The point `P` divides the interval joining `A(–1, –2)` to `B(9, 3)` internally in the ratio `4 : 1`. Find the coordinates of `P`. (2 marks)
Mechanics, EXT2* M1 2012 HSC 13c
A particle is moving in a straight line according to the equation
`x = 5 + 6 cos 2t + 8 sin 2t`,
where `x` is the displacement in metres and `t` is the time in seconds.
- Prove that the particle is moving in simple harmonic motion by showing that `x` satisfies an equation of the form `ddot x = -n^2 (x\ - c)`. (2 marks)
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- When is the displacement of the particle zero for the first time? (3 marks)
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Geometry and Calculus, EXT1 2012 HSC 13b
- Find the horizontal asymptote of the graph
`qquad qquad y=(2x^2)/(x^2 + 9)`. (1 mark) - Without the use of calculus, sketch the graph
`qquad qquad y=(2x^2)/(x^2 + 9)`,
showing the asymptote found in part (i). (2 marks)
Functions, EXT1 F1 2012 HSC 12b
Let `f(x) = sqrt(4x-3)`
- Find the domain of `f(x)`. (1 mark)
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- Find an expression for the inverse function `f^(-1) (x)`. (2 marks)
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- Find the points where the graphs `y = f(x)` and `y=x` intersect. (1 mark)
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- On the same set of axes, sketch the graphs `y = f(x)` and `y = f^(-1) (x)` showing the information found in part (iii). (2 marks)
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Calculus, EXT1 C2 2012 HSC 11d
Use the substitution `u = 2 - x` to evaluate `int_1^2 x (2 - x)^5\ dx`. (3 marks)
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Functions, EXT1 F1 2012 HSC 11c
Solve `x/(x - 3) < 2`. (3 marks)
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Calculus, EXT1 C2 2012 HSC 11a
Evaluate `int_0^3 1/(9 + x^2)\ dx`. (3 marks)
Calculus, EXT1 C2 2012 HSC 9 MC
What is the derivative of `cos^(–1) (3x)`?
- `1/(3 sqrt(1 - 9x^2))`
- `(-1)/(3 sqrt(1 - 9x^2))`
- `3/sqrt(1 - 9x^2)`
- `(-3)/sqrt(1 - 9x^2)`
Functions, EXT1 F2 2012 HSC 3 MC
A polynomial equation has roots `alpha`, `beta`, `gamma` where
`alpha + beta + gamma = -2`, `alphabeta + alphagamma + betagamma = 3`, `alphabetagamma = 1`.
Which polynomial equation has the roots `alpha`, `beta`, and `gamma`?
- `x^3 + 2x^2 + 3x + 1 = 0`
- `x^3 + 2x^2 + 3x- 1 = 0`
- `x^3- 2x^2 + 3x + 1 = 0`
- `x^3- 2x^2 + 3x- 1 = 0`
Algebra, EXT1 2012 HSC 1 MC
Which expression is a correct factorisation of `x^3 - 27`?
- ` (x - 3)( x^2 - 3x + 9)`
- `(x - 3)( x^2 - 6x + 9)`
- `(x - 3)( x^2 + 3x + 9)`
- `(x - 3)( x^2 + 6x + 9)`
Functions, EXT1 F1 2011 HSC 1c
Solve `(4-x)/x <1`. (3 marks)
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Probability, 2ADV S1 2009 HSC 5b
On each working day James parks his car in a parking station which has three levels. He parks his car on a randomly chosen level. He always forgets where he has parked, so when he leaves work he chooses a level at random and searches for his car. If his car is not on that level, he chooses a different level and continues in this way until he finds his car.
- What is the probability that his car is on the first level he searches? (1 mark)
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- What is the probability that he must search all three levels before he finds his car? (1 mark)
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- What is the probability that on every one of the five working days in a week, his car is not on the first level he searches? (1 mark)
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Plane Geometry, 2UA 2009 HSC 4c
In the diagram, `Delta ABC` is a right-angled triangle, with the right angle at `C`. The midpoint of `AB` is `M`, and `MP _|_ AC`.
Copy or trace the diagram into your writing booklet.
- Prove that `Delta AMP` is similar to `Delta ABC`. (2 marks)
- What is the ratio of `AP` to `AC`? (1 mark)
- Prove that `Delta AMC` is isosceles. (2 marks)
- Show that `Delta ABC` can be divided into two isosceles triangles. (1 mark)
- Copy or trace this triangle into your writing booklet and show how to divide it into four isosceles triangles. (1 mark)
Functions, 2ADV F1 2009 HSC 3b
Calculus, 2ADV C4 2009 HSC 2b
- Find `int 5\ dx`. (1 mark)
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- Find `int 3/((x - 6)^2)\ dx`. (2 marks)
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- Evaluate `int_1^4 x^2 + sqrtx\ dx`. (3 marks)
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Calculus, 2ADV C2 2009 HSC 2ai
Differentiate with respect to `x`:
`x sin x` (2 marks)
Trigonometry, 2ADV T2 2009 HSC 1e
Find the exact value of `theta` such that `2 cos theta = 1`, where `0 <= theta <= pi/2`. (2 marks)
Calculus, 2ADV C1 2009 HSC 1d
Find the gradient of the tangent to the curve `y = x^4- 3x` at the point `(1, –2)`. (2 marks)
Functions, 2ADV F1 2009 HSC 1a
Sketch the graph of `y-2x = 3`, showing the intercepts on both axes. (2 marks)
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Calculus, 2ADV C1 2010 HSC 7b
The parabola shown in the diagram is the graph `y = x^2`. The points `A (–1,1)` and `B (2, 4)` are on the parabola.
- Find the equation of the tangent to the parabola at `A`. (2 marks)
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- Let `M` be the midpoint of `AB`.
There is a point `C` on the parabola such that the tangent at `C` is parallel to `AB`.
Show that the line `MC` is vertical. (2 marks)
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- The tangent at `A` meets the line `MC` at `T`.
Show that the line `BT` is a tangent to the parabola. (2 marks)
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Linear Functions, 2UA 2010 HSC 3a
In the diagram `A`, `B` and `C` are the points `( –2, –4 ),\ (12,6)` and `(6,8)` respectively.
The point `N (2,2)` is the midpoint of `AC`. The point `M` is the midpoint of `AB`.
- Find the coordinates of `M`. (1 mark)
- Find the gradient of `BC`. (1 mark)
- Prove that `Delta ABC` is similar to `Delta AMN`. (2 marks)
- Find the equation of `MN`. (2 marks)
- Find the exact length of `BC`. (1 mark)
- Given that the area of `Delta ABC` is `44` square units, find the perpendicular distance from `A` to `BC`. (1 mark)
Functions, EXT1* F1 2010 HSC 2b
Solve the inequality `x^2 − x − 12 < 0`. (2 marks)
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Calculus, 2ADV C2 2010 HSC 2a
Differentiate `cosx/x` with respect to `x`. (2 marks)
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Calculus, 2ADV C2 2010 HSC 1e
Differentiate `x^2 tan x` with respect to `x`. (2 marks)
Financial Maths, 2ADV M1 2011 HSC 9d
- Rationalise the denominator in the expression `1/(sqrtn + sqrt(n+1))` where `n` is an integer and `n >= 1`. (1 mark)
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- Using your result from part (i), or otherwise, find the value of the sum
`1/(sqrt1+ sqrt2) + 1/(sqrt2 + sqrt3) + 1/(sqrt3 + sqrt4) + ... + 1/(sqrt99 + sqrt100)` (2 marks)
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Calculus, 2ADV C2 2012 HSC 12aii
Differentiate with respect to `x`.
`(cos x)/(x^2)`. (2 marks)
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Functions, 2ADV F1 2010 HSC 1b
Find integers `a` and `b` such that `1/(sqrt5\ - 2) = a + b sqrt5`. (2 marks)
Functions, 2ADV F1 2010 HSC 1a
Solve `x^2 = 4x`. (2 marks)
Probability, 2ADV S1 2012 HSC 13c
Two buckets each contain red marbles and white marbles. Bucket `A` contains 3 red and 2 white marbles. Bucket `B` contains 3 red and 4 white marbles.
Chris randomly chooses one marble from each bucket.
- What is the probability that both marbles are red? (1 mark)
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- What is the probability that at least one of the marbles is white? (1 mark)
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- What is the probability that both marbles are the same colour? (2 marks)
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Calculus, 2ADV C3 2011 HSC 7a
Let `f(x) = x^3-3x + 2`.
- Find the coordinates of the stationary points of `y = f(x)`, and determine their nature. (3 marks)
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- Hence, sketch the graph `y = f(x)` showing all stationary points and the `y`-intercept. (2 marks)
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Calculus, 2ADV C4 2011 HSC 6c
The diagram shows the graph `y = 2 cos x` .
- State the coordinates of `P`. (1 mark)
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- Evaluate the integral `int_0^(pi/2) 2 cos x\ dx`. (2 marks)
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- Indicate which area in the diagram, `A`, `B`, `C` or `D`, is represented by the integral
`int_((3pi)/2)^(2pi) 2 cos x\ dx`. (1 mark)
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- Using parts (ii) and (iii), or otherwise, find the area of the region bounded by the curve `y = 2 cos x` and the `x`-axis, between `x = 0` and `x = 2pi` . (1 mark)
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- Using the parts above, write down the value of
`int_(pi/2)^(2pi) 2 cos x\ dx`. (1 mark)
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Plane Geometry, 2UA 2011 HSC 6a
The diagram shows a regular pentagon `ABCDE`. Sides `ED` and `BC` are produced to meet at `P`.
- Find the size of `/_CDE`. (1 mark)
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- Hence, show that `Delta EPC` is isosceles. (2 marks)
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Calculus, 2ADV C2 2011 HSC 4a
Differentiate `x/sinx` with respect to `x`. (2 marks)
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Linear Functions, 2UA 2011 HSC 3c
The diagram shows a line `l_1`, with equation `3x + 4y - 12 = 0`, which intersects the `y`-axis at `B`.
A second line `l_2`, with equation `4x - 3y = 0`, passes through the origin `O` and intersects `l_1` at `E`.
- Show that the coordinates of `B` are `(0, 3)`. (1 mark)
- Show that `l_1` is perpendicular to `l_2`. (2 marks)
- Show that the perpendicular distance from `O` to `l_1` is `12/5`. (1 mark)
- Using Pythagoras’ theorem, or otherwise, find the length of the interval `BE`. (1 mark)
- Hence, or otherwise, find the area of `Delta BOE`. (1 mark)
Calculus, 2ADV C1 2011 HSC 2c
Find the equation of the tangent to the curve `y = (2x + 1)^4` at the point where `x = –1`. (3 marks)
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Functions, 2ADV F1 2011 HSC 1b
Simplify `(n^2 - 25)/(n - 5)`. (1 mark)
Functions, 2ADV F1 2011 HSC 1f
Rationalise the denominator of `4/(sqrt5\ - sqrt3)`.
Give your answer in the simplest form. (2 marks)
Functions, 2ADV F1 2011 HSC 1e
Solve `2 -3x <= 8`. (2 marks)
Functions, 2ADV F1 2011 HSC 1a
Evaluate `root(3)(651/(4pi))` correct to four significant figures. (2 marks)
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Trigonometry, 2ADV T1 2012 HSC 13a
The diagram shows a triangle `ABC`. The line `2x + y = 8` meets the `x` and `y` axes at the points `A` and `B` respectively. The point `C` has coordinates `(7, 4)`.
- Calculate the distance ` AB `. (2 marks)
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- It is known that `AC = 5` and `BC = sqrt 65 \ \ \ `(Do NOT prove this)
Calculate the size of `angle ABC` to the nearest degree. (2 marks)
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- The point `N` lies on `AB` such that `CN` is perpendicular to `AB`.
Find the coordinates of `N`. (3 marks)
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Integration, 2UA 2012 HSC 12d
At a certain location a river is `12` metres wide. At this location the depth of the river, in metres, has been measured at `3` metre intervals. The cross-section is shown below.
- Use Simpson's rule with the five depth measurements to calculate the approximate area of the cross-section (3 marks)
- The river flows at 0.4 metres per second.
- Calculate the approximate volume of water flowing through the cross-section in 10 seconds. (1 mark)
Functions, EXT1* F1 2012 HSC 11b
Solve `|\ 3x -1\ | < 2` (2 marks)
Functions, 2ADV F1 2012 HSC 11a
Factorise `2x^2 - 7x +3` (2 marks)
Calculus, 2ADV C1 2012 HSC 11c
Find the equation of the tangent to the curve `y = x^2` at the point where `x = 3`. (2 marks)
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Functions, 2ADV F1 2012 HSC 2 MC
Which of the following is equal to `1/(2sqrt5\-sqrt3)`?
- `(2sqrt5\-sqrt3)/7`
- `(2sqrt5 + sqrt3)/7`
- `(2sqrt5\-sqrt3)/17`
- `(2sqrt5 + sqrt3)/17`
Functions, 2ADV F1 2012 HSC 1 MC
What is `4.097 84` correct to three significant figures?
- `4.09`
- `4.10`
- `4.097`
- `4.098`
Linear Functions, 2UA 2013 HSC 12b
The points `A(–2, –1)`, `B(–2, 24)`, `C(22, 42)` and `D(22, 17)` form a parallelogram as shown. The point `E(18, 39)` lies on `BC`. The point `F` is the midpoint of `AD`.
- Show that the equation of the line through `A` and `D` is `3x- 4y + 2 = 0`. (2 marks)
- Show that the perpendicular distance from `B` to the line through `A` and `D` is `20` units. (1 mark)
- Find the length of `EC`. (1 mark)
- Find the area of the trapezium `EFDC`. (2 marks)
Calculus, 2ADV C3 2013 HSC 12a
The cubic `y = ax^3 + bx^2 + cx + d` has a point of inflection at `x = p`.
Show that `p= - b/(3a)`. (2 marks)
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Calculus, 2ADV C2 2013 HSC 11c
Differentiate `(sinx -1)^8`. (2 marks)
Functions, 2ADV F1 2013 HSC 11a
Evaluate `ln3` correct to three significant figures. (1 mark)
Functions, 2ADV F1 2013 HSC 1 MC
What are the solutions of `2x^2-5x-1 = 0`?
- `x = (-5 +-sqrt17)/4`
- `x = (5 +-sqrt17)/4`
- `x = (-5 +-sqrt33)/4`
- `x = (5 +-sqrt33)/4`
Financial Maths, STD2 F1 2010 HSC 23d
Warrick has a net income of $590 per week. He has created a budget to help manage his money.
- Find the value of `X`, the amount that Warrick allocates towards electricity each week. (1 mark)
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- Warrick has an unexpectedly high telephone and internet bill. For the last three weeks, he has put aside his savings as well as his telephone and internet money to pay the bill.
How much money has he put aside altogether to pay the bill? (1 mark)
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- The bill for the telephone and internet is $620. It is due in two weeks time. Warrick realises he has not put aside enough money to pay the bill.
How could Warrick reallocate non-essential funds in his budget so he has enough money to pay the bill? Justify your answer with suitable reasons and calculations. (3 marks)
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Measurement, STD2 M7 2011 HSC 24a
Part of the floor plan of a house is shown. The plan is drawn to scale.
- What is the width of the stairwell, in millimetres? (1 mark)
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- What are the internal dimensions of the bathroom, in millimetres? (1 mark)
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- What is the length `AB`, the internal length of the rumpus room, in millimetres? (1 mark)
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Probability, 2UG 2011 HSC 26a
The two spinners shown are used in a game.
Each arrow is spun once. The score is the total of the two numbers shown by the arrows.
A table is drawn up to show all scores that can be obtained in this game.
- What is the value of `X` in the table? (1 mark)
- What is the probability of obtaining a score less than 4? (1 mark)
- On Spinner `B`, a 2 is obtained. What is the probability of obtaining a score of 3? (1 mark)
- Elise plays a game using the spinners with the following financial outcomes.
⇒ Win `$12` for a score of `4`
⇒ Win nothing for a score of less than `4`
⇒ Lose `$3` for a score of more than `4`
It costs `$5` to play this game. Will Elise expect a gain or a loss and how much will it be?
Justify your answer with suitable calculations. (3 marks)
Probability, STD2 S2 2011 HSC 25c
At another school, students who use mobile phones were surveyed. The set of data is shown in the table.
- How many students were surveyed at this school? (1 mark)
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- Of the female students surveyed, one is chosen at random. What is the probability that she uses pre-paid? (1 mark)
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Ten new male students are surveyed and all ten are on a plan. The set of data is updated to include this information.
- What percentage of the male students surveyed are now on a plan? Give your answer to the nearest per cent. (1 mark)
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Statistics, STD2 S1 2011 HSC 25b
The graph below displays data collected at a school on the number of students
in each Year group, who own a mobile phone.
- Which Year group has the highest percentage of students with mobile phones? (1 mark)
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- Two students are chosen at random, one from Year 9 and one from Year 10.
Which student is more likely to own a mobile phone?
Justify your answer with suitable calculations. (2 marks)
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- Identify a trend in the data shown in the graph. (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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Algebra, STD2 A4 2013 HSC 30a
Wind turbines, such as those shown, are used to generate power.
In theory, the power that could be generated by a wind turbine is modelled using the equation
`T = 20\ 000w^3`
where | `T` is the theoretical power generated, in watts |
`w` is the speed of the wind, in metres per second. |
- Using this equation, what is the theoretical power generated by a wind turbine if the wind speed is 7.3 m/s ? (1 mark)
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In practice, the actual power generated by a wind turbine is only 40% of the theoretical power.
- If `A` is the actual power generated, in watts, write an equation for `A` in terms of `w`. (1 mark)
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The graph shows both the theoretical power generated and the actual power generated by a particular wind turbine.
- Using the graph, or otherwise, find the difference between the theoretical power and the actual power generated when the wind speed is 9 m/s. (1 mark)
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A particular farm requires at least 4.4 million watts of actual power in order to be self-sufficient.
- What is the minimum wind speed required for the farm to be self-sufficient? (1 mark)
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A more accurate formula to calculate the power (`P`) generated by a wind turbine is
`P = 0.61 xx pi xx r^2 × w^3`
where | `r` is the length of each blade, in metres |
`w` is the speed of the wind, in metres per second. |
Each blade of a particular wind turbine has a length of 43 metres.The turbine operates at a wind speed of 8 m/s.
- Using the formula above, if the wind speed increased by 10%, what would be the percentage increase in the power generated by this wind turbine? (3 marks)
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Financial Maths, STD2 F1 2011 HSC 23a
Sri has a gross salary of $56 350. She has tax deductions of $350 for union fees, $2000 in work-related expenses and $250 in donations to charities.
The Medicare levy is 1.5% of her taxable income.
Calculate Sri’s Medicare levy. (3 marks)
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