Stage 1Orientation0%
Understand the official syllabus, paper structure, marks and preparation resources.
Stage 2Foundation0%
Build the concepts, terminology and prerequisite knowledge needed for the subject.
Stage 3Official Syllabus Coverage0%
Cover every official FPSC topic in a practical sequence.
Stage 4Consolidation0%
Prepare short notes, comparisons, evidence, diagrams, facts, formulas or timelines.
Stage 5Exam Application0%
Apply knowledge through the paper-specific practice format.
Stage 6Past Papers0%
Attempt topic-wise and full-length past-paper practice.
Stage 7Revision and Mock Examination0%
Complete layered revision, weak-area review and a timed mock paper.
Paper 1 · 100 marks
Subject overview
PAPER: PURE MATHEMATICS (100 MARKS) Section-A (40- marks)
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.
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.
Real Numbers. Limits. Continuity. Differentiability. Indefinite integration. Mean value theorems. Taylor’s theorem, Indeterminate forms. Asymptotes. Curve tracing. Definite integrals. Func tions of several variables. Partial derivatives. Maxima and m
Real Numbers. Limits. Continuity. Differentiability. Indefinite integration. Mean value theorems. Taylor’s theorem, Indeterminate forms. Asymptotes. Curve tracing. Definite integrals. Func tions 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.
Modern Algebra
Ring, Subrings, Integral domains, Quotient fields, Isomorphism theorems, Field extension and finite fields.
Ring, Subrings, Integral domains, Quotient fields, Isomorphism theorems, Field extension and finite fields.
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 sphe re, the ellipsoid, the paraboloid and
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 sphe re, the ellipsoid, the paraboloid and the hyperboloid in Cartesian and spherical polar coordinates. Section-C (20-marks)
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.
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.
Calculus & Analytic Geometry
Solution of a system of homogeneous and non -homogeneous linear equations. Properties of determinants. Section-B (40- marks)
Solution of a system of homogeneous and non -homogeneous linear equations. Properties of determinants. Section-B (40- marks)
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.
- Suggested ReadingsAdvanced Calculus Kaplan, W.
- Suggested ReadingsAnalytic Function Theory Vol.1 Hille, E.
- Suggested ReadingsCalculus Anton H.,Biven I and Davis, S.
- Suggested ReadingsComplex Analysis Goodstein G.R.G.
- Suggested ReadingsComplex Variables Murray R. Spiegel
- Suggested ReadingsCalculus with Analytic Geometry Yusuf, S.M.
- Suggested ReadingsCalculus and Analytic Geometry Zia ul Haq
- Suggested ReadingsElements of Complex Analysis Pennisi, L.L.
- Suggested ReadingsTheory of Groups Majeed, A.
- Suggested ReadingsMathematical Methods Yusuf, S.M.
- Suggested ReadingsMathematical Techniques Karamat H.Dar
- Suggested ReadingsMathematical Analysis Apostal, T.M.
- Suggested ReadingsThe Theory of Groups Macdonald, I.N.
- Suggested ReadingsTopics in Algebra Herstein, I.N.
My Subjects
Stage 1Orientation0%
Understand the official syllabus, paper structure, marks and preparation resources.
Stage 2Foundation0%
Build the concepts, terminology and prerequisite knowledge needed for the subject.
Stage 3Official Syllabus Coverage0%
Cover every official FPSC topic in a practical sequence.
Stage 4Consolidation0%
Prepare short notes, comparisons, evidence, diagrams, facts, formulas or timelines.
Stage 5Exam Application0%
Apply knowledge through the paper-specific practice format.
Stage 6Past Papers0%
Attempt topic-wise and full-length past-paper practice.
Stage 7Revision and Mock Examination0%
Complete layered revision, weak-area review and a timed mock paper.
Paper 1 · 100 marks
Subject overview
PAPER: PURE MATHEMATICS (100 MARKS) Section-A (40- marks)
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.
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.
Real Numbers. Limits. Continuity. Differentiability. Indefinite integration. Mean value theorems. Taylor’s theorem, Indeterminate forms. Asymptotes. Curve tracing. Definite integrals. Func tions of several variables. Partial derivatives. Maxima and m
Real Numbers. Limits. Continuity. Differentiability. Indefinite integration. Mean value theorems. Taylor’s theorem, Indeterminate forms. Asymptotes. Curve tracing. Definite integrals. Func tions 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.
Modern Algebra
Ring, Subrings, Integral domains, Quotient fields, Isomorphism theorems, Field extension and finite fields.
Ring, Subrings, Integral domains, Quotient fields, Isomorphism theorems, Field extension and finite fields.
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 sphe re, the ellipsoid, the paraboloid and
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 sphe re, the ellipsoid, the paraboloid and the hyperboloid in Cartesian and spherical polar coordinates. Section-C (20-marks)
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.
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.
Calculus & Analytic Geometry
Solution of a system of homogeneous and non -homogeneous linear equations. Properties of determinants. Section-B (40- marks)
Solution of a system of homogeneous and non -homogeneous linear equations. Properties of determinants. Section-B (40- marks)
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.
- Suggested ReadingsAdvanced Calculus Kaplan, W.
- Suggested ReadingsAnalytic Function Theory Vol.1 Hille, E.
- Suggested ReadingsCalculus Anton H.,Biven I and Davis, S.
- Suggested ReadingsComplex Analysis Goodstein G.R.G.
- Suggested ReadingsComplex Variables Murray R. Spiegel
- Suggested ReadingsCalculus with Analytic Geometry Yusuf, S.M.
- Suggested ReadingsCalculus and Analytic Geometry Zia ul Haq
- Suggested ReadingsElements of Complex Analysis Pennisi, L.L.
- Suggested ReadingsTheory of Groups Majeed, A.
- Suggested ReadingsMathematical Methods Yusuf, S.M.
- Suggested ReadingsMathematical Techniques Karamat H.Dar
- Suggested ReadingsMathematical Analysis Apostal, T.M.
- Suggested ReadingsThe Theory of Groups Macdonald, I.N.
- Suggested ReadingsTopics in Algebra Herstein, I.N.