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Probability, STD1 S2 2020 HSC 26

Barbara plays a game of chance, in which two unbiased six-sided dice are rolled. The score for the game is obtained by finding the difference between the two numbers rolled. For example, if Barbara rolls a 2 and a 5, the score is 3.

The table shows some of the scores.
 


 

  1. Complete the six missing values in the table to show all possible scores for the game.   (1 mark)
  2. What is the probability that the score for a game is NOT 0?  (2 marks)

Show Answers Only
  1.  

     
  2. `frac{5}{6}`
Show Worked Solution

a.     

♦ Mean mark part (b) 47%.
b.       `Ptext{(not zero)}` `= frac{text(numbers) ≠ 0}{text(total numbers)}`
    `= frac{30}{36}`
    `= frac{5}{6}`

 
\(\text{Alternate solution (b)}\)

b.       `Ptext{(not zero)}` `= 1 – Ptext{(zero)}`
    `= 1 – frac{6}{36}`
    `= frac{5}{6}`

Statistics, STD1 S3 2020 HSC 22

A group of students sat a test at the end of term. The number of lessons each student missed during the term and their score on the test are shown on the scatterplot.
 


 

  1. Describe the strength and direction of the linear association observed in this dataset.  (2 marks)

  2. Calculate the range of the test scores for the students who missed no lessons.  (1 mark)

  3. Draw a line of the best fit in the scatterplot above.  (1 mark)
  4. Meg did not sit the test. She missed five lessons.

     

    Use the line of the best fit drawn in part (c) to estimate Meg's score on this test. (1 mark)

  5. John also did not sit the test and he missed 16 lessons.

     

    Is it appropriate to use the line of the best fit to estimate his score on the test? Briefly explain your answer. (1 mark)

Show Answers Only

a.    \(\text{Strength : strong}\)

\(\text{Direction : negative} \)

b.    \(\text{Range}\ = \text{high}-\text{low}\ = 100-80=20\)
 

c.   

d. 


 

e.    \(\text{John’s missed days are too extreme and the LOBF is not}\)

\(\text{appropriate. The model would estimate a negative score for}\)

\(\text{John which is impossible.}\)

Show Worked Solution

a.    \(\text{Strength : strong}\)

\(\text{Direction : negative} \)

♦ Mean mark (a) 45%.
♦♦ Mean mark (b) 31%.

b.    \(\text{Range}\ = \text{high}-\text{low}\ = 100-80=20\)
 

c.   

d. 


 
\(\therefore\ \text{Meg’s estimated score = 40}\)
 

e.    \(\text{John’s missed days are too extreme and the LOBF is not}\)

\(\text{appropriate. The model would estimate a negative score for}\)

\(\text{John which is impossible.}\)

♦ Mean mark (e) 38%.

Algebra, STD1 A3 2020 HSC 19

Each year the number of fish in a pond is three times that of the year before.

  1. The table shows the number of fish in the pond for four years.
    \begin{array} {|l|c|c|c|c|}
    \hline
    \rule{0pt}{2.5ex}\textit{Year}\rule[-1ex]{0pt}{0pt} & \ \ \ 2020\ \ \  & \ \ \ 2021\ \ \  & \ \ \ 2022\ \ \  & \ \ \ 2023\ \ \ \\
    \hline
    \rule{0pt}{2.5ex}\textit{Number of fish}\rule[-1ex]{0pt}{0pt} & 100 & & & 2700\\
    \hline
    \end{array}

    Complete the table above showing the number of fish in 2021 and 2022.   (2 marks)
     

  2. Plot the points from the  table in part (a) on the grid.   (2 marks)
     
  3. Which model is more suitable for this dataset: linear or exponential? Briefly explain your answer.   (2 marks)

Show Answers Only

a.   

\begin{array} {|l|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\textit{Year}\rule[-1ex]{0pt}{0pt} & \ \ \ 2020\ \ \  & \ \ \ 2021\ \ \  & \ \ \ 2022\ \ \  & \ \ \ 2023\ \ \ \\
\hline
\rule{0pt}{2.5ex}\textit{Number of fish}\rule[-1ex]{0pt}{0pt} & 100 & 300 & 900 & 2700\\
\hline
\end{array}

b.   
       

c.     The more suitable model is exponential.

A linear dataset would graph a straight line which is not the case here.

An exponential curve can be used to graph populations that grow at an increasing rate, such as this example.

Show Worked Solution

a.        

\begin{array} {|l|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\textit{Year}\rule[-1ex]{0pt}{0pt} & \ \ \ 2020\ \ \  & \ \ \ 2021\ \ \  & \ \ \ 2022\ \ \  & \ \ \ 2023\ \ \ \\
\hline
\rule{0pt}{2.5ex}\textit{Number of fish}\rule[-1ex]{0pt}{0pt} & 100 & 300 & 900 & 2700\\
\hline
\end{array}

b.  

c.     The more suitable model is exponential.

A linear dataset would graph a straight line which is not the case here.

An exponential curve can be used to graph populations that grow at an increasing rate, such as this example.

♦ Mean mark (c) 31%.

Statistics, STD1 S3 2020 HSC 4 MC

The table shows the average brain weight (in grams) and average body weight (in kilograms) of nine different mammals.

\begin{array} {|l|c|c|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \textit{Brain weight (g)} \rule[-1ex]{0pt}{0pt} & 0.7 & 0.4 & 1.9 & 2.4 & 3.5 & 4.3 & 5.3 & 6.2 & 7.8 \\
\hline
\rule{0pt}{2.5ex} \textit{Body weight (kg)} \rule[-1ex]{0pt}{0pt} & 0.02 &0.06 & 0.05 & 0.34 & 0.93 & 0.97 & 0.43 & 0.33 & 0.22 \\
\hline
\end{array}

Which of the following is the correct scatterplot for this dataset?
 

 

 

  

 
Show Answers Only

`C`

Show Worked Solution

`text{Consider data point} \ (1.9, 0.05)`

`→ \ text{Eliminate} \ A \ text{(too high)}`

`→ \ text{Eliminate} \ D \ text{(should be below 2nd data point)}`
 

`text{Consider data point} \ (2.4, 0.34)`

`→ \ text{Eliminate} \ B \ text{(not on graph)}` 

`=> \ C`

Functions, EXT1 F2 2020 HSC 11a

Let  `P(x) = x^3 + 3x^2-13x + 6`.

  1. Show that  `P(2) = 0`.  (1 mark)

  2. Hence, factor the polynomial  `P(x)`  as  `A(x)B(x)`, where  `B(x)`  is a quadratic polynomial.  (2 marks)

Show Answers Only
  1. `text(See Worked Solutions)`
  2. `P(x) = (x-2)(x^2 + 5x – 3)`
Show Worked Solution
i.    `P(2)` `= 8 + 12-26 + 6`
    `= 0`

 

ii.   

`:. P(x) = (x-2)(x^2 + 5x – 3)`

Measurement, STD2 M1 2020 HSC 25

A composite solid consists of a triangular prism which fits exactly on top of a cube, as shown.
 

Find the surface area of the composite solid.   (3 marks)

Show Answers Only

`424 \ text{cm}^2`

Show Worked Solution

`text{S.A. of 1 face of cube} = 8 xx 8 = 64 \ text{cm}^2`

`text{Height of triangle} = 11 – 8 = 3 \ text{cm}`

`therefore \ text{S.A. (triangular prism)}` `= 2 xx ( frac{1}{2} xx 8 xx 3 ) + 2 xx (5 xx 8)`
  `= 24 + 80`
  `= 104 \ text{cm}^2`

 

`therefore \ text{Total S.A.}` `= 5 xx 64 + 104`
  `= 424 \ text{cm}^2`

Functions, 2ADV F1 2020 HSC 24

The circle of  `x^2-6x + y^2 + 4y-3 = 0`  is reflected in the `x`-axis.

Sketch the reflected circle, showing the coordinates of the centre and the radius.  (3 marks)

Show Answers Only

Show Worked Solution
`x^2-6x + y^2 + 4y-3` `= 0`
`x^2-6x + 9 + y^2 + 4y + 4-16` `= 0`
`(x-3)^2 + (y + 2)^2` `= 16`

 
`=>\  text{Original circle has centre (3, − 2), radius = 4}`

`text(Reflect in)\ xtext(-axis):`

♦ Mean mark 48%.

`text{Centre (3, − 2) → (3, 2)}`
 

Financial Maths, STD2 F4 2020 HSC 21

The inflation rate over the year from January 2019 to January 2020 was 2%.

The cost of a school jumper in January 2020 was $122.

Calculate the cost of the jumper in January 2019 assuming that the only change in the cost of the jumper was due to inflation.   (2 marks)

Show Answers Only

`$119.61`

Show Worked Solution
`FV` `=PV(1+r)^n`
`122` `=C_(2019)(1+0.02)^1`
`C_2019 xx 1.02` `= 122`
`C_2019` `= frac(122)(1.02)`
  `= $119.61`

Financial Maths, STD2 F1 2020 HSC 20

The table shows the income tax rates for the 2019 – 2020 financial year.

\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 – \$90 000}\rule[-1ex]{0pt}{0pt} & \text{\$3572 plus 32.5 cents for each \$1 over \$37 000}\\
\hline
\rule{0pt}{2.5ex}\text{\$90 001 – \$180 000}\rule[-1ex]{0pt}{0pt} & \text{\$20 797 plus 37 cents for each \$1 over \$90 000}\\
\hline
\rule{0pt}{2.5ex}\text{\$180 001 and over}\rule[-1ex]{0pt}{0pt} & \text{\$54 097 plus 45 cents for each \$1 over \$180 000}\\
\hline
\end{array}

For the 2019 – 2020 financial year, Wally had a taxable income of $122 680. During the year, he paid $3000 per month in Pay As You Go (PAYG) tax.

Calculate Wally's tax refund, ignoring the Medicare levy.   (3 marks)

Show Answers Only

`$3111.40`

Show Worked Solution
`text(Tax paid)` `=12 xx 3000`
  `=$36\ 000`

 
`text(Tax payable on $122 680)`

`=20\ 797 + 0.37(122\ 680-90\ 000)`

`=20\ 797 + 0.37(32\ 680)`

`=$32\ 888.60`
 

`:.\ text(Tax refund)` `=36\ 000-32\ 888.60`  
  `=$3111.40`  

Algebra, STD2 A4 2020 HSC 19

A fence is to be built around the outside of a rectangular paddock. An internal fence is also to be built.

The side lengths of the paddock are `x` metres and `y` metres, as shown in the diagram.
 

 
A total of 900 metres of fencing is to be used. Therefore  `3x + 2y = 900`.
 
The area, `A`, in square metres, of the rectangular paddock is given by  `A =450x - 1.5x^2`.

The graph of this equation is shown.
  

  1. If the area of the paddock is `30 \ 000\ text(m)^2`, what is the largest possible value of `x`?   (1 mark)

  2. Find the values of `x` and `y` so that the area of the paddock is as large as possible.   (2 marks)

  3. Using your value from part (b), find the largest possible area of the paddock.   (1 mark)

Show Answers Only
  1. `200 \ text(m)`
  2. `x = 150 \ text(m and) \ y = 225 \ text(m)`
  3. `33 \ 750 \ text(m)^2`
Show Worked Solution

a.     `text(From the graph, an area of)\ 30\ 000\ text(m)^2`

♦ Mean mark part (a) 39%.

  `text(can have an)\ x text(-value of)\ \ x=100 or 200\ text(m.)`

`:. x_text(max) = 200 text(m)`
 

b.    `A_text(max) \ text(occurs when) \ \ x = 150`

♦♦ Mean mark part (b) 34%.

`text(Substitute)\ \ x=150\ \ text(into)\ \ 3x + 2y = 900:`

`3 xx 150 + 2y` `= 900`
`2y` `= 450`
`y` `= 225`

 
`therefore \ text(Maximum area when) \ \ x = 150 \ text(m  and) \ \ y = 225 \ text(m)`

♦ Mean mark part (c) 40%.
c.    `A_(max)` `= xy`
    `= 150 xx 225`
    `= 33 \ 750 \ text(m)^2`

Measurement, STD2 M6 2020 HSC 16

Consider the triangle shown.
 


 

  1. Find the value of `theta`, correct to the nearest degree.   (2 marks)

  2. Find the value of `x`, correct to one decimal place.   (2 marks)

Show Answers Only
  1. `39^@`
  2. `12.1 \ text{(to 1 d.p.)}`
Show Worked Solution
a.      `tan theta` `= frac{8}{10}`
  `theta` `= tan ^(-1) frac{8}{10}`
    `= 38.659…`
    `= 39^@ \ text{(nearest degree)}`

 

b.     `text{Using Pythagoras:}`

`x` `= sqrt{8^2 + 10^2}`
  `= 12.806…`
  `= 12.8 \ \ text{(to 1 d.p.)}`

Algebra, STD2 A2 2020 HSC 10 MC

A plumber charges a call-out fee of $90 as well as $2 per minute while working.

Suppose the plumber works for `t` hours.

Which equation expresses the amount the plumber charges ($`C`) as a function of time (`t` hours)?

  1.  `C = 2 + 90t`
  2.  `C = 90 + 2t`
  3.  `C = 120 + 90t`
  4.  `C = 90 + 120t`
Show Answers Only

`D`

Show Worked Solution

♦ Mean mark 42%.

`text(Hourly rate)\ = 60 xx 2=$120`

`:. C = 90 + 120t`

`=>D`

Statistics, STD2 S1 2020 HSC 7 MC

Which histogram best represents a dataset that is positively skewed?

 
 
Show Answers Only

`A`

Show Worked Solution

♦♦ Mean mark 32%.

`text(Positive skew occurs when the tail on the)`

`text{histogram is longer on the right-hand}`

`text{(positive) side.}`

`=> \ A`

Measurement, STD2 M1 2020 HSC 5 MC

A plant stem is measured to be 16.0 cm, correct to one decimal place.

What is the percentage error in this measurement?

  1.  0.3125%
  2.  0.625%
  3.  3.125%
  4.  6.25%
Show Answers Only

`A`

Show Worked Solution

♦ Mean mark 41%.

`text{Absolute error} = 1/2 xx \text{precision} = 1/2 xx 0.1 = 0.05\ text{cm}`

`%  text(error)` `= frac(0.05)(16.0) xx 100`
  `= 0.3125%`

 
`=> \ A`

Financial Maths, STD2 F4 2020 HSC 4 MC

Joan invests $200. She earns interest at 3% per annum, compounded monthly.

What is the future value of Joan's investment after 1.5 years?

  1. $209.07
  2. $209.19
  3. $279.51
  4. $311.93
Show Answers Only

`B`

Show Worked Solution

`text(Monthly interest rate) \ = frac(0.03)(12)`

`n \ = \ 1.5 xx 12 = 18`
  

`text(FV)` `= text(PV) \ (1 + r)^n`
  `= 200 (1 + frac(0.03)(12))^18`
  `= $209.19`

 
`=> \ B`

Functions, 2ADV F1 2020 HSC 1 MC

Which inequality gives the domain of  `y = sqrt(2x-3)`?

  1. `x < 3/2`
  2. `x > 3/2`
  3. `x <= 3/2`
  4. `x >= 3/2`
Show Answers Only

`D`

Show Worked Solution

`text(Domain exists when:)`

`2x-3` `>= 0`
`2x` `>= 3`
`x` `>= 3/2`

  
`=>D`

Statistics, STD1 S3 2019 HSC 27

A set of bivariate data is collected by measuring the height and arm span of eight children. The graph shows a scatterplot of these measurements.
 

  1. On the graph, draw a line of best fit by eye.  (1 mark)

  2. Robert is a child from the class who was absent when the measurements were taken. He has an arm span of 147 cm. Using your line of best fit from part (a), estimate Robert’s height.  (1 mark)

Show Answers Only
  1.   
  2. `text(Robert’s height ≈ 151.1 cm)`
Show Worked Solution

a.     
       

♦ Mean mark (a) 38%.

b.   `text(Robert’s height ≈ 151.1 cm)`

`text{(Answers can vary slightly depending on line of best fit drawn).}`

Probability, STD1 S2 2019 HSC 24

The faces on a biased six-sided die are labelled 1, 2, 3, 4, 5 and 6. The die was rolled twenty times. The relative frequency of rolling a 6 was 30% and the relative frequency of rolling a 2 was 15%. The number 3 was the only other number rolled in the twenty rolls.

How many times was the number 3 rolled in the twenty rolls of the die?  (3 marks)

 
Show Answers Only

`11`

Show Worked Solution

`text(Number of 6’s) = 30/100 xx 20 = 6`

`text(Number of 2’s) = 15/100 xx 20 = 3`

`:.\ text(Number of 3’s)` `= 20 – (6 + 3)`
  `= 11`

Measurement, STD1 M1 2019 HSC 15

The diagram shows a shape made up of a square of side length 8 cm and a semicircle.
  


 

Find the area of the shape to the nearest square centimetre.  (3 marks)

Show Answers Only

`89\ text(cm²  (nearest cm²))`

Show Worked Solution

♦♦ Mean mark 26%.

`text(Area)` `=\ text(Area of square + Area of semicircle)`
  `= 8 xx 8 + 1/2 xx pi xx 4^2`
  `= 89.13…`
  `= 89\ text(cm²  (nearest cm²))`

Functions, EXT1 F2 2019 HSC 11d

Find the polynomial  `Q(x)`  that satisfies  `x^3 + 2x^2-3x-7 = (x-2) Q(x) + 3`.  (2 marks)

Show Answers Only

`Q(x ) = x^2 + 4x + 5`

Show Worked Solution
`(x-2) ⋅ Q(x) + 3` `= x^3 + 2x^2-3x-7`
`(x-2) ⋅ Q(x)` `= x^3 + 2x^2-3x-10`

 

`:. Q(x ) = x^2 + 4x + 5`

Financial Maths, STD2 F4 2019 HSC 37

A new car is bought for $24 950. Each year the value of the car is depreciated by the same percentage.

The table shows the value of the car, based on the declining-balance method of depreciation, for the first three years.

\begin{array} {|c|c|}
\hline
\rule{0pt}{2.5ex}\textit{End of year}\rule[-1ex]{0pt}{0pt} & \textit{Value}\\
\hline
\rule{0pt}{2.5ex}1\rule[-1ex]{0pt}{0pt} & \$21\ 457.00 \\
\hline
\rule{0pt}{2.5ex}2\rule[-1ex]{0pt}{0pt} & \$18\ 453.02 \\
\hline
\rule{0pt}{2.5ex}3\rule[-1ex]{0pt}{0pt} & \$15\ 869.60 \\
\hline
\end{array}

What is the value of the car at the end of 10 years?  (3 marks)

Show Answers Only

`$5521.47`

Show Worked Solution

`text(Find the depreciation rate:)`

`S` `= V_0(1-r)^n`
`21\ 457` `= 24\ 950(1-r)^1`
`1-r` `= (21\ 457)/(24\ 950)`
`1-r` `= 0.86`
`r` `= 0.14`

 
`:.\ text(Value after 10 years)`

`= 24\ 950(1-0.14)^10`

`= 5521.474…`

`= $5521.47\ \ (text(nearest cent))`

Algebra, STD2 A2 2019 HSC 34

The relationship between British pounds `(p)` and Australian dollars `(d)` on a particular day is shown in the graph.
 

  1. Write the direct variation equation relating British pounds to Australian dollars in the form  `p = md`. Leave `m` as a fraction.  (1 mark)

  2. The relationship between Japanese yen `(y)` and Australian dollars `(d)` on the same day is given by the equation  `y = 76d`.

     

    Convert 93 100 Japanese yen to British pounds.  (2 marks)

Show Answers Only
  1. `p = 4/7 d`
  2. `93\ 100\ text(Yen = 700 pounds)`
Show Worked Solution

a.   `m = text(rise)/text(run) = 4/7`

♦ Mean mark 42%.

`p = 4/7 d`

 

b.   `text(Yen to Australian dollars:)`

`y` `=76d`
`93\ 100` `= 76d`
`d` `= (93\ 100)/76`
  `= 1225`

 
`text(Aust dollars to pounds:)`

`p` `= 4/7 xx 1225`
  `= 700\ text(pounds)`

 
`:. 93\ 100\ text(Yen = 700 pounds)`

Algebra, STD2 A4 2019 HSC 33

The time taken for a car to travel between two towns at a constant speed varies inversely with its speed.

It takes 1.5 hours for the car to travel between the two towns at a constant speed of 80 km/h.

  1. Calculate the distance between the two towns.  (1 mark)

  2. By first plotting four points, draw the curve that shows the time taken to travel between the two towns at different constant speeds.  (3 marks)

Show Answers Only
  1. `120\ text(km)`
  2.  
Show Worked Solution
a.    `D` `= S xx T`
    `= 80 xx 1.5`
    `= 120\ text(km)`

 
b.   

\begin{array} {|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ s\ \  \rule[-1ex]{0pt}{0pt} & 20 & 40 & 60 & 80 \\
\hline
\rule{0pt}{2.5ex} t \rule[-1ex]{0pt}{0pt} & 6 & 3 & 2 & 1.5 \\
\hline
\end{array}

Probability, STD2 S2 2019 HSC 25

A bowl of fruit contains 17 apples of which 9 are red and 8 are green.

Dennis takes one apple at random and eats it. Margaret also takes an apple at random and eats it.

By drawing a probability tree diagram, or otherwise, find the probability that Dennis and Margaret eat apples of the same colour.  (3 marks)

Show Answers Only

`8/17`

Show Worked Solution

`P(text(same colour))` `= P(R R) + P(GG)`
  `= 9/17 xx 8/16 + 8/17 xx 7/16`
  `= 72/272 + 56/272`
  `= 8/17`
♦♦ Mean mark 35%.

Algebra, STD2 A2 2019 HSC 14 MC

Last Saturday, Luke had 165 followers on social media. Rhys had 537 followers. On average, Luke gains another 3 followers per day and Rhys loses 2 followers per day.

If  `x`  represents the number of days since last Saturday and  `y`  represents the number of followers, which pair of equations model this situation?

A.  `text(Luke:)\ \ y = 165x + 3`

 

`text(Rhys:)\ \ y = 537x - 2`
B. `text(Luke:)\ \ y = 165 + 3x`

 

`text(Rhys:)\ \ y = 537 - 2x`
C. `text(Luke:)\ \ y = 3x + 165`

 

`text(Rhys:)\ \ y = 2x - 537`
D. `text(Luke:)\ \ y = 3 + 165x`

 

`text(Rhys:)\ \ y = 2 - 537x`
Show Answers Only

`B`

Show Worked Solution

`text(Luke starts with 165 and adds 3 per day:)`

`y = 165 + 3x`

`text(Rhys starts with 537 and loses 2 per day:)`

`y = 537 – 2x`

`=> B`

Financial Maths, STD2 F4 2019 HSC 13 MC

The graph show the future values over time of  `$P`, invested at three different rates of compound interest.
 


 

Which of the following correctly identifies each graph?

A. B.
C. D.
Show Answers Only

`C`

Show Worked Solution

`text(Values increase quicker)`

`text(- higher compounding interest rate)`

`text(- same rate but more frequent compounding period)`

`:. W = 10text(% quarterly)`

`X = 10text(% annually)`

`Y = 5text(% annually)`

 
`=> C`

Measurement, STD2 M6 2019 HSC 12 MC

An owl is 7 metres above ground level, in a tree. The owl sees a mouse on the ground at an angle of depression of 32°.

How far must the owl fly in a straight line to catch the mouse, assuming the mouse does not move?

  1.  3.7 m
  2.  5.9 m
  3.  8.3 m
  4.  13.2 m
Show Answers Only

`D`

Show Worked Solution

`text(Let)\ \ OM = text(Flight distance)`

♦ Mean mark 36%.

`sin32°` `= 7/(OM)`
`:. OM` `= 7/(sin32°)`
  `= 13.2\ text(m)`

 
`=> D`

Algebra, STD2 A1 2019 HSC 11 MC

Which of the following correctly expresses `y` as the subject of the formula  `3x-4y-1 = 0`?

  1.  `y = 3/4 x-1`
  2.  `y = 3/4 x + 1`
  3.  `y = (3x-1)/4`
  4.  `y = (3x + 1)/4`
Show Answers Only

`C`

Show Worked Solution

♦ Mean mark 50%.

`3x-4y-1` `= 0`
`4y` `= 3x-1`
`:. y` `= (3x-1)/4`

 
`=> C`

Financial Maths, STD2 F1 2019 HSC 9 MC

What is the interest earned, in dollars, if $800 is invested for `x` months at a simple interest rate of 3% per annum?

  1. `2x`
  2. `24x`
  3. `200x`
  4. `2400x`
Show Answers Only

`A`

Show Worked Solution

♦♦♦ Mean mark 20%!

`text(Interest)` `= 800 xx x/12 xx 3/100`
  `= 2x`

 
`=> A`

Measurement, STD2 M1 2019 HSC 8 MC

A person's weight is measured as 79.3 kg.

What is the absolute error of this measurement?

  1. 10 grams
  2. 50 grams
  3. 100 grams
  4. 500 grams
Show Answers Only

`B`

Show Worked Solution

♦ Mean mark 46%.

`text(A)text(bsolute error)` `= 1/2 xx\ text(precision)`
  `= 1/2 xx 0.1\ text(kg)`
  `= 1/2 xx 100\ text(grams)`
  `= 50\ text(grams)`

 
`=> B`

Financial Maths, STD2 F1 2019 HSC 7 MC

Julia earns $28 per hour. Her hourly pay rate increases by 2%.

How much will she earn for a 4-hour shift with this increase?

  1. $2.24
  2. $28.56
  3. $112
  4. $114.24
Show Answers Only

`D`

Show Worked Solution
`text(Hourly rate)` `= 28 xx 1.02`
  `= $28.56`

 

`:.\ text(Shift earnings)` `= 4 xx 28.56`
  `= $114.24`

`=> D`

Financial Maths, STD2 F4 2019 HSC 3 MC

Chris opens a bank account and deposits $1000 into it. Interest is paid at 3.5% per annum, compounding annually.

Assuming no further deposits or withdrawals are made, what will be the balance in the account at the end of two years?

  1. $1070.00
  2. $1071.23
  3. $1822.50
  4. $2070.00
Show Answers Only

`=> B`

Show Worked Solution
`FV` `= PV(1 + r)^n`
  `= 1000(1 + 0.035)^2`
  `= $1071.23`

 
`=> B`

Measurement, STD2 M6 2019 HSC 22

Two right-angled triangles, `ABC` and `ADC`, are shown.
 

Calculate the size of angle `theta`, correct to the nearest minute.  (3 marks)

Show Answers Only

`41°4^{′}\ \ text{(nearest minute)}`

Show Worked Solution

`text(Using Pythagoras in)\ DeltaACD:`

Mean mark 51%.

`AC^2` `= 2.5^2 + 6^2`
  `= 42.25`
`:.AC` `= 6.5\ text(cm)`

 
`text(In)\ DeltaABC:`

`costheta` `= 4.9/6.5`
`theta` `= cos^(−1)\ 4.9/6.5`
  `= 41.075…`
  `= 41°4^{′}31^{″}`
  `= 41°5^{′}\ \ text{(nearest minute)}`

Probability, STD2 S2 2019 HSC 20

A roulette wheel has the numbers 0, 1, 2, …, 36 where each of the 37 numbers is equally likely to be spun.
 

 
If the wheel is spun 18 500 times, calculate the expected frequency of spinning the number 8.  (2 marks)

Show Answers Only

`500`

Show Worked Solution

`P(8) = 1/37`

`:.\ text(Expected Frequency (8))`

`= 1/37 xx 18\ 500`

`= 500`

Measurement, STD2 M1 2019 HSC 16

A bowl is in the shape of a hemisphere with a diameter of 16 cm.
 

What is the volume of the bowl, correct to the nearest cubic centimetre?  (2 marks)

Show Answers Only

`1072\ text(cm)^3`

Show Worked Solution
`V` `= 1/2 xx 4/3pir^3`
  `= 1/2 xx 4/3 xx pi xx 8^3`
  `= 1072.3…`
  `= 1072\ text{cm}^3\ text{(nearest cm}^3 text{)}`

Statistics, STD2 S1 SM-Bank 1 MC

A survey asked the following question for students born in Australia:

"Which State or Territory were you born in?"

How would the responses be classified?

  1. Categorical, ordinal
  2. Categorical, nominal
  3. Numerical, discrete
  4. Numerical, continuous
Show Answers Only

`B`

Show Worked Solution

`text{The data is categorical (not numerical) since}`

`text(the name of a State is required.)`

`text(This data cannot be ordered.)`

`=> B`

Measurement, STD2 M1 2008 HSC 21 MC

A sphere and a closed cylinder have the same radius.

The height of the cylinder is four times the radius.

What is the ratio of the volume of the cylinder to the volume of the sphere?

  1. `2 : 1`
  2. `3 : 1`
  3. `4 : 1`
  4. `8 : 1`
Show Answers Only

`B`

Show Worked Solution

♦♦ Mean mark 33%.

`V_text(cylinder)` `: V_text(sphere)`
`pir^2h` `: 4/3pir^3`
`underbrace(pir^2 4r)_(h = 4r)` `: 4/3pir^3`
`4pir^3` `: 4/3pir^3`
`3` `: 1`

  
`=> B`

Plane Geometry, 2UA 2018 HSC 13b

In `Delta ABC`, sides `AB` and `AC` have length 3, and `BC` has length 2. The point `D` is chosen on `AB` so that `DC` has length 2.
 

  1. Prove that `Delta ABC` and `Delta CBD` are similar.  (2 marks)

  2. Find the length `AD`.  (2 marks)

Show Answers Only
  1. `text(Proof)\ \ text{(See Worked Solutions)}`
  2. `5/3`
Show Worked Solution

i.    `text(Prove)\ \ Delta ABC\ text(|||)\ Delta CBD`

`Delta ABC\ text{is isosceles:}`

`/_ ABC = /_ ACB qquad text{(angles opposite equal sides)}`

`Delta CBD\ text{is isosceles:}`

`/_ CBD = /_ CDB qquad text{(angles opposite equal sides)}`

 
`text{Since}\ \ /_ ABC =  /_ CBD`

`:. Delta ABC\ text(|||)\ Delta CDB qquad text{(equiangular)}`
 

ii.   `text(Using ratios of similar triangles)`

`(DB)/(CB)` `= (BC)/(AC)`
`{(3-AD)}/2` `= 2/3`
`3-AD` `= 4/3`
`:. AD` `= 5/3`

 

Plane Geometry, 2UA 2018 HSC 12c

The diagram shows the square `ABCD`. The point `E` is chosen on `BC` and the point `F` is chosen on `CD` so that  `EC = FC`.
 

  1. Prove that `Delta ADF` is congruent to `Delta ABE`.   (2 marks)

  2. The side length of the square is 14 cm and `EC` has length 4 cm. Find the area of  `AECF`.   (2 marks)

Show Answers Only
  1. `text(Proof)\ \ text{(See Worked Solutions)}`
  2. `56\ text(cm)^2`
Show Worked Solution

i.    `AB = AD\ \ text{(sides of a square)}`

`DF = DC-CF`

`BE = BC-CE`

`text{Since}\ CE = CF\ \ text{(given), and}\ DC = BC\ \ text{(sides of a square)}`

`=> DF = BE`

`=> /_ ADF = /_ ABE = 90^@`

`:. Delta ADF \equiv Delta ABE\ \ text{(SAS)}`

 

ii.   `text(Area of)\ Delta ABE` `= 1/2 xx 14 xx 10`
    `= 70\ text(cm)^2`

 
`:.\ text(Area of)\ AECF`

`= text(Area of)\ ABCD-(2 xx 70)`

`= (14 xx 14)-140`

`= 56\ text(cm)^2`

Measurement, STD2 M6 2018 HSC 30c

The diagram shows two triangles.

Triangle `ABC` is right-angled, with  `AB = 13 text(cm)`  and  `/_ABC = 62°`.

In triangle  `ACD, \ AD = x\ text(cm)`  and  `/_DAC = 40°`. The area of triangle  `ACD`  is 30 cm².
 

 
What is the value of `x`, correct to one decimal place?  (3 marks)

Show Answers Only

`8.1\ text{cm  (1 d.p.)}`

Show Worked Solution

`text(Find)\ AC:`

♦ Mean mark 39%.

`sin62°` `= (AC)/13`
`AC` `= 13 xx sin62°`
  `= 11.478…`

 
`text(Using the sine rule in)\ DeltaACD :`

`text(Area)` `= 1/2 xx AC xx AD xx sin40°`
`30` `= 1/2 xx 11.478… xx x xx sin40°`
`:.x` `= (30 xx 2)/(11.478… xx sin40°)`
  `= 8.13…`
  `= 8.1\ text{cm  (1 d.p.)}`

Financial Maths, STD2 F1 2018 HSC 30b

Last year, Luke’s taxable income was `$87\ 000` and the tax payable on this income was `$19\ 822`. This year, Luke’s taxable income has increased by `$16\ 800`.

  1. Use the table to calculate the tax payable by Luke this year.  (2 marks)
     

  2. How much extra money will Luke have this year, after paying tax, as a result of the increase in his taxable income? Ignore the Medicare levy.  (2 marks)

Show Answers Only
  1. `$26\ 038`
  2. `$10\ 584`
Show Worked Solution

a.    `text(Taxable income) = 87\ 000 + 16\ 800 = $103\ 800`

`:.\ text(Tax payable)` `= 19\ 822 + 0.37 (103\ 800-87\ 000)`
  `= $26\ 038`

 

b.    `text(Net income from last year)`

`= 87\ 000-19\ 822`

`= $67\ 178`
 

`text(Net income in current year)`

`= 103\ 800-26\ 038`

`= $77\ 762`
 

`:.\ text(Extra money)` `= 77\ 762-67\ 178`
  `= $10\ 584`

Measurement, STD2 M1 2018 HSC 30a

A cylindrical water tank has a radius of 9 metres and a capacity of 1.26 megalitres.
 

What is the height of the water tank? Give your answer in metres, correct to two decimal places.  (3 marks)

Show Answers Only

`4.95\ text{m}`

Show Worked Solution

`text{Converting megalitres to m³  (using 1 m³ = 1000 L):}`

♦ Mean mark 48%.

`1.26\ text(ML)` `= (1.26 xx 10^6)/(10^3)`
  `= 1.26 xx 10^3\ text(m)^3`
  `= 1260\ text(m)^3`

 

`V` `= pir^2h`
`1260` `= pi xx 9^2 xx h`
`h` `= 1260/(pi xx 9^2)`
  `= 4.951…`
  `= 4.95\ text{m  (2 d.p.)}`

Algebra, STD2 A1 2018 HSC 28b

Solve the equation  `(2x)/5 + 1 = (3x + 1)/2`, leaving your answer as a fraction.  (3 marks)

Show Answers Only

`5/11`

Show Worked Solution

♦ Mean mark 35%.

`underbrace{(2x)/5 + 1}_text(multiply x10)` `=underbrace{(3x + 1)/2}_text(multiply x10)`
`4x + 10` `= 15x + 5`
`11x` `= 5`
`x` `= 5/11`

Measurement, STD2 M1 2018 HSC 27c

A shade shelter is to be constructed in the shape of half a cylinder with open ends. The diameter is 3.8 m and the length is 10 m.
 

 
The curved roof is to be made of plastic sheeting.

What area of plastic sheeting is required, to the nearest m²?  (2 marks)

Show Answers Only

`60\ text(m²  (nearest m²))`

Show Worked Solution

`text(Flatten out the half cylinder,)`

`text(Width)` `= 1/2 xx text(circumference)`
  `= 1/2 xx pi xx 3.8`
  `= 5.969…`

 

`:.\ text(Sheeting required)` `= 10 xx 5.969…`
  `= 59.69…`
  `= 60\ text(m²  (nearest m²))`

Financial Maths, STD2 F4 2018 HSC 26h

A car is purchased for $23 900.

The value of the car is depreciated by 11.5% each year using the declining-balance method.

What is the value of the car after three years?  (2 marks)

Show Answers Only

`$16\ 566\ \ (text(nearest dollar))`

Show Worked Solution
`S` `= V_0(1-r)^n`
  `= 23\ 900(1-0.115)^3`
  `= 23\ 900(0.885)^3`
  `= 16\ 566.383…`
  `= $16\ 566\ \ (text(nearest dollar))`

Probability, STD2 S2 2018 HSC 26a

Jeremy rolled a biased 6-sided die a number of times. He recorded the results in a table.
  

\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \text{Number} \rule[-1ex]{0pt}{0pt} & \ \ 1 \ \ & \ \ 2 \ \  & \ \ 3 \ \  & \ \ 4 \ \  & \ \ 5 \ \  & \ \ 6 \ \ \\
\hline
\rule{0pt}{2.5ex} \text{Frequency} \rule[-1ex]{0pt}{0pt} & \ \ 23 \ \ & \ \ 19 \ \  & \ \ 48 \ \  & \ \ 20 \ \  & \ \ 21 \ \  & \ \ 19 \ \ \\
\hline
\end{array} 

What is the relative frequency of rolling a 3?  (1 mark)

Show Answers Only

\(\dfrac{8}{25}\)

Show Worked Solution
♦ Mean mark 40%.

\(\text{Rel Freq}\) \(=\dfrac{\text{number of 3’s rolled}}{\text{total rolls}}\)
  \(=\dfrac{48}{150}\)
  \(=\dfrac{8}{25}\)

Functions, 2ADV F1 2018 HSC 3 MC

What is the `x`-intercept of the line  `x + 3y + 6 = 0`?

  1. `(-6, 0)`
  2. `(6, 0)`
  3. `(0, -2)`
  4. `(0, 2)`
Show Answers Only

`A`

Show Worked Solution

`x text(-intercept occurs when)\ y = 0:`

`x + 0 + 6` `= 0`
`x` `= -6`

 
`:. x text{-intercept is}\  (-6, 0)`

`=>  A`

Linear Functions, 2UA 2018 HSC 2 MC

The point  `R(9, 5)`  is the midpoint of the interval  `PQ`, where `P` has coordinates  `(5, 3).`
 

What are the coordinates of  `Q`?

  1. `(4, 7)`
  2. `(7, 4)`
  3. `(13, 7)`
  4. `(14, 8)`
Show Answers Only

`C`

Show Worked Solution

`text(Using the midpoint formula):`

`(x_Q + x_P)/2` `= x_R` `(y_Q + y_P)/2` `= y_R`
`(x_Q + 5)/2` `= 9` `(y_Q + 3)/2` `= 5`
`x_Q` `= 13` `y_Q` `= 7`

 
`:. Q\ text(has coordinates)\ (13, 7).`

`=>  C`

Probability, STD2 S2 2018 HSC 20 MC

During a year, the maximum temperature each day was recorded. The results are shown in the table.
  


  

From the days with a maximum temperature less than 25°C, one day is selected at random.

What is the probability, to the nearest percentage, that the selected day occurred during winter?

  1. 19%
  2. 25%
  3. 32%
  4. 77%
Show Answers Only

`text(C)`

Show Worked Solution
`text{P(winter day)}` `= (text(winter days < 25))/text(total days < 25) xx 100`
  `= 71/223 xx 100`
  `= 31.8…%`

`=>\ text(C)`

Measurement, STD2 M1 2018 HSC 18 MC

The length of a window is measured as 2.4 m.

Which calculation will give the percentage error for this measurement?

  1. `0.05/2.4 xx 100`
  2. `0.05/100 xx 2.4`
  3. `0.5/2.4 xx 100`
  4. `0.5/100 xx 2.4`
Show Answers Only

`A`

Show Worked Solution

`text{Absolute error}\ =1/2 xx text{precision}\ = 1/2 xx 0.1 = 0.05\ text{m}`

`text{% error}` `=\ frac{text{absolute error}}{text{measurement}} xx 100%`  
  `=0.05/2.4 xx 100%`  

 
`=>A`

Measurement, STD2 M1 2018 HSC 13 MC

A rectangular pyramid has base side lengths `3x` and `4x`. The perpendicular height of the pyramid is `2x`. All measurements are in metres.
 

What is the volume of the pyramid in cubic metres?

  1. `8x^3`
  2. `9x^3`
  3. `12x^3`
  4. `24x^3`
Show Answers Only

`A`

Show Worked Solution
`text(Volume)` `= 1/3Ah`
  `= 1/3(4x xx 3x xx 2x)`
  `= 8x^3`

 
`=>A`

Measurement, STD2 M6 2018 HSC 12 MC

The diagram shows a triangle with side lengths 8 m, 9 m and 10m.
 


 

What is the value of `theta`, marked on the diagram, to the nearest degree?

  1. 49°
  2. 51°
  3. 59°
  4. 72°
Show Answers Only

`text(D)`

Show Worked Solution

`text(Using the cosine rule:)`

`costheta` `= (8^2 + 9^2 – 10^2)/(2 xx 8 xx 9)`
  `= 0.3125`
`:.theta` `= cos^(−1)(0.3125)`
  `= 71.790…^@`

 
`=>D`

Statistics, STD2 S1 2018 HSC 11 MC

A set of data is summarised in this frequency distribution table.
 

 
Which of the following is true about the data?

  1. Mode = 7, median = 5.5
  2. Mode = 7, median = 6
  3. Mode = 9, median = 5.5
  4. Mode = 9, median = 6
Show Answers Only

`text(B)`

Show Worked Solution

`text{Mode = 7  (highest frequency of 9)}`

`text(Median = average of 15th and 16th data points.)`

`:.\ text(Median = 6)`

`=>\ text(B)`