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Domino Servite School

Accreditation Number SCH 003461 PA Registration Number 122581

Trials

September 2011

Mathematics Paper I

Time: 3 hours Total: 150

Examiners: H Pretorius Moderator: A van Tonder G Scholefield

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PLEASE READ THE FOLLOWING WITH CARE

1. This paper consists of 11 pages. Please check that your paper is complete.

2. Read the questions carefully.

3. Answer all the questions.

4. In Question 5 there is a graph that must be drawn on the graph paper provided. The

graph paper must be handed in together with your answer script. Make sure your name

is written on the graph paper.

5. Number your answers exactly as the questions are numbered.

6. You may use an approved non-programmable and non-graphical calculator, unless a

specific question prohibits the use of a calculator.

7. Round off your answers to one decimal digit where necessary, unless specified

differently, or in financial maths where you must round off to the second place after

the decimal comma.

8. All the necessary working details must be clearly shown.

9. It is in your own interest to write legibly and to present your work neatly.

10. You must write in black or blue pen, except when drawing a sketch or graph.

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Question 1

a) Solve for x:

i)

(3)

ii) (3)

iii) (3)

b) If ( ) √

, find ( ). Leave your answer with positive indices. (3)

c) Find the average gradient of the function , between the points

where and (4)

d) If ( ) is a factor of , find the value of (3)

[19]

Question 2

a) A bank advertises the following options for investments. Interest rates are quoted as

per annum rates, compounded monthly.

Nominal

R0 - R4 999 0,00%

R5 000 - R19 999 2,50%

R20 000 - R49 999 3,75%

R50 000 - R99 999 3,75%

R100 000 - R499 999 4,00%

R500 000 and above 4,25%

Steven invests R80 000.

i) What interest rate will he earn? (1)

ii) After how many months will his investment first exceed R110 000? (4)

iii) He withdraws money, leaving R100 000 in the account. How much

would be in the account at the end of another three years? (2)

iv) What is the effective rate of interest on a deposit of R600 000? (3)

b) AgriBank states that a client may borrow an amount to purchase a house,

provided that his loan repayments will not result in a cost of more than 20%

of his gross monthly salary (calculated before tax, etc.). Interest on the loan is

charged at 6,8% per annum compounded monthly.

Dirk earns R15 500 per month before tax.

i) What is the maximum amount Dirk can pay per month on his loan? (2)

ii) He chooses to repay the loan at R3 000 per month for 20 years.

What is the value of a house he can afford to buy? (5)

[17]

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Question 3

In the figure the functions and are drawn. The equations of the functions are:

and

,

The functions meet at A ( )

Write down:

a) the coordinates of the y-intercept of . (1)

b) the value of (2)

c) the value of (2)

d) the equation of , such that is the reflection of about the axis (2)

e) the equation of , the inverse of in the form (2)

[9]

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Question 4

a) In a corner spectator stand, similar to that seen alongside, at

the Liverpool Football Club stadium, Anfield, the row of

chairs nearest ground-level can seat 138 people. The next

row can seat 141 people, the next 144 and so on in an

arithmetic sequence.

i) If there are 18 rows of seats in the first section, what

is the total number of people who could be seated in

this section? (4)

ii) If rows could be removed or added, how many rows would there be if the

last row seats 234 people? (3)

b) The Department of Basic Education has published the following data reflecting the

number of pupils in state schools in Grade 12 for the years 2005 to 2008.

Year Number of Gr 12 Pupils

2005 55 193

2006 57 953

2007 60 851

2008 63 894

i) Show that the data closely follows a geometric progression where the

constant ratio is 1,05 (i.e. ). (2)

ii) Use this information and an appropriate formula to predict the number of

pupils in Grade 12 in 2012. (3)

iii) In any year the number of Grade 12 pupils is 95% of those in Grade 11,

which in turn is 95% of those in Grade 10 and so on. Approximately how

many pupils were in Grade 8 in 2007? (4)

c) The picture is that of a Yellowwood, the national tree of South

Africa.

The tree grows rapidly for the first eight years to a height of

ten metres. Thereafter the increase in height per year in metres

follows the geometric sequence:

4 ; 3 ; 2,25 ; .......

What is the maximum height the tree can reach? (5)

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x2 4 6 8 10 12 14

y

2

4

6

8

10

12

14

fg

A

Question 5

a) Hassan Pty Ltd has a contract to produce Liverpool Football Club replica hoodies and

shirts as shown below.

Each week the company can produce a maximum combined total of 10 000 hoodies

and shirts. It takes two workers to produce a hoodie, but one person can produce a

shirt, per week. The company employs 12 000 people. The marketing department

cannot sell more than 4 000 hoodies per week, nor more than 9 000 shirts per week.

If the number of hoodies produced per week is and the number of shirts is one of

the constraint inequalities for the information above is:

Two more implicit constraints are and .

i) Write down three more constraint inequalities. (4)

ii) On the graph paper provided, and using a scale of 1000 units per centimetre

on both axes, sketch the inequalities graphically and show the feasible region

clearly. (5)

iii) The profits on the sale of each hoodie and shirt are R15 and R10 respectively.

By writing down and using an expression for total profit in terms of and

calculate the maximum possible profit. (4)

b) The graphs of two linear programming inequalities are shown alongside:

The equations of the lines are:

and

i) What is the gradient of ? (1)

ii) The equation of a cost function is:

For what values of would the

cost function indicate minimum

cost to be at A where the lines

meet? (4)

[18]

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Question 6

The diagram below shows the graph of the equation:

( )

Points A and D are the x-intercepts, B and D are turning points, and C is the y-intercept.

It is given that the coordinates of D are ( ). Find the coordinates of:

a) C (1)

b) A (4)

c) Use calculus methods to find the coordinates of B. (4)

d) A normal is a line which is perpendicular to a tangent at the point of contact of the

tangent to a curve. Find:

(i) the gradient of the tangent to at C. (2)

(ii) the equation of the normal to at C. (2)

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Temperature (in ˚C)

Time (in minutes)

Question 7

The graphical representation given below represents Janet making herself a cup of coffee in

Pretoria. Her kettle switches off as soon as the water begins boiling. After she made her

coffee she began studying, forgetting to drink her coffee.

Note the following: A is the point (0; 58,25), B is the point (0; 21), C is the point (7; 96) and

E is the point (7; 0)

Also note that this is a piece-wise function with the equation given as:

a) What was the temperature of the water when Janet poured it into the kettle? (1)

b) What is the boiling point temperature of water in Pretoria? (1)

c) Does Janet take her coffee white (with milk) or black (without milk)? Motivate your

answer. (2)

d) The mathematical constant e can be approximated by the value 2,718 (just as the

mathematical constant π can be approximated by the value 3,14). Using this value,

calculate at what time the coffee had a temperature of 50˚C. (4)

[8]

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Question 8

When it opened on 27 November 2009, the 300 metres high Beipanjiang highway bridge

became the third high crossing of the Beipanjiang River (China) to be opened in just 8 years.

The design is fairly typical for a Chinese designed suspension bridge. It has a stiffened truss

deck span of 636 metres strung between two H-frame concrete towers. The most unusual

aspect of the bridge is the east tower with a total height of 160 metres. The bridge extends

approximately 100 metres below the truss deck. The bridge currently ranks 8th

among the

world’s 10 highest bridges. Source: http://highestbridges.com/wiki/index.php?title=Beipanjiang_River_2009_Bridge

Below you can see a simplified line drawing of the bridge.

a) The smallest distance between the parabolic arch connecting the two towers and the

deck is 3 metres. In other words, at its lowest point, the arch is 3 metres above the

truss deck.

How high above the bottom of the gorge (i.e. the river bed) is the lowest point of the

parabolic arch? (1)

b) Find the equation of the parabolic arch. (6)

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The 2009 Bepanjiang Highway Bridge under construction

330 m

636 m

60 m

Let this point be the origin (0; 0)

100 m

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Question 9

A lap pool (a pool used to train swimmers) is designed to be seven times as long as it is wide.

The area of the sides and bottom is 90 m2.

a) Let x represent the width and h the height in metres. Show that the function that

models the volume can be given by ( )

. (3)

b) What is the domain of this real life problem? (4)

c) What is the width of the pool if the volume of water it can hold is at a maximum? (3)

d) What is the height of the pool if the volume of water it can hold is at a maximum? (2)

[12]

Question 10

a) Use completion of the square to find the maximum value of this ( ) in terms of if:

( ) (4)

b) The last digit in is 4 because

i) Write down the last digit in each of these numbers:

(1)

ii) Now find the last digit in the number: (4)

c) Find the sum of the sequence:

.............. (5)

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Question 11

On the next page you are given two columns of graphs. In the first column you are given the

graphs of gradient functions. In the second column you are given the graphs of functions.

You have to match each gradient function in column 1 with the graph of its corresponding

function in column 2.

Note: It is possible that there is no match, and then you just write “no match”. [12]

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First Column Second Column