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GRAPHS, FUR1 2021 VCAA 5 MC

A manufacturer makes and sells heaters.

The fixed cost to manufacture the heaters is $16 000 per month.

Each heater costs $52 to produce.

The selling price of each heater is $280.

The minimum number of heaters needed to be sold per month in order to make a profit is

  1. 48
  2. 49
  3. 57
  4. 70
  5. 71
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MATRICES, FUR1 2021 VCAA 6 MC

A fitness centre offers four different exercise classes: aerobics , boxfit , cardio and dance .

A customer's choice of fitness class is expected to change from week to week according to the transition matrix , shown below.


 

An equivalent transition diagram has been constructed below, but the labeling is not complete.
 

The proportion for one of the transitions is labelled .

The value of is

  1. 10%
  2. 15%
  3. 20%
  4. 25%
  5. 30%
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MATRICES, FUR1 2021 VCAA 2 MC

Every Friday, the same number of workers from a large office building regularly purchase their lunch from one of two locations: the deli, , or the cafe, .

It has been found that:

    • of the workers who purchase lunch from the deli on one Friday, 65% will return to purchase from the deli on the next Friday
    • of the workers who purchase lunch  from the cafe on one Friday, 55% will return to purchase from the cafe on the next Friday.

A transition matrix that can be used to describe this situation is

A. 
       		 B. 
       		  
     
C. 
       		 D. 
       		  
     
E. 
       		    
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Calculus, MET1 2021 VCAA 8

The gradient of a function is given by  .

The graph of the function has a single stationary point at  .

  1. Find the rule of the function.   (3 marks)

  2. Determine the nature of the stationary point.   (2 marks)

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a.   
 
   

 

 
 
 
   

 

♦ Mean mark part (b) 37%.

b. 
      

Graphs, MET1 2021 VCAA 4

  1. Sketch the graph of    on the axes below. Label asymptotes with their equations and axis intercepts with their coordinates.   (3 marks)

     

     

  2. Find the values of    for which  .   (1 mark)

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a. 

b.

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a.   

♦ Mean mark part (b) 32%.

b.   

Probability, MET1 2021 VCAA 6

An online shopping site sells boxes of doughnuts.

A box contains 20 doughnuts. There are only four types of doughnuts in the box. They are:

    • glazed, with custard
    • glazed, with no custard
    • not glazed, with custard
    • not glazed, with no custard

It is known that, in the box:

    •   of the doughnuts are with custard
    •   of the doughnuts are not glazed
    •   of the doughnuts are glazed, with custard
  1. A doughnut is chosen at random from the box.
  2. Find the probability that it is not glazed, with custard.  (1 mark)

  3. The 20 doughnuts in the box are randomly allocated to two new boxes, Box A and Box B.
  4. Each new box contains 10 doughnuts.
  5. One of the two new boxes is chosen at random and then a doughnut from that box is chosen at random.
  6. Let be the number of glazed doughnuts in Box A.
  7. Find the probability, in terms of , that the doughnut comes from box B given that it is glazed.  (2 marks)

  8. The online shopping site has over one million visitors per day.
  9. It is know that half of these visitors are less than 25 years old.
  10. Let    be the random variable representing the proportion of visitors who are less than 25 years old in a random sample of five visitors.
  11. Find  . Do not use a normal approximation.  (3 marks)

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a.   

 
   

 
b.   

♦♦♦ Mean mark part (b) 20%.

 
   
   
   

 
c.   

♦ Mean mark part (c) 40%.

 
   
   
   

CORE, FUR1 2021 VCAA 22 MC

Joanna deposited $12 000 in an investment account earning interest at the rate of 2.8% per annum, compounding monthly.

She would like this account to reach a balance of $25 000 after five years.

To achieve this balance, she will make an extra payment into the account each month, immediately after the interest is calculated.

The minimum value of this payment is closest to

  1. $113.85
  2. $174.11
  3. $580.16
  4. $603.22
  5. $615.47
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CORE, FUR1 2021 VCAA 16 MC

The number of visitors to a regional animal park is seasonal.

Data is collected and deseasonalised before a least squares line is fitted.

The equation of the least squares line is

deseasonalised number of visitors = 2349 – 198.5 × month number

where month number 1 is January 2020.

The seasonal indices for the 12 months of 2020 are shown in the table below.
 

The actual number of visitors predicted for February 2020 was closest to

  1. 1562
  2. 1697
  3. 1952
  4. 2245
  5. 2440
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Mean mark 52%.


 


 

CORE, FUR1 2021 VCAA 15 MC

The table below shows the number of visitors to an art gallery during the summer, autumn, winter and spring quarters for the years 2017 to 2019.

The quarterly average is also shown for each of these years.
 

   

The seasonal index for summer is closest to

  1. 1.077
  2. 1.081
  3. 1.088
  4. 1.092
  5. 1.096
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CORE, FUR1 2021 VCAA 10 MC

Oscar walked for nine consecutive days. The time, in minutes, that Oscar spent walking on each day is shown in the table below.
 

At least squares line is fitted to the data.

The equation of this line predicts that on day 10 the time Oscar spends walking will be the same as the time he spent walking on

  1. day 3
  2. day 4
  3. day 6
  4. day 8
  5. day 9
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CORE, FUR1 2021 VCAA 9 MC

The heights of females living in a small country town are normally distributed:

    • 16% of the females are more than 160 cm tall.
    • 2.5% of the females are less than 115 cm tall.

The mean and the standard deviation of this female population, in centimetres, are closest to

  1. mean = 135               standard deviation = 15
  2. mean = 135               standard deviation = 25
  3. mean = 145               standard deviation = 15
  4. mean = 145               standard deviation = 20
  5. mean = 150               standard deviation = 10
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CORE, FUR1 2021 VCAA 1-3 MC

The percentaged segmented bar chart below shows the age (under 55 years, 55 years and over) of visitors at a travel convention, segmented by preferred travel destination (domestic, international).
 

Part 1

The variables age (under 55 years, 55 years and over) and preferred travel destination (domestic, international) are

  1. both categorial variables.
  2. both numerical variables.
  3. a numerical variable and a categorical variable respectively.
  4. a categorical variable and a numerical variable respectively.
  5. a discrete variable and a continuous variable respectively.

 
Part 2

The data displayed in the percentaged segmented bar chart supports the contention that there is an association between preferred travel destination and age because

  1. more visitors favour international travel.
  2. 35% of visitors under 55 years favour international travel.
  3. 45% of visitors 55 years and over favour domestic travel.
  4. 65% of visitors under 55 years favour domestic travel while 45% of visitors 55 years and over favour domestic travel.
  5. the percentage of visitors who prefer domestic travel is greater than the percentage of visitors who prefer international travel.

 
Part 3

The results could also be summarised in a two-way frequency table.

Which one of the following frequency tables could match the pecentaged segmented bar chart?

 

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♦ Mean mark 50%.

CORE, FUR1 2021 VCAA 7-8 MC

800 participants auditioned for a stage musical. Each participant was required to complete a series of ability tests for which they received an overall score.

The overall scores were approximately normally distributed with a mean score of 69.5 points and a standard deviation of 6.5 points.
 

Part 1

Only the participants who scored at least 76.0 points in the audition were considered successful,

Using the 68-95-99.7% rule, how many of the participants were considered unsuccessful?

  1. 127
  2. 128
  3. 272
  4. 672
  5. 673

 
Part 2

To be offered a leading role in the stage musical, a participant must achieve a standardised score of at least 1.80

Three participants' names and their overall scores are given in the table below.

Which one of the following statements is true?

  1. Only Amy was offered a leading role.
  2. Only Cherie was offered a leading role.
  3. Only Brian was not offered a leading role
  4. Both Brian and Cherie were offered leading roles.
  5. All three participants were offered leading roles.
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CORE, FUR1 2021 VCAA 4 MC

The boxplots below show the distribution of the length of fish caught in two different ponds, Pond A and Pond B.
 

Based on the boxplots above, it can be said that.

  1. 50% of the fish caught in Pond A are the same length as the fish caught in Pond B.
  2. 50% of the fish caught in Pond B are longer than all of the fish caught in Pond A.
  3. 50% of the fish caught in Pond B are shorter than all of the fish caught in Pond A.
  4. 75% of the fish caught in Pond A are shorter than all of the fish caught in Pond B.
  5. 75% of the fish caught in Pond B are longer than all of the fish caught in Pond A.
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Algebra, MET1 2021 VCAA 5

Let    and  .

  1. The graphs of and have a common horizontal axis intercept at .
  2. Find the coordinates of the other horizontal axis intercept of the graph of .   (2 marks)

  3. Let the graph of be a transformation of the graph of where the transformations have been applied in the following order:
    • dilation by a factor of    from the vertical axis (parallel to the horizontal axis)
    • translation by two units to the right (in the direction of the positive horizontal axis
  4. State the rule of and the coordinates of the horizontal axis intercepts of the graph of .   (2 marks)

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a.   

 
 
 

 


 

b.   

♦♦ Mean mark part (b) 25%.


 


 

 
 
 

 

Functions, MET1 2021 VCAA 3

Consider the function 

  1. State the range of .   (1 mark)

  2. State the period of .   (1 mark)

  3. Solve    for  .   (3 marks)

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a.   


 

b.   
 

c.   
 
 
 

 

Vectors, EXT1 V1 SM-Bank 30

A man attempts to swim north across a river at a speed of 4.5 km/h.

If the river's current is moving at 11 km/h due east, find the bearing  (nearest degree) and speed (to 1 decimal place) that the man will actually be swimming.  (3 marks)

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Mechanics, EXT2 M1 2021 HSC 14b

An object of mass 5 kg is on a slope that is inclined at an angle of 60° to the horizontal. The acceleration due to gravity is   and the velocity of the object down the slope is .

As well as the force due to gravity, the object is acted on by two forces, one of magnitude newtons and one of magnitude newtons, both acting up the slope.

  1. Show that the resultant force down the slope is    newtons.  (2 marks)

  2. There is one value of such that the object will slide down the slope at a constant speed.
  3. Find this value of in , correct to 1 decimal place, given that  .  (2 marks)

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i.   

 
 
 

 

ii.   

 

 
 
 
 

Mechanics, EXT2 M1 2021 HSC 15c

An object of mass 1 kg is projected vertically upwards with an initial velocity of m/s. It experiences air resistance of magnitude newtons where is the velocity of the object, in m/s, and is a positive constant. The height of the object above its starting point is metres. The time since projection is seconds and acceleration due to gravity is m/s².

  1. Show that the time for the object to reach its maximum height is    seconds.  (3 marks)

  2. Find an expression for the maximum height reached by the object, in terms of , , and .  (3 marks)

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i.   
 
   
   
   
 
   

 


  


 

ii.   

Mean mark part (ii) 54%.
 

 

 
 

Calculus, EXT2 C1 2021 HSC 13c

  1. The integral    is defined by    for integers  .
    Show that    for   .  (2 marks)

  2. The diagram shows two regions.

Region is bounded by    and    between    and  .

Region is bounded by    and    between    and  .
 
   
 
The volume of the solid created when the region between the curve    and the  -axis, between    and  , is rotated about the -axis is given by  .

The volume of the solid of revolution formed when region is rotated about the -axis is .

The volume of the solid of revolution formed when region is rotated about the -axis is .

Using part (i), or otherwise, show that the ratio  is .  (4 marks)

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i.   
 
 

 

ii.   
   
   
   
   

 

 
 
 
 
 
 
 
 
 

Vectors, EXT2 V1 2021 HSC 12e

The diagram shows the pyramid    where    is a square. The diagonals of the square bisect each other at .
 

  1. Show that     (1 mark)

    Let be the point such that  .

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

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

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i.    


 

 
 
 
ii.    

 
 
 
iii. 
♦ Mean mark 48%.

 

λ

Complex Numbers, EXT2 N2 2021 HSC 11d

  1. Find the two square roots of  , giving the answers in the form  , where and are real numbers.  (2 marks)

  2. Hence, or otherwise, solve    giving your solutions in the form    where and are real numbers. (2 marks)

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i.     


 

 
 

 

 


 

 

ii.   

 
   
   

 

Mechanics, EXT2 V1 2021 HSC 8 MC

A particle is travelling from  on the curve joining to . At a particular time, the particle is at point and has velocity  , as shown in the diagram. 
 
The speed of the particle is increasing.
 
Which of the following diagrams shows an acceleration,  , which would allow the particle to follow the curve to ?
 

 
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Trigonometry, EXT1 T2 2021 HSC 13d

  1. The numbers  ,    and    are related by the equations    and  ,  where    is a constant.
  2. Show that  .  (2 marks)

  3. Hence, or otherwise, solve    for  .  (2 marks)

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i.   
   
   
   
   

 

ii.   

♦ Mean mark 50%.
 
 

 

Functions, EXT1 F1 2021 HSC 12d

A function is defined by    for    in the domain  .

  1. Sketch the graph of    showing the  -  and  -intercepts.  (2 marks)

  2. Find the equation of the inverse function,  ,  and state its domain.  (3 marks)

  3. Sketch the graph of   .   (1 mark)

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  4.  
     
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i.   


 

 

ii.   

 

 

 

iii.

Calculus, EXT1 C1 2021 HSC 12b

A bottle of water, with temperature 5°C, is placed on a table in a room. The temperature of the room remains constant at 25°C. After minutes, the temperature of the water, in degrees Celsius, is .

The temperature of the water can be modelled using the differential equation

   (Do NOT prove this.)

where is the growth constant

After 8 minutes, the temperature of the water is 10°C.

  1. By solving the differential equation, find the value of when the temperature of the water reaches 20°C. Give your answer to the nearest minute.  (3 marks)

  2. Sketch the graph of as a function of .  (1 mark)

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  2.  
       
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i.   
 
 
 

 

 

 

 

 

 
 

 

ii.

♦ Mean mark part (ii) 37%.

   

Vectors, EXT1 V1 2021 HSC 13b

When an object is projected from a point metres above the origin with initial speed m/s at an angle of    to the horizontal, its displacement vector, seconds after projection, is

.     (Do NOT prove this.)

A person, standing in an empty room which is 3 m high, throws a ball at the far wall of the room. The ball leaves their hand 1 m above the floor and 10 m from the far wall. The initial velocity of the ball is 12 m/s at an angle of 30° to the horizontal.

Show that the ball will NOT hit the ceiling of the room but that it will hit the far wall without hitting the floor.  (4 marks)

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Calculus, EXT1 C1 2021 HSC 11e

A spherical bubble is moving up through a liquid. As it rises, the bubble gets bigger and its radius increases at the rate of 0.2 mm/s.

At what rate is the volume of the bubble increasing when its radius reaches 0.6 mm? Express your answer in mm³/s rounded to one decimal place.  (2 marks)

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³

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  ³

Probability, STD1 S2 2021 HSC 20

In a bag, there are six playing cards, 2, 4, 6, 8, Queen and King. The Queen and King are known as picture cards.

Two of these cards are chosen randomly. All the possible outcomes are shown.
 

   
 

  1. What is the probability that the two cards chosen include one or both picture cards?  (1 mark)

  2. What is the probability that the two cards chosen do NOT include any picture cards?  (1 mark)

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a.   

 

b.   
   

Financial Maths, STD1 F1 2021 HSC 19

Yin purchased a car for $20 000. The value of the car decreases according to a linear model. The graph shows the value of the car, $, against the time, t months, since it was purchased.
 


 

  1. By how much does the value of the car decrease every 10 months?  (1 mark)

  2. Find the value of the car after 5 years.  (1 mark)

  3. Identify ONE problem with using this model to determine the value of Yin’s car over time.  (1 mark)

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a.   
   

 

b.   

♦♦♦ Mean mark part (c) 11%.

 

c.   

Statistics, STD1 S3 2021 HSC 18

People are placed into groups to complete a puzzle. There are 9 different groups.

The table shows the number of people in each group and the amount of time, in minutes, for each group to complete the puzzle.

  1. Complete the scatterplot by adding the last four points from the table.  (2 marks)
     
       
  2. Add a line of best fit by eye to the graph in part (a).  (1 mark)
  3. The graph in part (a) shows the association between the time to complete the puzzle and the number of people in the group.
  4. Identify the form (linear or non-linear), the direction and the strength of the association.  (3 marks)

  5. Calculate the mean of the time taken to complete the puzzle for the three groups of size 7 observed in the dataset.  (1 mark)

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  1.  
       
  2.  
       
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a.

b.


 

c.   

♦ Mean mark (c) 50%.


 

d.