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Following is the outline of Central Board of Secondary Education CBSE / NCERT prescribed syllabus for Standard Mathematics ( Conventional Maths Syllabi ) for Classes 1 - I, 2 - II, 3 - III, 4 - IV, 5 - V ( Classes 1st, 2nd, 3rd, 4th, 5th ) :
What is Long, What is Round, Counting in Groups, How Much Can You Carry, Counting in Tens, Patterns, Footprints, Jugs and Mugs, Tens and Ones, My Funday, Add our Points, Lines and Lines, Give and Take, The Longest Step, Birds Come, Birds Go, How Many Ponytails
Where to Look From, Fun with Numbers, Give and Take, Long and Short, Shapes and Designs, Fun with Give and Take, Time Goes On, Who is Heavier, How Many Times, Play with Patterns, Jugs and Mugs, Can We Share, Smart Charts, Rupees and Paise
Building with Bricks, Long and Short, A Trip to Bhopal, Tick-Tick-Tick, The Way The World Looks, The Junk Seller, Jugs and Mugs, Carts and Wheels, Halves and Quarters, Play with Patterns, Tables and Shares, How Heavy? How Light?, Fields and Fences, Smart Charts, Building with Bricks, Long and Short, A Trip to Bhopal, Tick-Tick-Tick, The Way The World Looks, The Junk Seller, Jugs and Mugs, Carts and Wheels, Halves and Quarters, Play with Patterns, Tables and Shares, How Heavy? How Light?, Fields and Fences, Smart Charts
Following is the outline of Central Board of Secondary Education CBSE / NCERT prescribed syllabus for conventional Mathematics. This syllabi covers standard Maths for different Classes, viz., 6 - VI, 7 - VII, 8 - VIII, 9 - IX, 10 - X, 11 - XI, 12 - XII ( Classes 6th, 7th, 8th, 9th, 10th, 11th, 12th ) :
Class 6 - VI ( 6th )
A Note For The Teachers, Knowing Our Numbers, Whole Numbers, Playing With Numbers, Laying With Numbers, Basic Geometrical Ideas, Understanding Elementary Shapes, Integers, Fractions, Decimals, Data Handling, Mensuration, Algebra, Ratio And Proportion, Symmetry, Practical Geometry, Brain-Teasers
Class 7 - VII ( 7th )
Integers, Fractions and Decimals, Data Handling, Simple Equations, Lines and Angles, The Triangle and its Properties, Congruence of Triangles, Comparing Quantities, Rational Numbers, Practical Geometry, Perimeter and Area, Algebraic Expressions, Exponents and Powers, Symmetry, Visualising Solid Shapes
Class 8 - VIII ( 8th )
Rational Numbers, Linear Equations in One Variable, Understanding Quadrilaterals, Practical Geometry, Data Handling, Squares and Square Roots, Cubes and Cube Roots, Comparing Quantities, Algebraic Expressions and Identities, Visualising Solid Shapes, Mensuration, Exponents and Powers, Direct and Inverse Proportions, Factorisation, Introduction to Graphs, Playing with Numbers
Class 9 - IX ( 9th )
NUMBER SYSTEMS, Irrational Numbers, Real Numbers and their Decimal Expansions, Representing Real Numbers on the Number Line, Operations on Real Numbers, Laws of Exponents for Real Numbers, POLYNOMIALS, Polynomials in One Variable, Zeroes of a Polynomial, Remainder Theorem, Factorisation of Polynomials, Algebraic Identities, COORDINATE GEOMETRY, Cartesian System, Plotting a Point in the Plane if its Coordinates are given, LINEAR EQUATIONS IN TWO VARIABLES, Linear Equations, Solution of a Linear Equation, Graph of a Linear Equation in Two Variables, Equations of Lines Parallel to x-axis and y-axis, INTRODUCTION TO EUCLID’S GEOMETRY, Introduction, Euclid’s Definitions, Axioms and Postulates, Equivalent Versions of Euclid’s Fifth Postulate, LINES AND ANGLES, Basic Terms and Definitions, Intersecting Lines and Non-intersecting Lines, Pairs of Angles, Parallel Lines and a Transversal, Lines Parallel to the same Line, Angle Sum Property of a Triangle, TRIANGLES, Congruence of Triangles, Criteria for Congruence of Triangles, Some Properties of a Triangle, Some More Criteria for Congruence of Triangles, Inequalities in a Triangle, QUADRILATERALS, Angle Sum Property of a Quadrilateral, Types of Quadrilaterals, Properties of a Parallelogram, Another Condition for a Quadrilateral to be a Parallelogram, The Mid-point Theorem, AREAS OF PARALLELOGRAMS AND TRIANGLES, Figures on the same Base and Between the same Parallels, Parallelograms on the same Base and between the same Parallels, Triangles on the same Base and between the same Parallels, CIRCLES, Circles and its Related Terms : A Review, Angle Subtended by a Chord at a Point, Perpendicular from the Centre to a Chord, Circle through Three Points, Equal Chords and their Distances from the Centre, Angle Subtended by an Arc of a Circle, Cyclic Quadrilaterals, CONSTRUCTIONS, Basic Constructions, Some Constructions of Triangles, HERON’S FORMULA, Area of a Triangle – by Heron’s Formula, Application of Heron’s Formula in finding Areas of Quadrilaterals, SURFACEAREAS AND VOLUMES, Surface Area of a Cuboid and a Cube, Surface Area of a Right Circular Cylinder, Surface Area of a Right Circular Cone, Surface Area of a Sphere, Volume of a Cuboid, Volume of a Cylinder, Volume of a Right Circular Cone, Volume of a Sphere, STATISTICS, Collection of Data, Presentation of Data, Graphical Representation of Data, Measures of Central Tendency, PROBABILITY, Probability – an Experimental Approach, PROOFS IN MATHEMATICS, Mathematically Acceptable Statements, Deductive Reasoning, Theorems, Conjectures and Axioms, What is a Mathematical Proof?, INTRODUCTION TO MATHEMATICAL MODELLING, Review of Word Problems, Some Mathematical Models, The Process of Modelling, its Advantages and Limitations
Class 10 - X ( 10th )
Real Numbers ( Introduction, Euclid’s Division Lemma, The Fundamental Theorem of Arithmetic, Revisiting Irrational Numbers, Revisiting Rational Numbers and Their Decimal Expansions, Summary ) ; Polynomials ( Introduction, Geometrical Meaning of the Zeroes of a Polynomial, Relationship between Zeroes and Coefficients of a Polynomial, Division Algorithm for Polynomials, Summary ) ; Pair of Linear Equations in Two Variables ( Introduction, Pair of Linear Equations in Two Variables, Graphical Method of Solution of a Pair of Linear Equations, Algebraic Methods of Solving a Pair of Linear Equations, Substitution Method, Elimination Method, Cross-Multiplication Method, Equations Reducible to a Pair of Linear Equations in Two Variables ) ; Quadratic Equations ( Introduction, Quadratic Equations, Solution of a Quadratic Equation by Factorisation, Solution of a Quadratic Equation by Completing the Square, Nature of Roots ) ; Arithmetic Progressions ( Introduction, Arithmetic Progressions, nth Term of an AP, Sum of First n Terms of an AP ) ; Triangles ( Introduction, Similar Figures, Similarity of Triangles, Criteria for Similarity of Triangles, Areas of Similar Triangles, Pythagoras Theorem ) ; Coordinate Geometry ( Introduction, Distance Formula, Section Formula, Area of a Triangle ) ; Introduction to Trigonometry ( Introduction, Trigonometric Ratios, Trigonometric Ratios of Some Specific Angles, Trigonometric Ratios of Complementary Angles, Trigonometric Identities ) ; Some Applications of Trigonometry ( Introduction, Heights and Distances ) ; Circles ( Introduction, Tangent to a Circle, Number of Tangents from a Point on a Circle ) ; Constructions ( Introduction, Division of a Line Segment, Construction of Tangents to a Circle ) ; Areas Related to Circles ( Introduction, Perimeter and Area of a Circle — A Review, Areas of Sector and Segment of a Circle, Areas of Combinations of Plane Figures ) ; Surface Areas and Volumes ( Introduction, Surface Area of a Combination of Solids, Volume of a Combination of Solids, Conversion of Solid from One Shape to Another, Frustum of a Cone ) ; Statistics ( Introduction, Mean of Grouped Data, Mode of Grouped Data, Median of Grouped Data, Graphical Representation of Cumulative Frequency Distribution ) ; Probability ( Introduction, Probability — A Theoretical Approach ) ; Appendix A1 : Proofs in Mathematics ( A1.1 Introduction, A1.2 Mathematical Statements Revisited, A1.3 Deductive Reasoning, A1.4 Conjectures, Theorems, Proofs and Mathematical Reasoning, A1.5 Negation of a Statement, A1.6 Converse of a Statement, A1.7 Proof by Contradiction, A1.8 ) ; Appendix A2 : Mathematical Modelling ( A2.1 Introduction, A2.2 Stages in Mathematical Modelling, A2.3 Some Illustrations, A2.4 Why is Mathematical Modelling Important? )
Class 11 - XI ( 11th )
Sets ( Introduction, Sets and their Representations, The Empty Set, Finite and Infinite Sets, Equal Sets, Subsets, Power Set, Universal Set,
Venn Diagrams, Operations on Sets, Complement of a Set, Practical Problems on Union and Intersection of Two Sets ) ; Relations and Functions ( Introduction, Cartesian Product of Sets, Relations, Functions ) ; Trigonometric Functions ( Introduction, Angles, Trigonometric Functions, Trigonometric Functions of Sum and Difference of Two Angles, Trigonometric Equations ) ; Principle of Mathematical Induction ( Introduction, Motivation, The Principle of Mathematical Induction ) ; Complex Numbers and Quadratic Equations ( Introduction, Complex Numbers, Algebra of Complex Numbers, The Modulus and the Conjugate of a Complex Number, Argand Plane and Polar Representation, Quadratic Equations ) ; Linear Inequalities ( Introduction, Inequalities, Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation, Graphical Solution of Linear Inequalities in Two Variables, Solution of System of Linear Inequalities in Two Variables ) ; Permutations and Combinations ( Introduction, Fundamental Principle of Counting, Permutations, Combinations ) ; Binomial Theorem ( Introduction, Binomial Theorem for Positive Integral Indices, General and Middle Terms ) ; Sequences and Series ( Introduction, Sequences, Series, Arithmetic Progression (A.P.), Geometric Progression (G.P.), Relationship Between A.M. and G.M., Sum to n terms of Special Series ) ; Straight Lines ( Introduction, Slope of a Line, Various Forms of the Equation of a Line, General Equation of a Line, Distance of a Point From a Line ) ; Conic Sections ( Introduction, Sections of a Cone, Circle, Parabola, Ellipse, Hyperbola ) ; Introduction to Three Dimensional Geometry ( Introduction, Coordinate Axes and Coordinate Planes in Three Dimensional Space ) ; Coordinates of a Point in Space ( Distance between Two Points, Section Formula ) ; Limits and Derivatives ( Introduction, Intuitive Idea of Derivatives, Limits, Limits of Trigonometric Functions, Derivatives ) ; Mathematical Reasoning ( Introduction, Statements, New Statements from Old, Special Words/Phrases, Implications, Validating Statements ) ; Statistics ( Introduction, Measures of Dispersion, Range, Mean Deviation, Variance and Standard Deviation, Analysis of Frequency Distributions ) ; Probability ( Introduction, Random Experiments, Event, Axiomatic Approach to Probability ) ; Appendix 1: Infinite Series ( A.1.1 Introduction, A.1.2 Binomial Theorem for any Index, A.1.3 Infinite Geometric Series, A.1.4 Exponential Series, A.1.5 Logarithmic Series ) ; Appendix 2: Mathematical Modelling ( A.2.1 Introduction, A.2.2 Preliminaries, A.2.3 What is Mathematical Modelling )
Class 12 - XII ( 12th ) - Part - 1 ( I )
Relations and Functions ( Introduction, Types of Relations, Types of Functions, Composition of Functions and Invertible Function, Binary Operations ) ; Inverse Trigonometric Functions ( Introduction, Basic Concepts, Properties of Inverse Trigonometric Functions ) ; Matrices ( Introduction, Matrix, Types of Matrices, Operations on Matrices, Transpose of a Matrix, Symmetric and Skew Symmetric Matrices, Elementary Operation (Transformation) of a Matrix, Invertible Matrices ) ; Determinants ( Introduction, Determinant, Properties of Determinants, Area of a Triangle, Minors and Cofactors, Adjoint and Inverse of a Matrix, Applications of Determinants and Matrices ) ; Continuity and Differentiability ( Introduction, Continuity, Differentiability, Exponential and Logarithmic Functions, Logarithmic Differentiation, Derivatives of Functions in Parametric Forms, Second Order Derivative, Mean Value Theorem ) ; Application of Derivatives ( Introduction, Rate of Change of Quantities, Increasing and Decreasing Functions, Tangents and Normals, Approximations, Maxima and Minima ) ; Appendix 1: Proofs in Mathematics ( A.1.1 Introduction, A.1.2 What is a Proof? ) ; Appendix 2: Mathematical Modelling ( A.2.1 Introduction, A.2.2 Why Mathematical Modelling?, A.2.3 Principles of Mathematical Modelling )
Class 12 - XII ( 12th ) - Part - 2 ( II )
Integrals ( Introduction, Integration as an Inverse Process of Differentiation, Methods of Integration, Integrals of some Particular Functions, Integration by Partial Fractions, Integration by Parts, Definite Integral, Fundamental Theorem of Calculus, Evaluation of Definite Integrals by Substitution, Some Properties of Definite Integrals ) ; Application of Integrals ( Introduction, Area under Simple Curves, Area between Two Curves ) ; Differential Equations ( Introduction, Basic Concepts, General and Particular Solutions of a Differential Equation, Formation of a Differential Equation whose General Solution is given, Methods of Solving First order, First Degree Differential Equations ) ; Vector Algebra ( Introduction, Some Basic Concepts, Types of Vectors, Addition of Vectors, Multiplication of a Vector by a Scalar, Product of Two Vectors ) ; Three Dimensional Geometry ( Introduction, Direction Cosines and Direction Ratios of a Line, Equation of a Line in Space, Angle between Two Lines, Shortest Distance between Two Lines, Plane, Coplanarity of Two Lines, Angle between Two Planes, Distance of a Point from a Plane, Angle between a Line and a Plane ) ; Linear Programming ( Introduction, Linear Programming Problem and its Mathematical Formulation, Different Types of Linear Programming Problems ) ; Probability ( Introduction, Conditional Probability, Multiplication Theorem on Probability, Independent Events, Bayes' Theorem, Random Variables and its Probability Distributions, Bernoulli Trials and Binomial Distribution )
Following is the outline of Central Board of Secondary Education CBSE / NCERT prescribed syllabus for Applied Mathematics ( Applied Maths ) syllabi for Classes 11 - XI, 12 - XII ( Classes 11th, 12th ) :
Class 11 - XI ( 11th ) - Applied Maths, Applied Mathematics
1. Number Theory :
a. Prime Numbers: Intersecting properties of prime number without proof, Ramanujan’s work on Prime number, Encryption and prime number
b. Ratio, Proportion and Logarithms: Business Application related to Ratio and Proportion. Practical Applications of Logarithms and Anti Logarithms
2. Interpretation of Data :
Interpretation of Data represented in the form of charts, graphs, Frequency distribution, Histogram, Pie-chart etc.
3. Analysis of Data :
Arithmetic Mean, Median, Mode, Geometric and Harmonic Mean, Range, Mean deviation, Standard Deviation, Variance, coefficient of variation, skewness.
4. Commercial Mathematics :
Profit and Loss, Simple interest, compound interest, depreciation, Effective rate of interest, present value, net present value, future value, annuities.
5. Set Theory :
Set and their representations, Empty set, Finite and Infinite sets, Equal sets, subsets, power set, universal set, Venn diagrams, union and intersections of sets, complement of set.
6. Relation and Function :
Pictorial representation of a function, domain, co-domain and range of function, Function as special type of Relation, it’s Domain and range.
7. Algebra :
a. Complex Number: Concept of iota, imaginary numbers, arithmetic operation on complex number.
b. Sequence and Series: Introduction of sequences, series, Arithmetic and Geometric Progression. Relationship between AM and GM, sum of n terms etc.
c. Permutations and Combinations: Basic concepts of Permutations and Combinations, Factorial, permutations, results, combinations with standard results, Binomial Theorem (statement only).
8. Trigonometry :
Trigonometric identities, calculation of Height and distance involving angles of all degrees till 90.
Class 12 - XII ( 12th ) - Applied Maths, Applied Mathematics
1. Fundamentals of Calculus :
Basics of Limits & continuity, differentiation of non-trigonometric functions, Basic applications of derivatives in finding Marginal cost, Marginal Revenues etc. Increasing and Decreasing Functions, Maxima / Minima. Integration as reverse process of differentiation, integration of simple algebraic functions.
2. Algebra :
Introduction of Matrices, Algebra of Matrices, Determinants of Square matrices (Application only).
3. Logical Reasoning :
Number series, Coding, decoding and odd man out, direction tests, blood relations, syllogism, Binary numbers, logical operations and truth table.
4. Commercial Mathematics :
Calculating EMI, calculations of Returns, Compound annual growth rate (CAGR), Stocks, Shares, Debenture, valuation of Bonds, GST, Concept of Banking.
5. Probability :
Introduction to probability of an event, Mutually exclusive events, conditional probability, Law of Total probability. Basic application of Probability Distribution (Binomial Distribution, Poisson Distribution and Normal Distribution).
6. Two dimensional Geometry :
Slope of a line, equation of a line in point slope form, slope intercept form and two point form.
7. Linear Programming :
Introduction, related terminology such as constraints, objective function, optimization, different types of LP, mathematical formulation of LP problem, graphical method of solution for problems in two variables.
8. Analysis of time based Data :
a. Index numbers: meaning and uses of index number, construction of index numbers, construction of consumer price indices.
b. Time series & trend analysis: Component of time series, additive models, Finding trend by moving average method.
Apart from the above outline of recently proposed syllabus of Applied Mathematics, there are several projects that are real-life based and could be taken up to meet the requirements of the governing Boards, viz., CBSE and other Boards.
For examples :
Algorithmic approach of Sieve of Erastosthene’s, Ramanujan’s theory of prime numbers: Use of prime numbers in coding and decoding of messages, Bertrnad’s postulate, etc..
Following is the outline of General College Level Engineering Maths for various Technical courses. This Engineering Mathematics syllabus also cover various Engineering Courses, Polytechnic Courses, Diploma Courses & Related College level courses :
Algebra of matrices, inverse, rank, system of linear equations, symmetric, skew symmetric and orthogonal matrices. Hermitian, skew-Hermitian and unitary matrices. Eigenvalues and eigenvectors, diagonalisation of matrices, Cayley-Hamilton Theorem.
Functions of single variable, limit, continuity and differentiability, Mean value theorems, Indeterminate forms and L'Hospital rule, Maxima and minima, Taylor's series, Fundamental and mean value-theorems of integral calculus. Evaluation of definite and improper integrals, Beta and Gamma functions, functions of two variables, limit, continuity, partial derivatives, Euler's theorem for homogeneous functions, total derivatives, maxima and minima, Lagrange method of multipliers, double and triple integrals and their applications, sequence and series, tests for convergence, power series, Fourier Series, Half range sine and cosine series.
Analytic functions, Cauchy-Riemann equations, Application in solving potential problems, Line integral, Cauchy's integral theorem and integral formula (without proof), Taylor's and Laurent' series, Residue theorem (without proof) and its applications.
Gradient, divergence and curl, vector identities, directional derivatives, line, surface and volume integrals, Stokes, Gauss and Green's theorems (without proofs) applications.
Ordinary Differential Equations:
First order equation (linear and nonlinear), Second order linear differential equations with variable coefficients, Variation of parameters method, higher order linear differential equations with constant coefficients, Cauchy- Euler's equations, power series solutions, Legendre polynomials and Bessel's functions of the first kind and their properties.
Partial Differential Equations:
Separation of variables method, Laplace equation, solutions of one dimensional heat and wave equations.
Probability and Statistics:
Definitions of probability and simple theorems, conditional probability, Bayes Theorem, random variables, discrete and continuous distributions, Binomial, Poisson, and normal distributions, correlation and linear regression.
Solution of a system of linear equations by L-U decomposition, Gauss-Jordan and Gauss-Seidel Methods, Newton's interpolation formulae, Solution of a polynomial and a transcendental equation by Newton-Raphson method, numerical integration by trapezoidal rule, Simpson's rule and Gaussian quadrature, numerical solutions of first order differential equation by Euler's method and 4th order Runge-Kutta method.
Differential Calculus - I:
Leibnitz’s theorem, Partial derivatives, Euler’s theorem for homogeneous functions, Total derivatives, Change of variables, Curve tracing: Cartesian and Polar coordinates.
Differential Calculus - II:
Taylor’s and Maclaurin’s Theorems, Expansion of function of several variables, Jacobian, Approximation of errors, Extrema of functions of several variables, Lagrange’s method of multipliers ( Simple applications)
Inverse of a matrix by elementary transformations, Rank of a matrix ( Echelon & Normal form), Linear dependence, Consistency of linear system of equations and their solution,. Characteristic equation, Eigen values and eigen vectors, Cayley-Hamilton Theorem,A brief introduction to Vector Spaces,Subspaces. Rank & Nullity. Linear transformations.
Double and triple integrals, Change of order of integration, Change of variables, Application of integration to lengths, Volumes and Surface areas – Cartesian and Polar coordinates. Beta and Gamma functions, Dirichlet’s integral and applications.
Point function, Gradient,Divergence and Curl and their physical interpretations, Vector identities, Directional derivatives. Line,Surface and Volume integrals, Applications of Green’s, Stoke’s and Gauss divergence theorems (without proofs)
Linear differential equations of nth order with constant coefficients, Complementary function and Particular integral, Simultaneous linear differential equations, Solution of second order differential equations by changing dependent & independent variables, Normal form, Method of variation of parameters, Applications to engineering problems (without derivation).
Series Solution and Special Functions:
Series solution of second order ordinary differential equations with variable coefficient (Frobenius method), Bessel and Legendre equations and their series solutions, Properties of Bessel function and Legendre polynomials.
Laplace transform, Existence theorem, Laplace transforms of derivatives and integrals, Initial and final value theorems, Unit step function, Dirac- delta function, Laplace transform of periodic function, Inverse Laplace transform, Convolution theorem, Application to solve simple linear and simultaneous differential equations.
Fourier Series and Partial Differential Equations:
Periodic functions, Fourier series of period 2, Euler’s Formulae, Functions having arbitrary periods, Change of interval, Even and odd functions, Half range sine and cosine series, Harmonic analysis. Solution of first order partial differential equations by Lagrange’s method, Solution of second order linear partial differential equations with constant coefficients.
Applications of Partial Differential Equations:
Classification of second order partial differential equations, Method of separation of variables for solving partial differential equations, Solution of one and two dimensional wave and heat conduction equations, Laplace equation in two dimension, Equation of transmission lines.
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