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CORE, FUR2 2017 VCAA 4

The eggs laid by the female moths hatch and become caterpillars.

The following time series plot shows the total area, in hectares, of forest eaten by the caterpillars in a rural area during the period 1900 to 1980.

The data used to generate this plot is also given.
 

The association between area of forest eaten by the caterpillars and year is non-linear.

A log10 transformation can be applied to the variable area to linearise the data.

  1. When the equation of the least squares line that can be used to predict log10 (area) from year is determined, the slope of this line is approximately 0.0085385
  2. Round this value to three significant figures.   (1 mark)
  3. Perform the log10 transformation to the variable area and determine the equation of the least squares line that can be used to predict log10 (area) from year.
  4. Write the values of the intercept and slope of this least squares line in the appropriate boxes provided below.
  5. Round your answers to three significant figures.  (2 marks)

The least squares line predicts that the log10 (area) of forest eaten by the caterpillars by the year 2020 will be approximately 2.85

  1. Using this value of 2.85, calculate the expected area of forest that will be eaten by the caterpillars by the year 2020.
  2.  i. Round your answer to the nearest hectare.   (1 mark)

  3. ii. Give a reason why this prediction may have limited reliability.   (1 mark)

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a. 

b. 

c.i. 

c.ii.
       

Show Worked Solution

a.   

♦ Mean marks of part (a) and (b) 44%.

 

b.   

 

♦♦ Mean mark part (c)(i) 29%.
COMMENT: When the question specifies using the value 2.85, use it!

c.i.   
 
   
   

 

c.ii.   

 

Calculus, 2ADV C3 SM-Bank 7

The graph of    for    is shown below.
 


 

  1. Calculate the area between the graph of   and the -axis.  (2 marks)

  2. For in the interval , show that the gradient of the tangent to the graph of    is  .  (1 mark)

The edges of the right-angled triangle are the line segments and , which are tangent to the graph of  , and the line segment , which is part of the horizontal axis, as shown below.

Let be the angle that makes with the positive direction of the horizontal axis.
 


 

  1. Find the equation of the line through and in the form  , for  .  (3 marks)

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Show Worked Solution
i.  
   
   
   
   
   

 

ii.  
 
   
   

 

iii. 

♦♦♦ Mean mark (Vic) part (iii) 20%.
MARKER’S COMMENT: Most successful answers introduced a pronumeral such as    to solve.

 

 

   
 
   

 

 

Algebra, MET2 2017 VCAA 4

Let  . Part of the graph of  is shown below.
 

  1. The transformation    maps the graph of    onto the graph of  .

     

    State the values of and .   (2 marks)

  2. Find the rule and domain for  , the inverse function of  .   (2 marks)

  3. Find the area bounded by the graphs of  and  .   (3 marks)

  4. Part of the graphs of  and  are shown below.
     

         
     
    Find the gradient of  and the gradient of    at  .   (2 marks)

The functions of  , where  , are defined with domain such that  .

  1. Find the value of such that  .  (1 mark)

  2. Find the rule for the inverse functions  of  , where  .   (1 mark)

  3. i. Describe the transformation that maps the graph of  onto the graph of  .   (1 mark)

    ii. Describe the transformation that maps the graph of  onto the graph of  .   (1 mark)

  4. The lines and are the tangents at the origin to the graphs of    and    respectively.
  5. Find the value(s) of for which the angle between and is 30°.   (2 marks)

  6. Let be the value of for which    has only one solution.
  7.  i. Find .   (2 marks)

  8. ii. Let    be the area bounded by the graphs of    and    for all  .
  9.     State the smallest value of such that  .   (1 mark)

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a. 

b. 

c.  ²

d.

e. 

f. 

g.i. 

g.ii. 

h. 

i.i. 

i.ii. 

Show Worked Solution

a.   

   

 

 

b.   

 

 

c.   

MARKER’S COMMENT: Specifically recommends using    in this type of integral to avoid errors in stating the integral.

 

  ²

 

d.   
 

 

e.   

 

f.   

 

♦♦ Mean mark part (g)(i) 31%.

g.i.   
 

 

 

♦♦ Mean mark part (g)(ii) 30%.

g.ii.   
 

 

 

h.   

♦♦♦ Mean mark part (h) 13%.

 

 

 

i.i   

♦♦♦ Mean mark part (i)(i) 4%.

 

 

i.ii 

♦♦♦ Mean mark part (i)(ii) 2%.

 

Probability, MET2 2017 VCAA 3

The time Jennifer spends on her homework each day varies, but she does some homework every day.

The continuous random variable , which models the time, in minutes, that Jennifer spends each day on her homework, has a probability density function , where

 

 

  1. Sketch the graph of on the axes provided below.  (3 marks)


     

       
     

  2. Find  .  (2 marks)

  3. Find  .  (2 marks)

  4. Find such that  , correct to four decimal places.  (2 marks)

  5. The probability that Jennifer spends more than 50 minutes on her homework on any given day is . Assume that the amount of time spent on her homework on any day is independent of the time spent on her homework on any other day.

     

    1. Find the probability that Jennifer spends more than 50 minutes on her homework on more than three of seven randomly chosen days, correct to four decimal places.  (2 marks)

    2. Find the probability that Jennifer spends more than 50 minutes on her homework on at least two of seven randomly chosen days, given that she spends more than 50 minutes on her homework on at least one of those days, correct to four decimal places.  (2 marks)

Let be the probability that on any given day Jennifer spends more than minutes on her homework.

Let be the probability that on two or three days out of seven randomly chosen days she spends more than minutes on her homework.

  1. Express as a polynomial in terms of .  (2 marks)

    1. Find the maximum value of , correct to four decimal places, and the value of for which this maximum occurs, correct to four decimal places.  (2 marks)

    2. Find the value of for which the maximum found in part g.i. occurs, correct to the nearest minute.  (2 marks)


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  1.  

Show Worked Solution

a.   

MARKER’S COMMENT: Many did not draw graph along -axis between 0 and 20 and for  .

 

b.   

 

c.   

 

d.   

♦ Mean mark part (d) 36%.

 

 

e.i.   

 

e.ii.   
   
   

 

f.   

♦ Mean mark part (f) 36%.

 
 
   
 

 

g.i.   

♦♦ Mean mark part (g)(i) 30%.

 

 

g.ii.   

♦♦♦ Mean mark part (g)(ii) 8%.

 

 

 

Algebra, MET2 2017 VCAA 2

Sammy visits a giant Ferris wheel. Sammy enters a capsule on the Ferris wheel from a platform above the ground. The Ferris wheel is rotating anticlockwise. The capsule is attached to the Ferris wheel at point . The height of above the ground, , is modelled by  , where is the time in minutes after Sammy enters the capsule and is measured in metres.

Sammy exits the capsule after one complete rotation of the Ferris wheel.
 


 

  1. State the minimum and maximum heights of above the ground.   (1 mark)

  2. For how much time is Sammy in the capsule?   (1 mark)

  3. Find the rate of change of with respect to and, hence, state the value of at which the rate of change of is at its maximum.   (2 marks)

As the Ferris wheel rotates, a stationary boat at , on a nearby river, first becomes visible at point . is 500 m horizontally from the vertical axis through the centre of the Ferris wheel and angle , as shown below.
 

   
 

  1. Find in degrees, correct to two decimal places.   (1 mark)

Part of the path of is given by  , where and are in metres.

  1. Find .   (1 mark)

As the Ferris wheel continues to rotate, the boat at is no longer visible from the point  onwards. The line through and is tangent to the path of , where angle .
 

   
 

  1. Find the gradient of the line segment in terms of and, hence, find the coordinates of , correct to two decimal places.   (3 marks)

  2. Find in degrees, correct to two decimal places.   (1 mark)

  3. Hence or otherwise, find the length of time, to the nearest minute, during which the boat at is visible.   (2 marks)

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Show Worked Solution
a.   
     

 

b.   

 

c.   

♦ Mean mark 50%.
MARKER’S COMMENT: A number of commons errors here – 2 answers given, calc not in radian mode, etc …

 

   

 

d.   

♦ Mean mark 36%.
MARKER’S COMMENT: Choosing degrees vs radians in the correct context was critical here.

 

 

e.   

 

f.   

 

♦♦♦ Mean mark part (f) 18%.
MARKER’S COMMENT: Many students were unable to use the rise over run information to calculate the second gradient.

 

 

 

 
 

 

 

♦♦♦ Mean mark part (g) 7%.

g.   
   
 

 

h.   

♦♦♦ Mean mark 5%.

 

 
 

Calculus, MET1 2017 VCAA 9

The graph of    is shown below.
 

  1. Calculate the area between the graph of and the -axis.   (2 marks)

  2. For in the interval , show that the gradient of the tangent to the graph of is  .   (1 mark)

The edges of the right-angled triangle are the line segments and , which are tangent to the graph of , and the line segment , which is part of the horizontal axis, as shown below.

Let be the angle that makes with the positive direction of the horizontal axis, where  .

  1. Find the equation of the line through and in the form  , for  .   (2 marks)

  2. Find the coordinates of when  .   (4 marks)

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Show Worked Solution
a.  
   
   
   
   
   

 

♦♦ Mean mark part (b) 35%.
MARKER’S COMMENT: Establishing the common denominator in the working was required!
b.  
 
   
   

 

c. 

♦♦♦ Mean mark part (c) 20%.
MARKER’S COMMENT: Most successful answers introduced a pronumeral such as to solve.

 

 

   
 
   

 

 

 

d. 

♦♦♦ Mean mark part (d) 17%.

   
 
   

 

 

 

 

Functions, MET1 2017 VCAA 7

Let  .

  1.  State the range of .   (1 mark)

  2.  Let  .
    1. Find the largest possible value of such that the range of is a subset of the domain of .   (2 marks)

    2. For the value of found in part b.i., state the range of .   (1 mark) 

  3. Let  .
  4. State the range of .   (1 mark)

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Show Worked Solution

a. 
 

 

b.i. 

♦ Mean mark 38%.
 


 


 

b.ii. 

♦♦♦ Mean mark 20%.


 

c. 

♦♦ Mean mark 30%.
 
 

 

Financial Maths, STD2 F1 SM-Bank 5

Alex is buying a used car which has a sale price of  $13 380. In addition to the sale price there are the following costs:

2014 27a1

  1. Stamp Duty for this car is calculated at $3 for every $100, or part thereof, of the sale price.  
    Calculate the Stamp Duty payable.   (1 mark)

  2. Alex wishes to take out comprehensive insurance for the car for 12 months.

     

    The cost of comprehensive insurance is calculated using the following:
     
          2014 27a2
    Find the total amount that Alex will need to pay for comprehensive insurance.   (3 marks)

  3. Alex has decided he will take out the comprehensive car insurance rather than the less expensive non-compulsory third-party car insurance.

  4.  

    What extra cover is provided by the comprehensive car insurance?   (1 mark)

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Show Worked Solution
♦♦♦ Mean mark 12%
IMPORTANT: “or part thereof ..” in the question requires students to round up to 134 to get the right multiple of $3 for their calculation.
i.   
 

 

ii.  

 

 

 
 

 

 
 

 

 

 

♦ Mean mark 34%.
iii.  
 

Measurement, STD2 M1 2017 HSC 30e

A solid is made up of a sphere sitting partially inside a cone.

The sphere, centre , has a radius of 4 cm and sits 2 cm inside the cone. The solid has a total height of 15 cm. The solid and its cross-section are shown.
 


 

Using the formula    where  is the radius of the cone's circular base and is the perpendicular height of the cone, find the volume of the cone, correct to the nearest cm³?  (3 marks)

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Show Worked Solution

 

 
 
 

Proof, EXT2 P2 2017 HSC 16c

A 2 by grid is made up of two rows of square tiles, as shown.
 

The tiles of the 2 by grid are to be painted so that tiles sharing an edge are painted using different colours. There are different colours available, where  .

It is NOT necessary to use all the colours.

Consider the case of the 2 by 2 grid with tiles labelled A, B, C and D, as shown.
 

There are ways to choose colours for the first column containing tiles A and B. Do NOT prove this.

  1. Assume the colours for tiles A and B have been chosen. There are two cases to consider when choosing colours for the second column. Either tile C is the same colour as tile B, or tile C is a different colour from tile B.

     

    By considering these two cases, show that the number of ways of choosing colours for the second column is  .  (2 marks)

  2. Prove by mathematical induction that the number of ways in which the 2 by grid can be painted is  , for  .  (2 marks)

  3. In how many ways can a 2 by 5 grid be painted if 3 colours are available and each colour must now be used at least once?  (2 marks)

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Show Worked Solution
i.   

♦♦♦ Mean mark 20%.

 

 

 

ii.   

♦♦♦ Mean mark 22%.

 

 

 

 

iii.   

♦♦♦ Mean mark 22%.

 
 

 

Harder Ext1 Topics, EXT2 2017 HSC 16a

  Let  , where  .

  1. Show that  , for any integer .  (1 mark)
  3. Let  , where is a positive integer.
  4. By summing the series, prove that  

    .  (3 marks)

  6. Deduce, from parts (i) and (ii), that.  (2 marks)

  7. Show that  
     is independent of .  (1 mark)

 

 

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Show Worked Solution

(i)   

 
 

 

(ii)   

♦♦ Mean mark 34%.

 
 
 
 
 
♦ Mean mark part (iii) 48%.

 

(iii)   
   
   
   

 

 
 
 
 

♦♦♦ Mean mark part (iv) 28%.

 

(iv)   

 

Conics, EXT2 2017 HSC 15c

The ellipse with equation  , where  , has eccentricity .

The hyperbola with equation  , has eccentricity .

The value of is chosen so that the hyperbola and the ellipse meet at  , as shown in the diagram.

  1. Show that  
  2. .  (2 marks)
  4. If the two conics have the same foci, show that their tangents at are perpendicular.  (3 marks)

 

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Show Worked Solution

(i)   

 

 

(ii)   

♦ Mean mark 51%.

 

 

 
 

 

   
   
 
 

 

 

 

MATRICES, FUR1 2017 VCAA 7 MC

At a fish farm:

    • young fish () may eventually grow into juveniles () or they may die ()
    •  juveniles () may eventually grow into adults () or they may die ()
    • adults () eventually die ().

The initial state of this population, , is shown below.
 

 

Every month, fish are either sold or bought so that the number of young, juvenile and adult fish in the farm remains constant.

The population of fish in the fish farm after months, , can be determined by the recurrence rule
 


 

where is a column matrix that shows the number of young, juvenile and adult fish bought or sold each month and the number of dead fish that are removed.

Each month, the fish farm will

  1. sell 1650 adult fish.
  2. buy 1750 adult fish.
  3. sell 17 500 young fish.
  4. buy 50 000 young fish.
  5. buy 10 000 juvenile fish.
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Show Worked Solution

 

 

 

GRAPHS, FUR1 2017 VCAA 8 MC

The shaded area in the graph below shows the feasible region for a linear programming problem.

The objective function is given by

Which one of the following statements is not true?

  1. When and , the minimum value of is at point .
  2. When and , the maximum value of is at point .
  3. When and , the minimum value of is at point .
  4. When and , the maximum value of is at point .
  5. When and , the maximum value of is at point .
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Show Worked Solution

 

NETWORKS, FUR1 2017 VCAA 8 MC

The flow of oil through a series of pipelines, in litres per minute, is shown in the network below.
 

 
The weightings of three of the edges are labelled .

Five cuts labelled A–E are shown on the network.

The maximum flow of oil from the source to the sink, in litres per minute, is given by the capacity of

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GEOMETRY, FUR1 2017 VCAA 8 MC

Three circles of radius 50 mm are placed so that they just touch each other.
The region enclosed by the circles is shaded in the diagram below.

The area of the shaded region, in square millimetres, is closest to

  1.  
  2.  
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Show Worked Solution

 
 
 

GEOMETRY, FUR1 2017 VCAA 7 MC

A triangle has:

• one side, , of length 4 cm
• one side, , of length 7 cm
• one angle, , of 26°.

Which one of the following angles, correct to the nearest degree, could not be another angle in triangle ?

  1.  
  2.  
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CORE, FUR1 2017 VCAA 24 MC

Xavier borrowed $245 000 to pay for a house.

For the first 10 years of the loan, the interest rate was 4.35% per annum, compounding monthly.

Xavier made monthly repayments of $1800.

After 10 years, the interest rate changed.

If Xavier now makes monthly repayments of $2000, he could repay the loan in a further five years.

The new annual interest rate for Xavier’s loan is closest to

  1.  0.35%
  2.  4.1%
  3.  4.5%
  4.  4.8%
  5.  18.7%
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Show Worked Solution

 

 

 

 

Statistics, NAP-I3-CA09

The graph shows the origin and type of all vehicles in a city.
 

 
Which statement is most accurate based on the graph?

 
 There are more utility vehicles than trucks and vans.
 
 Utility vehicles are the most common type of vehicles
 
 There are more Australian vehicles than European vehicles.
 
 There are more Asian vehicles than European vehicles.
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Show Worked Solution

Statistics, NAP-A4-CA03

A dealership sells new and used cars.

The graph shows the price of 2 similar cars and their age in years
 

 
Which one of these statements is true?

 
 Car Q is older and less expensive than Car P.
 
 Car P is newer and less expensive than Car Q.
 
 Car P is older and more expensive than Car Q.
 
 Car Q is newer and more expensive than Car P.
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Quadratic, EXT1 2017 HSC 14b

Let    be a point on the parabola  .

The tangent to the parabola at meets the parabola  , , at and . Let be the midpoint of .

  1. Show that the coordinates of and satisfy
  2. .  (2 marks)

  3. Show that the coordination of are  .  (2 marks)
  4. Find the value of so that the point always lies on the parabola  .  (2 marks)

 

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Show Worked Solution

(i)   

 

 

 

(ii)   

♦♦ Mean mark 30%.
 
 

 

 

 
 

 

(iii)   

♦♦ Mean mark 23%.
 
 

Mechanics, EXT2* M1 2017 HSC 13c

A golfer hits a golf ball with initial speed at an angle to the horizontal. The golf ball is hit from one side of a lake and must have a horizontal range of 100 m or more to avoid landing in the lake.
 

     

Neglecting the effects of air resistance, the equations describing the motion of the ball are

,

where is the time in seconds after the ball is hit and is the acceleration due to gravity in . Do NOT prove these equations.

  1. Show that the horizontal range of the golf ball is
     
          metres.  (2 marks)

  2. Show that if    then the horizontal range of the ball is less than 100 m.  (1 mark)

It is now given that    and that the horizontal range of the ball is 100 m or more.

  1. Show that  .  (2 marks)

  2. Find the greatest height the ball can achieve.  (2 marks)

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Show Worked Solution

i.   

 

 
 

 

ii.   

♦ Mean mark 44%.

 

 

iii.   

 

iv.   

♦♦ Mean mark 35%.
 

 

 
 

 

 
 
 
 

Plane Geometry, 2UA 2017 HSC 16c

In the triangle , the point is the mid-point of . The point lies on and

.

The line that passes through the point and is parallel to meets produced at .

Copy or trace this diagram into your writing booklet.

  1. Prove that is isosceles.  (3 marks)
  2. The point is chosen on so that is parallel to .
     
    Show that bisects .  (2 marks)

 

 

 

 

 

 

 

 

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Show Worked Solution
(i)  

♦♦♦ Mean mark 21%.

 

 

 

(ii)  

♦♦ Mean mark 31%.

Calculus, 2ADV C3 2017 HSC 16a

John’s home is at point and his school is at point . A straight river runs nearby.

The point on the river closest to is point , which is 5 km from .

The point on the river closest to is point , which is 7 km from .

The distance from to is 9 km.

To get some exercise, John cycles from home directly to point on the river, km from , before cycling directly to school at , as shown in the diagram.
 


  

The total distance John cycles from home to school is km.

  1. Show that  .   (1 mark)

  2. Show that if  , then  .   (3 marks)

  3. Find the value of that makes  .   (2 marks)

  4. Explain why this value of gives a minimum for .   (1 mark)

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Show Worked Solution
i.  

 
 

 

ii. 

♦ Mean mark 41%.

 

 

iii. 

♦♦ Mean mark 29%.
♦♦♦ Mean mark 9%.

 

iv.  

Measurement, 2UG 2017 HSC 30e

A solid is made up of a sphere sitting partially inside a cone.

The sphere, centre , has a radius of 4 cm and sits 2 cm inside the cone. The solid has a total height of 15 cm. The solid and its cross-section are shown.
 


 

What is the volume of the cone, correct to the nearest cm³?  (3 marks)

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³³

Show Worked Solution

♦♦ Mean mark 34%.

 

 
 
  ³³

Statistics, STD2 S5 2017 HSC 29d

All the students in a class of 30 did a test.

The marks, out of 10, are shown in the dot plot.
 


 

  1. Find the median test mark.  (1 mark)

  2. The mean test mark is 5.4. The standard deviation of the test marks is 4.22.

     

    Using the dot plot, calculate the percentage of the marks which lie within one standard deviation of the mean.  (2 marks)

  3. A student states that for any data set, 68% of the scores should lie within one standard deviation of the mean. With reference to the dot plot, explain why the student’s statement is NOT relevant in this context.  (1 mark)

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Show Worked Solution
♦ Mean mark 50%.
i.   
   
   

 

ii.   

♦♦ Mean mark 34%.

 

iii.   

♦♦♦ Mean mark 13%.

Measurement, 2UG 2017 HSC 29a

A new 200-metre long dam is to be built.
The plan for the new dam shows evenly spaced cross-sectional areas.
 


 

  1. Using TWO applications of Simpson’s rule, show that the volume of the dam is approximately  44 333 m³.  (2 marks)
  2. It is known that the catchment area for this dam is 2 km².
    Calculate how much rainfall is needed, to the nearest mm, to fill the dam.  (2 marks)

 

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Show Worked Solution
(i)   
   
   
    ³

 

(ii)   ²²

♦♦♦ Mean mark 11%.
²
  ²
 
 
 

Algebra, 2UG 2017 HSC 28a

Temperature can be measured in degrees Celsius () or degrees Fahrenheit ().

The two temperature scales are related by the equation  .

  1. Calculate the temperature in degrees Fahrenheit when it is  −20 degrees Celsius.  (1 mark)
  2. Solve the following equations simultaneously, using either the substitution method or the elimination method.  (2 marks)

     

  3. The graphs of   and    are shown below.
  4.  


  5. What does the result from part (ii) mean in the context of the graph?  (1 mark)     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Show Answers Only

 

 

Show Worked Solution
(i)  
   

 

(ii)  
 

 

♦♦ Mean mark 31%.
MARKER’S COMMENT: An area that requires attention.

 

♦♦♦ Mean mark 20%.

 

(iii)  

Number and Algebra, NAP-J2-19

A company has 3706 computers in a warehouse.

Another 423 computers are delivered to the warehouse.

Which of these could be used to calculate the correct number of computers in the warehouse?

 
 3000 + 4000 + 700 + 20 + 6 + 3
 
 3000 + 700 + 400 + 20 + 60 + 3
 
 3000 + 700 + 400 + 20 + 6 + 3
 
 3000 + 400 + 70 + 20 + 6 + 3
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