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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
 
 

 

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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♦ 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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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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♦ Mean mark 49%.

ii.   
 

 

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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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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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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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    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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Trig Ratios, EXT1 2015 HSC 12c Answer1

 

 

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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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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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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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♦ 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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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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♦ Mean mark 98%!

 

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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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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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♦ Mean mark 43%.


 

 

 

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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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Financial Maths, 2ADV M1 2018 HSC 14d

An artist posted a song online. Each day there were    downloads, where  is the number of days after the song was posted.

  1. Find the number of downloads on each of the first 3 days after the song was posted.  (1 mark)

  2. What is the total number of times the song was downloaded in the first 20 days after it was posted?  (2 marks)

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♦ Mean mark 38%.

Calculus, EXT1* C1 2018 HSC 13c

The population of a country grew exponentially between 1910 and 2010. This population can be modelled by the equation  , where    is the population of the country in millions, is the time in years after 1910 and is a positive constant. The population of the country in 1960 was 184 million.

  1. Show that the value of is 0.0139, correct to 4 decimal places.  (2 marks)

  2. Assuming that this model continues to be valid after 2010, estimate the population of the country in 2020 to the nearest million.  (2 marks)

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Calculus, 2ADV C3 2018 HSC 13a

Consider the curve  .

  1. Find the stationary points and determine their nature.  (3 marks)

  2. Given that the point  (2,16)  lies on the curve, show that it is a point of inflection.  (2 marks)

  3. Sketch the curve, showing the stationary points, the point of inflection and the and intercepts.  (2 marks)

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Calculus, 2ADV C1 2018 HSC 12d

The displacement of a particle moving along the -axis is given by

where is the displacement from the origin in metres and is the time in seconds, for .

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

  2. At which times is the particle stationary?  (2 marks)

  3. Find the position of the particle when the acceleration is zero.  (2 marks)

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Trigonometry, 2ADV T1 2018 HSC 12a

A ship travels from Port A on a bearing of 050° for 320 km to Port B. It then travels on a bearing of 120° for 190 km to Port C.
 


 

  1. What is the size of  ?  (1 mark)

  2. What is the distance from Port A to Port C ? Answer to the nearest 10 kilometres.  (2 marks)

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Measurement, STD2 M1 2018 HSC 28a

A field is bordered on one side by a straight road and on the other side by a river, as shown. Measurements are taken perpendicular to the road every 7.5 metres along the road.
 

 
Use four applications of the Trapeziodal rule to find an approximation to the area of the field. Answer to the nearest square metre.  (3 marks)

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Statistics, STD2 S5 2018 HSC 27e

Joanna sits a Physics test and a Biology test.

  1. Joanna’s mark in the Physics test is 70. The mean mark for this test is 58 and the standard deviation is 8.

     

    Calculate the -score for Joanna’s mark in this test.  (1 mark)

  2. In the Biology test, the mean mark is 64 and the standard deviation is 10.

     

    Joanna’s -score is the same in both the Physics test and the Biology test.

     

    What is her mark in the Biology test?  (2 marks)

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ii.   
 
 

Algebra, STD2 A4 2018 HSC 27d

The graph displays the cost () charged by two companies for the hire of a minibus for hours.
 


  

Both companies charge $360 for the hire of a minibus for 3 hours.

  1. What is the hourly rate charged by Company A?  (1 mark)

  2. Company B charges an initial booking fee of $75.

     

    Write a formula, in the form of  , for the cost of hiring a minibus from Company B for hours.  (2 marks)

  3. A minibus is hired for 5 hours from Company B.

     

    Calculate how much cheaper this is than hiring from Company A.  (2 marks)

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Financial Maths, STD2 F4 2018 HSC 26h

A car is purchased for $23 900.

The value of the car is depreciated by 11.5% each year using the declining-balance method.

What is the value of the car after three years?  (2 marks)

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Algebra, STD2 A1 2018 HSC 26b

Clark’s formula, given below, is used to determine the dosage of medicine for children.
 

 
For a particular medicine, the adult dosage is 325 mg and the correct dosage for a specific child is 90 mg.

How much does the child weigh, to the nearest kg?  (2 marks)

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Algebra, STD2 A2 2018 HSC 5 MC

The driving distance from Alex's home to his work is 20 km. He drives to and from work five times each week. His car uses fuel at the rate of 8 L/100 km.

How much fuel does he use driving to and from work each week?

  1.  16 L
  2.  20 L
  3.  25 L
  4.  40 L
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