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Financial Maths, 2ADV M1 2006 HSC 1f

Find the limiting sum of the geometric series  `13/5 + 13/25 + 13/125 + …`  (2 marks)

Show Answers Only

`13/4`

Show Worked Solution

`13/5 + 13/25 + 13/125`

`=>\ text(GP where)\ \ a=13/5,\ text(and)`

`r = T_2/T_1 = 13/25 ÷ 13/5 = 1/5`

`text(S)text(ince)\ |\ r\ | < 1`

`S_oo` `= a/(1-r)`
  `= (13/5)/(1 – 1/5)`
  `= 13/5 xx 5/4`
  `= 13/4`

Data, 2UG 2005 HSC 27a

The area graph shows sales figures for Shoey’s shoe store.

2UG-2005-27a

  1. Approximately how many school shoes were sold in January?   (1 mark)
  2. For which month does the graph indicate that the same number of school shoes and business shoes was sold?   (1 mark)
  4. Identify ONE trend in this graph, and suggest a valid reason for this trend.   (2 marks)
Show Answers Only
  1. `15\ 000`
  2.  `text(April)`
  3. `text(Possible trends include)`
  4. `text(- Boot sales peak in the Winter months of Jun-Aug)`
  5. `\ \ text(because of the colder weather.)`
    `text(- School shoe sales peak in January as children prepare)`
    `\ \ text(to return to school for the new year.)`
Show Worked Solution

(i)   `text(School shoes sold)`

`≈ 18\ 000 − 3000`

`≈ 15\ 000`

 

(ii)  `text(April)`

 

(iii) `text(Possible trends include)`

`text(- Boot sales peak in the Winter months of Jun-Aug)`

`\ \ text(because of the colder weather.)`

`text(- School shoe sales peak in January as children prepare)`

`\ \ text(to return to school for the new year.)`

Measurement, 2UG 2004 HSC 26b

The location of Sorong is `text(1°S 131°E)` and the location of Darwin is `text(12°S 131°E)`.

  1. What is the difference in the latitudes of Sorong and Darwin?  (1 mark)
  2. The radius of Earth is approximately `text(6400 km.)`

  3. Show that the great circle distance between Sorong and Darwin is approximately `text(1200 km)`.  (2 marks)

 

Show Answers Only
  1. `11°`
  2. `text(Proof)\ \ text{(See Worked Solutions)}`
Show Worked Solution

(i)  `text(Difference in latitudes)`

`= 12^@ – 1^@`

`= 11^@`

 

(ii)   `text(S)text(ince longitude is the same,)`

`text(Distance between Sarang and Darwin)`

`= 11/360 xx 2 pi r`

`= 11/360 xx 2 xx pi xx 6400`

`= 1228.7…\ text(km)`

`~~ 1200\ text(km … as required.)`

Data, 2UG 2006 HSC 23d

The graph shows the amounts charged by Company `A` and Company `B` to deliver parcels of various weights.

2UG-2006-23d

  1. How much does Company `A` charge to deliver a `3` kg parcel?  (1 mark)
  2. Give an example of the weight of a parcel for which both Company `A` and Company `B` charge the same amount.  (1 mark)
  4. For what weight(s) is it cheaper to use Company `A`?  (2 marks)
  5. What is the rate per kilogram charged by Company `B` for parcels up to `8` kg?  (1 mark)
Show Answers Only
  1. `$6`
  2. `text(4 kg or 7 kg)`
  3. `x > 7`
  4. `text($1.50 per kg)`
Show Worked Solution

(i)   `$6`

 

(ii)  `text(4 kg or 7 kg)`

 

(iii)  `text(Let)\ x = text(weights where Company)\ A`

`text{is cheaper (kg).}`

`4 < x <= 6`

`x > 7`

 

(iv)   `text(Company)\ B\ text(charges $12 for 8 kg)`

`:.\ text(Rate)` `= 12/8`
  `= $1.50\  text(per kg)`

Data, 2UG 2006 HSC 23b

This radar chart was used to display the average daily temperatures each month for two different towns.

2UG-2006-23b

  1. What is the average daily temperature of Town `B` for April?  (1 mark)
  2. In which month do the average daily temperatures of the two towns have the greatest difference?  (1 mark)
  4. In which months is the average daily temperature in Town `B` higher than in Town `A`?  (1 mark)
Show Answers Only
  1. `18.5^@\ \ text{(approx)}`
  2. `text(February)`
  3. `text(May, June, July, August and September)`
Show Worked Solution

(i)   `18.5^@\ \ \ text{(approx)}`

 

(ii)   `text(February)`

 

(iii)  `text(May, June, July, August and September)`

Measurement, STD2 M6 2005 HSC 25b

2UG-2005-25b

  1. Use Pythagoras’ theorem to show that `ΔABC` is a right-angled triangle.  (1 mark)

  2. Calculate the size of `∠ABC` to the nearest minute.  (2 marks)

Show Answers Only
  1. `text(Proof)`
  2. `67°23^{′}`
Show Worked Solution

i.   `ΔABC\ text(is right-angled if)\ \ a^2 + b^2 = c^2`

`a^2 + b^2` `= 5^2 + 12^2`
  `= 169`
  `= 13^2`
  `= c^2…\ text(as required.)`

MARKER’S COMMENT: Know your calculator process for producing an angle in minutes/seconds. Note >30 “seconds” rounds up to the higher “minute”.

 
ii.  `sin ∠ABC = 12/13`

`:.∠ABC` `= 67.38…°`
  `=67°22^{′}48^{″}`
  `= 67°23^{′}\ \ \ text{(nearest minute)}`

Financial Maths, STD2 F1 2005 HSC 25a

Reece is preparing his annual budget for 2006.

His expected income is:

• $90 every week as a swimming coach
• Interest earned from an investment of $5000 at a rate of 4% per annum.

His planned expenses are:

• $30 every week on transport
• $12 every week on lunches
• $48 every month on entertainment.

Reece will save his remaining income. He uses the spreadsheet below for his budget.
 

2UG-2005-25a
 

  1. Determine the values of `X`, `Y` and `Z`. (Assume there are exactly 52 weeks in a year.)  (3 marks)

At the beginning of 2006, Reece starts saving.

  1. Will Reece have saved enough money during 2006 for a deposit of $2100 on a car if he keeps to his budget? Justify your answer with suitable calculations.  (2 marks)

Show Answers Only
  1. `X =$200, Y = $1560\ text(and) \ Z = $576`
  2. `text(Reece will have saved have saved enough for a $2100 deposit.)`
Show Worked Solution

a.   `text(Interest on Investment) = X`

`X` `= 5000 xx 4 text(%)`
  `= $200`

 
`text(Transport =)\ Y`

`Y` `= 52 × 30`
  `= $1560`

 
`text(Entertainment =)\ Z`

`Z` `= 48 × 12`
  `= $576`

 

b.    `text(Total Income)` `= 4680 + 200`
    `= $4880`
`text(Total Expenses)` `= 1560 + 624 + 576`
  `= $2760`
`text(Savings)` `= 4880 − 2760`
  `= $2120`

 
`:.\ text(Reece will have saved enough for)`

`text(a $2100 deposit.)` 

Probability, STD2 S2 2005 HSC 23c

Moheb owns five red and seven blue ties. He chooses a tie at random for himself and puts it on. He then chooses another tie at random, from the remaining ties, and gives it to his brother.

  1. What is the probability that Moheb chooses a red tie for himself?  (1 mark)

Copy the tree diagram into your writing booklet.
 

2UG-2005-23c
 

  1. Complete your tree diagram by writing the correct probability on each branch.  (2 marks)
  2. Calculate the probability that both of the ties are the same colour.  (2 marks)

Show Answers Only
  1. `5/12`
  2.  
  3. `31/66`
Show Worked Solution
i. `P(R)` `= (#\ text(red ties))/(#\ text(total ties))`
    `= 5/12`

 

ii.  

 

iii. `Ptext((same colour))`

`= P(text(RR)) + P(text(BB))`

`= 5/12 × 4/11\ \ +\ \ 7/12 × 6/11`

`= 20/132 + 42/132`

`= 31/66`

Algebra, STD2 A4 2004 HSC 26a

  1. The number of bacteria in a culture grows from 100 to 114 in one hour.

     

    What is the percentage increase in the number of bacteria?   (1 mark)

  2. The bacteria continue to grow according to the formula  `n = 100(1.14)^t`, where `n` is the number of bacteria after `t` hours.

     

    What is the number of bacteria after 15 hours?   (1 mark)

\begin{array} {|l|c|}
\hline
\rule{0pt}{2.5ex} \text{Time in hours $(t)$} \rule[-1ex]{0pt}{0pt} & \;\; 0 \;\;  &  \;\; 5 \;\;  & \;\; 10 \;\;  & \;\; 15 \;\; \\
\hline
\rule{0pt}{2.5ex} \text{Number of bacteria ( $n$ )} \rule[-1ex]{0pt}{0pt} & \;\; 100 \;\;  &  \;\; 193 \;\;  & \;\; 371 \;\;  & \;\; ? \;\; \\
\hline
\end{array}

  1. Use the values of `n` from  `t = 0`  to  `t = 15`  to draw a graph of  `n = 100(1.14)^t`.

     

    Use about half a page for your graph and mark a scale on each axis.   (4 marks)

  2. Using your graph or otherwise, estimate the time in hours for the number of bacteria to reach 300.   (1 mark)

Show Answers Only
  1. `text(14%)`
  2. `714`
  3. `text(Proof)\ \ text{(See Worked Solutions)}`
  4. `text(8.4 hours)`
Show Worked Solution

i.   `text(Percentage increase)`

COMMENT: Common ADV/STD2 content in new syllabus.

`= (114 -100)/100 xx 100`

`= 14text(%)`

 

ii.  `n = 100(1.14)^t`

`text(When)\ \ t = 15,`

`n` `= 100(1.14)^15`
  `= 713.793\ …`
  `= 714\ \ \ text{(nearest whole)}`

 

iii. 

 

iv.  `text(Using the graph)`

`text(The number of bacteria reaches 300 after)`

`text(approximately 8.4 hours.)`

Data, 2UG 2004 HSC 24a

The following graphs have been constructed from data taken from the Bureau of Meteorology website. The information relates to a town in New South Wales.

The graphs show the mean 3 pm wind speed (in kilometres per hour) for each month of the year and the mean number of days of rain for each month (raindays).

2004 24a

  1. What is the mean 3 pm wind speed for September?  (1 mark)
  2. Which month has the lowest mean 3 pm wind speed?  (1 mark)
  3. In which three-month period does the town have the highest number of raindays?  (1 mark)
  4. Briefly describe the pattern relating wind speed with the number of raindays for this town. Refer to specific months.  (2 marks)

 

Show Answers Only
  1. `text(15 km/h)`
  2. `text(February)`
  3. `text(Jan – Mar)`
  4. `text(The mean number of rain days tends to be higher)`
    `text(when the wind speed is lower and vice versa.)`
    `text(For example, the highest number of mean rain)`
    `text(days is in Feb, which is also the month of the lowest)`
    `text(mean wind speed.)`

 

 

Show Worked Solution

(i)    `text(15 km/h)`

(ii)   `text(February)`

(iii)  `text(Jan – Mar)`

(iv)  `text(The mean number of rain days tends to be higher)`

`text(when the wind speed is lower and vice versa.)`

`text(For example, the highest number of mean rain)`

`text(days is in Feb, which is also the month of the lowest)`

`text(mean wind speed.)`

Measurement, STD2 M1 2005 HSC 23b

A clay brick is made in the shape of a rectangular prism with dimensions as shown.
 

  1. Calculate the volume of the clay brick.  (1 mark)

Three identical cylindrical holes are made through the brick as shown. Each hole has a radius of 1.4 cm.  
 

  1. What is the volume of clay remaining in the brick after the holes have been made? (Give your answer to the nearest cubic centimetre.)  (3 marks)

  2. What percentage of clay is removed by making the holes through the brick? (Give your answer correct to one decimal place.)  (1 mark)

Show Answers Only
  1. `text(1512 cm)^3`
  2. `text{1364 cm}^3`
  3. `text{9.8%}`
Show Worked Solution
a.    `V` `= l × b × h`
    `= 21 × 8 × 9`
    `= 1512\ text(cm)^3`

 

b.  `text(Volume of each hole)`

`= pir^2h`

`= pi × 1.4^2 × 8`

`= 49.260…\ text(cm)^3`

 

`:.\ text(Volume of clay still in brick)`

`= 1512 − (3 × 49.260…)`

`= 1364.219…`

`= 1364\ text{cm}^3\ text{(nearest whole)}`

 

c.  `text(Percentage of clay removed)`

`= ((3 × 49.260…))/1512 × 100`

`= 9.773…`

`= 9.8 text{%   (1 d.p.)}`

Measurement, 2UG 2005 HSC 12 MC

The shaded region represents a block of land bounded on one side by a road.

2UG-2005-12MC

What is the approximate area of the block of land, using Simpson’s rule?

(A)   `680\ text(m²)`

(B)   `760\ text(m²)`

(C)   `840\ text(m²)`

(D)   `1360\ text(m²)`

Show Answers Only

`A`

Show Worked Solution

2UG-2005-12MC Answer

`A` `≈ h/3[y_0 + 4y_1 + y_2]`
  `≈ 20/3 [19 + (4 × 15) + 23]`
  `≈ 20/3 [102]`
  `≈ 680\ text(m²)`

`=>  A`

Probability, STD2 S2 2005 HSC 11 MC

The diagram shows a spinner.
 


 

The arrow is spun and will stop in one of the six sections.

What is the probability that the arrow will stop in a section containing a number greater
than 4?

  1.    `2/5`
  2.    `2/3`
  3.    `1/3`
  4.    `1/2`
Show Answers Only

`D`

Show Worked Solution

`P\ text((number greater than 4))`

`= P(7) + P (9)`

`= 2/6 + 1/6`

`= 1/2`

`=>  D`

Financial Maths, STD2 F4 2004 HSC 25a

Tai uses the declining balance method of depreciation to calculate tax deductions for her business. Tai’s computer is valued at $6500 at the start of the 2003 financial year. The rate of depreciation is 40% per annum.

  1. Calculate the value of her tax deduction for the 2003 financial year.  (1 mark)

  2. What is the value of her computer at the start of the 2006 financial year?  (2 marks)

Show Answers Only
  1. `$2600`
  2. `$1404`
Show Worked Solution
i.  `text(Tax deduction)` `= 40 text(%) xx $6500`
  `= $2600`

 

ii. `text(Using)\  S = V_0(1 – r)^n,`

`text(Value at the start of 2006 FY)`

`= 6500(1 – 0.4)^3`

`= $1404`

Algebra, 2UG 2004 HSC 23b

Kirbee is shopping for computer software. Novirus costs `$115` more than
Funmaths. Let `x` dollars be the cost of Funmaths.

  1. Write an expression involving `x` for the cost of Novirus.  (1 mark)
  2. Novirus and Funmaths together cost `$415`. Write an equation involving
  3. `x` and solve it to find the cost of Funmaths.  (2 marks)
Show Answers Only
  1. `N = x + 115`
  2. `text(Funmaths costs $150)`
Show Worked Solution

(i)   `text(Let)\ \ N = text(cost of Novirus)`

`N = x + 115`

 

(ii)  `N + x = 415\ …\ (1)`

`text(Substitute)\ \ N = x + 115\ text{into (1) above}`

`x + 115 + x` `= 415`
`2x` `= 300`
`x` `= 150`

`:.\ text(Funmaths costs $150)`

Linear Functions, 2UA 2004 HSC 2a

The diagram shows the points  `A(text(−1) , 3)`  and  `B(2, 0)`.

The line  `l`  is drawn perpendicular to the  `x`-axis through the point  `B`.
 

Linear Functions, 2UA 2004 HSC 2a 
 

  1. Calculate the length of the interval  `AB`.   (1 mark)
  2. Find the gradient of the line  `AB`.   (1 mark)
  3. What is the size of the acute angle between the line  `AB`  and the line  `l`?   (1 mark)
  4. Show that the equation of the line  `AB`  is  `x + y − 2 = 0`.    (1 mark)
  5. Copy the diagram into your writing booklet and shade the region defined by  `x + y − 2 <= 0`.   (1 mark)
  6. Write down the equation of the line  `l`.   (1 mark)
  7. The point  `C`  is on the line  `l`  such that  `AC`  is perpendicular to  `AB`. Find the coordinates of  `C`.   (2 marks)

 

Show Answers Only
  1. `3 sqrt2\ text(units)`
  2. `-1`
  3. `45^@`
  4. `text(Proof)\ \ text{(See Worked Solutions)}`
  5. Linear Functions, 2UA 2004 HSC 2a Answer
  6. `x = 2`
  7. `C\ (2, 6)`
Show Worked Solution
(i)    `A(-1,3)\ \ \ \ \ B(2,0)`
`AB` `= sqrt( (x_2 – x_1)^2 + (y_2 – y_1)^2 )`
  `= sqrt( (2+1)^2 + (0-3)^2 )`
  `= sqrt(9+9)`
  `= sqrt 18`
  `= 3 sqrt 2\ text(units)`

 

(ii)   `text(Gradient of)\ AB` `= (y_2 – y_1)/(x_2 – x_1)`
    `= (0 – 3)/(2 + 1)`
    `= – 1`

 

(iii)   `text(S) text(ince Gradient)\ AB = – 1`

`/_ABO = 45^@`

`:.\ text(Angle between)\ AB\ text(and)\ l`

`= 90 – 45`

`= 45^@`

 

(iv)   `text(Equation of)\ AB\ text(has)\ m = -1,\ text(through)\ \ (2,0)`
`y – y_1` `= m(x – x_1)`
`y – 0` `= -1 (x – 2)`
`y` `= -x + 2`

 

`:.\ x + y – 2 = 0\ \ \ …\ text(as required)`

 

(v)

Linear Functions, 2UA 2004 HSC 2a Answer

`text{Origin (0,0) satisfies the inequality}`

`:.\ text(Shaded area is below)\ x + y – 2 = 0`

 

(vi)   `text(Equation of)\ l`
  `x = 2`

 

(vii)   `m_(AC) xx m_(AB)` `= -1\ \ \ (AC⊥AB)`
  `m_(AC) xx -1` `= -1`
  `m_(AC)` `= + 1`

 

`text(Equation of)\ AC\ text(has)\ \ m = 1,\ text(through)\ (-1, 3),`

`y – y_1` `= m(x – x_1)`
`y – 3` `= 1 (x + 1)`
`y` `= x + 4`

 

`=>C\ text(lies on)\ y = x + 4`

`text(When)\ \ x = 2,\ \ y = 6`

`:. C\ (2, 6)`

Functions, EXT1* F1 2004 HSC 1f

Find the values of  `x`  for which  `|\ x + 1\ |<= 5`.  (2 marks)

Show Answers Only

`-6 <= x <= 4`

Show Worked Solution

`text(Solution 1)`

`|\ x + 1\ |<= 5`

`-5` `≤x+1≤5`
`:.-6` `≤x≤4` 

 

`text(Solution 2)`

`|\ x + 1\ |<= 5`

`(x+1)^2` `<= 5^2`
` x^2 + 2x + 1` `<= 25`
 `x^2 + 2x – 24` `<= 0`
`(x + 6)(x – 4)` `<= 0`

Algebra, 2UA 2004 HSC 1f Answer

`:.\ -6 <= x <= 4`

Measurement, STD2 M6 2006 HSC 9 MC

What is the area of this triangle, to the nearest square metre?
 

 

  1. `text(152 m²)`
  2. `text(283 m²)`
  3. `text(328 m²)`
  4. `text(351 m²)`
Show Answers Only

`C`

Show Worked Solution

`text(Using the Sine rule)`

`A` `= 1/2 ab\ sin C`
  `= 1/2 xx 39 xx 47 xx sin 21^@`
  `=\ text(328.44… m²)`

 
`=>  C`

Probability, 2ADV S1 2004 HSC 1e

A packet contains 12 red,  8 green, 7 yellow and 3 black jellybeans.

One jellybean is selected from the packet at random.

What is the probability that the selected jellybean is red or yellow?  (2 marks)

Show Answers Only

`19/30`

Show Worked Solution

`text(12 R,  8 G,  7 Y,  3 B)`

`text(Total jellybeans) = 30`

`P text{(R or Y)}` `=\ text{(# Red + Yellow)}/text(Total jellybeans)`
  `= (12 + 7)/30`
  `= 19/30`

Functions, 2ADV F1 2004 HSC 1c

Solve   `(x-5)/3-(x+1)/4 = 5`.   (2 marks)

Show Answers Only

`83`

Show Worked Solution
`(x-5)/3-(x+1)/4` `= 5`
`12((x-5)/3)-12((x+1)/4)` `= 12 xx 5`
`4x-20-3x-3` `= 60`
`x-23` `= 60`
`:. x` `= 83`

Financial Maths, STD2 F1 2006 HSC 5 MC

A salesman earns $200 per week plus $40 commission for each item he sells.

How many items does he need to sell to earn a total of $2640 in two weeks?

  1. 33
  2. 56
  3. 61
  4. 66
Show Answers Only

`B`

Show Worked Solution

`text(Let items sold) = n`

`text{Wages over 2 weeks}\ (w)`

`= (2 xx 200) + 40n`

`= 400 + 40n`
 

`text(Find)\ n\ text(when)\ w = 2640:`

`2640` `= 400 + 40n`
`40n` `= 2240`
`n` `= 56`

 
`=>  B`

Measurement, STD2 M6 2006 HSC 3 MC

The angle of depression of the base of the tree from the top of the building is 65°. The height of the building is 30 m.

How far away is the base of the tree from the building, correct to one decimal place?
 


 

  1. 12.7 m
  2. 14.0 m
  3. 33.1 m
  4. 64.3 m
Show Answers Only

`B`

Show Worked Solution
 

`text(Let)\ d =\ text(distance from base to tree)`

`tan25^@` `=d/30`  
`:.d` `=30 xx tan25^@`  
  `=13.98…\ text{m}`  

 
`=>  B`

Probability, STD2 S2 2006 HSC 1 MC

The probability of an event occurring is `9/10.`

Which statement best describes the probability of this event occurring?

  1.    The event is likely to occur.
  2.    The event is certain to occur.
  3.    The event is unlikely to occur.
  4.    The event has an even chance of occurring.
Show Answers Only

`A`

Show Worked Solution

`text(The event is highly likely to occur)`

`text(but not certain.)`

`=>  A`

Probability, STD2 S2 2005 HSC 23a

There are 100 tickets sold in a raffle. Justine sold all 100 tickets to five of her friends. The number of tickets she sold to each friend is shown in the table.
 

  1. Justine claims that each of her friends is equally likely to win first prize.

     

    Give a reason why Justine’s statement is NOT correct.   (1 mark)

  2. What is the probability that first prize is NOT won by Khalid or Herman?   (2 marks)

Show Answers Only
  1. `text(The claim is incorrect because each of her friends)`
    `text(bought a different number of tickets and therefore)`
    `text(their chances of winning are different.)`
  2. `69/100`
Show Worked Solution

i.    `text(The claim is incorrect because each of her friends bought)`

`text(a different number of tickets and therefore their chances of)`

`text(winning are different.)`

 

ii.  `text(Number of tickets not sold to K or H)`

`= 45 + 10 + 14`

`= 69`
 

`:.\ text(Probability 1st prize NOT won by K or H)`

`= 69/100`

Measurement, STD2 M1 2004 HSC 23a

The diagram shows the shape of Carmel’s garden bed. All measurements are in
metres.

  1. Show that the area of the garden bed is 57 square metres.   (2 marks)

  2. Carmel decides to add a 5 cm layer of straw to the garden bed.

     

    Calculate the volume of straw required. Give your answer in cubic metres.   (2 marks)

  3. Each bag holds 0.25 cubic metres of straw.

     

    How many bags does she need to buy?   (2 marks)

  4. A straight fence is to be constructed joining point A to point B.

     

    Find the length of this fence to the nearest metre.   (2 marks)

Show Answers Only
  1. `text(Proof)\ \ text{(See Worked Solutions)}`
  2. `text(2.85 m³)`
  3. `text(She needs to buy 12 bags)`
  4. `8\ text{m  (nearest metre)}`
Show Worked Solution
a.    `text(Area of)\ Delta ABC` `= 1/2 xx b xx h`
  `= 1/2 xx 10 xx 5.1`
  `= 25.5\ text(m²)`
`text(Area of)\ Delta ACD` `= 1/2 xx 10 xx 6.3`
  `= 31.5\ text(m²)`

 

`:.\ text(Total Area)` `= 25.5 + 31.5`
  `= 57\ text(m² … as required)`

 

b.    `V` `= Ah`
  `= 57 xx 0.05`
  `= 2.85\ text(m³)`

 

c.    `text(Bags to buy)` `= 2.85/0.25`
  `= 11.4`

 
`:.\ text(She needs to buy 12 bags.)`

 

d.   `text(Using Pythagoras,)`

`AB^2` `= 6.0^2 + 5.1^2`
  `= 36 + 26.01`
  `= 62.01`
`AB` `= 7.874…`
  `=8\ text{m  (nearest metre)}`

Measurement, STD2 M7 2005 HSC 4 MC

The diagram is a scale drawing of a butterfly.
 

2UG-2005-4MC 
 

What is the actual wingspan of the butterfly?

  1.    2.5 cm
  2.    3 cm
  3.    15 cm
  4.    8.75 cm
Show Answers Only

`B`

Show Worked Solution

`text(Wingspan is 3 times the scale distance)`

`text(that equals 1 cm.)`

`:.\ text(Wingspan)` `= 3 × 1`
  `= 3\ text(cm)`

`=> B`

Statistics, STD2 S1 2004 HSC 12 MC

This box-and-whisker plot represents a set of scores.
 

What is the interquartile range of this set of scores?

  1. 1
  2. 2
  3. 3
  4. 5
Show Answers Only

`C`

Show Worked Solution

`text{Q}_1 = 8, \ text{Q}_3 = 11`

`text{IQR}` `= text{Q}_3-text{Q}_1`
  `= 11-8`
  `= 3`

 
`=> C`

Measurement, STD2 M6 2004 HSC 9 MC

What is the area of the triangle to the nearest square metre?
 

 

  1. `text(102 m²)`
  2. `text(153 m²)`
  3. `text(172 m²)`
  4. `text(178 m²)`
Show Answers Only

`C`

Show Worked Solution

`text(Using sine rule,)`

`text(Area)` `= 1/2 ab sin C`
  `= 1/2 xx 30 xx 20 xx sin 35^@`
  `=172.072…\ text(m²)`

`=> C`

Statistics, STD2 S1 2004 HSC 8 MC

This sector graph shows the distribution of 116 prizes won by three schools: X, Y and Z.
 

 
How many prizes were won by School X?

  1.   26
  2.   32
  3.   81
  4.   99
Show Answers Only

`B`

Show Worked Solution

`text(Centre angle of School X sector)`

`= 100^@\ text{(by measurement)}`
 

`:.\ text(Prizes won by school X)`

`= 100/360 xx 116`

`= 32.22\ …`

`=> B`

Statistics, STD2 S1 2004 HSC 6-7 MC

Use the set of scores  1, 3, 3, 3, 4, 5, 7, 7, 12  to answer Questions 6 and 7.
 

Question 6

What is the range of the set of scores?

  1. 6
  2. 9
  3. 11
  4. 12

 

Question 7

What are the median and the mode of the set of scores?

  1. Median 3, mode 5
  2. Median 3, mode 3
  3. Median 4, mode 5
  4. Median 4, mode 3
Show Answers Only

`text(Question 6:)\ C`

`text(Question 7:)\ D`

Show Worked Solution

`text(Question 6)`

`text(Range)` `= text(High) – text(Low)`
  `= 12 – 1`
  `= 11`

`=> C`

 

`text(Question 7)`

`text(9 scores)`

`:.\ text(Median)` `= (9 + 1) / 2`
  `=5 text(th score)`
  `= 4`

`text(Mode) = 3`

`=> D`

Algebra, STD2 A1 2004 HSC 3 MC

If  `K = Ft^3`, `F = 5`  and  `t = 0.715`, what is the value of `K` correct to three significant figures?

  1. `1.82`
  2. `1.827`
  3. `1.828`
  4. `1.83`
Show Answers Only

`D`

Show Worked Solution
`K` `= Ft^3`
  `= 5 xx (0.715)^3`
  `= 1.8276…`
  `= 1.83\ \ text{(3 sig figures)}`

 
`=> D`

Algebra, STD2 A2 2004 HSC 2 MC

Susan drew a graph of the height of a plant.
  

What is the gradient of the line?

  1. `1`
  2. `5`
  3. `7.5`
  4. `10`
Show Answers Only

`B`

Show Worked Solution

`text(2 points on graph)\ \ (0, 10),\ (1, 15)`

`text(Gradient)` `= (y_2-y_1) / (x_2-x_1)`
  `= (15-10) / (1-0)`
  `= 5`

`=> B`

CORE*, FUR1 2009 VCAA 4 MC

A delivery truck when new was valued at  $65 000.

The truck’s value depreciates at a rate of 22 cents per kilometre travelled.

After it has travelled a total distance of 132 600 km, the value of the truck will be

A.   `$14\ 300`

B.   `$22\ 100`

C.   `$22\ 516`

D.   `$29\ 172`

E.   `$35\ 828`

Show Answers Only

`E`

Show Worked Solution
`text(Value of truck)` `= $65\ 000 − 132\ 600 xx 0.22`
  `= $35\ 828`

`=>  E`

CORE*, FUR1 2009 VCAA 2 MC

An amount of $6500 is borrowed at a simple interest rate of 3.5% per annum.

The total interest paid over the period of the loan is $910.

The period of the loan is closest to

A.   2.5 years.

B.   3.5 years.

C.   3.8 years.

D.   4 years.

E.   4.9 years.

Show Answers Only

`D`

Show Worked Solution

`I = 910, \ P = 6500, \ r = 3.5`

`I` `= (PrT)/100`
`910` `= ((6500)(3.5)(T))/100`
`:. T` `= (910 xx 100)/(6500xx3.5)`
  `= 4\ \ text(years)`

 
`=>  D`

CORE*, FUR1 2008 VCAA 1 MC

A plumber quoted $300, excluding GST (Goods and Services Tax), to complete a job.

A GST of 10% is added to the price.

The full price for the job will be

A.       $3

B.     $30

C.   $303

D.   $310

E.   $330

Show Answers Only

`E`

Show Worked Solution
`text(GST)` `= text(10%) xx 300`
  `= 30`
`:.\ text(Total price)` `= 300+ 30`
  `= 330`

`=>  E`

CORE*, FUR1 2007 VCAA 5 MC

A new kitchen in a restaurant cost $50 000. Its value is depreciated over time using the reducing balance method.

The value of the kitchen in dollars at the end of each year for ten years is shown in the graph below.
 


 

Which one of the following statements is true?

A.  The kitchen depreciates by $4000 annually.

B.  At the end of five years, the kitchen's value is less than $20 000.

C.  The reducing balance depreciation rate is less than 5% per annum.

D.  The annual depreciation rate increases over time.

E.  The amount of depreciation each year decreases over time.

Show Answers Only

`E`

Show Worked Solution

`text(Consider) \ E,`

`text(The amount of depreciation each year is given by)`

`text(the decrease in the)\ y text(-value compared to the)`

`text(previous year. This clearly decreases over time, making)`

`E\ text(true.)`

`text(All other answers can be shown to be incorrect.)`

`=>  E`

CORE*, FUR1 2005 VCAA 3 MC

Tamara’s bank statement for September has been damaged by spilt ink as shown below.

2005VCAA-Business -3

Tamara’s Pay/Salary was deposited on 30 September.

What is the value of this deposit?

A.   `$441.95`

B.   `$442.20`

C.   `$444.65`

D.   `$785.70`

E.   `$788.15` 

Show Answers Only

`B`

Show Worked Solution

2005VCAA-Business-Sol-3

`text(From table)`

`text(Balance after interest on 01 Sept)`

`= 2143.50 + 2.45`

`= 2145.95`

`text(Balance after Telstra payment)`

`= 2145.95 − 616.40`

`= 1529.55`

`:.\ text(Salary)` `= 1971.75 − 1529.55`
  `= $442.2`

`=>  B`

CORE*, FUR1 2011 VCAA 3 MC

A van is purchased for $56 000.

Its value depreciates at a rate of 42 cents for each kilometre that it travels.

The value of the van after it has travelled 32 000 km is

A.   `$13\ 440`

B.   `$26\ 880`

C.   `$29\ 120`

D.   `$32\ 480`

E.   `$42\ 560`

Show Answers Only

`E`

Show Worked Solution
`text(Depreciation)` `= 0.42 xx 32\ 000`
  `= 13\ 440`

 

`:.\ text(Value of car)` `= 56\ 000 – 13\ 440`
  `= $42\ 560`

`=> E`

CORE*, FUR1 2011 VCAA 2 MC

An amount of $22 000 is invested for three years at an interest rate of 3.5% per annum, compounding annually.

The value of the investment at the end of three years is closest to

A.   `$2310`

B.   `$9433`

C.   `$24\ 040`

D.   `$24\ 392`

E.   `$31\ 433`

Show Answers Only

`D`

Show Worked Solution

`P = 22\ 000,\ \ R = 1.035,\ \ n = 3`

`A` `=PR^n`
  `= 22\ 000 xx 1.035^3`
  `= 24\ 391.79…`

`=>  D`

CORE*, FUR1 2011 VCAA 1 MC

An electrician charges $68 per hour to complete a job.

A Goods and Services Tax (GST) of 10% is added to the charge.

Including GST, the cost of a job that takes three hours is

A.       $6.80

B.     $20.40

C.   $204.00

D.   $210.80

E.   $224.40

Show Answers Only

`E`

Show Worked Solution
`text(C) text(ost for 3 hrs)` `= 68 xx 3`
  `= 204`
`text(GST)` `= 204 xx 10text(%)`
  `= 20.4`
`:.\ text(Total cost)` `= 204 + 20.4`
  `= 224.4`

`=> E`

CORE*, FUR1 2012 VCAA 4 MC

Mei’s starting salary is $65 000 per annum.

After the first year her salary will increase by 2.8%.

After the second year her salary will increase by a further 3.5%.

After this second increase, her salary will be closest to

A.   $66 820

B.   $68 690

C.   $69 030

D.   $69 160

E.   $69 630

Show Answers Only

`D`

Show Worked Solution

`text(Salary after 1st year)`

`= 65\ 000 + 2.8text(%) xx 65\ 000`

`= $66\ 820`

`text(Salary after 2nd year)`

`= $66\ 820 + 3.5text(%) xx 66\ 820`

`= $69\ 158.70`

`=> D`

CORE*, FUR1 2012 VCAA 1 MC

The selling price of a large tin of paint is $215.

After a 25% discount, the selling price of the tin of paint will become

A.     $43.00

B.     $53.75

C.   $161.25

D.   $190.00

E.   $195.00

Show Answers Only

`C`

Show Worked Solution
`text(Price after discount)` `= 215 – 25text(%) xx 215`
  `= $215 -53.75`
  `= $161.25`

`=>   C`

CORE, FUR1 2014 VCAA 5 MC

A bank approves a $90 000 loan for a customer.

The loan is to be repaid fully over 20 years in equal monthly payments.

Interest is charged at a rate of 6.95% per annum on the reducing monthly balance.

To the nearest dollar, the monthly payment will be

A.     $478

B.     $692

C.     $695

D.   $1409

E.   $1579

Show Answers Only

`C`

Show Worked Solution

`text(Monthly payments)\ = 20 xx 12 = 240`

`text(Annual interest rate)\ = 6.95text(%)`

`text(By TVM Solver:)`

`N` `=20 xx 12=240`
`I(%)` `= 6.95`
`PV` `= -90000`
`PMT` `= ?`
`FV` `= 0`
`text(P/Y)` `= text(C/Y) = 12`

 
`:. PMT = $695.07`

`=>  C` 

GRAPHS, FUR1 2014 VCAA 6 MC

The Domestics Cleaning Company provides household cleaning services.

For two hours of cleaning, the cost is $55.

For four hours of cleaning, the cost is $94.

The rule for the cost of cleaning services is

`text(cost) = a + b xx text(hours)`

where `a` is a fixed charge, in dollars, and `b` is the charge per hour of cleaning, in dollars per hour.

Using this rule, the cost for five hours of cleaning is

A.   `$19.50`

B.   `$97.50`

C.   `$99.50`

D.   `$113.50`

E.   `$121.50`

Show Answers Only

`D`

Show Worked Solution
`text(C)text(ost)` `= a + b xx text(hours)`
`55` `= a + 2b\ \ \ …\ (1)`
`94` `= a + 4b\ \ \ …\ (2)`

`(2) – (1)`

`39` `= 2b`
`b` `= 19.5`

 

`text(Substitute)\ b = 19.5\ text(into)\ (1)`

`a + (2 xx 19.5)` `= 55`
`a` `= 16`

`:.\ text(C)text(ost of 5 Hours)`

`= 16 + (5 xx 19.5)`

`= text($) 113.50`

`=>  D`

GRAPHS, FUR1 2014 VCAA 1 MC

The graph below shows the altitude, in metres, of a balloon over a six-hour flight.

Over the six-hour period, the length of time, in hours, where the altitude of the balloon was at least 1500 m is

A.   `3`

B.   `4`

C.   `5`

D.   `6`

E.   `7`

Show Answers Only

`B`

Show Worked Solution

`text{The graph is higher than 1500 metres (on the}\ y text(-axis)`

`text{between the 1 hour mark and the 5 hour mark).}`

`:.\ text(It spends 4 hours at or above 1500 m.)`

`=> B`

CORE*, FUR1 2014 VCAA 4 MC

The cost of hiring a plasterer is $86.00 per hour plus GST of 10%.

The cost of hiring a plasterer for four hours, including GST, is

A.   $120.40

B.   $309.60

C.   $344.00

D.   $352.60

E.   $378.40

Show Answers Only

`E`

Show Worked Solution

`text(C) text{ost per hour (incl. GST)}`

`= $86.00 + 10text(%) xx $86.00`

`= $86.00 + $8.60`

`= $94.60`

`:.\ text(C) text(ost for 4 hours)` `=4 xx $94.60`
  `= $378.40`

`=>  E` 

CORE*, FUR1 2014 VCAA 1 MC

This month, a business charges $1500 to install a water tank.

Next month, the charge will increase by 3.5%.

The charge next month will be

A.       `$45.00`

B.       `$52.50`

C.   `$1545.00`

D.   `$1552.50`

E.   `$1950.00` 

Show Answers Only

`D`

Show Worked Solution

`text(Next month’s charge)`

`= 1500 + 3.5text(%) xx 1500`

`= 1500 + 52.5`

`= $1552.5`

`=>  D`

Measurement, 2UG MM6 SM-Bank 02 MC

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°`

Show Answers Only

`D`

Show Worked Solution
`text(Arc Length)` `= theta/360 xx 2 xx pi xx r`
`12`  `= theta/360 xx 2 xx pi xx 10`
`theta/360`  `= 12/(20 pi)`
`theta`  `= (12 xx 360)/(20 pi)`
  `= 68.75…°`

`=>  D`

GRAPHS, FUR1 2007 VCAA 4 MC

Paul makes rulers. There is a fixed cost of $60 plus a manufacturing cost of $0.20 per ruler.

Last week Paul was able to break even by selling his rulers for $1 each.

The number of rulers Paul sold last week was

A.     `50`

B.     `75`

C.     `90`

D.   `120`

E.   `150`

Show Answers Only

`B`

Show Worked Solution

`text(Let)\ n = text(amount of rulers made)`

`text(Total cost making rulers) = 60 + 0.2n`

`text(Revenue at $1/ruler)\ = 1 xx n = n`

`:.\ text(Break-even at $1 per ruler occurs when)`

`60 + 0.2n` `= n`
`0.8n` `= 60`
`n` `= 75`

`:.\ text(Paul sold 75 rulers.)`

`=>  B`

GRAPHS, FUR1 2007 VCAA 2 MC

A builder's fee, `C` dollars, can be determined from the rule  `C = 60 + 55n`, where `n` represents the number of hours worked.

According to this rule, the builder's fee will be

A.   $60 for 1 hour of work.

B.   $110 for 2 hours of work.

C.   $500 for 8 hours of work.

D.   $550 for 10 hours of work.

E.   $1150 for 10 hours of work. 

Show Answers Only

`C`

Show Worked Solution

`C = 60 + 55n`

`text(Consider)\ A\ text(when)\ n = 1`

`C` `= 60 + 55(1)`
  `= 115`

`:.\ text(Eliminate)\ A`

`text(Consider)\ B\ text(when)\ n = 2`

`C` `= 60 + 55(2)`
  `= 170`

`:. text(Eliminate)\ B`

`text(Consider)\ C\ text(when)\ n = 8`

`C` `= 60 + 55(8)`
  `= 500`

`:.\ text(True)`

`D\ text(and)\ E\ text(can be similarly ruled out.)`

`=>  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

GRAPHS, FUR1 2009 VCAA 5-6 MC ab

GRAPHS, FUR1 2009 VCAA 5-6 MC cd

GRAPHS, FUR1 2009 VCAA 5-6 MC e

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.

GRAPHS, FUR1 2009 VCAA 1-3 MC 1  

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.

GRAPHS, FUR1 2009 VCAA 1-3 MC 2

 

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 5 MC

A block of land is triangular in shape.

The three sides measure 36 m, 58 m and 42 m.

To calculate the area, Heron’s formula is used.

The correct application of Heron’s formula for this triangle is

  1. `text(Area) = sqrt(136\ (136 − 36) (136 − 58) (136 − 42))`
  2. `text(Area) =sqrt(136\ (136 −18) (136 − 29) (136 − 21))`
  3. `text(Area) =sqrt(68\ (68 − 36) (68 − 58) (68 − 42))`
  4. `text(Area) = sqrt(68\ (68 −18) (68 − 29) (68 − 21))`
  5. `text(Area) = sqrt(68\ (136 − 36) (136 − 58) (136 − 42))` 
Show Answers Only

`C`

Show Worked Solution
`s` `= (36 + 58 + 42)/2`
  `= 68`

 

`:.\ text(Area)` `= sqrt(s(s – a) xx (s – b) xx (s – c))`
  `= sqrt(68 (68 – 36) xx (68 – 58) xx (68 – 42))`

 
`=>  C`

GEOMETRY, FUR1 2006 VCAA 2 MC

GEOMETRY, FUR1 2006 VCAA 2 MC 

The length of  `RT`  in the triangle shown is closest to

A.   `17\ text(cm)`

B.   `33\ text(cm)`

C.   `45\ text(cm)`

D.   `53\ text(cm)`

E.   `57\ text(cm)`

Show Answers Only

`D`

Show Worked Solution

`text(Using sine rule:)`

`(RT)/sin108^@` `= 45/sin54^@`
 `RT` `= (45 xx sin108^@)/sin54^@`
  `= 52.90…`

 
`=>  D`

GEOMETRY, FUR1 2007 VCAA 2 MC

GEOMETRY, FUR1 2007 VCAA 2 MC
 

For an observer on the ground at `A`, the angle of elevation of a weather balloon at `B` is 37°.

`C` is a point on the ground directly under the balloon. The distance `AC` is 2200 m.

To the nearest metre, the height of the weather balloon above the ground is

A.   `1324\ text(m)`

B.   `1658\ text(m)`

C.   `1757\ text(m)`

D.   `2919\ text(m)`

E.   `3655\ text(m)`

Show Answers Only

`B`

Show Worked Solution
`tan 37^@` `= (BC)/2200`
`:. BC`  `= 2200 xx tan 37^@`
  `= 1657.8…\ text(m)`

 
`=>  B`