AUCKLAND GRAMMAR SCHOOL
MATHEMATICS
Forms 3A − 3C
Final Examination Term 4, 2011
NAME:
FORM:
Time Allowed:
2 Hours
Total Marks:
175 marks
INSTRUCTIONS: Answer all questions. Use pencil for diagrams and graphs.
MARKS
Section A: Statistics
11 marks
Section B: Number & Sets
31 marks
Section C: Algebra
63 marks
Section D: Measurement
24 marks
Section E:
Geometry
22 marks
Section F:
Relations and Graphs
24 marks
Show all your working clearly to back up your answers.
Section ACalculators
- Statisticsare allowed in this examination
1. A survey of a small primary school on the North Shore showed the following information for the
predicted winner of the Rugby World Cup.
Predicted Winner of the Rugby
World Cup 2011
Aus
NZ
France
England
Scotland
Wales
South Africa
a) What percentage of the students voted for South Africa?
.…………………. [1]
b) If 180 students chose New Zealand, how many students were surveyed altogether
…………………. [2]
2. Joe’s mean mark for his 5 best subjects is 80%. If he includes French, his average drops to 77%
across 6 subjects. What mark did he get for French?
.…………………. [3]
Page 2
3. Lance was training for a 100km cycle ride. He trained for 5 hours each week and recorded the
total distance covered each week over a six week period.
Here is a summary of this data.
Training Log
160
140
Distance, km
120
100
80
60
40
20
0
Week
1
2
3
4
5
6
Distance
95
110
120
135
145
150
a. Calculate his average speed for week 1.
.………………….[1]
b. What was the percentage increase in his average speed from week 1 to week 6?
.………………….[2]
c. What was his average speed for the entire 6 weeks?
.………………….[2]
Page 3
Section B - Number & Sets
1. Evaluate the following:
(Give any fractional answers in their simplest form.)
a) 6
3
14 7
b) 𝟗
c)
𝟐
𝟐𝟓
𝟏
𝟐
𝟑
𝟑
𝟏
𝟕
𝟒
𝟖
….…………………. [1]
𝟓
𝟐
….…………………. [1]
…………..…………. [1]
2. Express
𝟏𝟐
𝟏𝟓
as a percentage.
……..………………. [1]
3. Arrange in order, from largest to smallest:
, 60%, 0.65,
………………………..…..………………. [2]
4. Roster the elements of these sets:
a) {factors of 81}
………..….……………………… [1]
b) {multiples of 3 less than 11}
………….…….………………… [1]
c)
𝒙: 𝟑
𝒙
𝟏, 𝒙 ∈ 𝕎
………..………………………… [1]
d)
𝟐𝒙
𝟑
: 𝟑
𝒙
𝟏, 𝒙 ∈ ℕ
….…………….…………………. [1]
Page 4
5. If Tom splits two and a quarter slabs of chocolate between his friends so that each gets three
eighths of a slab, how many friends does he have?
……………………. [2]
6. Fifteen of the forty animals in an animal shelter are dogs and the rest are cats. What percentage
are cats?
……..………………. [2]
7. The Herald reported that 1,620,000 people in Auckland watched the finals of the Rugby World
Cup live.
a) If this number represents 90% of Auckland, what is the total population of Auckland?
……..………………. [2]
b) The number of 1,620,000 was rounded to the nearest 1000 people.
i. What is the least number of people that actually watched the game?
……..………………. [1]
ii. What is the greatest number of people that actually watched the game?
……..………………. [1]
Page 5
8. Jake wants to buy a new pair of jeans.
a) He sees a pair at Just Jeans priced at $85 (including GST) but now reduced by 20% in
their spring sale. How much would he have to pay?
…..…………………. [2]
b) The ones he sees in Farmers cost $69 including GST of 15%. What is the price exclusive
of GST?
…..…………………. [2]
𝟑
9. Tom arrived late at school and consequently missed the first of period 1, which was Art. He
𝟏𝟎
then realised that he’d left his paint brushes and acrylics in his locker, so he was delayed by a
𝟏
further
of the lesson. If each lesson lasts 80 minutes,
𝟏𝟔
a) How many minutes of Art did Tom miss?
……………………………[2]
b) What fraction of the Art lesson was he present in class for?
……………………………[1]
Page 6
10. Shade in the region 𝑨 ∩ 𝑩
A
B
[1]
Shade in the region 𝑨 ∪ 𝑩′
A
B
[1]
11. For their end of year party, the boys in the Distance Squad were asked to bring cakes, drinks and
sandwiches. 58 brought cakes, 70 drinks and 46 brought sandwiches. 10 brought all three, 23
brought both cakes and drinks, 35 brought both drinks and sandwiches, 24 brought only cakes
and 14 forgot to bring anything.
Display this information in the Venn diagram below.
[1]
Use it to find how many boys:
a) brought cakes and drinks only.
…………………………. [1]
b) brought just sandwiches.
…………………………. [1]
c) are in the Distance Squad.
…………………………. [1]
Page 7
Section C - Algebra
1. Simplify the following:
a) 7𝑦 5𝑥 3𝑥 2𝑦
8𝑥
……..………………. [1]
b)
……..………………. [2]
c)
𝑥
15𝑥
6𝑥
……..………………. [2]
d)
……..………………. [2]
e)
……..………………. [2]
f)
g)
𝑥
3 𝑥
4
3 𝑥
……..………………. [2]
2
……..………………. [2]
Page 8
h)
𝑦
2
2𝑦
6
……..………………. [2]
i) √4𝑎
8𝑎𝑏
4𝑏
……..………………. [2]
2. Given that 𝑝
2, 𝑞
a) 3𝑝 7𝑞
3 𝑎𝑛𝑑 𝑟
8, evaluate:
……..………………. [1]
b)
𝑝𝑞 𝑟
……..………………. [2]
c)
……..………………. [2]
Page 9
3. Fully factorise the following:
a) 72
18𝑥
……..………………. [1]
b) 12𝑦
21𝑥𝑦
9𝑥𝑦
……..………………. [2]
c) 𝑥
5𝑥
14
d) 2𝑥
5𝑥
……..………………. [2]
……..………………. [2]
12𝑥
e) 2𝑥
18
……..………………. [2]
f) 𝑝
𝑞
……..………………. [1]
g) 16𝑥
𝑦
……..………………. [2]
Page 10
4. Solve the following equations:
13
a)
b) 6 𝑥
3
0
5 𝑥
……..………………. [2]
8
……..………………. [2]
2
c)
……..………………. [2]
d) 𝑥
3𝑥
40
0
…….……..………………. [2]
e)
𝑥
2 𝑥
3
6
…….........………………. [3]
Page 11
5.
a) Two sides of a rectangular garden bed are to
be lined with logs as shown. Three logs are
needed for each side but 1m is cut off the
third log to make sure the logs don’t extend
beyond the garden. The width of the garden
is 3m and its area is 42m2.
If each uncut log is x m, what is the value of
x?
……..………………. [3]
b) An ornamental pond is in the form of a right
angled triangle. If the two shorter sides are
𝟐𝒙 𝟑 𝒂𝒏𝒅 𝒙 𝟒, calculate the length of
the shortest side, given that the area of the
flower bed is 15 m2
……..………………. [4]
Page 12
6. The first four triangular numbers are 1, 3, 6, and 10.
𝒏 𝒏 𝟏
They all satisfy the formula 𝒕𝒏
where 𝒕𝒏 is the nth triangular number.
𝟐
a) What is the 20th triangular number?
……..………………. [2]
b) Which triangular number is 36?
……..………………. [3]
7. The Li children (Tang and Wei) and Smith children (Amy, Len and Callum) have been collecting
football cards. Tang has four times as many cards as Wei, Amy has two more cards than Tang,
Len has six cards less than Amy and Callum has half as many as Len. The Smith children have
eleven more cards than the Li children.
Write an equation and use it to find the number of cards Amy has.
……..………………. [3]
8. Form and solve an equation to find the three consecutive even numbers that add to 54.
……..………….........……. [3]
Page 13
Section D - Measurement
1. The rhombus shown below has sides of length 5cm and diagonals of length 6cm (BD) and 8cm
(AC).
Calculate
a) the area of the rhombus.
……..………………. [2]
b) the perimeter of the rhombus
……..………………. [1]
2. Peter took 10 minutes to ride to school which is 3.5km away from his house. One of his friends
plays a practical joke on him and lets all the air out of his tyres, so he has to walk home now. If
he walks at one seventh of the speed he rides, how long does he take to get home?
……..………………. [2]
Page 14
3. The hollow prism shown below was formed by cutting a rectangular hole out of a solid block of
metal
a) What is the volume of the prism?
……..………………. [3]
b) What is the mass of the prism in grams if it is made of a metal which has a density of
8000kg per m3?
……..………………. [2]
c) The engineer wants to coat both the inside and outside in a resin. What is the total
surface area to be covered?
……….………….[4]
Page 15
4. Calculate the area of the following regular hexagon
[2]
5. Liza has asked Henry to fetch her some water from a tap. He fills up his cylindrical bucket which
has a base area of 1,200 cm2 and a height of 30cm.
a) If the water from the tap flows out at 18 litres per minute how long does he take to fill
one bucket of water?
……….…………. [2]
b) He stumbles on his way to collect a second bucket of water and makes a hole in the
bottom of the bucket. This hole loses water at a rate of 500ml per minute, how much
longer will it now take for the water to reach the top of the bucket? Give your answer as
a fraction of a minute.
……….…………. [2]
Page 16
6. A “BB” bullet is fired vertically upwards into the air. Its speed, s, (in metres per second) is given
by the formula 𝑠 40 16𝑡, and its height, h (in metres above ground) is given by the formula
1 where t is the time in seconds since the “BB” gun was fired.
ℎ 2 20𝑡 4𝑡
Find:
a) the speed of the bullet when t=1.5 seconds
……….…………. [1]
b) how many seconds pass before the speed of the bullet is zero.
……….…………. [1]
c) the height of BB bullet when t=0 seconds. Explain what your answer represents.
……….…………. [2]
Section E - Geometry
1. Calculate the labelled angles, giving reasons for your answers:
a)
[4]
Page 17
b)
[4]
2.
Prove that 𝑨𝑬𝑪
𝟗𝟎°. Clearly state any geometrical reasons you have used.
[4]
Page 18
3. Given the following details:
R represents reflection in the line 𝑦
1
2
S represents translation by the vector
1
T represents rotation of 90° about the origin.
a) Draw and label the mirror line 𝑦
1 on the grid below.
[1]
b) Show and label the result of the following transformations acting on the flag A:
1. T(A)
[1]
2. RS(A)
[2]
c) Triangle 𝑩′ is the result of the following transformations 𝑩
𝑹𝑻𝑻 𝑩
Find the coordinates of the original triangle 𝑩
(
,
) (
,
) (
,
)
[3]
4. Here is a regular polygon.
a) What name is given to this shape?
[1]
b) Calculate the size of each interior angle of this shape.
[2]
Page 19
Section F - Relations & Graphs
1. Below each graph write the letter of the equation that matches:
a) Straight lines
y
y
4
4
2
−4
y
2
x
−2
2
4
4
6
−4
−2
2
−2
−2
−4
−4
−6
−6
………………
2
x
4
6
−4
B: 4𝑦
F: 𝑦
1.5𝑥 12
2𝑥 4
2
−4
−6
C: 6𝑥
G: 𝑥
………………
4𝑦 24
2.5
D: 𝑦
H: 𝑦
2.5
𝑥
b) Parabolas
6
4
−2
………………
A: 𝑦 4𝑥 4
E: 2𝑥 3𝑦 12
−2
4
[3]
y
y
4
4
y
4
2
2
2
−4
−4
−2
2
−2
2
4
y
y
2
−4
−4
………………
6
−6
−4
−2
2
6
−2
4
−4
2
−6
4
4
−2
𝑥
𝑥
−4
8
2
4
−2
………………
4
−2
2
………………
y
2𝑥
𝑥
3
−2
−4
………………
A: 𝑦
E: 𝑦
I:𝑦
−4
−2
−2
−4
4
B: 𝑦
F: 𝑦
J: 𝑦
−2
2
4
………………
C: 𝑦
𝑥
2
𝑥
2
G:
𝑦
𝑥
1 𝑥 𝑥 3
K: 𝑦
𝑥
3𝑥
𝑥
𝑥 2
−8
………………
D: 𝑦
𝑥 3 𝑥 1)
2𝑥 3
H: 𝑦 𝑥
4
L: 𝑦
𝑥
[6]
Page 20
6
2. Use the axes below to:
a) Plot the straight line which shows that the sum of two numbers 𝒙 𝒂𝒏𝒅 𝒚 is equal to
five.
[2]
b) State the gradient of this graph
Gradient = ………………………………………[1]
y
6
4
2
x
−6
−4
−2
2
4
6
8
−2
−4
𝟏
c) Complete the table below and hence draw the graph of 𝒚
𝟑
𝒙
𝑥
𝑥
𝒚
𝟏
𝒙
𝟐
𝟐
0
7
0
2
5
𝟐 𝒙
𝟓
𝟏
𝟎
𝟏
3
4
6
𝟐
𝟐
𝒙
𝟑
𝟐 𝒙
𝟓
𝟒
6
1
𝟑
5
6
[3+3]
d) Give the co-ordinates of the point/s where your two graphs intersect.
……………………………………………………………………………………… [2]
3. Complete the statements:
a) 𝑥, 𝑦 : 𝑦 𝑥 3
. . .
2 ,
,
1,
, 0,
, 1,
, 2 ,
. . .
[1]
b)
𝑥, 𝑦 : 𝑥
3𝑦
1
…
, 3
,
5,
,
, 1
, 0,
,
,1 ...
[2]
c) . . .
3,5 ,
2,4 ,
1,3 , 0,2 , 1,1 , 2,0 . . .
𝑥, 𝑦 : __________________________
[1]
Page 21