## Junior Lecturers /Degree Lecturers /Polytechnic Lecturers Syllabus and Scheme/Pattern of Written Examination

PARTA Written Exam | Subject | Marks | Questions | Duration (Minutes) |
---|---|---|---|---|

PAPER 1 | General Studies & Mental Ability | 150 | 150 | 150 |

PAPER 2 | Concerned Subject | 300 | 150 | 150 |

PARTB: ORAL TEST(Interview) | 50 | |||

TOTAL | 500 |

## Mathematics Syllabus

### Real Analysis :

- Finite, countable and uncountable sets – Real Number system R – infimum and supremum of a subset of R – Bolzano – Weierstrass theorem. Sequences, convergence, limit superior and limit inferior of sequences, sub sequences, Heine Borel Theorem. Infinite series – Tests of convergence.
- Continuity and uniform continuity of real valued functions of real variable. Monotonic functions and functions of bounded variation. Differentiability and mean value theorems.
- Riemann integrability. Sequences and Series of functions.

### Metric Spaces :

- Metric spaces – completeness, compactness and connectedness – continuity and uniform continuity of functions from one metric space into another. Topological spaces – base and subbase – continuous function.

### Elementary Number Theory :

- Primes and composite numbers – Fundamental Theorem of arithmetic – divisibility – congruences – Fermat’s theorem – Wilson’s Theorem – Euler’s – function.

### Group Theory :

- Groups, subgroups, normal subgroups – quotient groups – homomorphisms and isomorphism theorems – permutation groups, cyclic groups, Cayley’s theorem. Sylow’s theorems and their applications.

### Ring Theory :

- Rings, integral domains, fields – subrings and ideals – Quotient rings – homomorphisms– Prime ideals and maximal ideals – polynomial rings – Irreducibility of polynomials–Euclidean domains and principal ideal domains.

### Vector Spaces :

- Vector Spaces, Subspaces – Linear dependence and independence of vectors – basis and dimension – Quotient spaces – Inner product spaces – Orthonormal basis – Gram –Schmidt process.

### Matrix Theory :

- Linear transformations – Rank and nullity – change of bases. Matrix of a linear transformation – singular and non-singular matrices – Inverse of matrix – Eigenvalues and eigenvectors of matrix and of linear transformation – Cayley – Hamilton’s theorem.

### Complex Analysis :

- Algebra of complex numbers – the complex plane – Complex functions and their Analyticity – Cauchy-Riemann equations – Mobius transformations. Power Series.
- Complex Integration – Cauchy’s theorem – Morera’s Theorem – Cauchy’s integral formula – Liouville’s theorem – Maximum modules principle – Schwarz’s lemma–Taylor’s series – Laurents series.
- Calculus of residues and evaluation of integrals.

### Ordinary Differential Equation :

- Ordinary Differential Equation (ODE) of first order and first degree – Different methods of solving them – Exact Differential equations and integrating factors.
- ODE of first order and higher degree – equations solvable for p, x and y – Clairaut’s equations – Singular Solutions.
- Linear differential equations with constant coefficients and variable coefficients –variation of parameters.

### Partial Differential Equations :

- Formation of differential equations (PDE) – Lagrange and Charpit methods for solving first order – PDE’s – Cauchy problem for first order PDE’s Classification of second order PDE’s– General solution of higher order PDE’s with constant coefficients.