CSS Pure Mathematics Syllabus — FPSC PDF Download

Pure Mathematics scares most candidates, and the fear makes sense. The CSS Pure Mathematics syllabus covers real analysis, abstract algebra, linear algebra, and number theory. It carries 100 marks and demands proofs, not tricks. Our count covers papers from 2009 to 2025. Real numbers, limits, continuity, and differentiability top the table. They drew 28 questions and 7.4% weight. Group theory follows close. Subgroups and Lagrange’s theorem drew 25 questions. Isomorphism theorems drew 22. Vector spaces and linear transformations add 20 questions each. Notice the dead zone, too. Conjugate elements and commutator subgroups hold zero appearances. Spend your hours elsewhere. Write full proofs in practice, line by line. Examiners cut marks for skipped steps and hand-waving. Choose this subject only with a mathematics degree behind you. Nobody grows proof muscle in a few months.

PURE MATHEMATICS CSS Syllabus (100 MARKS)

Section-A (40- marks)

I. Modern Algebra  Group, subgroups, Lagranges 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 and finite fields.  Vector spaces, Linear independence, Bases, Dimension of a finitely generated space. Linear transformations, Matrices and their algebra. Reduction of matrices to their echelon form. Rank and nullity of a linear transformation.  Solution of a system of homogeneous and 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. Jacobnians, Double and triple integration (techniques only).Applications of Beta and Gamma functions. Areas and Volumes. Riemann-Stieltje’s integral. Improper integrals and their conditions of existences. Implicit function theorem.  Conic sections in Cartesian coordinates, Plane polar coordinates and their use to represent the straight line and conic sections. Cartesian and spherical polar coordinates in three dimensions. The plane, the sphere, the ellipsoid, the paraboloid and the hyperboloid in Cartesian and spherical polar coordinates.

Section-C (20-marks)

III. Complex Variables Function of a complex variable; Demoiver’s theorem and its applications. Analytic functions, Cauchy’s theorem. Cauchy’s integral formula, Taylor’s and Laurent’s series. Singularities. Cauchy residue theorem and contour integration. Fourier series and Fourier transforms.

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PURE MATHEMATICS Frequency Analysis (2009–2025)

PURE MATHEMATICS Top 5 Most Repeated Topics (2009–2025)

1. Indefinite integration. Mean value theorems. Taylor’s theorem, Indeterminate forms. (32 Questions)
2. Real Numbers. Limits. Continuity. Differentiability. (28 Questions)
3. Singularities. Cauchy residue theorem and contour integration. (28 Questions)
4. Group, subgroups, Lagranges theorem, Cyclic groups, Normal subgroups, Quotient groups. (25 Questions)
5. Fundamental theorem of homomorphism. Isomorphism theorems of groups, Inner automorphisms. (22 Questions)