Find the angle between the vectors `underset~r = ((3),(-2),(-1))` and `underset~s = ((2),(1),(1))`, giving the angle in degrees correct to 1 decimal place. (3 marks)
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Aussie Maths & Science Teachers: Save your time with SmarterEd
Find the angle between the vectors `underset~r = ((3),(-2),(-1))` and `underset~s = ((2),(1),(1))`, giving the angle in degrees correct to 1 decimal place. (3 marks)
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`70.9^@`
`underset~r = ((3),(- 2),(-1)) \ , \ |underset~r| \ = sqrt{3^2 + (-2)^2+(-1)^2} = sqrt14`
`underset~s = ((2),(1),(1)) \ , \ |underset~s| \ = sqrt{2^2 + 1^2 + 1^2} = sqrt6`
`underset~r * underset~s` | `= ((3),(-2),(-1)) ((2),(1),(1)) = 6-2-1 = 3` |
`underset~r * underset~s` | `= |underset~a| |underset~b| \ cos theta` |
`3` | `= sqrt14 sqrt6 \ cos theta` |
`cos theta` | `= 3/sqrt84` |
`theta` | `= cos^(-1) (3/sqrt84)` |
`= 70.9^@ \ text{(1 d.p.)}` |
Prove that the vectors `4 underset ~i + 5 underset ~j - 2 underset ~k` and ` −5 underset ~i + 6 underset ~j + 5underset ~k`, are perpendicular. (2 marks)
`text{See Worked Solution}`
`underset ~a ⋅ underset ~b` | `= ((4),(5),(-2))((-5),(6),(5))` |
`=4 xx (−5) + 5 xx 6 + (−2) xx 5` | |
`= -20+30+10` | |
`=0` |
`text(S)text(ince)\ \ underset ~a ⋅ underset ~b =0 \ =>\ \ underset ~a _|_ underset ~b`
Explain how TWO specific personal hygiene practices reduce the risk of infection. (4 marks)
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“
Answers can include any TWO of the following:
→ Washing hands after handling garbage to remove any pathogens from the skin.
→ Covering your mouth when coughing to reduce the chance of pathogens spreading via water droplets.
→ Covering cuts and sores with bandaids or bandages to reduce the chance of infecting others through the transfer of blood and puss which, as well as covering up potential portals of entry for pathogens, will also protect the individual.
→ Daily showers with body wash and shampoo to remove pathogens from skin and scalp.
Drinking water contaminated with dissolved lead (a heavy metal) can cause a serious disease.
Classify this disease as either infectious or non-infectious. Justify your answer. (2 marks)
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“
→ This disease is non-infectious as it is not caused by a pathogen and therefore cannot be transferred from host to host.
→ It can only be obtained through digestion of lead.
Exposure to radiation such as X-rays may change the sequence of bases in DNA.
What is this called?
`A`
→ Any permanent change to the DNA sequence is referred to as a mutation.
`=>A`
Specifications for a Ø10 steel bar require it to have a tolerance of `pm`0.05 mm.
What is the permitted range of diameters for this bar?
`C`
Upper limit = 10.00 + 0.05 = 10.05 mm
Lower limit = 10.00 – 0.05 = 9.95 mm
`=>C`
The diagram shows a self-driving electric vehicle.
Innovations in global positioning systems (GPS) and sensor technologies are used in the operation of this vehicle.
Describe how both of these innovations are used in the control of the vehicle. (3 marks)
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→ Sensors can be used to prevent collisions through the detection of objects in the path or vicinity of the vehicle.
→ GPS shows the vehicle’s position on the Earth’s surface using triangulation.
→ Sensors can be used to prevent collisions through the detection of objects in the path or vicinity of the vehicle.
→ GPS shows the vehicle’s position on the Earth’s surface using triangulation.
Which of the following body systems is involved in detecting and responding to environmental changes?
`D`
→ The nervous system detects and responds to environmental change using sensory nerves.
`=>D`
What is the name of the process that enables organisms to maintain a relatively stable internal environment?
`C`
→ Homeostasis is the process of keeping a fairly stable internal environment in response to change.
`=>C`
Which of the following greatly enhanced scientific understanding of the effects of trace elements?
`B`
→ AAS allows trace elements to be detected at much lower concentrations than previous techniques.
`=>B`
Which indicator in the table would be best for distinguishing between lemon juice (pH = 2.3) and potato juice (pH = 5.8)?
`B`
If Methyl orange is used:
→ Lemon juice would be red, potato juice yellow
`=>B`
During routine maintenance, ultrasonic testing is performed on some aircraft components such as aircraft landing gear.
What is the reason for performing this test?
`C`
`=>C`
A projectile is fired horizontally from a platform.
Measurements of the distance travelled by the projectile from the base of the platform are made for a range of initial velocities. \begin{array}{|c|c|} --- 0 WORK AREA LINES (style=lined) --- --- 6 WORK AREA LINES (style=lined) ---
\hline
\rule{0pt}{2.5ex}\textit{Initial velocity}& \textit{Distance travelled from} \\
\textit{of projectile}\ \text{(ms\(^{-1}\))} \rule[-1ex]{0pt}{0pt}& \textit{base of platform}\ \text{(m)} \\
\hline
\rule{0pt}{2.5ex} 1.4 \rule[-1ex]{0pt}{0pt}&1.0\\
\hline
\rule{0pt}{2.5ex} 2.3 \rule[-1ex]{0pt}{0pt}& 1.7\\
\hline
\rule{0pt}{2.5ex} 3.1 \rule[-1ex]{0pt}{0pt}& 2.2\\
\hline
\rule{0pt}{2.5ex} 3.9 \rule[-1ex]{0pt}{0pt}& 2.3 \\
\hline
\rule{0pt}{2.5ex} 4.2 \rule[-1ex]{0pt}{0pt}& 3.0 \\
\hline
\end{array}
a. b. 2.5 m
a. t = 0.714 s.
b.
\(s_x\)
\(=u(x) t\)
\(t\)
\(\dfrac{s_x}{u_x} \text{(gradient)}\)
Using the line of best fit, gradient = 0.714.
\(s_y\)
\(=u_y t+\frac{1}{2} a_y t^2\)
\(=0+0.5 \times 9.8 \times(0.714)^2\)
\(=2.5 \ \text{m}\)
Height = 2.5 m
An astronaut with a mass of 75 kg lands on Planet X where her weight is 630 N.
What is the acceleration due to gravity (in m s\(^{-2}\)) on Planet X ?
\(B\)
\(F\) | \(=mg\) | |
\(g\) | \(=\dfrac{F}{m}\) | |
\(=\dfrac{630}{75}\) | ||
\(=8.4\ text{m s}^{-2}\) |
\(\Rightarrow B\)
Tom is 25 years old, and likes to keep fit by exercising.
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a. | `text{Max heart rate}` | `=220-25` |
`=195\ text{bpm}` |
b. `text{50% max heart rate}\ = 0.5 xx 195 = 97.5\ text{bpm}`
`text{85% max heart rate}\ = 0.85 xx 195 = 165.75\ text{bpm}`
`:.\ text{Tom should aim for between 98 and 166 bpm in exercise.}`
The cost of hiring a campervan is $210 per day. There is also a charge of $0.35 per km travelled.
A family hired a campervan for 9 days and travelled 2700 km.
How much did the family pay in total? (2 marks)
Cost = $2835
`text{Cost}` | `= 210 xx 9 + 2700 xx 0.35` | |
`= $2835` |
Let `J_(n)=int_(0)^(1)x^(n)e^(-x)\ dx`, where "n" is a non-negative integer.
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i. | `J_0` | `=int_0^1 e^(-x)\ dx` |
`=[-e^(-x)]_0^1` | ||
`=-e^(-1)+1` | ||
`=1-1/e` |
ii. `text{Show}\ \ J_n<=1/(n+1)`
`text{Note:}\ e^(-x)<1\ \ text{for}\ \ x in [0,1]`
`J_n` | `=int_0^1 x^n e^(-x)\ dx` | |
`leq int_0^1 x^n \ dx` | ||
`leq 1/(n+1)[x^(n+1)]_0^1` | ||
`leq 1/(n+1)(1^(n+1)-0)` | ||
`leq 1/(n+1)\ \ text{… as required}` |
iii. `text{Show}\ \ J_n=nJ_(n-1)-1/e`
`u` | `=x^n` | `v′` | `=e^(-x)` |
`u′` | `=nx^(n-1)` | `v` | `=-e^(-x)` |
`J_n` | `=[-x^n * e^(-x)]_0^1-int_0^1 nx^(n-1)*-e^(-x)\ dx` | |
`=(-1^n * e^(-1)+0^n e^0)+nint_0^1 x^(n-1)*e^(-x)\ dx` | ||
`=nJ_(n-1)-1/e` |
iv. `text{Prove}\ \ J_(n)=n!-(n!)/(e)sum_(r=0)^(n)(1)/(r!)\ \ text{for}\ \ n >= 0`
`text{If}\ \ n=0:`
`text{LHS} = 1-1/e\ \ text{(see part (i))}`
`text{RHS} = 0!-0!/e (1/(0!)) = 1-1/e(1)=\ text{LHS}`
`:.\ text{True for}\ \ n=0.`
`text{Assume true for}\ \ n=k:`
`J_(k)=k!-(k!)/(e)sum_(r=0)^(k)(1)/(r!)`
`text{Prove true for}\ \ n=k+1:`
`text{i.e.}\ \ J_(k+1)=(k+1)!-((k+1!))/(e)sum_(r=0)^(k+1)(1)/(r!)`
`J_(k+1)` | `=(k+1)J_k-1/e\ \ text{(using part (iii))}` | |
`=(k+1)(k!-(k!)/(e)sum_(r=0)^(k)(1)/(r!))-1/e` | ||
`=(k+1)!-((k+1)!)/(e)sum_(r=0)^(k)(1)/(r!)-1/e xx ((k+1)!)/((k+1)!)` | ||
`=(k+1)!-((k+1)!)/e(\ sum_(\ r=0)^(k)(1)/(r!)+1/((k+1)!))` | ||
`=(k+1)!-((k+1)!)/e(\ sum_(\ r=0)^(k+1)(1)/(r!))` |
`=>\ text{True for}\ \ n=k+1`
`:.\ text{S}text{ince true for}\ n=1,\ text{by PMI, true for integers}\ n>=1`
v. `0<=J_n<= 1/(n+1)\ \ \ text{(part (ii))}`
`lim_(n->oo) 1/(n+1)=0\ \ => \ lim_(n->oo) J_n=0`
`text{Using part (iv):}`
`J_n/(n!)` | `=1-1/e sum_(r=0)^(n)(1)/(r!)` | |
`1/e sum_(r=0)^(n)(1)/(r!)` | `=1-J_n/(n!)` | |
`sum_(r=0)^(n)(1)/(r!)` | `=e-(eJ_n)/(n!)` | |
`lim_(n->oo)(\ sum_(\ r=0)^(n)(1)/(r!))` | `=lim_(n->oo)(e-(eJ_n)/(n!))` | |
`=e-0` | ||
`=e` |
Which of the following is an example of a non-infectious disease?
`D`
→ Virus, bacteria and fungi are all pathogens; disease carriers which can be transmitted between hosts.
→ Gene mutations are changes in DNA and cannot be transmitted to others by contact or vectors.
`=>D`
Outline ONE way that a pathogen can pass from person to person. (2 marks)
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→ A pathogen such as a bacteria, protozoa or fungus can be passed from person to person by direct transmission
→ This involves skin to skin contact, such as sexual intercourse or shaking hands.
Other modes of transmission include:
→ Object contamination.
→ Waterborne, foodborne or airborne.
→ Animal faeces or nasal secretions.
→ Bodily fluids/respiratory droplets.
→ A pathogen such as a bacteria, protozoa or fungus can be passed from person to person by direct transmission
→ This involves skin to skin contact, such as sexual intercourse or shaking hands.
Other modes of transmission include:
→ Object contamination.
→ Waterborne, foodborne or airborne.
→ Animal faeces or nasal secretions.
→ Bodily fluids/respiratory droplets.
Outline how ONE telecommunications engineering innovation has influenced traditional voice communication systems. (2 marks)
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Smartphones:
→ Have replaced landlines, enable constant communication from anywhere.
→ They also support video conferencing and access to information/the internet.
Other answers could include:
→ Digital radio: Has less interference and higher sound quality.
→ Videoconferencing: Enables visual as well as audio communication.
→ Cybersecurity: Protects your information when communicating with others.
Smartphones:
→ Have replaced landlines, enable constant communication from anywhere.
→ They also support video conferencing and access to information/the internet.
Other answers could include:
→ Digital radio: Has less interference and higher sound quality.
→ Videoconferencing: Enables visual as well as audio communication.
→ Cybersecurity: Protects your information when communicating with others.
Which structural formula represents pentan-2-one?
`D`
The ketone functional group is on the second carbon atom.
`=>D`
Which of the following only contains tasks performed by the aeronautical engineer?
`A`
By Elimination:
→ Engineers do not assist in air traffic control (eliminate `B` and `D`).
→ Engineers do not schedule flights (eliminate `C`).
`=>A`
Exposure to arsenic in drinking water has been associated with the onset of many diseases. The World Health Organisation recommends arsenic levels in drinking water should be below 10 `mu`g L-1 .
An epidemiological study involving 58 406 young adults was conducted over an 11-year period in one country to investigate young-adult mortality due to chronic exposure to arsenic in local drinking water. Each individual's average exposure and cumulative exposure to arsenic over the time of the study were calculated. Age, sex, education and socioeconomic status were taken into account during the analysis of the results.
The graphs show survival rates for males and females over the 11-year period associated with different average levels of exposure to arsenic in drinking water.
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a. Successful answers should include two of the following:
→ The identification of the age and sex of participants
→ Socioeconomic status
→ Arsenic exposure
→ Large sample size
→ The length of the study period
b. Consider the less than 90 `mu`g L-1 group:
→ The survival rate for both males and females was highest in this “control” group. This is despite the group being exposed to more arsenic than recommended by WHO.
→ In both males and females, increased exposure to arsenic led to lower survival rates.
→ This gradual survival decrease is best seen in males.
→ In females, all doses over 90 `mu`g L-1 lead to similar survival decrease, suggesting other factors, such as a gene or diet, are interacting with the dosage of arsenic.
→ Over the 11 year period, survival progressively declined, supporting the hypothesis.
→ It is important to note, however, that despite the large sample size and time period the study was conducted, survival only dropped by 0.1%.
a. Successful answers should include two of the following:
→ The identification of the age and sex of participants
→ Socioeconomic status
→ Arsenic exposure
→ Large sample size
→ The length of the study period
b. Consider the less than 90 `mu`g L-1 group:
→ The survival rate for both males and females was highest in this “control” group. This is despite the group being exposed to more arsenic than recommended by WHO.
→ In both males and females, increased exposure to arsenic led to lower survival rates.
→ This gradual survival decrease is best seen in males.
→ In females, all doses over 90 `mu`g L-1 lead to similar survival decrease, suggesting other factors, such as a gene or diet, are interacting with the dosage of arsenic.
→ Over the 11 year period, survival progressively declined, supporting the hypothesis.
→ It is important to note, however, that despite the large sample size and time period the study was conducted, survival only dropped by 0.1%.
Outline a change in technology that has led to improved fuel efficiency in cars. (2 marks)
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Successful answers should discuss one of the following:
→ Lighter/more streamlined cars are generally more fuel efficient. This has been achieved through improved manufacturing methods and improved materials (ie replacing steel bumpers with lighter polymers).
→ Improved materials technology used to produce aluminium engine blocks instead of cast iron to decrease weight.
→ Mathematical modelling to increase aerodynamics of a car.
→ Hybrid cars and regenerative braking.
→ Shift towards fuel injection and higher compression ratio.
→ Cruise control maintains better average fuel economy.
→ Improved efficient gearboxes and differentials with low friction and torque.
→ Improvements to lubrication technology to reduce internal friction.
Successful answers should discuss one of the following:
→ Lighter/more streamlined cars are generally more fuel efficient. This has been achieved through improved manufacturing methods and improved materials (ie replacing steel bumpers with lighter polymers).
→ Improved materials technology used to produce aluminium engine blocks instead of cast iron to decrease weight.
→ Mathematical modelling to increase aerodynamics of a car.
→ Hybrid cars and regenerative braking.
→ Shift towards fuel injection and higher compression ratio.
→ Cruise control maintains better average fuel economy.
→ Improved efficient gearboxes and differentials with low friction and torque.
→ Improvements to lubrication technology to reduce internal friction.
For real numbers `a,b >= 0` prove that `(a+b)/(2) >= sqrt(ab)`. (2 marks)
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`text{Proof (See Worked Solutions)}`
`text{S}text{ince}\ \ (sqrta-sqrtb)^2>=0:`
`a-2sqrt(ab)+b` | `>=0` | |
`a+b` | `>=2sqrt(ab)` | |
`:.(a+b)/2` | `>=sqrt(ab)\ \ text{… as required}` |
Express `(3-i)/(2+i)` in the form `x+iy`, where `x` and `y` are real numbers. (2 marks)
`1-i`
`(3-i)/(2+i)` | `=(3-i)/(2+i) xx (2-i)/(2-i)` | |
`=(6-3i-2i+i^2)/(2^2-i^2)` | ||
`=(5-5i)/5` | ||
`=1-i` |
A public education campaign was developed with the aim of lowering the incidence of skin cancer in the population.
The campaign was adopted Australia wide and is illustrated in the poster.
Which is the best method to measure the effectiveness of the campaign?
`C`
A direct measurement of any reduction in cancer incidence would prove campaign effectiveness.
`=>C`
Malaria is a disease in humans caused by a single-celled Plasmodium species. It is transmitted by female mosquitoes.
Which of the following is true for malaria?
`C`
The Plasmodium is the pathogen that causes malaria disease, while the mosquito transmits the pathogen and is therefore the vector.
`=>C`
A patient visited an audiologist for a hearing test. The audiologist tested both ears at specific frequencies. The volumes at which each frequency could be heard are shown.
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a.
b. Right ear is within normal hearing range.
Left ear has a deficit and cannot hear at a normal level.
c. Effective technology: Bone Conduction Implants
Bone conduction implants would prove to be the most effective technology to restore hearing to this patient.
Bone conduction implants detect sound waves via a microphone, relaying them to a sound processor that converts the waves into vibrations which are then directly transferred to the cochlea. This process bypasses the ear blockage, therefore restoring hearing to the patient.
a.
b. → Right ear is within normal hearing range.
→ Left ear has a deficit and cannot hear at a normal level.
c. Effective technology: Bone Conduction Implants
→ Bone conduction implants would prove to be the most effective technology to restore hearing to this patient.
→ Bone conduction implants detect sound waves via a microphone, relaying them to a sound processor that converts the waves into vibrations which are then directly transferred to the cochlea.
→ This process bypasses the ear blockage, therefore restoring hearing to the patient.
Some large cities, such as Sydney, have bridges which span large bodies of water.
Outline TWO ways in which the construction of such bridges affects society. (2 marks)
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→ Bridges spanning large bodies of water allow for faster travel than other existing transport such as ferries.
→ This results in greater work productivity due to faster commuting. People will also have greater opportunity to work and recreate on both sides of the bridge.
Other advantages could include:
→ Greater use of major infrastructure like hospitals, schools and stadiums
→ Train lines can be constructed on the bridge to allow for extra mass transit
→ Reduction in existing transport on the harbour such as ferries
→ Potential increase in leisure time
→ Bridges spanning large bodies of water allow for faster travel than other existing transport such as ferries.
→ This results in greater work productivity due to faster commuting. People will also have greater opportunity to work and recreate on both sides of the bridge.
Other advantages could include:
→ Greater use of major infrastructure like hospitals, schools and stadiums
→ Train lines can be constructed on the bridge to allow for extra mass transit
→ Reduction in existing transport on the harbour such as ferries
→ Potential increase in leisure time
The flight recorder, commonly known as the 'black box', was developed in Australia in the 1950s to assist in air crash investigations.
How does a black box assist in an air crash investigation?
`C`
`=>C`
For the vectors `underset~u= underset~i- underset~j` and `underset~v=2 underset~i+ underset~j`, evaluate each of the following.
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i. `underset~u= ((1),(-1)),\ \ underset~v= ((2),(1))`
`underset~u+3 underset~v` | `=((1),(-1))+3((2),(1))` | |
`=((1+3xx2),(-1+3xx1))` | ||
`=((7),(2))` |
ii. | `underset~u * underset~v` | `=((1),(-1))*((2),(1))` |
`=1xx2+(-1)xx1` | ||
`=1` |
A scientist is studying the growth of bacteria. The scientist models the number of bacteria, `N`, by the equation
`N(t)=200e^(0.013 t)`,
where `t` is the number of hours after starting the experiment.
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a. | `N(0)` | `=200e^0` |
`=200\ text{bacteria}` |
b. `text{Find}\ N\ text{when}\ \ t=24:`
`N(24)` | `=200e^(0.013xx24)` | |
`=273.23…` | ||
`=273\ text{bacteria (nearest whole)}` |
c. | `N` | `=200e^(0.013 t)` |
`(dN)/dt` | `=0.013xx200e^(0.013t)` | |
`=2.6e^(0.013t)` |
`text{Find}\ \ (dN)/dt\ \ text{when}\ \ t=24:`
`(dN)/dt` | `=2.6e^(0.013xx24)` | |
`=3.550…` | ||
`=3.55\ text{bacteria/hr (to 2 d.p.)}` |
Cards are stacked to build a 'house of cards'. A house of cards with 3 rows is shown.
A house of cards requires 3 cards in the top row, 6 cards in the next row, and each successive row has 3 more cards than the previous row.
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a. `a=3, \ d=3`
`S_n` | `=n/2[2a+(n-1)d]` | |
`S_12` | `=12/2(2xx3 + 11xx3)` | |
`=6(6+33)` | ||
`=234\ \ text{… as required}` |
b. `text{Find}\ \ n\ \ text{given}\ \ S_n=828:`
`828` | `=n/2[6+(n-1)3]` | |
`1656` | `=n(3+3n)` | |
`=3n^2+3n` |
`3n^2+3n-1656` | `=0` | |
`n^2+n-552` | `=0` | |
`(n+24)(n-23)` | `=0` |
`:. n=23\ text{rows}\ \ (n>0)`
Consider the following dataset.
`{:[13,16,17,17,21,24]:}`
Which row of the table shows how the median and mean are affected when a score of 5 is added to the dataset?
`D`
`text{Mean decreases.}`
`text{Median remains 17.}`
`=>D`
The formula `C=100 n+b` is used to calculate the cost of producing laptops, where `C` is the cost in dollars, `n` is the number of laptops produced and `b` is the fixed cost in dollars.
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a. `text{Find}\ \ C\ \ text{given}\ \ n=1943 and b=20\ 180`
`C` | `=100 xx 1943 + 20\ 180` | |
`=$214\ 480` |
b. `text{Find}\ \ n\ \ text{given}\ \ C=97\ 040 and a=26`
`C` | `=100 n+a n+20\ 180` | |
`97\ 040` | `=100n + 26n +20\ 180` | |
`126n` | `=76\ 860` | |
`n` | `=(76\ 860)/126` | |
`=610 \ text{laptops}` |
The table below shows the distances, in kilometres, between a number of towns.
Tom is 25 years old, and likes to keep fit by exercising.
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a. | `text{Max heart rate}` | `=220-25` |
`=195\ text{bpm}` |
b. `text{50% max heart rate}\ = 0.5 xx 195 = 97.5\ text{bpm}`
`text{85% max heart rate}\ = 0.85 xx 195 = 165.75\ text{bpm}`
`:.\ text{Tom should aim for between 98 and 166 bpm in exercise.}`
A marble is rolled off a horizontal bench and falls to the floor.
Rolling the marble at a slower speed would
`B`
Vertical distance from floor to bench is constant → time of flight stays the same
Slower horizontal velocity → range decreases
`=>B`
Which pair of components must be equal for a chemical system to be at equilibrium?
`A`
→ Rate of forward = rate of reverse reaction (dynamic equilibrium)
`=>A`
A patient felt tired, weak and had a swollen neck. After following the doctor's advice to eat more foods containing iodised salt, her symptoms disappeared.
What was the most likely cause of the patient's symptoms?
`C`
Since iodised salt made symptoms disappear, the patient most likely had a deficiency of iodine.
`=>C`
The polynomial `p(z) = z^3 + alpha z^2 + beta z + gamma`, where `z ∈ C` and `alpha, beta, gamma ∈ R`, can also be written as `p(z) = (z - z_1)(z - z_2)(z - z_3)`, where `z_1 ∈ R` and `z_2, z_3 ∈ C`.
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i. `text(By conjugate root theory)`
`z_2 = barz_3`
ii. `text(Let)\ \ z_1 = a + bi, \ z_2 = a – bi`
`|z_2 + z_3| = |2a| = 0 \ => \ a = 0`
`|z_2 – z_3| = |2b| = 6 \ => \ b = ±3`
`text(Using)\ \ p(2) = -13`
`(2 – z_1)(2 – 3i)(2 + 3i)` | `= -13` |
`(2 – z_1)(4 + 9)` | `= -13` |
`2 – z_1` | `= -1` |
`z_1` | `= 3` |
`p(z)` | `= (z – 3)(z – 3i)(z + 3i)` |
`= (z – 3)(z^2 + 9)` | |
`= z^3 – 3z^2 + 9z – 27` |
`:. alpha = –3, \ beta = 9, \ gamma = –27`
z
Christy sells blocks of land at the housing estate.
Her pay is based on the number of blocks that she sells each week.
The maximum number of blocks sold each week is 20.
The graph below shows Christy's weekly pay, in dollars, for the number of blocks sold.
John also sells blocks of land. He is paid $1000 for each block that he sells.
a. `6 \ text{blocks}`
b. `text{Blocks sold for equal pay: 3, 6, 11, 15, 20}`
c. `y <= 3/5 x`
The graph below shows the height, in metres, of a drone flying above a new housing estate over a six-minute period of time.
a. `4 \ text{minutes}`
b. `text{At} \ t = 0 , \ text{height} = 0 \ text{m}`
`text{At} \ t = 2 , \ text{height} = 200 \ text{m}`
`:. \ text{Average rate of change}`
`= {200 – 0}/{2}`
`= 100 \ text{metres per minute}`
The game of squash is played with a special ball that has a radius of 2 cm.
Squash balls may be sold in cube-shaped boxes.
Each box contains one ball and has a side length of 4.1 cm, as shown in the diagram below.
a. | `V` | `= 4/3 pi r^3` |
`= 4/3 xx pi xx 2^3` | ||
`= 33.5103 …` | ||
`= 33.51 \ text{cm}^3 \ text{(to 2 d.p.)}` |
b. `text{Volume of cube} = 4.1^3 = 68.921 \ text{cm}^3`
`:. \ text{Empty space}` | `= 68.921 – 33.51` |
`= 35.41 \ text{cm}^3 \ text{(to 2 d.p.)}` |
c. | `text(S.A.)` | `= 6 xx 4.1 xx 4.1` |
`= 100.86 \ text{cm}^2` |
d. `text{Length} = 17 div 4.1 = 4.146 … \ \ to 4 \ text{boxes}`
`text{Width} = 12.5 div 4.1 = 3.048 … \ to 3 \ text{boxes}`
`text{Height} = 8.5 div 4.1 = 2.073 … \ \ to 2\ text{boxes}`
`:. \ text{Max boxes}` | `= 4 xx 3 xx 2` |
`= 24` |
Maggie's house has five rooms, `A, B, C, D` and `E`, and eight doors.
The floor plan of these rooms and doors is shown below. The outside area, `F`, is shown shaded on the floor plan.
The floor plan is represented by the graph below.
On this graph, vertices represent the rooms and the outside area. Edges represent direct access to the rooms through the doors.
One edge is missing from the graph.
a.
b. `text{Degree} = 2`
c.i. `FABEDCF\ text(or)\ FCDEBAF`
c.ii. `text{Hamiltonian cycle}`
The main computer system in Elena's office has broken down.
The five staff members, Alex (`A`), Brie (`B`), Chai (`C`), Dex (`D`) and Elena (`E`), are having problems sending information to each other.
Matrix `M` below shows the available communication links between the staff members.
`qquadqquadqquadqquadqquadqquadqquadqquadqquad text(receiver)`
`qquadqquadqquadqquadqquadqquadqquad \ \ \ A \ \ B \ \ C \ \ D \ \ E`
`M= \ text{sender} \ \ {:(A),(B),(C),(D),(E):} [(0,1,0,0,1),(0,0,1,1,0),(1,0,0,1,0),(0,1,0,0,0),(0,0,0,1,0)] `
In this matrix:
`qquadqquadqquadqquadqquadqquadqquadqquadqquad text(receiver)`
`qquadqquadqquadqquadqquadqquadqquad \ \ \ A \ \ B \ \ C \ \ D \ \ E`
`M= \ text{sender} \ \ {:(A),(B),(C),(D),(E):} [(0,0,1,2,0),(0,1,0,1,0),(0,1,0,0,0),(0,0,1,1,0),(0,1,0,0,0)] `
Only one pair of individuals has two different two-step communication links.
List each two-step communication link for this pair. (1 mark)
a. `text{B (sender) to D (receiver)} => 1`
`text{D (sender) to B (receiver)} => 1`
`:. \ text{B and D can send information to each other}`
b. `text{Elena} to text{Dex} to text{Brie} to text{Chai}`
c. `text{The two 2-step links are from Alex to Dex.}`
`text{These are:}`
`text{Alex} to text{Brie} to text{Dex}`
`text{Alex} to text{Elena} to text{Dex}`
Elena imports three brands of olive oil: Carmani (`C`) Linelli (`L`) and Ohana (`O`).
The number of 1 litre bottles of these oils sold in January 2021 is shown in matrix `J` below.
`J = {:[(2800),(1700),(2400)]:} {:(C),(L),(O):}`
`k =` |
|
a. `3 xx 1`
b. `text{All brands} \ ↑ 5%`
`:. k = 1.05`
In the sport of heptathlon, athletes compete in seven events.
These events are the 100 m hurdles, high jump, shot-put, javelin, 200 m run, 800 m run and long jump.
Fifteen female athletes competed to qualify for the heptathlon at the Olympic Games.
Their results for three of the heptathlon events – high jump, shot-put and javelin – are shown in Table 1.
a. `3 \ text{High jump, shot-put and javelin}`
`text{Athlete number is not a numerical variable}`
b. `text{High jump mean}`
`= (1.76 + 1.79 + 1.83 + 1.82 + 1.87 + 1.73 + 1.68 + 1.82 +`
`1.83 + 1.87 + 1.87 + 1.80 + 1.83 + 1.87 + 1.78) ÷ 15`
`= 1.81`
c. | `z text{-score} (14.50)` | `= {14.50 – 13.74}/{1.43}` |
`= 0.531 …` | ||
`= 0.5 \ text{(to 1 d.p.)}` |
d. `P (z text{-score} > -1 ) = 84text(%)`
e. `text{If the} \ Q_3 \ text{value is also the highest value in the data set,}`
`text{there is no whisker at the upper end of a boxplot.}`
f. `text{Javelin (ascending):}`
`38.12, 39.22, 40.62, 40.88, 41.22, 41.32, 42.33, 42.41, `
`42.51, 42.65, 42.75, 42.88, 45.64, 45.68, 46.53`
`Q_1 = 40.88 \ \ , \ Q_3 = 42.88 \ \ , \ \ IQR = 42.88 – 40.88 = 2`
`text{Upper Fence}` | `= Q_3 + 1.5 xx IQR` |
`= 42.88 + 1.5 xx 2` | |
`= 45.88` |
`:. \ text{Minimum distance = 45.89 m (longer than upper fence value)}`
The graph below shows the average number of sunlight hours per day for each month of a particular year.
The number of months for which the average number of sunlight hours per day was recorded as being below four hours is
`B`
`text{3 points are below 4 (on}\ y text{-axis)}`
`=> B`
Five friends ate fruit for morning tea.
The bipartite graph below shows which types of fruit each friend ate.
Which one of the following statements is not true?
`C`
`text{Consider option C}`
`text{Van ate strawberry and orange}`
`:. text{Statement is not true.}`
`=> C`
Deepa invests $500 000 in an annuity that provides an annual payment of $44 970.55
Interest is calculated annually.
The first five lines of the amortisation table are shown below.
Part 1
The principal reduction associated with payment number 3 is
Part 2
The number of years, in total, for which Deepa will receive the regular payment of `$44\ 970.55` is closest to
`text(Part 1:)\ C`
`text(Part 2:)\ B`
`text{Part 1}`
`text{Principal reduction}` | `= 449\ 060.08 – 422\ 051.93` |
`= 27\ 008.15` |
`=> C`
`text{Part 2}`
`text{Interest rate} = {20\ 000}/{500\ 000} = 4text{% p.a.}`
`text{Find}\ N\ text{by TVM solver:}`
`N` | `= ?` |
`I(%)` | `= 4` |
`PV` | `= -500\ 000` |
`PMT` | `= 44\ 970.55` |
`FV` | `= 0` |
`text(P/Y)` | `= text(C/Y) = 1` |
`:. N = 15.000`
`=> B`
Find the angle between the vectors `underset~a = ((2),(0),(4))` and `underset~b = ((-3),(1),(2))`, giving the angle in degrees correct to 1 decimal place. (3 marks)
`83.1^@`
`underset~a = ((2),(0),(4)) \ , \ |underset~a| \ = sqrt{2^2 + 4^2} = sqrt20`
`underset~b = ((-3),(1),(2)) \ , \ |underset~b| \ = sqrt{(-3)^2 + 1^2 + 2^2} = sqrt14`
`underset~a * underset~b` | `= ((2),(0),(4)) ((-3),(1),(2)) = – 6 + 0 + 8 = 2` |
`underset~a * underset~b` | `= |underset~a| |underset~b| \ cos theta` |
`2` | `= sqrt20 sqrt14 \ cos theta` |
`cos theta` | `= 2/sqrt280` |
`theta` | `= cos^(-1) (1/sqrt70)` |
`= 83.1^@ \ text{(1 d.p,)}` |
Find `overset5 underset{n=1}∑ (i)^n`. (2 marks)
`i`
`overset5 underset{n=1}∑ (i)^n` | `= i + i^2 + i^3 + i^4 + i^5` |
`= i – 1 – i + 1 + i` | |
`= i` |
The complex numbers \(z=2 e^{i\small{\dfrac{\pi}{2}}}\) and \(w=6 e^{i \small{\dfrac{\pi}{6}}}\) are given.
Find the value of \(zw\) , giving the answer in the for \(r e^{i \theta}\). (2 marks)
\(12 e^{i 2 \small{\dfrac{\pi}{3}}}\)
\(zw\) | \(=2 e^{i \small{\dfrac{\pi}{2}}} \cdot 6 e^{i \small{\dfrac{\pi}{6}}}\) | |
\(=12 e^{i \small{\dfrac{\pi}{2}}+i \small{\dfrac{\pi}{6}}}\) | ||
\(=12 e^{i \small{\dfrac{2 \pi}{3}}}\) |
Which of the following is a vector equation of the line joining the points `A (4, 2, 5)` and `B (–2, 2, 1)`?
`B`
`overset->{AB}` | `= ((-2),(2),(1)) – ((4),(2),(5)) = ((-6),(0),(-4))` | |
`underset~r` | `= ((4),(2),(5)) + λ_1 ((-6),(0),(-4))` | |
`= ((4), (2), (5)) + λ_2 ((3),(0),(2))` |
`=>\ B`
Which expression is equal to `int x^5 e^{7x} dx`?
`A`
`u = x^5` | `v^{′} = e^{7x}` | |
`u^{′} = 5x^4` | `v = 1/7 e^{7x}` |
`int uv^{′}\ dx` | `= uv-int u^{′}v \ dx` | |
`= 1/7 x^5 e^{7x}-5/7 int x^4 e^{7x}\ dx` |
`=>\ A`
Four cubes are placed in a line as shown on the diagram.
Which of the following vectors is equal to `overset->{AB} + overset->{CQ}`
`B`
`overset->{AB} \ + \ overset->{CQ}` | `= overset->{CD} + overset->{DP}` | |
`= overset->{CP}` |
`=>\ B`
Find `(underset~i + 6underset~j) + (2underset~i - 7underset~j)`. (1 mark)
`3underset~i – underset~j`
`((1),(6)) + ((2),(-7)) = ((3),(-1)) = 3underset~i – underset~j`