Intra-University Math Olympiad

01. Math Olympiad Syllabus for Group-A:

Algebra:
Real and Complex Number Systems; Polynomials and Algebraic Equations; Matrices and Determinants, Graphical representation of Algebraic and Trigonometric Functions

Analysis: Functions and Limit; Differentiation and Integration (Finding Area).

Analytic Geometry: Coordinates and Straight Lines

Combinatory: Permutations and Combinations

Vector Analysis: Unit Vector, Dot Product, Cross Product, Projection, Gradient, Divergence & Curl,

02. Math Olympiad Syllabus for Group-B

Differential Calculus
1. Functions, Limit, Continuity and Differentiability, Physical meaning of derivative of a function, Indeterminate Forms.
2. Differentiation, Successive differentiation and Leibniz theorem
3. General Theorems and Expansions: Rolle’s Theorem, Mean Value Theorem, Taylor's Theorem and Maclaurian's Theorem.
4. Partial Differentiation, Euler’s formula, Maxima and minima
5. Indefinite integral: Physical meaning of integration of a function, method of Substitution, Integration by parts, special trigonometric functions and rational and partial fractions, different techniques of integration.

Integral Calculus
1. Definite integral: Fundamental theorem, general properties, and evaluations of definite integral and reduction formula, definite integral as the limit of a sum, Integration by method of successive reduction, Gamma and Beta Function.
2. Multiple Integral: Jacobian theorem, Double Integral, Change of order of integration, triple Integral, Physical Application of double and triple integral. Quadrature, Determination of length of curves, Finding Area of a region,
3. Integration by Revolution: Areas of surfaces of revolution, Volumes of solids of revolution. Solving Real world problems through calculus.

Geometry
1. Two Dimensional Geometry: Change of Axes, Pair of straight lines, General equation of second degree presents a pair of straight lines, Properties of Pair of straight lines, System of circles. conics
2. Three-dimensional Geometry: Rectangular co-ordinate System, Direction cosines, Direction ratios, Projections, Equation of planes, Different forms of planes.
3. Straight lines in three dimension, Angle between two lines, Angle between a lines and a plane, coplanar lines and Shortest distance, Spheres.

Differential equations
1. First order differential equation: Definition, solution of first order and first degree differential equation with initial conditions, Solution of Linear differential Equation, Separable Equations, homogeneous equations, Bernoulli Equation (imp), Exact Differential equations, Integrating Factors, Boundary Value Problems (imp)
2. Higher order Differential equations with constant coefficients: Solution of higher order homogeneous differential equations, Solution of non homogeneous differential equations, Auxiliary Equations, Complementary function and particular integral
3. Linear & Non-Linear Partial Differential Equations: Elimination of arbitrary constants and functions, Lagrange’s method, Charpit’s method. Solving linear partial differential equations with constant coefficients, Complementary function and particular integrals
4. Physical Applications (Modeling): Solution of Practical (Real world) problems using differential equations such as Growth and Decay Problems, Temperature Problems, Falling Body Problems, Dilution Problems, Electrical Circuits problems, Orthogonal Trajectories, Spring Problems, Buoyancy Problems, Classifying Solutions etc./modeling

Matrix
1. Matrix Terminology: Vector presentation by matrix, different types of matrices, algebraic operations on matrices, Transpose of a Matrix, Adjoint and inverse of a matrix, augmented matrix ,row operation method, rank of Matrices, Mathematical Problems using Matrix, distinguish between determinant and matrix, Normal Vector, Ortho-normal Vectors, Orthogonality, Gram-Schmidt Ortho-normalization Process, co-variance matrix,
2. Linear System of Equations & Vector Spaces: Echelon form, consistency and inconsistency, solution of homogeneous and non- homogeneous linear system of equations , Vector Spaces, subspaces, basis and dimension, linearly dependent and independent vectors
3. Characteristic equation: Eigen values, eigenvectors, Graphical presentation of Eigen vectors, Caley-Hamilon theorem, and similar matrices, diagonalization, and Characteristics roots. Adjacency Matrix with graphical representation, Geometrical Application of Matrices.
4. Matrix Decomposition: Singular Value Decomposition (SVD), LU Decomposition

Vector analysis
1. Vector analysis: Scalar and vectors, operation of vectors, vector addition and multiplication - their applications. Vector components in spherical and cylindrical systems, Scalar Field, Vector Field, Derivative of vectors and mathematical problems
2. Del operator: Del operator, gradient, divergence and curl and their physical significance.
3. Vector Integration: Line Integrals, physical significance of Vector integration and Problems, Plane Polar Coordinates, Cylindrical Polar Coordinates, Spherical Polar Coordinates
4. Vector’s Theorem :Greens, Gauss & Stocks theorem and their applications,

Complex function
1. Complex Valued Functions: Complex Number, Demoivre's Theorem, Complex mapping, Linear Transformation: translation, magnification and rotation, Non-linear transformations: inversion, bilinear. Set theory: Function, Relation etc.
2. Complex Differentiation: Differentiation of a complex function, Analytic function, Singularities, the Cauchy-Riemann Equations, harmonic functions
3. Complex Integration: Complex Path Integrals, closed contour, Cauchy’s Theorem, The Residue Theorem, Poles