Grade 12maths.stithian.com/New CAPS 2018 Prelim Papers/Education... · 2018. 9. 22. · MATHEMATICS: PAPER 1 Grade 12 10 September 2018 Examiner: Mr J. Moodliar 150 marks Moderator:

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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



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)


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)




(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)


(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)




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)


(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



SECTION B (67 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)


(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)



(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)



(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)




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)




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.


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.




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)


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 INFORMATION SHEET:

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Grade 12maths.stithian.com/New CAPS 2018 Prelim Papers/Education... · 2018. 9. 22. · MATHEMATICS: PAPER 1 Grade 12 10 September 2018 Examiner: Mr J. Moodliar 150 marks Moderator: