Aussie Maths & Science Teachers: Save your time with SmarterEd
A manufacturer makes and sells heaters.
The fixed cost to manufacture the heaters is $16 000 per month.
Each heater costs $52 to produce.
The selling price of each heater is $280.
The minimum number of heaters needed to be sold per month in order to make a profit is
A line passes through the points (8, 0), (0, 6) and (5, `m`).
The value of `m` is
Which one of the following represents a line with the same slope as `y = 2x + 3`?
A child's toy has the following design.
Find the area of the shaded region to the nearest square centimetre. (2 marks)
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The side length of an equilateral triangle is 4 cm, as shown in the diagram below.
Which one of the following is not a correct calculation for the area of this triangle?
`text{By Pythagoras:}`
`h = sqrt{4^2 -2^2} = sqrt12 = 2 sqrt3`
| `text{Area}` | `= 2 xx (1/2 xx 2 xx 2 sqrt3)` |
| `= 4 sqrt3 \ text{cm}^2` |
`text{Option C is the only option} ≠ 4 sqrt3`
`=> C`
Which one of the following cities is closest to the Greenwich meridian?
Consider the directed network below.
The number of vertices that cannot be reached from `X` is
Ramon and Norma are names that contain the same letters but in different order.
The permutation matrix that can change `[(R),(A),(M),(O),(N)]` into `[(N),(O),(R),(M),(A)]` is
| A. `[(0,0,0,0,1),(1,0,0,0,0),(0,0,1,0,0),(0,1,0,0,0),(0,0,0,1,0)]` | B. `[(0,0,0,0,1),(1,0,0,0,0),(0,0,1,0,0),(0,1,0,0,0),(0,0,0,1,0)]` | |
| C. `[(1,0,0,0,0),(0,0,0,1,0),(0,0,0,0,1),(0,0,1,0,0),(0,1,0,0,0)]` | D. `[(0,0,0,0,1),(0,0,0,1,0),(1,0,0,0,0),(0,1,0,0,0),(0,0,1,0,0)]` | |
| E. `[(0,0,0,0,1),(0,0,0,1,0),(1,0,0,0,0),(0,0,1,0,0),(0,1,0,0,0)]` |
A fitness centre offers four different exercise classes: aerobics `(A)`, boxfit `(B)`, cardio `(C)` and dance `(D)`.
A customer's choice of fitness class is expected to change from week to week according to the transition matrix `P`, shown below.
`qquadqquadqquadqquadqquadqquad text(this week)`
`P = {:(qquad\ A quadquadqquad \ B quadquad \ C quadqquad \ D),([(0.65,0, 0.20, 0.10),(0,0.65,0.10,0.30),(0.20,0.10,0.70,0),(0.15,0.25,0,0.60)]{:(A),(B),(C),(D):} qquad text(next week)):}`
An equivalent transition diagram has been constructed below, but the labeling is not complete.
The proportion for one of the transitions is labelled `w`.
The value of `w` is
`text{The 25% edge label must be} \ B \ text{(this week) to} \ D \ text{(next week)}`
`text{No edge exists between} \ C \ text{and} \ D`
`=> \ text{top left vertex is} \ C \ text{and top right vertex is} \ A.`
| `w` | `= A \ text{this week,} \ D \ text{next week}` |
| `= 0.15` |
`=> B`
`ax + 4y = 10`
`18x + by = 6`
The set of simultaneous linear equations above does not have a unique solution when
Every Friday, the same number of workers from a large office building regularly purchase their lunch from one of two locations: the deli, `D `, or the cafe, `C`.
It has been found that:
A transition matrix that can be used to describe this situation is
| A. `qquad text(this Friday)` `{:(qquad\ D quadquad \ C quad),([(0.55,0.35),(0.45,0.65)]{:(D),(C):} qquad text(next Friday)):}` |
B. `qquad text(this Friday)` `{:(qquad\ D quadquad \ C quad),([(0.65,0.45),(0.45,0.55)]{:(D),(C):} qquad text(next Friday)):}` |
|
| C. `qquad text(this Friday)` `{:(qquad\ D quadquad \ C quad),([(0.65,0.55),(0.45,0.55)]{:(D),(C):} qquad text(next Friday)):}` |
D. `qquad text(this Friday)` `{:(qquad\ D quadquad \ C quad),([(0.65,0.45),(0.35,0.55)]{:(D),(C):} qquad text(next Friday)):}` |
|
| E. `qquad text(this Friday)` `{:(qquad\ D quadquad \ C quad),([(0.65,0.55),(0.35,0.45)]{:(D),(C):} qquad text(next Friday)):}` |
The gradient of a function is given by `(dy)/(dx) = sqrt(x + 6)-x/2-3/2`.
The graph of the function has a single stationary point at `(3, 29/4)`.
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| a. | `dy/dx` | `= sqrt(x + 6)-x/2-3/2` |
| `y` | `=int sqrt(x + 6)-x/2-3/2\ dx` | |
| `=2/3(x+6)^(3/2)-x^2/4-3/2x + c` |
`text{Graph passes through}\ (3, 29/4)`
| `29/4` | `=2/3*9^(3/2)-3^2/4-3/2 * 3+c` | |
| `29/4` | `=18-9/4-9/2+c` | |
| `c` | `=29/4-18+9/4+9/2` | |
| `=-4` |
`:. y=2/3(x+6)^(3/2)-x^2/4-3/2x-4`
b.
`:.(3,29/4)\ text{is a maximum}.`
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a.
b. `x in [1, 2)`
a. `text(Asymptotes:)`
`x=2`
`text(As)\ \ x→ +-oo, \ y→1\ \ =>\ text(Asymptote at)\ \ y=1`
`ytext(-intercept at)\ (0,2)`
`xtext(-intercept at)\ (4,0)`
b. `text(By inspection of the graph:)`
`1-2/(x-2) >=3\ \ text(for)\ \ x in [1, 2)`
An online shopping site sells boxes of doughnuts. A box contains 20 doughnuts. There are only four types of doughnuts in the box. They are: It is known that, in the box: --- 5 WORK AREA LINES (style=lined) --- --- 5 WORK AREA LINES (style=lined) --- --- 6 WORK AREA LINES (style=lined) ---
a. `text(Create a 2-way table:)` `text{Pr(glazed)} = 6/20` `X\ ~\ text{Bi}(5, 0.5)`Show Worked Solution
`text{Pr(not glazed with custard)}`
`=8/20`
`=2/5`
b. `A_text{glazed} = g \ => \ B_text{glazed} = 6-g`♦♦♦ Mean mark part (b) 20%.
`text{Pr}(B | text{glazed})`
`= (text{Pr}(B ∩ text{glazed}))/text{Pr(glazed)}`
`=(1/2 xx (6-g)/10)/(6/20)`
`=(6-g)/20 xx 20/6`
`=(6-g)/6`
c. `text(Let)\ \ X=\ text(number of visitors < 25 years)`♦ Mean mark part (c) 40%.
`text{Pr}(hatP>=0.8)`
`= text{Pr}(X>=4)`
`=\ ^5C_4 * (1/2)^4(1/2) + (1/2)^5`
`=5/32 + 1/32`
`=3/16`
Joanna deposited $12 000 in an investment account earning interest at the rate of 2.8% per annum, compounding monthly.
She would like this account to reach a balance of $25 000 after five years.
To achieve this balance, she will make an extra payment into the account each month, immediately after the interest is calculated.
The minimum value of this payment is closest to
The number of visitors to a regional animal park is seasonal.
Data is collected and deseasonalised before a least squares line is fitted.
The equation of the least squares line is
deseasonalised number of visitors = 2349 – 198.5 × month number
where month number 1 is January 2020.
The seasonal indices for the 12 months of 2020 are shown in the table below.
The actual number of visitors predicted for February 2020 was closest to
The table below shows the number of visitors to an art gallery during the summer, autumn, winter and spring quarters for the years 2017 to 2019.
The quarterly average is also shown for each of these years.
The seasonal index for summer is closest to
The time series plot below shows the points scored by a basketball team over 40 games.
The nine-median smoothed points scored for game number 10 is closest to
Oscar walked for nine consecutive days. The time, in minutes, that Oscar spent walking on each day is shown in the table below.
At least squares line is fitted to the data.
The equation of this line predicts that on day 10 the time Oscar spends walking will be the same as the time he spent walking on
The heights of females living in a small country town are normally distributed:
The mean and the standard deviation of this female population, in centimetres, are closest to
The percentaged segmented bar chart below shows the age (under 55 years, 55 years and over) of visitors at a travel convention, segmented by preferred travel destination (domestic, international).
Part 1
The variables age (under 55 years, 55 years and over) and preferred travel destination (domestic, international) are
Part 2
The data displayed in the percentaged segmented bar chart supports the contention that there is an association between preferred travel destination and age because
Part 3
The results could also be summarised in a two-way frequency table.
Which one of the following frequency tables could match the pecentaged segmented bar chart?
`text(Part 1)`
`text{Preferred travel destination → categorical (nominal) variable}`
`text{Age → categorical (ordinal) variable}`
`=> A`
`text(Part 2)`
`text(Only option)\ D\ text(highlights a change in preference for domestic travel)`
`text(between the two age categories.)`
`=>D`
`text(Part 3)`
`text(Converting the frequency table data into percentages,)`
`text(consider option)\ A:`
`91/140 xx 100 = 65text(%),\ \ 49/140 xx 100 = 35text(%)`
`90/200 xx 100 = 45text(%), \ \ 110/200 xx 100 = 55text(%)`
`=> A`
800 participants auditioned for a stage musical. Each participant was required to complete a series of ability tests for which they received an overall score.
The overall scores were approximately normally distributed with a mean score of 69.5 points and a standard deviation of 6.5 points.
Part 1
Only the participants who scored at least 76.0 points in the audition were considered successful,
Using the 68-95-99.7% rule, how many of the participants were considered unsuccessful?
Part 2
To be offered a leading role in the stage musical, a participant must achieve a standardised score of at least 1.80
Three participants' names and their overall scores are given in the table below.
Which one of the following statements is true?
The relationship between resting pulse rate, in beats per minute, and age group (15-20 years, 21-30 years, 31-50 years, over 50 years) is best displayed using
The stem plot below shows the height, in centimetres, of 20 players in a junior football team.
A player with a height of 179 cm is considered an outlier because 179 cm is greater than
The boxplots below show the distribution of the length of fish caught in two different ponds, Pond A and Pond B.
Based on the boxplots above, it can be said that.
Let `f:R -> R, \ f(x) = x^2 - 4` and `g:R -> R, \ g(x) = 4(x - 1)^2 - 4`.
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Consider the function `g: R -> R, \ g(x) = 2sin(2x).`
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Evaluate `f′(4)`, where `f(x) = xsqrt(2x + 1)`. (2 marks)
A man attempts to swim north across a river at a speed of 4.5 km/h.
If the river's current is moving at 11 km/h due east, find the bearing (nearest degree) and speed (to 1 decimal place) that the man will actually be swimming. (3 marks)
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`underset~u = (0, 4.5)`
`underset~v = (11, 0)`
`underset~u + underset~v = ((0),(4.5)) + ((11),(0)) = ((11),(4.5))`
| `tan theta` | `= 11/4.5` |
| `theta` | `= tan^{-1} (11/4.5)` |
| `theta` | `= 67.75` |
| `theta` | `~~ 068^@` |
| `| underset~u + underset~v |` | `= sqrt{11^2 + 4.5^2}` |
| `= 11.88 ..` | |
| `= 11.9 \ text{km/h}\ \ text{(1 d.p.)}` |
`:. \ text{Swimmer’s speed is 11.9 km/h on a bearing of} \ 068^@`
An object of mass 5 kg is on a slope that is inclined at an angle of 60° to the horizontal. The acceleration due to gravity is `g\ text(m s)^(-2)` and the velocity of the object down the slope is `v\ text(m s)^(-1)`.
As well as the force due to gravity, the object is acted on by two forces, one of magnitude `2v` newtons and one of magnitude `2v^2` newtons, both acting up the slope.
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i.
| `F_s` | `=\ text(force down slope)` |
| `= 5g cos30 – 2v^2 – 2v` | |
| `= 5g · sqrt3/2 – 2v^2 – 2v` | |
| `= (5sqrt3)/2 g – 2v^2 – 2v` |
ii. `text(Constant speed occurs if)\ \ a = 0`
`F_s = ma = 0`
| `2v^2 + 2v – (5sqrt3)/2 xx 10` | `= 0` |
| `2v^2 + 2v – 25sqrt3` | `= 0` |
| `v` | `= (-2 + sqrt(2^2 + 4 · 2 · 25sqrt3))/(2 xx 2)` |
| `= (-2 + sqrt(4 + 200sqrt3))/4` | |
| `= (-1 + sqrt(1 + 50 sqrt3))/2` | |
| `= 4.179…` | |
| `= 4.2\ \ text(ms)^-1\ text{(1 d.p.)}` |
An object of mass 1 kg is projected vertically upwards with an initial velocity of `u` m/s. It experiences air resistance of magnitude `kv^2` newtons where `v` is the velocity of the object, in m/s, and `k` is a positive constant. The height of the object above its starting point is `x` metres. The time since projection is `t` seconds and acceleration due to gravity is `g` m/s².
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Evaluate `int_0^(pi/2) 1/(3 + 5cosx)\ dx`. (4 marks)
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Region `A` is bounded by `y = 1` and `x^2 + y^2 = 1` between `x = 0` and `x = 1`.
Region `B` is bounded by `y = 1` and `y = ln x` between `x = 1` and `x = e`.
The volume of the solid created when the region between the curve `y = f(x)` and the `x`-axis, between `x = a` and `x = b`, is rotated about the `x`-axis is given by `V = pi int_a^b [f(x)]^2\ dx`.
The volume of the solid of revolution formed when region `A` is rotated about the `x`-axis is `V_A`.
The volume of the solid of revolution formed when region `B` is rotated about the `x`-axis is `V_B`.
Using part (i), or otherwise, show that the ratio `V_A : V_B` is `1:3`. (4 marks)
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Use an appropriate substitution to evaluate `int_(sqrt10)^(sqrt13) x^3 sqrt{x^2 - 9}\ dx`. (3 marks)
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The diagram shows the pyramid `ABCDS` where `ABCD` is a square. The diagonals of the square bisect each other at `H`.
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Let `G` be the point such that `overset->{GA} + overset->{GB} + overset->{GC} + overset->{GD} + overset->{GS} = underset~0`.
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Prove by mathematical induction that `sqrt{n!} > 2^n` , for integers `n ≥ 9`. (3 marks)
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Which diagram best shows the curve described by the position vector
`underset~r (t) = -5 text{cos}(t) underset~i + 5 text{sin}(t) underset~j + t underset~k` for `0 ≤ t ≤ 4 pi` ?
Which polynomial could have `2 + i` as a zero, given that `k` is a real number?
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The region enclosed by `y = 2 - |x|` and `y = 1 - 8/(4 + x^2)` is shaded in the diagram.
Find the exact value of the area of the shaded region. (3 marks)
A function is defined by `f(x) = 4-(1-x/2)^2` for `x` in the domain `(-∞, 2]`.
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i. `f(x) = 4-(1-x/2)^2`
`text(At)\ \ x = 2, \ f(x) = 4`
`text(At)\ \ x = 0, \ f(x) = 3`
`xtext(-intercept:)\ \ 1-x/2 = 2 \ -> \ x = -2`
ii. `text(Inverse function: swap)\ \ x ↔ y`
`x = 4-(1-y/2)^2`
| `(1-y/2)^2` | `= 4-x` |
| `1-y/2` | `= ±sqrt(4-x)` |
| `y/2` | `= 1 ± sqrt(4-x)` |
| `y` | `= 2 +- 2sqrt(4-x)` |
| `:. f^(-1)(x)` | `=2-2sqrt(4-x),\ \ \ (y <= 2)` |
| `text(Domain)\ \ f^(-1)(x)` | `= text(Range)\ f(x)` |
| `= x ∈ (-∞, 4]` |
iii.
A bottle of water, with temperature 5°C, is placed on a table in a room. The temperature of the room remains constant at 25°C. After `t` minutes, the temperature of the water, in degrees Celsius, is `T`.
The temperature of the water can be modelled using the differential equation
`(dT)/(dt) = k(T - 25)` (Do NOT prove this.)
where `k` is the growth constant
After 8 minutes, the temperature of the water is 10°C.
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| i. | `(dT)/(dt)` | `= k(T – 25)` |
| `(dt)/(dT)` | `= 1/(k(T – 25))` | |
| `t` | `= 1/k int 1/(T – 25)\ dT` | |
| `kt` | `= ln |T – 25| + c` |
`text(When)\ \ t = 0, T = 5 \ => \ c = -ln 20`
| `kt` | `= ln |T – 25| – ln 20` |
| `= ln |(T – 25)/20|` |
`text(When)\ \ t = 8, T = 10`
| `8k` | `= ln (15/20)` |
| `= 1/8 ln(3/4)` |
`text(Find)\ \ t\ \ text(when)\ \ T = 20:`
| `1/8 ln (3/4)t` | `= ln |(20 – 25)/20|` |
| `t` | `= (8 ln(1/4))/(ln(3/4))` |
| `= 38.55…` | |
| `= 39\ text{minutes (0 d.p.)}` |
ii.
When an object is projected from a point `h` metres above the origin with initial speed `V` m/s at an angle of `theta^@` to the horizontal, its displacement vector, `t` seconds after projection, is
`underset~r(t) = (Vtcostheta)underset~i + (-5t^2 + Vtsintheta + h)underset~j`. (Do NOT prove this.)
A person, standing in an empty room which is 3 m high, throws a ball at the far wall of the room. The ball leaves their hand 1 m above the floor and 10 m from the far wall. The initial velocity of the ball is 12 m/s at an angle of 30° to the horizontal.
Show that the ball will NOT hit the ceiling of the room but that it will hit the far wall without hitting the floor. (4 marks)
`r(t) = (Vtcos theta)underset~i + (-5t^2 + Vtsintheta + h)underset~j`
`r′(t) = (Vcostheta)underset~i + (-10t + Vsintheta)underset~j`
`text(Max height occurs when)\ \ dot y = 0:`
| `10t` | `= Vsintheta` |
| `t` | `= (12 xx sin30^@)/10` |
| `= 0.6\ text(sec)` |
`text(Find)\ \ y\ \ text(when)\ \ t = 0.6:`
| `y` | `= -5(0.6)^2 + 12 xx 0.6 xx sin30^@ + 1` |
| `= 2.8\ text(m < 3 m)` |
`:.\ text(Ball will not hit ceiling.)`
`text(Find time of flight when)\ \ y = 0:`
| `-5t^2 + 6t + 1` | `= 0` |
| `5t^2 – 6t – 1` | `= 0` |
| `t` | `= (6 +- sqrt((-6)^2 + 4 · 5 · 1))/10` |
| `= (6 + sqrt56)/10` | |
| `= 1.348…` |
`text(Find)\ \ x\ \ text(when)\ \ t = 1.348:`
| `x` | `= 12 xx 1.348 xx cos 30` |
| `= 14.0 > 10` |
`:.\ text(Ball will hit the wall on the full.)`
A 2-metre-high sculpture is to be made out of concrete. The sculpture is formed by rotating the region between `y = x^2, y = x^2 + 1` and `y = 2` around the `y`-axis.
Find the volume of concrete needed to make the sculpture. (3 marks)
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| `V` | `= pi int_0^2 y\ dy-pi int_1^2 y-1\ dy` |
| `= pi [(y^2)/2]_0^2-pi [(y^2)/2-y]_1^2` | |
| `= pi(2-0)-pi[(2-2)-(1/2-1)]` | |
| `= 2pi-pi(1/2)` | |
| `= (3pi)/2\ text(u³)` |
Use mathematical induction to prove that
`1/(1 xx 2 xx 3) + 1/(2 xx 3 xx 4) + … + 1/(n(n + 1)(n + 2)) = 1/4 - 1/(2(n + 1)(n + 2))`
for all integers `n ≥ 1`. (3 marks)
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The roots of `x^4 - 3x + 6 = 0` are `alpha, beta, gamma` and `delta`.
What is the value of `1/alpha + 1/beta + 1/gamma + 1/delta`? (2 marks)
By factorising, or otherwise, solve `2sin^3x + 2sin^2x - sinx - 1 = 0` for `0 <= x <= 2pi`. (3 marks)
A spherical bubble is moving up through a liquid. As it rises, the bubble gets bigger and its radius increases at the rate of 0.2 mm/s.
At what rate is the volume of the bubble increasing when its radius reaches 0.6 mm? Express your answer in mm³/s rounded to one decimal place. (2 marks)
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A committee containing 5 men and 3 women is to be formed from a group of 10 men and 8 women.
In how many different ways can the committee be formed? (1 mark)
Which graph represents the function `y = sin^(-1) (sin x)`?
Which curve best represents the graph of the function `f(x) = -a sin x + b cos x` given that the constants `a` and `b` are both positive?
| A. |  |
B. |  |
| C. |  |
D. |  |
For the two vectors `overset->(OA)` and `overset->(OB)` it is know that
`overset->(OA) · overset->(OB) < 0`
Which of the following statements MUST be true?
In a bag, there are six playing cards, 2, 4, 6, 8, Queen and King. The Queen and King are known as picture cards.
Two of these cards are chosen randomly. All the possible outcomes are shown.
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Yin purchased a car for $20 000. The value of the car decreases according to a linear model. The graph shows the value of the car, $`V`, against the time, t months, since it was purchased.
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People are placed into groups to complete a puzzle. There are 9 different groups.
The table shows the number of people in each group and the amount of time, in minutes, for each group to complete the puzzle.
\begin{array} {|l|c|c|c|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \textit{Number of people} \rule[-1ex]{0pt}{0pt} & 2 & 2 & 3 & 5 & 5 & 7 & 7 & 7 & 8 \\
\hline
\rule{0pt}{2.5ex} \textit{Time taken (min)} \rule[-1ex]{0pt}{0pt} & 28 & 30 & 26 & 19 & 21 & 12 & 13 & 11 & 8 \\
\hline
\end{array}
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a.
b.
c. `text(Form: linear)`
`text{Direction: negative}`
`text{Strength: strong}`
| d. | `text{Mean time (7 people)}` | `= (12 + 13 + 11)/3` |
| `= 12\ text(minutes)` |
It costs $2.45 for a car to travel on a toll road. Due to inflation, the cost is to increase by 3% each year.
How much will it cost for a car to travel on the toll road in 5 years time? (2 marks)
The fuel consumption for a car is 6.7 litres/100 km. On a road trip, the car travels a distance of 1560 km and the fuel cost is $1.45 per litre.
What is the total fuel cost for the trip? (2 marks)
