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Functions, 2ADV F1 SM-Bank 25

Damon owns a swim school and purchased a new pool pump for $3250.

He writes down the value of the pool pump by 8% of the original price each year.

  1.  Construct a function to represent the value of the pool pump after years.   (1 mark)

  2.  Draw the graph of the function and state its domain and range.   (2 marks)

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Functions, 2ADV F1 EQ-Bank 22

Worker A picks a bucket of blueberries in hours. Worker B picks a bucket of blueberries in hours.

  1.  Write an algebraic expression for the fraction of a bucket of blueberries that could be picked in one hour if A and B worked together.  (2 marks)

  2.  What does the reciprocal of this fraction represent?  (1 mark)

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i.   

COMMENT: Note that the question asks for “a fraction”.


 


 

ii.   

Calculus, EXT1 C1 EQ-Bank 12

A tank is initially full. It is drained so that at time seconds the volume of water, , in litres, is given by

  for 

  1.  How much water was initially in the tank?  (1 mark)

  2.  After how many seconds was the tank one-quarter full?  (1 mark)

  3.  At what rate was the water draining out the tank when it was one-quarter full?  (2 marks)

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  2.  
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iii.   
 

 

 

 

Functions, EXT1 F1 SM-Bank 3

A circle has centre and radius 3.

  1.  Describe, with inequalities, the region that consists of the interior of the circle and more than 2 units above the -axis.  (2 marks)

  2.  Sketch the region.  (1 mark)

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  2.  

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i.   


 

COMMENT: The broken line on the graph represents an excluded boundary.

 

ii.   

Functions, EXT1 F1 2010 HSC 3b*

Let  .  The diagram shows the graph  .
 

 Inverse Functions, EXT1 2010 HSC 3b

  1. The graph has two points of inflection.  

     

    Find the    coordinates of these points.   (3 marks)

  2. Explain why the domain of    must be restricted if    is to have an inverse function.    (1 mark)

  3. Find a formula for    if the domain of    is restricted to  .   (2 marks)

  4. State the domain of  .    (1 mark)

  5. Sketch the curve  .    (1 mark)

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  5.  
  6. Inverse Functions, EXT1 2010 HSC 3b Answer
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COMMENT: It is also valid to show that is an even function and if a P.I. exists at , there must be another P.I. at .
 

 

 

ii.  
 
 
 

 

iii.  

 

 

 


 

♦ Parts (iv) and (v) were poorly answered with mean marks of 39% and 49% respectively.
iv.  

 

v. 

Inverse Functions, EXT1 2010 HSC 3b Answer

Functions, EXT1 F1 2004 HSC 5b*

The diagram below shows a sketch of the graph of  , where    for  .
 

Inverse Functions, EXT1 2004 HSC 5b

  1. On the same set of axes, sketch the graph of the inverse function,  .  (1 mark)
  2. State the domain of  .  (1 mark)

  3. Find an expression for    in terms of  .  (2 marks)

  4. The graphs of    and    meet at exactly one point  .

     

    Let  α  be the -coordinate of  . Explain why  α  is a root of the equation  .  (1 mark)

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i.  

Inverse Functions, EXT1 2004 HSC 5b Answer

ii.   


 

iii. 

 

 

 

iv.   

 

α

 

Functions, 2ADV F1 SM-Bank 11

Given   and 

  1.  Find integers and such that     (2 marks)

  2.  State the domain for which    is defined.   (2 marks)

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a.  
   
   

 
 

 

b.   

♦♦♦ Mean mark 13%.
MARKER’S COMMENT: “Very poorly answered” with a common response of    that ignored the information from part (a).


 

 vcaa-2011-meth-4ii


 

Calculus, 2ADV C4 2013* HSC 15a

The diagram shows the front of a tent supported by three vertical poles. The poles are 1.2 m apart. The height of each outer pole is 1.5 m, and the height of the middle pole is 1.8 m. The roof hangs between the poles.

2013 15a

The front of the tent has area  ². 

  1. Use the trapezoidal rule to estimate  .    (1 mark)

  2. Does the Trapezoidal rule give a higher or lower estimate of the actual area? Justify your answer.  (1 mark)

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  1. ²
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    ²

 

ii.       

Probability, 2ADV S1 2015 MET1 8

For events and from a sample space,  and  .

  1.  Calculate  .  (1 mark)

  2.  Calculate  , where denotes the complement of .  (1 mark)

  3.  If events and are independent, calculate  .  (1 mark)

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i.  

 

ii.    met1-2015-vcaa-q8-answer
 
 

 

iii.  

♦♦ Mean mark 28%.
MARKER’S COMMENT: A lack of understanding of independent events was clearly evident.

 

 

Probability, 2ADV S1 2009 MET1 5

Four identical balls are numbered 1, 2, 3 and 4 and put into a box. A ball is randomly drawn from the box, and not returned to the box. A second ball is then randomly drawn from the box.

  1. What is the probability that the first ball drawn is numbered 4 and the second ball drawn is numbered 1?  (1 mark)

  2. What is the probability that the sum of the numbers on the two balls is 5?  (1 mark)

  3. Given that the sum of the numbers on the two balls is 5, what is the probability that the second ball drawn is numbered 1?  (2 marks)

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ii.  

 

iii.  

♦ Mean mark 46%.

Probability, 2ADV S1 2012 MET1 4

On any given day, the number of telephone calls that Daniel receives is a random variable with probability distribution given by

  1. Find the mean of  .   (2 marks)

  2. What is the probability that Daniel receives only one telephone call on each of three consecutive days?   (1 mark)

  3. Daniel receives telephone calls on both Monday and Tuesday.

     

    What is the probability that Daniel receives a total of four calls over these two days?   (3 marks)

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i.   
   
   

 

ii.  
   

 

iii.  

♦ Mean mark 36%.

Calculus, 2ADV C4 2017* HSC 14b

  1. Find the exact value of  .  (1 mark)

  2. Using the Trapezoidal rule with three function values, find an approximation to the integral  leaving your answer in terms of and .  (2 marks)

  3. Using parts (i) and (ii), show that  .  (1 mark)

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i.  
   
   

 

ii.  

 
 

 

♦ Mean mark part (iii) 49%.

(iii)   
 
   

Calculus, 2ADV C4 2012* HSC 12d

At a certain location a river is 12 metres wide. At this location the depth of the river, in metres, has been measured at 3 metre intervals. The cross-section is shown below.
 

2012 12d

  1. Use the Trapezoidal rule with the five depth measurements to calculate the approximate area of the cross-section.   (3 marks)

  2. The river flows at 0.4 metres per second.
  3. Calculate the approximate volume of water flowing through the cross-section in 10 seconds.   (1 mark)
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  2. ³  
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  ²

 

♦ Mean mark 49%.

ii.   
 

 

  ³

Calculus, 2ADV C1 2008 HSC 6b

The graph shows the velocity of a particle,    metres per second, as a function of time,    seconds.

  1. What is the initial velocity of the particle?   (1 mark)

  2. When is the velocity of the particle equal to zero?    (1 mark)

  3. When is the acceleration of the particle equal to zero?    (1 mark)

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iii. 

Calculus, 2ADV C4 2016* HSC 14a

The diagram shows the cross-section of a tunnel and a proposed enlargement.

hsc-2016-14a

The heights, in metres, of the existing section at 1 metre intervals are shown in Table

hsc-2016-14ai

The heights, in metres, of the proposed enlargement are shown in Table

hsc-2016-14aii

Use the Trapezoidal rule with the measurements given to calculate the approximate increase in area.   (3 marks)

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²

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  ²

Trigonometry, 2ADV’ T1 2010 HSC 5a

A boat is sailing due north from a point    towards a point    on the shore line.

The shore line runs from west to east.

In the diagram,    represents a tree on a cliff vertically above  , and    represents a landmark on the shore. The distance    is 1 km.

From    the point    is on a bearing of 020°, and the angle of elevation to    is 3°.

After sailing for some time the boat reaches a point  , from which the angle of elevation to    is 30°.
 

5a
 

  1. Show that  .   (3 marks) 

  2. Find the distance . Give your answer to 1 decimal place.   (1 mark)

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ii.   
 
 
 
 

Trigonometry, 2ADV’ T1 2008 HSC 6a

From a point    due south of a tower, the angle of elevation of the top of the tower  , is 23°. From another point  , on a bearing of 120° from the tower, the angle of elevation of    is 32°. The distance    is 200 metres.
 

Trig Ratios, EXT1 2008 HSC 6a 
 

  1. Copy or trace the diagram into your writing booklet, adding the given information to your diagram.  (1 mark)

  2. Hence find the height of the tower. Give your answer to the nearest metre.  (3 marks)

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  1.  
    Trig Ratios, EXT1 2008 HSC 6a Answer

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Trig Ratios, EXT1 2008 HSC 6a Answer 

 

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Trigonometry, 2ADV’ T1 2015 HSC 12c

A person walks 2000 metres due north along a road from point to point . The point is due east of a mountain , where is the top of the mountain. The point is directly below point and is on the same horizontal plane as the road. The height of the mountain above point is metres.

From point , the angle of elevation to the top of the mountain is 15°.

From point , the angle of elevation to the top of the mountain is 13°.
 

Trig Ratios, EXT1 2015 HSC 12c
 

  1. Show that  .  (1 mark)

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

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i.    

Trig Ratios, EXT1 2015 HSC 12c Answer1

 

 

ii.   

 

 

Trig Ratios, EXT1 2015 HSC 12c Answer2 
 
 
 

Complex Numbers, EXT2 N2 2018 HSC 15b

  1. Use De Moivre's theorem and the expansion of to show that
     

     
                          (2 marks)

  2. Hence, show that
     
    .  (3 marks)

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ii.   
   

 

Harder Ext1 Topics, EXT2 2018 HSC 14d

Three people, , and , play a series of n games, where  . In each of the games there is one winner and each of the players is equally likely to win.

  1.  What is the probability that player wins every game?  (1 mark)
  2.  Show that the probability that and win at least one game each but never wins, is
     
         .  (1 mark)
     
  3.  Show that the probability that each player wins at least one game is 
     
         .  (2 marks)
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i.   

 

ii.   

♦ Mean mark 36%.


 

 

iii.   

♦♦♦ Mean mark 19%.


 

Mechanics, EXT2 M1 2018 HSC 14b

A falling particle experiences forces due to gravity and air resistance. The acceleration of the particle is  , where and are positive constants and is the speed of the particle. (Do NOT prove this.)

Prove that, after falling from rest through a distance, , the speed of the particle will be  .  (3 marks)

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Harder Ext1 Topics, EXT2 2018 HSC 13d

The points    and    lie on the rectangular hyperbola  .

The line has equation  . (Do NOT prove this.)

The and intercepts of are and respectively, as shown in the diagram.
 


  

  1. Show that  .  (3 marks)

The point    lies on the parabola  . The tangent to the parabola at intersects the rectangular hyperbola    at and and has equation  . (Do NOT prove this.) The point is the midpoint of the interval . One such case is shown in the diagram.
 

  1. Using part (i), or otherwise, show that lies on the parabola  .  (2 marks)
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ii.    

♦ Mean mark part (ii) 50%.


 

`= M_(RS)= ((at)/2, (−at^2)/2)“

 

 
 

Functions, EXT1′ F1 2018 HSC 12d

The diagram shows the graph of the function  .
  


 

Draw a separate half-page graph for each of the following functions, showing all asymptotes and intercepts.

  1.    (1 mark)

  2.    (2 marks)

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ii.

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ii.   

Graphs, EXT2 2018 HSC 12b

A curve has equation  .

  1. Use implicit differentiation to show that  .  (2 marks)
  2. Hence, or otherwise, find the coordinates of the points on the curve where  .  (2 marks)
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i.   

♦ Mean mark 98%!

 

ii.   

 

 

Complex Numbers, EXT2 N2 2018 HSC 11d

The points , and on the Argand diagram represent the complex numbers , and respectively.

The points , , and form a square as shown on the diagram.
 

 
It is given that  .

  1.  Find  .  (1 mark)

  2.  Find  .  (1 mark)

  3.  Find  .  (1 mark)

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ii.   
   
   

 

iii.   
   

Trigonometry, 2ADV T3 2018 HSC 15a

The length of daylight, , is defined as the number of hours from sunrise to sunset, and can be modelled by the equation

,

where is the number of days after 21 December 2015, for  .

  1. Find the length of daylight on 21 December 2015.   (1 mark)

  2. What is the shortest length of daylight?  (1 mark)

  3. What are the two values of    for which the length of daylight is 11?  (2 marks)

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ii.   

♦ Mean mark 43%.


 

 

 

iii.   


 

   
 
     

 

Functions, EXT1′ 2018 HSC 5 MC

The diagram shows the graph    for  . The region bounded by  ,  ,    and    is rotated about the line    to form a solid.
 

 
Which integral represents the volume of the solid formed?

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