GUJCET 12-Hour Crash

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.

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.

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).