Algebra and Trigonometry: Simple inequalities, Relations between roots and Co-efficient of a polynomial equation, Cardon’s method for solving a cubic equation. Summation of series, Hyperbolic functions.

Abstract and Linear Algebra: Groups, Subgroups, cyclic groups, normal subgroups. Elementary ideas of rings, integral domains and fields with examples. Rank of a matrix, Solution of a system of linear equations by rank method, consistency and inconsistency of a system of linear equations. Cayley Hamilton’s theorem.

Real Analysis: Convergences and Divergences of an infinite series. Tests for convergence. Comparison tests, radical tests, ratio and higher ratio tests (Raabe’s test, Gauss’s test, log test and higher log tests). Continuity and Differentiability.

Geometry: Polar equation of conics. Equations of cylinder, cone and central conicoids.

Calculus: Successive Differentiation, Leibnitz’s Theorem, Taylor’s and Maclaurin’s theorems. Pedal equations Curvature, asymptotes. Maxima and Minima for a function of two variables. Method of Undetermined multipliers. Partial Differentiation, Euler’s Theorem, change of variables, Taylor’s series for two variables, envelopes. Reduction Formulae, Definite Integrals. Applications in finding length, areas, volumes and  surfaces.

Complex Analysis: Continuity, differentiability and analyticity of a function of a complex variable, Cauchy Riemann equation, harmonic functions. Milne Thomson’s method’ to construct analytic function.

Differential Equations: Higher order differential equations with constant Co-efficients, complementary function and particular integrals. Cauchy homogeneous forms, Solution by change of independent variables. Variation of parameters.

Vectors and Mechanics: Scalar and vector point functions, gradient, divergence and curl, conservative field, solenoidal and irrotational vectors. directional derivatives. Common Catenary, Parabolic motion, Differential equation of a central orbit in polar and pedal forms.