Chapter 9 — Two Full Practice Papers

Topic: Solving a linear equation

A cyclist's distance from home, in kilometres, is given by the expression $6x - 4$, where $x$ is the number of hours since the ride began. The cyclist is 26 kilometres from home. What is the value of $x$?

A) $\frac{11}{3}$

B) $5$

C) $\frac{15}{2}$

D) $30$

Show the worked solution

Answer: B

Explanation

$$6x = 30 \quad\Longrightarrow\quad x = 5$$

Check: $30 - 4 = 26$ ✓ A subtracted on the wrong side, C used the wrong coefficient, D stopped one step early.

Takeaway: Add before dividing, and finish the last step.

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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) $8$

B) $210$

C) $270$

D) $72$

Show the worked solution

Answer: D

Explanation

$$0.30 \times 240 = 72$$

A divided, B subtracted 30, C added 30. A percentage of something is always a multiplication.

Takeaway: "$r$% of $b$" means $\frac{r}{100} \times b$.

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Topic: Slope from two points

A biologist recorded the length of a seedling on two occasions. After 1 week the seedling was 2 centimetres long, and after 5 weeks it was 14 centimetres long. Treating length as a linear function of time, what is the rate of growth, in centimetres per week?

A) $\frac{8}{3}$

B) $\frac{1}{3}$

C) $-3$

D) $3$

Show the worked solution

Answer: D

Explanation

$$m = \frac{14 - 2}{5 - 1} = \frac{12}{4} = 3$$

B is upside down, C has the sign flipped, A added.

Takeaway: Rise over run, same order top and bottom.

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Topic: Evaluating a function

The function $f$ is defined by $f(x) = x^2 - 3x$, where $f(x)$ is the profit, in hundreds of dollars, from selling $x$ hundred units of a product. What is the value of $f(5)$?

A) $-5$

B) $15$

C) $25$

D) $10$

Show the worked solution

Answer: D

Explanation

$$f(5) = 25 - 15 = 10$$

B and C each computed one term and stopped. A slipped a sign.

Takeaway: Substitute into every term, then combine.

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Topic: Expanding brackets

A manufacturer models the area of a rectangular panel, in square centimetres, as the product $(x - 4)(x + 6)$, where $x$ is a length in centimetres. Before the panel can be costed, the expression must be written without brackets. Which of the following expressions is equivalent to it for every value of $x$?

A) $x^{2} + 10 x - 24$

B) $x^{2} - 2 x - 24$

C) $x^{2} - 24$

D) $x^{2} + 2 x - 24$

Show the worked solution

Answer: D

Explanation

$$x^2 + 6x - 4x - 24 = x^2 + 2x - 24$$

B has $-2x$; the $+6$ is bigger than the $-4$, so the middle is positive. C skipped the middle. A multiplied rather than added them.

Takeaway: Four products, then collect the middle two.

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Topic: A linear equation with a fraction

A water tank is being filled. The volume in the tank, in litres, is given by $\dfrac{x}{4} + 3$, where $x$ 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

$$\frac{x}{4} = 8 \quad\Longrightarrow\quad x = 32$$

Check: $8 + 3 = 11$ ✓ Enter 32.

Takeaway: Subtract before multiplying up.

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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 $4 \times 21 = 84$. The three given total 61, so the fourth is $84 - 61 = 23$. Enter 23.

Takeaway: Turn a mean into a total straight away.

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Topic: A system by substitution

At a school fair, the number of adult tickets sold, $y$, was 2 fewer than three times the number of child tickets, $x$, so that $y = 3x - 2$. The organisers also found that $2x + y = 13$. What is the value of $x$?

A) $\frac{11}{5}$

B) $\frac{13}{5}$

C) $7$

D) $3$

Show the worked solution

Answer: D

Explanation

$$2x + (3x - 2) = 13 \quad\Longrightarrow\quad 5x = 15 \quad\Longrightarrow\quad x = 3$$

Then $y = 7$. Check both: $y = 9 - 2 = 7$ ✓ and $6 + 7 = 13$ ✓ C gave $y$.

Takeaway: Substitute, solve, then check which variable was asked for.

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Topic: Exponent rules

In a computer science calculation, the number of operations required is given by the expression $\dfrac{(x^3)^4}{x^5}$, where $x$ is a positive integer describing the size of the input. Before the expression can be compared with another algorithm's, it must be written as a single power of $x$. Which of the following is equivalent to it?

A) $x^{2}$

B) $x^{7}$

C) $x^{12}$

D) $x^{60}$

Show the worked solution

Answer: B

Explanation

$(x^3)^4 = x^{12}$, then dividing subtracts: $x^{12-5} = x^7$.

A added instead of multiplying at the first step. C skipped the division. D multiplied everything.

Takeaway: Power of a power multiplies; division subtracts.

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Topic: Ratio

The ratio of cats to dogs at a shelter is $5 : 3$. There are 48 animals in total. How many are dogs?

A) $3$

B) $30$

C) $16$

D) $18$

Show the worked solution

Answer: D

Explanation

$5 + 3 = 8$ parts, so one part is $48 \div 8 = 6$. Dogs are 3 parts: $3 \times 6 = 18$.

Check: cats $= 30$, and $18 + 30 = 48$ ✓ B gave the cats.

Takeaway: Add the parts, find one part, then multiply.

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Topic: Solving a quadratic by factoring

A rectangle's area, in square metres, is given by $x^2 - 2x - 15$, where $x$ is its width in metres. For what width is the area equal to zero square metres? Give the greatest such value of $x$.

A) $-3$

B) $3$

C) $-5$

D) $5$

Show the worked solution

Answer: D

Explanation

$(x - 5)(x + 3) = 0$, so $x = 5$ or $x = -3$. The greatest is 5.

A gave the smaller. B and C flipped the signs.

Takeaway: Factor, then read which root the question wants.

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Topic: Pythagoras

A rectangular garden plot is 8 metres wide and 15 metres long. A gardener wants to run a straight path from one corner of the plot to the opposite corner. What is the length of that path, in metres?

A) $23$

B) $\sqrt{161}$

C) $\frac{23}{2}$

D) $17$

Show the worked solution

Answer: D

Explanation

$$64 + 225 = 289 \quad\Longrightarrow\quad c = 17$$

The 8–15–17 triple. A added the legs, B subtracted.

Takeaway: Squares add, then take the root.

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Topic: An absolute-value equation

A machine is calibrated to fill bottles to 7 millilitres, and a bottle is rejected if its contents differ from 7 millilitres by exactly 4 millilitres or more. A bottle containing $x$ millilitres satisfies $|x - 7| = 4$. What is the greatest possible value of $x$?

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

$x - 7 = 4$ gives $x = 11$; $x - 7 = -4$ gives $x = 3$. The greatest is 11. Enter 11.

Takeaway: Two cases, then read the wording.

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Topic: A unit rate

A bottling plant runs one machine that fills 45 bottles every 3 minutes, and the supervisor needs to plan how many crates to have ready at the end of each shift segment. Assuming the machine continues at the same rate and does not stop, how many bottles will it fill in 8 minutes?

There are no options — work the answer out and enter it yourself, in the form the grid accepts.

Show the worked solution

Answer: 120

Explanation

$45 \div 3 = 15$ bottles a minute, so $15 \times 8 = 120$. Enter 120.

Takeaway: Reduce to a per-one rate, then scale.

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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. Two of its values were recorded in the table below.

Units sold, $x$	2	5
Profit, $y$	3	15

The $y$-intercept of the graph is the profit when nothing at all is sold, which for a business with fixed costs to cover may well be negative. What is the $y$-intercept of that graph?

A) $3$

B) $-5$

C) $11$

D) $5$

Show the worked solution

Answer: B

Explanation

$$3 = 4(2) + b = 8 + b \quad\Longrightarrow\quad b = -5$$

Check: at $x = 2$, $8 - 5 = 3$ ✓ C added instead of subtracting.

Takeaway: Substitute the point into $y = mx + b$ and solve for $b$.

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Topic: Exponential growth

An ecologist has been monitoring a population of 500 fish in a lake and estimates that the population grows by 20% during each year, with the growth compounding from one year to the next. Assuming the rate continues, what will the population be at the end of 2 years?

A) $600$

B) $700$

C) $620$

D) $720$

Show the worked solution

Answer: D

Explanation

$$500 \times 1.2 = 600, \qquad 600 \times 1.2 = 720$$

A did one year only. B added 20% of the original twice, which ignores compounding — the second year's 20% is of 600, not 500. C 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.

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Topic: Simplifying a rational expression

An engineer simplifies the expression $\dfrac{x^2 - 16}{x + 4}$ that arises in a calculation, where $x \ne -4$. Which of the following expressions is equivalent to it for every allowed value of $x$?

A) $-4$

B) $x + 4$

C) $x^{2} - 4$

D) $x - 4$

Show the worked solution

Answer: D

Explanation

$$\frac{(x+4)(x-4)}{x+4} = x - 4$$

B kept the wrong factor. C and A cancelled terms, not factors.

Takeaway: Factor first, cancel whole brackets only.

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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) $\frac{3}{8}$

B) $\frac{9}{14}$

C) $\frac{7}{12}$

D) $\frac{5}{14}$

Show the worked solution

Answer: B

Explanation

The pool is restricted to Spanish students, so the denominator is 70:

$$P = \frac{45}{70} = \frac{9}{14}$$

A used all 120. D counted the other 25.

Takeaway: "Chosen from those who…" fixes the denominator.

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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 $m$ miles is given by the function $C = 45 + 0.30m$. Which of the following is the best interpretation of the number $0.30$ 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 $m$ is the rate: 30 cents per mile. A describes the 45. C would be $54.

Takeaway: In $b + mx$, the multiplied number is the rate — check the units.

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Topic: A system solved by elimination

A caterer buys $x$ trays of sandwiches and $y$ trays of pastries. Two orders are recorded: the first satisfies $4x + 3y = 27$ and the second satisfies $4x - y = 7$. What is the value of $y$?

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

Both have $4x$, so subtract: $4y = 20$, giving $y = 5$. Then $x = 3$.

Check: $12 + 15 = 27$ ✓ and $12 - 5 = 7$ ✓ Enter 5.

Takeaway: Matching coefficients, subtract. Bracket the second equation so the double negative is visible.

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Topic: The discriminant

In the equation $x^2 + 10x + c = 0$, $c$ is a constant. The graph of the related function $y = x^2 + 10x + c$ touches the $x$-axis at exactly one point. What is the value of $c$?

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 $b^2 - 4ac = 0$:

$$100 - 4c = 0 \quad\Longrightarrow\quad c = 25$$

Then $x^2 + 10x + 25 = (x+5)^2$ ✓ Enter 25.

Takeaway: Exactly one solution ⇒ discriminant zero ⇒ perfect square.

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Topic: A circle equation

A circular sprinkler is installed in a field. The set of points it can reach is described in the $xy$-plane by the equation $(x + 2)^2 + (y - 5)^2 = 36$, where distances are measured in metres. What is the radius, in metres, of the region the sprinkler reaches?

A) $6$

B) $36$

C) $18$

D) $\left( -2, \ 5\right)$

Show the worked solution

Answer: A

Explanation

The right side is $r^2 = 36$, so $r = 6$. The centre is $(-2, 5)$, which is option D — a different question.

Takeaway: Take the square root of the right-hand side.

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Topic: Similar triangles

An architect is preparing two scale models of the same building, one larger than the other. The two models are similar solids, and the areas of their corresponding faces are in the ratio $9 : 25$. What is the ratio of their corresponding side lengths?

A) $\frac{9}{25}$

B) $\frac{3}{5}$

C) $\frac{81}{625}$

D) $\frac{5}{3}$

Show the worked solution

Answer: B

Explanation

Areas scale by the square of the length ratio, so take square roots:

$$\sqrt{\frac{9}{25}} = \frac{3}{5}$$

A left the area ratio unchanged. C squared instead of rooting. D took the root and then wrote it upside down.

Takeaway: Lengths $k$, areas $k^2$, volumes $k^3$. Go backwards with a root.

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Topic: A composite function

Two functions are defined by $f(x) = x + 3$ and $g(x) = 2x$. An engineer applies $f$ to an input and then applies $g$ to the result. What is the value of $g(f(4))$?

A) $11$

B) $8$

C) $24$

D) $14$

Show the worked solution

Answer: D

Explanation

Inside first: $f(4) = 7$. Then $g(7) = 14$.

A is $f(g(4)) = 11$, the other order — always offered.

Takeaway: $g(f(x))$ means $f$ first. Work outward from the inner bracket.

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Topic: An inequality in context

A lift holds at most 500 kg. An operator weighing 75 kg loads boxes of 35 kg each. What is the greatest number of boxes that can be carried in one trip?

A) $14$

B) $13$

C) $15$

D) $12$

Show the worked solution

Answer: D

Explanation

$$35n + 75 \le 500 \quad\Longrightarrow\quad 35n \le 425 \quad\Longrightarrow\quad n \le 12.1$$

You cannot carry part of a box, so round down to 12.

Check: $12 \times 35 + 75 = 495 \le 500$ ✓ and $13 \times 35 + 75 = 530 > 500$ ✓

A forgot the operator. B rounded up — a count of physical objects always rounds down against a maximum.

Takeaway: Include everything that adds to the load, then round down.

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Topic: Trigonometry

A ramp meets the ground at an acute angle $\theta$, forming a right triangle with the ground and a vertical support. For this angle, $\cos\theta = \frac{3}{5}$. What is the value of $\sin\theta$?

A) $\frac{3}{5}$

B) $\frac{4}{5}$

C) $\frac{5}{3}$

D) $\frac{3}{4}$

Show the worked solution

Answer: B

Explanation

$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac35$, so the adjacent side is 3 and the hypotenuse 5. The third side is $\sqrt{25 - 9} = 4$, and that is the opposite side:

$$\sin\theta = \frac{4}{5}$$

C is greater than 1, which no sine can be.

Takeaway: Draw the 3–4–5 triangle from the ratio given, then read off the one you need.

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Topic: Working backwards through a percentage

After a 25% increase, a subscription costs $60 a year. What was the price before the increase, 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: 48

Explanation

The new price is 125% of the old:

$$1.25 \times \text{old} = 60 \quad\Longrightarrow\quad \text{old} = \frac{60}{1.25} = 48$$

Check: 25% of 48 is 12, and $48 + 12 = 60$ ✓

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.

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Paper 1 — Module 2

27 questions · 35 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.

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Topic: Solving a linear equation

A diver's depth below the surface, in metres, is modelled by the expression $9 - 2x$, where $x$ is the number of minutes since the ascent began. At what value of $x$ is the diver 1 metre below the surface?

A) $-4$

B) $5$

C) $4$

D) $-5$

Show the worked solution

Answer: C

Explanation

$$-2x = -8 \quad\Longrightarrow\quad x = 4$$

Check: $9 - 8 = 1$ ✓ A kept the minus after dividing two negatives.

Takeaway: A negative divided by a negative is positive.

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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 $f(x) = 4x + 7$, where $x$ is the distance in kilometres. What is the value of $f(0)$, and what does it represent?

A) $0$

B) $7$

C) $4$

D) $11$

Show the worked solution

Answer: B

Explanation

$$f(0) = 4(0) + 7 = 7$$

A assumed an input of 0 gives an output of 0. C gave the slope.

Takeaway: $f(0)$ is the $y$-intercept — the constant term.

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Topic: Area of a rectangle

A school is laying artificial turf on a rectangular play area that measures 12 metres in length and 7 metres in width. The turf is sold by the square metre. How many square metres are needed to cover the area exactly once?

A) $19$

B) $38$

C) $42$

D) $84$

Show the worked solution

Answer: D

Explanation

$$A = 12 \times 7 = 84$$

B is the perimeter. C halved, which belongs to a triangle.

Takeaway: Area multiplies; perimeter adds. Rectangles have no halving step.

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Topic: An equivalent expression

A shop's daily takings, in dollars, are given by the expression $3(2x - 5) + 4x$, where $x$ is the number of hours the shop is open. Which of the following expressions is equivalent to it for every value of $x$?

A) $10 x - 15$

B) $10 x - 5$

C) $6 x - 11$

D) $6 x - 1$

Show the worked solution

Answer: A

Explanation

$$3(2x - 5) = 6x - 15, \qquad 6x - 15 + 4x = 10x - 15$$

B left the $-5$ undistributed. C dropped the $x$ from $4x$.

Takeaway: Distribute across every term, then collect like with like.

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Topic: A proportion

On a scale drawing, a length of $x$ centimetres represents a real distance in the same ratio as 10 is to 4. This relationship is written $\dfrac{x}{6} = \dfrac{10}{4}$. What is the value of $x$?

A) $\frac{12}{5}$

B) $\frac{5}{2}$

C) $15$

D) $\frac{20}{3}$

Show the worked solution

Answer: C

Explanation

Cross-multiply: $4x = 60$, so $x = 15$.

Check: $\frac{15}{6} = 2.5$ and $\frac{10}{4} = 2.5$ ✓

Takeaway: Cross-multiply, then check both fractions come to the same decimal.

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Topic: A quadratic solved by factoring

The height above the ground of a thrown ball, in metres, is zero when $x^2 + 4x - 21 = 0$, where $x$ is the time in seconds since it was thrown. Since time cannot be negative, only a positive solution is meaningful. What is the value of $x$?

There are no options — work the answer out and enter it yourself, in the form the grid accepts.

Show the worked solution

Answer: 3

Explanation

Two numbers multiplying to $-21$ and adding to $4$: that is $7$ and $-3$.

$$(x + 7)(x - 3) = 0 \quad\Longrightarrow\quad x = -7 \text{ or } x = 3$$

The condition $x > 0$ selects 3. Enter 3.

Takeaway: Solve both roots, then let the stated condition choose.

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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:

$$\frac{12}{80} \times 100 = 15$$

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.

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Topic: Parallel lines

In the $xy$-plane, a road is drawn as the line $y = -3x + 8$. A second road is drawn parallel to the first. What is the slope of the second road?

A) $3$

B) $-3$

C) $\frac{1}{3}$

D) $8$

Show the worked solution

Answer: B

Explanation

Parallel lines have equal slopes, so the slope is $-3$.

C flipped and negated, which gives a perpendicular slope, $\frac13$. D read the intercept.

Takeaway: Parallel copies the slope exactly, sign included.

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Topic: Volume of a rectangular prism

A packing crate has a rectangular base and vertical sides, measuring 5 metres by 4 metres by 3 metres. A supplier needs to know how much space the crate encloses in order to quote for shipping. What is the volume of the crate, in cubic metres?

A) $60$

B) $12$

C) $94$

D) $47$

Show the worked solution

Answer: A

Explanation

$$V = 5 \times 4 \times 3 = 60$$

C is the surface area. Check what the question asked for — a volume is in cubic units, a surface area in square ones.

Takeaway: Volume multiplies all three dimensions.

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Topic: Median with an even count

A researcher recorded the number of birds visiting a feeder on each of six consecutive mornings. The counts are listed in the table below in the order they were taken.

Morning	Mon	Tue	Wed	Thu	Fri	Sat
Birds	4	9	2	11	6	8

What is the median of these values?

A) $\frac{20}{3}$

B) $6$

C) $7$

D) $8$

Show the worked solution

Answer: C

Explanation

Sorted: $2, 4, 6, 8, 9, 11$. Six values, so average the middle two:

$$\frac{6 + 8}{2} = 7$$

A is the mean. B and D picked one middle value without averaging.

Takeaway: Sort first. Even count means average the middle pair.

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Topic: A negative exponent

In a calculation involving the intensity of a signal, a physicist writes the factor $5^{-2}$. What is the value of this expression?

A) $-25$

B) $-10$

C) $10$

D) $\frac{1}{25}$

Show the worked solution

Answer: D

Explanation

$$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$$

A negative exponent gives a small positive number, never a negative one.

Takeaway: Negative exponent means reciprocal, not negative.

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Topic: A system with no solution

Two rental firms price a job differently. The first charges according to $y = 4x + 1$ and the second according to $y = kx - 3$, where $x$ is the number of hours and $k$ is a constant. For what value of $k$ do the two firms never charge the same amount?

A) $-4$

B) $4$

C) $1$

D) $-3$

Show the worked solution

Answer: B

Explanation

No solution means parallel: equal slopes, different intercepts. So $k = 4$, and the intercepts 1 and $-3$ differ ✓

C matched intercepts, which would make the lines cross, not miss.

Takeaway: Parallel and distinct. Check the intercept after matching the slope.

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Topic: A system solved by elimination

A carpenter buys $x$ boxes of screws and $y$ boxes of nails for two separate jobs. The first job's order satisfies $5x + 2y = 26$ and the second job's order satisfies $3x - 2y = 6$, where both quantities are counted in boxes. What is the value of $x$?

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

The $y$ terms are $+2y$ and $-2y$, so adding the equations removes them:

$$8x = 32 \quad\Longrightarrow\quad x = 4$$

Then $5(4) + 2y = 26$ gives $y = 3$. Check both: $20 + 6 = 26$ ✓ and $12 - 6 = 6$ ✓ Enter 4.

Takeaway: Opposite signs on a variable, add the equations. Then check both originals.

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Topic: The vertex of a parabola

The cost per unit of manufacturing an item, in dollars, is modelled by the function $f(x) = x^2 + 6x + 11$, where $x$ is measured in thousands of units produced. The function has a minimum value. What is that minimum value?

There are no options — work the answer out and enter it yourself, in the form the grid accepts.

Show the worked solution

Answer: 2

Explanation

$$x = -\frac{6}{2} = -3, \qquad f(-3) = 9 - 18 + 11 = 2$$

Check: $f(-4) = 16 - 24 + 11 = 3$ and $f(-2) = 4 - 12 + 11 = 3$ — higher on both sides ✓

The question asked for the minimum value, which is the $y$, not the $-3$. Enter 2.

Takeaway: Find $x = -\frac{b}{2a}$, substitute, and check whether the question wants where or what.

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Topic: Interpreting a scatterplot model

A paediatric clinic recorded the age and height of a group of children and drew a line of best fit through the resulting scatterplot. The line has equation $y = 6x + 78$, where $x$ is age in years and $y$ 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 $x = 0$. B describes the 6. The model says nothing about sample size or age range.

Takeaway: Constant = value at zero; coefficient = rate. Check the units.

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Topic: An exponential decay model

A laboratory is monitoring a sample that decays to 85% of its mass during each hour that passes, beginning from a mass of 400 grams. The technician wants a function that gives the mass remaining after any whole number of hours. Which function gives the mass, in grams, after $h$ hours?

A) $400 \left(\frac{23}{20}\right)^{h}$

B) $400 - \frac{17 h}{20}$

C) $400 \left(\frac{17}{20}\right)^{h}$

D) $400 \left(\frac{3}{20}\right)^{h}$

Show the worked solution

Answer: C

Explanation

"Decays to 85%" gives the factor directly: $b = 0.85$.

$$M = 400(0.85)^h$$

D used $0.15$, the fraction lost. B is linear.

Takeaway: "Decays to 85%" means $b = 0.85$; "decays by 15%" means the same thing. Read which word is used.

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Topic: A line through two points

Two readings from a linear model were recorded in the table below, where $x$ is measured in minutes from a chosen reference time.

$x$	$-1$	3
$y$	4	16

What is the $y$-intercept of the line through those two points?

A) $4$

B) $3$

C) $7$

D) $1$

Show the worked solution

Answer: C

Explanation

$$m = \frac{16 - 4}{3 - (-1)} = \frac{12}{4} = 3$$

So $y = 3x + b$. Substitute $(-1, 4)$: $4 = -3 + b$, giving $b = 7$.

Check with the other point: $3(3) + 7 = 16$ ✓

A read a coordinate off a given point; B gave the slope.

Takeaway: Slope first, then substitute either point for $b$ — and confirm with the point you did not use.

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Topic: A radical equation

In an engineering formula the quantity $\sqrt{3x + 1}$ is required to equal 4. Solving the resulting equation gives candidate values, and any value that does not satisfy the original equation must be discarded. What is the solution to $\sqrt{3x + 1} = 4$?

A) $5$

B) $\frac{4}{3}$

C) $\frac{16}{3}$

D) $\frac{17}{3}$

Show the worked solution

Answer: A

Explanation

Square both sides:

$$3x + 1 = 16 \quad\Longrightarrow\quad 3x = 15 \quad\Longrightarrow\quad x = 5$$

Check in the original: $\sqrt{3(5) + 1} = \sqrt{16} = 4$ ✓ — and with radicals the check is compulsory, because squaring can invent solutions.

B 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.

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Topic: An arc of a circle

A landscape designer is edging a curved section of a circular garden bed of radius 9 metres. The curved section corresponds to a central angle of $40°$ measured at the centre of the circle, and the edging is sold by the metre. What is the length of that curved section, in metres?

A) $9 \pi$

B) $2 \pi$

C) $4 \pi$

D) $18 \pi$

Show the worked solution

Answer: B

Explanation

$\frac{40}{360} = \frac19$ of the circle. The whole circumference is $2\pi(9) = 18\pi$, so the arc is

$$\frac19 \times 18\pi = 2\pi$$

A is the sector area, a different quantity — an arc is a length. C 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.

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Topic: A system in context

A charity received two donations on the same day. Together the two amounts came to 54 units of currency, and the larger donation exceeded the smaller by 12 units. The treasurer needs to record each amount separately in the accounts. What is the smaller of the two donations?

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 $2x = 66$, so the larger is 33 and the smaller is $54 - 33 = 21$.

Check: $21 + 33 = 54$ ✓ and $33 - 21 = 12$ ✓ Enter 21.

Takeaway: The smaller of two numbers is $\frac{\text{sum} - \text{difference}}{2}$.

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Topic: A perfect square

A designer needs the expression $x^2 - 14x + c$ to be a perfect square, so that it can be written as the square of a single bracket. What value must the constant $c$ take?

There are no options — work the answer out and enter it yourself, in the form the grid accepts.

Show the worked solution

Answer: 49

Explanation

A perfect square is $(x + a)^2 = x^2 + 2ax + a^2$. Here $2a = -14$, so $a = -7$ and

$$c = a^2 = 49$$

Check: $(x - 7)^2 = x^2 - 14x + 49$ ✓ Enter 49.

Takeaway: Halve the middle coefficient, then square it.

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Topic: Compound percentage

An $800 investment earns 25% interest each year, and the interest is compounded annually so that each year's interest is calculated on the balance at the start of that year rather than on the original deposit. What is the value of the investment after 2 years?

A) $1000$

B) $1200$

C) $1250$

D) $2000$

Show the worked solution

Answer: C

Explanation

$$800 \times 1.25 = 1000, \qquad 1000 \times 1.25 = 1250$$

A did one year only. B 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. D grew for one year and then doubled.

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.

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Topic: Function transformation

The graph of the function $y = f(x)$ models the height of a wave and passes through the point $(3, 8)$. The whole graph is then redrawn as $y = f(x) - 5$. Through which point must the new graph pass?

A) $\left( 3, \ 3\right)$

B) $\left( -2, \ 8\right)$

C) $\left( 8, \ 3\right)$

D) $\left( 3, \ 13\right)$

Show the worked solution

Answer: A

Explanation

The $-5$ is outside the bracket, so it moves the graph down 5: $(3, 8) \to (3, 3)$.

B moved sideways, which only happens inside the bracket.

Takeaway: Outside the bracket moves vertically, in the direction the sign says.

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Topic: Probability from a table

A bag used in a probability demonstration holds counters that are identical apart from their colour. The counts are recorded in the table below.

Colour	Red	Blue	Green	Total
Number of counters	5	7	8	20

A student draws one counter at random without looking into the bag. What is the probability that the counter drawn is not green?

A) $\frac{2}{5}$

B) $\frac{2}{3}$

C) $\frac{1}{3}$

D) $\frac{3}{5}$

Show the worked solution

Answer: D

Explanation

There are 20 counters, of which $5 + 7 = 12$ are not green:

$$\frac{12}{20} = \frac{3}{5}$$

Or take the complement: $1 - \frac{8}{20} = \frac{12}{20}$ ✓

A answered the opposite question.

Takeaway: "Not" means subtract from 1, or count the ones you want directly. Both should agree.

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Topic: Solving for a variable in terms of others

The perimeter $P$ of a rectangular field is related to its length $l$ and its width $w$ by the formula $P = 2l + 2w$. A surveyor knows the perimeter and the length and needs the width. Which expression gives $w$ in terms of $P$ and $l$?

A) $P - 2 l$

B) $\frac{P}{2} - l$

C) $\frac{P}{2} - 2 l$

D) $\frac{P}{2} + l$

Show the worked solution

Answer: B

Explanation

$$2w = P - 2l \quad\Longrightarrow\quad w = \frac{P - 2l}{2}$$

A skipped the division. C divided only the $P$.

Takeaway: Whatever you divide by must divide every term.

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Topic: A perpendicular slope

Line $\ell$ has equation $y = \dfrac{4}{7}x - 5$. What is the slope of a line perpendicular to $\ell$?

A) $\frac{4}{7}$

B) $\frac{7}{4}$

C) $- \frac{4}{7}$

D) $- \frac{7}{4}$

Show the worked solution

Answer: D

Explanation

Flip and negate: $-\frac74$. Check: $\frac47 \times -\frac74 = -1$ ✓

B and C each did one half of the job.

Takeaway: Negative reciprocal is two steps. Multiply the two slopes; you should get exactly $-1$.

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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 $w$. Then the length is $2w + 3$, and the perimeter is

$$2(2w + 3) + 2w = 42$$ $$4w + 6 + 2w = 42 \quad\Longrightarrow\quad 6w = 36 \quad\Longrightarrow\quad w = 6$$

Check: width 6, length 15, perimeter $2(15 + 6) = 42$ ✓

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.

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Paper 2 — Module 1

27 questions · 35 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.

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Topic: Solving a linear equation

A courier charges a fixed collection fee plus an amount for each parcel. For one job the total charge, in dollars, satisfies $5x + 12 = 47$, where $x$ is the number of parcels collected. What is the value of $x$?

A) $\frac{59}{5}$

B) $7$

C) $35$

D) $\frac{47}{5}$

Show the worked solution

Answer: B

Explanation

$$5x = 35 \quad\Longrightarrow\quad x = 7$$

Check: $35 + 12 = 47$ ✓ C stopped one step early.

Takeaway: Subtract, then divide, then finish.

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Topic: Simplifying an expression

A factory's daily output, in units, is given by the expression $7x - 2(x - 5)$, where $x$ is the number of machines running. Which of the following expressions is equivalent to it for every value of $x$?

A) $5 x + 10$

B) $5 x - 10$

C) $9 x - 10$

D) $5 x - 5$

Show the worked solution

Answer: A

Explanation

$$-2(x - 5) = -2x + 10, \qquad 7x - 2x + 10 = 5x + 10$$

B kept the constant negative — but $-2 \times -5 = +10$.

Takeaway: A minus before a bracket flips every sign inside it.

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Topic: Reading a graph's intercept

A tank is draining. The volume remaining, in litres, is modelled by the line $y = -2x + 9$, where $x$ is the number of minutes since draining began. At what value of $y$ does this line cross the $y$-axis?

A) $-2$

B) $\frac{9}{2}$

C) $9$

D) $0$

Show the worked solution

Answer: C

Explanation

On the $y$-axis, $x = 0$, so $y = 9$.

B is the $x$-intercept, found by setting $y = 0$.

Takeaway: $y$-axis means $x = 0$. Each axis zeroes the other variable.

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Topic: Solving a two-step linear equation

A tailor charges a fitting fee and then an amount per metre of fabric. For one order the amount owed, in dollars, satisfies $7x - 9 = 33$, where $x$ is the number of metres used. What is the value of $x$?

A) $\frac{24}{7}$

B) $6$

C) $42$

D) $\frac{33}{7}$

Show the worked solution

Answer: B

Explanation

$$7x = 42 \quad\Longrightarrow\quad x = 6$$

Check: $42 - 9 = 33$ ✓ A subtracted where it should have added; C stopped a step early.

Takeaway: Undo the subtraction first, then the multiplication — and finish.

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Topic: A quadratic with two roots

A projectile is at ground level at the times that satisfy $x^2 - 9x + 20 = 0$, where $x$ is measured in seconds. What is the sum of the two solutions of this equation?

A) $-9$

B) $9$

C) $20$

D) $1$

Show the worked solution

Answer: B

Explanation

Sum of roots $= -\frac{b}{a} = 9$. (The roots are 4 and 5.)

C is the product, $\frac{c}{a} = 20$.

Takeaway: Sum $=-\frac ba$, product $=\frac ca$.

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Topic: A percentage increase

A town's population rises from 4500 to 5400. By what percent did it rise?

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:

$$\frac{900}{4500} \times 100 = 20$$

Using 5400 as the base would give 16.7%, the standard wrong answer. Enter 20.

Takeaway: Percentage change divides by the original.

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Topic: An exponent equation

A colony of bacteria triples in each observation period. The population is found to have grown by a factor of 81. What is the value of $x$?

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

Write 81 as a power of 3, so both sides share a base:

$$81 = 3 \times 3 \times 3 \times 3 = 3^4$$

The equation becomes $3^x = 3^4$. With the same base on both sides, the exponents must be equal:

$$x = 4$$

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.

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Topic: A system by elimination

A market stall sells $x$ bags of apples and $y$ bags of pears. On one day the stall recorded $3x + y = 14$, and on another it recorded $x - y = 2$. What is the value of $x$?

A) $4$

B) $2$

C) $16$

D) $\frac{7}{2}$

Show the worked solution

Answer: A

Explanation

The $y$ terms are $+y$ and $-y$, so add the equations: $4x = 16$, giving $x = 4$. Then $y = 2$.

Check: $12 + 2 = 14$ ✓ and $4 - 2 = 2$ ✓

B gave $y$. C stopped at $4x = 16$. D added only the left sides, using $4x = 14$.

Takeaway: Opposite signs, add the equations.

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Topic: Volume of a cylinder

A cylindrical water butt has a radius of 5 decimetres and a height of 4 decimetres. A gardener wants to know how much water it holds when full. What is the volume of the water butt, in cubic decimetres?

A) $20 \pi$

B) $40 \pi$

C) $100 \pi$

D) $400 \pi$

Show the worked solution

Answer: C

Explanation

$$V = \pi r^2 h = \pi(25)(4) = 100\pi$$

A used $r$ rather than $r^2$; B is a surface area.

Takeaway: Square the radius, not the height.

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Topic: Function notation

The function $g$ is defined by $g(x) = 2x^2 + 1$, where $g(x)$ gives the energy, in joules, stored by a spring displaced $x$ centimetres from its resting position. What is the value of $g(-3)$?

A) $-17$

B) $37$

C) $-35$

D) $19$

Show the worked solution

Answer: D

Explanation

$$g(-3) = 2(9) + 1 = 19$$

$(-3)^2 = 9$, not $-9$. A made that error.

Takeaway: Square inside brackets: a negative squared is positive.

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Topic: Solving for one variable in terms of another

A caterer's budget satisfies $3x + 4y = 24$, where $x$ is the number of trays of sandwiches and $y$ the number of trays of fruit. The caterer wants to express the number of fruit trays in terms of the number of sandwich trays. Which expression gives $y$ in terms of $x$?

A) $6 - \frac{3 x}{4}$

B) $\frac{3 x}{4} - 6$

C) $\frac{3 x}{4} + 6$

D) $24 - 3 x$

Show the worked solution

Answer: A

Explanation

$$4y = 24 - 3x \quad\Longrightarrow\quad y = \frac{24 - 3x}{4}$$

Check at $x = 4$: the original gives $12 + 4y = 24$, so $y = 3$; the answer gives $\frac{24-12}{4} = 3$ ✓

D never divided by 4.

Takeaway: Whatever divides must divide every term. Check with one value.

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Topic: Angles in a triangle

A surveyor measures two of the three interior angles of a triangular plot of land and finds them to be $35°$ and $95°$. What is the measure, in degrees, of the third angle of the plot?

A) $50$

B) $130$

C) $230$

D) $40$

Show the worked solution

Answer: A

Explanation

$$180 - 35 - 95 = 50$$

Check: $35 + 95 + 50 = 180$ ✓

B added the two given angles. C used $360°$, the total for a quadrilateral. D assumed a right angle the question never mentions.

Takeaway: Triangle angles total $180°$. Add all three back as a check.

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Topic: An exponential model

A microbiologist is growing a culture that begins with 60 cells and doubles in number every 3 hours, and the growth continues at that rate without interruption. The culture is left in the incubator for 12 hours. How many cells are there at the end of that period?

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

$\frac{12}{3} = 4$ periods, so the factor is $2^4 = 16$:

$$60 \times 16 = 960$$

Check by stepping: $60 \to 120 \to 240 \to 480 \to 960$ ✓ Enter 960.

Takeaway: Divide the time by the period length to count the periods.

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Topic: Scaling an area

A print shop is enlarging a rectangular poster so that both its length and its width are doubled, keeping the same shape. The manager needs to know how much more ink the larger poster will require. By what factor does the 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: 4

Explanation

$$(2l)(2w) = 4lw$$

Both dimensions double, so the area is multiplied by $2^2 = 4$. Try it: a $3 \times 5$ rectangle has area 15; a $6 \times 10$ one has area 60 ✓ Enter 4.

Takeaway: Lengths $\times k$, areas $\times k^2$, volumes $\times k^3$.

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Topic: Solving for a variable in terms of another

A printing job satisfies the relationship $5x - 2y = 20$, where $x$ is the number of colour pages and $y$ the number of black-and-white pages. Which expression gives $y$ in terms of $x$?

A) $\frac{5 x}{2} - 10$

B) $10 - \frac{5 x}{2}$

C) $\frac{5 x}{2} + 10$

D) $5 x - 20$

Show the worked solution

Answer: A

Explanation

$$-2y = 20 - 5x \quad\Longrightarrow\quad y = \frac{5x - 20}{2}$$

Check at $x = 4$: the original gives $20 - 2y = 20$, so $y = 0$; and the answer gives $\frac{20 - 20}{2} = 0$ ✓

D never divided by 2.

Takeaway: Whatever divides must divide every term. Check with one value.

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