Sketch the graph of the function `f(x) = 2arccos x`. Clearly indicate the domain and range of the function. (2 marks)
Combinatorics, EXT1 A1 2011 HSC 2c
Find an expression for the coefficient of `x^2` in the expansion of `(3x - 4/x)^8`. (2 marks)
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Polynomials, EXT1 2011 HSC 2b
The function `f(x) = cos2x\ - x` has a zero near `x = 1/2`.
Use one application of Newton’s method to obtain another approximation to this zero. Give your answer correct to two decimal places. (3 marks)
Mechanics, EXT2* M1 2012 HSC 14b
A firework is fired from `O`, on level ground, with velocity `70` metres per second at an angle of inclination `theta`. The equations of motion of the firework are
`x = 70t cos theta\ \ \ \ `and`\ \ \ y = 70t sin theta\ – 4.9t^2`. (Do NOT prove this.)
The firework explodes when it reaches its maximum height.
- Show that the firework explodes at a height of `250 sin^2 theta` metres. (2 marks)
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- Show that the firework explodes at a horizontal distance of `250 sin 2 theta` metres from `O`. (1 mark)
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- For best viewing, the firework must explode at a horizontal distance between 125 m and 180 m from `O`, and at least 150 m above the ground.
For what values of `theta` will this occur? (3 mark)
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Plane Geometry, EXT1 2012 HSC 14a
The diagram shows a large semicircle with diameter `AB` and two smaller semicircles with diameters `AC` and `BC`, respectively, where `C` is a point on the diameter `AB`. The point `M` is the centre of the semicircle with diameter `AC`.
The line perpendicular to `AB` through `C` meets the largest semicircle at the point `D`. The points `S` and `T` are the intersections of the lines `AD` and `BD` with the smaller semicircles. The point `X` is the intersection of the lines `CD` and `ST`.
Copy or trace the diagram into your writing booklet.
- Explain why `CTDS` is a rectangle. (1 mark)
- Show that `Delta MXS` and `Delta MXC` are congruent. (2 marks)
- Show that the line `ST` is a tangent to the semicircle with diameter `AC`. (1 mark)
Geometry and Calculus, EXT1 2012 HSC 13d
The concentration of a drug in the blood of a patient `t` hours after it was administered is given by
`C(t) = 1.4te^(–0.2t),`
where `C(t)` is measured in `text(mg/L)`.
- Initially the concentration of the drug in the blood of the patient increases until it reaches a maximum, and then it decreases. Find the time when this maximum occurs. (3 marks)
- Taking `t = 20` as a first approximation, use one application of Newton’s method to find approximately when the concentration of the drug in the blood of the patient reaches `0.3\ text(mg/L)`. (2 marks)
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)
Trigonometry, EXT1 T1 2012 HSC 13a
Write `sin(2 cos ^(-1) (2/3))` in the form `a sqrtb`, where `a` and `b` are rational. (2 mark)
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Quadratic, EXT1 2012 HSC 12d
Let `A(0, –k)` be a fixed point on the `y`-axis with `k > 0`. The point `C(t, 0)` is on the `x`-axis. The point `B(0, y)` is on the `y`-axis so that `Delta ABC` is right-angled with the right angle at `C`. The point `P` is chosen so that `OBPC` is a rectangle as shown in the diagram.
- Show that `P` lies on the parabola given parametrically by (2 marks)
- `x = t\ \ ` and`\ \ y = (t^2)/k`.
- Write down the coordinates of the focus of the parabola in terms of `k`. (1 mark)
Statistics, EXT1 S1 2012 HSC 12c
Kim and Mel play a simple game using a spinner marked with the numbers 1, 2, 3, 4 and 5.
The game consists of each player spinning the spinner once. Each of the five numbers is equally likely to occur.
The player who obtains the higher number wins the game.
If both players obtain the same number, the result is a draw.
- Kim and Mel play one game. What is the probability that Kim wins the game? (1 mark)
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- Kim and Mel play six games. What is the probability that Kim wins exactly three games? (2 marks)
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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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Proof, EXT1 P1 2012 HSC 12a
Use mathematical induction to prove that `2^(3n)\ – 3^n` is divisible by `5` for `n >= 1`. (3 marks)
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Combinatorics, EXT1 A1 2012 HSC 11f
- Use the binomial theorem to find an expression for the constant term in the expansion of
`(2x^3 - 1/x)^12`. (2 marks)
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- For what values of `n` does `(2x^3 - 1/x)^n` have a non-zero constant term? (1 mark)
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Combinatorics, EXT1 A1 2012 HSC 11e
In how many ways can a committee of 3 men and 4 women be selected from a group of 8 men and 10 women? (1 mark)
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Plane Geometry, EXT1 2012 HSC 10 MC
Functions, EXT1 F2 2012 HSC 8 MC
When the polynomial `P(x)` is divided by `(x + 1)(x-3)`, the remainder is `2x + 7`.
What is the remainder when `P(x)` is divided by `x-3`?
- `1`
- `7`
- `9`
- `13`
Mechanics, EXT2* M1 2012 HSC 6 MC
A particle is moving in simple harmonic motion with displacement `x`. Its velocity `v` is given by
`v^2 = 16(9 − x^2)`.
What is the amplitude, `A`, and the period, `T`, of the motion?
(A) `A = 3\ \ \ text(and)\ \ \ T = pi/2`
(B) `A = 3\ \ \ text(and)\ \ \ T = pi/4`
(C) `A = 4\ \ \ text(and)\ \ \ T = pi/3`
(D) `A = 4\ \ \ text(and)\ \ \ T = (2pi)/3`
Combinatorics, EXT1 A1 2012 HSC 5 MC
How many arrangements of the letters of the word `OLYMPIC` are possible if the `C` and the `L` are to be together in any order?
(A) `5!`
(B) `6!`
(C) `2 xx 5!`
(D) `2 xx 6!`
Trigonometry, EXT1 T1 2012 HSC 4 MC
Calculus, 2ADV C3 2009 HSC 10
`text(Let)\ \ f(x) = x - (x^2)/2 + (x^3)/3`
- Show that the graph of `y = f(x)` has no turning points. (2 marks)
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- Find the point of inflection of `y = f(x)`. (1 mark)
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- i. Show that `1 - x + x^2 - 1/(1 + x) = (x^3)/(1 + x)` for `x != -1`. (1 mark)
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ii. Let `g(x) = ln (1 + x)`.
Use the result in part c.i. to show that `f prime (x) >= g prime (x)` for all `x >= 0`. (2 marks)
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- Sketch the graphs of `y = f(x)` and `y = g(x)` for `x >= 0`. (2 marks)
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- Show that `d/(dx) [(1 + x) ln (1 + x) - (1 + x)] = ln (1 + x)`. (2 marks)
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- Find the area enclosed by the graphs of `y = f(x)` and `y = g(x)`, and the straight line `x = 1`. (2 marks)
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Trigonometry, 2ADV T3 2009 HSC 7b
Between 5 am and 5 pm on 3 March 2009, the height, `h`, of the tide in a harbour was given by
`h = 1 + 0.7 sin(pi/6 t)\ \ \ text(for)\ \ 0 <= t <= 12`
where `h` is in metres and `t` is in hours, with `t = 0` at 5 am.
- What is the period of the function `h`? (1 mark)
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- What was the value of `h` at low tide, and at what time did low tide occur? (2 marks)
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- A ship is able to enter the harbour only if the height of the tide is at least 1.35 m.
Find all times between 5 am and 5 pm on 3 March 2009 during which the ship was able to enter the harbour. (3 marks)
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Calculus, EXT1* C3 2009 HSC 6a
Trigonometry, 2ADV T1 2009 HSC 5c
The diagram shows a circle with centre `O` and radius 2 centimetres. The points `A` and `B` lie on the circumference of the circle and `/_AOB = theta`.
- There are two possible values of `theta` for which the area of `Delta AOB` is `sqrt 3` square centimetres. One value is `pi/3`.
Find the other value. (2 marks)
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- Suppose that `theta = pi/3`.
(1) Find the area of sector `AOB` (1 mark)
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(2) Find the exact length of the perimeter of the minor segment bounded by the chord `AB` and the arc `AB`. (2 marks)
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Functions, 2ADV F1 2009 HSC 5a
In the diagram, the points `A` and `C` lie on the `y`-axis and the point `B` lies on the `x`-axis. The line `AB` has equation `y = sqrt3x − 3`. The line `BC` is perpendicular to `AB`.
- Find the equation of the line `BC`. (2 marks)
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- Find the area of the triangle `ABC`. (2 marks)
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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)
Quadratic, 2UA 2009 HSC 4b
Find the values of `k` for which the quadratic equation
`x^2 - (k + 4)x + (k + 7) = 0`
has equal roots. (3 marks)
Integration, 2UA 2009 HSC 3d
The diagram shows a block of land and its dimensions, in metres. The block of land is bounded on one side by a river. Measurements are taken perpendicular to the line `AB`, from `AB` to the river, at equal intervals of `50\ text(m)`.
Use Simpson’s rule with six subintervals to find an approximation to the area of the block of land. (3 marks)
Functions, EXT1* F1 2009 HSC 3c
Shade the region in the plane defined by `y >= 0` and `y <= 4-x^2`. (2 marks)
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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, EXT1 C2 2012 HSC 7 MC
Which expression is equal to `int sin^2 3x\ dx`?
(A) `1/2 (x\ - 1/3 sin 3x) + C`
(B) `1/2 (x\ + 1/3 sin 3x) + C`
(C) `1/2 (x\ - 1/6 sin 6x) + C`
(D) `1/2 (x + 1/6 sin 6x) + C`
Trigonometry, 2ADV T3 2010 HSC 8c
The graph shown is `y = A sin bx`.
- Write down the value of `A`. (1 mark)
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- Find the value of `b`. (1 mark)
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On the same set of axes, draw the graph `y = 3 sin x + 1` for `0 <= x <= pi`. (2 marks)
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Trigonometry, 2ADV T1 2010 HSC 6b
The diagram shows a circle with centre `O` and radius 5 cm.
The length of the arc `PQ` is 9 cm. Lines drawn perpendicular to `OP` and `OQ` at `P` and `Q` respectively meet at `T`.
- Find `/_POQ` in radians. (1 mark)
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- Prove that `Delta OPT` is congruent to `Delta OQT`. (2 marks)
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- Find the area of the shaded region. (2 marks)
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Calculus, 2ADV C3 2010 HSC 6a
Let `f(x) = (x + 2)(x^2 + 4)`.
- Show that the graph `y=f(x)` has no stationary points. (2 marks)
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- Find the values of `x` for which the graph `y=f(x)` is concave down, and the values for which it is concave up. (2 marks)
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- Sketch the graph `y=f(x)`, indicating the values of the `x` and `y` intercepts. (2 marks)
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Calculus, 2ADV C4 2010 HSC 5b
- Prove that `sec^2 x + secx tanx = (1 + sinx)/(cos^2x)`. (1 mark)
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- Hence prove that `sec^2 x + secx tanx = 1/(1 - sinx)`. (1 mark)
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- Hence, use the identity `int sec ax tan ax\ dx=1/a sec ax` to find the exact value of
`int_0^(pi/4) 1/(1 - sinx)\ dx`. (2 marks)
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Probability, 2ADV S1 2010 HSC 4c
There are twelve chocolates in a box. Four of the chocolates have mint centres, four have caramel centres and four have strawberry centres. Ali randomly selects two chocolates and eats them.
- What is the probability that the two chocolates have mint centres? (1 mark)
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- What is the probability that the two chocolates have the same centre? (1 mark)
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- What is the probability that the two chocolates have different centres? (1 mark)
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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, 2ADV F1 2010 HSC 1g
Let `f(x) = sqrt(x-8)`. What is the domain of `f(x)`? (1 mark)
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Plane Geometry, 2UA 2011 HSC 9a
Calculus, EXT1* C3 2012 HSC 14b
Calculus, 2ADV C3 2012 HSC 14a
A function is given by `f(x) = 3x^4 + 4x^3-12x^2`.
- Find the coordinates of the stationary points of `f(x)` and determine their nature. (3 marks)
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- Hence, sketch the graph `y = f(x)` showing the stationary points. (2 marks)
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- For what values of `x` is the function increasing? (1 mark)
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- For what values of `k` will `f(x) = 3x^4 + 4x^3-12x^2 + k = 0` have no solution? (1 mark)
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Calculus, EXT1* C3 2011 HSC 8b
The diagram shows the region enclosed by the parabola `y = x^2`, the `y`-axis and the line `y = h`, where `h > 0`. This region is rotated about the `y`-axis to form a solid called a paraboloid. The point `C` is the intersection of `y = x^2` and `y = h`.
The point `H` has coordinates `(0, h)`.
- Find the exact volume of the paraboloid in terms of `h`. (2 marks)
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- A cylinder has radius `HC` and height `h`.
What is the ratio of the volume of the paraboloid to the volume of the cylinder? (1 mark)
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Trigonometry, 2ADV T1 2011 HSC 8a
In the diagram, the shop at `S` is 20 kilometres across the bay from the post office at `P`. The distance from the shop to the lighthouse at `L` is 22 kilometres and `/_ SPL` is 60°.
Let the distance `PL` be `x` kilometres.
- Use the cosine rule to show that `x^2-20x- 84 = 0`. (1 mark)
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- Hence, find the distance from the post office to the lighthouse. Give your answer correct to the nearest kilometre. (2 mark)
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Functions, 2ADV F1 2010 HSC 1c
Write down the equation of the circle with centre `(-1, 2)` and radius 5. (1 mark)
Calculus, 2ADV C4 2010 HSC 2di
Find `int sqrt(5x +1) \ dx .` (2 marks)
Calculus, 2ADV C4 2010 HSC 2e
Given that `int_0^6 ( x + k )\ dx = 30`, and `k` is a constant, find the value of `k`. (2 marks)
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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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FS Comm, 2UG SM-Bank 01
Patrick has purchased an `8 \ text(GB)` USB. The average size of each file he saves is `492 \ text(KB)`.
Find how many files Patrick can expect to save on his USB (to the nearest whole number). (2 marks)
Measurement, STD2 M7 SM-Bank 8
A patient is to receive 1.8 L of pain killer medication by intravenous drip that will take 1.5 hours to administer.
Given 1 mL = 4 drops, calculate the amount of drops per minute the machine must be set on. (2 marks)
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Algebra, STD2 A1 SM-Bank 1 MC
The blood alcohol content (`BAC`) of a male's blood is given by the formula;
`BAC_text(male) = (10N - 7.5H)/(6.8M)` , where
`N` is the number of standard drinks consumed,
`H` is the number of hours drinking and
`M` is the person's mass in kgs.
Calculate the `BAC` of a male who consumed 4 standard drinks in 3.5 hours and weighs 68 kgs, correct to 2 decimal places.
(A) 1.03
(B) 0.03
(C) 0.04
(D) 0.01
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
Integration, 2UA 2011 HSC 5c
The table gives the speed `v` of a jogger at time `t` in minutes over a 20-minute period. The speed `v` is measured in metres per minute, in intervals of 5 minutes.
The distance covered by the jogger over the 20-minute period is given by
`int_0^20 v\ dt`.
Use Simpson’s rule and the speed at each of the five time values to find the approximate distance the jogger covers in the 20-minute period. (3 marks)
Statistics, 2ADV 2011 HSC 5b
Kim has three red shirts and two yellow shirts. On each of the three days, Monday, Tuesday and Wednesday, she selects one shirt at random to wear. Kim wears each shirt that she selects only once.
- What is the probability that Kim wears a red shirt on Monday? (1 mark)
- What is the probability that Kim wears a shirt of the same colour on all three days? (1 mark)
- What is the probability that Kim does not wear a shirt of the same colour on consecutive days? (2 marks)
Calculus, 2ADV C4 2011 HSC 4d
- Differentiate `y=sqrt(9 - x^2)` with respect to `x`. (2 marks)
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- Hence, or otherwise, find `int (6x)/sqrt(9 - x^2)\ dx`. (2 marks)
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Calculus, 2ADV C4 2011 HSC 4c
The gradient of a curve is given by `dy/dx = 6x-2`. The curve passes through the point `(-1, 4)`.
What is the equation of the curve? (2 marks)
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Quadratic, 2UA 2011 HSC 3b
A parabola has focus `(3, 2)` and directrix `y = –4`. Find the coordinates of the vertex. (2 marks)
Trigonometry, 2ADV T2 2011 HSC 2b
Find the exact values of `x` such that `2sin x = - sqrt3`, where `0 <= x <= 2pi`. (2 marks)
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Calculus, 2ADV C4 2012 HSC 13b
The diagram shows the parabolas `y = 5x - x^2` and `y = x^2 - 3x`. The parabolas intersect at
the origin `O` and the point `A`. The region between the two parabolas is shaded.
- Find the `x`-coordinate of the point `A` (1 mark)
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- Find the area of the shaded region. (3 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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