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Financial Maths, STD2 F1 SM-Bank 5

Alex is buying a used car which has a sale price of  $13 380. In addition to the sale price there are the following costs:

2014 27a1

  1. Stamp Duty for this car is calculated at $3 for every $100, or part thereof, of the sale price.  
    Calculate the Stamp Duty payable.   (1 mark)

  2. Alex wishes to take out comprehensive insurance for the car for 12 months.

     

    The cost of comprehensive insurance is calculated using the following:
     
          2014 27a2
    Find the total amount that Alex will need to pay for comprehensive insurance.   (3 marks)

  3. Alex has decided he will take out the comprehensive car insurance rather than the less expensive non-compulsory third-party car insurance.

  4.  

    What extra cover is provided by the comprehensive car insurance?   (1 mark)

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♦♦♦ Mean mark 12%
IMPORTANT: “or part thereof ..” in the question requires students to round up to 134 to get the right multiple of $3 for their calculation.
i.   
 

 

ii.  

 

 

 
 

 

 
 

 

 

 

♦ Mean mark 34%.
iii.  
 

Measurement, STD2 M1 2017 HSC 30e

A solid is made up of a sphere sitting partially inside a cone.

The sphere, centre , has a radius of 4 cm and sits 2 cm inside the cone. The solid has a total height of 15 cm. The solid and its cross-section are shown.
 


 

Using the formula    where  is the radius of the cone's circular base and is the perpendicular height of the cone, find the volume of the cone, correct to the nearest cm³?  (3 marks)

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Proof, EXT2 P2 2017 HSC 16c

A 2 by grid is made up of two rows of square tiles, as shown.
 

The tiles of the 2 by grid are to be painted so that tiles sharing an edge are painted using different colours. There are different colours available, where  .

It is NOT necessary to use all the colours.

Consider the case of the 2 by 2 grid with tiles labelled A, B, C and D, as shown.
 

There are ways to choose colours for the first column containing tiles A and B. Do NOT prove this.

  1. Assume the colours for tiles A and B have been chosen. There are two cases to consider when choosing colours for the second column. Either tile C is the same colour as tile B, or tile C is a different colour from tile B.

     

    By considering these two cases, show that the number of ways of choosing colours for the second column is  .  (2 marks)

  2. Prove by mathematical induction that the number of ways in which the 2 by grid can be painted is  , for  .  (2 marks)

  3. In how many ways can a 2 by 5 grid be painted if 3 colours are available and each colour must now be used at least once?  (2 marks)

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i.   

♦♦♦ Mean mark 20%.

 

 

 

ii.   

♦♦♦ Mean mark 22%.

 

 

 

 

iii.   

♦♦♦ Mean mark 22%.

 
 

 

Harder Ext1 Topics, EXT2 2017 HSC 16a

  Let  , where  .

  1. Show that  , for any integer .  (1 mark)
  3. Let  , where is a positive integer.
  4. By summing the series, prove that  

    .  (3 marks)

  6. Deduce, from parts (i) and (ii), that.  (2 marks)

  7. Show that  
     is independent of .  (1 mark)

 

 

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(i)   

 
 

 

(ii)   

♦♦ Mean mark 34%.

 
 
 
 
 
♦ Mean mark part (iii) 48%.

 

(iii)   
   
   
   

 

 
 
 
 

♦♦♦ Mean mark part (iv) 28%.

 

(iv)   

 

Conics, EXT2 2017 HSC 15c

The ellipse with equation  , where  , has eccentricity .

The hyperbola with equation  , has eccentricity .

The value of is chosen so that the hyperbola and the ellipse meet at  , as shown in the diagram.

  1. Show that  
  2. .  (2 marks)
  4. If the two conics have the same foci, show that their tangents at are perpendicular.  (3 marks)

 

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(i)   

 

 

(ii)   

♦ Mean mark 51%.

 

 

 
 

 

   
   
 
 

 

 

 

MATRICES, FUR1 2017 VCAA 7 MC

At a fish farm:

    • young fish () may eventually grow into juveniles () or they may die ()
    •  juveniles () may eventually grow into adults () or they may die ()
    • adults () eventually die ().

The initial state of this population, , is shown below.
 

 

Every month, fish are either sold or bought so that the number of young, juvenile and adult fish in the farm remains constant.

The population of fish in the fish farm after months, , can be determined by the recurrence rule
 


 

where is a column matrix that shows the number of young, juvenile and adult fish bought or sold each month and the number of dead fish that are removed.

Each month, the fish farm will

  1. sell 1650 adult fish.
  2. buy 1750 adult fish.
  3. sell 17 500 young fish.
  4. buy 50 000 young fish.
  5. buy 10 000 juvenile fish.
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GRAPHS, FUR1 2017 VCAA 8 MC

The shaded area in the graph below shows the feasible region for a linear programming problem.

The objective function is given by

Which one of the following statements is not true?

  1. When and , the minimum value of is at point .
  2. When and , the maximum value of is at point .
  3. When and , the minimum value of is at point .
  4. When and , the maximum value of is at point .
  5. When and , the maximum value of is at point .
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NETWORKS, FUR1 2017 VCAA 8 MC

The flow of oil through a series of pipelines, in litres per minute, is shown in the network below.
 

 
The weightings of three of the edges are labelled .

Five cuts labelled A–E are shown on the network.

The maximum flow of oil from the source to the sink, in litres per minute, is given by the capacity of

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GEOMETRY, FUR1 2017 VCAA 8 MC

Three circles of radius 50 mm are placed so that they just touch each other.
The region enclosed by the circles is shaded in the diagram below.

The area of the shaded region, in square millimetres, is closest to

  1.  
  2.  
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GEOMETRY, FUR1 2017 VCAA 7 MC

A triangle has:

• one side, , of length 4 cm
• one side, , of length 7 cm
• one angle, , of 26°.

Which one of the following angles, correct to the nearest degree, could not be another angle in triangle ?

  1.  
  2.  
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CORE, FUR1 2017 VCAA 24 MC

Xavier borrowed $245 000 to pay for a house.

For the first 10 years of the loan, the interest rate was 4.35% per annum, compounding monthly.

Xavier made monthly repayments of $1800.

After 10 years, the interest rate changed.

If Xavier now makes monthly repayments of $2000, he could repay the loan in a further five years.

The new annual interest rate for Xavier’s loan is closest to

  1.  0.35%
  2.  4.1%
  3.  4.5%
  4.  4.8%
  5.  18.7%
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Statistics, NAP-I3-CA09

The graph shows the origin and type of all vehicles in a city.
 

 
Which statement is most accurate based on the graph?

 
 There are more utility vehicles than trucks and vans.
 
 Utility vehicles are the most common type of vehicles
 
 There are more Australian vehicles than European vehicles.
 
 There are more Asian vehicles than European vehicles.
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Statistics, NAP-A4-CA03

A dealership sells new and used cars.

The graph shows the price of 2 similar cars and their age in years
 

 
Which one of these statements is true?

 
 Car Q is older and less expensive than Car P.
 
 Car P is newer and less expensive than Car Q.
 
 Car P is older and more expensive than Car Q.
 
 Car Q is newer and more expensive than Car P.
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Quadratic, EXT1 2017 HSC 14b

Let    be a point on the parabola  .

The tangent to the parabola at meets the parabola  , , at and . Let be the midpoint of .

  1. Show that the coordinates of and satisfy
  2. .  (2 marks)

  3. Show that the coordination of are  .  (2 marks)
  4. Find the value of so that the point always lies on the parabola  .  (2 marks)

 

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(i)   

 

 

 

(ii)   

♦♦ Mean mark 30%.
 
 

 

 

 
 

 

(iii)   

♦♦ Mean mark 23%.
 
 

Mechanics, EXT2* M1 2017 HSC 13c

A golfer hits a golf ball with initial speed at an angle to the horizontal. The golf ball is hit from one side of a lake and must have a horizontal range of 100 m or more to avoid landing in the lake.
 

     

Neglecting the effects of air resistance, the equations describing the motion of the ball are

,

where is the time in seconds after the ball is hit and is the acceleration due to gravity in . Do NOT prove these equations.

  1. Show that the horizontal range of the golf ball is
     
          metres.  (2 marks)

  2. Show that if    then the horizontal range of the ball is less than 100 m.  (1 mark)

It is now given that    and that the horizontal range of the ball is 100 m or more.

  1. Show that  .  (2 marks)

  2. Find the greatest height the ball can achieve.  (2 marks)

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i.   

 

 
 

 

ii.   

♦ Mean mark 44%.

 

 

iii.   

 

iv.   

♦♦ Mean mark 35%.
 

 

 
 

 

 
 
 
 

Plane Geometry, 2UA 2017 HSC 16c

In the triangle , the point is the mid-point of . The point lies on and

.

The line that passes through the point and is parallel to meets produced at .

Copy or trace this diagram into your writing booklet.

  1. Prove that is isosceles.  (3 marks)
  2. The point is chosen on so that is parallel to .
     
    Show that bisects .  (2 marks)

 

 

 

 

 

 

 

 

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(i)  

♦♦♦ Mean mark 21%.

 

 

 

(ii)  

♦♦ Mean mark 31%.

Calculus, 2ADV C3 2017 HSC 16a

John’s home is at point and his school is at point . A straight river runs nearby.

The point on the river closest to is point , which is 5 km from .

The point on the river closest to is point , which is 7 km from .

The distance from to is 9 km.

To get some exercise, John cycles from home directly to point on the river, km from , before cycling directly to school at , as shown in the diagram.
 


  

The total distance John cycles from home to school is km.

  1. Show that  .   (1 mark)

  2. Show that if  , then  .   (3 marks)

  3. Find the value of that makes  .   (2 marks)

  4. Explain why this value of gives a minimum for .   (1 mark)

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i.  

 
 

 

ii. 

♦ Mean mark 41%.

 

 

iii. 

♦♦ Mean mark 29%.
♦♦♦ Mean mark 9%.

 

iv.  

Measurement, 2UG 2017 HSC 30e

A solid is made up of a sphere sitting partially inside a cone.

The sphere, centre , has a radius of 4 cm and sits 2 cm inside the cone. The solid has a total height of 15 cm. The solid and its cross-section are shown.
 


 

What is the volume of the cone, correct to the nearest cm³?  (3 marks)

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³³

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♦♦ Mean mark 34%.

 

 
 
  ³³

Statistics, STD2 S5 2017 HSC 29d

All the students in a class of 30 did a test.

The marks, out of 10, are shown in the dot plot.
 


 

  1. Find the median test mark.  (1 mark)

  2. The mean test mark is 5.4. The standard deviation of the test marks is 4.22.

     

    Using the dot plot, calculate the percentage of the marks which lie within one standard deviation of the mean.  (2 marks)

  3. A student states that for any data set, 68% of the scores should lie within one standard deviation of the mean. With reference to the dot plot, explain why the student’s statement is NOT relevant in this context.  (1 mark)

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♦ Mean mark 50%.
i.   
   
   

 

ii.   

♦♦ Mean mark 34%.

 

iii.   

♦♦♦ Mean mark 13%.

Measurement, 2UG 2017 HSC 29a

A new 200-metre long dam is to be built.
The plan for the new dam shows evenly spaced cross-sectional areas.
 


 

  1. Using TWO applications of Simpson’s rule, show that the volume of the dam is approximately  44 333 m³.  (2 marks)
  2. It is known that the catchment area for this dam is 2 km².
    Calculate how much rainfall is needed, to the nearest mm, to fill the dam.  (2 marks)

 

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(i)   
   
   
    ³

 

(ii)   ²²

♦♦♦ Mean mark 11%.
²
  ²
 
 
 

Algebra, 2UG 2017 HSC 28a

Temperature can be measured in degrees Celsius () or degrees Fahrenheit ().

The two temperature scales are related by the equation  .

  1. Calculate the temperature in degrees Fahrenheit when it is  −20 degrees Celsius.  (1 mark)
  2. Solve the following equations simultaneously, using either the substitution method or the elimination method.  (2 marks)

     

  3. The graphs of   and    are shown below.
  4.  


  5. What does the result from part (ii) mean in the context of the graph?  (1 mark)     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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(i)  
   

 

(ii)  
 

 

♦♦ Mean mark 31%.
MARKER’S COMMENT: An area that requires attention.

 

♦♦♦ Mean mark 20%.

 

(iii)  

Number and Algebra, NAP-J2-19

A company has 3706 computers in a warehouse.

Another 423 computers are delivered to the warehouse.

Which of these could be used to calculate the correct number of computers in the warehouse?

 
 3000 + 4000 + 700 + 20 + 6 + 3
 
 3000 + 700 + 400 + 20 + 60 + 3
 
 3000 + 700 + 400 + 20 + 6 + 3
 
 3000 + 400 + 70 + 20 + 6 + 3
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Statistics, NAP-J3-CA17

The bottles in Renee's fridge are pictured below.

Renee decides to make a graph where each bar represents one type of bottle in her fridge.
  

Renee makes an error when creating the graph.

What should Renee do to correct the error?

 
 Make each category bar a different colour.
 
 Change the title to 'Number of bottles in the fridge by volume'.
 
 Change the 'Number of bottles' label to 'Volume of bottles'.
 
 Remove the 'Juice' category since orange juice and apple juice are already shown.
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Number, NAP-J3-CA14

Madison uses the number sentence  15 × 12 = 180  to solve a problem.

Which of the following could be the problem?

 
 Madison buys 15 showbags. How much does each showbag cost?
 
 Madison spends $15 on 180 showbags. How much does she spend?
 
 Madison buys 15 showbags that cost $12 each. How many showbags does she buy?
 
 Madison buys 12 showbags that cost $15 each. How much does she spend?
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Number, NAP-J3-CA09

Dinesh is a teacher and buys pencils for his class.

He purchases the pencils in two different sized packets.

Dinesh buys 10 packet A's and 4 packet B's.

He then divides all the pencils equally among his 9 students.

How many pencils does each student receive?

 
 
 
 
 
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Number, NAP-J3-CA06 SA

There are 48 Year 7 students at a high school.

Each student is asked if they own a bike or not and the results are recorded.

of the students said they owned a bike.

How many Year 7 students at the school own a bike?
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