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St John’s College

UPPER V

Mathematics: Paper II

Learning Outcomes 3 and 4

August 2010 Time: 3 hours

Examiner: SLS / BH Marks: 150

Moderator: DG

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY

1. This question paper consists of 12 pages:

This examination script of 9 pages.

A one page ANSWER SHEET

A Formula Sheet.

Please check that your paper is complete.

2. Answer ALL the questions.

3. Please note that diagrams are not necessarily drawn to scale.

4. All necessary working details must be shown.

5. Approved non-programmable and non-graphical calculators may be used,

unless otherwise stated.

6. Answers must be rounded off to two decimal digits, unless otherwise stated.

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

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3;6 C

Y

X

3;2R

1;4 A

O

QUESTION 1

)1;4( A , (2; 3)R and (6; 3)C are points in the Cartesian plane.

a) Calculate the length of RA , leaving the answer in simplest surd form. (2)

b) Calculate the coordinates of F, the midpoint of RC. (2)

c) Determine the size of , correct to one decimal place. (3)

d) Determine the equation of the line AR. (3)

e) Prove that ΔARC is a right-angled triangle. (2)

f) Find point S, if ARCS forms a parallelogram. (3)

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QUESTION 2

a) If 04cos5 and 0tan , use a diagram to calculate the value of )sin( (4)

b) Evaluate the following expression without using a calculator and showing all working:

2

2

cos 135 2sin15 .cos195

cos 15 sin15 .cos15 .tan15

(6)

c) Prove the following identity:

122 2

sin .costan

1 cos sin

x xx

x x

(3)

d) Solve for x if 5

15tan x where ]180;90[ x (3)

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QUESTION 3

Zakumi was the official mascot for

the 2010 FIFA World Cup. He is a

leopard that loves to play football and

his main priority was to turn the 2010

World Cup into one joyful and

unforgettable party!

Assume the Zakumi sticker used on

the axes below has the identical front

and back.

a) Write down the rule for the transformation from A to A . (1)

b) Write down the rule for the transformation from A to A . (3)

c) The lowest point of the 2010 emblem lies at (1; 3)B . Write down the coordinates

of B if the emblem is rotated 90 anti-clockwise about the origin. (2)

d) The bottom left point of the South African flag is given by ( 3; 2)C and the area

of the flag is 1,5 square units. The flag is then enlarged through the origin by a

scale factor of 3. Determine the coordinates of C and the area of the enlarged flag.

(4)

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QUESTION 4

Below are the ages of the starting fifteen Springbok players for the first 2010 tri-nations

rugby match which we lost 32-12 against New Zealand.

25 25 26 26 27 27 27 27 27 28 29 29 30 32 33

a) Calculate the mean age of the starting fifteen. (2)

b) Calculate the standard deviation of the starting fifteen. (3)

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In addition, the ages of the NZ starting lineup for the first tri-nations match of 2010

produces the following measures:

28,12x 2,88n

c) How many of the NZ rugby players would you expect to be older than 31? (3)

d) Draw a box-and-whisker plot of the ages of the Springbok rugby team above the

NZ box-and-whisker on the given ANSWER SHEET. (4)

e) NZ comfortably beat the Springboks in the first match of the tri-nations this year.

Comment on the potential influence of the two teams’ age distribution on their

chances of winning the World Cup in 2011. (2)

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QUESTION 5

The following trigonometric functions have been drawn on the interval [ 90 ; 360 ]x .

( ) cos2f x x b ( ) sin( 30 )g x a x

a) Determine the value of a and b. (2)

b) Demonstrate that f and g do not intersect at 135°. (3)

c) State the period of f. (1)

d) Determine the equation of h(x), the reflection of f in the x-axis. (2)

e) Determine the values of [0 ; 360 ]x , where ( ) ( ) 0f x g x . (3)

f) If f(x) is shifted 60° to the right, give the equation of the function thus created. (2)

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QUESTION 6

A local gym is trying to decide which of two weight loss exercise programmes it will run for its

clients. The gym conducts a pilot study with Programme A and Programme B to investigate the

weight lost by 60 men in each programme. The cumulative frequency graph below is drawn

against the weight lost by each man in the two groups over the course of a month.

Use the graph to answer the following questions.

a) How much weight did the four men at point M lose? (1)

b) Estimate the median weight loss of Programme A. (2)

c) Estimate the lower quartile of Programme B. (2)

d) Estimate how many men lost more than 6 kgs on Programme B. (2)

e) In which programme did a greater number of men lose more than 8kg in the

month? (2)

f) Write a brief statement discussing which programme is more successful. Justify

your claims (3)

12

Weight lost (kg)

Fre

qu

ency

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QUESTION 7

Below is an image of a starfish set on Cartesian axes. The centre of the starfish has been

placed at the origin. You may assume the starfish is perfectly symmetrical.

a) Solve the angle of inclination of line OA, where O is the origin. Give this answer

correct to 1 decimal place. (2)

b) Calculate the co-ordinates of point B, as represented in the diagram above. (5)

c) How many axes of reflection exist in the starfish? (2)

d) Below is a diagram that shows the starfish as a regular pentagon.

(1) Determine the length of OA. (2)

(2) Hence, or otherwise, determine the area of the pentagon that contains the

starfish. (4)

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2

2

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QUESTION 8

In a rugby match, Bryan Habana scores a try at point T. A conversion kick must then be

taken by Morne Steyn at any point on TC, where TC is perpendicular to TP. In the

diagram, TP = d, PQ = x, ˆTCP and ˆPCQ .

a) Determine an expression for angle ˆCPQ in terms of (2)

b) Prove that cos( )

sinx

CP

(4)

c) If d = 10m, 5,6x m and the distance TC = 26m, solve (5)

(Solve angles accurate to 1 decimal place and distances to 3 decimal places)

11

QUESTION 9

a) Prove that sin( ) sin( ) 2cos .sinA B A B A B (3)

b) Prove that xxx 3sin4sin33sin (6)

c) Choose one of the identities, either question (a) or (b) above to assist you in solving the

general solution of sin3 sin 0x x (7)

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QUESTION 10

Two of the cogs that form part of the clock in the St John’s

College tower are represented in the diagram below. We

will represent the two cogs by a smaller circle with centre

O, the origin, and a larger circle with centre M. The point

of contact of the two circles is at point P(-3; 2). The radius

of the larger circle is 2 13 .

a) Determine the equation of the smaller circle centred at the origin. (2)

b) Determine the equation of the line OM . (2)

c) Determine the equation of the common tangent, t, to both circles. (4)

d) If x a at point M, write b in terms of a. (1)

e) Determine the equation of the larger circle. (8)

f) Which cog will Mr. Young have to replace first? (1)

18

QUESTION 11

Instead of writing a function (or relation) as y in terms x [usually ( )y f x ], we write both

y and x in terms of a new variable, which we call t. These are called parametric

equations.

Jules Lissajous, a nineteenth century French physicist, created parametric equations and used

their graphs to help him study harmonic motion. The graphs created have become known as

Lissajous figures. Lissajous figures are widely used to study signal strength in wireless

communications.

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EXAMPLE: Two rather fetching Lissajous figures are illustrated below.

sin(3 )x t

cos(2,5 )y t

sin(4 )x t

cos(7 )y t

We create Lissajous figures using the following parametric equations

sin( )x at K cos( )y bt

Use the Lissajous figure represented in the graph below to answer the questions that follow:

Note that the curve is symmetrical about the x-axis.

a) What is the value of 0 , 90t at (0; – 0,707) ? (2)

b) Find the value of the missing variables for (1) ; (2) and (3) in the table below. (4)

t x y

0° 1 1

(1) 0,5 0

90° -1 0

(2) 120°

180° 1 -1

(3) 270°

c) Solve a, b and K (4)

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NAME: SET:

ANSWER SHEET

QUESTION 4 (d)

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MATHEMATICS

INFORMATION SHEET

a

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2

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1

1

1 ; 1r

r

ara

i

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11

1 ; 11 r , 0r

cnbnaTn 2

snn

fnTTn2

2111

where f is the first term of the first difference

and s is the second difference

h

xfhxfxf

h

0lim

inPA 1 inPA 1

niPA 1 niPA 1

i

ixF

n11

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2;

2

2121 yyxxM

cxmy )–(– 11 xxmyy

12

12

–

–

xx

yym tanm

222 )–()–( rbyax

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:ABCIn C

c

B

b

A

a

sinsinsin

Acbcba cos.2–222

CbaABCarea sin.2

1

sin.coscos.sin)(sin sin.cos–cos.sin)–(sin

sin.sin–cos.cos)(cos sin.sincos.cos)–(cos

1cos2

sin21

sincos

2cos

2

2

22

cos.sin22sin

sincos;sincos; AAAA xyyxyx

n

xx

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)()(

Sn

AnAP

) (–)()() ( BandAPBPAPBorAP

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