CATHOLIC COLLEGE BENDIGOGENERAL MATHS (FURTHER) EXAMINATION

Friday 16th November, 2012Reading time: 1:50pm to 2:00pmWriting time: 2:00pm to 3:30pm

QUESTION AND ANSWER BOOK

Structure of bookletSection Number of questions Number of questions

to be answeredMarks

Multiple Choice 25 25 50Bivariate Data 3 3 10Trigonometry 3 3 10

Linear Programming 2 2 10Measurement 2 2 10

Total 90

Directions to StudentsMaterialsQuestion and answer booklet.Students may bring 4 handwritten, stapled, A4 pages as a resource. Calculator and writing materials are also required. The use of a grey lead pencil is recommended.

The taskPlease ensure that you write your name and circle the teacher’s name in the space provided at the top of the Multiple Choice answer page and this booklet.All Multiple Choice questions should be answered on the Multiple Choice answer sheetAll Short answer questions should be answered in the spaces provided.Write answers clearly and neatly in pen or dark pencil.

Name: ____________

Teacher: Mr Forster Mr Lahtz

Mr McConvill Mr McKee

Mr Scott

Multiple Choice Section: Each question is worth 2 marks

Section A: Bivariate Statistics

Question 1

The lengths and diameters (in mm) of a sample of jellyfish selected from a beach on the Queensland coast were recorded and displayed in the scatterplot below. The least squares regression line for this data is shown.

The jellyfish that had a diameter of 10mm had a length of

a) 7 mm b) 8 mmc) 10 mmd) 11 mme) 13 mm

Question 2

When studying the age (in days) and weights (in grams) of the Cowell Honeyeaters, it was found that they would have the following linear relationship

Weight=117.32+4.89 × Age

Determine the weight of the Cowell Honeyeater if it was 80 days old

a) 4.89 gramsb) 117.32 gramsc) 484.23 gramsd) 508.52 gramse) 9776.8 grams

Question 3

The Burkett Institute conducted a study into the general health of a person and the amount of cigarettes that they smoked a day. Their preliminary results show the following information

Average number of cigarettes: 10Correlation Coefficient: -0.823

Coefficient of Determination: 0.677

The relationship between the general health and amount of cigarettes could be said to be

a) Strong positiveb) Strong negativec) Moderate positived) Moderate negativee) No relationship

Question 4

For a set of bivariate data that involves the variables x and y, with y as the dependent variabler=–0.644 , x=5.30 , y=5.60 , sx=3.06 , s y=3.20

The equation of the least squares regression line is closest toa) y=9.15 – 0.67 xb) y=9.15+0.67 xc) y=2.05 – 0.62 xd) y=2.05 – 0.67 xe) y=2.05+0.67 x

This information is relevant for Question 5 and 6

Kate, a year 11 Maths student, looked at the amount of study she completed before each test and her mark on that test. The results are as below.

Hours Study 4 2 6 1 4 0 2 5Mark 78 56 99 40 82 43 52 90

Question 5

The least squares regression line that could be fitted to this data is

a) Mark=35.5+10.67 × Hours Studyb) Mark=−3.088+0.090× Hours Studyc) Mark=10.67+35.5× Hours Studyd) Hours Study=35.5+10.67 ×Marke) Hours Study=−3.088+0.090× Mark

Question 6

What percentage of the change in her test score is due to the change in the amount of time she studied?

a) 0.980 %b) 0.962 %c) 98.0 %d) 96.2 %e) 10.67 %

Question 7Eighteen students sat for a 15 question multiple-choice test. The number of errors made by each student on the test was plotted against the time they reported studying for the test

The equation for the least squares regression line isnumber of errors=8.8 –0.120 × study time

and the coefficient of determination is 0.8198.The value of Pearson’s product moment correlation coefficient, r, for this data, correct to two decimal places, is

a) - 0.91b) - 0.82c) 0.67d) 0.82e) 0.91

Section B: Trigonometry

Question 1

The value of x in the system of triangles above isa) 15°b) 25°c) 40°d) 50°e) 140°

Question 2

For the right-angled triangle ABC, with BC = 16 cm and AC = 25 cm, the size of angle BAC is closest toa) 7°b) 25°c) 33°d) 38°e) 40°

50°x°

25°

Question 3

When Millicent and Sophie were out walking a bush track to get home from McDonalds, they found that they got split up. However, later when Millicent was on top of a cliff, she could see Sophie. The cliff was 10m high and Sophie was located 30 m from the base of the cliff.

What was the angle of depression that Millicent was looking when she spotted Sophie?a) 0.34°b) 18.43°c) 19.47°d) 71.56°e) 70.53°

Question 4

The length of RT in the triangle shown is closest toa) 17 cmb) 33 cmc) 45 cmd) 53 cme) 57 cm

Sophie

Millicent

30 m

10 m

Question 5

Two ships are observed from point O. At a particular time their positions A and B are as shown.The distance between the ships at this time is

a) 3.0 kmb) 3.2 kmc) 4.5 kmd) 9.7 kme) 10.4 km

Question 6

The area of the above triangle is

a) 443.75 m2

b) 630.97 m2

c) 868.17 m2

d) 951.62 m2

e) 1050 m2

30 m

70 m

65°

Section C: Linear Programming

Question 1: The point (2,7) is found in the required region of which of the following inequalities

a) y>7b) x≤ 0c) 3 x+2 y ≥10d) y ≤2 xe) 2≥ y

Question 2: For the pair of simultaneous equations2x – 3y = 7 and 3x = 5 – y

the solution is

a) x=−2 , y=−1b) x=−1 , y=−3c) x=−1 , y=2d) x=2 , y=−3e) x=2 , y=−1

Question 3

The graph above has the inequality of

a) y ≤2b) y ≥2c) y<2d) y>2e) y=2

Required Region

y

x

2

Question 4The following inequalities define a region in the x-y plane, where the required region is shaded.

x ≥ 0x ≤ 2y ≥ 03x – y ≥ 3x + y ≤ 3

Which one of the following diagrams represents this region?

a) b)

c) d)

e)

Question 5In the diagram below, the shaded region (with boundaries included) represents the feasible region for a linear programming problem with the objective function Z = 5x + 3y.

The maximum value of Z for this feasible region occurs at the point with coordinates

a) (0, 80)b) (40, 60)c) (60, 50)d) (70, 30)e) (85, 0)

Question 6

The local toy shop sells 2 handmade dolls, the Jenna and the Keeley. The Jenna takes 3 hours to make and the Keeley takes 30 mins. Let J represent the number of Jenna dolls and K represent the number of Keeley dolls. Which of the following inequalities represents the amount of dolls that can be made in one full day?

a) 3J + 12

K

b) 3J + 12

K ≤ 24

c) 3J +30 Kd) 3J +30 K ≤ 24

e)13

J+2 K ≤ 24

Section D: Shape and Measurement

Question 1

108 cm is equal to

a) 10.8 mb) 10800 mmc) 1080 mmd) 0.0108 kme) 1.08 cm

Question 2

Cole and Cole Builders are building a wooden garden gate for a customer. The design of the gate is sketched below

How many linear metres of wood are needed for the frame of the gate?

a) 1.44 mb) 2 mc) 4 md) 5.44 m

1.2 m

0.8 m

e) 6.08 m

Question 3

The Cakebread Bakery makes cakes and bread. Their world famous sponge cake is covered (top and sides) with icing sugar. The cake must be circular in shape with a diameter of 24 cm and a height of 15cm.

What area of the cake would be covered in icing sugar?

a) 452.39 cm2

b) 1130.97 cm2

c) 1583.36 cm2

d) 2035.75 cm2

e) 4071.50 cm2

Question 4

The following two triangles are similar.

Therefore the value of x is

a) 4.5 cmb) 6.5 cmc) 7.5 cmd) 82 cme) 104 cm

x

7.5 cm 26 cm 18 cm

30 cm

Question 5

The Daley Newspaper is a tabloid paper that always ensured it has a big image on the front page. The paper’s definition of big is that it has to cover at least 75% of the area. The proposed front page was sent to the editors.

Does this meet the requirements set by the paper?

a) Yes as it covers 87.5%b) Yes as it covers 171%c) Yes as it covers 58.33%d) No as it covers 58.33%e) No as it covers 66.67%

40cm

60cm

70cm 80cmIMAGE

Daley Newspaper

Question 6

Dah Dee designs the flower bed of her garden in the shape of a D. It is a semicircle, with a smaller semicircle taken out of the middle. She raises the bed up 15 cm from the rest of the garden and therefore the garden bed looks like the diagram below.

The volume of dirt needed to construct this garden bed is

a) 0.2356 m3

b) 0.4712 m3

c) 0.7363 m3

d) 1.8850 m3

e) 47.12 m3

15 cm

2.5 m 1.5 m

Short Answer Section

Section A: Bivariate Statistics (10 Marks)

Question 1: One of the local scout troops, Aaron Troop, has 10 members of the troop. As part of their regular activities they need to sell cookies and collect merit badges.

The amount of cookies sold and the number of badges each member of the troop has is listed below.

Troop Member Number of merit badges

Number of cookies sold

Darren 2 4Jordan 3 3Daniel 1 3Jack 5 8Oliver 3 6Johns 7 11Martin 2 6Michael 0 4Harry 10 9Lucas 6 7

a) Which member has the most merit badges? (1 Mark)

b) Troop leader, Aaron, was constructing a scatterplot of the data, however discovered he had left off Michael. Complete the scatterplot by including his data point. (1 Mark)

Series 1

1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Merit Badges

Number of cookies

c) Using your calculator and using the merit badges as the independent variable, determine, in terms of the variables, the least squares regression line that could be fitted to the scatterplot (2 Marks)

Question 2: Another local troop, Bruns Troop, discovered that when looking at the same data there was a coefficient of determination (r2) of 0.734. It was also discovered that the least squares regression line for their troop has the equation of

Cookies sold=3.861+0.93 ×Number of Merit badges

a) Interpret the coefficient of determination (2 Marks)

b) Interpret the gradient of the least squares regression equation (1 Mark)

Question 3: A third troop, Clark Troop, discovered that upon some basic statistical analysis found the following

Average number of Merit badges = 3.73Average number of cookies sold = 6.15

Standard Deviation of Merit Badges = 2.87Standard Deviation of Cookies sold = 2.98Pearson’s Correlation Coefficient = 0.812

a) Interpret the strength of the relationship (1 Mark)

b) Using the information above, determine the least squares regression line for the Clark troop. (2 Marks)

Section B: Trigonometry (10 Marks)

Question 1: The fierce pirate Captain Bradybell hid some gold on a remote island. The captain dropped anchor and then travelled 200m South and then 500 m East to reach the island.

a) On the diagram above draw in the path the pirates travel to get to the island (1 Mark)b) Determine the angle of the triangle formed at the ship (2 Marks)

c) Calculate the bearing of the island from the ship. (1 Mark)

Captain Bradybell’s ship

Remote island

Question 2: The Captain sent second mate Delves on a hike. She was told to walk on a bearing of 75°T for 150 m, while the Captain headed on a bearing of 115°T for 200 m where she found the patch of sand she was looking for.

a) Show that the angle between Captain Bradybell and second mate Delves is 40°. (1 Mark)

b) When the Captain found the gold, what was the distance between the Captain and second mate Delves? (2 Marks)

Delves

200 m

150 m

Captain

Question 3: The gold was buried in a small patch of sand in the middle of the island. The small patch is marked by 3 huge palm trees that are 15 m, 17 m and 20 m apart.

a) Calculate the area of sand that Captain Bradybell must search to find her gold. (2 Marks)

b) With the ship’s crew able to search the area at 10 m2 per hour, how long, to 1 decimal place, will it take search the entire area? (1 Mark)

17 m

20 m

15 m

Section C: Linear Programming (10 Marks)

Question 1: The Good Munchies Factory makes Lemon Facesuckers and Butterscotch Toothbusters on one of their machines. You have been provided with the following information:

1. The machine can operate for 50 hours a week and takes 1 hour to makes a box of Lemon Facesuckers and 30 mins to make a box of Butterscotch Toothbusters.

2. At least 20 boxes of Butterscotch Toothbusters must be produced in a week.3. The machine must make at least 10 more boxes of Butterscotch Toothbusters as it does

Lemon Facesuckers.

Let L be the number of hours that the machine makes Lemon Facesuckers. Let B be the number of hours that the machine makes Butterscotch Toothbusters.

Based on conditions 1 and 3, you have established the following two inequations:

Inequality 1: L+ 12

B ≤ 50

Inequality 3: B ≤L+10

a) What is the maximum number of hours the machine can operate for in 4 weeks? (1 Mark)

b) Write an inequality that represents condition 2 (1 Mark)

f(x)=20

f(x)=-2x+100

10 20 30 40 50 60 70 80 90 100

10

20

30

40

50

60

70

80

90

100

L

B

c) Sketch the missing inequality on the graph, shade all three inequalities and indicate the feasible region. (3 Marks)

Question 2: A rival confectionary company, Mouthwatering Methods, also produces Lemon Facesuckers and Butterscotch Toothbusters. The Mouthwatering Methods company produces their boxes according to the following inequalities

Inequality 1 : L≥ 10

Inequality 2 :20L−10B ≤ 110

Inequality 3 :B ≤ L+40

When graphed it produces the following graph, where the feasible region is unshaded.

a) Determine the 3 corner points of the feasible region. (1 Mark)

b) A box of Lemon Facesuckers sells for $22, while a box of Butterscotch Toothbusters sells for $17. Calculate the income at the corner points and determine the highest income. (2 marks)

At Mouthwatering Methods, a box of Lemon Facesuckers costs $8 to make, while a bag of Butterscotch Toothbusters costs $11 to make. The Good Munchies Factory made a profit of $990.

c) Determine how much more profit Mouthwatering Methods makes than Good Munchies Factory. (2 mark)

Section D: Measurement (10 Marks)

Question 1: The Ghanda Mars Foundation is looking at putting the first person on Mars. Their world leading scientists have just unveiled the rocket ship.

a) What 2 shapes are used to create the rocket? (1 Mark)

b) What is the surface area of the rocket ship? (2 Marks)

c) Determine the height of the rocket. (2 Marks)

A model of the rocket is given to all family members that worked on the project. The model was for

a scale factor of 1

200 (i.e. 1 centimetre represented 2 metres).

d) Determine the height, in centimetres, of the model rocket (1 Mark)

8 m

20 m60 m

Question 2: Lead astronaut Madam Nally Gallagher needs to carry her oxygen pack on her back when she walks on the Mars surface. One of the cylinders has a height of 50 cm and a radius of 10cm.

a) What is the width of the oxygen pack? (1 Mark)

b) Determine, in cm3, the volume of the 2 cylinders. (1 Marks)

c) Assuming that Nally uses 500 mL of oxygen a day, how many days could she stay on the Mars surface without needing to change her oxygen pack? (2 Marks)

50 cm

10 cm