SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

NETWORKS, FUR2 2017 VCAA 4

The rides at the theme park are set up at the beginning of each holiday season.

This project involves activities A to O.

The directed network below shows these activities and their completion times in days.

  1. Write down the two immediate predecessors of activity I.   (1 mark)

  2. The minimum completion time for the project is 19 days.

     

     i.  There are two critical paths. One of the critical paths is A–E–J–L–N.
    Write down the other critical path.   (1 mark)

    ii.  Determine the float time, in days, for activity F.   (1 mark)

  3. The project could finish earlier if some activities were crashed.

     

    Six activities, B, D, G, I, J and L, can all be reduced by one day.

     

    The cost of this crashing is $1000 per activity.

     

     i.  What is the minimum number of days in which the project could now be completed?   (1 mark)

    ii.  What is the minimum cost of completing the project in this time?   (1 mark)

Show Answers Only

a.     

b.i.   

b.ii.   

c.i.   

c.ii.    

Show Worked Solution

a.   
  

b.i.   

♦♦ Mean mark part (b)(i) 44% and part (b)(ii) 28%.
  

b.ii.   
   

  
c.i.   

♦♦ Mean mark part (c)(i) 35%.

 


  

c.ii.   

♦♦♦ Mean mark part (c)(ii) 15%.

 

NETWORKS, FUR2 2017 VCAA 3

While on holiday, four friends visit a theme park where there are nine rides.

On the graph below, the positions of the rides are indicated by the vertices.

The numbers on the edges represent the distances, in metres, between rides.
 

  1. Electrical cables are required to power the rides.
    These cables will form a connected graph.
    The shortest total length of cable will be used.
  2. i. Give a mathematical term to describe a graph that represents these cables.   (1 mark)

  3. ii. Draw in the graph that represents these cables on the diagram below.   (1 mark)

     

Show Answers Only

a.i.   

a.ii.   
Show Worked Solution

a.i.   

a.ii.   

NETWORKS, FUR2 2017 VCAA 2

Bai joins his friends Agatha, Colin and Diane when he arrives for a holiday in Seatown.

Each person will plan one tour that the group will take.

Table 1 shows the time, in minutes, it would take each person to plan each of the four tours.
 

 
The aim is to minimise the total time it takes to plan the four tours.

Agatha applies the Hungarian algorithm to Table 1 to produce Table 2.

Table 2 shows the final result of all her steps of the Hungarian algorithm.
 


 

  1. In Table 2 there is a zero in the column for Colin.
    When all values in the table are considered, what conclusion about minimum total planning time can be made from this zero?   (1 mark)

  2. Determine the minimum total planning time, in minutes, for all four tours.   (1 mark)

Show Answers Only

a.   

b.   

Show Worked Solution

a.   

♦ Mean mark part (a) 37%.
MARKER’S COMMENT: The answer “Colin will plan Tour 2 because he is the fastest” received no marks.

 

b.   

 

MATRICES, FUR2 2017 VCAA 3

Senior students at a school choose one elective activity in each of the four terms in 2018.

Their choices are communication (), investigation (), problem-solving () and service ().

The transition matrix shows the way in which senior students are expected to change their choice of elective activity from term to term.
 


 

Let be the state matrix for the number of senior students expected to choose each elective activity in Term .

For the given matrix , a matrix rule that can be used to predict the number of senior students in each elective activity in Terms 2, 3 and 4 is
 


 

  1. How many senior students will not change their elective activity from Term 1 to Term 2?   (1 mark)

  2. Complete , the state matrix for Term 2, below.   (1 mark)


             

  3. Of the senior students expected to choose investigation () in Term 3, what percentage chose service () in Term 2?   (2 marks)

  4. What is the maximum number of senior students expected in investigation () at any time during 2018?   (1 mark)

Show Answers Only
  2.  
Show Worked Solution

a.   

♦ Mean mark 47%.

 

b.   
   

 

c.   
   

♦♦♦ Mean mark 13%.
MARKER’S COMMENT: A poorly understood and answered question worthy of careful attention.

 
 

 

d.   
   

♦ Mean mark 43%.

MATRICES, FUR2 2017 VCAA 2

Junior students at a school must choose one elective activity in each of the four terms in 2018.

Students can choose from the areas of performance (), sport () and technology ().

The transition diagram below shows the way in which junior students are expected to change their choice of elective activity from term to term.

 

  1. Of the junior students who choose performance () in one term, what percentage are expected to choose sport () the next term?   (1 mark)

Matrix lists the number of junior students who will be in each elective activity in Term 1.
 


 

  1. 306 junior students are expected to choose sport () in Term 2.
     
    Complete the calculation below to show this.   (1 mark)


  2. In Term 4, how many junior students in total are expected to participate in performance () or sport () or technology ()?   (1 mark)

Show Answers Only
  2.  
Show Worked Solution

a.   
 

b. 

♦♦ Mean mark part (c) 30%.

MARKER’S COMMENT: No matrix calculations were required here.


c.   

 
 

MATRICES, FUR2 2017 VCAA 1

A school canteen sells pies (), rolls () and sandwiches ().

The number of each item sold over three school weeks is shown in matrix .

 

  1. In total, how many sandwiches were sold in these three weeks?   (1 mark)

  2. The element in row and column of matrix is .
  3. What does the element indicate?  (1 mark)

  4. Consider the matrix equation



    where = cost of one pie, = cost of one roll and = cost of one sandwich.
  5.  i. What is the cost of one sandwich?   (1 mark)

The matrix equation below shows that the total value of all rolls and sandwiches sold in these three weeks is $915.60

Matrix in this equation is of order .

  1. ii. Write down matrix .   (1 mark)

Show Answers Only
Show Worked Solution
a.   
   

 
b.   


 

c.i.   
   

 

 

c.ii.   

CORE, FUR2 2017 VCAA 5

Alex is a mobile mechanic.

He uses a van to travel to his customers to repair their cars.

The value of Alex’s van is depreciated using the flat rate method of depreciation.

The value of the van, in dollars, after years, , can be modelled by the recurrence relation shown below.

  1. Recursion can be used to calculate the value of the van after two years.

     

    Complete the calculations below by writing the appropriate numbers in the boxes provided.   (2 marks)


    1. By how many dollars is the value of the van depreciated each year?   (1 mark)
    2. Calculate the annual flat rate of depreciation in the value of the van.
    3. Write your answer as a percentage.   (1 mark)
  3. The value of Alex’s van could also be depreciated using the reducing balance method of depreciation.
  4. The value of the van, in dollars, after years, , can be modelled by the recurrence relation shown below.

     

           

    At what annual percentage rate is the value of the van depreciated each year?   (1 mark)

Show Answers Only

a.

b.i.  

b.ii.

c.  

Show Worked Solution

a.   

  
b.i.   
  

b.ii.   
   

 

c.   
   

CORE, FUR2 2017 VCAA 6

Alex sends a bill to his customers after repairs are completed.

If a customer does not pay the bill by the due date, interest is charged.

Alex charges interest after the due date at the rate of 1.5% per month on the amount of an unpaid bill.

The interest on this amount will compound monthly.

  1. Alex sent Marcus a bill of $200 for repairs to his car.

     

    Marcus paid the full amount one month after the due date.

     

    How much did Marcus pay?   (1 mark)

Alex sent Lily a bill of $428 for repairs to her car.

Lily did not pay the bill by the due date.

Let be the amount of this bill months after the due date.

  1. Write down a recurrence relation, in terms of , and , that models the amount of the bill.   (2 marks)

  2. Lily paid the full amount of her bill four months after the due date.

     

    How much interest was Lily charged?

     

    Round your answer to the nearest cent.   (1 mark)

Show Answers Only
Show Worked Solution
a.   
   
   

♦ Mean mark part (b) 47%.
MARKER’S COMMENT: A recurrence relation has the initial value written first. Know why    is incorrect.

 

b.   

 

c.   
   

♦♦ Mean mark part (c) 29%.

 

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)

Show Answers Only

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)

Show Answers Only
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.

 

 

   
 
   

 

 

CORE, FUR2 2017 VCAA 3

The number of male moths caught in a trap set in a forest and the egg density (eggs per square metre) in the forest are shown in the table below.
 

  1. Determine the equation of the least squares line that can be used to predict the egg density in the forest from the number of male moths caught in the trap.
  2. Write the values of the intercept and slope of this least squares line in the appropriate boxes provided below.
  3. Round your answers to one decimal place.  (2 marks)

     

         

  4. The number of female moths caught in a trap set in a forest and the egg density (eggs per square metre) in the forest can also be examined.

     

    A scatterplot of the data is shown below.
     


     
    The equation of the least squares line is

     

                  egg density = 191 + 31.3 × number of female moths

    1. Draw the graph of this least squares line on the scatterplot (provided above).   (1 mark)

    2. Interpret the slope of the regression line in terms of the variables egg density and number of female moths caught in the trap.   (1 mark)

    3. The egg density is 1500 when the number of female moths caught is 55.
    4. Determine the residual value if the least squares line is used to predict the egg density for this number of female moths.   (1 mark)
    5. The correlation coefficient is 
    6. Determine the percentage of the variation in egg density in the forest explained by the variation in the number of female moths caught in the trap.
    7. Round your answer to one decimal place.   (1 mark)
Show Answers Only

a.   

b.i.  

b.ii. 

b.iii. 

b.iv. 

Show Worked Solution

a.   

 

b.i.   

♦♦ Mean mark 26%.
MARKER’S COMMENT: Not well answered! Many students did not realise the graph started at 10 on the -axis and many did not use a ruler!

 

♦ Mean mark 39%.
MARKER’S COMMENT: Students must clearly refer to the increase in egg density for every one-unit increase in female moths.

b.ii.   

 

 

 

b.iii.   

♦ Mean mark 48%.

 

 

b.iv.   

♦ Mean mark 47%.

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)

Show Answers Only

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)


Show Answers Only

  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)

Show Answers Only
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, MET2 2017 VCAA 1

Let  . Part of the graph of is shown below.
 

  1. Find the coordinates of the turning points.   (2 marks)

  2.   and   are two points on the graph of .

     

    1. Find the equation of the straight line through and .   (2 marks)

    2. Find the distance .   (1 mark)

Let  .

  1. Let  and be two points on the graph of .

     

    1. Find the distance in terms of .   (2 marks)

    2. Find the values of such that the distance is equal to  .   (1 mark)

  2. The diagram below shows part of the graphs of and  . These graphs intersect at the points with the coordinates and .
  3.  
       
  4.  
    1. Find the value of in terms of .   (1 mark)

    2. Find the area of the shaded region in terms of .   (2 marks)

Show Answers Only
    2. ²
Show Worked Solution
a.   

 


 

b.i.  

 

MARKER’S COMMENT: Students skilled in the use of technology will be much more efficient and minimise errors here.
b.ii.   
   
   

 

c.i.   
   
   

 

c.ii.   


 

d.i.   


 

d.ii.   
    ²

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)

Show Answers Only
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%.

   
 
   

 

 

 

 

Probability, MET1 2017 VCAA 8

For events and from a sample space, and .  Let  .

  1. Find   in terms of .  (1 mark)

  2. Find   in terms of .  (2 marks)

  3. Given that  , state the largest possible interval for .  (2 marks)

Show Answers Only

Show Worked Solution

a.   
   
   

 

b. 

♦ Mean mark 40%.
MARKER’S COMMENT: The most successful answers used a Venn diagram or table.

 

c. 

♦ Mean mark 37%.

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)

Show Answers Only
Show Worked Solution

a. 
 

 

b.i. 

♦ Mean mark 38%.
 


 


 

b.ii. 

♦♦♦ Mean mark 20%.


 

c. 

♦♦ Mean mark 30%.
 
 

 

Probability, MET1 2017 VCAA 5

For Jac to log on to a computer successfully, Jac must type the correct password. Unfortunately, Jac has forgotten the password. If Jac types the wrong password, Jac can make another attempt. The probability of success on any attempt is . Assume that the result of each attempt is independent of the result of any other attempt. A maximum of three attempts can be made.

  1. What is the probability that Jac does not log on to the computer successfully?   (1 mark)

  2. Calculate the probability that Jac logs on to the computer successfully. Express your answer in the form , where and are positive integers.   (1 mark)

  3. Calculate the probability that Jac logs on to the computer successfully on the second or on the third attempt. Express your answer in the form , where and are positive integers.   (2 marks)

Show Answers Only

Show Worked Solution

a. 

 

b. 

 

c. 

Probability, MET1 2017 VCAA 4

In a large population of fish, the proportion of angel fish is .

Let  be the random variable that represents the sample proportion of angel fish for samples of size drawn from the population.

Find the smallest integer value of such that the standard deviation of is less than or equal to .  (2 marks)

Show Answers Only

Show Worked Solution

 
 

 

 

Calculus, MET1 2017 VCAA 3

Let  

  1. Show that  .   (1 mark)

  2. Sketch the graph of   on the axes below. Label the axis intercepts and any stationary points with their coordinates.   (3 marks)

Show Answers Only
  2.  
Show Worked Solution

a. 

 

b.  


 

 

Calculus, MET1 2017 VCAA 2

Let  

  1. Find  .   (2 marks)

  2. Hence, calculate  . Express your answer in the form  , where is a positive integer.   (2 marks)

Show Answers Only
Show Worked Solution

a. 

 

 

b. 

MARKER’S COMMENT: Too many students ignored the hence instruction and were not able to form an integral.

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)

Show Answers Only
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.  
 

Harder Ext1 Topics, EXT2 2017 HSC 15b

Consider the curve  , for    and  , where is a positive constant.

  1. Show that the equation of the tangent to the curve at the point    is given by  .  (2 marks)
  2. The tangent to the curve at the point meets the and axes at   and  respectively. Show that  , where is the origin.  (3 marks)

 

Show Answers Only
Show Worked Solution

(i)   

 

 

 

(ii) 

 

 

 

 
 
 
 
 
 

Financial Maths, STD2 F1 SM-Bank 7

A dress was on sale with 25% discount.

As a regular customer, Kate received a further 10% on the already discounted price.

What was the overall percentage discount Kate received?  (2 marks)

Show Answers Only

Show Worked Solution

 
 
 

 

 


 

Financial Maths, STD2 F1 SM-Bank 12

A golf shop is having a Boxing Day sale.

  1. What is the percentage discount on a putter which is reduced from $120.00 to $102.00?  (1 mark)

  2. The same discount applies storewide. What discount amount is applicable to a box of golf balls whose original price was $25.00?  (1 mark)

  3. What is the sale price of a golf bag which originally cost $160.00  (1 mark)

Show Answers Only
Show Worked Solution
i.    
   
   

 

ii.   
   

 

iii.   
   
   

Financial Maths, STD2 F1 SM-Bank 2

George makes a single deposit of $9000 into an account that pays simple interest.

After 4 years, George's account has a balance of $10 350.

What simple interest rate did George receive on his investment?  (2 marks)

Show Answers Only

Show Worked Solution
 

 

 
 

Financial Maths, STD2 F1 SM-Bank 6

Michelle intends to keep a car purchased for $17 000 for 15 years. At the end of this time its value will be $3500.

  1. By what amount, in dollars, would the car’s value depreciate annually if Michelle used the flat rate method of depreciation?  (1 mark)

  2. Determine the annual flat rate of depreciation correct to one decimal place.  (1 mark) 

Show Answers Only
Show Worked Solution
i.   
   

 

 

ii.   

Financial Maths, STD2 F1 SM-Bank 5

Khan paid $900 for a printer.

This price includes 10% GST (goods and services tax).

  1. Determine the price of the printer before GST was added.

     

    Write your answer correct to the nearest cent.  (2 marks)

  2. Khan is able to depreciate the full $900 purchase price of his printer for taxation purposes.

     

    Under flat rate depreciation the printer will be valued at $300 after five years.

     

    Calculate the annual depreciation in dollars.  (1 mark)

Show Answers Only
Show Worked Solution

i.   

COMMENT: Reverse GST questions regularly cause problems for many students.
 
 

 

ii.   

Financial Maths, STD2 F1 SM-Bank 3

A company purchased a machine for $60 000.

For taxation purposes the machine is depreciated over time using the straight line depreciation method.

The machine is depreciated at a flat rate of 10% of the purchase price each year.

  1. By how many dollars will the machine depreciate annually?  (1 mark)

  2. Calculate the value of the machine after three years.  (1 mark)

  3. After how many years will the machine be $12 000 in value?  (1 mark)

Show Answers Only
Show Worked Solution
i.   
   

 

ii.   

 
 

 

iii.   

 

Financial Maths, STD2 F1 SM-Bank 10

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)

  2. Use calculations to show that the bike depreciates in value by $1500 each year.  (1 mark)

  3. Assume that the bike’s value continues to depreciate by $1500 each year. Determine its value five years after it was purchased.  (1 mark)

Show Answers Only
Show Worked Solution

i.   
 

ii.   

 

 

iii. 

 
 

Financial Maths, STD2 F1 SM-Bank 1 MC

Rae paid  $40 000  for new office equipment at the start of the 2013 financial year.

At the start of each following financial year, she used flat rate depreciation to revalue her equipment.

At the start of the 2016 financial year she revalued her equipment at  $22 000.

The annual flat rate of depreciation she used, as a percentage of the purchase price, was

  1. 11.25%
  2. 15%
  3. 17.5%
  4. 35%
Show Answers Only

Show Worked Solution

♦ Mean mark 50%.

Calculus, EXT2 C1 2017 HSC 15a

Let  , for  

  1. Find the value of .  (1 mark)

  2. Using integration by parts, or otherwise, show that for  
     
         .  (3 marks)

  3. Find the value of  .  (1 mark)

Show Answers Only
Show Worked Solution
i.  
   
   
   
   

 

ii. 

♦ Mean mark 48%.

   
 
 
 

 

 

iii.  
   
   
   

Mechanics, EXT2 2017 HSC 14c

A smooth double cone with semi-vertical angle    is rotating about its axis with constant angular velocity .

Two particles, each of mass , are sitting on the cone as it rotates, as shown in the diagram.

Particle 1 is inside the cone at vertical distance above the apex, , and moves in a horizontal circle of radius .

Particle 2 is attached to the apex by a light inextensible string so that it sits on the cone at vertical distance below the apex. Particle 2 also moves in a horizontal circle of radius .

The acceleration due to gravity is .

  1. The normal reaction force on Particle 1 is .
    By resolving into vertical and horizontal components, or otherwise, show that  .  (2 marks)
  2. The normal reaction force on Particle 2 is and the tension in the string is  .

  3. By considering horizontal and vertical forces, or otherwise, show that

  4. .  (2 marks)
  6. Show that  .  (2 marks)

 

Show Answers Only
Show Worked Solution
(i)  

 

(ii)  

 

 

 

 

(iii) 

♦♦ Mean mark 34%.

Calculus, EXT2 C1 2017 HSC 14a

It is given that  .

  1. Find and so that  .  (1 mark)

  2. Hence, or otherwise, show that for any real number
     
         .  (2 marks)

  3. Find the limiting value as    of
     
         .  (1 mark)

Show Answers Only
Show Worked Solution

i. 

 

 


 

ii. 

 

iii. 

 
 

Harder Ext1 Topics, EXT2 2017 HSC 14b

Two circles, and , intersect at the points and . Point is chosen on the arc of  as shown in the diagram.

The line segment produced meets  at .

The line segment produced meets  at .

The line segment produced meets  at .

The line segment produced meets the line segment at .

Copy or trace the diagram into your writing booklet.

  1. State why .  (1 mark)
  2. Show that .  (1 mark)
  3. Hence, or otherwise, show that and are concyclic points.  (3 marks)
Show Answers Only
Show Worked Solution

(i) 

 

 

(ii) 

 

 

 

 

 

(iii)  
   

 

♦ Mean mark 47%.

 

 

Measurement, STD2 M1 SM-Bank 4

Steel rods are manufactured in the shape of equilateral triangular prisms.
 


 

  1. Find the volume of the prism (answer correct to 1 decimal place).  (2 marks)

  2. The mass of steel is 7850 kg/m³. Use this information to find the mass of the steel rod correct to the nearest gram.  (2 marks)

Show Answers Only
  1. ³
Show Worked Solution

i.  

 

 
 
  ³

 

ii. 

 

³³

³  
  ³

 

 
 
 

Measurement, STD2 M7 SM-Bank 1

Bikram runs a hot yoga studio.

If it costs 34 cents for 1-kilowatt (1000 watts) for 1 hour, how much does it cost him to run three 3200-watt heaters from 9:00 am to 12:30 pm on a single day? (Give your answer to the nearest cent)  (2 marks)

Show Answers Only

Show Worked Solution

 

 
 

Measurement, STD2 M1 SM-Bank 7

Oscar and Nadine select a meal each from the table below.
 

 
Oscar has a hamburger and Nadine has the lamb souvlaki.

After dinner, Oscar goes for a run where he expends energy at 25 kJ/minute.

At the same time, Nadine goes for a brisk walk where she expends 12 kJ/minute.

Who will expend the kilojoule intake from their dinner the quickest, and by how many minutes?  (3 marks)

Show Answers Only

Show Worked Solution

 

 

 

 

Measurement, STD2 M1 SM-Bank 8

A cannon ball is made out of steel and has a diameter of 23 cm.

  1. Find the volume of the sphere in cubic centimetres (correct to 1 decimal place).  (2 marks)

  2. It is known that the mass of the steel used is 8.2 tonnes/m³. Use this information to find the mass of the cannon ball to the nearest gram.  (2 marks)

Show Answers Only
  1. ³
Show Worked Solution

i.   

 
 
  ³

 

ii.   ³³

³
  ³

 

 

 

Algebra, STD2 A4 SM-Bank 4

Penny is a baker and makes meat pies every day.

The cost of making pies, ,  can be calculated using the equation

Penny sells the pies for $5.75 each, and her income is calculated using the equation

  1. On the graph, draw the graphs of    and  .  (2 marks)

  2. On the graph, label the breakeven point and the loss zone.  (2 marks)

Show Answers Only

(i) and (ii)

Show Worked Solution
i.   

 

ii. 

Volumes, EXT2 2017 HSC 13d

The trapezium whose vertices are    and    forms the base of a solid.

Each cross-section perpendicular to the -axis is an isosceles triangle. The height of the isosceles triangle with base , is 1. The height of the isosceles triangle with base , is 2.  The cross-section through the point is the isosceles triangle , where lies on the line , as shown in the diagram.

Find the volume of the solid.  (4 marks)

Show Answers Only

³

Show Worked Solution

♦ Mean mark 51%.

 

 

 

 
 

 

 
 
  ³

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)

 

Show Answers Only
Show Worked Solution

(i)   

 

 

(ii)   

♦ Mean mark 51%.

 

 

 
 

 

   
   
 
 

 

 

 

Functions, EXT1′ F2 2017 HSC 12d

Let be a polynomial.

  1. Given that    is a factor of , show that
     
    .  (2 marks)

  2. Given that the polynomial    has a factor  , find the value of .  (2 marks)

Show Answers Only
Show Worked Solution
i.  
 
   

 

 

 

ii.  
 
   
   

 

 

CORE, FUR2 2017 VCAA 2

The back-to-back stem plot below displays the wingspan, in millimetres, of 32 moths and their place of capture (forest or grassland).

 

  1. Which variable, wingspan or place of capture, is a categorical variable?  (1 mark)

  2. Write down the modal wingspan, in millimetres, of the moths captured in the forest.  (1 mark)

  3. Use the information in the back-to-back stem plot to complete the table below.  (2 marks)

     

     

  4. Show that the moth captured in the forest that had a wingspan of 52 mm is an outlier.  (2 marks)

  5. The back-to-back stem plot suggests that wingspan is associated with place of capture.
  6. Explain why, quoting the values of an appropriate statistic.  (2 marks)

Show Answers Only

  3.    

     

Show Worked Solution

a.   

 

b.   

 

c.   

 

 

d.   
 

 

 

 

 

e.   

CORE, FUR2 2017 VCAA 1

The number of eggs counted in a sample of 12 clusters of moth eggs is recorded in the table below.
     

  1. From the information given, determine
  2.  i. the range   (1 mark)

  3. ii. the percentage of clusters in this sample that contain more than 170 eggs.   (1 mark)

In a large population of moths, the number of eggs per cluster is approximately normally distributed with a mean of 165 eggs and a standard deviation of 25 eggs.

  1. Using the 68–95–99.7% rule, determine
  2.  i. the percentage of clusters expected to contain more than 140 eggs.   (1 mark)

  3. ii. the number of clusters expected to have less than 215 eggs in a sample of 1000 clusters.   (1 mark)

  4. The standardised number of eggs in one cluster is given by  
  5. Determine the actual number of eggs in this cluster.   (1 mark)

Show Answers Only

a.   

b.i.  

b.ii. 

c. 

Show Worked Solution

a.i. 

a.ii. 

 

 

b.i.   
   

 

b.ii.   
   

 

 

c.   
 
 
   

Functions, EXT1′ F1 2017 HSC 12a

Consider the function  .

  1.  Show that    is increasing for all .  (1 mark)

  2.  Show that    is an odd function.  (1 mark)

  3.  Describe the behaviour of    for large positive values of .  (1 mark)

  4.  Hence sketch the graph of  .  (1 mark)

  5.  Hence, or otherwise, sketch the graph of  .  (1 mark)

Show Answers Only
  4.  
  5.  

Show Worked Solution

i. 


 

 
 

 


 

 
 

 

 

ii.  
     
   
   

 

 

iii. 

 

iv.  

 

v.  

Volumes, EXT2 2017 HSC 11e

The region bounded by the lines    and the -axis is rotated about the -axis.

Use the method of cylindrical shells to find an integral whose value is the volume of the solid of revolution formed. Do NOT evaluate the integral.  (2 marks)

Show Answers Only

Show Worked Solution