# General 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 syllabus is for **1st Annual 2015 and onward** organized by University of Sargodha (UoS), Sargodha.

## Paper A

- 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. Newton-Raphson formula. 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. General theorems (without proofs) and indeterminate forms. L’ Hospital rule of functions. Increasing and decreasing functions.

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

Translation and rotation of axes. Second degree equation with reference to conic section. Properties of conics. Polar equations of conics. Tangents and normals. Parametric representation of curves. Pedal Equations. Asymptotes. Extrema and its application. Singular points. Curvature. Evolutes and envelopes.

### SECTION-III (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. Numerical integration (rectangular, trapezoidal and 1/3 Simpson’s rules). Co-ordinates in three dimension. Rectangular, cylindrical and spherical co-ordinates. Equations of plan, straight line, sphere, cylinder, cone, ellipsoid, hyperboloid and paraboloid. Longitude and latitudes. Spherical triangle and direction of Qibla.

## Exercise wise paper pattern

### Section I

Chapter # 1 (Calculus) | |
---|---|

Ex 1.1 (Q.1 to 15) | Solution of inequalities |

Ex 1.2, 1.3 | Theory of limit, Continuity |

Chapter 2 (Calculus) | |

Ex 2.1 (Q.1 to 9) | Differentiability |

Ex 2.1 | Application of derivative to business, economics, physics etc. |

Ex 2.2 | Derivatives |

Ex 2.3 | Related rates |

Ex 2.4 | Newton-Raphson formula |

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 |

Ex 2.6 | Geometrical meaning of partial derivative for functions of two variables |

Chapter 9 (Calculus) | |

Ex 9.1 | Euler’s theorem |

Ex 9.2, 9.3 | Differentials, Increments and differentials, Chain Rule |

Chapter 3 (Calculus) | |

Ex 3.1 | Increasing and decreasing functions |

Ex 3.1, 3.2 | General theorems (without proofs) |

Ex 3.3 | Indeterminate forms, L'Hospital rule of functions |

### Section II

Chapter 6(Calculus) | |
---|---|

Ex 6.1 (Q.1 to 13) | Properties of conics, Translation and rotation of axes |

Ex 6.2 | Tangent and normal |

Ex 6.3, 6.4 | Polar equations of conics |

Ex 6.6 | Tangent and normal in polar coordinates. Pedal Equations |

Ex 6.7 | Parametric representation of curves |

Ex 6.1, 6.2 | Second-degree equation with reference to conic section |

Chapter 7 (Calculus) | |

Ex 7.1 | Asymptotes |

Ex 7.2 | Extrema and its application |

Ex 7.3 | Singular points |

Ex 7.7, 7.8, 7.9 | Curvature, Evolutes and envelopes |

### SECTION III

Chapter 4 (Calculus) | |
---|---|

Ex 4.1 | Indefinite integrals, Antiderivatives |

Ex 4.2, 4.3, 4.4, 4.5, 4.6 | Methods of integration |

Chapter 5 (Calculus) | |

Ex 5.1 | Definite integral as limit of sum |

Ex 5.2 | Fundamental theorem .Properties of definite integral |

Ex 5.3 | Improper integrals |

Ex 5.4 | Reduction formulas |

Ex 5.5 | Numerical integration 1/3 Simpson’s rules rectangular rules ,trapezoidal rules |

Chapter 10 (Calculus) | Double and Triple integral (Simple cases) |

Chapter 7 (Calculus) | |

Ex 7.5 | Area between curves |

Ex 7.6 | Length of arc, Intrinsic equations |

Chapter 8 (Calculus) | |

Ex 8.1 | Co-ordinates in three dimension |

Ex 8.2, 8.3 | Equation of straight line, Equations of plane |

Ex 8.10 | Equation of cylinder, Equation of cone |

Ex 8.11 | Equation of sphere |

Ex 8.12 | Equation of ellipsoid, paraboloid, hyperboloid |

Ex 8.13 | Spherical triangle, Longitude and latitudes, Direction of Qibla |

## Paper B

- NOTE: attempt two questions from each section.

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

Vectors in three-dimensions. Scalar and vector products with applications. Scalar and vector triple products. Differentiation and integration of vector functions. Gradient, divergence and curl. Sequences. Bounded Sequences. Convergence and divergence of sequences. 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. Complex numbers and their properties. De moivre’s theorem and its applications. Circular, logarithmic and hyperbolic functions. Separation into real and imaginary parts.

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

Algebra of matrices. Determinants with properties. 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. Network flow problems. Vector spaces and sub spaces. Linearly dependent and independent vectors. Bases and dimension. Linear transformation. Matrix of linear transformation (relevant theorems of bases and linear transformation with out proofs).

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

Basic concepts of differential equations. Classification and formation of DEs. Various methods of solutions of first order ODE (linear and non-linear). The Bernoulli’s, Ricatti and Clairaut’s equations. Singular solutions. Orthogonal trajectories. Application of first order ODE in problems of decay and growth, population dynamics, logistic equations. Linear DE of higher order (homogeneous and non-homogeneous). Solution by: D-operator and undetermined co-officients Methods. Reduction of order and variation of parameters methods for 2nd order linear DE. Cauchy-Euler equation.

**Note:** *A student has to take B-course of Mathematics (both I & II- papers) as an additional course to be eligible to do M.Sc. mathematics as a regular or a private candidate.*

## Exercise wise paper pattern

### SECTION-I

Chapter 2 (Vector) | |
---|---|

Ex 2.1 | Vectors in three-dimensions |

Ex 2.1 | Scalar Products with applications |

Ex 2.2 | Vector products with applications |

Ex 2.3, 2.4 | Scalar triple products, Vector triple products |

Chapter 3 (Vector) | |

Ex 3.1, 3.2 | Differentiation of vector functions |

Ex 3.3 | Integration of vector functions |

Chapter 4 (Vector) | |

Ex 4.1, 4.2, 4.3, 4.4 | Gradient, Divergence, Curl |

Chapter 8 (Method) | |

Ex 8.1 | Sequences, Bounded Sequences, Nth-term test |

Ex 8.2 | Convergence and divergence of sequences |

Ex 8.2 | Comparison test, Integral test |

Ex 8.3 | Ratio test, Root test |

Ex 8.4 | Convergence and divergence of alternating series |

Chapter 1 (Method) | |

Ex 1.1 | Complex numbers and their properties |

Ex 1.2 | Separation into real and imaginary parts |

Ex 1.2 | De moiré’s theorem and its applications (root of complex number) |

Ex 1.3, 1.4 | Circular function, Hyperbolic functions, Logarithmic function |

### SECTION-II

Chapter 3 (Method) | |
---|---|

Ex 3.1 | Algebra of matrices |

Ex 3.2 | Co-factors of a matrix, Minors of a matrix, Adjoint of a matrix |

Ex 3.2 | Inverse of a matrix, Elementary row operation |

Ex 3.2 | Elementary column operations, Echelon form of matrix, Rank of matrix |

Chapter 4 (Method) | |

Ex 4.1 | Solution of the system of linear equations by use of matrices |

Ex 4.1 | Network flow problems |

Ex 4.1 | Homogeneous equations, Non-homogeneous equations |

Chapter 5 (Method) | |

Ex 5.1 | Determinants with properties |

Ex 5.2 | Determinants with properties |

Chapter 6 (Method) | |

Ex 6.1 | Vector spaces and sub spaces |

Ex 6.2 | Linearly dependent and independent vectors |

Ex 6.2 | Basis and dimension |

Ex 6.3 | Linear transformation |

Ex 6.4 | Matrix of linear transformation (relevant theorems of bases and linear transformation with out proofs) |

### SECTION-III

Chapter 9 (Method) | |
---|---|

Ex 9.1 | Basic concepts of differential equations |

Ex 9.1 | Classification and formation of DEs |

Ex 9.2 | Various methods of solutions of first order ODE (linear and non-linear) |

Ex 9.3 | Homogeneous equations |

Ex 9.3 | Differential equation reduceable to homogeneous form |

Ex 9.4, 9.5, 9.6 | Exact equationIntegrating factorsBernoulli’s equation |

Ex 9.7 | Orthogonal trajectories |

Ex 9.8, 9.9 | Clairaut’s equations, Ricatti equations, Singular solutions |

Chapter 10 (Method) | |

Ex 10.1 | Homogeneous Linear DE of higher order |

Ex 10.2 | Non-homogeneous Linear DE of higher order |

Ex 10.3 | Solution by undetermined coefficients methods |

Ex 10.4 | Cauchy-Euler equation |

Ex 10.5 | Reduction of order methods for second order linear DE |

Ex 10.6 | Variation of parameters methods for second order linear DE |

Ex 10.8 | Solution by D-operator |

Ex 10.11 | Application of first order ODE in problems of decay and growth, population dynamics, logistic equations |

## Recommended Books

- Calculus by H.Anton. John Wiley and Sons New York.
- Calculus By C.H Edwards and D.E. Penney. Prentiee Hall. Ine. (1998)
- Calculus By S.I. Grossman. Academic Press Ine (London) Ltd. (1984)
- Calculus and Analytic Geometry by S.M. Yousaf. Illmi Kitab Khana. Urdu Bazar Lahore
- Calculus and analytic geometry by G.B Thomas and R.I. Finney. 9th Edition (1997), Adison-Wesley Publishing Company. Lahore.
- Elementary Linear Algebra by C.H. Edwards. Jr and Davide penney. Prentic Hall international Ine.
- Mathematical Techniques by K. H. Dar. Irfan-ul-Haq and M.A. Jajja. The Carvan Book House. Kachehry Road Lahore.
- Mathematics Methods by S.M. Yousaf. Illmi Kitab Khana. Urdu Bazar Lahore.
- Theory of Differential Equations by Dennis G. Zill. Books / Cole Thomson Learning Academic Resource Center. U.S.A.
- Elementary Vector Analysis. By Dr. Munawar Hussain. S. M. Hafeez. M. A. Saeed and Ch. Bashir Ahmed. The Caravan Book, House Kachahry Road, Lahore.