Calculus, EXT1* C1 2013 HSC 10 MC
A particle is moving along the `x`-axis. The displacement of the particle at time `t` seconds is `x` metres.
At a certain time, `dot x = -3\ text(ms)^(-1)` and `ddot x = 2\ text(ms)^(-2)`.
Which statement describes the motion of the particle at that time?
(A) The particle is moving to the right with increasing speed.
(B) The particle is moving to the left with increasing speed.
(C) The particle is moving to the right with decreasing speed.
(D) The particle is moving to the left with decreasing speed.
Financial Maths, STD2 F4 2011 HSC 28b
Norman and Pat each bought the same type of tractor for $60 000 at the same time. The value of their tractors depreciated over time.
The salvage value `S`, in dollars, of each tractor, is its depreciated value after `n` years.
Norman drew a graph to represent the salvage value of his tractor.
- Find the gradient of the line shown in the graph. (1 mark)
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- What does the value of the gradient represent in this situation? (1 mark)
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- Write down the equation of the line shown in the graph. (1 mark)
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- Find all the values of `n` that are not suitable for Norman to use when calculating the salvage value of his tractor. Explain why these values are not suitable. (2 marks)
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Pat used the declining balance formula for calculating the salvage value of her tractor. The depreciation rate that she used was 20% per annum.
- What did Pat calculate the salvage value of her tractor to be after 14 years? (2 marks)
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- Using Pat’s method for depreciation, describe what happens to the salvage value of her tractor for all values of `n` greater than 15. (1 mark)
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Statistics, STD2 S1 2011 HSC 25a
A study on the mobile phone usage of NSW high school students is to be conducted.
Data is to be gathered using a questionnaire.
The questionnaire begins with the three questions shown.
- Classify the type of data that will be collected in Q2 of the questionnaire. (1 mark)
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- Write a suitable question for this questionnaire that would provide discrete ordinal data. (1 mark)
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- An initial study is to be conducted using a stratified sample.
Describe a method that could be used to obtain a representative stratified sample. (1 mark)
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- Who should be surveyed if it is decided to use a census for the study? (1 mark)
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Algebra, STD2 A2 2010 HSC 27c
The graph shows tax payable against taxable income, in thousands of dollars.
- Use the graph to find the tax payable on a taxable income of $21 000. (1 mark)
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- Use suitable points from the graph to show that the gradient of the section of the graph marked `A` is `1/3`. (1 mark)
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- How much of each dollar earned between $21 000 and $39 000 is payable in tax? (1 mark)
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- Write an equation that could be used to calculate the tax payable, `T`, in terms of the taxable income, `I`, for taxable incomes between $21 000 and $39 000. (2 marks)
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Statistics, STD2 S1 2010 HSC 26b
A new shopping centre has opened near a primary school. A survey is conducted to determine the number of motor vehicles that pass the school each afternoon between 2.30 pm and 4.00 pm.
The results for 60 days have been recorded in the table and are displayed in the cumulative frequency histogram.
- Find the value of Χ in the table. (1 mark)
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- On the cumulative frequency histogram above, draw a cumulative frequency polygon (ogive) for this data. (1 mark)
- Use your graph to determine the median. Show, by drawing lines on your graph, how you arrived at your answer. (1 mark)
- Prior to the opening of the new shopping centre, the median number of motor vehicles passing the school between 2.30 pm and 4.00 pm was 57 vehicles per day.
What problem could arise from the change in the median number of motor vehicles passing the school before and after the opening of the new shopping centre?
Briefly recommend a solution to this problem. (2 marks)
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Measurement, STD2 M7 2013 HSC 30c
Joel mixes petrol and oil in the ratio 40 : 1 to make fuel for his leaf blower.
- Joel pours 5 litres of petrol into an empty container to make fuel for his leaf blower.
How much oil should he add to the petrol to ensure that the fuel is in the correct ratio? (1 mark)
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- Joel has 4.1 litres of fuel left in his container after filling his leaf blower.
He wishes to use this fuel in his lawnmower. However, his lawnmower requires the petrol and oil to be mixed in the ratio 25 : 1.
How much oil should he add to the container so that the fuel is in the correct ratio for his lawnmower? (3 marks)
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Measurement, 2UG 2011 HSC 24c
A ship sails 6 km from `A` to `B` on a bearing of 121°. It then sails 9 km to `C`. The
size of angle `ABC` is 114°.
Copy the diagram into your writing booklet and show all the information on it.
- What is the bearing of `C` from `B`? (1 mark)
- Find the distance `AC`. Give your answer correct to the nearest kilometre. (2 marks)
- What is the bearing of `A` from `C`? Give your answer correct to the nearest degree. (3 marks)
Probability, STD2 S2 2009 HSC 28d
In an experiment, two unbiased dice, with faces numbered 1, 2, 3, 4, 5, 6 are rolled 18 times.
The difference between the numbers on their uppermost faces is recorded each time. Juan performs this experiment twice and his results are shown in the tables.
Juan states that Experiment 2 has given results that are closer to what he expected than the results given by Experiment 1.
Is he correct? Explain your answer by finding the sample space for the dice differences and using theoretical probability. (4 marks)
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Statistics, STD2 S4 2009 HSC 28b
The height and mass of a child are measured and recorded over its first two years.
\begin{array} {|l|c|c|}
\hline \rule{0pt}{2.5ex} \text{Height (cm), } H \rule[-1ex]{0pt}{0pt} & \text{45} & \text{50} & \text{55} & \text{60} & \text{65} & \text{70} & \text{75} & \text{80} \\
\hline \rule{0pt}{2.5ex} \text{Mass (kg), } M \rule[-1ex]{0pt}{0pt} & \text{2.3} & \text{3.8} & \text{4.7} & \text{6.2} & \text{7.1} & \text{7.8} & \text{8.8} & \text{10.2} \\
\hline
\end{array}
This information is displayed in a scatter graph.
- Describe the correlation between the height and mass of this child, as shown in the graph. (1 mark)
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- A line of best fit has been drawn on the graph.
Find the equation of this line. (2 marks)
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Measurement, STD2 M6 2009 HSC 27b
A yacht race follows the triangular course shown in the diagram. The course from `P` to `Q` is 1.8 km on a true bearing of 058°.
At `Q` the course changes direction. The course from `Q` to `R` is 2.7 km and `/_PQR = 74^@`.
- What is the bearing of `R` from `Q`? (1 mark)
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- What is the distance from `R` to `P`? (2 marks)
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- The area inside this triangular course is set as a ‘no-go’ zone for other boats while the race is on.
What is the area of this ‘no-go’ zone? (1 mark)
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Statistics, STD2 S1 2009 HSC 26a
In a school, boys and girls were surveyed about the time they usually spend on the internet over a weekend. These results were displayed in box-and-whisker plots, as shown below.
- Find the interquartile range for boys. (1 mark)
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- What percentage of girls usually spend 5 or less hours on the internet over a weekend? (1 mark)
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- Jenny said that the graph shows that the same number of boys as girls usually spend between 5 and 6 hours on the internet over a weekend.
Under what circumstances would this statement be true? (1 mark)
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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 standard form, what is the approximate mass in grams of one microbe? (2 marks)
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Financial Maths, STD2 F4 2013 HSC 28d
Adhele has 2000 shares. The current share price is $1.50 per share. Adhele is paid a dividend of $0.30 per share.
Statistics, STD2 S1 2013 HSC 27c
A retailer has collected data on the number of televisions that he sold each week in 2012.
He grouped the data into classes and displayed the data using a cumulative frequency histogram and polygon (ogive).
- Use the cumulative frequency polygon to determine the interquartile range. (2 marks)
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- Oscar said that the retailer sold 300 televisions in 6 of the weeks in 2012.
Is he correct? Give a reason for your answer. (1 mark)
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Probability, STD2 S2 2013 HSC 26c
The probability that Michael will score more than 100 points in a game of bowling is `31/40`.
- A commentator states that the probability that Michael will score less than 100 points in a game of bowling is `9/40`.
Is the commentator correct? Give a reason for your answer. (1 mark)
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- Michael plays two games of bowling. What is the probability that he scores more than 100 points in the first game and then again in the second game? (1 mark)
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Financial Maths, STD2 F4 2010* HSC 22 MC
Statistics, STD2 S1 2010 HSC 21 MC
Probability, 2UG 2010 HSC 14 MC
A restaurant serves three scoops of different flavoured ice-cream in a bowl. There are five flavours to choose from.
How many different combinations of ice-cream could be chosen?
(A) `10`
(B) `15`
(C) `30`
(D) `60`
Algebra, STD2 A2 2009 HSC 24d
A factory makes boots and sandals. In any week
• the total number of pairs of boots and sandals that are made is 200
• the maximum number of pairs of boots made is 120
• the maximum number of pairs of sandals made is 150.
The factory manager has drawn a graph to show the numbers of pairs of boots (`x`) and sandals (`y`) that can be made.
- Find the equation of the line `AD`. (1 mark)
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- Explain why this line is only relevant between `B` and `C` for this factory. (1 mark)
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- The profit per week, `$P`, can be found by using the equation `P = 24x + 15y`.
Compare the profits at `B` and `C`. (2 marks)
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Probability, STD2 S2 2009 HSC 23b
A personal identification number (PIN) is made up of four digits. An example of a PIN is
- When all ten digits are available for use, how many different PINs are possible? (1 mark)
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- Rhys has forgotten his four-digit PIN, but knows that the first digit is either 5 or 6.
- What is the probability that Rhys will correctly guess his PIN in one attempt? (1 mark)
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Measurement, STD2 M1 2009 HSC 12 MC
How many square centimetres are in 0.0075 square metres?
(A) 0.75
(B) 7.5
(C) 75
(D) 7500
Algebra, STD2 A4 2012 HSC 30c
In 2010, the city of Thagoras modelled the predicted population of the city using the equation
`P = A(1.04)^n`.
That year, the city introduced a policy to slow its population growth. The new predicted population was modelled using the equation
`P = A(b)^n`.
In both equations, `P` is the predicted population and `n` is the number of years after 2010.
The graph shows the two predicted populations.
- Use the graph to find the predicted population of Thagoras in 2030 if the population policy had NOT been introduced. (1 mark)
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- In each of the two equations given, the value of `A` is 3 000 000.
What does `A` represent? (1 mark)
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- The guess-and-check method is to be used to find the value of `b`, in `P = A(b)^n`.
(1) Explain, with or without calculations, why 1.05 is not a suitable first estimate for `b`. (1 mark)
(2) With `n = 20` and `P = 4\ 460\ 000`, use the guess-and-check method and the equation `P = A(b)^n` to estimate the value of `b` to two decimal places. Show at least TWO estimate values for `b`, including calculations and conclusions. (2 marks)
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- The city of Thagoras was aiming to have a population under 7 000 000 in 2050. Does the model indicate that the city will achieve this aim?
Justify your answer with suitable calculations. (2 marks)
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Algebra, STD2 A4 2012 HSC 30b
A golf ball is hit from point `A` to point `B`, which is on the ground as shown. Point `A` is 30 metres above the ground and the horizontal distance from point `A` to point `B` is 300 m.
The path of the golf ball is modelled using the equation
`h = 30 + 0.2d-0.001d^2`
where
`h` is the height of the golf ball above the ground in metres, and
`d` is the horizontal distance of the golf ball from point `A` in metres.
The graph of this equation is drawn below.
- What is the maximum height the ball reaches above the ground? (1 mark)
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- There are two occasions when the golf ball is at a height of 35 metres.
What horizontal distance does the ball travel in the period between these two occasions? (1 mark)
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- What is the height of the ball above the ground when it still has to travel a horizontal distance of 50 metres to hit the ground at point `B`? (1 mark)
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- Only part of the graph applies to this model.
Find all values of `d` that are not suitable to use with this model, and explain why these values are not suitable. (2 marks)
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Statistics, STD2 S1 2009 HSC 21 MC
The mean of a set of ten scores is 14. Another two scores are included and the new mean is 16.
What is the mean of the two additional scores?
(A) 4
(B) 16
(C) 18
(D) 26
Algebra, STD2 A4 2013 HSC 22 MC
Leanne wants to build a rectangular vegetable garden in her backyard. She has 20 metres of fencing and will use a wall as one side of the garden. The plan for her garden is shown, where `x` metres is the width of her garden.
Which equation gives the area, `A`, of the vegetable garden?
(A) `A=10x-x^2`
(B) `A=10x-2x^2`
(C) `A=20x-x^2`
(D) `A=20x-2x^2`
Statistics, STD2 S5 2013 HSC 20 MC
There are 60 000 students sitting a state-wide examination. If the results form a normal distribution, how many students would be expected to score a result between 1 and 2 standard deviations above the mean?
You may assume for normally distributed data that:
-
- 68% of scores have `z`-scores between – 1 and 1
- 95% of scores have `z`-scores between – 2 and 2
- 99.7% of scores have `z`-scores between – 3 and 3.
(A) `8100`
(B) `16\ 200`
(C) `20\ 400`
(D) `28\ 500`
Quadratic, 2UA 2012 HSC 16c
Calculus, 2ADV C3 2008 HSC 10b
The diagram shows two parallel brick walls `KJ` and `MN` joined by a fence from `J` to `M`. The wall `KJ` is `s` metres long and `/_KJM=alpha`. The fence `JM` is `l` metres long.
A new fence is to be built from `K` to a point `P` somewhere on `MN`. The new fence `KP` will cross the original fence `JM` at `O`.
Let `OJ=x` metres, where `0<x<l`.
- Show that the total area, `A` square metres, enclosed by `DeltaOKJ` and `DeltaOMP` is given by
`A=s(x-l+l^2/(2x))sin alpha`. (3 marks)
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- Find the value of `x` that makes `A` as small as possible. Justify the fact that this value of `x` gives the minimum value for `A`. (3 marks)
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- Hence, find the length of `MP` when `A` is as small as possible. (1 mark)
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Calculus, 2ADV C3 2009 HSC 9b
An oil rig, `S`, is 3 km offshore. A power station, `P`, is on the shore. A cable is to be laid from `P` to `S`. It costs $1000 per kilometre to lay the cable along the shore and $2600 per kilometre to lay the cable underwater from the shore to `S`.
The point `R` is the point on the shore closest to `S`, and the distance `PR` is 5 km.
The point `Q` is on the shore, at a distance of `x` km from `R`, as shown in the diagram.
- Find the total cost of laying the cable in a straight line from `P` to `R` and then in a straight line from `R` to `S`. (1 mark)
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- Find the cost of laying the cable in a straight line from `P` to `S`. (1 mark)
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- Let `$C` be the total cost of laying the cable in a straight line from `P` to `Q`, and then in a straight line from `Q` to `S`.
Show that `C=1000(5-x+2.6sqrt(x^2+9))`. (2 marks)
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- Find the minimum cost of laying the cable. (4 marks)
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- New technology means that the cost of laying the cable underwater can be reduced to $1100 per kilometre.
Determine the path for laying the cable in order to minimise the cost in this case. (2 marks)
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Calculus, 2ADV C3 2011 HSC 10b
A farmer is fencing a paddock using `P` metres of fencing. The paddock is to be in the shape of a sector of a circle with radius `r` and sector angle `theta` in radians, as shown in the diagram.
- Show that the length of fencing required to fence the perimeter of the paddock is
`P=r(theta+2)`. (1 mark)
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- Show that the area of the sector is `A=1/2 Pr-r^2`. (1 mark)
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- Find the radius of the sector, in terms of `P`, that will maximise the area of the paddock. (2 marks)
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- Find the angle `theta` that gives the maximum area of the paddock. (1 mark)
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- Explain why it is only possible to construct a paddock in the shape of a sector if
`P/(2(pi+1)) <\ r\ <P/2` (2 marks)
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Mechanics, EXT2* M1 2009 HSC 6a
Two points, `A` and `B`, are on cliff tops on either side of a deep valley. Let `h` and `R` be the vertical and horizontal distances between `A` and `B` as shown in the diagram. The angle of elevation of `B` from `A` is `theta`, so that `theta=tan^-1(h/R)`.
At time `t=0`, projectiles are fired simultaneously from `A` and `B`. The projectile from `A` is aimed at `B`, and has initial speed `U` at an angle of `theta` above the horizontal. The projectile from `B` is aimed at `A` and has initial speed `V` at an angle `theta` below the horizontal.
The equations of motion for the projectile from `A` are
`x_1=Utcos theta` and `y_1=Utsin theta-1/2 g t^2`,
and the equations for the motion of the projectile from `B` are
`x_2=R-Vtcos theta` and `y_2=h-Vtsin theta-1/2 g t^2`, (DO NOT prove these equations.)
- Let `T` be the time at which `x_1=x_2`.
Show that `T=R/((U+V)\ cos theta)`. (1 mark)
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- Show that the projectiles collide. (2 marks)
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- If the projectiles collide on the line `x=lambdaR`, where `0<lambda<1`, show that
`V=(1/lambda-1)U`. (1 mark)
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Mechanics, EXT2* M1 2010 HSC 6b
A basketball player throws a ball with an initial velocity `v` m/s at an angle of `theta` to the horizontal. At the time the ball is released its centre is at `(0,0)`, and the player is aiming for the point `(d,h)` as shown on the diagram. The line joining `(0,0) ` and `(d,h)` makes an angle `alpha` with the horizontal, where `0<alpha<pi/2`.
Assume that at time `t` seconds after the ball is thrown its centre is at the point `(x,y)`, where
`x=vtcos theta`
`y=vt sin theta-5 t^2`. (DO NOT prove this.)
- If the centre of the ball passes through `(d,h)` show that
`v^2=(5d)/(cos theta sin theta-cos^2 theta tan alpha)` (3 marks)
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(2) What happens to `v` as `theta\ ->pi/2` ? (1 mark)
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- For a fixed value of `alpha`, let `F(theta)=cos theta sin theta-cos^2 theta tan alpha`.
Show that `F prime(theta)=0` when `tan2theta tan alpha=-1` (2 marks)
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- Using part (a)(ii)* or otherwise show that `F prime(theta)=0`, when `theta=alpha/2+pi/4`. (1 mark)
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*Please note for the purposes of this question, part (a)(ii) showed that when `tanA tanB=-1`, then `A-B=pi/2`
- Explain why `v^2` is a minimum when `theta=alpha/2+pi/4` (2 marks)
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Mechanics, EXT2* M1 2013 HSC 13c
Points `A` and `B` are located `d` metres apart on a horizontal plane. A projectile is fired from `A` towards `B` with initial velocity `u` m/s at angle `alpha` to the horizontal.
At the same time, another projectile is fired from `B` towards `A` with initial velocity `w` m/s at angle `beta` to the horizontal, as shown on the diagram.
The projectiles collide when they both reach their maximum height.
The equations of motion of a projectile fired from the origin with initial velocity `V` m/s at angle `theta` to the horizontal are
`x=Vtcostheta` and `y=Vtsintheta-g/2 t^2`. (DO NOT prove this.)
- How long does the projectile fired from `A` take to reach its maximum height? (2 marks)
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- Show that `usinalpha=w sin beta`. (1 mark)
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- Show that `d=(uw)/(g)sin(alpha+beta)`. (2 marks)
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Calculus, 2ADV C3 2012 HSC 16b
The diagram shows a point `T` on the unit circle `x^2+y^2=1` at an angle `theta` from the positive `x`-axis, where `0<theta<pi/2`.
The tangent to the circle at `T` is perpendicular to `OT`, and intersects the `x`-axis at `P`, and the line `y=1` intersects the `y`-axis at `B`.
- Show that the equation of the line `PT` is `xcostheta+ysin theta=1`. (2 marks)
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- Find the length of `BQ` in terms of `theta`. (1 mark)
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- Show that the area, `A`, of the trapezium `OPQB` is given by
`A=(2-sintheta)/(2costheta)` (2 marks)
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- Find the angle `theta` that gives the minimum area of the trapezium. (3 marks)
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Calculus, 2ADV C3 2013 HSC 14b
Two straight roads meet at `R` at an angle of 60°. At time `t=0` car `A` leaves `R` on one road, and car `B` is 100km from `R` on the other road. Car `A` travels away from `R` at a speed of 80 km/h, and car `B` travels towards `R` at a speed of 50 km/h.
The distance between the cars at time `t` hours is `r` km.
- Show that `r^2=12\ 900t^2-18\ 000t+10\ 000`. (2 marks)
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- Find the minimum distance between the cars. (3 marks)
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Calculus, EXT1* C1 2013 HSC 16b
Trout and carp are types of fish. A lake contains a number of trout. At a certain time, 10 carp are introduced into the lake and start eating the trout. As a consequence, the number of trout, `N`, decreases according to
`N=375-e^(0.04t)`,
where `t` is the time in months after the carp are introduced.
The population of carp, `P`, increases according to `(dP)/(dt)=0.02P`.
- How many trout were in the lake when the carp were introduced? (1 mark)
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- When will the population of trout be zero? (1 mark)
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- Sketch the number of trout as a function of time. (1 marks)
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- When is the rate of increase of carp equal to the rate of decrease of trout? (3 marks)
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- When is the number of carp equal to the number of trout? (2 marks)
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Calculus, 2ADV C3 2011 HSC 7b
The velocity of a particle moving along the `x`-axis is given by
`v=8-8e^(-2t)`,
where `t` is the time in seconds and `x` is the displacement in metres.
- Show that the particle is initially at rest. (1 mark)
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- Show that the acceleration of the particle is always positive. (1 mark)
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- Explain why the particle is moving in the positive direction for all `t>0`. (2 marks)
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- As `t->oo`, the velocity of the particle approaches a constant.
Find the value of this constant. (1 mark)
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- Sketch the graph of the particle's velocity as a function of time. (2 marks)
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Financial Maths, 2ADV M1 2008 HSC 9b
Peter retires with a lump sum of $100 000. The money is invested in a fund which pays interest each month at a rate of 6% per annum, and Peter receives a fixed monthly payment `$M` from the fund. Thus the amount left in the fund after the first monthly payment is `$(100\ 500-M)`.
- Find a formula for the amount, `$A_n`, left in the fund after `n\ ` monthly payments. (2 marks)
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- Peter chooses the value of `M` so that there will be nothing left in the fund at the end of the 12th year (after 144 payments). Find the value of `M`. (3 marks)
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Proof, EXT2* P2 2013 HSC 14a
- Show that for `k>0,\ \ 1/(k+1)^2-1/k+1/(k+1)<0`. (1 mark)
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- Use mathematical induction to prove that for all integers `n>=2`,
`1/1^2+1/2^2+1/3^2+\ …\ +1/n^2<2-1/n`. (3 marks)
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Calculus, 2ADV C4 2010 HSC 3b
- Sketch the curve `y=lnx`. (1 mark)
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- Use the trapezoidal rule with 3 function values to find an approximation to `int_1^3 lnx\ dx` (2 marks)
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- State whether the approximation found in part (ii) is greater than or less than the exact value of `int_1^3 lnx\ dx`. Justify your answer. (1 mark)
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Probability, 2ADV S1 2013 HSC 15d
Pat and Chandra are playing a game. They take turns throwing two dice. The game is won by the first player to throw a double six. Pat starts the game.
- Find the probability that Pat wins the game on the first throw. (1 mark)
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- What is the probability that Pat wins the game on the first or on the second throw? (2 marks)
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- Find the probability that Pat eventually wins the game. (2 marks)
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