Band 3 – Page 54

Calculus, EXT1 C2 2008 HSC 1b

Differentiate `cos^(–1) (3x)` with respect to `x`. (2 marks)

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`(-3)/sqrt(1 – 9x^2)`

Show Worked Solution

`y`	`= cos^(-1) (3x)`
`dy/dx`	`= – 1/sqrt(1 – (3x)^2) * d/(dx) (3x)`
	`= (-3)/sqrt(1 – 9x^2)`

Functions, EXT1 F2 2008 HSC 1a

The polynomial `x^3` is divided by `x + 3`. Calculate the remainder. (2 marks)

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`-27`

Show Worked Solution

`P(-3)`	`= (-3)^3`
	`= -27`

`:.\ text(Remainder when)\ x^3 -: (x + 3) = -27`

MARKER’S COMMENT: “Grave concern” that many who found `P(-3)=-27` stated the remainder was 27.

Statistics, STD2 S1 2008 HSC 26d

The graph shows the predicted population age distribution in Australia in 2008.

How many females are in the 0–4 age group? (1 mark)

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What is the modal age group? (1 mark)

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How many people are in the 15–19 age group? (2 marks)

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Give ONE reason why there are more people in the 80+ age group than in the 75–79 age group. (1 mark)

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Show Answers Only

`600\ 000`
`35-39`
`1\ 450\ 000`
`text(The 80+ group includes all people over 80)`
`text(and is not restricted by a 5-year limit.)`

Show Worked Solution

i.	`text{# Females (0-4)}`	`= 0.6 xx 1\ 000\ 000`
		`= 600\ 000`

ii.

`text(Modal age group)\ =`

`text(35 – 39)`

iii.	`text{# Males (15-19)}`	`= 0.75 xx 1\ 000\ 000`
		`= 750\ 000`

`text{# Females (15-19)}`	`= 0.7 xx 1\ 000\ 000`
	`= 700\ 000`

`:.\ text{Total People (15-19)}`	`= 750\ 000 + 700\ 000`
	`= 1\ 450\ 000`

iv.	`text(The 80+ group includes all people over 80)`
	`text(and is not restricted by a 5-year limit.)`

Probability, STD2 S2 2008 HSC 26a

Cecil invited 175 movie critics to preview his new movie. After seeing the movie, he conducted a survey. Cecil has almost completed the two-way table.

Determine the value of `A`. (1 mark)

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A movie critic is selected at random.

What is the probability that the critic was less than 40 years old and did not like the movie? (2 marks)

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Cecil believes that his movie will be a box office success if 65% of the critics who were surveyed liked the movie.

Will this movie be considered a box office success? Justify your answer. (1 mark)

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Show Answers Only

`58`
`6/25`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

i. `text{Critics liked and}\ >= 40`

`= 102-65`

`= 37`

`:. A = 37+31=68`

ii. `text{Critics did not like and < 40}`

`= 175-65-37-31`

`= 42`

`:.\ P text{(not like and < 40)}`

`= 42/175`

`= 6/25`

iii. `text(Critics liked) = 102`

`text(% Critics liked)`	`= 102/175 xx 100`
	`= 58.28…%`

`:.\ text{Movie NOT a box office success (< 65% critics liked)}`

Financial Maths, STD2 F1 2008 HSC 24a

Bob is employed as a salesman. He is offered two methods of calculating his income.

\begin{array} {|l|}
\hline
\rule{0pt}{2.5ex}\text{Method 1: Commission only of 13% on all sales}\rule[-1ex]{0pt}{0pt} \\
\hline
\rule{0pt}{2.5ex}\text{Method 2: \$350 per week plus a commission of 4.5% on all sales}\rule[-1ex]{0pt}{0pt} \\
\hline
\end{array}

Bob’s research determines that the average sales total per employee per month is $15 670.

Based on his research, how much could Bob expect to earn in a year if he were to choose Method 1? (2 marks)

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If Bob were to choose a method of payment based on the average sales figures, state which method he should choose in order to earn the greater income. Justify your answer with appropriate calculations. (3 marks)

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Show Answers Only

`$24\ 445.20`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

i. `text(Method 1)`

`text(Yearly sales)`	`= 12 xx 15\ 670`
	`= 188\ 040`

`:.\ text(Earnings)`	`= text(13%) xx 188\ 040`
	`= $24\ 445.20`

ii. `text(Method 2)`

`text(In 1 Year, Weekly Wage)`	`= 350 xx 52`
	`= 18\ 200`

`text(Commission)`	`= text(4.5%) xx 188\ 040`
	`= 8461.80`

`text(Total earnings)`	`= 18\ 200 + 8461.80`
	`= $26\ 661.80`

`:.\ text(Bob should choose Method 2.)`

Statistics, STD2 S1 2008 HSC 23a

You are organising an outside sporting event at Mathsville and have to decide which month has the best weather for your event. The average temperature must be between 20°C and 30°C, and average rainfall must be less than 80 mm.

The radar chart for Mathsville shows the average temperature for each month, and the table gives the average rainfall for each month.

If you consider only the temperature data, there are a number of possible months for holding the event. Name ONE of these months. (1 mark)

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If both rainfall and temperature data are considered, which month is the best month for the sporting event? (1 mark)

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`text(One of Feb, Mar, Nov, Dec)`
`text(November)`

Show Worked Solution

i. `text(One of Feb, Mar, Nov, Dec)`

ii. `text(November)`

Measurement, STD2 M6 2008 HSC 14 MC

Danni is flying a kite that is attached to a string of length 80 metres. The string makes an angle of 55° with the horizontal.

How high, to the nearest metre, is the kite above Danni’s hand?

46 m
66 m
98 m
114 m

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`B`

Show Worked Solution

`sin 55^@`	`= h/80`
`h`	`= 80 xx sin 55^@`
	`= 65.532…\ text(m)`

`=> B`

Financial Maths, STD2 F1 2008 HSC 7 MC

Luke’s normal rate of pay is $15 per hour. Last week he was paid for 12 hours, at time-and-a-half.

How many hours would Luke need to work this week, at double time, to earn the same amount?

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`D`

Show Worked Solution

`text(Amount earned last week)`

`= 12 xx 1.5 xx 15`

`= $270`

`text(Double time rate)`	`= 2 xx 15`
	`= $30 text(/hr)`

`:.\ text(# Hours at double time)`

`= 270/30`

`= 9\ text(hrs)`

`=> D`

Financial Maths, STD2 F1 2008 HSC 6 MC

Using the tax table, what is the tax payable on $43 561?

$5424.40
$10 824.40
$16 224.40
$17 424.40

Show Answers Only

`B`

Show Worked Solution

`text(Tax Payable)`

`= 5400 + 0.4 (43\ 561 – 30\ 000)`

`= 5400 + 5424.40`

`= 10\ 824.40`

`=> B`

Algebra, 2UG 2008 HSC 1 MC

Which expression is equivalent to `12k^3 ÷ 4k`?

`3k^2 `
`3k^3`
`8k^2`
`8k^3`

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`A`

Show Worked Solution

`12k^3 -: 4k`	`=(12k^3)/(4k)`
	`=3k^2`

`=> A`

Quadratic, 2UA 2008 HSC 4c

Consider the parabola `x^2 = 8(y\ – 3)`.

Write down the coordinates of the vertex. (1 mark)
Find the coordinates of the focus. (1 mark)
Sketch the parabola. (1 mark)
Calculate the area bounded by the parabola and the line `y = 5`. (3 marks)

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(i) `(0,3)`

(ii) `(0,5)`

(iii)

(iv) `10 2/3\ text(u²)`

Show Worked Solution

(i)

`text(Vertex)\ = (0,3)`

(ii)	`text(Using)\ \ \ x^2`	`= 4ay`
	`4a`	`= 8`
	`a`	`= 2`

`:.\ text(Focus) = (0,5)`

(iii)

(iv)

`text(Intersection when)\ y = 5`

`=> x^2`	`= 8 (5-3)`
`x^2`	`= 16`
`x`	`= +- 4`

`text(Find shaded area)`

`x^2`	`= 8 (y-3)`
`y – 3`	`= x^2/8`
`y`	`= x^2/8 +3`

`text(Area)`	`= int_-4^4 5\ dx – int_-4^4 x^2/8 + 3\ dx`
	`= int_-4^4 5 – (x^2/8 + 3)\ dx`
	`= int_-4^4 2 – x^2/8\ dx`
	`= [2x – x^3/24]_-4^4`
	`= [(8 – 64/24) – (-8 + 64/24)]`
	`= 5 1/3 – (- 5 1/3)`
	`= 10 2/3\ text(u²)`

Calculus, 2ADV C2 2008 HSC 2aiii

Differentiate with respect to `x`:

`sinx/(x+4)`. (2 marks)

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`(cosx (x+4) – sin x)/((x + 4)^2)`

Show Worked Solution

`y = sinx/(x + 4)`

`u`	`= sinx`	`\ \ \ \ \ u’`	`= cos x`
`v`	`= x + 4`	`v’`	`= 1`

`dy/dx`	`= (u’v – uv’)/v^2`
	`= (cos x (x + 4) – sin x)/(x+4)^2`

Calculus, 2ADV C2 2008 HSC 2aii

Differentiate with respect to `x`:

`x^2 log_e x` (2 marks)

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`x + 2x log_e x`

Show Worked Solution

`y`	`= x^2 log_e x`
`dy/dx`	`= x^2 * 1/x + 2x * log_e x`
	`= x + 2x log_e x`

Differentiation, 2UA 2008 HSC 2ai

Differentiate with respect to `x`:

`(x^2 + 3)^9` (2 marks)

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`18x (x^2 + 3)^8`

Show Worked Solution

`y`	`= (x^2 + 3)^9`
`dy/dx`	`= 9 (x^2 + 3)^8 2x`
	`= 18x (x^2 + 3)^8`

Calculus, 2ADV C4 2008 HSC 3b

Differentiate `log_e (cos x)` with respect to `x`. (2 marks)

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Hence, or otherwise, evaluate `int_0^(pi/4) tan x\ dx`. (2 marks)

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`- tan x`
`- log_e (1/sqrt2)\ \ text(or)\ \ 0.35\ \ text{(2 d.p.)}`

Show Worked Solution

i.	`y`	`= log_e (cos x)`
	`dy/dx`	`= (- sin x)/(cos x)`
		`= – tan x`

ii.	`int_0^(pi/4) tan x\ dx`
	`= – [log_e (cos x)]_0^(pi/4)`
	`= – [log_e(cos (pi/4)) – log_e (cos 0)]`
	`= – [log_e (1/sqrt2) – log_e 1]`
	`= – [log_e (1/sqrt2) – 0]`
	`= – log_e (1/sqrt2)`
	`= 0.346…`
	`= 0.35\ \ text{(2 d.p.)}`

Linear Functions, 2UA 2008 HSC 3a

In the diagram, `ABCD` is a quadrilateral. The equation of the line `AD` is `2x- y- 1 = 0`.

Show that `ABCD` is a trapezium by showing that `BC` is parallel to `AD`. (2 marks)
The line `CD` is parallel to the `x`-axis. Find the coordinates of `D`. (1 mark)
Find the length of `BC`. (1 mark)
Show that the perpendicular distance from `B` to `AD` is `4/sqrt5`. (2 marks)
Hence, or otherwise, find the area of the trapezium `ABCD`. (2 marks)

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`text(Proof)\ \ text{(See Worked Solutions)}`
`(3,5)`
`sqrt 5\ text(units)`
`text(Proof)\ \ text{(See Worked Solutions)}`
`8\ text(u²)`

Show Worked Solution

(i)

`text(Show)\ BC \ text(||)\ AD`

`B(0,3),\ \ C(1,5)`

`m_(BC)`	`= (y_2 – y_1)/(x_2 – x_1)`
	`= (5 – 3)/(1 – 0)`
	`= 2`

`text(Equation)\ \ AD\ \ text(is)\ \ 2x – y – 1 = 0`

`y`	`= 2x – 1`
`m_(AD)`	`= 2`

`:. BC\ text(||) \ AD`

`:. ABCD\ text(is a trapezium)`

(ii)	`text(Given)\ CD\ text(\|\|) \ x text(-axis)`
	`text(Equation)\ CD\ text(is)\ y = 5`
	`D\ text(is intersection of)`

`y`	`= 5,\ \ and`
`2x – y – 1`	`= 0`
`:. 2x – 5 – 1`	`=0`
`2x`	`=6`
`x`	`=3`

`:.\ D`

`= (3,5)`

(iii)

`B(0,3),\ \ C(1,5)`

`text(dist)\ BC`	`= sqrt ( (x_2 – x_1)^2 + (y_2 – y_1)^2 )`
	`= sqrt ( (1-0)^2 + (5-3)^2 )`
	`= sqrt (1 + 4)`
	`= sqrt 5\ text(units)`

(iv)

`text(Show)\ _|_\ text(dist of)\ B\ text(to)\ AD\ text(is)\ 4/sqrt5`

`B (0,3)\ \ \ \ \ 2x – y – 1 = 0`

`_\|_\ text(dist)`	`= \| (ax_1 + by_1 + c)/sqrt (a^2 + b^2) \|`
	`= \|( 2(0) – 1(3) -1 )/sqrt (2^2 + (-1)^2) \|`
	`= \| -4/sqrt5 \|`
	`= 4/sqrt 5\ \ \ text(… as required.)`

(v)	`text(Area)`	`= 1/2 h (a + b)`
		`= 1/2 xx 4/sqrt5 (BC + AD)`

`BC = sqrt5\ \ text{(part (iii))}`

`A(0,–1),\ \ D(3,5)`

`text(dist)\ AD`	`= sqrt ( (3-0)^2 + (5+1)^2 )`
	`= sqrt (9 + 36)`
	`= sqrt 45`
	`= 3 sqrt 5`

`:.\ text(Area)\ ABCD`	`= 1/2 xx 4/sqrt5 (sqrt5 + 3 sqrt 5)`
	`= 2 / sqrt5 (4 sqrt 5)`
	`= 8\ text(u²)`

Linear Functions, 2UA 2008 HSC 2b

Let `M` be the midpoint of `(-1, 4)` and `(5, 8)`.

Find the equation of the line through `M` with gradient `-1/2`. (2 marks)

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`x + 2y-14 = 0`

Show Worked Solution

`(-1,4)\ \ \ (5,8)`

`M`	`= ( (x_1 + x_2)/2, (y_1 + y_2)/2)`
	`= ( (-1 + 5)/2, (4 + 8)/2)`
	`= (2, 6)`

`text(Equation through)\ (2,6)\ text(with)\ m = -1/2`

`y-y_1`	`= m (x-x_1)`
`y-6`	`= -1/2 (x-2)`
`2y-12`	`= -x + 2`
`x + 2y-14`	`= 0`

Financial Maths, 2ADV M1 2008 HSC 1f

Find the sum of the first 21 terms of the arithmetic series 3 + 7 + 11 + ... (2 marks)

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`903`

Show Worked Solution

`S`	`= 3 + 7 + 11 + …`
`a`	`= 3`
`d`	`= 7 – 3 = 4`

`:. S_21`	`= n/2 [2a + (n – 1) d]`
	`= 21/2 [2 xx 3 + (21 – 1)4]`
	`= 21/2 [6 + 80]`
	`= 903`

Functions, 2ADV F1 2008 HSC 1e

Expand and simplify `(sqrt3-1)(2 sqrt3 + 5)`. (2 marks)

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`1 + 3 sqrt 3`

Show Worked Solution

`(sqrt 3-1)(2 sqrt 3 + 5)`

`= 2 xx 3 + 5 sqrt 3-2 sqrt 3-5`

`= 1 + 3 sqrt 3`

Functions, 2ADV F1 2008 HSC 1d

Solve `|\ 4x - 3\ | = 7`. (2 marks)

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`x = 5/2\ \ text(or)\ x = -1`

Show Worked Solution

`|\ 4x – 3\ | = 7`

`4x – 3`	`= 7`	`\ \ \ \ \ -(4x – 3)`	`= 7`
`4x`	`= 10`	`-4x + 3`	`= 7`
`x`	`= 5/2`	`-4x`	`= 4`
		`x`	`= -1`

`:. x=5/2\ \ text(or)\ \ -1`

Functions, 2ADV F1 2008 HSC 1a

Evaluate `2 cos (pi/5)` correct to three significant figures. (2 marks)

Show Answers Only

`1.62\ text{(3 sig)}`

Show Worked Solution

`2 cos (pi/5)`	`= 1.6180…`
	`= 1.62\ text{(3 sig)}`

Financial Maths, STD2 F5 SM-Bank 2

The table below shows the present value of an annuity with a contribution of $1.

Fiona pays $3000 into an annuity at the end of each year for 4 years at 2% p.a., compounded annually. What is the present value of her annuity? (1 mark)

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If John pays $6000 into an annuity at the end of each year for 2 years at 4% p.a., compounded annually, is he better off than Fiona? Use calculations to justify your answer. (2 marks)

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`$11\ 423.10`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

i. `text(Table factor when)\ \ n = 4,\ r = text(2%) \ => \ 3.8077`

`:.\ PVA\ text{(Fiona)}`	`= 3000 xx 3.8077`
	`= $11\ 423.10`

ii. `text(Table factor when)\ n = 2, r = text(4%)`

`=> 1.8861`

`:.\ PVA\ text{(John)}`	`= 6000 xx 1.8861`
	`= $11\ 316.60`

`:.\ text(Fiona will be better off because her)\ PVA`

`text(is higher.)`

Mechanics, EXT2* M1 2014 HSC 14a

The take-off point `O` on a ski jump is located at the top of a downslope. The angle between the downslope and the horizontal is `pi/4`. A skier takes off from `O` with velocity `V` m s⁻¹ at an angle `theta` to the horizontal, where `0 <= theta < pi/2`. The skier lands on the downslope at some point `P`, a distance `D` metres from `O`.

The flight path of the skier is given by

`x = Vtcos theta,\ y = -1/2 g t^2 + Vt sin theta`, (Do NOT prove this.)

where `t` is the time in seconds after take-off.

Show that the cartesian equation of the flight path of the skier is given by

`y = x tan theta - (gx^2)/(2V^2) sec^2 theta`. (2 marks)

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Show that

`D = 2 sqrt 2 (V^2)/(g) cos theta (cos theta + sin theta)`. (3 marks)

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Show that

`(dD)/(d theta) = 2 sqrt 2 (V^2)/(g) (cos 2 theta - sin 2 theta)`. (2 marks)

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Show that `D` has a maximum value and find the value of `theta` for which this occurs. (3 marks)

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Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

i. `text(Show)\ \ y = x tan theta – (gx^2)/(2V^2) sec^2 theta`

`x`	`= Vt cos theta`
`t`	`= x/(V cos theta)\ \ \ …\ text{(1)}`

`text(Subst)\ text{(1)}\ text(into)\ y = -1/2 g t^2 + Vt sin theta`

`y`	`= -1/2 g (x/(Vcostheta))^2 + V sin theta (x/(Vcostheta))`
	`= (-gx^2)/(2V^2 cos^2 theta) + x * (sin theta)/(cos theta)`
	`= x tan theta – (gx^2)/(2V^2) sec^2 theta\ \ \ text(… as required.)`

ii. `text(Show)\ D = 2 sqrt 2 (V^2)/g\ cos theta (cos theta + sin theta)`

♦♦ Mean mark 28%
MARKER’S COMMENT: The key to finding `P` and solving for `D` is to realise you need the intersection of the Cartesian equation in (i) and the line `y=–x`.

`text(S)text(ince)\ P\ text(lies on line)\ y = -x`

`-x`	`=x tan theta – (gx^2)/(2V^2) sec^2 theta`
`-1`	`=tan theta – (gx)/(2V^2) sec^2 theta`
`(gx)/(2V^2) sec^2 theta`	`= tan theta + 1`
`x (g/(2V^2))`	`=(sin theta)/(cos theta) * cos^2 theta + 1 * cos^2 theta`
`x`	`=(2V^2)/g\ (sin theta cos theta + cos^2 theta)`
	`=(2V^2)/g\ cos theta (cos theta + sin theta)`

`text(Given that)\ \ cos(pi/4)`	`= x/D = 1/sqrt2`
`text(i.e.)\ \ D`	`= sqrt 2 x`

`:.\ D = 2 sqrt 2 (V^2)/g\ cos theta (cos theta + sin theta)`

`text(… as required.)`

iii. `text(Show)\ (dD)/(d theta) = 2 sqrt 2 (V^2)/g\ (cos 2 theta – sin 2 theta)`

`D`	`= 2 sqrt 2 (V^2)/g\ (cos^2 theta + cos theta sin theta)`
`(dD)/(d theta)`	`= 2 sqrt 2 (V^2)/g\ [2cos theta (–sin theta) + cos theta cos theta + (– sin theta) sin theta]`
	`= 2 sqrt 2 (V^2)/(g) [(cos^2 theta – sin^2 theta) – 2 sin theta cos theta]`
	`= 2 sqrt 2 (V^2)/g\ (cos 2 theta – sin 2 theta)\ \ \ text(… as required)`

iv. `text(Max/min when)\ (dD)/(d theta) = 0`

♦ Mean mark 47%
IMPORTANT: Note that students can attempt every part of this question, even if they couldn’t successfully prove earlier parts.

`2 sqrt 2 (V^2)/g\ (cos 2 theta – sin 2 theta)`	`= 0`
`cos 2 theta – sin 2 theta`	`= 0`
`sin 2 theta`	`= cos 2 theta`
`tan 2 theta`	`= 1`
`2 theta`	`= pi/4`
`theta`	`= pi/8`

`(d^2D)/(d theta^2)`	`= 2 sqrt 2 (V^2)/g\ [-2 sin 2theta – 2 cos 2 theta]`
	`= 4 sqrt 2 (V^2)/g\ (-sin 2 theta – cos 2 theta)`

`text(When)\ \ theta = pi/8:`

`(d^2 D)/(d theta^2)`	`= 4 sqrt 2 (V^2)/g\ (- sin (pi/4) – cos (pi/4))`
	`= 4 sqrt 2 (V^2)/g\ (- 1/sqrt2 – 1/sqrt2) < 0`
	` =>\ text(MAX)`

`:.\ D\ text(has a maximum value when)\ theta = pi/8`

Calculus, EXT1 C1 2014 HSC 13b

One end of a rope is attached to a truck and the other end to a weight. The rope passes over a small wheel located at a vertical distance of 40 m above the point where the rope is attached to the truck.

The distance from the truck to the small wheel is `L\ text(m)`, and the horizontal distance between them is `x\ text(m)`. The rope makes an angle `theta` with the horizontal at the point where it is attached to the truck.

The truck moves to the right at a constant speed of `text(3 m s)^(-1)`, as shown in the diagram.

Using Pythagoras’ Theorem, or otherwise, show that `(dL)/(dx) = cos theta`. (2 marks)

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Show that `(dL)/(dt) = 3 cos theta`. (1 mark)

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Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

`text(Show)\ \ (dL)/(dx) = cos theta`

`text(Using Pythagoras,)`

`L^2`	`=40^2 + x^2`
`L`	`=(40^2 + x^2)^(1/2)`
`(dL)/(dx)`	`=1/2 * 2x * (40^2 + x^2)^(-1/2)`
	`=x/ sqrt((40^2 + x^2))`
	`=x/L`
	`=cos theta\ \ \ text(… as required.)`

ii. `text(Show)\ \ (dL)/(dt) = 3 cos theta`

`(dL)/(dt)`	`= (dL)/(dx) * (dx)/(dt)`
	`= cos theta * 3`
	`= 3 cos theta\ \ \ text(… as required)`

Calculus, EXT1 C1 2014 HSC 12f

Milk taken out of a refrigerator has a temperature of 2° C. It is placed in a room of constant temperature 23°C. After `t` minutes the temperature, `T`°C, of the milk is given by

`T = A-Be ^(-0.03t)`,

where `A` and `B` are positive constants.

How long does it take for the milk to reach a temperature of 10°C? (3 marks)

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`16\ text(mins)`

Show Worked Solution

`T = A – Be^(-0.03t)`

`text(Room temperature constant @)\ 23°`

`=>A = 23`

`:.T = 23 – Be^(-0.03 t)`

`text(At)\ t = 0,\ T = 2`

`2 = 23 – Be^0`

`=>B=21`

`:. T = 23 – 21e^(-0.03t)`

`text(Find)\ \ t\ \ text(when)\ \ T = 10`

`10`	`= 23 – 21e^(-0.03 t)`
`21e^(-0.03 t)`	`=13`
`e^(-0.03t)`	`=13/21`
`ln e^(-0.03t)`	`=ln (13/21)`
`-0.03 t`	`=ln (13/21)`
`:. t`	`=(ln (13/21))/(-0.03)`
	`=15.9857…`
	`=16\ text(mins)\ \ \ text{(nearest minute)}`

Functions, EXT1 F1 2014 HSC 11e

Solve `(x^2 + 5)/x > 6`. (3 marks)

--- 8 WORK AREA LINES (style=lined) ---

Show Answers Only

`0<x<1\ \ text(or)\ \ x>5`

Show Worked Solution

`(x^2 + 5)/x`	`> 6`
`x^2 ((x^2 + 5)/x)`	`> 6x^2`
`x^3 + 5x`	`> 6x^2`
`x^3 – 6x^2 + 5x`	`> 0`
`x (x^2 – 6x + 5)`	`> 0`
`x (x – 5)(x – 1)`	`> 0`

`:.\ 0 < x < 1\ \ text(or)\ \ x > 5`

Trigonometry, EXT1 T1 2014 HSC 11c

Sketch the graph `y = 6 tan^(-1)x`, clearly indicating the range. (2 marks)

Show Answers Only

Show Worked Solution

`y = 6 tan^(-1)x`

Quadratics, EXT1 2014 HSC 11a

Solve `(x + 2/x)^2 - 6 (x + 2/x) + 9 = 0`. (3 marks)

Show Answers Only

`x = 1\ text(or)\ 2`

Show Worked Solution

`(x + 2/x)^2 – 6(x + 2/x) + 9 = 0`

`text(Let)\ (x + 2/x) = X`

`:.\ X^2 – 6X + 9`	`= 0`
`(X – 3)^2`	`= 0`
`X`	`= 3`

`:.\ x + 2/x`	`= 3`
`x^2 + 2`	`= 3x`
`x^2 – 3x + 2`	`= 0`
`(x – 1)(x – 2)`	`= 0`

`:.\ x = 1\ text(or)\ 2`

Calculus, EXT1 C2 2014 HSC 6 MC

What is the derivative of `3 sin^(-1)\ x/2`?

`6/sqrt(4 - x^2)`
`3/sqrt(4 - x^2)`
`3/(2sqrt(4 - x^2))`
`3/(4sqrt(4 - x^2))`

Show Answers Only

`B`

Show Worked Solution

`y = 3 sin^(-1)\ x/2`

`dy/dx = 3 xx 1/sqrt(4 – x^2)`

`=> B`

Combinatorics, EXT1 A1 2014 HSC 3 MC

What is the constant term in the binomial expansion of `(2x - 5/(x^3))^12`?

`((12),(3)) 2^9 5^3`
`((12),(9)) 2^3 5^9`
`-((12),(3)) 2^9 5^3`
`-((12),(9)) 2^3 5^9`

Show Answers Only

`C`

Show Worked Solution

`text(General term)`

`T_k`	`= ((12),(k)) (2x)^(12-k) * (-1)^k *(5x^(-3))^k`
	`= ((12),(k)) 2^(12-k) * x^(12-k) * (-1)^k * 5^k * x^(-3k)`
	`= ((12),(k)) (-1)^k * 2^(12-k) * 5^k * x^(12-4k)`

`text(Constant term when)`

`12 – 4k`	`= 0`
`k`	`= 3`

`:.\ text(Constant term)`

`=((12),(3)) (-1)^3 * 2^9 * 5^3`

`= – ((12),(3)) * 2^9 * 5^3`

`=> C`

Trigonometry, EXT1 T3 2014 HSC 2 MC

Which expression is equal to `cos x - sin x`?

`sqrt 2 cos (x + pi/4)`
`sqrt 2 cos (x - pi/4)`
`2 cos (x + pi/4)`
`2 cos (x - pi/4)`

Show Answers Only

`A`

Show Worked Solution

`R cos (x + alpha) = R cos x cos alpha – R sinx sin alpha`

`:.\ R cosx cos alpha – R sinx sin alpha = cos x – sin x`

`R cos alpha = 1,\ \ \ \ R sin alpha = 1`

`R^2`	`= 1^2 + 1^2`
`R`	`= sqrt 2`

`:.\ sqrt 2 cos alpha`	`= 1`
`cos alpha`	`= 1/sqrt2`
`alpha`	`= pi/4`

`:.\ sqrt 2 cos (x + pi/4) = cosx – sinx`

`=> A`

Geometry and Calculus, EXT1 2009 HSC 4b

Consider the function `f(x) = (x^4 + 3x^2)/(x^4 + 3)`.

Show that `f(x)` is an even function. (1 mark)
What is the equation of the horizontal asymptote to the graph `y = f(x)`? (1 mark)
Find the `x`-coordinates of all stationary points for the graph `y = f(x)`. (3 marks)
Sketch the graph `y = f(x)`. You are not required to find any points of inflection. (2 marks)

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`y = 1`
`x = 0, – sqrt 3, sqrt 3`

Show Worked Solution

(i)	`f(x)`	`= (x^4 + 3x^2)/(x^4 + 3)`
	`f(–x)`	`= ((–x)^4 + 3(-x)^2)/((–x)^4 + 3)`
		`= (x^4 + 3x^2)/(x^4 + 3)`
		`= f(x)`

`:.\ text(Even function.)`

(ii)	`y`	`= (x^4 + 3x^2)/(x^4 + 3)`
		`= (1 + 3/(x^2))/(1 + 3/(x^4))`

`text(As)\ \ `	`x`	`-> oo`
	`y`	`-> 1`

`:.\ text(Horizontal asymptote at)\ \ y = 1`

(iii)

`f(x) = (x^4 + 3x^2)/(x^4 + 3)`

`u`	`= x^4 + 3x^2\ \ \ \ \ `	`v`	`= x^4 + 3`
`u prime`	`= 4x^3 + 6x\ \ \ \ \ `	`v prime`	`= 4x^3`

`f prime (x)`	`= (u prime v\ – u v prime)/(v^2)`
	`= ((4x^3 + 6x)(x^4 + 3)\ – (x^4 + 3x^2)4x^3)/((x^4 + 3)^2)`
	`= (4x^7 + 12x^3 + 6x^5 + 18x\ – 4x^7\ – 12x^5)/((x^4 + 3)^2)`
	`= (-6x^5 + 12x^3 + 18x)/((x^4 + 3)^2)`
	`= (-6x(x^4\ – 2x^2\ – 3))/((x^4 + 3)^2)`

`text(S.P. when)\ x = 0\ \ \ text(or) \ \ x^4\ – 2x^2\ – 3 = 0`

`text(Let)\ X = x^2`

MARKER’S COMMENT: Many students did not realise the denominator of `f′(x)` could be ignored when equating `f′(x)=0`.

`X^2\ – 2X\ – 3`	`= 0`
`(X\ – 3)(X + 1)`	`= 0`

`X = 3\ \ text(or)\ \ -1`

`:. x^2`	`= 3`	`text(or)\ \ \ \ \ `	`x^2 = -1`
`x`	`= +- sqrt3\ \ \ \ `		`text{(no solution)}`

`:.\ text(SPs occur when)\ \ x = 0, – sqrt3, sqrt3`

(iv)	`text(When)\ x = 0,\ `	`y = 0`
	`text(When)\ x = sqrt3,\ \ \ `	`y = ((sqrt3)^4 + 3(sqrt3)^2)/((sqrt3)^4 + 3) = 3/2`

Trig Calculus, EXT1 2009 HSC 3b

On the same set of axes, sketch the graphs of
1. `y = cos 2x` and `y = (x + 1)/2`, for `–pi <= x <= pi`. (2 marks)
Use your graph to determine how many solutions there are to the equation `2 cos 2x = x + 1` for `–pi <= x <= pi`. (1 mark)
One solution of the equation `2 cos 2x = x + 1` is close to `x = 0.4`. Use one application of Newton’s method to find another approximation to this solution. Give your answer correct to three decimal places. (3 marks)

Show Answers Only

`text(See sketch in Worked Solutions)`
`text(3 solutions)`
`0.398\ text{(3 d.p.)}`

Show Worked Solution

(i)

(ii) `text(3 solutions)`

(iii) `2 cos 2x = x + 1`

MARKER’S COMMENT: Better responses defined `f(x)` and `f′(x)` and evaluated each for `x=0.4` before calculating Newton’s formula, as done in the Worked Solution.

`f(x)`	`= 2 cos 2x\ – x\ – 1`
`f prime (x)`	`= -4 sin 2x\ – 1`

`=>f(0.4)`	`= 2 cos 0.8\ – 0.4\ – 1`
	`=-0.0065865 …`
`=> f prime(0.4)`	`= -4 sin 0.8\ – 1`
	`=-3.869424 …`

`text(Find)\ x_1\ text(where)`

`x_1`	`= 0.4\ – (f(0.4))/(f prime(0.4))`
	`= 0.4\ – ((-0.0065865 …)/(-3.869424 …))`
	`= 0.39829…`
	`= 0.398\ \ text{(3 d.p.)}`

Functions, EXT1 F2 2009 HSC 2a

The polynomial `p(x) = x^3-ax + b` has a remainder of `2` when divided by `(x-1)` and a remainder of `5` when divided by `(x + 2)`.

Find the values of `a` and `b`. (3 marks)

--- 8 WORK AREA LINES (style=lined) ---

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`a`	`= 4`
`b`	`= 5`

Show Worked Solution

`p(x)`	`= x^3-ax + b`
`P(1)`	`= 2`
`1-a + b`	`= 2`
`b`	`= a+1\ \ \ …\ text{(1)}`
`P (-2)`	`= 5`
`-8 + 2a + b`	`= 5`
`2a + b`	`= 13\ \ \ …\ text{(2)}`

`text(Substitute)\ \ b = a+1\ \ text(into)\ \ text{(2)}`

`2a + a+1`	`= 13`
`3a`	`= 12`
`:. a`	`= 4`
`:. b`	`= 5`

Trig Calculus, EXT1 2009 HSC 1e

Differentiate `x cos^2 x`. (2 marks)

Show Answers Only

`-2x cos x sin x + cos^2x`

Show Worked Solution

`y`	`= x cos^2 x`
`dy/dx`	`= x * – sin x * 2 cos x + 1 * cos^2 x`
	`= -2x cos x sin x + cos^2 x`

L&E, EXT1 2009 HSC 1b

Let `f(x) = ln (x - 3)`. What is the domain of `f(x)`? (1 mark)

Show Answers Only

`x > 3`

Show Worked Solution

`f(x) = ln (x – 3)`

`text(S)text(ince)\ \ \ `	`x – 3`	`> 0`
	`x`	`> 3`

Plane Geometry, 2UA 2014 HSC 15b

In `Delta DEF`, a point `S` is chosen on the side `DE`. The length of `DS` is `x`, and the length of `ES` is `y`. The line through `S` parallel to `DF` meets `EF` at `Q`. The line through `S` parallel to `EF` meets `DF` at `R`.

The area of `Delta DEF` is `A`. The areas of `Delta DSR` and `Delta SEQ` are `A_1` and `A_2` respectively.

Show that `Delta DEF` is similar to `Delta DSR`. (2 marks)
Explain why `(DR)/(DF) = x/(x + y)`. (1 mark)
Show that
1. `sqrt ((A_1)/A) = x/(x + y)`. (2 marks)
Using the result from part (iii) and a similar expression for
1. `sqrt ((A_2)/A)`, deduce that `sqrt A = sqrt (A_1) + sqrt (A_2)`. (2 marks)

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Corresponding sides of similar)\ Delta text(s)`
`text(are in the same ratio.)`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

(i) `text(Need to show)\ Delta DEF\ text(|||) \ Delta DSR`

`/_FDE\ text(is common)`

`/_DSR = /_DEF = theta\ \ text{(corresponding angles,}\ RS\ text(||)\ FE text{)}`

`:.\ Delta DEF \ text(|||) \ Delta DSR\ \ text{(equiangular)}`

(ii) `(DR)/(DF) = (DS)/(DE) = x/(x + y)`

`text{(Corresponding sides of similar triangles)}`

♦♦ Mean mark 27%
COMMENT: The critical step in solving part (iii) is realising that you need the areas of 2 non-right angled triangles and therefore the formula `A=½\ ab sin C` is required.

(iii) `text(Show)\ sqrt((A_1)/A) = x/(x + y)`

`text(Using Area)`	`= 1/2 ab sin C`
`A_1`	`= 1/2 xx DR xx x xx sin alpha`
`A`	`= 1/2 xx DF xx (x + y) xx sin alpha`

`(A_1)/A`	`= (1/2 * DR * x * sin alpha)/(1/2 * DF * (x + y) * sin alpha)`
	`= (DR * x)/(DF * (x + y)`
	`= (x * x)/((x + y)(x + y))\ \ \ \ text{(using part(ii))}`
	`= (x^2)/((x + y)^2)`
`:.\ sqrt ((A_1)/A)`	`= x/((x + y))\ \ \ text(… as required.)`

(iv) `text(Consider)\ Delta DFE\ text(and)\ Delta SQE`

`/_FED`

`= theta\ text(is common)`

`/_FDE`

`= /_QSE = alpha\ \ `

`text{(corresponding angles,}\ DF\ text(||)\ QS text{)}`

`:.\ Delta DFE\ text(|||)\ Delta SQE\ \ text{(equiangular)}`

`(QE)/(FE) = (SE)/(DE) = y/(x +y)`

`text{(corresponding sides of similar triangles)}`

♦♦ Mean mark 26%

`A_2`	`= 1/2 xx QE xx y xx sin theta`
`A`	`= 1/2 xx FE xx (x + y) xx sin theta`
`(A_2)/A`	`= (QE * y)/(FE * (x + y))`
	`= (y^2)/((x + y)^2)`
`sqrt ((A_2)/A)`	`= y/((x + y))`

`text(Need to show)\ sqrt A = sqrt (A_1) + sqrt (A_2)`

`sqrt(A_2)/sqrtA`	`= y/((x + y))`
`sqrt (A_2)`	`= (sqrtA * y)/((x + y))`

`text(Similarly, from part)\ text{(iii)}`

`sqrt (A_1) = (sqrtA * x)/((x + y))`

`sqrt (A_1) + sqrt (A_2)`	`= (sqrt A * x)/((x + y)) + (sqrt A * y)/((x + y))`
	`= (sqrt A (x + y))/((x + y))`
	`= sqrt A\ \ \ text(… as required.)`

Calculus, 2ADV C4 2014 HSC 12d

The parabola `y = −2x^2 + 8x` and the line `y = 2x` intersect at the origin and at the point `A`.

Find the `x`-coordinate of the point `A`. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
Calculate the area enclosed by the parabola and the line. (3 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

`x=3`
`9\ text(u²)`

Show Worked Solution

`text(Need to find)\ x\ text(-co-ord of)\ A`

`y`	`= 2x\ \ \ \ \ …\ text{(i)}`
`y`	`= -2x^2 + 8x\ \ \ \ \ …\ text{(ii)}`

`text(Subst)\ y = 2x\ text(from)\ text{(i)}\ text(into)\ text{(ii)}`

`-2x^2 + 8x`	`= 2x`
`-2x^2 + 6x`	`= 0`
`-2x (x\ – 3)`	`= 0`

`:.\ x = 0\ text(or)\ 3`

`:.\ x\ text(-coordinate of)\ A\ text(is 3)`

ii.	`text(Area)`	`= int_0^3 (-2x^2 + 8x)\ dx\ – int_0^3 2x\ dx`
		`= int_0^3 (-2x^2 + 6x)\ dx`
		`= [-2/3x^3 + 3x^2]_0^3`
		`= [(-2/3(3^3) + 3(3^2))\ – (0 + 0)]`
		`= -18 + 27`
		`= 9\ text(u²)`

Probability, 2ADV S1 2014 HSC 12c

A packet of lollies contains 5 red lollies and 14 green lollies. Two lollies are selected at random without replacement.

Draw a tree diagram to show the possible outcomes. Include the probability on each branch. (2 marks)

--- 6 WORK AREA LINES (style=lined) ---
What is the probability that the two lollies are of different colours? (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

`70/171`

Show Worked Solution

ii. `P text{(different colours)}`

`= P(RG) + P(GR)`

`= 5/19 xx 14/18 + 14/19 xx 5/18`

`= 70/342 + 70/342`

`= 140/342`

`= 70/171`

Financial Maths, 2ADV M1 2014 HSC 14d

At the beginning of every 8-hour period, a patient is given 10 mL of a particular drug.

During each of these 8-hour periods, the patient’s body partially breaks down the drug. Only `1/3` of the total amount of the drug present in the patient’s body at the beginning of each 8-hour period remains at the end of that period.

How much of the drug is in the patient’s body immediately after the second dose is given? (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
Show that the total amount of the drug in the patient’s body never exceeds 15 mL. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`13.33\ text{mL (2 d.p.)}`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

i. `text(Let)\ \ A =\ text(Amount of drug in body)`

`text(Initially)\ A = 10`

`text(After 8 hours)\ \ \ A`	`=1/3 xx 10`
`text(After 2nd dose)\ \ A`	`= 10 + 1/3 xx 10\ text(mL)`
	`=13.33\ text{mL (2 d.p.)}`

ii. `text(After the 3rd dose)`

`A_3`	`= 10 + 1/3 (10 + 1/3 xx 10)`
	`= 10 + 1/3 xx 10 + (1/3)^2 xx 10`

` =>\ text(GP where)\ a = 10,\ r = 1/3`

`text(S)text(ince)\ \ |\ r\ | < 1:`

`S_oo`	`= a/(1\ – r)`
	`= 10/(1\ – 1/3)`
	`= 10/(2/3)`
	`= 15`

`:.\ text(The amount of the drug will never exceed 15 mL.)`

Quadratic, 2UA 2014 HSC 14b

The roots of the quadratic equation `2x^2 + 8x + k = 0` are `alpha` and `beta`.

Find the value of `alpha + beta`. (1 mark)
Given that `alpha^2 beta + alpha beta^2 = 6`, find the value of `k`. (2 marks)

Show Answers Only

`-4`
`-3`

Show Worked Solution

(i)	`2x^2 + 8x + k = 0`
	`alpha + beta = (-b)/a = (-8)/2 = -4`

(ii)	`alpha^2 beta + alpha beta^2`	`= 6`
	`alpha beta (alpha + beta)`	`= 6`

`text(S)text(ince)\ \ alpha beta = c/a = k/2`

`=> k/2 (–4)`	`= 6`
`-2k`	`= 6`
`k`	`= -3`

Calculus, EXT1* C1 2014 HSC 13b

A quantity of radioactive material decays according to the equation

`(dM)/(dt) = -kM`,

where `M` is the mass of the material in kg, `t` is the time in years and `k` is a constant.

Show that `M = Ae^(–kt)` is a solution to the equation, where `A` is a constant. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
The time for half of the material to decay is 300 years. If the initial amount of material is 20 kg, find the amount remaining after 1000 years. (3 marks)

--- 10 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`1.98\ text(kg)\ text{(2 d.p.)}`

Show Worked Solution

i.	`M`	`= Ae^(-kt)`
	`(dM)/(dt)`	`= -k * Ae^(-kt)`
		`= -kM\ \ text(… as required)`

ii.

`text(At)\ \ t = 0,\ M = 20`

`=> 20`	`= Ae^0`
`A`	`= 20`
`:.\ M`	`= 20 e^(-kt)`

`text(At)\ \ t = 300,\ M = 10`

TIP: Many students find it efficient to save the exact value of `k` in the memory function of their calculator for these questions.

`=> 10`	`= 20 e^(-300 xx k)`
`e^(-300k)`	`= 10/20`
`ln e^(-300k)`	`= ln 0.5`
`-300 k`	`= ln 0.5`
`k`	`= – ln 0.5/300`
	`= 0.00231049…`

`text(Find)\ M\ text(when)\ t = 1000`

`M`	`= 20 e^(-1000k)`
	`= 20 e^(-1000 xx 0.00231049…)`
	`= 20 xx 0.099212…`
	`= 1.9842…`
	`= 1.98\ text(kg)\ text{(2 d.p.)}`

Linear Functions, 2UA 2014 HSC 12b

The points `A(0, 4)`, `B(3, 0)` and `C(6, 1)` form a triangle, as shown in the diagram.

Show that the equation of `AC` is `x + 2y − 8 = 0`. (2 marks)
Find the perpendicular distance from `B` to `AC`. (2 marks)
Hence, or otherwise, find the area of `Delta ABC`. (2 marks)

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`sqrt5\ text(units)`
`15/2\ text(u²)`

Show Worked Solution

(i) `text(Need to show equine of)\ AC\ text(is)\ x + 2y\ – 8 = 0`

`A (0,4)\ \ \ \ \ C (6,1)`

`m_(AC)`	`= (y_2\ – y_1)/(x_2\ – x_1)`
	`= (1\ – 4)/(6\ – 0)`
	`= – 1/2`

`text(Equation of line)\ m = -1/2,\ text(through)\ (0,4)`

`text(Using)\ \ y\ – y_1`	`= m (x\ – x_1)`
`y\ – 4`	`= -1/2 (x\ – 0)`
`y\ – 4`	`= -1/2 x`
`2y\ – 8`	`= – x`

`x + 2y\ – 8 = 0\ \ \ text(… as required)`

(ii) `_|_\ text(distance of)\ B (3,0)\ text(from)\ AC`

`_\|_\ text(dist)`	`= \| (ax_1 + by_1 + c)/sqrt(a^2 + b^2) \|`
	`= \| (1(3) + 2(0)\ – 8)/sqrt(1^2 + 2^2) \|`
	`= \| (-5)/sqrt5 \|`
	`= 5/sqrt5 xx sqrt5/sqrt5`
	`= sqrt 5\ text(units)`

(iii) `text(Area)\ Delta ABC = 1/2 b h`

`b = AC`

`text(dist)\ AC`	`= sqrt( (x_2\ – x_1)^2 + (y_2\ – y_1)^2 )`
	`= sqrt( (0\ – 6)^2 + (4\ – 1)^2 )`
	`= sqrt (36 + 9)`
	`= sqrt 45`
	`= 3 sqrt 5\ text(units)`

`h = sqrt 5\ \ \ text{(from part (ii))}`

`:.\ text(Area)\ Delta ABC`	`= 1/2 xx 3 sqrt 5 xx sqrt 5`
	`= 15/2\ text(u²)`

Functions, 2ADV F1 2014 HSC 6 MC

Which expression is a factorisation of `8x^3 + 27`?

`(2x - 3)(4x^2 + 12x - 9)`
`(2x + 3)(4x^2 - 12x + 9)`
`(2x - 3)(4x^2 + 6x - 9)`
`(2x + 3)(4x^2 - 6x + 9)`

Show Answers Only

`D`

Show Worked Solution

`8x^3 + 27`

COMMENT: Factorising a cubic is only examinable with scaffolding, as provided here by expanding the answer options.

`= (2x)^3 + 3^3`

`= (2x + 3)(4x^2 – 6x + 9)`

`=> D`

Calculus, 2ADV C4 2014 HSC 4 MC

Which expression is equal to `int e^(2x)\ dx`?

`e^(2x) + C`
`2e^(2x) + C`
`(e^(2x))/2 +C`
`(e^(2x + 1))/(2x + 1) + C`

Show Answers Only

`C`

Show Worked Solution

`int e^(2x)\ dx`

`= 1/2 e^(2x) + C`

`=> C`

Functions, 2ADV F2 2014 HSC 2 MC

Which graph best represents `y = (x - 1)^2`?

Show Answers Only

`B`

Show Worked Solution

`y = (x- 1)^2\ →\ text(vertex)\ (1, 0)`

`=> B`

Functions, 2ADV F1 2014 HSC 1 MC

What is the value of `(pi^2)/6`, correct to 3 significant figures?

`1.64`
`1.65`
`1.644`
`1.645`

Show Answers Only

`A`

Show Worked Solution

`(pi^2)/6`	`= 1.6449…`
	`= 1.64\ text{(3 sig. figures)}`

`=> A`

Trigonometry, 2ADV T1 2014 HSC 11g

The angle of a sector in a circle of radius 8 cm is `pi/7` radians, as shown in the diagram.

Find the exact value of the perimeter of the sector. (2 marks)

Show Answers Only

`(8pi)/7 + 16\ text(cm)`

Show Worked Solution

`text(Arc length)`	`= theta/(2pi) xx 2 pi r`
	`= pi/7 xx 8`
	`= (8pi)/7\ text(cm)`

`text(S)text(ector perimeter)`	`= text(arc) + 2 xx text(radius)`
	`= (8pi)/7 + 16\ text(cm)`

Calculus, 2ADV C1 2014 HSC 11c

Differentiate `x^3/(x + 1)`. (2 marks)

--- 5 WORK AREA LINES (style=lined) ---

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`(x^2(2x + 3))/((x + 1)^2)`

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`y = (x^3)/(x + 1)`

`text(Using)\ \ \ dy/dx = (u prime v\ – uv prime)/(v^2)`

`u`	`=x^3`	`\ \ \ \ \ v`	`=(x+1)`
`u ′`	`=3x^2`	`v′`	`=1`

`dy/dx`	`= (3x^2 (x + 1)\ – x^3 (1))/((x + 1)^2)`
	`= (3x^3 + 3x^2\ – x^3)/((x + 1)^2)`
	`= (2x^3 + 3x^2)/((x + 1)^2)`
	`= (x^2 (2x + 3))/((x + 1)^2)`

Functions, 2ADV F1 2014 HSC 11b

Factorise `3x^2 + x − 2`. (2 marks)

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`(3x- 2)(x + 1)`

Show Worked Solution

`3x^2 + x – 2`

`= (3x- 2)(x + 1)`

Functions, 2ADV F1 2014 HSC 11a

Rationalise the denominator of `1/(sqrt5-2)`. (2 marks)

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`sqrt5 + 2`

Show Worked Solution

`1/(sqrt5-2) xx (sqrt5 + 2)/(sqrt5 + 2)`

`= (sqrt5 + 2)/((sqrt5)^2-2^2)`

`= sqrt5 + 2`

Statistics, STD2 S1 2014 HSC 29c

Terry and Kim each sat twenty class tests. Terry’s results on the tests are displayed in the box-and-whisker plot shown in part (i).

Kim’s 5-number summary for the tests is 67, 69, 71, 73, 75.

Draw a box-and-whisker plot to display Kim’s results below that of Terry’s results. (1 mark)
What percentage of Terry’s results were below 69? (1 mark)

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Terry claims that his results were better than Kim’s. Is he correct?

Justify your answer by referring to the summary statistics and the skewness of the distributions. (4 marks)

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`text(See Worked Solutions)`
`text(50%)`
`text(See Worked Solutions)`

Show Worked Solution

♦ Mean mark 39%

ii. `text(50%)`

iii. `text(Terry’s results are more positively skewed than)`

♦♦ Mean mark 29%
COMMENT: Examiners look favourably on using language of location in answers, particularly the areas they have specifically pointed students towards (skewness in this example).

`text(Kim’s and also have a higher limit high.)`

`text(However, Kim’s results are more consistent,)`

`text(showing a tighter IQR. They also have a)`

`text(significantly higher median than Terry’s and)`

`text(are evenly skewed.)`

`:.\ text(Kim’s results were better.)`

Algebra, STD2 A2 2014 HSC 27b

Xuso is comparing the costs of two different ways of travelling to university.

Xuso’s motorcycle uses one litre of fuel for every 17 km travelled. The cost of fuel is $1.67/L and the distance from her home to the university car park is 34 km. The cost of travelling by bus is $36.40 for 10 single trips.

Which way of travelling is cheaper and by how much? Support your answer with calculations. (2 marks)

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`text(Motorcycle is $0.30 cheaper per 1-way trip)`

Show Worked Solution

`text(Compare cost of a 1-way trip)`

`text(Motorcycle)`

`text(Fuel used) = 34/17 = 2\ text(L)`

`text(C)text(ost) = 2 xx $1.67 = $3.34`

`text(Bus)`

`text(C)text(ost) = 36.40/10 = $3.64`

`text(Difference) = $3.64\ – 3.34\ = $0.30`

`:.\ text(Motorcycle is $0.30 cheaper per 1-way trip.)`

Financial Maths, STD2 F1 2014 HSC 13 MC

Jane sells jewellery. Her commission is based on a sliding scale of 6% on the first $2000 of her sales, 3.5% on the next $1000, and 2% thereafter.

What is Jane’s commission when her total sales are $5670?

$188.40
$208.40
$321.85
$652.05

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`B`

Show Worked Solution

`text(Commission)`

`= (2000 xx text(6%)) + (1000 xx text(3.5%)) + (5670-3000) xx text(2%)`

`= (2000 xx 0.06) + (1000 xx 0.035) + (2670 xx 0.02)`

`= 120 + 35 + 53.40`

`= 208.40`

`=> B`

Financial Maths, STD2 F4 2014 HSC 9 MC

A car is bought for $19 990. It will depreciate at 18% per annum.

Using the declining balance method, what will be the salvage value of the car after 3 years, to the nearest dollar?

$8968
$9195
$11 022
$16 392

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`C`

Show Worked Solution

`S`	`= V_0 (1-r)^n`
	`= 19\ 990 (1-18/100)^3`
	`= 19\ 990 (0.82)^3`
	`= $11\ 021.85`

`=> C`

Probability, STD2 S2 2014 HSC 8 MC

A group of 150 people was surveyed and the results recorded.

A person is selected at random from the surveyed group.

What is the probability that the person selected is a male who does not own a mobile?

`28/150`
`45/150`
`28/70`
`45/70`

Show Answers Only

`A`

Show Worked Solution

`P`	`= text(number of males without mobile)/text(number in group)`
	`= 28/150`

`=> A`

Probability, STD2 S2 2014 HSC 6 MC

A cafe menu has 3 entrees, 5 main courses and 2 desserts. Ariana is choosing a three-course meal consisting of an entree, a main course and a dessert.

How many different three-course meals can Ariana choose?

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`D`

Show Worked Solution

`text(# Combinations)`	`= 3 xx 5 xx 2`
	`= 30`

`=> D`

Measurement, STD2 M1 2014 HSC 2 MC

A measurement of 72 cm is increased by 20% and then the result is decreased by 20%.

What is the new measurement, correct to the nearest centimetre?

46 cm
69 cm
72 cm
104 cm

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`B`

Show Worked Solution

STRATEGY: Students confident in this area could save time in the calculations as follows: `72 xx 1.2 xx 0.8“ = 69.12`

`text(72 increased by 20%)`

`= 72 + (text(20%) xx 72) = 86.4\ text(cm)`

`text(86.4 decreased by 20%)`

`= 86.4\ – (text(20%) xx 86.4) = 69.12\ text(cm)`

`=> B`

Quadratic, EXT1 2014 HSC 13c

The point `P(2at, at^2)` lies on the parabola `x^2 = 4ay` with focus `S`.

The point `Q` divides the interval `PS` internally in the ratio `t^2 :1`.

Show that the coordinates of `Q` are
`x = (2at)/(1 + t^2)` and `y = (2at^2)/(1 + t^2)`. (2 marks)
Express the slope of `OQ` in terms of `t`. (1 mark)
Using the result from part (ii), or otherwise, show that `Q` lies on a fixed circle of radius `a`. (3 marks)

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`text(Proof)\ \ text{(See Worked Solutions)}`
`m_(OQ) = t`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

(i) `text(Show)\ \ Q = ((2at)/(1 + t^2), (2at^2)/(1 + t^2))`

`P (2at, at^2),\ S (0, a)`

`PS\ text(is divided internally in ratio)\ t^2: 1`

`Q`	`= ((nx_1 + mx_2)/(m + n), (ny_1 + my_2)/(m + n))`
	`= ((1(2at) + t^2(0))/(t^2 + 1), (1(at^2) + t^2 (a))/(t^2 + 1))`
	`= ((2at)/(1 + t^2), (2at^2)/(1 + t^2))\ \ text(… as required.)`

(ii)	`m_(OQ)`	`= (y_2\ – y_1)/(x_2\ – x_1)`
		`= ((2at^2)/(1 + t^2))/((2at)/(1 + t^2)) xx (1+t^2)/(1+t^2)`
		`= (2at^2)/(2at)`
		`= t`

(iii)	`text(Show)\ Q\ text(lies on a fixed circle radius)\ a`
	`text(S)text(ince)\ Q\ text(passes through)\ (0, 0)`

`=>\ text(If locus of)\ Q\ text(is a circle, it has)`

`text(diameter)\ QT\ text(where)\ T(0, 2a)`

`text(Show)\ \ QT _|_ OQ`

♦♦ Mean mark 22%

`text{(} text(angles on circum. subtended by)`

`text(a diameter are)\ 90^@ text{)}`

`m_(OQ) = t\ \ \ \ text{(see part (ii))}`

`text(Find)\ m_(QT),\ \ text(where:)`

`Q((2at)/(1 + t^2), (2at^2)/(1 + t^2)),\ \ \ \ \ T (0,2a)`

`m_(QT)`	`= (y_2\ – y_1)/(x_2\ – x_1)`
	`= ((2at^2)/(1 + t^2)\ – 2a)/((2at)/(1 + t^2)\ – 0)`
	`= (2at^2\ – 2a (1 + t^2))/(2at)`
	`= – (2a)/(2at)`
	`= – 1/t`

`m_(QT) xx m_(OT) = -1/t xx t = -1`

`=> QT _|_ OQ`

`=>O,\ T,\ Q\ text(lie on a circle.)`

`:.\ text(Locus of)\ Q\ text(is a fixed circle,)`

`text(centre)\ (0, a),\ text(radius)\ a`

Quadratic, EXT1 2009 HSC 2c

The diagram shows points `P(2t, t^2)` and `Q(4t, 4t^2)` which move along the parabola `x^2 = 4y`. The tangents to the parabola at `P` and `Q` meet at `R`.

Show that the equation of the tangent at `P` is `y = tx\ - t^2.` (2 marks)
Write down the equation of the tangent at `Q`, and find the coordinates of the point `R` in terms of `t`. (2 marks)
Find the Cartesian equation of the locus of `R`. (1 mark)

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`text(Proof)\ \ text{(See Worked Solutions)}`
`y = 2tx\ – 4t^2,\ R (3t, 2t^2)`
`y = 2/9 x^2`

Show Worked Solution

(i)	`x^2`	`= 4y`
	`y`	`= (x^2)/4`
	`dy/dx`	`= x/2`

IMPORTANT: Deriving this equation was specifically asked in 2009 and 2011, as well as being required in 2014. KNOW IT! Completing this in 60 seconds gains 2 full minutes on allocated time.

`text(At)\ \ x = 2t,`

`dy/dx = (2t)/2 = t`

`:.\ text(T)text(angent has)\ \ m = t\ \ text(and passes)`

`text(through)\ \ (2t, t^2).`

`y – t^2`	`= t (x – 2t)`
`y`	`= tx\ – 2t^2 + t^2`
	`= tx\ – t^2\ \ text(… as required)`

(ii)

`text(T)text(angent at)\ \ P (2t, t^2)\ \ text(is)\ \ y = tx\ – t^2`

`=>Q(4t, 4t^2) -= Q(2(2t), (2t)^2)`

`:.\ text(Equation of the tangent at)\ \ Q,`

`y`	`= 2tx\ – (2t)^2`
	`= 2tx\ – 4t^2`

`text(T)text(angents intersect at)\ \ R`

`y`	`= tx\ – t^2\ \ \ \ \ …\ text{(1)}`
`y`	`= 2tx\ – 4t^2\ \ \ \ \ …\ text{(2)}`

`text(Intersection when)\ text{(1)} = text{(2)}`

`tx\ – t^2`	`= 2tx\ – 4t^2`
`tx`	`= 3t^2`
`x`	`= 3t`

`text(Substitute)\ \ x = 3t\ \ text(into)\ text{(1)}`

`y`	`= t (3t)\ – t^2`
	`= 2t^2`

`:.\ R (3t, 2t^2)`

(iii)

`text(Find locus of)\ \ R`

`x`	`= 3t`	`\ \ \ \ \ …\ text{(1)}`
`y`	`= 2t^2\ \ \ \ \ …\ text{(2)}`

`text{From (1),}\ \ \ \ t = x/3`

`text(Substitute)\ \ t = x/3\ text(into)\ text{(2)}`

`y`	`= 2 (x/3)^2`
	`= 2/9 x^2`

`:.\ text(Locus of)\ \ R\ \ text(is)\ \ y = 2/9 x^2`

`_\|_\ text(dist)`	`= \| (ax_1 + by_1 + c)/sqrt (a^2 + b^2) \|`
	`= \|( 2(0) – 1(3) -1 )/sqrt (2^2 + (-1)^2) \|`
	`= \| -4/sqrt5 \|`
	`= 4/sqrt 5\ \ \ text(… as required.)`