Page 1 of 28
ST. DAVID’S MARIST INANDA
MATHEMATICS
PAPER 1 PRELIMINARY EXAMINATION GRADE 12
10 SEPTEMBER 2021
EXAMINER: MS N VAZZANA MARKS: 150 MODERATOR: MRS S RICHARD TIME: 3 HOURS
NAME:_____________________________________________________________ PLEASE PUT A CROSS NEXT TO YOUR TEACHER’S NAME
Mrs Black Mrs Kennedy Mrs Nagy Mrs Richard Mr Sokana Mr Vicente Ms Vazzana
INSTRUCTIONS:
✓ This paper consists of 28 pages. Please check that your paper is complete.
✓ Please answer all questions on the Question Paper,
✓ You may use an approved non-programmable, non-graphical calculator unless otherwise
stated.
✓ Answers must be rounded off to two decimal places, unless otherwise stated.
✓ It is in your interest to show all your working details.
✓ Work neatly. Do NOT answer in pencil. Write using a dark pen, preferably black.
✓ Diagrams are not drawn to scale.
SECTION A Q1 [20]
Q2 [18]
Q3 [6]
Q4 [9]
Q5 [5]
Q6 [8]
Q7 [12]
TOTAL [78]
LEARNER’S MARKS
SECTION B Q8 [8]
Q9 [8]
Q10 [12]
Q11 [10]
Q12 [15]
Q13 [7]
Q14 [12]
TOTAL [72]
LEARNER’S MARKS
TOTAL: /150
Page 2 of 28
SECTION A QUESTION 1 [20 Marks]
a) Solve for 𝑥 in each of the following equations:
i. 𝑥2 − 4𝑥 = 12
(3)
ii. 𝑥+2
𝑥+1−
3
𝑥−2=
1
𝑥+1
(4)
Page 3 of 28
iii. 𝑙𝑜𝑔(𝑥+1)(2𝑥 − 3) = 1, stating restrictions where necessary.
(4)
iv. log(𝑥 + 1) + 𝑙𝑜𝑔𝑥 = 2, round off your answers to TWO decimal
places.
(5)
Page 4 of 28
b) Determine the value(s) of p so that 3𝑥2 + 2𝑥 − 𝑝 + 1 = 0 has real roots.
(4)
Page 5 of 28
QUESTION 2 [18 Marks]
a) The 4th term of an arithmetic series is 108 and the 11th term is 80. Find the
common difference and the first term of the series.
(4)
b) Calculate the value of: ∑ (−3)𝑝
(4)
𝑝 = 4
21
Page 6 of 28
c) For what value(s) of x will the following infinite geometric series converge?
(1+3x)+(1-9x2)+…
(4)
Page 7 of 28
d) All the terms of a geometric series are positive. The sum of the first two terms
is 34 and the sum to infinity is 162. Determine the common ratio.
(6)
Page 8 of 28
QUESTION 3 [6 Marks]
Given 𝑃(𝐴) = 0,3 and 𝑃(𝐵) = 0,5. Calculate 𝑃(𝐴 𝑜𝑟 𝐵) if:
a) A and B are mutually exclusive events.
(2)
b) A and B are independent events.
(4)
Page 9 of 28
QUESTION 4 [9 Marks]
a) Jackie deposited R25 000 into a savings account with an interest rate of 18%
p.a. compounded quarterly. Jackie withdrew R8000 from the account 2 years
after depositing the initial amount. She deposited another R4000 into this
account 3,5 years after the initial deposit. How much money will Jackie have 5
years after the initial deposit was made?
(4)
Page 10 of 28
b) Mohammed has about 10 years to go to his retirement and he decides to set
up a savings annuity to enhance his pension. He decides to pay R10 000 on a
monthly basis. He plans to make his first payment on 1 November 2021 and
his final payment will be on 1 October 2031. The interest rate is 8% p.a
compounded monthly. Calculate the amount of money that will be available to
Mohammed on 1 November 2031 when he retires.
(5)
Page 11 of 28
QUESTION 5 [5 Marks]
The graph of a parabola ℎ has 𝑥 intercepts 𝑥 = −3 and 𝑥 = 5. The line 𝑘(𝑥) = 6 is a
tangent to ℎ at P, the turning point of ℎ.
Sketch the graph of ℎ on the axes below, clearly showing ALL intercepts
with axes and the turning point of h.
(5)
Page 12 of 28
QUESTION 6 [8 Marks]
The diagram below shows the graphs of the following functions:
𝑓(𝑥) =1
4𝑥2; 𝑥 ≥ 2 and 𝑔(𝑥) = (
1
3)
𝑥
a) Write down the range of 𝑓−1
(2)
b) Determine the inverse of 𝑓 in the form 𝑦 =
(2)
Page 13 of 28
c) Determine the inverse of 𝑔 in the form 𝑦 =
(1)
d) Sketch the graph of 𝑔−1 on the axes below
(3)
Page 14 of 28
QUESTION 7 [12 Marks]
a) Determine 𝑓′(𝑥) by first principles if 𝑓(𝑥) = −5𝑥2 − 𝑥 + 1
(5)
Page 15 of 28
b) Determine 𝐷𝑥[ (2𝑥 − 1)(𝑥 + 5)]
(3)
c) The gradient of the curve 𝑦 = 2𝑥3 +𝑎
√𝑥 at the point where 𝑥 = 1 is 8.
Determine the value of 𝑎.
(4)
Page 16 of 28
SECTION B
QUESTION 8 [8 Marks]
Grant can get a loan of R40 000 from a friend, Ameer, who allows him to make 8
half-yearly installments, starting only in two years’ time, making the first instalment
at the end of the second year. The interest charged is 9,5% p.a. compounded
monthly.
a) Convert the monthly interest rate of 9,5% p.a. to an annual interest rate
compounded semi-annually.
(3)
b) Calculate the half-yearly repayments he must make.
(5)
Page 17 of 28
QUESTION 9 [8 Marks]
A rectangular box has a length of 5𝑥 metres, breadth of 9 − 2𝑥 metres and its height
of 𝑥 metres.
a) Show that the volume of the box is given by 𝑉 = 45𝑥2 − 10𝑥3
(2)
b) Determine the maximum volume of the box.
(6)
Page 18 of 28
QUESTION 10 [12 Marks]
a) Determine: lim 𝑥→0
4+
3
𝑥1
𝑥−7
(4)
b) If 𝑦 = (𝑥3 − 1)2 , show that 𝑑𝑦
𝑑𝑥= 6𝑥2√𝑦
(4)
Page 19 of 28
c) Given 𝑓(𝑥) = 2𝑥3 − 2𝑥2 + 4𝑥 − 1. Determine the interval on which f is
concave up.
(4)
Page 20 of 28
QUESTION 11 [10 Marks]
a) The letters of the words KINGDOM are rearranged to form other
arrangements of all the letters.
i. How many arrangements are possible?
(2)
ii. How many arrangements are possible if the letters K and G can’t be
next to each other
(3)
iii. What is the probability that if an arrangement is chosen at random, the
letters K and G will be separated?
(1)
Page 21 of 28
b) In a class there are 15 girls and 7 boys. Two students are chosen to represent
the class in a meeting. Determine the probability that if two students are
chosen at random one will be a boy and the other will be a girl.
(4)
Page 22 of 28
QUESTION 12 [15 Marks]
a) Graphs 𝑓(𝑥) = −𝑥2 − 4𝑥 and 𝑔(𝑥) = 2𝑥 − 6 are sketched below. Point D
is the turning point of 𝑓. A and O are the x intercepts of 𝑓 and point C is
the x intercept of 𝑔 respectively. Point B is the y intercept of 𝑔.
i. Determine the area of ∆ 𝐴𝐷𝑂
(4)
Page 23 of 28
ii. For what value(s) of k will −𝑥2 − 4𝑥 = 2𝑥 + 𝑘 have two real roots that
are opposite in sign?
(2)
b) The graph of ℎ(𝑥) = 𝑥3 − 4𝑥2 + 4𝑥 has a local minimum at (𝑎; 0) and a
local maximum at (𝑏; 𝑐).
Determine the values of 𝑎, 𝑏 and 𝑐
(6)
Page 24 of 28
c) The sketch below shows the graph of 𝑦 = 𝑔′(𝑥). It is further given that
𝑔(0) = −5 and 𝑔(−2) = 0.
i. Write down the co-ordinates of the point of inflection of g
(1)
ii. Write down the co-ordinates of the point where g cuts the x axis
(1)
iii. For which values of x is the graph of g decreasing?
(1)
Page 25 of 28
QUESTION 13 [7 Marks]
Rectangles are cut from a strip of fabric of constant width and arranged in a row.
The first rectangle has a length of 10cm. The length of each subsequent rectangle is
85% of the length of the previous rectangle.
a) Determine the length of fabric required if the strip consists of 20 rectangles.
(4)
b) Determine the longest strip of fabric that can be used.
(3)
10𝑐𝑚
Page 26 of 28
QUESTION 14 [12 Marks]
The path of a slide can be modelled by the equation 𝑓(𝑥) =1
2𝑥2 − 2𝑥 + 2 between
0 ≤ 𝑥 ≤ 𝑎. A child comes off the slide and into the water following the path modelled
by 𝑔(𝑥) = −1
2𝑥2 + 4𝑥 + 𝑘 for 𝑥 ≥ 𝑎. ( the graph represented by the dotted line)
The child leaves the slide at an angle of 45° to the horizontal.
a) Determine the value of 𝑎
(3)
Page 27 of 28
b) Determine the value of 𝑘
(3)
c) How high above the water does the child reach before entering the water?
(3)
Page 28 of 28
d) Determine the horizontal distance the child travels by the time she starts on
the slide (at 𝑃) before entering the water.
(3)
TOTAL: 150 Marks