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Number, NAPX-p168014v02 SA

Catherine owns a walk-in wardrobe with 13 different types of bags.

There are the same number of bags of each type.

She has a total of 65 bags.

How many bags of each type does she have?

 

Show Answers Only

`5 \ text{bags}`

Show Worked Solution
`text{Bags of each type}` `= frac{65}{13}`
  `= 5 \ text{bags}`

Number, NAPX-p168014v01 SA

Richard has a bookshelf that he splits into 8 different categories.

He has a total of 152 books.

He has the same number of books in each category.

How many books in each category does he own?

 

Show Answers Only

`19 \ text{books}`

Show Worked Solution
`text{Books in each category}` `= frac{152}{8}`
  `= 19 \ text{books}`

Geometry, NAPX-p168012v02

On a tourist map, two landmarks are measured to be 12.2 centimetres apart.

On the map, 1 centimetre represents 5 kilometres.

What is the actual distance between the landmarks?

 
      2.44 km
 
    24.4 km
 
      6.1 km
 
    61.0 km
Show Answers Only

`61.0 \ text{m}`

Show Worked Solution
`text{Actual Distance}` `= 12.2 xx 5`
  `= 61.0 \ text{km}`

Geometry, NAPX-p168012v01

Two stores are 8.2 cm apart on a map.

On the map, 1 cm represents 2 km.

What is the actual distance between the stores?

 
     4.1 km
 
     1.64 km
 
   16.4 km
 
   41.0 km
Show Answers Only

`16.4 \ text{m}`

Show Worked Solution
`text{Actual Distance}` `= 8.2 xx 2`
  `= 16.4 \ text{km}`

Algebra, NAPX-p168006v02

Izzie and Miranda saved $66 altogether.

Miranda saved half as much money as Izzie.

How much money did Izzie saved?

 
    $11
 
    $22
 
    $30
 
    $44
Show Answers Only

`$44`

Show Worked Solution

`text(Strategy 1)`

`text{Let} \ \ 2x  = text{money saved by Izzie}`

`text{Let} \ \  x  = text{money saved by Miranda}`

`2x + x` `=66`  
`3x` `=66`  
`x` `=22`  

 
`:.\ text(Izzie saved $44)`

 

`text(Strategy 2)`

`text{Check each option:}`

`frac{1}{2} xx 30 + 30 = 45 \ text{(Incorrect)}`

`frac{1}{2} xx22 + 22 = 33 \ text{(Incorrect)}`

`frac{1}{2} xx 11 + 11 = 16.5 \ text{(Incorrect)}`

`frac{1}{2} xx 44 + 44 = 66 \ text{(Correct)}`

 

`therefore \ text{Izzie saved $44.}`

Algebra, NAPX-p168006v01

Adam and Kenneth bought a bike together that cost $150.

Adam paid twice as much money as Kenneth.

How much money did Kenneth paid?

 
    $50
 
    $75
 
   $100
 
   $120
Show Answers Only

`$50`

Show Worked Solution

`text(Strategy 1)`

`text{Let} \ \ x  = text{money paid by Kenneth}`

`text{Let} \ \ 2 x  = text{money paid by Adam}`

`x + 2x` `=150`  
`3x` `=150`  
`x` `=50`  

 
`:.\ text(Kenneth paid $50).`

 

`text(Strategy 2)`

`text{Check each option:}`

`2(50) + 50 = 150 \ text{(Correct)}`

`2(75) + 75 = 225 \ text{(Incorrect)}`

`2(100) + 100 = 300 \ text{(Incorrect)}`

`2(120) + 120 = 360 \ text{(Incorrect)}`

Number, NAPX-p169007v02 SA

Eisha won a jar of candy with 366 pieces of candy in it at her school fete.

She gave a hand full each to five of her friends.

Eisha then counted the remaining candy and found she had 232 pieces left.

How many pieces of candy did Eisha give to her friends?

Show Answers Only

`text(134)`

Show Worked Solution
`text(Pieces given to friends)` `= 366 – 232`
  `=134`

Measurement, NAPX-p169002v02

Anne was driving from her house to the supermarket to buy some groceries.

She drove a total distance of 2 kilometres and 54 metres.

Which of these shows the distance, in metres, that Anne drove to get to the supermarket?

 
 2.054 m
 
 254 m
 
 2054 m
 
 2540 m

Show Answers Only

`2054\ text(m)`

Show Worked Solution

`text(Conversion: 1 kilometre = 1000 metres)`

`text{Distance}` `= (2 × 1000) + 54`  
  `=2000 + 54`  
  `=2054\ text(m)`  

Measurement, NAPX-p169002v01

John walked from his office to his home.

He walked a total of 1 kilometre and 62 metres.

Which of these shows the distance John walked in metres.

 
   106.2 m
 
   162 m
 
 1062 m
 
 1620 m

Show Answers Only

`1062\ text(m)`

Show Worked Solution

`text(Conversion: 1 kilometre = 1000 metres)`

`text{Distance walked = 1000 + 62 = 1062 m}`

Probability, NAPX-p168996v02

A standard deck of 52 cards is made up four suits - Hearts, Diamonds, Clubs and Spades - that have 13 cards each.

Lara has a standard deck of cards and draws a King without looking and returns it to the deck.

She repeats this three times and draws a King each time.

If she randomly draws a card for the 4th time, which of the following is true?

 
 She is certain to draw a King.
 
 She is more likely to draw a King than a Spade.
 
 She is less likely to draw a King than a Queen.
 
 She is more likely to draw a Heart than a King.

Show Answers Only

`text(She is more likely to draw a Heart than a King.)`

Show Worked Solution

`Ptext{(Heart)} = 13/52 = 1/4`

`Ptext{(Spade)} = 13/52 = 1/4`

`Ptext{(King)} = 4/52 = 1/13`

`Ptext{(Queen)} = 4/52 = 1/13`
 

`:.\ text(She is more likely to draw a Heart than a King.)`

Probability, NAPX-p168996v01

Aaron rolls a fair dice four times.

Each time he rolls the dice he gets a three.

Aaron rolls the dice for the fifth time.

Which of the following is true?

 
 He is less likely to roll a two than a three.
 
 He is more likely to roll a three than a one.
 
 He has an equal chance of rolling a three as a six.
 
 He is certain to roll a three.

Show Answers Only

`text(He has an equal chance of rolling a three as a six)`

Show Worked Solution

`text(All numbers in a fair die have equal chance of appearing.)`

`:.\ text(He has an equal chance of rolling a three as a six.)`

Number, NAPX-p111573v02

`320 ÷`
 
 `= 40`

Which of the following numbers make the number sentence above correct?

 
   8
 
 16
 
 20
 
 80

Show Answers Only

`8`

Show Worked Solution

`text(Check each option:)`

`text(Option 1 -)\ 320 ÷ 8 = 40\ (text(Correct))`

`text(Option 2 -)\ 320 ÷ 16 = 20`

`text(Option 3 -)\ 320 ÷ 80 = 4`

`text(Option 4 -)\ 320 ÷ 20 = 16`
 

`:.\ text(8 makes the number sentence correct.)`

Number, NAPX-p111573v01

`7 xx`
 
 `= 161`

What number will make the number sentence correct?

 
       13
 
       23
 
     154
 
   1127

Show Answers Only

`13`

Show Worked Solution

`text(Check each option)`

`text(Option 1 -)\ 7 xx 13 = 91`

`text(Option 2 -)\ 7 xx 23 = 161\ (text(Correct))`

`text(Option 3 -)\ 7 xx 154 = 1078`

`text(Option 4 -)\ 7 xx 161 = 1127`

Number, NAPX-p115150v02

Sheryl has 90 cupcakes to sell to her customers.

She packed the cupcakes in boxes that fit 4 cupcakes inside.

How many full boxes of cupcakes can she sell?

 
 21
 
 23
 
 22
 
 20

Show Answers Only

`22`

Show Worked Solution

`text(Number of boxes)`

`= 90 ÷ 4`

`= 22\ text(remainder 2)`
 

`:.\ text(Sheryl can sell 22 boxes full of cupcakes.)`

Number, NAPX-p115150v01

Darren bought 141 eggs from the market.

He decided to put the eggs in cartons of 12.

How many full cartons of eggs does he have?

 
   9
 
 10
 
 11
 
 12

Show Answers Only

`11`

Show Worked Solution

`text(Number of cartons of eggs)`

`= 141 -: 12`

`= 11\ text(remainder 9)`
 

`:.\ text(Darren can make 11 full cartons of eggs.)`

Measurement, NAPX-p167325v02

Leo painted the patterns shown below.

All the triangles that make up the shape have the same area.

Which of the following has the largest area painted grey?

 
   
 
   
 
   
 

Show Answers Only

Show Worked Solution

`text(Largest area painted → most triangles painted grey.)`

`text(Checking each option:)`

`text(Option 1 – 12 triangles painted)`

`text(Option 2 – 10 triangles painted)`

`text(Option 3 – 12 triangles painted)`

`text{Option 4 – 14 triangles painted (largest area)}`

Statistics, NAPX-p167389v02

The graph below shows the number of people in a supermarket at 15-minute intervals during a 4 hour period.
 


 

What time were the least amount of people in the supermarket?

 
   11:15 AM
 
   12:00 PM
 
   12:30 PM
 
    1:45 PM

Show Answers Only

`11:15 \ text{AM}`

Show Worked Solution

`text(Each data point represents 15 minutes.)`

`:.\ text{The lowest data point in the graph is at 11:15 AM}`

Statistics, NAPX-p167389v01

The graph below shows the number of vehicles in a parking station at 30 minute intervals over a 6 hour period.
 


 

What time were the highest number of vehicles in the parking station?

 
   5:30 PM
 
   4:30 PM
 
   3:30 PM
 
   5:00 PM

Show Answers Only

`4:30 \ text(PM)`

Show Worked Solution

`text{The highest data point in the graph is between 4:00 PM and 5:00 PM.}`

`therefore \ text{Highest number was at 4:30 PM.}`

Number, NAPX-p167381v02

The image below shows a game of noughts and crosses in progress.

What fraction of the available boxes are not filled in?

 
   `frac{5}{9}`
 
   `frac{3}{8}`
 
   `frac{1}{2}`
 
   `frac{4}{9}`

Show Answers Only

`frac{4}{9}`

Show Worked Solution
`text{Fraction}` `= frac{ text{Squares not filled} }{ text{Total squares} }`  
  `= frac{4}{9}`  

Number, NAPX-p167381v01

The image below is the plan of a farming plot.
 


 

Squares containing plants represent land that is growing crops.

Which fraction shows the amount of land that is used for growing crops?

 
   `frac{4}{5}`
 
   `frac{5}{8}`
 
   `frac{1}{2}`
 
   `frac{3}{8}`

Show Answers Only

`frac{5}{8}`

Show Worked Solution
`text{Fraction}` `= frac{ text{Squares growing crops} }{ text{Total squares} }`  
  `= frac{10}{16}`  
  `= frac{5}{8}`  

Geometry, NAPX-p100504v02

Kendrick was driving on Sterling Road in the direction of the arrow on the map below.

 

Which direction is Kendrick heading?

 
   North-East
 
   North-West
 
   South-East
 
   South-West

Show Answers Only

`text(North-East)`

Show Worked Solution

`text{Kendrick was heading in the North-East direction.}`

Geometry, NAPX-p100504v01

Alexa was walking on Ruby Street in the direction of the arrow shown below.
 


 

In what direction is Alexa heading?

 
  North-East
 
  North-West
 
  South-East
 
  South-West

Show Answers Only

`text(South-East)`

Show Worked Solution

`text{Alexa was heading in the South-East direction.}`

Measurement, NAPX-p167365v02

Which shape has an area of exactly 12 square units?

 
   A
 
   B
 
   C
 
   D

Show Answers Only

`B`

Show Worked Solution

`text{Area of triangle} = text{b} xx text{h}`

`text{Check each option:}`

`text{A.} \ \ 5 xx 3 = 15 `

`text{B.} \ \ 2 xx 6 = 12 \ text{(Correct)}`

`text{C.} \ \ 8 xx 1 = 8`

`text{D.} \ \ 9 xx 2 = 18`

`therefore \ text{square B has an area of exactly 12 square units.}`

Measurement, NAPX-p167365v01

What shape has an area greater than 11 square units?

 
   A
 
   B
 
   C
 
   D

Show Answers Only

`A`

Show Worked Solution

`text{Area of triangle} = frac{1}{2} xx text{b} xx text{h}`

`text{Check each option:}`

`text{A.} \ frac{1}{2} xx 6 xx 4 = 12 \ text{(Correct)}`

`text{B.} \ frac{1}{2} xx 3 xx 3 = 4.5`

`text{C} \ frac{1}{2} xx 6 xx 3 = 9`

`text{D} \ frac{1}{2} xx 4 xx 4 = 8`

`therefore \ text{triangle A has an area greater than 11 square units.}`

Measurement, NAPX-p124624v04

Barnsy owned a carp that was one quarter of a metre long.

How long was Barnsy's carp in millimetres?

 
   4000
 
   2000
 
   400
 
   250
 
   50
Show Answers Only

`250`

Show Worked Solution

`1 \ text{metre} = 1000 \ text{millimetres}`

`frac{1}{4} \ text{metre}` ` = frac{1}{4} xx 1000`
  `= 250 \ text{millimetres}`

Measurement, NAPX-p116865v04

Scott expects his wine to be delivered on the 28th March.

He is told by the distributor that its delivery will be delayed by 5 days.
 

 
What day of the week will the wine now be delivered?

Monday Tuesday Wednesday Saturday
 
 
 
 
Show Answers Only

`text{Tuesday}`

Show Worked Solution

`text{28th March is a Thursday.}`

`therefore \ text{5 days later is a Tuesday.}`

Measurement, NAPX-p116865v03

Marty expects his gold mining drench to be delivered on the 28th of June.

Delays at customs mean it will be delivered 6 days later.
 


 

Which day of the week should the drench now be delivered?

Saturday Sunday Monday Tuesday
 
 
 
 
Show Answers Only

`text{Monday}`

Show Worked Solution

`text{28th June is a Tuesday.}`

`therefore \ text{6 days later is a Monday.}`

Measurement, NAPX-p116876v04

Flanno builds the shape below using identical small blocks.
 


 
What is the smallest number of blocks Flanno needs to build a cube?
  10   11  12   13
 
 
 
 
Show Answers Only

`10`

Show Worked Solution

`text{The smallest cube will be} \ 3 xx 3 xx 3`

`text{Bottom row = 1}`

`text{2nd row = 5}`

`text{Top row = 7}`

`therefore \ text{13 blocks needed}`

Measurement, NAPX-p116876v03

Ella builds the shape below using identical small blocks.
 


 
What is the smallest number of blocks Ella needs to build a cube?
  8   9   10   11
 
 
 
 
Show Answers Only

`10`

Show Worked Solution

`text{The smallest cube will be} \ 3 xx 3 xx 3`

`text{Bottom row = 2}`

`text{2nd row = 3}`

`text{Top row = 5}`

`therefore \ text{10 blocks needed}`

Algebra, MET1 2015 VCAA 7b

Solve  `3e^t = 5 + 8e^(−t)`  for `t`.  (3 marks)

Show Answers Only

`log_e(8/3)`

Show Worked Solution

`3e^t – 5 – 8e^(−t) = 0`

♦ Part (b) mean mark 44%.
MARKER’S COMMENT: Many students could not create a quadratic, and others who did made mistakes by trying to solve using the quadratic formula rather than simple factorising.

`text(Multiply both sides by)\ e^t:`

`3e^(2t) – 5e^t – 8 = 0`

`text(Let)\ \ y = e^t`

`log_2((6 – x)/(4 – x))` `= 2`
`3y^2 – 5y – 8` `= 0`
`(3y – 8)(y + 1)` `= 0`
`y` `=8/3\ quadquadquadquadquad text(or)\ \ \ \ ` `y` `=-1\ \ text{(No solution)}`
`e^t` `= 8/3`  
`:. t` `= log_e(8/3)\ \ \ `    

Mechanics, SPEC2 2020 VCAA 5

Two objects, each of mass `m` kilograms, are connected by a light inextensible strings that passes over a smooth pulley, as shown below. The object on the platform is initially at point A and, when it is released, it moves towards point C. The distance from point A to point C is 10 m. The platform has a rough surface and, when it moves along the platform, the object experiences a horizontal force opposing the motion of magnitude `F_1` newtons in the section AB and a horizontal force opposing the motion of magnitude `F_2` newtons when it moves in the section BC.
 

  1. On the diagram above, mark all forces that act on each object once the object on the platform has been released and the system is in motion.  (2 marks)

The force `F_1` is given by  `F_1 = kmg, \ k ∈ R^+`.

  1.  i. Show that an expression for the acceleration, in `text(ms)^(−2)`, of the object on the platform, in terms of `k`, as it moves from point A to point B is given by  `(g(1 - k))/2`.  (2 marks)
  2. ii. The system will only be in motion for certain values of `k`.
  3.     Find these values of `k`.  (1 mark)

Point B is midway between points A and C.

  1. Find, in terms of `k`, the time taken, is seconds, for the object on the platform to reach point B.  (2 marks)
  2. Express, in terms of `k`, the speed `v_B`, in `text(ms)^(−1)`, of the object on the platform when it reaches point B.  (2 marks)
  3. When the object on the platform is at point B, the string breaks. The velocity of the object at point B is  `v_B = 2.5\ text(ms)^(−1)`. The force that opposes motion from point B to point C is  `F_2 = 0.075 mg + 0.4 mv^2`, where `v` is the velocity of the object when it is a distance of `x` metres from point B. The object on the platform comes to rest before point C.
  4. Find the object's distance from point C when it comes to rest. Give your answer in metres, correct to two decimal places.  (4 marks)
Show Answers Only
  1.  
  2.  i. `text(See Worked Solutions)`
  3. ii. `k ∈ (0, 1), k ∈ R^+`
  4. `2sqrt(5/(g(1 – k)))`
  5. `v_text(B) = sqrt(5g(1 – k))`
  6. `3.15\ text(m)`
Show Worked Solution
a.   

 

b. i.   `text(Horizontally:)`

`ma = T – F_1 = T – kmg\ …\ (1)`

`text(Vertically:)`

`ma = mg – T\ …\ (2)`
 

`text(Add)\ \ (1) + (2) :`

`2ma = mg – kmg`

`:. a` `= (g – kg)/2`
  `= (g(1 – k))/2`
♦ Mean mark (b)(ii) 45%.

 

b. ii.   `text(System in motion when)\ a > 0`

`(g(1 – k))/2 > 0`

`:. k ∈ (0, 1), \ k ∈ R^+`

 

c.   `text(AB) = 5\ (text(given)), u = 0\ (text(given))`

`text(Find)\ t\ text(when)\ s = 5:`

`s = ut + 1/2at^2`

`5 = 0 + 1/2 · (g(1 – k))/2 · t^2`

`t^2` `= 20/(g(1 – k))`
`t` `= sqrt(20/(g(1 – k)))`
  `= 2sqrt(5/(g(1 – k)))`

 

d.   `text(At B,)\ s = 5`

`v_text(B)^2` `= u^2 + 2as`
  `= 0 + 2 · (g(1 – k))/2 · 5`
  `= 5g(1 – k)`

 
`:. v_text(B) = sqrt(5g(1 – k))`

 

e.   `text(Acceleration is against the direction of motion.)`

♦♦ Mean mark (e) 35%.
`a` `= −F/m`
  `= −0.075g – 0.4v^2`
  `= −0.4(0.1875g + v^2)`

 

`d/(dx)(1/2 v^2)` `= −0.4(0.1875g + v^2)`
`d/(dx)(v^2)` `= −0.8(0.1875g + v^2)`
`(dx)/(d(v^2))` `= −1.25(1/(0.1875g + v^2))`
`:. x` `= −1.25 int_(2.5^2)^0 1/(0.1875 + v^2)\ dv^2`
  `= 1.85\ text(m)`

 

`:.\ text(Distance from C)` `= 5 – 1.85`
  `= 3.15\ text(m)`

Complex Numbers, EXT2 N2 2020 SPEC2 2

Two complex numbers, `u` and `v`, are defined as  `u = −2 - i`  and  `v = −4 - 3i`.

  1. Express the relation  `|z - u| = |z - v|`  in the cartesian form  `y = mx + c`, where  `m, c ∈ R`.  (3 marks)

  2. Plot the points that represent `u` and `v` and the relation `|z - u| = |z - v|` on the Argand diagram below.  (2 marks)
     
         
     
  3. State a geometrical interpretation of the graph of  `|z - u| = |z - v|`  in relation to the points that represent `u` and `v`.  (1 mark)

  4.  i. Sketch the ray given by  `text(Arg)(z - u) = pi/4`  on the Argand diagram in part b.  (1 mark)

  5. ii. In Cartesian form, write down the function that describes the ray  `text(Arg)(z - u) = pi/4`.  (1 mark)

Show Answers Only
  1. `y = −x – 5`
  2.  
  3. `|z – u| = |z – v|\ text(is the perpendicular bisector of the line joining)\ u and v.`
  4.  i.
  5. ii. `f: (−2, ∞) ->, f(x) = x + 1`
Show Worked Solution

a.   `text(Let)\ \ z = x + iy`

`z – u = x + 2 + iy + i`

`z – v = x + 4 + iy + 3i`

`|z – u| = |z – v|`

`(x + 2)^2 + (y + 1)^2` `= (x + 4)^2 + (y + 3)^2`
`x^2 + 4x + 4 + y^2 + 2y + 1` `= x^2 + 8x + 16 + y^2 + 6y + 9`
`-4y` `= 4x + 20`
`y` `= −x – 5`

 

b.   

 

c.   `|z – u| = |z – v|\ text(is the graph of the perpendicular bisector of the)`

`text(line joining)\ u and v.`

 

d.i.   

 

d.ii.   `text(Arg)(z – u) = pi/4 =>\ text(gradient) = 1, ytext(-intercept at)\ (0, 1)`

`:. f: (−2, ∞) -> RR, \ f(x) = x + 1`

Complex Numbers, SPEC2 2020 VCAA 2

Two complex numbers, `u` and `v`, are defined as  `u = -2-i`  and  `v = −4-3i`.

  1. Express the relation  `|z-u| = |z-v|`  in the cartesian form  `y = mx + c`, where  `m, c ∈ R`.   (3 marks)

  2. Plot the points that represent `u` and `v` and the relation `|z-u| = |z-v|` on the Argand diagram below.   (2 marks)

     
         
     

  3. State a geometrical interpretation of the graph of  `|z-u| = |z-v|`  in relation to the points that represent `u` and `v`.   (1 mark)

  4.  i. Sketch the ray given by  `text(Arg)(z-u) = pi/4`  on the Argand diagram in part b.   (1 mark)

  5. ii. Write down the function that describes the ray  `text(Arg)(z-u) = pi/4`, giving the rule in cartesian form.   (1 mark)

  6. The points representing `u` and `v` and  `−5i`  lie on the circle given by  `|z-z_c| = r`, where `z_c` is the centre of the circle and `r` is the radius.
  7. Find `z_c` in the form  `a + ib`, where  `a, b ∈ R`, and find the radius `r`.   (3 marks)

Show Answers Only
  1. `y = −x-5`
  2.  
  3. `|z-u| = |z-v|\ text(is the perpendicular bisector of the line joining)\ u and v.`
  4.  i.
  5. ii. `f: (−2, ∞) ->, f(x) = x + 1`
  6. `z_c = −5/3-10/3 i and r = (5sqrt2)/3`
Show Worked Solution

a.   `text(Let)\ \ z = x + iy`

`z-u = x + 2 + iy + i`

`z-v = x + 4 + iy + 3i`

`|z-u| = |z-v|`

`(x + 2)^2 + (y + 1)^2` `= (x + 4)^2 + (y + 3)^2`
`x^2 + 4x + 4 + y^2 + 2y + 1` `= x^2 + 8x + 16 + y^2 + 6y + 9`
`-4y` `= 4x + 20`
`y` `= −x-5`

 

b.   

 

c.   `|z-u| = |z-v|\ text(is the graph of the perpendicular bisector of the)`

`text(line joining)\ u and v.`

 

d.i.   

♦♦ Mean mark (d.ii.) 25%.

 

d.ii.   `text(Arg)(z-u) = pi/4 =>\ text(gradient) = 1, ytext(-intercept at)\ (0, 1)`

`:. f: (−2, ∞) -> RR, \ f(x) = x + 1`

 

e.   `|z_c-u| = |z_c-v| = |z_c-(−5i)|`

♦ Mean mark (e) 40%.
`z_c-u` `= (a + 2) + (b + 1)i`
`z_c-v` `= (a + 4) + (b + 3)i`
`z_c +5i` `= a + (b + 5)i`

 
`a^2 + (b + 5)^2 = (a + 2)^2 + (b + 1)^2\ …\ (1)`

`a^2 + (b + 5)^2 = (a + 4)^2 + (b + 3)^2\ …\ (2)`

`a = −5/3, b = −10/3\ \ text{(by CAS)}`
 

`:.z_c = −5/3-10/3 i`

`:.r` `= |z_c-(−5i)|`
  `= sqrt((−5/3)^2 + (−10/3 + 5)^2)`
  `= (5sqrt2)/3`

Vectors, SPEC2 2020 VCAA 1

A particle moves in the  `x\ – y`  plane such that its position in terms of `x` and `y` metres at `t` seconds is given by the parametric equations

`x = 2sin(2t)`

`y = 3cos(t)`

where `t >= 0`

  1. Find the distance, in metres, of the particle from the origin when  `t = pi/6`.   (2 marks)

  2.   i. Express  `(dy)/(dx)`  in terms of `t` and, hence, find the equation of the tangent to the path of the particle at  `t = pi`  seconds.   (3 marks)

  3.  ii. Find the velocity, `underset ~ v`, in `text(ms)^(−1)`, of the particle when  `t = pi`.   (2 marks)

  4. iii. Find the magnitude of the acceleration, in `text(ms)^(−2)`, when  `t = pi`.   (2 marks)

  5. Find the time, in seconds, when the particle first passes through the origin.   (1 mark)

  6. Express the distance, `d` metres, travelled by the particle from  `t = 0`  to  `t = pi/6`  as a definite integral and find this distance correct to three decimal places.   (2 marks)

Show Answers Only
  1. `sqrt39/2\ text(metres)`
  2.   i. `y = −3`
  3.  ii. `4underset~i`
  4. iii. `3\ text(ms)^(−2)`
  5. `pi/2\ \ (text(1st time))`
  6. `1.804\ \ (text(to 3 d.p.))`
Show Worked Solution

a.   `text(At)\ \ t = pi/6,`

`x = 2sin\ pi/3 = sqrt3`

`y = 3cos\ pi/6 = (3sqrt3)/2`

`:.\ text(Distance)` `= sqrt((sqrt3)^2 + ((3sqrt3)/2)^2)`
  `= sqrt39/2\ text(metres)`

 

b.i.   `(dx)/(dt) = 4cos(2t),\ \ (dy)/(dt) = −3sin(t)`

`(dy)/(dx)` `= (dy)/(dt) · (dt)/(dx)`
  `= (−3sin(t))/(4cos(2t))`

 
`text(When)\ t = pi :`

`(dy)/(dx)` `= (−3sin(pi))/(4cos(2pi))=0`

 
`text(Equation of tangent where)\ \ m = 0,\ text(through)\ (0, −3):`

`y = −3`
  

b.ii.   `underset~r(t)` `= 2sin(2t)underset~i + 3cos(t)underset~j`
  `underset~v(t)` `= 4cos(2t)underset~i-3sin(t)underset~j`
  `underset~v(pi)` `= 4cos(2pi)underset~i-3sin(pi)underset~i`
    `= 4underset~i`

 

b.iii.  `underset~a(t) = −8sin(2t)underset~i-3cos(t)underset~j`

♦ Mean mark (b)(iii) 48%.
`underset~a(pi)` `= −8sin(2pi)underset~i-3cos(pi)underset~j`
  `= 3underset~j`
`|underset~a(pi)|` `= sqrt(0^2 + 3^2)`
  `= 3\ text(ms)^(−2)`

 

c.   `text(Find)\ t\ text(when)\ \ x = 2sin(2t) = 0\ \ text(and)\ \ y = 3cos(t) = 0:`

`t = pi/2\ \ (text(1st time))`

 

d.    `text(Distance)` `= int_0^(pi/6) sqrt(((dx)/(dt))^2 + ((dy)/(dt))^2)\ dt`
    `= int_0^(pi/6) sqrt((4cos(2t))^2 + (−3sin(t))^2)\ dt`
    `~~ 1.804\ \ (text(to 3 d.p.))`

Number, NAPX-p116764v04

Pringle has 55 twenty-cent coins in his money pouch.

How much money does Pringle have?

 
    $9.50
 
   $11.00
 
   $20.50
 
   $55.00
Show Answers Only

`$11.00`

Show Worked Solution

`text(5 twenty-cent coins = $1.00)`

`:.\ text(55 twenty-cent coins)` `=11 xx $1.00`  
  `=$11.00`  

Number, NAPX-p116764v03

Sheena has 26 twenty-cent coins in her piggy bank.

How much money does Sheena have?

 
   $26.20
 
   $5.20
 
   $5.00
 
   $4.40
Show Answers Only

$5.20

Show Worked Solution

`text(5 twenty-cent coins = $1.00)`

`:.\ text(26 twenty-cent coins)` `=5 xx $1.00 + 20\ text(cents)`  
  `=$5.20`  

Measurement, NAPX-p116903v03

Wilde compared the weight of four different solid figures by using a balancing scale.
 

          

 
Which of the four objects is the heaviest?

 
 
 
 
Show Answers Only

Show Worked Solution

`text{By inspection:}`

`text(Triangle → heavier than the hexagon.)`

`text(Triangle → same weight as the oval.)`

`text(Star → heavier than the oval.)`
 

`:.\ text(Star is the heaviest.)`

Measurement, NAPX-p116903v02

Banjo compared the weight of four different solid figures by using a balancing scale.
 

      
 

Which of the four objects is the lightest?

 
 
 
 
Show Answers Only

Show Worked Solution

`text{By inspection:}`

`text(Sphere → lighter than the cube.)`

`text(Sphere → same weight as the cylinder.)`

`text(Cylinder → heavier than the pentagonal prism.)`
 

`:.\ text(Pentagonal prism is the lightest.)`

Measurement, NAPX-p116893v03

Mince decided to paint some triangles red in the four figures below.

All triangles are the same size.

Which of these figures has the smallest area that is painted red.

 
 
 
 

Show Answers Only

Show Worked Solution

`text{Smallest area painted red = lowest number of triangles painted red.}`

`text{Check each option:}`

`text{1st Figure  – 9 triangles}`

`text{2nd Figure – 9 triangles}`

`text{3rd Figure – 10 triangles}`

`text{4th Figure – 7 triangles}`

`therefore \ text{4th Figure has the smallest area painted red}`

Measurement, NAPX-p116893v02

Jane painted some hexagons in the four figures pictured below.

All hexagons are the same size.

Which of these figures has the largest area that is painted green.

 
 
 
 
Show Answers Only

Show Worked Solution

`text{Largest area painted = figure with the most hexagons painted green.}`

`text{Check each option:}`

`text{1st Figure – 11 hexagons}`

`text{2nd Figure – 9 hexagons}`

`text{3rd Figure – 13 hexagons}`

`text{4th Figure – 9 hexagons}`

`therefore \ text{3rd Figure has the largest area painted green.}`

Number, NAPX-p116838v04

In a bakery Michelle bought one slice of cake, one donut and a croissant.
 


 

She gave the cashier $10.00.

How much change should Michelle receive?

$5.10 $5.60 $5.80 $6.00
 
 
 
 
Show Answers Only

`$6.00`

Show Worked Solution
`text{Total cost}` `= 1.35 + 1.20 + 1.45 `
  `= $4.00`

 

`text{Change}` `= 10 – 4`
  `= $6.00`

Number, NAPX-p116838v03

Alexander went shopping and bought the following items.
 


 

He gave the cashier $50.

How much change should Alexander receive?

$17.00 $18.20 $31.80 $33.00
 
 
 
 
Show Answers Only

`$17.00`

Show Worked Solution
`text{Total cost}` `= 9.50 + 12.10 + 11.40 `
  `= $33.00`

 

`text{Change}` `= 50 – 33.00 `
  `= $17.00`

Measurement, NAPX p124557v03

Timothy started with 53 millilitres of oil in a beaker.

He saw that there was some unused oil in a test tube and returned it to the beaker.


 

How much of the chemical solution was poured int0 the test tube?

 
  13 millilitres
 
  23 millilitres
 
  27 millilitres
 
  37 millilitres
Show Answers Only

`27 \ text{millilitres}`

Show Worked Solution
`text{Oil in the test tube }` `= text{New volume} – text{Original volume}`
  `= 80 – 53`
  `= 27 \ text{millilitres}`

Measurement, NAPX p124557v02

A chemist started with 685 millilitres of a solution.

He then poured some of the solution into a test tube.
 


 

How much of the chemical solution was poured into the test tube?

 
  90 millilitres
 
  135 millilitres
 
  145 millilitres
 
  185 millilitres
Show Answers Only

`185 \ text{millilitres}`

Show Worked Solution
`text{Solution in test tube }` `= text{New volume} – text{Original volume}`
  `= 685 – 500`
  `= 185 \ text{millilitres}`

Statistics, NAPX-p116802v04

In a neighborhood 10 villagers were asked if how many pets they own.

The results were: 3, 2, 1, 1, 1, 4, 2, 4, 5, 2

Select the dot plot that displays the data recorded

 
 
 
 
Show Answers Only

Show Worked Solution

`text{A dot represents the number of persons that owns N pets}`

`text{From the data above:}`

`text{Owns one pet = 3}`

`text{Owns two pets = 3}`

`text{Owns three pets = 1}`

`text{Owns four pets = 2}`

`text{Owns five pets = 1}`

`therefore \ text{the correct dot plot is:}`
 

Statistics, NAPX-p116802v03

10 people were asked how often they went to the doctor in the last 12 months.

Their responses were: 2, 3, 3, 4, 1, 2, 5, 1, 1, 2

Choose the dot plot which correctly displays the data.

 
    
 
    
       
 
    
 
    
Show Answers Only

Show Worked Solution

`text{From the data above:}`

`text{1 visit =  3 people}`

`text{2 visits =  3 people}`

`text{3 visits =  2 people}`

`text{4 visits =  1 person}`

`text{5 visits =  1 person}`
 

   

Geometry, NAPX-p116732v03

A bridge is drawn below.
 

   
 

The angle `y°` is marked from the foot of the bridge to the diagonal support.

The value of angle `y°`  is

 
  Less than 90°
 
  More than 180°
 
  Equal to 90°
 
  More than 90° but less than 180°
Show Answers Only

`text{Less than} \ 90^@`

Show Worked Solution

`text{The angle from the foot to the diagonal support is less than} \ 90^@`

Statistics, NAPX-p110558v02

A charity event was held for three weeks. On the first week $5000 was raised, on the second week $3500 was raised and on the last week $2000 was raised.

In the tables below, O = $500

Which table correctly shows the amount of donations raised in the 3 weeks?

 
  
 
  
 
  
 
    
Show Answers Only

Show Worked Solution

`text{O represents $500 in donations.}`

`text{1st week:} \ 5000 \div 500 = 10`

`therefore \ text{Ten O symbols needed.}`
 

`text{2nd week:} \ 3500 \div 500 = 7`

`therefore \ text{Seven O symbols needed.}`
 

`text{3rd week:} \ 2000 \div 500 = 4`

`therefore \ text{Four O symbols needed.}`

Statistics, NAPX-p110558v01

In a certain school, the number of Year 7 students is 120, Year 8 students is 80, and Year 9 students is 100.

In the tables below, V = 20 students

Which table correctly shows the number of students per year level?

 
    
 
    
 
    
 
    
Show Answers Only

Show Worked Solution

`text{V represents 20 students.}`

`text{Year 7 students:} \ 120 \div 20 = 6`

`therefore \ text{Six V symbols needed}`
 

`text{Year 8 students:} \ 80 \div 20 = 4`

`therefore \ text{Four V symbols needed}`
 

`text{Year 9 students:} \ 100 \div 20 = 5`

`therefore \ text{Five V symbols needed}`

Geometry, NAPX-p116721v03

Sarah drew a straight line on a shape.

The line divided the shape into two rectangles.

Which of these could have been Sarah’s shape?

 
 
 
 
Show Answers Only

Show Worked Solution

`text{Drawing any horizontal straight line on this shape divides it}`

`text{into two rectangles.}`

Algebra, NAPX-p116672v03

Rudy uses the number sentence  24 – 6 = 18

Which of the following problems can he solve with this number sentence?

 
   Rudy shared 24 slices of cake equally between 6 of his friends.
  How many slices does each of his friends get?
 
   Rudy has 24 slices of cake and gave 6 slices to his friends.
  How many slices does Rudy have left?
 
   Rudy took 6 slices of cake and shared them between 24 of his friends.
  How many slices does each friend receive?
 
   Rudy has 24 slices of cake and gave 18 of them to his friends.
  How many slices does Rudy have left?
Show Answers Only

`text{Rudy has 24 slices of cake and gave 6 slices to his friends.}`

`text{How many slices does Rudy have left?}`

Show Worked Solution

`text{Rudy has 24 slices of cake and gave 6 slices to his friends.}`

`text(Expressing this as an equation:)`

`text(Slices Rudy has left)` `=24 -6`  
`18` `=20 – 6`  

Algebra, NAPX-p116672v02

Shawn uses the number sentence  20 × 6 = 120

Which of the following problems can he solve with this number sentence?

 
   Shawn buys 20 toys and gave 6 of them to his nephew.
  How many toys does Shaun's nephew have now?
 
   Shawn buys 20 toys and received another 6 toys from friends.
  How many toys does Shaun have now?
 
   Shawn buys 20 toys and divides them up between his 6 nephews.
  How many toys does each nephew get?
 
   Shawn has 20 nephews and gives 6 toys to each one.
  Altogether, how many toys does Shaun give to his nephews?
Show Answers Only

`text{Shawn has 20 nephews and gives 6 toys to each one.}`

Show Worked Solution

`text{Shawn has 20 nephews and gives 6 toys to each one.}`

`text(Expressing this as an equation:)`

`text(Total toys to nephews)` `=20 xx 6`  
`120` `=20 xx 6`