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Inequalities, SMB-010

In this inequality `n` is a whole number.

`8/n < 5/8`

What is the smallest possible value for `n` to make this inequality true?  (2 marks)

Show Answers Only

`13`

Show Worked Solution
`8/n` `< 5/8`
`5n` `> 64`
`n` `> 64/5`
  `> 12.8`

 

`:. text(Smallest)\ \ n = 13.`

Inequalities, SMB-009

For the expression `x^2 > 5x`, what is the smallest positive whole number  `x`  can be that makes the expression correct?  (2 marks)

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`6`

Show Worked Solution

`text(If)\ \ x = 6,`

`6^2` `> 5 xx 6`
`36` `> 30\ \ \ =>\ text{correct}`

 

`text(All positive whole numbers smaller than 6 result in:)`

`x^2 = 5x\ \ \ text{(for}\ x=5 text{)}`

`x^2 < 5x\ \ \ text{(for}\ x=4,3,2, …text{)}`

Inequalities, SMB-007 MC

The weight (`w` kilograms) and age (`a` years) of a turtle are related by the following inequality:

\begin{array} {|l|}
\hline
\rule{0pt}{2.5ex}w < 8a-13\ \  \text{for all values of}\ a\ \text{between 1 and 10}\rule[-1ex]{0pt}{0pt} \\
\hline
\end{array}

Which pair of values satisfy this inequality?

  1. `w = 4 and a = 2`
  2. `w = 12 and a = 3`
  3. `w = 18 and a = 4`
  4. `w = 40 and a = 6`
Show Answers Only

`C`

Show Worked Solution

`text(Test each option by trial and error.)`

`text(Consider)\ \ w = 18,\ a = 4,`

`18 < 8 xx 4-13`

`18 < 19\ \ text{(correct)}`
 

`:. w = 18,\ a = 4\ text(satisfies the equation).`

`=>C`

Indices, SMB-030

Find the reciprocal of   `1/a + 1/b -c/(ab)`.   (3 marks)

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`(ab)/(a+b-c)`

Show Worked Solution
`1/a + 1/b -c/(ab)` `=b/(ab)+a/(ab)-c/(ab)`
  `=(b+a-c)/(ab)`

 
`text(Reciprocal of)\ \ x = x^(-1)`

`:.\ text(Reciprocal of)\ \ (b+a-c)/(ab)=>((b+a-c)/(ab))^(-1)=(ab)/(a+b-c)`

Indices, SMB-022

Simplify  `((4p^2)/8)^(-3)`  and express as a fraction involving non-negative indices only.  (3 marks)

Show Answers Only

`8/(p^6)`

Show Worked Solution
`((4p^2)/8)^(-3)` `= ((p^2)/2)^(-3)`
  `=p^(2xx -3)/(2^(-3))`
  `= (p^(-6))/(2^(-3))= (2^(3))/(p^(6))`
  `= 8/(p^6)`

Indices, SMB-021

Simplify  `((3y^3)/2)^(-2)`  and express as a fraction involving non-negative indices only.  (3 marks)

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`4/(9y^6)`

Show Worked Solution
`((3y^3)/2)^(-2)` `=(3^(-2)xxy^((3xx-2)))/(2^(-2))`
  `= (2^(2)xxy^(-6))/(3^(2))`
  `= 4/(9y^6)`

Indices, SMB-020

Express  `4a^(-5) -: 12a^4`  as a fraction involving non-negative indices only.  (1 mark)

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`1/(3a^9)`

Show Worked Solution
`4a^(-5) -: 12a^4` `=(4a^(-5))/(12a^4)`
  `= a^((-5-4))/3`
  `= (a^(-9))/3`
  `=1/(3a^9)`

Algebra, STD1 A3 2019 HSC 9 MC

The container shown is initially full of water.
 

Water leaks out of the bottom of the container at a constant rate.

Which graph best shows the depth of water in the container as time varies?
 

A. B.
C. D.
Show Answers Only

`D`

Show Worked Solution

`text(Depth will decrease slowly at first and accelerate.)`

`=> D`

Measurement, STD2 M1 2022 HSC 34

A composite solid is shown. The top section is a cylinder with a height of 3 cm and a diameter of 4 cm. The bottom section is a hemisphere with a diameter of 6 cm. The cylinder is centred on the flat surface of the hemisphere.
 


 

Find the total surface area of the composite solid in cm², correct to 1 decimal place.  (4 marks)

Show Answers Only

`122.5\ text{cm}^2`

Show Worked Solution
`text{S.A. of Cylinder}` `=pir^2+2pirh`  
  `=pi(2^2)+2pi(2)(3)`  
  `=16pi\ text{cm}^2`  

 

`text{S.A. of Hemisphere}` `=1/2 xx 4pir^2`  
  `=2pi(3^2)`  
  `=18pi\ text{cm}^2`  

 

`text{Area of Annulus}` `=piR^2-pir^2`  
  `=pi(3^2)-pi(2^2)`  
  `=5pi\ text{cm}^2`  

 

`text{Total S.A.}` `=16pi+18pi+5pi`  
  `=39pi`  
  `=122.522…`  
  `=122.5\ text{cm}^2\ \ text{(to 1 d.p.)}`  

♦ Mean mark 50%.

Measurement, STD1 M5 2021 HSC 26

The diagrams show two similar shapes. The dimensions of the small shape are enlarged by a scale factor of 1.5 to produce the large shape.
 


 

Calculate the area of the large shape.  (3 marks)

Show Answers Only

`279\ text(cm)^2`

Show Worked Solution

`text(Dimension of larger shape:)`

♦♦ Mean mark 32%.

`text(Width) = 16 xx 1.5 = 24\ text(cm)`

`text(Height) = 9 xx 1.5 = 13.5\ text(cm)`

`text(Triangle height) = 2.5 xx 1.5 = 3.75\ text(cm)`

`:.\ text(Area)` `= 24 xx (13.5-3.75) + 1/2 xx 24 xx 3.75`
  `= 279\ text(cm)^2`

Measurement, STD2 M1 2021 HSC 16

The volume, `V`, of a sphere is given by the formula

`V = frac{4}{3} pi r^3,`

where `r` is the radius of the sphere.

A tank consists of the bottom half of a sphere of radius 2 metres, as shown.
 

Find the volume of the tank in cubic metres, correct to one decimal place.   (2 marks)

Show Answers Only

`16.8\ text{m}^3`

Show Worked Solution
 `V` `= frac{1}{2} times frac{4}{3} pi r^3`
  `= frac{1}{2} times frac{4}{3} times pi times 2^3`
  `= 16.755…`
  `= 16.8\ text{m}^3\ \ text{(1 d.p.)}`

Measurement, STD1 M5 2020 HSC 28

Two similar right-angled triangles are shown.
 


 

The length of side `AB` is 8 cm and the length of side `EF` is 4 cm.

The area of triangle `ABC` is 20 cm2.

Calculate the length in centimetres of side `DF` in Triangle II, correct to two decimal places.   (4 marks)

Show Answers Only

`7.55\ \text{cm}`

Show Worked Solution

`text{Consider} \ Δ ABC :`

`text{Area}` `= frac{1}{2} xx AB xx BC`
`20` `= frac{1}{2} xx 8 xx BC`
`therefore \ BC` `= 5`

 

`text{Using Pythagoras in} \ Δ ABC :`

♦♦♦ Mean mark 11%.

`AC = sqrt(8^2 + 5^2) = sqrt89`

 

`text{S} text{ince} \ Δ ABC\ text{|||}\ Δ DEF,`

`frac{AC}{BC}` `= frac{DF}{EF}`
`frac{sqrt89}{5}` `= frac{DF}{4}`
`therefore \ DF` `= frac{4 sqrt89}{5}`
  `= 7.547 …`
  `= 7.55 \ text{cm (to 2 d.p.)}`

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{)}`

Measurement, STD2 M1 2012 HSC 25 MC

The solid shown is made of a cylinder with a hemisphere (half a sphere) on top.
 

What is the total surface area of the solid, to the nearest square centimetre?

  1. 628 cm²
  2. 679 cm²
  3. 729 cm²
  4. 829 cm²
Show Answers Only

`B`

Show Worked Solution

`text(Total surface area)`

`= pir^2 + 2pirh + 1/2 xx 4pir^2`

`= pi xx 4^2 + 2pi xx 4 xx 21 + 1/2 xx 4pi xx 4^2`

`= 678.58…\ text(cm²)`

`=> 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`

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 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)
 

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)}`

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?

  1. 33%
  2. 18%
  3. 68%
  4. 87%
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%.

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)

Show Answers Only

`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`
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`

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`