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Trigonometry, SMB-004

Without the use of a calculator, express the following as a fraction

  1. `sin 30^{@}`   (1 mark)

  2. `sin 60^{@}`   (2 marks)

Show Answers Only
  1. `sin 30^{@} = 1/2`
  2. `sin 60^{@} = sqrt3/2`
Show Worked Solution

i.    `sin 30^{@} = text{opp}/text{hyp} = 1/2`

ii.   `text{Let}\ \ x =\ text{unknown side}`

`text{By Pythagoras:}`

`2^2` `=x^2 + 1^2`  
`x^2` `=4-1`  
`x` `=sqrt3\ \ (x>0)`  

 
`sin 60^{@} = text{opp}/text{hyp} = sqrt3/2`

Trigonometry, SMB-003

Express the following as a fraction

  1. `sin theta`  (1 mark)

  2. `tan theta`  (1 mark)

Show Answers Only
  1. `sin theta = 7/25`
  2. `tan theta = 7/24`
Show Worked Solution

i.    `sin theta = text{opp}/text{hyp} = 7/25`

ii.   `tan theta = text{opp}/text{adj} = 7/24`

Trigonometry, SMB-002

Express the following as a simplified fraction

  1. `cos theta`  (1 mark)

  2. `tan theta`  (1 mark)

Show Answers Only
  1. `cos theta = 3/5`
  2. `tan theta = 4/3`
Show Worked Solution

i.    `cos theta = text{adj}/text{hyp} = 6/10 = 3/5`

ii.   `tan theta = text{opp}/text{adj} = 8/6 = 4/3`

Trigonometry, SMB-001

Express the following as a fraction

  1. `cos theta`  (1 mark)

  2. `sin theta`  (1 mark)

Show Answers Only
  1. `cos theta = 12/13`
  2. `sin theta = 5/13`
Show Worked Solution

i.    `cos theta = text{adj}/text{hyp} = 12/13`

ii.   `sin theta = text{opp}/text{hyp} = 5/13`

Numbers of Any Magnitude, SMB-024 MC

The first three days of the Brisbane cricket test had the following attendances:

\begin{array} {|l|c|}
\hline
\rule{0pt}{2.5ex}\text{Day 1}\rule[-1ex]{0pt}{0pt} & 20\ 156\\
\hline
\rule{0pt}{2.5ex}\text{Day 2}\rule[-1ex]{0pt}{0pt} & 18\ 397\\
\hline
\rule{0pt}{2.5ex}\text{Day 3}\rule[-1ex]{0pt}{0pt} & 29\ 981\\
\hline
\end{array}

What was the total crowd over the first 3 days, to the nearest 1000?

  1. `68\ 000`
  2. `69\ 000`
  3. `70\ 000`
  4. `71\ 000`
Show Answers Only

`B`

Show Worked Solution

`20\ 156 + 18\ 397 + 29\ 981`

`= 68\ 534`

`= 69\ 000\ text{(nearest 1000)}`
 

`=>B`

Numbers of Any Magnitude, SMB-023

A red blood cell and a white blood cell have a combined total mass of `4.2 xx 10^-11` grams.

If the red blood cell has a mass of `2 xx 10^-12` grams, what is the mass of the white blood cell?  (2 marks)

Show Answers Only

`4.0 xx 10^-11\ text(grams)`

Show Worked Solution
`text{Mass}` `=4.2 xx 10^-11-2 xx 10^-12`  
  `= 4.0 xx 10^-11\ text{(by calculator)}`  

Numbers of Any Magnitude, SMB-021

An insect expert estimates that an anthill 30 cm high contains `10^8` ants.

He also estimates that a 25 cm high anthill contains `10^7` ants.

If his estimations are correct, how many more ants live in the 30 cm high anthill than the 25 cm high one?  (2 marks)

Show Answers Only

`text(90 million)`

Show Worked Solution

`text(The extra ants in the 30 cm high anthill)`

`= 100\ 000\ 000-10\ 000\ 000`

`=90\ 000\ 000`

`=90\ text(million)`

Numbers of Any Magnitude, SMB-015

The width of red blood cell is 8 nanometres or `8 xx 10^{-9}` metres.

How many red blood cells, lined up side by side in straight line, would it take to form a 1 cm distance.  (2 marks)

Show Answers Only

`1.25 xx 10^6`

Show Worked Solution

`text{Convert cm to metres:}`

`text{100 cm = 1 m}\ => \ text{1 cm = 0.01 m}`

`text{Number of cells}` `=0.01/(8xx10^{-9})`  
  `=1\ 250\ 000`  
  `=1.25 xx 10^6`  

Measurement, STD2 M1 2009 HSC 25b

The mass of a sample of microbes is 50 mg. There are approximately  `2.5 × 10^6` microbes in the sample.

In scientific notation, what is the approximate mass in grams of one microbe?   (2 marks)

Show Answers Only

`2 xx 10^-8\ text(grams)`

Show Worked Solution
♦♦♦ Mean mark 20%.
IMPORTANT: Can you solve: 8 apples weigh 1kg, what does 1 apple weigh? This is exactly the same concept.
`text(We need to convert 50 mg into grams)`
`50\ text(mg) = 50/1000 = 0.05\ text(g) = 5 xx 10^-2\ text(grams)`

 

`:.\ text(Mass of 1 microbe)` `= text(mass of sample)/text(# microbes)`
  `= (5 xx 10^-2)/(2.5 xx 10^6)`
  `= 2 xx 10^-8\ text(grams)`

Numbers of Any Magnitude, SMB-008 MC

A person's height is measured as 1.7 metres.

What is the absolute error of this measurement?

  1. 1 centimetre
  2. 5 centimetres
  3. 10 centimetres
  4. 50 centimetres
Show Answers Only

`B`

Show Worked Solution

`text{1 metre = 100 cm}\ => \ text{0.1 metre = 10 cm}`

`text(A)text(bsolute error)` `= 1/2 xx\ text(precision)`
  `= 1/2 xx 0.1\ text(m)`
  `= 1/2 xx 10\ text(cm)`
  `= 5\ text(cm)`

 
`=> B`

Numbers of Any Magnitude, SMB-007

The capacity of a bottle is measured as 1.35 litres correct to the nearest 10 millilitres.

What is the percentage error for this measurement, correct to two significant figures?   (2 marks)

Show Answers Only

`text(0.37%)`

Show Worked Solution

`text{1.35 litres = 1350 milliliters (mL)}`

`text(A) text(bsolute error) = 1/2 xx\ text{precision} = 1/2xx10=5\ text(mL)`

`:.\ text(% error)` `= 5/1350 xx 100`
  `=0.3703… %`
  `=0.37%\ text{(to 2 sig.fig.)}`

Numbers of Any Magnitude, SMB-006

A dinosaur fossil is measured to be 1.3 metres in length.

What is the percentage error in this measurement, giving your answer correct to two decimal places?  (2 marks)

Show Answers Only

`3.85%`

Show Worked Solution

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

`%  text(error)` `= frac(0.05)(1.3) xx 100`
  `= 3.84615%`
  `=3.85%\ text{(to 2 d.p.)}`

Numbers of Any Magnitude, SMB-005

A puppy's weight is measured at 5.2 kilograms, to the nearest 100 grams.

Calculate the percentage error in this measurement, correct to one significant figure?  (3 marks)

Show Answers Only

`1%`

Show Worked Solution

`text{Using  1 kilogram = 1000 grams}`

`=>\ text{5.2 kilograms = 5200 grams}`

`text{Absolute error}\ =1/2 xx text{precision}\ = 1/2 xx 100 = 50\ text{g}`

`text{% error}` `=\ frac{text{absolute error}}{text{measurement}} xx 100%`  
  `=50/5200 xx 100%`  
  `=0.961… %`  
  `=1%\ text{(to 1 sig. fig.)}`  
TIP: Convert all measurements to either grams or kilograms before applying formula.

Numbers of Any Magnitude, SMB-004

The height of palm tree is measured at 8 metres, to the nearest metre.

What is the percentage error in this measurement?  (2 marks)

Show Answers Only

`6.25%`

Show Worked Solution

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

`text{% error}` `=\ frac{text{absolute error}}{text{measurement}} xx 100%`  
  `=0.5/8 xx 100%`  
  `=6.25%`  

Numbers of Any Magnitude, SMB-003

The width of a hockey field is measured to be 45 metres, correct to the nearest metre.

What is the upper limit for the width of the hockey field?  (2 marks)

Show Answers Only

`45.5\ text(m)`

Show Worked Solution

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

`=>\ text{True length could lie 0.5 metres either side of 45 m measurement.}`

`:.\ text(Upper limit)` `= 45+0.5`
  `= 45.5\ text(m)`

Numbers of Any Magnitude, SMB-002 MC

The width of a soccer field is measured to be 50.60 metres, correct to the nearest centimetre.

What is the lower limit for the length of the netball court?

  1. 50.55 m
  2. 50.59 m
  3. 50.595 m
  4. 50.599 m
Show Answers Only

`C`

Show Worked Solution

`text{1 cm = 0.01 metre}`

`text{Absolute error}\ =1/2 xx text{precision}\ = 1/2 xx 0.01 = 0.005\ text{m}`

`:.\ text(Lower limit)` `= 50.60-0.005`
  `= 50.595\ text(m)`

 
`=>C`

Numbers of Any Magnitude, SMB-001

A cockroach is measured in a school science experiment and its length is recorded as 5.2 cm.

What is the upper limit of accuracy of this measurement?  (2 marks)

Show Answers Only

`5.25\ text(cm)`

Show Worked Solution

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

`=>\ text{True length could lie 0.05 cm either side of 5.2 cm measurement.}`

`text(Upper limit)` `= 5.2 + 0.05`
  `= 5.25\ text(cm)`

Quadratics, SMB-002

By completing the table of values, sketch the graph of  `y=x^2+3`.  (3 marks)

\begin{array} {|l|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} x \rule[-1ex]{0pt}{0pt} & -2 & -1 & \ \ 0\ \  & \ \ 1\ \  & \ \ 2\ \  \\
\hline
\rule{0pt}{2.5ex} y \rule[-1ex]{0pt}{0pt} &  &   &  & 4 &  \\
\hline
\end{array}


Show Answers Only

\begin{array} {|l|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} x \rule[-1ex]{0pt}{0pt} & -2 & -1 & \ \ 0\ \  & \ \ 1\ \  & \ \ 2\ \  \\
\hline
\rule{0pt}{2.5ex} y \rule[-1ex]{0pt}{0pt} & 7 & 4  & 3 & 4 & 7 \\
\hline
\end{array}

Show Worked Solution

\begin{array} {|l|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} x \rule[-1ex]{0pt}{0pt} & -2 & -1 & \ \ 0\ \  & \ \ 1\ \  & \ \ 2\ \  \\
\hline
\rule{0pt}{2.5ex} y \rule[-1ex]{0pt}{0pt} & 7 & 4  & 3 & 4 & 7 \\
\hline
\end{array}

Quadratics, SMB-001

By completing the table of values, sketch the graph of  `y=x^2-2`.  (3 marks)

\begin{array} {|l|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} x \rule[-1ex]{0pt}{0pt} & -2 & -1 & \ \ 0\ \  & \ \ 1\ \  & \ \ 2\ \  \\
\hline
\rule{0pt}{2.5ex} y \rule[-1ex]{0pt}{0pt} &  & -1  &  &  &  \\
\hline
\end{array}


Show Answers Only

Show Worked Solution

\begin{array} {|l|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} x \rule[-1ex]{0pt}{0pt} & -2 & -1 & \ \ 0\ \  & \ \ 1\ \  & \ \ 2\ \  \\
\hline
\rule{0pt}{2.5ex} y \rule[-1ex]{0pt}{0pt} & 2 & -1  & 0 & -1 & 2 \\
\hline
\end{array}

Exponentials, SMB-002

  1. Complete the table of values below given  `y=2^x-1`  (2 marks)

\begin{array} {|l|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} x \rule[-1ex]{0pt}{0pt} & -2 & -1 & \ \ 0\ \  & \ \ 1\ \  & \ \ 2\ \  \\
\hline
\rule{0pt}{2.5ex} y \rule[-1ex]{0pt}{0pt} &  & -\frac{1}{2} &  &  & 3\\
\hline
\end{array}

  1. Draw the graph of  `y=2^x-1`  on the Cartesian plane below.  (2 marks)


Show Answers Only

i.   

\begin{array} {|l|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} x \rule[-1ex]{0pt}{0pt} & -2 & -1 & \ \ 0\ \  & \ \ 1\ \  & \ \ 2\ \  \\
\hline
\rule{0pt}{2.5ex} y \rule[-1ex]{0pt}{0pt} & -\frac{3}{4} & -\frac{1}{2} & 0 & 1 & 3\\
\hline
\end{array}

ii. 

Show Worked Solution

i.   

\begin{array} {|l|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} x \rule[-1ex]{0pt}{0pt} & -2 & -1 & \ \ 0\ \  & \ \ 1\ \  & \ \ 2\ \  \\
\hline
\rule{0pt}{2.5ex} y \rule[-1ex]{0pt}{0pt} & -\frac{3}{4} & -\frac{1}{2} & 0 & 1 & 3\\
\hline
\end{array}

ii.   

 

Cartesian Plane, SMB-003 MC

The coordinates of point `A` are `(7, 5)`.
 

`AB` is parallel to the `y`-axis.

If  `AB = 3, BC = 4 and AC = 5`, what are the coordinates of point `C`?

  1. `(3,2)`
  2. `(2,3)`
  3. `(1,5)`
  4. `(5,1)`
Show Answers Only

`A`

Show Worked Solution

`text(Coordinates of)\ C\ text(are):`

`(7-4, 5-3) = (3, 2)`

`=>A`

Cartesian Plane, SMB-002 MC

A Cartesian plane is shown.
 

Select the correct statement below.

  1. `O` is located where `x = 0` and `y < 0.`
  2. `P` is located where `x < 0` and `y < 0.`
  3. `Q` is located where `x = 0` and `y > 0.`
  4. `R` is located where `x < 0` and `y < 0.`
Show Answers Only

`D`

Show Worked Solution

`R\ text(occurs when)`

`x < 0 and y < 0`

`=>D`

Equations, SMB-012 MC

`y = 2x-3`

`y = 4x + 1`

Which value of `x` satisfies both of these equations?

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

`A`

Show Worked Solution

`text(If)\ x = −2,`

`2(-2)-3` `= -7`
`4(-2) + 1` `= -7`

 
`:. x = -2\ \ text(satisfies both.)`

`=>A`

Equations, SMB-011

The daily energy requirement, `E` (kilojoules), for a person of mass `m` (kilograms) is calculated using the rule  `E = 7m + 7300`.

For Elijah, `E = 7755`.

What is Elijah's mass?  (2 marks)

Show Answers Only

`65\ text{kgs}`

Show Worked Solution
`7755` `= 7m + 7300`
`7 m` `= 455`
`m` `= 455/7`
  `= 65\ text(kilograms)`

Equations, SMB-002

`2 (4x-2) + 1 +`
?
 `= 9x-3`

What term makes this equation true for all values of `x`?  (2 marks)

Show Answers Only
?
 `= x`
Show Worked Solution
`2(4x-2) + 1 +`
?
 `= 9x-3`
`8x-4 + 1 +`
?
 `= 9x-3`
`8x-3 +`
?
 `= 9x-3`
 
?
 `= x`