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Integration anti-derivative is an operation inverse to differentiation. From the historical point of view, the concept of integration originated earlier than the concept of differentiation. In fact the concept of integration owes its origin to the problem of finding areas of plane regions, surface areas and volumes of solid bodies etc.
Archimedes, Eudoxus and others developed it as a numerical value equal to the area under the curve of a function for some interval. Later on, link between apparently two different concepts of differentiation and integration was established by well-known mathematician Newton and Leibnitz in 17th century.
This relation is known as fundamental theorem of integral calculus. In the 19th century, Cauchy and Riemann developed the concept of Riemann integration. All 12th class Maths book solution given here are well solved each problem sums by the experienced Maths teacher. NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions covers various topics such as concepts of inverse trigonometric functions, inverse trigonometric functions properties, and miscellaneous examples.
There are two exercises which explain ranges of trigonometric functions and observe the behaviour in the graphical representation. NCERT Solutions for Class 12 Maths Chapter 2 for all exercises are solved by Mathematics teacher and solutions are well-structured so that it would be easier for students to learn and understand.
All the solutions are based on 12th Class Maths Book Solution. NCERT Solutions for Class 12 Maths Chapter 3 Matrices you can get the definition of Matrices, various types of matrices, equality of matrices, operations on matrices such as the addition of matrices and multiplication of matrices by a scalar, properties of matrices addition, properties of scalar multiplication, multiplication of matrices, properties of multiplication of matrices, transpose of matrices, properties of the transpose of the matrices, symmetric and elementary operation or transformation of a matrix, skew symmetric matrices, inverse of a matrix with by elementary operations, and at last, it covers miscellaneous examples.
There are four Exercises in this chapter All solutions are based on 12th Class Maths Book Solution. You can learn and study here about the definition and meaning of determinants.
There are six exercises in this chapter which explains remarks based on the order of determinants, properties of determinants, finding the area of a triangle using determinants, minors, adjoint, and cofactors of determinants, systems of linear equation in two or three variables with the use of inverse of matrices. Class 12 Maths Chapter 5, Continuity and Differentiability textbook has eight exercises which are based on 12th Class Maths Book Solution, in the 1st exercise you just learn about the definition of continuity.
And then you proceed with continuity, algebra of continuous function, definition and meaning of differentiability, derivative of implicit functions, derivatives of composite functions, derivative of inverse trigonometric functions, logarithmic differentiation, Concept of exponential and logarithmic functions, derivatives of functions in parametric forms, exponential and logarithmic functions, second order derivatives, mean value theorem with many miscellaneous examples.
NCERT Solutions for Class 12 Maths Chapter 6 Applications of Derivatives has five exercises which cover the definition of derivatives, tangents and normal, and rate of change of quantities, increasing and decreasing functions, use of derivatives in approximation maxima and minima, approximations, first derivative test, function in a closed interval and miscellaneous examples.
All the solutions are based on the latest 12th Class Maths Book Solution. Definition of the indefinite integral, inverse process of differentiation, geometrical interpretation of indefinite integral, comparison between differentiation and integration, properties of indefinite integral, methods of integration such as integration by substitution, integration by parts, integration using partial fractions, integration of some integral functions, integration using trigonometric identities, definition and meaning of definite integrals, definite integral the limit of a sum and at last miscellaneous examples.
And clear basic concepts related to differential equations and many more with step by step solutions so that students can easily understand. It covers all the solutions based on 12th Class Maths Book Solution including the degree of a differential equation, order of a differential equation, general and particular solutions of a differential equation, homogeneous differential equations, linear differential equations, procedure involved formation of a differential equation etc.
And also it covers some miscellaneous examples. And you can know how to find position vector, some basic concepts related to vector algebra etc. You can also learn about algebraic as well as geometric properties. And multiplication of a vector by a scalar, r, vector joining two points, components of a vector, section formula, product of two vector etc.
This chapter 10 has four exercises. NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry, you can get know about Three Dimensional Geometry, the student can constitute topics related to three dimensional geometry, direction cosines and direction ratios of line, equation of a line through a given point and parallel direction cosines of a line passing through two points, relation between the direction cosines of a line, equation of a line in space, plane passing through the intersection of two given planes, angle between two planes, coplanarity of two lines and many more.
And last it contains miscellaneous examples. There are three exercises in this chapter which is based on 12th Class Maths Book Solution. In this chapter. Here this chapter covered linear programming problem and its mathematical formulation, the mathematical formulation of the problem, different types of all linear programming problems graphical method, feasible and infeasible solutions and at last there are some miscellaneous examples given.
Multiplication rule of Probability. You can understand here the concept of random variables and its probability in this chapter. Other than this chapter also covers mean of random variable, variance of random variable, and binomial distribution and Bernoulli trials with miscellaneous examples. As Class 12 Maths is an important subject that plays a vital role in your future career. Mathematics is a subject you cannot get good marks in the exam by reading and memorizing the formulae. So you need to practice well to succeed.
It mathematical sense the two different words having a different meaning. Students get confused with these terms before going deep into this topic.
Remember that all functions are relations, but not all relations are functions. In this topic, you can find a definition of functions and relations, and its Special functions, different types of relations. Summary: Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions, the inverse of a function. Binary operations. In this NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions, we are going to know about Inverse trigonometric functions, which are simply defined as the inverse functions of the basic trigonometric functions which include sine, cosine, tangent, cotangent, secant, and with its cosecant functions.
In this topic, we know how to use inverse functions in trigonometry to get the angle with any of the trigonometry ratios. Trigonometric Functions is very important and mostly used in the field of engineering, physics, geometry and navigation.
Trigonometric functions are especially applicable to the right angle triangle. There are a total of six inverse trig functions including Arcsine, Arccosine, Arctangent, Arccotangent, Arcsecant, Arccosecant. Summary: Definition, range, domain, principal value branch.
Graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions. Matrix is a rectangular array of real or complex numbers in the form of m horizontal lines which is called row and n vertical lines which are called columns.
The matrix is just an arrangement of certain quantities. The elements which are used in a matrix may be real or complex numbers. When the elements of a matrix are real, then it is known as a real matrix.
Summary: Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric, and skew symmetric matrices. Operation on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix restrict to square matrices of order 2. Concept of elementary row and column operations.
Invertible matrices and proof of the uniqueness of inverse, if it exists; Here all matrices will have real entries. The determinant is a scalar value which can be calculated from the elements of a square matrix.
When a system of simultaneous linear equations is started to solve, then Development of determinants took place. The shape of every determinant is square.
If a determinant is of order n then it contains n rows and n columns. Class 12 Maths Chapter 4 Exercise 4. This chapter is the extension of differentiation of functions which you have studied in Class XI. You must have learnt to differentiate certain functions like polynomial functions and trigonometric functions. This chapter explains the very important concepts of continuity, differentiability and relations between them.
You will be learning about the differentiation of inverse trigonometric functions. Furthermore, you will be getting acquainted with a new class of functions called exponential and logarithmic functions. There are a total of eight exercises in this chapter so you will have to dedicate some extra time and effort to get well-versed with this chapter.
Class 12 Maths Chapter 5 Exercise 5. In Chapter 5, you will learn how to find the derivative of composite functions, implicit functions, exponential functions, inverse trigonometric functions, and logarithmic functions. In this chapter, you will study the applications of the derivative in various disciplines such as engineering, science, and many other fields. For instance, we will learn how the derivative can be used to determine the rate of change of quantities and to find the equations of tangent and normal to a curve at a point and many more usage of derivatives.
We will also use the derivative to find intervals of increasing or decreasing functions. There are five exercises in total where you will get a detailed insight into the Application of Derivatives. Finally, the derivative is used to find an approximate value of certain quantities. Class 12 Maths Chapter 6 Exercise 6. Differential Calculus is based on the idea of a derivative. Derivative came into existence for the problem of defining tangent lines to the graphs of functions and calculating the slope of such lines.
Integral Calculus eases the problem of defining and calculating the area of the region bounded by the graphs of the function. With a total of eleven exercises, you will have to learn every topic and their related questions with sheer concentration.
Class 12 Maths Chapter 7 Exercise 7. Here, in this chapter, you will study some specific application of integrals to find the area given under the simple curves, also, the area between lines and arcs of circles, parabolas and ellipses standard forms only.
There are two exercises in this chapter in which you will also deal with finding the area bounded by the above-said curves. Class 12 Maths Chapter 8 Exercise 8. In Differential Equations, you will be studying some basic concepts related to the differential equation, general and particular solutions of a differential equation, formation of differential equations, number of methods to solve a question based on first order - first-degree differential equation and some applications of differential equations in different areas.
There are a total of six exercises in this chapter for the students. Differential equations are implemented in a plethora of applications in all the other subjects and areas. Hence, if you study the Differential equations chapter in a detailed manner, it will help you to gain a detailed insight into all modern scientific investigations.
Class 12 Maths Chapter 9 Exercise 9. Did you know those quantities that involve only one value magnitude , which is a real number is called scalars?
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