MATHEMATICS
EXAMINATION GUIDELINES
GRADE 12
2021

1. INTRODUCTION
   The Curriculum and Assessment Policy Statement (CAPS) for Mathematics outlines the nature and purpose of the subject Mathematics. This guides the philosophy underlying the teaching and assessment of the subject in Grade 12.
   The purpose of these Examination Guidelines is to:
   • Provide clarity on the depth and scope of the content to be assessed in the Grade 12 National Senior Certificate Examination in Mathematics
   • Assist teachers to adequately prepare learners for the examinations
     This document deals with the final Grade 12 external examinations. It does not deal in any depth with the school-based assessment (SBA), performance assessment tasks (PATs) or final external practical examinations as these are clarified in a separate PAT document which is updated annually.
     These guidelines should be read in conjunction with:
   • The National Curriculum Statement (NCS) Curriculum and Assessment Policy Statement (CAPS): Mathematics
   • The National Protocol of Assessment: An addendum to the policy document, the National Senior Certificate: A qualification at Level 4 on the National Qualifications Framework (NQF), regarding the National Protocol for Assessment (Grades R–12)
   • National policy pertaining to the programme and promotion requirements of the National Curriculum Statement, Grades R to 12
     Included in this document is a list of Euclidean Geometry reasons, both in English and Afrikaans, which should be used as a guideline when teaching learners Euclidean Geometry.
     The Information Sheet for Paper 1 and 2 is included in this document.
2. ASSESSMENT IN GRADE 12
   All candidates will write two external papers as prescribed.
   2.1 Format of Question Papers for Grade 12

   Paper	Topics	Duration	Total	Date	Marking
   1	Patterns and sequences Finance, growth and decay Functions and graphs Algebra, equations and inequalities Differential Calculus Probability	3 hours	150	October/November	Externally
   2	Euclidean Geometry Analytical Geometry Statistics and regression Trigonometry	3 hours	150	October/November	Externally

   Questions in both Papers 1 and 2 will assess performance at different cognitive levels with an emphasis on process skills, critical thinking, scientific reasoning and strategies to investigate and solve problems in a variety of contexts.
   An Information Sheet is included on p. 15.
   2.2 Weighting of Topics per Paper for Grade 12

   PAPER 1	MARKS	PAPER 2	MARKS
   Algebra, Equations and Inequalities Number Patterns Functions and Graphs Finance, Growth and Decay Differential Calculus Counting Principle and Probability	25 25 25 25 25 25	Statistics and Regression Analytical Geometry Trigonometry Euclidean Geometry	20 40 50 40
   TOTAL	150	TOTAL	150

   2.3 Weighting of Cognitive Levels
   Papers 1 and 2 will include questions across four cognitive levels. The distribution of cognitive levels in the papers is given below.

   Cognitive Level	Description of Skills to be Demonstrated	Weighting	Approximate Number of Marks in a 150-mark Paper
   Knowledge	Recall Identification of correct formula on the information sheet (no changing of the subject) Use of mathematical facts Appropriate use of mathematical vocabulary Algorithms Estimation and appropriate rounding of numbers	20%	30 marks
   Routine Procedures	Proofs of prescribed theorems and derivation of formulae Perform well-known procedures Simple applications and calculations which might involve few steps Derivation from given information may be involved Identification and use (after changing the subject) of correct formula Generally similar to those encountered in class	35%	52–53 marks
   Complex Procedures	Problems involve complex calculations and/or higher-order reasoning There is often not an obvious route to the solution Problems need not be based on a real-world context Could involve making significant connections between different representations Require conceptual understanding Learners are expected to solve problems by integrating different topics.	30%	45 marks
   Problem Solving	Non-routine problems (which are not necessarily difficult) Problems are mainly unfamiliar Higher-order reasoning and processes are involved Might require the ability to break the problem down into its constituent parts Interpreting and extrapolating from solutions obtained by solving problems based in unfamiliar contexts.	15%	22–23 marks

3. ELABORATION OF CONTENT/TOPICS
   The purpose of the clarification of the topics is to give guidance to the teacher in terms of depth of content necessary for examination purposes. Integration of topics is encouraged as learners should understand Mathematics as a holistic discipline. Thus questions integrating various topics can be asked.
   FUNCTIONS
   • Candidates must be able to use and interpret functional notation. In the teaching process learners
     must be able to understand how f (x) has been transformed to generate f (-x) , - f (x) ,
     f (x + a) f (x) + a , af (x) and x = f (y) where a∈R.
   • Trigonometric functions will ONLY be examined in PAPER 2.
     
     NUMBER PATTERNS, SEQUENCES AND SERIES
   • The sequence of first differences of a quadratic number pattern is linear. Therefore, knowledge of linear patterns can be tested in the context of quadratic number patterns.
   • Recursive patterns will not be examined explicitly.
   • Links must be clearly established between patterns done in earlier grades.
     
     FINANCE, GROWTH AND DECAY
   • Understand the difference between nominal and effective interest rates and convert fluently between them for the following compounding periods: monthly, quarterly and half-yearly or semi-annually.
   • With the exception of calculating i in the Fv and Pv formulae, candidates are expected to calculate the value of any of the other variables.
   • Pyramid schemes will NOT be examined in the examination.
     
     ALGEBRA
   • Solving quadratic equations by completing the square will NOT be examined.
   • Solving quadratic equations using the substitution method (k-method) is examinable.
   • Equations involving surds that lead to a quadratic equation are examinable.
   • Solution of non-quadratic inequalities should be seen in the context of functions.
   • Nature of the roots will be tested intuitively with the solution of quadratic equations and in all theprescribed functions.
     
     DIFFERENTIAL CALCULUS
   • The following notations for differentiation c an be used: f '(x) , x D_x ,dy or y '
                                                                                                                dx
   • In respect of cubic functions, candidates are expected to be able to:
   • Determine the equation of a cubic function from a given graph.
   • Discuss the nature of stationary points including local maximum, local minimum and points of inflection.
   • Apply knowledge of transformations on a given function to obtain its image.
   • Candidates are expected to be able to draw and interpret the graph of the derivative of a function.
   • Surface area and volume will be examined in the context of optimisation.
   • Candidates must know the formulae for the surface area and volume of the right prisms. These formulae will NOT be provided on the formula sheet
   • If the optimisation question is based on the surface area and/or volume of the cone, sphere and/or pyramid, a list of the relevant formulae will be provided in that question. Candidates will be expected to select the correct formula from this list.
     
     PROBABILITY
   • Dependent events are examinable but conditional probabilities are not part of the syllabus.
   • Dependent events in which an object is not replaced are examinable.
   • Questions that require the learner to count the different number of ways that objects may be arranged in a circle and/or the use of combinations are not in the spirit of the curriculum.
   • In respect of word arrangements, letters that are repeated in the word can be treated as the same (indistinguishable) or different (distinguishable). The question will be specific in this regard.
     
     EUCLIDEAN GEOMETRY AND MEASUREMENT
   • Measurement can be tested in the context of optimisation in calculus and two- and three-dimensional trigonometry.
   • Composite shapes could be formed by combining a maximum of TWO of the stated shapes.
   • The following proofs of theorems are examinable:
     • The line drawn from the centre of a circle perpendicular to a chord bisects the chord;
     • The line drawn from the centre of a circle that bisects a chord is perpendicular to the chord;
     • The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circle (on the same side of the chord as the centre);
     • The opposite angles of a cyclic quadrilateral are supplementary;
     • The angle between the tangent to a circle and the chord drawn from the point of contact is equal to the angle in the alternate segment;
     • A line drawn parallel to one side of a triangle divides the other two sides proportionally;
     • Equiangular triangles are similar.
   • Corollaries derived from the theorems and axioms are necessary in solving riders:
     • Angles in a semi-circle
     • Equal chords subtend equal angles at the circumference
     • Equal chords subtend equal angles at the centre
     • In equal circles, equal chords subtend equal angles at the circumference
     • In equal circles, equal chords subtend equal angles at the centre.
     • The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle of the quadrilateral.
     • If the exterior angle of a quadrilateral is equal to the interior opposite angle of the quadrilateral, then the quadrilateral is cyclic.
     • Tangents drawn from a common point outside the circle are equal in length.
   • The theory of quadrilaterals will be integrated into questions in the examination.
   • Concurrency theory is excluded.
     
     TRIGONOMETRY
   • The reciprocal ratios cosec θ, sec θ and cot θ can be used by candidates in the answering of problems but will not be explicitly tested.
   • The focus of trigonometric graphs is on the relationships, simplification and determining points of intersection by solving equations, although characteristics of the graphs should not be excluded.
     
     ANALYTICAL GEOMETRY
   • Prove the properties of polygons by using analytical methods.
   • The concept of collinearity must be understood.
   • Candidates are expected to be able to integrate Euclidean Geometry axioms and theorems into Analytical Geometry problems.
   • The length of a tangent from a point outside the circle should be calculated.
   • Concepts involved with concurrency will not be examined.
     
     STATISTICS
   • Candidates should be encouraged to use the calculator to calculate standard deviation, variance and the equation of the least squares regression line.
   • The interpretation of standard deviation in terms of normal distribution is not examinable.
   • Candidates are expected to identify outliers intuitively in both the scatter plot as well as the box and whisker diagram. In the case of the box and whisker diagram, observations that lie outside the interval (lower quartile – 1,5 IQR; upper quartile + 1,5 IQR) are considered to be outliers. However, candidates will not be penalised if they did not make use of this formula in identifying outliers.
4. ACCEPTABLE REASONS: EUCLIDEAN GEOMETRY
   In order to have some kind of uniformity, the use of the following shortened versions of the theorem statements is encouraged.
   4.1 ACCEPTABLE REASONS: EUCLIDEAN GEOMETRY (ENGLISH)

   THEOREM STATEMENT	ACCEPTABLE REASON(S)
   LINES
   The adjacent angles on a straight line are supplementary.	∠s on a str line
   If the adjacent angles are supplementary, the outer arms of these angles form a straight line.	adj ∠s supp
   The adjacent angles in a revolution add up to 360º	∠s round a pt OR ∠s in a rev
   Vertically opposite angles are equal.	vert opp ∠s =
   If AB || CD, then the alternate angles are equal.	alt ∠s; AB || CD
   If AB || CD, then the corresponding angles are equal.	corresp ∠s; AB || CD
   If AB || CD, then the co-interior angles are supplementary.	co-int ∠s; AB || CD
   If the alternate angles between two lines are equal, then the lines are parallel.	alt ∠s =
   If the corresponding angles between two lines are equal, then the lines are parallel.	corresp ∠s =
   If the co-interior angles between two lines are supplementary, then the lines are parallel.	coint ∠s supp
   TRIANGLES
   The interior angles of a triangle are supplementary.	∠ sum in Δ OR sum of ∠s in Δ OR Int ∠s Δ
   The exterior angle of a triangle is equal to the sum of the interior opposite angles.	ext ∠ of Δ
   The angles opposite the equal sides in an isosceles triangle are equal.	∠s opp equal sides
   The sides opposite the equal angles in an isosceles triangle are equal.	sides opp equal ∠s
   In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.	Pythagoras OR Theorem of Pythagoras
   If the square of the longest side in a triangle is equal to the sum of the squares of the other two sides then the triangle is right-angled.	Converse Pythagoras OR Converse Theorem of Pythagoras
   If three sides of one triangle are respectively equal to three sides of another triangle, the triangles are congruent.	SSS
   If two sides and an included angle of one triangle are respectively equal to two sides and an included angle of another triangle, the triangles are congruent.	SAS OR S∠S
   If two angles and one side of one triangle are respectively equal to two angles and the corresponding side in another triangle, the triangles are congruent.	AAS OR ∠∠S
   If in two right-angled triangles, the hypotenuse and one side of one triangle are respectively equal to the hypotenuse and one side of the other, the triangles are congruent	RHS OR 90°HS
   THEOREM STATEMENT	ACCEPTABLE REASON(S)
   The line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to half the length of the third side	Midpt Theorem
   The line drawn from the midpoint of one side of a triangle, parallel to another side, bisects the third side.	line through midpt || to 2nd side
   A line drawn parallel to one side of a triangle divides the other two sides proportionally.	line || one side of Δ OR prop theorem; name || lines
   If a line divides two sides of a triangle in the same proportion, then the line is parallel to the third side.	line divides two sides of Δ in prop
   If two triangles are equiangular, then the corresponding sides are in proportion (and consequently the triangles are similar).	||| Δs OR equiangular Δs
   If the corresponding sides of two triangles are proportional, then the triangles are equiangular (and consequently the triangles are similar).	Sides of Δ in prop
   If triangles (or parallelograms) are on the same base (or on bases of equal length) and between the same parallel lines, then the triangles (or parallelograms) have equal areas.	same base; same height OR equal bases; equal height
   CIRCLES
   The tangent to a circle is perpendicular to the radius/diameter of the circle at the point of contact.	tan ⊥ radius tan ⊥ diameter
   If a line is drawn perpendicular to a radius/diameter at the point where the radius/diameter meets the circle, then the line is a tangent to the circle.	line ⊥ radius OR converse tan ⊥ radius OR converse tan ⊥ diameter
   The line drawn from the centre of a circle to the midpoint of a chord is perpendicular to the chord.	line from centre to midpt of chord
   The line drawn from the centre of a circle perpendicular to a chord bisects the chord.	line from centre ⊥ to chord
   The perpendicular bisector of a chord passes through the centre of the circle;	perp bisector of chord
   The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circle (on the same side of the chord as the centre)	∠ at centre = 2 ×∠ at circumference
   The angle subtended by the diameter at the circumference of the circle is 90.	∠s in semi-circle OR diameter subtends right angle OR ∠ in ½
   If the angle subtended by a chord at the circumference of the circle is 90º, then the chord is a diameter.	chord subtends 90º OR converse ∠s in semi-circle
   Angles subtended by a chord of the circle, on the same side of the chord, are equal	∠s in the same seg
   If a line segment joining two points subtends equal angles at two points on the same side of the line segment, then the four points are concyclic.	line subtends equal ∠s OR converse ∠s in the same seg
   Equal chords subtend equal angles at the circumference of the circle.	equal chords; equal ∠s
   Equal chords subtend equal angles at the centre of the circle.	equal chords; equal ∠s
   Equal chords in equal circles subtend equal angles at the circumference of the circles.	equal circles; equal chords; equal ∠s
   THEOREM STATEMENT	ACCEPTABLE REASON(S)
   Equal chords in equal circles subtend equal angles at the centre of the circles.	equal circles; equal chords; equal ∠s
   The opposite angles of a cyclic quadrilateral are supplementary	opp ∠s of cyclic quad
   If the opposite angles of a quadrilateral are supplementary then the quadrilateral is cyclic.	opp ∠s quad supp OR converse opp ∠s of cyclic quad
   The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.	ext ∠ of cyclic quad
   If the exterior angle of a quadrilateral is equal to the interior opposite angle of the quadrilateral, then the quadrilateral is cyclic.	ext ∠ = int opp ∠ OR converse ext ∠ of cyclic quad
   Two tangents drawn to a circle from the same point outside the circle are equal in length	Tans from common pt OR Tans from same pt
   The angle between the tangent to a circle and the chord drawn from the point of contact is equal to the angle in the alternate segment.	tan chord theorem
   If a line is drawn through the end-point of a chord, making with the chord an angle equal to an angle in the alternate segment, then the line is a tangent to the circle.	converse tan chord theorem OR ∠ between line and chord
   QUADRILATERALS
   The interior angles of a quadrilateral add up to 360.	sum of ∠s in quad
   The opposite sides of a parallelogram are parallel.	opp sides of ||m
   If the opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram.	opp sides of quad are ||
   The opposite sides of a parallelogram are equal in length.	opp sides of ||m
   If the opposite sides of a quadrilateral are equal , then the quadrilateral is a parallelogram.	opp sides of quad are = OR converse opp sides of a parm
   The opposite angles of a parallelogram are equal.	opp ∠s of ||m
   If the opposite angles of a quadrilateral are equal then the quadrilateral is a parallelogram.	opp ∠s of quad are = OR converse opp angles of a parm
   The diagonals of a parallelogram bisect each other.	diag of ||m
   If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.	diags of quad bisect each other OR converse diags of a parm
   If one pair of opposite sides of a quadrilateral are equal and parallel, then the quadrilateral is a parallelogram.	pair of opp sides = and ||
   The diagonals of a parallelogram bisect its area.	diag bisect area of ||m
   The diagonals of a rhombus bisect at right angles.	diags of rhombus
   The diagonals of a rhombus bisect the interior angles.	diags of rhombus
   All four sides of a rhombus are equal in length.	sides of rhombus
   All four sides of a square are equal in length.	sides of square
   The diagonals of a rectangle are equal in length.	diags of rect
   The diagonals of a kite intersect at right-angles.	diags of kite
   A diagonal of a kite bisects the other diagonal.	diag of kite
   A diagonal of a kite bisects the opposite angles	diag of kite

5. INFORMATION SHEET
   
6. GENERAL GUIDELINES FOR MARKING
   
   • If a learner makes more than one attempt at answering a question and does not cancel any of them out, only the first attempt will be marked irrespective of which of the attempt(s) may be the correct answer.
   • Consistent Accuracy marking regarding calculations will be followed in the following cases:
     
     • Subquestion to subquestion: When a certain variable is incorrectly calculated in one subquestion and needs to be substituted into another subquestion, full marks can be awarded for the subsequent subquestions provided the methods used are correct and the calculations are correct.
     • Assuming values/answers in order to solve a problem is unacceptable.
7. CONCLUSION
   This Examination Guidelines document is meant to articulate the assessment aspirations espoused in the CAPS document. It is therefore not a substitute for the CAPS document which teachers should teach to.
   Qualitative curriculum coverage as enunciated in the CAPS cannot be over-emphasised.