Algebra

Syllabus points

• Sets: empty/finite/infinite; algebra of sets; De Morgan; practical Venn problems
• Cartesian product, ordered pairs, relations, domain/codomain, equivalence relation
• Functions as mappings; invertible, even/odd, into/onto/one-one
• Special functions: polynomial, trig, exp, log, power, absolute value, greatest integer
• Sum, difference, product and composition of functions

Do / check

• Prove equivalence vs just check reflexive once
• Sketch floor/absolute piecewise before solving equations
• Domain of compositions written before simplifying

Syllabus points

• Complex numbers: +, ×, conjugate, polar form, |z|, Arg, triangle inequality, cube roots of unity, geometry
• Fundamental theorem of algebra; quadratic equations; roots–coefficients; symmetric functions of roots
• AP/GP, AM/GM, finite AP/GP sums, infinite GP, ∑n, ∑n², ∑n³
• Logarithms and properties
• Permutations and combinations
• Binomial theorem (positive integral index) and binomial coefficient properties

Do / check

• Always fix principal Arg branch
• Write root-sum/product before expanding
• For P&C, state the counting model in one line

Syllabus points

• Matrices as arrays; equality; +, scalar ×, matrix product; transpose
• Elementary row/column transformations
• Determinant, adjoint, inverse (order ≤ 3) and properties
• Diagonal, symmetric, skew-symmetric matrices
• Simultaneous linear equations in 2 or 3 variables

Do / check

• Reduce Mⁿ using characteristic equation before expanding
• Check AB≠BA traps in MC options

Syllabus points

• Sample space; impossible/simple/compound events
• Addition and multiplication rules; conditional probability; independence
• Total probability; Bayes’ theorem; P&C-based probability
• Mean, median, mode; mean deviation; SD and variance (grouped/ungrouped)
• Same mean, different variance comparisons; random variable mean & variance

Do / check

• Draw tree / partition before Bayes
• Distinguish independence from mutually exclusive