CBCS PATTERN SYLLABUS
Mat/ Sem IV/ C-8 – Analysis II
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
Convergence of improper integrals, comparison tests, absolute convergence, Abel’s and Dirichlet’s tests, Frullani’s Integrals. Definition & convergence of Beta & Gamma functions and their properties, duplication formula, inter-relation. 3 Questions
GROUP - B
Evaluation of double and triple integrals. Multiple Integrals of Dirichlet’s form, Liouville’s extension, change of order of integration and change of variables. 3 Questions
GROUP - C
Vector integration: Line integral, surface integral, volume integral, Green’s theorem in R2, Stoke ‘s theorem, Gauss Divergence theorem. 3 Questions
Books Recommended:
1. Elements of Real Analysis – Shanti Narayan & M D Raisinghania.
2. Mathematical Analysis – J N Sharma & A R Vashishtha.
CBCS PATTERN SYLLABUS
Mat/ Sem IV/ C 9 – Mechanics I
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
Reduction of system of coplanar forces, equation of resultant, condition for equilibrium. Astatic centre. 1 Question
Laws, angles and cone of friction, equilibrium on a rough inclined plane, particle constrained to move on a rough curve under any given forces. 2 Questions
GROUP - B
Kinematics in two dimension: Tangential, normal, radial, transverse velocities and acceleration. Angular velocity and acceleration. Rectilinear motion and simple pendulum. S.H.M., compounding of two S.H.M. Repulsive motion. Motion under inverse square law. 3 Questions
Rectilinear Motion (Kinetics): Newton’s law, Work, K.E., work energy principle, Impulse, Torque and angular momentum, conservation of energy, momentum and angular momentum, Hooke’s law, extension of an elastic string: Horizontal & vertical case. 3 Questions
Books Recommended:
1. Degree level Mechanics – Singh & Sen
CBCS PATTERN SYLLABUS Mat / Sem IV / C 10 - Ring Theory
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. Each question will be of 10 marks.
Ring: Definition and examples, commutative ring, ring with unity, unit in a ring, Matrix ring, Boolean ring, Ring of continuous functions. Direct product of rings, Properties of rings, subrings. 2 Questions
Nilpotent element, idempotent element, zero divisors, integral domain, division ring and field. Characteristic of a ring. 1 Question
Ideal, ideal generated by a subset of a ring, simple ring, factor rings, operations on ideals, prime and maximal ideals. 2 Questions
Ring homomorphisms, properties of ring homomorphisms, Isomorphism theorems I, II and III, field of quotients. 2 Questions
Polynomial rings over commutative rings, division algorithm and consequences, principal ideal domains, factorization of polynomials, reducibility tests, irreducibility tests, Eisenstein’s criterion. 2 Questions
Books Recommended:
1. John B. Fraleigh, A First Course in Abstract Algebra, 7th Ed., Pearson, 2002.
2. Joseph A. Gallian, Contemporary Abstract Algebra, 4th Ed., Narosa Publishing House, New Delhi, 1999.
3. C Musili, Introduction to Rings and Modules, 2nd edition, Narosa Publishing House.
4. Modern Algebra – Surjee
CBCS PATTERN SYLLABUS
Mat/ Sem4/ GE 4 – MATHEMATICS – IV
Instruction for Generic Elective: -
Eleven Questions will be set. Candidates will be required to answer Eight Questions.
Question no. 1 will be Compulsory consisting of 10 short answer type covering entire syllabus uniformly. Each question will be of 3 marks. Out of remaining 10 questions will be required to answer 7 questions selecting at least one from each group. Each question will be of 10 marks.
GROUP - A
REAL ANALYSIS IV
Riemann Integration, definition, Oscillatory sum and integrability condition. Integrability of monotonic and continuous functions. Fundamental theorem of integral calculus. 2 Questions
SET THEORY II
Partial order relation and relate concepts of u.b., l.b., inf., sup, maximal element, minimal element and lattice (definition and examples only), statement of Zorn’s lemma. 1 Question
COMPLEX VARIABLE II
Functions of complex variables limit, Continuity, derivative, Cauchy-Riemann Equations, Analytic function, Harmonic function. Import of some standard transformations e.g., w=z+c, w=cz, w=1/z, w=(az+b) / (cz+d) bilinear). Conformal transformation as transformation effected by analytic function. Special conformal transformation w=z2, w=ez, w=sin z. 2 Questions
GROUP - B
ABSTRACT ALGEBRA II
Matrices, operations on matrices, matrix algebra, kinds of matrices, Transpose, adjoint and inverse of a matrix, solution of system of linear equations. 2 Questions
DIFFERENTIAL EQUATIONS II
Linear Equation with constant co-efficients, Homogenous linear equations with variable coefficients. Simultaneous equation s dx dy dz PQR = = and total differential equation P dx + Q dy + R dz = 0 together with their geometric significance. 3 Questions
SEMESTER 5
