Grade 12 Mathematics

Number patterns, sequences and series

Weeks: Weeks 1-4

Revise number patterns with constant second differences (leading to quadratic general terms). Study arithmetic and geometric sequences and series, sigma notation, and sum formulae: Sn = n/2[2a + (n - 1)d], Sn = n/2(a + l), Sn = a(r^n - 1)/(r - 1) for r != 1, and Sn = a/(1 - r) for -1 < r < 1 (r != 1).

Functions

Weeks: Weeks 5-7

Formal definition of a function and the inverse of a function, including restrictions to obtain a one-to-one function. Determine and sketch inverses for y = ax + q, y = ax^2, y = a/x, and y = b^x (b > 0, b != 1). Focus on domain and range, intercepts, turning points, minima/maxima, asymptotes, symmetry, average gradient, and intervals of increase/decrease. Revise exponential laws and the definition of a logarithm: y = log_b x <=> x = b^y, including graphs for 0 < b < 1 and b > 1.

Trigonometry

Weeks: Weeks 8-11

Compound angle identities: sin(α ± β) = sin α cos β ± sin β cos α, cos(α ± β) = cos α cos β ∓ sin α sin β, sin 2α = 2 sin α cos α, cos 2α = cos^2 α - sin^2 α = 2 cos^2 α - 1 = 1 - 2 sin^2 α. Revise proofs of sine, cosine and area rules, and solve 2D/3D problems using these rules.

Euclidean geometry

Weeks: Weeks 1-2

Revise conditions for polygons to be similar and prove results based on earlier grades: midpoint theorem, equiangular triangles, proportional sides, and Pythagoras by similar triangles.

Analytical geometry

Weeks: Weeks 3-4

Revise line equations through two points, through a point with given gradient, and perpendicular/parallel lines. Use inclination (m = tan theta, 0 <= theta <= 180 degrees), circle equation (x - a)^2 + (y - b)^2 = r^2, and tangents to circles.

Differential calculus, including polynomials

Weeks: Weeks 5-8

Factorise third-degree polynomials; apply remainder and factor theorems. Use limits to define the derivative: f'(x) = lim(h->0) (f(x + h) - f(x)) / h. Apply first principles to f(x) = ax^2 + bx + c, f(x) = ax^3, f(x) = a/x, and f(x) = c. Use power and combination rules: d/dx[f(x) ± g(x)] = d/dx[f(x)] ± d/dx[g(x)] and d/dx[kf(x)] = k d/dx[f(x)]. Determine equations of tangents, introduce the second derivative for concavity, sketch cubic graphs (stationary and inflection points), find x-intercepts, and solve optimisation and rate-of-change problems (including motion).

Finance, growth and decay (continuation)

Weeks: Weeks 1-3

Revise simple and compound growth/decay: A = P(1 + i n) and A = P(1 + i)^n (also A = P(1 - i)^n [unclear]). Solve problems including straight-line depreciation and depreciation on a reducing balance, present and future value of annuities, and use logarithms to calculate n or time period. Critically analyse investment and loan options, including pyramid.

Statistics

Weeks: Weeks 4-5

Revise histograms, frequency polygons, ogives, variance and standard deviation of ungrouped data, symmetry/skewness, and outliers. Use statistical summaries, scatterplots, regression (least squares), and correlation to analyse bivariate data with interpolation/extrapolation and skewness discussion.

Counting and probability

Weeks: Weeks 6-7

Revise probability identities, addition rule for mutually exclusive events, complement rule, and dependent/independent events with product rule. Use Venn diagrams and formulas for three events, apply tree diagrams for consecutive/simultaneous events, and solve probability problems using counting principles and two-way tables.

Revision

Weeks: Week 8

Consolidate term content in preparation for the trial examination.

Trial examination

Weeks: Weeks 9-11

Trial examination covering all core Grade 12 Mathematics CAPS topics.

Revision and final examination

Revision culminates in the NSC final examination.

Final examination: Paper 1 focus and marks

Paper 1 (150 marks, 3 hours): Algebraic expressions, equations and inequalities (25), Number patterns (25), Functions and graphs (35), Finance, growth and decay (15), Differential calculus (35), Counting principle and probability (15).

Final examination: Paper 2 focus and marks

Paper 2 (150 marks, 3 hours): Statistics (20), Analytical geometry (40), Euclidean geometry (40), Trigonometry (50).