St John’s CollegePreliminary Examination

July 2015Mathematics Paper 1

Examiner: D Grigoratos Time: 3 hrs Moderator: W Young Marks: 150

NAME: ___________________________________________________________

TEACHER: GE WLY KJ DG DC BT

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY

1. Write your name and indicate your teacher’s initials on both the question paper and Answer Booklet.

2. This question paper consists of 10 pages, which includes a separate Information Sheet. This information sheet must be handed in at the end of the exam. Please do not write on it. Check that your paper is complete.

3. Answer the questions in the Answer Booklet provided, except Question 7 and Question 9 (b) which must be answered on the question paper.

4. Read the questions carefully. Number your answers exactly as the questions are numbered and answer the questions in the correct order.

5. You may only use an approved non-programmable and non-graphical calculator.

6. Round off your answers to one decimal digit where necessary.

7. All the necessary working details must be clearly shown. Equations may not be solved solely with a calculator.

7. It is essential that you present your work neatly and logically.

Question 1 2 3 4 5 6 7 8 9 10 11 Total

Out of 18 17 11 14 14 9 5 4 23 25 10 150

Mark

Page 1 of 10

SECTION A

QUESTION 1 18marks

(a) Solve for

(1) (3)

(2) (3)

(3) (3)

(4) (5)

(b) Simplify (4)

QUESTION 2 17marks

(a) Write down the nth term of the sequence (2)

(b) Given the arithmetic series:

Determine:

(1) the 100th term (2)

(2) the value(s) of if (4)

(c) The sum of the first terms of a series is given by the formula

(1) Determine the sum of the first 6 terms. (1)

(2) Determine the first 3 terms of the sequence. (3)

(d) Calculate: (5)

Page 2 of 10

QUESTION 3 11marks

(a) Determine from first principles if (4)

(b) Determine :

(1) (3)

(2) (4)

QUESTION 4 14marks

(a) Given that find:

(1) the points on the curve of where the tangents are parallel to the x-axis. (4)(2) the equation of the tangent to the curve of at the point where (4)

(b) Given

(1) Write in the form of and hence find the turning point. (4)(2) If the graph of is shifted units left and units down, determine the

equation of the new graph in the form (2)

QUESTION 5 14marks

(a) Spencer is a young man with ambition and sets his target at accumulating R1 000 000 in his savings account. The account gives him per annum compounded monthly and he is determined to invest R 9 500 per month for as long as it takes to become a millionaire.

How long will it take Spencer to realise his dream? (Answer to the nearest month) (5)

(b) Kamogelo needs to buy 8 industrial sock knitting machines for his business. The cost of a single machine, manufactured in Europe is € 12 750.00 The current rate of exchange between the rand and the euro is R13,50 = €1.00. Kamogelo needs to take out a loan to finance this project.

(1) Calculate the rand amount required to purchase all eight machines. (2)

(2) The bank agrees to give Kamogelo the loan he requires at 11.8% per annum over a period of six years. However, he would like to repay the loan through equal quarterly payments starting in three months time.Show that the effective annual rate of 11.8% per annum is converted to a nominal rate of 11, 3% per annum compounded quarterly. (3)

Page 3 of 10

(3) Calculate Kamogelo’s quarterly payments over the period of six years. (4)

SECTION A: 74 marks

SECTION B

QUESTION 6 9marks

(a) A linear function satisfies the following: and

Determine the inverse function in the form (4)

(b) Determine (3)

(c)

The graph of , parallel to the is shown above.

State which one of the sketches below, is a possible graph of .

(2)

Page 4 of 10

QUESTION 7 5 marks

The function has the following properties:

when , and

On the grid provided below, motivate by means of a sketch whether the y-intercept is positive or negative

Page 5 of 10

QUESTION 8 4marks

A man contributed to a retirement fund which guaranteed a pension payment of per month for a period of 5 years after retirement. He retired on 31st March 1985.

If he died before he had received 5 years pension payments, his beneficiaries would receive the balance outstanding. The man died on the 31st December 1988.

How much did his beneficiaries receive if the fund offered 7,8% per annum compounded monthly?

QUESTION 9 23marks

(a) The graph of is given below.

(1) Find the values of at the turning points. (4)

(2) Give the domain of (2)

(3) Find the co-ordinates of the points of inflection and hence state the

values of for which the graph is concave down. (4)

(4) State whether is a one-to-one function or many-to-one function (1)

Page 6 of 10

(b) Given Calculate:

(1) and (2)

(2) if (2)

(3) Sketch, on the grid provided below, the graph of indicating the co-ordinates of the points found in parts (1) and (2) (4)

(4) Using the graph determine the value(s) of for which (2)

(5) On the same set of axes, sketch the graph of (2)

Page 7 of 10

QUESTION 10 25marks

(a) Refer to the figure showing the graph of the cubic function

(1) Show that (5)

(2) Determine the co-ordinates of D (4)

(3) If the equation has only two real roots, what are the possible

values of (2)

(4) Give the intercepts of (2)

Page 8 of 10

(b) A zoo wishes to build a fence around a crocodile enclosure.

The sketch below indicates the shape of the enclosure in terms of and where

the straight sections are to be metres long and the circular end sections are of

radius metres.

The zoo requires the area of the enclosure to be

(1) Use the area of the enclosure to show that (2)

(2) Show that the length of the fence required is given as (3)

(3) What will be the radius of the circular ends that will ensure the shortest total length of this fence in order to minimise the building costs? (5)

(4) Hence find the value of and describe the shape of the enclosure. (2)

Page 9 of 10

QUESTION 11 10marks

(a) The probability that I go for a run this evening is 0,3 and the probability that I will have pasta for dinner tonight is 0,6.

Calculate the probability that I will go for a run or eat pasta if these events are independent. (4)

(b) A squad of 160 police trainees were asked to volunteer for the following refresher courses:

Crime scene management High speed driving course Weapons training

The Venn diagram below indicates their attendance.

(1) Calculate the probability that a trainee, chosen at random would like to attend both a crime scene management and a high speed driving course. (1)

(2) Prove that the events of choosing a crime scene management course and a high speed driving course are independent. (3)

(3) Ten new police officers are now included, none of whom wish to do any courses. Will this affect the independence of C and H? Explain (2)

SECTION B: 76 marks

Page 10 of 10

37

8

19

24

23

810

31

HC

W