UPSEE 2-Month Plan

Complex numbers as a+ib, algebra of complex numbers, modulus and argument, De Moivre's theorem, cube roots of unity, quadratic equations with real and complex roots, discriminant, and nature of roots.

Types of matrices, matrix operations (addition, multiplication, transpose), adjoint and inverse of matrices, determinant evaluation (up to 3×3), properties of determinants, and solving linear equations using matrices.

Fundamental principle of counting, permutation (linear and circular), combination, Pascal's triangle, binomial theorem (general and middle term), binomial expansion for positive integer indices, and arrangement problems.

Arithmetic progression (AP), geometric progression (GP), arithmetic-geometric progression (AGP), harmonic progression (HP), sum of n terms, infinite series convergence, and AM-GM inequality applications.

Positive integral index binomial expansion, general and middle terms, Pascal's triangle, binomial coefficient properties, and applications in finding coefficients and approximations.

Trigonometric ratios, identities (basic and conditional), signs in quadrants, allied angles, sum-to-product and product-to-sum formulas, multiple and submultiple angles, and solving trigonometric equations.

Cartesian coordinate system, distance formula, section formula, area of triangle, slope-intercept form, general equation of line, angle between lines, perpendicular and parallel conditions, and solving linear equations graphically.

Circle (equation, tangents, normals), parabola (standard forms, focal properties), ellipse (eccentricity, latus rectum), hyperbola (asymptotes, rectangular hyperbola), and standard equations with transformations.

Units and dimensions, SI base units, errors in measurement, significant figures, vernier calipers, screw gauge, and dimensional analysis — foundational concepts for all physics problem-solving.

Motion in one and two dimensions — displacement, velocity, acceleration, equations of motion, projectile motion, relative velocity, and circular motion with numerical applications.

Newton's three laws, friction (static and kinetic), circular motion dynamics, tension, spring force, and momentum conservation — core mechanics for engineering entrance exams.

Work done by constant and variable forces, kinetic and potential energy, work-energy theorem, conservation of mechanical energy, power, and collisions (elastic and inelastic).

Torque, moment of inertia, angular momentum, radius of gyration, theorems of moment of inertia, rotational kinematics, and conservation of angular momentum.

Newton's law of gravitation, acceleration due to gravity, variation of g with altitude and depth, Kepler's laws, orbital velocity, escape velocity, and gravitational potential energy.

Elasticity (stress-strain, Hooke's law, Young's modulus), fluid mechanics (Pascal's law, Bernoulli's principle, viscosity, surface tension, capillarity), and Archimedes' principle.

Heat, temperature, calorimetry, heat transfer (conduction, convection, radiation), first and second law of thermodynamics, specific heat capacity, and thermodynamic processes.

Dual nature of matter, de Broglie relation, Heisenberg uncertainty principle, quantum numbers, orbital shapes (s, p, d, f), Aufbau principle, Hund's rule, Pauli's exclusion principle, and electronic configurations.

Ionic and covalent bonding, VSEPR theory, hybridization (sp, sp², sp³, sp³d, sp³d²), valence bond theory, MOT (bond order, HOMO-LUMO), dipole moment, and resonance structures.

Modern periodic table, periodic trends — atomic radius, ionization enthalpy, electron gain enthalpy, electronegativity, valence electrons, metallic/non-metallic character across periods and groups.

Gas laws (Boyle's, Charles's, Avogadro's), ideal gas equation, kinetic theory of gases, van der Waals equation, liquefaction of gases, critical temperature, and solid-state (lattice, crystal systems, Bragg's law).

System and surroundings, first law of thermodynamics (internal energy, work, heat), enthalpy, Hess's law, spontaneity, Gibbs free energy, second law of thermodynamics, and entropy change calculations.

Reversible reactions, equilibrium constant (Kp, Kc), Le Chatelier's principle, factors affecting equilibrium, relationship between Kp and Kc, ionic equilibrium in solutions, pH, buffers, and solubility product.

Rate of reaction, rate law and order, molecularity, zero, first, and second-order reactions, Arrhenius equation, activation energy, catalyst role, and half-life calculations.

Oxidation-reduction concepts, balancing redox equations, electrochemical cells (galvanic, electrolytic), Nernst equation, standard electrode potentials, Faraday's laws of electrolysis, and corrosion prevention.