Determine the y-coordinate of the point with a residual of

Question

Section One

This section has seven (7) questions. Answer all questions. Write your answers in the spaces provided.

Question 1

Five university lecturers (L₁, L₂, L₃, L₄ and L₅) have been allocated four rooms (R₁, R₂, R₃ and R₄) to teach in. Only one lecturer will teach in a room at any time. Because some of the lecturers require specialist equipment, not all the rooms can be used by all the lecturers, as shown in the graph below.

(a) What is the name of such a graph shown above, where the vertices can be split into two groups so that each edge joins a vertex from one group to a vertex in the other group?

(b) How many lecturers can use room R₄?

(c) How many rooms can lecturer L₂ use?

(d) Briefly explain whether

   (i) all five lecturers can teach at the same time?

   (ii) all four rooms could be in use at the same time?

Question 2

The first four terms of an arithmetic sequence are shown on the graph below.

(a) Deduce a rule for the n^th term of this sequence.

(b) Given that the k^th term of this sequence is 399, determine the value of k.

Question 3

Graph G is shown below with vertices A, B, C, D and E.

(a) Redraw graph G to clearly show that it is planar.

(b) State the number of loops graph G contains.

(c) Show that Euler's formula does not apply to graph G.

Question 4

The average maximum temperature, T °C, was recorded for ten weather stations, together with the altitude of the station, metres. The data is shown in the table below.

Altitude, h	163	282	330	369	450	541	624	762	889	926
Temperature, T	29	27	23	24	21	24	20	18	14	15

(a) Construct a scatterplot on the axes below that can be used to identify whether an association exists between altitude and temperature.

(b) Describe the features of the scatterplot that indicate a strong, negative and linear association exists between altitude and temperature.

(c) Estimate a value for

(i) the temperature at an altitude of metres.

(ii) the value of the correlation coefficient between the two variables.

Question 5

A digraph is shown below.

(a) State, with justification, whether the digraph contains

(i) a walk of length 6.

(ii) a Hamiltonian path.

(b) Using column and row headings in the order A - B - C, construct the adjacency matrix Mfor the digraph and explain what the number in the first row and third column of matrix M³ represents.

Question 6
A medical study measured the body mass index (BMI) and plasma ghrelin levels (PGL) of a group of patients. The results were displayed in the scatterplot below, together with the leastsquares line of best fit and the correlation coefficient between the variables.

(a) How many patients in the study with a PGL of more than 300 had a BMIbetween 18.5 and 24.5?

(b) Determine the lower and upper predicted BMI for patients with a PGL between 270 and 390.

(c) Comment on the claim that a high level of plasma ghrelin causes a patient to have a low body mass index.

Question 7

A connected planar graph Phas four faces and four vertices.

(a) Determine the number of edges graph G₂ has.

(b) In each of the following, use the additional condition only within that part of the question.

(i) Draw P so that it is simple.

(ii) Draw graph P so that it contains a Hamiltonian path but not a Hamiltonian cycle.

(iii) Draw graph P so that it contains a Eulerian trail.

 

Section Two

This section has ten (10) questions. Answer all questions. Write your answers in the spaces provided.

Question 8

(a) The first three terms, in order, of a geometric sequence are 425, 340 and 272.

    (i) Deduce a rule for the nth term of this sequence.

    (ii) Calculate the 6^th term of the sequence.

(b) The first three terms, in order, of an arithmetic sequence are 1.8, 4.1 and 6.4.

(i) The rule for the nth term of this sequence is T_n = an + b. Determine the values of a and b.

(ii) Calculate the 275^th term of the sequence.

Question 9

A public relations company was tasked with determining whether a person's support for a sugary drinks tax could be associated with their interest in the news.

The company carried out a telephone survey, where people could respond to two questions as shown in the following table:

Question	Choice of response
Are you interested in the news?	Yes or No
Do you support a sugary drinks tax?	Yes, No or Undecided

The responses to the telephone survey are summarised in this table:

		Support for a sugary drinks tax
		Yes	No	Undecided
Interest in news	Yes	556	367	233
	No	196	136	146

(a) Calculate the number of people who

    (i) answered yes to being interested in the news.

    (ii) responded to the survey.

(b) Complete the two-way table below to show the associated row percentages for the previous table, rounding percentages to the nearest whole number.

		Support for a sugary drinks tax
		Yes	No	Undecided
Interest in news	Yes
	No

(c) What percentage of those who are not interested in the news do not support a sugary drinks tax?

(d) Give an example to show whether or not there is an association between Interest in news and Support for a Sugary Drinks Tax.

Question 10

The graph below represents a network of cycle tracks. The weight on each edge is the length, in km, of that track.

(a) Determine the shortest route from A to H, stating the route and its length.

(b) Two sections of track, ?? to E and E to G, are closed for repairs and cannot be used. What effect, if any, do these closures have on the length of the shortest route from A to H?

Question 11

A sequence of four connected graphs is shown below.

(a) Complete the missing entries in the table below, where the vertex sum is the sum of the degrees of all the vertices in a graph.

Graph (n)	1	2	3	4
Vertices (V)	3	4
Vertex sum (S)	6	10

Assume the sequence of graphs continues indefinitely.

(b) A graph in the sequence has 11 faces. How many vertices does it have?

(c) Deduce the n^th term rule for S_n, the vertex sum of graph n.

Question 12

A tomato grower added varying amounts of a liquid fertiliser (xml) to the irrigation systems of twelve greenhouses and observed the resulting yield of tomatoes per plant (y kg). A sample of the data recorded is shown in the table and scatterplot below.

x (mL)	6	7	8	10	11	14	15	16	18	19	21	22
y (kg)	19	22	21	24	26	23	28	25	25	30	31	32

(a) name the response variable.

(b) For this data, calculate
    (i) the correlation coefficient.

    (ii) the values of a and b in the equation of the least-squares line y = ax+ b.

(c) What percentage of the variation in the yield per plant can be explained by the variation in the amount of liquid fertiliser added?

(d) If the amount of liquid fertiliser added to the irrigation system in a greenhouse was increased by one millilitre, what increase in the yield of tomatoes per plant can be expected? Explain your answer.

(e) Use the equation of the least-squares line to calculate the value of y when x= 4 and when x= 24.

(f) Use your answers to part (f) to draw the least-squares line on the scatterplot.

(g) Estimate the yield of tomatoes per plant when 13 ml of liquid fertiliser is added to the irrigation system and comment on the reliability of this value.

Question 13

The roads in a suburb are represented in the graph below, where the number on each edge is the length, in km, of the road.

(a) The network contains a semi-Eulerian trail that starts from G.

    (i) Explain what semi-Eulerian means.
    (ii) At which vertex does the trail end?
    (iii) Determine the number of edges in the trail.

   (iv) How many times does the trail pass through vertex D?

(b) A worker needs to leave , travel along each road once to inspect its surface and then return to . Determine the minimum distance the worker must travel.

Question 14

When Atarcoin, a new cryptocurrency was launched, one Atarcoin was valued at $4.00. After one week of trading, the value of Atarcoin had increased to $5.00, and after another week had increased to $6.25.

(a) Show that the value of Atarcoin increased by 25% each week.

The value of Atarcoin, V_n in dollars, n weeks after its launch date, can be modelled by the recurrence relation V_n+1 = 1.25V_n, V₀ = 4.

(b) Calculate the value of Atarcoin ten weeks after its launch date.

(c) At the end of which week did the value of Atarcoin first exceed $100?

(d) Graph V_n against n on the axes below.

The value of Atarcoin peaked at the end of week 38, and from that time onwards, its value fell by 30% each week.

(e) Determine the value of Atarcoin at the end of week 39.

Question 15

The temperature, °C, of an industrial oven n minutes after it is turned on can be modelled by

T_n+1 = 0.88T_n + 21.6, T₀ = 21

(a) Use the recurrence relation to complete the table of values below, rounding the temperature to the nearest °C.

n	0	5	10	15	20
Tn

(b) Sketch a graph of the temperature of the oven for the first hour on the axes below. Make sure you add a suitable scale to the vertical axis.

Question 16

A linear model was fitted to a set of data, resulting in a correlation coefficient of r = -0.93 and a least-squares line with equation y^ = 7.18 - 0.25x. A residual plot for the linear model is shown below.

(a) Calculate, and add to the plot above, the residual for the point x= 19, y = 2.23.

(b) Use the residual plot to comment on the appropriateness of fitting a linear model to the data.

(c) Determine the y-coordinate of the point with a residual of 0.16 on the above plot.

Question 17

Isla is visiting a city that has four museums: A, B, C and D. The weights on the edges of the following graph represent the time, in minutes, that it takes to walk between the museums.

(a) List, in the order visited, a set of vertices that form a Hamiltonian cycle in the graph.

(b) Determine the shortest time it would take to leave A, walk to the other museums and return to A.

The time to walk from Isla's hotel, H, to museums A, B, C and D is 24, 10, 31 and 25 minutes respectively.

(c) Add vertex H and this information to the graph above.