# 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.

• NOTE: attempt two questions from each section.

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.

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.

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.

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
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
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
• NOTE: attempt two questions from each section.

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.

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).

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.

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