Mathematics Grade 12 Examination Guidelines 2021

Introduction and Purpose

• The Curriculum and Assessment Policy Statement (CAPS) for Mathematics outlines the nature and purpose of the subject, guiding teaching and assessment philosophy for Grade 12.
• The Examination Guidelines provide clarity on the depth and scope of content assessed in the Grade 12 National Senior Certificate (NSC) Examination.
• These guidelines assist teachers in adequately preparing learners for external examinations.
• This document focuses on the final Grade 12 external examinations and does not cover School-Based Assessment (SBA), Performance Assessment Tasks (PATs), or final external practical examinations in depth.
• It should be read alongside the National Curriculum Statement (NCS), the National Protocol of Assessment (NQF Level 4), and national policy regarding promotion requirements for Grades R–12.

Assessment Format and Weighting in Grade 12

• All candidates write two external papers.
• Questions assess performance across distinct cognitive levels with an emphasis on process skills, critical thinking, and strategies for solving problems in various contexts.

Paper 1 Overview

• Topics: Algebra, Equations and Inequalities; Number Patterns; Functions and Graphs; Finance, Growth and Decay; Differential Calculus; Counting Principle and Probability.
• Duration: 3 hours.
• Total Marks: 150 marks.
• Marking: Externally.
• Date: October/November.

Paper 2 Overview

• Topics: Statistics and Regression; Analytical Geometry; Trigonometry; Euclidean Geometry.
• Duration: 3 hours.
• Total Marks: 150 marks.
• Marking: Externally.
• Date: October/November.

Topic Weighting per Paper

• Paper 1:     * Algebra, Equations, and Inequalities: $25 \pm 3$ marks.     * Number Patterns: $25 \pm 3$ marks.     * Functions and Graphs: $35 \pm 3$ marks.     * Finance, Growth and Decay: $15 \pm 3$ marks.     * Differential Calculus: $35 \pm 3$ marks.     * Probability: $15 \pm 3$ marks.
• Paper 2:     * Statistics and Regression: $20 \pm 3$ marks.     * Analytical Geometry: $40 \pm 3$ marks.     * Trigonometry: $50 \pm 3$ marks.     * Euclidean Geometry: $40 \pm 3$ marks.

Weighting of Cognitive Levels

• Knowledge (20% | Approx. 30 marks):     * Includes recall, identification of correct formulas on the information sheet (without changing the subject), use of mathematical facts, appropriate vocabulary, algorithms, estimation, and rounding.
• Routine Procedures (35% | Approx. 52–53 marks):     * Includes proofs of prescribed theorems, derivation of formulas, simple applications involving few steps, identification and use of formulas (after changing the subject), and problems similar to those encountered in class.
• Complex Procedures (30% | Approx. 45 marks):     * Problems involving complex calculations or higher-order reasoning. No obvious route to the solution. Integration of different topics and conceptual understanding are required.
• Problem Solving (15% | Approx. 22–23 marks):     * Non-routine, mainly unfamiliar problems involving higher-order reasoning. Requires breaking problems into parts and interpreting or extrapolating solutions from unfamiliar contexts.

Elaboration of Content: Paper 1 Topics

Functions

• Candidates must interpret functional notation.
• Transformation of $f(x)$ to generate $f(-x)$, $-f(x)$, $f(x+a)$, $f(x)+a$, $af(x)$, and $x=f(y)$, where $a \in \mathbb{R}$.
• Trigonometric functions are strictly examined in Paper 2 only.

Number Patterns, Sequences, and Series

• Quadratic number patterns features a linear sequence of first differences; linear pattern knowledge can be tested within quadratic contexts.
• Recursive patterns are not explicitly examined.
• Links must be established with patterns from earlier grades.

Finance, Growth, and Decay

• Fluency in converting between nominal and effective interest rates for compounding periods: monthly, quarterly, and semi-annually.
• Candidates must be able to calculate any variable except for $i$ in the $Fv$ and $Pv$ formulas.
• Pyramid schemes are excluded from the examination.

Algebra

• Completing the square to solve quadratic equations is NOT examined.
• The substitution method ($k$-method) is examinable.
• Surd equations leading to quadratic equations are examinable.
• Non-quadratic inequalities should be solved in the context of functions.
• Nature of roots is tested intuitively with quadratic equations and prescribed functions.

Differential Calculus

• Accepted notations: $f'(x)$, $D_x$, $\frac{dy}{dx}$, or $y'$.
• Cubic functions: Candidates must determine equations from graphs, discuss stationary points (local max, local min, points of inflection), and apply transformations.
• Graphs of derivatives: Candidates must draw and interpret the derivative's graph.
• Optimization: Tested through surface area and volume of right prisms. Formulas for these will NOT be on the formula sheet.
• Cones, spheres, and pyramids: Necessary formulas will be provided within the specific question if required for optimization.

Probability

• Dependent events are examinable; however, conditional probabilities are not in the syllabus.
• Dependent events without replacement are examinable.
• The following are excluded: arranging objects in a circle and the use of combinations.
• Word arrangements: The question will specify if repeated letters are treated as indistinguishable or distinguishable.

Elaboration of Content: Paper 2 Topics

Euclidean Geometry and Measurement

• Measurement is tested within calculus optimization and 2D/3D trigonometry.
• Composite shapes are limited to a maximum of TWO stated shapes.
• Examinable Theorem Proofs:     * Perpendicular from center to chord bisects the chord.     * Line from center bisecting chord is perpendicular to the chord.     * Angle at center is double the angle at circumference.     * Opposite angles of a cyclic quadrilateral are supplementary.     * Tan-chord theorem: Angle between tangent and chord equals angle in alternate segment.     * Line parallel to one side of a triangle divides other sides proportionally.     * Equiangular triangles are similar.
• Necessary Corollaries/Axioms:     * Angles in a semi-circle.     * Equal chords subtend equal angles at the circumference/center.     * Exterior angles of cyclic quadrilaterals.     * Tangents from a common point are equal.
• Concurrency theory is excluded.

Trigonometry

• Reciprocal ratios (cosec, sec, cot) may be used by candidates but are not explicitly tested.
• Focus is on relationships, simplification, and determining points of intersection.

Analytical Geometry

• Proving properties of polygons using analytical methods.
• Understanding collinearity.
• Integration of Euclidean Geometry axioms/theorems into analytical problems.
• Calculating the length of a tangent from a point outside the circle.

Statistics

• Encouraged use of calculators for standard deviation, variance, and least squares regression line.
• Interpretation of standard deviation regarding normal distribution is excluded.
• Outliers are identified intuitively or via the formula: outside the interval $(Q_1 - 1.5 \times IQR, Q_3 + 1.5 \times IQR)$.

Acceptable Reasons: Euclidean Geometry (English)

Lines

• ∠s on a str line; adj ∠s supp
• ∠s round a pt OR ∠s in a rev
• vert opp ∠s =
• alt ∠s; AB || CD; corresp ∠s; AB || CD; co-int ∠s; AB || CD

Triangles

• ∠ sum in Δ OR sum of ∠s in Δ OR Int ∠s Δ
• ext ∠ of Δ
• ∠s opp equal sides; sides opp equal ∠s
• Pythagoras OR Theorem of Pythagoras
• SSS; SAS; AAS; RHS
• Midpt Theorem; line || one side of Δ
• ||| Δs; equiangular Δs; Sides of Δ in prop

Circles

• tan ∘ radius; tan ∘ diameter
• line from centre to midpt of chord; line from centre ∘ to chord
• ∠ at centre = 2 × ∠ at circumference
• ∠s in semi-circle; diameter subtends right angle
• ∠s in the same seg
• opp ∠s of cyclic quad; ext ∠ of cyclic quad
• Tans from common pt
• tan chord theorem

Quadrilaterals

• sum of ∠s in quad
• opp sides of ||m; opp ∠s of ||m; diag of ||m
• sides of rhombus; diags of rhombus; sides of square; diags of rect
• diags of kite; diag of kite bisects opposite angles

Information Sheet Formulas

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Finance: $A = P(1+ni)$; $A = P(1-ni)$; $A = P(1-i)^n$; $A = P(1+i)^n$
• Sequences: $T_n = a + (n-1)d$; $S_n = \frac{n}{2}[2a + (n-1)d]$; $T_n = ar^{n-1}$; $S_n = \frac{a(r^n - 1)}{r-1}$; $S_{\infty} = \frac{a}{1-r}$
• Annuities: $F = \frac{x[(1+i)^n - 1]}{i}$; $P = \frac{x[1 - (1+i)^{-n}]}{i}$
• Calculus: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
• Analytical Geometry: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$; $M(\frac{x_1+x_2}{2}; \frac{y_1+y_2}{2})$; $y = mx+c$; $y - y_1 = m(x-x_1)$; $m = \frac{y_2 - y_1}{x_2 - x_1}$; $m = \text{tan}(\theta)$; $(x-a)^2 + (y-b)^2 = r^2$
• Trigonometry: $\frac{a}{\text{sin}(A)} = \frac{b}{\text{sin}(B)} = \frac{c}{\text{sin}(C)}$; $a^2 = b^2 + c^2 - 2bc\cos(A)$; $\text{Area} = \frac{1}{2}ab\text{sin}(C)$
• Compound/Double Angles: $\text{sin}(\alpha \pm \beta) = \text{sin}\alpha\text{cos}\beta \pm \text{cos}\alpha\text{sin}\beta$; $\text{cos}(\alpha \pm \beta) = \text{cos}\alpha\text{cos}\beta \mp \text{sin}\alpha\text{sin}\beta$; $\text{cos}(2\alpha) = \text{cos}^2\alpha - \text{sin}^2\alpha = 1 - 2\text{sin}^2\alpha = 2\text{cos}^2\alpha - 1$
• Statistics: $\bar{x} = \frac{\sum x}{n}$; $\sigma^2 = \frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n}$; $P(A) = \frac{n(A)}{n(S)}$; $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$; $\hat{y} = a + bx$; $b = \frac{\sum(x - \bar{x})(y - \bar{y})}{\sum(x - \bar{x})^2}$

General Marking Guidelines

• Multiple Attempts: If a learner provides multiple attempts without canceling any, only the first attempt is marked.
• Consistent Accuracy (CA):     * Marks are awarded for subsequent subquestions if an initial error is carried through, provided the method remains correct.     * Assuming values/answers to solve a problem is strictly unacceptable.