# Syllabus & Paper Pattern for General Mathematics (Split Program)

There was one examination after two years for BA/BSc Program from University of Punjab (PU), Lahore but from this year (2016), PU has made changes in its examination policies for the said program. The BA/BSc Program has been split into two parts. Syllabus is break into two part year wise. After the each year of the program candidate has to appeared in examination instead of appearing after two year. In this regards syllabus of General Mathematics has been split into two part. After completion of 1st year of BSc, student has to appear in Paper A (known as General Mathematics-I or Calculus (Differential and Integral Calculus)) of General Mathematics. After the completion of 2nd year of BSc, student has to appear in Paper B (known as General Mathematics-II or Mathematical Methods: (Geometry, Infinite Series, Complex Numbers, Linear Algebra, Differential Equations)) of General Mathematics.

#### Calculus (Differential and Integral Calculus)

Note: Attempt six questions by selecting two questions from Section I, two questions from Section II, one question from Section III and one question from Section IV.

Section I (Attempt 2 questions out of 4)

Preliminaries: Real numbers and the real line; Functions and their graphs; Shifting and scaling graphs; Solution of equations involving absolute values; Inequalities

Limit and Continuity: Limit of a function, left hand and right hand limits, Theorems of limits (without proofs); Continuity, Continuous functions

Derivatives and its Applications: Differentiate functions; Differentiation of polynomial, rational and transcendental functions; Intermediate value theorem, Rolle’s theorem (without proofs); Mean value theorems and applications (without proofs); Higher derivatives, Leibniz’s theorem (without proofs); L’Hospitals Rule; Application of Taylor’s and Maclaurin’s theorem with their remainders

• Chapter 01: Real numbers, limits and continuity
• Chapter 02: The Derivative
• Chapter 03: General Theorem, Intermediate Forms → Calculus

Section II (Attempt 2 questions out of 4)

Integration and Definite Integrals: Techniques of evaluating indefinite integrals; Integration by substitutions, Integration by parts; Change of variable in indefinite integrals; Definite integrals, Fundamental theorem of calculus; Reduction formulas for algebraic and trigonometric integrands; Improper integrals, Gamma functions; Numerical integration

Plane Analytic Geometry: Conic section and quadratic equations; Classifying conic section by eccentricity; Translation and rotation of axis; Properties of circle, parabola, ellipse, hyperbola, Polar coordinates, conic sections in polar coordinates; Graphing in polar coordinates; Tangents and normal, pedal equations, parametric representations of curves

• Chapter 04: Techniques of Integration
• Chapter 05: The Definite Integral
• Chapter 06: Plane Curves I → Calculus

Section III (Attempt 1 question out of 2)

Applications of Integration: Asymptotes. Relative extrema, points of inflection and concavity; Singular, poirts, tangents at the origin; Graphing of Cartesian and polar curves;Area under the curve, area between two curves; Arc length aid intrinsic equations; Curvature, radius and centre of curvature; Involute and volute, envelope

• Chapter 07: Plane Curves II → Calculus

Section IV (Attempt 1 question out of 2)

Functions of Several Variables and Multiple Integrals: Limit and continuity of a function of two variables; The partial derivative, Computing partial derivatives algebraically; The second-order partial derivative; Tangent planes and normal lines; Maxima and minima of a function of two variables; Double integral in rectangular and polar form; Triple integral in rectangular, Cylindrical and spherical coordinates; Substitution in multiple integrals

• Chapter 09: Functions of Several Variables
• Chapter 10: Multiple Integrals → Calculus

#### Mathematical Methods: (Geometry, Infinite Series, Complex Numbers, Linear Algebra, Differential Equations)

Note: Attempt six questions by selecting two questions from Section I, one question from Section II, one question from Section III and two questions from Section IV.

Section I (Attempt 2 questions out of 4)

Complex Numbers: Complex Numbers and their properties; Polar form, argand diagram, separating into real and imaginary parts; De Moivre’s theorem and its applications; Elementary functions: circular, logarithmic, hyperbolic, exponential functions; Series solution by using complex numbers

Sequence and Series: Sequences, Infinite series: Convergence of sequence and series; The integral test, Comparison tests, Ratio test, Root test; Alternative series, Absolute and conditional convergence; Power series, Interval and radius of convergence

• Chapter 01: Complex Numbers
• Chapter 08: Infinite Series → Method

Section II (Attempt 2 questions out of 4)

Vectors: Introduction to vector algebra; Scalar and vector product; Scalar triple product and vector triple product; Applications to geometry; Vector equation of a line and plane; Partial derivatives of vector point functions; Scalar and vector fields; The gradient, divergence and curl

Analytic Geometry of Three Dimensions: Rectangular coordinates system in a space; Cylindrical and spherical coordinate system; Direction ratios and direction cosines of a line; Equation of straight lines and planes in three dimensions; Shortest distance between skew lines; Equation of sphere, cylinder, cone, ellipsoids, paraboloids, hyperboloids; Quadric and ruled surfaces; Spherical trigonometry, Direction of Qibla

• Vector Analysis → Elementary Vector Analysis
• Chapter 08: Analytic Geometry of Three Dimensions → Calculus

Section III (Attempt 1 question out of 2)

Matrices, Determinants, System of Linear Equations, and Vector Spaces: Algebra of Matrices, types of matrices; Determinant of square matrix, inverses of matrices; Rank of a matrix; Introduction to systems of linear equations; Cramer’s rule, Gaussian elimination and Gauss Jordan method; Solution of homogenous and non-homogenous linear equations; Vector spaces and subspaces; Linear combination Linear independence, Bases and dimension

• Chapter 03: Matrices
• Chapter 04: System of Linear Equations
• Chapter 05: Determinants
• Chapter 06: Vector Spaces → Method

Section IV (Attempt 1 question out of 2)

First Order Differential Equations: Formation of differential equation; Separable equations, Homogeneous and non-homogeneous equations; Linear and nonlinear equations; Exact and non-exact equations and integrating factors; Orthogonal trajectory, Bernoulli, Ricatti, Clairaut’s equations

Higher Order Linear Differential Equations: Fundamental solutions of linear homogenous equations; Operator method, Method of undetermined coefficients; Cauchy Euler’s equation; Variation of parameters

• Chapter 09: First Order Differential Equations
• Chapter 10: Higher Order Linear Differential Equations → Method

#### Legends

Calculus = Calculus with Analytic Geometry Published by Ilmi Kitab Khana, Lahore. : Notes of Calculus with Analytic Geometry

Method = Mathematical Method Published by Ilmi Kitab Khana, Lahore. : Notes of Mathematical Method

Vectors = Elementary Vector Analysis published by The Caravan Book House, Lahore. : Notes of Vector Analysis | Vector Analysis by Hameed Ullah: Notes

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