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Integration, EXT2 2016 HSC 14a

  1. Show that  `int sin^3 x\ dx = 1/3 cos^3 x - cos x + C.`  (1 mark)
  2. Using a graphical approach, or otherwise, explain why

     

    `int_0^pi cos^(2n - 1) x\ dx = 0`, for all positive integers `n.`  (1 mark)

  3. The diagram shows the region `R` enclosed by `y = sin^3 x` and the `x`-axis for `0 <= x <= pi.`

    ext2-hsc-2016-14a
     

     

    Using the method of cylindrical shells and the results in parts (i) and (ii), find the exact volume of the solid formed when `R` is rotated about the `y`-axis.  (3 marks)

Show Answers Only
  1. `text(See Worked Solutions)`
  2. `text(See Worked Solutions)`
  3. `(4pi^2)/3 \ u^3`
Show Worked Solution
i.    `int sin^3x\ dx` `= int sinx(1 – cos^2x)dx`
    `= int sinx\ dx – int (sin x · cos^2 x)\ dx`
    `= −cos x + 1/3cos^3x – c\ \ …\ text(as required)`

 

ii.   `int_0^pi cos^(2n – 1)x\ dx`

`text(Graphically, the integral can be represented)`

`text{below since (2n − 1) is always odd.}`

ext2-hsc-2016-14a-answer

`text(S)text(ince it is symmetrical about)\ \ x = pi/2,`

`int_0^pi cos^(2n – 1)x\ dx = 0`

 

iii.    `V` `= 2pi int_0^pi xy\ dx`
    `= 2pi int_0^pi (xsin^3x)\ dx`

 

`text(Using integration by parts:)`

`u` `= x` `du` `= dx`
`dv` `= sin^3x` `v` `= 1/3 cos^3x – cos x`

 

`I = [1/3 xcos^3x – xcosx]_0^pi – int_0^pi (1/3cos^3x – cosx)\ dx`

`text(S)text(ince)\ int_0^pi(1/3cos^3x – cosx)\ dx = 0\ \ text{(from part(ii)),}`

`I` `= [(1/3picos^3pi – picospi) – 0] – 0`
  `= (−pi/3 + pi)`
  `= (2pi)/3`

 

`:. V` `= 2pi · (2pi)/3`
  `= (4pi^2)/3 \ u^3`

Functions, 2ADV F1 2016 HSC 12a*

The diagram shows points `A(1, 0), B(2, 4)` and `C(6, 1).` The point `D` lies on `BC` such that `AD _|_ BC.`
 

hsc-2016-12a
 

  1. Show that the equation of `BC` is  `3x + 4y-22 = 0`.   (2 marks)

  2. Determine the gradient of `AD`.   (1 mark)

Show Answers Only

i.    `text(Proof)\ \ text{(See Worked Solutions)}`

ii.   `m_(AD)=4/3`

Show Worked Solution

i.    `B (2, 4),\ \ C (6, 1)`

`m_(BC) = (y_2-y_1)/(x_2-x_1) = (1-4)/(6-2) =-3/4`
 

`text(Equation of)\ \ BC,\ \ m=-3/4\ \ text(through)\ \ (2, 4):`

`y-y_1` `= m(x-x_1)`
`y-4` `=-3/4 (x-2)`
`4y-16` `= -3x + 6`
`3x + 4y-22` `= 0\ text(… as required.)`

 
ii.   `\text{Perpendicular lines:}\ m_1 xx m_2 = -1`

`m_(BC) =-3/4\ \ =>\ \ m_(AD)=4/3\ (BC _|_ AD)`

Conics, EXT2 2016 HSC 12d

  1. Show that the equation of the normal to the hyperbola `xy = c^2,\ \ c != 0`, at `P (cp, c/p)` is given by `px - y/p = c (p^2 - 1/p^2).`  (2 marks)
  2. The normal at `P` meets the hyperbola again at `Q (cq, c/q).`
    Show that     `q = -1/p^3.`  (3 marks)
Show Answers Only
  1. `text(See Worked Solutions)`
  2. `text(See Worked Solutions)`
Show Worked Solution

i.   `xy = c^2, c != 0`

`y` `= (c^2)/x`
`yprime` `= −(c^2)/(x^2)`

 

`text(At)\ P(cp, c/p),`

`yprime = −(c^2)/((cp)^2) = −1/(p^2)`

 

`:. text(Normal has)\ \ m = p^2,\ text(through)\ P(cp,c/p)`

`y – c/p` `= p^2(x – cp)`
  `= p^2x – cp^3`
`p^2x – y` `= cp^3 – c/p`
`:. px – y/p` `= c(p^2 – 1/(p^2))\ …\ text(as required)`

 

ii.   `text(Intersection of normal and hyperbola:)`

`px – y/p` `= c(p^2 – 1/(p^2))\ …\ (1)`
`xy` `= c^2\ …\ (2)`

 

`text(Substitute)\ \ y = (c^2)/x\ text{from  (2)  into  (1)}`

`px – ((c^2)/x)/p` `= c(p^2 – 1/(p^2))`
`px – (c^2)/(px)` `= c(p^2 – 1/(p^2))`
`px^2 – (c^2)/p` `= c(p^2 – 1/(p^2))x`

`px^2 – c(p^2 – 1/(p^2))x – (c^2)/p = 0`

`text(Using)\ sum\ text(roots) = −b/a,`

`cp + cq` `= (c(p^2 – 1/(p^2)))/p`
`:. q` `= ((p^2 – 1/(p^2)))/p – p`
  `= (p^2)/p – 1/(p^3) – p`
  `= −1/(p^3)\ \ …\ text(as required.)`

Complex Numbers, EXT2 N2 2016 HSC 12c

Let  `z = cos theta + i sin theta.`

  1. By considering the real part of `z^4`, show that `cos 4 theta` is
  2. `qquad cos^4 theta-6 cos^2 theta sin^2 theta + sin^4 theta.`   (2 marks)
  3. Hence, or otherwise, find an expression for  `cos 4 theta`  involving only powers of `cos theta.`   (1 mark)

Show Answers Only

i.    `text(See Worked Solutions)`

ii.   `8cos^4theta-8cos^2theta + 1`

Show Worked Solution

i.   `z = costheta + isintheta`

`z^4` `= (costheta + isintheta)^4`
 

`= cos^4theta + 4cos^3theta*(isintheta) + 6cos^2theta*(isintheta)^2 +`

`4costheta*(isintheta)^3 + (isintheta)^4`
 

`= cos^4theta + 4icos^3thetasintheta-6cos^2thetasin^2theta -`

`4icosthetasin^3theta + sin^4theta`

 

`z^4 = cos4theta + isin4theta\ \ text{(by De Moivre)}`
 

`text(Equating real parts:)`

`cos4theta = cos^4theta-6cos^2thetasin^2theta + sin^4theta\ …\ text(as required)`

 

ii.    `cos4theta` `= cos^4theta-6cos^2theta(1-cos^2theta) + (1-cos^2theta)^2`
    `= cos^4theta-6cos^2theta + 6cos^4theta + 1-2cos^2theta + cos^4theta`
    `= 8cos^4theta-8cos^2theta + 1`

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`

Graphs, EXT2 2016 HSC 11c

Find  `(dy)/(dx)`  for the curve given by  `x^3 + y^3 = 2xy`, leaving your answer in terms of `x` and `y.`  (2 marks)

Show Answers Only

`(2y – 3x^2)/(3y^2 – 2x)`

Show Worked Solution

`x^3 + y^3 = 2xy`

`3x^2 + 3y^2 · dy/dx` `= 2y + 2x · dy/dx`
`3y^2 · dy/dx – 2x *dy/dx` `= 2y – 3x^2`
`dy/dx(3y^2 – 2x)` `= 2y – 3x^2`
`:. dy/dx` `= (2y – 3x^2)/(3y^2 – 2x)`

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)`

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{cm}^2`

Show Worked Solution

`SA= 4pir^2= 4 xx pi xx 5^2= 314.15…= 314\ text(cm)^2  text{(nearest cm}^2 text{)}`

Algebra, STD2 A1 2016 HSC 2 MC

Which of the following equations has  `x = 5`  as the solution?

  1. `x - 5 = 10`
  2. `5 - x = 10`
  3. `x/2 = 10`
  4. `2x = 10`
Show Answers Only

`D`

Show Worked Solution
`2x` `= 10`
`:. x` `= 5`

 
`=> D`

Calculus, 2ADV C1 2016 HSC 11b

Differentiate  `(x + 2)/(3x-4).`   (2 marks)

Show Answers Only

`(-10)/(3x-4)^2`

Show Worked Solution

`y = (x + 2)/(3x-4)`

`text(Using the quotient rule:)`

`(g/h)^{′}` `= (g^{′} h-g h^{′})/h^2`
`y prime` `= (1 (3x-4)-(x + 2) · 3)/(3x-4)^2`
  `= (-10)/(3x-4)^2`

Functions, MET1 2006 VCAA 4

For the function  `f: [-pi, pi] -> R, f(x) = 5 cos (2 (x + pi/3))`

  1. write down the amplitude and period of the function.   (2 marks)

  2. sketch the graph of the function `f` on the set of axes below. Label axes intercepts with their coordinates.

     

    Label endpoints of the graph with their coordinates.   (3 marks)


VCAA 2006 meth 4b

Show Answers Only
  1. `text(Amplitude) = 5;\ \ \ text(Period) = pi`
  2.  
Show Worked Solution

a.   `text(Amplitude) = 5`

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

 

b.  

`text(Shift)\ \ y = 5 cos (2x)\ \ text(left)\ \ pi/3\ \ text(units).`

`text(Period) = pi`

`text(Endpoints are)\ \ (-pi, -5/2) and (pi,-5/2)`

Calculus, MET1 2014 VCAA 1a

If  `y = x^2sin(x)`, find  `(dy)/(dx)`.   (2 marks)

Show Answers Only

`2xsin(x) + x^2cos(x)`

Show Worked Solution

`text(Using product rule:)`

MARKER’S COMMENT: Factorising your answer is not necessary.
`(fg)^{prime}` `= f^{prime}g + fg^{prime}`
`:. (dy)/(dx)` `= 2xsin(x) + x^2cos(x)`

CORE*, FUR2 2007 VCAA 1

Khan wants to buy some office furniture that is valued at $7000.

    1. A store requires 25% deposit. Calculate the deposit.   (1 mark)
    2. The balance is to be paid in 24 equal monthly instalments. No interest is charged.
    3. Determine the amount of each instalment. Write your answer in dollars and cents.   (1 mark)

Another store offers the same $7000 office furniture for $500 deposit and 36 monthly instalments of $220.

 

    1. Determine the total amount paid for the furniture at this store.   (1 mark)
    2. Calculate the annual flat rate of interest charged by this store.
    3. Write your answer as a percentage correct to one decimal place.   (2 marks)

A third store has the office furniture marked at $7000 but will give 15% discount if payment is made in cash at the time of sale.

  1. Calculate the cash price paid for the furniture after the discount is applied.   (1 mark) 

Show Answers Only

  1. i.  `$1750`
  2. ii.`$218.75`
  3. i.  `$8420`
  4. ii.  `7.3text{%}`
  5. `$5950`

Show Worked Solution

a.i.    `text(Deposit)` `= 25text(%) xx 7000`
    `= $1750`

 
a.ii.   `text(Installment amount)`

`= ((7000-1750))/24`

`= $218.75`
 

b.i.    `text(Total paid)` `= 500 + 36 xx 220`
    `= $8420`

 

b.ii.   `text(Total interest paid)`

`= 8420-7000`

`= $1420`

 

`I` `= (PrT)/100`
`1420` `= (6500 xx r xx 3)/100`
`:. r` `= (1420 xx 100)/(6500 xx 3)`
  `= 7.282…`
  `= 7.3text{%  (1 d.p.)}`

 

c.    `text(Cash price)` `= 7000-15text(%) xx 7000`
    `= 7000-1050`
    `= $5950`

GRAPHS, FUR2 2007 VCAA 1

The Goldsmith family are going on a driving holiday in Western Australia.

On the first day, they leave home at 8 am and drive to Watheroo then Geraldton.

The distance––time graph below shows their journey to Geraldton.

GRAPHS, FUR2 2007 VCAA 1

At 9.30 am the Goldsmiths arrive at Watheroo.

They stop for a period of time.

  1. For how many minutes did they stop at Watheroo?  (1 mark)

After leaving Watheroo, the Goldsmiths continue their journey and arrive in Geraldton at 12 pm.

  1. What distance (in kilometres) do they travel between Watheroo and Geraldton?  (1 mark)
  2. Calculate the Goldsmiths'’ average speed (in km/h) when travelling between Watheroo and Geraldton.  (1 mark)

The Goldsmiths leave Geraldton at 1 pm and drive to Hamelin. They travel at a constant speed of 80 km/h for three hours. They do not make any stops.

  1. On the graph above, draw a line segment representing their journey from Geraldton to Hamelin.  (1 mark)

 

Show Answers Only
  1. `text(30 minutes)`
  2. `190 \ text(km)`
  3. `95\ text(km/hr)`
  4.  
    GRAPHS, FUR2 2007 VCAA 1 Answer
Show Worked Solution

a.   `text(30 minutes)`

 

b.   `text(Distance travelled)`

`= 310 – 120`

`= 190\ text(km)`

 

c.   `text(Time taken = 2 hours)`

`:.\ text(Average speed)` `= 190/2`
  `= 95\ text(km/hr)`

 

d.    GRAPHS, FUR2 2007 VCAA 1 Answer

 

GRAPHS, FUR2 2008 VCAA 1

Tiffany’s pulse rate (in beats/minute) during the first 60 minutes of a long-distance run is shown in the graph below.

GRAPHS, FUR2 2008 VCAA 1

  1. What was Tiffany’s pulse rate (in beats/minute) 15 minutes after she started her run?  (1 mark)
  2. By how much did Tiffany’s pulse rate increase over the first 60 minutes of her run?

     

    Write your answer in beats/minute.  (1 mark)

  3. The recommended maximum pulse rate for adults during exercise is determinded by subtracting the person’s age in years from 220.

     

    1. Write an equation in terms of the variables maximum pulse rate and age that can be used to determine a person’s recommended maximum pulse rate from his or her age.  (1 mark)

The target zone for aerobic exercise is between 60% and 75% of a person’s maximum pulse rate.

Tiffany is 20 years of age.

  1. Determine the values between which Tiffany’s pulse rate should remain so that she exercises within her target zone.

     

    Write your answers correct to the nearest whole number.  (1 mark)

Show Answers Only
  1. `110`
  2. `80`
    1. `text(Maximum pulse rate = 220 − age)`
    2. `text(Between 120 and 150)`
Show Worked Solution

a.   `text(110 beats/min)`

 

b.   `text(Increase of pulse rate)`

`=\ text(Rate at 60 min − initial rate)`

`= 150 – 70`

`= 80 \ text(beats/min)`

 

c.i.   `text(Maximum pulse rate = 220 − age)`

 

c.ii.   `text(Tiffany’s maximum pulse rate)`

`= 200 – 20`

`= 200\ text(beats/min)`

`=>\ text(Lower range) = 60text(%) xx 200 = 120`

`=>\ text(Higher range) = 75text(%) xx 200 = 150`

 

`:.\ text(Target range is 120 − 150 beats/min.)`

 

CORE*, FUR2 2009 VCAA 1

The recommended retail price of a golf bag is $500. Rebecca sees the bag discounted by $120 at a sale.

  1. What is the price of the golf bag after the $120 discount has been applied?   (1 mark)

  2. Find the discount as a percentage of the recommended retail price.   ( 1 mark)

Show Answers Only
  1. `$380`
  2. `text(24%)`
Show Worked Solution
a.    `text(Price)` `= 500-120`
    `= $380`

 

b.    `text(Discount)` `= 120/500 xx 100text(%)`
    `= 24text(%)`

 

GRAPHS, FUR2 2009 VCAA 1

Fair Go Airlines offers air travel between destinations in regional Victoria. 

Table 1 shows the fares for some distances travelled.

GRAPHS, FUR2 2009 VCAA 11

  1. What is the maximum distance a passenger could travel for $160?  (1 mark)

The fares for the distances travelled in Table 1 are graphed below.

GRAPHS, FUR2 2009 VCAA 13

  1. The fare for a distance longer than 400 km, but not longer than 550 km, is $280.

     

    Draw this information on the graph above.  (1 mark)

Fair Go Airlines is planning to change its fares.

A new fare will include a service fee of $40, plus 50 cents per kilometre travelled.

An equation used to determine this new fare is given by

fare = `40 + 0.5` × distance.

  1. A passenger travels 300 km.

     

    How much will this passenger save on the fare calculated using the equation above compared to the fare shown in Table 1?  (1 mark)

  2. At a certain distance between 250 km and 400 km, the fare, when calculated using either the new equation or Table 1, is the same.

     

    What is this distance?  (2 marks)

  3. An equation connecting the maximum distance that may be travelled for each fare in Table 1 on page 16 can be written as

     

             fare = `a` + `b` × maximum distance.

     

    Determine `a` and `b`.  (2 marks)

Show Answers Only
  1. `text(250 km)`
  2.  
    GRAPHS, FUR2 2009 VCAA 1 Answer
  3. `$30`
  4. `360\ text(km)`
  5. `b = 2/5\ text(and)\ a = 60`
Show Worked Solution

a.   `text(250 km)`

 

b.   

GRAPHS, FUR2 2009 VCAA 1 Answer

 

c.    `text(New fare)` `= 40 + 0.5 xx 300`
    `= $190`

 

`text(Fare from the table = $220`

`:.\ text(Passenger will save $30.)`

 

d.   `text(In table 1, a fare of $220 applies for travel)`

`text(between 250 – 400 km.)`

`:. 220` `= 40 + 0.5d`
`0.5d` `= 180`
`d` `= 360\ text(km)`

 

e.   `text(Equations in required form are:)`

`100` `= a + b xx 100\ \ …(1)`
`160` `= a + b xx 250\ \ …(2)`

 

`text(Subtract)\ (2) – (1)`

`60` `= 150b`
`:. b` `= 60/150 = 2/5`

 

`text(Substitute)\ b = 2/5\ text{into (1)}`

`100` `= a + 2/5 xx 100`
`:. a` `= 60`

 

GEOMETRY, FUR2 2010 VCAA 1

In the plan below, the entry gate of an adventure park is located at point `G`.

A canoeing activity is located at point `C`.

The straight path `GC` is 40 metres long.

The bearing of `C` from `G` is 060°.

 

Geometry anad Trig, FUR2 2011 VCAA 1_1
 

  1. Write down the size of the angle that is marked `x^@` in the plan above.  (1 mark)
  2. What is the bearing of the entry gate from the canoeing activity?  (1 mark)
  3. How many metres north of the entry gate is the canoeing activity?  (1 mark)

`CW` is a 90 metre straight path between the canoeing activity and a water slide located at point `W`.

`GW` is a straight path between the entry gate and the water slide.

The angle `GCW` is 120°.

 

GEOMETRY, FUR2 2010 VCAA 12
 

    1. Find the area that is enclosed by the three paths, `GC`, `CW` and `GW`.

       

      Write your answer in square metres, correct to one decimal place.  (1 mark)

    2. Show that the length of path `GW` is 115.3 metres, correct to one decimal place.  (1 mark)

Straight paths `CK` and `WK` lead to the kiosk located at point `K`.

These two paths are of equal length.

The angle `KCW` is 10°.

 

GEOMETRY, FUR2 2010 VCAA 13

    1. Find the size of the angle `CKW`.  (1 mark)
    2. Find the length of path `CK`, in metres, correct to one decimal place.  (1 mark)
Show Answers Only
  1. `120^@`
  2. `240^@`
  3. `20\ text(m)`
    1. `1558.8\ text{m²  (1 d.p.)}`
    2. `text(See Worked Solutions.)`
    1. `160^@`
    2. `45.7\ text{m  (1 d.p.)}`
Show Worked Solution
a.    `x^@ + 60^@` `= 180^@`
   `:. x^@` `= 120^@`

 

b.   `text(Bearing of)\ G\ text(from)\ C`

MARKER’S COMMENT: True bearings (3 figure) are preferred to quadrant bearings although S60°W was accepted.

`= 360 – 120`

`= 240^@`

 

c.   

GEOMETRY, FUR2 2010 VCAA 1 Answer1

`text(Let)\ d = text(distance north of)\ C\ text(from)\ G`

`cos60^@` `= d/40`
`:. d` `= 40 xx cos60`
  `= 20\ text(m)`

 

d.i.    `A` `= 1/2ab sinC`
    `= 1/2 xx 40 xx 90 xx sin120^@`
    `= 1558.84…`
    `= 1558.8\ text{m²  (1 d.p.)}`

 

d.ii.   `text(Using the cosine rule,)`

`GW` `= sqrt(40^2 + 90^2 + 2 xx 40 xx 90 xx cos120^@)`
  `= sqrt(13\ 300)`
  `= 115.32…`
  `= 115.3\ text{(1 d.p.)  …as required.}`

 

e.i.   `DeltaCKW\ text(is isosceles)`

`:. angleCKW` `= 180 – (2 xx 10)`
  `= 160^@`

 

e.ii.   `text(Using the sine rule,)`

`(CK)/(sin10^@)` `= 90/(sin160^@)`
`:. CK` `= (90 xx sin10^@)/(sin160^@)`
  `= 45.69…`
  `= 45.7\ text{m  (1 d.p.)}`

CORE*, FUR2 2011 VCAA 1

Tony plans to take his family on a holiday.

The total cost of $3630 includes a 10% Goods and Services Tax (GST).

  1. Determine the amount of GST that is included in the total cost.  (1 mark)

During the holiday, the family plans to visit some theme parks.

The prices of family tickets for three theme parks are shown in the table below.

BUSINESS, FUR2 2011 VCAA 1

  1. What is the total cost for the family if it visits all three theme parks?  (1 mark)

If Tony purchases the Movie Journey family ticket online, the cost is discounted to $202.40

  1. Determine the percentage discount.  (1 mark)
Show Answers Only
  1. `$330`
  2. `$462`
  3. `text(8%)`
Show Worked Solution

a.   `text(Let $)P\ text(be the cost ex-GST)`

`P + 10text(%)P` `= 3630`
`1.1P` `= 3630`
`P` `= 3630/1.1`
  `= $3300`
   
`:.\ text(GST)` `= 10text(%) xx $3300`
  `= $330`

 

b.   `text(C)text(ost to visit all 3 parks)`

`= 82 + 220 + 160`

`= $462`

 

c.    `text(Savings)` `= 220 – 202.40`
    `= 17.60`
`:.\ text(Discount)` `= (17.60)/220`
  `= 0.08`
  `= 8text(%)`

CORE*, FUR2 2012 VCAA 1

A club purchased new equipment priced at $8360. A 15% deposit was paid.

  1. Calculate the deposit.   (1 mark)

    1. Determine the amount of money that the club still owes on the equipment after the deposit is paid.   (1 mark)
    2. The amount owing will be fully repaid in 12 installments of $650.
    3. Determine the total interest paid.   (1 mark)

Show Answers Only

  1. `$1254`
  2. i.  `$7106`
  3. ii. `$684`

Show Worked Solution

a.    `text(Deposit)` `= 15text(%) xx 8360`
    `= $1254`

 

b.i.    `text(Amount still owed)` `= 8360-1254`
    `= $7106`

 

b.ii.    `text(Total repayments)` `= 12 xx 650`
    `= $7800`

 
`:.\ text(Total interest paid)`

`= 7800-7106`

`= $694`

NETWORKS, FUR1 2007 VCAA 3 MC

Consider the following graph.
 

 
 

An adjacency matrix that could be used to represent this graph is

A.   `[(0,2,0,1), (2,0,1,1), (0,1,0,1), (1,1,1,0)]` B.   `[(0,2,0,1), (0,0,1,1), (0,0,0,1), (0,0,0,0)]`
       
C.   `[(0,1,0,1), (2,0,0,1), (0,1,0,1), (1,1,1,0)]` D.   `[(0,2,0,1), (0,1,1,1), (0,1,1,1), (0,1,1,1)]`
       
E.   `[(1,2,0,1), (2,1,0,1), (0,1,1,0), (0,0,1,1)]`    
Show Answers Only

`A`

Show Worked Solution

`=>  A`

NETWORKS, FUR1 2010 VCAA 3 MC

`{:(\ quad A\ quad B\ quad C\ quad D\ quad E), ([(0, 1, 0, 0, 1), (1, 0, 1, 0, 1), (0, 1, 0, 1, 2), (0, 0, 1, 0, 1), (1, 1, 2, 1, 0)] {:(A), (B), (C), (D), (E):}):}` 

 

A graph that can be drawn from the adjacency matrix above is

vcaa-networks-fur1-2010-3ai

vcaa-networks-fur1-2010-3aii

vcaa-networks-fur1-2010-3aiii

Show Answers Only

`B`

Show Worked Solution

`text(From the matrix, we can see that)\ C and E`

`text(have 2 parallel edges.)`

`:.\ text(Eliminate choices)\ C, D,\ text(and)\ E.`

 

`text(An edge exists between)\ B and E.`

`:.\ text(Eliminate)\ A.`

`=>  B`

NETWORKS, FUR1 2006 VCAA 2 MC

The following directed graph represents a series of one-way streets with intersections numbered as nodes 1 to 8.
 

networks-fur1-2006-vcaa-2-mc-1
 

All intersections can be reached from

A.   intersection 4

B.   intersection 5

C.   intersection 6

D.   intersection 7

E.   intersection 8 

Show Answers Only

`B`

Show Worked Solution

`rArr B`

NETWORKS, FUR2 2006 VCAA 1

George, Harriet, Ian, Josie and Keith are a group of five musicians. 

They are forming a band where each musician will fill one position only. 

The following bipartite graph illustrates the positions that each is able to fill.

 

NETWORKS, FUR2 2006 VCAA 1
 

  1. Which musician must play the guitar?   (1 mark)

  2. Complete the table showing the positions that the following musicians must fill in the band.   (2 marks)

     

       
      NETWORKS, FUR2 2006 VCAA 11

Show Answers Only
  1. `text(George)`
  2.  
    networks-fur2-2006-vcaa-1-answer
Show Worked Solution

a.    `text(Harriet must play the drums, which means that)`

`text(George will play the guitar.)`

 

b.    networks-fur2-2006-vcaa-1-answer

NETWORKS, FUR1 2015 VCAA 5 MC

The graph below represents a friendship network. The vertices represent the four people in the friendship network: Kwan (`K`), Louise (`L`), Milly (`M`) and Narelle (`N`).

An edge represents the presence of a friendship between a pair of these people. For example, the edge connecting `K` and `L` shows that Kwan and Louise are friends.
 

 
Which one of the following graphs does not contain the same information?

 

 

 

Show Answers Only

`D`

Show Worked Solution

`=> D`

NETWORKS, FUR2 2009 VCAA 3

The city of Robville contains eight landmarks denoted as vertices `N` to `U` on the network diagram below. The edges on this network represent the roads that link the eight landmarks.
 


  

  1. Write down the degree of vertex `U`.  (1 mark)

  2. Steven wants to visit each landmark, but drive along each road only once. He will begin his journey at landmark `N`.
    1. Michael was the best player in 2014 and he considered purchasing cricket equipment that was valued at $750.
    2. At which landmark must he finish his journey?   (1 mark)
    3. Regardless of which route Steven decides to take, how many of the landmarks (including those at the start and finish) will he see on exactly two occasions?   (1 mark)
  3. Cathy decides to visit each landmark only once.
    1. Suppose she starts at `S`, then visits `R` and finishes at `T`.
    2. Write down the order Cathy will visit the landmarks.   (1 mark)

    3. Suppose Cathy starts at `S`, then visits `R` but does not finish at `T`.
    4. List three different ways that she can visit the landmarks.   (1 mark)

Show Answers Only

  1. `4`
    1. `P\ text{(the other odd degree vertex)}`
    2. `5`
    1. `(SR)QPONU(T)`
    2. `text(Other paths are)`
      `(SR)QPUTNO`
      `(SR)QPONTU`
      `(SR)TUNOPQ`
      `(SR)UTNOPQ`
      `text{(only 3 paths required)}`

Show Worked Solution

a.   `4`

b.i.   `P\ text{(the other odd degree vertex)}`

b.ii.   `5 (N, T, R, P, U)`

c.i.   `(SR)QPONU(T)`

c.ii.   `text(Other paths are)`

`(SR)QPUTNO`

`(SR)QPONTU`

`(SR)TUNOPQ`

`(SR)UTNOPQ`

`text{(only 3 paths required)}`

NETWORKS, FUR2 2011 VCAA 1

Aden, Bredon, Carrie, Dunlop, Enwin and Farnham are six towns.

The network shows the road connections and distances between these towns in kilometres.

 

  1. In kilometres, what is the shortest distance between Farnham and Carrie?   (1 mark)

  2. How many different ways are there to travel from Farnham to Carrie without passing through any town more than once?   (1 mark)

An engineer plans to inspect all of the roads in this network.

He will start at Dunlop and inspect each road only once.

  1. At which town will the inspection finish?   (1 mark)

Another engineer decides to start and finish her road inspection at Dunlop.

If an assistant inspects two of the roads, this engineer can inspect the remaining six roads and visit each of the other five towns only once.

  1. How many kilometres of road will the assistant need to inspect?   (1 mark)

Show Answers Only
  1. `200\ text(km)`
  2. `6`
  3. `text(Bredon)`
  4. `240\ text(km)`
Show Worked Solution

a.   `text{Farnham to Carrie (shortest)}`

`= 60 + 140`

`= 200\ text(km)`
  

b.   `text(Different paths are)`

`FDC, FEDC, FEBC,`

`FEABC, FDEBC,`

`FDEABC`

`:. 6\ text(different ways)`
  

c.   `text(A possible path is)\ DFEABCDEB\ text(and will finish)`

`text{at Bredon (the other odd-degree vertex).}`
  

d.   `text(If the engineer’s path is)`

`DFEABCD,`

`text(Distance assistant inspects)`

`= 110 + 130`

`= 240\ text(km)`

NETWORKS, FUR2 2013 VCAA 1

The vertices in the network diagram below show the entrance to a wildlife park and six picnic areas in the park: `P1`, `P2`, `P3`, `P4`, `P5` and `P6`.

The numbers on the edges represent the lengths, in metres, of the roads joining these locations.

 

 

  1. In this graph, what is the degree of the vertex at the entrance to the wildlife park?   (1 mark)

  2. What is the shortest distance, in metres, from the entrance to picnic area `P3`?   (1 mark)

  3. A park ranger starts at the entrance and drives along every road in the park once.
  4. i. At which picnic area will the park ranger finish?   (1 mark)

  5. ii. What mathematical term is used to describe the route the park ranger takes?   (1 mark)

  6. A park cleaner follows a route that starts at the entrance and passes through each picnic area once, ending at picnic area `P1`.

     

    Write down the order in which the park cleaner will visit the six picnic areas.   (1 mark)

Show Answers Only
  1. `3`
  2. `1000\ text(m)`
  3. i. `P4`
    ii. `text(Euler path)` 
  4. `E-P5-P4-P6-P3-P2-P1`
Show Worked Solution

a.   `3`
  

b.   `text( Shortest distance)`

`= E-P1-P3`

`= 600 + 400`

`= 1000\ text(m)`
  

c.i.   `text(A route could be)`

`E-P1-P2-P3-P4-P5`

`-E-P6-P1-P3-P6-P4`

`:.\ text(Finish at)\ P4\ \ text{(the other odd degree vertex)}`
  

c.ii.   `text(Euler path)`
  

d.   `E-P5-P4-P6-P3-P2-P1`

MATRICES, FUR2 2006 VCAA 1

A manufacturer sells three products, `A`, `B` and `C`, through outlets at two shopping centres, Eastown (`E`) and Noxland (`N`). 

The number of units of each product sold per month through each shop is given by the matrix `Q`, where

`{:((qquadqquadqquad\ A,qquadquadB,qquad\ C)),(Q=[(2500,3400,1890),(1765,4588,2456)]{:(E),(N):}):}`

  1. Write down the order of matrix `Q`.   (1 mark)

The matrix `P`, shown below, gives the selling price, in dollars, of products `A`, `B`, `C`.

`P = [(14.50),(21.60),(19.20)]{:(A),(B),(C):}`

  1.   i. Evaluate the matrix `M`, where `M = QP`.   (1 mark)

  2.  ii. What information does the elements of matrix `M` provide?   (1 mark)

  3. Explain why the matrix `PQ` is not defined.   (1 mark)

Show Answers Only
  1. `2 xx 3`
    1. `M = QP = [(135\ 320.5),(171\ 848.5)]`
    2. `text(The total of selling products)\ A, B,and C`
      `text(at each of Eastown and Noxland.)`
  3. `PQ\ text(is not defined because the number of)`
    `text(columns in)\ P !=\ text(the number of rows in)\ Q.`
Show Worked Solution

a.   `2 xx 3`
 

b.i.    `M` `= QP`
    `= [(2500,3400,1890),(1765,4588,2456)][(14.50),(21.60),(19.20)]`
    `= [(135\ 320.5),(171\ 848.5)]`

 
b.ii.  `text(The total revenue from selling products)\ A, B,`

   `text(and)\ C\ text(at each of Eastown and Noxland.)`
 

c.   `PQ\ text(is not defined because the number of)`

`text(columns in)\ P !=\ text(the number of rows in)\ Q.`

NETWORKS, FUR2 2010 VCAA 1

The members of one team are Kristy (`K`), Lyn (`L`), Mike (`M`) and Neil (`N`). 

In one of the challenges, these four team members are only allowed to communicate directly with each other as indicated by the edges of the following network.
 

Network, FUR2 2011 VCAA 1
 

The adjacency matrix below also shows the allowed lines of communication.

`{:(quadKquadLquadMquadN),([(0,1,0,0),(1,0,1,0),(0,f,0,1),(0,g,1,0)]{:(K),(L),(M),(N):}):}`

 

  1. Explain the meaning of a zero in the adjacency matrix.   (1 mark)

  2. Write down the values of `f` and `g` in the adjacency matrix.   (1 mark)

Show Answers Only
  1. `text(No direct communication is allowed.)`
  2. `f = 1, g = 0`
Show Worked Solution

a.   `text(No direct communication is allowed.)`
  

b.   `f = 1, g = 0`

MATRICES, FUR2 2009 VCAA 1

Three types of cheese, Cheddar (`C`), Gouda (`G`) and Blue (`B`), will be bought for a school function.

The cost matrix `P` lists the prices of these cheeses, in dollars, at two stores, Foodway and Safeworth.
 

`P = [(6.80, 5.30, 6.20),(7.30, 4.90, 6.15)]{:(text(Foodway)),(text(Safeworth)):}`
 

  1. What is the order of matrix `P`?   (1 mark)

The number of packets of each type of cheese needed is listed in the quantity matrix `Q`.
 

`Q = [(8),(11),(3)]{:(C),(G),(B):}`
 

    1. Evaluate the matrix  `W = PQ`.   (1 mark)
    2. At which store will the total cost of the cheese be lower?   (1 mark)

Show Answers Only

  1. `2 xx 3`
    1. `W = PQ = [(131.30),(130.75)]`
    2. `text(Safeworth)`

Show Worked Solution

a.   `2 xx 3`

 

b.i.    `W` `=PQ`
    `= [(6.80,5.30,6.20),(7.30,4.90,6.15)][(8),(11),(3)]`
    `= [(131.30),(130.75)]`

 

b.ii.   `text(Safeworth)`

MATRICES, FUR2 2010 VCAA 1

In a game of basketball, a successful shot for goal scores one point, two points, or three points, depending on the position from which the shot is thrown.

`G`  is a column matrix that lists the number of points scored for each type of successful shot.

`G = [(1),(2),(3)]`

In one game, Oscar was successful with

    • 4 one-point shots for goal
    • 8 two-point shots for goal
    • 2 three-point shots for goal.
  1. Write a row matrix, `N`, that shows the number of each type of successful shot for goal that Oscar had in that game.   (1 mark)

  2. Matrix `P` is found by multiplying matrix `N` with matrix `G` so that  `P = N xx G`
  3. Evaluate matrix `P`.   (1 mark)

  4. In this context, what does the information in matrix `P` provide?   (1 mark)

Show Answers Only
  1. `N = [(4, 8, 2)]`
  2. `P = [26]`
  3. `text(The total points scored by Oscar in the game.)`
Show Worked Solution

a.   `N = [(4, 8, 2)]`
 

b.    `P` `= NG`
    `= [(4, 8, 2)][(1),(2),(3)]`
    `= [26]`

 
c.   `text(The total points scored by Oscar in the game.)`

MATRICES, FUR2 2011 VCAA 1

The diagram below shows the feeding paths for insects (`I`), birds (`B`) and lizards (`L`). The matrix `E` has been constructed to represent the information in this diagram. In matrix `E`, a 1 is read as "eat" and a  0  is read as "do not eat".
 

MATRICES, FUR2 2011 VCAA 11

  1. Referring to insects, birds or lizards
  2.  i. what does the 1 in column `B`, row `L`, of matrix `E` indicate?   (1 mark)

  3. ii. what does the row of zeros in matrix `E` indicate?   (1 mark)

The diagram below shows the feeding paths for insects (`I`), birds (`B`), lizards (`L`) and frogs (`F`).

The matrix `Z` has been set up to represent the information in this diagram.

Matrix `Z` has not been completed.
 

MATRICES, FUR2 2011 VCAA 12

  1. Complete the matrix `Z` above by writing in the seven missing elements.   (1 mark)

Show Answers Only

a.i.    `text(Birds eat lizards)`

a.ii.   `text(Insects, birds or lizards do not eat birds)`

b.
      `{:((qquadqquadquadI,B,L,F)),(Z = [(0,1,1,1),(0,0,0,0),(0,1,0,0),(0,1,1,0)]{:(I),(B),(L),(F):}):}`

Show Worked Solution

a.i.   `text(The 1 represents that birds eat lizards.)`
 

a.ii.   `text(It indicates that insects, birds or lizards DO NOT)`

   `text(eat birds.)`
 

b.    `{:((qquadqquadquadI,B,L,F)),(Z = [(0,1,1,1),(0,0,0,0),(0,1,0,0),(0,1,1,0)]{:(I),(B),(L),(F):}):}`

MATRICES, FUR2 2013 VCAA 1

Five trout-breeding ponds, `P`, `Q`, `R`, `X` and `V`, are connected by pipes, as shown in the diagram below.
 

Matrices, FUR2 2013 VCAA 1 

The matrix `W` is used to represent the information in this diagram.

`{:({:\ qquadqquadqquadPquadQquad\ Rquad\ Xquad\ V:}),(W = [(0,1,1,1,0), (1,0,0,1,0),(1,0,0,1,0),(1,1,1,0,1),(0,0,0,1,0)]):}{:(),(P),(Q),(R),(X),(V):}`

In matrix `W`

•  the 1 in column 1, row 2, for example, indicates that a pipe directly connects pond `P` and pond `Q`

•  the 0 in column 1, row 5, for example, indicates that pond `P` and pond `V` are not directly connected by a pipe.

  1. Find the sum of the elements in row 3 of matrix `W`.   (1 mark)

  2. In terms of the breeding ponds described, what does the sum of the elements in row 3 of matrix `W` represent?   (1 mark)

The pipes connecting pond `P` to pond `R` and pond `P` to pond `X` are removed.

Matrix `N` will be used to show this situation. However, it has missing elements.

  1. Complete matrix `N` below by filling in the missing elements in row 1 and column 1.   (1 mark)

        
             Matrices, FUR2 2013 VCAA 1_c

Show Answers Only
  1. `2`
  2. `text(The sum means that 2 other)`
    `text(ponds connect directly to pond)\ R.`
  3.  
    `{:({:qquadqquadqquadPquadQquadRquadXquadV:}),(N = [(0,1,0,0,0),(1,0,0,1,0),(0,0,0,1,0),(0,1,1,0,1),(0,0,0,1,0)]):}{:(),(P),(Q),(R),(X),(V):}`
Show Worked Solution

a.   `1+ 0 + 0 +1+ 0 = 2`
 

b.   `text(The sum means that 2 other ponds)`

`text(connect directly to pond)\ R.`
 

c.    `{:({:qquadqquadqquadPquadQquad\ Rquad\ Xquad\ V:}),(N = [(0,1,0,0,0),(1,0,0,1,0),(0,0,0,1,0),(0,1,1,0,1),(0,0,0,1,0)]):}{:(),(P),(Q),(R),(X),(V):}`

MATRICES, FUR1 2009 VCAA 1 MC

`3[[2,1],[0,3]] + 2[[−1,0],[2,−7]]` equals
 

A.     `[[4,3],[4,−5]]`

 

B.   `6[[1,1],[2,−4]]`

 

C.     `[[4,3],[4,2]]`

 

D.   `5[[1,1],[2,−4]]`

 

E.     `[[3,6],[7,4]]`

Show Answers Only

`A`

Show Worked Solution

`3[(2,1),(0,3)] + 2[(−1,0),(2,−7)]`

`= [(6,3),(0,9)] + [(−2,0),(4,−14)]`

`= [(4,3),(4,−5)]`

`=>  A`

MATRICES, FUR1 2007 VCAA 2 MC

The number of tourists visiting three towns, Oldtown, Newtown and Twixtown, was recorded for three years. 

The data is summarised in the table below.

MATRICES, FUR1 2007 VCAA 2 MC

The `3 xx 1` matrix that could be used to show the number of tourists visiting the three towns in the year 2005 is

A.   `[(975, 1002, 1390)]` B.   `[(1002, 1081, 1095)]`
   
C.   `[(975), (1002), (1390)]` D.   `[(1002), (1081), (1095)]`
   
E.   `[(975, 1002, 1390), (2105, 1081, 1228), (610, 1095, 1380)]`  

 

Show Answers Only

`D`

Show Worked Solution

`=>  D`

MATRICES, FUR1 2011 VCAA 2 MC

If  `A=[(0,1),(1,0)],\ B=[(1),(0)]`  and  `C= [(0),(1)]`, then  `AB + 2C` equals

 

A.   `[(0),(3)]`

 

B.   `[(3),(0)]`

 

C.   `[(1),(2)]`

 

D.   `[(2),(0)]`

 

E.   `[(2),(3)]`

Show Answers Only

`A`

Show Worked Solution
`AB + 2C` `= [(0,1),(1,0)][(1),(0)] + 2[(0),(1)]`
  `= [(0),(1)] + [(0),(2)]`
  `= [(0),(3)]`

`=> A`

MATRICES, FUR1 2011 VCAA 1 MC

The matrix below shows the airfares (in dollars) that are charged by Zeniff Airlines to fly between Adelaide (`A`), Melbourne (`M`) and Sydney (`S`).

`{:(qquadqquadquadtext(from)),({:(qquadA,\ M,\ S):}),([(0,85,89),(85,0,99),(97,101,0)]):}{:(),(),(A),(M),(S):}{:(),(),(),(qquadtext(to)),():}`

 

The cost to fly from Melbourne to Sydney with Zeniff Airlines is

A.     `$85`

B.     `$89`

C.     `$97`

D.     `$99`

E.   `$101`

Show Answers Only

`E`

Show Worked Solution

`=> E`

GRAPHS, FUR2 2012 VCAA 1

The cost, `C`, in dollars, of making `n` phones, is shown by the line in the graph below.

GRAPHS, FUR2 2012 VCAA 1

    1. Calculate the gradient of the line, `C`, drawn above.  (1 mark)
    2. Write an equation for the cost, `C`, in dollars, of making `n` phones.  (1 mark)

  2. The revenue, `R`, in dollars, obtained from selling `n` phones is given by  `R = 150n`.

    1. Draw this line on the graph above.  (1 mark)
    2. How many phones would need to be sold to obtain $54 000 in revenue?  (1 mark)
  3. Determine the number of phones that would need to be made and sold to break even.  (1 mark) 
Show Answers Only
    1. `50`
    2. `C = 20\ 000 + 50n`
    1. `text(See Worked Solutions)`
    2. `360`
  3. `200`
Show Worked Solution

a.i.   `text{Using (0, 20 000) and (300, 35 000)}`

`text(Gradient)` `= (y_2 – y_1)/(x_2 – x_1)`
  `= (35\ 000 – 20\ 000)/(300 – 0)`
  `= 50`

 

a.ii.   `C = 50n + 20\ 000`

 

b.i.    GRAPHS, FUR2 2012 VCAA 1 Answer

 

b.ii.    `54\ 000` `= 150n`
  `:. n` `= (54\ 000)/150`
    `= 360`

`:. 360\ text(planes need to be sold.)`

 

c.   `text(Breakeven occurs when)`

`text(Revenue)` `=\ text(C)text(osts)`
`150n` `= 50n + 20\ 000`
`100n` `= 20\ 000`
`:. n` `= 200\ text(phones)`

CORE*, FUR2 2013 VCAA 1

Hugo is a professional bike rider.

The value of his bike will be depreciated over time using the flat rate method of depreciation.

The graph below shows his bike’s initial purchase price and its value at the end of each year for a period of three years.
 

  1. What was the initial purchase price of the bike?   (1 mark)
    1. Show that the bike depreciates in value by $1500 each year.   (1 mark)
    2. Assume that the bike’s value continues to depreciate by $1500 each year.
    3. Determine its value five years after it was purchased.   (1 mark)

The unit cost method of depreciation can also be used to depreciate the value of the bike.

In a two-year period, the total depreciation calculated at $0.25 per kilometre travelled will equal the depreciation calculated using the flat rate method of depreciation as described above.

  1. Determine the number of kilometres the bike travels in the two-year period.   (1 mark)

Show Answers Only

  1. `$8000`
  2. i.  `text(See Worked Solutions)`
    ii. `$500`
  3. `12\ 000\ text(km)`

Show Worked Solution

a.   `$8000`
  

b.i.   `text(Value after 1 year) = $6500`

`:.\ text(Annual depreciation)` `= 8000-6500`
  `= $1500`

  
b.ii.   `text(Value after)\ n\ text(years)`

`= 8000-1500n`

`:.\ text(After 5 years,)`

`text(Value)` `= 8000-1500 xx 5`
  `= $500`

  
c.   `text(After 2 years,)`

`text(Depreciation)` `= 2 xx 1500`
  `= $3000`

  
`:.\ text(Distance travelled)`

`= 3000/0.25`

`= 12\ 000\ text(km)`

GRAPHS, FUR2 2013 VCAA 2

Students at the camp can participate in two different watersport activities: canoeing and surfing.

The cost of canoeing is $30 per hour and the cost of surfing is $20 per hour.

The budget allows each student to spend up to $200, in total, on watersport activities.

The way in which a student decides to spend the $200 is described by the following inequality.

30 × hours canoeing + 20 × hours surfing ≤ 200

  1. Hillary wants to spend exactly two hours canoeing during the camp.

     

    Calculate the maximum number of hours she could spend surfing.  (1 mark)

  2. Dennis would like to spend an equal amount of time canoeing and surfing.

     

    If he spent a total of $200 on these activities, determine the maximum number of hours he could spend on each activity.  (1 mark)

Show Answers Only
  1. `7`
  2. `4\ text(hours)`
Show Worked Solution

a.   `text(Hillary spends 2 hrs canoeing)`

`text(Let)\ x = text(hours spent surfing)`

`30 xx 2 + 20x` `= 200`
`20x` `= 140`
`x` `= 7`

`:.\ text(Maximum hours surfing = 7)`

 

b.   `text(Let)\ t = text(time spent canoeing and surfing)`

`30 xx t + 20 xx t` `= 200`
`50t` `= 200`
`t` `= 4\ text(hours)`

`:.\ text(Dennis could spend a maximum)`

`text(of 4 hours on each activity.)`

GRAPHS, FUR2 2013 VCAA 1

The distance-time graph below shows the first two stages of a bus journey from a school to a camp.

GRAPHS, FUR2 2013 VCAA 1

  1. At what constant speed, in kilometres per hour, did the bus travel during stage 1 of the journey?  (1 mark)
  2. For how many minutes did the bus stop during stage 2 of the journey?  (1 mark)

The third stage of the journey is missing from the graph.

During stage 3, the bus continued its journey to the camp and travelled at a constant speed of 60 km/h for one hour.

  1. Draw a line segment on the graph above to represent stage 3 of the journey.  (1 mark)
  2. Find the average speed of the bus over the three hours.

    Write your answer in kilometres per hour.  (1 mark)

The distance, `D` km, of the bus from the school, `t` hours after departure is given by

`D = {(100t                      0 ≤ t ≤ 1.5), (150                        1.5 ≤ t ≤ 2), (60t + k            2≤ t ≤ 3):}`

  1. Determine the value of `k`.  (1 mark)
Show Answers Only
  1. `text(100 km/hr)`
  2. `text(30 minutes)`
  3. `text(See Worked Solutions)`
  4. `70\ text(km/hr)`
  5. `30`
Show Worked Solution

a.   `text(100 km/hr)`

 

b.   `text(30 minutes)`

 

c.    GRAPHS, FUR2 2013 VCAA 1 Answer

 

d.   `text(Average speed over 3 hours)`

♦ Mean mark of parts (c) and (d) consumed was 46%.

`= (text(Total distance))/3`

`= 210/3`

`= 70\ text(km/hr)`

 

e.   `text(When)\ t = 2,`

`D = 150,\ text(and)`

`D = 60 xx 2 + k`

`:. 120 + k` `= 150`
`:. k` `= 30`

GRAPHS, FUR2 2015 VCAA 1

Ben is flying to Japan for a school cultural exchange program.

The graph below shows the cost of a particular flight to Japan, in dollars, on each day in February.

Graphs, FUR2 2015 VCAA 1

  1. What is the cost, in dollars, of this flight to Japan on 19 February?  (1 mark)
  2. On how many days in February is the cost of this flight to Japan more than $1000?  (1 mark) 
Show Answers Only
  1. `$1200`
  2. `8`
Show Worked Solution

a.   `$1200`

b.   `8`

Algebra, MET2 2014 VCAA 1

The population of wombats in a particular location varies according to the rule  `n(t) = 1200 + 400 cos ((pi t)/3)`, where `n` is the number of wombats and `t` is the number of months after 1 March 2013.

  1. Find the period and amplitude of the function `n`.   (2 marks)

  2. Find the maximum and minimum populations of wombats in this location.   (2 marks)

  3. Find  `n(10)`.   (1 mark)

  4. Over the 12 months from 1 March 2013, find the fraction of time when the population of wombats in this location was less than  `n(10)`.   (2 marks)

Show Answers Only
  1. `text(Period) = text(6 months);\ text(Amplitude) = 400`
  2. `text(Max) = 1600;\ text(Min) = 800`
  3. `1000`
  4. `1/3`
Show Worked Solution

a.   `text(Period) = (2pi)/n = (2pi)/(pi/3) = 6\ text(months)`

MARKER’S COMMENT: Expressing the amplitude as [800,1600] in part (a) is incorrect.

`text(A)text(mplitude) = 400`
  

b.   `text(Max:)\ 1200 + 400 = 1600\ text(wombats)`

`text(Min:)\ 1200-400 = 800\ text(wombats)`
  

c.   `n(10) = 1000\ text(wombats)`
   

d.    `text(Solve)\ n(t)` `= 1000\ text(for)\ t ∈ [0,12]`

`t= 2,4,8,10`

`text(S)text(ince the graph starts at)\ \ (0,1600),`

♦ Mean mark 48%.

`=> n(t) < 1000\ \ text(for)`

`t ∈ (2,4)\ text(or)\ t ∈ (8,10)`

`:.\ text(Fraction)` `= ((4-2) + (10-8))/12`
  `= 1/3\ \ text(year)`

GRAPHS, FUR2 2006 VCAA 1

Harry operates a mobile pet care service. The call-out fee charged depends on the distance he has to travel to tend to a pet. The call-out fees for distances up to 30 km are shown on the graph below.

GRAPHS, FUR2 2006 VCAA 1

  1. According to this graph

     

    1. what is the call-out fee to travel a distance of 20 km?  (1 mark)
    2. what is the maximum distance travelled for a call-out fee of $10?  (1 mark)

A call-out fee of $50 is charged to travel distances of more than 30 km but less than or equal to 40 km.

  1. Draw this information on the graph above.  (1 mark) 
Show Answers Only
    1. `$30`
    2. `5\ text(km)`
  2.  
Show Worked Solution

a.i.   `$30`

 

a.ii.   `text(5 km)`

 

b.   

Calculus, MET2 2015 VCAA 1

Let  `f: R -> R,\ \ f(x) = 1/5 (x-2)^2 (5-x)`. The point  `P(1, 4/5)`  is on the graph of  `f`, as shown below.

The tangent at `P` cuts the y-axis at `S` and the x-axis at `Q.`

VCAA 2015 1ai

  1. Write down the derivative  `f^{prime} (x)` of `f (x)`.   (1 mark)

  2.  i. Find the equation of the tangent to the graph of  `f` at the point  `P(1, 4/5)`.   (1 mark)

    ii. Find the coordinates of points `Q` and `S`.   (2 marks)

  3. Find the distance `PS` and express it in the form  `sqrt b/c`, where `b` and `c` are positive integers.  (2 marks)

VCAA 2015 1di

  1. Find the area of the shaded region in the graph above.   (3 marks)

Show Answers Only
  1. `-3/5(x-4)(x-2)`
  2.  i.`y = -9/5x + 13/5`
    ii. `S (0, 13/5), \ \ Q (13/9, 0)`
  3. `sqrt 106/5`
  4. `108/5\ text(units²)`
Show Worked Solution
a.    `f(x)` `= 1/5 (x-2)^2 (5-x)`
  `f^{prime}(x)` `=1/5 xx 2(x-2)(5-x)-1/5 (x-2)^2`
    `= -3/5(x-4)(x-2)`

 

b.i.   `text(Solution 1)`

`y = -9/5x + 13/5qquad[text(CAS:tangentLine)\ (f(x),x,1)]`
  

`text(Solution 2)`

`m_text(tan) = -9/5,\ \ text(through)\ \ (1, 4/5)`

`y-4/5` `=-9/5 (x-1)`
`y` `=-9/5 x +13/5`

  
b.ii.   `text(At)\ S,\ \ x=0`

`y =-9/5 xx 0 + 13/5 = 13/5`

`:. S(0,13/5)`
  

`text(At)\ Q,\ \ y=0`

`0` `=-9/5x + 13/5`
`x` `=13/5 xx 5/9=13/9`

 
`:. Q(13/9,0)`
  

c.   `P(1, 4/5),\ \ S(0,13/5)`

`text(dist)\ PS` `=sqrt((x_2-x_1)^2 + (y_2-y_1)^2)`
  `= sqrt((1-0)^2 + (4/5-13/5)^2)`
  `= (sqrt106)/5`

 

d.   `text(Find intersection pts of)\ SQ\ text(and)\ f(x),`

MARKER’S COMMENT: Many students complicated their answer by splitting up the area.

`text{Solve (using technology):}`

`1/5 (x-2)^2 (5-x) =-9/5x + 13/5`

`x = 1,7`

`:.\ text(Area)` `= int_1^7(f(x)-(-9/5x + 13/5))dx`
  `= 108/5\ text(u²)`

GEOMETRY, FUR2 2007 VCAA 1-3

Question 1

Tessa is a student in a woodwork class.

The class will construct geometrical solids from a block of wood.

Tessa has a piece of wood in the shape of a rectangular prism.

This prism, `ABCDQRST`, shown in Figure 1, has base length 24 cm, base width 28 cm and height 32 cm.

GEOMETRY, FUR2 2007 VCAA 11 

On the front face of Figure 1, `ABRQ`, Tessa marks point `W` halfway between `Q` and `R` as shown in Figure 2 below. She then draws line segments `AW` and `BW` as shown.

GEOMETRY, FUR2 2007 VCAA 12

  1. Determine the length, in cm, of `QW`.  (1 mark)
  2. Calculate the angle `WAQ`. Write your answer in degrees, correct to one decimal place.  (1 mark)
  3. Calculate the angle `AWB` correct to one decimal place.  (1 mark)
  4. What fraction of the area of the rectangle `ABRQ` does the area of the triangle `AWB` represent?  (1 mark)

 

Question 2

Tessa carves a triangular prism from her block of wood.

Using point `V`, halfway between `T` and `S` on the back face, `DCST`, of Figure 1, she constructs the triangular prism shown in Figure 3.

GEOMETRY, FUR2 2007 VCAA 2

  1. Show that, correct to the nearest centimetre, length `AW` is 34 cm.  (1 mark)
  2. Using length `AW` as 34 cm, find the total surface area, in cm², of the triangular prism `ABCDWV` in Figure 3.  (2 marks)

 

Question 3

Tessa's next task is to carve the right rectangular pyramid `ABCDY` shown in Figure 4 below.

She marks a new point, `Y`, halfway between points `W` and `V` in Figure 3. She uses point `Y` to construct this pyramid.

GEOMETRY, FUR2 2007 VCAA 3

  1. Calculate the volume, in cm³, of the pyramid `ABCDY` in Figure 4.  (1 mark)
  2. Show that, correct to the nearest cm, length `AY` is 37 cm.  (2 marks)
  3. Using `AY` as 37 cm, demonstrate the use of Heron's formula to calculate the area, in cm², of the triangular face `YAB`.  (2 marks)
Show Answers Only

Question 1

  1. `12\ text(cm)`
  2. `20.6^@`
  3. `41.2^@\ text{(1 d.p.)}`
  4. `1/2`

Question 2

  1. `34\ text(cm)`
  2. `3344\ text(cm²)`

Question 3

  1. `7168\ text(cm³)`
  2. `37\ text(cm)`
  3. `420\ text(cm²)`
Show Worked Solution

`text(Question 1)`

a.    `QW` `= 1/2 xx QR`
    `= 1/2 xx 24`
    `= 12\ text(cm)`

 

b.   VCAA GEO FUR2 2007 1bi
`tan\ /_ WAQ` `= 12/32`
`:. /_ WAQ` `= tan^-1\ 3/8`
  `= 20.55…`
  `= 20.6^@\ text{(1 d.p.)}`

 

c.    `/_ AWB` `= 2 xx /_ WAG`
    `= 2 xx 20.6`
    `= 41.2^@\ text{(1 d.p.)}`

 

d.   `text(Area)\ ABRQ` `= 24 xx 32`
    `= 768\ text(cm²)`
  `text(Area)\ Delta AWB` `= 2 xx text(Area)\ Delta AQW`
    `= 2 xx 1/2 xx 12 xx 32`
    `= 384\ text(cm²)`

 

`:.\ text(Fraction of area)`

`= 384/768`

`= 1/2`

 

`text(Question 2)`

a.   VCAA GEO FUR2 2007 2ai

`text(Using Pythagoras in)\ Delta AWX,`

`AW` `= sqrt (12^2 + 32^2)`
  `= sqrt 1168`
  `= 34.176…`
  `= 34\ text{cm  (nearest cm)  …  as required.}`

 

b.   `text(Total S.A. of triangular prism)`

`= text(area of base) + text(area of sides) + text(area of ends)`

`= 24 xx 28 + 2 xx (34 xx 28) + 2 xx (1/2 xx 24 xx 32)`

`= 672 + 1904 + 768`

`= 3344\ text(cm²)`

 

`text(Question 3)`

a.    `V` `= 1/3 xx b xx h`
    `= 1/3 xx 24 xx 28 xx 32`
    `= 7168\ text(cm³)`

 

b.   VCAA GEO FUR2 2007 3bi

`text(Using Pythagoras in)\ Delta ABC,`

`AC` `= sqrt (24^2 + 28^2)`
  `= 36.878…`

 

`text(Let)\ Z\ text(be the midpoint of)\ AC`

`:. AZ = (36.878…)/2 = 18.439…`

VCAA GEO FUR2 2007 3bii

`text(Using Pythagoras in)\ Delta AYZ,`

`AY` `= sqrt (32^2 + (18.439…)^2)`
  `= sqrt (1364)`
  `= 36.9…`
  `= 37\ text{cm  (nearest cm)  …  as required}`

 

c.   `text(Using Heron’s formula,)`

`s` `= (37 + 37 + 24)/2=49`

 

`:.\ text(Area of)\ Delta YAB`

`= sqrt (49 (49 – 24) (49 – 37) (49 – 37))`

`= sqrt (49 xx 25 xx 12 xx 12)`

`= 420\ text(cm²)`

MATRICES, FUR2 2008 VCAA 1

Two subjects, Biology and Chemistry, are offered in the first year of a university science course. 

The matrix `N` lists the number of students enrolled in each subject.

`N = [(460),(360)]{:(text(Biology)),(text(Chemistry)):}`

The matrix `P` lists the proportion of these students expected to be awarded an `A`, `B`, `C`, `D` or `E` grade in each subject.

`{:((qquadqquadqquadA,qquad\ B,qquadquadC,qquad\ D,qquadE)),(P = [(0.05,0.125,0.175,0.45,0.20)]):}`

  1. Write down the order of matrix `P`.   (1 mark)

  2. Let the matrix  `R = NP`.
  3.  i. Evaluate the matrix `R`.   (1 mark)

  4. ii. Explain what the matrix element `R_24`  represents.   (1 mark)

  5. Students enrolled in Biology have to pay a laboratory fee of $110, while students enrolled in Chemistry pay a laboratory fee of $95.
  6.  i. Write down a clearly labelled row matrix, called `F`, that lists these fees.   (1 mark)

  7. ii. Show a matrix calculation that will give the total laboratory fees, `L`, paid in dollars by the students enrolled in Biology and Chemistry. Find this amount.   (1 mark)

Show Answers Only
  1. `1 xx 5`
    1. `[(23,57.5,80.5,207,92),(18,45,63,162,72)]`
    2. `text(See Worked Solutions)`
    1.  `text(See Worked Solutions)`
    2. `text(See Worked Solutions)`
Show Worked Solution

a.   `1 xx 5`
 

b.i.   `R` `= NP`
    `= [(460), (360)] [0.05 quad 0.125 quad 0.175 quad 0.45 quad 0.20]`
    `= [(23,57.5,80.5,207,92),(18,45,63,162,72)]`

 
b.ii. `R_24\ text(represents the number of chemistry students)`

  `text(expected to get a)\ D.`
 

c.i.   `{:(qquad qquad qquad B quad qquad C),(F = [(110, 95)]):}`

 
c.ii.  `text(Total laboratory fees)`

`L = [(110, 95)] [(460), (360)] = [84\ 800]`

 

MATRICES, FUR2 2012 VCAA 1

Matrix `F` below shows the flight connections for an airline that serves four cities, Anvil (`A`), Berga (`B`), Cantor (`C`) and Dantel (`D`).

`{:(qquad qquad qquad qquad quad text(from)),((qquad qquad quad\ A,B,C,D)),(F = [(0\ ,1\ ,0\ ,0),(1,0,1,0),(0,0,0,1),(0,1,0,0)]):}{:(),(),(A),(B),(C),(D):}{:(),(qquad text(to)):}`

In this matrix, the `1` in column `C` row `B`, for example, indicates that, using this airline, you can fly directly from Cantor to Berga. The `0` in column `C` row `D`, for example, indicates that you cannot fly directly from Cantor to Dantel.

  1. Complete the following sentence.
  2. On this airline, you can fly directly from Berga to ________ and ________.   (1 mark)

  3. List the route that you must follow to fly from Anvil to Cantor.   (1 mark)

  4. Evaluate the matrix product `G = KF`, where `K = [1,1,1,1]`.   (1 mark)

     

  5. In the context of the problem, what information does matrix `G` contain?   (1 mark)

Show Answers Only
  1. `text(Anvil and Dantel)`
  2. `text(Anvil – Berga – Dantel – Cantor)`
  3. `[1,2,1,1]`
  4. `text(Matrix)\ G\ text(contains the information of the total)`
    `text(amount of direct flights out of each of 4 cities,)`
    `text(that go to another city in the same group.)`
Show Worked Solution

a.   `text(Anvil and Dantel)`
 

b.   `text(Anvil – Berga – Dantel – Cantor)`
 

c.   `G` `= KF`
    `= [1,1,1,1][(0,1,0,0),(1,0,1,0),(0,0,0,1),(0,1,0,0)]`
    `= [1,2,1,1]`

 
d.    `text(Matrix)\ G\ text(contains the information of the total)`

`text(amount of direct flights out of each of 4 cities,)`

`text(that go to another city in the same group.)`

CORE, FUR2 2008 VCAA 1

In a small survey, twenty-five Year 8 girls were asked what they did (walked, sat, stood, ran) for most of the time during a typical school lunch time. 

Their responses are recorded below.
 

`{:(text(sat),text(stood),text(sat),text(ran),text(sat)),(text(walked ),text(walked ),text(sat),text(walked ),text(ran)),(text(sat),text(walked),text(walked ),text(walked),text(ran)),(text(walked),text(ran),text(walked),text(ran),text(walked)),(text(ran),text(sat),text(ran),text(ran),text(walked)):}`

 

Use the data to

  1. complete the following frequency table  (1 mark)

     

     

         Core, FUR2 2008 VCAA 1

  2. determine the percentage of Year 8 girls who ran for most of the time during a typical school lunch time.  (1 mark) 
Show Answers Only
  1.  
    Core, FUR2 2008 VCAA 1 Answer
  2. `text(32%)`
Show Worked Solution
a.    Core, FUR2 2008 VCAA 1 Answer

 
b.   `text(Percentage who ran)`

`=8/25 xx 100text(%)`

`= 32text(%)`

MATRICES, FUR2 2015 VCAA 1

Students in a music school are classified according to three ability levels: beginner (`B`), intermediate (`I`) or advanced (`A`).

Matrix `S_0`, shown below, lists the number of students at each level in the school for a particular week.

`S_0 = [(20), (60), (40)] {:(B), (I), (A):}`

  1. How many students in total are in the music school that week?   (1 mark)

The music school has four teachers, David (`D`), Edith (`E`), Flavio (`F`) and Geoff (`G`).

Each teacher will teach a proportion of the students from each level, as shown in matrix `P` below.

`{:(qquadqquadqquad{:(Dquad,Eqquad,Fqquad,G):}),(P = [(0.25,0.5,0.15,0.1)]):}`

The matrix product, `Q = S_0P`, can be used to find the number of students from each level taught by each teacher.

  1.  i. Complete matrix `Q`, shown below, by writing the missing elements in the shaded boxes.   (1 mark) 

 Matrices, FUR2 2015 VCAA 1

  1. ii. How many intermediate students does Edith teach?  (1 mark)

The music school pays the teachers $15 per week for each beginner student, $25 per week for each intermediate student and $40 per week for each advanced student.

These amounts are shown in matrix `C` below.

`{:(qquad qquad quad{:(B quad, I quad,A):}),(C = [(15, 25, 40)]):}`

The amount paid to each teacher each week can be found using a matrix calculation.

  1.  i. Write down a matrix calculation in terms of `Q` and `C` that results in a matrix that lists the amount paid to each teacher each week.   (1 mark)

  2. ii. How much is paid to Geoff each week?   (1 mark)

Show Answers Only
  1. `120`
    1. `[(5, 10, 3, 2), (15, 30, 9, 6), (10, 20, 6, 4)]`
    2. `30`
    1. `C xx Q`
    2. `$340`
Show Worked Solution

a.   `20 + 60 + 40 = 120`
 

b.i.   `Q` `= S_0 P`
    `= [(20), (60), (40)] [(0.25, 0.5, 0.15, 0.1)]`
    `= [(5, 10, 3, 2), (15, 30, 9, 6), (10, 20, 6, 4)]`

 
b.ii.   `text(Edith teaches 30 intermediate students.)`

 
c.i.   `C xx Q`

 
c.ii.   `CQ = [(15, 25, 40)] [(5, 10, 3, 2), (15, 30, 9, 6), (10, 20, 6, 4)]`

`:.\ text(Geoff’s pay)`

`= 15 xx 2 + 25 xx 6 + 40 xx 4`

`= $340`

CORE, FUR2 2009 VCAA 1

Table 1 shows the number of rainy days recorded in a high rainfall area for each month during 2008.
 

CORE, FUR2 2009 VCAA 11

The dot plot below displays the distribution of the number of rainy days for the 12 months of 2008.
 

CORE, FUR2 2009 VCAA 12

  1. Circle the dot on the dot plot that represents the number of rainy days in April 2008.   (1 mark)

  2. For the year 2008, determine

     

  3.  i. the median number of rainy days per month.   (1 mark)
  4. ii. the percentage of months that have more than 10 rainy days. Write your answer correct to the nearest per cent.   (1 mark) 
Show Answers Only
  1.  
    CORE, FUR2 2009 VCAA 12 Answer
    1. `15.5`
    2. `text(92%)`
Show Worked Solution
a.    CORE, FUR2 2009 VCAA 12 Answer

 

b.i.    `text(Median)` `= text{(6th + 7th)}/2`
    `=(15+16)/2`
    `=15.5`

 

b.ii.   `text(Months with more than 10 rainy days)`

`=11/12 xx text(100%)`

`=91.66…`

`=92text(%)\ \ text{(nearest %)}`