Aussie Maths & Science Teachers: Save your time with SmarterEd
A scientist is estimating the population of a bee colony.
The scientist captures \(p\) bees but only marks \(k\%\) of them and releases them all back into the colony. The next day, the scientist captures \(q\) bees and finds that \(r\) bees were marked.
Assuming all bees have an equal chance of being captured, which expression gives the best estimate of the total bee population?
Maya plans to go on an overseas trip in 5 years' time. Maya expects that the trip will cost \(\$14\ 000\). To save for the trip, Maya began by contributing \(\$750\) every 6 months into an annuity. After making a number of payments, Maya realised that the annuity would not reach \(\$14\ 000\) in 5 years. Maya decides to double the contribution until the trip.
The interest rate was 2% per annum compounding half yearly.
| Rate Period | 0.01 | 0.02 | 0.03 | 0.04 |
| 2 | 2.0100 | 2.0200 | 2.0300 | 2.0400 |
| 4 | 4.0604 | 4.1216 | 4.1836 | 4.2465 |
| 6 | 6.1520 | 6.3081 | 6.4684 | 6.6330 |
| 8 | 8.2857 | 8.5830 | 8.8923 | 9.2142 |
| 10 | 10.4622 | 10.9497 | 11.4639 | 12.0061 |
| 12 | 12.6825 | 13.4121 | 14.1920 | 15.0258 |
What is the maximum number of \(\$750\) contributions Maya could make before she will need to double her payments to reach \(\$14\ 000\)? (4 marks)
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A group of students participated in a coding challenge. The times taken to complete the challenge are normally distributed with a mean of 60 minutes and a standard deviation of 10 minutes.
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A rectangular tub has length 50 cm, width 20 cm and unknown height \((h)\).
Ten identical cylinders are arranged inside the tub in two rows of five.
Each cylinder has diameter 10 cm and height \(h\) cm.
Exactly 3 litres of water is poured into the tub to fill all the spaces around the cylinders.
What is the height of the tub, correct to the nearest whole number? (4 marks)
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A ship sails 8 km from \(A\) to \(B\) on a bearing of 070°. It then sails from \(B\) to \(C\).
Point \(D\) lies on the line \(CE\) and is 5 km due east of \(B\), and \(E\) is 8.7 km due south of \(C\) and directly east of \(A\).
What is the true bearing of \(A\) from \(C\), correct to the nearest degree? (4 marks)
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The network shows the lengths of the paths between some exhibits at a zoo.
Nikolai is visiting the penguins and wants to visit the lions. The shortest path Nikolai can take is Penguins—Giraffes—Snakes—Lions.
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The floor plan of a house has been drawn using a scale of \(1:100\).
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An amount of $3500 is invested for 3 years at an interest rate of 5.75% per annum, compounded annually.
Find the simple interest rate per annum that will give the same future value. Give your answer as a percentage, correct to 2 decimal places. (4 marks)
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The stem-and-leaf plot shows the age of 24 volunteers at a charity event.
\(\begin{array}{c|l}
\text{Stem} & \quad \quad \text{Leaf} \\
\hline \rule{0pt}{2.5ex}1 & \ 3\ \ \ \ 5 \\
2 & \ 0\ \ \ \ 2\ \ \ \ 5 \\
3 & \ 0\ \ \ \ 2\ \ \ \ 4\ \ \ \ 6 \\
4 & \ 1\ \ \ \ 3\ \ \ \ 3\ \ \ \ 7\ \ \ \ 8\ \ \ \ 9 \\
5 & \ 0\ \ \ \ 2\ \ \ \ 2\ \ \ \ \textbf{X} \\
6 & \ 1\ \ \ \ 4\ \ \ \ 6\ \ \\
7 & \ 0\ \ \ \ 2\ \ \\
\end{array} \qquad \qquad \boxed{\begin{array}{l} \text{KEY:} \\ 3\,|\,2=32 \end{array}}\)
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Discuss how environmental conditions and group strengths and weaknesses influence the strategies and tactics used in ONE group sport. (8 marks)
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Refer to the information below as well as your own knowledge to answer the question.
The information relates to the treatment of a sports injury.
Evaluate the coach’s advice and propose a suitable rehabilitation program for the football player. (12 marks)
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Refer to the information below as well as your own knowledge to answer the question.
City X is an urban area with a large number of young people, aged 10–24. The infographic provides information about City X.
To what extent could the application of Sustainable Development Goal 3: Good Health and Wellbeing and Sustainable Development Goal 4: Quality Education assist in improving the health status of young people in City X? (12 marks)
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Explain how TWO approaches to health used in other OECD countries could be used in Australia to improve health outcomes. Provide examples to support your answer. (8 marks)
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Analyse how the principles of progressive overload and training thresholds could be applied to a cyclist’s aerobic training program to improve performance. Provide examples to support your answer. (8 marks)
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Estelle deposits a single lump sum into an account that earns 3% per annum compound interest.
Present value interest factors for an annuity of $1 for various interest rates \((r)\) and numbers of periods \((N)\) are shown in the table.
From this account, Estelle plans to make the following withdrawals.
Find the smallest lump sum Estelle must deposit so that both sets of withdrawals can be made. (3 marks)
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Farah wants to buy a commercial espresso machine for her cafe at a cost of $4200. She has decided to pay off whatever she owes in a single amount 75 days after the purchase, and is weighing up two ways to fund it.
Advise Farah on the cheaper option, backing up your recommendation with the relevant calculations. (4 marks)
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The mass of a sample of hungry cat fleas is 0.675 kilograms. There are approximately `1.5 × 10^6` hungry cat fleas in the sample.
In standard form, what is the approximate mass in grams of one hungry cat flea? (2 marks)
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Yasmin is planning to purchase a Newcastle Knights corporate box for $5000. She intends to repay the entire amount 90 days after making the purchase.
She is comparing two credit options.
Determine which option Yasmin should choose, showing calculations to support your answer. (4 marks)
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An experimental rocket is at a height of 5000 m, ascending at a speed of \(50\sqrt{2}\) m s\(^{-1}\) at an angle of 45° to the horizontal, when its engine stops. The rocket is then subject to gravity and to air resistance proportional to its velocity. Take \(g\) = 10 m s\(^{-2}\).
The velocity vector of the rocket, \(t\) seconds after the engine stops, is
\(\mathbf{v}(t) = 50e^{-0.2t}\,\mathbf{i} + (100e^{-0.2t}-50)\mathbf{j}.\) (Do NOT prove this.)
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As a projectile of mass \(m\) kilograms travels through air, it experiences a frictional force. The magnitude of this force is proportional to the speed \(v\) of the projectile. The constant of proportionality is the positive number \(k\). The position of the particle at time \(t\) is denoted by \(\displaystyle \binom{x}{y}\). The acceleration due to gravity is \(g \ \text{m s}^{-2}\).
Based on Newton's laws of motion, which equation models the motion of this projectile?
Consider a sequence of rectangles with side lengths \(a_{ n }\) and \(b_{ n }\).
The first rectangle has \(a_1=2\) and \(b_1=1\).
For integers \(n \geq 1,\ \ a_{n+1}=\dfrac{a_n+b_n}{2}\) and \(b_{n+1}=\dfrac{2}{a_{n+1}}.\)
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Sienna buys a new car and pays a total of $2,200 in stamp duty.
Stamp duty is calculated on the vehicle as follows:
Determine the market value of Sienna's car. (2 marks)
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The total cost of a tradesperson's invoice, including 10% GST, is $462.
What was the cost of the job before GST was added?
Zoe purchased a new ute for $36 000. The straight-line depreciation model used by her accountant assumes the ute decreases in value by $4800 each year.
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Maya purchased a motorbike for $12 000. The value of the motorbike decreases according to a linear model. The graph shows the value of the motorbike, $\(V\), against the time, \(t\) months, since it was purchased.
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Two cyclists, Aiko and Ben, are riding along the same straight track in the same direction.
Aiko starts 2000 m ahead of Ben. Aiko rides at a constant speed of 250 metres/minute and Ben rides at a constant speed of 500 metres/minute.
Let \(d\) = distance from the starting point in metres, and
\(t\) = time in minutes.
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a. \(d=500t\)
b.
c. \(8\ \text{minutes}\)
d. \(\text{Aiko travelled 2000 m, Ben travelled 4000 m.}\)
a. \(d=500t\)
b. \(\text{Table of values:}\)
\(\begin{array}{|c|c|c|c|c|c|c|} \hline t & 0 & 2 & 4 & 6 & 8 & 10 \\ \hline \text{Aiko} & 2000 & 2500 & 3000 & 3500 & 4000 & 4500 \\ \hline \text{Ben} & 0 & 1000 & 2000 & 3000 & 4000 & 5000 \\ \hline \end{array}\)
c. \(\text{From the graph, the lines intersect at }\ t=8.\)
\(\therefore\ \text{Ben catches Aiko after 8 minutes}\)
d. \(\text{When}\ \ t=8, d=4000:\)
\(\text{Aiko started 2000 m ahead, so the distance travelled}\)
\(=4000-2000=2000\ \text{m}\)
\(\text{Ben started at the origin, so the distance travelled =4000 m}\)
Two water tanks sit side by side on a farm.
Tank A has a capacity of 1000 litres and is full. It is being emptied at a constant rate of 60 litres per minute.
At the same time, Tank B is empty and is being filled at a constant rate of 40 litres per minute.
Let \(V\) = volume of water in litres, and
\(t\) = time in minutes.
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a. \(V=40t\)
b. \(\text{Table of values:}\)
\(\begin{array}{|c|c|c|c|c|c|c|c|} \hline t & 0 & 2 & 4 & 6 & 8 & 10 & 12 \\ \hline \text{Tank A} & 1000 & 880 & 760 & 640 & 520 & 400 & 280 \\ \hline \text{Tank B} & 0 & 80 & 160 & 240 & 320 & 400 & 480 \\ \hline \end{array}\)
c. \(10\ \text{minutes}\)
d. \(400\ \text{litres}\)
a. \(V=40t\)
b. \(\text{Table of values:}\)
\(\begin{array}{|c|c|c|c|c|c|c|c|} \hline t & 0 & 2 & 4 & 6 & 8 & 10 & 12 \\ \hline \text{Tank A} & 1000 & 880 & 760 & 640 & 520 & 400 & 280 \\ \hline \text{Tank B} & 0 & 80 & 160 & 240 & 320 & 400 & 480 \\ \hline \end{array}\)
c. \(\text{From the graph, the lines intersect at }\ t=10.\)
\(\therefore\ \text{Both tanks contain the same volume after 10 minutes}\)
d. \(\text{Method 1: Graphically}\)
\(\text{From the graph, when }\ t=10,\ \text{the volume in both tanks = 400 litres}\)
\(\text{Method 2: Algebraically}\)
\(\text{When }\ t=10:\)
\(\text{Tank A volume:}\ \ V=1000-60\times10=400\ \text{L}\)
\(\text{Tank B volume:}\ \ V=40\times10=400\ \text{L}\ \checkmark\)
\(\therefore\ \text{Each tank contains } 400\ \text{litres}\)
The orthographic views of a security camera cover, manufactured from sheet metal, are shown. Complete a half pattern development of the camera cover. (6 marks)
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Describe the process of artificial age hardening of aluminium alloys. (3 marks)
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Consider the function \(f:\left[0, \dfrac{5 \pi}{2}\right] \rightarrow R, f(x)=\sin (x)+1\).
The graph of \(y=f(x)\) is shown below.
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a. \(f\left(\dfrac{2 \pi}{3}\right)=\dfrac{2+\sqrt{3}}{2}\)
b. \(x=\dfrac{\pi}{6}, \dfrac{5 \pi}{6}, \dfrac{13 \pi}{6}\)
c. \(k=2 \pi, \ \ a=\dfrac{\pi}{2}\)
d. \(y=-\dfrac{x}{2}+\dfrac{\pi}{3}+\dfrac{\sqrt{3}}{2}+1\)
e.i. \(x_2=5.2\)
e.ii.
f.i. \(f^{\prime}(p)=\cos (p)\)
\(\text{Equation of tangent at}\ (p, \sin (p)+1):\)
| \(y-(\sin (p)+1)\) | \(=\cos (p)(x-p)\) |
| \(t(x)\) | \(=\cos (p)(x-p)+\sin (p)+1\) |
f.ii. \(y \text{-int (min)}=1-2 \pi \ \text { (at } p=2 \pi)\)
\(y \text{-int (max)}=1+\pi \ \text { (at } p= \pi)\)
f.iii. \(p=2.38 \ \text{or} \ 7.04\)
g.i. \(g(x)=a x^3+b x^2+c x+d, \ f(x)=\sin (x)+1\)
\(\text{Given} \ \ f(0)=g(0):\)
\(d=\sin (0)+1=1\)
\(g^{\prime}(x)=3 a x^2+2 b x+c, \ f^{\prime}(x)=\cos (x)\)
\(\text{Given} \ \ f^{\prime}(0)=g^{\prime}(0):\)
\(c=1\)
g.ii. \(\text {Area}=1.53\)
g.iii. \(r=\pi, \ b=-\dfrac{1}{\pi}\)
a. \(f(x)=\sin (x)+1\)
\(f\left(\dfrac{2 \pi}{3}\right)=\sin \left(\dfrac{2 \pi}{3}\right)+1=\dfrac{2+\sqrt{3}}{2}\)
b. \(\text{Solve \(\ f(x)=\dfrac{3}{2} \ \) for \(x\) (by CAS):}\)
\(\sin (x)+1=\dfrac{3}{2} \ \Rightarrow \ \sin (x)=\dfrac{1}{2}\)
\(\text{Solve} \ \ \sin (x)=\dfrac{1}{2} \ \text { for } \ x \in\left[0, \dfrac{5 \pi}{2}\right]:\)
\(x=\dfrac{\pi}{6}, \dfrac{5 \pi}{6}, \dfrac{13 \pi}{6}\)
c. \(f(x+k)=f(x) \ \ \text{for} \ \ x \in[0, a]\)
\(\text{By inspection of graph:}\)
\(k=2 \pi, \ \ a=\dfrac{\pi}{2}\)
d. \(f(x)=\sin (x)+1 \ \Rightarrow \ f^{\prime}(x)=\cos (x)\)
\(f^{\prime}\left(\dfrac{2 \pi}{3}\right)=-\dfrac{1}{2}\)
\(\text{Find equation of line} \ \ m_2=-\dfrac{1}{2} \ \ \text{through}\ \ \left(\dfrac{2 \pi}{3}, \dfrac{2+\sqrt{3}}{2}\right):\)
\(y=-\dfrac{x}{2}+\dfrac{\pi}{3}+\dfrac{\sqrt{3}}{2}+1\)
e.i. \(x_0=\dfrac{2 \pi}{3}\)
\(x_1=\dfrac{2 \pi}{3}-\dfrac{f\left(\dfrac{2 \pi}{3}\right)}{f^{\prime}\left(\dfrac{2 \pi}{3}\right)}=5.8264 \ldots\)
\(x_2=5.8264 \ldots-\dfrac{f(5.8264)}{f^{\prime}(5.8264)}=5.2 \ \text{(1 d.p.)}\)
e.ii.
f.i. \(f^{\prime}(p)=\cos (p)\)
\(\text{Equation of tangent at}\ (p, \sin (p)+1):\)
| \(y-(\sin (p)+1)\) | \(=\cos (p)(x-p)\) |
| \(t(x)\) | \(=\cos (p)(x-p)+\sin (p)+1\) |
f.ii. \(t(x)=\cos (p) x+\sin (p)+1-p \times \cos (p)\)
\(y\text{-intercept}=\sin (p)+1-p \times \cos (p)\)
\(\text{Find max/min of} \ y\text{-int for} \ p \in\left[0, \dfrac{5 \pi}{2}\right] \ \ \text{(by CAS):}\)
\(y \text{-int (min)}=1-2 \pi \ \text { (at } p=2 \pi)\)
\(y \text{-int (max)}=1+\pi \ \text { (at } p= \pi)\)
f.iii. \(x \text{-intercept of} \ f(x) \ \text{occurs at} \ \ x=\dfrac{3 \pi}{2}\)
\(\text{Solve} \ \ t\left(\dfrac{3 \pi}{2}\right)=0 \ \ \text {for}\ p:\)
\(p=2.38 \ \text{or} \ 7.04\)
g.i. \(g(x)=a x^3+b x^2+c x+d, \ f(x)=\sin (x)+1\)
\(\text{Given} \ \ f(0)=g(0):\)
\(d=\sin (0)+1=1\)
\(g^{\prime}(x)=3 a x^2+2 b x+c, \ f^{\prime}(x)=\cos (x)\)
\(\text{Given} \ \ f^{\prime}(0)=g^{\prime}(0):\)
\(c=1\)
g.ii. \(g(x)=a x^3+b x^2+x+1 \ \Rightarrow \ g^{\prime}(x)=3 a x^2+2 b x+1\)
\(\text{Given \(\ g(2 \pi)=f(2 \pi)=1\ \) and \(\ \ g^{\prime}(2 \pi)=f^{\prime}(2 \pi)=1\)}\)
\(\text{Solve \(\ g(2 \pi)=1 \ \) and \(\ g^{\prime}(2 \pi)=1\) simultaneously for \(a, b\):}\)
\(a=\dfrac{1}{2 \pi^2}, \ b=-\dfrac{3}{2 \pi}\)
\(\text{Find intersection of}\ f(x)\ \text{and}\ g(x)\ \text{(by CAS)}:\)
\(f(x)=g(x)\ \ \Rightarrow\ \ x=\pi\)
\(\text {Area}=\displaystyle \int_0^\pi f(x)-g(x)\, d x+\int_\pi^{2 \pi} g(x)-f(x)\, d x=1.53\)
g.iii. \(a=0, c=1, d=1\)
\(g(x)=b x^2+x+1, \ f(x)=\sin (x)+1\)
\(g^{\prime}(x)=2 b x+1, \ f^{\prime}(x)=\cos (x)\)
\(\text{Solve simultaneous equations for \(b\) and \(r\):}\)
\(br^2+r=\sin (r)\ \ldots\ (1)\)
\(2 b r+1=\cos (r)\ \ldots\ (2)\)
\(r=\pi, \ b=-\dfrac{1}{\pi}\)
A data signal and a carrier wave are combined to form the modulated signal shown.
What type of modulation is depicted?
An image of a screw jack is shown. The tommy bar is used to operate the screw jack
The screw jack has a pitch of 8 mm, and the force on the tommy bar is applied 400 mm from the axis of the screw.
What is the velocity ratio (VR) of the screw jack?
Consider the function \(h(x)=a\, \log _e(b x)\), where \(a, b \in R\).
Given that its derivative \(h^{\prime}(x)\) has range \((0, \infty)\), which of the following must be true?
Let \(A\) be a point on the line \(y=x+c\) and \(B\) be a point on the curve \(y=\log _e(x-1)\).
If \(A\) and \(B\) are placed such that the line segment \(A B\) has the minimum possible length, and this length is \(\sqrt{2}\), the value of \(c\) must be
\(f(x)=x+c, \ g(x)=\log _e(x-1)\)
\(\text {Shortest distance between line and curve is perpendicular distance.}\)
\(\text{Consider the example when}\ \ c=0:\)
\(\text{At point \(B\), gradient \(=1\)}\)
\(\text{Solve} \ \ g^{\prime}(x)=1:\)
\(\dfrac{1}{x-1}=1 \ \Rightarrow \ x=2\)
\(\text{Find \(\perp\) distance \((d)\) between \((2,0)\) and \((x, y)\) which lies on \(f(x)\):}\)
\(d=\sqrt{(x-2)^2+(f(x)-0)^2}=\sqrt{(x-2)^2+(x+c)^2}\)
\(\text{Minimise} \ d\ \text{(by CAS)}\ \Rightarrow \ x=\dfrac{2-c}{2}\)
\(d\left(\dfrac{2-c}{2}\right)=\sqrt{2} \ \Rightarrow \ c=0\ \ (c \neq-4)\)
\(\Rightarrow D\)
Consider the function \(h(x)=a\, \log _e(b x)\), where \(a, b \in R \backslash\{0\}\).
Given that its derivative \(h^{\prime}(x)\) has range \((0, \infty)\), which of the following must be true?
An image of a shaft and its end view are shown.
Which drawing correctly represents section \(\text{A–A}\)?
A tank initially contains 5 kg of salt dissolved in 3000 litres of water. Salty water that contains 0.1 kg of salt per litre of water enters the tank at a rate of 20 litres per minute. The solution is kept thoroughly mixed and drains from the tank via a tap at the same rate of 20 litres per minute.
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The position vectors of particles \(P\) and \(Q\) at time \(t\) seconds are given by
\begin{align*}
{\underset{\sim}{r}}_P(t)=\left(t^3+a t^2\right) \underset{\sim}{i}-\underset{\sim}{j} \ \ \text {and} \ \ {\underset{\sim}{r}}_Q(t)=(b t+2 t) \underset{\sim}{i}+\left(2 t^2+c t+t\right) \underset{\sim}{j},
\end{align*}
where \(t \geq 0\) and \(a, b, c \in R\).
The particles collide when \(t=1\).
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When the particles collide, their velocities are at right angles to each other.
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Consider the functions
\(f: R \backslash\{1\} \rightarrow R, f(x)=\dfrac{w^2}{(x-1)^2}\)
and
\(g: R \rightarrow R, g(x)=(x-w)^2\)
where \(w \in R\).
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Frances is constructing a home gym. This project requires 12 activities, \(A\) to \(L\), to be completed.
The activity network below shows each activity and its completion time in days.
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a. \(\text{Scan network forwards/backwards:}\)
\(\text{Critical paths: }ADHJL, ADIKL\)
b. \(\text{LST for Activity}\ E\)
\(=20-4-4-3\)
\(=9 \ \text{days}\)
| c. | \(\begin{array}{|l|c|c|c|c|c|} \hline \rule{0pt}{2.5ex}\ \ \text{Activity} \ \ \rule[-1ex]{0pt}{0pt}& \ \ B \ \ &\ \ C \ \ & \ \ E \ \ & \ \ F \ \ & \ \ G \ \ \\ \hline \rule{0pt}{2.5ex} \ \ \text{Float} \ \ \rule[-1ex]{0pt}{0pt}& 5 & 1 & 5 & 6 & 1 \\ \hline \end{array}\) |
\(\therefore\ \text{Activity \(F\) has the longest float time.}\)
d. \(\text{Current time = 25 days}\)
\(\text{New time = 22 days.}\)
\(\text{Reduce:} \ A(2 \ \text {days}), H(1 \ \text{day}), K(1 \ \text{day})\)
\(\text{Minimum cost}=2 \times 500+1 \times 300+1 \times 100=\$1400\)
An early learning centre runs seven different activities during its 40-day holiday program.
The activities are cooking \((C)\), drama \((D)\), gardening \((G)\), lunch \((L)\), music \((M)\), reading \((R)\) and sport \((S)\).
The timetabled order of the activities for day one of the holiday program is shown below.
\begin{array} {|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ \text{9 am} \ \ \rule[-1ex]{0pt}{0pt} & \text{10 am} \rule[-1ex]{0pt}{0pt} & \text{11 am} \rule[-1ex]{0pt}{0pt} & \text{12 pm} \rule[-1ex]{0pt}{0pt} & \ \ \text{1 pm} \ \ \rule[-1ex]{0pt}{0pt} & \ \ \text{2 pm} \ \ \rule[-1ex]{0pt}{0pt} & \ \ \text{3 pm} \ \ \\
\hline
\rule{0pt}{2.5ex} \textit{C} \rule[-1ex]{0pt}{0pt} & \textit{D} \rule[-1ex]{0pt}{0pt} & \textit{G} \rule[-1ex]{0pt}{0pt} & \textit{L} \rule[-1ex]{0pt}{0pt} & \textit{M} \rule[-1ex]{0pt}{0pt} & \textit{R} \rule[-1ex]{0pt}{0pt} & \textit{S}\\
\hline
\end{array}
The timetabled order of the activities for day one is also shown in matrix \(X\) below.
\begin{aligned}
X = & \begin{bmatrix}
C \\
D \\
G \\
L \\
M \\
R\\
S \\
\end{bmatrix}
\end{aligned}
Matrix \(P\), shown below, is a permutation matrix used to determine the timetabled order of activities from one day to the next.
\begin{aligned}
P = & \begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix}
\end{aligned}
A column matrix containing the timetabled order of activities on one day is multiplied by matrix \(P\) to determine the timetabled order of activities for the next day.
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An early learning centre offers a 10-week activity program for four-year-old children. There are 27 children enrolled in the program. They participate in three different activities over the 10 weeks. The activities are cooking \((C)\), gardening \((G)\) and music \((M)\).
The transition matrix \(K\), shown below, gives the expected proportion of children in the program who will change activities from one week to the next.
\begin{aligned}
& \quad \quad \ \ \ \textit{this week} \\
& \quad \ \ C \quad \quad G \ \quad \ \ M \\
K = & \begin{bmatrix}
0 & 0.76 & 0.36 \\
0.55 & 0 & 0.64 \\
0.45 & 0.24 & 0\\
\end{bmatrix}\begin{array}{l}
C\\
G\\
M
\end{array} \ \textit{next week} \\
\end{aligned}
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The early learning centre contains three rooms, Nursery \((N)\), Toddler \((T)\) and Pre-kinder \((P)\) .
From one year to the next, children can move between rooms, stay in the same room, or may leave \((L)\) the centre. The following transition matrix, \(M\), shows the expected proportion of children who will move between categories or stay in the same category from one year to the next.
\begin{aligned}
& \quad \quad \quad \quad \quad \textit{this year} \\
& \quad \quad \ \ \ \ N \quad \quad \ \ T \quad \quad P \quad \ L \\
M&=\begin{bmatrix}
0.25 & 0 & 0 & 0 \\
0.625 & 0.25 & 0 & 0 \\
0 & 0.625 & 0.1 & 0 \\
0.125 & 0.125 & 0.9 & 1
\end{bmatrix}\begin{array}{l}
N\\
T \\
P \\
P
\end{array}\quad \textit{next year}
\end{aligned}
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Using profits from his recent film, Declan invests $650 000 into a 10-year annuity.
The annuity earns interest at 6.4% per annum, compounding quarterly.
Declan receives a regular quarterly payment from the annuity.
Halfway through the 10-year annuity, Declan writes a recurrence relation to represent the quarter-to-quarter balance for the remainder of the annuity.
Let \(D_n\) be the balance of Declan's annuity \(n\) quarters after the halfway point of the annuity.
Complete the recurrence relation below in terms of \(D_0, D_{n+1}\) and \(D_n\) that can model this balance. (2 marks)
\(D_0=\begin{array}{|c|}\hline \rule{0pt}{4ex} \quad \quad \quad \quad \quad & \quad \quad \quad \quad \quad \\ \hline \end{array}, D_{n+1}=1.016 \times D_n+\begin{array}{|c|}\hline \rule{0pt}{4ex} \quad \quad \quad \quad \quad & \quad \quad \quad \quad \quad \\ \hline \end{array}\)
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Declan is a filmmaker and content creator.
He has taken out a reducing balance loan to fund a new production.
Interest is calculated monthly and Declan makes monthly repayments.
Three rows of the amortisation table for Declan’s loan are shown below.
\begin{array}{|c|c|c|c|c|}
\hline
\hline \rule{0pt}{2.5ex}\quad \textbf{Payment} \quad & \quad\textbf{Payment} \quad & \quad\textbf{Interest} \quad& \textbf{Principal} & \quad\textbf{Balance}\quad\\
\textbf{number} & \textbf{(\$)} \rule[-1ex]{0pt}{0pt}& \textbf{(\$)} & \quad\textbf{reduction (\$)} \quad& \textbf{(\$)}\\
\hline \hline \rule{0pt}{2.5ex}0 \rule[-1ex]{0pt}{0pt}& 0.00 & 0.00 & 0.00 & 850\,000.00 \\
\hline \hline \rule{0pt}{2.5ex}1\rule[-1ex]{0pt}{0pt} & 15\,730.88 & 2975.00 & 12\,755.88 & 837\,244.12 \\
\hline \hline \rule{0pt}{2.5ex}2 \rule[-1ex]{0pt}{0pt}& 15\,730.88 & 2930.35 & 12\,800.53 & 824\,443.59 \\
\hline
\end{array}
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\begin{array}{|c|c|c|c|c|}
\hline
\hline \rule{0pt}{2.5ex}\quad \textbf{Payment} \quad & \quad\textbf{Payment} \quad & \quad\textbf{Interest} \quad& \textbf{Principal} & \quad\textbf{Balance}\quad\\
\textbf{number} & \textbf{(\$)} \rule[-1ex]{0pt}{0pt}& \textbf{(\$)} & \quad\textbf{reduction (\$)} \quad& \textbf{(\$)}\\
\hline \hline \rule{0pt}{2.5ex}0 \rule[-1ex]{0pt}{0pt}& 0.00 & 0.00 & 0.00 & 850\,000.00 \\
\hline \hline \rule{0pt}{2.5ex}1\rule[-1ex]{0pt}{0pt} & 15\,730.88 & 2975.00 & 12\,755.88 & 837\,244.12 \\
\hline \hline \rule{0pt}{2.5ex}2 \rule[-1ex]{0pt}{0pt}& 15\,730.88 & 2930.35 & 12\,800.53 & 824\,443.59 \\
\hline \hline \rule{0pt}{2.5ex}3 \rule[-1ex]{0pt}{0pt}& 15\,730.88 & & & \\
\hline
\end{array}
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The time series plot below shows the number of homes sold in a town each month over a four‑year period.
Month 1 is January 2016 and month 48 is December 2019.
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\begin{array}{|c|c|}
\hline \rule{0pt}{2.5ex}\quad \textbf{Year} \quad \rule[-1ex]{0pt}{0pt}& \textbf{Total number of sales}\\
\hline \rule{0pt}{2.5ex}2016 \rule[-1ex]{0pt}{0pt}& 361 \\
\hline \rule{0pt}{2.5ex}2017 \rule[-1ex]{0pt}{0pt}& 354 \\
\hline \rule{0pt}{2.5ex}2018 \rule[-1ex]{0pt}{0pt}& 358 \\
\hline \rule{0pt}{2.5ex}2019 \rule[-1ex]{0pt}{0pt}& 357 \\
\hline
\end{array}
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The scatterplot below shows the sale price of a home, in dollars, against the distance of the home from the city centre of Melbourne, in kilometres, distance from city centre.
The sample consists of three‑bedroom homes sold between 2016 and 2018
The equation of the least squares line for the data in the scatterplot is
sale price\(=1\,765\,353-35\,054 \times\)distance from city centre
The coefficient of determination is 0.0806
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\begin{array}{|l|l|}
\hline \rule{0pt}{2.5ex}\text {strength} \quad \quad \rule[-1ex]{0pt}{0pt}& \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \\
\hline \rule{0pt}{2.5ex}\text {direction} \rule[-1ex]{0pt}{0pt}& \\
\hline
\end{array}
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a. \(\text{Distance from city centre}\)
b. \(r=-0.284\)
c. \(\text{sale price}=1\,765\,353\)
d. \(\text{Extrapolation as 2 km lies outside the explanatory variable data range.}\)
e. \(\text{Strength: weak. Direction: negative}\)
f.i. \(\text{Sale price (est)}=1\,765\,353-35\,054 \times 15.5=1\,222\,016\)
\(\text{Actual sale price}=1\,250\,000\)
\(\text{Residual}=1\,250\,000-1\,222\,016=27\,984\)
f.ii.
a. \(\text{Distance from city centre}\)
b. \(\text{Since slope is negative}\)
\(r=-\sqrt{0.0806}=-0.284\)
c. \(\text{Find sale price when distance fran city centre}=0:\)
\(\text{sale price}=1\,765\,353\)
d. \(\text{Extrapolation as 2 km lies outside the explanatory variable data range.}\)
\(\text{(Note: interpolation/extrapolation should be referenced to the}\)
\(\text{explanatory variable range).}\)
e. \(\text{Strength: weak}\)
\(\text{Direction: negative}\)
f.i. \(\text{Sale price (est)}=1\,765\,353-35\,054 \times 15.5=1\,222\,016\)
\(\text{Actual sale price}=1\,250\,000\)
\(\text{Residual}=1\,250\,000-1\,222\,016=27\,984\)
f.ii.
Analyse the relationship between training thresholds and TWO physiological adaptations. In your answer, provide examples of both aerobic and resistance training. (8 marks)
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A 0.010 L aliquot of an acid was titrated with 0.10 mol L\(^{-1} \ \ce{NaOH}\), resulting in the following titration curve.
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a. \(\text{Strategy 1:}\)
\(\text{From the shape of the titration curve, the acid was weak.}\)
\(\text{Equivalence point}\ \ \Rightarrow \ \ \ce{NaOH}\ \text{added}\ = 0.24\ \text{L} \)
\(\text{Halfway to the equivalent point = 0.012 L}, \ce{[ HA ]=\left[ A^{-}\right]}\)
\(\text{Here the pH} \approx 4.4 , \text{or}\ \ce{\left[H^{+}\right] is 4.0 \times 10^{-5}}\).
\(K_a=10^{-\text{pH}}=\ce{\left[H+\right]} \times \dfrac{\ce{\left[A^{-}\right]}}{\ce{[HA]}}\)
\(\text{At this pH,} \ \ce{\left[A^{-}\right]=[HA] so} \ K_a=4.0 \times 10^{-5}\).
\(\text{Strategy 2:}\)
\(\text{Equivalence point is at} \ \ce{0.024 L NaOH added}\).
\(\text{Shape of curve shows acid is monoprotic.}\)
\(\ce{[HA] \times 0.010=0.1 \times 0.024}\)
\(\ce{[HA]=0.24 mol L^{-1}}\)
\(\text{pH at start is approximately 2.5}\)
\(\text{So,} \ \ce{\left[H+\right]=3.16 \times 10^{-3}}\)
\(K_a=\dfrac{\ce{\left[H^{+}\right]\left[A^{-}\right]}}{\ce{[HA]}} \quad \ce{\left[A^{-}\right]=\left[H^{+}\right]}\)
\(\ce{[HA]=0.24-3.16 \times 10^{-3}=0.237}\)
\(K_a=\dfrac{\left(3.16 \times 10^{-3}\right)^2}{0.237}=4.2 \times 10^{-5}\)
b. pH of the final solution < 13:
a. \(\text{Strategy 1:}\)
\(\text{From the shape of the titration curve, the acid was weak.}\)
\(\text{Equivalence point}\ \ \Rightarrow \ \ \ce{NaOH}\ \text{added}\ = 0.24\ \text{L} \)
\(\text{Halfway to the equivalent point = 0.012 L}, \ce{[ HA ]=\left[ A^{-}\right]}\)
\(\text{Here the pH} \approx 4.4 , \text{or}\ \ce{\left[H^{+}\right] is 4.0 \times 10^{-5}}\).
\(K_a=10^{-\text{pH}}=\ce{\left[H+\right]} \times \dfrac{\ce{\left[A^{-}\right]}}{\ce{[HA]}}\)
\(\text{At this pH,} \ \ce{\left[A^{-}\right]=[HA] so} \ K_a=4.0 \times 10^{-5}\).
\(\text{Strategy 2:}\)
\(\text{Equivalence point is at} \ \ce{0.024 L NaOH added}\).
\(\text{Shape of curve shows acid is monoprotic.}\)
\(\ce{[HA] \times 0.010=0.1 \times 0.024}\)
\(\ce{[HA]=0.24 mol L^{-1}}\)
\(\text{pH at start is approximately 2.5}\)
\(\text{So,} \ \ce{\left[H+\right]=3.16 \times 10^{-3}}\)
\(K_a=\dfrac{\ce{\left[H^{+}\right]\left[A^{-}\right]}}{\ce{[HA]}} \quad \ce{\left[A^{-}\right]=\left[H^{+}\right]}\)
\(\ce{[HA]=0.24-3.16 \times 10^{-3}=0.237}\)
\(K_a=\dfrac{\left(3.16 \times 10^{-3}\right)^2}{0.237}=4.2 \times 10^{-5}\)
b. pH of the final solution < 13:
The following three solids were added together to 1 litre of water:
Which precipitate(s), if any, will form? Justify your answer with appropriate calculations. (5 marks)
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Consider the possibility of an electron and a positron colliding in a particle accelerator to produce a proton and an antiproton, as shown in the equation below.
\(\text{electron}+ \text{positron}\ \rightarrow \ \text{proton}+ \text {antiproton}\)
Which statement makes the correct conclusion about the possibility of such a reaction, and provides a plausible reason for this conclusion?
The escape velocity from the surface of a planet, which has no atmosphere, is \(v\). A mass is launched at 45° to the planet's surface at \(v\).
What will be the subsequent motion of the mass?
A satellite with velocity \(v\), is in a geostationary orbit as shown in Figure 1.
At point \(Y\), the satellite explodes and splits into two pieces \(m_{ a }\) and \(m_{ b }\), of identical mass. As a result of the explosion, the velocity of one piece, \(m_{ a }\), changes from \(v\) to \(2 v\) as shown in Figure 2.
Analyse the subsequent motion of BOTH \(m_{ a }\) and \(m_{ b }\) after the explosion. Include reference to relevant conservation laws and formulae in your answer. (8 marks)
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Analyse the role of experimental evidence and theoretical ideas in developing the Standard Model of matter. (6 marks)
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Analyse the consequences of the theory of special relativity in relation to length, time and motion. Support your answer with reference to experimental evidence. (8 marks)
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0.1 mol of solid sodium acetate is dissolved in 500 mL of 0.1 mol L\(^{-1}\) \(\ce{HCl}\) in a beaker. This solution has a pH of 4.8 .
500 mL of distilled water is then added to the beaker.
What is the pH of the final solution?
The concentration of silver ions in a solution is determined by titrating it with aqueous sodium chloride, using yellow potassium chromate as the indicator.
Which row of the table correctly identifies the colour change at the endpoint and the more soluble salt?
\begin{align*}
\begin{array}{l}
\rule{0pt}{2.5ex} \ & \\
\ \rule[-1ex]{0pt}{0pt}& \\
\rule{0pt}{2.5ex}\textbf{A.}\rule[-1ex]{0pt}{0pt}\\
\rule{0pt}{2.5ex}\textbf{B.}\rule[-1ex]{0pt}{0pt}\\
\rule{0pt}{2.5ex}\textbf{C.}\rule[-1ex]{0pt}{0pt}\\
\rule{0pt}{2.5ex}\textbf{D.}\rule[-1ex]{0pt}{0pt}\\
\end{array}
\begin{array}{|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ \text{Colour change} \ \ & \text{More soluble salt} \\
\text{at endpoint}\rule[-1ex]{0pt}{0pt}& \text{} \\
\hline
\rule{0pt}{2.5ex}\text{Red to yellow}\rule[-1ex]{0pt}{0pt}& \ce{Ag2CrO4}\\
\hline
\rule{0pt}{2.5ex}\text{Yellow to red}\rule[-1ex]{0pt}{0pt}& \ce{Ag2CrO4}\\
\hline
\rule{0pt}{2.5ex}\text{Red to yellow}\rule[-1ex]{0pt}{0pt}& \ce{AgCl} \\
\hline
\rule{0pt}{2.5ex}\text{Yellow to red}\rule[-1ex]{0pt}{0pt}& \ce{AgCl} \\
\hline
\end{array}
\end{align*}
Consider the point \(B\) with three-dimensional position vector \(\underset{\sim}{b}\) and the line \(\ell: \underset{\sim}{a}+\lambda \underset{\sim}{d}\), where \(\underset{\sim}{a}\) and \(\underset{\sim}{d}\) are three-dimensional vectors, \(\abs{\underset{\sim}{d}}=1\) and \(\lambda\) is a parameter.
Let \(f(\lambda)\) be the distance between a point on the line \(\ell\) and the point \(B\).
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A particle of mass 1 kg is projected from the origin with a speed of 50 ms\(^{-1}\), at an angle of \(\theta\) below the horizontal into a resistive medium.
The position of the particle \(t\) seconds after projection is \((x, y)\), and the velocity of the particle at that time is \(\underset{\sim}{v}=\displaystyle \binom{\dot{x}}{\dot{y}}\).
The resistive force, \(\underset{\sim}{R}\), is proportional to the velocity of the particle, so that \(\underset{\sim}{R}=-k \underset{\sim}{v}\), where \(k\) is a positive constant.
Taking the acceleration due to gravity to be 10 ms\(^{-2}\), and the upwards vertical direction to be positive, the acceleration of the particle at time \(t\) is given by:
\(\underset{\sim}{a}=\displaystyle \binom{-k \dot{x}}{-k \dot{y}-10}\). (Do NOT prove this.)
Derive the Cartesian equation of the motion of the particle, given \(\sin \theta=\dfrac{3}{5}\). (5 marks)
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Consider the equation
\(z^n \cos\left[n \theta\right]+z^{n-1} \cos \left[(n-1) \theta\right]+z^{n-2} \cos \left[(n-2) \theta\right]+\cdots+z\, \cos\left[\theta\right]=1\)
where \(z \in \mathbb{C} , \theta \in \mathbb{R} \), and \(n\) is a positive integer.
Using a proof by contradiction and the triangle inequality, or otherwise, prove that all the solutions to the equation lie outside the circle \(\abs{z}=\dfrac{1}{2}\) on the complex plane. (4 marks)
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