CBCS PATTERN SYLLABUS Mat/Sem V/ C 11 – Analysis -III 
Instruction: -
Ten questions will be set. Candidates will be required to answer Seven Questions.
Question no. 1 will be Compulsory consisting of 10 short answer type covering entire syllabus uniformly. Each question will be of 2 marks. Out of remaining 9 questions candidates will be required to answer 6 questions selecting at least one from each group. Each question will be of 10 marks.
GROUP - A 4 Questions
Metric spaces:
Definition and examples of metric spaces. Sequences in metric space, Cauchy sequence, complete metric space. Open and closed balls, neighbourhood, open set, interior of a set. Limit point of a set, closed set, diameter of a set, Cantor’s theorem. Subspaces, dense sets, perfect sets. Baire’s Category theorem. Continuous mappings, sequential criterion and characterizations of continuity by open sets, Homeomorphism.
GROUP - B 5 Questions
Complex Analysis:
Complex numbers, Continuity and differentiability of functions of complex variable, Analytic functions, Cauchy- Riemann differential equations in Cartesian and polar forms. Conformal representation: Transformation, Jacobian, conformal transformation, some general transformations, bilinear transformation. critical points, fixed points, cross ratio, preservance of cross ratio, fixed points of bilinear transformation.
Books Recommended:
1. Introduction to Topology – G F Simmons.
2. Metric Spaces – P K Jain & Khalil Ahmad.
3. Complex variable – J N Sharma.
CBCS PATTERN SYLLABUS
Mat / Sem V / C 12 - Linear Algebra
Instruction: -
Ten questions will be set. Candidates will be required to answer Seven Questions.
Question no. 1 will be Compulsory consisting of 10 short answer type covering entire syllabus uniformly. Each question will be of 2 marks. Out of remaining 9 questions candidates will be required to answer 6 questions selecting at least one from each group. Each question will be of 10 marks.
GROUP - A
Vector spaces, subspaces, algebra of subspaces, linear combination of vectors, linear span, linear dependence and linear independence, basis and dimension, coordinate vector of a vector relative to a basis. Complement of a subspace, direct sum and quotient space. 3 Questions
GROUP - B
Linear transformations, null space, range, rank and nullity of a linear transformation, Sylvester’s law of nullity. Matrix representation of a linear transformation, algebra of linear transformations. Isomorphism, isomorphism theorems, invertibility and isomorphism, change of coordinate matrix. 3 Questions
GROUP - C
Linear functional, dual spaces, dual basis, double dual, transpose of a linear transformation and its matrix in the dual basis. Characteristic polynomial and characteristic values of a linear operator, diagonalizability, Cayley-Hamilton theorem and its applications. 3 Questions
Books Recommended:
1. Linear Algebra – K Hoffman & R Kunze.
2. Higher Algebra – S K Mapa.
3. Linear Algebra – A R Vashishtha.
CBCS PATTERN SYLLABUS
Mat / Sem V /DSE1- NUMBER THEORY
Instruction for Generic Elective: -
Eleven Questions will be set. Candidates will be required to answer Seven Questions.
Question no. 1 will be Compulsory consisting of 10 short answer type covering entire syllabus uniformly. Each question will be of 2 marks. Out of remaining 9 questions will be required to answer 6. Each question will be of 10 marks.
Divisibility and primes, H.C.F., Euclid’s Algorithm, unique factorization, perfect numbers. 2 Questions
Residue class, complete and reduced residue system, congruences and their properties, Fermat’s theorem, Wilson’s theorem. 2 Questions
Arithmetical functions, Euler’s and Mobius function, Mobius inversion formula. 2 Questions
The Diophantine equations: ax + by = c , x^2 +y^2 = z^2 1 Question
Algebraic Congruence, solution by inspection, Solution of ax ≡ b (mod c), system of linear congruences, Chinese remainder theorem. 1 Question
Farey sequence, continued fractions, Pell’s equation.
1 Question Books Recommended:
1. Number Theory – G H Hardy & E M Wright.
2. Number Theory – S G telang.
3. Number Theory – Harikisan
4. Number Theory – S. B. Malik
CBCS PATTERN SYLLABUS
Mat/ Sem V/ DSE 2 – SPECIAL FUNCTION
Instruction for Generic Elective: -
Eleven Questions will be set. Candidates will be required to answer Seven Questions.
Question no. 1 will be Compulsory consisting of 10 short answer type covering entire syllabus uniformly. Each question will be of 2 marks. Out of remaining 9 questions will be required to answer 6 questions selecting at least one from each group. Each question will be of 10 marks
. GROUP - A
Series Solution:
Ordinary point, singular point(regular), general methods and forms of series solution (Indicial equation –Frobenius method) [ N.B.: Results of analysis regarding validity of series solution are taken to be granted] 2 Question
Bessel’s equation:
Solution, recurrence formula for ( ) n J x , Generating function for ( ) n J x , equations reducible to Bessel’s equation, Orthogonality of Bessel’s function 1 Question
Legendre’s equation:
Solution, Rodrigue’s formula, Legendre’s polynomials, generating function for ( ) P x n , orthogonality of Legendre’s polynomials 1 Questions
Hypergeometric Functions:
Special cases, integral representation, summation theorem. 1 Question
GROUP - B
Laplace Transform:
Definition, Laplace Transform of elementary functions, properties, uniqueness and inverse Laplace Transform, Laplace Transform of derivatives and integrals, Multiplication by n t , division by t. Convolution theorem, Application of Laplace transform to differential equations. 4 Questions
Books Recommended:
1. Advance differential equations – M D Raisinghania.
2. Differential Equations – J N Sharma
3. Laplace Transform – Goyal & Gupta
SEMESTER 6
