Aussie Maths & Science Teachers: Save your time with SmarterEd
The hyperbola with equation `xy = 8` is the hyperbola `x^2 - y^2 = k` referred to different axes.
What is the value of `k?`
`text(Distance)\ OA = 4.`
`text(Rotate)\ xy = 8quad\ text(clockwise 45°)`
`text{Intersection is (4, 0)}`
| `4^2 – 0^2` | `= k` |
| `:.k` | `= 16` |
`=> D`
The value of `E` varies directly with the square of `S`.
It is known that `E = 20` when `S = 10`.
What is the value of `E` when `S = 40`?
A company manufactures phones. The company’s income equation and cost equation are drawn on the same graph.
Which region of the graph is the profit zone?
(A) `W`
(B) `X`
(C) `Y`
(D) `Z`
Which expression is equivalent to `4 + log_2 x?`
(A) `log_2 (2x)`
(B) `log_2 (16 + x)`
(C) `4 log_2 (2x)`
(D) `log_2 (16x)`
In a raffle, `30` tickets are sold and there is one prize to be won.
What is the probability that someone buying `6` tickets wins the prize?
(A) `1/30`
(B) `1/6`
(C) `1/5`
(D) `1/4`
`{:(\ quad A\ quad B\ quad C\ quad D\ quad E), ([(0, 1, 0, 0, 1), (1, 0, 1, 0, 1), (0, 1, 0, 1, 2), (0, 0, 1, 0, 1), (1, 1, 2, 1, 0)] {:(A), (B), (C), (D), (E):}):}`
A graph that can be drawn from the adjacency matrix above is
The number of vertices with an odd degree in the network above is
A. `1`
B. `2`
C. `3`
D. `4`
E. `5`
The network shows the activities that are needed to complete a particular project.
Part 1
The total number of activities that need to be completed before activity `L` may begin is
A. `2`
B. `4`
C. `6`
D. `7`
E. `8`
Part 2
The duration of every activity is initially 5 hours. For an extra cost, the completion times of both activity `F` and activity `K` can be reduced to 3 hours each.
If this is done, the completion time for the project will be
A. decreased by 2 hours.
B. decreased by 3 hours.
C. decreased by 4 hours.
D. decreased by 6 hours.
E. unchanged.
In the network shown, the number of vertices of even degree is
A. `2`
B. `3`
C. `4`
D. `5`
E. `6`
In the digraph above, all vertices are reachable from every other vertex.
All vertices would still be reachable from every other vertex if we remove the edge in the direction from
A. `Q` to `U`
B. `R` to `S`
C. `S` to `T`
D. `T` to `R`
E. `V` to `U`
The number of people attending the morning, afternoon and evening sessions at a cinema is given in the table
below. The admission charges (in dollars) for each session are also shown in the table.
A column matrix that can be used to list the number of people attending each of the three sessions is
A. `[25,56,124]`
B. `[[25],[56],[124]]`
C. `[12,15,20]`
D. `[[12],[15],[20]]`
E. `[[25,56,124],[12,15,20]]`
A system of three simultaneous linear equations is written in matrix form as follows.
`[(1, – 2, 0), (1, 0, 3), (0, 2, – 1)] [(x), (y), (z)] = [(4), (11), (– 5)]`
One of the three linear equations is
A. `x - 2y + z = 4`
B. `x + y + 3z = 11`
C. `2x - y = – 5`
D. `x + 3z = 11`
E. `3y - z = – 5`
There are 30 children in a Year 6 class. Each week every child participates in one of three activities: cycling (C), orienteering (O) or swimming (S).
The activities that the children select each week change according to the transition matrix below.
`{:({:qquadqquadqquadqquadtext(this week):}),(qquadqquadqquad\ Cqquad\ OqquadquadS),(T = [(0.5,0.3, 0.3), (0.1,0.6,0.2), (0.4,0.1,0.5)]{:(C), (O), (S):}qquadtext(next week)):}`
From the transition matrix it can be concluded that
A. in the first week of the program, ten children do cycling, ten children do orienteering and ten children do swimming.
B. at least 50% of the children do not change their activities from the first week to the second week.
C. in the long term, all of the children will choose the same activity.
D. orienteering is the most popular activity in the first week.
E. 50% of the children will do swimming each week.
`2 xx [(2,8), (4, -1), (3,5)] -[(3,7), (4,2), (2,3)] quad text (equals)`
The order of matrix `X` is `3 xx 2.`
The element in row `i ` and column `j` of matrix `X` is `x_(ij)` and it is determined by the rule
`x_(ij) = i + j`
The matrix `X` is
A fast-food stand at the football sells pies (`P`) and chips (`C`).
Each week, 300 customers regularly buy either a pie or chips, but not both, from this stand.
For the first five weeks, the customers’ choice of pie or chips is expected to change weekly according to the transition matrix `T_1`, where
`{:({:qquadqquadqquad\ text(this week):}),(qquadqquadqquadquadPqquadqquadC),(T_1 = [(0.65,0.25),(0.35,0.75)]{:(P),(C):}qquadtext(next week)):}`
After the first five weeks, due to expected cold weather, the customers’ choice of pie or chips is expected to change weekly according to the transition matrix `T_2`, where
`{:({:qquadqquadqquad\ text(this week):}),(qquadqquadqquadquadPqquadqquadC),(T_2 = [(0.85,0.25),(0.15,0.75)]{:(P),(C):}qquadtext(next week)):}`
In week 1, 150 customers bought a pie and 150 customers bought chips.
Let `S_1` be the state matrix for week 1.
The number of customers expected to buy a pie or chips in week 8 can be found by evaluating
A. `T_2^7S_1`
B. `T_1^8S_1`
C. `T_2^3(T_1^4S_1)`
D. `T_1^3(T_2^4S_1)`
E. `T_1^3(T_2^5S_1)`
A large population of birds lives on a remote island. Every night each bird settles at either location A or location B.
It was found on the first night that the number of birds at each location was the same.
On each subsequent night, a percentage of birds changed the location at which they settled.
This movement of birds is described by the transition matrix
`{:(qquadqquadAqquadB),({: (A), (B):} [(0.8, 0), (0.2, 1)]):}`
Assume this pattern of movement continues.
In the long term, the number of birds that settle at location A will
A. not change.
B. gradually decrease to zero.
C. eventually settle at around 20% of the island’s bird population.
D. eventually settle at around 80% of the island’s bird population.
E. gradually increase.
Australians go on holidays either within Australia or overseas.
Market research shows that
A transition matrix that could be used to describe this situation is
| A. | `[(0.95),(0.20)]` | B. | `[(0.95),(0.05)] + [(0.20),(0.80)]` |
| C. | `[(0.95,0.95),(0.20,0.20)]` | D. | `[(0.95,0.20),(0.05,0.80)]` |
| E. | `[(0.95,0.80),(0.05,0.20)]` |
Kerry sat for a multiple-choice test consisting of six questions.
Each question had four alternative answers, `A`, `B`, `C`, or `D.`
He selected `D` for his answer to the first question.
He then determined the answers to the remaining questions by following the transition matrix
`{:(qquadqquadqquadqquadqquadqquadqquadqquad\ text(This question)),({:qquadqquadqquadqquadqquadqquadqquadqquadqquadAquadBquadCquadD:}),(text(Next question)qquad{: (A), (B), (C), (D):}[(1,0,1,0), (0,0,0,1), (0,1,0,0), (0,0,0,0)]):}`
The answers that he gave to the six test questions, starting with `D`, were
Each year, a family always goes on its holiday to one of three places; Portland (`P`), Quambatook (`Q`) or Rochester (`R`).
They never go to the same place two years in a row. For example, if they went to Portland one year, they would not go to Portland the next year; they would go to Quambatook or Rochester instead.
A transition matrix that can be used to model this situation is
An international mathematics competition is conducted in three sections – Junior, Intermediate and Senior.
There are money prizes for gold, silver and bronze levels of achievement in each of these sections.
Table 1 shows the number of students who were awarded prizes in each section.
Table 2 shows the value, in dollars, of each prize.
A matrix product that gives the total value of all the Silver prizes that were awarded is
If `A = [(8, 4), (5, 3)]` and the product `AX = [(5, 6), (8, 10)]` then `X` is
| A. `[(24, -14), (13, -7.5)]` | B. `[(-4.25, -5.5), (9.75, 12.5)]` |
| C. `[(-3.75, 7), (-6.5, 12)]` | D. `[(25, 11), (-19.5, -8.5)]` |
| E. `[(0.625, 1.5), (1.6, 3.333)]` |
The number of tourists visiting three towns, Oldtown, Newtown and Twixtown, was recorded for three years.
The data is summarised in the table below.
The `3 xx 1` matrix that could be used to show the number of tourists visiting the three towns in the year 2005 is
| A. `[(975, 1002, 1390)]` | B. `[(1002, 1081, 1095)]` |
| C. `[(975), (1002), (1390)]` | D. `[(1002), (1081), (1095)]` |
| E. `[(975, 1002, 1390), (2105, 1081, 1228), (610, 1095, 1380)]` |
Each night, a large group of mountain goats sleep at one of two locations, `A` or `B`.
On the first night, equal numbers of goats are observed to be sleeping at each location.
From night to night, goats change their sleeping locations according to a transition matrix `T`.
It is expected that, in the long term, more goats will sleep at location `A` than location `B`.
Assuming the total number of goats remains constant, a transition matrix `T` that would predict this outcome is
If `A=[(0,1),(1,0)],\ B=[(1),(0)]` and `C= [(0),(1)]`, then `AB + 2C` equals
A. `[(0),(3)]`
B. `[(3),(0)]`
C. `[(1),(2)]`
D. `[(2),(0)]`
E. `[(2),(3)]`
The matrix below shows the airfares (in dollars) that are charged by Zeniff Airlines to fly between Adelaide (`A`), Melbourne (`M`) and Sydney (`S`).
`{:(qquadqquadquadtext(from)),({:(qquadA,\ M,\ S):}),([(0,85,89),(85,0,99),(97,101,0)]):}{:(),(),(A),(M),(S):}{:(),(),(),(qquadtext(to)),():}`
The cost to fly from Melbourne to Sydney with Zeniff Airlines is
A. `$85`
B. `$89`
C. `$97`
D. `$99`
E. `$101`
`A` and `B` are square matrices such that `AB = BA = I`, where `I` is an identity matrix.
Which one of the following statements is not true?
A. `ABA = A`
B. `AB^2A = I`
C. `B\ text(must equal)\ A`
D. `B\ text(is the inverse of)\ A`
E. `text(both)\ A\ text(and)\ B\ text(have inverses)`
Students from Year 7 and Year 8 in a school sold trees to raise funds for a school trip.
The number of large, medium and small trees that were sold by each year group is shown in the table below.
The large trees were sold for $32 each, the medium trees were sold for $26 each and the small trees were sold for $18 each.
A matrix product that can be used to calculate the amount, in dollars, raised by each year group by selling trees is
`[(0,0,0,0),(2,1,1,3),(0,0,0,0),(0,0,0,0)][(0,0,2,0),(0,0,1,0),(0,0,1,0),(0,0,3,0)]\ \ text(is equal to)`
A circle has 5 equally spaced points on its circumference, as shown.
How many different triangles using 3 of these points as vertices can be drawn? (3 marks)
A. `60`
B. `20`
C. `15`
D. `10`
Wendy buys one type of flower each day.
She chooses from tulips (`T`), roses (`R`), carnations (`C`), irises (`I`) and daisies (`D`).
The type of flower she buys on one day depends on the type of flower she bought the previous day, according to a transition matrix.
Today, Wendy bought tulips.
The transition matrix that, starting tomorrow, ensures Wendy buys flowers in alphabetical order (`C`, `D`, `I`, `R`, `T`) is
The numbers of adult and child tickets purchased for five performances of a stage show are shown in the table below.
Which one of the following matrix calculations can be used to determine both the total number of adult tickets and the total number of child tickets purchased for all five performances?
A. `[(1),(1),(1),(1),(1)][(142,128,89,104,115),(24,31,24,18,23)]`
B. `[(1,1),(1,1),(1,1),(1,1),(1,1)][(142,128,89,104,115),(24,31,24,18,23)]`
C. `[(142,128,89,104,115),(24,31,24,18,23)][(1,1,1,1,1),(1,1,1,1,1)]`
D. `[(1,1)][(142,128,89,104,115),(24,31,24,18,23)]`
E. `[(142,128,89,104,115),(24,31,24,18,23)][(1),(1),(1),(1),(1)]`
Four matrices are shown below.
`W = [(3),(6),(2)]` `X = [(4,1,5),(2,0,6)]` `Y = [(7,1)]` `Z = [(8,5,0),(1,9,3),(4,2,7)]`
Which one of the following matrix products is not defined?
Matrix `B` below shows the number of photography (`P`), art (`A`) and cooking (`C`) books owned by Steven (`S`), Trevor (`T`), Ursula (`U`), Veronica (`V`) and William (`W`).
`{:(qquadqquadqquadPquadAquadC),(B = [(8,5,4),(1,4,5),(3,3,4),(4,2,2),(1,4,1)]{:(S),(T),(U),(V),(W):}):}`
The element in row `i` and column `j` of matrix `B` is `b_(ij)`.
The element `b_32` is the number of
A. art books owned by Trevor.
B. art books owned by Ursula.
C. art books owned by Veronica.
D. cooking books owned by Ursula.
E. cooking books owned by Trevor.
The graphs of the functions with rules `f (x)` and `g (x)` are shown below.
Which one of the following best represents the graph of the function with rule `g(-f(x))?`
**Won't be asked in METHODS from 2016 onwards due to absolute value signs.
For which one of the following functions is the equation `|\ f(x + y) - f(x - y)\ | = 4 sqrt {f(x)\ f(y)}` true for all `x in R` and `y in R?`
A. `f(x) = x^2`
B. `f(x) = |\ 2x\ |`
C. `f(x) = e^x`
D. `f(x) = x^3`
E. `f(x) = x`
Regular customers at a hairdressing salon can choose to have their hair cut by Shirley, Jen or Narj.
The salon has 600 regular customers who get their hair cut each month.
In June, 200 customers chose Shirley (S) to cut their hair, 200 chose Jen (J) to cut their hair and 200 chose Narj (N) to cut their hair.
The regular customers’ choice of hairdresser is expected to change from month to month as shown in the transition matrix, `T`, below.
`{:(qquad qquad qquad qquad text(this month)), (qquad qquad qquad quad {:(S, quad quad J, qquad N):}), (T = [(0.75, 0.10, 0.10), (0.10, 0.75, 0.15), (0.15, 0.15, 0.75)] {:(S), (J), (N):} qquad text(next month)):}`
In the long term, the number of regular customers who are expected to choose Shirley is closest to
A. `150`
B. `170`
C. `185`
D. `195`
E. `200`
Two hundred and fifty people buy bread each day from a corner store. They have a choice of two brands of bread: Megaslice (M) and Superloaf (S).
The customers’ choice of brand changes daily according to the transition diagram below.
On a given day, 100 of these people bought Megaslice bread while the remaining 150 people bought Superloaf bread.
The number of people who are expected to buy each brand of bread the next day is found by evaluating the matrix product
A Student Representative Council (SRC) consists of five members. Three of the members are being selected to attend a conference.
In how many ways can the three members be selected?
(A) `10`
(B) `20`
(C) `30`
(D) `60`
Stockholm is located at `text(59°N 18°E)` and Darwin is located at `text(13°S 131°E)`.
What is the time difference between Stockholm and Darwin? (Ignore time zones and daylight saving.)
(A) `184` minutes
(B) `288` minutes
(C) `452` minutes
(D) `596` minutes
Which of the following graphs best represents the equation `y = x^3 + 1`?
What is the value of the derivative of `y = 2 sin 3x - 3 tan x` at `x = 0`?
(A) `-1`
(B) `0`
(C) `3`
(D) `-9`
What is `0.005\ 233\ 59` written in scientific notation, correct to 4 significant figures?
This is a sketch of a sector of a circle.
Find the value of `theta` to the nearest degree.
(A) `47°`
(B) `48°`
(C) `68°`
(D) `69°`
Mapupu and Minoha are two towns on the equator.
The longitude of Mapupu is `text(16°E)` and the longitude of Minoha is `text(52°W)`.
How far apart are these two towns if the radius of Earth is approximately `6400\ text(km)`?
(A) `4000\ text(km)`
(B) `7600\ text(km)`
(C) `1\ 447\ 600\ text(km)`
(D) `2\ 734\ 400\ text(km)`
