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Trigonometry, MET2-NHT 2019 VCAA 2

The wind speed at a weather monitoring station varies according to the function

where is the speed of the wind, in kilometres per hour (km/h), and    is the time, in minutes, after 9 am.

  1. What is the amplitude and the period of  ?   (2 marks)

  2. What are the maximum and minimum wind speeds at the weather monitoring station?   (1 mark)

  3. Find  , correct to four decimal places.   (1 mark)

  4. Find the average value of    for the first 60 minutes, correct to two decimal places.   (2 marks)

A sudden wind change occurs at 10 am. From that point in time, the wind speed varies according to the new function

                   

where    is the speed of the wind, in kilometres per hour, is the time, in minutes, after 9 am and  . The wind speed after 9 am is shown in the diagram below.
 

  1. Find the smallest value of , correct to four decimal places, such that    and    are equal and are both increasing at 10 am.   (2 marks)

  2. Another possible value of was found to be 31.4358

     

    Using this value of , the weather monitoring station sends a signal when the wind speed is greater than 38 km/h.

     

    i.  Find the value of at which a signal is first sent, correct to two decimal places.   (1 mark)

     

    ii. Find the proportion of one cycle, to the nearest whole percent, for which  .   (2 marks)

  3. Let    and  .
     
    The transformation    maps the graph of    onto the graph of  .

     

    State the values of  , , and , in terms of where appropriate.   (3 marks)

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  2.  

  6. i.

     

    ii.

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a.   

 

b.   

 

c.   
 

 

d.   

 
 

 

e.   

 

f.i. 


 

f.ii. 

 
 

 

g.   

 


 


 

 

Graphs, MET2-NHT 2019 VCAA 1

Parts of the graphs of    and    are shown on the axes below.
 


 

The two graphs intersect at three points,  (–2, 0),  (1, 0)  and  (, ). The point  (, )  is not shown in the diagram above.

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

  2. Find the values of such that  .   (1 mark)

  3. State the values of for which
    1.    (1 mark)

    2.    (1 mark)

  4. Show that    for all  .   (1 mark)

  5. Find the values of such that    has exactly one negative solution.   (2 marks)

  6. Find the values of such that    has no solutions.   (1 mark)

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  3. i. 
    ii.
  5.  
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a.   


 

 

 

b.   

 

c.i. 

c.ii. 

 

d.     
 

 

 
 
 

 

e.     
 

 

 

f.   

Trigonometry, EXT1 T3 EQ-Bank 4

The current flowing through an electrical circuit can be modelled by the function

  1.  Express the function in the form  .  (2 marks)

  2.  Find the time at which the current first obtains it maximum value.   (1 mark)

  3.  Sketch the graph of  . Clearly show its range and label the coordinates of its first maximum value. Do not label -intercepts.   (1 mark)

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  2.  
  3.  
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i.   

 
   

 

 

 

 

ii.   

 

 

iii.   

Probability, 2ADV S1 2019 MET2-N 9 MC

At the start of a particular week, Kim has three red apples and two green apples. She eats one apple everyday. On Monday, Tuesday and Wednesday of that week, she randomly selects an apple to eat. In this three-day period, the probability that Kim does not eat an apple of the same colour on any two consecutive days is

  1.  
  2.  
  3.  
  4.  
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CORE, FUR1-NHT 2019 VCAA 19 MC

Consider the recurrence relation shown below.

This recurrence relation could be used to determine the value of

  1. a perpetuity with a payment of $2000 per quarter.
  2. an annuity with withdrawals of $2000 per quarter.
  3. an annuity investment with additional payments of $2000 per quarter.
  4. an item depreciating at a flat rate of 1.3% of the purchase price per quarter.
  5. a compound interest investment earning interest at the rate of 1.3% per annum.
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CORE, FUR1-NHT 2019 VCAA 16 MC

A small cafe is open every day of the week except Tuesday.

The table below shows the daily seasonal indices for labour costs at this cafe.

Excluding the day that the cafe is closed, the long-term average weekly labour cost is $9786.

The expected daily labour cost on a Wednesday is closest to

  1.  $1127
  2.  $1315
  3.  $1631
  4.  $1733
  5.  $2022
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CORE, FUR1-NHT 2019 VCAA 14 MC

The time series plot below shows the daily number of visitors to a historical site over a two-week period.
 

This time series plot is to be smoothed using seven-median smoothing.

The smoothed number of visitors on day 4 is closest to

  1. 120
  2. 140
  3. 145
  4. 150
  5. 160
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CORE, FUR1-NHT 2019 VCAA 8 MC

The variables recovery time after exercise (in minutes) and fitness level (below average, average, above average) are

  1. both numerical.
  2. both categorical.
  3. an ordinal variable and a nominal variable respectively.
  4. a numerical variable and a nominal variable respectively.
  5. a numerical variable and an ordinal variable respectively.
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CORE, FUR1-NHT 2019 VCAA 5-7 MC

The birth weights of a large population of babies are approximately normally distributed with a mean of 3300 g and a standard deviation of 550 g.

Part 1

A baby selected at random from this population has a standardised weight of 

Which one of the following calculations will result in the actual birth weight of this baby?
 

 

Part 2

Using the 68–95–99.7% rule, the percentage of babies with a birth weight of less than 1650 g is closest to

  1. 0.14%
  2. 0.15%
  3. 0.17%
  4. 0.3%
  5. 2.5%

 

Part 3

A sample of 600 babies was drawn at random from this population.

Using the 68–95–99.7% rule, the number of these babies with a birth weight between 2200 g and 3850 g is closest to

  1. 111
  2. 113
  3. 185
  4. 408
  5. 489
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Probability, MET2-NHT 2019 VCAA 9 MC

At the start of a particular week, Kim has three red apples and two green apples. She eats one apple everyday. On Monday, Tuesday and Wednesday of that week, she randomly selects an apple to eat. In this three-day period, the probability that Kim does not eat an apple of the same colour on any two consecutive days is

  1.  
  2.  
  3.  
  4.  
  5.  
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Combinatorics, EXT1 A1 2019 MET1 8

A fair standard die is rolled 50 times. Let be a random variable with binomial distribution that represents the number of times the face with a six on it appears uppermost.

  1. Write down the expression for  , where  .  (1 mark)

  2. Show that  .  (2 marks)

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a. 

 

b.  
   
   
   

Calculus, 2ADV C4 2019 MET1 7

The shaded region in the diagram below is bounded by the vertical axis, the graph of the function with rule    and the horizontal line segment that meets the graph at  , where  .
 


 

Let    be the area of the shaded region.

Show that  .  (3 marks)

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Statistics, EXT1 S1 2019 MET1 6

Jacinta tosses a coin five times.

  1. Assuming that the coin is fair and given that Jacinta observes a head on the first two tosses, find the probability that she observes a total of either four or five heads.  (2 marks)

  2. Albin suspects that a coin is not actually a fair coin and he tosses it 18 times.

     

    Albin observes a total of 12 heads from the 18 tosses.

  3. Let  = probability of obtaining a head.
  4. Find the range of in which 95% of observations are expected to lie within.  (2 marks)

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  2.  
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a.   

 
 

 

b.   

Calculus, 2ADV C3 2019 MET1 4

Given the function  ,

  1. State the domain and range of .  (1 mark)

  2. i.  Find the equation of the tangent to the graph of  at  .  (2 marks)

     

    ii. On the axes below, sketch the graph of the function  , labelling any asymptote with its equation.

     

        Also draw the tangent to the graph of    at  .  (4 marks)
     

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  1.  

  2. i. 

     

    ii.

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a.  
 

 

b.i.  
 
 

 

 

b.ii.  

Probability, MET1-NHT 2019 VCAA 8

A fair standard die is rolled 50 times. Let be a random variable with binomial distribution that represents the number of times the face with a six on it appears uppermost.

  1. Write down the expression for  , where  .  (1 mark)

  2. Show that  .  (2 marks)

  3. Hence, or otherwise, find the value of for which    is the greatest.  (2 marks)

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a. 

 

b.  
   
   
   

 

c. 

 

Calculus, MET1-NHT 2019 VCAA 7

The shaded region in the diagram below is bounded by the vertical axis, the graph of the function with rule    and the horizontal line segment that meets the graph at  , where  .
 


 

Let    be the area of the shaded region.

  1. Show that  .   (3 marks)

  2. Determine the range of values of .   (2 marks)

    1. Express in terms of  , for a specific value of , the area bounded by the vertical axis, the graph of    and the horizontal axis.   (2 marks)

    2. Hence, or otherwise, find the area described in part c.i.   (1 mark)

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a.   

 
 
 

 

b.   


 

 

c.i. 

   
   
   
   

 

 

c.ii. 

 
 
 
 

Calculus, MET1-NHT 2019 VCAA 4

A function has rule  .

  1. State the maximal domain of and the range of over its maximal domain.   (2 marks)

  2. i.  Find the equation of the tangent to the graph of at .   (2 marks)

     

    ii. On the axes below, sketch the graph of the function , labelling any asymptote with its equation.

  3.     Also draw the tangent to the graph of at  .   (4 marks)
      

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  1.  

  2. i. 
    ii. 
     
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a.  
 

 

b.i.  
 
 

 

 

b.ii.  

Calculus, MET1-NHT 2019 VCAA 3

  1. Evaluate    and  .   (2 marks)

  2. Show that  .   (1 mark)

  3. Use your answers to part a. and part b. to evaluate    in the form  , where  and are positive integers.   (1 mark)

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a.  
   
   

 

 
 

 

b.  
   
   

 

c.  
   
   
   
   

Statistics, EXT1 S1 EQ-Bank 23

A light manufacturer knows that 6% of the light bulbs it produces are defective.

Light bulbs are supplied in boxes of 20 bulbs. Boxes are supplied in pallets of 120 boxes.

Calculate the probability that

  1. A box of light bulbs contains exactly 3 defective bulbs.  (1 mark)

  2. A box of light bulbs contains at least 1 defective bulb.  (1 mark)

  3. A pallet contains between 90 and 95 (inclusive) boxes with at least 1 defective bulb (use the probability table attached).  (3 marks)

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i.   

 

 

ii.   
   
   

 

iii.   


 


 


 

 
 

Statistics, EXT1 S1 EQ-Bank 22

It is known that 43% of voters in a city voted for Labor in the election.

If 250 voters from the city are selected at random, use a suitable approximation and the attached probability table to find the probability that at least 120 of those chosen voted for Labor.  (3 marks)

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Statistics, EXT1 S1 EQ-Bank 20

Netball Australia records show that 10% of all registered players are over the age of 25.

  1. A random survey of 100 netball players was carried out to find out how many were over 25 years of age.

     

    Assuming the sample proportion is normally distributed, calculate the expected mean and standard deviation of this group.  (2 marks)

  2. Using the probability table attached, estimate the probability that at least 15 players surveyed will be over 25 years of age.  (2 marks)

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i.  
 
   
   
 
   

 

ii. 

 
 
 

Statistics, EXT1 S1 EQ-Bank 21

A biased coin has a 0.6 chance of landing on heads. The coin is tossed 15 times.

  1. Calculate the probability of obtaining 7, 8 or 9 heads using binomial probability distribution.
  2. Give your answer correct to 3 decimal places.  (2 marks)

  3. Calculate the probability of obtaining 7, 8 or 9 heads using normal approximation to the binomial distribution and the probability table attached.
  4. Give your answer correct to 3 decimal places.  (2 marks)

  5. Could this binomial distribution be reasonably approximated with a normal distribution? Support your answer with a brief calculation.  (1 mark)

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i.   
   
   

 

ii.   

 

COMMENT: A quick sketch of the normal distribution curve is often helpful when using probability tables in this context.
 

 
 
 

 

iii.   

COMMENT: is also common in assessing if normal approximation is reasonable.

Combinatorics, EXT1 A1 EQ-Bank 14

A delivery company has 1095 packages to deliver on a given day.

It has 17 delivery vans that will deliver all packages. If one van delivers more packages than all other vans, the company pays the driver a $100 bonus.

What is the minimum number of packages a van could deliver and still win the $100 bonus.  (2 marks)

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COMMENT: Note that “By PHP” refers to by pigeonhole principle.


 

Combinatorics, EXT1 A1 EQ-Bank 13

A sock drawer contains blue, white and green socks.

If individual socks are randomly chosen from the drawer, what is the minimum number that must be selected to ensure there are at least three pairs?   (2 marks)

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COMMENT: Note that “By PHP” refers to by pigeonhole principle.


 

Combinatorics, EXT1 A1 SM-Bank 11

A multiple choice quiz asks students 4 questions. Each question has three possible answers, a, b or c, and students must attempt each question.

How many students must do the quiz to ensure that at least two sets of answers are identical?    (2 marks)

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COMMENT: Note that “By PHP” refers to by pigeonhole principle.


 

Statistics, SPEC2-NHT 2019 VCAA 6

A paint company claims that the mean time taken for its paint to dry when motor vehicles are repaired is 3.55 hours, with a standard deviation of 0.66 hours.

Assume that the drying time for the paint follows a normal distribution and that the claimed standard deviation value is accurate.

  1. Let the random variable    represent the mean time taken for the paint to dry for a random sample of 36 motor vehicles.

     

    Write down the mean and standard deviation of  .   (2 marks)

At a car crash repair centre, it was found that the mean time taken for the paint company's paint to dry on randomly selected vehicles was 3.85 hours. The management of this crash repair centre was not happy and believed that the claim regarding the mean time taken for the paint to dry was too low. To test the paint company's claim, a statistical test was carried out.

  1. Write down suitable null and alternative hypotheses and respectively to test whether the mean time taken for the paint to dry is longer than claimed.   (1 mark)

  2. Write down an expression for the    value of the statistical test and evaluate it correct to three decimal places.   (2 marks)

  3. Using a 1% level of significance, state with a reason whether the crash repair centre is justified in believing that the paint company's claim of mean time taken for its paint to dry of 3.55 hours is too low.  (1 mark)

  4. At the 1% level of significance, find the set of sample mean values that would support the conclusion that the mean time taken for the paint to dry exceeded 3.55 hours. Give your answer in hours, correct to three decimal places.  (2 marks)

  5. If the true time taken for the paint to dry is 3.83 hours, find the probability that the paint company's claim is not rejected at the 1% level of significance, assuming the standard deviation for the paint to dry is still 0.66 hours. Give your answer correct to two decimal places.  (1 mark)

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a.   
 
 

 

b.   

 

c.   
 
 
 

 

d.   

 

e.   

 

f.   
 
 

Mechanics, SPEC2-NHT 2019 VCAA 5

A pallet of bricks weighing 500 kg sits on a rough plane inclined at an angle of  α to the horizontal, where  α. The pallet is connected by a light inextensible cable that passes over a smooth pulley to a hanging container of mass kilograms in which there is 10 L of water. The pallet of bricks is held in equilibrium by the tension newtons in the cable and a frictional resistance force of 50 newtons acting up and parallel to the plane. Take the weight force exerted by 1 L of water to be newtons.
  


 

  1. Label all forces acting on both the pallet of bricks and the hanging container on the diagram above, when the pallet of bricks is in equilibrium as described.   (1 mark)
  2. Show that the value of is 80.  (3 marks)

Suddenly the water is completely emptied from the container and the pallet of bricks begins to slide down the plane. The frictional resistance force of 50 newtons acting up the plane continues to act on the pallet.

  1. Find the distance, in metres, travelled by the pallet after 10 seconds.  (3 marks)
  2. When the pallet reaches a velocity of  , water is poured back into the container at a constant rate of 2 L per second, which in turn retards the motion of the pallet moving down the plane. Let    be the time, in seconds, after the container begins to fill. 
  3.   i. Write down, in terms of  , an expression for the total mass of the hanging container and the water it contains after seconds. Give your answer in kilograms.  (1 mark)
  4.  ii. Show that the acceleration of the pallet down the plane is given by    for  .  (2 marks)
  5. iii. Find  the velocity of the pallet when  . Give your answer in metres per second, correct to one decimal place.  (2 marks)
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  1.  
  4.   i.
     ii.
    iii.
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a.

 
b.   


 


 

αα
 

 

c.   

α
 
 

 

d.i.   
 

d.ii.   

 

d.iii.   

 

Calculus, SPEC2-NHT 2019 VCAA 3


 

The vertical cross-section of a barrel is shown above. The radius of the circular base (along the -axis) is 30 cm and the radius of the circular top is 70 cm. The curved sides of the cross-section shown are parts of the parabola with rule  . The height of the barrel is 50 cm.
 
a.  i.   Show that the volume of the barrel is given by  .  (1 marks)
     ii.  Find the volume of the barrel in cubic centimetres.  (1 marks)    

The barrel is initially full of water. Water begins to leak from the bottom of the barrel such that    cubic centimetres per second, where after    seconds the depth of the water is    centimetres, the volume of water remaining in the barrel is    cubic centimetres and the uppermost surface area of the water is    square centimetres.
 
b.     Show that  ?  (2 marks)
c.      Find    in terms of  . Express your answer in the form  , where    and    are positive integers.  (3 marks)
d.     Using a definite integral in terms of  , find the time, in hours, correct to one decimal place, taken for the barrel to empty.  (2 marks)

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a.  i.

      ii. 

b.   

c.   

d.   

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a.i.
 
 
     
 

 

a.ii.
   
   

 

b.   

 
 

 

c.   

 
 
 

  
 
d.   

 
 

Calculus, SPEC2-NHT 2019 VCAA 2

Consider the function with rule  .

  1. State the equations of the asymptotes of the graph of .  (2 marks)

  2. State the coordinates of the stationary points and the point of inflection. Give your answers correct to two decimal places.  (2 marks)

  3. Sketch the graph of from    to    (endpoint coordinates are not required) on the set of axes below, labeling the turning points and the point of inflection with their coordinates correct to two decimal places. Label the asymptotes with their equations.  (3 marks)

Consider the function with rule    where  .

  1. For what values of will have no stationary points?  (2 marks)

  2. For what value of will the graph of have a point of  inflection located on the -axis?  (1 marks)

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i.     

 ii.   
iii.

iv. 

v. 

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i.   


  

ii.   


 


 

iii.

 

iv.   

 

 

v.   

Statistics, EXT1 S1 EQ-Bank 13

On average, batsmen playing cricket in a T20 competition play a scoring shot two out of every three balls.

In a regular season, a total of 1200 overs that each contain six balls, are bowled.

Estimate how many overs would have at least five scoring shots.  (3 marks)

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Statistics, EXT1 S1 EQ-Bank 12

A manufacturer of pool cue tips knows that 4% of pool cue tips produced in its factory need to be scrapped.

A random sample of 10 cue tips produced in the factory is examined.

  1. What is the probability that exactly 3 cue tips need to be scrapped?

     

    Give your answer correct to 3 decimal places.  (1 mark)

  2. What is the probability that, at most, 2 cue tips need to be scrapped?

     

    Give your answer correct to 3 decimal places.  (2 marks)

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i.     

 
 

 

ii.  
   
   
   

Vectors, EXT1 V1 2019 SPEC2-N 4

A snowboarder at the Winter Olympics leaves a ski jump at an angle of degrees to the horizontal, rises up in the air, performs various tricks and then lands at a distance down a straight slope that makes an angle of 45° to the horizontal, as shown below.

Let the origin of a cartesian coordinate system be at the point where the snowboarder leaves the jump, with a unit vector in the positive direction being represented by    and a unit vector in the positive direction being represented by  . Distances are measured in metres and time is measured in seconds.

The position vector of the snowboarder    seconds after leaving the jump is given by


 


 

  1. Find the angle  .    (2 marks)

  2. Find the speed, in metres per second, of the snowboarder when she leaves the jump at .    (1 mark)

  3. Find the maximum height above reached by the snow boarder. Give your answer in metres, correct to one decimal place.    (2 marks)

  4. Show that the time spent in the air by the snowboarder is    seconds.    (3 marks)

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a.   

 

 

b.    
   
   

 
c.   

 
   

 
d.   

 

Vectors, SPEC2-NHT 2019 VCAA 4

A snowboarder at the Winter Olympics leaves a ski jump at an angle of degrees to the horizontal, rises up in the air, performs various tricks and then lands at a distance down a straight slope that makes an angle of 45° to the horizontal, as shown below.

Let the origin of a cartesian coordinate system be at the point where the snowboarder leaves the jump, with a unit vector in the positive direction being represented by    and a unit vector in the positive direction being represented by  . Distances are measured in metres and time is measured in seconds.

The position vector of the snowboarder    seconds after leaving the jump is given by


 


 

  1. Find the angle  .    (2 marks)

  2. Find the speed, in metres per second, of the snowboarder when she leaves the jump at .   (1 mark)

  3. Show that the time spent in the air by the snowboarder is    seconds.   (3 marks)

  4. Find the total distance the snowboarder travels while airborne. Give your answer in metres, correct to two decimal places.   (2 marks)

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a.   

 

 

b.    
   
   

 
c.   

 
   

 
d.   

 


 
e.   

 

Complex Numbers, SPEC2-NHT 2019 VCAA 1

In the complex plane, is the with equation .

  1.  Verify that the point (0, 0) lies on .   (1 marks)

  2.  The line    can also be expressed in the form  , where  .

     

     Find  in cartesian form.   (2 marks)

  3.  Find, in cartesian form, the points(s) of intersection of    and the graph of  .   (2 marks)

  4.  Sketch    and the graph of    on the Argand diagram below.   (2 marks)

     
     
         
     

  5.  Find the area of the sector defined by the part of    where  , the graph of    where  , and imaginary axis where  .   (1 marks)

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a.   

 

b. 
 
 
 
 
 
 


c.


 

 

d.   

 

e.


 

f.