Aussie Maths & Science Teachers: Save your time with SmarterEd
Find the value of `x` and `y`, given
`underset ~r = ((5), (-1), (2)) + lambda ((x), (y), (-3))`
and `underset ~r` is perpendicular to both `underset ~v` and `underset ~w`, where
`underset ~v = ((1), (2), (1)) + mu_1 ((3), (-3), (-1))` and `underset ~w = ((-3), (1), (1)) + mu_2 ((-4), (-1), (-2))`. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
Given `f(x) = 2 sinx` and `g(x) = x`, sketch `y = f(x) + g(x)` for `0 <= x <= 2pi`.
Clearly label endpoints and all local maximum and minimums. (3 marks)
| `y` | `= 2sinx + x` |
| `(dy)/(dx)` | `= 2cosx + 1` |
| `(dy)/(dx)` | `= 0 \ => \ cosx = −1/2 => x = (2pi)/3, (4pi)/3` |
`text(MAX at)\ ((2pi)/3, 3.83)`
`text(MIN at)\ ((4pi)/3, 2.46)`
--- 8 WORK AREA LINES (style=lined) ---
--- 4 WORK AREA LINES (style=lined) ---
| i. | `y` | `= sinx + cosx` |
| `(dy)/(dx)` | `= cosx – sinx` | |
| `(dy)/(dx)` | `= 0\ \ text(when)\ \ sinx = cosx` |
COMMENT: Question style consistent with NESA exemplar questions in official Topic Guidance.
`:. text(SP’s when)\ \ x = pi/4, −(3pi)/4`
ii. `text(Max/min when)\ \ (dy)/(dx) = 0`
`x = pi/4, −(3pi)/4`
| `y_(text(max))` | `= sin\ pi/4 + cos\ pi/4` |
| `= 2 xx 1/sqrt2` | |
| `= sqrt2` |
`:.\ text(By symmetry, range)\ {−sqrt2 <= y <= sqrt2}`
The following diagram shows the graph of `y = g(x)`.
Draw separate one-third page sketches of the graphs of the following:
--- 8 WORK AREA LINES (style=lined) ---
--- 8 WORK AREA LINES (style=lined) ---
| i. | |
| ii. | |
A high school conducted a survey asking students what their favourite Summer sport was.
The Pareto chart shows the data collected.
--- 2 WORK AREA LINES (style=lined) ---
--- 2 WORK AREA LINES (style=lined) ---
Members of the association will travel to a conference in cars and minibuses:
The constraints above can be represented by the following three inequalities.
`text(Inequality 1) qquad qquad x + y <= 8`
`text(Inequality 2) qquad qquad x >= 3`
`text(Inequality 3) qquad qquad y >= 2`
A maximum of 60 people can attend the conference.
Use this information to write Inequality 4. (1 mark)
The graph below shows the four lines representing Inequalities 1 to 4.
Also shown on this graph are four of the integer points that satisfy Inequalities 1 to 4. Each of these integer points is marked with a cross (✖).
Each car will cost $70 to hire and each minibus will cost $100 to hire.
The cost of hiring each minibus remained $100.
All original constraints apply.
If the increase in the cost of hiring each car is more than `k` dollars, then the maximum cost of transporting members to this conference can only occur when using six cars and two minibuses.
Determine the value of `k`. (1 mark)
a. `5x + 10y <= 60`
| b. |  |
c. `text(Consider the line)\ \ 5x + 10y = 60\ \ text(on the graph)`
`text(Touches the feasible region at)\ (4, 4)\ text(only)`
`:.\ text (C) text(ost of 60 members)`
`= 4 xx 70 + 4 xx 100`
`= $680`
d. `text(Coordinates that allow 55 to travel) \ => \ (5, 3) and (3, 4)`
`text(C) text(ost)\ (5, 3) = 5 xx 70 + 3 xx 100 = $650`
`text(C) text(ost)\ (3, 4) = 3 xx 70 + 4 xx 100 = $610`
`:.\ text(Minimum cost is $610)`
e. `text(Objective function): \ C = ax + 100y`
`text(Max cost occurs at)\ \ (6, 2)\ \ text(when)\ \ a > 100`
`text{(i.e. graphically when the slope of}\ \ C = ax + 100y`
`text(is steeper than)\ \ x + y= 8 text{)}`
| `:. k + 70` | `= 100` |
| `k` | `= 30` |
Each branch within the association pays an annual fee based on the number of members it has.
To encourage each branch to find new members, two new annual fee systems have been proposed.
Proposal 1 is shown in the graph below, where the proposed annual fee per member, in dollars, is displayed for branches with up to 25 members.
Complete the inequality by writing the appropriate symbol and number in the box provided. (1 mark)
| 3 ≤ number of members |
|
Proposal 2 is modelled by the following equation.
annual fee per member = – 0.25 × number of members + 12.25
Write down all values of the number of members for which this is the case. (1 mark)
a. `3`
b. `3 <= text(number of members) <= 10`
| c. |  |
d. `text(Same annual fee occurs when graphs intersect.)`
`text(Member numbers): 9, 17, 25`
The graph below shows the membership numbers of the Wombatong Rural Women’s Association each year for the years 2008–2018.
The following diagram shows a crane that is used to transfer shipping containers between the port and the cargo ship.
The length of the boom, `BC`, is 25 m. The length of the hoist, `AB`, is 15 m.
Round your answer to the nearest degree. (1 mark)
`qquad` 
The base of the crane (`Q`) is 20 m from a shipping container at point `R`. The shipping container will be moved to point `P`, 38 m from `Q`. The crane rotates 120° as it moves the shipping container anticlockwise from `R` to `P`.
What is the distance `RP`, in metres?
Round your answer to the nearest metre. (1 mark)
Four chains connect the shipping container to a hoist at point `M`, as shown in the diagram below.
`qquad` 
The shipping container has a height of 2.6 m, a width of 2.4 m and a length of 6 m.
Each chain on the hoist is 4.4 m in length.
What is the vertical distance, in metres, between point `M` and the top of the shipping container?
Round your answer to the nearest metre. (2 marks)
| a.i. | `AC` | `= sqrt(BC^2 – AB^2)` |
| `= sqrt(25^2 – 15^2)` | ||
| `= sqrt 400` | ||
| `= 20` |
| a.ii. | `tan\ /_ ACB` | `= 15/20` |
| `/_ ACB` | `= tan^(-1) (3/4)` | |
| `= 36.86…` | ||
| `~~ 37^@` |
b. `text(Using cosine rule:)`
| `RP^2` | `= 20^2 + 38^2 – 2 xx 20 xx 38 xx cos 120^@` |
| `= 2604` | |
| `:. RP` | `= 51.029…` |
| `~~ 51\ text(metres)` |
| c. |  |
`text(Find)\ h => text(need to find)\ x`
`text(Consider the top of the container)`
| `x` | `= sqrt(3^2 + 1.2^2)` |
| `~~ 3.2311` |
| `:. h` | `= sqrt(4.4^2 – 3.2311^2)` |
| `= 2.98…` | |
| `~~ 3\ text(metres)` |
A cargo ship travels from Magadan (60° N, 151° E) to Sydney (34° S, 151° E).
Find the shortest great circle distance between Magadan and Sydney.
Round your answer to the nearest kilometre. (1 mark)
There is a two-hour time difference between Sydney and Perth at that time of year.
How many hours did it take the cargo ship to travel from Sydney to Perth? (1 mark)
| a. |  |
`text{Sydney’s angle with the equator (34°S) is smaller than Magadan’s}`
`text{(60°N) resulting in a shorter distance to the equator.)`
| b. | `text(Distance)` | `= (34 + 60)/360 xx 2pi xx 6400` |
| `~~ 10\ 500\ text(km)` |
c. `text(Perth is 2 hours behind Sydney)`
`text(6 am Sydney = 4 am Perth)`
| `:.\ text(Travel time)` | `= 4\ text{am (1 Jun)} – 10\ text{am (11 Jun)}` |
| `= 6\ text(hours) + 10\ text(days)` | |
| `= 6 + 10 xx 24` | |
| `= 246\ text(hours)` |
The following diagram shows a cargo ship viewed from above.
The shaded region illustrates the part of the deck on which shipping containers are stored.
Each shipping container is in the shape of a rectangular prism.
Each shipping container has a height of 2.6 m, a width of 2.4 m and a length of 6 m, as shown in the diagram below.
Each barrel is 1.25 m high and has a diameter of 0.73 m, as shown in the diagram below.
Each barrel must remain upright in the shipping container
`qquad qquad` 
What is the maximum number of barrels that can fit in one shipping container? (1 mark)
Fencedale High School is planning to renovate its gymnasium.
This project involves 12 activities, `A` to `L`.
The directed network below shows these activities and their completion times, in weeks.
The minimum completion time for the project is 35 weeks.
--- 2 WORK AREA LINES (style=lined) ---
--- 1 WORK AREA LINES (style=lined) ---
--- 2 WORK AREA LINES (style=lined) ---
It is possible to reduce the completion time for activities `C, D, G, H` and `K` by employing more workers.
What is the minimum time, in weeks, that the renovation project could take? (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
a. `text(Scanning forwards and backwards:)`
 |
`text(Critical path:)\ ABDFGIKL`
`:. 8\ text(activities)`
b. `text(LST for activity)\ E = 12\ text{weeks (i.e. start of 13th week)}`
c. `text(Consider float times of all activities not on critical path.)`
`J-5, H-1, E-1, C-1`
`:.\ text(Activity)\ J\ text(has the largest float time.)`
d. `text(Critical path after reducing)\ CDGHK\ text(by 2 weeks is)`
`ABDFGIKL.`
| `:.\ text(Minimum time)` | `= 2 + 4 + 7 + 1 + 2 + 2 + 5 + 6` |
| `= 29\ text(weeks)` |
Fencedale High School has six buildings. The network below shows these buildings represented by vertices. The edges of the network represent the paths between the buildings.
a. `text(Office)`
b.i. `text(Hamiltonian cycle)`
`text{(note a Hamiltonian path does not start and finish}`
`text{at the same vertex.)}`
b.ii. `text(One of many solutions:)`
On Sunday, matrix `V` is used when calculating the expected number of visitors at each location every hour after 10 am. It is assumed that the park will be at its capacity of 2000 visitors for all of Sunday.
Let `L_0` be the state matrix that shows the number of visitors at each location at 10 am on Sunday.
The number of visitors expected at each location at 11 am on Sunday can be determined by the matrix product
`{:(qquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquad text(this hour)),(qquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquad qquad qquad \ A qquad quad F qquad \ G \ quad quad W),({:V xx L_0 qquad text(where) qquad L_0 = [(500), (600), (500), (400)]{:(A),(F),(G),(W):}, qquad text(and):} qquad V = [(0.3,0.4,0.6,0.3),(0.1,0.2,0.1,0.2),(0.1,0.2,0.2,0.1),(0.5,0.2,0.1,0.4)]{:(A),(F),(G),(W):}\ text(next hour)):}`
Which location is expected to have more than 600 visitors at 11 am on Sunday? (1 mark)
State matrix `R_n` contains the number of visitors at each location `n` hours after 10 am on Sunday, after the safety restrictions have been enforced.
Matrix `R_1` can be determined from the matrix recurrence relation
`qquad qquad qquad R_0 = [(500),(600),(500),(400)]{:(A),(F),(G),(W):}, qquad qquad R_1 = V xx R_0 + B_1`
where matrix `B_1` shows the required movement of visitors at 11 am.
`qquad qquad R_2 = VR_1 + B_2`
where matrix `B_2` shows the required movement of visitors at 12 noon.
Determine the state matrix `R_2`. (1 mark)
The theme park has four locations, Air World `(A)`, Food World `(F)`, Ground World `(G)` and Water World `(W)`.
The number of visitors at each of the four locations is counted every hour.
By 10 am on Saturday the park had reached its capacity of 2000 visitors and could take no more visitors.
The park stayed at capacity until the end of the day
The state matrix, `S_0`, below, shows the number of visitors at each location at 10 am on Saturday.
`S_0 = [(600), (600), (400), (400)] {:(A),(F),(G),(W):}`
Let `S_n` be the state matrix that shows the number of visitors expected at each location `n` hours after 10 am on Saturday.
The number of visitors expected at each location `n` hours after 10 am on Saturday can be determined by the matrix recurrence relation below.
`{:(qquad qquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquad text( this hour)),(qquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquad qquad qquad quad A qquad quad F qquad \ G \ quad quad W),({:S_0 = [(600), (600), (400), (400)], qquad S_(n+1) = T xx S_n quad quad qquad text(where):}\ T = [(0.1,0.2,0.1,0.2),(0.3,0.4,0.6,0.3),(0.1,0.2,0.2,0.1),(0.5,0.2,0.1,0.4)]{:(A),(F),(G),(W):}\ text(next hour)):}`
`S_1 = [(\ text{______}\ ), (\ text{______}\ ), (300),(\ text{______}\ )]{:(A),(F),(G),(W):}`
Matrix `V`, below, shows the proportion of visitors moving from one location to another each hour after 10 am on Sunday.
`qquad qquad {:(qquadqquadqquadqquadqquadtext(this hour)),(qquad qquad qquad \ A qquad quad F qquad \ G \ quad quad W),(V = [(0.3,0.4,0.6,0.3),(0.1,0.2,0.1,0.2),(0.1,0.2,0.2,0.1),(0.5,0.2,0.1,0.4)]{:(A),(F),(G),(W):}\ text(next hour)):}`
Matrix `V` is similar to matrix `T` but has the first two rows of matrix `T` interchanged.
The matrix product that will generate matrix `V` from matrix `T` is
`qquad qquad V = M xx T`
where matrix `M` is a binary matrix.
Write down matrix `M`. (1 mark)
`qquad qquad qquad M = [( , , , , , , , , ), ( , , , , , , , , ), ( , , , , , , , , ), ( , , , , , , , , ), ( , , , , , , , , )]`
For `n = 0, 1, 2,`...
let `I_n = int_0 ^{(pi)/(4)} tan^(n) theta d theta`.
--- 5 WORK AREA LINES (style=lined) ---
--- 6 WORK AREA LINES (style=lined) ---
For `n >= 0`, let `I_n = int_0 ^{(pi)/(4)} tan^(2n) theta d theta`.
--- 6 WORK AREA LINES (style=lined) ---
--- 5 WORK AREA LINES (style=lined) ---
Use integration by parts to evaluate `int_1^e x^7 log_e x dx`. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
If `ab` is divisible by 3, prove by contrapositive that `a` or `b` is divisible by 3. (3 marks)
--- 8 WORK AREA LINES (style=lined) ---
The car park at a theme park has three areas, `A, B` and `C`.
The number of empty `(E)` and full `(F)` parking spaces in each of the three areas at 1 pm on Friday are shown in matrix `Q` below.
`{:(qquad qquad qquad \ E qquad F),(Q = [(70, 50),(30, 20),(40, 40)]{:(A),(B),(C):}quad text(area)):}`
Drivers must pay a parking fee for each hour of parking.
Matrix `P`, below, shows the hourly fee, in dollars, for a car parked in each of the three areas.
`{:(qquad qquad qquad qquad qquad text{area}), (qquad qquad qquad A qquad quad quad B qquad qquad C), (P = [(1.30, 3.50, 1.80)]):}`
`qquad qquad qquad P xx L = [207.00]`
where matrix `L` is a `3 xx 1` matrix.
Write down matrix `L`. (1 mark)
The number of whole hours that each of the 110 cars had been parked was recorded at 1 pm. Matrix `R`, below, shows the number of cars parked for one, two, three or four hours in each of the areas `A, B` and `C`.
`{:(qquadqquadqquadqquadquadtext(area)),(quad qquadqquadquad \ A qquad B qquad C),(R = [(3, 1, 1),(6, 10, 3),(22, 7,10),(19, 2, 26)]{:(1),(2),(3),(4):}\ text(hours)):}`
Complete the matrix `R^T` below. (1 mark)
`qquad R^T = [( , , , , , , , , ), ( , , , , , , , , ), ( , , , , , , , , ), ( , , , , , , , , ), ( , , , , , , , , )]`
If `a^2-4a + 3` is even, `a ∈ Ζ`,
prove by contrapositive that `a` is odd. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
If `x, y, z ∈ R` and `x ≠ y ≠ z`, then
--- 5 WORK AREA LINES (style=lined) ---
--- 6 WORK AREA LINES (style=lined) ---
Phil invests $200 000 in an annuity from which he receives a regular monthly payment.
The balance of the annuity, in dollars, after `n` months, `A_n`, can be modelled by the recurrence relation
`A_0 = 200\ 000, qquad A_(n + 1) = 1.0035\ A_n - 3700`
At some point in the future, the annuity will have a balance that is lower than the monthly payment amount.
Round your answer to the nearest cent. (1 mark)
What monthly payment could Phil have received from this perpetuity? (1 mark)
Four cards are placed on a table with a letter on one face and a shape on the other.
 |
 |
 |
 |
You are given the rule: "if N is on a card then a circle is on the other side."
Which cards need to be turned over to check if this rule holds?
Phil is a builder who has purchased a large set of tools.
The value of Phil’s tools is depreciated using the reducing balance method.
The value of the tools, in dollars, after `n` years, `V_n` , can be modelled by the recurrence relation shown below.
`V_0 = 60\ 000, qquad qquad V_(n + 1) = 0.9 V_n`
--- 1 WORK AREA LINES (style=lined) ---
After how many years will Phil replace these tools? (1 mark)
--- 1 WORK AREA LINES (style=lined) ---
Let `V_n` be the value of the tools after `n` years, in dollars.
Write down a recurrence relation, in terms of `V_0, V_(n + 1)` and `V_n`, that could be used to model the value of the tools using this flat rate depreciation. (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
Phil is a builder who has purchased a large set of tools.
The value of Phil’s tools is depreciated using the reducing balance method.
The value of the tools, in dollars, after `n` years, `V_n` , can be modelled by the recurrence relation shown below.
`V_0 = 60\ 000, qquad V_(n + 1) = 0.9 V_n`
After how many years will Phil replace these tools? (1 mark)
Let `V_n` be the value of the tools after `n` years, in dollars.
Write down a recurrence relation, in terms of `V_0, V_(n + 1)` and `V_n`, that could be used to model the value of the tools using this flat rate depreciation. (1 mark)
The total rainfall, in millimetres, for each of the four seasons in 2015 and 2016 is shown in Table 5 below.
Use the values in Table 5 to find the seasonal indices for summer, autumn and spring.
Write your answers in Table 6, rounded to two decimal places. (2 marks)
Use the appropriate seasonal index from Table 6 to deseasonalise the total rainfall for winter in 2017.
Round your answer to the nearest whole number. (1 mark)
| a. |  |
`text{Average rainfall (2015)}`
`= (142 + 156 + 222 + 120)/4`
`= 160`
`text{Average rainfall (2016)}`
`= (135 + 153 + 216 + 96)/4`
`= 150`
| `text{SI (Summer)}` | `= 1/2 (142/160 + 135/150)=0.89` |
| `text{SI (Autumn)}` | `= 1/2(156/160 + 153/150)=1.00` |
| `text{SI (Spring)}` | `= 1/2 (120/160 + 96/150)=0.70` |
| b. | `text{Winter (deseasonalised)}` | `= 262/1.41` |
| `~~ 186\ text(mm)` |
The scatterplot below shows the atmospheric pressure, in hectopascals (hPa), at 3 pm (pressure 3 pm) plotted against the atmospheric pressure, in hectopascals, at 9 am (pressure 9 am) for 23 days in November 2017 at a particular weather station.
A least squares line has been fitted to the scatterplot as shown.
The equation of this line is
pressure 3 pm = 111.4 + 0.8894 × pressure 9 am
Round your answer to the nearest whole number. (1 mark)
Round your answer to the nearest whole number. (1 mark)
Round your answer to one decimal place. (1 mark)
What is this assumption? (1 mark)
Explain why. (1 mark)
The relative humidity (%) at 9 am and 3 pm on 14 days in November 2017 is shown in Table 3 below.
A least squares line is to be fitted to the data with the aim of predicting the relative humidity at 3 pm (humidity 3 pm) from the relative humidity at 9 am (humidity 9 am).
Round both values to three significant figures and write them in the appropriate boxes provided.
| humidity 3 pm = |
|
+ |
|
× humidity 9 am (1 mark) |
Round your answer to three decimal places. (1 mark)
Prove that `log_2 11` is irrational. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
If `n` is a positive integer,
prove `sqrt(10n+2)` is always irrational. (3 marks)
--- 8 WORK AREA LINES (style=lined) ---
Prove that `root3 2` is irrational. (3 marks)
--- 8 WORK AREA LINES (style=lined) ---
Prove that `sqrt3` is irrational. (3 marks)
--- 8 WORK AREA LINES (style=lined) ---
The five-number summary for the distribution of minimum daily temperature for the months of February, May and July in 2017 is shown in Table 2.
The associated boxplots are shown below the table.
Explain why the information given above supports the contention that minimum daily temperature is associated with the month. Refer to the values of an appropriate statistic in your response. (2 marks)
The parallel boxplots below show the maximum daily temperature and minimum daily temperature, in degrees Celsius, for 30 days in November 2017.
For November 2017
| i. | the interquartile range for the minimum daily temperature was |
|
°C (1 mark) |
| ii. | the median value for maximum daily temperature was |
|
°C higher than the |
| median value for minimum daily temperature (1 mark) |
| iii. | the number of days on which the maximum daily temperature was less than the median value for |
| minimum daily temperature was |
|
(1 mark) |
Determine the number of days in November 2017 for which this temperature difference is expected to be greater than 9.4 °C. (1 mark)
Table 1 shows the day number and the minimum temperature, in degrees Celsius, for 15 consecutive days in May 2017.
The incomplete ordered stem plot below has been constructed using the data values for days 1 to 10.
(Answer on the stem plot above.) (1 mark)
Use this stem plot to determine
a. `text(Day number)`
| b. |  |
c.i. `15\ text(data points)`
`Q_1 = 12.2\ text{(4th data point)}`
| c.ii. | `text(% Days)` | `= 3/15 xx 100` |
| `= 20%` |
The feasible region for a linear programming problem is shaded in the diagram below.
The equation of the objective function for this problem is of the form
`P = mx + ny`, where `m > 0` and `n > 0`
The dotted line passing through the points `V` and `W` in the diagram has the same slope as the objective function for this problem.
The minimum value of the objective function can be determined by calculating its value at
Giovanni makes and sells toy train sets.
Each toy train set costs $40.00 to make and sells for $65.00
Giovanni also has a fixed cost for making a batch of toy train sets.
Last Sunday, Giovanni sold a batch of 35 toy train sets for a profit of $250.
Giovanni’s fixed cost for this batch of train sets is
The graph below shows a straight line that crosses the `x`-axis at `(-10, 0)`, passes through the point `(-6, 8)` and crosses the `y`-axis at `(0, q)`.
What is the value of `q`?
A shop sells two types of discs: compact discs (CDs) and digital video discs (DVDs).
CDs are sold for $7.00 each and DVDs are sold for $13.00 each.
Bonnie bought a total of 16 discs for $178.00
How many DVDs did Bonnie buy?
It can be shown that
`qquad (8(1-x))/((2-x^2)(2-2x+x^2)) = (4-2x)/(2-2x+x^2)-(2x)/(2-x^2)` (do NOT prove this)
Use this result to evaluate `int_0^1 (8(1-x))/((2-x^2)(2-2x+x^2))\ dx` (4 marks)
--- 6 WORK AREA LINES (style=lined) ---
--- 2 WORK AREA LINES (style=lined) ---
All cities in China are in the same time zone.
Wuhan (31° N, 114° E) and Chengdu (31° N, 104° E) are both cities in China.
On one day in January, the sun set in Wuhan at 5.50 pm.
Assuming that 15° of longitude equates to a one-hour time difference, the time the sun set in Chengdu on the same day is
In triangle `ABC`, the point `D` lies on the line `AC`, as shown in the diagram below.
The length of the line `BC` is equal to the length of the line `BD`.
Which one of the following statements is true?
`text(Let)\ \ angle ACB = x`
`angle BDC = x\ \ \ text{(angles opposite equal side in isosceles Δ)}`
| `angle ACB + angle ADB` | `=x+180-x` | |
| `=180` |
`=> C`
Triangle `M`, shown below, has side lengths of 3 cm, 4 cm and 5 cm.
Four other triangles have the following side lengths:
The triangles that are similar to triangle `M` are
Town B is located on a bearing of 060° from Town A.
The diagram that could illustrate this is
| A. |  |
B. |  |
| C. |  |
D. |  |
| E. |  |
`=> A`
The map below shows all the road connections between five towns, `P, Q, R, S` and `T`.
The road connections could be represented by the adjacency matrix
| A. | `{:(qquad qquad P\ \ Q\ \ R\ \ \ S\ \ \ T), ({:(P),(Q),(R),(S),(T):}[(1, 3, 0, 2, 2),(3,0,1,1,1),(0,1,0,1,0),(2,1,1,0,2),(2,1,0,2,0)]):}` | B. | `{:(qquad qquad P\ \ Q\ \ R\ \ \ S\ \ \ T), ({:(P),(Q),(R),(S),(T):}[(1, 2, 0, 2, 2),(2,0,1,1,1),(0,1,0,1,0),(2,1,1,0,2),(2,1,0,2,0)]):}` |
| C. | `{:(qquad qquad P\ \ Q\ \ R\ \ \ S\ \ \ T), ({:(P),(Q),(R),(S),(T):}[(0, 3, 0, 2, 2),(3,0,1,1,1),(0,1,0,1,0),(2,1,1,0,2),(2,1,0,2,0)]):}` | D. | `{:(qquad qquad P\ \ Q\ \ R\ \ \ S\ \ \ T), ({:(P),(Q),(R),(S),(T):}[(0, 2, 0, 2, 2),(2,0,1,1,1),(0,1,0,1,0),(2,1,1,1,2),(2,1,0,2,0)]):}` |
| E. | `{:(qquad qquad P\ \ Q\ \ R\ \ \ S\ \ \ T), ({:(P),(Q),(R),(S),(T):}[(1, 2, 0, 2, 2),(2,0,1,1,1),(0,1,0,1,0),(2,1,1,1,1),(2,1,0,1,0)]):}` |
The following diagram shows the distances, in metres, along a series of cables connecting a main server to seven points, `A` to `G`, in a computer network.
Calculate the minimum length of cable, in metres, required to ensure that each of the seven points is connected to the main server directly or via another point. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
`text(Using Prim’s Algorithm:)`
`text{Edge 1: main server to D (15)}`
`text{Edge 2: DE (28)}`
`text{Edge 3: EG (30)}`
`text{Edge 4: GF (32)}`
`text{Edge 5: EA (35)}`
`text{Edge 6: AB (28)}`
`text{Edge 7: GC (35)}`
`:.\ text(Minimum length)`
`= 15 + 28 + 30 + 32 + 35 + 28 + 35`
`= 203\ text(m)`
The following diagram shows the distances, in metres, along a series of cables connecting a main server to seven points, `A` to `G`, in a computer network.
The minimum length of cable, in metres, required to ensure that each of the seven points is connected to the main server directly or via another point is
`text(Using Prim’s Algorithm:)`
`text{Edge 1: main server to D (15)}`
`text{Edge 2: DE (28)}`
`text{Edge 3: EG (30)}`
`text{Edge 4: GF (32)}`
`text{Edge 5: EA (35)}`
`text{Edge 6: AB (28)}`
`text{Edge 7: GC (35)}`
`:.\ text(Minimum length)`
`= 15 + 28 + 30 + 32 + 35 + 28 + 35`
`= 203\ text(m)`
`=> B`
Two graphs, labelled Graph 1 and Graph 2, are shown below.
Which one of the following statements is not true?
The flow of water through a series of pipes is shown in the network below
The numbers on the edges show the maximum flow through each pipe in litres per minute.
The capacity of Cut `Q`, in litres per minute, is
The flow of water through a series of pipes is shown in the network below
The numbers on the edges show the maximum flow through each pipe in litres per minute.
The capacity of Cut `Q`, in litres per minute, is
Consider the graph below.
The minimum number of extra edges that are required so that an Eulerian circuit is possible in this graph is
`text(Eulerian circuit requires all vertex degrees to be even.)`
`:.\ text(Need 2 extra edges.)`
`=> C`
The communication matrix below shows the direct paths by which messages can be sent between two people in a group of six people, `U` to `Z`.
`{:(qquad qquad qquad qquad qquad qquad qquad qquad text(receiver)),(qquad qquad qquad qquad qquad quad U quad V quad W \ \ \ X \ \ \ Y quad \ Z), (text(sender) qquad {:(U),(V),(W),(X),(Y), (Z):}[(0, 1, 1, 0, 1, 1), (1, 0, 1, 0, 1, 0), (1, 1, 0, 1, 0, 1), (0, 1, 0, 0, 1, 1), (0, 0, 1, 1, 0, 1), (1, 1, 0, 1, 1, 0)]):}`
A 1 in the matrix shows that the person named in that row can send a message directly to the person named in that column.
For example, the 1 in row 4, column 2 shows that `X` can send a message directly to `V`.
In how many ways can `Y` get a message to `W` by sending it directly to one other person?
--- 1 WORK AREA LINES (style=lined) ---
--- 5 WORK AREA LINES (style=lined) ---
How many numbers greater than 6000 can be formed with the digits 1, 4, 5, 7, 8 if no digit is repeated. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
The variable point `(3t, 2t^2)` lies on a parabola.
Find the Cartesian equation for this parabola. (2 marks)
A water park is open from 9 am until 5 pm.
There are three activities, the pool `(P)`, the slide `(S)` and the water jets `(W)`, at the water park.
Children have been found to change their activity at the water park each half hour, as shown in the transition matrix, `T`, below.
`{:(qquad qquad qquad quadtext(this half year)),(qquad qquad qquad quad P qquad qquad S qquad quad W), (T = [(0.80,0.20,0.40),(0.05,0.60,0.10),(0.15,0.20,0.50)] {:(P),(S),(W):} text( next half year)):}`
A group of children has come to the water park for the whole day.
The percentage of these children who are expected to be at the slide `(S)` at closing time is closest to