(A World of Education…an Education for the World!)

GRADE 12 - MATHEMATICS PAPER 1EXAMINER:Mrs K. HulmeDATE:13 July 2009

MODERATOR: Mr I. L. AtteridgeTOTAL:150

TIME:3 hrs

CANDIDATE’S NAME:______________________________________________________________

CANDIDATE’S MATHS TEACHER:______________________________________________________________

INSTRUCTIONS TO CANDIDATES:

1. Answer all questions in the answer book provided.

2. Rule a right hand margin on each page and rule off after each question.

3. All written work must be done using blue or black ink. Diagrams and graphs must be drawn neatly using pencil.

4. No correction fluids may be used

5. Non-programmable calculators may be used unless otherwise stated.

6. Round off to TWO decimal places unless otherwise stated.

7. It is in your own interests to work neatly and to show all necessary steps in calculations.

THIS EXAMINATION CONSISTS OF 11 PAGES

Grade 12 Mathematics Paper 1

13 July 2009

Page 1 of 11

SECTION A:

QUESTION 1:

a)Determine the 7th term of the following sequence: (2)

((2))b)Determine the value for the following expression:

c)The fifth term of an arithmetic sequence is and the twelfth term is .

(1)Determine the common difference and the first term(6)

(2)Determine the sum of the first 40 terms(3)

[13]

QUESTION 2:

Emile inherits R2 500 000,00 from an unknown great-uncle on his 21st birthday. He buys himself a Honda S2000 for R400 000,00 and invests the rest at 11,00% p.a. compounded quarterly. On his 30th birthday, he withdraws R100 000,00 to fly his Grade 12 Maths teacher to his graduation from Cambridge University in the United Kingdom. 15 years after his initial investment, the interest rate changes to 10,75% p.a. compounded monthly. Use a time-line to calculate the value of his investment when he retires on his 60th birthday.

[7]

QUESTION 3:

Solve the following equations or inequalities:

a)(to two decimal places)(3)

b) (4)

c)(6)

d)(5)

[18]

QUESTION 4:

a)Match the following graphs with their correct equations:

(1)

(Possible equations:A.B.C.D. )

(2)

(3)

(4)

(4)

b)The following graphs represent the functions

for

(1)Write down the values of and (2)

(2)What is the period of ?(1)

(3)For which values of is ?(3)

c)

In the preceding figure, the graphs of , and are given. The graphs of and are symmetrical with respect to the -axis and they intersect at P on the -axis. The graph of cuts the -axis at Q and is symmetrical to about the line .

(1)If and , calculate the value of (2)

(2)Write down the coordinates of P and Q(2)

(3)Write down the equations that define , and (3)

(4)Calculate without using a calculator(3)

[20]

QUESTION 5:

NB – Be very careful with your setting-out and use of mathematical notation

a) If , determine the derivative of from first principles(6)

b)Determine:

(1) if (2)

(2) (4)

c)The distance which a body travels in seconds is metres. If the relationship between and is given by ; determine

(1)the average speed of the body for the time between and (3)

(2)the instantaneous speed of the body when (2)

[17]

SECTION B:

QUESTION 6:

a)Consider the graph of the arithmetic sequence shown below:

(1)Write a formula for (4)

(2)What are the coordinates of the 46th point?(2)

(3)Find the sum of the heights from the horizontal axis of the first 46 points of the sequence’s graph(2)

b)

http://www.soccerworldcupafrica.co.za/images/stadia/soccer_city_stadium-main.jpg

economic multiplier

Definition

Estimated number by which the amount of a capital investment (or a change in some other component of aggregate demand) is multiplied to give the total amount by which the national income is increased. This multiplier takes all direct and indirect benefits from that investment (or from the change in demand) into account.

http://www.businessdictionary.com/definition/economic-multiplier.html

In 2010, the final of the FIFA World Cup will be held at Soccer City near Soweto. The stadium seats 94 000 people. Suppose 50 000 people visit Soweto during the days immediately before the Final, and each spends R500,00. In the month after the event, the people of Soweto spend 60% of the income from the visitors. The next month, 60% is spent again, and so on.

(1) What is the initial amount the visitors spent?

(2) In the long run, how much money does this single event seem to add to the Soweto economy?

(3) The ratio of the long-run amount to the initial amount is called the economic multiplier. What is the economic multiplier in this example?

(4) If the initial amount spent by the visitors is R10 000 000,00 and the economic multiplier is 1,8; what percentage of the initial amount is spent again and again in the local economy?(10)

[19]

QUESTION 7:

TABLE SHOWING PRIME INTEREST RATE HISTORY

(interest rate p.a. compounded monthly)

28 May 09

11.00%

 

11 Dec 06

12.50%

 

13 Sep 02

17.00%

 

02 Jul 99

18.00%

04 May 09

12.00%

 

13 Oct 06

12.00%

 

14 Jun 02

16.00%

 

03Mar 99

19.00%

24 Mar 09

13.00%

 

03 Aug 06

11.50%

 

18 Mar 02

15.00%

 

02 Apr 99

20.00%

06 Feb 09

14.00%

 

12 Jun 06

11.00%

 

16 Jan 02

14.00%

 

02 Mar 99

21.00%

12 Dec 08

15.00%

 

18 Apr 05

10.50%

 

01 Oct 01

13.00%

 

02 Feb 99

22.00%

13 Jun 08

15.50%

 

16 Aug 04

11.00%

 

16 Jul 01

13.50%

 

04 Jan 99

22.75%

11 Apr 08

15.00%

 

15 Dec 03

11.50%

 

18 Jun 01

13.75%

 

 

 

07 Dec 07

14.50%

 

20 Oct 03

12.00%

 

01 Feb 00

14.50%

 

 

 

15 Oct 07

14.00%

 

15 Sept 03

13.50%

 

04 Oct 99

15.50%

 

 

 

20 Aug 07

13.50%

 

15 Aug 03

14.50%

 

16 Aug 99

16.50%

 

 

 

11 Jun 07

13.00%

 

13 Jun 03

15.50%

 

14 Jul 99

17.50%

 

 

 

http://www.propertyloans.co.za/interest-rate-history/

In January, 2009, Fern Dain planned to take out a loan of R650 000,00 in order to buy a house. The interest rate was 15,00% p.a. as announced by the South African Reserve Bank on 12 December 2008, compounded monthly. The loan had to be repaid over a period of 15 years by means of equal monthly payments.

a) What were the monthly repayments at that stage?(3)

b)What is the total amount that she would have paid over the 15 years?(1)

Fern only got around to taking out the loan in June, 2009. The interest rate had been changed on 28 May 2009.

c)What were her monthly repayments using the new interest rate?(2)

d)What is the total amount that she will have paid, assuming no more changes to the interest rate over the period of the loan?(1)

e)Fern decided that she could afford the monthly repayments quoted in January 2009 {i.e. the answer to question a)}. How long will it take her to pay back the loan?(4)

[11]

QUESTION 8:

During nesting season, two different bird species inhabit a region in Mpumalanga with area m2. Mrs Pike estimates that this ecological region can provide 72 000 kg of food during the season. Each nesting pair of Crowned Plovers needs 40 kg of food during a specified time period and 120 m2 of land. Each nesting pair of Guinea Fowl needs 70 kg of food and 90 m2 of land.

Let represent the number of pairs of Crowned Plovers and represent the number of pairs of Guinea Fowl.

a)Describe the meaning of the constraints and (1)

b)Describe the meaning of the constraint (1)

c)What is the remaining constraint?(2)

d)Graph this system of inequalities on the set of axes provided in the Answer Booklet and identify each vertex of the feasible region(6)

e)Maximise the total number of nesting pairs, N, by considering the function

(3)

[13]

QUESTION 9:

a)Draw the inverse of , shown below. Answer this question in the Answer Booklet.

(3)

b) Your head gets larger as you grow. Most of the growth comes in the first few years of life, and there is very little additional growth after you reach adolescence. The estimated percentage of adult size for your head is given by the formula ; where is your age in years and is the percentage of the average adult size.

(1)Complete the table of data given in the Answer Booklet(2)

(2)Why do you think the table stopped at 50?(1)

(3)Graph this function on the set of axes given in the Answer Booklet(5)

(4) Describe the transformations of the graph of that produce the graph in the previous question(3)

(5)A 3-year-old child’s head is what percentage of the adult size?(2)

(6)After how many years and months is a child’s head 80% of an adult’s head?(3)

[19]

QUESTION 10:

a)The following diagram represents the curve . L is an -intercept. M and N(2 ; 0) are the turning points.

Calculate the values of and (6)

b)Refer to the following diagram:

Find the maximum area A that a rectangle inscribed in a semi-circle can have, in terms of radius . Hint: find A2 and maximise it.(7)

[13]

-2

-2

2

2

x

y

0

-3

-2

-1

1

2

3

90

30

60

-30

-60

-90

O

y = x

x

y

h

f

g

Q

P

64

2

5

10

15

(1 ; 4)

(2 ; 9)

(3 ; 14)

(4 ; 19)

n

u

n

20

5

5

x

y

O

N(2 ; 0)

M

L

r

(x ; y)