CBCGS [2017 - current]
CBGS [2013 - 2016]
Old [2000 - 2012]
Topics with syllabus and resources
100.00 Module 1 The Laplace Transform
- The Laplace transform:- Definition and properties (without proofs); all standard transform methods for elementary functions including hyperbolic functions; Heaviside unit step function, Dirac delta function; the error function; evaluation of integrals using Laplace transforms; inverse Laplace transforms using partial fractions and H(t-a); convolution (no proof).
200.00 Module 2 Matrices
- Eigenvalues and eigenspaces of 2x2 and 3x3 matrices;
- existence of a basis and finding the dimension of the eigenspace (no proofs);
- nondiagonalisable matrices; minimal polynomial;
- Cayley - Hamilton theorem (no proof); quadratic forms;
- orthogonal and congruent reduction of a quadratic form in 2 or 3 variables; rank, index, signature; definite and indefinite forms.
300.00 Module 3 Complex Analysis
- Complex analysis:-
- Cauchy-Riemann equations (only in Cartesian coordinates) for an analytic function (no proof);
- harmonic function;
- Laplace’s equation;
- harmonic conjugates and orthogonal trajectories (Cartesian coordinates); to find f(z) when u+v or u - v are given;
- Milne-Thomson method; cross-ratio (no proofs);
- conformal mappings; images of straight lines and circles.
400.00 Module 4 Cauchy’S Formula and Theorem
- Complex Integration Cauchy’s integral formula; poles and residues;
- Cauchy’s residue theorem;
- applications to evaluate real integrals of trigonometric functions;
- integrals in the upper half plane; the argument principle.
500.00 Module 5 Statistics
- (No theory questions expected in this module) Mean, median, variance, standard deviation;
- binomial, Poisson and normal distributions;
- correlation and regression between 2 variables.
600.00 Module 6 Optimization
- Non-linear programming:-
- Lagrange multiplier method for 2 or 3 variables with at most 2 constraints;
- conditions on the Hessian matrix (no proof);
- Kuhn-Tucker conditions with at most 2 constraints.