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Binomial, EXT1 2016 HSC 14b

Consider the expansion of  , where is a positive integer.

  1. Show that  .  (1 mark)
  2. Show that  .  (1 mark)
  3. Hence, or otherwise, show that  .  (2 marks)
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i.    

 

ii.  

 

iii. 

♦♦♦ Mean mark 14%.
 

 

 

 

Mechanics, EXT2* M1 2016 HSC 13b

The trajectory of a projectile fired with speed    at an angle    to the horizontal is represented by the parametric equations

  and   ,

where is the time in seconds.

  1. Prove that the greatest height reached by the projectile is  .  (2 marks)

A ball is thrown from a point above the horizontal ground. It is thrown with speed at an angle of to the horizontal. At its highest point the ball hits a wall, as shown in the diagram.
 

     ext1-2016-hsc-q13
 

  1. Show that the ball hits the wall at a height of above the ground.  (2 marks)

The ball then rebounds horizontally from the wall with speed . You may assume that the acceleration due to gravity is .

  1. How long does it take the ball to reach the ground after it rebounds from the wall?  (2 marks)

  2. How far from the wall is the ball when it hits the ground?  (1 mark)

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i.   
 

 

 

 

ii.  

 

♦♦ Mean mark part (iii) 35%.
iii.   ext1-hsc-2016-13bi

 

MARKER’S COMMENT: Many students struggled to solve: .

 

 

iv.  

♦ Mean mark 43%.

Calculus, EXT1 C1 2016 HSC 12a

The diagram shows a conical soap dispenser of radius 5 cm and height 20 cm.
 

     ext1-2016-hsc-q12
 

At any time seconds, the top surface of the soap in the container is a circle of radius cm and its height is cm.

The volume of the soap is given by  .

  1.  Explain why  .  (1 mark)

  2.  Show that  .  (1 mark)

The dispenser has a leak which causes soap to drip from the container. The area of the circle formed by the top surface of the soap is decreasing at a constant rate of  ².
 

  1.  Show that  .  (2 marks)

  2.  What is the rate of change of the volume of the soap, with respect to time, when ?  (2 marks)

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i.   ext1-hsc-2016-12a

 

 

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Statistics, EXT1 S1 2016 HSC 11f

A darts player calculates that when she aims for the bullseye the probability of her hitting the bullseye is    with each throw.

  1. Find the probability that she hits the bullseye with exactly one of her first three throws.  (1 mark)

  2. Find the probability that she hits the bullseye with at least two of her first six throws.  (2 marks)

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ii.  

Financial Maths, 2ADV M1 2016 HSC 14b

A gardener develops an eco-friendly spray that will kill harmful insects on fruit trees without contaminating the fruit. A trial is to be conducted with 100 000 insects. The gardener expects the spray to kill 35% of the insects each day and that exactly 5000 new insects will be produced each day.

The number of insects expected at the end of the th day of the trial is

  1. Show that  .  (2 marks)

  2. Show that  .  (1 mark)

  3. Find the expected insect population at the end of the fourteenth day, correct to the nearest 100.  (1 mark)

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i.  
 
   
   

 

ii. 
 
   
 
   
   
   
   

 

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Integration, 2UA 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
 

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

hsc-2016-14aii

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

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²

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Polynomials, EXT2 2016 HSC 15c

  1. Use partial fractions to show that

      (2 marks)

  2. Suppose that for a positive integer

     

    Show that    (3 marks)

  3. Hence, or otherwise, find the limiting sum of

     

      (2 marks)

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Calculus, EXT1* C1 2016 HSC 13c

A radioactive isotope of Curium has a half-life of 163 days. Initially there are 10 mg of Curium in a container.

The mass in milligrams of Curium, after days, is given by

where and are constants.

  1. State the value of .  (1 mark)

  2. Given that after 163 days only 5 mg of Curium remain, find the value of .  (2 marks)

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i. 

 

ii. 

Calculus, 2ADV C3 2016 HSC 13a

Consider the function  

  1. Find the two stationary points and determine their nature.  (4 marks)

  2. Sketch the graph of the function, clearly showing the stationary points and the and intercepts.  (2 marks)

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i.  

ii.
 ext2-hsc-2016-13bi

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ii.  ext2-hsc-2016-13bi

Calculus, 2ADV C4 2016 HSC 12d

  1. Differentiate   . (1 mark)

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

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ii. 

Trigonometry, 2ADV T1 2016 HSC 12c

Square tiles of side length 20 cm are being used to tile a bathroom.

The tiler needs to drill a hole in one of the tiles at a point which is 8 cm from one corner and 15 cm from an adjacent corner.

To locate the point the tiler needs to know the size of the angle shown in the diagram.
 

 hsc-2016-12c

 
Find the size of the angle to the nearest degree.  (3 marks)

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α

 

 

 
 

Algebra, STD2 A2 2016 HSC 29e

The graph shows the life expectancy of people born between 1900 and 2000.
 


  1. According to the graph, what is the life expectancy of a person born in 1932?  (1 mark)

  2. With reference to the value of the gradient, explain the meaning of the gradient in this context.  (2 marks)

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i.   

ii.   

 
 

 

♦♦ Mean mark (ii) 33%.

Complex Numbers, EXT2 N2 2016 HSC 12c

Let 

  1. By considering the real part of  , show that    is
     
      (2 marks)

     

  2. Hence, or otherwise, find an expression for    involving only powers of   (1 mark)

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ii.   
   
   

Conics, EXT2 2016 HSC 12a

The diagram shows an ellipse.

ext2-hsc-2016-12a

  1. Write an equation for the ellipse.  (1 mark)
  2. Find the eccentricity of the ellipse.  (1 mark)
  3. Write the coordinates of the foci of the ellipse.  (1 mark)
  4. Write the equations of the directrices of the ellipse.  (1 mark)
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ii.   
 
 
 

 

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Functions, EXT1′ F1 2016 HSC 11d

The diagram shows the graph of  
 

ext2-hsc-2016-11d

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

  1.    (2 marks)

  2.    (2 marks)

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i.   

ext2-hsc-2016-11d-answer2
ii.   

ext2-hsc-2016-11d-answer4
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i.    ext2-hsc-2016-11d-answer2

 

ii.    ext2-hsc-2016-11d-answer4

Complex Numbers, EXT2 N1 2016 HSC 11a

Let  

  1.  Express    in modulus-argument form.  (2 marks)

  2.  Show that    is real.  (1 mark)

  3.  Find a positive integer such that    is purely imaginary.  (1 mark)

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ii.   
   
   

 

 

iii.  

Mechanics, EXT2 2016 HSC 8 MC

A small object of mass kg sits on a rotating conical surface at metres from the axis and with as shown in the diagram.

The surface is rotating about its axis with angular velocity The forces acting on the object are gravity, a normal reaction force and a frictional force , which prevents the object from sliding down the surface.

 ext2-hsc-2016-8a

Which pair of statements is correct?

A.
 
B.
 
C.
 
D. 
 
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Measurement, STD2 M1 2016 HSC 28b

The cost of buying a new heater is $990. It has an energy consumption of 505 kWh per year.

Energy is charged at the rate of $0.35 kWh.

How much will it cost in total to purchase and then run this heater for five years?  (2 marks)

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Financial Maths, STD2 F4 2016 HSC 27d

Marge borrowed $19 000 to buy a used car. Interest on the loan was charged at 4.8% pa at the end of each month. She made a repayment of $436 at the end of every month. The table below sets out her monthly repayment schedule for the first four months of the loan. 
 

2ug-2016-hsc-q27_1

  1. Some values in the table are missing. Write down the values for and .  (2 marks)

  2. Calculate the value of .  (2 marks)

  3. Marge repaid this loan over four years.

     

    What is the total amount that Marge repaid?  (1 mark)

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i.   
 

 

 

ii.   

 
♦♦ Mean mark part (iii) 28%.
COMMENT: Read carefully whether total paid or total interest paid is required.

 

iii.   

Algebra, STD2 A2 2016 HSC 26c

Peta’s car uses fuel at the rate of 5.9 L /100 km for country driving and 7.3 L /100 km for city driving. On a trip, she drives 170 km in the country and 25 km in the city.

Calculate the amount of fuel she used on this trip.  (2 marks)

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Probability, MET1 2009 VCAA 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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a.  

 

b.  

 

c.  

♦ Mean mark 46%.

Calculus, MET1 2010 VCAA 9

Part of the graph of    is shown below.

vcaa-2010-meth-9a

  1. Find the derivative of  .   (1 mark)

  2. Use your answer to part a. to find the area of the shaded region in the form    where and are non-zero real constants.   (3 marks)

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a.  

 

 

b.  

♦♦ Mean mark 33%.
MARKER’S COMMENT: The most common error was not dividing everything through by 2. Be careful!

 

 
 
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