Band 4 – Page 78

GRAPHS, FUR1 2008 VCAA 6 MC

At the local bakery, the cost of four donuts and six buns is $14.70.

The cost of three donuts and five buns is $11.90.

At this bakery, the cost of one donut and two buns will be

A. `$2.80`

B. `$3.80`

C. `$3.85`

D. `$4.55`

E. `$4.85`

Show Answers Only

`D`

Show Worked Solution

`text(Let)\ x = text(cost of doughnuts)`

`text(Let)\ y = text(cost of buns)`

`4x + 6y = $14.70\ \ \ …\ (1)`

`3x + 5y = $11.90\ \ \ …\ (2)`

`text(Multiply)\ (1) xx 3\ text(and)\ (2) xx 4`

`12x + 18y = 44.10\ \ \ …\ (3)`

`12x + 20y = 47.60\ \ \ …\ (4)`

`(4) − (3)`

`2y`	`= 3.50`
`y`	`= $1.75`

`text(Substitute)\ \ y= $1.75\ text(into)\ (1)`

`4x + 6(1.75)`	`= 14.70`
`4x`	`= 4.20`
`x`	`= 1.05`

`:.\ text(C)text(ost of 1 doughnut and 2 buns)`

`= 1.05 + 2 xx 1.75`

`= $4.55`

`=> D`

GRAPHS, FUR1 2008 VCAA 3 MC

The graph below shows the time `t`, in hours, taken to travel 100 km at an average speed of `s` km/h.

Which statement is false?

As average speed increases, the time taken to travel 100 km decreases.
It will take 2 hours to travel 100 km at an average speed of 50 km/h.
The relationship between time and average speed is linear.
When travelling at an average speed of 20 km/h, the 100 km journey takes 5 hours to complete.
A formula that relates `s` and `t` is `t = 100/s, \ s > 0`

Show Answers Only

`C`

Show Worked Solution

`text(Using the formula)`

`text(distance = speed)\ xx\ text(time,)`

`A, B, D\ text(and)\ E\ text(are correct.)`

`text(However, as shown by the curved graph, the)`

`text(relationship between)\ t\ text(and)\ s\ text(is NOT linear.)`

`=> C`

Algebra, STD2 A4 SM-Bank 27

Fiona and John are planning to hold a fund-raising event for cancer research. They can hire a function room for $650 and a band for $850. Drinks will cost them $25 per person.

Write a formula for the cost ($C) of holding the charity event for `x` people. (1 mark)

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The graph below shows the planned income and costs if they charge $50 per ticket. Estimate the number of guests they need to break even. (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
How much profit will Fiona and John make if 80 people attend their event? (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

`$C = 1500 + 25x`
`60`
`$500`

Show Worked Solution

i.	`text(Fixed C) text(osts)`	`= 650 + 850`
		`= $1500`

`text(Variable C) text(osts) = $25x`

`:.\ $C = 1500 + 25x`

ii.	`text(From the graph)`
	`text(C) text(osts = Income when)\ x = 60`
	`text{(i.e. where graphs intersect)}`

iii. `text(When)\ \ x = 80:`

`text(Income)`	`= 80 xx 50`
	`= $4000`

`$C`	`= 1500 + 25 xx 80`
	`= $3500`

`:.\ text(Profit)`	`= 4000 – 3500`
	`= $500`

Statistics, STD2 S5 SM-Bank 4 MC

The length of a type of ant is approximately normally distributed with a mean of 4.8 mm and a standard deviation of 1.2 mm.

A standardised ant length of `z\ text(= −0.5)` corresponds to an actual ant length of

A. ` text(2.4 mm)`

B. `text(3.6 mm)`

C. `text(4.2 mm)`

D. `text(5.4 mm)`

Show Answers Only

`C`

Show Worked Solution

`z`	`= \ \ (x – mu)/sigma`
`-0.5`	`= \ \ (x – 4.8)/1.2`
`-0.6`	`= \ \ x – 4.8`
`x`	`= \ \ 4.2\ text(mm)`

`=>C`

Quadratic, 2UA SM-Bank 01

Solve `9^x-10(3^x)+9=0` (2 marks)

Show Answers Only

`x=0\ \ text(or)\ \ 2`

Show Worked Solution

IMPORTANT: Students should recognise this equation as a quadratic, and substitute `3^x` with a variable such as `X`.

`9^x-10(3^x)+9`	`=0`
`(3^2)^x-10(3^x)+9`	`=0`
`(3^x)^2-10(3^x)+9`	`=0`
`text(Let)\ X=3^x`
`:.\ X^2-10X+9`	`=0`
`(X-9)(X-1)`	`=0`

`X`	`=9`	`\ \ \ \ \ \ \ \ \ \ `	`X`	`=1`
`3^x`	`=9`	`\ \ \ \ \ \ \ \ \ \ `	`3^x`	`=1`
`x`	`=2`	`\ \ \ \ \ \ \ \ \ \ `	`x`	`=0`

`:.x=0\ \ text(or)\ \ 2`

Financial Maths, 2ADV M1 SM-Bank 11

Express the recurring decimal `0.323232...` as a fraction. (2 marks)

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Show Answers Only

`32/99`

Show Worked Solution

`0.3232…`	`=32/100+32/10^4+32/10^6+…`
	`=32/100(1+1/10^2+1/10^4+…)`
`=>\ text(GP where)\ a=1,\ \ r=1/10^2`
	`=32/100(a/(1-r))`
	`=32/100(1/(1-1/100))`
	`=32/100(1/(99/100))`
	`=32/99`

GRAPHS, FUR1 2009 VCAA 8 MC

Brian, a landscaping contractor, charges by the hour for his company’s services.

To complete a particular job, he will have to use three workers and pay each of them $20 per hour. The fixed costs for the job are $150 and it will take four hours to complete the job.

To break even on this job, his hourly charge to the client should be

A. `$38.25`

B. `$57.50`

C. `$97.50`

D. `$127.50`

E. `$132.50`

Show Answers Only

`C`

Show Worked Solution

`text(Fixed Cost)`	`= $150`
`text(Labour Cost)`	`= $20 xx 3 xx 4`
	`= $240`
`text(Total Cost)`	`= 150 + 240`
	`= $390`

`:.\ text(Breakeven Hourly Charge)`

	`=(text(Total C) text(ost))/text(Job Hours)`
	`= 390/4`
	`= $97.50`

`=> C`

GRAPHS, FUR1 2009 VCAA 7 MC

A school’s squash and volleyball teams plan to enter a sports competition.

A squash team requires at least 4 players.

A volleyball team requires at least 6 players.

No more than 25 students from any one school can enter the competition.

Let `x` be the number of squash players sent by the school to the competition.
Let `y` be the number of volleyball players sent by the school to the competition.

The constraints above define the feasible region shaded in the graph below.

A fee is charged for all players entering the competition. Squash players are charged $5 and volleyball players are charged $4.

Given the above constraints, the maximum cost for the school’s squash and volleyball teams to enter the competition is

A. `$44`

B. `$104`

C. `$119`

D. `$121`

E. `$144`

Show Answers Only

`C`

Show Worked Solution

`text(S)text(ince squash players pay a higher fee, the)`

`text(maximum cost occurs when the maximum)`

`text(number of squash players enter, which is 19)`

`text{(from the graph).}`

`:.\ text(Max cost)`	`= 19 xx $5 + 6 × $4`
	`= $119`

`=> C`

GRAPHS, FUR1 2009 VCAA 5-6 MC

Kathy is a tutor who offers tutorial sessions for English and History students.

Part 1

An English tutorial session takes 1.5 hours.

A History tutorial session take 30 minutes.

Kathy has no more than 15 hours available in a week for tutorial sessions.

Let `x` represent the number of English tutorial sessions Kathy has each week.
Let `y` represent the number of History tutorial sessions Kathy has each week.

An inequality representing the constraint on Kathy’s tutorial time each week (in hours) is

A. `1.5x + 30y = 15`

B. `1.5x + 30y >= 15`

C. `1.5x + 30y <= 15`

D. `1.5x + 0.5y >= 15`

E. `1.5x + 0.5y <= 15`

Part 2

Kathy prefers to have no more than 18 tutorial sessions in total each week.

She prefers to have at least 4 English tutorial sessions.

She also prefers to have at least as many History tutorial sessions as English tutorial sessions.

Let `x` represent the number of English tutorial sessions Kathy has each week.
Let `y` represent the number of History tutorial sessions Kathy has each week.

The shaded region that satisfies all of these constraints is

Show Answers Only

`text(Part 1:)\ E`

`text(Part 2:)\ D`

Show Worked Solution

`text(Part 1)`

`text(English tutorial time)`	`= 1.5x\ text(hours)`
`text(History tutorial time)`	`= 30y\ text(minutes)`
	`= 0.5y\ text(hours)`

`text(Kathy has no more than 15 hours available,)`

`:.\ 1.5x + 0.5y <= 15`

`=> E`

`text(Part 2)`

`text(The constraints can be given by:)`

`x+y`	`<=18`
`y`	`>=4`
`y`	`>=x`

`=> D`

GRAPHS, FUR1 2009 VCAA 1-3 MC

The graph below shows the water temperature in a fish tank over a 12-hour period.

Part 1

Over the 12-hour period, the temperature of the tank is increasing most rapidly

A. during the first 2 hours.

B. from 2 to 4 hours.

C. from 4 to 6 hours.

D. from 6 to 8 hours.

E. from 8 to 10 hours.

Part 2

The fish tank is considered to be a safe environment for a type of fish if the water temperature is maintained between 24°C and 28°C.

Over the 12-hour period, the length of time (in hours) that the environment was safe for this type of fish was closest to

A. `1.5`

B. `5.0`

C. `7.0`

D. `8.5`

E. `10.5`

Part 3

The graph below can be used to determine the cost (in cents) of heating the fish tank during the first five hours of heating.

The cost of heating the tank for one hour is

A. `4\ text(cents.)`

B. `5\ text(cents.)`

C. `15\ text(cents.)`

D. `20\ text(cents.)`

E. `100\ text(cents.)`

Show Answers Only

`text(Part 1:)\ B`

`text(Part 2:)\ C`

`text(Part 3:)\ A`

Show Worked Solution

`text(Part 1)`

`text(The steepest part of the curve from the choices)`

`text(given is the 2 to 4 hour period. Therefore, the)`

`text(temperature increases the most during this)`

`text(period.)`

`=> B`

`text(Part 2)`

`text{The curve is between 24 and 28 (on the}`

` y\ text{axis) during the following periods:}`

`3.5\ text(to)\ 7.75`	`= 4.25\ text(hours)`
`9.5\ text(to)\ 12`	`= 2.5\ text(hours) `

`:.\ text(Environment was safe for 6.75 hours)`

`=> C`

`text(Part 3)`

`text(C)text(ost of heating for 5 hours = 20 cents)`

`:.\ text(C)text(ost of 1 hour)`	`= 20/5`
	`= 4\ text(cents) `

`=> A`

GEOMETRY, FUR1 2006 VCAA 7 MC

In the diagram, `AD = 9\ text(cm), AC = 24\ text(cm) and DB = 27\ text(cm.)`

Line segments `AC` and `DE` are parallel.

The length of `DE` is

A. `6\ text(cm)`

B. `8\ text(cm)`

C. `12\ text(cm)`

D. `16\ text(cm)`

E. `18\ text(cm)`

Show Answers Only

`E`

Show Worked Solution

`/_CAD = /_EDB\ \ \ text{(} text(corresponding angles,)\ AC\ text(||)\ DE text{)}`

`/_DBE\ text(is common)`

`:. Delta ACB\ text(|||)\ Delta DEB\ \ text{(equiangular)}`

`:.\ (DE)/(DB)`	`= (AC)/(AB)`	`\ \ \ text{(corresponding sides}` `\ \ \ \ text{of similar triangles)}`
`(DE)/27`	`= 24/36`
`DE`	`= (24 xx 27)/36`
	`= 18\ text(cm)`

`=> E`

GEOMETRY, FUR1 2006 VCAA 6 MC

The rectangular box shown in this diagram is closed at the top and at the bottom.

It has a volume of 6 m³.

The base dimensions are 1.5 m × 2 m.

The total surface area of this box is

A. `10\ text(m²)`

B. `13\ text(m²)`

C. `13.5\ text(m²)`

D. `20\ text(m²)`

E. `27\ text(m²)`

Show Answers Only

`D`

Show Worked Solution

`V`	`= text(Area of base) xx text(height)\ (h)`
`6`	`= 1.5 xx 2 xx h`
`6`	`= 3h`
`:. h`	`= 2\ text(m)`

`:.\ text(Surface Area of box)`

`= 2 xx (1.5 xx 2) + 2 xx (1.5 xx 2) + 2(2 xx 2)`

`= 6 + 6 + 8`

`= 20\ text(m²)`

`=> D`

Measurement, 2UG 2007 HSC 28c

A piece of plaster has a uniform cross-section, which has been shaded, and has dimensions as shown.

Use two applications of Simpson’s rule to approximate the area of the cross-section. (3 marks)
The total surface area of the piece of plaster is `7480.8\ text(cm²)`.
Calculate the area of the curved surface as shown on the diagram. (2 marks)

Show Answers Only

`50.4\ text(cm²)`
`3500\ text(cm²)`

Show Worked Solution

(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)`

GEOMETRY, FUR1 2006 VCAA 4 MC

The building shown in the diagram is 8 m wide and 24 m long.

The side walls are 4 m high.

The peak of the roof is 6 m vertically above the ground.

In cubic metres, the volume of this building is

A. `384`

B. `576`

C. `960`

D. `1152`

E. `4608`

Show Answers Only

`C`

Show Worked Solution

`text(Area of front of house)`

`= (4 xx 8) + 1/2 (8 xx 2)`

`= 32 + 8`

`= 40\ text(m²)`

`V`	`= Ah`
	`= 40 xx 24`
	`= 960\ text(m³)`

`=> C`

GEOMETRY, FUR1 2007 VCAA 4 MC

A steel beam used for constructing a building has a cross-sectional area of 0.048 m² as shown.

The beam is 12 m long.

In cubic metres, the volume of this steel beam is closest to

A. `0.576`

B. `2.5`

C. `2.63`

D. `57.6`

E. `2500`

Show Answers Only

`A`

Show Worked Solution

`text(Volume)`	`= A xx h`
	`= 0.048 xx 12`
	`= 0.576\ text(m³)`

`=> A`

GEOMETRY, FUR1 2007 VCAA 3 MC

A rectangle is 3.79 m wide and has a perimeter of 24.50 m.

Correct to one decimal place, the length of the diagonal of this rectangle is

A. `9.2\ text(m)`

B. `9.3\ text(m)`

C. `12.2\ text(m)`

D. `12.3\ text(m)`

E. `12.5\ text(m)`

Show Answers Only

`B`

Show Worked Solution

`text(Perimeter)`	`= 24.50\ text(m)`
`24.50`	`= 2x + (2 xx 3.79)`
`2x`	`= 16.92`
`x`	`= 8.46\ text(m)`

`text(Using Pythagoras,)`

`d^2`	`= x^2 + 3.79^2`
	`= 8.46^2 + 3.79^2`
	`= 85.9357`
`:.\ d`	`= 9.27…\ text(m)`

`=> B`

GEOMETRY, FUR1 2008 VCAA 8 MC

A regular hexagon has side length 3.0 cm and height 5.2 cm as shown in the diagram above.

The area (in cm²) of the hexagon is closest to

A. `11.7`

B. `13.5`

C. `15.6`

D. `18.0`

E. `23.4`

Show Answers Only

`E`

Show Worked Solution

`text(Area of rectangle)`	`= 3.0 xx 5.2`
	`= 15.6\ text(cm²)`

`text(Using Pythagoras to find)\ h:`

`3.0^2`	`= 2.6^2 + h^2`
`h^2`	`= 9 – 6.76`
	`= 2.24`
`h`	`= 1.496…`

`text(Area of)\ \ Delta ABC`

`= 1/2 xx b xx h`

`= 1/2 xx 5.2 xx 1.496…`

`= 3.891…\ text(cm²)`

`:.\ text(Area of hexagon)`

`= 15.6 + (2 xx 3.891…)`

`= 23.38…\ text(cm²)`

`=> E`

GEOMETRY, FUR1 2008 VCAA 7 MC

Sand is poured out of a truck and forms a pile in the shape of a right circular cone. The diameter of the base of the pile of sand is 2.6 m. The height is 1.2 m.

The volume (in m³) of sand in the pile is closest to

A. `2.1`

B. `3.1`

C. `6.4`

D. `8.5`

E. `25.5`

Show Answers Only

`A`

Show Worked Solution

`text(Radius of base)`	`= 1/2\ text(diameter)`
	`= 1.3\ text(m)`

`V`	`= 1/3 pi r^2 h`
	`= 1/3 xx pi xx 1.3^2 xx 1.2`
	`= 2.12 …\ text(m³)`

`=> A`

GEOMETRY, FUR1 2008 VCAA 4 MC

The solid cylindrical rod shown above has a volume of 490.87 cm³. The length is 25.15 cm.

The radius (in cm) of the cross-section of the rod, correct to one decimal place, is

A. `2.5`

B. `5.0`

C. `6.3`

D. `12.5`

E. `19.6`

Show Answers Only

`A`

Show Worked Solution

`V = pir^2h`

`text(Where length) =h = 25.15\ text(cm)`

`:.\ 490.87`	`= pi xx r^2 xx 25.15`
`r^2`	`= 490.87/(pi xx 25.15)`
	`= 6.2126…`
`:. r`	`= 2.492…\ text(cm)`

`=> A`

FS Comm, 2UG SM-Bank 03

Zilda needs to download an `8.3 text(MB)` file from the internet. If the download transfer rate is `5 text(kbps)`, how long will it take her to download the file (to the nearest minute)? (3 marks)

Show Answers Only

`232\ text(mins)\ text{(nearest minute)}`

Show Worked Solution

`8.3 text(MB file)`

`= 8.3 xx 2^20\ text(bytes)`

`= 8.3 xx 2^20 xx 8\ text(bits)`

`text(Download speed is 5kbps)`

`= 5000\ text(bps)`

`:.\ text(Time to download)`

`= (8.3 xx 2^20 xx 8)/5000`

`= 13925.08…\ text(seconds)`

`= 232.08\ text(mins)`

`= 232\ text(mins)\ text{(nearest minute)}`

Algebra, MET1 SM-Bank 28

Consider the simultaneous linear equations below.

`4x - 2y = 18`

`3x + ky = 10` (3 marks)

where `k` is a real constant.

What are the values of `k` where no solutions exist? (3 marks)
What values of `k` do the simultaneous equations have a unique solution? (1 mark)

Show Answers Only

`k=-3/2`
`k\ in R text(\) {-3/2}`

Show Worked Solution

a.	`4x – 2y`	`=18`
	`y`	`=2x-9\ \ …\ (1)`

`=> m_1 = 2,\ \ c_1=-9`

`3x +ky`	`=10`
`y`	`=-3/k x +10/k\ \ …\ (2)`

`=> m_2 = – 3/k,\ \ c_2=10/k`

`text(No solution if)\ \ m_1=m_2, and c_1!=c_2.`

`-3/k`	`=2`
`k`	`=- 3/2`

`text(When)\ \ k=-3/2, c_1!=c_2.`

`:.\ text(No solution when)\ \ k=-3/2.`

b. `text(A unique solution exists when)\ \ m_1 != m_2,`

`k in R\ text(\) {-3/2}`

Algebra, 2UG AM3 SM-Bank 05

Find the value of `R_1` if

`1/R = 1/R_1 + 1/R_2`, `R = 1.12` and `R_2 = 2.24` (2 marks)

Show Answers Only

`R_1 = 2.24`

Show Worked Solution

`1/R`	`= 1/R_1 + 1/R_2`
`1/1.12`	`= 1/R_1 + 1/2.24`
`1/R_1`	`= 1/1.12 – 1/2.24`
	`= 2/2.24 – 1/2.24`
	`= 1/2.24`
`:.\ R_1 = 2.24`

Algebra, STD2 A2 2007 HSC 27b

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.

Write an equation for the total cost (`$c`) of purchasing and running these four light globes for one year in terms of `d`. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---
Find the value of `d` (correct to three decimal places) if the total cost of running these four light globes for one year is $250. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
If the use of the light globes increases to ten hours per night every night of the year, does the total cost double? Justify your answer with appropriate calculations. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
The manufacturer’s specifications state that the expected life of the light globes is normally distributed with a standard deviation of 170 hours.

What is the mean life, in hours, of these light globes if 97.5% will last up to 5000 hours? (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

`$c = 24 + 7300d`
`0.031\ $ text(/hr)\ text{(3 d.p.)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(4660 hours.)`

Show Worked Solution

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.)`

Algebra, STD2 A4 2007 HSC 27a

A rectangular playing surface is to be constructed so that the length is 6 metres more than the width.

Give an example of a length and width that would be possible for this playing surface. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
Write an equation for the area (`A`) of the playing surface in terms of its length (`l`). (1 mark)

--- 1 WORK AREA LINES (style=lined) ---

A graph comparing the area of the playing surface to its length is shown.
Why are lengths of 0 metres to 6 metres impossible? (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
What would be the dimensions of the playing surface if it had an area of 135 m²? (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Company `A` constructs playing surfaces.
Draw a graph to represent the cost of using Company `A` to construct all playing surface sizes up to and including 200 m².

Use the horizontal axis to represent the area and the vertical axis to represent the cost. (2 marks)

--- 8 WORK AREA LINES (style=lined) ---
Company `B` charges a rate of $360 per square metre regardless of size.
Which company would charge less to construct a playing surface with an area of 135 m²

Justify your answer with suitable calculations. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(One possibility is a length of 10 m, and a width)`

`text{of 4 m (among many possibilities).}`
`A = l (l – 6)\ \ text(m²)`
`text(Given the length must be 6 m more than)`

`text(the width, it follows that the length)`

`text(must be greater than 6 m.)`
`text(15 m × 9 m)`
`text(Proof)\ \ text{(See worked solutions)}`

Show Worked Solution

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.)`

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.`

Financial Maths, STD2 F1 2007 HSC 26b

Myles is in his third year as an apprentice film editor.

Myles purchased film-editing equipment for $5000.

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)

--- 4 WORK AREA LINES (style=lined) ---
Myles earns $800 per week. Calculate his taxable income for this year if the only allowable deduction is the amount of depreciation of his film-editing equipment in the third year of use. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
Use this tax table to calculate Myles’s tax payable. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(9.1%)`
`$41\ 145`
`$8443.50`

Show Worked Solution

i.	`S = V_0 – Dn`
	`S = $3635,\ \ \ V_0 = 5000,\ \ \ n = 3`

`3635`	`= 5000 – D xx 3`
`3D`	`= 1365`
`D`	`= $455`

`:.\ text(Rate of depreciation per year)`

`= 455/5000 xx 100`

`= 9.1 text(%)`

ii.	`text(Income per year)`	`= 52 xx 800`
		`= $41\ 600`

`text(Taxable income)`	`=\ text(Income – Deductions)`
	`= 41\ 600 – 455`
	`= $41\ 145`

iii.	`text(Tax payable)`	`= 4500 + 0.3(41\ 145 – 28\ 000)`
		`= 4500 + 3943.50`
		`= $8443.50`

Measurement, STD2 M6 2007 HSC 26a

The diagram shows information about the locations of towns `A`, `B` and `Q`.

It takes Elina 2 hours and 48 minutes to walk directly from Town `A` to Town `Q`.

Calculate her walking speed correct to the nearest km/h. (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
Elina decides, instead, to walk to Town `B` from Town `A` and then to Town `Q`.

Find the distance from Town `A` to Town `B`. Give your answer to the nearest km. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---
Calculate the bearing of Town `Q` from Town `B`. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

`5\ text(km/hr)\ text{(nearest km/hr)}`
`18\ text(km)\ text{(nearest km)}`
`236^@`

Show Worked Solution

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^@`

GEOMETRY, FUR1 2011 VCAA 3 MC

The radius of a circle is 6.5 centimetres.

A square has the same area as this circle.

The length of each side of the square, in centimetres, is closest to

A. `6.4`

B. `10.2`

C. `11.5`

D. `23.0`

E. `33.2`

Show Answers Only

`C`

Show Worked Solution

`text(Area of circle)`	`= pi r^2`
	`= pi xx 6.5^2`
	`= 132.73…\ text(cm²)`

`text(Area of square)`	` = l^2`
` l^2`	` = 132.73…`
`:. l`	` = 11.52…\ text(cm)`

`=> C`

GEOMETRY, FUR1 2011 VCAA 2 MC

The point `Q` on building `B` is visible from the point `P` on building `A`, as shown in the diagram above.

Building `A` is 16 metres taller than building `B`.

The horizontal distance between point `P` and point `Q` is 23 metres.

The angle of depression of point `Q` from point `P` is closest to

A. `35°`

B. `41°`

C. `44°`

D. `46°`

E. `55°`

Show Answers Only

`A`

Show Worked Solution

`tan theta`	`= 16/23`
`theta`	`= 34.8^@`

`:.\ text(Angle of depression from)\ P\ text(to)\ Q = 34.8^@`

`text{(alternate,}\ PS\ text(||)\ RQ text{)}`

`=> A`

GEOMETRY, FUR1 2011 VCAA 5 MC

A right pyramid, shown below, has a rectangular base with length 4 m and width 3 m.

The height of the pyramid is 2 m.

The angle `VCO` that the sloping edge `VC` makes with the base of the pyramid, to the nearest degree, is

A. `22°`

B. `27°`

C. `34°`

D. `39°`

E. `45°`

Show Answers Only

`D`

Show Worked Solution

`text(Pyramid’s base)`

`text(Using Pythagoras,)`

`(AC)^2`	`= 4^2 + 3^2`
	`= 25`
`AC`	`= 5`
`:.\ OC`	`= 2.5`

`text(In)\ Delta VOC,`

`tan\ theta`	`= 2/2.5`
`theta`	`= 38.65…^@`

`=> D`

GEOMETRY, FUR1 2013 VCAA 7 MC

A greenhouse is built in the shape of a trapezoidal prism, as shown in the diagram above.

The cross-section of the greenhouse (shaded) is an isosceles trapezium. The parallel sides of this trapezium are 4 m and 10 m respectively. The two equal sides are each 5 m.

The length of the greenhouse is 12 m.

The five exterior surfaces of the greenhouse, not including the base, are made of glass.

The total area, in m², of the glass surfaces of the greenhouse is

A. `196`

B. `212`

C. `224`

D. `344`

E. `672`

Show Answers Only

`C`

Show Worked Solution

`text(Area of trapezoid)\ =\ 1/2\ text(h)\ (a+b)`

`text(Using Pythagoras,)`

`h^2 + 3^2`	`= 5^2`
`h^2`	`= 16`
`h`	`= 4`

`∴\ text{Area (trapezoid)}= 1/2 xx 4\ (10 + 4)= 28\ text(m²)`

`text(Area of side)`	`= 12 xx 5`	`= 60\ text(m²)`
`text(Area of roof)`	`= 4 xx 12`	`= 48\ text(m²)`

`∴\ text(Total area of glass surface)`

`= 2 xx (28 + 60) + 48`

`= 224\ text(m²)`

`=> C`

GEOMETRY, FUR1 2013 VCAA 6 MC

In triangle `MNR`, point `P` lies on side `MR` and point `Q` lies on side `NR`.

The lines `PQ` and `MN` are parallel.

The length of `RQ` is 4 cm, the length of `QN` is 6 cm and the length of `PQ` is 5 cm.

The length of `MN`, in cm, is equal to

A. `7.5`

B. `8.3`

C. `12.0`

D. `12.5`

E. `15.0`

Show Answers Only

`D`

Show Worked Solution

`/_ PRQ\ \ text(is common)`

`/_ RPQ = /_ RMN\ \ (text{corresponding}\ PQ\ text{||}\ MN)`

`:. Delta RPQ\ text(|||)\ Delta RMN\ text{(equiangular)}`

`∴ (MN)/(RN)`	`= (PQ)/(RQ)`	`\ \ \ \ text{(corresponding sides of}` `\ \ \ \ \ text{similar triangles)}`
`(MN)/10`	`= 5/4`
`:.MN`	`= 12.5\ text(cm)`

`=> D`

GEOMETRY, FUR1 2013 VCAA 5 MC

The scale used on a map is `1:50\ 000`.

On this map, a distance of 4 km would be represented by

A. `2.0\ text(cm)`

B. `5.0\ text(cm)`

C. `8.0\ text(cm)`

D. `12.5\ text(cm)`

E. `20.0\ text(cm)`

Show Answers Only

`C`

Show Worked Solution

`text(Map scale is)\ 1:50\ 000`

STRATEGY: Converting 4 km into metres makes the units of calculation much more manageable here.

`∴\ text(On the map, 4 km)`

`= 4000/(50\ 000)\ text(m)`

`= 0.08\ text(m)`

`= 8\ text(cm)`

`=> C`

GEOMETRY, FUR1 2008 VCAA 2-3 MC

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`

Show Answers Only

`text(Part 1:)\ D`

`text(Part 2:)\ D`

Show Worked Solution

`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`

♦ Mean mark (Part 1) 43%.
MARKER’S COMMENT: The low mean mark for this question flagged a fundamental lack of understanding in this area.

`= 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`

Statistics, STD2 S5 2007 HSC 25d

The results of two class tests are normally distributed. The means and standard deviations of the tests are displayed in the table.

Stuart scored 63 in Test 1 and 62 in Test 2. He thinks that he has performed better in Test 1. Do you agree? Justify your answer using appropriate calculations. (2 marks)

--- 5 WORK AREA LINES (style=lined) ---
If 150 students sat for Test 2, how many students would you expect to have scored less than 64? (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`126`

Show Worked Solution

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`

Probability, STD2 S2 2007 HSC 25a

Give an example of an event that has a probability of exactly `3/4`. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(Choosing a red ball out of a bag that)`

`text(contains 3 red balls and 1 green ball.)`

`text {(} text(An infinite amount of examples are)`

`text(possible) text{).}`

Show Worked Solution

`text(Choosing a red ball out of a bag that contains)`

`text(3 red balls and 1 green ball.)`

`text {(} text(An infinite amount of examples are)`

`text(possible) text{).}`

Statistics, STD2 S1 2007 HSC 24d

Barry constructed a back-to-back stem-and-leaf plot to compare the ages of his students.

Write a brief statement that compares the distribution of the ages of males and females from this set of data. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
What is the mode of this set of data? (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
Liam decided to use a grouped frequency distribution table to calculate the mean age of the students at Barry’s Ballroom Dancing Studio.

For the age group 30 - 39 years, what is the value of the product of the class centre and the frequency? (2 marks)

--- 3 WORK AREA LINES (style=lined) ---
Liam correctly calculated the mean from the grouped frequency distribution table to be 39.5.

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)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(More males attend than females and a higher proportion)`
`text(of those are younger males, with the distribution being)`
`text(positively skewed. Female attendees are generally older)`
`text(and have a negatively skewed distribution.)`
`text(Mode) = 64\ \ \ text{(4 times)}`
`172.5`
`text(The difference in the answers is due to the class)`
`text(centres used in group frequency tables distorting)`
`text(the mean value from the exact data.)`

Show Worked Solution

i.	`text(More males attend than females and a higher proportion)`
	`text(of those are younger males, with the distribution being)`
	`text(positively skewed. Female attendees are generally older)`
	`text(and have a negatively skewed distribution.)`

ii.

`text(Mode) = 64\ \ \ text{(4 times)}`

iii.	`text(Class centre)`	`= (30 + 39)/2`
		`= 34.5`
	`text(Frequency) = 5`

`:.\ text(Class centre) xx text(frequency)`

`= 34.5 xx 5`

`= 172.5`

iv.	`text(The difference in the answers is due to the class)`
	`text(centres used in group frequency tables distorting)`
	`text(the mean value from the exact data.)`

Algebra, STD2 A2 2007 HSC 24c

Sandy travels to Europe via the USA. She uses this graph to calculate her currency conversions.

After leaving the USA she has US$150 to add to the A$600 that she plans to spend in Europe.

She converts all of her money to euros.How many euros does she have to spend in Europe? (3 marks)

--- 6 WORK AREA LINES (style=lined) ---
If the value of the euro falls in comparison to the Australian dollar, what will be the effect on the gradient of the line used to convert Australian dollars to euros? (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

`480\ €`
`text(If the value of the euro falls against the)`
`text(A$, then 1 A$ will buy more euros than)`
`text(before and the gradient used to convert)`
`text{the currencies will steepen (increase).}`

Show Worked Solution

i. `text(From graph:)`

`75\ text(US$)`	`=\ text(100 A$)`
`=> 150\ text(US$)`	`=\ text(200 A$)`

`:.\ text(Sandy has a total of 800 A$)`

`text(Converting A$ to €:)`

`text(100 A$)`	`= 60\ €`
`:.\ text(800 A$)`	`= 8 xx 60`
	`= 480\ €`

ii. `text(If the value of the euro falls against the)`

`text(A$, then 1 A$ will buy more euros than)`

`text(before and the gradient used to convert)`

`text{the currencies will steepen (increase).}`

Statistics, STD2 S1 2007 HSC 24a

Consider the following set of scores:

`3, \ 5, \ 5, \ 6, \ 8, \ 8, \ 9, \ 10, \ 10, \ 50.`

Calculate the mean of the set of scores. (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
What is the effect on the mean and on the median of removing the outlier? (2 marks)

--- 5 WORK AREA LINES (style=lined) ---

Show Answers Only

`11.4`
`text{If the outlier (50) is removed, the mean}`

`text(would become lower.)`
`text(Median will NOT change.)`

Show Worked Solution

i. `text(Total of scores)`

`= 3 + 5 + 5 + 6 + 8 + 8 + 9 + 10 + 10 +50`

`= 114`

`:.\ text(Mean) = 114/10 = 11.4`

ii. `text(Mean)`

`text{If the outlier (50) is removed, the mean}`

`text(would become lower.)`

`text(Median)`

`text(The current median (10 data points))`

`= text(5th + 6th)/2 = (8 + 8)/2 = 8`

`text(The new median (9 data points))`

`=\ text(5th value)`

`= 8`

`:.\ text(Median will NOT change.)`

Measurement, STD2 M7 2007 HSC 23c

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.

How many crocodiles were captured in total during the second stage of the study? (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
Calculate the estimate for the total population of crocodiles in this area. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(45 crocodiles)`
`text(60 crocodiles)`

Show Worked Solution

i.	`text(Let)\ C_1`	`=\ text(crocodiles captured in stage 1)`
	`C_2`	`=\ text(crocodiles captured in stage 2)`

`C_1`	`=\ text(40%)\ xx C_2`
`18`	`=\ text(40%)\ xx C_2`
`:.\ C_2`	`= 18/0.4 = 45\ text(crocodiles)`

COMMENT: Std2 sample exam questions from NESA included capture/recapture as examinable content within M7 Rates and Ratios.

ii.

`text(Capture) = 24/text(Population)`

`text(Recapture) = 18/45`

`24/text(Population)`	`= 18/45`
`:.\ text(Population)`	`= (24 xx 45)/18`
	`= 60`

GEOMETRY, FUR1 2009 VCAA 6-7 MC

`ABCD` is a sloping rectangular roof above a horizontal rectangular ceiling, `TCDR`.

`AB`	`= DC`	`= 12\ text(metres)`
`RD`	`= TC`	`= 3.8\ text(metres)`
`AR`	`= BT`	`= 1.5\ text(metres)`

Part 1

The angle of depression of `D` from `A` is closest to

A. `21.5^@`

B. `23.3^@`

C. `66.7^@`

D. `68.5^@`

E. `111.5^@`

Part 2

The angle `ACR` is closest to

A. `6.80^@`

B. `6.84^@`

C. `7.13^@`

D. `18.80^@`

E. `21.54^@`

Show Answers Only

`text(Part 1:)\ A`

`text(Part 2:)\ A`

Show Worked Solution

`text(Part 1)`

`text(Let)\ theta =`

`text(Angle of Depression of)\ D\ text(from)\ A`

`/_ADR`	`= theta\ text( (alternate,)\ AB \ text (\|\|)\ RD)`
`tan theta`	`= 1.5 / 3.8`
`theta`	`= 21.54…^@`
`=> A`

`text(Part 2)`

`text(Find)\ x,`

`text(Using Pythagoras in)\ Delta RCD,`

`x^2`	`= 3.8^2 + 12^2`
	`= 158.44`
`x`	`= 12.587…\ text(m)`

`text(In)\ Delta ACR,`

`tan /_ ACR`	`= 1.5 / (12.587…)`
`/_ ACR`	`= 6.79…^@`

`=> A`

Measurement, STD2 M1 2007 HSC 23b

A cylindrical water tank, of height 2 m, is placed in the ground at a school.

The radius of the tank is 3.78 metres. The hole is 2 metres deep. When the tank is placed in the hole there is a gap of 1 metre all the way around the side of the tank.

When digging the hole for the water tank, what volume of soil was removed? Give your answer to the nearest cubic metre. (3 marks)

--- 6 WORK AREA LINES (style=lined) ---
Sprinklers are used to water the school oval at a rate of 7500 litres per hour.

The water tank holds 90 000 litres when full.

For how many hours can the sprinklers be used before a full tank is emptied? (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
Water is to be collected in the tank from the roof of the school hall, which has an area of 400 m².

During a storm, 20 mm of rain falls on the roof and is collected in the tank.

How many litres of water were collected? (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`144\ text(m³)\ \ text{(nearest m³)}`
`text(12 hours)`
`8000\ text(litres)`

Show Worked Solution

i. `V = pi r^2 h\ \ \ \ text(where)`

`h = 2\ text(and)\ r = 4.78\ text(m)`

`:.\ V`	`= pi xx 4.78^2 xx 2`
	`= 143.56…`
	`= 144\ text(m³)\ \ text{(nearest m³)}`

ii. `text(Total water) = 90\ 000\ text(litres)`

`text(Usage) = 7500\ \ text(litres/hr)`

`:.\ text(Hours before it is empty)`

`= (90\ 000)/7500`

`= 12\ text(hours)`

iii. `text(Water collected)`

`= 400 xx 0.020`

`= 8\ text(m²)`

`= 8000\ text(litres)`

GEOMETRY, FUR1 2009 VCAA 3 MC

The locations of three towns, `Q`, `R` and `T`, are shown in the diagram above.

Town `T` is due south of town `R`.

The angle `TRQ` is `48^@`.

The bearing of town `R` from town `Q` is

A. `048^@`

B. `132^@`

C. `138^@`

D. `228^@`

E. `312^@`

Show Answers Only

`E`

Show Worked Solution

`∠RQA = 48^@ text{(} text(alternate)\ RT\ text(||)\ QA text{})`

`∴ text(Bearing of)\ R\ text(from)\ Q`

`= 360 – 48`

`= 312^@`

`=> E`

GEOMETRY, FUR1 2009 VCAA 5 MC

A right triangular prism has a volume of 160 cm³.

A second right triangular prism is made with the same width, twice the height and three times the length of the prism shown.

The volume of the second prism (in cm³) is

A. `320`

B. `640`

C. `960`

D. `1280`

E. `1920`

Show Answers Only

`C`

Show Worked Solution

`text(Volume of existing prism)\ (V)`

	`= 1/2 xx b xx h xx l`
	`= 160 \ text(cm³)`

`text(Volume of new prism)\ (V_1)`

	`= 1/2 xx b xx 2h xx 3l`
	`= 6 xx 1/2 xx b xx h xx l`
	`= 6 xx V`
	`= 6 xx 160`
	`= 960 \ text(cm³)`

`=> C`

Financial Maths, STD2 F1 2007 HSC 23a

Lilly and Rose each have money to invest and choose different investment accounts.

The graph shows the values of their investments over time.

How much was Rose’s original investment? (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
At the end of 6 years, which investment will be worth the most and by how much? (2 marks)

--- 4 WORK AREA LINES (style=lined) ---
Lilly’s investment will reach a value of $20 000 first.
How much longer will it take Rose’s investment to reach a value of $20 000? (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

`$5000`
`text(Rose’s is worth $2000 more.)`
`text(It takes Lilly 14 years to reach $20 000 and it takes)`

`text{Rose 1 year longer (15 years) to reach the same value}`

Show Worked Solution

i. `$5000\ text{(} y text(-intercept) text{)}`

ii. `text(After 6 years,)`

`text(Lilly’s investment)`	`= $9000`
`text(Rose’s investment)`	`= $11\ 000`
`:.\ text(Rose’s is worth $2000 more.)`

iii. `text(It takes Lilly 14 years to reach $20 000 and it)`

`text{takes Rose 1 year longer (15 years) to reach the}`

`text(same value.)`

CORE*, FUR1 2009 VCAA 7 MC

The difference equation `u_(n + 1) = 4u_n - 2` generates a sequence.

If `u_2 = 2`, then `u_4` will be equal to

A. 4

B. 8

C. 22

D. 40

E. 42

Show Answers Only

`C`

Show Worked Solution

`u_(n+1)`	`= 4u_n – 2`
`∴ u_3`	`= 4u_2 – 2`
	`= 4 xx 2 – 2\ \ text{(given}\ u_2 = 2 text{)}`
	`= 6`
`∴ u_4`	`= 4u_3 – 2`
	`= 4 xx 6 – 2`
	`= 22`

`=> C`

GEOMETRY, FUR1 2014 VCAA 8 MC

The distance, `AC`, across a small lake can be calculated using the measurements shown in the diagram below.

In this diagram, `BCA` and `BDE` are right-angled triangles, where `CB = 40.4\ text(m), BD = 10\ text(m)` and `BE = 12\ text(m).`

The distance between the points `A` and `C`, in metres, is closest to

A. `22.4`

B. `26.8`

C. `33.6`

D. `48.5`

E. `177.8`

Show Answers Only

`B`

Show Worked Solution

`/_ ABC = /_ DBE\ \ text{(vertically opposite angles)}`

`/_ ACB = /_ BDE = 90°\ \ text{(given)}`

`∴ Delta ABC\ \ text(|||)\ \ Delta EBD\ \ \ \ text{(equiangular)}`

`∴ (AB)/40.4`	`= 12/10`	`\ \ \ \ \ text{(corresponding sides}` `\ \ \ \ \ \ text{of similar triangles)}`
`AB`	`= (12 xx 40.4)/10`
	`= 48.48\ \ text(m)`

`text(Using Pythagoras in)\ Delta ABC:`

`48.48^2`	`= x^2 + 40.4^2`
`x^2`	`= 718.1504`
`x`	`= 26.79\ text(m)`

`=>B`

CORE*, FUR1 2011 VCAA 7 MC

Let `P_2011` be the number of pairs of shoes that Sienna owns at the end of 2011.

At the beginning of 2012, Sienna plans to throw out the oldest 10% of pairs of shoes that she owned in 2011.

During 2012 she plans to buy 15 new pairs of shoes to add to her collection.

Let `P_2012` be the number of pairs of shoes that Sienna owns at the end of 2012.

A rule that enables `P_2012` to be determined from `P_2011` is

A. `P_2012 = 1.1 P_2011 + 15`

B. `P_2012 = 1.1 (P_2011 + 15)`

C. `P_2012 = 0.1 P_2011 + 15`

D. `P_2012 = 0.9 (P_2011 + 15)`

E. `P_2012 = 0.9 P_2011 + 15`

Show Answers Only

`E`

Show Worked Solution

`text(By throwing out 10%, Sienna keeps 90% of her)`

`text{her 2011 shoes (or 0.9} \ P_2011 text{) and then adds 15.}`

`:. P_(2012) = 0.9\ P_2011 + 15`

`=> E`

PATTERNS, FUR1 2011 VCAA 4 MC

The number of bees in a colony was recorded for three months and the results are displayed in the table below.

If this pattern of increase continues, which one of the following statements is not true.

A. There will be nine times as many bees in the colony in month 5 than in month 3.

B. In month 4, the number of bees will equal 270.

C. In month 6, the number of bees will equal 7290.

D. In month 8, the number of bees will exceed 20 000.

E. In month 10, the number of bees will be under 200 000.

Show Answers Only

`C`

Show Worked Solution

`text(Sequence is 10, 30, 90, …)`

`text(GP where)\ \ \ a`	` = 10, and`
`r`	` = t_2/t_1=30 / 10 = 3`

`text(In A,)\ \ T_5 = 10 xx 3^4 and T_3 = 10 xx 3^2`

`:. T_5 = T_3 xx 3^2\ \ text{(True)}`

`text(In B,)\ \ T_4 = 10 xx 3^3 = 270\ \ text{(True)}`

`text(In C,)\ \ T_6 = 10 xx 3^5 = 2430\ \ text{(NOT true)}`

`text(In D,)\ \ T_8 = 10 xx 3^7 = 21\ 870\ \ text{(True)}`

`text(In E,)\ \ T_10 = 10 xx 3^9 = 196\ 830\ \ text{(True)}`

`=> C`

CORE*, FUR1 2011 VCAA 3 MC

The graph above shows the first five terms of a sequence.

Let `A_n` be the `n`th term of the sequence.

A difference equation that generates the terms of this sequence is

A. `A_(n+1) = 2A_n - 2`	`\ \ \ \ text(where) \ \ \ \ `	`A_1 =8`
B. `A_(n+1) = 3A_n`	`\ \ \ \ text(where) \ \ \ \ `	`A_1 =8`
C. `A_(n+1) = -2A_n`	`\ \ \ \ text(where) \ \ \ \ \ \ \ \ `	`A_1 =8`
D. `A_(n+1) = -1 / 2 A_n`	`\ \ \ \ text(where) \ \ \ \ `	`A_1 =8`
E. `A_(n+1) = -A_n - 1`	`\ \ \ \ text(where) \ \ \ \ `	`A_1 =8`

Show Answers Only

`D`

Show Worked Solution

`A_1 = 8, \ \ A_2 = –4, \ \ A_3 = 2\ \ text{(from graph)}`

`text(This sequence is geometric where)`

`r=t_2/t_1=- 1/2`

`:.\ text(Difference equation is)\ \ \ A_(n+1) = -1/2 A_n`

`=> D`

PATTERNS, FUR1 2011 VCAA 2 MC

The first three terms of an arithmetic sequence are –3, –7, –11 . . .

An expression for the `n`th term of this sequence, `t_n`, is

A. `t_n = 1 - 4n`

B. `t_n = 1 - 8n`

C. `t_n = -3 - 4n`

D. `t_n = -3 + 4n`

E. `t_n = -7 + 4n`

Show Answers Only

`A`

Show Worked Solution

`text(Sequence is –3, –7, –11, …)`

`text(AP where)\ \ \ a`	`= –3, and`
`d`	`= –7 – (–3) = –4`

`t_n`	` = a + (n – 1) d`
	` = –3 + (n – 1) (–4)`
	` = –3 – 4n + 4`
	` = 1 – 4n`

`=> A`

GEOMETRY, FUR1 2012 VCAA 7 MC

`PQR` is a triangle with side lengths `x, 10` and `y`, as shown below.

In this triangle, angle `RPQ = 37°` and angle `QRP = 42°.`

Which one of the following expressions is correct for triangle `PQR`?

A. `x = 10/(sin 37°)`

B. `y = 10/(tan 37°)`

C. `x = 10 × (sin 42°)/(sin 37°)`

D. `y = 10 × (sin 37°)/(sin 101°)`

E. `10^2 = x^2 + y^2 - 2xy cos 42°`

Show Answers Only

`C`

Show Worked Solution

`∠ PQR`	`= 180 – (37 + 42)\ \ \ \ text {(angle sum of}\ ΔPQR text{)}`
	`= 101°`

`text (Using the sine rule:)`

`y/sin 101= x/sin 42= 10/sin 37`

`:. x= 10 × (sin 42)/(sin 37)`

`rArr C`

GEOMETRY, FUR1 2012 VCAA 3 MC

A rectangular sheet of cardboard has length 50 cm and width 20 cm.

This sheet of cardboard is made into an open-ended cylinder by joining the two shorter sides, with no overlap.

This is shown in the diagram below.

The radius of this cylinder, in cm, is closest to

A. `6.4`

B. `8.0`

C. `15.6`

D. `15.9`

E. `17.8`

Show Answers Only

`B`

Show Worked Solution

`text (Circumference) = 50\ text(cm)`

`2pi r`	`= 50`
`:. r`	`= 50/(2pi)`
	`= 7.95…\ text(cm)`

`rArr B`

Measurement, STD2 M2 2007 HSC 20 MC

Kim lives in Perth. He wants to watch an ice hockey game being played in Toronto starting at 10.00 pm on Wednesday.

Toronto is 13 hours behind Perth.

What is the time in Perth when the game starts?

9.00 am on Wednesday
7.40 pm on Wednesday
12.20 am on Thursday
11.00 am on Thursday

Show Answers Only

`D`

Show Worked Solution

`:.\ text(Time in Perth)`

`=\ text{10 pm (Wed) + 13 hours}`

`=\ text(11 am on Thursday)`

`=> D`

Probability, STD2 S2 2007 HSC 16 MC

Leanne copied a two-way table into her book.

Leanne made an error in copying one of the values in the shaded section of the table.

Which value has been incorrectly copied?

The number of males in full-time work
The number of males in part-time work
The number of females in full-time work
The number of females in part-time work

Show Answers Only

`D`

Show Worked Solution

`text(By checking row and column total, the number)`

`text(of females part-time work is incorrect)`

`=> D`

Algebra, 2UG 2007 HSC 14 MC

Which expression is equivalent to `3x^2 (x + 8) + x^2`?

(A) `3x^3 + x^2 + 8`

(B) `3x^3 + 25x^2`

(C) `4x^3 + 32x^2`

(D) `24x^3 + x^2`

Show Answers Only

`B`

Show Worked Solution

`3x^2 (x + 8) + x^2`

`= 3x^3 + 24x^2 + x^2`

`= 3x^3 + 25x^2`

`=> B`

Probability, 2UG 2007 HSC 13 MC

The positions of President, Secretary and Treasurer of a club are to be chosen from a committee of `5` people.

In how many ways can the three positions be chosen?

(A) `3`

(B) `10`

(C) `60`

(D) `125`

Show Answers Only

`C`

Show Worked Solution

`text(3 positions and order matters)`

`:.\ text(# Combinations)`	`= 5 xx 4 xx 3`
	`= 60`

`=> C`

Financial Maths, STD2 F4 2007 HSC 12 MC

The value of a car is depreciated using the declining balance method.

Which graph best illustrates the value of the car over time?

Show Answers Only

`C`

Show Worked Solution

`text(Declining balance depreciates quicker in absolute)`

`text(terms in the early stages, and slower as time goes)`

`text(on and the balance owing decreases.)`

`=> C`

Measurement, 2UG 2007 HSC 11 MC

`P` and `Q` are points on the circumference of a circle with centre `O` and radius `3` cm.

What is the length of the arc `PQ`, in centimetres, correct to three significant figures?

(A) `1.57`

(B) `3.14`

(C) `4.71`

(D) `18.8`

Show Answers Only

`B`

Show Worked Solution

`text(Length of Arc)\ PQ`	`= 60/360 xx 2 pi r`
	`= 1/6 xx 2pi (3)`
	`= pi`
	`= 3.14\ text(cm)`

`=> B`

Probability, STD2 S2 2007 HSC 10 MC

Each time she throws a dart, the probability that Mary hits the dartboard is `2/7`.

She throws two darts, one after the other.

What is the probability that she hits the dartboard with both darts?

`1/21`
`4/49`
`2/7`
`4/7`

Show Answers Only

`B`

Show Worked Solution

`P text{(hits)} = 2/7`

`P text{(hits twice)}`	`= 2/7 xx 2/7`
	`= 4/49`

`=> B`

Statistics, STD2 S4 2007 HSC 9 MC

Which of the following would be most likely to have a positive correlation?

The population of a town and the number of schools in that town
The price of petrol per litre and the number of litres of petrol sold
The hours training for a marathon and the time taken to complete the marathon
The number of dogs per household and the number of televisions per household

Show Answers Only

$A$

Show Worked Solution

$\text{Positive correlation means that as one variable increases,}$

$\text{the other tends to increase also.}$

$\Rightarrow A$