M. Malan
St udy G u id e
St udy G u id e
Via Afrika Mathematics
Grade 12
, Exponents and Surds
Contents
Introduction ......................................................................................................................... 3
Chapter 1 Number patterns, sequences and series ........................................ 4
OVERVIEW ............................................................................................................................... 4
Unit 1 Arithmetic sequences and series
Unit 2 Geometric sequences and series
Unit 3 The sum to n terms(Sn): Sigma notation
Unit 4 Convergence and sum to infinity
Mixed exercises .................................................................................................................... 8
Chapter 2 Functions ..................................................................................... 10
OVERVIEW ............................................................................................................................... 10
Unit 1 The definitions of a function
Unit 2 The inverse of a function
Unit 3 The inverse of y = ax + q
Unit 4 The inverse of the quadratic function y = ax2
Mixed exercises .................................................................................................................... 21
Chapter 3 Logarithms ................................................................................... 24
OVERVIEW ............................................................................................................................... 24
Unit 1 The definition of a logarithm
Unit 2 Solving exponential equations using logarithms
Unit 3 The graph of y = logbx where b > 1 and 0 < b < 1
Mixed exercises .................................................................................................................... 28
Chapter 4 Finance, growth and decay ............................................................ 29
OVERVIEW ............................................................................................................................... 29
Unit 1 Future value annuities
Unit 2 Present value annuities
Unit 3 Calculating the period
Unit 4 Analysing investments and loans
Mixed exercises .................................................................................................................... 35
Chapter 5 Compound angles ......................................................................... 37
OVERVIEW ............................................................................................................................... 37
Unit 1 Deriving a formula for cos(𝛼 − 𝛽)
Unit 2 Formula for cos(𝛼 + 𝛽) and 𝑠𝑖𝑖(𝛼 ± 𝛽)
Unit 3 Double angles
Unit 4 Identities
Unit 5 Equations
Unit 6 Trigonometric graphs and compound angles
Mixed exercises .................................................................................................................... 46
Chapter 6 Solving problems in three dimensions ........................................... 48
OVERVIEW ............................................................................................................................... 48
Unit 1 Problems in three dimensions
Unit 2 Compound angle formulae in three dimensions
Mixed exercises .................................................................................................................... 51
© Via Afrika ›› Mathematics Grade 12 1
, Exponents and Surds
Chapter 7 Polynomials .................................................................................. 53
OVERVIEW ............................................................................................................................... 53
Unit 1 The Remainder Theorem
Unit 2 The Factor Theorem
Mixed exercises .................................................................................................................... 57
Chapter 8 Differential calculus ...................................................................... 58
OVERVIEW ............................................................................................................................... 58
Unit 1 Limits
Unit 2 The gradient of a graph at a point
Unit 3 The derivative of a function
Unit 4 The equation of a tangent to a graph
Unit 5 The graph of a cubic function
Unit 6 The second derivative (concavity)
Unit 7 Applications of differential calculus
Mixed exercises .................................................................................................................... 70
Chapter 9 Analytical geometry ...................................................................... 72
OVERVIEW ............................................................................................................................... 72
Unit 1 Equation of a circle with centre at the origin
Unit 2 Equation of a circle centred off the origin
Unit 3 The equation of the tangent to the circle
Mixed exercises .................................................................................................................... 77
Chapter 10 Euclidean geometry ..................................................................... 81
OVERVIEW ............................................................................................................................... 81
Unit 1 Proportionality in triangles
Unit 2 Similarity in triangles
Unit 2 Theorem of Pythagoras
Mixed exercises .................................................................................................................... 94
Chapter 11 Statistics: regression and correlation ........................................... 97
OVERVIEW ............................................................................................................................... 97
Unit 1 Symmetrical and skewed data
Unit 2 Scatter plots and correlation
Mixed exercises .................................................................................................................... 106
Chapter 12 Probability .................................................................................. 108
OVERVIEW ............................................................................................................................... 108
Unit 1 Solving probability problems
Unit 2 The counting principle
Unit 3 The counting principle and probability
Mixed exercises .................................................................................................................... 112
ANSWERS TO MIXED EXERCISES ...................................................................................................... 113
EXEMPLAR PAPER 1 ..................................................................................................................... 151
EXEMPLAR PAPER 2 .................................................................................................................... 168
© Via Afrika ›› Mathematics Grade 12 2
, Exponents and Surds
Introduction to Via Afrika Mathematics Grade 12 Study Guide
Woohoo! You made it! If you’re reading this it means that you made it through Grade 11,
and are now in Grade 12. But I guess you are already well aware of that…
It also means that your teacher was brilliant enough to get the Via Afrika Mathematics
Grade 12 Learner’s Book. This study guide contains summaries of each chapter, and should
be used side-by-side with the Learner’s Book. It also contains lots of extra questions to
help you master the subject matter.
Mathematics – not for spectators
You won’t learn anything if you don’t involve yourself in the subject-matter actively. Do
the maths, feel the maths, and then understand and use the maths.
Understanding the principles
• Listen during class. This study guide is brilliant but it is not enough. Listen to your
teacher in class as you may learn a unique or easy way of doing something.
• Study the notation, properly. Incorrect use of notation will be penalised in tests
and exams. Pay attention to notation in our worked examples.
• Practise, Practise, Practise, and then Practise some more. You have to practise
as much as possible. The more you practise, the more prepared and confident you
will feel for exams. This guide contains lots of extra practice opportunities.
• Persevere. We can’t all be Einsteins, and even old Albert had difficulties learning
some of the very advanced Mathematics necessary to formulate his theories. If you
don’t understand immediately, work at it and practise with as many problems from
this study guide as possible. You will find that topics that seem baffling at first,
suddenly make sense.
• Have the proper attitude. You can do it!
The AMA of Mathematics
ABILITY is what you’re capable of doing.
MOTIVATION determines what you do.
ATTITUDE determines how well you do it.
“Pure Mathematics is, in its way, the poetry of logical ideas.” Albert Einstein
© Via Afrika ›› Mathematics Grade 12 3
, 1 Chapter
Number patterns, sequences and series
Overview
Unit 1 Page 10
Arithmetic sequences and • Formula for an arithmetic
series sequence
Unit 2 Page 14
Geometric sequences and • Formula for the nth term
Chapter 1 Page 8 series of a sequence
Number patterns,
sequences and Unit 3 Page 18
series The sum to 𝑖 terms (𝑆𝑛 ): Sigma • The sum to 𝑖 terms in an
notation arithmetic sequence
• The sum to 𝑖 terms in a
geometric sequence
Unit 4 Page 28
Convergence and sum to infinity • Convergence
REMEMBER YOUR STUDY APPROACH SHOULD BE:
1 Work through all examples in this chapter of your Learner’s Bok.
2 Work through the notes in this chapter of this study guide.
3 Do the exercises at the end of the chapter in the Learner’s Book.
4 Do the mixed exercises at the end of this chapter in this study guide.
© Via Afrika ›› Mathematics Grade 12 4
, 1 Chapter
Number patterns, sequences and series
TABLE 1: SUMMARY OF SEQUENCES AND SERIES
TYPE GENERAL TERM: 𝑇𝑛 SUM OF TERMS: 𝑆𝑛 EXAMPLES
A) 2 ; 5 ; 8 ; 11 ; ...
Arithmetic Sequence (AS) 𝑇𝑛 = 𝑎 + (𝑖 − 1)𝑑 𝑆𝑛
𝑖
(also named the linear = [2𝑎 + (𝑖 − 1)𝑑] 𝑑= +3 +3 +3
𝑎 = 𝑓𝑖𝑟𝑠𝑡 𝑡𝑐𝑟𝑚 𝑇1 2
sequence) or
𝑖 𝑇𝑛 = 2 + (𝑖 − 1)(3)
𝑑 = 𝑐𝑜𝑖𝑠𝑡𝑎𝑖𝑡 𝑑𝑖𝑓𝑓. 𝑆𝑛 = [𝑎 + 𝑙] = 2 + 3𝑖 − 3
Constant 2
st 𝑑 = 𝑇2 − 𝑇1 = 3𝑖 − 1
1 difference where
𝑜𝑟 𝑇3 − 𝑇2 etc.
𝑙 = the last term of
B) 1 ; -4 ; -9 ; ...
the sequence
𝑑 = -5 -5
𝑇𝑛 = 1 + (𝑖 − 1)(−5)
= 1 − 5𝑖 + 5
= −5𝑖 + 6
𝑎(𝑟 𝑛 − 1) A) 2 ; -4 ; 8 ; -16 ; ...
𝑆𝑛 =
Geometric Sequence (GS) 𝑇𝑛 = 𝑎𝑟 𝑛−1 𝑟−1
(also named exponential 𝑟 = x-2 x-2 x-2
𝑎 = 𝑓𝑖𝑟𝑠𝑡 𝑡𝑐𝑟𝑚 𝑇1 Or
sequence) 𝑛
𝑎(1 − 𝑟 )
𝑆𝑛 = 𝑇𝑛 = 2(−2)𝑛−1
𝑟 = 𝑐𝑜𝑖𝑠𝑡𝑎𝑖𝑡 1−𝑟
NOT CONVERGING as 𝑟 < −1
𝑟𝑎𝑡𝑖𝑜
𝑎
Or 𝑆∞ =
Constant 𝑇2 𝑇3
1−𝑟
B) 3 ;
3
;
3
;
3
; ...
ratio 𝑟= 𝑜𝑟 Where −1 < 𝑟 < 1 2 4 8
𝑇1 𝑇2
(Converging series) 1 1 1
𝑟= x x x
2 2 2
1 𝑛−1
𝑇𝑛 = 3 � �
2
CONVERGING as −1 < 𝑟 < 1
𝑇𝑛 = 𝑎𝑖2 + 𝑏𝑖 + 𝑐 3 ; 8 ; 16 ; 27 ; ...
Quadratic Sequence (QS)
𝑓= 1st difference 𝑓: 5 8 11
𝑠= 2nd difference
𝑠: 3 3
Constant Determine 𝑎, 𝑏 and 𝑐
nd
2 using simultaneous Setup three equations using
difference equations (see the first three terms:
example) 𝑇1 = 3:
3=𝑎+𝑏+𝑐 …(1)
Alternatively: 𝑇2 = 8:
𝑎 =𝑠÷2 8 = 4𝑎 + 2𝑏 + 𝑐 …(2)
𝑏 = 𝑓1 − 3𝑎 𝑇3 = 16:
𝑐 = 𝑇1 − 𝑎 − 𝑏 16 = 9𝑎 + 3𝑏 + 𝑐 …(3)
where Solving simultaneously leads
𝑓1 = first term of first to:
differences 𝑇𝑛 = 32𝑖2 + 12𝑖 + 1
© Via Afrika ›› Mathematics Grade 12 5
, 1
Chapter
Number patterns, sequences and series
TYPES OF QUESTIONS YOU STRATEGY TO ANSWER THIS TYPE EXAMPLE(S) OF THIS TYPE OF
CAN EXPECT OF QUESTION QUESTION
Identify any of the following Determine whether sequence has a See Table 1 above
three types of sequences: • constant 1st difference (AS)
Arithmetic (AS), Geometric • constant ratio (GS)
(GS) and Quadratic (QS) • constant 2nd difference (QS)
Determine the formula for the You need to find: See Table 1 above
general term, 𝑇𝑛 , of AS, GS • 𝑎 and 𝑑 for an AS
and QS (from Grade 11) • 𝑎 and 𝑟 for a GS
• 𝑎, 𝑏 and 𝑐 for a QS
Determine any specific term Substitute the value of 𝑖 into 𝑇𝑛 See Text Book :
for a sequence e.g. 𝑇30 Example 1, nr. 1 d and 2 d, p.8
(AS)
Example 1, nr. 1 b, 3 b, p.11
(AS)
Example 1, nr. 1, p. 15 (GS)
Determine the number of Substitute all known variables into See Text Book:
terms in a sequence, 𝑖, for an the general term to get an equation Example 1, nr.1 c, p.8
AS, GS and QS or with 𝒔 as the only unknown. Solve Example 1, nr.1 c, p.11
the position, 𝑖, of a specific for 𝑖. Example 1, nr. 3, p.15
given term or when the sum OR
of the series is given Substitute all known variables into
the 𝑆𝑛 -formula to get an equation
with 𝒔 as the only unknown. Solve
for 𝑖. Example 2, nr.3, p.20
Example 3, nr. 2, p.24
Remember:
𝑖 must be a natural number
(not negative, not a fraction)
When given two sets of For each set of information given, See Text Book:
information, make use of substitute the values of 𝑖 and 𝑇𝑛 or Example 1, nr. 3, p.11 (AS)
simultaneous equations to 𝑖 and 𝑆𝑛 . Example 1, nr.2, p.15 (AS)
solve: Example 3, nr.3, p.24 (GS)
𝒂 and 𝒅 (for an AS) You then have 2 equations which
𝒂 and 𝒓 (for a GS) you can solve simultaneously (by
substitution)
Determine the value of a For AS use constant difference: The first three terms of an AS
variable (𝑥) when given a 𝑇3 − 𝑇2 = 𝑇2 − 𝑇1 are given by
sequence in terms of 𝑥. 2𝑥 − 4; 𝑥 − 3; 8 − 2𝑥
For GS use constant ratio: Determine 𝑥:
𝑇2 𝑇3 8 − 2𝑥 − (𝑥 − 3) = 𝑥 − 3 −
= (2𝑥 − 4)
𝑇1 𝑇2
∴𝑥=5
© Via Afrika ›› Mathematics Grade 12 6
, 1
Chapter
Number patterns, sequences and series
For a series given in sigma Remember: ∑𝑛𝑘=1 𝑇𝑘 has 𝑖 terms
notation: The “counter” indicates the number (counter 𝑘 runs from 1 to 𝑖)
• Determine the number of of terms in the series ∑𝑛𝑘=0 𝑇𝑘 has (𝑖 + 1) terms
terms (counter runs from 0 to 𝑖; so
one term extra)
∑𝑛𝑘=5 𝑇𝑘 has (𝑖 − 4) terms
( four terms not counted )
• Determine the value of Remember the expression next to See Text Book:
the series, in other words, the ∑-sign is the general term, 𝑇𝑛 . Example 1, p.19
𝑆𝑛 . This will help you to determine 𝑎
and 𝑑 or 𝑟.
Determine the general term, 𝑇𝑘 and Example 1, p.19
Write a given series in sigma number of terms, 𝑖 and substitute
notation. into ∑𝑛
𝑘=1 𝑇𝑘
Determine the sum, 𝑺𝒔 , of an In some cases you have to first See Text Book:
AS and a GS (when the determine the number of terms, 𝑖
number of terms are given or using 𝑇𝑛 . Example 2, nr.1 & 2, p.20
not given) Substitute the values of 𝑎, 𝑖 and 𝑑/𝑟 Example 3, nr. 1, p.24
into the formula for 𝑆𝑛
Determine whether a GS is Converging if −1 < 𝑟 < 1
converging or not
Determine 𝑆∞ for a Substitute vales of 𝑎 and 𝑟 See Text Book:
converging GS Into formula for 𝑆∞ Example 1, nr. 1, p.29
Determine the value of a Determine 𝑟 in terms of 𝑥 See Text Book:
variable (𝑥) for which a series and use −1 < 𝑟 < 1 Example 1, nr. 3, p.29
will converge,
e.g. (2𝑥 + 1) + (2𝑥 + 1)2 +…
Apply your knowledge of Generate a sequence of terms from See Text Book:
sequences and series on an the information given. Identify the Exercise 5, nr. 6, p.30
applied example (often type of sequence.
involving diagram/s)
© Via Afrika ›› Mathematics Grade 12 7
, 1
Chapter
Number patterns, sequences and series
Mixed Exercise on sequences and series
1 Consider the following sequence: 5; 9; 13; 17; 21; …
a Determine the general term.
b Which term is equal to 217?
2 a T5 of a geometric sequence is 9 and T9 is 729. Determine the constant ratio.
b Determine T10.
3 The following is an arithmetic sequence: 2 x − 4 ; 5 x ; 7 x − 4
a Determine the value of x .
b Determine the first 3 terms.
4 Consider the following sequence: 2 ; 7 ; 15 ; 26 ; 40 ; …
a Determine the general term.
b Which term is equal to 260?
5 How many terms are there in the following sequence?
17 ; 14 ; 11 ; 8 ; … ; -2785
6 Tom links balls with rods in arrangements as shown below:
Arrangement 1 Arrangement 2 Arrangement 3 Arrangement 4
1 ball, 4 rods 4 balls, 12 rods 9 balls, 24 rods 16 balls 40 rods
a Determine the number of balls in the nth arrangement.
b Determine the number of rods in the nth arrangement.
7 Determine the following:
30 10
1
a ∑k =1
(8 − 5k) b ∑
k =2
4
(2)𝑘−1
8 Write the following in sigma notation: 1+5+9+…+21
9 The 5th term of an arithmetic sequence is zero and the 13th term is equal to 12.
Determine:
a the constant difference and the first term.
b the sum of the first 21 terms.
© Via Afrika ›› Mathematics Grade 12 8
, 1Chapter
Number patterns, sequences and series
10 The first two terms of a geometric sequence are: (𝑥 + 3) and (𝑥 2 − 9)
a For which value of 𝑥 is this a converging sequence?
b Calculate the value of 𝑥 if the sum of the series to infinity is 13.
99+97+95+⋯+1
11 Calculate the value of:
299+297+295+⋯+201
12 𝑆𝑛 = 3𝑖2 − 2𝑖. Determine 𝑇9 .
13 The first four terms of a geometric sequence are 7; 𝑥 ; 𝑦 ; 189.
a Determine the values of 𝑥 and 𝑦.
b If the constant ratio is 3, make use of a suitable formula to determine the number of
terms in the sequence that will give a sum of 206 668.
© Via Afrika ›› Mathematics Grade 12 9
