PLUMSTEAD HIGH SCHOOL
GRADE 12
NATIONAL
SENIOR CERTIFICATE
MATHEMATICS PAPER 1
JUNE 2023
MARKS: 150
Examiner: Mr. R. Mukondwa
TIME: 3 hours
Moderator: Mr. C. Philander
Name of Candidate
Name of Class Teacher
**FOR OFFICIAL USE ONLY**
Questio
n
MARKER
Marks
Marker’s
Initials
MODERATOR
Marks
Moderator’s Initials
1
2
3
4
5
6
7
8
9
10
11
TOTAL
This question paper consists of 10 printed pages and one information sheet.
2
INSTRUCTIONS AND INFORMATION
Read the following instructions carefully before answering the questions.
1.
This question paper consists of 11 questions.
2.
Answer ALL the questions.
3.
Number the answers correctly according to the numbering system used in this
question paper.
4.
Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in
determining your answers.
5.
Answers only will not necessarily be awarded full marks.
6.
You may use an approved scientific calculator (non-programmable and nongraphical), unless stated otherwise.
7.
If necessary, round off answers to TWO decimal places, unless stated otherwise.
8.
Diagrams are NOT necessarily drawn to scale.
9.
An information sheet with formulae is included at the end of the question paper.
10.
Write neatly and legibly.
3
QUESTION 1
1.1
1.2
Solve for 𝑥;
1.1.1
(2𝑥 − 5)(𝑥 − 2) = 0
(2)
1.1.2
3𝑥 2 + 8𝑥 = −2 (correct to 2 decimal places)
(4)
1.1.3
2𝑥 = √2𝑥 + 3 + 9
(4)
1.1.4
3𝑥 2 − 2𝑥 ≤ 0
(4)
Solve for x and y simultaneously
𝑥 = 2𝑦 + 3 and 𝑥 2 − 𝑥𝑦 − 𝑦 2 = 1
1.3
1.4
Without solving the equation; discuss the nature of the roots of the equation
2𝑥 2 + 6 = −7𝑥
(6)
(3)
Calculate the exact value of
√22023
√22021 − √22019
(4)
[27]
QUESTION 2
2.1
2.2
Given the quadratic sequence: 2; 3; 6; 11; …
2.1.1
Determine the nth term of the sequence
(4)
2.1.2
Determine the 20th term.
(2)
2.1.3
Which term of the sequence equals 578?
(4)
The third term of an arithmetic series is 4 and the tenth term is -17.
2.2.1
2.2.2
Determine the first term and the common difference of this
sequence.
Calculate the sum of the first 25 terms of the above mentioned
sequence.
(4)
(2)
[16]
4
QUESTION 3
3.1
Evaluate:
∞
∑ 36−𝑖
𝑖=5
3.2
Joanne saves R1 on the first day of the month. Each day after that she saves
twice as much as the day before. How many days would it take her to save
R1 000 000?
(3)
(4)
[7]
QUESTION 4
5
Given: ℎ(𝑥) = 𝑥−4 + 3
4.1
Write down the equations of the asymptotes of h.
(2)
4.2
Determine the coordinates of the x-intercept of h.
(3)
4.3
Sketch the graph of h, clearly showing the asymptotes and the intercepts with
the axes.
(3)
4.4
Determine the equation of the axis of symmetry that has a positive gradient.
(2)
4.5
Write down the domain of g given that 𝑔(𝑥) = ℎ(𝑥 + 3)
(2)
[12]
5
QUESTION 5
In the diagram below, A and B are the x-intercepts of the graph of
𝑓(𝑥) = 𝑥 2 + 2𝑥 − 3.
A straight line, g, through A cuts f at C(-4; 5) and the y-axis at (0; 1).
M is a point on f and N is a point on g such that MN is parallel to the y-axis.
MN cuts the x-axis at T.
5.1
Show that 𝑔(𝑥) = −𝑥 + 1
(2)
5.2
Calculate the coordinates of A and B
(3)
5.3
Determine the range of f.
(3)
5.4
Given that MN = 6:
5.4.1
5.4.2
5.5
Determine the length of OT if T lies on the positive x-axis. Show
ALL your working.
(4)
Hence write down the coordinates of N.
(2)
Determine the values of x for which 𝑓(𝑥) ≤ 𝑔(𝑥)
(2)
[16]
6
QUESTION 6
Sketched below is the graph of 𝑓(𝑥) = 𝑎−𝑥 , where 𝑎 > 0 passing through the point
(-1; 3).
Use the sketch and the given information to answer the following questions.
6.1
Determine the value of a.
(2)
6.2
Write down the equation of 𝑓 −1 in the form 𝑦 = ⋯
(2)
6.3
Draw a neat sketch graph of the inverse of 𝑓, showing the co-ordinates of
any intercepts with the axes, as well as the co-ordinates of the point on the
curve where 𝑥 = 3.
(3)
6.4
Determine the value(s) of x for which 𝑓 −1 (𝑥) > 1
(2)
[9]
7
QUESTION 7
7.1
7.2
7.3
Suppose Thomas borrows R2 000 with the agreement that he will pay back
R3 000 exactly 2 years from now. Calculate the interest rate charged per
annum, if interest is compounded monthly.
(3)
The price of an article depreciates by a fixed rate, 14,5% p.a., according to
the method of reducing balance. The original price was R180 500, and after
n years the price is R 90 500. Determine the value of n.
(3)
Mary is an excellent financial planner. She buys a photocopier for he
business office at a cost of R125 000. She knows that she will have to
replace the photocopier after 5 years with a better model. After doing her
research she finds out that;
• Depreciation on this photocopier can be calculated at 15% per annum
straight-line.
• Inflation on the price of photocopiers can be calculated at 9% per
annum.
After 5 years Mary plans to sell the copier she has and top up money to buy
a brand new one.
7.3.1
Calculate the price at which she will sell the photocopier after 5
years.
(2)
7.3.2
Calculate the price of a new photocopier in 5 years time.
(2)
7.3.3
How much extra money will she need to purchase a new
photocopier in 5 years time.
(1)
7.3.4
Mary is offered an investment account paying an interest of 11%
per annum compounded monthly. How much money must she
invest now so she has enough money to buy a new copier in 5
years time.
(3)
[14]
QUESTION 8
8.1
Given 𝑓(𝑥) = −6𝑥 2 + 5. Determine 𝑓′(𝑥) from first principles.
8.2
Determine;
8.2.1
8.2.2
8.3
𝑑𝑦
𝑑𝑥
(5)
if 𝑦 = 4𝑥 3 − 2𝑥 2 + 5𝑥 + 7
3
(2)
2
𝐷𝑥 [ √𝑥 5 + 4√𝑥 − 𝑥]
(3)
3
Determine the equation of the tangent to the graph of 𝑓(𝑥) = 4𝑥 − 𝑥 at the
point (−1; −3)
(4)
[14]
8
QUESTION 9
The sketch below shows the graph of 𝑓(𝑥) = −𝑥 3 + 4𝑥 2 + 11𝑥 − 30.
E and F are stationary points of f.
A(-3; 0), B and C are the x-intercepts of f.
9.1
Calculate the coordinates of B and C.
(4)
9.2
Determine the coordinates of E and F, the stationary points of f.
(5)
9.3
Determine the x-coordinate of the point of inflection.
(3)
9.4
Using the graph, determine the values of x for which
9.4.1
𝑓(𝑥) ≤ 0
(2)
9.4.2
𝑓 ′ (𝑥) > 0
(2)
[16]
9
QUESTION 10
A box with a square base is to be made to contain a volume of 108cm3. The box
must be open at the top.
Let the base be xcm and the height h as shown in the diagram below.
10.1
Show that the total outer surface area of the box is given by;
𝐴(𝑥) = 𝑥 2 +
10.2
432
𝑥
Determine the dimensions of the box for which the outer surface area is a
minimum.
(3)
(4)
[7]
10
QUESTION 11
11.1 Suppose we are given two mutually exclusive events, X and Y such that
𝑃(𝑋) = 0,2 and 𝑃(𝑋 𝑜𝑟 𝑌) = 0,5
Determine 𝑃(𝑌).
(2)
11.2 A bag contains five R2 coins and eight R5 coins. Themba then picks two
coins from the bag. What is the probability that he will end up with exactly
R7?
(4)
11.3 A survey was conducted amongst leaners at Plumstead High School. The number of
learners arriving late for school and the number of learners using public transport are
recorded in the table below
Late for School
On Time
Total
Use Public
Transport
150
450
600
Don’t use Public
Transport
100
300
400
Total
250
750
a
11.3.1
How many learners were surveyed at the school
11.3.2
Suppose a learner is selected at random. Calculate the
probability that a chosen learner;
11.3.3
(1)
a) Is late for school
(1)
b) Uses public transport.
(1)
Are the events “Uses public transport” and “Late for school”
independent of each other. Justify your answer with a suitable
calculation.
(3)
[12]
TOTAL MARKS 150
11
INFORMATION SHEET
x=
− b  b 2 − 4ac
2a
A = P(1 − i) n
A = P( 1 + ni )
A = P(1 − ni)
Tn = a + ( n − 1 )d
S n = n2 [2a + (n − 1)d ]
Tn = ar
F=
(

)
a rn −1
Sn =
r −1
n −1
; r 1

x (1 + i ) − 1
i
n
P=
a
; −1  r  1
1− r
S =

x 1 − (1 + i )
i
−n
A = P(1 + i) n

f ( x + h) − f ( x )
h
h→ 0
f ' ( x) = lim
 x1 + x 2 y1 + y 2
;
2
 2
d = ( x 2 − x1 ) 2 + ( y 2 − y1 ) 2
y = mx + c
M 
y − y1 = m( x − x1 )
m=
y 2 − y1
x 2 − x1



m = tan 
(x − a)2 + ( y − b)2 = r 2
In ABC:
area ABC =
a
b
c
=
=
sin A sin B sin C
a 2 = b 2 + c 2 − 2bc.cos A
1
ab. sin C
2
sin( +  ) = sin . cos  + cos . sin 
sin( − ) = sin . cos  − cos . sin 
cos ( +  ) = cos . cos  − sin . sin 
cos ( − ) = cos . cos  + sin . sin 
cos 2  − sin 2 

cos 2 = 1 − 2 sin 2 
2 cos 2  − 1
sin 2 = 2 sin . cos 
2
 (xi − x )
n
x=
 fx
2 =
n
P( A ) =
n( A )
n (S )
ŷ = a + bx
i =1
n
P(A or B) = P(A) + P(B) – P(A and B)
b=
 ( x − x )( y − y )
 ( x − x )2