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GEOMETRY, FUR2 2011 VCAA 4

A lighthouse tower, shaded on Figure 5 below, is in the shape of a truncated cone.

It has circular cross-sections that decrease uniformly from a radius of 3.5 metres at ground level to a radius of 2 metres at the walkway.

The height of the lighthouse tower is 18 metres.

The angle marked as is the angle that the outer wall of the lighthouse tower makes with the horizontal at ground level.

GEOMETRY, FUR2 2011 VCAA 41

  1. Determine the size of the angle .

     

    Write your answer in degrees correct to one decimal place.  (1 mark)

The lighthouse tower is part of a cone. The height of this cone is metres and the base radius is 3.5 metres, as shown in Figure 6.

GEOMETRY, FUR2 2011 VCAA 42

    1. Determine , the height of this cone, in meters.  (2 marks)
    2. Determine the volume of the lighthouse tower.

       

      Write your answer to the nearest cubic metre.  (1 mark)

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    2. ³
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a.   VCAA GEO FUR2 2011 4i
 
 
♦♦♦ Mean mark for all parts (combined) 17%.
MARKER’S COMMENT: A common mistake had the base of this triangle equal to 3.5m.

 

b.i.   VCAA GEO FUR2 2011 4ii
 
 
  ii.   VCAA GEO FUR2 2011 4iii

³

 

³

 

³³

MATRICES, FUR2 2014 VCAA 3

There are three candidates in the election: Ms Aboud (), Mr Broad () and Mr Choi ().

Mr Choi may need to withdraw from the election at the end of May.

Matrix , shown below, shows the percentage of voters who change their preferred candidate, from month to month, before Mr Choi would withdraw from the election.

Matrix , shown below, shows the percentage of voters who change their preferred candidate, from May to June, after Mr Choi would withdraw from the election.

Consider the voters who preferred Mr Broad in May and who were expected to prefer Mr Choi in June.

  1. What percentage of these voters are now expected to prefer Mr Broad in June?   (1 mark)

The state matrix that indicates the number of voters who are expected to have a preference for each candidate in January, , is given below.

  1. If Mr Choi withdraws, how many votes is Mr Broad expected to receive in the election in June?
  2. Write your answer, correct to the nearest vote.   (2 marks)
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a.  

♦♦♦ Mean mark for (a)-(b) was 13% (combined).
MARKER’S COMMENT: In (a), identifying the sub-group expected to change their vote proved very difficult for most students.


  

MARKER’S COMMENT: Few students correctly modified the transition matrix from to in this question.
b.   
   
   
   

 

 
 
 

 

CORE, FUR2 2006 VCAA 3

The heights (in cm) and ages (in months) of the 15 boys are shown in the scatterplot below.
 

Core, FUR2 2006 VCAA 3
 

  1. Fit a three median line to the scatterplot. Circle the three points you used to determine this three median line.  (2 marks)
  2. Determine the equation of the three median line. Write the equation in terms of the variables height and age and give the slope and intercept correct to one decimal place.  (2 marks)
  3. Explain why the three median line might model the relationship between height and age better than the least squares regression line.  (1 mark)
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  1.   
    Core, FUR2 2006 VCAA 3 Answer

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a.    Core, FUR2 2006 VCAA 3 Answer
♦ Average mean mark for all parts 36%.

b.  

 
 

 

 

 

 
 

 

 

c.   

CORE, FUR2 2008 VCAA 5

The number of hours spent doing homework each week (homework hours) and the number of hours spent watching television each week (television hours) were recorded for a group of 20 Year 12 students. 

The results are displayed in the table below and a scatterplot constructed as shown.
 

CORE, FUR2 2008 VCAA 5

The relationship between homework hours per week and television hours per week is clearly nonlinear. 

A reciprocal transformation applied to the variable, television hours, can be used to linearise the scatterplot.

  1. Apply this reciprocal transformation to the data and determine the equation of the least squares regression line that allows homework hours to be predicted from the reciprocal of television hours.
  2. Write the coefficients correct to two decimal places.  (2 marks)

  3. If a student spends 12 hours per week watching television, use the least squares regression line to predict the number of hours that the student spends doing homework. Give your answer correct to one decimal place.  (1 mark)

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a.  

♦♦♦ Avg mean mark for both parts 16%.
MARKER’S COMMENT: A majority of students didn’t understand the term “reciprocal”.

 

b.   
   
   

CORE, FUR2 2011 VCAA 4

The average age of women at first marriage in years (average age) and average yearly income in dollars per person (income) were recorded for a group of 17 countries.

The results are displayed in Table 2. A scatterplot of the data is also shown.
 

2011 4-1

The relationship between average age and income is nonlinear.

A log transformation can be applied to the variable income and used to linearise the scatterplot.

  1. Apply this log transformation to the data and determine the equation of the least squares regression line that allows average age to be predicted from log (income).
  2. Write the coefficients for this equation, correct to two decimal places, in the spaces provided.   (2 marks)

    2011 4-2

  1. Use this equation to predict the average age of women at first marriage in a country with an average yearly income of $20 000 per person.
  2. Write your answer correct to one decimal place.   (1 mark) 
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a.  

♦♦ Parts (i) and (ii) had a combined mean mark 0f 34%.
MARKER’S COMMENT: Students struggled with the log transformation, incorrect data entry and choosing the correct dependent variable.

 

b.  

CORE, FUR2 2012 VCAA 2

The maximum temperature and the minimum temperature at this weather station on each of the 30 days in November 2011 are displayed in the scatterplot below.

CORE, FUR2 2012 VCAA 2

The correlation coefficient for this data set is  . 

The equation of the least squares regression line for this data set is

maximum temperature = × minimum temperature

  1. Draw this least squares regression line on the scatterplot above.   (1 mark)

  2. Interpret the vertical intercept of the least squares regression line in terms of maximum temperature and minimum temperature.   (1 mark)

  3. Describe the relationship between the maximum temperature and the minimum temperature in terms of strength and direction.   (1 mark)

  4. Interpret the slope of the least squares regression line in terms of maximum temperature and minimum temperature.   (1 mark)

  5. Determine the percentage of variation in the maximum temperature that may be explained by the variation in the minimum temperature.
  6. Write your answer, correct to the nearest percentage.   (1 mark)

On the day that the minimum temperature was 11.1 °C, the actual maximum temperature was 12.2 °C.

  1. Determine the residual value for this day if the least squares regression line is used to predict the maximum temperature.
  2. Write your answer, correct to the nearest degree.   (2 marks)

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a.  

 CORE, FUR2 2012 VCAA 2 Answer 

b.  

 

♦ Parts (i) to (vi) have an average mean mark of 41%.

c.   

 

d.  

 

 

 

e.   
   
   

 

f.  

MARKER’S COMMENT: Students had particular difficulty with this part, with many using the incorrect calculation of  12.2 – 11.1 = 1.1.
 
 
 

CORE, FUR2 2013 VCAA 4

The time series plot below shows the average pay rate, in dollars per hour, for workers in a particular country for the years 1991 to 2005.
 

CORE, FUR2 2013 VCAA 4

A three median line will be used to model the increasing trend in the average pay rate shown in this time series.

The explanatory variable to be used is year.

  1. Three medians will be used to draw the three median line.

     

    1. On the time series plot above, mark the location of each of the three medians with a cross ().  (2 marks)
    2. Draw the three median line on the time series plot.  (1 mark)
  2. Calculate the slope of the three median line.

     

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

  3. Interpret the slope of the three median line in terms of the relationship between the variables average pay rate and year.  (1 mark)
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  1. i. & ii.

  3.  

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a.i. & ii.   

♦ Mean mark 45%.
MARKER’S COMMENT: Few students found the correct middle point.

 

CORE, FUR2 2013 VCAA 4 Answer

♦♦ Parts (ii) and (iii) produced an overall mean mark of 32%.

 

b.   
   
   
   

 

MARKER’S COMMENT: Answering “an increase of 0.8 each year” received no marks (refer to actul units).

 

c.   

 

CORE, FUR2 2014 VCAA 4

The scatterplot below shows the population density, in people per square kilometre, and the area, in square kilometres, of 38 inner suburbs of the same city.

Core, FUR2 2015 VCAA 41

For this scatterplot,

  1. Describe the association between the variables population density and area for these suburbs in terms of strength, direction and form.   (1 mark)

  2. The mean and standard deviation of the variables population density and area for these 38 inner suburbs are shown in the table below.
     
    Core, FUR2 2015 VCAA 42
  3.   i. One of these suburbs has a population density of 3082 people per square kilometre.
  4.     Determine the standard -score of this suburb’s population density.
  5.     Write your answer, correct to one decimal place.   (1 mark)

Assume the areas of these inner suburbs are approximately normally distributed.

  1.  ii. How many of these 38 suburbs are expected to have an area that is two standard deviations or more above the mean?
  2.     Write your answer, correct to the nearest whole number.   (1 mark)

  3. iii. How many of these 38 inner suburbs actually have an area that is two standard deviations or more above the mean?   (1 mark)

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Show Worked Solution
♦♦ Mean mark 30%.
MARKER’S COMMENT: The most common error was to describe the association as positive.
a.   
   

 

 

b.i   
   
   
   

 

b.ii.   

 

b.iii.   

♦♦ Very few students answered this part correctly.

²

 

GRAPHS, FUR1 2008 VCAA 9 MC

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

GRAPHS, FUR1 2008 VCAA 9 MC

Which objective function, , has its maximum value at the point ?

A.   

B.   

C.   

D.   

E.   

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♦♦♦ Mean mark 17%.
MARKER’S COMMENT: Even though (option E) has a larger value at than option A, its maximum value occurs at the vertex (0, 60).

 

GRAPHS, FUR1 2012 VCAA 9 MC

The cost of posting a parcel that weighs 500 grams or less depends on its weight, as shown on the graph below.

graphs 2012 VCAA 9mc

The total cost of posting three separate parcels weighing respectively 100 grams, 150 grams and 210 grams is $11.

The total cost of posting three separate parcels weighing respectively 60 grams, 110 grams and 350 grams is $8.

The total cost of posting two separate parcels weighing respectively 170 grams and 500 grams is

A.     

B.     

C.     

D.   

E.    

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GRAPHS, FUR1 2013 VCAA 8 MC

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

An objective function    has its value maximised at both vertex and vertex .

The values of and could be

A.  

B.  

C.  

D.  

E.  

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Show Worked Solution
♦♦ Mean mark 35%.

 

 

GEOMETRY, FUR1 2015 VCAA 8 MC

A composite shape is made up of a parallelogram and a triangle, as shown below.

GEO & TRIG, FUR1 2015 VCAA 8 MC

Which one of the following is always true?

A.  

B.  

C.  

D.  

E.  

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GEO & TRIG, FUR1 2015 VCAA 8 MC Answer1 

♦♦ Mean mark 25%.

 

CORE*, FUR1 2015 VCAA 8 MC

Cindy took out a reducing balance loan of $8400 to finance an overseas holiday.

Interest was charged at a rate of 9% per annum, compounding quarterly.

Her loan is to be fully repaid in six years, with equal quarterly payments.

After three years, Cindy will have reduced the balance of her loan by approximately

A.     9%

B.   35%

C.   43%

D.   50%

E.   57%

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Show Worked Solution

♦♦ Mean mark 33%.

 

 

 

 
 

CORE, FUR1 2010 VCAA 11 MC

A student uses the following data to construct the scatterplot shown below.
 

CORE, FUR1 2015 VCAA 11 MC
 

A reciprocal transformation is applied to the -axis and is used to linearise the scatterplot.

With as the dependent variable, the slope of the least squares regression line that is fitted to the linearised plot is closest to

A.   

B.     

C.         

D.        

E.      

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Show Worked Solution

q11 2010

♦♦ Mean mark 35%.

CORE*, FUR1 2013 VCAA 9 MC

The following information relates to the repayment of a home loan of $300 000.
 
• The loan is to be repaid fully with monthly payments of $2500.
• Interest compounds monthly.
• After the first monthly payment has been made, the amount owing on the loan is $299 000.

Which one of the following statements is true?

  1. After two months, $297 995 is still owing on the loan.
  2. $1000 of interest has been paid in the first month.
  3. The loan will be fully repaid in less than 15 years.
  4. Halfway through the term of the loan, the amount still owing will be $150 000.
  5. Payments of $2750 rather than $2500 per month will reduce the time to repay the loan fully by more than three years. 
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♦♦ Mean mark 29%.

 
 

 

 

PATTERNS, FUR 1 2006 VCAA 9 MC

A healthy eating and gym program is designed to help football recruits build body weight over an extended period of time.

Roh, a new recruit who initially weighs 73.4 kg, decides to follow the program.

In the first week he gains 400 g in body weight.

In the second week he gains 380 g in body weight.

In the third week he gains 361 g in body weight.

 If Roh continues to follow this program indefinitely, and this pattern of weight gain remains the same, his eventual body weight will be closest to

A.   

B.   

C.   

D.   

E.   

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Show Worked Solution

♦♦ Mean mark 33%.

 

 
 

 

CORE, FUR1 2006 VCAA 11-13 MC

The following information relates to Parts 1, 2 and 3.

The table shows the seasonal indices for the monthly unemployment numbers for workers in a regional town.

Part 1

The seasonal index for October is missing from the table.

The value of the missing seasonal index for October is

A.   

B.   

C.   

D.   

E.   

 

Part 2

The actual number of unemployed in the regional town in September is 330.

The deseasonalised number of unemployed in September is closest to

A.   

B.   

C.   

D.   

E.   

 

Part 3

A trend line that can be used to forecast the deseasonalised number of unemployed workers in the regional town for the first nine months of the year is given by

deseasonalised number of unemployed = 373.3 – 3.38 × month number

where month 1 is January, month 2 is February, and so on.

The actual number of unemployed for June is predicted to be closest to

A.   

B.   

C.   

D.   

E.   

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Show Worked Solution

 
 

 

 
 

 

♦♦ Mean mark 29%.
MARKERS’ COMMENT: 59% of students correctly found the deseasonalised number but failed to convert it to the actual.

 
 

PATTERNS, FUR1 2013 VCAA 9 MC

Eleven speed bumps are placed on a road.

The speed bumps are placed so that the distance between consecutive speed bumps decreases according to an arithmetic sequence.

The distance between the first and last speed bumps is exactly 100 m.

The smallest distance between consecutive speed bumps is 2 m.

The largest distance, in m, between two consecutive speed bumps is

A.  

B.  

C.  

D. 

E.  

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Show Worked Solution

♦♦ Mean mark 33%.
MARKER’S COMMENT: The not often used formula provides the most efficient route to this solution, although can also be used after finding the value of .

 

 

Graphs, MET2 2015 VCAA 11 MC

The transformation that maps the graph of    onto the graph of    is a

  1. dilation by a factor of from the -axis.
  2. dilation by a factor of from the -axis.
  3. dilation by a factor of from the -axis.
  4. dilation by a factor of from the -axis.
  5. dilation by a factor of from the -axis.
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♦♦ Mean mark 24%.
 

 

Mechanics, EXT2* M1 2008 HSC 7

EXT1 2008 HSC 7
 

A projectile is fired from    with velocity    at an angle of inclination    across level ground. The projectile passes through the points    and  , which are both    metres above the ground, at times    and    respectively. The projectile returns to the ground at  .

The equations of motion of the projectile are

  and   . (Do NOT prove this.)

  1. Show that    AND   .  (2 marks)

Let  α  and  β. It can be shown that
 
        and   .  (Do NOT prove this.)

  1. Show that  .  (2 marks)

  2. Show that  .  (1 mark)

Let   and  .

  1. Show that    and  .  (2 marks)

Let the gradient of the parabola at    be .

  1. Show that  .  (3 marks)

  2. Show that  .  (2 marks)

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i.   

 

 
 

 

ii.   

 
 
 
 

 

iii. 

 
 
 

 

iv.   

 

 

 

 
 
 

 

v.   
 
 
 
 
 
 

 

vi.   

 
 
 

 

Binomial, EXT1 2008 HSC 6c

Let    and    be positive integers with  .

  1. Use the binomial theorem to expand  , and hence write down the term of
     
  2.  which is independent of  .  (2 marks)
     

  3. Given that
     
  4. ,
     
    apply the binomial theorem and the result of part(i) to find a simpler expression for
     
  5. .  (3 marks)

 

 

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Show Worked Solution

(i)  

 

(ii)  


 


 


 

Calculus, 2ADV C3 2004 HSC 10b

2004 10b

 
The diagram shows a triangular piece of land    with dimensions   metres,  metres and    metres, where  .

The owner of the land wants to build a straight fence to divide the land into two pieces of equal area. Let    and    be points on    and    respectively so that    divides the land into two pieces of equal area.

Let   metres, metres and   metres.

  1. Show that  .  (1 mark)

  2. Use the cosine rule in triangle    to show that
     
           (2 marks)

  3. Show that the value of    in the equation in part (ii) is a minimum when
     
         .  (4 marks)

  4. Show that the minimum length of the fence is   metres, where  . 

     

    (You may assume that the value of    given in part (iii) is feasible.)  (2 marks)

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i.    
 

 

 

ii. 

   

 

iii.  
 
   

 

 

 

iv. 

 
 
 
 
 
 
 
 
 

 

Calculus, 2ADV C3 2004 HSC 9c

Consider the function  , for  .

  1. Show that the graph of    has a stationary point at  .  (2 marks)

  2. By considering the gradient on either side of  , or otherwise, show that the stationary point is a maximum.  (1 mark)

  3. Use the fact that the maximum value of    occurs at    to deduce that    for all  .  (2 marks)

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Show Worked Solution

i.     

 
 

 

 

 

ii.   

Geometry and Calculus, 2UA 2004 HSC 9c Answer

 

 

 

iii.  
   

MARKER’S COMMENT: Very challenging for most students. Successful students recognised the link to part (ii).

 

Calculus, EXT1* C1 2004 HSC 9b

A particle moves along the -axis. Initially it is at rest at the origin. The graph shows the acceleration, , of the particle as a function of time    for  .
 

2004 9b
 

  1. Write down the time at which the velocity of the particle is a maximum.  (1 marks)

  2. At what time during the interval    is the particle furthest from the origin? Give brief reasons for your answer.  (2 marks)

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i.   

 

 

ii. 

Calculus, 2ADV C4 2004 HSC 8b

The diagram shows the graph of the parabola  . The points    and    are on the parabola, and is the point where the tangent to the parabola at intersects the directrix.
 

2004 8s
 

  1. Write down the equation of the directrix of the parabola  .  (1 mark)
  2. Find the equation of the tangent to the parabola at the point .  (2 marks)
  3. Show that is the point  .  (1 mark)
  4. Given that the equation of the line is  , find the area bounded by the line and the parabola.  (2 marks)
  5. Hence, or otherwise, find the shaded area bounded by the parabola, the tangent at and the line .  (3 marks)
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  4. ²
  5. ²
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(i)   
   
 
 

 

 

(ii)  
 
 
 
   

 

 

(iii)  

 

 

(iv)  
 

 

²

 

(v)  

   
 

 

 
 
 
 
 
 
 

 

 

²

  ²

Polynomials, EXT2 2015 HSC 16b

Let    be a positive integer.

  1. By considering  , show that
  3.  
  4.  
  5. Let  ,  for    (2 marks)

  6.  
  7. Show that
    1.   (2 marks)

  9.  
  10. By considering the roots of  , find the value of
    1.   (3 marks)

    2.  
  11. Prove that
    1.   (2 marks)

 

 

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  4.  
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(i)   

 

 

(ii)  

 

♦♦♦ Mean mark 8%!
STRATEGY: Note that finding the correct roots gained 2 full marks in this part.

 

(iii) 

 

 

 

♦♦ Mean mark 14%.

 

(iv) 

 
 
 

Harder Ext 1 Topics, EXT2 2015 HSC 16a

  1. A table has rows and columns, creating cells as shown.
  2. Counters are to be placed randomly on the table so that there is one counter in each cell. There are identical black counters and identical white counters.
  4. Show that the probability that there is exactly one black counter in each column is   (2 marks)

  5. The table is extended to have rows and columns. There are counters, where are identical black counters and the remainder are identical white counters. The counters are placed randomly on the table with one counter in each cell.

  6. Let be the probability that each column contains exactly one black counter.

  7. Show that   (2 marks)
  10. Find   (2 marks)
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(i) 

 

 

 

(ii) 

♦♦ Mean mark 16%.

 

 

♦♦♦ Mean mark 5%.
COMMENT: Lowest State mean mark on the 2015 exam.
(iii)  
   
   
   
   

Proof, EXT2 P1 2015 HSC 15b

Suppose that    and    is a positive integer.

  1. Show that    (2 marks) 

  2. Hence, or otherwise, show that 
     
            (2 marks)

  3. Hence, explain why
     
          (1 mark)

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Show Worked Solution
i.   
 
 

 

♦♦ Mean mark 21%.
STRATEGY: The conversion of the middle term from a fraction into a logarithm should flag the need for integration of each term.
ii.   
 
 
 

 

 

iii.   
 
♦♦ Mean mark 29%.
 

Harder Ext1 Topics, EXT2 2006 HSC 8b

For  , let  , where    is an integer and  

  1. The two points of inflection of    occur at    and  , where  . Find    and    in terms of    (4 marks)

  2. Show that
    1.   (2 marks)

  3. Using the following
  4. If    then   (DO NOT prove this)
    2. show that    (2 marks)

  5. What can be said about the ratio    as    (1 mark)

 

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(i)
 
 
   
   

 

 

 

 

(ii)   ­
­
­

 

(iii)  

 

 

(iv)   
 

Calculus, EXT2 C1 2006 HSC 7b

  1. Let  ,  where  .  
     
    Show that     (3 marks)

  2. Hence find the exact value of
     
           (2 marks)

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Show Worked Solution
STRATEGY: The choice of and was extremely important in being able to answer this question. Take note!
i.   
 

­
 
 
 
 

 

 

ii.   ­
­
­
­

Harder Ext1 Topics, EXT2 2006 HSC 4d

In the acute-angled triangle    is the midpoint of    is the midpoint of    and    is the midpoint of  . The circle through    and    also cuts    at    as shown in the diagram.

Copy or trace the diagram into your writing booklet.

  1. Prove that    is a parallelogram.  (1 mark)
  2. Prove that    (1 mark)
  3. Prove that    (2 marks)
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Show Worked Solution
(i)  


   

 

(ii)  

 

(iii)  
 

 

 

 

Harder Ext1 Topics, EXT2 2007 HSC 8c

The diagram shows a regular  -sided polygon with vertices  . Each side has unit length. The length    of the ‘diagonal’    where    is given by

    (Do NOT prove this.)

  1. Show, using the identity,
  2.     (Do NOT prove this.)
    1. that     (2 marks)

  5. Let    be the perimeter of the polygon and  
  6. . 
  7. Show that     (2 marks)

  8. Hence calculate the limiting value of    as    (1 mark)

 

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(i)  

 

(ii)   
   
   
MARKER’S COMMENT: Applying the result    was poorly done.

 

(iii)  
   
   
   
   

Complex Numbers, EXT2 N2 2007 HSC 8b

  1. Let    be a positive integer. Show that if    then
     
        .  (2 marks)

  2. By substituting    where  , into part (i), show that
     
       
     
              (3 marks)

  3. Suppose  .  Using part (ii), show that
     
          (2 marks)

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i.  

 
 
 

 

ii.   
   
 
   

 

 
 
 
   

 
 
 

 

iii. 

Calculus, EXT2 C1 2007 HSC 8a

  1. Using a suitable substitution, show that
     
           (1 mark)

  2. A function    has the property that  

     

    Using part (i), or otherwise, show that
          
           (2 marks)

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i. 


 

 
 
MARKER’S COMMENT: Integrating was poorly done in part (ii). Pay careful attention to this in the Worked Solution.

 

ii. 

 
 
 
 

Proof, EXT2 P1 2007 HSC 7a

  1. Show that    for     (2 marks)

  2. Let  .

     

    Show that the graph of    is concave up for     (2 marks)

  3. By considering the first two derivatives of  ,show that    for     (2 marks)

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i.   
 
MARKER’S COMMENT: A geometric proof using arc length and a right angled triangle caused problems as few students could deal with the case  .

 

 

 

ii.
 
 
 

 

 

iii. 

 


 

 

Harder Ext1 Topics, EXT2 2013 HSC 10 MC

A hostel has four vacant rooms. Each room can accommodate a maximum of four people.

In how many different ways can six people be accommodated in the four rooms?

  1. 4020
  2. 4068
  3. 4080
  4. 4096
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♦♦♦ Mean mark 15%.
STRATEGY: The group of 5 can be in any of the four rooms, and note that the remaining person can then be in any of the 3 remaining rooms.

 

Harder Ext1 Topics, EXT2 2009 HSC 8c

A game is being played by    people, , sitting around a table. Each person has a card with their own name on it and all the cards are placed in a box in the middle of the table. Each person in turn, starting with  , draws a card at random from the box. If the person draws their own card, they win the game and the game ends. Otherwise, the card is returned to the box and the next person draws a card at random. The game continues until someone wins.

Let    be the probability that    wins the game.

Let  .

  1. Show that    (1 mark)
  2. Let    be a fixed positive integer and let    be the probability that    wins in no more than    attempts.
  3. Use  
  4. to show that, if    is large,  is approximately equal to    (3 marks)

 

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(i)   

 

 

(ii)  
 
 
 

 

 

Proof, EXT2 P2 2009 HSC 8a

  1. Using the substitution  ­, or otherwise, show that
     
    ­­  (2 marks)

  2. Use mathematical induction to prove that, for integers  ,
     
    ­­  (3 marks) 

  3. Show that  ­  (2 marks)

  4. Hence find the exact value of

  5.  

      (2 marks)

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i.   

­­

 
 
 
 
 
 

 

ii.  

­
­
 
­
 

 

­­

­­

­­­
  ­­
  ­­
  ­
 
 
  ­
 

 

 

iii.   ­

­

­

­

 

 

  ­

­

 

iv.   

­

­

Complex Numbers, EXT2 N2 2009 HSC 7b

Let  

  1. Show that  , where    is a positive integer.  (2 marks)

  2. Let    be a positive integer. Show that
     

     
          (3 marks)

  3. Hence, or otherwise, prove that
     
       
     
    where    is a positive integer.  (2 marks)

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i.   
 
 
   
 
   

 

 

ii. 

 

iii. 

Harder Ext1 Topics, EXT2 2009 HSC 6c

The diagram shows a circle of radius  , centred at the origin, . The line    is tangent to the circle at  , the line    is horizontal, and    lies on the line  .

  1. Find the length of    in terms of    (1 mark)
  2. The point    moves such that  .
  3. Show that the equation of the locus of    is
    1.   (2 marks)

  4. Find the focus, , of the parabola in part (ii).   (2 marks)
  5. Show that the difference between the length    and the length    is independent of    (2 marks)
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(i)   
 
 
 

 

(ii) 

 

 

(iii) 

♦♦♦ “Few” students answered part (iii) correctly (exact data unavailable).
 

 

 

  

(iv) 

♦♦♦ “Few” students answered part (iv) correctly (exact data unavailable).
MARKER’S COMMENT: Very few used the efficient focus-directrix definition here.
 

 

 

 

 
 

 

Integration, EXT2 2010 HSC 8

Let

 and  ,

where    is an integer,  .  (Note that , .)

  1. Show that
  2.  for  .   (2 marks)

  3. Using integration by parts on  , or otherwise, show that
    1.  for  .   (1 mark)

  4. Use integration by parts on the integral in part (ii) to show that
    1.  for  .   (3 marks)

  5. Use parts (i) and (iii) to show that
    1.  for  .   (1 mark)

  6. Show that
    1. .   (2 marks)

  7. Use the fact that
  8.  for    to show that

    1. .   (1 mark)

  9. Show that
  10. .   (1 mark)

  12. From parts (vi) and (vii) it follows that
    1. .

  13. Use the substitution    in this inequality to show that
  14.  
    1. .   (2 marks)

  15. Use part (v) to deduce that
    1. .   (1 mark)

  16. What is
  17. ?   (1 mark)

 

 

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♦♦ Mean mark part (i) 21%.
(i)   
 
 
 
 
 
 
 
   

 

 
 
 

 

♦♦ Mean mark part (ii) 32%.

 

(ii)   
 
 
 
 
 
 
♦♦♦ Mean mark part (iii) 8%.

 

(iii)   
 
   
   
   
 
 

 

 
 
 
 
 
♦♦♦ Mean mark part (iv) 13%.

 

(iv)  
 
   
   
   
♦♦♦ Mean mark part (v) 9%.

 

(v)  
 
   

 
♦♦♦ Mean mark part (vi) 8%.

 

(vi) 
 

 

 

(vii)  
 
 

 

♦♦♦ Mean mark part (vii) 2%.

 

♦♦♦ Mean mark part (viii) 4%.
(viii)  
 
 

 

 
 
 

 

 
 

 

♦♦♦ Mean mark part (ix) 3%.

(ix) 

,

 

 

♦♦♦ Mean mark part (x) 19%.

 

(x) 

Polynomials, EXT2 2010 HSC 7c

Let  , where    is an odd integer,  .

  1. Show that    has exactly two stationary points.  (1 mark)
  2. Show that    has a double zero at  .  (1 mark)
  3. Use the graph    to explain why    has exactly one real zero other than  .  (2 marks)
  4. Let  α  be the real zero of    other than  .
  5. Given that    for , or otherwise, show that  α.  (2 marks)

  6. Deduce that each of the zeros of    has modulus less than or equal to  .  (2 marks)
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(i)  
   

 

 

(ii)   
 

 

 

(iii)  

♦♦ Mean mark part (iii) 26%.

 

 Polynomials, EXT2 2010 HSC 7c Answer1

♦♦♦ Mean mark part (iv) 3%.
(iv)   
   
   
   

 

 
 
 
 

 

α

 

(v)  

♦♦♦ Mean mark part (v) 2%.

 

 

α

   
   
   

 

α

Harder Ext1 Topics, EXT2 2010 HSC 7a

In the diagram    is a cyclic quadrilateral. The point    is on    such that  , and hence    is similar to  .

Harder Ext1 Topics, EXT2 2010 HSC 7ai

Copy or trace the diagram into your writing booklet.

  1. Show that    is similar to  .  (2 marks)
  2. Using the fact that  , 
  3. show that  .  (2 marks)
  5. A regular pentagon of side length    is inscribed in a circle, as shown in the diagram.

Harder Ext1 Topics, EXT2 2010 HSC 7aii

  1. Let    be the length of a chord in the pentagon.
  3. Use the result in part (ii) to show that  .  (2 marks)

 

 

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(i)    Harder Ext1 Topics, EXT2 2010 HSC 7a Answer
   

 

 

(ii)  

♦♦ Mean mark part (ii) 31%.

 

 

  

(iii) 

♦♦ Mean mark part (iii) 24%.

 

Harder Ext1 Topics, EXT2 2010 HSC 5c

A TV channel has estimated that if it spends    on advertising a particular program it will attract a proportion    of the potential audience for the program, where

and   is a given constant.

  1. Explain why
     has its maximum value when  .   (1 mark)

  2. Using
  3. , or otherwise, deduce that
  4.  
  5.   for some constant  .   (3 marks)
  6.  

  8. The TV channel knows that if it spends no money on advertising the program then the audience will be one-tenth of the potential audience.
  9. Find the value of the constant    referred to in part (c)(ii).   (1 mark)

  10. What feature of the graph
     is determined by the result in part (c)(i)?   (1 mark)

  11.  
  12. Sketch the graph
      (1 mark) 

 

 

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(i)  

♦♦ Mean mark part (i) 32%.

 

♦ Mean mark part (ii) 41%.

 

(ii)  

 

 

(iii)  
♦♦ Mean mark part (iv) 25%.

 

(iv)  

 

(v)  

♦♦ Mean mark part (v) 12%.

Harder Ext1 Topics, EXT2 2010 HSC 5c

Harder Ext1 Topics, EXT2 2010 HSC 4c

Let    be a real number, .

Show that, for every positive real number  , there is a positive real number    such that  .   (3 marks) 

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♦♦♦ Mean mark 13%.

 

 

 

 

Conics, EXT2 2010 HSC 3d

The diagram shows the rectangular hyperbola  , with  .

Conics, EXT2 2010 HSC 3d

The points  ,  and    are points on the hyperbola,
with  .

  1. The line    is the line through    perpendicular to  .
  2. Show that the equation of    is
    1. .   (2 marks)

  4. The line    is the line through    perpendicular to  .
  5. Write down the equation of  .   (1 mark)
  6. Let    be the point of intersection of the lines    and  .
  7. Show that    is the point  .   (2 marks)
  8. Give a geometric description of the locus of  .   (1 mark)
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(i) 

Conics, EXT2 2010 HSC 3d Answer1
 
 
 

 

 

 

 

(ii)  
   
   
   

 

 

 

 

(iii) 

 

 

 

(iv) 

♦♦ Mean mark part (iv) 1%.
MARKER’S COMMENT: Most students found the correct locus, although very few indicated that it was only the branch in the 1st quadrant.

 

 

Harder Ext1 Topics, EXT2 2010 HSC 3c

Two identical biased coins are each more likely to land showing heads than showing tails.

The two coins are tossed together, and the outcome is recorded. After a large number of trials it is observed that the probability that the two coins land showing a head and a tail is  .

What is the probability that both coins land showing heads?   (2 marks)

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♦♦♦ Mean mark 19%.

 

Polynomials, EXT2 2011 HSC 8c

Let be a root of the complex monic polynomial

Let    be the maximum value of  

  1. Show that
  2.   (2 marks)

  4. Hence, show that for any root    of  
    1.   (3 marks)

  6. Let  , where the real numbers    satisfy  for all  , and  
  8. Using parts (i) and (ii), or otherwise, show that    has no real solutions.  (3 marks)
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(i)  

♦♦♦ Mean mark part (i) 7%.

 
 

 

(ii) 

♦♦♦ Mean mark part (ii) 3%.

 

 

 

 

(iii)   
 

 

♦♦♦ Mean mark part (iii) 2%.

 

 

 

 
 

 

 

HSC 2011 8d

 

 

Harder Ext1 Topics, EXT2 2011 HSC 8b

A bag contains seven balls numbered from to . A ball is chosen at random and its number is noted. The ball is then returned to the bag. This is done a total of seven times.

  1. What is the probability that each ball is selected exactly once?  (1 mark)
  2. What is the probability that at least one ball is not selected?  (1 mark)
  3. What is the probability that exactly one of the balls is not selected?  (2 marks)
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(i)  

♦♦ Mean mark part (i) 32%.

 

(ii)  

 

(iii) 

♦♦♦ Mean mark part (iii) 2%.