Aussie Maths & Science Teachers: Save your time with SmarterEd
Shaun decides to buy a new sound system on a time-payment (hire-purchase) plan.
The sound system is priced at $3500.
Shaun pays a deposit of $500 and repayments of $80 per month for five years.
The flat rate of interest charged per annum, correct to one decimal place, is
A. 6.0%
B. 8.7%
C. 10.3%
D. 12.0%
E. 15.3%
Jane invests in an ordinary perpetuity to provide her with a weekly payment of $500.
The interest rate for the investment is 5.9% per annum.
Assuming there are 52 weeks per year, the amount that Jane needs to invest in the perpetuity is closest to
A. $26 000
B. $102 000
C. $154 000
D. $221 000
E. $441 000
$15 000 is invested for 12 months.
For the first six months the interest rate is 6.1% per annum compounding monthly
After six months the interest rate increases to 6.25% per annum compounding monthly.
The total interest earned by this investment over 12 months is closest to
A. $926
B. $935
C. $941
D. $953
E. $965
The transaction details for a savings account for the month of July 2012 are shown below.
Interest is calculated and paid monthly on the minimum monthly balance.
The annual rate of interest paid on this account is closest to
A. `3.5text(%)`
B. `4.3text(%)`
C. `4.7text(%)`
D. `4.9text(%)`
E. `5.2text(%)`
$100 000 is invested in a perpetuity at an interest rate of 6% per annum.
After 10 quarterly payments have been made, the amount of money that remains invested in the perpetuity is
A. $15 000
B. $40 000
C. $85 000
D. $94 000
E. $100 000
$10 000 is invested for five years. Interest is earned at a rate of 8% per annum, compounding quarterly.
Which one of the following calculations will give the total interest earned, in dollars, by this investment?
A. `10\ 000 xx 1.02^5-10\ 000`
B. `10\ 000 xx 1.02^20-10\ 000`
C. `10\ 000 xx 1.08^5 -10\ 000`
D. `10\ 000 xx 1.08^20-10\ 000`
E. `10\ 000 xx 1.02^20`
The vertical line that passes through the point `(3, 2)` has the equation
A. `x + y = 5`
B. `xy = 6`
C. `3y = 2x`
D. `y = 2`
E. `x = 3`
`=>E`
Xavier and Yvette share a job.
Yvette must work at least twice as many hours as Xavier.
They must work at least 40 hours each week, in total.
Xavier must work at least 10 hours each week.
Yvette can only work for a maximum of 30 hours each week.
Let `x` represent the number of hours that Xavier works each week.
Let `y` represent the number of hours that Yvette works each week.
In which one of the following graphs does the shaded area show the feasible region defined by these conditions?
A loan of $1000 is to be repaid with six payments of $180 per month.
The effective annual rate of interest charged is closest to
A. 8.0%
B. 13.7%
C. 16.0%
D. 27.4%
E. 30.9%
The four inequalities below were used to construct the feasible region for a linear programming problem.
`x >= 0`
`y >= 0`
`x + y <= 9`
`y <= 1/2 x`
A point that lies within this feasible region is
A. `(4, 4)`
B. `(5, 3)`
C. `(6, 2)`
D. `(6, 4)`
E. `(7, 3)`
The point of intersection of two lines is `(2, – 2)`.
One of these two lines could be
A. `x - y = 0`
B. `2x + 2y = 8`
C. `2x + 2y = 0`
D. `2x - 2y = 4`
E. `2x - 2y = 0`
Which one of the following statements about the line with equation `12x - 4y = 0` is not true?
A. the line passes through the origin
B. the line has a slope of 12
C. the line has the same slope as the line with the equation `12x - 4y = 12`
D. the point `(1, 3)` lies on the line
E. for this line, as `x` increases `y` increases
A gas-powered camping lamp is lit and the gas is left on for six hours. During this time the lamp runs out of gas.
The graph shows how the mass, `M`, of the gas container (in grams) changes with time, `t` (in hours), over this period.
Part 1
Assume that the loss in weight of the gas container is due only to the gas being burnt.
From the graph it can be seen that the lamp runs out of gas after
A. `1.5\ text(hours.)`
B. `3\ text(hours.)`
C. `4.5\ text(hours.)`
D. `6\ text(hours.)`
E. `220\ text(hours.)`
Part 2
Which one of the following rules could be used to describe the graph above?
A. `M={(332.5 − 25t \ \ \ \ \ text( for ) \ \ \ \ \ \ 0 ≤ t ≤ 4.5),(220 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ text(for ) \ \ \ 4.5 < t ≤ 6) :}`
B. `M={(332.5 − 25t \ \ \ \ \ text( for ) \ \ \ \ \ \ 0 ≤ t ≤ 4.5),(220t \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ text( for ) \ \ \ 4.5 < t ≤ 6) :}`
C. `M={(332.5 + 25t \ \ \ \ \ text( for ) \ \ \ \ \ \ 0 ≤ t ≤ 4.5),(220t \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ text( for ) \ \ \ 4.5 < t ≤ 6) :}`
D. `M={(332.5 − 12.5t \ \ text( for ) \ \ \ \ \ \ 0 ≤ t ≤ 4.5),(220t \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ text( for ) \ \ \ 4.5 < t ≤ 6) :}`
E. `M={(332.5 − 12.5t \ \ text( for ) \ \ \ \ \ \ 0 ≤ t ≤ 4.5),(220 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ text(for ) \ \ \ 4.5 < t ≤ 6) :}`
A region is defined by the following inequalities
`y >= -4x + 10`
`y - x >= 1`
A point that lies within this region is
A. `(1, 3)`
B. `(2, 1)`
C. `(3, 2)`
D. `(4, 6)`
E. `(5, 1)`
The graph above shows the relationship between `y` and `x^2`.
The relationship between `y` and `x` is
A. `y = 4x`
B. `y = 1/4 x`
C. `y = 1/4 x^2`
D. `y = 16x^2`
E. `y = 1/16 x^2`
A mixture contains two liquids, `A` and `B`.
Liquid `A` costs $2 per litre and liquid `B` costs $3 per litre.
Let `x` be the volume (in litres) of liquid `A` purchased.
Let `y` be the volume (in litres) of liquid `B` purchased.
Which graph below shows all possible volumes of liquid `A` and liquid `B` that can be purchased for exactly $12?
The time, in hours, that each student spent sleeping on a school night was recorded for `1550` secondary-school students. The distribution of these times was found to be approximately normal with a mean of 7.4 hours and a standard deviation of 0.7 hours.
How many students would you expect to spend more than 8.1 hours sleeping on a school night?
You may assume for normally distributed data that:
A. `16`
B. `248`
C. `1302`
D. `1510`
For the triangle shown, the value of `sin x^@` is given by
A. `(sin 125.1^@)/2`
B. `(5^2 + 4^2 − 8^2)/(2 xx 5 xx 4)`
C. `2 xx sin 125.1^@`
D. `(5^2 + 8^2 − 4^2)/(2 xx 5 xx 8)`
E. `(5 xx sin 125.1^@)/8`
A piece of plaster has a uniform cross-section, which has been shaded, and has dimensions as shown.
| (i) |  |
| `A` | `= h/3 [y_0 +4y_1 + y_2]\ text(… applied twice)` |
| `= 3.6/3 (5 + 4 xx 4.6 + 3.7) + 3.6/3 (3.7 + 4 xx 2.8 + 0)` | |
| `= 32.52 + 17.88` | |
| `= 50.4\ text(cm²)` |
(ii) `text(Total Area) = 7480.8\ text(cm²)`
| `text(Area of Base)` | `= 14.4 xx 200` |
| `= 2880\ text(cm²)` |
| `text(Area of End)` | `= 5 xx 200` |
| `= 1000\ text(cm²)` |
| `text(Area of sides)` | `= 2 xx 50.4\ \ \ text{(from (i))}` |
| `= 100.8\ text(cm²)` |
`:.\ text(Area of curved surface)`
`= 7480.8 – (2880 + 1000 + 100.8)`
`= 3500\ text(cm²)`
Two unbiased dice, `A` and `B`, with faces numbered `1`, `2`, `3`, `4`, `5` and `6` are rolled.
The numbers on the uppermost faces are noted. This table shows all the possible outcomes.
A game is played where the difference between the highest number showing and the lowest number showing on the uppermost faces is calculated.
The points `M`, `N` and `P` form the vertices of a triangular course for a yacht race.
`MN = MP = 4\ text(km.)`
The bearing of `N` from `M` is 070°
The bearing of `P` from `M` is 180°
Three people perform different calculations to determine the length of `NP` in kilometres.
| Graeme | `\ \ \ \ \ NP` | `= sqrt(16 + 16 − 2 xx 4 xx 4 xx cos110^@)` |
| Shelley | `NP` | `= 2 xx 4 xx cos 35^@` |
| Tran | `NP` | `= (4 xx sin110^@)/sin35^@` |
The correct length of `NP` would be found by
A. Graeme only.
B. Tran only.
C. Graeme and Shelley only.
D. Graeme and Tran only.
E. Graeme, Shelley and Tran.
`text(Using cosine rule in)\ \ Delta MNP:`
`NP = sqrt(4^2 + 4^2 − 2 xx 4 xx 4 xx cos110^@)`
`:.\ text(Graeme is correct)`
`text(Using sine rule in)\ \ Delta MNP:`
| `(NP)/sin110^@` | `= 4/sin35^@` |
| `NP` | `= (4 xx sin110^@)/sin35^@` |
`:.\ text(Tran is correct)`
`text(Let)\ \ Q\ text(be the midpoint of)\ NP`
`text(In)\ \ Delta PQM:`
| `cos35^@` | `= x/4` |
| `x` | `= 4 xx cos35^@` |
| `NP` | `= 2 xx 4 xx cos35^@` |
`:.\ text(Shelley is correct.)`
`=> E`
A closed cubic box of side length 36 cm is to contain a thin straight metal rod.
The maximum possible length of the rod is closest to
A. `36\ text(cm)`
B. `51\ text(cm)`
C. `62\ text(cm)`
D. `108\ text(cm)`
E. `216\ text(cm)`
Two hikers, Anton and Beth, walk in different directions from the same camp.
Beth walks for 12 km on a bearing of 135° to a picnic ground.
Anton walks for 6 km on a bearing of 045° to a lookout tower.
On what bearing (to the nearest degree) should Anton walk from the lookout tower to meet Beth at the picnic ground?
A. `063°`
B. `108°`
C. `153°`
D. `162°`
E. `180°`
`text(S)text(ince)\ T\ text(is on a bearing)\ 045^@\ text(from)\ O\ text(and)\ P`
`text(is on a bearing)\ 135^@,\ /_ TOP= 90^@`
`text(In)\ Delta TOP,`
| `tan /_ OTP` | `= 12/6 = 2` |
| `:. /_ OTP` | `= 63.43…^@` |
`text(Let)\ A\ text(be directly south of)\ T`
| `/_ OTA` | `= 45^@\ text{(alternate)}` |
| `:.\ /_ATP` | `= 63.43… − 45` |
| `= 18.43…^@` |
`:.\ text(Bearing of)\ P\ text(from)\ T`
`= 180 − 18.43…`
`= 161.57…^@`
`=> D`
A tent with semicircular ends is in the shape of a prism. The diameter of the ends is 1.5 m. The tent is 2.5 m long.
The total surface area (in m2) of the tent, including the base, is closest to
A. `5.5`
B. `7.7`
C. `8.8`
D. `11.4`
E. `15.3`
Consider the simultaneous linear equations below.
`4x - 2y = 18`
`3x + ky = 10` (3 marks)
where `k` is a real constant.
A clubhouse uses four long-life light globes for five hours every night of the year. The purchase price of each light globe is $6.00 and they each cost `$d` per hour to run.
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What is the mean life, in hours, of these light globes if 97.5% will last up to 5000 hours? (1 mark)
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i. `text(Purchase price) = 4 xx 6 = $24`
| `text(Running cost)` | `= text(# Hours) xx text(Cost per hour)` |
| `= 4 xx 5 xx 365 xx d` | |
| `= 7300d` | |
| `:.\ $c = 24 + 7300d` | |
ii. `text(Given)\ \ $c = $250`
| `250` | `= 24 + 7300d` |
| `7300d` | `= 226` |
| `d` | `= 226/7300` |
| `= 0.03095…` | |
| `= 0.031\ $ text(/hr)\ text{(3 d.p.)}` |
iii. `text(If)\ d\ text(doubles to 0.062)\ \ $text(/hr)`
| `$c` | `= 24 + 7300 xx 0.062` |
| `= $476.60` | |
| `text(S) text(ince $476.60 is less than)\ 2 xx $250\ ($500),` | |
| `text(the total cost increases to less than double)` | |
| `text(the original cost.)` | |
iv. `sigma = 170`
`z\ text(-score of 5000 hours) = 2`
| `z` | `= (x – mu)/sigma` |
| `2` | `= (5000 – mu)/170` |
| `340` | `= 5000 – mu` |
| `mu` | `= 4660` |
`:.\ text(The mean life of these globes is 4660 hours.)`
The first term of a geometric sequence is `a`, where `a < 0`.
The common ratio of this sequence, `r`, is such that `r < –1`.
Which one of the following graphs best shows the first `10` terms of this sequence?
There are 10 checkpoints in a 4500 metre orienteering course. Checkpoint 1 is the start and checkpoint 10 is the finish.
The distance between successive checkpoints increases by 50 metres as each checkpoint is passed.
Calculate the distance, in metres, between checkpoint 2 and checkpoint 3. (3 marks)
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The graph above shows consecutive terms of a sequence.
The sequence could be
A rectangular playing surface is to be constructed so that the length is 6 metres more than the width.
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A graph comparing the area of the playing surface to its length is shown.
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Company `A` constructs playing surfaces.
Use the horizontal axis to represent the area and the vertical axis to represent the cost. (2 marks)
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Justify your answer with suitable calculations. (1 mark)
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`text{of 4 m (among many possibilities).}`
`text(the width, it follows that the length)`
`text(must be greater than 6 m.)`
i. `text(One possibility is a length of 10 m, and a width)`
`text{of 4 m (among many possibilities).}`
ii. `text(Length) = l\ text(m)`
`text(Width) = (l – 6)\ text(m)`
| `:.\ A` | `= l (l – 6)` |
iii. `text(Given the length must be 6m more than the width,)`
`text(it follows that the length must be greater than 6 m)`
`text(so that the width is positive.)`
iv. `text(From the graph, an area of 135 m² corresponds to)`
`text(a length of 15 m.)`
`:.\ text(The dimensions would be 15 m × 9 m.)`
| v. | |
vi. `text(Company)\ A\ text(cost) = $50\ 000`
| `text(Company)\ B\ text(cost)` | `= 135 xx 360` |
| `= $48\ 600` |
`:.\ text(Company)\ B\ text(would charge $1400 less)`
`text(than Company)\ A.`
Myles is in his third year as an apprentice film editor.
After 3 years it has depreciated to $3635 using the straight-line method.
Calculate the rate of depreciation per year as a percentage. (2 marks)
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The diagram shows information about the locations of towns `A`, `B` and `Q`.
Calculate her walking speed correct to the nearest km/h. (1 mark)
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Find the distance from Town `A` to Town `B`. Give your answer to the nearest km. (2 marks)
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i. `text(2 hrs 48 mins) = 168\ text(mins)`
| `text(Speed)\ text{(} A\ text(to)\ Q text{)}` | `= 15/168` |
| `= 0.0892…\ text(km/min)` |
| `text(Speed)\ text{(in km/hr)}` | `= 0.0892… xx 60` |
| `= 5.357…\ text(km/hr)` | |
| `= 5\ text(km/hr)\ text{(nearest km/hr)}` |
| ii. |  |
`text(Using cosine rule)`
| `AB^2` | `= 15^2 + 10^2 – 2 xx 15 xx 10 xx cos 87^@` |
| `= 309.299…` | |
| `AB` | `= 17.586…` |
| `= 18\ text(km)\ text{(nearest km)}` |
`:.\ text(The distance from Town)\ A\ text(to Town)\ B\ text(is 18 km.)`
| iii |  |
| `/_CAQ` | `= 31^@\ \ \ text{(} text(straight angle at)\ A text{)}` |
| `/_AQD` | `= 31^@\ \ \ text{(} text(alternate angle)\ AC\ text(||)\ DQ text{)}` |
| `/_DQB` | `= 87 – 31 = 56^@` |
| `/_QBE` | `= 56^@\ \ \ text{(} text(alternate angle)\ DQ \ text(||)\ BE text{)}` |
`:.\ text(Bearing of)\ Q\ text(from)\ B`
`= 180 + 56`
`= 236^@`
A rectangular prism with a square base, `ABCD`, is shown above.
The diagonal of the prism, `AH`, is 8 cm.
The height of the prism, `HC`, is 4 cm.
The volume of this rectangular prism is
A. `64\ text(cm³)`
B. `96\ text(cm³)`
C. `128\ text(cm³)`
D. `192\ text(cm³)`
E. `256\ text(cm³)`
`text(Using Pythagoras in)\ Delta AHC,`
| `AH^2` | `= HC^2 + AC^2` |
| `8^2` | `= 4^2 + AC^2` |
| `AC^2` | `= 48` |
| `:. AC` | `= 6.928…\ text(cm)` |
`Delta ABC\ text(is a right-angled and isosceles.)`
`:. /_ CAB = /_ ACB = 45°`
| `sin 45°` | `= (CB)/(6.298…)` |
| `CB` | `= 6.928… xx sin 45°` |
| `= 4.898…\ text(cm)` |
| `:.\ text(Volume)` | `= (CB)^2 xx 4` |
| `= (4.898…)^2 xx 4` | |
| `= 96\ text(cm³)` |
`=> B`
There are four telecommunications towers in a city. The towers are called Grey Tower, Black Tower, Silver Tower and White Tower.
Grey Tower is 10 km due west of Black Tower.
Silver Tower is 10 km from Grey Tower on a bearing of 300°.
White Tower is 10 km due north of Silver Tower.
Correct to the nearest degree, the bearing of Black Tower from White Tower is
A. `051°`
B. `129°`
C. `141°`
D. `309°`
E. `321°`
`text(S)text(ince bearing of Silver from Grey is 300°)`
`/_ XGS = 60°, and /_SGO = 30°`
`text(In)\ Delta SGO,`
| `sin 30°` | `= a/10` |
| `a` | `= 10 xx sin 30` |
| `= 5` |
| `cos 30°` | `= b/10` |
| `b` | `= 10 xx cos 30` |
| `= 8.66…` |
| `tan theta` | `= (OB)/(OW)` |
| `= (10 + 8.66…)/(10 + 5)` | |
| `=1.244…` | |
| `theta` | `= 51.2…°` |
`:.\ text(Bearing of Black from White tower)`
`= 180 – 51.2…º`
`= 128.79…°`
`=> B`
A cafe sells two sizes of cupcakes with a similar shape.
The large cupcake is 6 cm wide at the base and the small cupcake is 4 cm wide at the base.
The price of a cupcake is proportional to its volume.
If the large cupcake costs $5.40, then the small cupcake will cost
A. `$1.60`
B. `$2.32`
C. `$2.40`
D. `$3.40`
E. `$3.60`
An orienteering course is triangular in shape and is marked by three points, `A`, `B` and `C`, as shown in the diagram below.
Part 1
In this course, the bearing of `B` from `A` is `050^@` and the bearing of `C` from `B` is `120^@`.
The bearing of `B` from `C` is
A. `060°`
B. `120°`
C. `240°`
D. `300°`
E. `310°`
Part 2
In this course, `B` is 7.0 km from `A`, ` C` is 8.0 km from `B` and `A` is 12.3 km from `C`.
The area (in km²) enclosed by this course is closest to
A. `21`
B. `24`
C. `25`
D. `26`
E. `28`
`text(Part 1)`
`text(Let)\ D\ text(be directly North of)\ C`
| `/_ BCD = 60^@\ \ \ ` | `text{(cointerior, North South}` `\ \ text{lines are parallel)}` |
`∴ text(Bearing of)\ B\ text(from)\ C`
`= 360 − 60`
`= 300^@`
`=> D`
`text(Part 2)`
`text(Using cosine rule in)\ Delta ABC,`
| `cos /_ BAC` | `= (7.0^2 + 12.3^2 − 8.0^2) / (2 xx 7.0 xx 12.3)` |
| `= 0.7914…` | |
| `/_ BAC` | `= 37.67…^@` |
`text(Using sine rule,)`
| `text(Area of)\ Delta ABC` | `= 1/2 xx 7.0 xx 12.3 xx sin 37.67…^@` |
| `= 26.31…\ text(km²)` |
`=> D`
The results of two class tests are normally distributed. The means and standard deviations of the tests are displayed in the table.
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i. `text(In Test 1,)\ \ mu = 60,\ sigma = 6.2`
| `z text(-score)\ (63)` | `= (x – mu)/sigma` |
| `= (63 – 60)/6.2` | |
| `= 0.483…` |
`text(In Test 2,)\ \ mu = 58,\ sigma = 6.0`
| `z text(-score)\ (62)` | `= (62 – 58)/6.0` |
| `= 0.666…` | |
`text(S) text(ince Stuart’s)\ z\ text(-score is higher in Test 2,)`
`text(his performance relative to the class is better)`
`text(despite his mark being slightly lower.)`
ii. `text(In Test 2)`
| `z text(-score)\ (64)` | `= (64 – 58)/6` |
| `= 1` |
`=> text(84% have)\ z text(-score) < 1`
`:.\ text(# Students expected below 64)`
`= text(84%) xx 150`
`= 126`
In a stack of 10 DVDs, there are 5 rated PG, 3 rated G and 2 rated M.
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Grant chooses two DVDs at random from the stack. Copy or trace the tree diagram into your writing booklet.
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i. `text(5 PG, 3 G, 2 M)`
`P text{(M)} = 2/10 = 1/5`
| ii. | |
iii. `P text{(same rating)}`
`= P text{(PG, PG)} + P text{(G, G)} + P text{(M, M)}`
`= (1/2 xx 4/9) + (3/10 xx 2/9) + (1/5 xx 1/9)`
`= 2/9 + 1/15 + 1/45`
`= 14/45`
The angle of depression from `J` to `M` is 75°. The length of `JK` is 20 m and the length of `MK` is 18 m.
Copy or trace this diagram into your writing booklet and calculate the angle of elevation from `M` to `K`. Give your answer to the nearest degree. (3 marks)
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`/_AJL = 90^@`
`/_MJL = 90 – 75 = 15^@`
`text(Using sine rule in)\ Delta MJK`
`text(Let)\ /_JMK = x^@`
| `20/sin x` | `= 18/sin15^@` |
| `18 sin x` | `= 20 xx sin 15^@` |
| `sin x` | `= (20 xx sin 15^@)/18 = 0.2875…` |
| `x` | `= 16.71…^@` |
| `/_JML = 75^@\ text{(} text(alternate angles,)\ ML \ text(||) \ AJ text{)}` | |
| `:.\ /_KML` | `= 75^@ – 16.71…` |
| `= 58.287…^@` | |
| `= 58^@\ \ \ text{(nearest degree)}` |
`:.\ text(Angle of Elevation from)\ M\ text(to)\ K\ text(is)\ 58^@.`
Barry constructed a back-to-back stem-and-leaf plot to compare the ages of his students.
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For the age group 30 - 39 years, what is the value of the product of the class centre and the frequency? (2 marks)
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Caitlyn correctly used the original data in the back-to-back stem-and-leaf plot and calculated the mean to be 38.2.
What is the reason for the difference in the two answers? (1 mark)
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A scientific study uses the ‘capture-recapture’ technique.
In the first stage of the study, 24 crocodiles were caught, tagged and released.
Later, in the second stage of the study, some crocodiles were captured from the same area. Eighteen of these were found to be tagged, which was 40% of the total captured during the second stage.
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The rectangle shown below is 54 cm high and 20 cm wide.
The rhombuses drawn inside the rectangle are all the same size and shape.
The size of the angle `theta`, in the shaded rhombus, is closest to
A. `34°`
B. `45°`
C. `56°`
D. `58°`
E. `67°`
`text(Diagonals of shaded rhombus are,)`
`54/3 = 18\text (cm), and\ 20/2 = 10\ text(cm)`
`:.\ text (Rhombus is 4 right-angled triangles.)`
| `theta` | `= 2x` |
| `tan x` | `= 5/9` |
| `x` | `= 29.054…°` |
| `:. theta` | `= 58.109…°` |
`=> D`
A hose with a circular cross-section is 85 metres long.
The outside diameter of the hose is 29 millimetres. Its walls are 2 millimetres thick.
One litre of water occupies a volume of 1000 cm3.
When the hose is full with water, the volume it holds (in litres) is closest to
A. `4`
B. `42`
C. `49`
D. `56`
E. `167`
A patient takes 15 milligrams of a prescribed drug at the start of each day.
Over the next 24 hours, 85% of the drug in his body is used. The remaining 15% stays in his body.
Let `D_n` be the number of milligrams of the drug in the patient’s body immediately after taking the drug at the start of the `n`th day.
A difference equation for determining `D_(n+1)`, the number of milligrams in the patient’s body immediately after taking the drug at the start of the `n+1`th day, is given by
| A. `D_(n + 1) = 85 D_n + 15` | `D_1 = 15` |
| B. `D_(n + 1) = 0.85 D_n + 15` | `D_1 = 15` |
| C. `D_(n + 1)= 0.15 D_n + 15` | `D_1 = 15` |
| D. `D_(n + 1)= 0.15 D_n + 0.85` | `D_1 = 15` |
| E. `D_(n + 1)= 15 D_n + 85` | `D_1 = 15` |
The `n`th term of a sequence is given by `t_n = 100 − 20n`, where `n = 1, 2, 3, 4\ . . .`
A difference equation that generates the same sequence is
| A. `t_(n +1)= 100 - 20t_n` | `t_1 = 80` |
| B. `t_(n+1) = 100t_n - 20` | `t_1 = 1` |
| C. `t_(n+1) = 80t_n` | `t_1 = 80` |
| D. `t_(n+1) = 100 - t_n` | `t_1 = 20` |
| E. `t_(n+1) = t_n - 20` | `t_1 = 80` |
On Monday morning, Jim told six friends a secret. On Tuesday morning, those six friends each told the secret to six other friends who did not know it. The secret continued to spread in this way on Wednesday, Thursday and Friday mornings.
The total number of people (not counting Jim) who will know the secret on Friday afternoon is
A. `259`
B. `1296`
C. `1555`
D. `7776`
E. `9330`
A toy train track consists of a number of pieces of track which join together.
The shortest piece of the track is 15 centimetres long and each piece of track after the shortest is 2 centimetres longer than the previous piece.
The total length of the complete track is 7.35 metres.
The length of the longest piece of track, in centimetres, is
A. `21`
B. `47`
C. `49`
D. `55`
E. `57`
The first three terms of an arithmetic sequence are `1, 3, 5 . . .`
The sum of the first `n` terms of this sequence, `S_n`, is
A. `S_n = n^2`
B. `S_n = n^2 - n`
C. `S_n = 2n`
D. `S_n = 2n - 1`
E. `S_n = 2n + 1`
For which one of the following geometric sequences is an infinite sum not able to be determined?
A. `4, 2, 1, 1 / 2\ . . .`
B. `1, 2, 4, 8\ . . .`
C. `–4, 2, –1, 1 / 2\ . . .`
D. `1, 1 / 2, 1 / 4, 1 / 8\ . . .`
E. `–1, 1/2, –1 / 4, 1 / 8\ . . .`
The difference equation
`t_(n+2) = t_(n+1) + t_n` where `t_1 = a` and `t_2 = 7`
generates a sequence with `t_5 = 27`.
The value of `a` is
A. 0
B. 1
C. 2
D. 3
E. 4
A cylinder of radius `R` and height `H` has volume `V.`
The volume of a cylinder with radius `3R` and height `3H` is
A. `3V`
B. `6V`
C. `9V`
D. `27V`
E. `81V`
This set of data is arranged in order from smallest to largest.
`5, \ 6, \ 11, \ x, \ 13, \ 18, \ 25`
The range is six less than twice the value of `x`.
Which one of the following is true?
Which of the following correctly expresses `T` as the subject of `B = 2pi (R + T/2)`?
Ms Wigginson decided to survey a sample of 10% of the students at her school.
The school enrolment is shown in the table.
She surveyed the same number of students in each year group.
How would the numbers of students surveyed in Year 10 and Year 11 have changed if Ms Wigginson had chosen to use a stratified sample based on year groups?
If pressure (`p`) varies inversely with volume (`V`), which formula correctly expresses `p` in terms of `V` and `k`, where `k` is a constant?
A cross-country race is run on a triangular course. The points `A, B` and `C` mark the corners of the course, as shown below.
The distance from `A` to `B` is 2050 m.
The distance from `B` to `C` is 2250 m.
The distance from `A` to `C` is 1900 m.
The bearing of `B` from `A` is 140°.
Part 1
The bearing of `C` from `A` is closest to
A. `032°`
B. `069°`
C. `192°`
D. `198°`
E. `209°`
Part 2
The area within the triangular course `ABC`, in square metres, can be calculated by evaluating
A. `sqrt (3100 xx 1200 xx 1050 xx 850)`
B. `sqrt (3100 xx 2250 xx 2050 xx 1900)`
C. `sqrt (6200 xx 4300 xx 4150 xx 3950)`
D. `1/2 xx 2050 xx 2250 xx sin\ (140^@)`
E. `1/2 xx 2050 xx 2250 xx sin\ (40^@)`
Sam takes a tablet containing 200 mg of medicine once every 24 hours.
Every 24 hours, 40% of the medicine leaves her body. The remaining 60% of the medicine stays in her body.
Let `D_n` be the number of milligrams of the medicine in Sam’s body immediately after she takes the `n`th tablet.
The difference equation that can be used to determine the number of milligrams of the medicine in Sam’s body immediately after she takes each tablet is shown below.
`D_(n + 1) = 0.60D_n + 200,\ \ \ \ \ \ D_1 = 200`
Which one of the following statements is not true?
A. The number of milligrams of the medicine in Sam’s body never exceeds 500.
B. Immediately after taking the third tablet, 392 mg of the medicine is in Sam’s body.
C. The number of milligrams of the medicine that leaves Sam’s body during any 24-hour period will always be less than 200.
D. The number of milligrams of the medicine that leaves Sam’s body during any 24-hour period is constant.
E. If Sam stopped taking the medicine after the fifth tablet, the amount of the medicine in her body would drop to below 200 mg after a further 48 hours.
The first term of a geometric sequence is `a`, where `a < 0`.
The common ratio of this sequence, `r`, is such that `r < –1`.
Which one of the following graphs best shows the first 10 terms of this sequence?
The time series plot below shows the hours of sunshine per day at a particular location for 16 consecutive days.
The three median method is used to fit a trend line to the data.
The slope of this trend line will be closest to
A. `–0.7`
B. `–0.2`
C. `0.0`
D. `0.2`
E. `0.7`
The time series plot below shows the growth in Internet use (%) in a country from 1989 to 1997 inclusive.
If a three-median line is fitted to the data it would show that, on average, the increase in Internet use per year was closest to
A. `0.33text(%)`
B. `0.36text(%)`
C. `0.41text(%)`
D. `0.45text(%)`
E. `0.49text(%)`
