Maths (Class 12 / Grade 12)
Maths XII is an online training course designed for the students of class 12 following the syllabus of Central Board of Secondary Education (CBSE). This is a self study package explains basic concepts in interesting and innovative manner with graphical user interface. Different topics are explained with flash animations, analogies and suitable illustrative examples. Interactive simulations are provided for clear understanding of the concepts. Frequently asked questions and quiz are provided with every topic for self evaluation of students. For quick revision of formulae, a brief summary has been given at the end of each chapter. This course is suitable for mathematics students, and for anyone interested in exploring the world of mathematics.
Unit 1.1: Relations
This chapter includes Ordered pair, Cartesian product, domain, co domain, Range, and Relations.
Unit 1.2: Functions
This chapter includes representation of functions, function as a relation, domain, range and co domain of functions, and different types of functions. Also One to one and onto functions, composite functions, and inverse of a function.
Unit 1.3: Binary Operations
Unit 1.4: Inverse Trigonometric Functions
This chapter includes definition and graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions and some solved examples.
Unit 2.1: Algebra of Matrices
This chapter includes concept of matrix, notation, order, equality, types of matrices, zero matrix, transpose of a matrix, symmetric and skew symmetric matrices. Addition, multiplication
and scalar multiplication of matrices, simple properties of addition, multiplication and scalar multiplication, Non - commutative of multiplication of matrices and solved examples.
Unit 2.2: Determinants
This chapter explains determinant of a square matrix (up to 3 x 3 matrices), properties of
determinants, minors, cofactors and applications of determinants in finding the
area of a triangle.
Unit 2.3: Adjoint and Inverse of Matrices
This chapter consists of Ad joint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.
Unit 3.1: Continuity
This chapter includes concept of continuous function, continuity at a point, discontinuity at a point, Classification of discontinuities, and examples.
Unit 3.2: Differentiation I
This chapter includes concept of differentiation, derivative at a point, derivative of a function,
Derivative of algebraic functions, trigonometric functions, composite functions with solved examples.
Unit 3.3: Differentiation II
This chapter extends differentiation to differentiation of inverse trigonometric functions including inverse of sin, cos, tan, cosec, sec, and cot functions. Also includes differentiation of exponential functions, logarithmic functions and examples.
Unit 3.4: Differentiation III
This chapter includes differentiation of implicit functions, differentiation of functions in parametric form, differentiation of a function with respect to another function and higher order derivatives.
Unit 3.5: Rolle’s Theorem and Lagrange’s Theorem
This includes Rolle’s theorem with proof and examples, and Lagrange’s theorem with proof and example.
Unit 3.6: Rate of Change of Quantities
This chapter explains the application of differentiation in rate of change of quantities.
It includes various examples of rate of change of distance, length, volume, cost etc.
Unit 3.7: Increasing and Decreasing Functions
This chapter explains the concept of increasing and decreasing functions, with the concept of strictly increasing and strictly decreasing functions and critical value.
Unit 3.8: Tangents and Normals
This chapter deals with the concept of slopes and tangents, equations of tangent and normal, angle of intersection of curves with solved examples.
Unit 3.9: Approximation by Differentials
This chapter teaches finding approximates values of functions using differentials. It also help us to find approximate measure of relative error and percentage error.
Unit 3.10: Maxima and Minima
This chapter deals with application of derivatives in finding maximum and minimum values of functions.
Unit 3.11: Fundamental Integration Formulae
In this chapter we learn the concept of Integration, The process of finding functions with given derivatives is Integration, i.e the inverse of Differentiation is integration, we shall also study some basic Integration formulae.
Unit 3.12: Integration by Substitutions
When the Integrand of the given function is not standard we use different methods to find the integrals of the given function. One such method is substitution which is illustrated in this chapter.
Unit 3.13: Integration by Parts
In this chapter we learn the method of Integration when the integrand is a product of two functions.
Unit 3.14: Integration by Partial Fractions
If the given function is a rational function, then partial fractions method is use to find the integrals.
Unit 3.15: Integration of Rational Functions
In this chapter we shall study some formulae to solve some standard rational functions.
Unit 3.16: Integration of Irrational Functions
In this chapter we shall study some formulae to solve some standard irrational functions.
Unit 3.17: Definite Integral as a Limit of a Sum
In this chapter we shall study about definite Integrals as a limit of a sum and to evaluate limits of the definite Integrals.
Unit 3.18: Definite Integral Using Indefinite Integral
In this chapter we shall study another method of evaluating limits of a definite integrals which is using indefinite integrals.
Unit 3.19: Area of Bounded Regions Using Definite Integrals
Major application of definite integrals is in finding the areas of bounded regions which is illustrated in this chapter.
Unit 3.20: Differential Equation and its Application
In this chapter we learn about differential equations, various forms of differential equation and application of differential equation in practical problems.
Unit 4.1: Vectors
Vectors are an essential part of mathematical background required by mathematicians, engineers, physicists and scientists. Vectors help in presentation of physical and geometrical ideas.
This chapter deals with algebra of vectors and its application to various types of geometrical and trigonometrical problems.
Unit 4.2: Scalar Product of Vectors
This chapter is about multiplication of vectors, scalar product and its geometrical interpretation, properties of scalar product, orthonormal vectors and solved examples.
Unit 4.3: Cross Product of Vectors
This chapter is about vector product of vectors or cross product, It covers geometrical interpretation of cross product, properties of cross product, scalar triple products and co planarity of vectors.
Unit 4.4: Straight Line in Space
In this chapter we study the concept of straight line in space or 3d coordinate; we study to find the direction cosines and direction ratios of a straight line in vector and Cartesian form. Also discuss the equation of a line under different conditions, angle between two lines, two planes, a line and a plane, shortest distance between two skew lines and distance of a point from a plane
Unit 4.5: Plane in Space
This chapter explains Plane in space, Angle between two planes, Cartesian form and vector form of plane, Equation of plane in normal form, equation of a plane passing through non collinear points, Intercept form of a plane, angle between line and a plane, distance between plane and a point with various examples.
Unit 5.1: Linear Programming
Definitions of related terminology such as constraints, objective function, optimization, different types of linear programming problems, graphical method of solution, for problems in two variable, feasible and infeasible regions, feasible and infeasible solutions, and optimum feasible solutions.
Unit 6.1: Probability
This chapter includes concept of Probability as random experiment, independent and dependent events, Conditional Probability, Properties of conditional Probabilities, and various types of examples.
Unit 6.2: Multiplication Theorem on Probability
This topic includes the law of total Probability, examples to understand total probability, Bayes’ theorem, with examples and unsolved examples for practice.
Unit 6.3: Binomial Distribution
This chapter includes Binomial Distribution and its necessary condition, concept of binomial variable and binomial distribution with examples, variance and standard deviation of binomial Distribution, various types of examples, and unsolved questions.
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