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L&E, 2ADV E1 SM-Bank 14

The spread of a highly contagious virus can be modelled by the function

Where is the number of days after the first case of sickness due to the virus is diagnosed and    is the total number of people who are infected by the virus in the first    days.

  1. Calculate  .  (1 mark)

  2. Find the value of    and interpret it result.  (2 marks)

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i.  
   
   

 

ii.   
   
   

 

Vectors, EXT1 V1 SM-Bank 23

A fireworks rocket is fired from an origin , with a velocity of 140 metres per second at an angle of    to the horizontal plane.
 


 

The position vector , from , of the rocket after    seconds is given by

The rocket explodes when it reaches its maximum height.

  1. Show the rocket explodes at a height of    metres.  (2 marks)

  2. Show the rocket explodes at a horizontal distance of    metres from .  (1 mark)

  3. For best viewing, the rocket must explode at a horizontal distance of 500 m and 800 m from , and at least 600 m above the ground.

     

    For what values of    will this occur.  (3 marks)

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i.   

 

 
 

 

ii.   

 
 

 

iii.   

 

 

 

 

 

 

 

 

Vectors, EXT1 V1 SM-Bank 22

Consider the rhombus    shown below, where    and    and    is a real constant.
 

  1. Find  .  (1 mark)

  2. Show that the diagonals of rhombus    are perpendicular.  (2 marks)

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i.   

 

 

ii.   

 
 

 

 
 

 

 
 
 

 

Vectors, EXT1 V1 SM-Bank 20

Consider the vector  , where    and    are unit vectors in the positive direction of the and axes respectively.

  1. Find the unit vector in the direction of  .    (1 mark)

  2. Find the acute angle that    makes with the positive direction of the -axis.   (1 mark)

  3. The vector  .

     

    Given that    is perpendicular to  , find the value of  .   (1 mark)

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i.   

 

ii.   

 
 
     

 

iii.   

Vectors, EXT1 V1 SM-Bank 19

Consider the following vectors

  1. Find  .  (1 mark)

  2. The points , and are vertices of a triangle. Prove that the triangle has a right angle at .  (2 marks)

  3. Find the length of the hypotenuse of the triangle.  (1 mark)

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i. 

COMMENT: Many teachers recommend column vector notation to simplify calculations and minimise errors – we agree!

 
 
 

 

ii.   
   
   
   

 

 

iii.   

 
 
 
 
 

Vectors, EXT1 V1 SM-Bank 15

Consider the vectors

  1. Calculate  .  (1 mark)

  2. Find the values of    for which  .  (2 marks)

  3. Find the value of    such that    is perpendicular to  .  (1 mark)

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i.   
   
   

 

ii.   

 

iii.   

Statistics, NAP-L3-CA02

Kim is a bricklayer and he recorded the number of bricks he layed each day for five days in a picture graph.
 


 

What is the difference between the number of bricks Kim laid on Monday and the number of bricks he laid on Thursday?

 
 
 
 
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Calculus, EXT2 C1 2019 HSC 15c

  1. Show that  .  (2 marks)


  2. Let  .

     


    Show that  .  (3 marks)


  3. Evaluate  .  (2 marks)

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i.   


 

 
 
 
 

 
ii.   

 

 
 
 
 

 

iii.   
 
   
   
   

 

 
 

Mechanics, EXT2 M1 2019 HSC 14b

A parachutist jumps from a plane, falls freely for a short time and then opens the parachute. Let t be the time in seconds after the parachute opens,   be the distance in metres travelled after the parachute opens, and    be the velocity of the parachutist in .

The acceleration of the parachutist after the parachute opens is given by

where is the acceleration due to gravity and is a positive constant.

  1. With an open parachute the parachutist has a terminal velocity of  .

     

    Show that  .  (1 mark)

    At the time the parachute opens, the speed of descent is .

  2. Show that it takes    seconds to slow down to a speed of .  (4 marks)

  3. Let    be the distance the parachutist travels between opening the parachute and reaching the speed .

     

     

    Show that  .  (3 marks)

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i.   

 

ii.   

 

 

 

 

 
 
 

 

iii.   
 
 
 
   
   

 

 
 
 
 
 
 

Functions, EXT1′ F1 2019 HSC 12d

Consider the function  .

  1.  Sketch the graph  .  (1 mark)

  2.  Sketch the graph  .  (2 marks)

  3.  Without using calculus, sketch the graph  .  (2 marks)

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i.   

 

ii.   

 

iii.   

Calculus, EXT1′ C3 2019 HSC 12b

The diagram shows two straight railway tracks that meet at an angle of    at the point .

Trains    and    are joined by a cable which is 70 m long.

At time    seconds, train    is    metres from    and train    is    metres from .

Train    is towing train    and is moving at a constant speed of  away from .

  1. Show that  .  (1 mark)
  2. What is the value of    when train    is 30 metres from    and train    is 50 metres from ?  (3 marks)

 

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(i)   

 

(ii)   
 

 

 

 
 

Algebra, STD1 A3 2019 HSC 30

A small business makes and sells bird houses.

Technology was used to draw straight-line graphs to represent the cost of making the bird houses and the revenue from selling bird houses . The -axis displays the number of bird houses and the -axis displays the cost/revenue in dollars.
 


 

  1. How many bird houses need to sold to break even?  (1 mark)

  2. By first forming equations for cost and revenue , determine how many bird houses need to be sold to earn a profit of $1900.  (3 marks)

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a.   

 

b.   

♦ Mean mark part (b) 47%.


 


 

Statistics, STD1 S3 2019 HSC 27

A set of bivariate data is collected by measuring the height and arm span of eight children. The graph shows a scatterplot of these measurements.
 

  1. On the graph, draw a line of best fit by eye.  (1 mark)

  2. Robert is a child from the class who was absent when the measurements were taken. He has an arm span of 147 cm. Using your line of best fit from part (a), estimate Robert’s height.  (1 mark)

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  1.   
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a.     
       

♦ Mean mark (a) 38%.

b.   

Financial Maths, STD1 F1 2019 HSC 26

Scott decided to have a shed built at his home. He agreed to the following costs:

  • Materials
– Timber $5400
– Roof $1800
– Nails $160
– Paint $375
  • $70 per hour for the builder when working Monday to Friday
  • $30 per hour for the labourer when working Monday to Friday
  • Builder and labourer paid time-and-a-half when working on Saturday.

It took six days to build the shed. The builder and labourer both worked from 8 am until 4 pm each day from Monday to Friday. On each of these days, they both took a 1 hour unpaid lunch break at 12 noon.

The builder and labourer also both worked 4 hours on Saturday, without a break, to finish the job.

What was the total cost to build the shed?  (4 marks)

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Probability, STD1 S2 2019 HSC 24

The faces on a biased six-sided die are labelled 1, 2, 3, 4, 5 and 6. The die was rolled twenty times. The relative frequency of rolling a 6 was 30% and the relative frequency of rolling a 2 was 15%. The number 3 was the only other number rolled in the twenty rolls.

How many times was the number 3 rolled in the twenty rolls of the die?  (3 marks)

 
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Measurement, STD1 M5 2019 HSC 20

The plan of the lower level of a small house is shown.


 

  1. How many windows are shown on the plan?  (1 mark)

  2. What is the actual perimeter, in metres, of the shaded part of the kitchen floor?  (2 marks)

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a.   

 

b.   
   

 

 

Measurement, STD1 M4 2019 HSC 18

The travel graph displays Nikau's car trip along a straight road from home and back again. The trip has been broken into four separate sections: , ,   and  .
 


 

  1. How far did Nikau travel in total?  (1 mark)

  2. In which section of the trip, , , and , did Nikau travel the fastest?  (1 mark)

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a.   
   

 

♦ Mean mark part (b) 42%.

b.   

Mechanics, EXT2* M1 2019 HSC 13d

The point    is on a sloping plane that forms an angle of 45° to the horizontal. A particle is projected from the point  . The particle hits a point    on the sloping plane as shown in the diagram.
 


 

The equation of the line    is  . The equations of motion of the particle are

where    is the time in seconds after projection. Do NOT prove these equations.

  1. Find the distance    between the point of projection and the point where the particle hits the sloping plane.  (2 marks)

  2. What is the size of the acute angle that the path of the particle makes with the sloping plane as the particle hits the point  ?  (3 marks)

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i.   
 

 

 

 
 

 

ii.   
 

 

 
 
 

 

♦ Mean mark part (ii) 39%.
 

 

 
 

 

Mechanics, EXT2* M1 2019 HSC 13c

A particle moves in a straight line. At time    seconds the particle has a displacement of m, a velocity of    and acceleration  .

Initially the particle has displacement  0 m  and velocity  . The acceleration is given by  . The velocity of the particle is always positive.

  1. Show that  .  (2 marks)

  2. Find an expression for as a function of  .   (2 marks)

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i.   
 
 
   

 

 

 

 

ii.   
 
 
   
   

 

♦ Mean mark part (ii) 46%.

Calculus, EXT1 C1 2019 HSC 12d

A refrigerator has a constant temperature of 3°C. A can of drink with temperature 30°C is placed in the refrigerator.

After being in the refrigerator for 15 minutes, the temperature of the can of drink is 28°C.

The change in the temperature of the can of drink can be modelled by  ,  where is the temperature of the can of drink, is the time in minutes after the can is placed in the refrigerator and is a constant.

  1. Show that  , where is a constant, satisfies

     

    .  (1 mark)

  2. After 60 minutes, at what rate is the temperature of the can of drink changing?   (3 marks)

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a.   

 
 

 

b.   

 

 

 
 
 
 

Mechanics, EXT2* M1 2019 HSC 12b

A particle is moving along the -axis in simple harmonic motion. The position of the particle is given by

for   

  1. Write    in the form  , where    and  .  (2 marks)

  2. Find the two values for    where the particle comes to rest.   (1 mark)

  3. When is the first time that the speed of the particle is equal to half of its maximum speed?  (2 marks)

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i.   

  

 

♦ Mean mark part (ii) 46%.

 

ii.   
 

 

iii.   

♦ Mean mark part (iii) 49%.

 

 

 

Statistics, EXT1 S1 2019 HSC 11f

Prize-winning symbols are printed on 5% of ice-cream sticks. The ice-creams are randomly packed into boxes of 8.

  1. What is the probability that a box contains no prize-winning symbols?  (1 mark)

  2. What is the probability that a box contains at least 2 prize-winning symbols?  (2 marks)

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i.   

 

ii.   

Financial Maths, 2ADV M1 2019 HSC 16a

A person wins $1 000 000 in a competition and decides to invest this money in an account that earns interest at 6% per annum compounded quarterly. The person decides to withdraw $80 000 from this account at the end of every fourth quarter. Let    be the amount remaining in the account after the th withdrawal.

  1.  Show that the amount remaining in the account after the withdrawal at the end of the eighth quarter is

     

    .  (2 marks)

  2.  For how many years can the full amount of $80 000 be withdrawn?  (3 marks)

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i. 

 
 
   
   
   

 

ii.   
   
 
   
   

♦♦♦ Mean mark part (ii) 19%.

 

 

Probability, 2ADV S1 2019 HSC 15d

The probability that a person chosen at random has red hair is 0.02

  1. Two people are chosen at random.

     

    What is the probability that at least ONE has red hair?  (2 marks)

  2. What is the smallest number of people that can be chosen at random so that the probability that at least ONE has red hair is greater than 0.4?  (2 marks)

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a.  
 

 

 

b. 

♦♦ Mean mark 24%.

 

 

Financial Maths, STD2 F4 2019 HSC 37

A new car is bought for $24 950. Each year the value of the car is depreciated by the same percentage.

The table shows the value of the car, based on the declining-balance method of depreciation, for the first three years.

What is the value of the car at the end of 10 years?  (3 marks)

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Algebra, STD2 A2 2019 HSC 34

The relationship between British pounds and Australian dollars on a particular day is shown in the graph.
 

  1. Write the direct variation equation relating British pounds to Australian dollars in the form  . Leave as a fraction.  (1 mark)

  2. The relationship between Japanese yen and Australian dollars on the same day is given by the equation  .

     

    Convert 93 100 Japanese yen to British pounds.  (2 marks)

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a.   

♦ Mean mark 42%.

 

b.