Calculus, 2ADV C2 2007 HSC 2ai
Differentiate with respect to `x`:
`(2x)/(e^x + 1)` (2 marks)
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Financial Maths, STD2 F4 2006 HSC 27c
Kai purchased a new car for $30 000. It depreciated in value by $2000 per year for the first three years.
After the end of the third year, Kai changed the method of depreciation to the declining balance method at the rate of 25% per annum.
- Calculate the value of the car at the end of the third year. (1 mark)
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- Calculate the value of the car seven years after it was purchased. (2 marks)
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- Without further calculations, sketch a graph to show the value of the car over the seven years.
Use the horizontal axis to represent time and the vertical axis to represent the value of the car. (3 marks)
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Probability, STD2 S2 2006 HSC 26c
A new test has been developed for determining whether or not people are carriers of the Gaussian virus.
Two hundred people are tested. A two-way table is being used to record the results.
- What is the value of `A`? (1 mark)
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- A person selected from the tested group is a carrier of the virus.
What is the probability that the test results would show this? (2 marks)
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- For how many of the people tested were their test results inaccurate? (1 mark)
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Calculus, EXT1* C3 2005 HSC 6c
The graphs of the curves `y = x^2` and `y = 12 - 2x^2` are shown in the diagram.
- Find the points of intersection of the two curves. (1 mark)
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- The shaded region between the curves and the `y`-axis is rotated about the `y`-axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed. (3 marks)
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Calculus, 2ADV C3 2005 HSC 6b
A tank initially holds 3600 litres of water. The water drains from the bottom of the tank. The tank takes 60 minutes to empty.
A mathematical model predicts that the volume, `V` litres, of water that will remain in the tank after `t` minutes is given by
`V = 3600(1 − t/60)^2,\ \ text(where)\ \ 0 ≤ t ≤ 60`.
- What volume does the model predict will remain after ten minutes? (1 mark)
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- At what rate does the model predict that the water will drain from the tank after twenty minutes? (2 marks)
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- At what time does the model predict that the water will drain from the tank at its fastest rate? (2 marks)
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L&E, 2ADV E1 2005 HSC 5a
Use the change of base formula to evaluate `log_3 7`, correct to two decimal places. (1 mark)
Calculus, 2ADV C3 2005 HSC 4b
A function `f(x)` is defined by `f(x) = (x + 3)(x^2- 9)`.
- Find all solutions of `f(x) = 0` (2 marks)
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- Find the coordinates of the turning points of the graph of `y = f(x)`, and determine their nature. (3 marks)
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- Hence sketch the graph of `y = f(x)`, showing the turning points and the points where the curve meets the `x`-axis. (2 marks)
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- For what values of `x` is the graph of `y = f(x)` concave down? (1 mark)
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Trigonometry, 2ADV T1 2005 HSC 4a
A pendulum is 90 cm long and swings through an angle of 0.6 radians. The extreme positions of the pendulum are indicated by the points `A` and `B` in the diagram.
- Find the length of the arc `AB`. (1 mark)
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- Find the straight-line distance between the extreme positions of the pendulum. (2 marks)
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- Find the area of the sector swept out by the pendulum. (1 mark)
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Plane Geometry, 2UA 2005 HSC 3c
In the diagram, `A`, `B` and `C` are the points `(6, 0), (9, 0)` and `(12, 6)` respectively. The equation of the line `OC` is `x - 2y = 0`. The point `D` on `OC` is chosen so that `AD` is parallel to `BC`. The point `E` on `BC` is chosen so that `DE` is parallel to the `x`-axis.
- Show that the equation of the line `AD` is `y = 2x - 12`. (2 marks)
- Find the coordinates of the point `D`. (2 marks)
- Find the coordinates of the point `E`. (1 marks)
- Prove that `ΔOAD\ text(|||)\ ΔDEC`. (2 marks)
- Hence, or otherwise, find the ratio of the lengths `AD` and `EC`. (1 marks)
Financial Maths, 2ADV M1 2005 HSC 3a
Evaluate `sum_(n = 3)^5 (2n + 1)`. (1 mark)
Calculus, 2ADV C3 2006 HSC 5a
A function `f(x)` is defined by `f(x) =2x^2(3 - x)`.
- Find the coordinates of the turning points of `y =f(x)` and determine their nature. ( 3 marks)
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- Find the coordinates of the point of inflection. (1 mark)
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- Hence sketch the graph of `y =f(x)`, showing the turning points, the point of inflection and the points where the curve meets the `x`-axis. (3 marks)
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- What is the minimum value of `f(x)` for `–1 ≤ x ≤4`? (1 mark)
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Calculus, 2ADV C3 2005 HSC 2d
Find the equation of the tangent to `y = log_ex` at the point `(e, 1)`. (2 marks)
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Trig Calculus, 2UA 2005 HSC 2cii
Evaluate `int_0^(pi/6) cos\ 3x\ dx`. (2 marks)
Calculus, 2ADV C1 2005 HSC 2bii
Differentiate with respect to `x`:
`x^2/(x − 1).` (2 marks)
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Trigonometry, 2ADV T2 2005 HSC 2a
Solve `cos\ theta =1/sqrt2` for `0 ≤ theta ≤ 2pi`. (2 marks)
Functions, EXT1* F1 2005 HSC 1e
Find the values of `x` for which `|\ x − 3\ | ≤ 1`. (2 marks)
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Functions, 2ADV F1 2005 HSC 1d
Express `((2x-3))/2-((x-1))/5` as a single fraction in its simplest form. (2 marks)
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Trig Calculus, 2UA 2005 HSC 1c
Find a primitive of `4 + sec^2\ x`. (2 marks)
Financial Maths, 2ADV M1 2006 HSC 1f
Find the limiting sum of the geometric series `13/5 + 13/25 + 13/125 + …` (2 marks)
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Functions, 2ADV F2 2006 HSC 1c
Sketch the graph of `y = |\ x + 4\ |`. (2 marks)
Data, 2UG 2005 HSC 27a
The area graph shows sales figures for Shoey’s shoe store.
- Approximately how many school shoes were sold in January? (1 mark)
- For which month does the graph indicate that the same number of school shoes and business shoes was sold? (1 mark)
- Identify ONE trend in this graph, and suggest a valid reason for this trend. (2 marks)
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)`.
- What is the difference in the latitudes of Sorong and Darwin? (1 mark)
- The radius of Earth is approximately `text(6400 km.)`
- Show that the great circle distance between Sorong and Darwin is approximately `text(1200 km)`. (2 marks)
Data, 2UG 2006 HSC 23d
The graph shows the amounts charged by Company `A` and Company `B` to deliver parcels of various weights.
- How much does Company `A` charge to deliver a `3` kg parcel? (1 mark)
- Give an example of the weight of a parcel for which both Company `A` and Company `B` charge the same amount. (1 mark)
- For what weight(s) is it cheaper to use Company `A`? (2 marks)
- What is the rate per kilogram charged by Company `B` for parcels up to `8` kg? (1 mark)
Data, 2UG 2006 HSC 23b
This radar chart was used to display the average daily temperatures each month for two different towns.
- What is the average daily temperature of Town `B` for April? (1 mark)
- In which month do the average daily temperatures of the two towns have the greatest difference? (1 mark)
- In which months is the average daily temperature in Town `B` higher than in Town `A`? (1 mark)
Measurement, STD2 M6 2005 HSC 25b
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.
- Determine the values of `X`, `Y` and `Z`. (Assume there are exactly 52 weeks in a year.) (3 marks)
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At the beginning of 2006, Reece starts saving.
- 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)
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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.
- What is the probability that Moheb chooses a red tie for himself? (1 mark)
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Copy the tree diagram into your writing booklet.
- Complete your tree diagram by writing the correct probability on each branch. (2 marks)
- Calculate the probability that both of the ties are the same colour. (2 marks)
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Algebra, STD2 A4 2004 HSC 26a
- 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)
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- 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)
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- 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)
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- Using your graph or otherwise, estimate the time in hours for the number of bacteria to reach 300. (1 mark)
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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).
- What is the mean 3 pm wind speed for September? (1 mark)
- Which month has the lowest mean 3 pm wind speed? (1 mark)
- In which three-month period does the town have the highest number of raindays? (1 mark)
- Briefly describe the pattern relating wind speed with the number of raindays for this town. Refer to specific months. (2 marks)
Measurement, STD2 M1 2005 HSC 23b
A clay brick is made in the shape of a rectangular prism with dimensions as shown.
- Calculate the volume of the clay brick. (1 mark)
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Three identical cylindrical holes are made through the brick as shown. Each hole has a radius of 1.4 cm.
- 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)
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- What percentage of clay is removed by making the holes through the brick? (Give your answer correct to one decimal place.) (1 mark)
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Measurement, 2UG 2005 HSC 12 MC
Probability, STD2 S2 2005 HSC 11 MC
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.
- Calculate the value of her tax deduction for the 2003 financial year. (1 mark)
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- What is the value of her computer at the start of the 2006 financial year? (2 marks)
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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.
- Write an expression involving `x` for the cost of Novirus. (1 mark)
- Novirus and Funmaths together cost `$415`. Write an equation involving
- `x` and solve it to find the cost of Funmaths. (2 marks)
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`.
- Calculate the length of the interval `AB`. (1 mark)
- Find the gradient of the line `AB`. (1 mark)
- What is the size of the acute angle between the line `AB` and the line `l`? (1 mark)
- Show that the equation of the line `AB` is `x + y − 2 = 0`. (1 mark)
- Copy the diagram into your writing booklet and shade the region defined by `x + y − 2 <= 0`. (1 mark)
- Write down the equation of the line `l`. (1 mark)
- The point `C` is on the line `l` such that `AC` is perpendicular to `AB`. Find the coordinates of `C`. (2 marks)
Functions, EXT1* F1 2004 HSC 1f
Find the values of `x` for which `|\ x + 1\ |<= 5`. (2 marks)
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Measurement, STD2 M6 2006 HSC 9 MC
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)
Calculus, 2ADV C1 2004 HSC 1b
Differentiate `x^4 + 5x^(−1)` with respect to `x`. (2 marks)
Functions, 2ADV F1 2004 HSC 1c
Solve `(x-5)/3-(x+1)/4 = 5`. (2 marks)
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?
- 33
- 56
- 61
- 66
Measurement, STD2 M6 2006 HSC 3 MC
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?
(A) The event is likely to occur.
(B) The event is certain to occur.
(C) The event is unlikely to occur.
(D) The event has an even chance of occurring.
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.
- 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)
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- What is the probability that first prize is NOT won by Khalid or Herman? (2 marks)
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Measurement, STD2 M1 2004 HSC 23a
The diagram shows the shape of Carmel’s garden bed. All measurements are in
metres.
- Show that the area of the garden bed is 57 square metres. (2 marks)
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- 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)
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- Each bag holds 0.25 cubic metres of straw.
How many bags does she need to buy? (2 marks)
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- 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)
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Measurement, STD2 M7 2005 HSC 4 MC
Statistics, STD2 S1 2004 HSC 12 MC
Measurement, STD2 M6 2004 HSC 9 MC
Statistics, STD2 S1 2004 HSC 8 MC
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?
- 6
- 9
- 11
- 12
Question 7
What are the median and the mode of the set of scores?
- Median 3, mode 5
- Median 3, mode 3
- Median 4, mode 5
- Median 4, mode 3
Measurement, STD2 M6 2004 HSC 5 MC
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?
(A) `1.82`
(B) `1.827`
(C) `1.828`
(D) `1.83`
Algebra, STD2 A2 2004 HSC 2 MC
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`
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.
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
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.
CORE*, FUR1 2005 VCAA 3 MC
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`
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