The importance of Science and its various sub-disciplines has been recognised and upheld since the beginning of civilisation. Physics, Chemistry, and Biology have been studied and explored in various forms and through a variety of intriguing methods,, in various civilisations across the globe. It can be said without a doubt that Mathematics is one of the most prominent and vital scientific disciplines. The relative importance of Mathematics as compared to other sciences can be a subject of intense debate. Mathematics and knowledge of the basic working principles and tenets that are a part of Mathematics and its operations are crucial to attaining a holistic and comprehensive understanding of subjects like Physics, Chemistry etc. Physics and Mathematics are often perceived as sister disciplines, and it can be comfortably pointed out that both of these disciplines are quite interdisciplinary concerning the importance of calculations, quantities, and other nuances.
Mathematics comprises a multitude of underlying themes and topics. These are quite distinct from each other and are simultaneously interconnected in a myriad of ways. Intricate calculations, comprehensive formulae and their derivatives, intriguing theorems, and thought-provoking postulates make up the entirety of the curriculum in Mathematics. The history of a major proportion of this versatile curriculum can be traced back to thousands of years of research and exploration. As a result, Mathematics can appear to be quite a complex, dynamic, and sophisticated subject at first glance. However, it is also noteworthy that it is a fundamental academic discipline.
The knowledge of Mathematics is vital for the smooth and unhindered functioning of human activities as well as of man-made machines and operations. Mathematical operations are also indispensable towards ensuring the efficient organizsation of daily life. Mathematics, as a discipline, has also made extensive contributions towards uncovering many intriguing mysteries of the world and the workings of various natures and natural phenomena. Mathematics has a vital role to play in almost all professional sectors as well as in creative occupations like Art, Literature, etc. A variety of modern art involves the use and practical application of concepts of Mathematics like Geometry and Proportions, etc.
Therefore, Mathematics is also a vital and indispensable academic discipline. The knowledge of Mathematics is imparted to students from the earliest stages of education and gradually grows more sophisticated and is classified as a rise in grades. Beginning with simple and easy-to-understand concepts like Addition, Subtraction, Multiplication, Division etc, students proceed to learn about complex themes such as Mensuration, Percentage, Trigonometry, Calculus, Geometry, Probability, etc. All of these themes are inseparable parts of the prescribed NCERT academic curriculum for Mathematics for the students of various classes.The entire NCERT academic curriculum for school students in mathematics has been carefully designed to lay a solid foundation for the students in order to prepare them for their future endeavors.This is precisely because Mathematics is a vital subject within a huge number of professions and occupations like Engineering, Architecture, Technology, Astronomy etc. Additionally, Mathematics as an academic discipline with a long history also has immense and unfathomable potential for research. Therefore, curious students may find Mathematics their ideal choice of subject for pursuing higher education and scientific research. Mathematics is also a crucial part of the prescribed NCERT academic curriculum for Class 12.
The prescribed NCERT academic syllabus for Mathematics in Class 12 is composed of thirteen distinct chapters. These chapters include themes such as Differential Equations, Integrals, Theorems of Calculus, Probability, etc. All of these themes are dynamic and vast academic topics that have been the subject of great interest over the years in various fields of academic research and study. Therefore, these themes have been comprehensively covered in the NCERT prescribed academic syllabus for Class 12.
The academic content on these themes includes various challenging and diverse assessments associated with different chapters in the form of exercises. Probability is a theme which is the focus of Chapter 13. Ex 13.5 Class 12 is one of the exercises that is part of Chapter 13. It is very crucial for students to be able to solve this exercise comprehensively to be adequately prepared for examinations. Reliable and authentic NCERT Solutions like the NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 can prove to be important reference materials in this context.
Extramarks is providing the NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 to aid the efforts of students. The NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 have been designed in effective collaboration with various renowned and knowledgeable subject experts. The NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 are versatile and user-friendly online learning resources which can be utilized as academic reference materials. The NCERT Solutions for Class 12 Maths, Chapter 13, Exercise 13.5 can also be used as practise problems for continuous revision.Furthermore, Extramarks’ NCERT Solutions for Class 12 Maths, Chapter 13, Exercise 13.5 has been prepared after extensive research.It has also been ensured, with regard to the NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 that there are no discrepancies in the solutions provided in the latest NCERT academic curriculum.
NCERT Solutions for Class 12 Maths Chapter 13 Probability (Ex 13.5) Exercise 13.5
A diverse number of constituting topics and themes make up the composition of Mathematics as an academic discipline. While these underlying themes are indeed interlinked and associated with each other in many interesting ways, they are also very different from each other at the same time. The academic curriculum of Mathematics comprises many elements such as complex and minute calculations, detailed formulae and their respective derivatives, sophisticated theorems, and conventionally established postulates, among others. A significant chunk of this dynamic composition can be put under a historical lens. This can lead to the discovery that these themes have sparked a huge amount of research, invention, and experimentation over the ages.
Accordingly, Mathematics has taken on the appearance of a complex, versatile, and elite subject and appears mysterious at first glance. The indispensability of Mathematics as an academic discipline, however, cannot be overlooked. Continuous and unbarred operation of a large proportion of human activities along with the smooth functioning of a great number of man-made crafts prerequisites basic knowledge of Mathematics. Mathematical knowledge holds great importance in humans’ daily lives and ensures that everything is in perfect, workable order. As an academic discipline, Mathematics has also been crucial in discovering answers to many secrets of the world, nature and natural phenomena, and the working of the universe itself.
Thirteen different chapters make up the entirety of the academic curriculum of Mathematics prescribed for Class 12 by the NCERT. Themes like Differential Equations, Integrals, Theorems of Calculus, Probability etc.Each one of these themes has been the subject of vast academic studyand research and has sparked great interest in the minds of intellectuals across the world. Therefore, the academic syllabus for Mathematics for Class 12 prescribed by NCERT has been compiled to include the nuances and fundamental tenets of these themes. This has been done to ensure that students are appropriately knowledgeable concerning these themes in time for their future academic and professional ventures.
Along with the academic elements, various comprehensive and competitive assessments have been included within the chapters covering various themes.The last chapter covers the theme of Probability. Exercise 13.5 is part of this chapter. To excel in this topic in the examinations, regular practise and revision of Exercise 13.5 with the help of the NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 is crucial. The NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 have been designed to serve this purpose. The NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 consists of comprehensive and easy-to-understand solutions for Exercise 13.5. Important sub-topics related to the broader theme of probability are also covered in the NCERT Solutions for Class 12 Maths Chapter 13 Exercise 13.5.
The Extramarks website is providing access to the NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 to complement the hard work of students. The NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 have been compiled with reference to the contributions of various well-known and proficient subject experts. The NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 are dynamic and student-friendly online learning resources which can be used as academic reference content. The NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 can also be utilized as practice assessments for regular revision. Further, the NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 curated by Extramarks has been prepared through intensive research. It has also been ensured with regard to the NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 that there are no significant gaps between the solutions provided and the latest NCERT academic curriculum.
The NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 have been prepared in a comprehensive, step-by-step format. The internet-friendly framework of the NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 is made to cater to the needs and demands of students. The NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 can also be used by teachers and parents as guideline-type reference materials. This is because the design of the NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 is compatible with the methods and styles of classroom teaching. The NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 being provided by the Extramarks learning platform are very objective in their approach. Students often find it difficult to efficiently manage their time during examinations. The solutions provided as a part of the NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 have been prepared in a time-efficient manner for ease of revision. Furthermore, the format of the NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 has formulae that are properly applied and highlighted to make it easier for students to follow the calculation procedure.Finally, the NCERT Solutions Class 12 Maths Exercise 13.5 could also be coupled with other learning resources available on the Extramarks website for efficient and convenient learning.
Access NCERT Solutions for Class 12 Maths Chapter 13 – Probability
NCERT Solutions for Class 12 Maths Chapter 13 Probability Exercise 13.5
Within the prescribed academic curriculum for Mathematics in Class 12 by the NCERT, Probability is a major theme. It is also quite a complex topic. It consists of many individual formulae and their particular derivatives. These formulae, including the symbols and equations used in their composition, have to be adequately comprehended and understood. This is important because, without appropriate comprehension and logical understanding of the specific structure of the given formulae, students would find it difficult to apply the formulae practically for problem-solving. However, merely by retaining formulae and figuring out their practical application, one cannot excel in Probability. To ace the mathematical concept of Probability, students would also need to familiarise themselves with the variety of theorems, functions, and uses of Probability associated with the practical world. Probability becomes a part of the academic syllabus of Mathematics in very early grades, whichis a testimony to its importance as a concept. Therefore, while it is essential to attain knowledge of the practical application of various formulae, it is equally vital to acquire theoretical knowledge of the relevance of Probability and the variety of facts that accompany this knowledge. The NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 can undoubtedly cater to this purpose.
Classroom teaching is a lengthy and complex process. There could be many reasons for this. The primary reason could be that each student is an individual in their own right.By virtue of this fact, students have their own methods of grasping things, expressing themselves, and learning or comprehending academic content that is delivered to them. However, all of these different individuals are brought together in a classroom setting for the convenience of imparting education. The time allotted for the teaching of each subject and topic is limited, and teachers have to also comply with a strict schedule within which they are expected to complete the prescribed syllabus.
In such a situation, it may become cumbersome and difficult to accommodate the individuality of students and their apparent diversity. Teachers may often find it difficult to accommodate all doubts and queries raised by students during school hours and may not be able to respond to all of them adequately. Students may be left with many unclarified doubts or unresolved queries. The accumulation of such doubts and queries can take a toll on the conceptual clarity of students. However, conceptual clarity is an imperative resource for a conceptually rich academic discipline like Mathematics. Inadequate conceptual clarity can turn into a major obstacle during examinations. ConsequentlyAs a result, the NCERT Solutions for Class 12 Maths, Chapter 13, Exercise 13.5, can be particularly useful in this regard.
The NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 available at the Extramarks learning platform are detailed and authentic. The high quality of the NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 has been ensured in order to maximise the convenience of students. The NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 have been carefully crafted with full consideration of the needs and demands of students. Students can effectively utilise the NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 to aid and encourage self-learning. The easy-to-understand and well-organised structure of the NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 can also help students resolve their doubts and clarify their basic concepts. With the assistance of the NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5, students can practise regularly. This will also help them match the pace of learning of their peers and boost their confidence. The NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 would also be greatly helpful to parents who tutor their children at home themselves. The NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 have been engineered in accordance with the provisions of the latest NCERT syllabus and classroom teaching methods to facilitate convenience in learning.
The Extramarks website is a reliable, reputed, and efficient learning platform. Extramarks provides high-quality, comprehensive ,and easy-to-understand academic content like the NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 to complement the efforts of students. The NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 are a part of the entire reserve of academic reference content available on the Extramarks website.The content has been meticulously curated and divided to meet the needs and levels of learning and information absorption of students of various ages Therefore, the Extramarks learning platform has other well-structured and dependable content like the NCERT Solutions Class 1, NCERT Solutions Class 2, NCERT Solutions Class 3, NCERT Solutions Class 4, NCERT Solutions Class 5, NCERT Solutions Class 6, NCERT Solutions Class 7, NCERT Solutions Class 8, NCERT Solutions Class 9, NCERT Solutions Class 10, NCERT Solutions Class 11 and NCERT Solutions Class 12. These solutions have been compiled and made available in a single place, i.e. at the Extramarks learning platform for the convenience of students and learners. This has been done so that students would not have to browse through the Internet for hours in search of easy access to quality academic content like the NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5.
Probability is an inseparable part of the academic curriculum of Mathematics prescribed for Class 12. It is a very interesting and thought-provoking concept. It is also a very versatile academic theme. Therefore, it has to be read, comprehended, and analysed at many levels. While on the one hand, students are expected to acquire knowledge of and retain important formulae, on the other hand, they also have to be made aware of the practical applications and uses of concepts of probability in daily life.This latter type of knowledge is critical in determining the context of many critical questions and appropriately solving them.The NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 cover both of these aspects of the concept of Probability. Remarkably, the concepts which constitute the theme of Probability have multiple uses in our daily lives. Some of them can be listed as follows – flipping a coin, choosing a card from the deck, throwing a die, winning a lottery, predicting the outcome of a cricket match, etc. All of these applications and practical questions involving them have been covered in the NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5.
Probability can be defined as the likelihood of the occurrence of any event. The earliest known form of Probability and Statistics can be found in the work of a Central Asian mathematician, called Al-Khalil. He authored a book titled Cryptographic Messages, which involved the first use of permutation and combination to list all the Arabic words with or without vowels. Probability is thatProbability is the branch of mathematics that deals with calculating the likelihood of an event occurring.he occurrence of any event could be a number between 0 and 1. Here, 0 indicates the impossibility of an event and 1 indicates certainty. There are a multitude of areas in which Probability theory and concepts are widely used. A few notable examples of fields of studies such as Statistics, Finance, Gambling, Artificial Intelligence (AI), Computer Science, Machine Learning, Game Theory and Philosophy. The Probability concept is also used intensively in fields such as Weather Forecasting, Sports Strategising, Insurance, etc. Therefore, it is very evident that Probability as a mathematical concept has unparalleled relevance. Its wide uses in daily life also make it a fascinating area of study. It is highly recommended that students use the NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 to gain a good understanding of the concept of Probability and its applications.
Extramarks is committed to providing high-quality, reliable, and authentic NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 to complement the sincere efforts and perseverance of students. The NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 cover important practical and everyday life applications of probability concepts and theories.The questions related to types of probabilities when throwing dice and various theories related to this phenomenon have been covered in detail in the NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5. The NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 along with other curated and high-quality reference material are available on the Extramarks website for the convenience of students. The NCERT Solutions Class 12 Maths Chapter 13 Exercise 13.5 would undoubtedly ensure that students are adequately prepared for their examinations.