in your Class 8 exam, a clear understanding of the basics, a command over formulae, and a sound practice regimen are all that you need. Here are some tips for using NCERT Solutions in your favour during exam preparation.

- Early preparations: Starting your preparations early, right from the beginning of the academic year, is the best way to avoid any last-minute haste. Doing so will help you provide enough time for each subject as well as for solving the NCERT Solutions well in advance. Solving the solution well ahead of time can help you in clearing your doubts related to them at the earliest.
- Review your syllabus: Going through your whole syllabus and highlighting the main topics to be studied is the first step towards studying right. Ensure to solve the Solutions of NCERT related to the topics you find important or difficult.
- Know the exam pattern: It is important to understand the exam pattern approved by the Board concerned with the exam you are going to appear for. By doing so, you will get an idea of the chapter that has more weightage and can help you concentrate more on the solutions of that chapter.
- Study regularly: Study the solutions regularly, along with the NCERT books, to be extremely familiar with the problem-solving method.
- Revision: As the final step, revise all the NCERT Solutions as the exam date gets nearer to brush up on the concepts and keep them fresh in your mind.

## Other NCERT Resources For Class 8

Ncert exemplars for class 8.

The solutions for all the questions present in the NCERT Exemplars books of Class 8 Maths and Science are given here. The NCERT Exemplars for Class 8 have been developed to help you in your academics. We at BYJU’S, are providing these NCERT Exemplar Solutions to facilitate easy learning for students. You can quickly go through the solutions and find out accurate answers. You can find the subject-wise links for Class 8 exemplar solutions in the table below.

## NCERT Books For Class 8

The NCERT aims to provide qualitative education to school children in India by preparing and publishing textbooks, study materials, educational kits, etc. The core of the schooling system is NCERT Books. NCERT books for Class 8 Maths and Science are provided at BYJU’S, which can be downloaded from the links given below.

## NCERT Syllabus For Class 8

The key to exam preparation is knowing the syllabus. Knowing the syllabus will help you understand the topics and concepts that will be covered in the academic session, in turn, helping you to make proper use of other NCERT resources.

## NCERT Solutions and Competitive Exams

The National Talent Search Examination Syllabus consists of the topics covered in Mathematics and Science, along with other subjects of Class 8 as well. Hence, solving NCERT Solutions for Class 8 will help you understand at least a few of the concepts mentioned in the NTSE Syllabus . Similarly, in the case of Olympiads, the Class 8 NCERT Solutions for Class 8 Maths plays an important role in the Maths Olympiad while Class 8 NCERT Solutions for Class 8 Science plays an important role in the Science Olympiad. Therefore, the more you solve the NCERT Solutions, the more familiar and thorough you get with the concepts, helping you excel in Competitive Exams like NTSE and Olympiads.

## NCERT Solutions in IIT-JEE Preparation

Getting into a reputed institute to pursue engineering is a dream for many students. Yearly, lakhs and lakhs of people attempt the IIT-JEE , out of which only a few succeed in making it to the reputed institutions. This is one of the principal reasons that you are advised to start the preparation early. Starting your preparation for IIT-JEE soon will give you an edge over other students. The earlier you start, the more time you can provide JEE with. You can give yourself ample time to get your basics right and to balance between school and coaching classes. You can also dedicate your time to preparations for JEE during the weekends, still doing a great job in polishing yourself for the IIT-JEE. You can revise the subjects on a regular basis and recollect the concepts with ease, even at the time of examination.

## Frequently Asked Questions on NCERT Solutions for Class 8

Can i score well using byju’s ncert solutions for class 8 for my exam preparation, how should i study the ncert solutions for class 8 to attain a remarkable score, what are the advantages of using the ncert solutions for class 8 from byju’s, leave a comment cancel reply.

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Math Assignment / Class X / Chapter 8 / Trigonometry

Extra questions of chapter 8 class 10 Trigonometric Functions with answer and hints to the difficult questions. Important and useful math assignment for the students of class 10

For better results

- Students should learn all the basic points of Trigonometry
- Student should revise NCERT book thoroughly with examples.
- Now revise this assignment. This assignment integrate the knowledge of the students.

ASSIGNMENT FOR 10 STANDARD TRIGONOMETRY

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## Lesson Plan Math Class 10 (Ch-1) | Real Numbers

## Lesson Plan Maths Class XII | For Maths Teacher

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## WEEK 8 Class 2 - Assignment

Due: Wed Apr 17 11:59 pm Late submissions accepted until Fri Apr 19 11:59 pm

Assignment by Julie Zelenski, with modifications by Nick Troccoli, Chris Gregg, Katie Creel and Brynne Hurst

Along with this assignment, we've posted a page of extra bits practice problems with solutions. There is also an online practice site . These are very helpful to confirm your understanding of bit operators, binary, and more as you get started with this assignment!

We'll cover the remaining shift ( << and >> ) bit operators and the GDB debugger in the lecture following the release of this assignment, both of which are needed to work on the coding portions of the assignment. However, prior to this lecture, you can complete the Case Study portion of the assignment and read through the coding problem descriptions to understand what is being asked in each part.

## Learning Goals

This assignment covers topics from the first few lectures and the first lab. You will be building your skills with:

- editing, compiling, testing, and debugging C programs under Unix
- writing code that manipulates bits and integers using the C bitwise and arithmetic operators
- working within/around the limits of integer representation
- understanding the ramifications of integer representations in real-world software

You shouldn't need material related to strings/pointers for this assignment - in other words, this assignment focuses just on the material related to bits and bytes.

This assignment asks you to complete three programs and one case study. Each program explores a particular aspect of bitwise or numeric manipulation:

- Maxwell's Demon has you model gas particles in chambers.
- UTF8 has you create unicode characters by constructing bit patterns.
- Saturated Arithmetic has you detect overflow in addition operations.
- The Ariane-5 Case Study has you examine the consequences of a real-world overflow bug.

The code you will write is short, but these small passages are mighty!

Recommended Timeline For Completion : As with all CS107 assignments, you must start early . There are three different programs to work on for the assignment, and our recommendation would be to try and complete one program every two to three days, with the assignment a bit over a week long in total.

- The working on assignments page contains info about the assignment process.
- The collaboration policy page outlines permitted assignment collaboration, emphasizing that you are to do your own independent thinking, design, coding, and debugging. If you are having trouble completing the assignment on your own, please reach out to the course staff; we are here to help!

To get started on the assignment, clone the starter project using the command

The starter project contains the following:

- readme.txt : a text file where you will answer questions for the assignment
- sat.c . util.c , utf8.c and Makefile : three partially-written programs that you will modify, and their Makefile for compiling
- maxwell_demon.c : the provided partial implementation of the Maxwell's Demon program. You do not need to modify this file.
- custom_tests : the file where you will add custom tests for your programs
- SANITY.ini , sanity.py and prototypes.h : files to configure and run Sanity Check. You can ignore these.
- maxwell_demon_soln , sat_soln and utf8_soln : executable solutions for the programs you will modify.
- tools : contains symbolic links to the sanitycheck and submit programs for testing and submitting your work. It also contains a new codecheck tool for checking for style and other common code issues.

Assignment Support: Through TA helper hours and the discussion forum, our focus will be on supporting you so that you can track down your own bugs. Please ask us how to best use tools (like the brand-new GDB!), what strategies to consider, and advice about how to improve your debugging process or track down your bug. We're happy to help you with these in order to help you drive your own debugging. For this reason, if you have debugging questions during helper hours, please make sure to gather information and explore the issue on your own first, and fill out the QueueStatus questions with this information. For instance, if your program is failing in certain cases, try to find a small replicable test case where the program fails; then, try using GDB to narrow down to what function/code block/etc. you think is causing the issue; then, further investigate just that area to try and find the bug. As another example, if your program is crashing, take a similar approach, but try using GDB and the backtrace command (check it out in lab!) to narrow in on the source of the crash. This information is important for the course staff to effectively help you with debugging. Starting with a future assignment, we will require this information when signing up for helper hours for debugging help , so please make sure to provide as much information as possible.

## Tools and Testing

We have added a new tool, codecheck , that can check your code for common style and other issues that are detectable at compile-time. This check is NOT exhaustive; it is only meant to identify common style and other code issues. Check the course style guide and assignment spec for more code guidelines. You can run Codecheck like the sanitycheck and submit tools, from the tools folder, by running tools/codecheck , which runs it on all .c files in the current directory. You can also run it on individual files, like this:

Note that Codecheck does check for magic numbers , but for assign1 it may not align exactly with the best approach, as it considers anything above a certain numeric value automatically a "magic number". For this assignment, it's not necessarily stylistically bad if Codecheck shows any issue messages about magic numbers. But, you should still be vigilant and evaluate each suggestion from Codecheck to decide how you can make your code as clean as possible.

You only need to run codecheck on the files you are modifying for this assignment: sat.c , util.c and utf8.c .

This assignment heavily emphasizes testing. For each program you write below, you should also add at least 3 to 5 additional tests of your own per program (total of at least 10) in the custom_tests file that show thoughtful effort to develop comprehensive test coverage. When you add a test, also document your work by including comments (on their own line, starting with # ) in the custom_tests file that remind you why you included each test and how the tests relate to one another. The tests supplied with the default SanityCheck are a start that you should build on, with the goal of finding and fixing any bugs before submitting, just like how a professional developer is responsible for vetting any code through comprehensive testing before adding it to a team repository.

The best way to approach testing on this assignment is, once you understand the expected program behavior but BEFORE you write any code, write some tests that cover various cases you can think of. This is because once you start writing code, you may start to think in terms of how your code works rather than how the code should work, meaning if you omit handling a case in your code, you may also omit covering that case in your testing. Thus, a good strategy is to write some tests before implementing anything, and then as you implement, you can add further tests. Use the tests as a way to gauge your progress and uncover bugs! We provide some testing recommendations in each problem section below.

How do you brainstorm what additional tests to consider? First review the cases supplied with default sanitycheck to see what is already covered. Then consider the full range of inputs/possibilities beyond those simple cases. If the variety of inputs is sufficiently small, you may be able to enumerate a test case for each possibility. In other situations, exhaustive coverage is not feasible, so you instead identify representative samples to validate the operation on specific cases. You should target ordinary cases as well as edge conditions and unusual inputs. For more advice, check out our guide to testing , from the assignments dropdown.

## Review and Comment

When you inherit code to build on, you should first carefully review it to understand its behavior and plan how your code will fit into the existing program. You are given about 25 lines for each of sat.c and utf8.c and will add about a dozen lines of your own for each of the 3 programming parts, so there is more starter code to review than new code to add.

Our provided code for the sat.c and utf8.c programs processes the command-line arguments, handles user error in how the program is invoked, and has a few important details, particularly in converting string arguments to numbers. The triangle program from assign0 used the atoi function, which is simpler to use but has no capability to detect malformed inputs. For these programs, we are using the more robust, but complex, strtol . Read the function documentation at man strtol and review how the function is used in convert_arg in sat.c and utf8.c .

Some questions you might consider as a self-test for the sat.c and utf8.c programs:

- How do the sample programs respond if invoked with missing arguments? excess arguments?
- When converting the user's string argument to a number, what base is passed to strtol ? In which bases does this allow the argument to be specified? How does the user's argument indicate which base to use?
- What does strtol do if it encounters an invalid character in the string to convert?
- How is the second argument to strtol used for error-detection? Why is it ok that the end variable is dereferenced when it is initialized to NULL ? If you did not want that error-detection, what do you pass as the second argument instead?
- What does strtol do if asked to convert a string whose value would be too large to be represented in a long ?
- Do you see anything unexpected or erroneous? We intend for our code to be bug-free; if you find otherwise, please let us know!

After reviewing the code, try experimenting with it! Build and run the programs, invoking them with different arguments to see how each is treated. Run the program under the debugger (check out our Debugger Guide under "Handouts") and step through code, printing intermediate values as you go. Observing the code in action is an excellent way to better understand it. Confirm your answers to the self-test questions above to verify your understanding.

We intentionally left off all the comments on the starter code, so you should add in documentation when implementing each program below. Add an appropriate overview comment for the entire program, document each of the functions, and add inline comments where necessary to highlight any particularly tricky details. (One exception: you do not need to read, modify or add comments to maxwell_demon.c ).

## 1. Maxwell's Demon

Maxwell's demon is a fun thought experiment about the second law of thermodynamics postulated by James Maxwell. In the original experiment idea, a "demon" controls a little door between two chambers of gas in a box. The demon can see individual molecules of gas, and it is quick enough to open and close the door such that it allows only slow particles to pass through the door in one direction, and only fast particles to pass through the door in the other direction. In this way, the slower particles congregate on one side, causing that side of the box to have a low temperature, and the fast particles congregate on the other side and create a high temperature. This thought experiment turned out to seemingly violate the second law of thermodynamics (though ultimately the second law is safe - read the full article for more information if you're interested!).

We have created a game maxwell_demon loosely based on Maxwell's Demon where the player acts as the "demon" to try and isolate particles in one of the chambers. You will implement 3 short functions in util.c to complete its implementation using bit operations.

The game screen looks as follows - there is a left chamber and a right chamber, with a barrier in the middle separating them. Each chamber is a grid, with particles at different locations within that grid. Gas particles bounce around in each of the chambers, and in this game all particles travel at the same speed.

We represent each of the two chambers as an array of 64-bit unsigned long s, where each unsigned long represents a row of the chamber. The unsigned long for a single row contains information about the gas particles (if any) in that row at that time.

The vertical barrier in the middle separating the two chambers is represented by a separate unsigned long , with a 0 at each place where there is an opening, and a 1 everywhere else - the least significant bit of the unsigned long represents the portion of the barrier at the bottom of the screen, and the more significant bits represent the portions of the barrier towards the top of the screen.

The provided code in maxwell_demon.c implements the game and animations already; your task is to implement 3 key helper functions in util.c , which are used by the provided code, to complete the implementation:

- barrier_from_openings : creates the unsigned long value for the vertical barrier based on the number of openings it should have.
- get_dir_value : gets the information for the gas particle at a given row and column number in one of the chambers.
- set_dir_value : sets the information for the gas particle at a given row and column number in one of the chambers.

(Bonus: this is not required, but if you want a particularly interesting bitwise challenge, trace through the already-implemented count_particles function in maxwell_demon.c for some cool uses of bit operations).

Before you begin, take a moment to run the sample solution (in the samples/ folder) and try out the maxwell_demon game for yourself! You can use the o and c keys to (o)pen and (c)lose the barrier, one step at a time. The object is to get all of the particles onto one side of the barrier. The difficulty changes as more particles are added, and as the simulation speeds up (you can type + to speed up, and - to slow down during the game). There are other configuration options as well - run ./maxwell_demon -h to view the full list. There is also a video below of the game in action:

For the functions below, you are not allowed to use any loops in your implementations. Instead, consider how bit operations can help you implement the necessary functionality.

## Step 1: barrier_from_openings

The barrier_from_openings function returns an unsigned long correctly representing a vertical barrier with the specified details:

- barrier_openings : the number of openings in the middle of the barrier (guaranteed to be a number between 0 and 63 inclusive, and <= num_rows )
- num_rows : the number of rows high the barrier is (guaranteed to be a number between 1 and 64 inclusive)

The function should return an unsigned long as follows:

- The least-significant num_rows bits should all be 1s, with the exception of the bits representing the opening, which should be 0s and centered within the num_rows 1 bits.
- The bits in the unsigned long outside of the num_rows bits should be 0s

As an example, barrier_from_openings(3, 20) would return 0000000000000000000000000000000000000000000011111111100011111111 . This would represent a barrier with 3 openings, 20 rows high. Another way to visualize this is as follows:

Note that here, the "extra" unneeded bits in the unsigned long (beyond the least significant 20) are all 0s.

If the opening size is even, then the opening should be exactly centered within the least-significant num_rows bits. If the opening size is odd (as above), the extra opening bit should be towards the least significant half (in other words, in the diagram above there are nine 1s above the opening and 8 1s below).

## Testing Recommendation

To test barrier_openings , you can run ./maxwell_demon -test-barrier barrier_openings num_rows , where barrier_openings and num_rows are the parameters to the function. You can test for the correct values by running the solution code, as in the following example:

In particular, make sure to check edge cases for various values; as a hint, make sure to test the full range of possible values of the two parameters! . Keep an eye out in particular for shifting a value by its bitwidth or more, which is undefined behavior.

## Step 2: get_dir_value

The get_dir_value function returns the bits encoding information for the particle at a given row and column index within a chamber:

- rows : an array of unsigned longs representing the rows in one of the two chambers
- row_idx : the index within rows that we want to fetch (guaranteed to be a valid index)
- col_idx : the index (from the left - i.e. col_idx = 0 refers to the leftmost index in the row) within the row that we want to fetch (guaranteed to be a valid index)

So how does an unsigned long encode information about a single row of particles? We might think we could use 1 bit per particle (1 if particle, 0 if no particle), for a total of 64 particles per row. However, in addition to storing whether there is a particle at a given index, we must also store a particle's current direction (e.g. north, south, southwest, etc.). For this reason, we will encode each particle using 4 bits ; this means each row fits 16 indexes of particles, aka 16 groups of 4 bits. Here is how those 4 bits are used to encode a particle:

- The highest-order bit encodes whether or not there is a particle at that index. 1 indicates there is a particle, and 0 indicates no particle.
- The lower three bits encode the direction of the particle, as follows:

Here are a few examples:

- 1000 would encode a particle traveling north
- 0XXX would encode no particle (the lower 3 bits do not matter in this case)
- 1100 would encode a particle traveling south

Thus, each row has 16 indexes, each 4 bits big. For example, a full row could be 0xd8009000000a000 in hex. In binary, this comes out to

This can be broken into 16 four-bit numbers, each representing one of the 16 indexes in this row. Index 0 is the leftmost group, and index 15 is the rightmost group:

rows is an array of these unsigned long encodings, which in total can represent an entire chamber. Neat!

get_dir_value should return the 4 bits that encode information about the specified row/column index. For example, if row_idx refers to the row with the above binary representation:

- if col_idx = 0 , the function would return 0 (the leftmost group of 4 bits)
- if col_idx = 1 , the function would return 13 ( 0b1101 , or 0xd ) (the second-leftmost group of 4 bits)
- if col_idx = 15 , the function would return 0 (the rightmost group of 4 bits)

To test get_dir_value , you can run ./maxwell_demon -test-get-dir row col_idx , specifying the 64-bit row value and the column number. Here are some examples (these are provided as example tests in custom_tests ):

In particular, make sure to check edge cases for various values; as a hint, make sure to test the full range of possible index values! . Keep an eye out in particular for shifting a value by its bitwidth or more, which is undefined behavior.

## Step 3: set_dir_value

This function is similar to the get_dir_value function, except instead of returning the bits at the specified index, it changes the 4 bits at the specified index to a new set of 4 bits and stores that updated value back in rows . Specifically, it should overwrite whatever 4 bits are already at that index and replace them with the new bits specified .

new_value 's 4 least-significant bits are the new bits to set at the specified row/col index; all other bits in new_value beyond these are guaranteed to be 0s. As an example, let's say the specified row has the following binary representation:

If col_idx = 12 and new_value = 0x6 , rows would be updated to contain the following new value at the specified row index (underlined), completely overwriting the existing 4 bits :

To test set_dir_value , you can run ./maxwell_demon -test-set-dir row col_idx new_value , specifying the 64-bit row value, the column number and the new value to set at that index. Here are some examples:

You can also test by running the full simulation program - if you run ./maxwell_demon -h you can see the list of options, including an option to set the "random seed" (which means that the game won't be randomly different each time; the same random seed will result in the same particle behavior), and using -d for debug mode to proceed one frame at a time.

A C char is a one-byte data type, capable of storing 2 8 different bit patterns. The ASCII encoding standard ( man ascii ) establishes a mapping for those patterns to various letters, digits and punctuation, but is limited to 256 options, so only a subset are included. Unicode is ASCII's more cosmopolitan cousin, defining a universal character set that supports glyphs from a wide variety of world languages, both modern and ancient ( Egyptian hieroglyphics, the original emoji? ). This large number of characters requires a different system than just a single char , however. For this part of the assignment, you will implement the most popular Unicode encoding, UTF-8, which according to recent measurements , is used by over 93% of all websites (with known character encodings)!

Unicode maps each character to a code point, which is a hexadecimal number. There are over 1 million different code points in the Unicode standard! A character's code point is commonly written in the format U+NNNN , which signifies the hexadecimal value 0xNNNN .

However, the Unicode standard does not specify how to best encode these code points in binary. There are a variety of different possible Unicode encodings which specify, given a code point, how to actually store it. Why multiple encodings? Each may have a different priority, whether it is compatibility with other non-Unicode encodings, the locale it is intended for, space efficiency, etc. Some questions that may have different answers depending on the encoding are: should smaller code points take up the same amount of space as larger code points? If not, how can we tell how long the encoding is? And is it possible to preserve compatibility with other encodings, such as ASCII?

One of the most popular Unicode encodings is UTF-8. UTF-8 is popular partially because it preserves backwards compatibility with ASCII - in fact, it was designed such that ASCII is a subset of UTF-8, meaning that the characters represented in the ASCII encoding have the same encoding in ASCII and UTF-8. The UTF-8 encoding represents a code point using 1-4 bytes, depending on the size of the code point. If it falls in the range U+0000 to U+007F , its UTF-8 encoding is 1 byte long. If it falls in the range U+0080 to U+07FF , its UTF-8 encoding is two bytes long. And if it falls in the range U+0800 to U+FFFF , its UTF-8 encoding is 3 bytes long (there is a 4 byte range, but we'll be ignoring that for this assignment). The way UTF-8 works is it splits up the binary representation of the code point across these UTF-8 encoded byte(s) .

The utf8 program accepts code points as command line arguments and prints out their UTF-8 encoding and the actual character that encoding represents.

( Note: if your Unicode characters are not printing or appear garbled, but the encoded hex bytes are correct, you may need to set your terminal to expect UTF-8. Changing the configuration is OS/terminal-specific, but within 5 minutes of Googling we found utf8 advice for Putty , MobaXterm and Xterm . The default Mac OS X Terminal.app works by default. Note that sanity check should still pass even if your terminal is showing the wrong character.)

Your task is to implement the to_utf8 function that takes a single Unicode code point and constructs its sequence of UTF-8 encoded bytes. This function is called once for every argument specified by the user.

The to_utf8 function has two parameters; an unsigned short code_point and a byte array utf8_bytes . The function constructs the UTF-8 representation for the given code point and writes the sequence of encoded bytes into the array utf8_bytes . The first byte (e.g. leading byte) should go at index 0, and the remaining bytes (if any) should follow at index 1, etc. The utf8_bytes array is provided by the client and is guaranteed to be large enough to hold a full 3 bytes, although only 1 or 2 may be needed. While we will talk about C arrays in more depth later, it turns out that if you pass an array as a parameter, modifying the elements of that array parameter will modify the elements of the original array passed in, so the function above can pass back the byte(s) it creates. The function returns the number of bytes written to the array (either 1, 2, or 3). As an example, if code_point were 0xfc , utf8_bytes would be updated to store 0xc3 at index 0, and 0xbc at index 1, and to_utf8 would return 2.

Your to_utf8 function will need to be written to handle each of the 3 code point ranges and construct the corresponding UTF-8 encoding . You are allowed to use comparison operators to distinguish this. Your code will need to use bitmasks and bitwise operators to manually construct each byte for the UTF-8 encodings.

Testing recommendations: there are some 65,000 possible code points you could feed to utf8 , but given that they all flow through just a few distinct paths, having a test case targeted to hit each of those paths, with a little extra attention at the boundaries, should be sufficient to give you confidence about the rest.

## 1 Byte Encodings

If a code point is in the range U+0000 to U+007F , inclusive, its UTF-8 encoding is 1 byte. A code point in this range can be represented using at most 7 bits - called its significant bits . The significant bits are the bits that could actually store something meaningful for the code point. It's 7 here because, for a code point from 0 to 0x7f , it would need at most 7 bits to store a value in that range, which are the 7 least significant, so those are all that we care about. The UTF-8 encoding is one byte with its most significant bit 0 , and its remaining 7 bits as the seven significant bits from the code point.

Example: for code point 0x001f (which is 0001 1111 ), its UTF-8 encoding would be 0001 1111 ( 0x001f ) and to_utf8 would return 1.

## 2 Byte Encodings

If a code point is in the range U+0080 to U+07FF , inclusive, its UTF-8 encoding is 2 bytes. A code point in this range can be represented using at most 11 significant bits. The UTF-8 encoding is:

- The first byte has a leading 110 , followed by the 5 most significant bits in the code point
- The second byte has a leading 10 , followed by the remaining 6 significant bits in the code point

Example: for code point 0x00fc (which is 1111 1100 ), the first byte would be 1100 0011 ( 0xc3 ), the second byte would be 1011 1100 ( 0xbc ) and to_utf8 would return 2.

## 3 Byte Encodings

If a code point is in the range U+0800 to U+FFFF , inclusive, its UTF-8 encoding is 3 bytes. A code point in this range can be represented using at most 16 significant bits. The UTF-8 encoding is:

- The first byte has a leading 1110 , followed by the 4 most significant bits in the code point
- The second byte has a leading 10 , followed by the 6 next most significant bits in the code point
- The third byte has a leading 10 , followed by the final 6 significant bits in the code point

Example: for code point 0x4751 (which is 0100 0111 0101 0001 ), the first byte would be 1110 0100 ( 0xe4 ), the second byte would be 1001 1101 ( 0x9d ), the third byte would be 1001 0001 ( 0x91 ) and to_utf8 would return 3.

Here is a table containing the design described above. The xxx bits store the binary representation of the code point.

In a single-byte sequence, the high-order bit is always 0 , and the other 7 bits are the value of the code point itself. This is so the first 128 Unicode code points use the same single byte representation as the corresponding ASCII character!

For the multi-byte sequences, the first byte is called the leading byte , and the subsequent byte(s) are continuation bytes. The high-order bits of the leading byte indicate the number of total bytes in the sequence through the number of 1 s (for instance, the leading byte of an encoding for a code point from U+0080 to U+07FF starts with 110 , indicating the entire encoding is 2 bytes long, since there are two 1 s). The high-order bits of a continuation byte are always 10 . The bit representation of the code point is then divided across the low-order bits of the leading and continuation bytes.

Walkthrough Example: (paraphrased from Wikipedia UTF-8 page )

Consider encoding the Euro sign, €.

- The Unicode code point for € is U+20AC .
- The code point is within the range U+0800 to U+FFFF and will require a three-byte sequence. There are 14 significant bits in the binary representation of 0x20AC.
- Hex code point 20AC is binary 00100000 10101100 . Two leading zero bits of padding are used to fill to 16 bits. These 16 bits will be divided across the three-byte sequence.
- Let's calculate the leading byte. High-order bits are fixed: three 1s followed by a 0 indicate the three-byte sequence. The low-order bits store the 4 highest bits of the code point. This byte is 1110 0010 . The code point has 12 bits remaining to be encoded.
- Next, let's calculate the first continuation byte. High-order bits are fixed 10 , low-order bits store the next 6 bits of the code point. This byte is 10 000010 . The code point has 6 bits remaining to be encoded.
- Finally, let's calculate the last continuation byte. High-order bits are fixed 10 , low-order bits store the last 6 bits of the code point. This byte is 10 101100 .
- The three-byte sequence this results in is 1110 0010 10 000010 10 101100 , which can be more concisely written in hexadecimal, as e2 82 ac . The underscores indicate where the bits of the code point were distributed across the encoded bytes.

## 3. Saturating Arithmetic

The sat program performs a synthetic saturating addition operation. Saturating addition clamps the result into the representable range. Instead of overflowing with wraparound as ordinary two's-complement addition does, a saturating addition returns the type's maximum value when there would be positive overflow, and minimum when there would be negative overflow. Saturating arithmetic is a common feature in 3D graphics and digital signal processing applications.

The sat program accepts different numbers of arguments. With just one argument, it prints the representable range of numbers with that bitwidth:

Any additional specified arguments are added together using saturated arithmetic - for instance, if we add 126 and 5, the result overflows and sticks to the maximum value of 127.

You are to implement these 3 functions for the sat program. You are encouraged to call any of these functions from any of the others as needed to share logic!

For these 3 functions, you may not use the following in your implementations:

- No relational operators or math.h functions . You are prohibited from making any use of the relational operators. This means no use of < > <= >= . You may use != == and all other operators (arithmetic, logical, bitwise, ...). You also should not call any function from the floating point math.h library (e.g no pow , no exp2 ). These restrictions are intended to guide you to implement the operation via bitwise manipulation.
- No special cases based on bitwidth. Whether the value of bitwidth is 4, 64, or something in between, your functions must use one unified code path to handle any/all values of bitwidth without special-case handling. You should not use if switch ?: to divide the code into different cases based on the value of bitwidth. This doesn't mean that you can't use conditional logic (such as to separately handle overflow or non-overflow cases), but conditionals that dispatch based on the value of bitwidth or make a special case out of one or more bitwidths are disallowed.
- No loops or recursion. No loops or recursion at all. (In particular, no loop increment by one and stop at max.)

A solution that violates any of these restrictions will receive zero credit , so please verify your approach is in compliance!

Testing recommendations : your custom tests should cover a variety of different bitwidths, including extreme values, and on sums within range as well as those that overflow.

## Step 1: signed_min

signed_min returns the smallest value representable in bitwidth bits. bitwidth is a number between 4 and 64, inclusive. Note, however, that the function returns a long ; your function needs to return the minimum value, but correctly represented in 64 bits, not just bitwidth bits. As an example, with a bitwidth of 4, the min is binary 1000 , which represents the value -8. We want to return the value -8 but represented using 64 bits, which would be 11111...1000 . As a hint, try writing out what the signed min values are for different bitwidths, and see if you notice a pattern in how to go from a given bitwidth to its signed min binary representation. Your goal is to determine an expression based on bitwidth that evaluates to the correct long value. Once you implement this function, you should be able to run sat with just the bitwidth argument and see the correct value outputted for the min.

## Step 2: signed_max

signed_max returns the largest value representable in bitwidth bits. It works the same as signed_min , but for the largest value instead of the smallest. Like signed_min , the function returns a long ; your function needs to return the maximum value, but correctly represented in 64 bits, not just bitwidth bits. As an example, with a bitwidth of 4, the max is binary 0111 , which represents the value 7. We want to return the value 7 but represented using 64 bits, which would be 000....0111 . As a hint, try writing out what the signed max values are for different bitwidths, and see if you notice a pattern in how to go from a given bitwidth to its signed max binary representation. You may also notice similarities with your signed_min implementation. Once you implement this function, you should be able to run sat with just the bitwidth argument and see the correct value outputted for the max.

## Step 3: sat_add

sat_add returns the saturated addition of the two operands which are guaranteed to fit in bitwidth bits. This means:

- If the sum of the operands does not overflow, we just return the sum
- If the sum of the operands overflows past the max, return the max
- If the sum of the operands overflows past the min, return the min

Remember that you may not use relational operators such as < or > .

The key idea is to try and develop a way to examine a value's binary representation to know whether it has overflowed its bitwidth. As a hint, try picking a bitwidth such as 4, and write out some addition operations that do and don't overflow. Then ask yourself - by looking at the operands and the sum, did this addition overflow? How did I know that? When is it possible for overflow to occur?

## 4. Case Study: Ariane-5

Once you have implemented your sat program, you will have a taste of functions that work around overflow (completing sat is not required to complete this part, but provides context). Overflow errors can cause many problems, which we saw some examples of in lecture. These errors are not very common, but they can be disastrous when they occur. Read through this article explaining an issue experienced by the European Space Agency, and answer the following questions in the readme.txt file.

View Ariane-5 Article

1) What was the cause of the Ariane 5 rocket failure, and why was it not detected before the rocket was launched? Write your answer in 3-5 sentences.

2) Imagine that you are developing software for Ariane 6. You notice that Ariane 6 takes the average of two different sensors' measurements of horizontal velocity (which are stored as int s) by adding them together and dividing by 2, as in the following code:

What issues might arise from this calculation? How would you test for other potential overflow issues that could arise before the launch date? Write your answer in 3-5 sentences. Note: There are many possible answers to this question. Your answer should rely on critical reflection about appropriate testing strategies.

## Submitting and Grading

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You should only need to modify the following files for this assignment: readme.txt , sat.c , util.c , utf8.c , custom_tests

We would also appreciate if you filled out this homework survey to tell us what you think once you submit. We appreciate your feedback!

Below is the tentative grading rubric. We use a combination of automated tests and human review to evaluate your submission. More details are given in our page linked to from the Assignments dropdown explaining How assignments are graded .

Readme Answers (10 points)

Code Functionality (80 points)

- Sanity cases . (30 points) The default sanity check tests are used for sanity grading.
- Comprehensive cases . (45 points) We will thoroughly test on a variety of an additional inputs ranging from simple to complex, including edge conditions where appropriate.
- Custom sanitycheck tests (5 points) Your custom_tests file should include at least 10 tests of your own, 3-5 per program, that show thoughtful effort to develop comprehensive testing coverage. Please include comments that explain your choices. We will run your custom tests against your submission as well as review the cases to assess the strength of your testing efforts.
- Clean compile . (2 points) We expect your code to compile cleanly without warnings.

Code Quality (buckets weighted to contribute ~10 points)

The grader's code review is scored into buckets to emphasize the qualitative features of the review over the quantitative. The styleguide is a great overall resource for good program style. Here are some highlights for this assignment:

- Bitwise manipulation . We expect you to show proficiency in bitwise manipulation through clean construction and use of masks and proper application of the bitwise operations.
- Algorithms. Your chosen approach should demonstrate that you understand how to leverage bit patterns and numeric representation to directly and efficiently accomplish the task. Avoid unnecessary divergence and special-case code; unify into one general-purpose path wherever possible.
- Style and readability. We expect your code to be clean and readable. We will look for descriptive names, defined constants (not magic numbers!), and consistent layout. Be sure to use the most clear and direct C syntax and constructs available to you.
- Documentation. You should document both the code you wrote and what we provided. We expect program overview and per-function comments that explain the overall design along with sparing use of inline comments to draw attention to noteworthy details or shed light on a dense or obscure passage. The audience for the comments is your C-savvy peer.

## Post-Assignment Check-in

How did the assignment go for you? We encourage you to take a moment to reflect on how far you've come and what new knowledge and skills you have to take forward. Once you finish this assignment, you will have completed your first C programs manipulating bits, and understood more about the limitations of integer representations! This assignment may definitely have a bit of a learning curve getting used to the write/compile/debug process in C and the command line. Celebrate this important milestone; the skills you develop now will pay off throughout the quarter!

To help you gauge your progress, for each assignment/lab, we identify some of its takeaways and offer a few thought questions you can use as a self-check on your post-task understanding. If you find the responses don't come easily, it may be a sign a little extra review is warranted. These questions are not to be handed in or graded. You're encouraged to freely discuss these with your peers and course staff to solidify any gaps in you understanding before moving on from a task. They could also be useful as review before exams.

- Consider all negative int s whose absolute value is an exact power of two (-8, -32 and so on). What do their bit patterns have in common?
- How can you determine if an integer bit pattern has a pair of consecutive 'on' bits? (Straightforward to do using a loop, but there is also a clever single expression...)
- How can integer limitations manifest themselves in real-world software?
- A treasure trove of bit wizardry curated by Sean Anderson, Stanford CS PhD.

To help approach this assignment, we've compiled some tips that may help with the various problems, as well as bit manipulation in general.

Don't use 0b -prefixed literals. There is no official support in C for numbers written in binary with 0b . We used this form in lecture to explicitly write out binary numbers, but it's not something you can use in standard C programs. Instead, you should... (see the next part)

Embrace hexadecimal. The base-16 system very conveniently translates to and from binary -- each byte is expressed in two hex digits, one hex digit for the 4 high-order bits, the other for the 4 low bits. Masks in hex or expressed via their construction are more evocative of the binary representation and help you avoid mistakes. For example, a mask to extract the sign bit from a 32-bit value can be cleanly expressed as 1 << 31 or 0x80000000 and both are quite readable. The mask also happens to be decimal 2147483648U , but who can tell? Would you notice if you had accidentally mistyped 2147486348U ? Say no to decimal, hex is your pal!

Learn your bitwise operators. It is wasteful to invoke the (expensive, floating-point) pow function to compute integer powers of two: 1 << 10 does a perfect job of computing pow(2, 10) and requires way fewer compute cycles. The bitwise operations are also the right tools to rearrange/isolate/change bits as opposed to the more indirect use of integer multiply/divide/mod with powers of two.

Aim for directness. There may be more direct ways to write certain bit manipulations. Here are a few:

No left-shift right-shift . In the code below, the first version right-shifts then immediately left-shifts back. This seems to just cycle the bits back to where they were originally, but in fact the right-shift will discard the lsb and the left-shift will then zero-fill. This is roundabout way to wipe the lsb, but the better approach is to just directly mask it off as in second version:

No redundant/unnecessary ops . Imagine your goal is to extract the 4 high-order bits of a byte and shift them down to the lower half-byte. You could do it with two steps (mask and shift), but in fact, the shift will already discard the low-order bits, so there's no need to mask them off first!

Know your identities/relationships . When putting together a bitwise construction, stop and consider whether there is a more direct way to accomplish the same thing, e.g.

At first you may find yourself using a longer expression than needed, but reviewing in a later pass, you can spot these issues and make them more concise. The goal is eventually to go straight to the most direct expressions from the beginning.

## Frequently Asked Questions

I'm getting the error warning: left shift count >= width of type . what does this mean.

Code such as this will produce that warning:

An integer constant defaults to type int , so shifting a 32-bit int by 32 positions would effectively shift off all the bits, and thus the compiler warns about it. Assigning the result to a variable of type long (64 bits) doesn't help; you have to start with a 64-bit value if you intend to compute a 64-bit result. Read the next question!

## What bitwidth and signedness does a numeric integer constant have?

An integer literal is a signed int by default. Suffixes can be added to the constant to dictate its type. A suffix of L indicates the type is long and U (either by itself or with L ) indicates it is unsigned. E.g. 1UL means a literal 1 that is an unsigned long.

## What are the rules for integer promotion in C?

When assigning from narrower bitwidth (e.g. char, short) to a wider one, the source bytes are copied to the low-order bytes of the destination, and the upper bytes are assigned based on the promotion rules. If the source type was unsigned, the upper bytes of destination are zero-filled. If the source type was signed, its sign bit (the msb) is replicated across the upper bytes of destination (this is called sign extension). It may help to trace through this and confirm for yourself how/why this is value-preserving. If an operation is performed on two operands of mixed type, integer promotion is implicitly applied to the narrower first. It also the case (and a bit surprising, too) that integer promotion is also applied to types narrower than int when an operation is performed on them, even if no other wider type is involved. For example, the bitshift and invert operations promote the operand to an int, so bit-shifting a char will promote to an int before shifting, bit-inverting of a char similarly promotes, and the default result type of these expressions will be int. If the bitwidth of the operand was larger than int, the bitwidth will be unchanged by these promotion rules.

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Step 1: signed_min. signed_min returns the smallest value representable in bitwidth bits. bitwidth is a number between 4 and 64, inclusive. Note, however, that the function returns a long; your function needs to return the minimum value, but correctly represented in 64 bits, not just bitwidth bits.

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