The scatterplot shows the number of ice-creams sold, \(y\), at a shop over a ten-day period, and the temperature recorded at 2 pm on each of these days. \(y=0.936 x-8.929\), where \(x\) is the temperature. --- 4 WORK AREA LINES (style=lined) --- --- 3 WORK AREA LINES (style=lined) ---
Probability, STD1 S2 2023 HSC 8 MC
Measurement, STD2 M6 2023 HSC 35
L&E, 2ADV E1 2008 HSC 7a
Solve `log_2 x-3/log_2 x=2` (3 marks)
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Measurement, STD2 M1 2009 HSC 25b
The mass of a sample of microbes is 50 mg. There are approximately `2.5 × 10^6` microbes in the sample.
In scientific notation, what is the approximate mass in grams of one microbe? (2 marks)
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Quadratics, SMB-015
The diagram shows the curve with equation `y = x^2-7x + 10`. The curve intersects the `x`-axis at points `A and B`. The point `C` on the curve has the same `y`-coordinate as the `y`-intercept of the curve.
- Find the `x`-coordinates of points `A and B.` (2 marks)
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- Write down the coordinates of `C.` (2 marks)
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Algebra, STD1 A3 2019 HSC 9 MC
The container shown is initially full of water.
Water leaks out of the bottom of the container at a constant rate.
Which graph best shows the depth of water in the container as time varies?
A. | B. | ||
C. | D. |
Algebra, STD1 A1 2019 HSC 34
Given the formula `C = (A(y + 1))/24`, calculate the value of `y` when `C = 120` and `A = 500`. (3 marks)
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Statistics, STD1 S3 2022 HSC 23
A teacher surveyed the students in her Year 8 class to investigate the relationship between the average number of hours of phone use per day and the average number of hours of sleep per day.
The results are shown on the scatterplot below.
- The data for two new students, Alinta and Birrani, are shown in the table below. Plot their results on the scatterplot. (2 marks)
\begin{array} {|l|c|c|}
\hline
& \textit{Average hours of} & \textit{Average hours of} \\ & \textit{phone use per day} & \textit{sleep per day} \\
\hline
\rule{0pt}{2.5ex} \text{Alinta} \rule[-1ex]{0pt}{0pt} & 4 & 8 \\
\hline
\rule{0pt}{2.5ex} \text{Birrani} \rule[-1ex]{0pt}{0pt} & 0 & 10.5 \\
\hline
\end{array}
- By first fitting the line of best fit by eye on the scatterplot, estimate the average number of hours of sleep per day for a student who uses the phone for an average of 2 hours per day. (2 marks)
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Probability, STD1 S2 2022 HSC 17
Each number from 1 to 30 is written on a separate card. The 30 cards are shuffled. A game is played where one of these cards is selected at random. Each card is equally likely to be selected.
Ezra is playing the game, and wins if the card selected shows an odd number between 20 and 30.
- List the numbers which would result in Ezra winning the game. (1 mark)
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- What is the probability that Ezra does NOT win the game? (2 marks)
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Measurement, STD1 M5 2022 HSC 5 MC
Functions, EXT1 F2 2022 HSC 3 MC
Let `P(x)` be a polynomial of degree 5. When `P(x)` is divided by the polynomial `Q(x)`, the remainder is `2x+5`.
Which of the following is true about the degree of `Q`?
- The degree must be 1.
- The degree could be 1.
- The degree must be 2.
- The degree could be 2.
Functions, 2ADV F1 2022 HSC 12
A student believes that the time it takes for an ice cube to melt (`M` minutes) varies inversely with the room temperature `(T^@ text{C})`. The student observes that at a room temperature of `15^@text{C}` it takes 12 minutes for an ice cube to melt.
- Find the equation relating `M` and `T`. (2 marks)
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- By first completing this table of values, graph the relationship between temperature and time from `T=5^@C` to `T=30^@ text{C}`. (2 marks)
\begin{array} {|c|c|c|c|}
\hline \ \ T\ \ & \ \ 5\ \ & \ 15\ & \ 30\ \\
\hline M & & & \\
\hline \end{array}
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Algebra, STD2 A4 2022 HSC 24
A student believes that the time it takes for an ice cube to melt (`M` minutes) varies inversely with the room temperature `(T^@ text{C})`. The student observes that at a room temperature of `15^@text{C}` it takes 12 minutes for an ice cube to melt.
- Find the equation relating `M` and `T`. (2 marks)
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- By first completing this table of values, graph the relationship between temperature and time from `T=5^@C` to `T=30^@ text{C}` (2 marks)
\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ T\ \ \rule[-1ex]{0pt}{0pt} & \ \ \ 5\ \ \ & \ \ 15\ \ \ & \ \ 30\ \ \\
\hline
\rule{0pt}{2.5ex} \ \ M\ \ \rule[-1ex]{0pt}{0pt} & & & \\
\hline
\end{array}
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Functions, 2ADV F1 2022 HSC 1 MC
Which of the following could be the graph of `y= -2 x+2`?
Measurement, STD2 M1 2022 HSC 34
A composite solid is shown. The top section is a cylinder with a height of 3 cm and a diameter of 4 cm. The bottom section is a hemisphere with a diameter of 6 cm. The cylinder is centred on the flat surface of the hemisphere.
Find the total surface area of the composite solid in cm², correct to 1 decimal place. (4 marks)
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Measurement, STD2 M1 2022 HSC 28
A dam is in the shape of a triangular prism which is 50 m long, as shown.
Both ends of the dam, `A B C` and `D E F`, are isosceles triangles with equal sides of length 25 metres. The included angles `B A C` and `E D F` are each `150^@`.
Calculate the number of litres of water the dam will hold when full. (4 marks)
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Measurement, STD2 M6 2022 HSC 26
The diagram shows two right-angled triangles, `ABC` and `ABD`,
where `AC=35 \ text{cm},BD=93 \ text{cm}, /_ACB=41^(@)` and `/_ADB=theta`.
Calculate the size of angle `theta`, to the nearest minute. (4 marks)
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Algebra, STD2 A4 2022 HSC 22
The formula `C=100 n+b` is used to calculate the cost of producing laptops, where `C` is the cost in dollars, `n` is the number of laptops produced and `b` is the fixed cost in dollars.
- Find the cost when 1943 laptops are produced and the fixed cost is $20 180. (1 mark)
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- Some laptops have some extra features added. The formula to calculate the production cost for these is
- `C=100 n+a n+20\ 180`
- where `a` is the additional cost in dollars per laptop produced.
- Find the number of laptops produced if the additional cost is $26 per laptop and the total production cost is $97 040. (2 marks)
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Financial Maths, STD2 F1 2022 HSC 21
A real estate agent's commission for selling houses is 2% for the first $800 000 of the sale price and 1.5% for any amount over $800 000.
Calculate the commission earned in selling a house for $1 500 000. (2 marks)
Algebra, STD2 A2 2022 HSC 14 MC
Which of the following correctly expresses `x` as the subject of `y=(ax-b)/(2)` ?
- `x=(2y+b)/(a)`
- `x=(y+b)/(2a)`
- `x=(2y)/(a)+b`
- `x=(y)/(2a)+b`
Financial Maths, STD2 F4 2022 HSC 11 MC
In ten years, the future value of an investment will be $150 000. The interest rate is 4% per annum, compounded half-yearly.
Which equation will give the present value `(PV)` of the investment?
- `PV=(150\ 000)/((1+0.04)^(10))`
- `PV=(150\ 000)/((1+0.04)^(20))`
- `PV=(150\ 000)/((1+0.02)^(10))`
- `PV=(150\ 000)/((1+0.02)^(20))`
Algebra, STD2 A4 2022 HSC 9 MC
An object is projected vertically into the air. Its height, `h` metres, above the ground after `t` seconds is given by `h=-5 t^2+80 t`.
For how long is the object at a height of 300 metres or more above the ground?
- 4 seconds
- 6 seconds
- 8 seconds
- 10 seconds
Measurement, STD2 M6 2022 HSC 8 MC
Which true bearing is the same as `text{S} 48^@ text{W}`?
- `132^@`
- `222^@`
- `228^@`
- `312^@`
Financial Maths, STD2 F1 2022 HSC 7 MC
Tian is paid $20.45 per hour, as well as a meal allowance of $16.20 per day.
What are Tian's total earnings if she works 9 hours per day for 5 days?
- `$329.85`
- `$936.45`
- `$1001.25`
- `$1649.25`
Functions, 2ADV F2 2021 HSC 19
Without using calculus, sketch the graph of `y = 2 + 1/(x + 4)`, showing the asymptotes and the `x` and `y` intercepts. (3 marks)
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Functions, EXT1 F2 2021 HSC 3 MC
What is the remainder when `P(x) = -x^3-2x^2-3x + 8` is divided by `x + 2`?
- `-14`
- `-2`
- `2`
- `14`
Probability, STD1 S2 2021 HSC 20
In a bag, there are six playing cards, 2, 4, 6, 8, Queen and King. The Queen and King are known as picture cards.
Two of these cards are chosen randomly. All the possible outcomes are shown.
- What is the probability that the two cards chosen include one or both picture cards? (1 mark)
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- What is the probability that the two cards chosen do NOT include any picture cards? (1 mark)
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Statistics, STD1 S1 2021 HSC 2 MC
A survey of which of the following would provide data that are both categorical and
nominal?
- Hair colour
- Height in centimetres
- Number of people present at a concert
- Size of coffee cup classified as small, medium or large
Networks, STD1 N1 2021 HSC 1 MC
Measurement, STD2 M6 2021 HSC 32
A right-angled triangle `XYZ` is cut out from a semicircle with centre `O`. The length of the diameter `XZ` is 16 cm and `angle YXZ` = 30°, as shown on the diagram.
- Find the length of `XY` in centimetres, correct to two decimal places. (2 marks)
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- Hence, find the area of the shaded region in square centimetres, correct to one decimal place. (3 marks)
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Algebra, STD2 A4 2021 HSC 24
A population, `P`, is to be modelled using the function `P = 2000 (1.2)^t`, where `t` is the time in years.
- What is the initial population? (1 mark)
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- Find the population after 5 years. (1 mark)
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- On the axes below, draw the graph of the population against time, showing the points at `t = 0` and at `t = 5`. (2 marks)
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Measurement, STD2 M6 2021 HSC 14 MC
Consider the diagram below.
What is the true bearing of `A` from `B`?
- `025^@`
- `065^@`
- `115^@`
- `295^@`
Measurement, STD2 M1 2021 HSC 16
Functions, 2ADV F1 2021 HSC 11
Solve `x+(x-1)/2 = 9` (2 marks)
Functions, 2ADV F1 2021 HSC 8 MC
Algebra, STD2 A4 2021 HSC 10 MC
Statistics, STD2 S1 2021 HSC 3 MC
Financial Maths, STD2 F4 2021 HSC 26
Nina plans to invest $35 000 for 1 year. She is offered two different investment options.
Option A: Interest is paid at 6% per annum compounded monthly.
Option B: Interest is paid at `r` % per annum simple interest.
- Calculate the future value of Nina's investment after 1 year if she chooses Option A. (2 marks)
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- Find the value of `r` in Option B that would give Nina the same future value after 1 year as for Option A. Give your answer correct to two decimal places. (2 marks)
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Financial Maths, STD2 F1 2021 HSC 19
Adam purchased some office furniture five years ago. It depreciated by $2300 each year based on the straight-line method of depreciation. The salvage value of the furniture is now $7500.
Find the initial value of the office furniture. (2 marks)
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Measurement, STD2 M1 2021 HSC 6 MC
Suppose `a=b/7`, where `b=22.`
What is the value of `a`, correct to three significant figures?
- 3.14
- 3.15
- 3.142
- 3.143
Financial Maths, STD2 F4 2021 HSC 4 MC
Three years ago an appliance was valued at $2467. Its value has depreciated by 15% each year, based on the declining-balance method.
What is the salvage value today, to the nearest dollar?
- $952
- $1110
- $1357
- $1515
Statistics, STD1 S1 2020 HSC 24
- The ages in years, of ten people at the local cinema last Saturday afternoon are shown.
\(38 \ \ 25 \ \ 38 \ \ 46 \ \ 55 \ \ 68 \ \ 72 \ \ 55 \ \ 36 \ \ 38\)
- The mean of this dataset is 47.1 years.
- How many of the ten people were aged between the mean age and the median age? (2 marks)
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- On Wednesday, ten people all aged 70 went to this same cinema.
- Would the standard deviation of the age dataset from Wednesday be larger than, smaller than or equal to the standard deviation of the age dataset given in part (a)? Briefly explain your answer without performing any calculations. (2 marks)
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Measurement, STD1 M5 2020 HSC 28
Probability, STD1 S2 2020 HSC 26
Barbara plays a game of chance, in which two unbiased six-sided dice are rolled. The score for the game is obtained by finding the difference between the two numbers rolled. For example, if Barbara rolls a 2 and a 5, the score is 3.
The table shows some of the scores.
- Complete the six missing values in the table to show all possible scores for the game. (1 mark)
- What is the probability that the score for a game is NOT 0? (2 marks)
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Statistics, STD1 S3 2020 HSC 22
A group of students sat a test at the end of term. The number of lessons each student missed during the term and their score on the test are shown on the scatterplot.
- Describe the strength and direction of the linear association observed in this dataset. (2 marks)
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- Calculate the range of the test scores for the students who missed no lessons. (1 mark)
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- Draw a line of the best fit in the scatterplot above. (1 mark)
- Meg did not sit the test. She missed five lessons.
Use the line of the best fit drawn in part (c) to estimate Meg's score on this test. (1 mark)
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- John also did not sit the test and he missed 16 lessons.
Is it appropriate to use the line of the best fit to estimate his score on the test? Briefly explain your answer. (1 mark)
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Algebra, STD1 A3 2020 HSC 19
Each year the number of fish in a pond is three times that of the year before.
- The table shows the number of fish in the pond for four years.
\begin{array} {|l|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\textit{Year}\rule[-1ex]{0pt}{0pt} & \ \ \ 2020\ \ \ & \ \ \ 2021\ \ \ & \ \ \ 2022\ \ \ & \ \ \ 2023\ \ \ \\
\hline
\rule{0pt}{2.5ex}\textit{Number of fish}\rule[-1ex]{0pt}{0pt} & 100 & & & 2700\\
\hline
\end{array}Complete the table above showing the number of fish in 2021 and 2022. (2 marks)
- Plot the points from the table in part (a) on the grid. (2 marks)
- Which model is more suitable for this dataset: linear or exponential? Briefly explain your answer. (2 marks)
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Statistics, STD1 S3 2020 HSC 4 MC
The table shows the average brain weight (in grams) and average body weight (in kilograms) of nine different mammals.
\begin{array} {|l|c|c|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \textit{Brain weight (g)} \rule[-1ex]{0pt}{0pt} & 0.7 & 0.4 & 1.9 & 2.4 & 3.5 & 4.3 & 5.3 & 6.2 & 7.8 \\
\hline
\rule{0pt}{2.5ex} \textit{Body weight (kg)} \rule[-1ex]{0pt}{0pt} & 0.02 &0.06 & 0.05 & 0.04 & 0.93 & 0.97 & 0.43 & 0.33 & 0.22 \\
\hline
\end{array}
Which of the following is the correct scatterplot for this dataset?
|
|
Networks, STD1 N1 2020 HSC 1 MC
Functions, EXT1 F2 2020 HSC 11a
Let `P(x) = x^3 + 3x^2-13x + 6`.
- Show that `P(2) = 0`. (1 mark)
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- Hence, factor the polynomial `P(x)` as `A(x)B(x)`, where `B(x)` is a quadratic polynomial. (2 marks)
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Measurement, STD2 M1 2020 HSC 25
Functions, 2ADV F1 2020 HSC 24
The circle of `x^2-6x + y^2 + 4y-3 = 0` is reflected in the `x`-axis.
Sketch the reflected circle, showing the coordinates of the centre and the radius. (3 marks)
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Financial Maths, STD2 F4 2020 HSC 21
The inflation rate over the year from January 2019 to January 2020 was 2%.
The cost of a school jumper in January 2020 was $122.
Calculate the cost of the jumper in January 2019 assuming that the only change in the cost of the jumper was due to inflation. (2 marks)
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Financial Maths, STD2 F1 2020 HSC 20
The table shows the income tax rates for the 2019 – 2020 financial year.
\begin{array} {|l|l|}
\hline
\rule{0pt}{2.5ex}\textit{ Taxable income}\rule[-1ex]{0pt}{0pt} & \textit{ Tax payable}\\
\hline
\rule{0pt}{2.5ex}\text{\$0 – \$18 200}\rule[-1ex]{0pt}{0pt} & \text{Nil}\\
\hline
\rule{0pt}{2.5ex}\text{\$18 201 – \$37 000}\rule[-1ex]{0pt}{0pt} & \text{19 cents for each \$1 over \$18 200}\\
\hline
\rule{0pt}{2.5ex}\text{\$37 001 – \$90 000}\rule[-1ex]{0pt}{0pt} & \text{\$3572 plus 32.5 cents for each \$1 over \$37 000}\\
\hline
\rule{0pt}{2.5ex}\text{\$90 001 – \$180 000}\rule[-1ex]{0pt}{0pt} & \text{\$20 797 plus 37 cents for each \$1 over \$90 000}\\
\hline
\rule{0pt}{2.5ex}\text{\$180 001 and over}\rule[-1ex]{0pt}{0pt} & \text{\$54 097 plus 45 cents for each \$1 over \$180 000}\\
\hline
\end{array}
For the 2019 – 2020 financial year, Wally had a taxable income of $122 680. During the year, he paid $3000 per month in Pay As You Go (PAYG) tax.
Calculate Wally's tax refund, ignoring the Medicare levy. (3 marks)
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Algebra, STD2 A4 2020 HSC 19
A fence is to be built around the outside of a rectangular paddock. An internal fence is also to be built.
The side lengths of the paddock are `x` metres and `y` metres, as shown in the diagram.
A total of 900 metres of fencing is to be used. Therefore `3x + 2y = 900`.
The area, `A`, in square metres, of the rectangular paddock is given by `A =450x - 1.5x^2`.
The graph of this equation is shown.
- If the area of the paddock is `30 \ 000 text(m)^2`, what is the largest possible value of `x`? (1 mark)
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- Find the values of `x` and `y` so that the area of the paddock is as large as possible. (2 marks)
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- Using your value from part (b), find the largest possible area of the paddock. (1 mark)
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Measurement, STD2 M6 2020 HSC 16
Algebra, STD2 A2 2020 HSC 10 MC
A plumber charges a call-out fee of $90 as well as $2 per minute while working.
Suppose the plumber works for `t` hours.
Which equation expresses the amount the plumber charges ($`C`) as a function of time (`t` hours)?
- `C = 2 + 90t`
- `C = 90 + 2t`
- `C = 120 + 90t`
- `C = 90 + 120t`
Statistics, STD2 S1 2020 HSC 7 MC
Which histogram best represents a dataset that is positively skewed?
Measurement, STD2 M1 2020 HSC 5 MC
A plant stem is measured to be 16.0 cm, correct to one decimal place.
What is the percentage error in this measurement?
- 0.3125%
- 0.625%
- 3.125%
- 6.25%
Financial Maths, STD2 F4 2020 HSC 4 MC
Joan invests $200. She earns interest at 3% per annum, compounded monthly.
What is the future value of Joan's investment after 1.5 years?
- $209.07
- $209.19
- $279.51
- $311.93
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