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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.
For every square matrix A of order m x n, there exists a number associated with it known as the determinant of a square matrix where the number of elements in a determinant of order n is n2. Summary: Determinant of a square matrix up to 3 x 3 matrices , properties of determinants, minors, co-factors and applications of determinants in finding the area of a triangle.
Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of a system of linear equations by examples, solving system of linear equations in two or three variables having unique solution using the inverse of a matrix. Continuity and Differentiability is one of the most important topics here you can understand the concepts like continuity at a point, continuity on an interval etc.
Differentiability of functional parameters is very difficult to learn. It can be seen that a function is continuous at a point if we are able to graph it without lifting the pen. If the function is undefined, then the function is discontinuous. Summary: Continuity and differentiability, a derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit functions.
Concept of exponential and logarithmic functions. Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second-order derivatives. Derivative can be defined as the rate of change of one quantity with respect to another.
The concept of derivatives has been used in many ways, such as when a matter of change of temperature or rate of change of shapes and sizes. Where the derivative function is used to calculate the highest and lowest point of the curve in a graph, with the help of the graph, you can also know its turning point as well. Simple problems that illustrate basic principles and understanding of the subject as well as real-life situations.
This is used to find our many useful quantities such as areas, volumes, displacement, etc. Integrals are related to usually definite integrals. Integration is one of the two major calculus in Mathematics. The idea of limit is used where algebra and geometry are implemented.
Integration is a step of adding or summing up the parts to find the whole, and the opposite step is known as differentiation. An integral that contains the upper and lower limits where Indefinite integrals are defined without upper and lower limits. Summary: Integration as the inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the types given in the syllabus and problems based on them.
Definite integrals as a limit of a sum, Fundamental Theorem of Calculus without proof. Basic properties of definite integrals and evaluation of definite integrals. Applications of the Integrals is applied in various fields to the calculation of areas like Mathematics, Science, and Engineering etc.
An integral is a function which is a given function is the derivative. It is used to find the areas of the two-dimensional and volumes of three-dimensional objects. The integral is also known as anti-derivative as it is the opposite process of differentiation. And the function of Differential Equations is the rate of change of a function at a point. Basically, Differential Equations is used in physics, engineering, biology, and another field.
The main study of solutions that satisfy the equations, and the properties of the solutions. The derivatives represent a rate of change, where differential equation describes a relationship between the quantities and the speed of change.
Summary: Definition, order and degree, general and particular solutions of a differential equation. Formation of differential equation whose general solution is given.
The solution of differential equations by the method of separation of variables solutions of homogeneous differential equations of the first order and first degree. Solutions of the linear differential equation of the type given in the syllabus. There are numbers of quantities that involve magnitude as well as direction. When the quantity has magnitude and direction, then it is known as vectors. And such types of quantities are known as Vector Quantities.
The main use of vectors is to describe the physical quantities having directions and magnitude associated with them. Summary: Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors equal, unit, zero, parallel and collinear vectors , position vector of a point, negative of a vector, components of a vector, the addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio.
Definition, Geometrical Interpretation, properties and application of scalar dot product of vectors, vector cross product of vectors, a scalar triple product of vectors. Dimensional Geometry is about the angle between line and plane, between two lines etc. To understand the different types of shapes and figures, Dimensional Geometry is introduced.
Summary: Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, coplanar and skew lines, the shortest distance between two lines.
Cartesian and vector equation of a plane. The angle between i two lines, ii two planes, iii a line and a plane. The distance of a point from a plane. The basic use of linear programming is to maximize or minimize the numerical value. Linear programming used in many Mathematics field and some other field such as economics, business, telecommunication, and manufacturing fields. Linear programming LP is defined as the problem of maximizing or minimizing a linear function.
Summary: Introduction, related terminology such as constraints, objective function, optimization, different Byjus Class 10 Maths Ncert Solutions Sdn types of linear programming L. It is a number of events to occur from the total number of event that occurs. The probability formula is defined as the likelihood of an event to happen be equal to the given ratio of the number which results and the total number of outcomes.
And where Conditional Probability is the likelihood of an event or outcome occurring is defined as on the occurrence of a previous event or outcome. Repeated independent Bernoulli trials and Binomial distribution.
Class 12th maths plays an important role for the preparation of many undergraduate competition exams. So well described Maths solution at Tiwari Academy help you to clear all doubts and concepts of all chapters. It is the best study material to score more marks in your class 12 board exam and competitive exams in future. There are total of 13 chapters which are given by the expert math teachers with extensive reviewed.
All Maths Class 12 NCERT Solutions are provided by the experienced Math after extensive research on each and every chapter to provide correct and authentic information to the students. These solutions for every chapter strictly adhere to the CBSE curriculum. Integration by substitution. Integration by parts Integration by successive reduction. Important Questions on 12th Maths Chapter 7 What is anti derivatives of a function?
If a function f is differentiable in an interval I, i. What is integration? The formula that gives all anti derivatives is called the indefinite integral of the function and such process of finding anti derivatives is called integration. What is Integral Calculus?
The development of integral calculus arises out of the efforts of solving the problems of the following types: a the problem of finding a function whenever its derivative is given, b the problem of finding the area bounded by the graph of a function under certain conditions.
These two problems lead to the two forms of the integrals, e. What is meant by integration constant C?
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