# Pure Mathematics (Paper A & B)

This paper consist of two papers of 100 marks each. One paper is called “Paper A” and the other is called “Paper B”.

• 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. 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)

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) .

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.

• NOTE: attempt two questions from each section.

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.

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.

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.

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. Set Theory and Logic by Stoll, Robert R.S. Chand & Co. New Delhi (1986)
10. Number Theory by Dr. Manzoor Hussain. The Carvan Book House. Kachehry Road, Lahore.
11. Elementary Linear Algebra (sixth edition) by Howard Anton And Chris Rorres. John Willey & Sons. Inc.