Mathematics — official units

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

Syllabus points

• Trig functions: periodicity and graphs
• Addition/subtraction; multiple and sub-multiple angle formulae
• General solution of trigonometric equations
• Inverse trigonometric functions (principal value) and elementary properties

Do / check

• Write principal range of inverse trig before solving
• Convert to sin/cos single form for equations

Syllabus points

• Cartesian coordinates, distance, section formulae, shift of origin
• Straight line forms; angle between lines; distance point–line; angle bisectors; concurrency
• Triangle centres: centroid, orthocentre, incentre, circumcentre
• Circle: forms, tangent/normal/chord, parametric; intersections
• Parabola, ellipse, hyperbola: standard form, foci, directrices, eccentricity, parametric, tangent/normal
• Locus problems

Do / check

• Standardise conic form before using a²/b² formulae
• For locus, eliminate the parameter cleanly

Syllabus points

• Distance between two points; direction cosines and ratios
• Equation of a straight line in space; skew lines; shortest distance
• Equation of a plane; distance of a point from a plane
• Angles: line–line, plane–plane, line–plane; coplanar lines

Do / check

• Sketch normals before computing angles
• Verify a known point lies on the plane

Syllabus points

• Limits and continuity (sum/difference/product/quotient); L’Hôpital
• Continuity of composites; intermediate value property
• Derivatives: sum/product/quotient/chain; standard function families
• Tangents and normals; increasing/decreasing; second derivatives; max/min
• Rolle’s and Lagrange’s MVT; geometric interpretation; implicit derivatives up to order two

Do / check

• State domain/continuity before differentiating piecewise
• Check endpoints separately for extrema on closed intervals

Syllabus points

• Indefinite integrals of standard functions
• Definite integrals as limit of sums; properties; FTC
• Integration by parts, substitution, partial fractions
• Area bounded by simple curves
• Formation of ODEs; homogeneous first-order; separation of variables; linear first-order DE

Do / check

• Use definite-integral properties before expanding
• Identify DE type before applying a method

Syllabus points

• Addition of vectors; scalar multiplication
• Dot and cross products
• Scalar and vector triple products and geometrical interpretations

Do / check

• Keep geometric meaning (area/volume/coplanarity) beside algebra