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Algebra, STD2 A4 EQ-Bank 8

Two friends, Sequoia and Raven, sold organic chapsticks at the the local market.

Sequoia sold her chapsticks for $4 and Raven sold hers for $3 each. In the first hour, their total combined sales were $20.

If Sequoia sold chapsticks and Raven sold chapsticks, then the following equation can be formed:

In the first hour, the friends sold a total of 6 chapsticks between them.

Find the number of chapsticks each of the friends sold during this time by forming a second equation and solving the simultaneous equations graphically.  (5 marks)

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Probability, STD2 S2 SM-Bank 1

A game consists of two tokens being drawn at random from a barrel containing 20 tokens. There are 17 red tokens and 3 black tokens. The player keeps the two tokens drawn.

  1.  Complete the probability tree by writing the missing probabilities in the boxes.  (2 marks)
     
     
       
     
  2.  What is the probability that a player draws at least one red token. Give your answer in exact form.  (2 marks)

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Functions, 2ADV F2 SM-Bank 1

  1.  Draw the graph  .  (1 mark)

  2.  Explain how the above graph can be transformed to produce the graph
     
             
     
    and sketch the graph, clearly identifying all intercepts.  (3 marks)

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Calculus, 2ADV C1 SM-Bank 3

The displacement metres from the origin at time seconds of a particle travelling in a straight line is given by

     when   

  1.  Calculate the velocity when  .  (1 mark)

  2.  When is the particle stationary?  (2 marks)

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Functions, 2ADV F1 EQ-Bank 27

The stopping distance of a car on a certain road, once the brakes are applied, is directly proportional to the square of the speed of the car when the brakes are first applied.

A car travelling at 70 km/h takes 58.8 metres to stop.

How far does it take to stop if it is travelling at 105 km/h?  (3 marks)

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Functions, 2ADV F1 EQ-Bank 26

Fuifui finds that for Giant moray eels, the mass of an eel is directly proportional to the cube of its length.

An eel of this species has a length of 25 cm and a mass of 4350 grams.

What is the expected length of a Giant moray eel with a mass of 6.2 kg? Give your answer to one decimal place.  (3 marks)

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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.   

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

 

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Calculus, EXT1 C3 SM-Bank 1

The region enclosed by the semicircle    and the -axis is to be divided into two pieces by the line  , when  .
 


 

The two pieces are rotated about the -axis to form solids of revolution. The value of is chosen so that the volumes of the solids are in the ratio .

Show that satisfies the equation  .  (3 marks)

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(i)   

 

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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  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 .
 

 

 

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♦ Parts (iv) and (v) were poorly answered with mean marks of 39% and 49% respectively.
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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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Inverse Functions, EXT1 2004 HSC 5b Answer

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Functions, 2ADV F1 SM-Bank 7

Let    for    and    for  all .

  1. Find  , where  .  (1 mark)

  2. State the domain and range of  .  (2 marks)

  3. Show that  .  (2 marks)
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ii.  

♦♦ Mean mark part (a)(ii) 30%.

 

 

MARKER’S COMMENT: Many students were unsure of how to present their working in this question. Note the layout in the solution.
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Calculus, 2ADV C4 2007* HSC 10a

An object is moving on the -axis. The graph shows the velocity, , of the object, as a function of time, . The coordinates of the points shown on the graph are  . The velocity is constant for  .
 


 

  1. The object is initially at the origin. During which time(s) is the displacement of the object decreasing?  (1 mark)

  2. If the object travels 7 units in the first 4 seconds, estimate the time at which the object returns to the origin. Justify your answer.  (2 marks)

  3. Sketch the displacement, , as a function of time.  (2 marks)

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Statistics, NAPX-K2-04 SA

Sharnie records the number of items she recycles in one week.

The tally table shows Sharnie's results.
 

 
How many items does Sharnie recycle in total?
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Probability, 2ADV S1 2017 MET1 8

For events and from a sample space,   and  .  Let  .

  1. Find   in terms of .  (1 mark)

  2. Find   in terms of .  (2 marks)

  3. Given that  , state the largest possible interval for .  (2 marks)

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ii. 

♦ Mean mark 40%.
MARKER’S COMMENT: The most successful answers used a Venn diagram or table.
 

 

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♦ Mean mark 37%.

Probability, 2ADV S1 2014 MET1 9

Sally aims to walk her dog, Mack, most mornings. If the weather is pleasant, the probability that she will walk Mack is , and if the weather is unpleasant, the probability that she will walk Mack is .

Assume that pleasant weather on any morning is independent of pleasant weather on any other morning.

  1. In a particular week, the weather was pleasant on Monday morning and unpleasant on Tuesday morning.

     

     

    Find the probability that Sally walked Mack on at least one of these two mornings.  (2 marks)

  2. In the month of April, the probability of pleasant weather in the morning was .
    1. Find the probability that on a particular morning in April, Sally walked Mack.  (2 marks)

    2. Using your answer from part b.i., or otherwise, find the probability that on a particular morning in April, the weather was pleasant, given that Sally walked Mack that morning.  (2 marks)

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b.i.  
 

met1-2014-vcaa-q9-answer1 
 

 

 

♦ Part (b)(ii) mean mark 38%.
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Probability, 2ADV S1 2007 MET1 11

There is a daily flight from Paradise Island to Melbourne. The probability of the flight departing on time, given that there is fine weather on the island, is 0.8, and the probability of the flight departing on time, given that the weather on the island is not fine, is 0.6.

In March the probability of a day being fine is 0.4.

Find the probability that on a particular day in March

  1. the flight from Paradise Island departs on time  (2 marks)

  2. the weather is fine on Paradise Island, given that the flight departs on time.  (2 marks)

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ii.  

♦♦ Mean mark 29%.
MARKER’S COMMENT: Students continue to struggle with conditional probability. Attention required here.
 

 

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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ii.    met1-2015-vcaa-q8-answer
 
 

 

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♦♦ 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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♦ Mean mark 46%.

Probability, 2ADV S1 2013 MET1 7

The probability distribution of a discrete random variable, , is given by the table below.

  1. Show that  .  (3 marks)

  2. Let  .

    1. Calculate  .  Answer in exact form.  (2 marks)

    2. Find  .  (1 mark)

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b.i.  
   
   
   

 

♦♦ Part (b)(ii) mean mark 32%.
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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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♦ Mean mark 36%.

Probability, 2ADV S1 2008 MET1 7

Jane drives to work each morning and passes through three intersections with traffic lights. The number of traffic lights that are red when Jane is driving to work is a random variable with probability distribution given by

  1. What is the mode of  ?  (1 mark)

  2. Jane drives to work on two consecutive days. What is the probability that the number of traffic lights that are red is the same on both days?  (2 marks)

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Calculus, 2ADV C4 2018* HSC 15c

The shaded region is enclosed by the curve    and the line  , as shown in the diagram. The line    meets the curve    at    and  . Do NOT prove this.
 


 

  1.  Use integration to find the area of the shaded region.  (2 marks)

  2. Use the Trapezoidal rule and four function values to approximate the area of the shaded region.  (2 marks)

The point is chosen on the curve    so that the tangent at is parallel to the line    and the -coordinate of is positive

  1.  Show that the coordinates of are  .  (2 marks)

  2.  Using the perpendicular distance formula  ,  find the area of  .  (2 marks)

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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 2011* HSC 5c

The table gives the speed of a jogger at time in minutes over a  20-minute period. The speed is measured in metres per minute, in intervals of 5 minutes.

The distance covered by the jogger over the 20-minute period is given by  .

Use the Trapezoidal rule and the speed at each of the five time values to find the approximate distance the jogger covers in the 20-minute period.   (3 marks)

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Calculus, 2ADV C4 SM-Bank 07

The diagram shows a block of land and its dimensions, in metres. The block of land is bounded on one side by a river. Measurements are taken perpendicular to the line  , from    to the river, at equal intervals of 50 m.
 

Integration, 2UA 2009 HSC 3d

 
Use the Trapezoidal rule with six subintervals to find an approximation to the area of the block of land.   (3 marks)

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