A car takes 6 hours to complete a journey when travelling at 60 km/h.
How long would the same journey take if the car were travelling at 100 km/h?
- 36 minutes
- 1 hour and 40 minutes
- 3 hours and 6 minutes
- 3 hours and 36 minutes
Algebra, 2UG 2012 HSC 14 MC
Which of the following expresses `2x^2 (5- x)-x(x-2)` in its simplest form?
(A) `–2x^3 + 9x^2 + 2x`
(B) `–2x^3- 9x^2- 2x `
(C) `9x^2- x + 2`
(D) `9x^2- x- 2`
Financial Maths, STD2 F4 2012 HSC 24 MC
A $400 000 loan can be repaid by making either monthly or fortnightly repayments.
The graph shows the loan balances over time using these two different methods of repayment.
The monthly repayment is $2796.86 and the fortnightly repayment is $1404.76.
What is the difference in the total interest paid using the two different methods of
repayment, to the nearest dollar?
- $51 596
- $166 823
- $210 000
- $234 936
Measurement, STD2 M6 2012 HSC 10 MC
What is the area of this triangle, to the nearest square metre?
- `33\ text(m²)`
- `37\ text(m²)`
- `42\ text(m²)`
- `44\ text(m²)`
Algebra, STD2 A2 2012 HSC 8 MC
Dots were used to create a pattern. The first three shapes in the pattern are shown.
The number of dots used in each shape is recorded in the table.
\begin{array} {|l|c|}
\hline
\rule{0pt}{2.5ex} \text{Shape $(S)$} \rule[-1ex]{0pt}{0pt} &\;\;\; 1 \;\;\; & \;\; \;2 \;\;\; & \;\;\; 3 \;\;\; \\
\hline
\rule{0pt}{2.5ex} \text{Number of dots $(N)$} \rule[-1ex]{0pt}{0pt} &\;\;\; 6 \;\;\; & \;\; \;8 \;\;\; & \;\; \;10\; \;\; \\
\hline
\end{array}
How many dots would be required for Shape 156?
- `316`
- `520`
- `624`
- `936`
Statistics, STD2 S1 2012 HSC 7 MC
The Pi Company has two bakeries. The radar chart displays the monthly sales for the bakeries.
What was the difference in sales in June between the two bakeries?
- $7.50
- $17.50
- $7500
- $17 500
Measurement, STD2 M1 2012 HSC 6 MC
What is the volume of this rectangular prism in cubic centimetres?
- 6 cm³
- 600 cm³
- 60 000 cm³
- 6 000 000 cm³
Measurement, STD2 M6 2011 HSC 4 MC
The angle of depression from a kookaburra’s feet to a worm on the ground is 40°. The worm is 15 metres from a point on the ground directly below the kookaburra’s feet.
How high above the ground are the kookaburra's feet, correct to the nearest metre?
- 10 m
- 11 m
- 13 m
- 18 m
Measurement, STD2 M6 2012 HSC 4 MC
Which expression could be used to calculate the value of `x` in this triangle?
- `29 xx cos40^@`
- `29 xx cos 50^@`
- `cos40^@/29`
- `cos50^@/29`
Statistics, STD2 S1 2012 HSC 1 MC
A set of 15 scores is displayed in a stem-and-leaf plot.
What is the median of these scores?
- 7
- 8
- 77
- 78
Measurement, STD2 M1 2013 HSC 19 MC
A logo is designed using half of an annulus.
What is the area of the logo, to the nearest cm²?
- `25\ text(cm²)`
- `33\ text(cm²)`
- `132\ text(cm²)`
- `143\ text(cm²)`
Measurement, STD2 M1 2013 HSC 16 MC
The shaded region shows a quadrant with a rectangle removed.
What is the area of the shaded region, to the nearest cm2?
- 38 cm²
- 52 cm²
- 61 cm²
- 70 cm²
Statistics, STD2 S1 2013 HSC 15 MC
The frequency histogram shows the number of goals scored by a football team in each game in a season.
What is the mean number of goals scored per game by this team?
- 4
- 4.5
- 5
- 5.5
Statistics, STD2 S1 2013 HSC 14 MC
The July sales prices for properties in a suburb were:
$552 000, $595 000, $607 000, $607 000, $682 000, and $685 000.
On 1 August, another property in the same suburb was sold for over one million dollars.
If the property had been sold in July, what effect would it have had on the mean and median sale prices for July?
- Both the mean and median would have changed.
- Neither the mean nor the median would have changed.
- The mean would have changed and the median would have stayed the same.
- The mean would have stayed the same and the median would have changed.
Financial Maths, STD2 F1 2013 HSC 13 MC
Polly borrowed $11 000. She repaid the loan in full at the end of two years with a lump sum of $12 000.
What annual simple interest rate was she charged?
- 4.17%
- 4.55%
- 8.33%
- 9.09%
Measurement, STD2 M1 2013 HSC 12 MC
A square pyramid fits exactly on top of a cube to form a solid.
What is the volume of the solid?
- 513 cm³
- 999 cm³
- 1242 cm³
- 1539 cm³
Probability, STD2 S2 2013 HSC 10 MC
Students studying vocational education courses were surveyed about their living arrangements.
One of these students is selected at random.
What is the probability that this student is male and living with his parent(s)?
- 31%
- 40%
- 56%
- 77%
Financial Maths, STD2 F1 2013 HSC 9 MC
Lynne invests $1000 for a term of 15 months. Simple interest is paid on the investment at a rate of 3.75% per annum.
How much will Lynne's investment be worth at the end of the term?
- $1046.88
- $1047.09
- $1296.88
- $1468.75
Statistics, STD2 S1 2013 HSC 6 MC
A survey was conducted where people were asked which of two brands of smartphones they preferred. The results were:
- 48% preferred Brand X
- 52% preferred Brand Y
A graph displaying the data is to be included in a magazine article. The editor of the magazine wishes to ensure that the graph is not misleading in any way.
Which graph should the editor choose to include in the article?
Algebra, 2UG 2013 HSC 5 MC
Which of the following is `(12T^2)/(3Txx2W)` in its simplest form?
- `(2T)/W`
- `2TW`
- `(8T)/W`
- `8TW`
Financial Maths, STD2 F1 2013 HSC 3 MC
Luke's normal rate of pay is $24.80 per hour. In one week he worked 14 hours at the normal rate, 4 hours at time-and-a-half, and 3 .5 hours at double time. He was also paid a wet weather allowance of $50 for the week.
What was his pay for the week?
- $583.20
- $620.40
- $669.60
- $719.60
Statistics, STD2 S4 2013 HSC 2 MC
Which graph best shows data with a correlation closest to 0.3?
Probability, STD2 S2 2013 HSC 1 MC
Which of the following events would be LEAST likely to occur?
- Tossing a fair coin and obtaining a head
- Rolling a standard six-sided die and obtaining a 3
- Randomly selecting the letter 'G' from the 26 letters of the alphabet
- Winning first prize in a raffle of 100 tickets in which you have 4 tickets
Quadratic, 2UA 2011 HSC 2a
The quadratic equation `x^2-6x+2=0` has roots `alpha` and `beta`.
- Find `alpha+beta`. (1 mark)
- Find `alpha beta`. (1 mark)
- Find `1/alpha+1/beta`. (1 mark)
Quadratic, 2UA 2012 HSC 11e
Find the coordinates of the focus of the parabola `x^2=16(y-2)`. (2 marks)
Quadratric, 2UA 2013 HSC 7 MC
A parabola has focus `(5,0)` and directrix `x=1`.
What is the equation of the parabola?
(A) `y^2=16(x-5)`
(B) `y^2=8(x-3)`
(C) `y^2=-16(x-5)`
(D) `y^2=-8(x-3)`
Show Worked Solution
`y^2=4ax`
`text{Focal length}=a=(5-1)/2=2`
`text(Vertex)=(3,0)`
| `:.\ y^2` | `=4xx2(x-3)` |
| `=8(x-3)` |
`=>B`
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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Show Worked Solution
| i. |  |
`A=text(Area)\ Delta OJK+text(Area)\ Delta OMP`
♦♦♦ Low mean marks highlighted (although exact data not available before 2009).
`text(Using sine rule)`
`text(Area)\ Delta OJK=1/2\ x s sin alpha`
`text(Area)\ DeltaOMP =>text(Need to find)\ \ MP`
| `/_OKJ` | `=/_MPO\ \ text{(alternate angles,}\ MP\ text(||)\ KJtext{)}` |
| `/_PMO` | `=/_OJK=alpha\ \ text{(alternate angles,}\ MP\ text(||)\ KJtext{)}` |
`:.\ DeltaOJK\ text(|||)\ Delta OMP\ \ text{(equiangular)}`
| `=>x/s` | `=(l-x)/(MP)\ ` | ` text{(corresponding sides of similar triangles)}` |
| `MP` | `=(l-x)/x *s` |
| `text(Area)\ Delta\ OMP` | `=1/2 (l-x)* MP * sin alpha` |
| `=1/2 (l-x)*((l-x))/x* s sin alpha` |
| `:. A` | `=1/2 x*s sin alpha+1/2 (l-x)*((l-x))/x* s sin alpha` |
| `=s sin alpha(1/2 x+1/2 (l-x)*((l-x))/x)` | |
| `=s sin alpha(1/2 x+(l-x)^2/(2x))` | |
| `=s sin alpha(1/2 x+l^2/(2x)-l+1/2 x)` | |
| `=s(x-l+l^2/(2x))sin alpha\ \ \ \ text(… as required)` |
ii. `text(Find)\ x\ text(such that)\ A\ text(is a minimum)`
MARKER’S COMMENT: Students who could not complete part (i) are reminded that they can still proceed to part (ii) and attempt to differentiate the result given.
Note that `l` and `alpha` are constants when differentiating.
Note that `l` and `alpha` are constants when differentiating.
| `A` | `=s(x-l+l^2/(2x))sin alpha` |
| `(dA)/(dx)` | `=s(1-l^2/(2x^2))sin alpha` |
`text(MAX/MIN when)\ (dA)/(dx)=0`
| `s(1-l^2/(2x^2))sin alpha` | `=0` |
| `l^2/(2x^2)` | `=1` |
| `2x^2` | `=l^2` |
| `x^2` | `=l^2/2` |
| `x` | `=l/sqrt2,\ \ \ x>0` |
`(d^2A)/(dx^2)=s((l^2)/(2x^3))sin alpha`
`text(S)text(ince)\ \ 0<alpha<90°\ \ =>\ sin alpha>0,\ \ l>0\ \ text(and)\ \ x>0`
`(d^2A)/(dx^2)>0\ \ \ =>text(MIN at)\ \ x=l/sqrt2`
iii. `text(S)text(ince)\ \ MP=((l-x))/x s\ \ text(and MIN when)\ \ x=l/sqrt2`
| `MP` | `=((l-l/sqrt2)/(l/sqrt2))s xx sqrt2/sqrt2` |
| `=((sqrt2 l-l))/l s` | |
| `=(sqrt2-1)s\ \ text(metres)` |
`:.\ MP=(sqrt2-1)s\ \ text(metres when)\ A\ text(is a MIN.)`
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 2010 HSC 5a
A rainwater tank is to be designed in the shape of a cylinder with radius `r` metres and height `h` metres.
The volume of the tank is to be 10 cubic metres. Let `A` be the surface area of the tank, including its top and base, in square metres.
- Given that `A=2pir^2+2pi r h`, show that `A=2 pi r^2+20/r`. (2 marks)
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- Show that `A` has a minimum value and find the value of `r` for which the minimum occurs. (3 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 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 in the Physical World, 2UA 2008 HSC 6b
The graph shows the velocity of a particle, `v` metres per second, as a function of time, `t` seconds.
- What is the initial velocity of the particle? (1 mark)
- When is the velocity of the particle equal to zero? (1 mark)
- When is the acceleration of the particle equal to zero? (1 mark)
- By using Simpson's Rule with five function values, estimate the distance travelled by the particle between `t=0` and `t=8`. (3 marks)
Show Worked Solution
(i) `text(Find)\ v \ text(when) t=0`
`v=20\ \ text(m/s)`
(ii) `text(Particle comes to rest at)\ t=10\ text{seconds (from graph)}`
(iii) `text(Acceleration is zero when)\ t=6\ text{seconds (from graph)}`
| (iv) |  |
MARKER’S COMMENT: Less errors were made by students using a table and the given formula. Note however, that the formula `A~~(b-a)/6xx` `[f(a)+4f((a+b)/2)+f(b)]` produced the most errors.
| `text(Area)` | `~~h/3[y_0+y_n+4text{(odds)}+2text{(evens)}]` |
| `~~h/3[y_0+y_4+4(y_1+y_3)+2(y_2)]` | |
| `~~2/3[20+60+4(50+80)+2(70)]` | |
| `~~2/3[740]` | |
| `~~493 1/3` |
`:.\ text{Distance travelled is 493 1/3 m (approx.)}`
Calculus, 2ADV C4 2009 HSC 7a
The acceleration of a particle is given by
`a=8e^(-2t)+3e^(-t)`,
where `x` is the displacement in metres and `t` is the time in seconds.
Initially its velocity is `text(– 6 ms)^(–1)` and its displacement is 5 m.
- Show that the displacement of the particle is given by
- `qquad x=2e^(-2t)+3e^-t+t`. (2 marks)
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- Find the time when the particle comes to rest. (3 marks)
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- Find the displacement when the particle comes to rest. (1 mark)
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Calculus, EXT1* C1 2010 HSC 7a
The acceleration of a particle is given by
`ddotx=4cos2t`,
where `x` is the displacement in metres and `t` is the time in seconds.
Initially the particle is at the origin with a velocity of `text(1 ms)^(–1)`.
- Show that the velocity of the particle is given by
`dotx=2sin2t+1`. (2 marks)
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- Find the time when the particle first comes to rest. (2 marks)
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- Find the displacement, `x`, of the particle in terms of `t`. (2 marks)
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Calculus, 2ADV C4 2012 HSC 15b
The velocity of a particle is given by
`v=1-2cost`,
where `x` is the displacement in metres and `t` is the time in seconds. Initially the particle is 3 m to the right of the origin.
- Find the initial velocity of the particle. (1 mark)
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- Find the maximum velocity of the particle. (1 mark)
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- Find the displacement, `x`, of the particle in terms of `t`. (2 marks)
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- Find the position of the particle when it is at rest for the first time. (2 marks)
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Calculus, EXT1 C1 2010 HSC 2b
The mass `M` of a whale is modelled by
`M=36-35.5e^(-kt)`
where `M` is measured in tonnes, `t` is the age of the whale in years and `k` is a positive constant.
- Show that the rate of growth of the mass of the whale is given by the differential equation
`qquad qquad (dM)/(dt)=k(36-M)` (1 mark)
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- When the whale is 10 years old its mass is 20 tonnes.
Find the value of `k`, correct to three decimal places. (2 marks)
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- According to this model, what is the limiting mass of the whale? (1 mark)
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Calculus, EXT1 C1 2011 HSC 5b
To test some forensic science students, an object has been left in the park. At 10am the temperature of the object is measured to be 30°C. The temperature in the park is a constant 22°C. The object is moved immediately to a room where the temperature is a constant 5°C.
The temperature of the object in the room can be modelled by the equation
`T=5+25e^(-kt)`,
where `T` is the temperature of the object in degrees Celcius, `t` is the time in hours since the object was placed in the room and `k` is a constant.
After one hour in the room the temperature of the object is 20°C.
- Show that `k=ln(5/3)` (2 marks)
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- In a similar manner, the temperature of the object in the park before it was discovered can be modelled by an equation in the form `T=A+Be^(-kt)`, with the same constant `k=ln(5/3)`.
Find the time of day when the object had a temperature of 37°C. (3 marks)
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Calculus, EXT1* C1 2008 HSC 5c
Light intensity is measured in lux. The light intensity at the surface of a lake is 6000 lux. The light intensity, `I` lux, a distance `s` metres below the surface of the lake is given by
`I=Ae^(-ks)`
where `A`, and `k` are constants.
- Write down the value of `A`. (2 marks)
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- The light intensity 6 metres below the surface of the lake is 1000 lux. Find the value of `k`. (2 marks)
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- At what rate, in lux per metre, is the light intensity decreasing 6 metres below the surface of the lake? (2 marks)
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Calculus, EXT1* C1 2009 HSC 6b
Radium decays at a rate proportional to the amount of radium present. That is, if `Q(t)` is the amount of radium present at time `t`, then `Q=Ae^(-kt)`, where `k` is a positive constant and `A` is the amount present at `t=0`. It takes 1600 years for an amount of radium to reduce by half.
- Find the value of `k`. (2 marks)
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- A factory site is contaminated with radium. The amount of radium on site is currently three times the safe level.
How many years will it be before the amount of radium reaches the safe level. (2 marks)
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Calculus, EXT1* C1 2010 HSC 8a
Assume that the population, `P`, of cane toads in Australia has been growing at a rate proportional to `P`. That is, `(dP)/(dt)=kP` where `k` is a positive constant.
There were 102 cane toads brought to Australia from Hawaii in 1935.
Seventy-five years later, in 2010, it is estimated that there are 200 million cane toads in Australia.
If the population continues to grow at this rate, how many cane toads will there be in Australia in 2035? (4 marks)
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Calculus, EXT1* C1 2011 HSC 10a
The intensity, `I`, measured in watt/m2, of a sound is given by
`I=10^-12xxe^(0.1L)`,
where `L` is the loudness of the sound in decibels.
- If the loudness of a sound at a concert is 110 decibels, find the intensity of the sound. Give your answer in scientific notation. (1 mark)
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- Ear damage occurs if the intensity of a sound is greater than `8.1xx10^-9` watt/m2.
What is the maximum loudness of a sound so that no ear damage occurs? (2 marks)
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- By how much will the loudness of a sound have increased if its intensity has doubled? (2 marks)
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Calculus, EXT1* C1 2012 HSC 14c
Professor Smith has a colony of bacteria. Initially there are 1000 bacteria. The number of bacteria, `N(t)`, after `t` minutes is given by
`N(t)=1000e^(kt)`.
- After 20 minutes there are 2000 bacteria.
Show that `k=0.0347` correct to four decimal places. (1 mark)
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- How many bacteria are there when `t=120`? (1 mark)
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- What is the rate of change of the number of bacteria per minute, when `t=120`? (1 mark)
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- How long does it take for the number of bacteria to increase from 1000 to 100 000? (2 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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Show Answers Only
- `text(374 trout)`
- `text{148 months (nearest month)}`
-
- `text{After 80 months (nearest month)}`
- `text{After 135 months (nearest month)}`
Show Worked Solution
i. `text(Carp introduced at)\ \ t=0`
MARKER’S COMMENT: A number of students did not equate `e^0=1` in this part.
`N=375-e^0=374`
`:.\ text(There was 374 trout when carp were introduced.)`
ii. `text(Trout population will be zero when)`
NOTE: The last line of the solution isn’t necessary but is included as good practice as a check that the answer matches the exact question asked.
| `N` | `=375-e^(0.04t)=0` |
| `e^(0.04t)` | `=375` |
| `0.04t` | `=ln375` |
| `t` | `=ln375/0.04` |
| `=148.173 …` | |
| `=148\ text{months (nearest month)}` |
`:.\ text(After 148 months, the trout population will be zero.)`
| iii. |  |
♦♦ Mean mark 33% for part (iii)
iv. `text(We need) \ |(dN)/(dt)|=(dP)/(dt)`
`text(Given)\ N=375-e^(0.04t)`
`(dN)/(dt)=-0.04e^(0.04t)`
`text(Find)\ P\ text(in terms of)\ t`
`text(Given)\ (dP)/(dt)=0.02P`
`=> P=Ae^(0.02t)`
♦♦♦ Mean mark 20%
COMMENT: Students who progressed to `0.2e^(0.02t)“=0.04e^(0.04t)` received 2 full marks in this part. Show your working!
COMMENT: Students who progressed to `0.2e^(0.02t)“=0.04e^(0.04t)` received 2 full marks in this part. Show your working!
`text(Find)\ A\ \ =>text(when)\ t=0,\ P=10`
| `10` | `=Ae^0` |
| `:.A` | `=10` |
| `=>P` | `=10xx0.02e^(0.02t)` |
| `=0.2e^(0.02t)` |
`text(Given that)\ \ (dP)/(dt)=|(dN)/(dt)|`
| `0.2e^(0.02t)` | `=0.04e^(0.04t)` |
| `5e^(0.02t)` | `=e^(0.04t)` |
| `e^(0.04t)/e^(0.02t)` | `=5` |
| `e^(0.04t-0.02t)` | `=5` |
| `lne^(0.02t)` | `=ln5` |
| `0.02t` | `=ln5` |
| `t` | `=ln5/0.02` |
| `=80.4719…` | |
| `=80\ text{months (nearest month)}` |
v. `text(Find)\ t\ text(when)\ N=P`
| `text(i.e.)\ \ 375-e^(0.04t)` | `=10e^(0.02t)` |
| `e^(0.04t)+10e^(0.02t)-375` | `=0` |
`text(Let)\ X=e^(0.02t),\ text(noting)\ \ X^2=(e^(0.02t))^2=e^(0.04t)`
♦♦♦ Mean mark 4%!
MARKER’S COMMENT: Correctly applying substitution to exponentials to form a solvable quadratic proved very difficult for almost all students.
MARKER’S COMMENT: Correctly applying substitution to exponentials to form a solvable quadratic proved very difficult for almost all students.
| `:.\ X^2+10X-375` | `=0` |
| `(X-15)(X+25)` | `=0` |
`X=15\ \ text(or)\ \ –25`
`text(S)text(ince)\ X=e^(0.02t)`
| `e^(0.02t)` | `=15\ \ \ \ (e^(0.02t)>0)` |
| `lne^(0.02t)` | `=ln15` |
| `0.02t` | `=ln15` |
| `t` | `=ln15/0.02` |
| `=135.4025…` | |
| `=135\ text(months)` |
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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Show Worked Solution
i. `text(Initial velocity when)\ \ t=0`
| `v` | `=8-8e^0` |
| `=0\ text(m/s)` | |
| `:.\ text(Particle is initially at rest.)` | |
MARKER’S COMMENT: Students whose working showed `e^(-2t)` as `1/e^(2t)`, tended to score highly in this question.
ii. `a=d/(dt) (v)=-2xx-8e^(-2t)=16e^(-2t)`
`text(S)text(ince)\ e^(-2t)=1/e^(2t)>0\ text(for all)\ t`.
`=>\ a=16e^(-2t)=16/e^(2t)>0\ text(for all)\ t`.
`:.\ text(Acceleration is positive for all)\ \ t>0`.
iii. `text{S}text{ince the particle is initially at rest, and ALWAYS}`
♦♦♦ Mean mark 22%
COMMENT: Students found part (iii) the most challenging part of this question by far.
COMMENT: Students found part (iii) the most challenging part of this question by far.
`text{has a positive acceleration.`
`:.\ text(It moves in a positive direction for all)\ t`.
iv. `text(As)\ t->oo`, `e^(-2t)=1/e^(2t)->0`
`=>8/e^(2t)->0\ text(and)`
`=>v=8-8/e^(2t)->8\ text(m/s)`
`:.\ text(As)\ \ t->oo,\ text(velocity approaches 8 m/s.)`
IMPORTANT: Use previous parts to inform this diagram. i.e. clearly show velocity was zero at `t=0` and the asymptote at `v=8`.
| v. |
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Calculus, 2ADV C4 2010 HSC 5c
The diagram shows the curve `y=1/x`, for `x>0`.
The area under the curve between `x=a` and `x=1` is `A_1`. The area under the curve between `x=1` and `x=b` is `A_2`.
The areas `A_1` and `A_2` are each equal to `1` square unit.
Find the values of `a` and `b`. (3 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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Financial Maths, 2ADV M1 2008 HSC 5b
Consider the geometric series
`5+10x+20x^2+40x^3+\ ...`
- For what values of `x` does this series have a limiting sum? (2 marks)
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- The limiting sum of this series is `100`.
Find the value of `x`. (2 marks)
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Financial Maths, 2ADV M1 2008 HSC 4b
The zoom function in a software package multiplies the dimensions of an image by 1.2. In an image, the height of a building is 50 mm. After the zoom function is applied once, the height of the building in the image is 60 mm. After the second application, it is 72 mm.
- Calculate the height of the building in the image after the zoom function has been applied eight times. Give your answer to the nearest mm. (2 marks)
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- The height of the building in the image is required to be more than 400 mm. Starting from the original image, what is the least number of times the zoom function must be applied? (2 marks)
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Proof, EXT1 P1 2011 HSC 6a
Use mathematical induction to prove that for `n>=1`,
`1xx5+2xx6+3xx7+\ …\ +n(n+4)=1/6n(n+1)(2n+13)`. (3 marks)
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L&E, 2ADV E1 2009 HSC 1f
Solve the equation `lnx=2`. Give you answer correct to four decimal places. (2 marks)
L&E, 2ADV E1 2010 HSC 4d
Let `f(x)=1+e^x`.
Show that `f(x)xxf(–x)=f(x)+f(–x)`. (2 marks)
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Calculus, 2ADV C4 2010 HSC 4b
The curves `y=e^(2x)` and `y=e^-x` intersect at the point `(0,1)` as shown in the diagram.
Find the exact area enclosed by the curves and the line `x=2`. (3 marks)
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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Show Worked Solutions
| i. |  |
MARKER’S COMMENT: Many students failed to illustrate important features in their graph such as the concavity, `x`-axis intercept and `y`-axis asymptote (this can be explicitly stated or made graphically clear).
| ii. `text(Area)` | `~~h/2[f(1)+2xxf(2)+f(3)]` |
| `~~1/2[0+2ln2+ln3]` | |
| `~~1/2[ln(2^2 xx3)]` | |
| `~~1/2ln12` | |
| `~~1.24\ \ text(u²)` `text{(to 2 d.p.)}` |
| iii. |  |
♦♦♦ Mean mark 12%.
MARKER’S COMMENT: Best responses commented on concavity and that the trapezia lay beneath the curve. Diagrams featured in the best responses.
MARKER’S COMMENT: Best responses commented on concavity and that the trapezia lay beneath the curve. Diagrams featured in the best responses.
`text(The approximation is less because the sides)`
`text{of the trapezia lie below the concave down}`
`text{curve (see diagram).}`
Calculus, 2ADV C4 2010 HSC 2dii
Find `intx/(4+x^2)\ dx`. (2 marks)
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Calculus, 2ADV C3 2010 HSC 2c
Find the gradient of the tangent to the curve `y=ln (3x)` at the point where `x=2`. (2 marks)
Calculus, 2ADV C4 2011 HSC 4b
Evaluate `int_e^(e^3) 5/x\ dx` (2 marks)
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Calculus, 2ADV C4 2012 HSC 12b
Find `int(4x)/(x^2+6)\ dx`. (2 marks)
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Calculus, 2ADV C4 2012 HSC 9 MC
What is the value of `int_1^4 1/(3x)\ dx`?
- `1/3ln3`
- `1/3ln4`
- `ln9`
- `ln12`
L&E, 2ADV E1 2012 HSC 7 MC
Let `a=e^x`
Which expression is equal to `log_e(a^2)`?
- `e^(2x)`
- `e^(x^2)`
- `2x`
- `x^2`