COMPLEX NUMBERS
MATHEMATICS SYLLABUS
1. Algebra
1.1 Logic and Set: introduction of Logic, statements, logical connectives, truth tables, basic laws of logic, theorems based on set operations.
1.2 Real numbers: field axioms, order axioms, interval, absolute value, geometric representation of real numbers.
1.3 Function: Review, domain & range of a function, Inverse function, composite function, functions of special type, algebraic (linear, quadratic & cubic), Trigonometric, exponential, logarithmic)
1.4 Curve sketching: odd and even functions, periodicity of a function, symmetry (about origin, x-and y-axis), monotonicity of a function, sketching graphs of polynomials and some rational functions, Trigonometric, exponential, logarithmic function (simple cases only)
1.5 Sequence and series: arithmetic, geometric, harmonic sequences and series and their properties A.M, G.M, H.M and their relations, sum of infinite geometric series. 1.6 Matrices and determinants: Transpose of a matrix and its properties, Minors and cofactors, Adjoint, Inverse matrix, Determinant of a square matrix, Properties of determinants (without proof)
1.7 Complex number: definition imaginary unit, algebra of complex numbers, geometric representation, absolute value (Modulus) and conjugate of a complex numbers and their properties, square root of a complex number, polar form of complex numbers.
2. Trigonometry
2.1 Properties of a triangle (Sine law, Cosine law, tangent law, Projection laws, Half angle laws).
2.2 Solution of triangle(simple cases)
3. Analytic Geometry
3.1 Straight Line: length of perpendicular from a given point to a given line. Bisectors of the angles between two straight lines. Pair of straight lines: General equation of second degree in x and y, condition for representing a pair of lines. Homogenous second-degree equation in x and y. angle between pair of lines. Bisectors of the angles between pair of lines.
3.2 Circle: Condition of tangency of a line at a point to the circle, Tangent and normal to a circle.
3.3 Conic section: Standard equation of parabola, equations of tangent and normal to a parabola at a given point.
4. Vectors
4.1 Vectors: collinear and non collinear vectors, coplanar and noncoplanar vectors, linear combination of vectors,
4.2 Product of vectors: scalar product of two vectors, angle between two vectors, geometric interpretation of scalar product, properties of scalar product, condition of perpendicularity.
5. Statistics and Probability
5.1 Measure of Dispersion: introduction, standard deviation), variance, coefficient of variation, Skewness (Karl Pearson and Bowley)
5.2 Probability: independent cases, mathematical and empirical definition of probability, two basic laws of probability (without proof).
6. Calculus
6.1 Limits and continuity: limits of a function, indeterminate forms. algebraic properties of limits (without proof), Basic theorems on limits of algebraic, trigonometric, exponential and logarithmic functions, continuity of a function, types of discontinuity, graphs of discontinuous function.
6.2 Derivatives: derivative of a function, derivatives of algebraic, trigonometric, exponential and logarithmic functions by definition (simple forms), rules of differentiation. derivatives of parametric and implicit functions, higher order derivatives, geometric interpretation of derivative, monotonicity of a function, interval of monotonicity, extreme of a function, concavity, points of inflection, derivative as rate of measure.
6.3 Anti-derivatives: anti-derivative. integration using basic integrals, integration by substitution and by parts methods, the definite integral, the definite integral as an area under the given curve, area between two curves.
7. Computational Methods
7.1 Computing Roots: Approximation & error in computation of roots in nonlinear equation, Algebraic and transcendental equations & their solution by bisection and Newton- Raphson Methods
8. Mechanics or Mathematics for economics and Finance
8.1 Statics: Forces and resultant forces, parallelogram law of forces, composition and resolution of forces, Resultant of coplanar forces acting on a point, Triangle law of forces and Lami's theorem.
8.2 Dynamics: Motion of particle in a straight line, Motion with uniform acceleration, motion under the gravity, motion down a smooth inclined plane. The concepts and theorem restated and formulated as application of calculus
8.3 Mathematics for economics and finance:
Mathematical Models and Functions, Demand and supply, Cost, Revenue, and profit functions, Elasticity of demand, supply and income , Budget and Cost Constraints, Equilibrium and break even
