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Students have to more focussed about their studies as this is the start of their future learning and career. Therefore, using these materials students can really score good marks in their final exams. All the solutions for all the chapters for class 11 Maths subject have been arranged in a proper manner.

Students can easily find answers for any question here which is mentioned in the book. These solutions are in accordance with syllabus. So students need not worry about the given content for the solutions. They have well prepared and structured solutions available here, by us.

Solving the questions present in the textbooks is not an easy task. It requires mathematical and logical skills to solve those problems with a lot of efforts. Class 11th Maths is a next level Maths, where students will learn a large variety of topics, which they are going to face in higher studies as well, such as in Class 12th.

As we all know, 12th is the most important class for all the students, as after this they move to their college level. But you can get admission in a good recognized institute or university, only when you have score the minimum qualifying marks.

And it is only then, you will be able to get good jobs or get placed directly from campus recruitment, conducted by colleges. Therefore, it very necessary for you to build a strong base in class 11 maths subject and understand the concepts deeply.

To make it easier for them to learn, we are providing here solutions for all the students of 11th standard, such that, they can clarify their doubts for all types of questions. Below are the introductions for each chapter and links for all the exercises, read them thoroughly.

In this Chapter, we will study about this approach called axiomatic approach of probability. In particular, while other philosophies of mathematics allow objects that can be proved to exist even though they cannot be constructed, intuitionism allows only mathematical objects that one can actually construct.

Intuitionists also reject the law of excluded middle i. Formalist definitions identify mathematics with its symbols and the rules for operating on them. Haskell Curry defined mathematics simply as "the science of formal systems". In formal systems, the word axiom has a special meaning different from the ordinary meaning of "a self-evident truth", and is used to refer to a combination of tokens that is included in a given formal system without needing to be derived using the rules of the system.

Several authors consider that mathematics is not a science because it does not rely on empirical evidence. Mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the other sciences.

Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics.

The opinions of mathematicians on this matter are varied. Many mathematicians [57] feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts ; others feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics.

In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the philosophy of mathematics. Mathematics arises from many different kinds of problems. At first these were found in commerce, land measurement , architecture and later astronomy ; today, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself.

For example, the physicist Richard Feynman invented the path integral formulation of quantum mechanics using a combination of mathematical reasoning and physical insight, and today's string theory , a still-developing scientific theory which attempts to unify the four fundamental forces of nature , continues to inspire new mathematics. Some mathematics is relevant only in the area that inspired it, and is applied to solve further problems in that area.

But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. A distinction is often made between pure mathematics and applied mathematics. However pure mathematics topics often turn out to have applications, e.

This remarkable fact, that even the "purest" mathematics often turns out to have practical applications, is what the physicist Eugene Wigner has named " the unreasonable effectiveness of mathematics ". As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: there are now hundreds of specialized areas in mathematics and the latest Mathematics Subject Classification runs to 46 pages.

For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty in a simple and elegant proof , such as Euclid 's proof that there are infinitely many prime numbers , and in an elegant numerical method that speeds calculation, such as the fast Fourier transform.

Hardy in A Mathematician's Apology expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic.

A theorem expressed as a characterization of the object by these features is the prize. The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.

And at the other social extreme, philosophers continue to find problems in philosophy of mathematics , such as the nature of mathematical proof. Most of the mathematical notation in use today was not invented until the 16th century.

Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. According to Barbara Oakley , this can be attributed to the fact that mathematical ideas are both more abstract and more encrypted than those of natural language.

Mathematical language can be difficult to understand for beginners because even common terms, such as or and only , have a more precise meaning than they have in everyday speech, and other terms such as open and field refer to specific mathematical ideas, not covered by their laymen's meanings.

Mathematical language also includes many technical terms such as homeomorphism and integrable that have no meaning outside of mathematics. Additionally, shorthand phrases such as iff for " if and only if " belong to mathematical jargon. There is a reason for special notation and technical vocabulary: mathematics requires more precision than everyday speech.

Mathematicians refer to this precision of language and logic as "rigor". Mathematical proof is fundamentally a matter of rigor. Mathematicians want their theorems to follow from axioms by means of systematic reasoning.

This is to avoid mistaken " theorems ", based on fallible intuitions, of which many instances have occurred in the history of the subject. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics. Today, mathematicians continue to argue among themselves about computer-assisted proofs.

Since large computations are hard to verify, such proofs may be erroneous if the used computer program is erroneous. Axioms in traditional thought were "self-evident truths", but that conception is problematic. Nonetheless mathematics is often imagined to be as far as its formal content nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change i. In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic , to set theory foundations , to the empirical mathematics of the various sciences applied mathematics , and more recently to the rigorous study of uncertainty.

While some areas might seem unrelated, the Langlands program has found connections between areas previously thought unconnected, such as Galois groups , Riemann surfaces and number theory. Discrete mathematics conventionally groups together the fields of mathematics which study mathematical structures that are fundamentally discrete rather than continuous. In order to clarify the foundations of mathematics , the fields of mathematical logic and set theory were developed.

Mathematical logic includes the mathematical study of logic and the applications of formal logic to other areas of mathematics; set theory is the branch of mathematics that studies sets or collections of objects. The phrase "crisis of foundations" describes the search for a rigorous foundation for mathematics that took place from approximately to The crisis of foundations was stimulated by a number of controversies at the time, including the controversy over Cantor's set theory and the Brouwer�Hilbert controversy.

Mathematical logic is concerned with setting mathematics within a rigorous axiomatic framework, and studying the implications of such a framework. Therefore, no formal system is a complete axiomatization of full number theory. Modern logic is divided into recursion theory , model theory , and proof theory , and is closely linked to theoretical computer science , [75] as well as to category theory.

In the context of recursion theory, the impossibility of a full axiomatization of number theory can also be formally demonstrated as a consequence of the MRDP theorem. Theoretical computer science includes computability theory , computational complexity theory , and information theory. Computability theory examines the limitations of various theoretical models of the computer, including the most well-known model�the Turing machine.

Complexity theory is the study of tractability by computer; some problems, although theoretically solvable by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with the rapid advancement of computer hardware.

The deeper properties of integers are studied in number theory , from which come such popular results as Fermat's Last Theorem. The twin prime conjecture and Goldbach's conjecture are two unsolved problems in number theory.

According to the fundamental theorem of algebra , all polynomial equations in one unknown with complex coefficients have a solution in the complex numbers, regardless of degree of the polynomial. Consideration of the natural numbers also leads to the transfinite numbers , which formalize the concept of " infinity ". Another area of study is the size of sets, which is described with the cardinal numbers. These include the aleph numbers , which allow meaningful comparison of the size of infinitely large sets.

Many mathematical objects, such as sets of numbers and functions , exhibit internal structure as a consequence of operations or relations that are defined on the set. Mathematics then studies properties of those sets that can be expressed in terms of that structure; for instance number theory studies properties of the set of integers that can be expressed in terms of arithmetic operations.

Moreover, it frequently happens that different such structured sets or structures exhibit similar properties, which makes it possible, by a further step of abstraction , to state axioms for a class of structures, and then study at once the whole class of structures satisfying these axioms.

Thus one can study groups , rings , fields and other abstract systems; together such studies for structures defined by algebraic operations constitute the domain of abstract algebra.

By its great generality, abstract algebra can often be applied to seemingly unrelated problems; for instance a number of ancient problems concerning compass and straightedge constructions were finally solved using Galois theory , which involves field theory and group theory. Another example of an algebraic theory is linear algebra , which is the general study of vector spaces , whose elements called vectors have both quantity and direction, and can be used to model relations between points in space.

This is one example of the phenomenon that the originally unrelated areas of geometry and algebra have very strong interactions in modern mathematics. Combinatorics studies ways of enumerating the number of objects that fit a given structure. The study of space originates with geometry �in particular, Euclidean geometry , which combines space and numbers, and encompasses the well-known Pythagorean theorem. Trigonometry is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions.

The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries which play a central role in general relativity and topology.

Quantity and space both play a role in analytic geometry , differential geometry , and algebraic geometry. Convex and discrete geometry were developed to solve problems in number theory and functional analysis but now are pursued with an eye on applications in optimization and computer science.

Within differential geometry are the concepts of fiber bundles and calculus on manifolds , in particular, vector and tensor calculus. Within algebraic geometry is the description of geometric objects as solution sets of polynomial equations, combining the concepts of quantity and space, and also the study of topological groups , which combine structure and space.

Lie groups are used to study space, structure, and change. Topology in all its many ramifications may have been the greatest growth area in 20th-century mathematics; it includes point-set topology , set-theoretic topology , algebraic topology and differential topology. In particular, instances of modern-day topology are metrizability theory , axiomatic set theory , homotopy theory , and Morse theory.

Other results in geometry and topology, including the four color theorem and Kepler conjecture , have been proven only with the help of computers.

Understanding and describing change is a common theme in the natural sciences , and calculus was developed as a tool to investigate it. Functions arise here as a central concept describing a changing quantity. The rigorous study of real numbers and functions of a real variable is known as real analysis , with complex analysis the equivalent field for the complex numbers.

Functional analysis focuses attention on typically infinite-dimensional spaces of functions. One of many applications of functional analysis is quantum mechanics. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equations.

Many phenomena in nature can be described by dynamical systems ; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Applied mathematics concerns itself with mathematical methods that are typically used in science , engineering , business , and industry.

Thus, "applied mathematics" is a mathematical science with specialized knowledge. The term applied mathematics also describes the professional specialty in which mathematicians work on practical problems; as a profession focused on practical problems, applied mathematics focuses on the "formulation, study, and use of mathematical models" in science, engineering, and other areas of mathematical practice.

In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake.

Thus, the activity of applied mathematics is vitally connected with research in pure mathematics. Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially with probability theory. Statisticians working as part of a research project "create data that makes sense" with random sampling and with randomized experiments ; [77] the design of a statistical sample or experiment specifies the analysis of the data before the data becomes available.

When reconsidering data from experiments and samples or when analyzing data from observational studies , statisticians "make sense of the data" using the art of modelling and the theory of inference �with model selection and estimation ; the estimated models and consequential predictions should be tested on new data. Statistical theory studies decision problems such as minimizing the risk expected loss of a statistical action, such as using a procedure in, for example, parameter estimation , hypothesis testing , and selecting the best.

In these traditional areas of mathematical statistics , a statistical-decision problem is formulated by minimizing an objective function , like expected loss or cost , under specific constraints: For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence.

Computational mathematics proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory ; numerical analysis includes the study of approximation and discretisation broadly with special concern for rounding errors.

Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic matrix and graph theory. Other areas of computational mathematics include computer algebra and symbolic computation.

Arguably the most prestigious award in mathematics is the Fields Medal , [80] [81] established in and awarded every four years except around World War II to as many as four individuals. The Fields Medal is often considered a mathematical equivalent to the Nobel Prize.

The Wolf Prize in Mathematics , instituted in , recognizes lifetime achievement, and another major international award, the Abel Prize , was instituted in The Chern Medal was introduced in to recognize lifetime achievement. These accolades are awarded in recognition of a particular body of work, which may be innovational, or provide a solution to an outstanding problem in an established field. A famous list of 23 open problems , called " Hilbert's problems ", was compiled in by German mathematician David Hilbert.

This list achieved great celebrity among mathematicians, and at least nine of the problems have now been solved. A new list of seven important problems, titled the " Millennium Prize Problems ", was published in Only one of them, the Riemann hypothesis , duplicates one of Hilbert's problems. A solution to any of these problems carries a 1 million dollar reward. From Wikipedia, the free encyclopedia. This article is about the field of study.

For other uses, see Mathematics disambiguation and Math disambiguation. Field of study. Main article: History of mathematics. Main article: Definitions of mathematics. Main article: Mathematical beauty. Isaac Newton left and Gottfried Wilhelm Leibniz developed infinitesimal calculus.

Main article: Mathematical notation. See also: Areas of mathematics and Glossary of areas of mathematics. Main article: Pure mathematics. Main articles: Arithmetic , Number system , and Number theory. Main article: Algebra. Main article: Geometry. Main article: Calculus. Main article: Applied mathematics. Main article: Statistics. Mathematics portal. International Mathematical Olympiad Lists of mathematics topics Mathematical sciences Mathematics and art Mathematics education National Museum of Mathematics Philosophy of mathematics Relationship between mathematics and physics Science, technology, engineering, and mathematics.

Therefore, Euclid's depiction in works of art depends on the artist's imagination see Euclid. Like research physicists and computer scientists, research statisticians are mathematical scientists. Many statisticians have a degree in mathematics, and some statisticians are also mathematicians.

Oxford English Dictionary. Oxford University Press. Archived from the original on November 16, Retrieved June 16, The science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis. ISBN Cengage Learning. Calculus is the study of change�how things change, and how quickly they change.

Applied Mathematics. Tata McGraw�Hill Education. The mathematical study of change, motion, growth or decay is calculus. Educational Studies in Mathematics. JSTOR S2CID Illustrious scholars have debated this matter until they were blue in the face, and yet no consensus has been reached about whether mathematics is a natural science, a branch of the humanities, or an art form.

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PMID Chicago Review Press. OCLC Archived from the original on March 31, Retrieved May 29, Archived from the original on September 16, Retrieved October 27, Archived from the original on September 7, New York: Dover Publications.

Archived from the original on April 13, Archived from the original on October 17, A history of Arabic astronomy : planetary theories during the golden age of Islam. New York University Press. Archived from the original on May 2, Archived from the original on July 25, The Story of Mathematics. Online Etymology Dictionary.

Archived from the original on March 7, OED Online , "Mathematics". Archived from the original on September 17, Boas, Jr.

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