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?
- 0.75
- 7.5
- 75
- 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?
- 4
- 16
- 18
- 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=10x-x^2`
- `A=10x-2x^2`
- `A=20x-x^2`
- `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.
- `8100`
- `16\ 200`
- `20\ 400`
- `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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