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Statistics, STD2 S4 2007 HSC 9 MC

Which of the following would be most likely to have a positive correlation?

  1. The population of a town and the number of schools in that town
  2. The price of petrol per litre and the number of litres of petrol sold
  3. The hours training for a marathon and the time taken to complete the marathon
  4. The number of dogs per household and the number of televisions per household
Show Answers Only

\(A\)

Show Worked Solution

\(\text{Positive correlation means that as one variable increases,}\)

\(\text{the other tends to increase also.}\)

\(\Rightarrow A\)

Measurement, STD2 M7 2007 HSC 4 MC

What scale factor has been used to transform Triangle `A` to Triangle `B`?
  

  1. `1/2`
  2. `3/4`
  3. `2`
  4. `3`
Show Answers Only

`A`

Show Worked Solution

`text(Take two corresponding sides)`

`text(In)\ Delta A:\ 3\ text(cm)`

`text(In)\ Delta B:\ 1 \frac{1}{2}\ text(cm)`

`:.\ text(Scale factor converting)\ Delta A\ text(to)\ Delta B = frac{1}{2}`

`=>  A`

Financial Maths, STD2 F1 2007 HSC 3 MC

Joe is about to go on holidays for four weeks. His weekly salary is $280 and his holiday loading is 17.5% of four weeks pay.

What is Joe’s total pay for the four weeks holiday?

  1. $196
  2. $329
  3. $1169 
  4. $1316 
Show Answers Only

`D`

Show Worked Solution
`text(Salary)\ text{(4 weeks)}` `= 4 xx 280`
  `= $1120`

 

`text(Holiday loading)` `= 1120 xx 17.5%`
  `= $196`

 

`:.\ text(Total pay)` `= 1120 + 196`
  `= $1316`

 
`=>  D`

Probability, STD2 S2 2007 HSC 2 MC

Each student in a class is given a packet of lollies. The teacher records the number of red lollies in each packet using a frequency table.
 

What is the relative frequency of a packet of lollies containing more than three red lollies?

  1.    `4/19`
  2.    `4/15`
  3.    `11/19`
  4.    `11/15`
Show Answers Only

`A`

Show Worked Solution

`text(# Packets with more than 3)`

`= 3 + 1 = 4`

`text(Total packets) = 19`

`:.\ text(Relative Frequency) = 4/19`

`=>  A`

Functions, EXT1 F2 2008 HSC 1a

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

Show Answers Only

`-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.

Financial Maths, STD2 F4 2008 HSC 27c

A plasma TV depreciated in value by 15% per annum. Two years after it was purchased it had depreciated to a value of $2023, using the declining balance method.

What was the purchase price of the plasma TV?   (2 marks)

Show Answers Only

`$2800`

Show Worked Solution

`S = V_0 (1-r)^n`

`2023` `= V_0 (1-0.15)^2`
`2023` `= V_0 (0.85)^2`
`V_0` `= 2023/0.85^2`
  `= 2800`

 

`:.\ text(The purchase price) = $2800`

Statistics, STD2 S1 2008 HSC 26d

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

 

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

  2. What is the modal age group?   (1 mark)

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

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

Show Answers Only
  1. `600\ 000`
  2. `35-39`
  3. `1\ 450\ 000`
  4. `text(The 80+ group includes all people over 80)`

  5.  

    `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.
 

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

  2. 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)

  3. 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)

Show Answers Only
  1. `58`
  2. `6/25`
  3. `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. 

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

  2. 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)

Show Answers Only
  1. `$24\ 445.20`
  2. `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 23f

Christina has completed three Mathematics tests. Her mean mark is 72%.

What mark (out of 100) does she have to get in her next test to increase her mean mark to 73%?   (2 marks)

Show Answers Only

`76`

Show Worked Solution

`text(Total marks in 3 tests)`

`= 3 xx 72`

`= 216`

`text(We need 4-test mean) = 73`

`text(i.e.)\ \ \ ` `text{Total Marks (4 tests)}-:4` `= 73`
  `text(Total Marks)\ text{(4 tests)}` `= 292`

 

`:.\ text(4th test score)` `= 292 – 216`
  `= 76`

Statistics, STD2 S1 2008 HSC 23e

In a survey, 450 people were asked about their favourite takeaway food. The results are displayed in the bar graph.

2008 23e
 
How many people chose pizza as their favourite takeaway food?   (2 marks)

Show Answers Only

`175`

Show Worked Solution

`text(Number of people who chose pizza)`

COMMENT: This question required measurement of the actual image on the exam. The same methodology works here.

`= text{Length of pizza section}/text{Total length of bar} xx 450`

`~~ 7/18 xx 450`

`~~ 175`
 

`:.\ 175\ text(people chose pizza.)`

Measurement, STD2 M7 2008 HSC 20 MC

A point `P` lies between a tree, 2 metres high, and a tower, 8 metres high. `P` is 3 metres away from the base of the tree.

From `P`, the angles of elevation to the top of the tree and to the top of the tower are equal.
 

What is the distance, `x`, from `P` to the top of the tower?

  1. 9 m
  2. 9.61 m
  3. 12.04 m
  4. 14.42 m
Show Answers Only

`D`

Show Worked Solution

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

`text(In smaller triangle:)`

`h^2` `= 2^2 + 3^2`
  `= 13`
`h` `= sqrt 13`
   
`x/sqrt13` `= 8/2\ \ \ text{(sides of similar Δs in same ratio)}`
`x` `= (8 sqrt 13)/2`
  `= 14.422…`

 
`=>  D`

Probability, STD2 S2 2008 HSC 16 MC

A bag contains some marbles. The probability of selecting a blue marble at random from this bag is  `3/8`.

Which of the following could describe the marbles that are in the bag?

  1.    `3`  blue,  `8`  red
  2.    `6`  blue,  `11`  red
  3.    `3`  blue,  `4`  red,  `4`  green
  4.    `6`  blue,  `5`  red,  `5`  green 
Show Answers Only

`D`

Show Worked Solution

`P(B) = 3/8`

`text(In)\ A,\ \ ` `P(B) = 3/11`
`text(In)\ B,\ \ ` `P(B) = 6/17 `
`text(In)\ C,\ \ ` `P(B) = 3/11`
`text(In)\ D,\ \ ` `P(B) = 6/16 = 3/8`

`=>  D`

Statistics, STD2 S1 2008 HSC 13 MC

The height of each student in a class was measured and it was found that the mean height was 160 cm.

Two students were absent. When their heights were included in the data for the class, the mean height did not change.

Which of the following heights are possible for the two absent students?

  1.    155 cm and 162 cm
  2.    152 cm and 167 cm
  3.    149 cm and 171 cm
  4.    143 cm and 178 cm
Show Answers Only

`C`

Show Worked Solution

`text(S) text(ince the mean doesn’t change)`

`=>\ text(2 absent students must have a)`

`text(mean height of 160 cm.)`

`text(Considering each option given,)`

`(149 + 171) -: 2 = 160`

`=>  C`

Statistics, STD2 S4 2008 HSC 12 MC

A scatterplot is shown.
 

Which of the following best describes the correlation between  \(R\)  and  \(T\)?

  1. Positive
  2. Negative 
  3. Positively skewed
  4. Negatively skewed
Show Answers Only

\(A\)

Show Worked Solution

\(\text{Correlation is positive.}\)

\(\text{NB. The skew does not directly relate to correlation.}\)

\(\Rightarrow  A\)

Measurement, STD2 M1 2008 HSC 11 MC

The diagram shows the floor of a shower. The drain in the floor is a circle with a diameter of 10 cm.

What is the area of the shower floor, excluding the drain?
 

 
 

  1. 9686 cm²
  2. 9921 cm²
  3. 9969 cm²
  4. 10 000 cm²
Show Answers Only

`B`

Show Worked Solution
COMMENT: Students should see that answers are all in cm², and therefore use cm as the base unit for their calculations. 
`text(Area)` `=\ text(Square – Circle)`
  `= (100 xx 100)-(pi xx 5^2)`
  `= 10\ 000-78.5398…`
  `= 9921.46…\ text(cm²)`

 
`=>  B`

Statistics, STD2 S1 2008 HSC 10 MC

The marks for a Science test and a Mathematics test are presented in box-and-whisker plots.
 

 Which measure must be the same for both tests?

  1. Mean
  2. Range
  3. Median
  4. Interquartile range
Show Answers Only

`D`

Show Worked Solution

`text(IQR)=text(Upper Quartile)-text(Lower Quartile)`

`text{In both box plots, IQR = 3 intervals (against bottom scale)}`

`=>  D`

Statistics, STD2 S1 2008 HSC 8 MC

What is the median of the following set of scores?
 

 
 

  1.    12
  2.    13
  3.    14
  4.    15
Show Answers Only

`C`

Show Worked Solution
`text(Median` `=(n+1)/2`
  `=(33+1)/2`
  `=\ text (17th score)`

 

`:.\ text(Median is 14)`

`=>  C`

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?

  1. 4
  2. 6
  3. 8
  4. 9
Show Answers Only

`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`

Measurement, STD2 M6 2008 HSC 5 MC

What is the size of the smallest angle in this triangle?
 

  1. `29^@` 
  2. `47^@`
  3. `58^@`
  4. `76^@`
Show Answers Only

`B`

Show Worked Solution

`text(Smallest angle is opposite smallest side.)`

` cos A` `= (b^2 + c^2-a^2)/(2bc)`
  `= (7^2 + 8^2-6^2)/(2 xx 7 xx 8)`
  `= 0.6875`
`A` `=cos ^(-1)(0.6875)`
`:.\ A` `= 46.567…^@`

 
`=>  B`

Statistics, STD2 S1 2008 HSC 3 MC

The stem-and-leaf plot represents the daily sales of soft drink from a vending machine.

If the range of sales is 43, what is the value of  2008 3 mc  ?

 
 

  1.    `4` 
  2.    `5`
  3.    `24`
  4.    `25`
Show Answers Only

`A`

Show Worked Solution

`text(Range = High) – text(Low) = 43`

`:.\ 67 – text(Low)` `= 43`
`text(Low)` `= 24`

`:.\ N = 4`

`=>  A`

Plane Geometry, 2UA 2008 HSC 4a

In the diagram, `XR` bisects `/_PRQ` and `XY\ text(||)\ QR`.

Prove that `Delta XYR` is an isosceles triangle.   (2 marks)

Show Answers Only

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

Show Worked Solution

`text{Since}\ XR\ text{bisects}\ /_PRQ`

`/_XRQ` `= /_YRX = theta`
`/_RXY` `= theta\ \ \ text{(} text(alternate angles,)\ XY\ text(||)\ QR text{)}`

 
`:.\ Delta XYR\ \ text(is isosceles)`

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)

Show Answers Only

`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`

Functions, 2ADV F1 2008 HSC 1c

Simplify  `2/n-1/(n+1)`.   (2 marks)

Show Answers Only

`(n + 2)/(n(n+1))`

Show Worked Solution

`2/n-1/(n+1)`

`= (2(n+1)-1(n))/(n(n+1))`

`= (2n + 2-n)/(n(n+1))`

`= (n+2)/(n(n+1))`

Functions, EXT1 F2 2014 HSC 9 MC

The remainder when the polynomial  `P(x) = x^4-8x^3-7x^2 + 3`  is divided by  `x^2 + x`  is  `ax + 3`.

What is the value of  `a`?

  1. `-14`
  2. `-11`
  3. `-2`
  4. `5`
Show Answers Only

`C`

Show Worked Solution

`P(x) = x^4-8x^3-7x^2 + 3`

`text(Given)\ \ P(x)` `= (x^2 + x) *Q(x) + ax + 3`
  `= x (x + 1) Q(x) + ax + 3`

 
`P(-1) = 1 + 8-7 + 3 = 5`

`:. -a + 3` `= 5`
`a` `= -2`

 
`=>  C`

Plane Geometry, EXT1 2014 HSC 1 MC

The points \(A\), \(B\) and \(C\) lie on a circle with centre \(O\), as shown in the diagram.

The size of \(\angle ACB\) is 40°.

 What is the size of \(\angle AOB\)?

  1. \(20^{\circ}\)
  2. \(40^{\circ}\)
  3. \(70^{\circ}\)
  4. \(80^{\circ}\)
Show Answers Only

\(D\)

Show Worked Solution

\(\angle AOB = 2 \times 40 = 80^{\circ}\)

\(\text{(angles at centre and circumference on arc}\ AB\text{)}\) 

\(\Rightarrow D\)

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)

Show Answers Only
`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`

Functions, 2ADV F1 2014 HSC 5 MC

Which equation represents the line perpendicular to  `2x-3y = 8`, passing through the point  `(2, 0)`?

  1. `3x + 2y = 4`
  2. `3x + 2y = 6`
  3. `3x-2y = -4`
  4. `3x-2y = 6`
Show Answers Only

`B`

Show Worked Solution
`2x-3y` `= 8`
`3y` `= 2x-8`
`y` `= 2/3x-8/3`
`m` `= 2/3`
`:.\ m_text(perp)` `= -3/2\ \ \ (m_1 m_2=-1\text( for)_|_text{lines)}`

 

`text(Equation of line)\ \ m = -3/2\ \ text(through)\ \ (2,0):`

`y-y_1` `= m (x-x_1)`
`y-0` `= -3/2 (x-2)`
`y` `= -3/2x + 3`
`2y` `= -3x + 6`
`3x + 2y` `= 6`

 
`=>  B`

Algebra, STD2 A4 2014 HSC 29a

The cost of hiring an open space for a music festival is  $120 000. The cost will be shared equally by the people attending the festival, so that  `C`  (in dollars) is the cost per person when  `n`  people attend the festival.

  1. Complete the table below by filling in the THREE missing values.   (1 mark)
    \begin{array} {|l|c|c|c|c|c|c|}
    \hline
    \rule{0pt}{2.5ex}\text{Number of people} (n) \rule[-1ex]{0pt}{0pt} & \ 500\ & \ 1000 \ & 1500 \ & 2000 \ & 2500\ & 3000 \ \\
    \hline
    \rule{0pt}{2.5ex}\text{Cost per person} (C)\rule[-1ex]{0pt}{0pt} &  &  &  & 60 & 48\ & 40 \ \\
    \hline
    \end{array}
  2. Using the values from the table, draw the graph showing the relationship between  `n`  and  `C`.   (2 marks)
     
  3. What equation represents the relationship between `n` and `C`?   (1 mark)

  4. Give ONE limitation of this equation in relation to this context.   (1 mark)

  5. Is it possible for the cost per person to be $94? Support your answer with appropriate calculations.   (1 mark)

Show Answers Only

i.   

\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Number of people} (n) \rule[-1ex]{0pt}{0pt} & \ 500\ & \ 1000 \ & 1500 \ & 2000 \ & 2500\ & 3000 \ \\
\hline
\rule{0pt}{2.5ex}\text{Cost per person} (C)\rule[-1ex]{0pt}{0pt} & 240 & 120 & 80 & 60 & 48\ & 40 \ \\
\hline
\end{array}
 

ii. 

iii.   `C = (120\ 000)/n`

`n\ text(must be a whole number)`
 

iv.   `text(Limitations can include:)`

  `•\ n\ text(must be a whole number)`

  `•\ C > 0`
 

v.   `text(If)\ C = 94:`

`94` `= (120\ 000)/n`
`94n` `= 120\ 000`
`n` `= (120\ 000)/94`
  `= 1276.595…`

 
`:.\ text(C)text(ost cannot be $94 per person,)`

`text(because)\ n\ text(isn’t a whole number.)`

Show Worked Solution

i.   

\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Number of people} (n) \rule[-1ex]{0pt}{0pt} & \ 500\ & \ 1000 \ & 1500 \ & 2000 \ & 2500\ & 3000 \ \\
\hline
\rule{0pt}{2.5ex}\text{Cost per person} (C)\rule[-1ex]{0pt}{0pt} & 240 & 120 & 80 & 60 & 48\ & 40 \ \\
\hline
\end{array}
 

ii. 

 

♦ Mean mark (iii) 48%

iii.   `C = (120\ 000)/n`

 

♦♦♦ Mean mark (iv) 7%
COMMENT: When asked for limitations of an equation, look carefully at potential restrictions with respect to both the domain and range.

iv.   `text(Limitations can include:)`

  `•\ n\ text(must be a whole number)`

  `•\ C > 0`

 

v.   `text(If)\ C = 94`

`=> 94` `= (120\ 000)/n`
`94n` `= 120\ 000`
`n` `= (120\ 000)/94`
  `= 1276.595…`
♦ Mean mark (v) 38%

 

`:.\ text(C)text(ost cannot be $94 per person,)`

`text(because)\ n\ text(isn’t a whole number.)`

Measurement, STD2 M1 2014 HSC 27c

The base of a water tank is in the shape of a rectangle with a semicircle at each end, as shown.

The tank is 1400 mm long, 560 mm wide, and has a height of 810 mm.  
  

What is the capacity of the tank, to the nearest litre?   (4 marks) 

Show Answers Only

`581\ text(L)`

Show Worked Solution

`V = Ah` 

♦ Mean mark 41%
STRATEGY: Adjusting measurements to metres makes the final conversion to litres simple.

`text(Finding Area of base)`

`text(Semi-circles have radius 280 mm) = 0.28\ text(m)`

`:.\ text(Area of 2 semicircles)`

`=2 xx 1/2 xx pi r^2`

`= pi xx (0.28)^2`

`= 0.2463…\ text(m)^2`
 

`text(Area of rectangle)`

`= l xx b`

`= (1.4-2 xx 0.28) xx 0.56`

`= 0.4704\ text(m)^2`

 

`:.\ text(Volume)` `= Ah`
  `= (0.2463… + 0.4704) xx 0.810`
  `= 0.580527…\ text(m)^3`
  `= 580.527…\ text(L)\ \ text{(using 1m³} = 1000\ text{L)}`
  `= 581\ text(L)\ text{(nearest L)}`

Algebra, STD2 A2 2014 HSC 26f

The weight of an object on the moon varies directly with its weight on Earth.  An astronaut who weighs 84 kg on Earth weighs only 14 kg on the moon.

A lunar landing craft weighs 2449 kg when on the moon. Calculate the weight of this landing craft when on Earth.   (2 marks)

Show Answers Only

 `14\ 694\ text(kg)`

Show Worked Solution

`W_text(moon) prop W_text(earth)`

`=> W_text(m) = k xx W_text(e)`

`text(Find)\ k\ text{given}\  W_text(e) = 84\ text{when}\ W_text(m) = 14`

`14` `= k xx 84`
`k` `= 14/84 = 1/6`

 

`text(If)\ W_text(m) = 2449\ text(kg),\ text(find)\ W_text(e):`

`2449` `= 1/6  xx W_text(e)`
`W_text(e)` `= 14\ 694\ text(kg)`

 

`:.\ text(Landing craft weighs)\ 14\ 694\ text(kg on earth)`

Algebra, STD2 A1 2014 HSC 26c

Solve the equation  `(5x + 1)/3-4 = 5-7x`.   (3 marks)

Show Answers Only

 `x = 1`

Show Worked Solution
`(5x + 1)/3-4` `= 5-7x`
`5x + 1-3(4)` `= 3(5-7x)`
`5x + 1-12` `= 15-21x`
`26x` `= 26`
`:. x` `= 1`

Measurement, STD2 M1 2014 HSC 25 MC

A grain silo is made up of a cylinder with a hemisphere (half a sphere) on top. The outside of the silo is to be painted.
  

 What is the area to be painted?

  1. `8143\ text(m²)`
  2. `11\ 762\ text(m²)`
  3. `12\ 667\ text(m²)`
  4. `23\ 524\ text(m²)`
Show Answers Only

`A`

Show Worked Solution

`text(Total Area) = text(Area of cylinder) + text(½ sphere)`

♦ Mean mark 40%
`text(Area of cylinder)` `= 2 pi rh`
  `= 2pi xx 24 xx 30`
  `= 4523.9`
`text(Area of ½ sphere)` `= 1/2 xx 4 pi r^2`
  `= 1/2 xx 4 pi xx 24^2`
  `= 3619.1`
`:.\ text(Total area)` `= 4523.9 + 3619.1`
  `= 8143\ text(m²)`

`=>  A`

Probability, STD2 S2 2014 HSC 16 MC

In Mathsville, there are on average eight rainy days in October.

Which expression could be used to find a value for the probability that it will rain on two consecutive days in October in Mathsville?

  1. `8/31 xx 7/30`
  2. `8/31 xx 7/31`
  3. `8/31 xx 8/30`
  4. `8/31 xx 8/31`
Show Answers Only

`D`

Show Worked Solution

`P text{(rains)} = 8/31\ \ \text{(independent event for each day)}`

`text{Since each day has same probability:}`

`P(R_1 R_2) = 8/31 xx 8/31`

`=>  D`

♦♦♦ Mean mark 16%.
Lowest mark of any MC question in 2014!

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? 

  1. $188.40
  2. $208.40
  3. $321.85
  4. $652.05
Show Answers Only

`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`

Measurement, STD2 M1 2014 HSC 12 MC

A path 1.5  metres wide surrounds a circular lawn of radius 3 metres. 

What is the approximate area of the path?

  1. 7.1 m²
  2. 21.2 m²
  3. 35.3 m²
  4. 56.5 m²
Show Answers Only

`C`

Show Worked Solution

`text(Area of annulus)`

`= pi (R^2-r^2)`

`= pi (4.5^2-3^2)`

`= pi (11.25)`

`=35.3\ text{m²  (1 d.p.)}`
 

`=>  C`

Measurement, STD2 M1 2014 HSC 10 MC

The top of the Sydney Harbour Bridge is measured to be 138.4 m above sea level. 

What is the percentage error in this measurement?

  1. 0.036%
  2. 0.050%
  3. 0.072%
  4. 0.289%
Show Answers Only

`A`

Show Worked Solution
 
♦ Mean mark 48%

`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%`  
  `=0.05/138.4 xx 100%`  
  `=0.036%`  

 
`=>  A`

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? 

  1. $8968
  2. $9195
  3. $11 022
  4. $16 392
Show Answers Only

`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?

  1. `28/150`
  2. `45/150` 
  3. `28/70` 
  4. `45/70` 
Show Answers Only

`A`

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

 
`=>  A`

Algebra, STD2 A2 2014 HSC 7 MC

Which of the following is the graph of   `y = 2x-2`? 
  


  

Show Answers Only

`D`

Show Worked Solution
♦ Mean mark 46%

`y = 2x-2`

`text(By elimination)`

`text(It has a)\ y\ text(intercept of)\ -2`

`=> text(Cannot be)\ B\ text(or)\ C`

 

`(-1, 0)text{ from}\ A\ text(doesn’t satisfy equation)`

`text(but)\ (1,0)\ text(from)\ D\ text(does)`

`=>  D`

Algebra, STD2 A4 2014 HSC 3 MC

The diagram shows the graph of an equation.
  

 Which of the following equations does the graph best represent?

  1. `y = 3/x + 1`
  2. `y = 3^x + 1`
  3. `y = 3x^2 + 1`
  4. `y = 3x^3 + 1`
Show Answers Only

`C`

Show Worked Solution

`text(Graph is a parabola that passes through)\ (0, 1).`

`=>  C`

Functions, EXT1 F2 2013 HSC 1 MC

The polynomial  `P(x) = x^3-4x^2-6x + k`  has a factor  `x-2`.

What is the value of  `k`? 

  1. `2` 
  2. `12`
  3. `20` 
  4. `36`  
Show Answers Only

`C`

Show Worked Solution

`P(x) = x^3-4x^2-6x  + k`

`text(S)text(ince)\ \ (x-2)\ \ text(is a factor,)\ \ P(2) = 0`

`2^3-4*2^2-6*2 + k` `= 0`
`8-16-12 + k` `= 0`
`k` `= 20`

 
`=>  C`

Functions, EXT1 F2 2010 HSC 2c

Let  `P(x) = (x + 1)(x-3) Q(x) + ax + b`, 

where  `Q(x)`  is a polynomial and  `a`  and  `b`  are real numbers.

The polynomial  `P(x)`  has a factor of  `x-3`.

When  `P(x)`  is divided by  `x + 1`  the remainder is  `8`. 

  1. Find the values of  `a`  and  `b`.  (2 marks)

  2. Find the remainder when  `P(x)`  is divided by  `(x + 1)(x-3)`.     (1 mark)

Show Answers Only
  1. `a = -2,\ b = 6`
  2. ` -2x + 6`
Show Worked Solution

i.  `P(x) = (x+1)(x-3)Q(x) + ax + b`

`(x-3)\ \ text{is a factor   (given)}`

`:. P (3)` `= 0`
`3a + b` `= 0\ \ \ …\ text{(1)}`

 
`P(x) ÷ (x+1)=8\ \ \ text{(given)}`

`:.P(-1)` `= 8`
`-a + b` `= 8\ \ \ …\ text{(2)}`

 
`text{Subtract  (1) – (2)}`

`4a` `= -8`
`a` `= -2`

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

`-6 + b` `= 0`
`b` `= 6`

 
`:. a= – 2, \ b=6` 
 

ii.  `P(x) -: (x + 1)(x-3)`

`= ((x+1)(x-3)Q(x)-2x + 6)/((x+1)(x-3))`

`= Q(x) + (-2x + 6)/((x+1)(x-3))`

 
`:.\ text(Remainder is)\ \ -2x + 6`

COMMENT: This question requires a fundamental understanding of the remainder theorem.

Functions, EXT1 F1 2010 HSC 1d

Solve  `3/(x+2) < 4`.   (3 marks)

Show Answers Only

 `x < -2\ \ text(or)\ \ x > -5/4`

Show Worked Solution

`text(Solution 1)`

`3/(x + 2) < 4`

`text(Multiply b.s. by)\ \ (x + 2)^2`

`3(x + 2)` `< 4(x + 2)^2`
`3x + 6` `< 4 (x^2 + 4x + 4)`
`3x + 6` `< 4x^2 + 16x + 16`
`4x^2 + 13x + 10` `> 0`
`(4x + 5)(x + 2)` `> 0`

 
`text(LHS)\ = 0\ \ text(when)\ \ x = -5/4\ \ text(or)\ \ -2`

Algebra, EXT1 2010 HSC 1d Answer

`text(From graph)`

`x < -2\ \ text(or)\ \  x > -5/4`

 
`text(Alternate Solution)`

`text(If)\ \ x + 2 > 0\ \ \ \ text{(i.e.}\ \ x > –2 text{)}`

`3` `< 4(x + 2)`
`3` `< 4x + 8`
`4x` `> -5`
`x` `> -5/4`

 
`text(If)\ \ x + 2 < 0\ \ \ \ text{(i.e.}\ x < –2 text{)}`

`3` `> 4 (x + 2)`
`3` `> 4x + 8`
`4x` `< -5`
`x` `< -5/4`
`:. x` `< –2\ \ \ \ text{(satisfies both)}`

 
`:.\ x < –2\ \ text(or)\ \ x > –5/4`

Functions, EXT1 F2 2011 HSC 2a

Let  `P(x) = x^3-ax^2 + x`  be a polynomial, where  `a`  is a real number.

When  `P(x)`  is divided by  `x-3`  the remainder is  `12`.

Find the remainder when  `P(x)`  is divided by  `x + 1`.    (3 marks)

Show Answers Only

`-4`

Show Worked Solution

`P(x) = x^3 – ax^2 + x`

`text(S)text(ince)\ \ P(x) -: (x – 3)\ \ text(has remainder 12,)`

`P(3) = 3^3-a xx 3^2 + 3` `=12`
`27-9a + 3` `= 12`
`9a` `= 18`
`a` `=2`

 
`:.\ P(x) = x^3-2x^2 + x`

 

`text(Remainder)\ \ P(x) -: (x + 1)\ \ text(is)\ \ P(–1)`

`P(-1)` `= (-1)^3-2(-1)^2-1`
  `= – 4`

Plane Geometry, EXT1 2012 HSC 10 MC

The points `A`, `B` and `P` lie on a circle centred at `O`. The tangents to the circle at `A` and `B` meet at the point `T`, and `/_ATB = theta`.

 What is `/_APB` in terms of  `theta`? 

  1. `theta/2`  
  2. `90^@-theta/2`
  3. `theta` 
  4. `180^@-theta` 
Show Answers Only

`B`

Show Worked Solution

`/_ BOA= 2 xx /_ APB`

`text{(angles at centre and circumference on arc}\ AB text{)}`

`/_TAO = /_ TBO = 90^@\ text{(angle between radius and tangent)}`

`:.\ theta + /_BOA` `= 180^@\ text{(angle sum of quadrilateral}\ TAOB text{)}`
`theta + 2 xx /_APB` `= 180^@`
`2 xx /_APB` `= 180^@-theta`
`/_APB` `= 90^@-theta/2`

 
`=>  B`

Functions, EXT1 F2 2012 HSC 8 MC

When the polynomial  `P(x)`  is divided by  `(x + 1)(x-3)`, the remainder is  `2x + 7`.  

What is the remainder when  `P(x)`  is divided by  `x-3`? 

  1. `1` 
  2. `7` 
  3. `9` 
  4. `13` 
Show Answers Only

`D`

Show Worked Solution

`text(Let)\ \ P(x) =A(x) * Q(x) + R(x)`

`text(where)\ \ A(x) = (x + 1)(x-3),\ text(and)\ \ R(x)=2x+7`

`text(When)\ \ P(x) -: (x-3),\ text(remainder) = P(3)`

`P(3)` `= 0 + R(3)`
  `= (2 xx 3) + 7`
  `= 13`

 
`=>  D`

Functions, EXT1 F1 2011 HSC 1c

Solve  `(4-x)/x <1`.  (3 marks)

Show Answers Only

 `x<0\ \ text(or)\ \ x>2`

Show Worked Solution

`text(Solution 1)`

`(4-x)/x < 1`

`text(If)\ x<0,\ \ \ \ \ 4-x` `> x`
`2x` `< 4`
`x` `<2`

`=> x<0\ \ \ text{(satisfies both)}`
 

`text(If)\ x>0,\ \ \ \ \ 4-x` `<x`
`2x` `>4`
`x` `>2`

`=> x>2\ \ \ text{(satisfies both)}`

`:.\ x < 0\ \ text(or)\ \ x > 2`

 
`text(Solution 2)`

`text(Multiply both sides by)\ \ x^2`

`x(4-x)` `< x^2`
`4x-x^2` `< x^2`
`2x^2-4x` `>0`
`2x(x-2)` `>0`

 

 EXT1 2011 1c

`text(From graph)`

`x<0\ \ text(or)\ \ x >2`

Functions, EXT1* F1 2009 HSC 3c

Shade the region in the plane defined by  `y >= 0`  and  `y <= 4-x^2`.   (2 marks)

Show Answers Only
  

 

`text(Shaded area is region where)`

`y >= 0\ text(and)\ y >= 4-x^2`

Show Worked Solution

COMMENT: This past “Advanced” HSC question now fits into the Ext1 (new) syllabus.

`text(Shaded area is region where)`

`y >= 0\ \ text(and)\ \ y >= 4-x^2`

Functions, 2ADV F1 2009 HSC 1a

Sketch the graph of  `y-2x = 3`, showing the intercepts on both axes.   (2 marks)

Show Answers Only

 

Show Worked Solution

`y-2x=3\ \ =>\ \ y=2x+3`

`ytext{-intercept}\ = 3`

`text{Find}\ x\ text{when}\ y=0:`

`0-2x=3\ \ =>\ \ x=-3/2`
 

Functions, 2ADV F1 2010 HSC 1g

Let  `f(x) = sqrt(x-8)`.  What is the domain of  `f(x)`?   (1 mark)

Show Answers Only

 `x >= 8`

Show Worked Solution
♦ Mean mark 49%.
MARKER’S COMMENT: `x>8` was a common incorrect answer.

`f(x) = sqrt(x-8)`

`text(Domain exists for:)`

`(x-8)` `>= 0`
`x` `>= 8`

Functions, 2ADV F1 2010 HSC 1c

Write down the equation of the circle with centre `(-1, 2)` and radius 5.   (1 mark)

Show Answers Only

 `text{Circle with centre}\  (-1,2),\ r = 5`

`(x + 1)^2 + (y-2)^2 = 25`

Show Worked Solution
 MARKER’S COMMENT: Expanding this equation is not necessary!

`text{Circle with centre}\ (-1, 2),\ r = 5`

`(x + 1)^2 + (y-2)^2 = 25`

Functions, 2ADV F1 2010 HSC 1a

Solve  `x^2 = 4x`.   (2 marks)

Show Answers Only

 `x = 0\ text(or)\ 4`

Show Worked Solution
`x^2` `= 4x`
`x^2-4x` `= 0`
`x(x-4)` `= 0`

 

`:.\ x = 0\ text(or)\ 4`

Plane Geometry, 2UA 2011 HSC 6a

The diagram shows a regular pentagon `ABCDE`. Sides `ED` and `BC` are produced to meet at `P`.
  

  1. Find the size of `/_CDE`.    (1 mark)

  2. Hence, show that `Delta EPC` is isosceles.    (2 marks)

Show Answers Only
  1. `108°`
  2. `text(Proof)\ \ text{(see Worked Solutions)}`
Show Worked Solution
i.  

`text(Angle sum of pentagon)=(5-2) xx 180°=540°`

`:.\ /_CDE` `= 540/5\ \ \ text{(regular pentagon has equal angles)}`
  `= 108°`
MARKER’S COMMENT: Very few students solved part (i) efficiently. Remember the general formula for the sum of internal angles equals (# sides – 2) x 90°.

 
ii.  `text(Show)\ Delta EPC\ text(is isosceles)`

`text(S)text(ince)\ ED=CD\ \ text{(sides of a regular pentagon)}`

`Delta ECD\ text(is isosceles)`

`/_DEC=1/2 xx (180-108)= 36^{\circ}\ \ \ text{(Angle sum of}\ Delta DEC text{)}`

`/_CDP=72^@\ \ \ (\angle PDE\ \text{is a straight angle})`

`/_DCP=72^@\ \ \ (\angle PCB\ \text{is a straight angle})`

`=> /_CPD= 180-(72 + 72)=36^{\circ}\ \ \ text{(angle sum of}\ Delta CPD text{)}`

`:.\ Delta EPC\ \text(is isosceles)\ \ \ text{(2 equal angles)}`