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Functions, 2ADV F1 2007 HSC 1b

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

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`x > 1`
 

Show Worked Solution
`2x-5` `> -3`
`2x` `> 2`
`x` `> 1`

 

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)}`

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?

  1. `257\ text(m)^3`
  2. `1027\ text(m)^3`
  3. `8983\ text(m)^3`
  4. `26\ 950\ text(m)^3`
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`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`

Probability, STD2 S2 2006 HSC 26c

A new test has been developed for determining whether or not people are carriers of the Gaussian virus.

Two hundred people are tested. A two-way table is being used to record the results.
 

  1.  What is the value of  `A`?  (1 mark)

  2.  A person selected from the tested group is a carrier of the virus.

     

     What is the probability that the test results would show this?  (2 marks) 

  3.  For how many of the people tested were their test results inaccurate?  (1 mark)

Show Answers Only
  1. `98`
  2. `37/43`
  3. `28`
Show Worked Solution
i.  `A` `= 200-(74 + 12 + 16)`
  `= 98`

 

ii.  `P` `= text(# Positive carriers)/text(Total carriers)`
  `= 74/86`
  `= 37/43`

 

iii.  `text(# People with inaccurate results)`

`= 12 + 16`

`= 28`

Probability, STD2 S2 2005 HSC 16 MC

On a television game show, viewers voted for their favourite contestant. The results were recorded in the two-way table.

\begin{array} {|l|c|c|}
\hline
\rule{0pt}{2.5ex} \rule[-1ex]{0pt}{0pt} & \textbf{Male viewers} & \textbf{Female viewers} \\
\hline
\rule{0pt}{2.5ex}\textbf{Contestant 1}\rule[-1ex]{0pt}{0pt} & 1372 & 3915\\
\hline
\rule{0pt}{2.5ex}\textbf{Contestant 2}\rule[-1ex]{0pt}{0pt} & 2054 & 3269\\
\hline
\end{array}

One male viewer was selected at random from all of the male viewers.

What is the probability that he voted for Contestant 1?

  1. `1372/(10\ 610)`
  2. `1372/5287`
  3. `1372/3426`
  4. `1372/2054`
Show Answers Only

`C`

Show Worked Solution

`text(Total male viewers)\ = 1372 + 2054= 3426`

  
`P\ text{(Male viewer chosen voted for C1)}`

`= text(Males who voted for C1)/text(Total male viewers)`

`= 1372/3426`
 

`=>  C`

Probability, STD2 S2 2004 HSC 25c

Lie detector tests are not always accurate. A lie detector test was administered to 200 people.

The results were:

• 50 people lied. Of these, the test indicated that 40 had lied;
• 150 people did NOT lie. Of these, the test indicated that 20 had lied.

  1. Complete the table using the information above   (2 marks)
      
        

  2. For how many of the people tested was the lie detector test accurate?   (1 mark)

  3. For what percentage of the people tested was the test accurate?   (1 mark)

  4. What is the probability that the test indicated a lie for a person who did NOT lie?   (1 mark)

Show Answers Only
  1. `text(See Worked Solutions)`
  2. `170`
  3. `text(85%)`
  4. `2/15`
Show Worked Solution

i.

ii.  `text(# Accurate readings)`

`= 40 + 130`

`= 170`
 

iii.  `text(Percentage of people with accurate readings)`

`= text(# Accurate readings)/text(Total readings) xx 100`

`= 170/200`

`= 85 text(%)`
 

iv.  `text{P(lie detected when NOT a lie)}`

`= 20/150`

`= 2/15`

Probability, STD2 S2 2007 HSC 16 MC

Leanne copied a two-way table into her book.
 

 

Leanne made an error in copying one of the values in the shaded section of the table.

Which value has been incorrectly copied?

  1. The number of males in full-time work
  2. The number of males in part-time work
  3. The number of females in full-time work
  4. The number of females in part-time work
Show Answers Only

`D`

Show Worked Solution

`text(By checking row and column total, the number)`

`text(of females part-time work is incorrect)`

`=>  D`

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)

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  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)}`

Functions, 2ADV F1 2008 HSC 1c

Simplify  `2/n-1/(n+1)`.   (2 marks)

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`(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))`

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 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`

Functions, 2ADV F1 2010 HSC 1a

Solve  `x^2 = 4x`.   (2 marks)

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 `x = 0\ text(or)\ 4`

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

 

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

Algebra, 2UG 2010 HSC 27a

Fully simplify  `(4x^2)/(3y) -: (xy)/5`.   (3 marks) 

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 `(20x)/(3y^2)`

Show Worked Solution
♦♦ Mean mark 33%.
MARKER’S COMMENT: Remember to “invert and multiply” when dividing a fraction by a fraction (see Worked Solution).
`(4x^2)/(3y) -: (xy)/5` `= (4x^2)/(3y) xx 5/(xy)`
  `= (20x^2)/(3xy^2)`
  `= (20x)/(3y^2)`

Probability, STD2 S2 2009 HSC 27c

In each of three raffles, 100 tickets are sold and one prize is awarded.

Mary buys two tickets in one raffle. Jane buys one ticket in each of the other two raffles.

Determine who has the better chance of winning at least one prize. Justify your response using probability calculations.   (4 marks)  

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`P(text(Mary wins) )` `= 2/100`
  `= 1/50`

 

`P(text(Jane wins at least 1) )` `= 1-P (text(loses both) )`
  `= 1-99/100 xx 99/100`
  `= 1-9801/(10\ 000)`
  `= 199/(10\ 000)`

 
`text{Since}\ \ 1/50 > 199/(10\ 000)`

`=>\ text(Mary has a better chance of winning.)`

Show Worked Solution
`P(text(Mary wins) )` `= 2/100`
  `= 1/50`

 

`P(text(Jane wins at least 1) )` `= 1-P (text(loses both) )`
  `= 1-99/100 xx 99/100`
  `= 1-9801/(10\ 000)`
  `= 199/(10\ 000)`

 
`text{Since}\ \ 1/50 > 199/(10\ 000)`

`=>\ text(Mary has a better chance of winning.)`

♦♦ Mean mark 31%.
MARKER’S COMMENT: Very few students calculated Jane’s chance of winning correctly. Note the use of “at least” in the question. Finding `1-P`(complement) is the best strategy here.

Algebra, 2UG 2011 HSC 12 MC

Which of the following expresses `(6x^2y)/3-:(2y)/5` in its simplest form?

  1. `5x^2`
  2. `30x^2y`
  3. `1/(5x^2)`
  4. `5/(4x^2y^2)`
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`A`

Show Worked Solution
ALGEBRA TIP: When you divide by a fraction, invert the divisor and multiply as shown in the Worked Solution.
`(6x^2y)/3-:(2y)/5` `=(6x^2y)/3xx5/(2y)`
  `=(30x^2y)/(6y)`
  `=5x^2`

 
`=>A`

Probability, STD2 S2 2012 HSC 12 MC

Two unbiased dice, each with faces numbered 1, 2, 3, 4, 5, 6, are rolled. 

What is the probability of a 6 appearing on at least one of the dice? 

  1. `1/6`  
  2. `11/36` 
  3. `25/36`  
  4. `5/6`  
Show Answers Only

`B`

Show Worked Solution

`P(text(at least 1 six))`

`= 1-P(text(no six)) xx P(text(no six))`

`=1-5/6 xx 5/6`

`=11/36`
 

`=>  B`

♦♦♦ Mean mark 25%
COMMENT: The term “at least” should flag that calculating the probability of `1-P text{(event not happening)}` is likely to be the most efficient way to solve.

Measurement, STD2 M1 2013 HSC 12 MC

A square pyramid fits exactly on top of a cube to form a solid.

2013 12 mc

What is the volume of the solid?

  1. 513 cm³
  2. 999 cm³
  3. 1242 cm³
  4. 1539 cm³
Show Answers Only

`B`

Show Worked Solution
`text(Volume )` `=text{Vol (cube)} +text{Vol (pyramid)}`
  `=l^3+1/3Ah`
  `=(9xx9xx9)+(1/3xx9xx9xx10)`
  `=999\ text(cm)^3`

 
`=>\ B`

L&E, 2ADV E1 2011 HSC 1c

Solve   `2^(2x+1)=32`.    (2 marks) 

Show Answer Only

`x=2`

Show Worked Solutions
MARKER’S COMMENT: Many students correctly solved this by taking the logarithms of both sides.
`2^(2x+1)` `=32`
`2^(2x+1)` `=2^5`
`2x+1` `=5`
`:. x` `=2`