Aussie Maths & Science Teachers: Save your time with SmarterEd
Rod is saving for a holiday. He deposits $3600 into an account at the end of every year for four years. The account pays 5% per annum interest, compounding annually.
The table shows future values of an annuity of $1.
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A sports car worth $150 000 is bought in December 2005.
In December each year, beginning in 2006, the value of the sports car is depreciated by 10% using the declining balance method of depreciation.
In which year will the depreciated value first fall below $120 000? (2 marks)
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In June, Ms Bigspender received a statement for her credit card account.
The account has no interest-free period. Compound interest is calculated daily and charged to her account on the statement date.
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The location of Sorong is `text(1°S 131°E)` and the location of Darwin is `text(12°S 131°E)`.
Three cards labelled `C`, `J` and `M` can be arranged in any order.
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A 130 cm long garden rake leans against a fence. The end of the rake is 44 cm from the base of the fence.
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| i. |
|
| `cos theta` | `= 44/130` |
| `= 70.216… ^@` | |
| `= 70^@\ \ \ text{(nearest degree)}` |
| ii. |
|
`text(Using cosine rule)`
| `x^2` | `= 130^2 + 44^2-2 xx 130 xx 44 xx cos 53^@` |
| `= 11\ 951.23…` | |
| `x` | `= 109.32…` |
| `= 109\ text{cm (nearest cm)}` |
The graph shows the amounts charged by Company `A` and Company `B` to deliver parcels of various weights.
Vicki wants to investigate the number of hours spent on homework by students at her high school.
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The responses are shown in the frequency table.
What is the mean amount of time spent on homework? (2 marks)
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a. `text(A valid method would be using a stratified sample.)`
`text(The number of students sampled in each year is)`
`text(proportional to the size of each year.)`
| b. |  |
| `text(Mean)` | `= text(Sum of Scores) / text(Total scores)` |
| `= 1455/200` | |
| `= 7.275\ text(hours)` |
List TWO ways in which this graph is misleading. (2 marks)
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Simplify `(ab^2)/w xx (4w)/(3b).` (2 marks)
Robyn plays a game in which she randomly chooses one of these five cards. She plays the game `60` times, replacing the card after each game.
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The sector graph shows the proportion of people, as a percentage, living in each region of Sumcity. There are 24 000 people living in the Eastern Suburbs.
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Jake used the information above to draw a column graph.
Identify this region and justify your answer. (2 marks)
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The formula `D = (2A)/15` is used to calculate the dosage of Hackalot cough medicine to be given to a child.
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The correct dosage of Hackalot cough medicine for Sam is 4 mL.
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`21\ \ \ 45\ \ \ 29\ \ \ 27\ \ \ 19\ \ \ 35\ \ \ 23\ \ \ 58\ \ \ 34\ \ \ 27` (2 marks)
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`:.\ text(Data is positively Skewed.)`
| a. | |
b. `text(10 scores)`
| `:.\ text(Median)` | `= text{(5th + 6th)}/2` |
| `= (27 + 29)/2` | |
| `= 28` |
c. `text(The data has a tail that stretches to the right)`
`:.\ text(Data is positively skewed.)`
Moheb owns five red and seven blue ties. He chooses a tie at random for himself and puts it on. He then chooses another tie at random, from the remaining ties, and gives it to his brother.
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Copy the tree diagram into your writing booklet.
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| i. | `P(R)` | `= (#\ text(red ties))/(#\ text(total ties))` |
| `= 5/12` |
| ii. | |
iii. `Ptext((same colour))`
`= P(text(RR)) + P(text(BB))`
`= 5/12 × 4/11\ \ +\ \ 7/12 × 6/11`
`= 20/132 + 42/132`
`= 31/66`
David is paid at these rates:
His time sheet for last week is:
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Aaron decides to borrow $150 000 over a period of 20 years at a rate of 7.0% per annum.
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How much interest would he save by repaying the loan over 15 years instead of 20 years? (2 marks)
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What is the percentage increase in the number of bacteria? (1 mark)
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What is the number of bacteria after 15 hours? (1 mark)
\begin{array} {|l|c|}
\hline
\rule{0pt}{2.5ex} \text{Time in hours $(t)$} \rule[-1ex]{0pt}{0pt} & \;\; 0 \;\; & \;\; 5 \;\; & \;\; 10 \;\; & \;\; 15 \;\; \\
\hline
\rule{0pt}{2.5ex} \text{Number of bacteria ( $n$ )} \rule[-1ex]{0pt}{0pt} & \;\; 100 \;\; & \;\; 193 \;\; & \;\; 371 \;\; & \;\; ? \;\; \\
\hline
\end{array}
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Use about half a page for your graph and mark a scale on each axis. (4 marks)
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i. `text(Percentage increase)`COMMENT: Common ADV/STD2 content in new syllabus.
`= (114 -100)/100 xx 100`
`= 14text(%)`
ii. `n = 100(1.14)^t`
`text(When)\ \ t = 15,`
| `n` | `= 100(1.14)^15` |
| `= 713.793\ …` | |
| `= 714\ \ \ text{(nearest whole)}` |
| iii. |  |
iv. `text(Using the graph)`
`text(The number of bacteria reaches 300 after)`
`text(approximately 8.4 hours.)`
Joe sells three different flavours of ice-cream from three different tubs in a cabinet. The flavours are chocolate, strawberry and vanilla.
The following graphs have been constructed from data taken from the Bureau of Meteorology website. The information relates to a town in New South Wales.
The graphs show the mean 3 pm wind speed (in kilometres per hour) for each month of the year and the mean number of days of rain for each month (raindays).
A clay brick is made in the shape of a rectangular prism with dimensions as shown.
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Three identical cylindrical holes are made through the brick as shown. Each hole has a radius of 1.4 cm.
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On a television game show, viewers voted for their favourite contestant. The results were recorded in the two-way table.
\begin{array} {|l|c|c|}
\hline
\rule{0pt}{2.5ex} \rule[-1ex]{0pt}{0pt} & \textbf{Male viewers} & \textbf{Female viewers} \\
\hline
\rule{0pt}{2.5ex}\textbf{Contestant 1}\rule[-1ex]{0pt}{0pt} & 1372 & 3915\\
\hline
\rule{0pt}{2.5ex}\textbf{Contestant 2}\rule[-1ex]{0pt}{0pt} & 2054 & 3269\\
\hline
\end{array}
One male viewer was selected at random from all of the male viewers.
What is the probability that he voted for Contestant 1?
A car bought for $50 000 is depreciated using the declining balance method.
Which graph best represents the salvage value of the car over time?
Using the formula `d = 5t^3 - 2`, Marcia tried to find the value of `t` when `d = 137`.
Here is her solution. She has made one mistake.
Which line does NOT follow correctly from the previous line?
Lie detector tests are not always accurate. A lie detector test was administered to 200 people.
The results were:
• 50 people lied. Of these, the test indicated that 40 had lied;
• 150 people did NOT lie. Of these, the test indicated that 20 had lied.
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i.
ii. `text(# Accurate readings)`
`= 40 + 130`
`= 170`
iii. `text(Percentage of people with accurate readings)`
`= text(# Accurate readings)/text(Total readings) xx 100`
`= 170/200`
`= 85 text(%)`
iv. `text{P(lie detected when NOT a lie)}`
`= 20/150`
`= 2/15`
Tai uses the declining balance method of depreciation to calculate tax deductions for her business. Tai’s computer is valued at $6500 at the start of the 2003 financial year. The rate of depreciation is 40% per annum.
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The normal distribution shown has a mean of 170 and a standard deviation of 10.
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| i. `ztext{-score(180)}` | `= (x – mu)/sigma` |
| `= (180 – 170) / 10` | |
| `= 1` |
| `ztext{-score(190)}` | `= (190 -170)/10` |
| `= 2` |
`:. 1 <= ztext(-score) <= 2`
| ii. |  |
`text(From the graph above,)`
`text(13.5% lies in the shaded area.)`
The diagram shows a radial survey of a piece of land.
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| i. |  |
`text(Let)\ E\ text(be directly east of)\ A`
`/_ EAQ = 45^@\ \ \ \ text{(given)}`
| `:. /_ PAQ` | `= 90^@ + 45^@` |
| `= 135^@` |
ii. `text(Bearing of)\ R\ text(from)\ A`
`= 90^@ + 45^@ + 50^@`
`= 185^@\ \ \ text{(or S 5°W)}`
iii. `text(Using cosine rule in)\ Delta PAB`
| ` cos\ /_ PAB` | `= (31^2 + 28^2 – 35^2) / (2 xx 31 xx 28)` |
| `= 0.2995…` | |
| `:. /_ PAB` | `= 72.570…^@` |
| `= 73^@\ \ \ \ text{(nearest degree)}` |
Calculate the height `(h\ text{metres})` of the tree in the diagram. All measurements are
in metres. (2 marks)
Kirbee is shopping for computer software. Novirus costs `$115` more than
Funmaths. Let `x` dollars be the cost of Funmaths.
In 2004 there were 13.5 million registered motor vehicles in Australia. The number of registered motor vehicles is increasing at a rate of 2.3% per year.
Which expression represents the number (in millions) of registered motor vehicles, if `y` represents the number of years after 2004?
What is the bearing of `A` from `B`?
`text(Bearing of)\ A\ text(from)\ B`
`= 180 +120`
`= 300^@`
`=> D`
The diagram shows the points `A(text(−1) , 3)` and `B(2, 0)`.
The line `l` is drawn perpendicular to the `x`-axis through the point `B`.
| (i) | `A(-1,3)\ \ \ \ \ B(2,0)` |
| `AB` | `= sqrt( (x_2 – x_1)^2 + (y_2 – y_1)^2 )` |
| `= sqrt( (2+1)^2 + (0-3)^2 )` | |
| `= sqrt(9+9)` | |
| `= sqrt 18` | |
| `= 3 sqrt 2\ text(units)` |
| (ii) | `text(Gradient of)\ AB` | `= (y_2 – y_1)/(x_2 – x_1)` |
| `= (0 – 3)/(2 + 1)` | ||
| `= – 1` |
| (iii) | `text(S) text(ince Gradient)\ AB = – 1` |
`/_ABO = 45^@`
`:.\ text(Angle between)\ AB\ text(and)\ l`
`= 90 – 45`
`= 45^@`
| (iv) | `text(Equation of)\ AB\ text(has)\ m = -1,\ text(through)\ \ (2,0)` |
| `y – y_1` | `= m(x – x_1)` |
| `y – 0` | `= -1 (x – 2)` |
| `y` | `= -x + 2` |
`:.\ x + y – 2 = 0\ \ \ …\ text(as required)`
(v)
`text{Origin (0,0) satisfies the inequality}`
`:.\ text(Shaded area is below)\ x + y – 2 = 0`
| (vi) | `text(Equation of)\ l` |
| `x = 2` |
| (vii) | `m_(AC) xx m_(AB)` | `= -1\ \ \ (AC⊥AB)` |
| `m_(AC) xx -1` | `= -1` | |
| `m_(AC)` | `= + 1` |
`text(Equation of)\ AC\ text(has)\ \ m = 1,\ text(through)\ (-1, 3),`
| `y – y_1` | `= m(x – x_1)` |
| `y – 3` | `= 1 (x + 1)` |
| `y` | `= x + 4` |
`=>C\ text(lies on)\ y = x + 4`
`text(When)\ \ x = 2,\ \ y = 6`
`:. C\ (2, 6)`
The mean of a set of 5 scores is 62.
What is the new mean of the set of scores after a score of 14 is added?
Peter rides his bike at a speed of 27 km/h.
What is this speed in m/s?
Kay randomly selected a marble from a bag of marbles, recorded its colour and returned it to the bag. She repeated this process a number of times.
Based on these results, what is the best estimate of the probability that Kay will choose a green marble on her next selection?
Which equation represents the relationship between `x` and `y` in this table?
Find integers `a` and `b` by showing working to expand and simplify
`(3-sqrt2)^2 = a-b sqrt2`. (2 marks)
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The radius of Mars is approximately `3\ 397\ 000` metres. Write this number in scientific notation, correct to two significant figures. (2 marks)
Marcella is planning to roll a standard six-sided die 60 times.
How many times would she expect to roll the number 4?
A set of scores is displayed in a stem-and-leaf plot.
What is the median of this set of scores?
Stan worked for 24 hours as shown on his pay slip.
What was his hourly rate of pay?
There are 100 tickets sold in a raffle. Justine sold all 100 tickets to five of her friends. The number of tickets she sold to each friend is shown in the table.
Give a reason why Justine’s statement is NOT correct. (1 mark)
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Rita purchased a camera for $880 while on holidays in Australia. This price included 10% GST. When she left Australia she received a refund of the GST.
What was Rita’s refund?
The diagram shows the shape of Carmel’s garden bed. All measurements are in
metres.
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Calculate the volume of straw required. Give your answer in cubic metres. (2 marks)
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How many bags does she need to buy? (2 marks)
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Find the length of this fence to the nearest metre. (2 marks)
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The table is used to calculate monthly loan repayments.
Samantha has borrowed $70 000 at 8% per annum for 15 years.
What is her monthly loan repayment?
During a ten-minute period, Kath is exercising and Jim is resting.
How much more air would Kath breathe than Jim during this time?
Simplify `2m^2 × 3m p^2`
Janet’s gross income last year was $60 000. She had allowable tax deductions of $5000. Janet paid 1.5% of her taxable income for the Medicare levy.
How much was Janet’s Medicare levy?
Which formula should be used to calculate the distance between Toby and Frankie?
Four radio stations reported the probability of rain as shown in the table.
Which radio station reported the highest probability of rain?
If `d = 6t^2`, what is a possible value of `t` when `d = 2400`?
Using the tax table, determine the tax payable on a taxable income of $47 000.
Use the set of scores 1, 3, 3, 3, 4, 5, 7, 7, 12 to answer Questions 6 and 7.
Question 6
What is the range of the set of scores?
Question 7
What are the median and the mode of the set of scores?
In 2005, Peter’s annual salary was $35 000.
At the start of each subsequent year, his annual salary increases by 4.75%.
In 2010 his salary will be closest to
A. $36 663
B. $42 140
C. $43 310
D. $44 140
E. $55 670
A new air-conditioning unit was purchased for $5000 on 1 January 2009.
On 1 January of each year after 2009 its value is depreciated by 20% using the reducing balance method.
The value of the air conditioner will be below $1500 for the first time on 1 January
A. 2012
B. 2013
C. 2014
D. 2015
E. 2016
Sam and Charlie each invest $5000 for three years.
Sam’s investment earns simple interest at the rate of 7.5% per annum.
Charlie’s investment earns interest at the rate of 7.5% per annum compounding annually.
At the conclusion of three years, correct to the nearest cent, Sam will have
A. $86.48 less than Charlie.
B. $86.48 more than Charlie.
C. $132.23 less than Charlie.
D. $132.23 more than Charlie.
E. the same as Charlie.
Sandra has purchased a $4200 plasma television under a hire-purchase agreement. She paid $600 deposit and will pay the balance in equal monthly instalments over one year.
A flat interest rate of 6% per annum is charged.
Part 1
The amount of each monthly instalment is
Part 2
The annual effective interest rate that Sandra pays under this agreement is closest to
Pia invests $800 000 in an ordinary perpetuity to provide an ongoing fortnightly pension for her retirement.
The interest rate for this investment is 5.8% per annum.
Assuming there are 26 fortnights per year, the amount she will receive at the end of each fortnight is closest to
A. $464
B. $892
C. $1422
D. $1785
E. $3867
