Helene started her walk to work at 2:15 pm.
She arrived at her work at 3:03 pm.
How long did Helene walk for?
`text(12 minutes)`  `text(48 minutes)`  `text(72 minutes)`  `text(88 minutes)` 




Aussie Maths Teachers: Save your time with SmarterMaths
Helene started her walk to work at 2:15 pm.
She arrived at her work at 3:03 pm.
How long did Helene walk for?
`text(12 minutes)`  `text(48 minutes)`  `text(72 minutes)`  `text(88 minutes)` 




`text(48 minutes)`
`text(One Strategy:)`
`text(2:15 pm to 3:00 pm = 45 minutes)`
`text(3:00 pm to 3:03 pm = 3 minutes)`
`text(Walking time = 45 + 3 = 48 minutes)`
How many days are there in 5 weeks?
7 days  25 days  35 days  50 days 




`35\ text(days)`
`text(Days in 5 weeks)`
`= 5 xx 7`
`= 35\ text(days)`
MarySue has $1.15 in 5cent pieces.
How many 5cent pieces does she have?
`17`  `21`  `23`  `565` 




`23`
`text(One strategy:)`
`text($1.15 = 115 cents)`
`10 xx 5¢ = 50¢`
`20 xx 5¢ = 100¢`
`⇒ 23 xx 5¢ = 115¢`
Kranskie delivers brochures by hand and is paid 10 cents for every brochure he delivers.
Kranskie delivers 79 brochures in his first hour of work.
How much money will he be paid for this?
`79¢`  `$7.90`  `$79`  `$790` 




`$7.90`
`79 xx 10\ text(cents)`
`= 790\ text(cents)`
`= $7.90`
In an AFL competition, Elie's team won 9 games and lost the other games.
Altogether she played 22 games.
Finish the subtraction sentence below to show the number of games she lost.
`\  9 =` 
`22  9 = 13`
`22  9 = 13`
Ernie has collected 93 bottles for recycling.
Grover has collected 88 bottles for recycling.
In total, how many bottles can Ernie and Grover deliver to the recycling depot?
`165`  `171`  `177`  `181` 




`181`
`text(One strategy:)`
`text(Total bottles)`  `=93+88` 
`=90 + 80 + 3 + 8`  
`=170 + 11`  
`=181` 
In which one of these numbers does the numeral 4 represent 4 tens?
`4389`  `438`  `401`  `7049` 




`7049`
`7047`
`7 → text(thousands)`
`0 → text(hundreds)`
`4 → text(tens)`
`9 → text(ones)`
What number is 13 less than 1005?
`902`  `908`  `992`  `1018` 




`992`
`text(One strategy:)`
`100513`  `=1005  10  3` 
`=9953`  
`=992` 
Find `x` given `100^(x2) = 1000^x`. (2 marks)
`4`
`100^(x2)`  `= 1000^x` 
`(10^2)^(x2)`  `= (10^3)^x` 
`10^(2x4)`  `= (10)^(3x)` 
`2x4`  `=3x` 
`:. x`  `= 4` 
Training program 1 has the cricket team starting from exercise station `S` and running to exercise station `O`.
For safety reasons, the cricket coach has placed a restriction on the maximum number of people who can use the tracks in the fitness park.
The directed graph below shows the capacity of the tracks, in number of people per minute.
A local fitness park has 10 exercise stations: `M` to `V`.
The edges on the graph below represent the tracks between the exercise stations.
The number on each edge represents the length, in kilometres, of each track.
The Sunny Coast cricket coach designs three different training programs, all starting at exercise station `S`.
Training program number 
Training details  
1  The team must run to exercise station `O`.  
2  The team must run along all tracks just once.  
3  The team must visit each exercise station and return to exercise station `S`. 
Complete the following sentence by filling in the boxes provided. (1 mark)
This track is between exercise station 

and exercise station 

a.  `text(Shortest distance)`  `= STUVO` 
`= 0.6 + 1.2 + 0.6 + 0.8`  
`= 3.2\ text(km)` 
b.i. `text(Eulerian trail)`
b.ii. `text(Station)\ P\ text{(only other vertex with}\ S\ text{to have odd degree)} `
c. `S and T`
A cricket team has 11 players who are each assigned to a batting position.
Three of the new players, Alex, Bo and Cameron, can bat in position 1, 2 or 3.
The table below shows the average scores, in runs, for each player for the batting positions 1, 2 and 3.
Batting position  
1  2  3  
Player  Alex  22  24  24  
Bo  25  25  21  
Cameron  24  25  19 
Each player will be assigned to one batting position.
To which position should each player be assigned to maximise the team’s score? Write your answer in the table below.
Player  Batting position  
Alex  
Bo  
Cameron 
`text{Bo (1), Cameron (2), Alex (3)}`
`text(Test different combinations:)`
`text(CBA) = 24 + 25 + 24 = 73`
`text(BCA) = 25 + 25 + 24 = 74`
`text(BAC) = 25 + 24 + 19 = 68`
`:.\ text{Combination for max score: Bo (1), Cameron (2), Alex (3)}`
Player  Batting position  
Alex  3  
Bo  1  
Cameron  2 
The Sunny Coast Cricket Club has five new players join its team: Alex, Bo, Cameron, Dale and Emerson.
The graph below shows the players who have played cricket together before joining the team.
For example, the edge between Alex and Bo shows that they have previously played cricket together.
Finn had not played cricket with any of these players before.
Represent this information on the graph above. (1 mark)
An offer to buy the Westmall shopping centre was made by a competitor.
One market research project suggested that if the Westmall shopping centre were sold, each of the three centres (Westmall, Grandmall and Eastmall) would continue to have regular shoppers but would attract and lose shoppers on a weekly basis.
Let `S_n` be the state matrix that shows the expected number of shoppers at each of the three centres `n` weeks after Westmall is sold.
A matrix recurrence relation that generates values of `S_n` is
`S_(n+1) = T xx S_n`
`{:(quad qquad qquad qquad qquad qquad qquad qquad text(this week)),(qquad qquad qquad qquad qquad qquad quad \ W qquad quad G qquad quad \ E),(text(where)\ T = [(quad 0.80, 0.09, 0.10),(quad 0.12, 0.79, 0.10),(quad 0.08, 0.12, 0.80)]{:(W),(G),(E):}\ text(next week,) qquad qquad S_0 = [(250\ 000), (230\ 000), (200\ 000)]{:(W),(G),(E):}):}`
Using values from the recurrence relation above, the graph below shows the expected number of shoppers at Westmall, Grandmall and Eastmall for each of the 10 weeks after Westmall is sold.
Give your answer correct to the nearest thousand. (1 mark)
Write down the week number in which this is expected to occur. (1 mark)
Round your answer to the nearest whole number. (1 mark)
a.  `S_1`  `= TS_0` 
`= [(0.80, 0.09, 0.10),(0.12, 0.79, 0.10),(0.08, 0.12, 0.80)][(250\ 000),(230\ 000),(200\ 000)]=[(240\ 700),(231\ 700),(207\ 600)]` 
b.  `text(Using the graph)`  
`text(Difference)`  `= 250\ 000  220\ 000`  
`= 30\ 000` 
c. `text(Testing options:)`
`S_6 = T^6S_0 = [(0.80, 0.09, 0.10),(0.12, 0.79, 0.10),(0.08, 0.12, 0.80)]^6[(250\ 000),(230\ 000),(200\ 000)] = [(text(__)), (233\ 708), (text(__))]`
`:.\ text(Maximum shoppers in Grandmall expected in week 6.)`
d. `text(Test with high integer)\ n:`
`S_50 = T^50S_0 > text(Westmall) = 218\ 884`
The preferred number of cafes `(x)` and sandwich bars `(y)` in Grandmall’s food court can be determined by solving the following equations written in matrix form.
`[(5, 9),(4, 7)][(x),(y)]=[(7), (6)]`
Use this information to explain why this matrix has an inverse. (1 mark)
`[(text( __), 9),(text( __), text( __)\ )]`
`text(the matrix has an inverse)`
a. `text(S) text(ince determinant) = 1 != 0,`
`>\ text(the matrix has an inverse)`
b. `[(7, 9),(4, 5)]`
c.  `[(x), (y)] = [(7, 9), (4, 5)][(7), (6)] = [(7),(2)]` 
`:.\ text(Preferred number of sandwich bars) = 2`
The three major shopping centres in a large city, Eastmall `(E)`, Grandmall `(G)` and Westmall `(W)`, are owned by the same company.
The total number of shoppers at each of the centres at 1.00 pm on a typical day is shown in matrix `V`.
`qquad qquad qquad {:(qquad qquad qquad \ E qquad qquad G qquad qquad \ W),(V = [(2300,2700,2200)]):}`
Each of these centres has three major shopping areas: food `(F)`, clothing `(C)` and merchandise `(M)`.
The proportion of shoppers in each of these three areas at 1.00 pm on a typical day is the same at all three centres and is given in matrix `P` below.
`qquad qquad qquad P = [(0.48), (0.27), (0.25)] {:(F),(C),(M):}`
If this were to happen, how many shoppers, in total, would be at Grandmall at this time? (1 mark)
`{:(quad qquad qquad qquad \ E qquad qquad G qquad qquad W), (Q = [(1104, \ text{___}, 1056 ), (621,\ text{___}, 594), (575, 675, 550)]{:(F),(C), (M):}):}`
What does the element `q_23` represent? (1 mark)
The average daily amount spent, in dollars, by each shopper in each of the three areas at Grandmall in 2019 is shown in matrix `A_2019` below.
`qquad qquad A_2019 = [(21.30), (34.00), (14.70)] {:(F),(C),(M):}`
On one particular day, 135 shoppers spent the average daily amount on food, 143 shoppers spent the average daily amount on clothing and 131 shoppers spent the average daily amount on merchandise.
Area  food  clothing  merchandise 
Expected change  increase by 5%  decrease by 15%  decrease by 1% 
The average daily amount, in dollars, expected to be spent in each area in 2020 can be determined by forming the matrix product
`qquad qquad A_2020 = K xx A_2019`
Write down matrix `K`. (1 mark)
a. `1 xx 3`
b.  `0.25 xx G\ text(shoppers in)\ M`  `= 700` 
`:. G\ text(shoppers in)\ M`  `= 700/0.25`  
`= 2800` 
c.i.  `{:(quad qquad qquad qquad \ E qquad qquad G qquad qquad W), (Q = [(1104, 1296, 1056 ), (621, 729, 594), (575, 675, 550)]{:(F),(C), (M):}):}` 
c.ii.  `q_23\ text(represents the number of people)` 
`text(in the clothing area of Westmall.)` 
d. `text(Total spent) = [(135, 143, 131)] [(21.30), (34.00), (14.70)] = [9663.20]`
e.  `A_2020`  `= K xx [(21.30), (34.00), (14.70)]` 
`:. K`  `= [(1.05, 0, 0),(0, 0.85, 0),(0, 0, 0.99)]` 
Training program 1 has the cricket team starting from exercise station `S` and running to exercise station `O`.
For safety reasons, the cricket coach has placed a restriction on the maximum number of people who can use the tracks in the fitness park.
The directed graph below shows the capacity of the tracks, in number of people per minute.
When considering the possible flow of people through this network, many different cuts can be made.
Samuel now invests $500 000 in an annuity from which he receives a regular monthly payment.
The balance of the annuity, in dollars, after `n` months, `A_n` , can be modelled by a recurrence relation of the form
`A_0 = 500\ 000, qquad A_(n+1) = kA_n  2000`
a.  `A_1 = 1.0024 xx 500\ 000  2000 = $499\ 200` 
`A_2 = 1.0024 xx 499\ 200  2000 = $498\ 398.08` 
b.  `text(Monthly interest rate)`  `= (1.0024  1) xx 100 = 0.24text(%)` 
`text(Annual interest rate)`  `= 12 xx 0.24 = 2.88text(%)` 
c.  `text(Perpetuity would occur when)`  
`k xx 500\ 000  2000`  `= 500\ 000`  
`k`  `= (502\ 000)/(500\ 000)`  
`= 1.004` 
Samuel opens a savings account.
Let `B_n` be the balance of this savings account, in dollars, `n` months after it was opened.
The monthtomonth value of `B_n` can be determined using the recurrence relation shown below.
`B_0 = 5000, qquad B_(n+1) = 1.003B_n`
If Samuel had deposited an additional $50 at the end of each month immediately after the interest was added, how much extra money would be in the savings account after one year?
Round your answer to the nearest dollar. (1 mark)
a.  `B_1`  `= 1.003 (5000)` 
`B_2`  `= 1.003^2 (5000)` 
`vdots`
`:. B_4`  `= 1.003^4 (5000)` 
`= $5060.27` 
b. `text(Monthly interest rate)`
`= (1.003  1) xx 100`
`= 0.3%`
c. `text(Extra)\ =\ text(value of annuity after 12 months)`
`text(By TVM solver:)`
`N`  `= 12` 
`I(%)`  `= 3.6` 
`PV`  `= 0` 
`PMT`  `= 50` 
`FV`  `= ?` 
`text(PY)`  `= text(CY) = 12` 
`FV = 609.84`
`:.\ text(Extra money) = $610`
Samuel owns a printing machine.
The printing machine is depreciated in value by Samuel using flat rate depreciation.
The value of the machine, in dollars, after `n` years, `Vn` , can be modelled by the recurrence relation
`V_0 = 120\ 000, qquad V_(n+1) = V_n  15\ 000`
Write down this rule for `V_n`. (1 mark)
a. `$15\ 000`
b.  `V_1`  `= 120\ 000  15\ 000 = $105\ 000` 
`V_2`  `= 105\ 000  15\ 000 = $90\ 000` 
c.  `text(Flat rate percentage`  `= (15\ 000)/ (120\ 000) xx 100` 
`= 12.5 text(%)` 
d. `V_n = 120\ 000  15\ 000n, \ n = 0, 1, 2, …`
The table below shows the mean age, in years, and the mean height, in centimetres, of 648 women from seven different age groups.
A scatterplot displaying this data shows an association between the mean height and the mean age of these women. In an initial analysis of the data, a line is fitted to the data by eye, as shown.
Using these two points, determine the equation of this line. Write the values of the intercept and the slope in the appropriate boxes below.
Round your answers to three significant figures. (1 mark)
mean height = 

+ 

× mean age 
The associated residual plot that was generated is shown below.
The residual plot indicates that the association between the mean height and the mean age of women is nonlinear.
The data presented in the table in part a is repeated below. It can be linearised by applying an appropriate transformation to the variable mean age.
Apply an appropriate transformation to the variable mean age to linearise the data. Fit a least squares line to the transformed data and write its equation below.
Round the values of the intercept and the slope to four significant figures. (2 marks)
a.  `text(Difference)`  `= 167.1  156.7` 
`= 10.4\ text(cm)` 
b. `text(Strong and negative.)`
c. `text(Slope) = (157  168)/(85  20) = 0.169`
`text(Equation of line)`
`y  168`  `= 0.1692 (x  20)` 
`y`  `= 0.169x + 171` 
`:.\ text(mean height) = 171  0.169 xx text(mean age)`
D. `text(By CAS)`
`text(mean height) = 167.9  0.001621 xx text{(mean age)}^2`
The scatterplot below shows body density, in kilograms per litre, plotted against waist measurement, in centimetres, for 250 men.
When a least squares line is fitted to the scatterplot, the equation of this line is
body density = 1.195 – 0.001512 × waist measurement
(Answer on the scatterplot above.)
Round your answer to two decimal places. (1 mark)
Show that, when this least squares line is fitted to the scatterplot, the residual, rounded to two decimal places, is –0.02 (1 mark)
Write down the value of the correlation coefficient `r`.
Round your answer to three decimal places. (1 mark)
Does this residual plot support the assumption of linearity that was made when fitting this line to this data? Briefly explain your answer. (1 mark)
a. `text(LSRL passes through)\ (60, 1.1043) and (130, 0.998)`
b.  `text(body density)`  `= 1.195  0.001512 xx 65` 
`= 1.09672`  
`= 1.10\ text{kg/litre (to 2 d.p.)}` 
c.  `text(A waist of 65 cm is outside the)` 
`text(range of the existing data set.)` 
`:.\ text(Extrapolating)`
d.  `text(Body density decreases by 0.001512 kg/litre)` 
`text(for each increase in waist size of 1 cm.)` 
e. `text{Body density (predicted)}`
`= 1.195  0.001512 xx 122`
`~~ 1.0105\ text(kg/litre)`
`text(Residual)`  `= text(Actual  predicted)` 
`~~ 0.995  1.0105`  
`~~ 0.0155`  
`~~ 0.02\ text{(to 2 d.p.)}` 
f.  `r`  `= sqrt(0.6783)` 
`=  0.8235…`  
`= 0.824\ text{(to 3 d.p.)}` 
g. `text(The residual plot has no pattern and is)`
`text(centred around zero.)`
`:.\ text(It supports the assumption of linearity)`
`text(of the LSRL.)`
The age, in years, body density, in kilograms per litre, and weight, in kilograms, of a sample of 12 men aged 23 to 25 years are shown in the table below.
Age (years) 
Body density 
Weight 

23  1.07  70.1  
23  1.07  90.4  
23  1.08  73.2  
23  1.08  85.0  
24  1.03  84.3  
24  1.05  95.6  
24  1.07  71.7  
24  1.06  95.0  
25  1.07  80.2  
25  1.09  87.4  
25  1.02  94.9  
25  1.09  65.3 
a.i.  `n = 12`  
`text(Median)`  `= (text{6th + 7th})/2`  
`= (24 + 24)/2`  
`= 24` 
a.ii.  `text(Mean)`  `= (∑\ text{body density})/12` 
`= 1.065\ text(kg/litre)` 
b.i. `text(Weight)`
b.ii. `text(Slope) = 0.00112\ text{(by CAS)}`
c.  `r`  `= 0.53847\ text{(by CAS)}` 
`r^2`  `= 0.289…` 
`:. 29 text(%)`
The neck size, in centimetres, of 250 men was recorded and displayed in the dot plot below.
`qquad`
Use the fivenumber summary to construct a boxplot, showing any outliers if appropriate, on the grid below. (2 marks)
a. `text(Mode) = 38\ text(cm)`
b.i.  `text(Expected number of men)`  `= (1  0.997) xx 250` 
`= 0.75`  
`= 1\ text{(nearest whole)}` 
b.ii.  `text(When)\ \ z = + 3`  
`text(Neck size limits)`  `= 38 + (2.3 xx 3)`  
`= 44.9 or 31.1` 
`:.\ text(1 man has neck size outside 3 s.d.)`
c. `IQR = 3936=3`
`text(Upper fence)`  `=Q_3 + 1.5 xx 3`  
`=39 + 4.5`  
`=43.5` 
`text(Lower fence)`  `=Q_1  1.5 xx 3`  
`=36  4.5`  
`=31.5` 
Body mass index (BMI), in kilograms per square metre, was recorded for a sample of 32 men and displayed in the ordered stem plot below.
a. `text(Positively skewed)`
b. `32\ text(data points)`
`text(Median)`  `= text(16th + 17th)/2` 
`= (24.5 + 24.6)/2`  
`= 24.55` 
c.  `text(Percentage)`  `= 12/32 xx 100` 
`= 37.5%` 
A shape of an arrow head is folded half a long the dotted line.
The folded shape can also be called a

Quadrilateral 

Triangle 

Hexagon 

Pentagon 
`text{Triangle}`
`text{The folded shape is a triangle}`
Which picture shows the pencils creating an angle of 30°?







Which picture shows that the book is opened at an angle of 180°?







`text{This book is opened at an angle of 180}^@`
Which letter has exactly one line of symmetry?




`text{The letter A has only one line of symmetry.}`
Which letter has two lines of symmetry?




`text{The letter X has two lines of symmetry.}`
`8 xx` 

` 13 = 75` 
What value would make the number sentence right?

11 

9 

13 

7 
`11`
`text{Check each option (multiply then add):}`
`8 xx 11 13 = 75 \ text{(Correct)}`
`8 3 xx 9 13 = 59 \ text{(Incorrect)}`
`8 3 xx 9 13 = 91 \ text{(Incorrect)}`
`8 xx 7 13 = 43 \ text{(Incorrect)}`
`12 + 3 xx ` 

`= 39` 
For this number sentence to be true, what is the missing value?

15 

11 

9 

7 
`9`
`text{Check each option (multiply then add):}`
`12 + 3 xx 15= 57 \ text{(Incorrect)}`
`12 + 3 xx 11= 57 \ text{(Incorrect)}`
`12 + 3 xx 9 = 57 \ text{(Correct)}`
`12 + 3 xx 7= 57 \ text{(Incorrect)}`
The return trip from a school to the museum is 15.97 kilometres.
How far does a student need to travel if they visit the museum 6 times?

89.53 km 

95.82 km 

57.39 km 

64.72 km 
`95.82 \ text{km}`
`:.\ text(Distance)`  `= 6 xx 15.97 \ text{km}`  
`= 95.82 \ text{km}` 
A return trip from your house to the grocery store is 9.82 kilometres.
If you go to the store 23 times, how far do you need to travel?

426.95 km 

234.21 km 

225.86 km 

382.23 km 
`225.86 \ text{km}`
`:.\ text(Distance)`  `= 23 xx 9.82 \ text{km}`  
`= 225.86 \ text{km}` 
The scales below are evenly balanced.
What is the mass of the small cube?
grams 
`26 grams`
`text{Mass of small cube}`  `= 44  18`  
`= 26 grams` 
The scales below are evenly balanced.
What is the mass of the small cube?
grams 
`13 grams`
`text{Mass of small cube}`  `= 32  19`  
`= 13 grams` 
Solve the equation `sqrt 3 sin (x) = cos (x)` for `x in [– pi, pi]`. (2 marks)
`x = pi/6,\ \ \  (5 pi)/6`
`text(Divide both sides by)\ \ cos(x) :`
`sqrt 3 sin x`  `=cos x` 
`sqrt 3 tan x`  `= 1` 
`tan x`  `= 1/sqrt 3` 
`=>\ text(Base angle)\ = pi/6` 
`:. x = pi/6\ \ text(or)\ \  (5 pi)/6,\ \ \ x in [– pi, pi].`
Two objects, each of mass `m` kilograms, are connected by a light inextensible strings that passes over a smooth pulley, as shown below. The object on the platform is initially at point A and, when it is released, it moves towards point C. The distance from point A to point C is 10 m. The platform has a rough surface and, when it moves along the platform, the object experiences a horizontal force opposing the motion of magnitude `F_1` newtons in the section AB and a horizontal force opposing the motion of magnitude `F_2` newtons when it moves in the section BC.
The force `F_1` is given by `F_1 = kmg, \ k ∈ R^+`.
Point B is midway between points A and C.
a. 
b. i. `text(Horizontally:)`
`ma = T  F_1 = T  kmg\ …\ (1)`
`text(Vertically:)`
`ma = mg  T\ …\ (2)`
`text(Add)\ \ (1) + (2) :`
`2ma = mg  kmg`
`:. a`  `= (g  kg)/2` 
`= (g(1  k))/2` 
b. ii. `text(System in motion when)\ a > 0`
`(g(1  k))/2 > 0`
`:. k ∈ (0, 1), \ k ∈ R^+`
c. `text(AB) = 5\ (text(given)), u = 0\ (text(given))`
`text(Find)\ t\ text(when)\ s = 5:`
`s = ut + 1/2at^2`
`5 = 0 + 1/2 · (g(1  k))/2 · t^2`
`t^2`  `= 20/(g(1  k))` 
`t`  `= sqrt(20/(g(1  k)))` 
`= 2sqrt(5/(g(1  k)))` 
d. `text(At B,)\ s = 5`
`v_text(B)^2`  `= u^2 + 2as` 
`= 0 + 2 · (g(1  k))/2 · 5`  
`= 5g(1  k)` 
`:. v_text(B) = sqrt(5g(1  k))`
e. `text(Acceleration is against the direction of motion.)`
`a`  `= −F/m` 
`= −0.075g  0.4v^2`  
`= −0.4(0.1875g + v^2)` 
`d/(dx)(1/2 v^2)`  `= −0.4(0.1875g + v^2)` 
`d/(dx)(v^2)`  `= −0.8(0.1875g + v^2)` 
`(dx)/(d(v^2))`  `= −1.25(1/(0.1875g + v^2))` 
`:. x`  `= −1.25 int_(2.5^2)^0 1/(0.1875 + v^2)\ dv^2` 
`= 1.85\ text(m)` 
`:.\ text(Distance from C)`  `= 5  1.85` 
`= 3.15\ text(m)` 
A pilot is performing at an air show. The position of her aeroplane at time `t` relative to a fixed origin `O` is given by
`underset~r_text(A) (t) = (450  150sin((pit)/6))underset~i + (400  200cos((pit)/6))underset~j`,
where `underset~i` is a unit vector in a horizontal direction, `underset~j` is a unit vector vertically up, displacement components are measured in metres and time `t` is measured in seconds where `t >= 0`.
A friend of the pilot launches an experimental jetpowered drone to take photographs of the air show. The position of the drone at time `t` relative to the fixed origin is given by `underset~r_text(D)(t) = (30t)underset~i + (−t^2 + 40t)underset~j`, where `t` is in seconds and `0 <= t <= 40, underset~i` is a unit vector in the same horizontal direction, `underset~j` is a unit vector vertically up, and displacement components are measured in metres.
a.  `underset~r_text(A)′(t)`  `= −25picos((pit)/60)underset~i + 100/3pisin((pit)/6)underset~j` 
`= (25pi)/3(−3cos((pit)/6) + 4sin((pit)/6))` 
`underset~r_text(A)′(t)`  `= (25pi)/3 sqrt(9cos^2((pit)/6) + 16sin^2((pit)/6))` 
`= (25pi)/3 sqrt(9 + 7sin^2((pit)/6))` 
`:. underset~r_text(A)′(t)_text(max)`  `= (25pi)/3 sqrt(9 + 7)` 
`= (100pi)/3\ text(ms)^(−1)` 
b. i. `x = 450  150 sin((pit)/6) \ => \ sin((pit)/6) = (450  x)/150`
`sin^2((pit)/6) = ((x  450)^2)/(22\ 500)`
`y = 400  200cos((pit)/6) \ => \ cos((pit)/6) = (400  y)/200`
`cos^2((pit)/6) = ((y  400)^2)/(40\ 000)`
`text(Using)\ \ sin^2theta + cos^2theta = 1:`
`((x  450)^2)/(22\ 500) + ((y  400)^2)/(40\ 000) = 1`
b. ii. `text(When)\ x = 450, \ y = 200, 600`
`text(When)\ y = 400, \ x = 300, 600`
`text(As)\ \ t > 1, \ x↓, \ y↑`
`:.\ text(Motion is clockwise.)`
c. `x = 30t \ => \ t = x/30`
`y`  `= −t^2 + 40t` 
`= −(x^2)/900 + 4/3 x` 
d. `text(The graph shows 2 points where the paths cross.)`
`text(Consider)\ (316, 310):`
`text(Drone passes through when)`
`t = 316/30 ~~ 10.53\ text(seconds)`
`text(Plane passes through when)`
`450  150sin((pit)/6) = 316 \ => \ t ~~ 12.85\ text{seconds (no contact)}`
`text(Similarly, consider)\ (600, 400):`
`text(Drone passes through when)`
`t = 600/30 = 20\ text(seconds)`
`text(Plane passes through when)`
`400  200cos((pit)/6) = 400 \ => \ t = 10\ text(or)\ ~~ 14.7\ text{seconds (no contact)}`
`:.\ text(Drone will not make contact with the plane.)`
Let `f(x) = x^2e^(−x)`.
Let `g(x) = x^n e^(−x)`, where `n ∈ Z`.
a. `f′(x) = 2xe^(−x)  x^2e^(−x)`
`text(SP's when)\ \ f′(x) = 0:`
`x^2e^(−x)`  `= 2xe^(−x)` 
`x`  `= 2\ \ text(or)\ \ 0` 
`f(0) = 0; \ f(2) = 4e^(−2)`
`:. text(SP's at)\ \ (0, 0) and (2, 4e^(−2))`
b. `text(As)\ \ x > ∞, \ f(x) > 0^+`
`:. text(Horizontal asymptote at)\ \ y = 0`
c. 
`text(POI when)\ \ f″(x) = 0`
`:. text(POI's:)\ (0.59, 0.19), \ (3.41, 0.38)`
d. `g′(x) = x^(n  1) e^(−x)(n  x)`
`g″(x) = x^(n  2) e^(−x)(x^2  2xn + n^2  n)`
e.i. `text(Solve:)\ \ x^2  2xn + n^2  n = 0`
`x = n ± sqrtn`
e.ii. 
Two complex numbers, `u` and `v`, are defined as `u = −2  i` and `v = −4  3i`.
a. `text(Let)\ \ z = x + iy`
`z  u = x + 2 + iy + i`
`z  v = x + 4 + iy + 3i`
`z  u = z  v`
`(x + 2)^2 + (y + 1)^2`  `= (x + 4)^2 + (y + 3)^2` 
`x^2 + 4x + 4 + y^2 + 2y + 1`  `= x^2 + 8x + 16 + y^2 + 6y + 9` 
`4y`  `= 4x + 20` 
`y`  `= −x  5` 
b. 
c. `z  u = z  v\ text(is the graph of the perpendicular bisector of the)`
`text(line joining)\ u and v.`
d.i. 
d.ii. `text(Arg)(z  u) = pi/4 =>\ text(gradient) = 1, ytext(intercept at)\ (0, 1)`
`:. f: (−2, ∞) > RR, \ f(x) = x + 1`
Two complex numbers, `u` and `v`, are defined as `u = −2  i` and `v = −4  3i`.
a. `text(Let)\ \ z = x + iy`
`z  u = x + 2 + iy + i`
`z  v = x + 4 + iy + 3i`
`z  u = z  v`
`(x + 2)^2 + (y + 1)^2`  `= (x + 4)^2 + (y + 3)^2` 
`x^2 + 4x + 4 + y^2 + 2y + 1`  `= x^2 + 8x + 16 + y^2 + 6y + 9` 
`4y`  `= 4x + 20` 
`y`  `= −x  5` 
b. 
c. `z  u = z  v\ text(is the graph of the perpendicular bisector of the)`
`text(line joining)\ u and v.`
d.i. 
d.ii. `text(Arg)(z  u) = pi/4 =>\ text(gradient) = 1, ytext(intercept at)\ (0, 1)`
`:. f: (−2, ∞) > RR, \ f(x) = x + 1`
e. `z_c  u = z_c  v = z_c  (−5i)`
`z_c  u`  `= (a + 2) + (b + 1)i` 
`z_c  v`  `= (a + 4) + (b + 3)i` 
`z_c +5i`  `= a + (b + 5)i` 
`a^2 + (b + 5)^2 = (a + 2)^2 + (b + 1)^2\ …\ (1)`
`a^2 + (b + 5)^2 = (a + 4)^2 + (b + 3)^2\ …\ (2)`
`a = −5/3, b = −10/3\ \ text{(by CAS)}`
`:.z_c = −5/3  10/3 i`
`:.r`  `= z_c  (−5i)` 
`= sqrt((−5/3)^2 + (−10/3 + 5)^2)`  
`= (5sqrt2)/3` 
A particle moves in the `x\ – y` plane such that its position in terms of `x` and `y` metres at `t` seconds is given by the parametric equations
`x = 2sin(2t)`
`y = 3cos(t)`
where `t >= 0`
a. `text(At)\ \ t = pi/6,`
`x = 2sin\ pi/3 = sqrt3`
`y = 3cos\ pi/6 = (3sqrt3)/2`
`:.\ text(Distance)`  `= sqrt((sqrt3)^2 + ((3sqrt3)/2)^2)` 
`= sqrt39/2\ text(metres)` 
b.i. `(dx)/(dt) = 4cos(2t),\ \ (dy)/(dt) = −3sin(t)`
`(dy)/(dx)`  `= (dy)/(dt) · (dt)/(dx)` 
`= (−3sin(t))/(4cos(2t))` 
`text(When)\ t = pi :`
`(dy)/(dx)`  `= (−3sin(pi))/(4cos(2pi))=0` 
`text(Equation of tangent where)\ \ m = 0,\ text(through)\ (0, −3):`
`y = −3`
b.ii.  `underset~r(t)`  `= 2sin(2t)underset~i + 3cos(t)underset~j` 
`underset~v(t)`  `= 4cos(2t)underset~i  3sin(t)underset~j`  
`underset~v(pi)`  `= 4cos(2pi)underset~i  3sin(pi)underset~i`  
`= 4underset~i` 
b.iii. `underset~a(t) = −8sin(2t)underset~i  3cos(t)underset~j`
`underset~a(pi)`  `= −8sin(2pi)underset~i  3cos(pi)underset~j` 
`= 3underset~j` 
`underset~a(pi)`  `= sqrt(0^2 + 3^2)` 
`= 3\ text(ms)^(−2)` 
c. `text(Find)\ t\ text(when)\ \ x = 2sin(2t) = 0\ \ text(and)\ \ y = 3cos(t) = 0:`
`t = pi/2\ \ (text(1st time))`
d.  `text(Distance)`  `= int_0^(pi/6) sqrt(((dx)/(dt))^2 + ((dy)/(dt))^2)\ dt` 
`= int_0^(pi/6) sqrt((4cos(2t))^2 + (−3sin(t))^2)\ dt`  
`~~ 1.804\ \ (text(to 3 d.p.))` 
Ana recycles 3 kilograms of plastic bottles in her first week of recycling.
She then recycles twice as much as she recycled the previous week for the next 5 weeks.
The total weight, in kilograms, of the plastic bottles she recycles in any given week will be

always even 

always odd 

sometimes even or sometimes odd 

none of the above 
`text{Always odd}`
`text{1st week: 3 kg (odd)}`
`text{2nd week: 3 + 6 = 9 kg (odd)}`
`text{3rd week: 9 + 18 = 27 kg (odd)}`
`vdots`
`text{Any week: odd + even = odd}`
Leo puts $15 into his piggy bank to start saving money.
He then puts the same amount into the piggy bank at the end of each month.
The total amount of money in his piggy bank will be which of the following:

Always an even number 

Always an odd number 

Sometimes odd sometimes even 

None of the above 
`text{Sometimes odd sometimes even}`
`text{Starting amount = $15 (odd)}`
`text{End of 1st month = 15 + 15 = $30 (even)}`
`text{End of 2nd month = 30 + 15 = $45 (odd)}`
`:.\ text{Total amount will be either an odd or even number.}`
Michael’s company is shown in the map.
Michael rides home 5 kilometres west, walks 3 kilometres south and walks another 1 kilometre west.
In which cell on the map is Michael’s home?
D1  E2  C1  D2 




`text{D1}`
Phil’s home is shown in the map below.
Everyday he drives 3 kilometres east and 2 kilometres south from home to work
In which cell on the map is Phil’s office?
A5  B5  E5  D5 




`text{E5}`
Louise donated money to two charities
She donated $125 to the first charity and donated $45 to the second charity.
Which number sentence could be used to find total amount of money Louise donated?

120 + 40 = 160 

130 + 50 = 180 

120 + 40 + 5 = 165 

130 + 50 – 10 = 170 
`130 + 50  10 = 170`
`text(Total amount donated to charity)`
`= 125 + 45`
`= 130  5 + 50  5`
`= 130 + 50  10`
`= 170`
Shane played two basketball games.
She scored 35 points in the first game.
In the second game she scored 25 points.
Which number sentence could be used to find the total number of points Shane scored in the first two games?
30 + 20 + 5 = 55  40 + 30 = 70  30 + 20 + 10 = 60  30 + 20 = 50 




`30 + 20 + 10 = 60`
`text(Total points in first 2 games)`
`= 35 + 25`
`= 30 + 5 + 20 + 5`
`= 30 + 20 + 10`
`= 60`
The scale pictured below is balanced.
What is the weight of the cube?
grams 
`31 \ text{grams}`
`text{S} text{ince the scale is balanced, weight on both sides = 44 grams}`
`text(Weight of cube)`  `= 44  13` 
`= 31\ text(grams)` 
The scale pictured below is balanced.
What is the weight of the rounded figure?
kilograms 
`5\ text{kg}`
`text{S} text{ince the scale is balanced, the weight on both sides = 24 kg.}`
`text{Missing weight}`  `=24  19`  
`= 5\ text(kg)` 
Gary folded this piece of paper along the dashed lines to make a solid figure.
Which of these models did Gary make?




`text{By folding the paper along the dashed lines we could obtain a figure of:}`
In a school, a group of students were given a list of 4 subjects and asked to pick their favourite.
A table of their responses is shown below.
How many students were asked this question altogether?
`44`
`text{Mathematics} = 5 + 5 + 5 + 2 = 17`
`text{Science} = 5 + 5 = 10`
`text{English} = 5 + 1 = 6`
`text{History} = 5 + 5 + 1 = 11`
`:.\ text{Total} = 17 + 10 + 6 + 11 = 44`