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Graphs, MET1 SM-Bank 27

The graph shown is  .

  1. Write down the value of  .   (1 mark)

  2. Find the value of  .   (1 mark)

  3. Copy or trace the graph into your writing booklet.

     

    On the same set of axes, draw the graph    for  .   (2 marks)

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a.   

b. 

  

 MARKER’S COMMENT: Graphs are consistently drawn too small by many students. Aim to make your diagrams 1/3 to 1/2 of a page. 
c.

Graphs, MET1 SM-Bank 20

The rule for   is    for  .  This function has an inverse,  .

  1. Sketch the graphs of    and    on the same set of axes. (Use the same scale on both axes.)   (2 marks)

  2. Find the rule for the inverse function  .    (2 marks)

  3. Evaluate  .    (1 mark)

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  1.  
    Inverse Functions, EXT1 2008 HSC 5a Answer
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a. 

Inverse Functions, EXT1 2008 HSC 5a Answer
b.  

 

 

 
 
 

 

 

c.   
   
   
   
   

Calculus, MET1 2016 VCAA 3

Let    where  .

  1. Sketch the graph of  . Label the axis intercepts with their coordinates and label any asymptotes with the appropriate equation.   (3 marks)
     

     

  2. Find the area enclosed by the graph of  , the lines    and  , and the -axis.   (2 marks)

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  1.  
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a.   

 

b.   
   
   
    ²

Probability, MET2 2009 VCAA 3

The Bouncy Ball Company (BBC) makes tennis balls whose diameters are normally distributed with mean 67 mm and standard deviation 1 mm. The tennis balls are packed and sold in cylindrical tins that each hold four balls. A tennis ball fits into such a tin if the diameter of the ball is less than 68.5 mm.

  1. What is the probability, correct to four decimal places, that a randomly selected tennis ball produced by BBC fits into a tin?  (2 marks)

BBC management would like each ball produced to have diameter between 65.6 and 68.4 mm.

  1. What is the probability, correct to four decimal places, that the diameter of a randomly selected tennis ball made by BBC is in this range?  (2 marks)

    1. What is the probability, correct to four decimal places, that the diameter of a tennis ball which fits into a tin is between 65.6 and 68.4 mm?  (1 mark)

    2. A tin of four balls is selected at random. What is the probability, correct to four decimal places, that at least one of these balls has diameter outside the desired range of 65.6 to 68.4 mm?  (2 marks)

BBC management wants engineers to change the manufacturing process so that 99% of all balls produced have diameter between 65.6 and 68.4 mm. The mean is to stay at 67 mm but the standard deviation is to be changed.

  1. What should the new standard deviation be (correct to two decimal places)?  (3 marks)

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a. 

b. 

c.i. 

c.ii. 

d. 

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a.  

 

b.  

 

c.i.  

♦ Mean mark 42%.

 

  ii. 

♦♦ Mean mark 30%.

 

d.  

♦♦ Mean mark 35%.

 vcaa-graphs-fur2-2009-3di

 

Calculus, MET2 2009 VCAA 1

Let  

The graph of    is shown below.

VCAA 2009 1a

  1. State the interval for which the graph of is strictly decreasing.   (2 marks)

  2. Points and are the points of intersection of    with the -axis. Point has coordinates  and point has coordinates .

     

    Find the length of such that the area of rectangle is equal to the area of the shaded region.   (2 marks)
     
    VCAA 2009 1c

  3. The points and are labelled on the diagram.

      
     

          VCAA 2009 1d

     

    1. Find , the gradient of the chord .   (Exact value to be given.)   (1 mark)

    2. Find    such that  .  (Exact value to be given.)   (2 marks)

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a. 

b. 

c.i.   

c.ii. 

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a.   vcaa-2009-1ai

 

 

b.   vcaa-2009-1ci
 
 

 

c.i.  
 

 

 c.ii.  
 

Calculus, MET2 2011 VCAA 3

  1. Consider the function  .

     

    1. Find     (1 mark)

    2. Explain why   for all .   (1 mark)

  2. The cubic function is defined by  , where , , and are real numbers.

     

    1. If has stationary points, what possible values can have?   (1 mark)

    2. If has an inverse function, what possible values can have?   (1 mark)

  3. The cubic function is defined by  .

     

    1. Write down a expression for  .   (2 marks)

    2. Determine the coordinates of the point(s) of intersection of the graphs of    and  .   (2 marks)

  4. The cubic function is defined by  , where and are real numbers.

     

    1. If has exactly one stationary point, find the value of .   (3 marks)

    2. If this stationary point occurs at a point of intersection of    and  , find the value of .   (3 marks)

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a.i.  
  

a.ii.  

♦ Mean mark 47%.

 

b.i.  

♦♦♦ Mean mark part (b)(i) 9%, and part (b)(ii) 20%.
MARKER’S COMMENT: Good exam strategy should point students to investigate earlier parts for direction. Here, part (a) clearly sheds light on a solution!

 

b.ii.  

 

c.i.  


  

c.ii. 

   

 


  

♦ Mean mark part (d)(i) 44%.
d.i.   
 
 
 
 

 

d.ii.  

♦♦♦ Mean mark part (d)(ii) 14%.

 

Probability, MET2 2016 VCAA 3*

A school has a class set of 22 new laptops kept in a recharging trolley. Provided each laptop is correctly plugged into the trolley after use, its battery recharges.

On a particular day, a class of 22 students uses the laptops. All laptop batteries are fully charged at the start of the lesson. Each student uses and returns exactly one laptop. The probability that a student does not correctly plug their laptop into the trolley at the end of the lesson is 10%. The correctness of any student’s plugging-in is independent of any other student’s correctness.

  1. Determine the probability that at least one of the laptops is not correctly plugged into the trolley at the end of the lesson. Give your answer correct to four decimal places.   (2 marks)

  2. A teacher observes that at least one of the returned laptops is not correctly plugged into the trolley.
  3. Given this, find the probability that fewer than five laptops are not correctly plugged in. Give your answer correct to four decimal places.   (2 marks)

The time for which a laptop will work without recharging (the battery life) is normally distributed, with a mean of three hours and 10 minutes and standard deviation of six minutes. Suppose that the laptops remain out of the recharging trolley for three hours.

  1. For any one laptop, find the probability that it will stop working by the end of these three hours. Give your answer correct to four decimal places.   (2 marks)

A supplier of laptops decides to take a sample of 100 new laptops from a number of different schools. For samples of size 100 from the population of laptops with a mean battery life of three hours and 10 minutes and standard deviation of six minutes,  is the random variable of the distribution of sample proportions of laptops with a battery life of less than three hours.

  1. Find the probability that . Give your answer correct to three decimal places. Do not use a normal approximation.   (3 marks)

It is known that when laptops have been used regularly in a school for six months, their battery life is still normally distributed but the mean battery life drops to three hours. It is also known that only 12% of such laptops work for more than three hours and 10 minutes.

  1. Find the standard deviation for the normal distribution that applies to the battery life of laptops that have been used regularly in a school for six months, correct to four decimal places.   (2 marks)

The laptop supplier collects a sample of 100 laptops that have been used for six months from a number of different schools and tests their battery life. The laptop supplier wishes to estimate the proportion of such laptops with a battery life of less than three hours.

  1. Suppose the supplier tests the battery life of the laptops one at a time.
  2. Find the probability that the first laptop found to have a battery life of less than three hours is the third one.   (1 mark)

The laptop supplier finds that, in a particular sample of 100 laptops, six of them have a battery life of less than three hours.

  1. Determine the 95% confidence interval for the supplier’s estimate of the proportion of interest. Give values correct to two decimal places.   (1 mark)

  2. The supplier also provides laptops to businesses. The probability density function for battery life, (in minutes), of a laptop after six months of use in a business is
     

     


     

  3. Find the mean battery life, in minutes, of a laptop with six months of business use, correct to two decimal places.   (1 mark)

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a.  

 

 
 

 

 b.   

MARKER’S COMMENT: Early rounding was a common mistake, producing 0.9312.

 

c.   

MARKER’S COMMENT: Some working must be shown for full marks in questions worth more than 1 mark.

 

d.   

♦ Mean mark part (d) 33%.

 
 
 

 

e.   

 

f.   
   

 

g.   

 

h.   
 

Calculus, MET2 2016 VCAA 2

Consider the function  

  1.  i. Given that  ,
  2.      show that  .   (1 mark)

  3. ii. Find the values of for which the graph of    has a stationary point.   (1 mark)

The diagram below shows part of the graph of  , the tangent to the graph at    and a straight line drawn perpendicular to the tangent to the graph at  . The equation of the tangent at the point with coordinates    is  .

The tangent cuts the -axis at . The line perpendicular to the tangent cuts the -axis at .
 


 

  1.   i. Find the coordinates of .   (1 mark)

  2.  ii. Find the equation of the line that passes through and and, hence, find the coordinates of .   (2 marks)

  3. iii. Find the area of triangle .   (2 marks)

  4. The tangent at is parallel to the tangent at . It intersects the line passing through and at .

     


     
     i. Find the coordinates of .   (2 marks)

  5. ii. Find the length of .   (3 marks)

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  1.  i.
  2. ii.
  3.   i.
  4. ii.
  5.      
  6. iii. ²
Show Worked Solution
a.i.   
   
   
 

 

  

 

a.ii.   


 

b.i.   


 

b.ii.   

   

   


 

b.iii.  
   
    ²

 

♦ Mean mark part (c)(i) 43%.
MARKER’S COMMENT: Many students gave an incorrect -value here. Be careful!
c.i.   
 
 
   

  

 

c.ii.  

 

♦♦ Mean mark 32%.

 

 

 

Calculus, MET2 2016 VCAA 1

Let  .

  1. Find the period and range of .   (2 marks)

  2. State the rule for the derivative function .   (1 mark)

  3. Find the equation of the tangent to the graph of at  .   (1 mark)

  4. Find the equations of the tangents to the graph of    that have a gradient of 1.   (2 marks)

  5. The rule of   can be obtained from the rule of  under a transformation , such that
      

     

     

    Find the value of and the value of .   (3 marks)

  6. Find the values of  , such that  .   (2 marks)

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a.   

MARKER’S COMMENT: Including round brackets rather than square ones was a common mistake.


  

b.   
 

c.  


 

d.   

♦ Mean mark part (d) 50%.

 

e.  

♦♦ Mean mark part (e) 27%.
   

 
   

 

f.   

♦ Mean mark part (f) 50%.

Graphs, MET2 2016 VCAA 8 MC

The UV index, , for a summer day in Melbourne is illustrated in the graph below, where is the number of hours after 6 am.
 

     
 

The graph is most likely to be the graph of

A.  

B.  

C.  

D.  

E.  

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CORE, FUR2 SM-Bank VCE 2

Spiro is saving for a car. He has an account with $3500 in it at the start of the year.

At the end of each month, Spiro adds another $180 to the account.

The account pays 3.6% interest per annum, compounded monthly.

    1. What is the interest rate per month?   (1 mark)
    2. Write a recurrence relation that models Spiro's investment, with representing the balance of his account after months.   (1 mark)
  2. What will be the balance of Spiro's account after 3 months?
  3. Write your answer correct to the nearest cent.   (1 mark)

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a.i.   
   

 

a.ii.   
   
   
     
     

  


  

b.   
 
   

CORE, FUR2 SM-Bank VCE 1

Joe buys a tractor under a buy-back scheme. This scheme gives Joe the right to sell the tractor back to the dealer through either a flat rate depreciation or unit cost depreciation.

  1. The recurrence relation below can be used to calculate the price Joe sells the tractor back to the dealer , based on the flat rate depreciation, after years
     

     

    1. Write the general rule to find the value of in terms of .?   (1 mark)
    2. Hence or otherwise, find the time it will take Joe's tractor to lose half of its value.   (1 mark)
  2. Joe uses the unit cost method to depreciate his tractor, he depreciates $2.75 per kilometre travelled.
    1. How many kilometres does Joe's tractor need to travel for half its value to be depreciated? Round your answer to the nearest kilometre?   (1 mark)
    2. Joe's tractor travels, on average, 2500 kilometres per year. Which method, flat rate depreciation or unit cost depreciation, will result in the greater annual depreciation? Write down the greater depreciation amount correct to the nearest dollar.   (1 mark)

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  1. i. 
    ii.
  2. i.
    ii.
         

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a.i.   
 
   
   
     

 

a.ii.   

 

b.i.   
   
   

  
b.ii.