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v1 Algebra, STD2 A2 2011 HSC 23b

Sticks were used to create the following pattern. 
  

The number of sticks used is recorded in the table.

\begin{array} {|l|c|}
\hline
\rule{0pt}{2.5ex} \text{Shape $(S)$} \rule[-1ex]{0pt}{0pt} & \;\;\; 1 \;\;\; & \;\;\; 2 \;\;\; & \;\;\; 3 \;\;\; \\
\hline
\rule{0pt}{2.5ex} \text{Number of sticks $(N)$}\; \rule[-1ex]{0pt}{0pt} & \;\;\; 6 \;\;\; & \;\;\; 10 \;\;\; & \;\;\; 14 \;\;\; \\
\hline
\end{array}

  1. Draw Shape 4 of this pattern.  (1 mark)

  2. How many sticks would be required for Shape 128?    (1 mark)

  3. Is it possible to create a shape in this pattern using exactly 609 sticks?

     

    Show suitable calculations to support your answer.    (2 marks)

Show Answers Only
  1. \(\text{See Worked Solutions.}\)
  2. \(514\)
  3. \(\text{No (See worked solution)}\)
Show Worked Solution

  i.    \(\text{Shape 4 is shown below:}\)

ii.    \(\text{Since}\ \ N=2+4S\)

♦ Mean mark 48%.
MARKER’S COMMENT: Students should attempt to find a “rule” in such questions, and use this formula to solve the question, as per the Worked Solution.  
\(\text{If }S\) \(=128\)
\(N\) \(=2+(4\times 128)\)
  \(=514\)

 

iii.    \(609\) \(=2+4S\)
  \(4S\) \(=607\)
  \(S\) \(=151.75\)

    
\(\text{Since}\ S\ \text{is not a whole number, 609 sticks}\)

\(\text{will not create a shape in this pattern.}\)

CORE, FUR2 2020 VCAA 7

Samuel owns a printing machine.

The printing machine is depreciated in value by Samuel using flat rate depreciation.

The value of the machine, in dollars, after `n` years, `Vn` , can be modelled by the recurrence relation

`V_0 = 120\ 000, qquad V_(n+1) = V_n-15\ 000`

  1. By what amount, in dollars, does the value of the machine decrease each year?   (1 mark)

  2. Showing recursive calculations, determine the value of the machine, in dollars, after two years.   (1 mark)

  3. What annual flat rate percentage of depreciation is used by Samuel?   (1 mark)

  4. The value of the machine, in dollars, after `n` years, `V_n`, could also be determined using a rule of the form `V_n = a + bn`.

     

    Write down this rule for `V_n`.   (1 mark)

Show Answers Only
  1. `$15\ 000`
  2. `$90\ 000`
  3. `12.5%`
  4. `V_n = 120\ 000-15\ 000n, n = 0, 1, 2, …`
Show Worked Solution

a. `$15\ 000`
  

b.   `V_1` `= 120\ 000-15\ 000 = $105\ 000`
  `V_2` `= 105\ 000-15\ 000 = $90\ 000`

 

c.   `text(Flat rate percentage` `= (15\ 000)/ (120\ 000) xx 100`
    `= 12.5 text(%)`

 

♦ Mean mark part d. 44%.

d.  `V_n = 120\ 000-15\ 000n, \ n = 0, 1, 2, …`

Complex Numbers, EXT2 N1 2019 HSC 11a

Let  `z = 1 + 3i`  and  `w = 2 - i`.

  1. Find  `z + bar w`.  (1 mark)

  2. Express  `z/w`  in the form  `x + iy`, where  `x` and  `y`  are real numbers.  (2 marks)

Show Answers Only
  1. `3 + 4i`
  2. `-1/5 + 7/5 i`
Show Worked Solution

i.   `z = 1 + 3i`

`w = 2 – i \ => \ bar w = 2 + i`

`z + bar w` `= 1 + 3i + 2 + i`
  `= 3 + 4i`

 

ii.    `z/w` `= (1 + 3i)/(2 – i) xx (2 + i)/(2 + i)`
    `= ((1 + 3i)(2 + i))/(2^2 – i^2)`
    `= (2 + i + 6i + 3i^2)/5`
    `= (-1 + 7i)/5`
    `= -1/5 + 7/5 i`

GRAPHS, FUR1 2018 VCAA 02 MC

Steven is a wedding photographer.

He charges his clients a fixed fee of $500, plus $250 per hour of photography.

The equation that represents the total amount, `$C`, Steven charges, for `t` hours of photography is

  1.  `C = 250t`
  2.  `C = 500t`
  3.  `C = 750t`
  4.  `C = 500 + 250t`
  5.  `C = 250 + 500t`
Show Answers Only

`D`

Show Worked Solution

`C = 500 + 250t`

`=> D`

Graphs, EXT2 2018 HSC 12b

A curve has equation  `x^2 + xy + y^2 = 3`.

  1. Use implicit differentiation to show that  `dy/dx = −(2x + y)/(x + 2y)`.  (2 marks)
  2. Hence, or otherwise, find the coordinates of the points on the curve where  `dy/dx = 0`.  (2 marks)
Show Answers Only
  1. `text(See Worked Solutions)`
  2. `(1,−2)\ \ text(and)\ \ (−1,2).`
Show Worked Solution

i.   `x^2 + xy + y^2 = 3`

♦ Mean mark 98%!

`2x + y + x · dy/dx + 2y · dy/dx=0`

`dy/dx(x + 2y)` `= −2x – y`
`:. dy/dx` `= −(2x + y)/(x + 2y)\ \ \ text(… as required)`

 

ii.   `text(When)\ \ dy/dx = 0,`

`−(2x + y)/(x + 2y)` `= 0`
`-2x-y` `=0`
`y` `= −2x\ \ \ \ (text(*))`

 

`text(Substituting into)\ \ x^2 + xy + y^2 = 3`

`x^2 + x(−2x) + (−2x)^2` `= 3`
`x^2 – 2x^2 + 4x^2` `= 3`
`3x^2` `= 3`
`x` `= +-1`
`y` `= +-2\ \ \ \ (text(*))`

 
`:.\ text(Coordinates are)\ \ (1,−2)\ \ text(and)\ \ (−1,2).`

Complex Numbers, EXT2 N1 2018 HSC 11a

Let  `z = 2 + 3i`  and  `w = 1 - i.`

  1. Find  `zw`.  (1 mark)

  2. Express  `barz  - 2/w`  in the form  `x + iy`, where `x` and `y` are real numbers.  (2 marks)

Show Answers Only
  1. `5 + i`
  2. `1 – 4i`
Show Worked Solution

i.   `z = 2 + 3iqquadw = 1 – i`

♦ Mean mark part (i) 97!%.

`zw` `= (2 + 3i)(1 – i)`
  `= 2 – 2i + 3i – 3i^2`
  `= 2 + i + 3`
  `= 5 + i`

 

ii.    `barz – 2/w` `= 2 – 3i – 2/(1 – i)`
    `= 2 – 3i – (2(1 – i))/((1 – i)(1 + i))`
    `= 2 – 3i – (1 + i)`
    `= 1 – 4i`

Functions, EXT1 F2 2018 HSC 11a

Consider the polynomial  `P(x) = x^3-2x^2-5x + 6`.

  1. Show that  `x = 1`  is a zero of  `P(x)`.  (1 mark)

  2. Find the other zeros.  (2 marks)

Show Answers Only
  1. `text(See Worked Solution)`
  2. `x = -2 and x = 3`
Show Worked Solution

i.   `P(1) = 1-2-5 + 6 = 0`

`:. x=1\ \ text(is a zero)`

 

ii.   `text{Using part (i)} \ => (x-1)\ text{is a factor of}\ P(x)`

`P(x) = (x-1)*Q(x)`
 

`text(By long division:)`

`P(x)` `= (x-1) (x^2-x-6)`
  `= (x-1) (x-3) (x + 2)`

 
`:.\ text(Other zeroes are:)`

`x = -2 and x = 3`

Graphs, MET1 2016 VCAA 5

Let  `f : (0, ∞) → R`, where  `f(x) = log_e(x)`  and  `g: R → R`, where  `g (x) = x^2 + 1`.

  1.   i. Find the rule for `h`, where  `h(x) = f (g(x))`.   (1 mark)

    ii. State the domain and range of `h`.   (2 marks)

  2. iii. Show that  `h(x) + h(-x) = f ((g(x))^2 )`.   (2 marks)

  3. iv. Find the coordinates of the stationary point of `h` and state its nature.   (2 marks)

     

  4. Let  `k: (-∞, 0] → R`  where  `k (x) = log_e(x^2 + 1)`.
  5.  i. Find the rule for  `k^(-1)`.   (2 marks)

  6. ii. State the domain and range of  `k^(-1)`.   (2 marks)

Show Answers Only
  1.   i. `log_e(x^2 + 1)`
  2.  ii. `[0,∞)`
  3. iii. `text(See Worked Solutions)`
  4. iv. `(0, 0)`
  5.  i. `-sqrt(e^x-1)`
  6. ii. `text(Domain)\ (k) = (-∞,0]`
  7.    `text(Range)\ (k) = [0,∞)`
Show Worked Solution
a.i.    `h(x)` `= f(x^2 + 1)`
    `= log_e(x^2 + 1)`

 

a.ii.   `text(Domain)\ (h) =\ text(Domain)\ (g) = R`

♦♦ Mean mark part (a)(ii) 30%.
  `text(For)\ x ∈ R` `-> x^2 + 1 >= 1`
  `-> log_e(x^2 + 1) >= 0`

`:.\ text(Range)\ (h) = [0,∞)`

 

MARKER’S COMMENT: Many students were unsure of how to present their working in this question. Note the layout in the solution.
a.iii.   `text(LHS)` `= h(x) + h(−x)`
    `= log_e(x^2 _ 1) + log_e((-x)^2 + 1)`
    `= log_e(x^2 + 1) + log_e(x^2 + 1)`
    `= 2log_e(x^2 + 1)`

 

`text(RHS)` `= f((x^2 + 1)^2)`
  `= 2log_e(x^2 + 1)`

 

`:. h(x) + h(-x) = f((g(x))^2)\ \ text(… as required)`

 

a.iv.   `text(Stationary points when)\ \ h^{prime}(x) = 0`

♦ Mean mark part (a)(iv) 41%.
MARKER’S COMMENT: Solving a fraction is zero

   `text(Using Chain Rule:)`

`h^{prime}(x)` `= (2x)/(x^2 + 1)`

`:.\ text(S.P. when)\ \ x=0`

 

`text(Find nature using 1st derivative test:)`

`:.\ text{Minimum stationary point at (0, 0)}.`

 

b.i.   `text(Let)\ \ y = k(x)`

♦ Mean mark (b)(i) 49%.
MARKER’s COMMENT: Many students failed to consider the restrictions on the domain in `k(x)` and only select the negative root.

  `text(Inverse: swap)\ x ↔ y`

`x` `= log_e(y^2 + 1)`
`e^x` `= y^2 + 1`
`y^2` `= e^x-1`
`y` `= ±sqrt(e^x-1)`

 

`text(But range)\ \ (k^(-1)) =\ text(domain)\ (k)`

`:.k^(-1)(x) =-sqrt(e^x-1)`

♦ Mean mark part (b)(ii) 44%.

 

b.ii.   `text(Range)\ (k^(-1)) =\ text(Domain)\ (k) = (-∞,0]`

  `text(Domain)\ (k^(-1)) =\ text(Range)\ (k) = [0,∞)`

Statistics, NAP-H1-04

Che's hockey team played 3 games.

The number of goals his team scored in each game is shown in the tally below.

The team scored 6 goals in Game 1.

What was the total number of goals scored by the team in all three games?

`9` `17` `20` `23`
 
 
 
 
Show Answers Only

`23\ text(goals)`

Show Worked Solution
`text(Total goals)` `=6+8+9`
  `=23`

Statistics, NAP-I1-04

The graph below shows the number of students that use different ways to get to school.
 

 

Which statement is true?

 
 More students walk to school than ride a skateboard.
 
 Less students ride in a car to school than ride a bike.
 
 The most common way to get to school is by car.
 
 The most common way to get to school is by bus.
Show Answers Only

`text(The most common way to get to school is by bus.)`

Show Worked Solution

`text(The most common way to get to school)`

`text(is by bus.)`

Measurement, NAP-I1-02

Malcolm and Barnaby were measuring their screwdrivers using paddle pop sticks.
 


 

Which of these statements is true?

 
 Malcolm's screwdriver is longer than Barnaby's screwdriver.
 
 Malcolm's screwdriver is shorter than Barnaby's screwdriver.
 
 Malcolm's screwdriver is the same length as Barnaby's screwdriver.
Show Answers Only

`text(Malcolm’s screwdriver is shorter)`

`text(than Barnaby’s screwdriver.)`

Show Worked Solution

`text(Malcolm’s screwdriver is 3 paddlepop sticks long.)`

`text(Barnaby’s screwdriver is 3.5 paddlepop sticks long.)`

`:.\ text(Malcolm’s screwdriver is shorter than)`

`text(Barnaby’s screwdriver.)`

S&A, EXT1 2016 HSC 1 MC

Which sum is equal to  `sum_(k = 1)^20 (2k + 1)`?

  1. `1 + 2 + 3 + 4 + … + 20`
  2. `1 + 3 + 5 + 7 + … + 41`
  3. `3 + 4 + 5 + 6 + … + 20`
  4. `3 + 5 + 7 + 9 + … + 41`
Show Answers Only

 `D`

Show Worked Solution

`sum_(k = 1)^20 (2k + 1)`

`= 3 + 5 + 7 + … + 41`

`=>   D`

Conics, EXT2 2016 HSC 12a

The diagram shows an ellipse.

ext2-hsc-2016-12a

  1. Write an equation for the ellipse.  (1 mark)
  2. Find the eccentricity of the ellipse.  (1 mark)
  3. Write the coordinates of the foci of the ellipse.  (1 mark)
  4. Write the equations of the directrices of the ellipse.  (1 mark)
Show Answers Only
  1. `(x^2)/9 + (y^2)/4 = 1`
  2. `sqrt5/3`
  3. `(−sqrt5,0)\ text(and)\ (sqrt5,0)`
  4. `x = −(9sqrt5)/5\ text(and)\ x = (9sqrt5)/5`
Show Worked Solution

i.   `(x^2)/9 + (y^2)/4 = 1`

 

ii.    `4` `= 9(1 – e^2)`
  `9e^2` `= 5`
  `e^2` `= 5/9`
  `:. e` `= sqrt5/3`

 

iii.   `text(Foci are)`

`(−sqrt5,0)\ text(and)\ (sqrt5,0)`

 

iv.   `text(Ellipse directrices:)`

`x = −(9sqrt5)/5\ \ text(and)\ \ x = (9sqrt5)/5`

Complex Numbers, EXT2 N1 2016 HSC 11a

Let  `z = sqrt 3 - i.`

  1.  Express  `z`  in modulus-argument form.  (2 marks)

  2.  Show that  `z^6`  is real.  (1 mark)

  3.  Find a positive integer `n` such that  `z^n`  is purely imaginary.  (1 mark)

Show Answers Only
  1. `z = sqrt3 – i = 2 text(cis)((−pi)/6)`
  2. `text(Proof)\ \ text{(See Worked Solutions)}`
  3. `n = 3`
Show Worked Solution

i.   `z = sqrt3 – i`

`|\ z\ | = sqrt((sqrt3)^2 + 1^2) = 2`

`:. z = sqrt3 – i` `= 2(sqrt3/2 – 1/2 i)`
  `= 2(cos(− pi/6) + isin(− pi/6))`
  `= 2 text(cis)(− pi/6)`

 

ii.    `z^6` `= 2^6(cos(− pi/6) + isin(− pi/6))^6`
    `= 64\ text(cis)(−pi)quadquadtext{(by De Moivre)}`
    `= −64`

 
`:. z^6\ text(is real.)`

 

iii.   `z^n = 2^n (cos(−(npi)/6) + isin(−(npi)/6))`

`z^n\ text(is purely imaginary when:)`

`cos(−(npi)/6)=0`

`text(Or more generally,)`

`(npi)/6` `= pi/2 + kpi`
`n` `= 6k + 3,quad(k ∈ ZZ)`
`:.n` `=3,\ \ (n>0)`

Probability, 2UA 2016 HSC 2 MC

In a raffle, `30` tickets are sold and there is one prize to be won.

What is the probability that someone buying `6` tickets wins the prize?

(A)  `1/30`

(B)  `1/6`

(C)  `1/5`

(D)  `1/4`

Show Answers Only

`C`

Show Worked Solution

`P(W) = 6/30 = 1/5`

`=>  C`

GRAPHS, FUR2 2009 VCAA 2

Luggage over 20 kg in weight is called excess luggage.

Fair Go Airlines charges for transporting excess luggage.

The charges for some excess luggage weights are shown in Table 2.

GRAPHS, FUR2 2009 VCAA 21

  1. Complete this graph by plotting the charge for excess luggage weight of 10 kg. Mark this point with a cross (×).  (1 mark)

    GRAPHS, FUR2 2009 VCAA 22

  2. A graph of the charge against (excess luggage weight)² is to be constructed.

     

    Fill in the missing (excess luggage weight)² value in Table 3 and plot this point with a cross (×) on the graph below.  (1 mark)

    GRAPHS, FUR2 2009 VCAA 23

GRAPHS, FUR2 2009 VCAA 24

  1. The graph above can be used to find the value of `k` in the equation below.

     

          charge = `k` × (excess luggage weight)²

     

    Find `k`.  (1 mark)

  2. Calculate the charge for transporting 12 kg of excess luggage.

     

    Write your answer in dollars correct to the nearest cent.  (1 mark)

Show Answers Only
  1.  
    GRAPHS, FUR2 2009 VCAA 2 Answer
  2.  
    GRAPHS, FUR2 2009 VCAA 2 Answer1
    GRAPHS, FUR2 2009 VCAA 2 Anwer2
  3. `0.45`
  4. `$64.80`
Show Worked Solution
a.   

GRAPHS, FUR2 2009 VCAA 2 Answer

 

b.    GRAPHS, FUR2 2009 VCAA 2 Answer1

GRAPHS, FUR2 2009 VCAA 2 Anwer2

 

c.   `text(Using the point)\ (100,45),`

`45` `= k xx 100`
`:. k` `= 0.45`

 

d.   `text(Charge for 12 kg excess)`

`= 0.45 xx 12^2`

`= $64.80`

NETWORKS, FUR1 2008 VCAA 1 MC

Steel water pipes connect five points underground.

The directed graph below shows the directions of the flow of water through these pipes between these points. 

 

networks-fur1-2008-vcaa-1-mc
 

The directed graph shows that water can flow from

A.   point 1 to point 2.

B.   point 1 to point 4.

C.   point 4 to point 1.

D.   point 4 to point 2.

E.   point 5 to point 2.

Show Answers Only

`=> C`

Show Worked Solution

`=> C`

NETWORKS, FUR2 2014 VCAA 1

Four members of a train club, Andrew, Brianna, Charlie and Devi, have joined one or more interest groups for electric, steam, diesel or miniature trains.

The edges of the bipartite graph below show the interest groups that these four train club members have joined.

 

NETWORKS, FUR2 2014 VCAA 1
 

  1. How many of these four members have joined the steam trains interest group?   (1 mark)

  2. Which interest group have both Brianna and Charlie joined?   (1 mark)

Show Answers Only
  1. `2`
  2. `text(Miniature trains)`
Show Worked Solution

a.   `2`

b.   `text(Miniature trains.)`

MATRICES, FUR1 2012 VCAA 1 MC

`2 xx [(2,8), (4, -1), (3,5)] -[(3,7), (4,2), (2,3)] quad text (equals)`

  1. `[(1,1), (0,-3), (4,2)]`
  2. `[(-2,2), (0,-6), (2,4)]`
  3. `[(1,9), (12,0), (8,13)]`
  4. `[(1,9), (4,-4), (4,7)]`
  5. `[(-1,1), (0,-3), (1,2)]`
Show Answers Only

`D`

Show Worked Solution

`2 xx [(2,8),(4,−1),(3,5)] – [(3,7),(4,2),(2,3)]`

`= [(4,16),(8,−2),(6,10)] – [(3,7),(4,2),(2,3)]`

`= [(1,9),(4,−4),(4,7)]`

`rArr D`

MATRICES, FUR1 2007 VCAA 1 MC

The matrix sum `[(0, -4), (2, 5)] + [(5, 4), (-2, 2)]` is equal to
 

A.   `[(5, 0), (0, 7)]` B.   `[(0, 0), (0, 7)]`
   
C.   `[(5, -4), (0, 7)]` D.   `[(0, 5, -4,4), (2, -2, 5, 2)]`
   
E.   `[(0, -4, 5, 4), (2, 5, -2, 2)]`  
Show Answers Only

`A`

Show Worked Solution

`[(0,−4),(2,5)] + [(5,4),(−2,2)]`

`= [(0 + 5,−4 + 4),(2 + −2,5 + 2)]`

`= [(5,0),(0,7)]`

`=>  A`

GRAPHS, FUR1 2011 VCAA 1-2 MC

The charges for posting letters that weigh 100 g or less are shown in the graph below.

Part 1

The charge for posting a 35 g letter is

A.   `$0.40`

B.   `$0.60`

C.   `$0.90`

D.   `$1.50`

E.   `$2.00`

 

Part 2

Two letters are posted.

The total postage charge cannot be

A.   `$0.80`

B.   `$1.20`

C.   `$1.40`

D.   `$2.10`

E.   `$3.00`

Show Answers Only

`text (Part 1:)\ B`

`text (Part 2:)\ C`

Show Worked Solution

`text(Part 1)`

`text(From the graph)`

`text(C) text(ost for 35g letter) = $0.60`

`=>  B`

 

`text(Part 2)`

`text(Charges are  $0.40, $0.60, $0.90, and $1.50)`

`text(Consider A,)`

`$0.40 xx 2 = $0.80\ \ \ text{(Not A)}`

`text(Similarly, B, D and E can be shown to be)`

`text(combinations of 2 other charges.)`

`text(Only C isn’t a possible combination of 2 charges.)`

`=>  C`

GRAPHS, FUR1 2012 VCAA 2 MC

At a convenience store, one doughnut costs $2.40 and one drink costs $3.00.

A customer purchased five doughnuts and a number of drinks at a total cost of $24.00.

The number of drinks purchased was

A.     `4`

B.     `5`

C.     `6`

D.     `9`

E.   `10`

Show Answers Only

`A`

Show Worked Solution

 `text(Let)\  x = text(number of drinks)`

`5 xx 2.40 + 3x` `= 24`
`3x` `= 12`
`:. x` `= 4`

`=>  A`

Graphs, MET2 2015 VCAA 1 MC

Let `f: R -> R,\ f(x) = 2sin(3x) - 3.`

The period and range of this function are respectively

  1. `text(period) = (2 pi)/3 and text(range) = text{[−5, −1]}`
  2. `text(period) = (2 pi)/3 and text(range) = text{[−2, 2]}`
  3. `text(period) = pi/3 and text(range) = text{[−1, 5]}`
  4. `text(period) = 3 pi and text(range) = text{[−1, 5]}`
  5. `text(period) = 3 pi and text(range) = text{[−2, 2]}`
Show Answers Only

`A`

Show Worked Solution

`text(Range:)\ [−3 – 2, −3 + 2]`

`= [−5,−1]`

`text(Period) = (2pi)/n = (2pi)/3`

`=>   A`

Complex Numbers, EXT2 N1 2015 HSC 11b

Consider the complex numbers  `z = -sqrt 3 + i`  and  `w = 3 (cos\ pi/7 + i sin\ pi/7).`

  1. Evaluate  `|\ z\ |.`   (1 mark)

  2. Evaluate  `text(arg)(z).`   (1 mark)

  3. Find the argument of  `z/w.`   (1 mark)

Show Answers Only
  1. `2`
  2. `(5 pi)/6`
  3. `(29 pi)/42`
Show Worked Solution
i.   `|\ z\ |` `= sqrt ((-sqrt3)^2 + 1^2)`
  `= 2`

 

ii.   `text(arg)\ (z) ­=` `tan^-1 (1- sqrt 3)`
`­=` `pi – pi/6`
`­=` `(5 pi)/6`

 

iii.   `text(arg) (z/w) ­=` `text(arg)\ z – text(arg)\ w`
`­=` `(5 pi)/6 – pi/7`
`­=` `(29 pi)/42`

Complex Numbers, EXT2 N1 2010 HSC 2b

  1. Express  `-sqrt3 − i`  in modulus–argument form.  (2 marks)

  2. Show that  `(-sqrt3 − i)^6`  is a real number.  (2 marks)

Show Answers Only
  1. `2text(cis)(-(5pi)/6)`
  2. `-64`
Show Worked Solution
i.    `|-sqrt3 − i\ |` `=sqrt((-sqrt3)^2+sqrt((-1)^2))`
    `=2`

Complex Numbers, EXT2 2010 HSC 2b 

`text(From the graph)`

`text{arg}(-sqrt3-i)=- (5pi)/6\ \ \ \ text{(for}\  –pi<theta<pi text{)}`

`:.-sqrt3 − i= 2text(cis)(-(5pi)/6)`

 

`text{Alternative Solution (to find the argument)}`

`-sqrt3-i` `= 2(- sqrt3/2 − 1/2 i)`
  `=2text(cis)(-(5pi)/6)`

 

ii.   `(-sqrt3 − i)^6` `= [2text(cis)(-(5pi)/6)]^6`
    `=2^6[cos((-5pi)/6 xx6) +i sin((-5 pi)/6 xx6)]\ \ \ \ text{(De Moivre)}`
    `= 2^6[cos(-5pi) + i sin(-5pi)]`
    `= 64(-1 + 0i)`
    `= -64`

Complex Numbers, EXT2 N2 2011 HSC 2b

On the Argand diagram, the complex numbers  `0, 1 + i sqrt 3 , sqrt 3 + i`  and  `z`  form a rhombus.
 


 

  1. Find  `z`  in the form  `a + ib`, where  `a`  and  `b`  are real numbers.  (1 mark)

  2. An interior angle, `theta`, of the rhombus is marked on the diagram.

     

    Find the value of `theta.`  (2 marks)

Show Answers Only
  1. `(1 + sqrt 3) + i(1 + sqrt 3)`
  2. `(5 pi)/6`
Show Worked Solution
i.   `z` `= 1 + i sqrt 3 + sqrt 3 + i`
  `= (1 + sqrt 3) + i (1 + sqrt 3)`

 

ii.   `text(arg)\ z = tan^-1 ((1 + sqrt 3)/(1 + sqrt 3)) = pi/4`

`text(arg)\ (sqrt 3 + i) = tan^-1 (1/sqrt 3) = pi/6`

`text(Difference) = pi/4 – pi/6 = pi/12`
 

`=>\ text(Opposite angles of a rhombus are equal)`

`=>\ text(The diagonals of a rhombus bisect the angles)`

`:.theta` `= pi – 2 xx pi/12`
  `= (5 pi)/6\ \ text{(angle sum of triangle)`

Complex Numbers, EXT2 N1 2011 HSC 2a

Let  `w = 2 - 3i`  and  `z = 3 + 4i.`

  1.  Find  `bar w + z.`   (1 mark)

  2.  Find  `|\ w\ |.`   (1 mark)

  3.  Express  `w/z`  in the form  `a + ib`, where  `a`  and  `b`  are real numbers.   (2 marks)

Show Answers Only
  1. `5 + 7i`
  2. `sqrt13`
  3. `-6/25 -17/25 i`
Show Worked Solution

i.   `w = 2 – 3i,\ \ z = 3 + 4i,\ \ bar w = 2 + 3i`

`bar w + z` `= 2 + 3i + 3 + 4i`
  `= 5 + 7i`

 

ii.   `|\ w\ |` `= sqrt (2^2 + 3^2)`
  `= sqrt 13`

 

 iii.   `w/z` `= (2 – 3i)/(3 + 4i) xx (3 – 4i)/(3 – 4i)`
  `= (6 – 8i – 9i – 12)/(9 + 16)`
  `= (-6 – 17i)/25`
  `= -6/25 -17/25 i`

Complex Numbers, EXT2 N1 2013 HSC 11a

Let  `z = 2- i sqrt 3`  and  `w = 1 + i sqrt 3.`

  1. Find  `z + bar w.`  (1 mark)
  2. Express `w` in modulus–argument form.  (2 marks)
  3. Write `w^24` in its simplest form.  (2 marks)
Show Answers Only
  1. `3 – i\ 2 sqrt 3`
  2. `2 text(cis) pi/3`
  3. `2^24`
Show Worked Solution

i.    `z = 2 – i sqrt 3\ ,\ \ w = 1 + i sqrt 3`

`bar w = 1 – i sqrt 3`

`z + bar w` `= 2 – i sqrt 3 + 1 – i sqrt 3`
  `= 3 – i\ 2 sqrt 3`

 

ii.   `|\ w\ |` `=sqrt(1^2 + (sqrt3)^2)=2`
 `:.w` `= 2 (1/2 + i sqrt 3/2)`
  `=2(cos\ pi/3 + i sin\ pi/3)`
  `= 2 text(cis) pi/3`
MARKER’S COMMENT: The directive “in its simplest form” required students to convert `text(cis)\ 8pi` to 1.

 

iii.   `w^24` `= 2^24 text(cis)\ (24 xx pi/3)`
  `= 2^24\ text(cis) \ 8 pi`
  `= 2^24`

GEOMETRY, FUR1 2009 VCAA 1 MC

GEOMETRY, FUR1 2009 VCAA 1 MC
 

The two triangles, `ABC` and `FGH`, are similar.

The length `GH` is

A.   `14\ text(cm)`

B.   `24\ text(cm)`

C.   `26\ text(cm)`

D.   `28\ text(cm)`

E.   `32\ text(cm)`

Show Answers Only

`D`

Show Worked Solution

`Delta ABC\ text(|||)\ Delta FGH`

`:. (GH)/(BC)` `=(GF)/(BA)`
`(GH)/(14)` `=(24)/(12)`
`:.GH` `= 28 \ text(cm)`

 
`=>  D`

CORE, FUR1 2012 VCAA 1-2 MC

The following bar chart shows the distribution of wind directions recorded at a weather station at 9.00 am on each of 214 days in 2011.
 

Part 1

According to the bar chart, the most frequently observed wind direction was

A.  south-east.

B.  south.

C.  south-west.

D.  west.

E.  north-west.

 
Part 2

According to the bar chart, the percentage of the 214 days on which the wind direction was observed to be east or south-east is closest to

A.  `10text(%)`

B.  `16text(%)`

C.  `25text(%)`

D.  `33text(%)`

E.  `35text(%)`

Show Answers Only

`text(Part 1:)\ E`

`text(Part 2:)\ B`

Show Worked Solution

`text(Part 1)`

`text{North-west (highest bar)}`

`=>E`

 

`text(Part 2)`

`text (# Days with East or South East wind)`

`= 10 + 25`

`= 35`

`:.\ text(% Days)` `= 35/text (Total Days) xx 100` 
  `= 35/214 xx 100`
  `= 16.355…text(%)`

`=> B`

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

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

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

\(\Rightarrow D\)

Combinatorics, EXT1 A1 2011 HSC 2e

Alex’s playlist consists of 40 different songs that can be arranged in any order.  

  1. How many arrangements are there for the 40 songs?    (1 mark)

  2. Alex decides that she wants to play her three favourite songs first, in any order.
  3. How many arrangements of the 40 songs are now possible?   (1 mark)

Show Answers Only
  1. `40!`
  2. `6 xx 37!`
Show Worked Solution
i.    `#\ text(Arrangements) = 40!`

 

ii.    `#\ text(Arrangements)` `= 3! xx 37!`
    `= 6 xx 37!`

Financial Maths, STD2 F1 2010 HSC 23d

Warrick has a net income of $590 per week. He has created a budget to help manage his money.

       2010 23d

  1. Find the value of `X`, the amount that Warrick allocates towards electricity each week.    (1 mark)

  2. Warrick has an unexpectedly high telephone and internet bill. For the last three weeks, he has put aside his savings as well as his telephone and internet money to pay the bill.

     

    How much money has he put aside altogether to pay the bill?    (1 mark)

  3. The bill for the telephone and internet is $620. It is due in two weeks time. Warrick realises he has not put aside enough money to pay the bill.

     

    How could Warrick reallocate non-essential funds in his budget so he has enough money to pay the bill? Justify your answer with suitable reasons and calculations.   (3 marks)

Show Answers Only
  1. `$25`
  2. `text(Warwick has put aside $240 to pay the bill.)`
  3. `text(Warwick could reallocate funds for Entertainment and`

  4.  

    `text(Clothes and Gifts to pay the bill.)`

Show Worked Solution
i.    `X` `= 590 ` `-(175 + 45 + 10 + 15 + 90`
      `+ 40 + 30 + 70 + 50 + 40)`
    `= 590-565`
    `= $25`

 

ii.    `text(Weekly Amount)` `= 40 + 40`
    `= 80`
`text{Total (3 weeks)}` `= 3 xx 80`
  `= 240`

 
`:.\ text(Warwick has put aside $240 to pay the bill.)`

 

(iii)   `text(Amount required less amount put aside)`
  `= 620\-240`
  `= $380`

 
`text(Extra 2 weeks of savings and telephone)`

`= 2 xx (40 + 40)`

`= $160`
 

`:.\ text(Funds to be reallocated)`

`= 380\-160`

`= 220\ text(over 2 weeks)`

`= $110\ text(per week)`
 

`text(Non essential items are Entertainment)`

`text(and Clothes and Gifts)`

`text(Amount)` `= 70 + 50`
  `= $120\ text(per week)`

 

`:.\ text(Warwick could reallocate funds for Entertainment)`

`text(and Clothes and Gifts to pay the bill.)`

Algebra, STD2 A2 2011 HSC 23b

Sticks were used to create the following pattern. 

The number of sticks used is recorded in the table.

\begin{array} {|l|c|}
\hline
\rule{0pt}{2.5ex} \text{Shape $(S)$} \rule[-1ex]{0pt}{0pt} & \;\;\; 1 \;\;\; & \;\;\; 2 \;\;\; & \;\;\; 3 \;\;\; \\
\hline
\rule{0pt}{2.5ex} \text{Number of sticks $(N)$}\; \rule[-1ex]{0pt}{0pt} & \;\;\; 5 \;\;\; & \;\;\; 8 \;\;\; & \;\;\; 11 \;\;\; \\
\hline
\end{array}

  1. Draw Shape 4 of this pattern.  (1 mark)

  2. How many sticks would be required for Shape 100?    (1 mark)

  3. Is it possible to create a shape in this pattern using exactly 543 sticks?

     

    Show suitable calculations to support your answer.    (2 marks)

Show Answers Only
  1. `text(See Worked Solutions.)`
  2. `302`
  3. `text(No)`
Show Worked Solution

  i.     `text(Shape 4 is shown below:)`

ii.     `text(S)text(ince)\ \ N=2+3S`

♦ Mean mark 48%.
MARKER’S COMMENT: Students should attempt to find a “rule” in such questions, and use this formula to solve the question, as per the Worked Solution.  
`text(If)\ \ S` `=100`,
`N` `=2+(3xx100)`
  `=302`

 

iii.    `543` `=2+3S`
  `3S` `=541`
  `S` `=180 1/3`

 

`text(S)text(ince S is not a whole number, 543 sticks)`

`text(will not create a figure.)`