MATHEMATICS 2018 PAPER 1

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MATHEMATICS 2018 PAPER 1 :

QUESTION/VRAAG 1

1.1.1 0342 xx

0)1)(3( xx

3x or 1x

factors

3x

1x

(3)

1.1.2 0155 2 xx

a

acbbx

2

42

)5(2

)1)(5(4255

10

55

x = 0,72 or x = 0,28

substitution into the correct

formula

x = 0,72

x = 0,28

(3)

0102030405060708090

10095

89

5864

82

22

67

ALGEBRA

Question 1

Question was well answered.

Learners who did not achieve

full marks factorised

incorrectly

Incorrect substitution into the quadratic formula

Incorrect formula

a

acbb

a

acbbx

2

4or

2

4 22

R Learners rounded off incorrectly


1.1.3

01032 xx

0)2)(5( xx

5or 2 xx

factors

5or 2 xx

(3)

1.1.4 43 xx 1689 2 xxx 016172 xx 0)1)(16( xx

1or16 xx NA

squaring both sides

standard form

factors

answer with

selection

(4)

1.2 )1......(192 2 xy

)2( ...... 23 yx

)3.......( 23 xy

19)23(2 2 xx

1946 2 xx

0369 2 xx

0123 2 xx 0)1)(13( xx

1 or 3

1 xx

5or1 yy

23 xy

substitution

standard form

factors

both x values

both y values

5 –2

– 2 5 – + +

learners battled to answer this question because they

lacked the knowledge of inequalities combined with

equal roots.

44 x

4 and4 xx

4 4 xx

Solution only given as

Solution only given as

Learners battled to answer this question because they

lacked the knowledge of inequalities The learners

struggled to write down the answer to the inequality

correctly.

THIS MUST BE A FOCUS OF TEACHING IN

GRADE 11 AND 12.

Incorrect squaring of the right hand side of the equation.

Encourage learners to write the right hand side as 4 4x x

rather than 2

4x

Learners did not test answers and therefore lost the mark for

discarding –1

Generally, learners have a good idea of the process required

to solve a set of simultaneous equations but lack the

accuracy of the algebra to complete the question properly.

Encourage learners to be more careful and accurate.


(6)


1.3

43

)4(3

643

3

333

9

x

x

x

85

645

645

p

p

p

p

x

p

x

5

3

5

3 3331

OR/OF

2

33

5

3.3p

x

p

x

527

33

64

3.64 33

827

4

54

1

433 x

85 p

p

x

527

33

answer

(4)

[23]

If the calculator is used to find factors, learners must still show the required

working out. Learners loose unnecessary marks by not showing steps.

There was a penalty for not taking the answer to the correct 2 decimal places.

Rounding off should be clearly understood by learners, and emphasised in

instructions in class assessments.

Correct formula need to be used. The numerator and denominator in the

formula must be separated correctly.

Basic mathematics must be re-enforced

Teach learners the rules about surd equations

Coefficient with means BOTH need to be squared

check the validity of all answers

Teach inequalities properly. Integrate the algebra with functions so that the

learners have a visual understanding of inequalities.

What does < and > mean…….

This was a problem solving question which was very poorly

done by all the candidates.

Teachers must try and expose their top candidates to as

many of these types of questions from all different aspects of

the curriculum.


QUESTION/VRAAG 2

2.1.1 42 answer

(1)

2.1.2 2a = 6 3a + b = 1 a + b + c = 2

a = 3 3(3) + b = 1 (3) + (–8) + c = 2

b = –8 c = 7

783 2 nnTn

a = 3

b = –8

c = 7

783 2 nnTn

(4)

2.1.3 7)20(8)20(3 2

20 T

= 1047

substitution

answer

(2)

2.2 427 nTn

140427 n

1827 n

26n

427 nTn

140427 n

26n

(3)

2.3 la

nSn

2 OR/OF dna

nSn 12

2

427352

nn

Sn 77702

nn

Sn

7772

nn

Sn

nnSn2

77

2

7 2

427352

nn

Sn or

77702

nn

Sn

nnSn2

77

2

7 2

equating

0102030405060708090

9081 82 81

5669

Linear and Quadratic Sequences

Question 2

Question 2.1 and 2.2 was generally well done.

Learners still confuse terms and Tn. Keep on

practising the correct terminology and

processes with the learners.

Learners struggled to understand what was

necessary in question 2.3. Please practise

more integrated questions with the learners

throughout the year.


7832

77

2

7 22 nnnn

0149313 2 nn

0)213)(7( nn

13

27 norn

NA

7n

standard form

factors

answer with

selection

(6)

[16]

Emphasise that n cannot be negative or a decimal

To find nT of a quadratic pattern refrain from teaching three simultaneous

equations. Learners make too many mistakes.

Use of English in Maths is of utmost importance

Understanding of the concepts is more important than merely doing routine

procedures

QUESTION/VRAAG 3

3.1

2

1r and 6S

r

aS

1

2

11

6

a

substitution

0102030405060708090

7884

52

18

49

Question 3


a = 3

answer

(2)

3.2 1 n

n arT

7

82

13

T

128

38 T

substitution

answer

(2)

3.3

8125,523

1

1

kn

k

8125,5...4

3

2

33

r

raS

n

n

1

)1(= 5,8125

2

11

2

113

n

= 5,8125

n

2

116 =5,8125

32

1

2

1

n

522 n n = 5

expansion

substitution

simplification

answer

(4)

3.4 p

k

k

20

1

1)2(3

p 192.3........4

3

2

33

20

1

)2(24k

k

= 202.24.....3612

=

192.3.....

4

3

2

334

= 4p

expansion

expansion

answer (3)

[11]

Learners struggle with sigma notation and working out how

to get r and a. They must be drilled in working with

exponents and encourage learners to write out the first three

terms of the series before interpreting the question.

Majority of the learners did not even attempt question 3.4


QUESTION/VRAAG 4

0

10

20

30

40

50

60

70

80 67

80 77

46

59

Inverse Functions

Question 4

4.1 Yes

The vertical line cuts the graph once/ One to one mapping

OR/OF For every x-value there is only on possible y value

answer

reason

(2)

4.2 R(–12 ; –6) answer

(1)

4.3 )12;6(substitute)( 2 axxf

2)6(12 a

3

1a

substitution

answer

(2)

4.4 2

3

1: xyf

21

3

1: yxf

xy 32

xy 3

Only xy 3 and 0x

changing x and y

xy 32

xy 3 and

0x (3)

[8]

For the most part this question was done well by the

learners except for 4.4

Learners did not take into consideration the negative root

was the only solution.

Encourage learners to write down the domain for inverse

functions as a matter of course.



QUESTION/VRAAG 5

0

10

20

30

40

50

60

70

42

70 68

17

50

Hyperbola

Question 5

5.1 Domain: 1; xRx answer

(1)

5.2

0

1

y

x

x = 1

y = 0

(2)

5.3

y intercept

shape

asymptote on positive

x-axis

(3)

5.4 1;0 xx

OR/OF

0 < x < 1 or 1 < x

OR/OF

x > 0

x 1

(2)

x

y

1

1 O

The basic knowledge of functions is still poorly answered.

Learners must be able to answer domain and range

questions. They ARE going to be asked. Be strict with

how you mark them during the year.

Learners must write down the equations of asymptotes

correctly and NOT use p = … and q = … Try to teach

conceptually so that learners can apply their knowledge

to ANY function question.

Interpretation in functions remains a very challenging

question for most learners as they do not know how to

translate what they are required to do to the picture.

Keep practising and explaining all different types of

examples throughout the Grade 10, 11 and 12 years.


0 ; 1 1 ;x

[8]


QUESTION/VRAAG 6

6.1

04

15

m

1m

1c

1)( xxg

substitution

c = 1

(2)

6.2 0322 xx

0)3)(1( xx

3or1 xx

A(–1 ; 0) B(3 ; 0)

factors

coordinates A

coordinates B (3)

6.3

2

31x

1x 32)( 2 xxxf

31212

y 4y 4y or );4[

x -value

substitution

answer

(3)

6.4.1 MN: )1()32( 2 xxxy

432 xx

436 2 xx

1030 2 xx

equation of MN

substituting y = 6

0102030405060708090

58

82

53

45

68

3826

46

Question 6

Learners struggled to answer this question.

Teach learners that “Show” means that they

must just work the expression out using

normal methods. The examiners have just

given them the answer.

Learners struggled to answer this question.

They knew to calculate the turning point but

then didn’t go further to answer the

question.

This was one of the worst answered questions

in the paper. Learners used measurement,

assumption and many other techniques

which got them to a correct answer via

incorrect methods.

Ensure learners do not assume and they must

use the given information to answer a

question.


)2)(5(0 xx

x = 5 or x = –2

OT = 2 or OT = 5

na

values of x

OT = 2

(4)

6.4.2 2substitute1 xxy

= (–2) + 1

= –1

N(–2 ; –1)

substituting x = –2

answer (2)

6.5 22)( xxf

122 x

2

3x

4

15

2

3

f

2

31

4

15xy

4

21 xy

22)( xxf

2

3x

4

15

2

3

f

substitution

answer

(5)

6.6

4

21k

answer

(1)

[20]

Emphasise both drawing and interpretation of functions / graphs.

Teach learners to analyse the information given

Basic graphs need a lot of attention especially in junior grades. The identification of the

TYPE of graph using the basic equation of the graph must be done on an ongoing basis.

Inequalities related to graphs need to be practised.

Learners to have a clear understanding of questions such as

0.;5;;0 xgxfxgxgxfxf

Spend enough time on graphs in grade 11 to ensure that basic concepts are understood.

Do NOT use the calculator as the main teaching tool to sketch graphs.

Emphasise the properties of functions such as:

Intercepts with the axes

Axes of symmetry

Asymptotes

Turning points

Concavity

The different notation related to domain and rage or inequalities need to be understood

by learners. What does > ; < mean. Interval notation such as ;;;

Transformations and functional notation must be understood by learners.

It is important for teachers to expose learners to high order thinking questions

throughout the year.

This was one of the worst answered questions

in the paper. Learners did not read that the

line was parallel to g(x) = x + 1 and just

equated the derivative to 0 not 1.

Interpretation of what was required was

problematic in both 6.5 and 6.6.

Inter


It is important for teachers to expose learners to questions that integrate topics.

Integration of topics must also be done in the teaching of the relevant sections.


QUESTION/VRAAG 7

0

20

40

60

80 49

20

77

49 47

Financial Mathematics

Question 7

7.1.1

i

ixF

n ]1)1[(

4

088.0

14

088,0100015

16

F

F = R283 972,28

4

088,0 and n = 16

substitution into

correct formula

answer

(3)

7.1.2

A = R283 972,28

4

4

088,01000100

= R 174 877,60

R283 972,28

4

4

088,01000100

answer

(3)

7.2.1 i

ixP

n

11

12

105,0

12

105,011

0005001

2012

x

70,97514Rx

12

105,0i

240n

substitution into

correct formula

answer

(4)

Learners really didn’t understand what was

being asked of them at all in this question.

Learners did not do well in this question as

they misinterpreted the wording “In three

months” for the first payment, so calculated

n incorrectly.

Learners also struggled with the

compounding quarterly concept.

This question was well done in the context of

the paper.


Ensure that you provide different real-life type examples for learners to work out what

will be withdrawal and what will be deposit and then how to work with the interest on

these amounts.

Ensure that you work with different compoundings in finance for ALL different types of

questions.

Emphasise the difference between i and r

Spend time teaching the learners to interrogate what they are doing all the time. Finance

should be taught with more insight. It is not merely the substitution of values into a

formula.

Answers need to be realistic. Learners have little concept of the validity of answers

Teach the correct use of calculators in order to prevent inaccurate final answers.

It should be standard practise in the classroom for learners to round correctly to two

decimal places in Financial Mathematics

Guard against early rounding

Basic algebraic rules such as multiplication and exponential laws should be taught

properly in earlier grades and revised continuously in grades 11 and 12.

Teachers must ensure that they are revising work from Grade 11 throughout the Grade

12 year with their learners.

Teachers need to reinforce why and when to use the future or present value formula.

Learners need to do MANY examples in real – life situations.

As language is a problem when answering finance questions due to the interpretation,

teachers should use the correct language both in class when teaching learners and in the

assessment the learners are required to do.

7.2.2 i

ixP

n

11

12

105,0

12

105,01170,97514

812

P

927,74 R969P

R14 975,70

96n

12

105,0i

substitution into

correct formula

answer

(5)

[15]

Learners made errors with the interest rate

and n in this calculation.

Some learners do not have any idea of how to

calculate outstanding balance. When

teaching it, stick to one method so to not

confuse learners.


QUESTION/VRAAG 8

8.1

h

xfhxfxf

h

)()(lim)('

0

h

xhxhx

h

552lim

222

0

h

hxh

h

)2(lim

0

=0

limh

)2( hx

x2

52 22 hxhx

52 x

factorise

0

limh

hx 2

x2

(5)

8.2.1

463 23 xxxy

1129 2 xxdx

dy

29x

x12

1 (3)

8.2.2

)1(2)1( xxxy

1

)1(2

x

xxy 1 if x

xy 2

2dx

dy

factors

simplification

xy 2

answer

(4)

[12]

Always ensure that you use the correct notation at all times when working with

Calculus.

0

20

40

60

8073

80

39

62

Calculus Rules

Question 8

Question 8.1 and question 8.2 were well done

by the learners.

Learners lost marks due to notation.

Learners did not understand that they

needed to isolate y before differentiating.


Make sure that different types of algebraic manipulation are done before using the basic

differentiation rule. Teachers must ensure they do many different types of this

manipulation with the learners in class.

Do not omit do expose learners to

It seems as if learners handled this question better when they simplified or even the

fraction first and then applied the limit as h 0 to the simplified fraction.

BASIC algebraic manipulation in earlier grades MUST be taught properly and revised

on an ongoing basis.

The drill and practise of the rules of differentiation is necessary. The original function

MUST be in terms where one can correctly identify the coefficient, variable and

exponent BEFORE the rules of differentiation can be applied.

Fractions and exponential laws should be emphasised when working with Calculus.

Do drill and practise exercises for first principles.

When differentiating by rule, learners need to understand what the original expression

consist of. Identify the fractions, exponents and in the original expression. Make sure

that learners KNOW how to simplify before differentiation.

The language of Calculus must be taught and understood

Derivative xf / gradient of a tangent at a point of contact

1f is the y value related to x = 1 on the graph of f

1/f is the gradient of the tangent to the graph of f at x = 1


QUESTION/VRAAG 9

9.1.1 2

1))(5()( xxxxg 2

1)(520 x

42

1 x

21 x 2)2)(5()( xxxg

)44)(5()( 2 xxxxg

2016)( 23 xxxxg

substitution

21 x

)44)(5( 2 xxx

2016)( 23 xxxxg

(4)

9.1.2 2016)( 23 xxxxg

1623)( 2 xxxg

01623 2 xx

0)2)(83( xx

2or3

8

xx

)81,50;67,2(Ror27

1372;

3

8R

)0;2(P

derivative

equating to zero

factors

co-ordinates of R

co-ordinates of P

(5)

9.1.3 26 xxg

026 x

3

1x

up concave

026 x

3

1x

conclusion

(3)

0

20

40

60

80

23

69

51

22

36

Question 9

Learners did not understand what to do with

the given information in question 9.1.

They did not know how to work with the given

equal root and no other given root.

Learners knew what to do but made careless

errors and did leave out the equating to zero.

Question 8.1 and question 8.2 were well done

by the learners.

Learners lost marks due to notation.

Learners did not know what to do with the

second derivative.

Learners do not understand the concept of

concavity.


OR/OF

26)( xxg

2)0( g

upconcave

OR/OF

26)( xxg

2)0( g

conclusion (3)

9.2

point of inflection, x = 3

y – intercept

concave up for 3x and

concave down for 3x

(3)

[15]

The concept of concavity needs to be taught more thoroughly and with understanding

rather than just teaching the learners that when they see the word concavity make the

second derivative equal to zero.

Teachers need to better equip themselves by understanding concavity better themselves.

Learners need to be exposed in various ways (worksheets, tests etc) to ALL different

ways of calculating the expression for any function. All properties can be tested when

determining the equation of a function.

x

y

O

27

3

This question was very poorly done by the

learners. They had little to no understanding

of what was being asked and what to do to

answer the question.


QUESTION/VRAAG 10

10.1

2

3

HG

AH

answer (1)

10.2 Area of a parallelogram height base

Area tBC5

2.

5

3

= tBC.25

6

tBC 5

Area = tt525

6

tttA5

6

25

6 2

5

6

25

12 ttA /

05

6

25

12 t

t5

2

t525

6

tt525

6

05

6

25

12 t

01020304050

44

13 13

Optimisation

Question 10

A

B C

D E

F G

H

This question was very poorly done and

mostly left out by learners.

Learners did not expect geometry concepts in

Paper 1. This is not an unfair question but

rather and unexpected question.

Please ensure that you expose learners to

different concepts across the two papers.

Integration of concepts is part of what can be

asked in exam questions.


03012 t

2

5or

12

30t

answer

(5)

[6]

QUESTION/VRAAG 11

11.1.1 8071675 answer

(2)

11.1.2 34567

2520

!2

!7

34567 or !2

!7

answer

(2)

11.2 172 = 14 172

answer

(3)

[7]

Please ensure that probability and counting principles are taught explicitly.

While they form small sections in the paper, these are sections where the learners, if

taught can achieve marks.

Teachers must ensure that they have understood these concepts first before teaching

them to learners.

0

5

10

15

20

25

30

35

4029

36

16

23

Counting Principles

Question 11

Learners mixed up the 5 spaces and the 7

digits that can be used.

Learners either knew what they were doing

or did not have any idea of how to answer the

questions.


QUESTION/VRAAG 12

12.1 P(A or B) = P(A) + P(B)

0,74 = 0,45 + y

y = 0,29

P(A or B) = P(A) + P(B)

substitution

answer

(3)

12.2

Let the number of gifts = x

The total number of gifts = 4x

118

7

14

1

4

x

x

x

x

118

7

14

1

4

1

x

x

118

28

14

1

x

x

118x – 118 = 112x – 28

x = 15

There are 15 mystery gift bags

4x

x

x

4 or

4

1

14

1

x

x

equating to 118

7

simplification

answer

(6)

[9]

Terminology and the related theory need more emphasis

Revision on tree – and Venn diagrams needs to be done

020406080

72

11 27

Probability

Question 12

S

G

S

S

G

G

This question was well done by learners.

Learners who didn’t achieve, did not write

down the rule correctly or did not know that

mutually exclusive events have no

intersection.

This question was extremely poorly done by

learners.

Learners did not know how to set up a

probability tree without knowing the total

number of options available.


Proper understanding of probability is required

Training should be provided BUT it is also the responsibility of the teacher to get

comfortable with this section by working through a number of available textbooks and

past papers.

Documents

MATHEMATICS 2018 PAPER 1