Answer all questions in the answer booklet provided. Please start each question on a new page.
Full marks are not necessarily awarded for a correct answer with no working. Answers must be
supported by working and/or explanations. Solutions found from a graphic display calculator should
be supported by suitable working. For example, if graphs are used to find a solution, you should sketch
these as part of your answer. Where an answer is incorrect, some marks may be given for a correct
method, provided this is shown by written working. You are therefore advised to show all working
Question
1. [Maximum mark: 13]
Scott purchases food for his dog in large bags and feeds the dog the same amount of
dog food each day. The amount of dog food left in the bag at the end of each day can be
modelled by an arithmetic sequence.
On a particular day, Scott opened a new bag of dog food and fed his dog. By the end of the
third day there were 115.5 cups of dog food remaining in the bag and at the end of the eighth
day there were 108 cups of dog food remaining in the bag.
(a) Find the number of cups of dog food
(i) fed to the dog per day;
(ii) remaining in the bag at the end of the first day. [4]
(b) Calculate the number of days that Scott can feed his dog with one bag of food. [2]
In 2021, Scott spent $625 on dog food. Scott expects that the amount he spends on dog food
will increase at an annual rate of 6.4%.
(c) Determine the amount that Scott expects to spend on dog food in 2025. Round your
answer to the nearest dollar. [3]
(d) (i) Calculate the value of \(\sum_{n-1}^{10}(625\times 1.064^{(n-1)})\)
(ii) Describe what the value in part (d)(i) represents in this context. [3]
(e) Comment on the appropriateness of modelling this scenario with a geometric sequence. [1]
Ans
Question
2. [Maximum mark: 15]
A cafe makes x litres of coffee each morning. The cafe’s profit each morning, C, measured
in dollars, is modelled by the following equation
\(C=\frac{x}{10}(k^{2}-\frac{3}{100}x^{2})\)
where k is a positive constant.
(a) Find an expression for \(\frac{dC}{dx}\) in terms of k and x . [3]
(b) Hence find the maximum value of C in terms of k . Give your answer in the form pk3,
where p is a constant. [4]
The cafe’s manager knows that the cafe makes a profit of $426 when 20 litres of coffee are
made in a morning.
(c) (i) Find the value of k .
(ii) Use the model to find how much coffee the cafe should make each morning to
maximize its profit. [3]
(d) Sketch the graph of C against x , labelling the maximum point and the x-intercepts
with their coordinates. [3]
The manager of the cafe wishes to serve as many customers as possible.
(e) Determine the maximum amount of coffee the cafe can make that will not result in a
loss of money for the morning. [2]
Ans
Question
3. [Maximum mark: 18]
The Voronoi diagram below shows four supermarkets represented by points with coordinates
A(0, 0), B(6, 0), C(0, 6) and D(2 , 2). The vertices X, Y, Z are also shown. All distances
are measured in kilometres.
(a) Find the midpoint of [BD]. [2]
(b) Find the equation of (XZ). [4]
The equation of (XY) is y = 2 − x and the equation of (YZ) is y = 0.5x + 3.5.
(c) Find the coordinates of X. [3]
The coordinates of Y are (−1 , 3) and the coordinates of Z are (7 , 7).
(d) Determine the exact length of [YZ]. [2]
(e) Given that the exact length of [XY] is \(\sqrt{32}\), find the size of XŶZ in degrees. [4]
(f) Hence find the area of triangle XYZ. [2]
A town planner believes that the larger the area of the Voronoi cell XYZ, the more people will
shop at supermarket D.
(g) State one criticism of this interpretation. [1]
Ans
Question
4. [Maximum mark: 15]
A student investigating the relationship between chemical reactions and temperature finds
the Arrhenius equation on the internet.
\(k=Ae^{-\frac{c}{T}}\)
This equation links a variable k with the temperature T, where A and c are positive
constants and T > 0.
(a) Show that \(\frac{dk}{dT}\) is always positive. [3]
(b) Given that \(\lim_{T\rightarrow \infty }k=A \ and \ \lim_{T\rightarrow 0}k=0,\),
sketch the graph of k against T. [3]
The Arrhenius equation predicts that the graph of ln k against \(\frac{1}{T}\)
is a straight line.
(c) Write down
(i) the gradient of this line in terms of c ;
(ii) the y-intercept of this line in terms of A. [4]
The following data are found for a particular reaction, where T is measured in Kelvin
and k is measured in cm3 mol