1. Mechanics & Dynamics
Probability of events, motion under gravity, uniform velocity and acceleration
Resultant of forces, parallel forces, friction (coefficient & angle)
Coplanar force systems, couples, and moments
Vector cross products, position and velocity vectors
2. Abstract Algebra (Group Theory & Ring Theory)
Binary operations, identity and inverse elements
Permutation groups — order, inverse of permutations
Subgroups, Lagrange’s theorem, cyclic subgroups
Group homomorphisms, kernel, one-one and onto maps
Rings: integral domains, fields (Z₅, Z₆, Z₃₆)
Ideals: maximal ideals, prime ideals, quotient rings
External direct product of rings (Z ⊕ Z)
Irreducibility of polynomials over Q and R
3. Linear Algebra
Vector spaces, subspaces, dimension, basis
Linear transformations: rank, nullity, kernel
Hom(V, V) dimension
Matrices: adjoint, determinant, rank, eigenvalues
System of linear equations (unique solution conditions)
4. Real Analysis
Sets, relations, functions (one-one, onto, surjection)
Sequences and series: convergence, limit points
p-series convergence, Cesàro mean
Closure of sets, bounded sets
Continuity and discontinuity of functions
Partial derivatives, Euler’s theorem (homogeneous functions)
Jacobians, Riemann integration (upper/lower sums)
5. Complex Analysis
Complex numbers: Argand plane, equilateral triangle condition
Analytic functions, Cauchy’s integral formula
Contour integration
Complex logarithms and trigonometric functions
6. Calculus & Differential Equations
Maxima/minima (AM-GM inequality applications)
Arc length (cardioid), definite integrals
Ordinary differential equations: order, degree, exact equations
Integrating factors, singular solutions, general solutions
Partial differential equations (basic)
7. Algebra & Number Theory
Roots of polynomial equations (Vieta’s formulas)
Factorials and trailing zeros
Divisibility properties
Power series and their inverses (logarithm series)
8. Trigonometry & Complex Numbers
De Moivre’s theorem applications
Gregory–Leibniz series (π/4)
Product of cosines
9. Coordinate Geometry
Conics: ellipse, parabola, hyperbola, circle (conditions)
Confocal conics, polar form of conics
Pair of straight lines (perpendicularity condition)
3D geometry: sphere, planes, direction cosines
10. Vector Calculus
Divergence of vector functions
Differentiation of vector functions
Parallel and perpendicular vector conditions
11. Operations Research & Numerical Methods
Linear programming: feasible solutions, optimal solutions, convex sets
Graph theory: trees, complete graphs (edges/lines count)
Newton-Raphson method (order of convergence)
Simpson’s one-third rule
12. Statistics & Probability
Variance of sequences
Weighted mean (binomial coefficients)
Probability (complementary events, independent events)
Course Purchase Query 