November 19, 2019
Amitraj
BCA 105: Basic Mathematics
UNIT-I
Sets & Relations : Sets and elements, Equal sets, Universal set & Empty set, Subsets, Venn diagrams, Basic operations on sets, Union & Intersection, Complements, Difference, Symmetric Difference, Fundamental Products, Algebra of sets and Duality, Finite Sets, Counting Principle, Classes of sets, Power sets, Partitions, Mathematical Induction, Cartesian Products of Sets, Relations, Pictorial representations of Relations, Composition of relations, Types of relations, Equivalence Relations, Partial ordering relations.
UNIT-II
Functions, Limit and Continuity : Functions, Kinds of Functions , Concept of real function, Domain and Range (simple cases), Composition Function, One-to-one, onto, into, invertible functions, Mathematical Functions , Exponential and Logarithmic Functions, Graph of functions ( plotting of linear function, absolute value function, parabolic functions, Sin(x), Cos(x), tan(x), reciprocal function, ex, log x, Signum function), Polar coordinates and graph, Limit of variable, Limit of function, Evaluation of limits of various types of functions, Continuity & Discontinuity at a point, Continuity over an interval.
Trignometrical Functions: Definitions, proofs for any angle θ, signs of ratios, ratios of some standard angles.
UNIT-III
Quadratic Equation: Solution of Quadratic Equations, Nature of Roots.
Co-ordinates and Loci: Cartesian co-ordinate system, Introduction to Polar co- ordinates, distance between two points, section formulae, Area of triangle, Locus and its Equation.
Straight Line: Equation of straight line parallel to an Axis, slope form, intercept form, through two point condition of concurrency of three lines.
Matrices and Determinants : Definition and Types of Matrices, Addition , Subtraction and Multiplication of a Matrices, Scalar Multiplication, Transpose of Matrix, Determinants, Determinants of square matrix of order 1, 2 and 3, Area of a triangle, Solution of system of linear equations by Cramer’s Rule, Minors and Cofactors , Adjoint of a Matrix, Inverse of a Matrix(up to order 3).
UNIT-IV
Differential Calculus: Derivative of a Function, Various Formulae-Product and Quotient Rule of Differentiation, Differentiation of Function of Function(chain rule), Trignometrical functions, Inverse Trignometrical functions, Exponential function, Logarithmic function, Implicit functions, Logarithmic Differentiation, Diffentiation of function w.r.t. another function, Higher Derivatives, Successive Differentiation, Liebnitz Theorem, Expansion of functions(up to 3 or 4 terms only) using Maclaurin’s and Taylor’s Theorem, Maxima and Minima (simple cases), Curve tracing (simple cases), Introduction to partial differentiation.
UNIT-V
Integral Calculus : Anti-Derivatives, Constant of integration, Indefinite integral, Elementary Integration Formulae, Methods of Integration, Integration by Substitution, Integration by parts, integration through partial fractions and rationalization, Concept of Definite integral, properties of definite integral, Integration of 0∫π/2 Sinn x Cosm x using Gamma function. Area of Bounded Region, Circle, Parabola, Ellipse in standard form between two ordinates and x- axis.
UNIT-I
Sets & Relations : Sets and elements, Equal sets, Universal set & Empty set, Subsets, Venn diagrams, Basic operations on sets, Union & Intersection, Complements, Difference, Symmetric Difference, Fundamental Products, Algebra of sets and Duality, Finite Sets, Counting Principle, Classes of sets, Power sets, Partitions, Mathematical Induction, Cartesian Products of Sets, Relations, Pictorial representations of Relations, Composition of relations, Types of relations, Equivalence Relations, Partial ordering relations.
UNIT-II
Functions, Limit and Continuity : Functions, Kinds of Functions , Concept of real function, Domain and Range (simple cases), Composition Function, One-to-one, onto, into, invertible functions, Mathematical Functions , Exponential and Logarithmic Functions, Graph of functions ( plotting of linear function, absolute value function, parabolic functions, Sin(x), Cos(x), tan(x), reciprocal function, ex, log x, Signum function), Polar coordinates and graph, Limit of variable, Limit of function, Evaluation of limits of various types of functions, Continuity & Discontinuity at a point, Continuity over an interval.
Trignometrical Functions: Definitions, proofs for any angle θ, signs of ratios, ratios of some standard angles.
UNIT-III
Quadratic Equation: Solution of Quadratic Equations, Nature of Roots.
Co-ordinates and Loci: Cartesian co-ordinate system, Introduction to Polar co- ordinates, distance between two points, section formulae, Area of triangle, Locus and its Equation.
Straight Line: Equation of straight line parallel to an Axis, slope form, intercept form, through two point condition of concurrency of three lines.
Matrices and Determinants : Definition and Types of Matrices, Addition , Subtraction and Multiplication of a Matrices, Scalar Multiplication, Transpose of Matrix, Determinants, Determinants of square matrix of order 1, 2 and 3, Area of a triangle, Solution of system of linear equations by Cramer’s Rule, Minors and Cofactors , Adjoint of a Matrix, Inverse of a Matrix(up to order 3).
UNIT-IV
Differential Calculus: Derivative of a Function, Various Formulae-Product and Quotient Rule of Differentiation, Differentiation of Function of Function(chain rule), Trignometrical functions, Inverse Trignometrical functions, Exponential function, Logarithmic function, Implicit functions, Logarithmic Differentiation, Diffentiation of function w.r.t. another function, Higher Derivatives, Successive Differentiation, Liebnitz Theorem, Expansion of functions(up to 3 or 4 terms only) using Maclaurin’s and Taylor’s Theorem, Maxima and Minima (simple cases), Curve tracing (simple cases), Introduction to partial differentiation.
UNIT-V
Integral Calculus : Anti-Derivatives, Constant of integration, Indefinite integral, Elementary Integration Formulae, Methods of Integration, Integration by Substitution, Integration by parts, integration through partial fractions and rationalization, Concept of Definite integral, properties of definite integral, Integration of 0∫π/2 Sinn x Cosm x using Gamma function. Area of Bounded Region, Circle, Parabola, Ellipse in standard form between two ordinates and x- axis.
