Chapter 9 — Two Full Practice Papers
Two complete Math sections, built to the real shape:
- Two modules of 27 questions, done one after the other — the form of the official paper practice tests. The digital test you sit has 22 questions a module; the study guide explains why this book uses the longer form
- Grid-ins at questions 6, 7, 13, 14, 20, 21 and 27 in each module — the positions they occupied in all sixteen official modules examined
- A mix of topics weighted the way the test weights them: about 35% Algebra, 35% Advanced Math, 15% Problem-Solving and Data Analysis, 15% Geometry and Trigonometry
- Difficulty rising through each module, roughly
Save these for the final weeks. They are most useful once you have worked through the chapters, because a mock taken too early mostly measures what you have not studied yet.
How to sit one
Give yourself 43 minutes per module, which is the official timing for a 27-question module — the same pace as the digital test's 22 questions in 35 minutes, about a minute and a half a question. Do Module 1, take a short break, then Module 2. Use a calculator — you are allowed one on every question.
Work on paper. Write the answer down for every question, including ones you guess: there is no penalty for a wrong answer, and a blank scores the same as a wrong one.
Do not check answers between modules.
Scoring
Count how many of the 54 you got right. That is your raw score. This table converts it to the 200–800 Math section scale, and it is the real conversion table published with official practice test 10 — not an estimate.
| Raw score | Math section score |
|---|---|
| 0 | 200 |
| 3 | 210–220 |
| 6 | 220–250 |
| 9 | 250–310 |
| 12 | 310–350 |
| 15 | 340–380 |
| 18 | 370–410 |
| 21 | 390–430 |
| 24 | 410–450 |
| 27 | 440–480 |
| 30 | 470–510 |
| 33 | 490–550 |
| 36 | 530–590 |
| 39 | 560–620 |
| 42 | 600–660 |
| 45 | 640–700 |
| 48 | 680–740 |
| 50 | 730–770 |
| 52 | 770–790 |
| 54 | 800 |
The score is a range because this simplified conversion is slightly less precise than the one the real test uses. Treat the middle of the range as your estimate.
Two things the table tells you that are worth reading off it. Every question is worth roughly the same, so the cheapest marks are the easy ones you rushed. And the middle of the range is where the gains are largest: moving from 27 right to 39 right is about 120 points, while moving from 45 to 51 is a similar gain for far more work.
Paper 1 — Module 1
27 questions · 43 minutes · calculator permitted
Topic: Solving a linear equation
A baker sells loaves of bread at a market for $6 each and pays $4 a day to rent the stall. On one day the money from the loaves, less the rent for the stall, came to $26. How many loaves did the baker sell that day?
A)
B)
C)
D)
Show the worked solution
Answer: B
Explanation
With loaves sold, the money from the loaves is dollars, and the rent takes 4 of them away:
Check: ✓
A is the money from the loaves, not the number of loaves. C ignored the rent and divided 26 by 6. D subtracted the 4 where it has to be added back.
Takeaway: Add before dividing, and finish the last step.
Topic: Percentage of a number
A regional council surveyed 240 households about recycling and found that 30% of them separate glass from other waste. The council plans to write to each of those households with information about a new collection service. How many households will receive the letter?
A)
B)
C)
D)
Show the worked solution
Answer: D
Explanation
C is the 70% who do not separate glass. B subtracted 30 and A added 30. A percentage of something is always a multiplication.
Takeaway: "% of " means
Topic: Slope from two points
A tank is being filled with water at a constant rate. The graph shows the depth of the water in the tank, in centimetres, during the first 6 minutes.
At what rate, in centimetres per minute, is the depth of the water increasing?
A)
B)
C)
D)
Show the worked solution
Answer: C
Explanation
Read the two marked points off the graph: after 1 minute the depth is 4 centimetres, and after 5 minutes it is 16 centimetres.
B is upside down, A has the sign flipped, D added.
Takeaway: Rise over run, same order top and bottom.
Topic: Evaluating a function
The function is defined by What is the value of
A)
B)
C)
D)
Show the worked solution
Answer: D
Explanation
A added the 15 instead of subtracting it. C and B each computed one term and stopped.
Takeaway: Substitute into every term, then combine.
Topic: Expanding brackets
Which of the following expressions is equivalent to
A)
B)
C)
D)
Show the worked solution
Answer: D
Explanation
B has the is bigger than the so the middle is positive. C skipped the middle. A multiplied rather than added them.
Takeaway: Four products, then collect the middle two.
Topic: A linear equation with a fraction
A water tank is being filled. The volume in the tank, in litres, is given by where is the number of seconds since filling began. After how many seconds does the tank hold 11 litres?
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 32
Explanation
Check: ✓ Enter 32.
Takeaway: Subtract before multiplying up.
Topic: Mean from a total
A teacher recorded the scores of four students on a recent class test and set them out in the table below. One script was mislaid before its score could be entered.
| Student | Amara | Ben | Chen | Dara |
|---|---|---|---|---|
| Score | 18 | 24 | 19 | ? |
The mean of all four scores is known to be 21. What was the fourth score?
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 23
Explanation
The four must total The three given total 61, so the fourth
is Enter 23.
Takeaway: Turn a mean into a total straight away.
Topic: A system by substitution
At a bake sale, a volunteer sold cakes and trays of cookies. The number of trays of cookies sold was 2 fewer than 3 times the number of cakes sold. Twice the number of cakes sold, added to the number of trays of cookies sold, is 13. How many cakes did the volunteer sell?
A)
B)
C)
D)
Show the worked solution
Answer: C
Explanation
The two sentences say and Substitute the first into the second:
Then Check both: ✓ and ✓ A reached and stopped. D lost the altogether, reaching B gave the trays of cookies.
Takeaway: Substitute, solve, then check which variable was asked for.
Topic: Exponent rules
Which of the following is equivalent to where
A)
B)
C)
D)
Show the worked solution
Answer: B
Explanation
then dividing subtracts:
A added instead of multiplying at the first step. C skipped the division. D multiplied everything.
Takeaway: Power of a power multiplies; division subtracts.
Topic: Ratio
An animal shelter houses only cats and dogs. The ratio of the number of cats to the number of dogs at the shelter is and the shelter houses 48 animals in total. How many of the animals are dogs?
A)
B)
C)
D)
Show the worked solution
Answer: C
Explanation
parts, so one part is Dogs are 3 parts:
Check: cats and ✓ A is the ratio number itself, not a count of animals. D gave the cats. B divided the 48 by 3 instead of by the 8 parts.
Takeaway: Add the parts, find one part, then multiply.
Topic: A quadratic from its graph
The graph of the quadratic function is shown in the -plane.
Which of the following could define
A)
B)
C)
D)
Show the worked solution
Answer: D
Explanation
The graph crosses the -axis at and at so and
A function that is zero at has the factor and one that is zero at 5 has the factor
Check one of them: at ✓
Why each wrong option is wrong:
- A, — is zero at 3 and at both signs are the wrong way round.
- B, — is zero at and at The graph has a positive zero.
- C, — is zero at 3 and at 5. The graph has a negative zero.
Takeaway: A zero at means a factor of so the sign in the bracket is the opposite of the sign on the axis.
Topic: Pythagoras
A rectangular garden plot is shown, with its dimensions in metres. A gardener wants to run a straight path from one corner of the plot to the opposite corner, along the dashed line.
What is the length of that path, in metres?
A)
B)
C)
D)
Show the worked solution
Answer: C
Explanation
The path is the hypotenuse of a right triangle whose legs are the two sides of the plot, 8 and 15:
The 8–15–17 triple. D added the legs, B subtracted their squares, A averaged them.
Takeaway: Squares add, then take the root.
Topic: An absolute-value equation
On a number line, the distance between and 7 is 4. What is the greatest possible value of
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 11
Explanation
The distance between and 7 is so
gives gives The greatest is 11.
Enter 11.
Takeaway: Two cases, then read the wording.
Topic: A unit rate
A machine in a bottling plant fills bottles at a constant rate of 45 bottles every 4 minutes. At this rate, how many minutes does the machine take to fill 300 bottles?
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 80/3 or 26.66 or 26.67
Explanation
bottles a minute, so 300 bottles take
The decimal never ends, and the grid holds five characters, so it must be
filled: enter 80/3, or 26.66 (cut off), or 26.67 (rounded). The shorter
26.7 is marked wrong.
Takeaway: Reduce to a per-one rate, then scale. A decimal that never ends has to fill the grid.
Topic: The equation of a line
A small business's profit, in thousands of dollars, is a linear function of the number of units it sells, The table shows two values.
| Units sold, | 2 | 5 |
|---|---|---|
| Profit, | 3 | 15 |
What is the -intercept of the graph of this function in the -plane?
A)
B)
C)
D)
Show the worked solution
Answer: A
Explanation
Check: at ✓ The slope is which is where the comes from.
B is the -value of the first point, not the value at D added the 8 instead of subtracting it. C has the sign wrong: is negative.
Takeaway: Substitute the point into and solve for
Topic: Exponential growth
A biologist estimates that a lake held 500 fish at the start of a study and that the number of fish in the lake increases by 20% each year. If the estimate is correct, how many fish will the lake hold 2 years after the start of the study?
A)
B)
C)
D)
Show the worked solution
Answer: D
Explanation
A did one year only. C added 20% of the original twice, which ignores compounding — the second year's 20% is of 600, not 500. B worked out the second year's growth correctly, 120, and then added it to 500 instead of 600.
Takeaway: Percentage growth compounds: multiply, do not add.
Topic: Simplifying a rational expression
Which of the following is equivalent to where
A)
B)
C)
D)
Show the worked solution
Answer: D
Explanation
B kept the wrong factor. C and A cancelled terms, not factors.
Takeaway: Factor first, cancel whole brackets only.
Topic: A two-way table
A school recorded which of its 120 students study Spanish and which study history, and set the counts out in the two-way table below.
| Studies history | Does not study history | Total | |
|---|---|---|---|
| Studies Spanish | 45 | 25 | 70 |
| Does not study Spanish | 20 | 30 | 50 |
| Total | 65 | 55 | 120 |
A student who studies Spanish is chosen at random. What is the probability that the student also studies history?
A)
B)
C)
D)
Show the worked solution
Answer: D
Explanation
The pool is restricted to Spanish students, so the denominator is 70:
B used all 120. C is the chance of studying Spanish at all, A counted the other 25.
Takeaway: "Chosen from those who…" fixes the denominator.
Topic: Interpreting a model
A car rental company charges a fixed amount to hire a vehicle and a further amount for each mile driven. The total cost, in dollars, of hiring a car and driving it miles is given by the function Which of the following is the best interpretation of the number in this context?
A) The fixed rental fee
B) The cost of each additional mile
C) The total cost of a 30-mile trip
D) The number of miles included free
Show the worked solution
Answer: B
Explanation
The number multiplied by is the rate: 30 cents per mile. A describes the 45. C would be $54. D invents something the model does not contain: a number of miles would not be multiplied by
Takeaway: In the multiplied number is the rate — check the units.
Topic: A system solved by elimination
A warehouse packs small boxes and large boxes, and every box of one size weighs the same. Together, 4 small boxes and 3 large boxes weigh 26 kilograms. The 4 small boxes weigh 8 kilograms more than one large box does. What is the weight, in kilograms, of one large box?
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 9/2 or 4.5
Explanation
Let be the weight of a small box and the weight of a large box, in kilograms. Each sentence gives one equation:
Write the second as Both equations now have so subtract the second from the first:
A large box weighs 4.5 kilograms.
Check: the second equation gives and then ✓
Enter 9/2 or 4.5.
Takeaway: One sentence, one equation. Matching terms, subtract — and bracket the second equation so the double negative is visible.
Topic: The discriminant
In the equation is a constant. The graph of the related function touches the -axis at exactly one point. What is the value of
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 25
Explanation
One solution means
Then ✓ Enter 25.
Takeaway: Exactly one solution ⇒ discriminant zero ⇒ perfect square.
Topic: A circle equation
A circular sprinkler is installed in a field. The set of points it can reach is described in the -plane by the equation where distances are measured in metres. What is the radius, in metres, of the region the sprinkler reaches?
A)
B)
C)
D)
Show the worked solution
Answer: A
Explanation
The right side is so B is not C halved the 36 instead of taking its square root. D is the centre, — a point, not a length, and the answer to a different question.
Takeaway: Take the square root of the right-hand side.
Topic: Similar figures: from an area ratio to a length ratio
Two scale models of the same building are similar solids. The areas of their corresponding faces are in the ratio What is the ratio of their corresponding side lengths?
A)
B)
C)
D)
Show the worked solution
Answer: B
Explanation
Areas scale by the square of the length ratio, so take square roots:
So the lengths are in the ratio A left the area ratio unchanged. C squared instead of rooting. D took the root and then wrote it upside down.
Takeaway: Lengths areas volumes Go backwards with a root.
Topic: A composite function
The functions and are defined by and What is the value of
A)
B)
C)
D)
Show the worked solution
Answer: B
Explanation
Inside first: Then
A is the other order — always offered. C added the two separate outputs, D multiplied the three numbers in sight.
Takeaway: means first. Work outward from the inner bracket.
Topic: An inequality in context
A trailer can carry at most 500 kilograms. It already holds a generator that weighs 75 kilograms. What is the greatest number of 35-kilogram sacks of cement that can be added?
A)
B)
C)
D)
Show the worked solution
Answer: A
Explanation
You cannot load part of a sack, so round down to 12.
Check: ✓ and ✓
C forgot the generator. B rounded up — a count of physical objects always rounds down against a maximum. D did both.
Takeaway: Include everything that adds to the load, then round down.
Topic: Trigonometry
A ramp, the level ground and a vertical support form the right triangle shown, where is the angle the ramp makes with the ground. The lengths marked are in metres.
What is the value of
A)
B)
C)
D)
Show the worked solution
Answer: C
Explanation
The side next to the angle is 3 and the hypotenuse is 5. By Pythagoras the third side, the one opposite is
A is the cosine, adjacent over hypotenuse. D is greater than 1, which no sine can be. B is adjacent over opposite, — neither a sine nor a cosine.
Takeaway: Find the missing side first, then read off the ratio you need: sine is opposite over hypotenuse.
Topic: Working backwards through a percentage
A streaming service raised the price of its annual subscription by 25%. After the increase, the subscription costs $60 a year. What was the price, in dollars, of the annual subscription before the increase?
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 48
Explanation
The new price is 125% of the old:
Check: 25% of 48 is 12, and ✓
Taking 25% off $60 gives $45, not $48 — which is exactly why working
backwards needs division, not a second percentage. Enter 48.
Takeaway: To undo a percentage change, divide by the multiplier.
Paper 1 — Module 2
27 questions · 43 minutes · calculator permitted
Take a short break before starting. On the real test the second module's difficulty depends on how you did in the first; here it simply continues.
Topic: Solving a linear equation
A diver's depth below the surface, in metres, is modelled by the expression where is the number of minutes since the ascent began. At what value of is the diver 1 metre below the surface?
A)
B)
C)
D)
Show the worked solution
Answer: C
Explanation
Check: ✓ B kept the minus after dividing two negatives. D added the 1 to the 9 instead of subtracting it, reaching A made both slips.
Takeaway: A negative divided by a negative is positive.
Topic: Reading a linear function
A delivery service charges a fixed handling fee plus an amount for each kilometre driven. The total charge, in dollars, is where is the distance in kilometres. What is the value of
A)
B)
C)
D)
Show the worked solution
Answer: C
Explanation
A assumed an input of 0 gives an output of 0. B gave the slope. D added the two numbers in the formula, which is The 7 is the handling fee: the charge before any distance is driven.
Takeaway: is the -intercept — the constant term.
Topic: Area of a rectangle
A school is laying artificial turf on the rectangular play area shown, whose dimensions are in metres. The turf is sold by the square metre.
How many square metres of turf are needed to cover the play area exactly once?
A)
B)
C)
D)
Show the worked solution
Answer: D
Explanation
A added the two sides. B is the perimeter. C halved, which belongs to a triangle.
Takeaway: Area multiplies; perimeter adds. Rectangles have no halving step.
Topic: An equivalent expression
Which of the following expressions is equivalent to
A)
B)
C)
D)
Show the worked solution
Answer: A
Explanation
B left the undistributed. C dropped the from D made both of those slips at once.
Takeaway: Distribute across every term, then collect like with like.
Topic: A proportion
A map of a nature reserve is drawn to scale so that every 4 centimetres on the map represent 10 kilometres on the ground. Two picnic sites are 6 centimetres apart on the map. What is the actual distance, in kilometres, between the two picnic sites?
A)
B)
C)
D)
Show the worked solution
Answer: D
Explanation
Kilometres over centimetres is the same for the map's scale and for the two picnic sites. With kilometres between them:
Cross-multiply: so
Check: and ✓
A cross-multiplied the wrong pair, B is the number of kilometres that one centimetre stands for, not the distance. C multiplied 4 by 10 and divided by 6.
Takeaway: Cross-multiply, then check both fractions come to the same decimal.
Topic: A quadratic solved by factoring
A ball is thrown upward from the top of a cliff. Its height above the ground, in metres, seconds after it is thrown is given by After how many seconds does the ball hit the ground?
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 5
Explanation
On the ground the height is zero:
Divide every term by to make it easy to factor:
A time of is before the ball was thrown, so it is rejected. The ball lands
after 5 seconds. Check: ✓ Enter 5.
Takeaway: "Hits the ground" means height zero. Solve both roots, then let the situation choose: a time cannot be negative.
Topic: Percentage change
A retailer reduced the price of a jacket at the end of the season. Before the reduction the jacket cost $80, and afterwards it cost $68. By what percent was the price reduced?
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 15
Explanation
The fall is $12. Percentage change is measured against the original:
Using 68 as the base would give 17.6%, which is the standard wrong answer.
Enter 15.
Takeaway: Percentage change divides by the original value, always.
Topic: Parallel lines
In the -plane, line is parallel to the line What is the slope of line
A)
B)
C)
D)
Show the worked solution
Answer: A
Explanation
Parallel lines have equal slopes, so the slope is C dropped the sign.
B flipped and negated, which gives a perpendicular slope, D read the intercept.
Takeaway: Parallel copies the slope exactly, sign included.
Topic: Volume of a rectangular prism
A packing crate is a rectangular prism with the measurements shown, in metres.
What is the volume of the crate, in cubic metres?
A)
B)
C)
D)
Show the worked solution
Answer: D
Explanation
A added the three edges. B multiplied only two of the three edges: that is the area of one face. C is half the surface area. Check what the question asked for — a volume is in cubic units, an area in square ones.
Takeaway: Volume multiplies all three dimensions.
Topic: Median with an even count
A researcher recorded the number of birds visiting a feeder on each of six consecutive mornings. The bar graph shows the six counts in the order they were taken.
What is the median of the six counts?
A)
B)
C)
D)
Show the worked solution
Answer: C
Explanation
Read the six heights off the graph: 4, 9, 2, 11, 6 and 8.
Sorted: Six values, so average the middle two:
B is the mean. A and D picked one middle value without averaging.
Takeaway: Sort first. Even count means average the middle pair.
Topic: A negative exponent
What is the value of
A)
B)
C)
D)
Show the worked solution
Answer: B
Explanation
A negative exponent gives a small positive number, never a negative one.
A squared the 5 and made it negative. D ignored the minus sign on the exponent. C multiplied 5 by 2.
Takeaway: Negative exponent means reciprocal, not negative.
Topic: A system with no solution
Two rental firms price a job differently. The first charges according to and the second according to where is the number of hours and is a constant. For what value of do the two firms never charge the same amount?
A)
B)
C)
D)
Show the worked solution
Answer: D
Explanation
No solution means parallel: equal slopes, different intercepts. So and the intercepts 1 and differ ✓
A has the sign of the slope wrong. C matched intercepts, which would make the lines cross, not miss. B copied the other line's intercept.
Takeaway: Parallel and distinct. Check the intercept after matching the slope.
Topic: A system solved by elimination
At a small museum, every adult ticket has the same price and every child ticket has the same price. A group that buys 5 adult tickets and 2 child tickets pays $26 in total. The cost of 3 adult tickets is $6 more than the cost of 2 child tickets. What is the price, in dollars, of one adult ticket?
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 4
Explanation
Let an adult ticket cost dollars and a child ticket dollars. The two sentences say
The terms are and so adding the equations removes them:
Then gives Check both: ✓ and
✓ Enter 4.
Takeaway: Opposite signs on a variable, add the equations. Then check both originals.
Topic: The vertex of a parabola
What is the minimum value of the function
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 7/4 or 1.75
Explanation
Check: and — higher on both sides ✓
The question asked for the minimum value, which is the not the
Enter 7/4 or 1.75.
Takeaway: Find substitute, and check whether the question wants where or what.
Topic: Interpreting a scatterplot model
A paediatric clinic recorded the age and the height of each child in a group. The scatterplot shows the data, together with a line of best fit whose equation is where is age in years and is height in centimetres.
Which of the following is the best interpretation of the number 78 in this context?
A) The predicted height at age 0
B) The predicted growth each year
C) The number of children measured
D) The oldest age in the study
Show the worked solution
Answer: A
Explanation
The constant is the value at B describes the 6. C and D are things the model says nothing about: a line of best fit describes a relationship, not the size of the sample or the range of ages.
Takeaway: Constant = value at zero; coefficient = rate. Check the units.
Topic: An exponential decay model
A laboratory holds a sample of a radioactive substance with a mass of 400 grams. During each hour that passes, the sample decays so that its mass at the end of the hour is 85% of its mass at the start of the hour. Which function gives the mass in grams, of the sample after hours?
A)
B)
C)
D)
Show the worked solution
Answer: C
Explanation
Each hour the mass is multiplied by "is 85% of" gives the factor directly,
A grows by 15% an hour instead of shrinking. B is linear: it removes 0.85 grams an hour. D used the fraction lost.
Takeaway: "Decays to 85%" means "decays by 15%" means the same thing. Read which word is used.
Topic: A line through two points
The table shows two points on a line in the -plane.
| 3 | ||
|---|---|---|
| 4 | 16 |
What is the -intercept of the line?
A)
B)
C)
D)
Show the worked solution
Answer: D
Explanation
So Substitute giving
Check with the other point: ✓
C read a coordinate off a given point; B gave the slope. A put in for the point losing its sign:
Takeaway: Slope first, then substitute either point for — and confirm with the point you did not use.
Topic: A radical equation
What is the solution to the equation
A)
B)
C)
D)
Show the worked solution
Answer: B
Explanation
Square both sides:
Check in the original: ✓ — and with radicals the check is compulsory, because squaring can invent solutions.
A added 1 to both sides rather than squaring. C forgot to subtract the 1, and D added it instead of subtracting.
Takeaway: Square, solve, then substitute back into the original.
Topic: An arc of a circle
The figure shows a circular garden bed with centre and a radius of 9 metres. Points and lie on the edge of the bed, and angle measures
What is the length, in metres, of the shorter arc from to
A)
B)
C)
D)
Show the worked solution
Answer: A
Explanation
of the circle. The whole circumference is so the arc is
C is the sector area, a different quantity — an arc is a length. B used 18 as the radius, which is the diameter. D is the whole circumference, with the fraction never applied.
Takeaway: Fraction of the circle, times the circumference for an arc, times the area for a sector.
Topic: A system in context
A charity received two donations. Together they came to $54, and the larger donation was $12 more than the smaller. What was the smaller donation, in dollars?
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 21
Explanation
Adding the two statements gives so the larger is 33 and the smaller is
Check: ✓ and ✓ Enter 21.
Takeaway: The smaller of two numbers is
Topic: A perfect square
The expression where is a constant, can be written in the form What is the value of
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 81
Explanation
A perfect square is Here so and
Check: ✓ Enter 81.
Takeaway: Halve the middle coefficient, then square it.
Topic: Compound percentage
An investment of $800 is placed in an account that earns interest at a rate of 25% per year, compounded annually, and no money is added to the account or withdrawn from it. What is the value of the investment, in dollars, 2 years after it is placed in the account?
A)
B)
C)
D)
Show the worked solution
Answer: D
Explanation
B did one year only. C used simple interest — 25% of the original $800 twice, which is $400 rather than $450. Compounding earns more, because the second year's interest is on $1000, not $800. A added $25 a year instead of 25%.
Takeaway: Compound interest multiplies by the factor once per period. Compounding always beats simple interest over more than one period, and the gap is the whole point of the question.
Topic: Function transformation
The graph of passes through the point Which point must the graph of pass through?
A)
B)
C)
D)
Show the worked solution
Answer: A
Explanation
The is outside the bracket, so it moves the graph down 5:
B moved sideways, which only happens inside the bracket. C swapped the coordinates. D moved the graph up 5 instead of down.
Takeaway: Outside the bracket moves vertically, in the direction the sign says.
Topic: Probability from a bar graph
A bag used in a probability demonstration holds red, blue and green counters that are identical apart from their colour. The bar graph shows how many counters of each colour the bag holds.
A student draws one counter at random without looking into the bag. What is the probability that the counter drawn is not green?
A)
B)
C)
D)
Show the worked solution
Answer: C
Explanation
The bars show 5 red, 7 blue and 8 green counters. That is 20 counters, of which are not green:
Or take the complement: ✓
B answered the opposite question. D compared the green counters with the other twelve instead of with all twenty. A assumed three colours means three equally likely outcomes; the counts are not equal.
Takeaway: "Not" means subtract from 1, or count the ones you want directly. Both should agree.
Topic: Solving for a variable in terms of others
The perimeter of a rectangular field is related to its length and its width by the formula A surveyor knows the perimeter and the length and needs the width. Which expression gives in terms of and
A)
B)
C)
D)
Show the worked solution
Answer: B
Explanation
A skipped the division. C divided only the D moved the across without changing its sign.
Takeaway: Whatever you divide by must divide every term.
Topic: A perpendicular slope
Line has equation What is the slope of a line perpendicular to
A)
B)
C)
D)
Show the worked solution
Answer: A
Explanation
Flip and negate: Check: ✓
C is the slope of a parallel line. D and B each did one half of the job.
Takeaway: Negative reciprocal is two steps. Multiply the two slopes; you should get exactly
Topic: Putting several steps together
A landscaper is marking out a rectangular flower bed. Its length is 3 metres more than twice its width, and the total distance around the outside of the bed is 42 metres. What is the width of the flower bed, in metres?
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 6
Explanation
Let the width be Then the length is and the perimeter is
Check: width 6, length 15, perimeter ✓
The question asked for the width, not the length — and with no options to
warn you, re-reading the final line is the whole safety net. Enter 6.
Takeaway: Define the letter, build the equation, solve, then check which of the two quantities was asked for. This is the last question of the module; do not rush it.
Paper 2 — Module 1
27 questions · 43 minutes · calculator permitted
Sit this one at least a week after Paper 1, and mark Paper 1 first — a mock is only useful if you have acted on what the last one told you.
Topic: Solving a linear equation
A courier charges a fixed collection fee of $12 plus $5 for each parcel. For one job the total charge was $47. How many parcels were collected?
A)
B)
C)
D)
Show the worked solution
Answer: A
Explanation
With parcels the charge is $5 each plus the $12 fee, so
Check: ✓ C added the 12 instead of subtracting it. D stopped one step early. B ignored the fee and divided the whole 47 by 5.
Takeaway: Subtract, then divide, then finish.
Topic: Simplifying an expression
Which of the following expressions is equivalent to
A)
B)
C)
D)
Show the worked solution
Answer: A
Explanation
B kept the constant negative — but C turned the subtraction of into an addition. D subtracted the and then carried the 5 down without multiplying it by anything.
Takeaway: A minus before a bracket flips every sign inside it.
Topic: Reading a graph's intercept
The graph of a line is shown in the -plane.
What is the -coordinate of the point where the line crosses the -axis?
A)
B)
C)
D)
Show the worked solution
Answer: A
Explanation
The line meets the -axis at the point so the -coordinate is
C is the slope: the line rises 3 for every 2 it moves to the right. D is where the line crosses the -axis, at B is the value of on the -axis, not of
Takeaway: On the -axis, Each axis zeroes the other variable, so read the -intercept where the line meets the vertical axis.
Topic: Solving a two-step linear equation
A sports club orders printed T-shirts that cost $7 each. The club has a voucher that takes $9 off the total cost of one order, and after the voucher is applied the club pays $33 for the order. How many T-shirts did the club order?
A)
B)
C)
D)
Show the worked solution
Answer: C
Explanation
With T-shirts the order costs dollars before the voucher and after it:
Check: ✓
A subtracted the 9 where it has to be added back; D is the cost before the voucher, not the number of T-shirts; B ignored the voucher.
Takeaway: Undo the subtraction first, then the multiplication — and finish.
Topic: A quadratic with two roots
What is the sum of the solutions to the equation
A)
B)
C)
D)
Show the worked solution
Answer: C
Explanation
Sum of roots (The roots are 4 and 5.)
A has the sign wrong: the sum is and is D is the product, B subtracted the roots,
Takeaway: Sum product
Topic: A percentage increase
The population of a town was 4500 at the start of a year and 5400 at the end of the year. By what percent did the population of the town increase over the year?
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 20
Explanation
The rise is 900, measured against the original 4500:
Using 5400 as the base would give 16.7%, the standard wrong answer. Enter 20.
Takeaway: Percentage change divides by the original.
Topic: An exponent equation
A colony of bacteria triples in each observation period. After how many observation periods has the population grown by a factor of 81?
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 4
Explanation
After periods the population has been multiplied by 3 a total of times, so
Write 81 as a power of 3, so both sides share a base:
The equation becomes With the same base on both sides, the exponents must be equal:
Enter 4.
Takeaway: Rewrite both sides over a common base, then match the exponents. The powers of 2, 3 and 5 are the ones this test reuses, so knowing them to the fifth power or so is worth the memorising.
Topic: A system by elimination
At a café, every sandwich has the same price and every coffee has the same price. A customer who buys 3 sandwiches and 1 coffee pays $14 in total. A sandwich costs $2 more than a coffee. What is the price, in dollars, of one sandwich?
A)
B)
C)
D)
Show the worked solution
Answer: D
Explanation
Let a sandwich cost dollars and a coffee dollars. The two sentences say
The terms are and so add the equations: giving Then
Check: ✓ and ✓
A gave the price of a coffee. B subtracted the 2 on the right instead of adding it, reaching C added only the left sides, using
Takeaway: Opposite signs, add the equations.
Topic: Volume of a cylinder
A cylindrical water butt has the inside radius shown, and it is filled with water to the height shown. Both measurements are in decimetres.
What is the volume of the water in the butt, in cubic decimetres?
A)
B)
C)
D)
Show the worked solution
Answer: D
Explanation
A used rather than C is a surface area. B is the area of the base, never multiplied by the height.
Takeaway: Square the radius, not the height.
Topic: Function notation
The function is defined by What is the value of
A)
B)
C)
D)
Show the worked solution
Answer: B
Explanation
not A made that error. D doubled the before squaring it, working out the square belongs to alone. C added the 1 before doubling, working out
Takeaway: Square inside brackets: a negative squared is positive.
Topic: Solving for one variable in terms of another
A caterer spends exactly $24 on trays of sandwiches and trays of fruit. Each tray of sandwiches costs $3 and each tray of fruit costs $4, so where is the number of trays of sandwiches and is the number of trays of fruit. Which of the following expresses in terms of
A)
B)
C)
D)
Show the worked solution
Answer: A
Explanation
Check at the original gives so the answer gives ✓
B has both signs reversed, which makes it the negative of the answer. C kept the positive when it crossed the equals sign. D never divided by 4.
Takeaway: Whatever divides must divide every term. Check with one value.
Topic: Angles in a triangle
Triangle is shown.
What is the measure, in degrees, of angle
A)
B)
C)
D)
Show the worked solution
Answer: A
Explanation
Check: ✓
C added the two given angles. D used the total for a quadrilateral. B subtracted one given angle from the other.
Takeaway: Triangle angles total Add all three back as a check.
Topic: An exponential model
A biologist starts a culture with 60 cells. The number of cells in the culture doubles every 3 hours. How many cells are in the culture 12 hours after the biologist starts it?
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 960
Explanation
periods, so the factor is
Check by stepping: ✓ Enter 960.
Takeaway: Divide the time by the period length to count the periods.
Topic: Scaling an area
A rectangular poster is enlarged so that both its length and its width become 1.5 times as great as before. By what factor does its area increase?
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 9/4 or 2.25
Explanation
Both dimensions are multiplied by 1.5, so the area is multiplied by
Try it: a rectangle has area 8; a one
has area 18, and ✓ Enter 2.25 or 9/4.
Takeaway: Lengths areas volumes
Topic: Converting radians to degrees
An angle measures radians. What is the measure of the angle, in degrees?
A)
B)
C)
D)
Show the worked solution
Answer: A
Explanation
radians is so put 180 where the is:
Check the size: is a little more than half of so the angle should be a little more than ✓
D multiplied the 3 by and never divided by 5. B took as a full turn, C used the fraction upside down, of
Takeaway: radians To turn radians into degrees, replace the with 180 and work out the fraction.
Topic: Probability
A bag used for a classroom demonstration holds 4 white marbles and 11 black marbles, identical apart from colour. A pupil draws a single marble at random without looking into the bag. What is the probability that the marble drawn is white?
A)
B)
C)
D)
Show the worked solution
Answer: B
Explanation
There are 15 marbles altogether, 4 of them white:
C used 11 as the denominator, which is the count of the other colour, not the total. D is the chance of drawing black. A treated "4 white" as one chance in four.
Takeaway: The denominator is the total number of outcomes, not the number of the ones you do not want.
Topic: Slope from a table
The table shows four values of the linear function which models the cost, in dollars, of running a machine for hours.
| Hours, | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cost, | 9 | 13 | 17 | 21 |
What is the slope of the graph of in the -plane?
A)
B)
C)
D)
Show the worked solution
Answer: C
Explanation
D and B each gave one of the two changes without dividing. A divided them the wrong way up.
Takeaway: Slope is a change divided by a change, not either one alone.
Topic: A perfect-square trinomial
The expression where is a constant, is a perfect square. What is the value of
A)
B)
C)
D)
Show the worked solution
Answer: D
Explanation
Halve the middle coefficient and square it:
Check: ✓ A halved and forgot to square. B copied the middle coefficient. C doubled the 12 where its half should be squared.
Takeaway: Halve, then square. Two steps.
Topic: A system of linear inequalities from its graph
In the -plane shown, the shaded region is the set of all points that satisfy a system of two inequalities. One boundary line is solid, and the points on it are solutions; the other is dashed, and the points on it are not.
Which of the following is the system?
A) and
B) and
C) and
D) and
Show the worked solution
Answer: A
Explanation
Take the two lines one at a time.
The solid line crosses the -axis at and rises 1 for every 1 it runs: The shading lies below it — at the line is at and the shaded point is lower. Solid means the line is included:
The dashed line crosses the -axis at 3 and falls 1 for every 1 it runs: The shading lies above it — at the line is at and is higher. Dashed means the line is not included:
Check the test point in both: ✓ and ✓
B shades above the solid line, where is not. C shades below the dashed line, where is not either. D has the right two sides but swaps which line is solid: it would leave out a point on the solid line, and take in a point on the dashed one.
Takeaway: One line at a time: its equation, then a test point for the side, then solid or dashed for the sign. A point in the region must pass both inequalities, so one test point checks an option completely.
Topic: A word problem with two unknowns
A cinema sells adult tickets at $14 each and child tickets at $9 each, and every ticket sold is one or the other. On one evening the cinema sold 60 tickets in total and took $690 at the box office. How many child tickets were sold that evening?
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 30
Explanation
Let and be the two counts:
Substituting
Check: 30 adults and 30 children give ✓ Enter 30.
Takeaway: One equation counts the items, the other counts the money. Then read which count the question asked for.
Topic: A model compared across two options
A phone company offers two data plans. Plan A costs $30 a month plus $2 for each gigabyte of data used. Plan B costs $50 a month, however much data is used. For how many gigabytes of data used in a month do the two plans cost the same?
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 10
Explanation
Below 10 GB plan A is cheaper; above it, plan B. Enter 10.
Takeaway: "Cost the same" means set the two expressions equal.
Topic: Composite functions
The functions and are defined by and What is the value of
A)
B)
C)
D)
Show the worked solution
Answer: B
Explanation
Inside first: Then
A is the other order. C is with never applied. D added the 4 instead of subtracting it.
Takeaway: means first — work outward.
Topic: A circle's centre
In the -plane, a circle has the equation What are the coordinates of the centre of the circle?
A)
B)
C)
D)
Show the worked solution
Answer: C
Explanation
has centre and the signs flip. gives is so
A copied both signs straight from the brackets. B and D each flipped only one of the two.
Takeaway: Read the bracket, then flip the sign. Both coordinates.
Topic: A linear inequality in context
A delivery van can carry a load of at most 900 kilograms, and the load includes the driver. The driver weighs 80 kilograms, and each crate that the van carries weighs 60 kilograms. Which inequality gives the possible values of the number of crates that the van can carry on one trip?
A)
B)
C)
D)
Show the worked solution
Answer: D
Explanation
The load is the crates plus the driver, and "at most 900" means
A left the driver out, B points the wrong way, C subtracts a weight that adds.
Takeaway: "At most" is Include everything that adds to the load.
Topic: Interpreting standard deviation
Two classes sat the same test. The table summarises their results.
| Class | Mean | Lowest score | Highest score |
|---|---|---|---|
| A | 70 | 68 | 72 |
| B | 70 | 45 | 95 |
Which statement about the two classes is true?
A) Class A has the greater standard deviation
B) Class B has the greater standard deviation
C) The standard deviations are equal, because the means are equal
D) Standard deviation cannot be judged from a range
Show the worked solution
Answer: B
Explanation
Standard deviation measures spread, not centre. Class A's scores sit within 4 marks of each other; class B's span 50. B is far more spread out.
A has the classes the wrong way round. C confuses the two ideas: equal means say nothing about spread. D is too cautious — with every score in class A inside a 4-mark band, no calculation is needed to see which class is more spread out.
Takeaway: Same mean, different spread is entirely possible — that is what standard deviation exists to describe.
Topic: A system with no solution
Two competing services quote for the same work. The first charges according to and the second according to where is the number of units ordered and is a constant. For what value of is there no order size at which the two quotes are equal?
A)
B)
C)
D)
Show the worked solution
Answer: D
Explanation
No solution means parallel: equal slopes, different intercepts. So and the intercepts and differ ✓
A has the sign of the slope wrong. C matched intercepts, which would make the lines cross rather than miss. B copied the other line's intercept.
Takeaway: Parallel and distinct. Check the intercept after matching the slope.
Topic: Putting several ideas together
In the -plane, the graph of where is a constant, passes through the point What is the value of
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: -5
Explanation
Substitute the point into the equation:
Check: ✓
This answer is negative. Enter -5, minus sign included — it counts as one
of the six characters a negative answer is allowed.
Takeaway: A point on a graph satisfies its equation. Substitute both coordinates and solve for the unknown constant.
Paper 2 — Module 2
27 questions · 43 minutes · calculator permitted
The last module. Work steadily, and keep ten seconds at the end of each grid-in to substitute your answer back.
Topic: Solving a linear equation
The paper tray of a printer holds 300 sheets at the start of a print run, and the printer uses 20 sheets each minute. The run is stopped when 120 sheets are left in the tray. For how many minutes did the printer run?
A)
B)
C)
D)
Show the worked solution
Answer: B
Explanation
After minutes the printer has used sheets, so are left:
Check: sheets used, and ✓ A divided the 120 sheets that were left by 20; that is how long the rest of the paper would last, not how long the printer had run. C divided all 300 sheets by 20, the time to empty the tray. D added the 120 where it should be subtracted, reaching
Takeaway: Write what is left as the start minus what was used. Then subtract first, and divide.
Topic: Translating a sentence into an equation
A number is increased by 6, and the result is then multiplied by 3, giving 48. Which equation represents this?
A)
B)
C)
D)
Show the worked solution
Answer: B
Explanation
Take it in the order the sentence gives. "A number is increased by 6" is "The result is multiplied by 3" multiplies all of that, and a bracket is how that is written:
(Solving it gives so — and 10 increased by 6, then tripled, really is 48.)
A multiplied the number first and added the 6 afterwards. C multiplied only the 6. D divided by 3 where the sentence multiplies.
Takeaway: When an operation is applied to "the result", it applies to everything so far. That is what a bracket is for.
Topic: Expanding a bracket
Which of the following expressions is equivalent to
A)
B)
C)
D)
Show the worked solution
Answer: A
Explanation
B added the 5 instead of subtracting it. C left the 3 undistributed. D multiplied the 3 by 4 and forgot the
Takeaway: Distribute to every term inside, then combine the constants.
Topic: The slope of a line
The graph of the linear function is shown in the -plane.
What is the slope of the graph?
A)
B)
C)
D)
Show the worked solution
Answer: D
Explanation
Read the two marked points off the graph: and From the first to the second the line falls 4 while it moves 2 to the right:
A read the -intercept. B lost the minus: the line falls from left to right, so its slope is negative. C is run over rise, the fraction upside down.
Takeaway: Slope is rise over run between any two points of the line. A line that falls from left to right has a negative slope.
Topic: Reading a linear model
A city taxi firm charges a fixed amount when the journey begins and then a further amount for every mile travelled, so that the total fare in dollars is for a journey of miles. Which of the following is the best interpretation of the number 4 in this context?
A) The fixed charge before any distance is travelled
B) The cost of each mile
C) The number of miles included in the base fare
D) The total cost of a 4-mile journey
Show the worked solution
Answer: A
Explanation
Set the fare is already $4 before any distance. That is the fixed charge.
B describes the 2.5. C confuses a charge with a distance: the 4 is in dollars. D would be $14.
Takeaway: Substitute zero to see what the constant means, and check the units — dollars cannot be miles.
Topic: A quadratic built from a rectangle
A gardener uses exactly 22 metres of fencing to enclose a rectangular vegetable plot on all four sides, with no fencing left over. The area of the plot is 30 square metres. What is the length, in metres, of the shorter side of the plot?
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 5
Explanation
Half the fencing, 11 metres, makes one length and one width. If one side is metres, the other is and the area is their product:
Two numbers multiplying to 30 and adding to and
The sides are 5 and 6 metres. The shorter is 5. Enter 5.
Takeaway: A perimeter gives the sum of the two sides and an area gives their product. Find both roots, then read which one the question asked for.
Topic: A linear equation with a fraction
A fuel tank was full at the start of a trip. During the trip 3 litres of fuel were used, and at the end of the trip 4 litres of fuel were left in the tank. How many litres of fuel does the tank hold when it is full?
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 35/2 or 17.5
Explanation
Let the full tank hold litres. It started the trip with litres, lost 3, and had 4 left:
Add 3 to both sides:
Multiply by 5, then divide by 2:
The tank holds 17.5 litres.
Check: of 17.5 is 7, and ✓ Enter 35/2 or 17.5.
Takeaway: Clear the constant first, then the fraction. Substituting back is your only safety net on a grid-in.
Topic: Exponent rules
Which of the following is equivalent to
A)
B)
C)
D)
Show the worked solution
Answer: B
Explanation
so
A left the 2 uncubed. C multiplied the coefficient by 3 rather than cubing it. D added the exponents, where a power of a power multiplies them.
Takeaway: The outside power applies to the coefficient too — and it cubes it, not triples it.
Topic: A system by substitution
A bookshop sells notebooks and folders. Each notebook has the same price and each folder has the same price. A folder costs $5 more than a notebook, and 3 notebooks and 2 folders cost $25 altogether. What is the price, in dollars, of one folder?
A)
B)
C)
D)
Show the worked solution
Answer: C
Explanation
Let a notebook cost dollars and a folder dollars. The two sentences say and Substitute the first into the second:
Then
Check: ✓ A stopped at the price of a notebook. B is the difference between the two prices. D divided the 25 by the 2 folders, as if the notebooks were free.
Takeaway: Substitute, solve, then finish by finding the variable asked for.
Topic: Area of a triangle
A triangular flower bed has a base of 18 metres and a perpendicular height of 11 metres. A gardener covers the whole bed with mulch that costs $4 for each square metre. What is the cost, in dollars, of the mulch?
A)
B)
C)
D)
Show the worked solution
Answer: C
Explanation
The mulch is paid for by the square metre, so the area comes first:
That is 99 square metres, at $4 each:
D forgot the — the usual slip — and paid for 198 square metres. A stopped at the area: 99 is a number of square metres, and the question asks for dollars. B multiplied the base by the height and stopped, with no half and no cost.
Takeaway: The half is part of the formula. It is on the reference sheet; copy it before substituting. Then read what is asked for: an area, or what the area costs.
Topic: An absolute-value equation
What is the sum of the solutions to the equation
A)
B)
C)
D)
Show the worked solution
Answer: B
Explanation
Their sum is
C and A each kept one case only. D is the right-hand side of the equation, not a value of
Takeaway: Absolute value gives two cases, and "sum of all possible values" needs both of them.
Topic: Reading an exponential model
A company buys a machine for its factory. The value in dollars, of the machine years after it was bought is modelled by the equation Which of the following is the best interpretation of the number in this context?
A) The machine loses 90% of its value each year
B) The machine loses 10% of its value each year
C) The machine gains 90% of its value each year
D) The machine is worth $0.90 after years
Show the worked solution
Answer: B
Explanation
The factor is what remains: so 10% is lost each year.
Check: a drop of 1200, which is 10% of 12000 ✓
A read the factor as the loss. C reads a factor below 1 as growth; it shrinks. D treats the 0.9 as a value in dollars — it multiplies the value, it is not the value.
Takeaway: The factor is what survives. Subtract from 1 for the percentage lost.
Topic: A three-part inequality
What is the greatest integer that satisfies
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 7
Explanation
Add 1 to all three parts: Divide by 3:
The greatest integer is 7, and the right end is inclusive so 7 is allowed.
Check: at the middle is 20, and ✓ At it is 23, which
is too big ✓ Enter 7.
Takeaway: Every step to all three parts. Then check whether the end you want is inclusive.
Topic: The discriminant
In the equation is a constant. The graph of the related function touches the -axis at exactly one point. What is the value of
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 49/4 or 12.25
Explanation
One solution means
Then ✓ Enter 49/4 or
12.25.
Takeaway: One solution ⇒ discriminant zero ⇒ perfect square.
Topic: Perpendicular lines
In the -plane, line is perpendicular to the line What is the slope of line
A)
B)
C)
D)
Show the worked solution
Answer: A
Explanation
Flip and negate: Check: ✓
C is the slope of a parallel line. D and B each did one half of the job.
Takeaway: Negative reciprocal means both steps. Multiply the two slopes; you should get exactly
Topic: A system built from a situation
A stationery shop sells pens at $3 each and notebooks at $7 each, and every item sold is one or the other. A customer buys 10 items altogether and pays $54 in total. How many notebooks did the customer buy?
A)
B)
C)
D)
Show the worked solution
Answer: B
Explanation
Let be pens and notebooks:
Substituting so and
Check: 4 pens and 6 notebooks give ✓
A is the pen count — both come out, and the question named notebooks. C is the total number of items. D divided all the money by the price of a pen, as if only pens were bought.
Takeaway: One equation counts the items, the other the money. Then read which count was asked for.
Topic: A function transformation
The graph of passes through the point Which point must the graph of pass through?
A)
B)
C)
D)
Show the worked solution
Answer: A
Explanation
The change is inside the bracket, so it is horizontal and goes opposite to the sign: moves the graph left 3.
Track the point: we want the input making the bracket equal 5, so and Then
B moved right, following the plus sign. C and D moved the point up and down, which is what a change outside the bracket does.
Takeaway: Inside the bracket, opposite to the sign. Track one known point.
Topic: An exponential decay factor
Ecologists have monitored the birds of one species in a coastal region for several years. The number of birds of that species in the region years after the monitoring began is modelled by the equation Which of the following statements is best supported by the model?
A) The population falls by 94% each year
B) The population falls by 800 each year
C) The population falls by 6% each year
D) The population grows by 6% each year
Show the worked solution
Answer: C
Explanation
The factor is what remains, so is lost: 6% a year.
Check: a drop of 480, and ✓
A read the factor as the loss. B would be a linear model. D has the size right and the direction wrong: a factor below 1 shrinks.
Takeaway: The factor is what survives. Subtract from 1 for the percentage lost.
Topic: Simplifying a rational expression
Which of the following is equivalent to where
A)
B)
C)
D)
Show the worked solution
Answer: A
Explanation
Take out the 3, then use the difference of two squares:
Check at the original is and ✓
B cancelled the wrong bracket. C cancelled the under and the 2 under the 12 — terms, not factors. D forgot that the 3 multiplies the whole bracket.
Takeaway: Common factor, then the pattern, then cancel whole brackets.
Topic: An inequality with the variable on both sides
A bowling alley has two ways to pay. A member pays a one-time fee of $48 and then $12 for each game. A non-member pays no fee and pays $20 for each game. What is the least number of games for which a member pays less in total than a non-member?
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 7
Explanation
After games a member has paid dollars and a non-member The member has paid less when
Subtract Divide by 8, which is positive, so nothing flips:
The inequality is strict: after 6 games both have paid 120 dollars, which is the same, not less. The least number of games is 7.
Check: at ✓ At is false ✓ Enter 7.
Takeaway: "Less than" is a strict inequality. At the break-even point nobody has paid less, so the answer is the next whole number up.
Topic: A factor of a polynomial
The function is defined by where is a constant. If is a factor of what is the value of
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 1
Explanation
A factor is zero exactly where the function is zero. So if is a factor, then
Check: with ✓ Enter 1.
Note the sign. The factor is and the number to substitute is — the value that makes the factor zero.
Takeaway: is a factor exactly when Substitute the value that makes the factor zero, and solve for the constant.
Topic: A linear model compared to a fixed cost
One gym charges a membership fee of $25 a month and a further $4 for each class attended. A second gym charges a flat $65 a month with no charge per class. At how many classes a month do the two gyms cost the same?
A)
B)
C)
D)
Show the worked solution
Answer: A
Explanation
Check: ✓ B ignored the monthly fee. C added it to the 65 instead of subtracting. D stopped at
Takeaway: "Cost the same" sets the two expressions equal. Subtract the fixed part before dividing.
Topic: The vertex of a parabola
A company runs a generator for part of each day. The daily running cost, in dollars, is modelled by where is the number of hours for which the generator runs that day. According to the model, what is the minimum daily running cost, in dollars?
A)
B)
C)
D)
Show the worked solution
Answer: B
Explanation
Check: and — higher on both sides ✓
C gave where the minimum is; the question asked what it is. D is the cost at A has the sign of the vertex wrong.
Takeaway: Find then substitute. Read which of the two numbers is wanted.
Topic: Scaling two dimensions of a solid
The radius of a right circular cone is tripled and its height is halved. The volume of the new cone is how many times the volume of the original cone?
A)
B)
C)
D)
Show the worked solution
Answer: B
Explanation
From the reference sheet, Put the new measurements in:
The volume is multiplied by The radius is squared in the formula, so tripling it multiplies the volume by 9; the height is not, so halving it multiplies by and
A multiplied by 3 and by forgetting that the radius is squared. C tripled the radius correctly and ignored the height. D doubled the height where the question halves it.
Takeaway: Give each changed dimension the power it has in the formula, then multiply the factors together.
Topic: Trigonometry in context
A builder is installing a wheelchair ramp that rises at an angle of to the horizontal ground, and the sloping surface of the ramp measures 9 metres from end to end. Building regulations require the vertical rise to be recorded. Which expression gives that height, in metres?
A)
B)
C)
D)
Show the worked solution
Answer: A
Explanation
The ramp is the hypotenuse and the height is opposite the angle. Opposite and hypotenuse means sine:
Sanity-check the size: the height must be less than the 9 m ramp. Sine of an acute angle is under 1, so multiplying shrinks it ✓ C divides, giving about 43 m — a ramp 9 m long cannot rise 43 m. B is the horizontal distance the ramp covers, the side adjacent to the angle. D would be right if the 9 m were measured along the ground instead of along the ramp.
Takeaway: Label hypotenuse, opposite and adjacent, then pick the ratio. Check the answer's size against the geometry.
Topic: A radical equation with an extraneous solution
What is the solution to the equation
A)
B)
C)
D)
Show the worked solution
Answer: C
Explanation
Square both sides to clear the root:
Bring everything to one side and factor:
So the squared equation has two solutions, and Both must now be tested in the original equation, because squaring turns a false statement into a true one — but
At ✓
At and ✗
The square-root symbol means the non-negative root, so it can never equal a negative number. is extraneous — it was manufactured by the squaring.
Why each wrong option is wrong:
- A, — kept the extraneous root without testing it. This is the whole point of the question: squaring is not a reversible step.
- B, — added the two roots of the squared equation. The question asks for the solution, not for a sum.
- D, — doubled the right-hand side instead of squaring the left, solving
Takeaway: whenever you square both sides, every solution you find is only a candidate. Substitute each one back into the original equation. A square root is never negative, so an equation like has no solution where
Topic: Several steps together
The rectangular practice area shown is twice as long as it is wide, and the distance around the outside of it is 54 metres.
What is the area of the practice area, in square metres?
There are no options — work the answer out and enter it yourself, in the form the grid accepts.
Show the worked solution
Answer: 162
Explanation
Let the width be the length is The perimeter:
So the width is 9 and the length 18. The question asks for the area:
Check the perimeter: ✓
Two traps in one question. The first is stopping at the second is giving
the perimeter-shaped answer. With no options to warn you, re-read the last line.
Enter 162.
Takeaway: Solve for the unknown, then do the extra step the question actually asks for. This is the last question of the paper — read it twice.