A-Course of Mathematics (Paper A & B)

This subject is consists of two papers of 100 marks each. One is called “Paper A” and other is called “Paper B”. This page is updated on February 15, 2015. This syllabus is for 1st Annual 2015 and onward organized by University of Sargodha, Sargodha.

  • NOTE: attempt two questions from each section.

SECTION-I (4/12: 17,17,17,17)

Theory of limit and continuity. Solution of Inequalities. Derivatives and its application to business, economics and physics etc. Differentials. Related rates. Higher order derivatives. Leibnitz’s theorem. Limits and continuity of functions of two variables. Partial differentiation and its geometrical meaning for functions of two variables. Euler’s theorem. Increments and differentials. Chain Rule. Extrema by 2nd order derivative test and by Lagrange multiplier method. General theorems and indeterminate forms. L’ Hospital rule and its applications. Increasing and decreasing functions. Intermediate value theorem and its immediate consequence (only statements)

SECTION-II (4/12: 16,16,16,16)

Translation and rotation of axes. Second degree equation with reference to conic section. Properties of conics. Tangents and normals (Cartesian Coordinates), Polar equations of conics. Sketching of Curves in polar coordinates, Tangents and normals (Polar Coordinates). Parametric representation of curves. Pedal Equations. Vector spaces and sub spaces. Linearly dependent and independent vectors. Bases and dimension. Linear transformations and matrix of linear transformation. (relevant theorems of bases and linear transformation without proofs) .

SECTION-III (4/12: 17,17,17,17)

Sequences. Bounded Sequences. Cauchy sequences. Convergence and divergence of sequences. Cauchy’s theorem. Nth-term test, comparison test, ratio test, root test and integral test for convergence and divergence of infinite series. Convergence and divergence of alternating series. Power series. Complex numbers and their properties. De moivre’s theorem and its applications. Circular, logarithmic and hyperbolic functions. Separation into real and imaginary parts.

SECTION-I

Chapter 1 (Calculas)
Ex 1.2, 1.3 Theory of limit and continuity
Ex 1.1 Q1 to 15 Solution of Inequalities
Chapter 2 (Calculus)
Ex 2.1 Derivatives and its application to business, economics and physics etc
Ex 2.3 Related rates,Differentials
Ex 2.5 Higher order derivatives Leibnitz’s theorem
Ex 2.6 Limits and continuity of functions of two variables
Ex 2.6 Partial differentiation and its geometrical meaning for functions of two variables
Chapter 9 (Calculus)
Ex 9.1, 9.2, 9.3 Euler’s theorem, Increments and differentials, Chain Rule
Ex 9.6, 9.7 Extrema by 2nd order derivative test and by Lagrange multiplier method
Chapter 3 (Calculus)
Ex 3.1 General theorems and indeterminate forms
Ex 3.1 Increasing and decreasing functions
Ex 3.3 L’ Hospital rule and its applications
Ex 3.3 Intermediate value theorem and its immediate consequence (only statements)

SECTION-II

Chapter 6 (Calculus)
Ex 6.1 Translation and rotation of axes
Ex 6.1 Second degree equation with reference to conic section
Ex 6.2 Properties of conics. Tangents and normals (Cartesian Coordinates)
Ex 6.3, 6.4 Sketching of Curves in polar coordinates,Polar equations of conics
Ex 6.5 Sketching of Curves in polar coordinates
Ex 6.6 Tangents and normals (Polar Coordinates)
Ex 6.7 Pedal Equations, Parametric representation of curves
Chapter 6 (Method)
Ex 6.1 Vector spaces and sub spaces, Bases and dimension
Ex 6.2 Linearly dependent and independent vectors
Ex 6.3 Linear transformations and matrix of linear transformation
Ex 6.1 to 6.4 Relevant theorems of bases and linear transformation without proofs

SECTION-III

Chapter 8 (Method)
Ex 8.1 Sequences, Bounded Sequences, Cauchy sequences
Ex 8.1 Convergence and divergence of sequences
Ex 8.2 Nth-term test, Cauchy’s theorem, Comparison test, Integral test for convergence and divergence of infinite series
Ex 8.3 Ratio test, Root test
Ex 8.4 Convergence and divergence of alternating series
Ex 8.5 Power series
Chapter 1 (Method)
Ex 1.1 Complex numbers and their properties
Ex 1.2 De moivre’s theorem and its applications
Ex 1.3, 1.4 Circular functions, Logarithmic and hyperbolic functions
Ex 1.5 Separation into real and imaginary parts
  • NOTE: attempt two questions from each section.

SECTION-I (4/12: 17,17,17,17)

Antiderivatives and indefinite integrals. Methods of integration. Definite integral as limit of sum. Fundamental theorem. Properties. Improper integrals. Reduction formulas. Double and triple integral (simple cases). Area between curves. Length of arc. Intrinsic equations. Asymptotes. Extrema and its application. Singular points. Curvature. Evolute and envelopes. Volume and surfaces of revolution.

SECTION-II (4/12: 16,16,16,16)

Definition and examples of metric spaces. Open and closed balls and sets. Neighborhoods. Limit points. Interior, exterior and boundary sets. Closure of a set. Complete metric spaces. Definition and examples of topological spaces. Basic properties. Neighborhoods. Limit points. Interior, exterior and boundary sets. Closure of a set. Divisibility. Euclid theorem. Greatest divisor. Least common multiple. Prime factorization theorem. Introduction to elementary logic. Predicate calculus. Methods of proofs.

SECTION-III (4/12: 17,17,17,17)

Definition and examples of a group. Order of an element of a group. Subgroup. Cyclic and permutation groups. Lagrange’s theorm. Rings and fields. Algebra of matrices. Co-factors, minors, adjoint and inverse of a matrix. Elementary row and column operations. Echelon form and rank of matrix. Solution of the system of linear equations(Homogeneous and non-homogeneous) by use of matrices. Net work flow problems. Determinants with properties.

SECTION-I

Chapter 4 (Calculus)
Ex 4.1 Antiderivatives and indefinite integrals
Ex 4.1 to EX 4.6 Methods of integration
Chapter 5 (Calculus)
Ex 5.1 Definite integral as limit of sum
Ex 5.2 Fundamental theorem, Properties
Ex 5.3 Improper integrals
Ex 5.4 Reduction formulas
Chapter 10 (Calculus)
Ex 10.1 Double and triple integral (simple cases)
Chapter 7 (Calculus)
Ex 7.1 Asymptotes
Ex 7.2 Extrema and its application
Ex 7.3 Singular points
Ex 7.5 Area between curves
Ex 7.6 Length of arc, Intrinsic equations
Ex 7.7, 7.8 Curvature, Evolute and envelopes
Chapter 9 (Calculus)
Ex 9.8 Volume and surfaces of revolution

SECTION-II

Study On Notes
Chapter 1 Definition and examples of metric spaces
Chapter 2 Open and closed balls and sets
Chapter 2 Neighborhoods, Limit points
Chapter 3 Interior, exterior and boundary sets, Closure of a set, Neighborhoods
Chapter 4 Complete metric spaces
Chapter 1 Definition and examples of topological spaces, Basic properties
Chapter 2 Limit points, Interior, exterior and boundary sets, Closure of a set
Chapter 1 Divisibility, Euclid theorem, Greatest divisor, Least common multiple
Chapter 2 Prime factorization theorem
Chapter 3 Introduction to elementary logic, Predicate calculus, Methods of proofs

SECTION-III

Chapter 2 (Method)
Ex 2.1 Definition and examples of a group
Ex 2.2 Order of an element of a group
Ex 2.2 Subgroup, Cyclic groups
Ex 2.3 permutation groups, Rings and fields
Ex 2.2 Lagrange’s theorm
Chapter 3 (Method)
Ex 3.1 Algebra of matrices
Ex 3.2 Co-factors, minors, adjoint and inverse of a matrix
Ex 3.2 Elementary row and column operations, Echelon form and rank of matrix
Chapter 4 (Method)
Ex 4 Solution of the system of linear equations (Homogeneous and non-homogeneous) by use of matrices, Net work flow problems
chapter 5 (Method)
Ex 5.1, 5.2 Determinants with properties
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