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MATHEMATICS: PAPER I Page 1 of 12
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MATHEMATICS: PAPER 1 Grade 12
10 September 2018
Examiner: Mr J. Moodliar 150 marks
Moderator: Mrs V. Rudman Time: 3 hrs
Name:
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY:
1. This question paper consists of 12 pages and 13 questions. An information sheet is supplied at the end of this question paper.
2. Read the questions carefully and plan the length of your answers in relation to the sections being covered in each question.
3. Number your answers exactly as the questions are numbered. 4. You may use an approved non-programmable and non-graphical calculator, unless
otherwise stated. 5. It is in your own interest to write legibly and to present your work neatly. 6. Round off your answers to ONE decimal digit unless otherwise stated. 7. All necessary working details must be clearly shown. Answers only will not necessarily
be awarded full marks.
Possible Marks Actual Marks
Marker Signature
Moderator Signature
Question 1 : 27
Question 2 : 11
Question 3 : 18
Question 4 : 13
Question 5: 5
Question 6: 9
Question 7: 11
Question 8: 25
Question 9: 10
Question 10: 5
Question 11: 4
Question 12: 7
Question 13: 5
TOTAL 150
MATHEMATICS: PAPER I Page 2 of 12
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SECTION A (83 marks)
QUESTION 1 (27 marks)
(a) Solve for 𝑥, leaving answers in simplest surd form where necessary: (1) 27𝑥 . 9𝑥−2 = 1 (4)
(2) √5 − 𝑥 − 𝑥 = 1 (6)
(3) (2𝑥 − 3)(𝑥 + 4) ≤ 6 (4)
(4)
x
n
n1
147)152(
(4)
(b) Solve for 𝑥 and 𝑦: 𝑥 − 3𝑦 = 1 𝑥2 − 2𝑥 + 9𝑦2 = 17 (5)
(c) Determine the nature of roots of the equation: 𝑥2 + 𝑝(𝑥 − 𝑝) − 1 = 0, for real values of 𝑝. (4)
QUESTION 2 (11 marks)
Consider the function 𝑓(𝑥) = −(𝑥 + 1)2 + 4
(a) Sketch the graph of 𝑓(𝑥) showing all intercepts. (3)
(b) Calculate the co-ordinates of the point on 𝑓 for which the tangent to 𝑓 will have a gradient of 1. (4)
(c) Write down the values of 𝑘 for which 𝑓(𝑥) – 𝑘 will always be a negative value. (2)
(d) Explain why the inverse of 𝑓 is not a function. (2)
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QUESTION 3 (18 marks)
(a) The sequence −1 ; 𝑝 ; 13 is a quadratic sequence. The sequence of the
first differences is 6 ; 𝑞 ; …
Determine the values of 𝑝 and 𝑞. (4)
(b) The twelfth term of an arithmetic sequence is 5, and the common difference between successive terms is 3. Determine which term has a value of 47. (5)
(c) 𝑆𝑛=4𝑛2 + 1 represents the sum of the first n terms of a particular series. Determine the value of the second term. (3)
(d) The watercress is an invasive plant species that invades rivers and wetlands in South Africa. Watercress was noted to cover an area of 8𝑚2 of a local pond when it was first spotted. Thereafter, the additional area of the pond it covered every month was 85%.
(1) Determine the area covered by the watercress in the seventh month. (3)
(2) Determine the total area covered by the watercress in the first seven months. (3)
QUESTION 4 (13 marks)
(a) R100 000 invested at 12% p.a., compounded monthly, grows to
R181 669,67. For how long was the money invested? (3)
(b) A couple takes a mortgage loan on a house. The plan is to repay
the loan monthly over a period of 30 years. The value of the loan is
R 500 000 and the interest is 9% per annum compounded monthly.
(1) Calculate the monthly payment. (4)
(2) What is the total amount that the house would eventually cost? (2)
(3) After 28 years the couple wants to clear the account. What would be the outstanding balance of the account? (4)
MATHEMATICS: PAPER I Page 4 of 12
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QUESTION 5 (5 marks)
For two independent events A and B, in the sample space S, it is given that 𝑃(𝐴) = 0.6
and 𝑝(𝐵) = 0.3. Round your answers off to two decimal places in this question:
(a) Determine 𝑃(𝐴 𝑎𝑛𝑑 𝐵). (1)
(b) Draw a Venn diagram to represent this information. (3)
(c) Determine 𝑃(𝐴 𝑎𝑛𝑑 𝐵’). (1)
QUESTION 6 (9 marks)
(a) Determine by 𝑓′(𝑥) by first principles if 𝑓(𝑥) = 7𝑥 − 2𝑥2. (5)
(b) Differentiate with respect to 𝑥 if = 2𝑥3−1
√𝑥 . (4)
TOTAL FOR SECTION A – 83 MARKS
MATHEMATICS: PAPER I Page 5 of 12
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SECTION B (67 marks)
QUESTION 7 (11 marks)
(a) The South African Traffic Service is doing a clamp down on speeding. During a
recent speeding trap they collected the following data:
Speeding Not Speeding TOTAL
Male 398 217 615
Female 205 180 385
TOTAL 603 397 1000
(1) Determine the probability that a driver selected at random is a female driver,
given that the driver is NOT speeding. (1)
(2) Are the events of being a male and speeding independent? Show all the
necessary calculations to substantiate your answer. (4)
(b) The letters of the word ‘SYLLABUS’ are used to form different eight letter codes.
Determine the probability that the code formed will start and end with the same
letters. (6)
QUESTION 8 (25 marks)
(a) Given 𝑓(𝑥) = 2𝑥 − 5 and 𝑔(𝑥) = 5
𝑥−2+ 3
(1) Determine 𝑓−1, the inverse of 𝑓 in the form y = ……… (2)
(2) Sketch the graph of 𝑓−1, clearly indicating all intercepts with the axes as well as
the asymptote. (4)
(3) For what values of 𝑥 is 𝑓−1 > 0? (1)
(4) The graph of ℎ(𝑥) is 𝑓(𝑥) translated horizontally so that it passes through the
point of intersection of the asymptotes of 𝑔(𝑥).
Determine the equation of ℎ(𝑥). (4)
MATHEMATICS: PAPER I Page 6 of 12
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(b) Given the curve of 3 2( ) 8 12f x x x x as shown in the sketch below:
(1) Give the coordinates of A. (2)
(2) Determine the coordinates of B, the turning point of 𝑓(𝑥). (3)
(3) Determine the 𝑥 coordinate of the point of inflection. (2)
MATHEMATICS: PAPER I Page 7 of 12
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(c) In the diagram below:
2 3 4p x x x
2 8q x x
Points A and C are the x -intercepts of p and q .
Points B and C are the points of intersection of p and q .
Point D is the y -intercept and point E is the turning point of p .
(1) For what values of 𝑘 will 𝑥2 − 3𝑥 − 4 = 2𝑥 + 𝑘 have two real roots that are opposite
in sign? (2)
(2) Solve for 𝑥, giving the exact solution if 𝑝′(𝑥)
𝑞(𝑥) ≥ 0. (5)
MATHEMATICS: PAPER I Page 8 of 12
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QUESTION 9 (10 marks)
In the diagram alongside, a cone is cut out of a cylindrical piece of wood. The cone and
the cylinder have equal radii and the height of the cylinder is 2 cm more than the height
of the cone.
The ratio of the radius to the height of the cylinder is 1 :3.
Volume of a right circular cone 21
3r h
All measurements are in cm.
r
h
H
2 cm
(a) Show that the volume of wood left over, after the cone has been cut out of the
cylinder is given by 3 22
23
V r r . (5)
(b) The rate of change of volume of the wood with respect to r when r p is
2498 cm cm . Determine p . (5)
MATHEMATICS: PAPER I Page 9 of 12
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QUESTION 10 (5 marks)
If I cut a pizza with a single cut then I get two pieces. If I cut a pizza with two single cuts,
as illustrated in the diagram, then I get three pieces. If I cut a pizza with three single cuts,
as illustrated in the diagram, then I get six pieces.
The number of pieces (𝑝) with n cuts, according to the cutting pattern illustrated in the
diagram, is given by the formula 𝑝 = 𝑎𝑛2 + 𝑏𝑛 + 𝑐.
What is the number of pieces I can get with six single cuts? Show ALL calculations.
QUESTION 11 (4 marks)
Andre is required in a test to fine the derivative of a function f (x). However, by mistake he
finds the inverse instead. He finds that:
𝑓−1(𝑥) = √𝑥 + 7
2
3
Find the correct answer to the problem.
MATHEMATICS: PAPER I Page 10 of 12
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QUESTION 12 (7 marks)
Given: 𝑓(𝑥) = 1
𝑥. The tangent at D where 𝑥 = 𝑎 is drawn.
(a) Show that the equation of the tangent at D is 𝑥 + 𝑎2𝑦 = 2𝑎. (4)
(b) Calculate the area of ∆𝑂𝐵𝐶. (3)
QUESTION 13 (5 marks)
The following series is given:
2 2 2 2 2 2 2 2 2 2 2 224 23 22 21 20 19 ......... 6 5 4 3 2 1
Without the use of a calculator and showing all necessary working, show that the sum of
the sequence is 300.
TOTAL FOR SECTION B – 67 MARKS
TOTAL FOR PRELIMINARY EXAMINATION P1: 150 MARKS
MATHEMATICS: PAPER I Page 11 of 12
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MATHEMATICS INFORMATION SHEET:
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2
4–– 2
n
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n1
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1 2
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dnaTn 1 dnan
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1 n
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niPA 1 niPA 1
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sinsinsin
Acbcba cos.2–222 CbaABCarea sin.2
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sin.coscos.sin)(sin sin.cos–cos.sin)–(sin
sin.sin–cos.cos)(cos sin.sincos.cos)–(cos
MATHEMATICS: PAPER I Page 12 of 12
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1cos2
sin21
sincos
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cos.sin22sin
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∑(𝑥−�̅�)2
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