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Grade 12 Mathematics
September Exam Guidelines 2018 (CAPS)
Two 150 mark papers each in 3 hours
PAPER 1 (Maximum 6 marks for bookwork)
Description Weighting of marks
Algebra, equations and inequalities 25 3
Patterns and sequences 25 3
Finance, growth and decay 15 3
Functions and graphs 35 3
Differential Calculus 35 3
Probability 15 3
TOTAL 150
PAPER 2 (Maximum 12 marks bookwork)
Description Weighting of marks
Statistics 20 ± 3
Analytical Geometry 40 ± 3
Trigonometry 40 ± 3
Euclidean Geometry and Measurement 50 ± 3
TOTAL 150
2018 Grade 12 September and Final Exam Guidelines Page 2 of 14
Cognitive level
requirements
and weighting
Description of skills to be demonstrated
Knowledge
20%
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
Routine
Procedures
35%
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
Complex
Procedures
30%
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.
Problem Solving
15%
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.
2018 Grade 12 September and Final Exam Guidelines Page 3 of 14
Paper 1
Algebraic Manipulation ( Grade 11)
Simplification of Algebraic Fractions
o multiplication, division and addition and subtraction
o denominators and/or numerators that need to be factorised
o common factor, difference of squares, trinomials, sum & difference of cubes
Fractions over fractions
Linear and Quadratic Equations and Inequalities (Grade 11)
Factorising
Fractions where denominators and/or numerators need to be factorised
Quadratic formula
Solving equations using the substitution method or k - method
Simultaneous Equations
Surd Equations (Solutions must be checked for extraneous answers.)
Modelling or problem solving questions, both linear and quadratic
Inequalities (including number lines, interval and inequality notation)
Nature of Roots will be tested intuitively with the solution of quadratic equations and in
all the prescribed functions
Classify roots:
o for non – real roots
o for real roots
o for real, equal, rational roots
o for real, unequal roots which are
rational if is a perfect square
irrational if is not a perfect square
Check for zero denominators and invalid solutions in equations with fractions
Linear Inequalities including number lines, interval and set-builder notation
Quadratic Inequalities solution illustrated with number line or graph
2018 Grade 12 September and Final Exam Guidelines Page 4 of 14
Patterns
Linear patterns
Patterns with a constant 1st difference form linear pattern, in the form
Quadratic patterns
constant second difference and can be expressed in the form
Find a formula for Tn
find n given Tn
find missing terms in sequence
constant 2nd difference = 2a , first 1st difference = 3a+b & first term = a + b + c
Mixed patterns
Exponential Patterns (Patterns with a constant ratio)
Determine, for any pattern:
o the general termo the term valueo the number of terms in a sequence of any pattern
Sequences and Series ( Grade 12)
Arithmetic Sequence
Geometric Sequence
; r≠1 ; −1<r<1
Convert fluently between Σ notation and expanded notation.
Proofs of the sum of arithmetic and geometric series are examinable.
Mixed patterns
Exponents (Grade 11)
Simplify expressions using the laws of exponents for rational exponents.
Add, subtract, multiply and divide simple surds
Exponential equations
Surds
Logs (Grade 12)
Definition of a logarithm: If
2018 Grade 12 September and Final Exam Guidelines Page 5 of 14
Complicated logarithm law simplification is not required
Solving logarithm equations and inequalities with the aid of graphs
Functions and Inverse Functions (Grade 12)
Definition of a function
Restrictions on domain to ensure inverse is a function
Revision of exponential function
Inverse functions of
o the straight line
o the parabola y = ax2
o the exponential function y = ax
o the logarithmic function y = logax
Log functions as inverses of exponential functions
Finance
Simple Interest Growth Formula
applications involving hire purchase
find interest rate, number of years or principle given the final amount
Simple Interest Decay or Straight line depreciation
Compound Interest Growth
applications involving inflation, population growth, exchange rates
find P, i, or n ( using logs)
the effect of different compounding intervals
effective and nominal interest rates
convert fluently between nominal and effective interest rates for:
monthly, quarterly, half-yearly/semi-annual compounding periods
time lines
Compound Interest Decay or depreciation on a reducing balance
and
where payment commences 1 time period from the present and ends at n.
Interest must be compounded at the same rate as the payments.
Calculate the value of any of the variables in the above formulae except i
2018 Grade 12 September and Final Exam Guidelines Page 6 of 14
Graphs (Grade 11)
Straight line
o y = mx + c
o
o y = k
o x = k
o Sketching and finding the equation
Parabola in 3 forms:
o y = ax2 + bx + c
o y = a(x−p)2+ q
o y = a( x−x1) (x −x2)
o Sketching and finding equations of parabolas
Intersection of straight line and parabola
Finding lengths, including the maximum length, between two given two graphs
Hyperbola:
Exponential:
Plotting and finding equations.
Intersections; graph interpretation
Knowledge and use of characteristics of ALL graphs
Domain ; range ; increasing and decreasing ; asymptotes
Function notation f(x)
Reading solutions to inequalities from graphs
Transformations
o translations ( vertical and horizontal shifts)
o reflections about the axes ( no inverses) of the 3 graphs
How f (x) has been transformed to generate
Inverse functions( Grade 12) f −1(x) or x = f(y)
o Inverse functions for
Real life applications
2018 Grade 12 September and Final Exam Guidelines Page 7 of 14
Polynomials (Grade 12)
Factorising and solving 3rd degree polynomial equations
by inspection
by applying remainder and factor theorems
Calculus (Grade 12)
Average gradient between two points
Intuitive understanding of limits
Differentiation from 1st Principles
o
o
o
o
Differentiate by using the rule
o Use of exponents and rearranging f(x) into the sum/difference of terms
o Different notations: or or
Equation of tangent at a point on a graph
Second derivative
o A point of inflection occurs at x = a if
and when x < a and when x > a
or
and when x < a and when x > a
o In summary, a point of inflection only occurs at x = a if concavity changes
from positive to zero to negative
or
from negative to zero to positive
o Concavity changes at the point of inflection
o A curve is concave down when
o A curve is concave up when
2018 Grade 12 September and Final Exam Guidelines Page 8 of 14
Sketching cubic functions
o Find x intercepts by solving
o Find x-coordinates of local max and local min stationary points by solving
o Find y-coordinates of local maximum and local minimum stationary points by
substituting x-values into original equation.
o Discuss the nature of stationary points including local maximum, local minimum and
points of inflection
o Apply knowledge of transformations to a given function to obtain its image
Calculus continued
Find the equation of a cubic functions
o usually 2 unknowns of the function
o given TP’s ; x-intercepts or 2 other points
o from a given graph
Discuss and/or answer questions about increasing and deceasing functions
Interpret derivative functions
o Draw a cubic function from the graph of its derivative
o Draw a parabola from the graph of its derivative
Solve practical problems involving
o optimisation (can overlap with measurement in Paper 2)
o rates of change
o the calculus of motion including velocity and acceleration
o volume and surface area of right prisms and cylinders, cones and spheres.
Formulae for optimisation questions
o the formulae for the surface area and volume of right prisms will NOT be provided
o if the optimisation question is based on the surface area of volume of a cone, sphere
and/or pyramid, a list of relevant formulae will be provided for that question.
o candidates will be expected to select the formula from the list provided
2018 Grade 12 September and Final Exam Guidelines Page 9 of 14
Probability
Probability Theory
P(A∪B) = P(A) + P(B) – P(A∩B)
A and B are mutually exclusive if P(A∩B) = 0 or if P(A∪B) = P(A) + P(B)
A and B are complementary if they are mutually exclusive and P(A) + P(B) = 1
P(not A) = P( ) = 1 – P(A)
A and B are independent if P(A)×P(B) = P(A∩B)
Venn Diagrams
Tree Diagrams for simultaneous events which are not necessarily independent
(Non-replacement of balls and cards are not independent events)
Two Way Contingency Tables - solve probability problems & test independence of events
Counting Principles (Grade 12 work completed in Grade 11)
The Basic Counting Principle
The number of possible outcomes for an event which has
choices for the 1st event, choices for the 2nd event and choices for the 3 event =
Number of arrangements
Without repeats, n elements can be arranged in n! ways, with repeats in nn ways.
Without repeats, n elements can be arranged into r slots in ways, with repeats in n r ways
n elements can be arranged in where a, b and c are the number of times different elements
are repeated, for example - in the word SLEEPIEST a = 3 ( 3 Es) b = 2 (2 Ss) and n = 9
so the number of arrangements is
Number of arrangement in a row where elements have to be in specific positions
( e.g. places next to each other or 1st and last etc.)
In respect of word arrangements, letters that are repeated in the word can be treated as
o the same (indistinguishable)
o different (distinguishable)
The question will be specific in this regard.
2018 Grade 12 September and Final Exam Guidelines Page 10 of 14
Paper 2
Trigonometry
Positive and Negative Angles in all 4 quadrants
Definitions of sinx, cosx and tanx on the Cartesian Plane
From one ratio to another. Algebraic examples included.
Numerical and Algebraic Reductions
o reductions: 180°± θ ; 360°± θ; – θ
o co – ratios: 90°± θ or θ ± 90°
Special Angles
o 30°; 45° and 60° as well as obtuse and reflex angles such as
o (using graphs or unit circles)
Compound Angles and Double Angles
o
o
o
o
o
o
o
o
o Deriving formulae from cos(A−B)
o Expanding and contracting formulae
Trigonometric equations using the general solution
o
o
o
o
o
o
2018 Grade 12 September and Final Exam Guidelines Page 11 of 14
Identities not on the formula sheet which must be known
o
o
Trigonometry Continued
Proving identities (with or without compound angles)
Finding the values of the angle(s) for which an identity is undefined
Trigonometric graphs:
o
o
o
o
o plotting graphs
o finding equations of graphs from sketches
o graph interpretation
o intersections of graphs by estimations and by calculation
Triangle Rules (Proofs are required for exams)
o Sine rule:
o Cos rule:
o Area rule: Applications of rules
o Numerical examples
o Algebraic proofs
o in 2 Dimensions
o Simple applications in 3 D
o Geometric figures
navigational problems
2018 Grade 12 September and Final Exam Guidelines Page 12 of 14
Coordinate Geometry Formulae
o distance
o midpoint M( x1+x2
2 ;y1+ y2
2 )
o gradientm=
y2− y1x2−x1
Equation of straight line
o y=mx+c or
y− y1=m( x−x1 )
o collinear points
o parallel and perpendicular lines
o perpendicular bisectors
o medians
o altitudes
Angle of inclination
The length of a tangent from a point outside a circle to the point of contact
Prove properties of quadrilaterals
Collinearity
Find the fourth vertex of a parallelogram
Equation of circles
o Finding the equation of a circle ( x−a )2+( y−b )2=r2
o Completing the square to find centre and radius of circle.
o Find equation of the tangent to a circle (radius perpendicular to tangent)
2018 Grade 12 September and Final Exam Guidelines Page 13 of 14
Euclidean Geometry
Examinable Proofs of theorems
o Line drawn from centre of circle perpendicular to chord bisects chord
o Angle subtended by arc at centre of circle is double the size of the angle subtended by the
same arc at the circle (on same side of chord as centre)
o Opposite angles of a cyclic quadrilateral are supplementary
o Angle between tangent and chord drawn from point of contact is equal to angle subtended
by chord in alternate segment
Geometry proofs (riders) using theorems , corollaries and converses
Proportionality and Similarity
Examinable Proofs
o A line drawn parallel to one side a triangle divides the other 2 sides proportionally
o Equiangular triangles are similar and their corresponding sides are in proportion
Corollaries derived from the theorems and axioms are necessary in solving:
o angles in semi-circles are always equal to
o equal chords subtend equal angles at the circumference
o in equal circles, equal chords subtend equal angles at the circumference
o in equal circles, equal chords subtend equal angles at the centre
o the exterior angle of a cyclic quad is equal to the interior opposite angle of the quadrilateral
o if the exterior angle of a quad is equal to the interior opposite angle then the quad is cyclic
o tangents drawn from a common point outside the circle are equal in length
The theory of quadrilaterals (quadrilateral properties) will be integrated into questions
o in Coordinate Geometry
o in Euclidean Geometry
2018 Grade 12 September and Final Exam Guidelines Page 14 of 14
Statistics Univariate numerical data – continuous and discontinuous
Histograms
Frequency polygons
Measures of central tendency
o mean
o median
o mode
Measures of dispersion
o quartiles
o range
o interquartile range
Five number summary
Box and whisker diagrams
o draw
o discuss distribution and skewness
o outliers are values that lie outside the interval (Q1 – 1,5 IQR; Q3 + 1,5 IQR)
Cumulative frequency graphs or ogives
o draw (remember to plot endpoints of intervals)
o read off 5 number summary
o applications involving percentiles, deciles etc.
Variance and Standard deviation ( using calculator for large data sets)
Bivariate numerical data
o scatter plots and curve/line of best fit
o least squares line (regression line)
o correlation
Measurement
right cylinders with A = and V =
right prisms with A = 2(lb + lh + bh) and V = lbh
spheres with A = and V =
hemispheres with A = if closed; A = if open & V =
cones with A = and V =
2018 Grade 12 September and Final Exam Guidelines Page 15 of 14
pyramids with A = sum of all faces and V =
