The number of students that could solve a Rubik's cube in less than 2 minutes were counted at 7 different primary schools.
The results were recorded in the graph below.
How many students, in total, could solve the cube in less than 2 minutes?
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Algebra, NAP-I3-NC07
Point
What are the new coordinates of
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Probability, NAP-I3-NC06 SA
Peter has a marble bag that contains 20 marbles that are either red or green in colour.
The probability of randomly picking a green marble is 70%.
What is the probability of randomly picking a red marble?
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Geometry, NAP-E4-CA04
A sculpture is pictured from 3 different angles below:
The sculpture is in the shape of
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a rectangle and a circle. |
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two spheres. |
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a cylinder and a circle. |
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a cylinder and a sphere. |
Measurement, NAP-C3-NC07
Michelle purchased the 4 items pictured below at the supermarket.
The total mass of Michelle's items is closest to
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Geometry, NAP-C3-NC06
An orientation course is set up with 4 drinking stations.
Runners start and then progress through the course in the order of the drinking stations labelled 1-4 in the diagram below.
At which drinks station do the runners make the greatest change of direction?
| Station 1 | Station 2 | Station 3 | Station 4 |
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Number, NAP-C3-NC05
John wants to find the answer to
Which of the following shows a way to get the same answer?
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Geometry, NAP-C3-NC04
Rahul draws a net, which is pictured below.
What 3D object will it make?
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Pentagonal prism |
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Pentagonal pyramid |
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Hexagonal pyramid |
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Hexagonal prism |
Probability, NAP-C3-NC03
Jenny and Sam play a board game with the spinner shown.
Jenny spins the arrow.
On which number is the arrow most likely to stop?
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Algebra, NAP-C3-CA03
At an apple orchard, apples are picked and put in a basket.
The table below shows the total number of apples in the basket after each minute.
How many apples are in the basket after 10 minutes?
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Statistics, NAP-C3-CA02
For three days, Terry counted the colour of cars he saw drive past his house.
Which column on the graph below shows the total number of Red cars?
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Number, NAP-E4-CA05
Zilda has 4 children.
Each child needs 6 pens when they start school for the new year.
Zilda has no pens when she goes to the shop.
The shop sells its pens in packets of 5.
How many packets does Zilda need to buy?
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Statistics, NAP-E4-CA03
Anne taught a kindergarten class and asked her students what their favorite fruit is.
Their answers were used to draw the pie chart below.
Around a quarter of her students preferred which fruit?
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Geometry, NAP-E4-CA02
A map of a park shows four different gates.
If Alan enters the park and is travelling in south-east direction, what gate did he enter the park by?
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Number, NAP-E4-CA01
The airforce buys 8 new fighter jets for $882.0 million.
Each jet costs the same amount.
What is the cost of 1 fighter jet?
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Algebra, NAP-E4-NC07
Which expression is equal to
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Number, NAP-E4-NC03
Bogdan grows large turnips in his garden.
One turnip he picks weighs 1.2 kg.
Bogdan cuts 300 grams off the turnip to cook with.
What fraction has he used for cooking?
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Number, NAP-E4-NC02
Cameron paid $1.44 for 9 pencils.
How much did each pencil cost?
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Number, NAP-F4-CA07
David buys a surfboard cover online that costs him $128.
The postage costs involved are listed in the table below.
The surfboard cover weighs 1.1 kg
What does David have to pay, in total, to purchase the surfboard and have it delivered?
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Geometry, NAP-F4-NC06
Which two lines below are lines of symmetry?
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Number, NAP-F4-NC04
Nigella placed a leg of lamb into her oven in the morning.
The 1st measurement of the lamb's temperature was −6°C.
Fifteen minutes later, a second temperature reading was taken, which measured 16°C.
What was the change in temperature between the two measurements?
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Number, NAP-F4-NC03
A circle is divided into 5 equal areas and labelled, as shown in the diagram below.
What percentage of the circle's area has been labelled with an odd number?
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Geometry, NAP-F4-CA05
9 cubes are used to make an obstacle, pictured below, that is part of a children's adventure playground.
Gary paints all sides of the step except for the bottom side which sits on the ground.
How many cube faces will Gary paint?
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Number, NAP-F4-CA04
The first three days of the Brisbane cricket test had the following attendances:
What was the total crowd over the first 3 days, to the nearest
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Algebra, NAP-F4-CA03
Find the value of
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Geometry, NAP-H4-NC04
Some tiles are missing from this pattern of tiles.
When complete the pattern has two lines of symmetry.
Which of these could be the missing part of the pattern?
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Show Answers Only
Show Worked Solution
Measurement, NAP-G4-NC04
A giant earthworm measures 2.1 metres.
How long is it in centimetres?
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Number, NAP-G4-NC02
Ben spent twice as much money as Mark.
If they spent a total of $90, how much did Ben spend?
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Number, NAP-G4-CA07
Two arrows point to numbers on a number line.
Which one of these numbers lies between the two arrows?
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Measurement and Geometry, NAP-G4-CA04
Petunia walks along the path from
How far does she walk in kilometres?
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Geometry, NAP-H4-NC06
Banjo drew the net of a rectangular prism.
One of the prism's faces is drawn incorrectly.
Which face did he draw incorrectly?
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Number, NAP-H4-NC02
Andreas has $8 to buy batteries for his toy racing car.
Each battery costs $1.60 and he buys 4 batteries.
Which expression shows how much money he has left?
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Algebra, NAP-H4-NC01
The table below shows a pattern.
What is the top number when the bottom number is 28?
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Geometry, NAP-H4-CA03
This is Albert's left glove.
Albert's right glove is a mirror image of his left glove.
Which of these is Albert's right glove?
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Show Answers Only
Show Worked Solution
Measurement, NAP-I4-NC02
Janus measures the width of his driveway to be 4 metres and 18 centimetres.
Which answer shows how Janus can write this measurement in metres?
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Measurement, NAP-I4-CA06
Peter left home at 9:15 in the morning and did not return until 5:25 in the afternoon.
How long was Peter away from his house?
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3 hours 50 minutes |
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4 hours 10 minutes |
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7 hours 50 minutes |
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8 hours 10 minutes |
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13 hours 20 minutes |
Number, NAP-I4-CA03
On a country property, 1 acre of land is recommended for every 4 sheep.
How many acres of land would be needed for 16 sheep?
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Number, NAP-I4-CA02
Which of the following is equal to 32?
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Quadratic, EXT1 2016 HSC 14c
The point
The tangent to the parabola
The normal to the parabola
- Show that the point
has coordinates . (1 mark) - Show that the locus of
lies on another parabola . (3 marks) - State the focal length of the parabola
. (1 mark)
It can be shown that the minimum distance between
- Find the values of
so that the distance between and is a minimum. (2 marks)
Binomial, EXT1 2016 HSC 14b
Consider the expansion of
- Show that
. (1 mark) - Show that
. (1 mark) - Hence, or otherwise, show that
. (2 marks)
Plane Geometry, EXT1 2016 HSC 13c
The circle centred at
Copy or trace the diagram into your writing booklet.
- Show that
is a cyclic quadrilateral. (2 marks) - Hence, or otherwise, prove that
is perpendicular to . (2 marks)
Show Worked Solution
| i. |  |
ii.
♦♦♦ Mean mark 14%.
Calculus, 2ADV C3 2016 HSC 16b
Some yabbies are introduced into a small dam. The size of the population,
where
- Show that the rate of growth of the size of the population is
. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
- Find the range of the function
, justifying your answer. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
- Show that the rate of growth of the size of the population can be written as
. (1 mark)
--- 4 WORK AREA LINES (style=lined) ---
- Hence, find the size of the population when it is growing at its fastest rate. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
Show Worked Solution
| i. | ||
ii.
♦♦♦ Mean mark (ii) 21%.
iii.
♦♦♦ Mean mark (iii) 18%.
iv.
♦♦♦ Mean mark (iv) 14%.
Proof, EXT2 P2 2016 HSC 16c
In a group of
A situation in which none of the
Let
- Tom is one of the
people. In some derangements Tom finds that he and one other person have each other's hat. Show that, for
, the number of such derangements is (1 mark)
--- 4 WORK AREA LINES (style=lined) ---
- By also considering the remaining possible derangements, show that, for
(2 marks)
--- 6 WORK AREA LINES (style=lined) ---
- Hence, show that
, for (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
- Given
and , deduce that , for (1 mark)
--- 5 WORK AREA LINES (style=lined) ---
- Prove by mathematical induction, or otherwise, that for all integers
(2 marks)
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Complex Numbers, EXT2 N2 2016 HSC 16b
- The complex numbers
and form the vertices of an equilateral triangle in the Argand diagram. Show that
(2 marks)
--- 8 WORK AREA LINES (style=lined) ---
- Give an example of non-zero complex numbers
and , so that and form the vertices of an equilateral triangle in the Argand diagram. (1 mark)
--- 3 WORK AREA LINES (style=lined) ---
Show Worked Solution
i.
♦♦ Mean mark 23%.
ii.
♦♦ Mean mark 33%.
Complex Numbers, EXT2 N2 2016 HSC 16a
- The complex numbers
and , where and , satisfy
By considering the real and imaginary parts of, or otherwise, show that and form the vertices of an equilateral triangle in the Argand diagram. (3 marks)
--- 12 WORK AREA LINES (style=lined) ---
- Hence, or otherwise, show that if the three non-zero complex numbers
and satisfy
AND
then they form the vertices of an equilateral triangle in the Argand diagram. (2 marks)
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Show Worked Solution
| i. | ||
♦♦♦ Mean mark 25%.
STRATEGY: A clear graphical image simplifies both parts of this question significantly.
STRATEGY: A clear graphical image simplifies both parts of this question significantly.
ii.
♦♦♦ Mean mark 5%.
Polynomials, EXT2 2016 HSC 15c
- Use partial fractions to show that
(2 marks) - Suppose that for
a positive integer
Show that
(3 marks) - Hence, or otherwise, find the limiting sum of
(2 marks)
Algebra, STD2 A4 2016 HSC 29b
The mass
- What is the value of
? (1 mark)
--- 1 WORK AREA LINES (style=lined) ---
- What is the daily growth rate of the pig’s mass? Write your answer as a percentage. (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
Probability, STD2 S2 2016 HSC 23 MC
A group of 485 people was surveyed. The people were asked whether or not they smoke. The results are recorded in the table.
A person is selected at random from the group.
What is the approximate probability that the person selected is a smoker OR is male?
- 33%
- 18%
- 68%
- 87%
Algebra, 2UG 2016 HSC 18 MC
The value of
It is known that
What is the value of
Statistics, STD2 S1 2016 HSC 7 MC
Which set of data is classified as categorical and nominal?
- blue, green, yellow
- small, medium, large
- 5.2 cm, 6 cm, 7.21 cm
- 4 people, 5 people, 9 people
Calculus, 2ADV C4 2016 HSC 9 MC
What is the value of
Show Worked Solution
♦♦♦ Mean mark 19%.
Trigonometry, 2ADV T2 2016 HSC 8 MC
How many solutions does the equation
Probability, MET1 2007 VCAA 6
Two events,
- Calculate
when . (1 mark)
--- 5 WORK AREA LINES (style=lined) ---
- Calculate
when and are mutually exclusive events. (1 mark)
--- 5 WORK AREA LINES (style=lined) ---
Show Worked Solution
a.
♦♦ Mean mark (a) 31%.
MARKER’S COMMENTS: Students who drew a Venn diagram or Karnaugh map were the most successful.
MARKER’S COMMENTS: Students who drew a Venn diagram or Karnaugh map were the most successful.
♦♦ Mean mark (b) 23%.
| b. |  |
Functions, MET1 2008 VCAA 10
Let
- Find the rule and domain of the inverse function
. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
- On the axes provided, sketch the graph of
for its maximal domain. (1 mark)
--- 0 WORK AREA LINES (style=lined) ---
- Find
in the form where , and are real constants. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
Show Answers Only
Show Worked Solution
a.
b.
♦ Mean mark part (b) 19%.
MARKER’S COMMENT: Few students were aware that a function of its own inverse function is the line
MARKER’S COMMENT: Few students were aware that a function of its own inverse function is the line
♦ Mean mark 34%.
| c. | ||
Calculus, MET1 2011 VCAA 10
The figure shown represents a wire frame where
Let
- Find
and in terms of and . (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
- Find the length,
cm, of the wire in the frame, including length , in terms of and . (1 mark)
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- Find
, and hence show that when . (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
- Find the maximum value of
if . (1 mark)
--- 5 WORK AREA LINES (style=lined) ---
Show Worked Solution
a.
| b. | ||
c.
♦ Mean mark 35%.
d.
♦♦♦ Mean mark 5%.
Functions, MET1 2011 VCAA 4
If the function
- find integers
and such that (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
- state the maximal domain for which
is defined. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
Show Worked Solution
| a. | ||
b.
♦♦♦ Mean mark 13%.
MARKER’S COMMENT: “Very poorly answered” with a common response of
MARKER’S COMMENT: “Very poorly answered” with a common response of
Calculus, MET1 2012 VCAA 10
Let
- i. Find, in terms of
, the -coordinate of the stationary point of the graph of (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
- ii. State the values of
such that the -coordinate of this stationary point is a positive number. (1 mark)
--- 4 WORK AREA LINES (style=lined) ---
- For a particular value of
, the tangent to the graph of at passes through the origin. - Find this value of
. (3 marks)
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Calculus, MET1 2014 VCAA 10
A line intersects the coordinate axes at the points
- When
, the line is a tangent to the graph of at the point with coordinates , as shown.
If
and are non-zero real numbers, find the values of and . (3 marks)
--- 9 WORK AREA LINES (style=lined) ---
- The rectangle
has a vertex at on the line. The coordinates of are , as shown.
- Find an expression for
in terms of . (1 mark)
--- 4 WORK AREA LINES (style=lined) ---
- Find the minimum total shaded area and the value of
for which the area is a minimum. (2 marks)
--- 7 WORK AREA LINES (style=lined) ---
- Find the maximum total shaded area and the value of
for which the area is a maximum. (1 mark)
--- 4 WORK AREA LINES (style=lined) ---
- Find an expression for
Show Worked Solution
a.
♦ Mean mark 48%.
MARKER’S COMMENT: The most common error incorrectly stated that (6,0) lay on the parabola.
MARKER’S COMMENT: The most common error incorrectly stated that (6,0) lay on the parabola.
| b.i. |
♦♦ Mean mark (b.i.) 32%.
♦♦♦ Mean mark (b.ii.) 18%.
| b.ii. | ||
b.iii.
♦♦♦ Mean mark (b.iii.) 8%.
Calculus, MET1 2015 VCAA 10
The diagram below shows a point,
The diagram also shows the tangent to the circle at
- Find the coordinates of
in terms of . (1 mark)
--- 5 WORK AREA LINES (style=lined) ---
- Find the gradient of the tangent to the circle at
in terms of . (1 mark)
--- 4 WORK AREA LINES (style=lined) ---
- The equation of the tangent to the circle at
can be expressed as - i. Point
, with coordinates , is on the line segment . - Find
in terms of . (1 mark)
--- 3 WORK AREA LINES (style=lined) ---
- ii. Point
, with coordinates , is on the line segment . - Find
in terms of . (1 mark)
--- 4 WORK AREA LINES (style=lined) ---
- Consider the trapezium
with parallel sides of length and . - Find the value of
for which the area of the trapezium is a minimum. Also find the minimum value of the area. (3 marks)
--- 10 WORK AREA LINES (style=lined) ---
Calculus, MET2 2010 VCAA 4
Consider the function
- Find the
-coordinate of each of the stationary points of and state the nature of each of these stationary points. (4 marks)
--- 7 WORK AREA LINES (style=lined) ---
In the following,
- Write down, in terms of
and , the possible values of for which is a stationary point of . (3 marks)
--- 5 WORK AREA LINES (style=lined) ---
- For what value of
does have no stationary points? (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
- Find
in terms of if has one stationary point. (2 marks)
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- What is the maximum number of stationary points that
can have? (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
- Assume that there is a stationary point at
and another stationary point where . - Find the value of
. (3 marks)
--- 4 WORK AREA LINES (style=lined) ---
Show Worked Solution
a.
b.
♦ Mean mark (b) 46%.
MARKER’S COMMENT: Issues with use of CAS caused significant difficulties in this question.
MARKER’S COMMENT: Issues with use of CAS caused significant difficulties in this question.
c.
♦ Mean mark part (c) 47%.
d.
♦♦♦ Mean mark (d) 16%.
♦ Mean mark (e) 37%.
e.
f.
♦♦♦ Mean mark (f) 9%.
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