PURE MATHEMATICS CSS SYLLABUS (100 MARKS)
Section–A (40 Marks)
I. Modern Algebra
- Group
- Subgroups
- Lagrange’s theorem
- Cyclic groups
- Normal subgroups
- Quotient groups
- Fundamental theorem of homomorphism
- Isomorphism theorems of groups
- Inner automorphisms
- Conjugate elements
- Conjugate subgroups
- Commutator subgroups
- Ring
- Subrings
- Integral domains
- Quotient fields
- Isomorphism theorems
- Field extension
- Finite fields
- Vector spaces
- Linear independence
- Bases
- Dimension of a finitely generated space
- Linear transformations
- Matrices and their algebra
- Reduction of matrices to echelon form
- Rank and nullity of a linear transformation
- Solution of a system of homogeneous linear equations
- Solution of a system of non-homogeneous linear equations
- Properties of determinants
Section–B (40 Marks)
II. Calculus & Analytic Geometry
- Real numbers
- Limits
- Continuity
- Differentiability
- Indefinite integration
- Mean value theorems
- Taylor’s theorem
- Indeterminate forms
- Asymptotes
- Curve tracing
- Definite integrals
- Functions of several variables
- Partial derivatives
- Maxima and minima
- Jacobians
- Double integration (techniques only)
- Triple integration (techniques only)
- Applications of Beta and Gamma functions
- Areas and volumes
- Riemann–Stieltjes integral
- Improper integrals and conditions of existence
- Implicit function theorem
- Conic sections in Cartesian coordinates
- Plane polar coordinates
- Representation of straight line and conic sections
- Cartesian coordinates in three dimensions
- Spherical polar coordinates in three dimensions
- Plane
- Sphere
- Ellipsoid
- Paraboloid
- Hyperboloid in Cartesian and spherical polar coordinates
Section–C (20 Marks)
III. Complex Variables
- Function of a complex variable
- De Moivre’s theorem and applications
- Analytic functions
- Cauchy’s theorem
- Cauchy’s integral formula
- Taylor’s series
- Laurent’s series
- Singularities
- Cauchy residue theorem
- Contour integration
- Fourier series
- Fourier transforms
FPSC Official PURE MATHEMATICS CSS SYLLABUS
Here is the link to the PURE MATHEMATICS CSS SYLLABUS on the FPSC official website
