# Mathematics Grade 12 P1 Free State Mock 2023 Paper & Memo

> Grade 12 Mathematics Paper 1 question paper and memo from the Free State Mock Exam 2023, 29 pages. Read online or get the free PDF download.

- Page: https://www.hlayiso.com/library/grade-12-mathematics-2023-free-state-mock-exam-question-paper-and-memorandum-paper-1/835364e6-4cb8-46e0-a1ac-7d6bf6c886d8
- Updated: 2026-09-28
- Subject: Mathematics
- Grade: Grade 12
- Document type: Question paper and memo
- Year: 2023
- Exam period: Free State Mock Exam
- Paper: 1
- Curriculum: CAPS
- Pages: 29

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Mathematics · Grade 12 · Free State Mock Exam · 2023. Question paper and memorandum, 29 pages. Read online or download the PDF.

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### Page 1

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education

Department of
Education

FREE STATE PROVINCE

Gr 12 MATHEMATICS P1

IAN

12611B

MARKS: 1530

This question paper consists of 9 pages and 1 information sheet.

X10

P2

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Grade 12 Prep. Exam.

INSTRUCTIONS AND INFORMATION

Read the following instructions carefully before answering the questions.

10.

tion paper consists of 10 questions.

sie] ALL the questions.

sr your answers correctly according to the numbering system used in this
question paper.

Clearly show ALL calculations, diagrams, graphs, et cetera that you have used
in determining your answers.

Answer only will NOT necessarily be awarded full marks.

An approved scientific calculator (non-programmable, non-graphic) may be used,
unless stated otherwise.

If necessary, round off answers to TWO decimal places, unless stated otherwise.
Diagrams are NOT necessarily drawn to scale.
An information sheet with formulae is included at the end of the question paper.

Write neatly and legibly.

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Grade 12 Prep. Exam.

QUESTION 1

1.1 Solve for x:

Al 1(2x+1)=0. (2)
1.12 1(2x+1)=4 (correct to two decimal places) (4)
11.3 x+v¥x-2=4 )
11.4 3x74+x<0 GB)

1.2 Solve for x and _y in the following simultaneous equations:
xy=8and 2x+y=17 (6)

1.3. Simplify the following WITHOUT USING A CALCULATOR :

atx? — V5x? xVV2ix? + V5x2 (3)

[23]

QUESTION 2

2.1 Consider the series: a+(a+d)+(a+2d)+...

2.1.1 Prove that the sum of the first n terms of this arithmetic series
will be S, = Fhe +(n-1)d]. (3)

2.1.2 Given: 2* +2,.2* +3.2* +... The sum of the first 20 terms of the series
is 1680. Calculate the value of x. (4)

nin

2.2 Given: S, =

, calculate 7, . (3)

2.3. Consider the series: 32 + (—16)+8+(—4)+...
Calculate the sum of the first 10 terms of the series. (3)
[13]

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Grade 12 Prep. Exam

QUESTION 3

3.1 Given the quadratic number pattern —4; —6; —10; —16; ...

3.1 etermine 7’. (4)
3. tween which consecutive terms of the pattern is the difference -100? (4)
3.2. Given that x - 3) = -3 . Calculate the value of k. (4)
n=l
[12]
QUESTION 4

Given the function: A(x) = 5 42.
x-

4.1 Write down the equations of the asymptotes of h . (2)
4.2 Calculate the x-intercept of h. (2)
43. Draw the graph of /. Clearly show all asymptotes and intercepts with the axes. (4)

4.4 Determine the equation of the axis of symmetry of #, in the form y= mx+c,

where m<0. (3)

4.5 Determine the range of — A(x) +1 (2)
4.6 Determine the values of x where h(x) < 0. (2)
[13]

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Grade 12 Prep. Exam.

QUESTION 5

In the diagram, the graphs of f(x) = -30 +2x+6 and g(x)=—x—-2 are drawn, C and E

are the y-inter of fand g, respectively. The parabola has a turning point at D and cuts the
x-axis at A an} is also the x-intercept of g. DF is a line parallel to the y-axis with F a
point on g.

D

|

PNE

5.1 Calculate the:

5.1.1 Coordinates of D. (3)
5.1.2 Distance DF. (3)
5.2 For which values of k will f(x) =k have two positive roots? (2)

5.3. Given that f(x) =A'(x). Determine the x-coordinates of the turning points of h. (3)

5.4. Determine the value(s) of x where {'(x)x g(x) <0. (2)
[13]

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Grade 12 Prep, Exam.

QUESTION 6

The diagram shows the graphs of f(x) = a" and g, the reflection of fin the y-axis.

af ;) li f The two graphs intersect at T.

Y

6.1 Write down the coordinates of T. (1)
6.2 Calculate the value of a. (2)
6.3. Determine the equation of g. (2)
6.4. Write down the equation of 7 (x); the inverse of f, in the form y= ... (2)
6.5 For which values of x will f'(x) <1? (3)

[10]

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Grade 12 Prep. Exam.

QUESTION 7

7A

How many years will it take to triple an investment if the interest is compounded

annuallyat a rate of 9,8% p.a.? (3)
7.2 Andi ds R64 000 for a holiday. He started to invest a fixed amount of
his si Let a rate of 8,5% p.a. compounded monthly, at the end of each month,
for tepryeers.
7.2.1 Calculate the monthly payment he will have to make to achieve this. (3)
7.2.2. Tf Andile has stopped his payment at the end of eight years, what will the
total of his investment be at the end of ten years? 3)
7.3. Madri took out a loan of R400 000 at an interest rate of 10,4% p.a. compounded
monthly. She repaid the loan at the end of the first month and every month for
15 years. Her monthly instalment is R4 396,83.
7.3.1 Calculate the outstanding balance after nine years. (3)
7.3.2 How much interest did she pay over the nine years? (3)
[15]
QUESTION 8
8.1 Determine the derivalive of f(%) =3-.° using FIRST PRINCIPLES. (5)
8.2 Determine:
2
8.2.1 D, E = vi (4)
x
dy . 3
82.2 =,if p=|x -l 3
@ if y=(e 1 @)
83 Given: f(x) = x* -12x-16

8.3.1 Calculate the:

(a) Coordinates of the turning points of f. : (5)

(b) _—_-x-intercepts of ff (3)
8.3.2 y=15x+ pis a tangent to the graph of f Calculate the x: inates of

the point(s) of contact. (4)
8.3.3. For which value(s) of x will the given function be concave up? i o

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Grade 12 Prep. Exam.

QUESTION 9

91 The diagram shows the straight-line 2, where h(x) = f'(x).

The x-intercept of h is 1.
The wing is true for function i f(1)=—3 and f(3)=0.

Y

Draw a sketch graph of the function f; clearly indicating all x-intercepts
and turning point(s). (3)

9.2 During an experiment the temperature, T in °C varies with time 7 in seconds,
to the equation T(t) = 60 + 27t — Pote [0:6].

Calculate:
9.2.1 The average change of temperature between 3 and 6 seconds. (3)
9.2.2 After how many seconds the temperature will be a maximum. (3)

[9]

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FS/September 2023
Grade 12 Prep. Exam.
QUESTION 10
10.1 Given: P(4) = 0,4 and P(B) =0,5
10.1.1 sulate P(A or B) if Aand B are mutually exclusive events. (2)
10.1. an ulate P(A or B) if Aand Bare independent events. (3)

10.2 A four-digit code must be set using the letters A, E, I, O, U and digits 0 to 9.
The letters may be repeated, but the digits may not be repeated.
The code must consist of two letters and two digits in that order, for example, UO19.
10.2.1 How many different codes are possible with the information given? (4)

10.2.2 What is the probability that a code that is picked randomly will start
with an A and be an even number? (4)

[13]

TOTAL: 150

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Grade 12 Prep. Exam.

INFORMATION SHEET
_ =btvb? —4ac
2a
A=P(1+ni) A=P(l-ni) A=P(l-i)" A=P(1+i)"
T, =a+(n~ 8, =F Pa+(n—d]
alr" -1
T, =ar"" _al" =) gril Sse -teret

fn " r-l 1—

pai -1 pasi-b+)"]

i i

™

i fth)-F()
"(x)= lim
fe) h>0 h
xX, +x, yy ty
d= ()-—m)+O,-V) Mj 1,-1.2
2 2
y=mxt+e y-y, =m(x-x,) m=22 71 m=tan@
%2—%y

(x-af +(y-bf =r?
inaspe. —* => = ©

sn A sinB sinC

a =b’ +c? —2be.cos A

area. AABC=-ab.sin C
sin(a + £) = sina.cos B + cosa.sin 8 sin(a - £)=sin @.cos 8 -cosa.sin B
cos(a + B)=cosa@.cos B -sina.sin B cos(a — f)= cos a@.cos f +sin a.sin 8

cos’ a-sin* @
cos2a@ =41-2sin’ @ sin 2@ = 2sina.cosa
2cos? a -1
_ vf >, - x)
x= o ==!
n n
n(A)

MA)="5) P(A or B) = P(A) + P(B) — P(A and B)
pratbhx p= =F)

DG@e-zy

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education

Department of
Education

FREE STATE PROVINCE

PREPARATORY EXAMINATION
VOORBEREIDENDE EKSAMEN

GRADE/GRAAD 12

MATHEMATICS P1
WISKUNDE V1

SEPTEMBER 2023

MARKS/PUNTE: 150

MARKING GUIDELINES
NASTENRIGLYNE

These marking guidelines consist of 17 pages.
Hierdie nasienriglyne bestaan uit 17 bladsye.

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Grade/Graad 12 Prep. Exam./Voorb. Eksam. | Marking Guidelines/Nasienriglyne

NOTE:
e Constant accuracy applies in the whole marking guideline.
e Ifa learner answers a question twice, mark only the first attempt.

e Ifa leame cancels a question, but does not redo it, mark that attempt.
al

iy
4000
NOTA: onay
e = Volgeh kuraatheid word in ALLE aspekte van die nasienriglyne toegepas.

e = As 'n kandidaat 'n vraag TWEE KEER beantwoord, sien slegs die EERSTE poging na.
e As 'n kandidaat 'n antwoord van 'n vraag doodtrek en nie oordoen nie, sien die
doodgetrekte poging na.

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Grade/Graad 12 Prep. Exam./Voorb. Eksam. | Marking Guidelines/Nasienriglyne

QUESTION/VRAAG 1
al
1.1.1 (x —1)(2x+1) =0
x=1 or/of x = —+ ¥x=1
2
Vxa-i
(2)
1.1.2 (x-I(2x+1)=4
2x? ~y-5-0 Y standard form
-C1)+¥C1) - 42-5) Y correct substitution in
2(2) correct formula
_1+V41 Y 1,85
4 Y -1,35
Penalise/mark -1 for
x=185 — or/of x=—1,35 incorrect rounding
(4)
1.1.3
x+vx-2=4 Vi
Vx-2=4- isolate the surd
* 2 * v squaring both sides
(vx=2) =(4-x)?
x—2=16—8x +x? Y standard form
0=x?-9x +18
0 = (x - 6)(x - 3) Y factors
Y answer with choice
x=6 or/of x=3
Not applicable/Nie van toepassing nie
(5)
11.4 3x7 +x<0
x(3x+1)<0 V factors

Critical values: 0 and/en -:

xe [-+;0| Or -F<x<0

Y critical values

Y answer

(3)

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Grade/Graad 12 Prep. Exam./Voorb. Eksam. | Marking Guidelines/Nasienriglyne
1.2 2x+y=17
oy =17-2x VY y=17-2x
Substitute in/verv. in xy =8 y .
iy x(I 7- 2x) =8 substitution
m4 0 =2x? -17x+8 Y standard form
foamy 0 = (2x-1)(x-8)
i 1 ¥ factors
Ox= 5 or x=8
\ Y both answers for x
»=17-2) — -17- Y both answers for y
y=l7 (+) ovof » 17-2(8)
y=16 7
OR/OF OR
xe 17-y
2
a in/erv. in xy =8 17-y
-y x=
——~ ly=8 2
( 2 p
ITp—y? =16 ¥ substitution
O=y? -17y +16 Y standard form
0=(y-16Xy-1) y
“y=l6 oof y=l factors
,-lc6 17-1 ¥ both answers for x
2 x=
or/of 2
xe 1 x=8 ¥ both answers for y
2
(6)
v
1.3 Vv2ix _ 5x2 x vax asx? one square root
= Wai — 5x? |V2tx? + V5x?) V
; difference between
= 21x" —5x° squares
=V16x" =
=4x v nse
@) Lay
Et
[23]

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Grade/Graad 12 Prep. Exam./Voorb. Eksam. | Marking Guidelines/Nasienriglyne

QUESTION/VRAAG 2

=a+(a+d)+(a4+2d)+...4a+(n—1)d

Ss
8, =a+(n-lId+a+(n-2)d+...+4

Y T, =a+(n-1)d

v reverse
=n(2a+(n-1
, = n2a+ (=I) vr
(3)
2.1.2 2° 4+2.2* +3.2* +...
“a=2* value a] v
dad! value d
20 (4/5 x x Y substitution in correct
1 =—(2|2 19)\2
680 = = (2(2")+ (19)2")) foeke
168 = 21.2*
g-2* ¥ Simplification (8 = 2”)
x=3 v answer
(4)
22 S, = mn
v
T, =5,-S, correct method
3 , ¥ substitution
_8 +8 7 +7
4 4 v answer
=18-14
a (3)
a(l-r")
23 " To? 7 ba : ;
1 values of a and r
32(1-(-3)
Sio Y substitution into correct

S, _341 21,31
10 = 46 or/of 21,

formula

(3)

[13]

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Grade/Graad 12 Prep. Exam./Voorb. Eksam. | Marking Guidelines/Nasienriglyne
QUESTIONVRAAG 3
3.1 -4;-6;-10;-16;... given/gegee
3.1.1 First difference/eerste verskil : -2 5-4; -6
Second differences/tweede verskil -2 ; -2 ¥ value of a
2a=-2 3(-1)+b=-2 Y¥ value of b
a=-l b=1
v value of c
-l+l+c=—+4
cu--44 Y T,=-n’ +n-4

(4)
ANSWER ONLY FULL
MARKS

3.1.2 T, =-2+(n-1X-2) ¥d=2
T, =-2n VT, =-2n
-100 =-2n v 50
v answer
“n=S0
Between/tussen
Ts) and/en T:,
OR
OR/OF
YT, ~T, =~100
Ta —T, =—100 Y substitution
=(n+1) +(n+1)-4-[n? +n-4]=-100 | “50
5 v
=n? —2n-ltn4l—44n?-n+4=-100 |” WT
—2n = -100
n=50
Between/tussen
Ts, and/en T;,
(4)

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Grade/Graad 12 Prep. Exam./Voorb. Eksam. | Marking Guidelines/Nasienriglyne

3.2
a=k->
2
3 v value of a and r
r=k-—
2
§ =
l-r
Ls k- 3 Y substitution in correct
-2=—__2_ formula
2
5 5, 5 3 Y — simplification shown
-St+ik- =k-=
3 3 2 2
-10+10k-15=6k-9
Y answer
4k =16
k=4
(4)
[12]

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Grade/Graad 12 Prep. Exam./Voorb. Eksam. | Marking Guidelines/Nasienriglyne
QUESTION/VRAAG 4
. 2
Given/gegee h(x)=—- +2
fo -
41 2 Vv x=2
2 v y=2
(2)
4.2 0=- +2
x-2 Vv y=0
—2(x-2)=-2
Vx=
x-2=1 x=3
x=3
(2)
43
v BOTH
asymptotes
stbcotiet tea Ean cPade ——! _ v x- intercept
¥ y- intercept
v shape
(4)
4.4
Y gradient -1
y-2=-l(x-2) ¥ substitute (2; 2)
ys-x+4 ¥ equation

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sics.com

Marking Guidelines/Nasienriglyne

FS/VS/September 2023

4.5 —hA(x)+1
2
= 722 Y new equation
-] v answer
2
eR;sy#-1 (2)
ANSWER ONLY
FULL MARKS
46
h(x) <0 Y critical values
“x € (2;3] Y notation
OR/OF 2<x<3
(2)
[15]

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Grade/Graad 12 Prep. Exam./Voorb. Eksam. | Marking Guidelines/Nasienriglyne

FS/VS/September 2023

QUESTION/VRAAG 5

Given/gegee f(x) = -ix +2x+6

g(x) =-x-2
f'(x) =0
5.1.1 y= 2 Q=-x+2 ¥ method for turning point
_ 1
{- } x=2 Vx
2 OR/OF ¥ x -value
| y- value
y= 5h -4x)+6
=2
* y= 1 (e248 (3)
. D(2:8) 2
D(2;8)
5.1.2 D(2;8) Vv coordinates F
v F(2;-4) v answer
DF = 12 units/eenhede GB)
5.2 6<k<8 vv answer
(2)
53 h'(x) = f(x)
1
0 =-—x° +2x+6 VY f(x)=h'(x) =0
O=x* —4x-12
0 =(x—6)(x +2) v Factors
A(-2:0) Y both x values
B(6;0)
“x =-2 . . -
for turning points of/vir
v= (3)
draaipunte van h
3.4 S'(x)* g(a) $0 Y criticalvalues
2.x €[-2;2] Y notation
(2)
[13]

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Grade/Graad 12 Prep. Exam./Voorb. Eksam. | Marking Guidelines/Nasienriglyne
QUESTIONVRAAG 6
6.1 | T(O; 1) v answer
fair Q)
had »y -@ Y substitute B
6.2 pe 8
| 3 v answer
ast
2
(2)
2 3)" v base
6.3 &(x) = (5) OR/OF g(x) = (3) Y exponent
(2)
3y Y swop x and y
a. _(3
6.4 Joix= (5) Y answer
“y=log,x
: (2)
6.5 log, x=1
2 3
vv x= >
yu} _
. 5 ;
lx) <1 critical values
£® 3 Y notation
xe(0;—] GB)
[10]

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Grade/Graad 12 Prep. Exam./Voorb. Eksam. | Marking Guidelines/Nasienriglyne

QUESTION/VRAAG 7
7A
A= P(l+i)"
Bx = x(1+ 0,098)" Y correct substitution in
3 = 1,098" correct formula
=| 3
no OB 108 Y use of logs (independent
n=11,751

It will take 12 years to triple the money/Dit
sal 12 jaar neem om die geld te

mark)
¥ 12 also accept 11,75

verdriedubbel. (3)
72
i)’ j= 22085
42.1 rp -Ml+)) -1 via
i n=120  bothiandn
0,085 \'°
y( + D | - Y substitution in correct
64000 = formula
0,085 Y answer
12
x = 34018
3)
96 ¥9%=n
8:
sous (1- 0,0 *) ] o4 | ¥ compound interest 24
722 F= 12 + 0,085 months
. 0,085 12 v answer
12
= 55135,69
GB)
73
0,104\~7
4 1-(1+-——-
P= 396,83] ( +? ) ¥2=nandi ==
73.1 , 0,104
= R234770,75 12 ¥ substittition into correct

formula

v answer;

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Grade/Graad 12 Prep. Exam./Voorb. Eksam.

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Marking

FS/VS/September 2023

uidelines/Nasienriglyne

Jan}
Jonni

—

ALTERNATIVE methodALTERNATIEWE
metode

74 108
4000001 + al

108
i 19693 (16) -|
2

7 0,104

12
= R234770,77

¥ loan with n = 108

Y F, with n= 108

v answer

(G3)

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7.3.2 During 9 years, she paid R474 857,64. Y R474 857,64
Her payment on the loan however is only Y R165 229,25 or
R165 229,25. R165 229,23
She paid interest of R309 628,39 over the 9 | Vv R309 628,39 or
years./Gedurende 9 jaar het sy R474 857,64 R306 628,41
betaal. Haar betaling op die lening is egter
slegs R165 229,25. Sy het rente van R309 (3)
628,39 oor die 9 jaar betaal
[15]
NO]

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Grade/Graad 12 Prep. Exam./Voorb. Eksam. | Marking Guidelines/Nasienriglyne

QUESTIONWVRAAG 8 Penalise 1 mark for incorrect notation in question 8
only/Penaliseer slegs 1 punt vir verkeerde notasie in vraag 8.
8.1 f(x) =3-x° ;
Y =3-x°-2xh-h
f(x+h)=3-(x+A) wes
=3-—x*? —2xh-h?
He) = Tey LEFF)
F(x) = fim
) v substitution in correct
. 3-x° -2xh-? -(3-x7 formula
= fim, 7
. 2
= lim 224 Y simplify
=) h
= Tim A(- 2x- h) ¥ factors
=) h
= lim (-2x-h) Y answer
=-2x (5)
8.2
Vv -1
8.2.1 oJ? -vs|
x Ve
i
= Ty?
= o,|2s x | V 2x2
- 1G
2x -=x yi x2
2
(4)
8.2.2
=(x3-1
* (° y Y x6 2x +1
yp=x’-2x' +1 Y 6x5
oD 6x5 — 6x? ¥ — 6x?
dx (3)

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### Page 27

Dawnloadednérom Stanmorephysics.com F9/VS/September 2023
Grade/Graad 12 Prep. Exam./Voorb. Eksam. | Marking Guidelines/Nasienriglyne
8.3 f(x) =x° -12x-16
8.3.1 (a) f(x) = 3x? -12 Y 3x7 -12
v=0
0 =3(v-2\x+2
(x—2\v+2) Y¥ factors
x=2
x=-2 ¥ (2; -32)
2. (2-32) ¥(-2; 0)
(-2;0) (5)
8.3.1(b) x -12x-16=0 v y=0
Y factors
2 2\x-4)=
(« * Ke * Ke ) ° Y BOTH answers
x=-2
x=4
(3)
8.3.2
15=3x? -12 Y derivative = 15
> Y standard form
0=3x° -27 V x=3
x =9 Y x=-3
x=33
(4)
8.3.3 S'"(x) = 6x
0=6x ¥ 6x=0
Con 0 ‘honk ; , Y values
oncave up/konkaaf op: x €(0;0) V notation
or writlen/ofgeskryfas x > 0
(3)
[27]

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### Page 28

Dewnleadednfnom Stanmoreplysics.com FS/VS/September 2023
Grade/Graad 12 Prep. Exam./Voorb. Eksam. | Marking Guidelines/Nasienriglyne

QUESTION/WVRAAG 9
9.1
lag
A v turning point
loi ,
im v shape
Y x- intercepts
0 1
(i 5)
(1:3)
(3)
9.2 T(t) =60+27t-t*
_ T(6)-T@) Y correct formula
9.2.1 Average change/ | Y substitution
6-114 v answer
Gemid. verandering = 3
=-36
(3)
9.2.2
0=27-30° Y 27-307
By? v=0
Stank Y answer
es
(3)
19] =

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### Page 29

Hewnleadednférom Stanmore

physics.com

FS/VS/September 2023

Grade/Graad 12 Prep. Exam./Voorb. Eksam. | Marking Guidelines/Nasienriglyne

QUESTIONVRAAG 10
10.1
P(A) = 0,4
P(B) = 0,5
10.1.1 P(A or/of B) = P(A) + P(B) Vv P(AandB)=0
— P(A and/en B) Y answer
= 0,4+0,5-0
=0,9 (2)
10.1.2 P(A or/of B) = P(A) + P(B) Y tule
— P(A and/en B) V P(A)x P(B) = P(AandB)
=0,4+ 0,5 -(0,4 x 0,5) v answer
=0.7
(3)
10.2
10.2.1 5x5x10x9 v5
= 2250 v5
v 10
v9
(4)
10.2.2 1x5x9x5 Y denominator 2250
2250 v 1x5
l ¥ 9x5
“10 v answer
=0,1
(4)
[13]

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