## Transcribed Text

Question One (5 marks)
On the same axes sketch the following graphs (labelling the graphs carefully with
their equations, all asymptotes with their equations and all axes intercepts with their
co-ordinates) Please use your own graph paper.
a) y=2x b) y=2x -3
Question Two (5 marks)
On the same axes sketch the following graphs (labelling the graphs carefully with
their equations, all asymptotes with their equations and all axes intercepts with their
co-ordinates) Please use your own paper.
a) y=log3x b) y= log3(x+2)
Question Three (5 marks)
The population of New Zealand can be modelled by:
P = 300000(1.02)t
where t is the number of years since the beginning of 1840.
a) What was the growth rate of the population (as a percent) (1 mark)
b) What was the initial population (in 1840) (1 mark)
c) In which year does this model predict the population of NZ first reach 2
million? (3 marks)
Question Four (6 marks)
Information on the population of insects in a region is being collected.
The population can be modelled by the equation:
I = 360 × (1.014)t
where I is the number of insects and t is the number of days since the beginning of
April (when the initial estimate was made).
a) What was the initial estimate of the number of insects?
(1 mark)
b) How many more insects are there after 7 days than first estimated?
(round to the nearest whole number)
(2 marks)
c) On what date would the number of insects first exceed 720?
(3 marks)
Question Five (2 marks)
A cleaning solution made up of cleaning concentrate and water is applied to a piece
of carpet in a testing situation. Initially, 2.4 kg of dirt was spread over and trampled
in to the carpet. After the first wash it is found that 0.8 kg of dirt is removed, leaving
1.6 kg of dirt. Each subsequent wash removes a further third of the remaining dirt.
a) Write a formula to find how much dirt, A kg, will remain after the nth wash of
the carpet. (1 mark)
b) Use this equation to find the amount of dirt that remains after 6 washes
(rounded correctly to 2 decimal places) (1 mark)
Question Six (4 marks)
Part of the surface of a pond is covered by a weed. The area can be modelled by the
equation:
A = 1.2× (1.012)t
where the area A is measured in square metres (m2), and t is the number of days
since the area of the weed was first measured.
a) By what percentage growth rate of the pond weed? (1 marks)
b) How many days, after it is first measured (to the nearest day), would it take the
area of weed to increase to 300% of the original area? (3 marks)
Question Seven (5 marks)
Genevieve is investigating paper sizes. She takes measurements and finds that an
A0 size piece of paper has an area of 1 m2. The length is 119 cm and the width is
84.1 cm. When an A0 size piece of paper is cut in half it is referred to as A1 size
paper, which has an area of 0.5 m2.
This pattern continues as shown in the diagram below.
a) Give the equation for the area, A m2, of a piece of AN size paper in terms
of N, where N is the number of cuts. (1 mark)
b) Use this equation to find the area of a piece of an A8 size piece of paper.
(Rounded to the nearest cm2) (2 marks)
c) The ratio of length to width of any piece of paper is always the same.
Use this information to find the width of a piece of A7 size paper.
(Round to the nearest mm) (2 marks)
A1
A2
A3
A4
119 cm
84.1 cm
Question Eight (8 marks)
The number of radioactive atoms N of a particular material present at time
t years from the start of 2016 is modelled by the equation N = 5000(2)−kt
(a) Determine the number of radioactive atoms initially present. (1 mark)
(b) It is found that N = 2500 when t = 5 years.
Determine the exact value of the constant k. (3 marks)
(c) Find the number of radioactive atoms when t = 15 years (1 marks)
(d) A different radioactive material has the following model of decay: N = 5000(2)−0.4t
The atmosphere is deemed to be safe when there are less than 50 radioactive
particles present.
Show that the time until the atmosphere is safe is given by the formula
t = 5
log10 2
years
(3 marks)

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