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Financial Maths, STD2 F4 2020 HSC 22

Nisa has a credit card on which interest at 17% per annum, compounded daily, is charged on the amount owing.

At the beginning of the month, Nisa owes $500 on her credit card. She makes no other purchases using the credit card, but fifteen days later, she repays $250.

Assuming that interest is charged for the fifteen days, calculate the amount owing on the credit card immediately after the $250 payment is made.   (3 marks)

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Financial Maths, STD2 F4 2020 HSC 21

The inflation rate over the year from January 2019 to January 2020 was 2%.

The cost of a school jumper in January 2020 was $122.

Calculate the cost of the jumper in January 2019 assuming that the only change in the cost of the jumper was due to inflation.   (2 marks)

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Financial Maths, STD2 F1 2020 HSC 20

The table shows the income tax rates for the 2019 – 2020 financial year.

For the 2019 – 2020 financial year, Wally had a taxable income of $122 680. During the year, he paid $3000 per month in Pay As You Go (PAYG) tax.

Calculate Wally's tax refund, ignoring the Medicare levy.   (3 marks)

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Probability, 2ADV S1 2020 HSC 14

History and Geography are two of the subjects students may decide to study. For a group of 40 students, the following is known.

    • 7 students study neither History nor Geography
    • 20 students study History
    • 18 students study Geography
  1. A student is chosen at random. By a using a Venn diagram, or otherwise, find the probability that the student studies both History and Geography.  (2 marks)

  2. A students is chosen at random. Given that the student studies Geography, what is the probability that the student does NOT study History?  (1 mark)

  3. Two different students are chosen at random, one after the other. What is the probability that the first student studies History and the second student does NOT study History?  (2 marks)

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a.   
 

 

♦ Mean mark (b) 49%.
b.   
   

 

c.   
   

Networks, STD2 N2 2020 HSC 18

The diagram represents a network with weighted edges.
 


 

  1. Draw a minimum spanning tree for this network and determine its length.   (3 marks)

  2. The network is revised by adding another vertex, . Edges and have weights of 12 and 10 respectively, as shown.
     

   
 

What is the length of the minimum spanning tree for this revised network?   (1 mark)

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Calculus, 2ADV C3 2020 HSC 25

A landscape gardener wants to build a garden in the shape of a rectangle attached to a quarter-circle. Let and be the dimensions of the rectangle in metres, as shown in the diagram.
 


 

The garden bed is required to have an area of 36 m² and to have a perimeter which is as small as possible. Let metres be the perimeter of the garden bed.

  1. Show that  .  (3 marks)

  2. Find the smallest possible perimeter of the garden bed, showing why this is the minimum perimeter.  (4 marks)

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a.   
 
 
 

♦ Mean mark part (a) 47%.

 

 
 
 

 

b.   

 

 

 

 

Statistics, 2ADV S3 2020 HSC 23

A continuous random variable, , has the following probability density function.
 


 

  1. Find the value of .  (2 marks)

  2. Find  . Give your answer correct to four decimal places.  (2 marks)

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♦ Mean mark part (b) 44%.
b.   
   
   
   
   
   

Calculus, 2ADV C4 2020 HSC 20

Kenzo is driving his car along a road while his friend records the velocity of the car, , in km/h every minute over a 5-minute period. The table gives the velocity    at time    hours.
 

 

The distance covered by the car over the 5-minute period is given by

.

Use the trapezoidal rule and the velocity at each of the six time values to find the approximate distance in kilometres the car has travelled in the 5-minute period. Give your answer correct to one decimal place.  (2 marks)

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Trigonometry, 2ADV T1 2020 HSC 15

Mr Ali, Ms Brown and a group of students were camping at the site located at . Mr Ali walked with some of the students on a bearing of 035° for 7 km to location . Ms Brown, with the rest of the students, walked on a bearing of 100° for 9 km to location .
 


 

  1. Show that the angle is 65°.  (1 mark)

  2. Find the distance .  (2 marks)

  3. Find the bearing of Ms Brown's group from Mr Ali's group. Give your answer correct to the nearest degree.  (2 marks)

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b.   

 
 
 

 
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Measurement, STD2 M7 2020 HSC 17

Ayla wishes to estimate the number of trees on a square block of land measuring 1000 m by 1000 m. She counts the number of trees on a 5 m by 5 m section of the block and finds there are 8 trees.

Based on this, estimate the number of tress on the entire square block of land.   (2 marks)

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Statistics, 2ADV S3 2020 HSC 9 MC

Suppose the weight of melons is normally distributed with a mean of and a standard deviation of .

A melon has a weight below the lower quartile of the distribution but NOT in the bottom 10% of the distribution.

Which of the following most accurately represents the region in which the weight of this melon lies?
 

A. B.
C. D.
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Statistics, 2ADV S3 2020 HSC 3 MC

John recently did a class test in each of three subjects. The class scores on each test were normally distributed.

The table shows the subjects and John's scores as well as the mean and standard deviation of the class scores on each test.
 

 
Relative to the rest of class, which row of the table below shows John's strongest subject and his weakest subject?
 

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Statistics, STD2 S5 2020 HSC 8 MC

John recently did a class test in each of three subjects. The class scores on each test were normally distributed.

The table shows the subjects and John's scores as well as the mean and standard deviation of the class scores on each test.

Relative to the rest of class, which row of the table below shows John's strongest subject and his weakest subject?

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Mechanics, EXT2 M1 EQ-Bank 2

A particle with mass moves horizontally against a resistance force , equal to    where is the particle's velocity.

Initially, the particle is travelling in a positive direction from the origin at velocity .

  1. Show that the particle's displacement from the origin, , can be expressed as
     
           (2 marks)

     

  2. Show that the time, , when the particle is travelling at velocity , is given by
     
           (4 marks)

     

  3. Express as a function of , and hence find the limiting values of and .  (2 marks)

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Mechanics, EXT2 M1 EQ-Bank 1

A canon ball of mass 9 kilograms is dropped from the top of a castle at a height of metres above the ground.

The canon ball experiences a resistance force due to air resistance equivalent to  , where is the speed of the canon ball in metres per second. Let    and the displacement, metres at time seconds, be measured in a downward direction.

  1. Show the equation of motion is given by
     
           (1 mark)

  2. Show, by integrating using partial fractions, that
     
           (5 marks)

     

  3. If the canon hits the ground after 4 seconds, calculate the height of the castle, to the nearest metre.  (3 marks)

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ii.   
   

 

 


 


 


 

 
 
 

 

 

 

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Measurement, STD2 M7 SM-Bank 19

A laundromat can wash 12 loads of laundry in one hour at full capacity.

A standard load of laundry weighs 7 kilograms.

Here is some information about two different washing machines.
 


 

Working at full capacity, how many litres of water would the laundromat expect to save in one hour by using the front loader instead of the top loader?   (2 marks)

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Functions, 2ADV F2 EQ-Bank 13

The curve    is subject to the following transformations

    • Translated 2 units in the positive -direction
    • Dilated in the positive -direction by a factor of 4
    • Reflected in the -axis

The final equation of the curve is  .

  1.  Find the equation of the graph after the dilation.  (1 mark)

  2.  Find the values of    and  .  (2 marks)

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ii.   
   

 

 

COMMENT: Using “swap” terminology for reflections in the y-axis is simpler and more intelligible for students in our view.

 

 

 

 
 

 

Statistics, STD2 S4 EQ-Bank 4

Ten high school students have their height and the length of their right foot measured.

The results are recorded in the table below.
 


 

  1. Using technology, calculate Pearson's correlation coefficient for the data. Give your answer to 3 decimal places.  (1 mark)

  2. Describe the strength of the association between height and length of right foot for these students.  (1 mark)

  3. Using technology, determine the least squares regression line that allows height to be predicted from right foot length.  (1 mark)

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i.   

COMMENT: Issues here? YouTube has short and excellent help videos – search your calculator model and topic – eg. “fx-82 correlation” .

 

 

ii.   

 

iii.   

Functions, 2ADV F2 EQ-Bank 16

  has been produced by three successive transformations: a translation, a dilation and then a reflection.

  1. Describe each transformation and state the equation of the graph after each transformation.  (2 marks)

  2. Sketch the graph.  (1 mark)

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ii.   

Functions, 2ADV F1 EQ-Bank 8

Jacques is a marine biologist and finds that the mass of a crab is directly proportional to the cube of the diameter of its shell.

If a crab with a shell diameter of 15 cm weighs 680 grams, what will be the diameter of a crab that weighs 1.1 kilograms? Give your answer to 1 decimal place.  (2 marks)

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Statistics, 2ADV S3 EQ-Bank 2

Let  denote a normal random variable with mean 0 and standard deviation 1.

The random variable has the probability density function

   where 

The diagram shows the graph of  .
 

 
 

  1. Complete the table of values for the given function, correct to four decimal places.  (1 mark)
     

     
  2. Use the trapezoidal rule and 5 function values in the table in part i. to estimate

     

            (2 marks)

  3. The weights of Rhodesian ridgebacks are normally distributed with a mean of 48 kilograms and a standard deviation of 6 kilograms.

     

    Using the result from part ii., calculate the probability of a randomly selected Rhodesian ridgeback weighing less than 36 kilograms.  (2 marks)

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ii. 

iii. 

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Statistics, 2ADV S2 EQ-Bank 3

The table below lists the average life span (in years) and average sleeping time (in hours/day) of 9 animal species.
 


 

  1. Using sleeping time as the independent variable, calculate the least squares regression line. (1 mark)

  2. A wallaby species sleeps for 4.5 hours, on average, each day.

     

    Use your equation from part i to predict its expected life span, to the nearest year.   (1 mark)

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i.   

COMMENT: Issues here? YouTube has short and excellent help videos – search your calculator model and topic – eg. “fx-82 regression line” .


 

ii.   

 

Calculus, 2ADV C4 EQ-Bank 5

The velocity of a particle moving along the -axis at metres per second at seconds, is shown in the graph below.
 


 

Initially, the displacement is equal to 12 metres.

  1. Write an equation that describes the displacement, , at time seconds.  (2 marks)

  2. Draw a graph that shows the displacement of the particle,   metres from the origin, at a time seconds between    and  . Label the coordinates of the endpoints of your graph.  (2 marks)

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Financial Maths, 2ADV M1 EQ-Bank 1

Ralph opens an annuity account and makes a contribution of $12 000 at the end of each year for 9 years.

For the first 8 years, the interest rate is 4% per annum, compounded annually.

For the 9th year, the interest rate decreases to 3% per annum, compounded annually.
 


 

Use the Future Value of an Annuity table to calculate the amount in the account immediately after the 9th contribution is made.  (3 marks)

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Statistics, 2ADV S2 EQ-Bank 2

The table below lists the average body weight (in kilograms) and average brain weight (in grams) of nine animal species.
 


 

A least squares regression line is fitted to the data using body weight as the independent variable.

  1. Calculate the equation of the least squares regression line. (1 mark)

  2. If dingos have an average body weight of 22.3 kilograms, calculate the predicted average brain weight of a dingo using your answer to part i.   (1 mark)

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i.   

COMMENT: Know this critical calculator skill!.

 

ii.   

 

Statistics, 2ADV S2 EQ-Bank 1

The arm spans (in cm) and heights (in cm) for a group of 13 boys have been measured. The results are displayed in the table below.
 

CORE, FUR2 2008 VCAA 4

The aim is to find a linear equation that allows arm span to be predicted from height.

  1. What will be the independent variable in the equation?  (1 mark)

  2. Assuming a linear association, determine the equation of the least squares regression line that enables arm span to be predicted from height. Write this equation in terms of the variables arm span and height. Give the coefficients correct to two decimal places.  (2 marks)

  3. Using the equation that you have determined in part b., interpret the slope of the least squares regression line in terms of the variables height and arm span.  (1 mark)

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a.  

COMMENT: Calculator skills for finding the least squares regression line were required in NESA sample exam – know this critical skill well!

 

b.   

 

c.   

Statistics, 2ADV S3 EQ-Bank 1

A probability density function can be used to model the lifespan of a termite, , in weeks, is given by
 


 

  1. Show that the value of    is  .  (2 marks)

  2. Find the cumulative distribution function.  (2 marks)

  3. Find the probability that a termite's lifespan is greater than 5 weeks.  (1 mark)

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Calculus, 2ADV C4 EQ-Bank 2

The population, , of Tasmanian Devils in a sanctuary is given by  , where    is the time in years after the sanctuary was established.

The devil population changes at a rate modelled by the function  .

Calculate the increase in the number of Tasmanian Devils at the end of the first 8 years. Give your answer correct to three significant figures.  (3 marks)

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