num-title-qs-hsc

Functions, 2ADV F1 2017 HSC 2 MC

Which expression is equal to `3x^2-x-2`?

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

Show Answers Only

`D`

Show Worked Solution

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

`=> D`

Functions, 2ADV F1 2017 HSC 1 MC

What is the gradient of the line `2x + 3y + 4 = 0`?

`-2/3`
`2/3`
`-3/2`
`3/2`

Show Answers Only

`A`

Show Worked Solution

`2x + 3y + 4`	`= 0`
`3y`	`= -2x-4`
`y`	`= -2/3 x-4/3`
`:.\ text(Gradient)`	`= -2/3`

`=> A`

Algebra, STD2 A4 2017 HSC 28e

A movie theatre has 200 seats. Each ticket currently costs $8.

The theatre owners are currently selling all 200 tickets for each session. They decide to increase the price of tickets to see if they can increase the income earned from each movie session.

It is assumed that for each one dollar increase in ticket price, there will be 10 fewer tickets sold.

A graph showing the relationship between an increase in ticket price and the income is shown below.

What ticket price should be charged to maximise the income from a movie session? (1 mark)

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What is the number of tickets sold when the income is maximised? (1 mark)

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The cost to the theatre owners of running each session is $500 plus $2 per ticket sold.

Calculate the profit earned by the theatre owners when the income earned from a session is maximised. (2 marks)

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Show Answers Only

`$14`
`140`
`$1180`

Show Worked Solution

i. `text(Graph is highest when increase = $6)`

♦ Mean mark 50%.

`:.\ text(Ticket price)`	`= 8 + 6`
	`= $14`

ii. `text(Solution 1)`

♦ Mean mark 45%.

`text(Tickets sold)`	`=200-(4 xx 10)`
	`=140`

`text(Solution 2)`

`text(Tickets)`	`= text(max income)/text(ticket price)`
	`= 1960/14`
	`= 140`

iii.	`text(C)text(ost)`	`= 140 xx $2 + $500`
		`= $780`

`:.\ text(Profit when income is maximised)`

`= 1960-780`

`= $1180`

Statistics, STD2 S1 2017 HSC 27a

Jamal surveyed eight households in his street. He asked them how many kilolitres (kL) of water they used in the last year. Here are the results.

`220, 105, 101, 450, 37, 338, 151, 205`

Calculate the mean of this set of data. (1 mark)

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What is the standard deviation of this set of data, correct to one decimal place? (1 mark)

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Show Answers Only

`200.875`
`127.4\ \ text{(1 d.p.)}`

Show Worked Solution

i.	`text(Mean)`	`= (220 + 105 + 101 + 450 + 37 + 338 + 151 + 205) ÷ 8`
		`= 200.875`

♦ Mean mark part (ii) 47%.
IMPORTANT: The population standard deviation is required here.

ii.	`text(Std Dev)`	`= 127.357…\ \ text{(by calc)}`
		`= 127.4\ \ text{(1 d.p.)}`

Measurement, STD2 M6 2017 HSC 26d

A sewer pipe needs to be placed into the ground so that it has a 2° angle of depression. The length of the pipe is 15 000 mm.

How much deeper should one end of the pipe be compared to the other end? Answer to the nearest mm. (2 marks)

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Show Answers Only

`523\ text{mm (nearest mm)}`

Show Worked Solution

`text(Let)\ \ x = text(depth needed)`

`sin 2^@`	`= x/(15\ 000)`
`x`	`= 15\ 000 xx sin 2^@`
	`= 523.49…`
	`= 523\ text{mm (nearest mm)}`

Measurement, STD2 M1 2017 HSC 25 MC

In the circle, centre `O`, the area of the quadrant is 100 cm².

What is the arc length `l`, correct to one decimal place?

8.9 cm
11.3 cm
17.7 cm
25.1 cm

Show Answers Only

`C`

Show Worked Solution

`text(Find)\ r:`

♦ Mean mark 44%.

`text(Area)`	`= 1/4 pir^2`
`100`	`= 1/4 pir^2`
`r^2`	`= 400/pi`
`:. r`	`= 11.283…\ text(cm)`

`text(Arc length)`	`= theta/360 xx 2pir`
	`= 90/360 xx 2pi xx 11.283…`
	`= 17.724…`
	`= 17.7\ text(cm)`

`=> C`

Probability, STD2 S2 2017 HSC 15 MC

The faces on a twenty-sided die are labelled $0.05, $0.10, $0.15, … , $1.00.

The die is rolled once.

What is the probability that the amount showing on the upper face is more than 50 cents but less than 80 cents?

A. `1/4`

B. `3/10`

C. `7/20`

D. `1/2`

Show Answers Only

`A`

Show Worked Solution

`text(Possible faces that satisfy are:)`

♦ Mean mark 50%.

`55text(c),60text(c),65text(c),70text(c),75text(c)`

`:.\ text(Probability)`	`= 5/20`
	`= 1/4`

`=>A`

Statistics, STD2 S4 2017 HSC 12 MC

Which of the data sets graphed below has the largest positive correlation coefficient value?

A.		B.
C.		D.

Show Answers Only

$C$

Show Worked Solution

$\text{Largest positive correlation occurs when both variables move}$

$\text{in tandem. The tighter the linear relationship, the higher the}$

$\text{correlation.}$

$\Rightarrow C$

$\text{(Note that B is negatively correlated)}$

Financial Maths, STD2 F1 2017 HSC 11 MC

A new car was bought for $19 900 and one year later its value had depreciated to $16 300.

What is the approximate depreciation, expressed as a percentage of the purchase price?

Show Answers Only

`A`

Show Worked Solution

`text(Net Depreciation)`	`= 19\ 900-16\ 300`
	`= $3600`

`:. %\ text(Depreciation)`	`= 3600/(19\ 900) xx 100`
	`= 18.09…text(%)`

`=>A`

Algebra, STD2 A1 2017 HSC 9 MC

What is the value of `x` in the equation `(5-x)/3 = 6`?

`-13`
`-3`
`3`
`13`

Show Answers Only

`A`

Show Worked Solution

`(5-x)/3`	`= 6`
`5-x`	`= 18`
`x`	`= 5-18`
	`= -13`

`=>A`

Financial Maths, STD2 F1 2017 HSC 6 MC

Tom earns a weekly wage of $1025. He also receives an additional allowance of $87.50 per day when handling toxic substances.

What is Tom’s income in a fortnight in which he handles toxic substances on 5 separate days?

$1112.50
$1462.50
$2225.00
$2487.50

Show Answers Only

`D`

Show Worked Solution

`text(Fortnightly wage)`	`= 2 xx 1025`
	`= $2050`

`text(Allowances)`	`= 5 xx 87.50`
	`= $437.50`

`:.\ text(Income)`	`= 2050 + 437.50`
	`= $2487.50`

`=>D`

Probability, STD2 S2 2017 HSC 5 MC

In a survey of 200 randomly selected Year 12 students it was found that 180 use social media.

Based on this survey, approximately how many of 75 000 Year 12 students would be expected to use social media?

A. 60 000

B. 67 500

C. 74 980

D. 75 000

Show Answers Only

`B`

Show Worked Solution

`text(Expected number)`	`= 180/200 xx 75\ 000`
	`= 67\ 500`

`=> B`

Algebra, STD2 A2 2017 HSC 3 MC

The graph shows the relationship between infant mortality rate (deaths per 1000 live births) and life expectancy at birth (in years) for different countries.

What is the life expectancy at birth in a country which has an infant mortality rate of 60?

68 years
69 years
86 years
88 years

Show Answers Only

$A$

Show Worked Solution

$\text{When infant mortality rate is 60, life expectancy}$

$\text{at birth is 68 years (see below).}$

$\Rightarrow A$

Statistics, STD2 S1 2017 HSC 1 MC

The box-and-whisker plot for a set of data is shown.

What is the median of this set of data?

Show Answers Only

`C`

Show Worked Solution

`text(Median = 30)`

`=> C`

Plane Geometry, EXT1 2016 HSC 4 MC

In the diagram, `O` is the centre of the circle `ABC`, `D` is the midpoint of `BC`, `AT` is the tangent at `A` and `∠ATB = 40^@`.

What is the size of the reflex angle `DOA`?

`80^@`
`140^@`
`220^@`
`280^@`

Show Answers Only

`C`

Show Worked Solution

`/_ ODT`	`=90^@\ \ text{(line through centre bisecting chord)}`
`/_OAT`	`= 90^@\ \ text{(tangent ⊥ to radius at point of contact)}`
`/_ DOA`	`= 360-(90 + 90 + 40)`
	`= 140^@`

`:. DOA\ \ text{(reflex)}`	`= 360-140`
	`= 220^@`

`=> C`

Functions, EXT1 F2 2016 HSC 2 MC

What is the remainder when `2x^3-10x^2 + 6x + 2` is divided by `x-2`?

`-66`
`-10`
`-x^3 + 5x^2-3x-1`
`x^3-5x^2 + 3x + 1`

Show Answers Only

`B`

Show Worked Solution

`P(2)`	`= 2 · 2^3-10 · 2^2 + 6 · 2 + 2`
	`= -10`

`=> B`

L&E, 2ADV E1 2016 HSC 14e

Write `log 2 + log 4 + log 8 + … + log 512` in the form `a log b` where `a` and `b` are integers greater than `1.` (2 marks)

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Show Answers Only

`45 log 2`

Show Worked Solution

`log 2 + log 4 + log 8 + … + log 512`

`= log 2^1 + log 2^2 + log2^3 + … + log 2^9`

`= log 2 + 2 log 2 + 3 log 2 + … + 9 log 2`

`= 45 log 2`

♦ Mean mark 40%.

TIP: Note that `log 2 = log_10 2`

Measurement, STD2 M1 2016 HSC 30c

A school playground consists of part of a circle, with centre `O`, and a rectangle as shown in the diagram. The radius `OB` of the circle is 45 m, the width `BC` of the rectangle is 20 m and `AOB` is 100°.

What is the area of the whole playground, correct to the nearest square metre? (5 marks)

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Show Answers Only

`6971\ text{m² (nearest m²)}`

Show Worked Solution

`text(In)\ DeltaOEB,`

`sin50^@`	`= (EB)/45`
`EB`	`= 45 xx sin50^@`
	`= 34.47…`

`:. AB`	`= 2 xx 34.47…`
	`= 68.944\ \ (text(3 d.p.))`

`cos50^@`	`= (OE)/45`
`:. OE`	`= 45 xx cos50^@`
	`= 28.925\ \ (text(3 d.p.))`

`text(Area of)\ DeltaOAB`

`= 1/2 xx AB xx OE`

`= 1/2 xx 68.944 xx 28.925`

`= 997.12\ text(m²)`

`text(Area)\ ABCD`	`= 20 xx 68.944`
	`= 1378.88\ text(m²)`

`text(Area of major sector)\ OAB`

`= pi xx 45^2 xx 260/360`

`= 4594.58\ text(m²)`

`:.\ text(Area of playground)`

`= 997.12 + 1378.88 + 4594.58`

`= 6970.58`

`= 6971\ text{m² (nearest m²)}`

Measurement, STD2 M1 2016 HSC 28e

A company makes large marshmallows. They are in the shape of a cylinder with diameter 5 cm and height 3 cm, as shown in the diagram.

Find the volume of one of these large marshmallows, correct to one decimal place. (2 marks)

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A cake is to be made by stacking 24 of these large marshmallows and filling the gaps between them with chocolate. The diagrams show the cake and its top view. The shading shows the gaps to be filled with chocolate.

What volume of chocolate will be required? Give your answer correct to the nearest whole number. (3 marks)

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Show Answers Only

`58.9\ text{cm}^3`
`193\ text{cm}^3`

Show Worked Solution

i.	`V`	`= pir^2h`
		`= pi xx 2.5^2 xx 3`
		`= 58.904…`
		`= 58.9\ text{cm³ (1 d.p.)}`

ii.

`text(Volume of rectangle)`

♦ Mean mark part (ii) 35%.

`= 15 xx 10 xx 6`

`= 900\ text(cm)^3`

`text(Volume of marshmallows in rectangle)`

`= 6 xx 2 xx 58.9`

`= 706.8\ text(cm)^3`

`:.\ text(Volume of chocolate)`

`= 900-706.8`

`= 193.2`

`= 193\ text{cm}^3 \ text{(nearest cm}^3 text{)}`

Financial Maths, STD2 F1 2016 HSC 26f

Theo is completing his tax return. He has a gross salary of $82 521 and income from a rental property totalling $10 920. He is claiming $13 420 in allowable deductions.

Determine Theo’s taxable income. (1 mark)

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Using the tax table below, calculate Theo’s tax payable. (2 marks)

\begin{array} {|l|l|}
\hline
\rule{0pt}{2.5ex}\textit{ Taxable income}\rule[-1ex]{0pt}{0pt} & \textit{ Tax payable}\\
\hline
\rule{0pt}{2.5ex}\text{\$0 – \$18 200}\rule[-1ex]{0pt}{0pt} & \text{Nil}\\
\hline
\rule{0pt}{2.5ex}\text{\$18 201 – \$37 000}\rule[-1ex]{0pt}{0pt} & \text{19 cents for each \$1 over \$18 200}\\
\hline
\rule{0pt}{2.5ex}\text{\$37 001 – \$80 000}\rule[-1ex]{0pt}{0pt} & \text{\$3572 plus 32.5 cents for each \$1 over \$37 000}\\
\hline
\rule{0pt}{2.5ex}\text{\$80 001 – \$180 000}\rule[-1ex]{0pt}{0pt} & \text{\$17 547 plus 37 cents for each \$1 over \$80 000}\\
\hline
\rule{0pt}{2.5ex}\text{\$180 001 and over}\rule[-1ex]{0pt}{0pt} & \text{\$54 547 plus 45 cents for each \$1 over \$180 000}\\
\hline
\end{array}

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In addition to the above tax, Theo must also pay a Medicare levy of $1600.42
Theo has already paid $20 525 as Pay As You Go (PAYG) tax.
Should Theo receive a tax refund or will he owe more tax? Justify your answer with calculations. (2 marks)

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Show Answers Only

`$80\ 021`
`$17\ 554.77`
`$1369.81\ text(refund)`

Show Worked Solution

i. `text(Taxable income)`

`= 82\ 521 + 10\ 920-13\ 420`

`= $80\ 021`

ii. `text(Tax payable)`

`= 17\ 547 + (80\ 021-80\ 000) xx 0.37`

`= $17\ 554.77`

iii. `text(Total tax payable)`

`= 17\ 554.77 + 1600.42`

`= $19\ 155.19`

`text(Tax paid > tax payable)`

`:.\ text(Refund)`	`= 20\ 525-19\ 155.19`
	`= $1369.81`

Financial Maths, STD2 F1 2016 HSC 26e

Jenny earns a yearly salary of $63 752. Her annual leave loading is 17.5% of four weeks pay.

Calculate her total pay for her four weeks of annual leave. (3 marks)

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Show Answers Only

`$5762.20`

Show Worked Solution

`text(4 weeks’ normal pay)`

`= 4/52 xx 63\ 752`

`= $4904`

`:.\ text(Annual leave pay)`

`= 4904(1 + 17.5/100)`

`= $5762.20`

Algebra, 2UG 2016 HSC 26b

Simplify `(8x^4y)/(24x^3y^5)`. (2 marks)

Show Answers Only

`x/(3y^4)`

Show Worked Solution

`(8x^4y)/(24x^3y^5)`	`=(x^((4-3))y^((1-5)))/3`
	`=(xy^(-4))/3`
	`=x/(3y^4)`

Measurement, STD2 M1 2016 HSC 26a

Calculate the surface area of a sphere with a radius of 5 cm, correct to the nearest whole number. (1 mark)

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Show Answers Only

`314\ text{cm²}`

Show Worked Solution

`SA`	`= 4pir^2`
	`= 4 xx pi xx 5^2`
	`= 314.15…`
	`= 314\ text{cm² (nearest cm²)}`

Algebra, STD2 A1 2016 HSC 24 MC

Which of the following correctly expresses `Q` as the subject of `e = iR + Q/C`?

`Q = Ce + CiR`
`Q = Ce-CiR`
`Q = (e + iR)/C`
`Q = (e-iR)/C`

Show Answers Only

`=> B`

Show Worked Solution

`e`	`= iR + Q/C`
`Q/C`	`= e-iR`
`:. Q`	`= C(e-iR)`
	`= Ce-CiR`

`=> B`

Probability, STD2 S2 2016 HSC 23 MC

A group of 485 people was surveyed. The people were asked whether or not they smoke. The results are recorded in the table.

A person is selected at random from the group.

What is the approximate probability that the person selected is a smoker OR is male?

Show Answers Only

`=> C`

Show Worked Solution

`P(text(Smoker or a male))`

`= (text(Total males + female smokers))/(text(Total surveyed))`

`= (264 + 68)/485`

`= 0.684…`

`=> C`

♦♦ Mean mark 34%.

Statistics, STD2 S1 2016 HSC 22 MC

The box-and-whisker plots show the results of a History test and a Geography test.

In History, 112 students completed the test. The number of students who scored above 30 marks was the same for the History test and the Geography test.

How many students completed the Geography test?

Show Answers Only

`=> C`

Show Worked Solution

`text{In History} \ => \ text{Q}_3 = 30\ \text{marks}`

`:.\ text{Scoring over 30}\ = 25text(%) xx 112 = 28\ \text{students}`

`text{In Geography} \ => \ text{Median}\ = 30\ \text{marks}`

`:.\ text{Students completing Geography}\ =2 xx 28 = 56\ \text{students}`

`=> C`

Financial Maths, STD2 F1 2016 HSC 20 MC

Isabella works a 35-hour week and is paid at an hourly rate of $18. Any overtime hours worked are paid at time-and-a-half. In a particular week, she earned $1008.

How many hours in total did Isabella work in this week to earn this amount?

37.3
42
49
56

Show Answers Only

`=> C`

Show Worked Solution

`text(Let)\ \ X=\ text(number of extra hours worked.)`

`text(Total wage)`	`= 35 xx 18 + X xx 27`
`1008`	`= 630 + 27X`
`27X`	`=378`
`X`	`=14`

`:.\ text(Total hours worked)\ =35+14=49`

`=> C`

Statistics, STD2 S1 2016 HSC 19 MC

A soccer referee wrote down the number of goals scored in 9 different games during the season.

`2, \ 3, \ 3, \ 3, \ 5, \ 5, \ 8, \ 9, \ ...`

The last number has been omitted. The range of the data is 10.

What is the five-number summary for this data set?

`2, 3, 5, 8.5, 12`
`2, 3, 5, 8.5, 10`
`2, 3, 5, 8, 12`
`2, 3, 5, 8, 10`

Show Answers Only

`=> A`

Show Worked Solution

`text{Since range is 10} \ => \ text{Last data point = 12}`

`text{Q}_1 = 3`

`text{Q}_3 = (8 + 9)/2 = 8.5`

`text(Median = 5)`

`=> A`

♦ Mean mark 46%.

Measurement, STD2 M7 2016 HSC 16 MC

The width (`W`) of a river can be calculated using two similar triangles, as shown in the diagram.

What is the approximate width of the river?

`17.8\ text(m)`
`19.3\ text(m)`
`23.2\ text(m)`
`24.9\ text(m)`

Show Answers Only

`=> A`

Show Worked Solution

`text{Triangles are similar (equiangular)}`

`text(Using similar ratios:)`

`W/(7.1)`	`= 20.3/8.1`
`:. W`	`= (20.3 xx 7.1)/8.1`
	`= 17.79…`

`=> A`

Algebra, STD2 A2 2016 HSC 14 MC

The graph shows a line which has an equation in the form `y = mx + c`.

Which of the following statements is true?

`m` is positive and `c` is negative
`m` is negative and `c` is positive
`m` and `c` are both positive
`m` and `c` are both negative

Show Answers Only

`=> A`

Show Worked Solution

`m` is the gradient and the line slopes to the right so `m` is positive.

`c` is the `y`-intercept which is negative.

`:.\ m` is positive and `c` is negative.

`=> A`

Measurement, STD2 M1 2016 HSC 12 MC

A container is in the shape of a triangular prism which has a capacity of 12 litres. The area of the base is 240 cm².

What is the distance, `h`, between the two triangular ends of the container?

5 cm
20 cm
25 cm
50 cm

Show Answers Only

`=> D`

Show Worked Solution

♦♦ Mean mark 35%.

`text{1 mL = 1 cm}^3\ \ =>\ \ text{1 L = 1000 cm}^3`

`text(Volume)`	`= Ah`
`12\ 000`	`= 240 xx h`
`h`	`= (12\ 000)/240`
	`= 50\ text(cm)`

`=> D`

Statistics, STD2 S1 2016 HSC 7 MC

Which set of data is classified as categorical and nominal?

blue, green, yellow
small, medium, large
5.2 cm, 6 cm, 7.21 cm
4 people, 5 people, 9 people

Show Answers Only

`A`

Show Worked Solution

`text(Categorical and nominal data is)`

♦♦ Mean mark 26%.

`text(qualitative and not ordered.)`

`=> A`

Algebra, STD2 A1 2016 HSC 5 MC

Which expression is equivalent to `2(3x-4) + 2`?

`6x-2`
`6x-4`
`6x-6`
`6x-10`

Show Answers Only

`C`

Show Worked Solution

`2(3x-4) + 2`

`= 6x-8 + 2`

`= 6x-6`

`=> C`

Measurement, STD2 M1 2016 HSC 1 MC

What is 208.345 correct to two significant figures?

208
210
208.34
208.35

Show Answers Only

`B`

Show Worked Solution

`208.345 = 210\ (2\ text(sig. fig.))`

♦♦ Mean mark 36%!!

`=> B`

Functions, 2ADV F1 2016 HSC 11a

Sketch the graph of `(x-3)^2 + (y + 2)^2 = 4.` (2 marks)

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

Show Answers Only

Show Worked Solution

`(x-3)^2 + (y + 2)^2 = 4\ \ text(is a circle),`

`text(centre)\ (3, -2),\ text(radius 2.)`

Functions, EXT1 F2 2007 HSC 2c

The polynomial `P(x) = x^2 + ax + b` has a zero at `x = 2`. When `P(x)` is divided by `x + 1`, the remainder is `18`.

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

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Show Answers Only

`a = -7\ \ text(and)\ \ b = 10`

Show Worked Solution

`P(x) = x^2 + ax + b`

`text(S)text(ince there is a zero at)\ \ x = 2,`

`P(2)`	`=0`
`2^2 + 2a + b`	`= 0`
`2a + b`	`= -4`	`…\ (1)`

`P(-1) = 18,`

`(-1)^2-a + b`	`= 18`
`-a + b`	`= 17`	`…\ (2)`

`text(Subtract)\ \ (1)-(2),`

`3a`	`= -21`
`a`	`= -7`

`text(Substitute)\ \ a = -7\ \ text{into (1),}`

`2(-7) + b`	`= -4`
`b`	`= 10`

`:.a = -7\ \ text(and)\ \ b = 10`

Functions, 2ADV F1 2007 HSC 1f

Find the equation of the line that passes through the point `(1, 3)` and is perpendicular to `2x + y + 4 = 0`. (2 marks)

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`x-2y + 7 = 0`

Show Worked Solution

`2x + y + 4`	`= 0`
`y`	`= -2x-4`

`=>\ text(Gradient) = -2`

`:. text(⊥ gradient) = 1/2\ \ \ (m_1 m_2=-1)`

`text(Equation of line)\ \ m = 1/2, \ text(through)\ (-1, 3)`

`y-y_1`	`= m (x-x_1)`
`y-3`	`= 1/2 (x + 1)`
`y`	`= 1/2 x + 7/2`
`2y`	`= x + 7`
`:. x-2y + 7`	`= 0`

Functions, 2ADV F1 2007 HSC 1b

Solve `2x-5> -3` and graph the solution on a number line. (2 marks)

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

Show Answers Only

`x > 1`

Show Worked Solution

`2x-5`	`> -3`
`2x`	`> 2`
`x`	`> 1`

Functions, EXT1 F2 2015 HSC 11f

Consider the polynomials `P(x) = x^3-kx^2 + 5x + 12` and `A(x) = x - 3`.

Given that `P(x)` is divisible by `A(x)`, show that `k = 6`. (1 mark)

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Find all the zeros of `P(x)` when `k = 6`. (2 marks)

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Show Answers Only

`text(See Worked Solutions)`
`3, 4, −1`

Show Worked Solution

i.	`P(x)`	`= x^3-kx^2 + 5x + 12`
	`A(x)`	`= x-3`

`text(If)\ P(x)\ text(is divisible by)\ A(x)\ \ =>\ \ P(3) = 0`

`3^3-k(3^2) + 5 xx 3 + 12`	`= 0`
`27-9k + 15 + 12`	`= 0`
`9k`	`= 54`
`:.k`	`= 6\ \ …\ text(as required)`

ii. `text(Find all roots of)\ P(x)`

`P(x)=(x-3)*Q(x)`

`text{Using long division to find}\ Q(x):`

`:.P(x)`	`= x^3-6x^2 + 5x + 12`
	`= (x-3)(x^2-3x − 4)`
	`= (x-3)(x-4)(x + 1)`

`:.\ text(Zeros at)\ \ \ x = -1, 3, 4`

Plane Geometry, EXT1 2015 HSC 3 MC

Two secants from the point `P` intersect a circle as shown in the diagram.

What is the value of `x`?

Show Answers Only

`B`

Show Worked Solution

`text{Property: products of intercepts of secants from external point are equal}`

`x(x + 3)`	`= 4(4 + 6)`
`x^2 + 3x`	`= 40`
`x^2 + 3x-40`	`= 0`
`(x-5)(x + 8)`	`= 0`

`:.x = 5,\ \ (x>0)`

`=>B`

Functions, EXT1 F2 2015 HSC 1 MC

What is the remainder when `x^3-6x` is divided by `x + 3`?

`-9`
`9`
`x^2-2x`
`x^2-3x + 3`

Show Answers Only

`A`

Show Worked Solution

`text(Remainder)`	`= P(-3)`
	`= (-3)^3-6(-3)`
	`= -27 + 18`
	`= -9`

`=> A`

Algebra, STD2 A1 2015 HSC 28d

The formula `C = 5/9 (F-32)` is used to convert temperatures between degrees Fahrenheit `(F)` and degrees Celsius `(C)`.

Convert 3°C to the equivalent temperature in Fahrenheit. (2 marks)

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`37.4\ text(degrees)\ F`

Show Worked Solution

`C`	`= 5/9(F-32)`
`F-32`	`= 9/5C`
`F`	`= 9/5C + 32`

`text(When)\ \ C = 3,`

`F`	`= (9/5 xx 3) + 32`
	`= 37.4\ text(degrees)\ F`

Algebra, STD2 A2 2015 HSC 27c

Ariana’s parents have given her an interest‑free loan of $4800 to buy a car. She will pay them back by paying `$x` immediately and `$y` every month until she has repaid the loan in full.

After 18 months Ariana has paid back $1510, and after 36 months she has paid back $2770.

This information can be represented by the following equations.

`x + 18y = 1510`

`x + 36y = 2770`

Graph these equations below and use to solve simultaneously for the values of `x` and `y`. (2 marks)

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How many months will it take Ariana to repay the loan in full? (2 marks)

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`x = 250, \ y = 70`
`text(65 months)`

Show Worked Solution

`:.\ text(Solution is)\ \ x = 250, \ y = 70`

ii. `text(Let)\ \ A = text(the amount paid back after)\ n\ text(months)`

`A = 250 + 70n`

♦ Mean mark 44%.

`text(Find)\ n\ text(when)\ A = 4800`

`250 + 70n`	`= 4800`
`70n`	`= 4550`
`n`	`= 65`

`:.\ text(It will take Ariana 65 months to repay)`

`text(the loan in full.)`

Measurement, STD2 M7 2015 HSC 27a

At a particular time during the day, a tower of height 19.2 metres casts a shadow. At the same time, a person who is 1.65 metres tall casts a shadow 5 metres long.

What is the length of the shadow cast by the tower at that time? (2 marks)

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`58\ text{m}`

Show Worked Solution

`text(Both triangles have right angles and a common)`

`text(angle to the ground.)`

`:.\ text{Triangles are similar (equiangular)}`

`text(Let)\ x =\ text(length of tower shadow)`

`x/5`	`= 19.2/1.65\ \ text{(corresponding sides of similar triangles)}`
`x`	`= (5 xx 19.2)/1.65`
	`= 58.1818…`
	`= 58\ text{m (nearest m)}`

Measurement, STD2 M1 2015 HSC 26f

Approximately 71% of Earth’s surface is covered by water. Assume Earth is a sphere with a radius of 6400 km.

Calculate the number of square kilometres covered by water. (2 marks)

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`3.7 xx 10^8\ text(km²)\ \ text{(nearest km²)}`

Show Worked Solution

`text(Surface area of Earth)`

`= 4pir^2`

`= 4pi xx 6400^2`

`:.\ text(Surface covered by water)`

`= text(71%) xx 4pi xx 6400^2`

`= 365\ 450\ 163.7…`

`= 3.7 xx 10^8\ text(km²)\ \ text{(nearest km²)}`

Probability, STD2 S2 2015 HSC 26e

The table shows the relative frequency of selecting each of the different coloured jelly beans from packets containing green, yellow, black, red and white jelly beans.

\begin{array} {|c|c|}
\hline
\rule{0pt}{2.5ex} \textit{Colour} \rule[-1ex]{0pt}{0pt} & \textit{Relative frequency} \\
\hline
\rule{0pt}{2.5ex} \text{Green} \rule[-1ex]{0pt}{0pt} & 0.32 \\
\hline
\rule{0pt}{2.5ex} \text{Yellow} \rule[-1ex]{0pt}{0pt} & 0.13 \\
\hline
\rule{0pt}{2.5ex} \text{Black} \rule[-1ex]{0pt}{0pt} & 0.14 \\
\hline
\rule{0pt}{2.5ex} \text{Red} \rule[-1ex]{0pt}{0pt} & \\
\hline
\rule{0pt}{2.5ex} \text{White} \rule[-1ex]{0pt}{0pt} & 0.24 \\
\hline
\end{array}

What is the relative frequency of selecting a red jelly bean? (1 mark)

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Based on this table of relative frequencies, what is the probability of NOT selecting a black jelly bean? (1 mark)

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Show Answers Only

$0.17$
$0.86$

Show Worked Solution

i. $\text{Relative frequency of red}$

$= 1-(0.32 + 0.13 + 0.14 + 0.24)$

$= 1-0.83$

$= 0.17$

ii. $P\text{(not selecting black)}$

$= 1-P\text{(selecting black)}$

$= 1-0.14$

$= 0.86$

Financial Maths, STD2 F4 2015 HSC 26d

A family currently pays $320 for some groceries.

Assuming a constant annual inflation rate of 2.9%, calculate how much would be paid for the same groceries in 5 years’ time. (2 marks)

Show Answers Only

`$369.17\ \ text{(nearest cent)}`

Show Worked Solution

`FV`	`= PV(1 + r)^n`
	`= 320(1.029)^5`
	`= $369.1703…`
	`= $369.17\ \ text{(nearest cent)}`

Algebra, STD2 A1 2015 HSC 24 MC

Consider the equation `(2x)/3-4 = (5x)/2 + 1`.

Which of the following would be a correct step in solving this equation?

`(2x)/3-3 = (5x)/2`
`(2x)/3 = (5x)/2 + 5`
`2x-4 = (15x)/2 + 3`
`(4x)/6-8 = 5x + 2`

Show Answers Only

`B`

Show Worked Solution

`(2x)/3-4`	`= (5x)/2 + 1`
`(2x)/3`	`= (5x)/2 + 5`

`=>B`

Measurement, STD2 M6 2015 HSC 22 MC

The area of the triangle shown is 250 cm².

What is the value of `x`, correct to the nearest whole number?

`11`
`18`
`22`
`24`

Show Answers Only

`D`

Show Worked Solution

`text(Using)\ \ \ A = 1/2ab\ sin\ C`

♦ Mean mark 42%.

`250`	`= 1/2 xx 30x\ sin\ 44^@`
`250`	`= 15x\ sin\ 44 ^@`
`:.x`	`= 250/(15\ sin\ 44^@)`
	`= 23.99…\ text(m)`

`=>D`

Financial Maths, STD2 F4 2015 HSC 17 MC

What amount must be invested now at 4% per annum, compounded quarterly, so that in five years it will have grown to $60 000?

$8919
$11 156
$49 173
$49 316

Show Answers Only

`C`

Show Worked Solution

`text(Using)\ \ FV = PV(1 + r)^n`

`r`	`= text(4%)/4`	`= text(1%) = 0.01\ text(per quarter)`
`n`	`= 5 xx 4`	`= 20\ text(quarters)`

`60\ 000`	`= PV(1 + 0.01)^(20)`
`:.PV`	`= (60\ 000)/1.01^(20)`
	`= $49\ 172.66…`

`⇒ C`

Probability, STD2 S2 2015 HSC 16 MC

The probability of winning a game is `7/10`.

Which expression represents the probability of winning two consecutive games?

`7/10 xx 6/9`
`7/10 xx 6/10`
`7/10 xx 7/9`
`7/10 xx 7/10`

Show Answers Only

`D`

Show Worked Solution

`text{Since the two events are independent:}`

`P text{(W)}`	`= 7/10`
`P text{(WW)}`	`= 7/10 xx 7/10`

`=>D`

Measurement, STD2 M1 2015 HSC 12 MC

The length of a fish was measured to be 49 cm, correct to the nearest cm.

What is the percentage error in this measurement, correct to one significant figure?

0.01%
0.5%
1%
2%

Show Answers Only

`C`

Show Worked Solution

♦ Mean mark 41%.

`text{Absolute error}\ =1/2 xx text{precision}\ = 1/2 xx 1 = 0.5\ text{cm}`

`text{% error}`	`=\ frac{text{absolute error}}{text{measurement}} xx 100%`
	`=0.5/49 xx 100%`
	`=1.020… %`
	`=1%\ \ text{(to 1 sig fig)}`

`=>C`

Measurement, STD2 M6 2015 HSC 9 MC

From the top of a cliff 67 metres above sea level, the angle of depression of a buoy is 42°.

How far is the buoy from the base of the cliff, to the nearest metre?

`60\ text(m)`
`74\ text(m)`
`90\ text(m)`
`100\ text(m)`

Show Answers Only

`B`

Show Worked Solution

♦ Mean mark 49%.
COMMENT: The angle of depression is a regularly examined concept. Make sure you know exactly what it refers to.

`text(Let)\ x\ text(= distance of buoy from cliff base)`

`tan\ 42^@`	`= 67/x`
`x\ tan\ 42^@`	`= 67`
`x`	`= 67/(tan\ 42^@)`
	`= 74.41…\ text(m)`

`⇒ B`

Measurement, STD2 M1 2015 HSC 8 MC

The Louvre Pyramid in Paris has a square base with side length 35 m and a perpendicular height of 22 m.

What is the volume of this pyramid, to the nearest m³?

`257\ text(m)^3`
`1027\ text(m)^3`
`8983\ text(m)^3`
`26\ 950\ text(m)^3`

Show Answers Only

`C`

Show Worked Solution

`V`	`= 1/3Ah`
`A`	`= 35 xx 35`
	`= 1225\ text(m)^2`

`:.V`	`= 1/3 xx 1225 xx 22`
	`= 8983.33…\ text(m)^3`

`=>C`

Statistics, STD2 S1 2015 HSC 6 MC

The times, in minutes, that a large group of students spend on exercise per day are presented in the box‑and‑whisker plot.

What percentage of these students spend between 40 minutes and 60 minutes per day on exercise?

Show Answers Only

`C`

Show Worked Solution

`text{Q}_1 = 40, \ text(Median) = 60`

`:.\ text(% Students between 40 and 60)`

`= 50text{%}-25text{%}`

`=25 text{%}`

`=>C`

Statistics, STD2 S1 2015 HSC 4 MC

On a school report, a student’s record of completing homework is graded using the following codes.

C = consistently
U = usually
S = sometimes
R = rarely
N = never

What type of data is this?

Categorical, ordinal
Categorical, nominal
Numerical, continuous
Numerical, discrete

Show Answers Only

`A`

Show Worked Solution

`text(The data has been grouped into categories and)`

`text(because each category can be ranked, it is ordinal.)`

`⇒ A`

Algebra, STD2 A1 2015 HSC 2 MC

Which of the following is `4x + 3y-x-5y` in its simplest form?

`3x - 2y`
`3x + 8y`
`5x - 2y`
`5x + 8y`

Show Answers Only

`A`

Show Worked Solution

`4x + 3y-x-5y`

`= 3x-2y`

`⇒ A`

Measurement, STD2 M1 2015 HSC 1 MC

What is 1 560 200 km written in standard form correct to two significant figures?

`1.56 × 10^4 \ text(km)`
`1.6 × 10^5 \ text(km)`
`1.56 × 10^6 \ text(km)`
`1.6 × 10^6 \ text(km)`

Show Answers Only

`D`

Show Worked Solution

♦♦ Mean mark 30%.
COMMENT: Incredibly, the first MC question in 2015 had the lowest mean mark of all MC questions in the exam!

`1\ 560\ 200`

`= 1.5602 xx 10^6`

`= 1.6 xx 10^6\ text(km)\ \ \ text{(2 sig fig)}`

`=> D`

Plane Geometry, 2UA 2004 HSC 2b

In the diagram, `ABC` is an isosceles triangle with `AB = AC` and `/_BAC = 38^@`. The line `BC` is produced to `D`.

Find the size of `/_ACD`. Give reasons for your answer. (2 marks)

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Show Answers Only

`109^@`

Show Worked Solution

`/_ABC`	`= 1/2 (180-38)\ \ \ text{(base angle of isosceles}\ Delta ABC text{)}`
	`= 71^@`

`:.\ /_ACD`	`= 71 + 38\ \ \ text{(exterior angle of}\ Delta ABC text{)}`
	`= 109^@`

Functions, 2ADV F1 2006 HSC 1e

Solve `3-5x <= 2`. (2 marks)

Show Answers Only

`x >= 1/5`

Show Worked Solution

`3-5x`	`<= 2`
`-5x`	`<= -1`
`x`	`>= 1/5`