Paper I
100 marksCandidates will be asked to attempt three questions from Section A and two questions from section B.
Section A
Limits, Continuity, Differentiability and its Applications, General theorems (Rolle's
Theorem, Mean value theorem), Asymptotes, Applications of Maxima and Minima.
Definite and Indefinite integrals and their Application, Quadrature, Rectification,
Numerical methods of Integration (Trapezoidal and Simpson rule), Multiple integrals and
their Applications. Areas and Volumes, Centre of Mass, Reimann-Stijles Integral,
Ordinary Differential Equations (O.D.Eqs) and their Applications in Rectilinear motion
and Growth/Decay problems. 2nd Order Differential Equations with Applications (Spring
Mass and Simple Harmonic Oscillator Problems).
Section B
Sequences and Series, Convergence tests, Power Series, Radius and Interval of
Convergence. Complex Analysis, Function of Complex Variable, Demoivre's Theorem
and its Application. Analytic Function, Singularities, Cauchy theorem, Cauchy Integral
formula.
Conic Sections in Cartesian coordinates, Plane Polar Coordinates and their use to
represent the straight line and Conic section. Vector equation for plane and space
curves. Tangents and Normals and Binormals, Curvature and torsion, Serre Frenet's
Formula.
Recommended reading
9 titles- 01Anton, H, Calculus: A New Horizon, Ed. 6, John Willey, New York, 1999.
- 02Thomas, G. B. Finney. A. R., Calculus, Ed. 9, John Willey, New York, 2005.
- 03Yusuf, S.M. Amin. M., Calculus with Analytic Geometry, Ilmi Kitab Khana, Lahore.
- 04Zill, D. G. , Cullen, M.R., Differential Equations with Boundary Value Problems, Ed. 3. PWS Publishing Co., 1997.
- 05Abell, Braselton, Modem Differential Equations Ed. 2, Thomas Learning Inc. USA, 2001.
- 06Curchill, R.V., Brawn J.W., Complex Variables and Applications, Ed. 5, McGraw Hill, New York, 1989.
- 07Ghori, Q.K., Mechanics. Ilmi Kitab Khan, Lahore.
- 08Weather burn, C.E., Differential Geometry, The English Languages Book Society and Cambridge Uni. Press. 1964.
- 09Guggenheinerar, H.W., Differential Geometry, McGraw Hill, 1990.
Paper II — Section A
100 marksGroups: Definition and examples of Groups, Order of a Group, Order of an element of
a Group, Abelian and non-Abelian Groups and Cyclic groups. Lagrange theorem and
applications, Normal subgroups, Characteristic Subgroups of a group, Normalizer in a
group, Centralizer in a group. Fundamental Theorem of Homomorphism, Isomorphism
theorems of groups, Automorphisms
Rings, Fields and Vector Spaces: Examples of Rings, Subrings, Integral domains,
Fields, Vector spaces, Linear independence/ dependence, Basis and dimension of
finitely generated spaces, Examples of Field extension and finite fields, Examples of
finite and infinite dimensional vector spaces.
Section B
Metric Spaces and Topological Spaces: Definition and Examples of Metric spaces
and topological spaces, Closed and Open Spheres, Interior, Exterior and Frontier of a
Set, Sequences in Metric Spaces, Convergence of Sequences. Definition and examples
of Normed Spaces. Inner product spaces, Gram-Schmidt Process of Orthonormalization
Matrices and Linear Algebra: Linear transformations, Matrices and their algebra,
Reduction of matrices to Echelon and Reduced Echelon form. Solution of a system of
homogenous and Non-Homogenous equations, Numerical methods of solving
equations (Gauss-Siedal method, Jaccobi method) Properties of Determinants,
Eigenvalues and Eigenvectors and the Diagonalization of the Symmetric Matrices.
Recommended reading
10 titles- 01Nicholson. W.K., Elementary Linear Algebra with Applications, Ed. 2, Prentice Hall, Englewood, USA.
- 02Herstein, I.N., Topics in Algebra, John Willey and Sons (New York) 1964.
- 03Rowen, L., Rings (I & II). Academic Press, Ins.
- 04Dar, K.H., First Step to Abstract Algebra, Feroz Sons Publishers, Lahore (1996).
- 05Yusuf, S.M., Majeed A., Amin. M., Mathematical Methods, Ilmi Kitab Khana, Lahore.
- 06Atkinson, K. E., An Introduction to Numerical Analysis, Ed. 2 John Willey, New York, 1989.
- 07Ahmad. F and Afzal. M, Numerical Analysis, National Book Foundation, Islamabad.
- 08Simmons, F.J., Topology, McGraw Hill Company, New York.
- 09Kreyszig, E., Introductory Functional Analysis with Applications, John Willey and Sons, New York, 1978.
- 10Majeed, A. Elements of Topology and Functional Analysis, Ilmi Kitab Khana, Lahore.
Others in Group C
You may take one subject from this group, so these are alternatives to Mathematics, not companions to it.