Mathematics

ashmir Public Services Commission (JKPSC)
Lecturer (10+2) Mathematics
Subject / Paper: Mathematics
Exam Cycle: 2025
Exam Type: Pen-and-Paper (OMR Based)
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COURSE OVERVIEW
Exam Name: Lecturer (10+2) Mathematics
Conducting Authority: Jammu and Kashmir Public Services Commission (JKPSC)
Level: Competitive
Examination Type: MCQs
No. of Questions: 100 MCQs
Total Marks: 100 Marks
Duration: 2 Hours
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LIST OF UNITS (Verbatim)
Unit I: REAL ANALYSIS
Unit II: COMPLEX ANALYSIS
Unit III: Linear Algebra
Unit IV: Number theory
Unit V: Abstract Algebra
Unit VI: MODULES & LATTICES
Unit VII: DIFFERENTIAL EQUATIONS
Unit VIII: TOPOLOGY
Unit IX: FUNCTIONAL ANALYSIS
Unit X: DIFFERENTIAL GEOMETRY
Unit XI: Probability and Statistics
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UNIT-WISE DETAILED SYLLABUS (Verbatim)

Unit I: REAL ANALYSIS

Topic 1: Elementary set theory, countable and uncountable sets, Real number system as a complete ordered field, Archimedean property, supremum and infimum
Subtopics:
○ Elementary set theory
○ countable and uncountable sets
○ Real number system as a complete ordered field
○ Archimedean property
○ supremum and infimum
Topic 2: Resume of sequences, series and Riemann integration, continuity, uniform continuity, differentiability
Subtopics:
○ Resume of sequences, series and Riemann integration
○ continuity
○ uniform continuity
○ differentiability
Topic 3: Fundamental theorem of integral calculus
Topic 4: classes of R-integrable functions
Topic 5: Functions of bounded variation
Topic 6: Cauchy’s general principle of uniform convergence
Topic 7: uniform convergence and integration
Topic 8: uniform convergence and differentiation
Topic 9: weierstras theorem
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Unit II: COMPLEX ANALYSIS

Topic 1: Complex numbers and functions. CR equations
Subtopics:
○ Complex numbers and functions
○ CR equations
Topic 2: analytic functions, entire functions, the order of an entire function
Subtopics:
○ analytic functions
○ entire functions
○ the order of an entire function
Topic 3: Necessary and sufficient condition for analyticity
Topic 4: harmonic functions, harmonic conjugate
Subtopics:
○ harmonic functions
○ harmonic conjugate
Topic 5: contour integration
Topic 6: Cauchy integral formula/ theorem
Topic 7: Liouvilles theorem
Topic 8: Taylor’s and Laurent’s theorems
Topic 9: classification of singularities, Riemann’s theorem, weierstras theorem on essential singularity
Subtopics:
○ classification of singularities
○ Riemann’s theorem
○ weierstras theorem on essential singularity
Topic 10: Calculus of residues
Topic 11: Cauchy’s residue theorem
Topic 12: Integration by the method of residues
Topic 13: maximum/ minimum modulus theorem
Topic 14: Schwarz lemma
Topic 15: Power series, Hadamard formula for the radius of convergence, a power series represents an analytic function within the circle of convergence
Subtopics:
○ Power series
○ Hadamard formula for the radius of convergence
○ a power series represents an analytic function within the circle of convergence
Topic 16: Rouch’s theorem, the fundamental theorem of algebra
Subtopics:
○ Rouch’s theorem
○ the fundamental theorem of algebra
Topic 17: Morea’s theorem, Hurwitz theorem
Subtopics:
○ Morea’s theorem
○ Hurwitz theorem
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Unit III: Linear Algebra

Topic 1: Vector spaces over real and complex fields
Topic 2: linear dependence and independence
Topic 3: subspaces, bases, dimensions
Subtopics:
○ subspaces
○ bases
○ dimensions
Topic 4: Linear transformations, rank and nullity
Subtopics:
○ Linear transformations
○ rank and nullity
Topic 5: matrix representation of linear transformations
Topic 6: linear functional
Topic 7: Algebra of matrices: row and column reduction, echelon form, congruence and similarity, rank, inverse, solutions of linear systems
Subtopics:
○ row and column reduction
○ echelon form
○ congruence and similarity
○ rank
○ inverse
○ solutions of linear systems
Topic 8: Eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem
Subtopics:
○ Eigenvalues and eigenvectors
○ characteristic polynomial
○ Cayley-Hamilton theorem
Topic 9: Special matrices: symmetric, skew-symmetric, Hermitian, skew-Hermitian, orthogonal, unitary, and their eigenvalues
Subtopics:
○ symmetric
○ skew-symmetric
○ Hermitian
○ skew-Hermitian
○ orthogonal
○ unitary
○ their eigenvalues
Topic 10: quadratic forms
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Unit IV: Number theory

Topic 1: Divisibility and Prime Numbers: Division algorithm, Euclidean algorithm, GCD, LCM, prime numbers, fundamental theorem of arithmetic, congruence’s, Chinese remainder theorem.
Subtopics:
○ Division algorithm
○ Euclidean algorithm
○ GCD
○ LCM
○ prime numbers
○ fundamental theorem of arithmetic
○ congruence’s
○ Chinese remainder theorem
Topic 2: Modular Arithmetic: properties of congruence’s, linear congruence’s, Fermat’s little theorem, Euler’s theorem, Wilson’s theorem.
Subtopics:
○ properties of congruence’s
○ linear congruence’s
○ Fermat’s little theorem
○ Euler’s theorem
○ Wilson’s theorem
Topic 3: Euler’s totient function, Mobius function, Mobius inversion formula, sigma function, number of divisors function
Subtopics:
○ Euler’s totient function
○ Mobius function
○ Mobius inversion formula
○ sigma function
○ number of divisors function
Topic 4: Diophantine Equations
Topic 5: Quadratic Residues and Reciprocity (Legendre symbol, quadratic reciprocity law, Jacobi symbol)
Subtopics:
○ Legendre symbol
○ quadratic reciprocity law
○ Jacobi symbol
Topic 6: Continued Fractions
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Unit V: Abstract Algebra

Topic 1: GROUPS: Groups, subgroups, abelian groups, normal subgroups, quotient groups, homomorphism of groups, cyclic groups
Subtopics:
○ Groups
○ subgroups
○ abelian groups
○ normal subgroups
○ quotient groups
○ homomorphism of groups
○ cyclic groups
Topic 2: Structure theorem for cyclic groups
Topic 3: permutation groups, Sn , An.
Subtopics:
○ permutation groups
○ Sn
○ An
Topic 4: class equations, automorphisms, inner automorphisms
Subtopics:
○ class equations
○ automorphisms
○ inner automorphisms
Topic 5: Cauchy’s and sylow’s theorem for Abelian groups
Topic 6: Cayley’s theorem
Topic 7: Sylow’s theorem and Cauchy’s theorem
Topic 8: Finite Abelian groups, Fundamental theorem on finite Abelian groups
Subtopics:
○ Finite Abelian groups
○ Fundamental theorem on finite Abelian groups
Topic 9: Composition series. The Jordan-Holder theorem for finite groups
Subtopics:
○ Composition series
○ The Jordan-Holder theorem for finite groups
Topic 10: RINGS: Definition and examples of rings
Topic 11: Integral domains
Topic 12: ideals, Principal ideal, Prime ideals and maximal ideals
Subtopics:
○ ideals
○ Principal ideal
○ Prime ideals and maximal ideals
Topic 13: Nil ideal, Radical ideal, Annihilator ideal
Subtopics:
○ Nil ideal
○ Radical ideal
○ Annihilator ideal
Topic 14: quotient rings
Topic 15: UFD, PID, ED
Subtopics:
○ UFD
○ PID
○ ED
Topic 16: Fields of quotients of an integral domains
Topic 17: Polynomial rings
Topic 18: Irreducible polynomials, Einstein’s criterion
Subtopics:
○ Irreducible polynomials
○ Einstein’s criterion
Topic 19: FIELDS: Fields, finite fields and sub fields
Subtopics:
○ Fields
○ finite fields and sub fields
Topic 20: Prime fields and their structure
Topic 21: Extensions of fields
Topic 22: Algebraic numbers and algebraic extensions of a field
Topic 23: Galois Theory
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Unit VI: MODULES & LATTICES

Topic 1: Definitions, fundamental concepts
Topic 2: chain conditions
Topic 3: Noetherian ring, Artinian ring
Subtopics:
○ Noetherian ring
○ Artinian ring
Topic 4: Hilbert’s basis theorem
Topic 5: Krulls intersection theorem
Topic 6: Prime and primary ideals
Topic 7: Sequences theorems
Topic 8: Partially ordered sets
Topic 9: Lattices
Topic 10: modular lattices
Topic 11: complemented modular lattice
Topic 12: operations on lattices: sublattices, ideals and homomorphisms
Subtopics:
○ sublattices
○ ideals and homomorphisms
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Unit VII: DIFFERENTIAL EQUATIONS

Topic 1: First order ODE’s
Topic 2: integrating factors
Topic 3: Wronskian of solutions
Topic 4: initial value problem
Topic 5: singular solutions
Topic 6: P-discriminant, C-discriminant
Subtopics:
○ P-discriminant
○ C-discriminant
Topic 7: Equations of the second degree with constant coefficients
Topic 8: Total differential equation. Pdx+Qdy+Rdz = 0
Topic 9: Necessary and sufficient conditions that such a differential equation may be integrable
Topic 10: Partial differential equation of the first order
Topic 11: Lagrange’s and charpits method for solving first order PDE’s
Topic 12: classification of second order PDE’s
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Unit VIII: TOPOLOGY

Topic 1: Metric spaces: Definition and examples, open sets, completeness, convergence, continuous mapping, completion of a metric space, Cantor’s intersection theorem
Subtopics:
○ Definition and examples
○ open sets
○ completeness
○ convergence
○ continuous mapping
○ completion of a metric space
○ Cantor’s intersection theorem
Topic 2: Contraction mapping, Banach’s contraction Principle
Subtopics:
○ Contraction mapping
○ Banach’s contraction Principle
Topic 3: Topological spaces: Definition and examples, Elementary properties, Kuratowski’s axioms, continuous mappings and their characterisation
Subtopics:
○ Definition and examples
○ Elementary properties
○ Kuratowski’s axioms
○ continuous mappings and their characterisation
Topic 4: Bases and subbases
Topic 5: concept of first countability, second countability, separability
Subtopics:
○ first countability
○ second countability
○ separability
Topic 6: Tychnoffs theorem
Topic 7: Lebasgue’s covering lemma
Topic 8: continuous maps on compact spaces
Topic 9: Separation axioms
Topic 10: compactness and connectedness in topological spaces
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Unit IX: FUNCTIONAL ANALYSIS

Topic 1: BANACH SPACES, Definition and examples
Topic 2: Quotient spaces
Topic 3: Dual of a normed linear space. Dual spaces
Topic 4: Hahn Banach Theorem
Topic 5: closed graph theorem
Topic 6: open mapping theorem
Topic 7: HILBERT SPACES, Definition and examples
Topic 8: Cauchy-Schwarz inequality
Topic 9: Bessel’s inequality
Topic 10: orthonormal systems
Topic 11: Riesz representation theorem
Topic 12: inner product spaces
Topic 13: Adjoint of a Hilbert space
Topic 14: Normal operators
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Unit X: DIFFERENTIAL GEOMETRY

Topic 1: Curves with torsion
Topic 2: arc length
Topic 3: curvature, S-Frenet formulae
Subtopics:
○ curvature
○ S-Frenet formulae
Topic 4: spherical curvature
Topic 5: spherical indicatrices
Topic 6: involutes and evolutes
Topic 7: helix, Bertrand curves
Subtopics:
○ helix
○ Bertrand curves
Topic 8: Helix
Topic 9: Envelopes of one and two parameter family of surfaces
Topic 10: Developable surfaces
Topic 11: Developable associated with a curve
Topic 12: Curvilinear coordinates
Topic 13: Fundamental magnitudes of first and second order, the two fundamental forms; curvature of normal section, Meunier’s theorem
Subtopics:
○ Fundamental magnitudes of first and second order
○ the two fundamental forms
○ curvature of normal section
○ Meunier’s theorem
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Unit XI: Probability and Statistics

Topic 1: Conditional probability
Topic 2: multiplication theorem on probability
Topic 3: independent events
Topic 4: Bayes’ Theorem; Theorem of total probability
Subtopics:
○ Bayes’ Theorem
○ Theorem of total probability
Topic 5: Random Variable and its probability distribution
Topic 6: Mean, variance and standard deviation of a random variable
Topic 7: Bernoulli trials and binomial distribution
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EXAM STRUCTURE / SCHEME OF EXAMINATION
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• Paper / Section details: [Verbatim]
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SYLLABUS SUMMARY (Auto-Generated)
• Total Units: 11
• Total Topics: 140
• Exam: Lecturer (10+2) Mathematics
• Subject: Mathematics
• Exam Year: 2025
✓ Extracted verbatim from the official uploaded document