Mathematics (Syllabus D) 4024 / 22 — October/November 2024

DifficultyMedium-Easy

Worked solution

Approach

To find the mass of mango as a percentage of the total mass, we first need to calculate the total mass of all the fruits in the bag. Then, we express the mass of the mango as a fraction of this total and multiply by $100$ to get the percentage.

Working

First, calculate the total mass of the mixed fruit:

\text{Total mass} = \text{Pineapple} + \text{Mango} + \text{Papaya}

\text{Total mass} = 96 + 84 + 60

\text{Total mass} = 240 \, \text{g}

Next, write the mass of the mango ( $84 \, \text{g}$ ) as a fraction of the total mass ( $240 \, \text{g}$ ):

\text{Fraction} = \frac{84}{240}

Convert this fraction to a percentage by multiplying by $100$ :

\text{Percentage} = \frac{84}{240} \times 100

Simplify the calculation:

\frac{84}{240} = \frac{7}{20}

\frac{7}{20} \times 100 = 7 \times \frac{100}{20} = 7 \times 5 = 35

Answer

The mass of mango is $35\%$ of the total mass.

35\%

Final answer

35

Detailed explanation

Walkthrough

The question asks for the mass of the mango relative to the entire contents of the bag, expressed as a percentage.

Step 1: Find the whole. We are given three separate masses. To compare any single part to the "total," we must first add these parts together. $96 + 84 + 60 = 240$ grams. This is our denominator.

Step 2: Form the ratio. The numerator is the specific part we are interested in: the mango, which is $84$ grams. So the ratio is $\frac{84}{240}$ .

Step 3: Convert to percentage. A percentage is simply a ratio out of $100$ . Multiplying the fraction by $100$ gives us the answer. $\frac{84}{240} \times 100 = 35\%$ .

Key Takeaways

When asked for a part as a percentage of a total, always ensure you have calculated the full total first. Do not just use the other numbers provided in the table; add them all up.

Common Mistakes

Forgetting to add all three weights to get the total mass (e.g., using $96+84=180$ as the total).
Calculating the percentage of pineapple or papaya instead of mango.
Arithmetic errors in the final multiplication or division.

Things to Be Careful About

Ensure you are answering the specific question asked (mango vs total). It is easy to rush and pick the wrong number from the table.
Check your addition carefully. $96+84=180$ , and $180+60=240$ .

Techniques used

calculate total mass from partsexpress one part as a fraction of the totalconvert fraction to percentage

(b)

Tom makes a drink by mixing juice and water in the ratio $3 : 7$ .
He makes $1.4$ litres of this drink.

Calculate the amount of juice Tom uses.
Give your answer in millilitres.

______ $\text{ml}$

DifficultyMedium-Easy

Worked solution

Approach

Tom mixes juice and water in the ratio $3 : 7$ . The total volume of the drink is given in litres, but the answer is required in millilitres. We should first convert the total volume to millilitres, then divide this amount according to the ratio parts.

Working

First, convert the total volume of the drink from litres to millilitres:

1.4 \, \text{litres} = 1.4 \times 1000 = 1400 \, \text{ml}

The ratio of juice to water is $3 : 7$ . This means there are $3 + 7 = 10$ equal parts (or shares) in total.

Calculate the value of one part:

\text{Value of one part} = \frac{1400}{10} = 140 \, \text{ml}

The amount of juice corresponds to $3$ parts. Calculate the volume of juice:

\text{Volume of juice} = 3 \times 140

\text{Volume of juice} = 420 \, \text{ml}

Answer

420 \, \text{ml}

Final answer

420

Detailed explanation

Walkthrough

We are given a total mixture volume and a ratio of ingredients. The key is to split the total volume into the same number of equal parts as defined by the ratio.

Step 1: Unit consistency. The final answer requires millilitres, but the input is in litres. $1$ litre equals $1000$ ml. So, $1.4$ litres becomes $1400$ ml.

Step 2: Total parts. The ratio $3:7$ means for every $3$ units of juice, there are $7$ units of water. The whole drink consists of $3 + 7 = 10$ units.

Step 3: Division. Divide the total volume ( $1400$ ml) by the total number of units ( $10$ ) to find the size of one unit. $1400 / 10 = 140$ ml.

Step 4: Final calculation. Juice is $3$ units. Multiply the size of one unit by $3$ : $3 \times 140 = 420$ ml.

Key Takeaways

Always check the units required for the final answer. If they differ from the given units, convert early. When dividing a total by a ratio, remember to add the ratio numbers together to find the total number of parts.

Common Mistakes

Dividing by only one of the ratio numbers (e.g., dividing $1400$ by $3$ or $7$ directly without adding them).
Forgetting to convert litres to millilitres.
Calculating the amount of water instead of juice.

Things to Be Careful About

The question explicitly asks for the answer in millilitres. Providing the answer in litres ( $1.4$ L total -> $0.42$ L juice) would likely lose accuracy marks depending on the specific marking scheme rules, though usually, unconverted units are penalized. Always match the requested unit.

Techniques used

convert litres to millilitresdivide a quantity into a given ratiocalculate the value of one share

(c)

The cost of a fruit drink is directly proportional to the amount of juice it contains.
A fruit drink containing $125\,\text{ml}$ of juice costs $1.50.

Calculate the cost of a fruit drink containing $175\,\text{ml}$ of juice.

$ ______

DifficultyMedium-Easy

Worked solution

Approach

The cost is directly proportional to the amount of juice. This means we can find the cost per ml of juice (the constant of proportionality) using the given pair of values, and then use that rate to calculate the cost for the new amount.

Working

Given:
Amount of juice = $125 \, \text{ml}$
Cost = $1.50

Find the cost per ml:

\text{Cost per ml} = \frac{\text{Cost}}{\text{Amount of juice}} = \frac{1.50}{125}

Now, calculate the cost for $175 \, \text{ml}$ of juice by multiplying the cost per ml by $175$ :

\text{New Cost} = 175 \times \frac{1.50}{125}

Simplify the calculation:

\frac{175}{125} = \frac{7}{5} = 1.4

\text{New Cost} = 1.4 \times 1.50

\text{New Cost} = 2.10

Alternatively, using cross-multiplication logic for direct proportion:

\frac{C_1}{V_1} = \frac{C_2}{V_2} \implies \frac{1.50}{125} = \frac{x}{175}

x = \frac{1.50 \times 175}{125} = 2.10

Answer

\$ 2.10

Final answer

2.10

Detailed explanation

Walkthrough

"Directly proportional" means that if one quantity doubles, the other doubles. We can model this with a constant rate. Here, the rate is dollars per ml.

Step 1: Find the unit rate. Divide the known cost by the known volume. $1.50 / 125$ . This tells us how much each ml costs.

Step 2: Apply the rate. Multiply this unit cost by the new volume ( $175$ ml).

Calculation tip: It is often easier to simplify the fraction $\frac{175}{125}$ before multiplying. Both are divisible by $25$ . $175/25 = 7$ and $125/25 = 5$ . So we are calculating $\frac{7}{5} \times 1.50$ . $1.50 / 5 = 0.30$ , and $0.30 \times 7 = 2.10$ .

Key Takeaways

For direct proportion problems, finding the "unit value" (cost per item, speed per hour, etc.) is a reliable method. Set up the relationship $\frac{y_1}{x_1} = \frac{y_2}{x_2}$ .

Common Mistakes

Inverting the ratio (multiplying by $\frac{125}{175}$ instead of $\frac{175}{125}$ ).
Arithmetic errors when dividing decimals.

Things to Be Careful About

Currency format. The answer is in dollars. While $2.1$ is numerically correct, $2.10$ reflects standard currency formatting. The mark scheme accepts $2.1[0]$ , so either is fine, but be aware of significant figures or decimal places if specified.

Techniques used

find the constant of proportionalityuse direct proportion to find a new value

(d)

Kofi has a bag containing nuts and raisins.
There are $285\,\text{g}$ of raisins in the bag.
The remaining $62\%$ of the mass in the bag is nuts.

Calculate the mass of nuts in the bag.

______ $\text{g}$

DifficultyMedium

Worked solution

Approach

We know the mass of raisins and the percentage of nuts. Since the bag contains only nuts and raisins, the percentage of raisins plus the percentage of nuts must equal $100\%$ . We can use the known mass of raisins and its corresponding percentage to find the total mass, or directly relate the raisin mass to the nut mass.

Working

The mass of nuts is $62\%$ of the total mass. Therefore, the mass of raisins represents the remaining percentage:

\text{Percentage of raisins} = 100\% - 62\% = 38\%

We are given that the mass of raisins is $285 \, \text{g}$ . So, $38\%$ of the total mass is $285 \, \text{g}$ .

Let $T$ be the total mass.

0.38 \times T = 285

Solve for the total mass $T$ :

T = \frac{285}{0.38}

To make the division easier, multiply numerator and denominator by $100$ :

T = \frac{28500}{38}

Calculate $T$ :

28500 \div 38 = 750 \, \text{g}

So, the total mass of the bag is $750 \, \text{g}$ .

Now, calculate the mass of nuts. We can do this by subtracting the mass of raisins from the total:

\text{Mass of nuts} = \text{Total mass} - \text{Mass of raisins}

\text{Mass of nuts} = 750 - 285 = 465 \, \text{g}

Alternatively, calculate $62\%$ of $750$ :

0.62 \times 750 = 465 \, \text{g}

Answer

465 \, \text{g}

Final answer

465

Detailed explanation

Walkthrough

This is a reverse percentage problem. Usually, we are given a total and asked to find a percentage of it. Here, we are given a part (raisins) and told what percentage the other part (nuts) is.

Step 1: Find the percentage of the known part. If nuts are $62\%$ , raisins must be $100 - 62 = 38\%$ .

Step 2: Link percentage to value. We know $38\%$ corresponds to $285$ g. This allows us to find $1\%$ or the full $100\%$ .

Step 3: Find the total. $285 / 0.38 = 750$ g.

Step 4: Find the target value. The question asks for the mass of nuts. Nuts are $62\%$ of the total. $0.62 \times 750 = 465$ g. Or simply $750 - 285 = 465$ g.

Key Takeaways

In problems involving parts of a whole, the percentages of all parts must add up to $100\%$ . If you know the percentage of one part, you immediately know the percentage of the rest.

Common Mistakes

Assuming the $285$ g corresponds to the $62\%$ (nuts) instead of the remaining portion (raisins). The text says " $285$ g of raisins" and " $62\%$ ... is nuts".
Calculating the total mass and stopping there.
Arithmetic errors in dividing by a decimal ( $0.38$ ).

Things to Be Careful About

Read carefully: " $285$ g of raisins" is the data point. " $62\%$ ... is nuts" is the constraint. Do not mix them up. $285$ is NOT $62\%$ of the total.

Techniques used

determine the missing percentageset up an equation relating a part to the totalsolve for the unknown part

(e)

The mass of mixed nuts and seeds in a bag is $500\,\text{g}$ , correct to the nearest $10\,\text{g}$ .
The mass of nuts in the bag is $350\,\text{g}$ , correct to the nearest $5\,\text{g}$ .

Calculate the upper bound and the lower bound of the mass of seeds in the bag.

Upper bound = ______ $\text{g}$
Lower bound = ______ $\text{g}$

DifficultyMedium

Worked solution

Approach

The mass of seeds is the total mass minus the mass of nuts. To find the upper bound of the mass of seeds, we need the largest possible total mass and the smallest possible mass of nuts. To find the lower bound of the mass of seeds, we need the smallest possible total mass and the largest possible mass of nuts.

Working

Step 1: Determine the bounds for the total mass.
The total mass is $500 \, \text{g}$ , correct to the nearest $10 \, \text{g}$ .
The accuracy is $10 \, \text{g}$ , so the error interval is half of this: $\pm 5 \, \text{g}$ .

\text{Upper Bound (Total)} = 500 + 5 = 505 \, \text{g}

\text{Lower Bound (Total)} = 500 - 5 = 495 \, \text{g}

Step 2: Determine the bounds for the mass of nuts.
The mass of nuts is $350 \, \text{g}$ , correct to the nearest $5 \, \text{g}$ .
The accuracy is $5 \, \text{g}$ , so the error interval is half of this: $\pm 2.5 \, \text{g}$ .

\text{Upper Bound (Nuts)} = 350 + 2.5 = 352.5 \, ext{g}

\text{Lower Bound (Nuts)} = 350 - 2.5 = 347.5 \, \text{g}

Step 3: Calculate the bounds for the mass of seeds.
Let $S$ be the mass of seeds. $S = \text{Total} - \text{Nuts}$ .

To maximize $S$ (Upper Bound), we take the maximum Total and minimum Nuts:

\text{UB}(S) = \text{UB(Total)} - \text{LB(Nuts)}

\text{UB}(S) = 505 - 347.5 = 157.5 \, \text{g}

To minimize $S$ (Lower Bound), we take the minimum Total and maximum Nuts:

\text{LB}(S) = \text{LB(Total)} - \text{UB(Nuts)}

\text{LB}(S) = 495 - 352.5 = 142.5 \, \text{g}

Answer

Upper bound = $157.5 \, \text{g}$
Lower bound = $142.5 \, \text{g}$

\text{Upper bound} = 157.5, \quad \text{Lower bound} = 142.5

Final answer

Upper bound = 157.5, Lower bound = 142.5

Detailed explanation

Walkthrough

Bounds problems require careful attention to whether you are adding/subtracting or multiplying/dividing quantities, and whether you want the maximum or minimum result.

Here, Seeds = Total - Nuts.

To get the biggest possible amount of seeds, imagine the bag is heavier than stated (Total is high) and the nuts are lighter than stated (Nuts is low). High minus Low gives the biggest difference.
$\text{UB(Seeds)} = \text{UB(Total)} - \text{LB(Nuts)}$

To get the smallest possible amount of seeds, imagine the bag is lighter than stated (Total is low) and the nuts are heavier than stated (Nuts is high). Low minus High gives the smallest difference.
$\text{LB(Seeds)} = \text{LB(Total)} - \text{UB(Nuts)}$

Calculating the Bounds:

Total: Nearest 10g. Half-step is 5g. Range: $[495, 505]$ .
Nuts: Nearest 5g. Half-step is 2.5g. Range: $[347.5, 352.5]$ .

Final Calculation:

UB: $505 - 347.5 = 157.5$
LB: $495 - 352.5 = 142.5$

Key Takeaways

For addition/subtraction of bounds:

Upper Bound of Sum/Difference = Max(A) +/- Min(B)
Lower Bound of Sum/Difference = Min(A) +/- Max(B)
Note the swap: to maximize a subtraction, you subtract the smaller number.

Common Mistakes

Using the same accuracy for both items (e.g., treating the nuts as nearest 10g).
Adding the bounds instead of subtracting them (e.g., doing $505 + 352.5$ ).
Reversing the logic for UB/LB (subtracting the upper bound of nuts from the upper bound of total yields an incorrect middle value).

Things to Be Careful About

Check the accuracy of each measurement individually. "Nearest 10g" implies a tolerance of $\pm 5$ . "Nearest 5g" implies a tolerance of $\pm 2.5$ . Mixing these up is a very common error.

Techniques used

determine upper and lower bounds for rounded valuescalculate bounds for a difference

Q2MediumStatisticsNumber

(a)

The table shows the age and value of 10 cars of the same model.

Age (years)	3	3	4	4	5	5	5	6	8	8
Value ($)	5500	6200	4200	4000	4000	3700	4500	3000	1500	2000

5M

(i)

Complete the scatter diagram.
The first 6 points have been plotted for you.

Fig. 1

DifficultyMedium-Easy

Worked solution

Approach

Identify the remaining $4$ data pairs from the table and plot them accurately on the scatter diagram.

The remaining data pairs to plot are:

(5, 4500), \quad (6, 3000), \quad (8, 1500), \quad (8, 2000)

Working

For $(5, 4500)$ : go to $5$ on the horizontal axis and up to halfway between $4000$ and $5000$ on the vertical axis.
For $(6, 3000)$ : go to $6$ on the horizontal axis and up to $3000$ on the vertical axis.
For $(8, 1500)$ : go to $8$ on the horizontal axis and up to halfway between $1000$ and $2000$ on the vertical axis.
For $(8, 2000)$ : go to $8$ on the horizontal axis and up to $2000$ on the vertical axis.

Answer

Plot the four points $(5, 4500)$ , $(6, 3000)$ , $(8, 1500)$ , and $(8, 2000)$ on the scatter diagram.

Final answer

4 points plotted at (5, 4500), (6, 3000), (8, 1500), and (8, 2000)

Detailed explanation

Walkthrough

The table lists $10$ pairs of values representing the age and value of $10$ cars. The first $6$ points plotted are $(3, 5500)$ , $(3, 6200)$ , $(4, 4200)$ , $(4, 4000)$ , $(5, 4000)$ , and $(5, 3700)$ .

The remaining $4$ points from the table are:

$(5, 4500)$
$(6, 3000)$
$(8, 1500)$
$(8, 2000)$

Using the grid:

Each major horizontal grid mark represents $1$ year, subdivided into small squares.
Each major vertical grid mark represents $1000, with $10$ small squares per $1000 (meaning each small square is $100).

Plot each of the $4$ points with a small neat cross or dot.

Key Takeaways

On a scatter diagram, each data item is plotted as a single point $(x, y)$ .
Always check the scale on both axes to ensure points are placed precisely.

Common Mistakes

Plotting at incorrect vertical grid intervals (e.g. misinterpreting the grid subdivisions for $500).
Forgetting to plot one of the duplicate $x$ -values (e.g. only plotting one point at $x = 8$ ).

Things to Be Careful About

Ensure all $4$ points are plotted clearly and accurately using a sharp pencil.

Techniques used

read coordinates from a tableplot points on a grid

(ii)

Draw a line of best fit.

(iii)

Use your line of best fit to find an estimate for the value of a car of this model that is 7 years old.

$ ______

(iv)

Jay has a car of this model that is 12 years old and he wants to find its value.

Explain why Jay should not use this scatter diagram to find an estimate for the value of this car.

______ because ______

(b)

Jay records the distances travelled by 50 cars.
The frequency table shows the results.

Distance ( $d$ thousand km)	$10 < d \leq 40$	$40 < d \leq 50$	$50 < d \leq 60$	$60 < d \leq 100$
Frequency	8	14	11	17

5M

(i)

Work out the fraction of the cars that have travelled more than $50\,000\,\text{km}$ .
Give your answer in its simplest form.

______

(ii)

Find the interval that contains the median.

______

DifficultyMedium-Easy

Worked solution

Approach

Find the position of the median item for $50$ cars and determine which class interval contains this position using cumulative frequency.

Working

Total frequency $n = 50$ .

The median position is:

\frac{50}{2} = 25\text{th value}

Calculate the cumulative frequencies:

Interval $10 < d \leq 40$ : $8$
Interval $40 < d \leq 50$ : $8 + 14 = 22$
Interval $50 < d \leq 60$ : $22 + 11 = 33$

Since the $22\text{nd}$ value is in $40 < d \leq 50$ , the $23\text{rd}$ through $33\text{rd}$ values lie in the interval $50 < d \leq 60$ .
Therefore, the $25\text{th}$ value lies in $50 < d \leq 60$ .

Answer

50 < d \leq 60

Final answer

50 < d <= 60

Detailed explanation

Walkthrough

The median is the middle value. For $n = 50$ , the median position is $\frac{50}{2} = 25$ (or $\frac{50 + 1}{2} = 25.5$ ).
Find the cumulative frequencies by running a cumulative total:
- Up to $40$ : $8$ cars
- Up to $50$ : $8 + 14 = 22$ cars
- Up to $60$ : $22 + 11 = 33$ cars
- Up to $100$ : $33 + 17 = 50$ cars
The $25\text{th}$ car falls after the $22\text{nd}$ car and before the $33\text{rd}$ car, which corresponds to the interval $50 < d \leq 60$ .

Key Takeaways

To find the median interval of grouped data, find $\frac{n}{2}$ and locate which group contains that cumulative frequency.

Common Mistakes

Picking the interval with the highest frequency ( $60 < d \leq 100$ ), which is the modal interval, not the median interval.
Picking the interval with the median label/middle row rather than finding the median of the cumulative frequency.

Things to Be Careful About

Write the interval exactly as printed in the table: $50 < d \leq 60$ .

Techniques used

calculate cumulative frequenciesidentify the median class interval

(iii)

Calculate an estimate of the mean distance travelled.

______ thousand km

DifficultyMedium

Worked solution

Approach

To estimate the mean of grouped data, find the midpoint of each class interval, multiply each midpoint by its corresponding frequency, sum these products, and divide by the total frequency ( $50$ ).

Working

Find the midpoint ( $x$ ) for each interval:

\begin{aligned} 10 < d \leq 40 &\implies x = \frac{10 + 40}{2} = 25 \\[6pt] 40 < d \leq 50 &\implies x = \frac{40 + 50}{2} = 45 \\[6pt] 50 < d \leq 60 &\implies x = \frac{50 + 60}{2} = 55 \\[6pt] 60 < d \leq 100 &\implies x = \frac{60 + 100}{2} = 80 \end{aligned}

Calculate $\sum f x$ :

\begin{aligned} \sum f x &= (8 \times 25) + (14 \times 45) + (11 \times 55) + (17 \times 80) \\[6pt] &= 200 + 630 + 605 + 1360 \\[6pt] &= 2795 \end{aligned}

Calculate the estimated mean:

\text{Estimated mean} = \frac{\sum f x}{\sum f} = \frac{2795}{50} = 55.9

Answer

55.9

Final answer

55.9

Detailed explanation

Walkthrough

For grouped continuous data, the exact values are unknown, so we use the midpoint of each interval as the representative value.

Find the midpoint of each interval:
- $\frac{10 + 40}{2} = 25$
- $\frac{40 + 50}{2} = 45$
- $\frac{50 + 60}{2} = 55$
- $\frac{60 + 100}{2} = 80$
Multiply each midpoint by its frequency ( $f \times x$ ):
- $8 \times 25 = 200$
- $14 \times 45 = 630$
- $11 \times 55 = 605$
- $17 \times 80 = 1360$
Sum the products ( $\sum fx$ ):

200 + 630 + 605 + 1360 = 2795

Divide by the total frequency ( $n = 50$ ):

\text{Mean} = \frac{2795}{50} = 55.9

Key Takeaways

Formula for the estimated mean of grouped data:

\bar{x} = \frac{\sum f x}{\sum f}

where $x$ represents the midpoint of each class interval and $f$ is the frequency.

Common Mistakes

Using interval widths instead of midpoints.
Using upper bounds or lower bounds instead of midpoints.
Dividing by $4$ (the number of classes) instead of $50$ (total frequency).

Things to Be Careful About

The intervals have unequal widths ( $30$ , $10$ , $10$ , $40$ ), so calculate each midpoint carefully rather than assuming a uniform step size.
The unit "thousand km" is already given on the answer line, so write $55.9$ rather than $55\,900$ .

Techniques used

calculate midpoints of class intervalscalculate sum of frequency times midpointcalculate estimated mean for grouped data

Q3Medium-HardMensurationNumberTrigonometry

(a)

Fig. 2

The diagram shows a tank.
The tank is a cuboid with length $1.2\,\text{m}$ , width $0.6\,\text{m}$ and height $h\,\text{m}$ .
The volume of the tank is $1.8\,\text{m}^3$ .

5M

(i)

Calculate the value of $h$ .

$h$ = ______

(ii)

Fuel is pumped into the empty tank at a rate of $0.2\,\text{m}^3$ per minute.

Calculate the time taken to fill the tank to $90\%$ of its volume.
Give your answer in minutes and seconds.

______ minutes ______ seconds

DifficultyMedium-Easy

Worked solution

Approach

First calculate $90\%$ of the total volume of $1.8\,\text{m}^3$ . Then divide this target volume by the pumping rate of $0.2\,\text{m}^3/\text{min}$ to obtain the time in minutes, and finally convert any decimal part of a minute into seconds.

Working

Calculate $90\%$ of the total volume:

\text{Volume to fill} = 1.8 \times \frac{90}{100} = 1.62\,\text{m}^3

Calculate the time taken in minutes:

\text{Time} = \frac{1.62}{0.2} = 8.1\,\text{minutes}

Convert the decimal minutes to minutes and seconds:

0.1\,\text{minutes} = 0.1 \times 60 = 6\,\text{seconds}

So the time taken is $8\,\text{minutes}\ 6\,\text{seconds}$ .

Answer

8\text{ minutes } 6\text{ seconds}

Final answer

8 minutes 6 seconds

Detailed explanation

Walkthrough

Find the required volume to fill, which is $90\%$ of the tank's total volume of $1.8\,\text{m}^3$ :

1.8 \times 0.90 = 1.62\,\text{m}^3

The fuel is pumped at a constant rate of $0.2\,\text{m}^3$ per minute. Divide the required volume by the pumping rate to get the time in minutes:

\text{Time} = \frac{1.62}{0.2} = 8.1\,\text{minutes}

Alternatively, find the total time to fill the tank ( $1.8 / 0.2 = 9\,\text{minutes}$ ) and take $90\%$ of that time:

9 \times 0.90 = 8.1\,\text{minutes}

Convert $8.1\,\text{minutes}$ into minutes and seconds. The integer part is $8\,\text{minutes}$ . The fractional part is $0.1\,\text{minutes}$ , which converts to seconds by multiplying by $60$ :

0.1 \times 60 = 6\,\text{seconds}

Thus, the total time is $8\,\text{minutes}$ and $6\,\text{seconds}$ .

Key Takeaways

$\text{Time} = \frac{\text{Volume}}{\text{Rate}}$ .
When converting a decimal minute to seconds, multiply the fractional part by $60$ , not $100$ .

Common Mistakes

Converting $8.1$ minutes directly to $8\text{ min } 10\text{ s}$ or $8\text{ min } 1\text{ s}$ instead of multiplying $0.1$ by $60$ .
Calculating time for $100\%$ volume instead of $90\%$ .

Things to Be Careful About

Ensure both the minutes and seconds slots are filled correctly with integers ( $8$ and $6$ ).

Techniques used

calculate percentage of a quantitydivide by rate to find timeconvert decimal minutes to minutes and seconds

(b)

Fig. 3

The diagram shows a tank with an open top.
The tank is a prism with trapezium $ABCD$ as its cross-section.
$AD = BC = 70\,\text{cm}$ , $CD = 80\,\text{cm}$ and $AB = 110\,\text{cm}$ .
The base of the tank is a square.

Calculate the total surface area of the outside of the tank.

______ $\text{cm}^2$

5M Similar

DifficultyMedium-Hard

Worked solution

Approach

The tank is an open-topped prism whose external surface consists of $5$ faces:

Two congruent trapezoidal cross-sections ( $ABCD$ and $EFGH$ ).
One square base ( $CDHG$ ).
Two congruent sloping rectangular sides ( $ADHE$ and $BCGF$ ).
(The top face $ABFE$ is open and not included.)

First, use Pythagoras' theorem on the symmetrical isosceles trapezium $ABCD$ to determine its perpendicular height $h$ .
Then calculate and sum the areas of the 5 exterior faces.

Working

Step 1: Find the perpendicular height of the trapezium $ABCD$

Trapezium $ABCD$ is an isosceles trapezium with parallel sides $AB = 110\,\text{cm}$ and $CD = 80\,\text{cm}$ , and non-parallel sides $AD = BC = 70\,\text{cm}$ .

The horizontal overhang on each side is:

\frac{AB - CD}{2} = \frac{110 - 80}{2} = \frac{30}{2} = 15\,\text{cm}

Using Pythagoras' theorem to find the perpendicular height $h$ :

h = \sqrt{70^2 - 15^2} = \sqrt{4900 - 225} = \sqrt{4675} \approx 68.37397\,\text{cm}

Step 2: Calculate the area of the two trapezoidal faces

\text{Area of one trapezium} = \frac{1}{2}(AB + CD) \times h = \frac{1}{2}(110 + 80) \times \sqrt{4675} = 95 \times \sqrt{4675} \approx 6495.527\,\text{cm}^2

\text{Area of both trapeziums} = 2 \times \left(95 \times \sqrt{4675}\right) = 190 \times \sqrt{4675} \approx 12991.054\,\text{cm}^2

Step 3: Calculate the area of the base and side faces

The base $CDHG$ is a square of side $80\,\text{cm}$ (since $CD = 80\,\text{cm}$ and the base is square, the prism length is $DH = CG = 80\,\text{cm}$ ):

\text{Area of base} = 80 \times 80 = 6400\,\text{cm}^2

The two sloping rectangular side faces ( $ADHE$ and $BCGF$ ) each have dimensions $70\,\text{cm} \times 80\,\text{cm}$ :

\text{Area of two side faces} = 2 \times (80 \times 70) = 2 \times 5600 = 11200\,\text{cm}^2

Step 4: Total surface area

\text{Total surface area} = 12991.054 + 6400 + 11200 = 30591.054\,\text{cm}^2

Rounding to 3 significant figures gives $30\,600\,\text{cm}^2$ (or to the nearest integer, $30\,591\,\text{cm}^2$ ).

Answer

30\,600\text{ or } 30\,591

Final answer

30600

Detailed explanation

Walkthrough

Identify the faces of the open-topped tank:
- The top rectangular face $ABFE$ is open, so it has no material and is excluded from the surface area calculation.
- The front face $ABCD$ and back face $EFGH$ are identical isosceles trapeziums.
- The bottom face $CDHG$ is a square base of dimensions $80\,\text{cm} \times 80\,\text{cm}$ . This confirms the length of the prism along the parallel edges ( $DH, CG, BF, AE$ ) is $80\,\text{cm}$ .
- The two inclined side faces $ADHE$ and $BCGF$ are rectangles with side lengths $70\,\text{cm}$ and $80\,\text{cm}$ .
Find the height of the cross-section:
- Dropping perpendiculars from $D$ and $C$ to line $AB$ divides $AB$ into three segments: a middle section equal to $CD = 80\,\text{cm}$ and two equal right-angled triangle bases at each end of length $\frac{110 - 80}{2} = 15\,\text{cm}$ .
- By Pythagoras' theorem:

h = \sqrt{70^2 - 15^2} = \sqrt{4900 - 225} = \sqrt{4675} \approx 68.374\,\text{cm}

Compute the area of the faces:
- Two trapeziums:

2 \times \left(\frac{110 + 80}{2} \times \sqrt{4675}\right) = 190 \times \sqrt{4675} \approx 12\,991.05\,\text{cm}^2

Square base:

80 \times 80 = 6400\,\text{cm}^2

Two sloping sides:

2 \times (80 \times 70) = 11\,200\,\text{cm}^2

Sum the areas:

\text{Total Area} = 12\,991.05 + 6400 + 11\,200 = 30\,591.05\,\text{cm}^2

Given to 3 significant figures, this is $30\,600\,\text{cm}^2$ .

Key Takeaways

To calculate the area of an isosceles trapezium when only the slant side and parallel lengths are given, use Pythagoras' theorem to find the perpendicular height.
Always check whether a container is closed or open-topped so that irrelevant faces are not added.
The base being a square gives the depth/length of the prism ( $80\,\text{cm}$ ).

Common Mistakes

Adding the top face ( $110 \times 80 = 8800\,\text{cm}^2$ ) when the question clearly states the tank has an "open top".
Using the slant height $70\,\text{cm}$ directly in the trapezium area formula instead of computing the perpendicular height $\sqrt{70^2 - 15^2}$ .
Calculating the base of the right triangle as $(110 - 80) = 30\,\text{cm}$ instead of dividing by $2$ to get $15\,\text{cm}$ .

Things to Be Careful About

Keep unrounded values in your calculator (e.g. $\sqrt{4675}$ ) during intermediate steps to prevent premature rounding errors.

Techniques used

use Pythagoras' theorem to find perpendicular height of an isosceles trapeziumcalculate area of a trapeziumcalculate area of rectangular base and inclined rectangular facessum areas of all external faces of an open-topped prism

Q4MediumNumber

(a)

Anya buys 4 shirts and 3 hats.
She pays $100 and receives $21.50 in change.
Each shirt costs the same amount.
Each hat costs $13.50.

Work out the cost of one shirt.

$ ______

DifficultyMedium-Easy

Worked solution

Approach

Work out the total amount Anya spent, subtract the money spent on hats, and divide the remainder by 4 to find the cost of one shirt.

Working

She pays $100 and receives $21.50 change, so the total spent is:

100 - 21.50 = 78.50

The 3 hats cost:

3 \times 13.50 = 40.50

The 4 shirts together cost:

78.50 - 40.50 = 38.00

So one shirt costs:

38.00 \div 4 = 9.50

Answer

The cost of one shirt is $9.50.

Final answer

$9.50

Detailed explanation

Walkthrough

Anya starts with $100 but gets $21.50 back, so the amount actually spent is found by subtracting the change from the money handed over. The hats are bought in threes at $13.50 each, so multiply $13.50 by 3 to find the total hat cost. Subtract this from the total spent, because the remaining $38.00 must be the total cost of the 4 shirts. Finally divide by 4, since all shirts cost the same amount, to find the cost of one shirt.

Key Takeaways

This question uses a single flow: total paid minus change gives total spent; total spent minus hat cost gives total shirt cost; division by the number of shirts gives one shirt. It also tests keeping money amounts as decimals with two places.

Common Mistakes

Using $100 as the total spent without subtracting the $21.50 change.
Forgetting to multiply the hat price by 3.
Subtracting the hats from $100 or $21.50 instead of from the total spent.
Dividing the whole total spent by 4 instead of dividing after subtracting the hats.
Writing the answer as 9.5 without the dollar sign or with too many decimal places; use $9.50 for money.

Things to Be Careful About

All quantities are in dollars, so no unit conversion is needed. The mark scheme awards M1 for either $100 − $21.50 or 3 × $13.50, and M2 for the combined calculation $100 − $21.50 − 3 × $13.50. Show these intermediate values so both available method marks are earned. Give the final answer as a money amount.

Techniques used

subtract the change from the amount paidmultiply the number of hats by the hat costsubtract the hat total from the total spentdivide the shirt total by the number of shirts

(b)

The exchange rate between dollars ($) and euros (€) is $1 = €0.91.
Anya buys a new camera for $150.
She sees the same camera for sale online for €140.

Calculate the difference between the price in dollars and the price in euros.
Give your answer in dollars, correct to the nearest cent.

$ ______

DifficultyMedium-Easy

Worked solution

Approach

Convert the online euro price into dollars, then subtract Anya's dollar price.

Working

Since $1 = €0.91, dividing an amount in euros by 0.91 converts it to dollars:

140 \div 0.91 = 153.846\ldots

To the nearest cent:

153.846\ldots \approx 153.85

The difference is:

153.85 - 150 = 3.85

Answer

The difference is $3.85.

Final answer

$3.85

Detailed explanation

Walkthrough

The exchange rate is given as $1 = €0.91, which means one dollar is worth 0.91 euros. To change a price in euros into dollars, use the reverse: for every 0.91 euros there is 1 dollar, so divide the euro amount by 0.91. Applying this to €140 gives 153.846... dollars. The question asks for the difference in dollars, so round this converted price to the nearest cent before subtracting, giving $153.85. Subtracting Anya's $150 price leaves $3.85.

Key Takeaways

Exchange rate conversions require dividing by the rate when going from the second currency back to the base currency. Always identify which currency is given and which is required before choosing whether to multiply or divide.

Common Mistakes

Multiplying €140 by 0.91 instead of dividing; this uses the exchange rate in the wrong direction.
Comparing €140 with $150 without converting to the same currency.
Rounding 153.846 to 154 before subtracting, giving $4.00 instead of $3.85; nearest cent rounding should be to two decimal places.
Omitting the dollar sign in the final answer.

Things to Be Careful About

The answer must be correct to the nearest cent, so give it to two decimal places. The mark scheme accepts $3.85 cao, with M1 for the conversion 140 ÷ 0.91. This is a calculator component, so decimal division is expected, but showing the division establishes the method mark.

Techniques used

convert euros to dollars using the exchange rateround to the nearest centsubtract the two dollar amounts

(c)

Anya invests $600 in a savings account.
The account pays compound interest at a rate of $r\%$ per year.
At the end of 3 years the total interest is $21.86.

Calculate the value of $r$ .

$r$ = ______

DifficultyMedium

Worked solution

Approach

Use the compound interest formula. Let $k = 1 + \frac{r}{100}$ , so the amount after 3 years is $600k^3$ . Solve for $k$ , then convert it back to $r$ .

Working

After 3 years the account contains:

600 + 21.86 = 621.86

So

600\left(1 + \frac{r}{100}\right)^3 = 621.86

Let $k = 1 + \frac{r}{100}$ . Then:

600k^3 = 621.86

Divide by 600:

k^3 = \frac{621.86}{600} = 1.036433\ldots

Take the cube root:

k = \sqrt[3]{1.036433\ldots} = 1.01200\ldots

Since $k = 1 + \frac{r}{100}$ :

\frac{r}{100} = 1.01200\ldots - 1 = 0.01200\ldots

r = 100 \times 0.01200\ldots = 1.2

Answer

$r = 1.2$ (the interest rate is 1.2% per year).

Final answer

1.2

Detailed explanation

Walkthrough

The total amount at the end of 3 years is the original $600 plus the $21.86 interest, i.e. $621.86. With compound interest at $r\%$ per year, the multiplier each year is $1 + \frac{r}{100}$ , so after three years the multiplier is $\left(1 + \frac{r}{100}\right)^3$ . Write $600\left(1+\frac{r}{100}\right)^3 = 621.86$ . To find the growth factor, divide both sides by 600; this gives $\left(1+\frac{r}{100}\right)^3 = 1.036433\ldots$ . Taking the cube root recovers $1+\frac{r}{100} = 1.01200\ldots$ , because the money is compounded for 3 years. Subtracting 1 leaves $0.01200\ldots$ , and multiplying by 100 converts this decimal to the percentage rate $r = 1.2$ .

Key Takeaways

Compound interest questions are best solved using the growth multiplier $k = 1 + \frac{r}{100}$ and the formula $A = Pk^n$ . To find an unknown rate, isolate $k^n$ , then take the $n$ th root. The problem also reinforces that interest is added to the principal to find the total amount.

Common Mistakes

Using $21.86 as the amount instead of $621.86.
Taking a square root instead of a cube root, because the investment is for 3 years.
Dividing $21.86 by 3 and treating that as the yearly interest; this ignores compounding.
Giving an answer of 1.012 or 0.012 as $r$ ; $r$ is the percentage, so the growth factor must be converted by subtracting 1 and multiplying by 100.
Stopping at $k = 1.012$ without finding $r$ .

Things to Be Careful About

The mark scheme awards M1 for $600k^3 = 621.86$ and M2 for the cube root $\sqrt[3]{\frac{621.86}{600}}$ ; the final answer $1.2$ is nfww, so all working must be shown. Use the cube root, not a square root, and be careful to use the total amount $621.86 rather than only the interest. Since this is a calculator component, the decimal evaluations are acceptable.

Techniques used

use the compound interest formulaset the growth multiplier as a single variabledivide by the principaltake the cube rootconvert the growth factor to a percentage

Q5MediumProbability

Fig. 4

Mandeep has these 9 number cards.

(a)

She takes one of the 9 cards at random, notes the number and replaces it.

Find the probability that the card shows an odd number.

______

(b)

Mandeep takes one of the 9 cards at random, notes the number and replaces it.
She then takes a second card at random.

Find the probability that both cards show the number 1.

______

DifficultyMedium-Easy

Worked solution

Approach

Since the first card is replaced, the two events are independent. Calculate the probability of drawing a 1 on the first draw, and multiply it by the probability of drawing a 1 on the second draw.

Working

The cards are: 1, 1, 2, 2, 2, 5, 6, 7, 8.

Probability of drawing a 1 on the first draw:
There are two 1s out of 9 cards.

P(\text{1st is 1}) = \frac{2}{9}

Because the card is replaced, the total number of cards remains 9, and there are still two 1s.

Probability of drawing a 1 on the second draw:

P(\text{2nd is 1}) = \frac{2}{9}

Since the events are independent:

P(\text{both are 1}) = \frac{2}{9} \times \frac{2}{9}

P(\text{both are 1}) = \frac{4}{81}

Answer

\frac{4}{81}

Final answer

4/81

Detailed explanation

Walkthrough

Mandeep draws a card, notes the number, and replaces it before drawing a second card. This means the two draws are independent events; the outcome of the first draw does not affect the probabilities of the second draw. First, find the probability of drawing a '1' on the first draw. There are two '1' cards out of nine total cards, so the probability is $\frac{2}{9}$ . Because the card is replaced, the second draw has the exact same conditions: two '1' cards out of nine total cards, giving a probability of $\frac{2}{9}$ . To find the probability of both events happening, multiply the individual probabilities: $\frac{2}{9} \times \frac{2}{9} = \frac{4}{81}$ .

Key Takeaways

When an item is replaced, the probabilities for subsequent draws remain unchanged.
For independent events, multiply their individual probabilities to find the combined probability.

Common Mistakes

Forgetting to replace the card in the denominator for the second probability (using 8 instead of 9).
Adding the probabilities instead of multiplying them.

Things to Be Careful About

The word 'replaces' is crucial; it means the events are independent.
Ensure the fraction $\frac{4}{81}$ cannot be simplified further.

Techniques used

calculate probability of first eventapply replacement to maintain constant probabilitiesmultiply independent probabilities

(c)

Mandeep takes two of the 9 cards at random without replacement.
She calculates the product of the two numbers shown.

Find the probability that the product is less than 5.

______

DifficultyMedium

Worked solution

Approach

Mandeep takes two cards without replacement. List all pairs of cards whose product is less than 5. Calculate the probability for each pair and sum them. Alternatively, count the number of favourable combinations and divide by the total number of combinations.

Working

The cards are: 1, 1, 2, 2, 2, 5, 6, 7, 8.

We need the product of two cards to be less than 5. The possible products less than 5 are 1, 2, and 4. This means the cards must be chosen from the set {1, 1, 2, 2, 2}.

Method 1: Counting combinations
Total number of ways to choose 2 cards from 9:

\frac{9 \times 8}{2} = 36

Favourable outcomes (pairs with product < 5):

Two 1s: $1 \times 1 = 1$ . There is $\binom{2}{2} = 1$ way.
One 1 and one 2: $1 \times 2 = 2$ . There are $2 \times 3 = 6$ ways.
Two 2s: $2 \times 2 = 4$ . There are $\binom{3}{2} = 3$ ways.

Total favourable combinations = $1 + 6 + 3 = 10$ .

P(\text{product} < 5) = \frac{10}{36} = \frac{5}{18}

Method 2: Using probabilities
We can pick two cards from the 5 cards that are 1 or 2. The probability of picking two such cards (without replacement) is:

P = \frac{5}{9} \times \frac{4}{8}

P = \frac{20}{72} = \frac{5}{18}

Alternatively, summing the specific ordered pairs:

P = \left(\frac{2}{9} \times \frac{1}{8}\right) + \left(\frac{2}{9} \times \frac{3}{8}\right) + \left(\frac{3}{9} \times \frac{2}{8}\right) + \left(\frac{3}{9} \times \frac{2}{8}\right)

P = \frac{2}{72} + \frac{6}{72} + \frac{6}{72} + \frac{6}{72} = \frac{20}{72} = \frac{5}{18}

Answer

\frac{5}{18}

Final answer

5/18

Detailed explanation

Walkthrough

Mandeep takes two cards without replacement, so the probabilities change after the first draw. We need the product of the two numbers to be less than 5. Looking at the available numbers (1, 1, 2, 2, 2, 5, 6, 7, 8), any card with a number 5 or greater will make the product 5 or more if paired with any other card (since the smallest other card is 1, and $5 \times 1 = 5$ ). Thus, both cards must be chosen from the five cards that show 1 or 2.

Using the combination approach: there are $\binom{9}{2} = 36$ total ways to pick 2 cards. The number of ways to pick 2 cards from the five cards that are 1 or 2 is $\binom{5}{2} = 10$ . All these 10 pairs have products less than 5 (specifically: $1\times1=1$ , $1\times2=2$ , $2\times2=4$ ). The probability is $\frac{10}{36}$ , which simplifies to $\frac{5}{18}$ .

Using the probability approach: the probability the first card is 1 or 2 is $\frac{5}{9}$ . Given that, there are 4 such cards left out of 8 total cards, so the probability the second card is also 1 or 2 is $\frac{4}{8}$ . Multiplying these gives $\frac{5}{9} \times \frac{4}{8} = \frac{20}{72} = \frac{5}{18}$ .

Key Takeaways

When without replacement, the total number of items decreases by 1 for the second draw.
Identifying the subset of items that satisfy a condition (product < 5) simplifies the calculation.
Combinations can be used to count outcomes efficiently.

Common Mistakes

Forgetting that the events are dependent (without replacement) and using $\frac{4}{9}$ instead of $\frac{4}{8}$ for the second draw.
Missing combinations like $2 \times 2 = 4$ .
Including pairs like $1 \times 5 = 5$ , which is not less than 5.

Things to Be Careful About

'Without replacement' means the denominator for the second probability is 8, not 9.
The inequality is strict ( $< 5$ ), so a product of exactly 5 is not included.
Ensure fractions are simplified to their lowest terms.

Techniques used

identify combinations that yield a product less than 5calculate probabilities without replacementsum the probabilities of mutually exclusive outcomes

Q6MediumAlgebra and Graphs

(a)

Complete the table for $y = \frac{x}{4}(2x^2 - x - 10)$ .

$x$	$-3$	$-2$	$-1$	0	1	2	3
$y$		0	1.75	0	$-2.25$	$-2$	3.75

(b)

Draw the graph of $y = \frac{x}{4}(2x^2 - x - 10)$ for $-3 \leq x \leq 3$ .

Fig. 5

DifficultyMedium-Easy

Worked solution

Approach

Plot all seven points from the completed table on the given grid, then join them with a smooth curve appropriate to a cubic function.

Working

The completed table is:

$x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$
$y$	$-8.25$	$0$	$1.75$	$0$	$-2.25$	$-2$	$3.75$

Answer

A smooth cubic curve passing through $(-3, -8.25)$ , $(-2, 0)$ , $(-1, 1.75)$ , $(0, 0)$ , $(1, -2.25)$ , $(2, -2)$ , $(3, 3.75)$ , with a local maximum near $(-1.1, 1.8)$ and a local minimum near $(1.5, -2.6)$ .

Final answer

Smooth cubic curve through (-3, -8.25), (-2, 0), (-1, 1.75), (0, 0), (1, -2.25), (2, -2), (3, 3.75) with local max near (-1.1, 1.8) and local min near (1.5, -2.6)

Detailed explanation

Walkthrough

The student has already computed the $y$ -value at $x = -3$ in part (a). All seven points are now known:

$(-3, -8.25)$
$(-2, 0)$
$(-1, 1.75)$
$(0, 0)$
$(1, -2.25)$
$(2, -2)$
$(3, 3.75)$

Plot each point on the grid. Then join them with a smooth curve. Since the function is a cubic $y = \frac{1}{2}x^3 - \frac{1}{4}x^2 - \frac{5}{2}x$ with a positive leading coefficient, the curve rises to the right and falls to the left. It has a local maximum (between $x = -2$ and $x = 0$ ) and a local minimum (between $x = 0$ and $x = 3$ ). Do NOT join the points with straight line segments — a cubic is a smooth curve.

Key Takeaways

When graphing a polynomial, always plot several points and join them with a smooth curve. The end behaviour of a cubic with positive leading coefficient is: $y \to -\infty$ as $x \to -\infty$ and $y \to +\infty$ as $x \to +\infty$ .

Common Mistakes

Joining points with straight lines instead of a smooth curve.
Misplotting $(-3, -8.25)$ — it is near the bottom of the grid.
Not recognising the turning points; the curve should clearly show a peak near $x \approx -1.1$ and a valley near $x \approx 1.5$ .
Forgetting that the curve is continuous — do not break it at the $x$ -intercepts.

Things to Be Careful About

The grid has $x$ -axis from $-3$ to $3$ and $y$ -axis from $-10$ to $6$ , with gridlines every $0.2$ on $x$ and every $0.5$ on $y$ . Read coordinates carefully. The mark scheme awards marks for correctly plotted points (B2FT for 6 or 7 correct, B1FT for 4 or 5) and then a mark for the smooth curve.

Techniques used

plot the seven table points on the griddraw a smooth cubic curve through the points

(c)

The equation $\frac{x}{4}(2x^2 - x - 10) = k$ has exactly two solutions.

Use your graph to find the possible values of $k$ .

$k$ = ______ or $k$ = ______

DifficultyMedium

Worked solution

Approach

The equation $\frac{x}{4}(2x^2 - x - 10) = k$ is equivalent to finding where the horizontal line $y = k$ meets the graph of $y = \frac{x}{4}(2x^2 - x - 10)$ . A horizontal line meets a cubic in exactly two points only when it passes through a turning point (local maximum or local minimum). So $k$ must equal the $y$ -value at the local maximum or the $y$ -value at the local minimum.

Working

From the graph drawn in part (b):

The local maximum occurs near $x \approx -1.1$ , where $y \approx 1.8$ .
The local minimum occurs near $x \approx 1.5$ , where $y \approx -2.6$ .

Reading from the graph:

k \approx 1.8 \quad \text{or} \quad k \approx -2.6

(Values read from the student's own graph are accepted within a reasonable tolerance.)

Answer

k \approx 1.8 \quad \text{or} \quad k \approx -2.6

Final answer

k ≈ 1.8 or k ≈ -2.6

Detailed explanation

Walkthrough

The equation $\frac{x}{4}(2x^2 - x - 10) = k$ asks: for which values of $k$ does the horizontal line $y = k$ intersect the cubic graph in exactly two points?

A horizontal line generally meets a cubic in 1 or 3 points. It meets in exactly 2 points only when it is tangent to the curve at a turning point — that is, when $k$ equals the $y$ -value at a local maximum or a local minimum. At these values, one of the three intersection points merges into a double root.

From the graph drawn in part (b), identify the highest point of the dip (local max) and the lowest point of the rise (local min):

Local maximum: near $x = -1.1$ , the graph peaks at approximately $y = 1.8$ .
Local minimum: near $x = 1.5$ , the graph bottoms out at approximately $y = -2.6$ .

So $k = 1.8$ or $k = -2.6$ (reading from the student's graph; the mark scheme allows a tolerance based on what the student actually plotted).

Key Takeaways

A horizontal line $y = k$ intersects a cubic in exactly two points if and only if $k$ equals the $y$ -value at a turning point. This is a key graphical interpretation of turning points.

Common Mistakes

Reading the $x$ -coordinate of the turning point instead of the $y$ -coordinate. The question asks for $k$ , which is a $y$ -value.
Giving only one value when two are required (both the max and min must be found).
Not reading from their own graph — the mark scheme follows through from the student's drawn curve.

Things to Be Careful About

The answer must be read from the student's own graph (strict follow-through). The mark scheme awards B1 for each turning point value, dependent on a correctly drawn positive cubic with visible max and min. Typical expected readings are approximately $k \approx 1.8$ and $k \approx -2.6$ .

Techniques used

recognise that exactly two solutions means the horizontal line touches a turning pointread the y-values at the local maximum and local minimum from the graph

(d)

By drawing a suitable line on the grid, find the solutions of $2x^3 - x^2 - 10x = 2x - 4$ .

$x$ = ______ , $x$ = ______ , $x$ = ______

DifficultyMedium

Worked solution

Approach

We need to solve $2x^3 - x^2 - 10x = 2x - 4$ . Divide both sides by $4$ to make the left-hand side match the function from the graph:

\frac{2x^3 - x^2 - 10x}{4} = \frac{2x - 4}{4}

\frac{x}{4}(2x^2 - x - 10) = \frac{x}{2} - 1

So we draw the line $y = \frac{x}{2} - 1$ on the same grid and read off the $x$ -coordinates where it intersects the cubic.

Working

The line $y = \frac{x}{2} - 1$ passes through:

$(0, -1)$
$(2, 0)$
$(-2, -2)$

Draw this straight line on the grid.

The line intersects the cubic at three points. Reading the $x$ -coordinates from the graph:

x \approx -2.5, \quad x \approx 0.3, \quad x \approx 2.5

(More precisely: $x \in [-2.6, -2.3]$ , $x \in [0.2, 0.4]$ , $x \in [2.3, 2.7]$ .)

Answer

x \approx -2.5, \quad x \approx 0.3, \quad x \approx 2.5

Final answer

x ≈ -2.5, x ≈ 0.3, x ≈ 2.5

Detailed explanation

Walkthrough

The equation to solve is $2x^3 - x^2 - 10x = 2x - 4$ . We want to use the graph of $y = \frac{x}{4}(2x^2 - x - 10)$ , so we rearrange the equation to make the left-hand side match this expression.

Divide both sides by $4$ :

\frac{2x^3 - x^2 - 10x}{4} = \frac{2x - 4}{4}

Factor the left side:

\frac{x(2x^2 - x - 10)}{4} = \frac{x}{2} - 1

This is $y = \frac{x}{2} - 1$ on the right-hand side. So we draw the straight line $y = \frac{x}{2} - 1$ on the same grid as the cubic and find where they intersect.

The line $y = \frac{x}{2} - 1$ has gradient $\frac{1}{2}$ and $y$ -intercept $-1$ . Key points on the line within the grid:

When $x = 0$ , $y = -1$ → point $(0, -1)$
When $x = 2$ , $y = 0$ → point $(2, 0)$
When $x = -2$ , $y = -2$ → point $(-2, -2)$
When $x = -3$ , $y = -2.5$ → point $(-3, -2.5)$
When $x = 3$ , $y = 0.5$ → point $(3, 0.5)$

Draw this line. It crosses the cubic at three points. Reading the $x$ -coordinates:

Left intersection: $x \approx -2.5$ (acceptable range $-2.6$ to $-2.3$ )
Middle intersection: $x \approx 0.3$ (acceptable range $0.2$ to $0.4$ )
Right intersection: $x \approx 2.5$ (acceptable range $2.3$ to $2.7$ )

Key Takeaways

To solve an equation graphically using a given curve, rearrange it so one side matches the curve's expression. The other side then becomes a line (or another curve) to draw. The solutions are the $x$ -coordinates of the intersection points.

Common Mistakes

Not dividing by 4 correctly, leading to the wrong line equation.
Drawing the line $y = 2x - 4$ instead of $y = \frac{x}{2} - 1$ (forgetting to divide both sides by 4).
Reading the $y$ -coordinates of intersections instead of the $x$ -coordinates.
Only finding two intersections when there are three.
Not showing the line drawn on the grid (the mark scheme requires M2 for the correct line).

Things to Be Careful About

The mark scheme awards M2 for drawing the correct line $y = \frac{x}{2} - 1$ (or equivalent forms like $y = px - 1$ with $p \neq 0$ ). A1 is awarded for two correct solutions (dependent on at least M1 being earned). After M0, SC1 is awarded for all three correct solutions. The acceptable ranges for the three $x$ -values are approximately $-2.6$ to $-2.3$ , $0.2$ to $0.4$ , and $2.3$ to $2.7$ . Working must be shown (www/nfww implied by the mark scheme structure).

Techniques used

rearrange the equation to match y = x/4(2x^2 - x - 10) on one sideidentify the straight line to draw on the graphread the x-coordinates of the intersections

Q7MediumAlgebra and Graphs

(a)

Solve.

4x + 7 = 16

$x$ = ______

(b)

Solve.

5(4 - y) = 30

$y$ = ______

DifficultyEasy

Worked solution

Approach

First, expand the bracket to remove the parentheses. Then, treat $y$ as the unknown variable and solve the resulting linear equation.

Working

Expand the bracket in the equation:

5(4 - y) = 30

Multiply the terms inside the bracket by $5$ :

20 - 5y = 30

Subtract $20$ from both sides:

-5y = 30 - 20

-5y = 10

Divide by $-5$ :

y = \frac{10}{-5}

y = -2

Alternative Method:
Divide both sides by $5$ first:

4 - y = \frac{30}{5}

4 - y = 6

Rearrange to solve for $y$ :

-y = 6 - 4

-y = 2 \implies y = -2

Answer

y = -2

Final answer

-2

Detailed explanation

Walkthrough

The equation is $5(4 - y) = 30$ . We can solve this by expanding the bracket first: $5 \times 4 = 20$ and $5 \times (-y) = -5y$ , giving $20 - 5y = 30$ . Then subtract $20$ from both sides to get $-5y = 10$ . Finally, divide by $-5$ to find $y = -2$ . Alternatively, noticing that $30$ is divisible by $5$ , we can divide the entire equation by $5$ immediately to get $4 - y = 6$ , which is often quicker.

Key Takeaways

Expanding brackets is a standard first step when variables are inside parentheses. Remember that multiplying a negative term (like $-y$ ) by a positive number results in a negative term.

Common Mistakes

Expanding incorrectly, e.g., writing $20 - y$ instead of $20 - 5y$ .
Sign errors when rearranging, such as thinking $4 - y = 6$ implies $y = 2$ (forgetting the negative sign).
Adding $5y$ to both sides but messing up the signs on the other side.

Things to Be Careful About

Watch out for negative signs. A common trap is forgetting that $-y = 2$ means $y = -2$ . The method mark (M1) is awarded for correct expansion or isolation steps like $4-y=6$ .

Techniques used

expand bracketsisolate the variable

(c)

Write down all the integers that satisfy this inequality.

-\frac{3}{2} \leq x < 3

______

DifficultyMedium-Easy

Worked solution

Approach

We need to list all integers $x$ that fall strictly between $-\frac{3}{2}$ and $3$ . Note the inequality symbols: $\leq$ means "greater than or equal to" and $<$ means "strictly less than".

Working

Convert the lower bound to a decimal or mixed number to make it easier to visualize on a number line:

-\frac{3}{2} = -1.5

The inequality is:

-1.5 \leq x < 3

This means $x$ must be greater than or equal to $-1.5$ AND strictly less than $3$ .

Let's check integers starting from the lower bound:

Is $-2$ included? No, because $-2 < -1.5$ .
Is $-1$ included? Yes, because $-1.5 \leq -1$ .
Is $0$ included? Yes.
Is $1$ included? Yes.
Is $2$ included? Yes.
Is $3$ included? No, because the inequality is $x < 3$ (strictly less than), so $3$ is excluded.

The integers are $-1, 0, 1, 2$ .

Answer

-1, 0, 1, 2

Final answer

-1, 0, 1, 2

Detailed explanation

Walkthrough

The problem asks for integers satisfying $-\frac{3}{2} \leq x < 3$ . First, recognize that $-\frac{3}{2}$ is $-1.5$ . The symbol $\leq$ includes the boundary, so any integer greater than $-1.5$ works. The first integer after $-1.5$ is $-1$ . The upper bound is $3$ , but the symbol $<$ excludes $3$ . So we count up from $-1$ until we reach $2$ . The list is $-1, 0, 1, 2$ .

Key Takeaways

Always convert fractions to decimals or visualize them on a number line to avoid confusion with negative numbers. Pay close attention to whether the inequality is strict ( $<$ or $>$ ) or inclusive ( $\leq$ or $\geq$ ).

Common Mistakes

Including $3$ in the answer set (the most common error due to missing the strict inequality).
Starting at $-2$ (thinking $-2$ is greater than $-1.5$ ).
Listing non-integers or skipping numbers.

Things to Be Careful About

The marking scheme awards partial credit (B1) for getting 3 out of 4 correct with no extras, or 4 correct with 1 extra. However, aiming for the exact set $-1, 0, 1, 2$ is necessary for full marks. Ensure you write "integers" clearly if the context wasn't already specified in the question stem, though here it is.

Techniques used

interpret inequality boundsidentify integers within a range

(d)

Rearrange the formula $y = \frac{4y - x}{3x}$ to make $x$ the subject.

$x$ = ______

DifficultyMedium

Worked solution

Approach

To make $x$ the subject, we must isolate $x$ on one side of the equation. Since $x$ appears in the numerator and denominator, we first eliminate the fraction by cross-multiplying.

Working

Given:

y = \frac{4y - x}{3x}

Cross-multiply (Method Mark M1 - elimination of fraction):

y(3x) = 4y - x

3xy = 4y - x

Group all terms containing $x$ on one side. Add $x$ to both sides (Method Mark M1 - isolation of terms):

3xy + x = 4y

Factorise $x$ out of the left-hand side (Method Mark M1 - factorising):

x(3y + 1) = 4y

Divide by $(3y + 1)$ to complete the rearrangement:

x = \frac{4y}{3y + 1}

Answer

x = \frac{4y}{3y + 1}

Final answer

4y/(3y+1)

Detailed explanation

Walkthrough

We start with $y = \frac{4y - x}{3x}$ . To get rid of the fraction, multiply both sides by the denominator $3x$ . This gives $3xy = 4y - x$ . Now, look at where $x$ is. It is multiplied by $3y$ on the left, and it is subtracted on the right. We need to bring them together. Add $x$ to both sides: $3xy + x = 4y$ . Now factor out $x$ : $x(3y + 1) = 4y$ . Finally, divide by the bracket $(3y + 1)$ to leave $x$ alone.

Key Takeaways

When the subject variable appears on both sides of an equation (even implicitly in different terms), you must group those terms together before you can isolate the variable. Factorisation is the key step to pull the variable out.

Common Mistakes

Forgetting to distribute the $y$ to both terms in the numerator during cross-multiplication (e.g., writing $3x = 4y - x$ ).
Moving $x$ to the wrong side and getting stuck.
Failing to factorise $3xy + x$ as $x(3y+1)$ , leaving it as $x = 4y - 3xy$ .

Things to Be Careful About

Check your final answer by substituting simple values back into the original and rearranged formulas if possible. The mark scheme requires three specific method marks: eliminating the fraction, isolating $x$ terms, and factorising/completing. All three must be shown.

Techniques used

eliminate the denominatorgroup terms containing the new subjectfactorise to isolate the subject

(e)

Simplify.

\frac{12x^2 - 3y^2}{2x^2 + 8x - xy - 4y}

______

DifficultyMedium

Worked solution

Approach

To simplify the algebraic fraction, we must factorise both the numerator and the denominator completely, then cancel any common factors.

Working

Numerator: $12x^2 - 3y^2$
First, take out the common factor of $3$ :

3(4x^2 - y^2)

Recognize $4x^2 - y^2$ as a difference of two squares ( $a^2 - b^2 = (a-b)(a+b)$ ):

3(2x - y)(2x + y)

Denominator: $2x^2 + 8x - xy - 4y$
Factorise by grouping. Group the first two terms and the last two terms:

(2x^2 + 8x) - (xy + 4y)

Factor out $2x$ from the first group and $y$ from the second:

2x(x + 4) - y(x + 4)

Now factor out the common binomial $(x + 4)$ :

(x + 4)(2x - y)

Combine and Cancel:

\frac{3(2x - y)(2x + y)}{(x + 4)(2x - y)}

Cancel the common factor $(2x - y)$ :

\frac{3(2x + y)}{x + 4}

Expand the numerator if required (optional, but equivalent):

\frac{6x + 3y}{x + 4}

Answer

\frac{6x + 3y}{x + 4}

Final answer

(6x+3y)/(x+4)

Detailed explanation

Walkthrough

The expression is a complex fraction. Step 1 is to factorise the top part (numerator). Notice both terms share a $3$ . Pull it out: $3(4x^2 - y^2)$ . Inside the bracket, we have a square minus a square, which splits into $(2x-y)(2x+y)$ . Step 2 is to factorise the bottom part (denominator). There are four terms, suggesting 'factorising by grouping'. Split it into $(2x^2+8x)$ and $(-xy-4y)$ . Factor out $2x$ from the first pair and $-y$ from the second. You will see $(x+4)$ appearing in both, allowing you to write $(x+4)(2x-y)$ . Step 3 is to cancel $(2x-y)$ from the top and bottom.

Key Takeaways

Always look for a highest common factor (HCF) before attempting more complex factorisation methods. Recognizing the 'difference of two squares' pattern is crucial for quick simplification. For four-term denominators, grouping is the standard technique.

Common Mistakes

Missing the HCF of $3$ in the numerator.
Incorrectly factorising $4x^2 - y^2$ as $(4x-y)(x+y)$ or similar.
Sign errors in the denominator grouping (e.g., factoring out $y$ instead of $-y$ leading to mismatched brackets).
Cancelling terms that are not factors (e.g., cancelling $2x$ from $2x^2$ and $8x$ individually).

Things to Be Careful About

The mark scheme offers partial credit (B1/B2) for seeing correct partial factorisations. Make sure your working clearly shows the factorised forms of the numerator and denominator separately before cancelling. The final answer must be fully simplified.

Techniques used

factorise the numerator by grouping or difference of two squaresfactorise the denominator by groupingcancel common factors

Q8Medium-EasyGeometryTrigonometryMensuration

Fig. 6

$ADB$ and $AEC$ are straight lines.
$BC$ is parallel to $DE$ .
$BC = 9.8\,\text{cm}$ , $BD = 2.7\,\text{cm}$ , $DE = 5.6\,\text{cm}$ and $CE = 3.9\,\text{cm}$ .

(a)

Complete the missing angles and reasons to show that triangle $ABC$ is similar to triangle $ADE$ .

In triangle $ABC$ and triangle $ADE$ ,

angle $BAC$ = angle ______ because common angle

angle $ABC$ = angle ______ because ______

angle $ACB$ = angle ______ because ______

As the three pairs of angles are equal, triangle $ABC$ is similar to triangle $ADE$ .

DifficultyEasy

Worked solution

Approach

Use the properties of parallel lines cut by a transversal and the fact that the two triangles share a vertex to identify the three pairs of equal angles.

Working

In triangle $ABC$ and triangle $ADE$ :

\angle BAC = \angle DAE

because it is the common angle.

Since $BC$ is parallel to $DE$ and $AB$ is a transversal:

\angle ABC = \angle ADE

because they are corresponding angles.

Since $BC$ is parallel to $DE$ and $AC$ is a transversal:

\angle ACB = \angle AED

because they are corresponding angles.

As all three pairs of angles are equal, triangle $ABC$ is similar to triangle $ADE$ by AAA.

Answer

angle $BAC$ = angle $DAE$ because common angle

angle $ABC$ = angle $ADE$ because corresponding angles

angle $ACB$ = angle $AED$ because corresponding angles

Final answer

DAE; ADE, corresponding angles; AED, corresponding angles

Detailed explanation

Walkthrough

To prove two triangles are similar, we need to show that their corresponding angles are equal. The first angle is obvious: both triangles share the angle at vertex $A$ , so $\angle BAC = \angle DAE$ .

For the other two angles, we use the fact that $BC$ is parallel to $DE$ . When a transversal line crosses two parallel lines, the corresponding angles are equal. Here, $AB$ acts as a transversal crossing $BC$ and $DE$ , making $\angle ABC$ and $\angle ADE$ corresponding angles. Similarly, $AC$ acts as a transversal, making $\angle ACB$ and $\angle AED$ corresponding angles.

With all three pairs of angles matching, the triangles are similar by the Angle-Angle-Angle (AAA) criterion.

Key Takeaways

Parallel lines cut by a transversal create equal corresponding angles.
Triangles that share a vertex and have parallel bases are similar.
Similarity can be proven using the AAA (Angle-Angle-Angle) criterion.

Common Mistakes

Writing the wrong angle for the common angle (e.g., writing $BAC$ instead of $DAE$ for the second blank).
Confusing corresponding angles with alternate interior angles or co-interior angles. Remember, corresponding angles are in the same relative position at each intersection.
Forgetting to write the reason "corresponding angles" or writing an incomplete reason.

Things to Be Careful About

Ensure the angles are written in the correct order to match the vertices of the similar triangles ( $ABC \sim ADE$ ).
The reasons must be exact: "common angle" and "corresponding angles".

Techniques used

identify the common angleidentify corresponding angles from parallel linesestablish similarity by AAA

(b)

Show that $AD = 3.6\,\text{cm}$ and $AE = 5.2\,\text{cm}$ .

DifficultyMedium-Easy

Worked solution

Approach

Since triangle $ABC$ is similar to triangle $ADE$ , the ratios of their corresponding sides are equal. Use this property to set up an equation for $AD$ and another for $AE$ , then solve each.

Working

The sides of triangle $ABC$ are $AB$ , $AC$ , and $BC$ . The corresponding sides of triangle $ADE$ are $AD$ , $AE$ , and $DE$ .

We are given:
$BC = 9.8\,\text{cm}$ , $BD = 2.7\,\text{cm}$ , $DE = 5.6\,\text{cm}$ , $CE = 3.9\,\text{cm}$ .

Since $D$ lies on $AB$ , $AB = AD + BD = AD + 2.7$ .
Since $E$ lies on $AC$ , $AC = AE + CE = AE + 3.9$ .

Using the ratio of corresponding sides $\frac{AD}{AB} = \frac{DE}{BC}$ :

\frac{AD}{AD + 2.7} = \frac{5.6}{9.8}

Cross-multiply to solve for $AD$ :

9.8 \times AD = 5.6 \times (AD + 2.7)

9.8 \, AD = 5.6 \, AD + 15.12

9.8 \, AD - 5.6 \, AD = 15.12

4.2 \, AD = 15.12

AD = \frac{15.12}{4.2} = 3.6\,\text{cm}

Similarly, using the ratio $\frac{AE}{AC} = \frac{DE}{BC}$ :

\frac{AE}{AE + 3.9} = \frac{5.6}{9.8}

Cross-multiply to solve for $AE$ :

9.8 \times AE = 5.6 \times (AE + 3.9)

9.8 \, AE = 5.6 \, AE + 21.84

4.2 \, AE = 21.84

AE = \frac{21.84}{4.2} = 5.2\,\text{cm}

Answer

$AD = 3.6\,\text{cm}$ and $AE = 5.2\,\text{cm}$

Final answer

AD = 3.6 cm, AE = 5.2 cm

Detailed explanation

Walkthrough

Because the triangles are similar, the ratio of any two corresponding sides is constant. We can equate the ratio of the smaller triangle's side to the larger triangle's corresponding side.

For $AD$ : The side $AD$ in the small triangle corresponds to side $AB$ in the large triangle. We know $AB = AD + 2.7$ and $DE = 5.6$ , $BC = 9.8$ . Setting up the proportion $\frac{AD}{AD + 2.7} = \frac{5.6}{9.8}$ allows us to solve for $AD$ .

Cross-multiplying gives $9.8 \cdot AD = 5.6 \cdot AD + 15.12$ . Subtracting $5.6 \cdot AD$ from both sides leaves $4.2 \cdot AD = 15.12$ , so $AD = 3.6$ .

For $AE$ : The side $AE$ corresponds to $AC = AE + 3.9$ . Setting up $\frac{AE}{AE + 3.9} = \frac{5.6}{9.8}$ and solving gives $9.8 \cdot AE = 5.6 \cdot AE + 21.84$ , so $4.2 \cdot AE = 21.84$ , meaning $AE = 5.2$ .

Key Takeaways

In similar triangles, the ratio of corresponding sides is equal.
When a side of the larger shape is split into two parts, express the full side as the sum of those parts before setting up the ratio.
Cross-multiplication and algebraic rearrangement are needed to solve for the unknown segment length.

Common Mistakes

Using the wrong side in the ratio (e.g., $\frac{AD}{BD}$ instead of $\frac{AD}{AB}$ ).
Forgetting to add the segment lengths to get the full side length of the larger triangle.
Arithmetic errors when cross-multiplying or dividing.

Things to Be Careful About

Ensure all lengths are in the same units (they are all in cm here).
Show clear algebraic steps for eliminating the fraction, as marks are awarded for the correct rearrangement leading to the answer.

Techniques used

set up proportion from similar trianglessolve linear equation for side length

(c)

Calculate angle $DAE$ .

Angle $DAE$ = ______

DifficultyMedium-Easy

Worked solution

Approach

In triangle $ADE$ , all three side lengths are now known: $AD = 3.6\,\text{cm}$ , $AE = 5.2\,\text{cm}$ , and $DE = 5.6\,\text{cm}$ . Use the cosine rule to calculate angle $DAE$ .

Working

The cosine rule states:

c^2 = a^2 + b^2 - 2ab \cos C

Rearranging to solve for $\cos C$ :

\cos C = \frac{a^2 + b^2 - c^2}{2ab}

Substitute the values for triangle $ADE$ , where we want to find $\angle DAE$ :

\cos(\angle DAE) = \frac{AD^2 + AE^2 - DE^2}{2 \times AD \times AE}

\cos(\angle DAE) = \frac{3.6^2 + 5.2^2 - 5.6^2}{2 \times 3.6 \times 5.2}

\cos(\angle DAE) = \frac{12.96 + 27.04 - 31.36}{37.44}

\cos(\angle DAE) = \frac{8.64}{37.44}

\cos(\angle DAE) \approx 0.230769

\angle DAE = \cos^{-1}(0.230769) \approx 76.659^\circ

Rounding to 3 significant figures:

\angle DAE = 76.7^\circ

Answer

Angle $DAE$ = $76.7^\circ$

Final answer

76.7 degrees

Detailed explanation

Walkthrough

We have a triangle with all three sides known ( $3.6$ , $5.2$ , $5.6$ ). When you know three sides and need to find an angle, the cosine rule is the appropriate tool.

The formula is $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$ , where $a$ is the side opposite the angle you want to find. Here, the side opposite $\angle DAE$ is $DE = 5.6$ . The adjacent sides are $AD = 3.6$ and $AE = 5.2$ .

Plugging in: $\cos(\angle DAE) = \frac{3.6^2 + 5.2^2 - 5.6^2}{2 \times 3.6 \times 5.2}$ . Calculating the numerator gives $12.96 + 27.04 - 31.36 = 8.64$ . The denominator is $37.44$ . Dividing gives approximately $0.23077$ . Taking the inverse cosine yields $76.659^\circ$ , which rounds to $76.7^\circ$ .

Key Takeaways

The cosine rule relates the lengths of the sides of a triangle to the cosine of one of its angles.
Use the form $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$ when finding an angle from three known sides.
Always identify the side opposite the angle you are finding and place it as the negative term in the numerator.

Common Mistakes

Using the wrong side as the negative term in the numerator (e.g., subtracting $5.2^2$ instead of $5.6^2$ ).
Forgetting to use the inverse cosine function ( $\cos^{-1}$ ) after finding the cosine value.
Rounding too early in the calculation, which can lead to an incorrect final angle.

Things to Be Careful About

The question asks for the angle, so you must use $\cos^{-1}$ , not just leave the cosine value.
Respect the accuracy demand: 3 significant figures is standard unless otherwise stated, giving $76.7^\circ$ .

Techniques used

apply cosine rule to find angle

(d)

Calculate the area of triangle $ABC$ .

______ $\text{cm}^2$

DifficultyMedium-Easy

Worked solution

Approach

Find the full side lengths $AB$ and $AC$ of triangle $ABC$ , then use the sine area formula $\text{Area} = \frac{1}{2}ab\sin C$ with the angle $\angle BAC$ (which is the same as $\angle DAE$ ) calculated in part (c).

Working

First, calculate the side lengths of triangle $ABC$ :

AB = AD + BD = 3.6 + 2.7 = 6.3\,\text{cm}

AC = AE + CE = 5.2 + 3.9 = 9.1\,\text{cm}

The angle $\angle BAC$ is the same as $\angle DAE$ , so $\angle BAC \approx 76.659^\circ$ .

Using the area formula for a triangle given two sides and the included angle:

\text{Area} = \frac{1}{2} \times AB \times AC \times \sin(\angle BAC)

\text{Area} = \frac{1}{2} \times 6.3 \times 9.1 \times \sin(76.659^\circ)

\text{Area} = 28.665 \times 0.973205

\text{Area} \approx 27.897\,\text{cm}^2

Rounding to 3 significant figures:

\text{Area} = 27.9\,\text{cm}^2

Answer

$27.9\,\text{cm}^2$

Final answer

27.9 cm^2

Detailed explanation

Walkthrough

To find the area of triangle $ABC$ , we can use the formula $\text{Area} = \frac{1}{2}ab\sin C$ , where $a$ and $b$ are two sides and $C$ is the included angle.

We already know $\angle BAC = 76.659^\circ$ from part (c). The sides forming this angle are $AB$ and $AC$ . From part (b), we know $AD = 3.6$ and $BD = 2.7$ , so $AB = 6.3$ . Similarly, $AE = 5.2$ and $CE = 3.9$ , so $AC = 9.1$ .

Substituting into the formula: $\text{Area} = \frac{1}{2} \times 6.3 \times 9.1 \times \sin(76.659^\circ)$ . This gives approximately $27.897$ , which rounds to $27.9\,\text{cm}^2$ .

Key Takeaways

The area of a triangle can be calculated using two sides and the sine of the included angle: $\frac{1}{2}ab\sin C$ .
Always ensure you are using the full side lengths of the triangle in question, not just the segments from the smaller similar triangle.
The angle at the common vertex is the same for both similar triangles.

Common Mistakes

Using the sides of the smaller triangle ( $AD$ and $AE$ ) instead of the larger triangle ( $AB$ and $AC$ ), which would give the area of triangle $ADE$ ( $9.11\,\text{cm}^2$ ).
Forgetting to add the segment lengths to get the full side lengths $AB$ and $AC$ .
Rounding the angle too early before taking the sine, which can cause a slight discrepancy in the final area.

Things to Be Careful About

Use the unrounded value of the angle ( $76.659^\circ$ ) when calculating the sine to maintain accuracy.
The final answer should be given to 3 significant figures as per standard accuracy demands, yielding $27.9\,\text{cm}^2$ .

Techniques used

calculate side lengths of larger triangleapply area formula using sine of included angle

Q9MediumGeometryTrigonometryMensuration

(a)

Fig. 7

$A, B, C$ and $D$ are points on a circle, centre $O$ .
$AOD$ and $OCE$ are straight lines.
$DE$ is a tangent to the circle at $D$ .
Angle $ABC = 126°$ and $DE = 10.5\,\text{cm}$ .

Calculate the radius of the circle.

______ $\text{cm}$

5M Similar

DifficultyMedium

Worked solution

Approach

Use circle theorems to find the angle $\angle DOC$ and confirm that $\angle ODE = 90^\circ$ . Then apply the tangent ratio in the right-angled triangle $ODE$ to calculate the radius $r$ .

Working

Since $ABCD$ is a cyclic quadrilateral, opposite angles sum to $180^\circ$ :

\angle ADC = 180^\circ - \angle ABC = 180^\circ - 126^\circ = 54^\circ

Since $AOD$ is a straight line passing through the centre $O$ , $\angle ODC = \angle ADC = 54^\circ$ . In $\triangle OCD$ , $OC = OD = r$ , so it is isosceles and $\angle OCD = \angle ODC = 54^\circ$ . The angle at the centre is:

\angle DOC = 180^\circ - 54^\circ - 54^\circ = 72^\circ

Alternatively, the reflex angle at the centre is $2 \times 126^\circ = 252^\circ$ , giving interior $\angle AOC = 360^\circ - 252^\circ = 108^\circ$ , and since $AOD$ is a straight line, $\angle DOC = 180^\circ - 108^\circ = 72^\circ$ .

Since $DE$ is a tangent to the circle at $D$ , the radius $OD$ is perpendicular to the tangent:

\angle ODE = 90^\circ

In right-angled triangle $ODE$ , $\angle DOE = \angle DOC = 72^\circ$ (as $OCE$ is a straight line). Using the tangent ratio:

\tan(72^\circ) = \frac{DE}{OD} = \frac{10.5}{r}

Solving for $r$ :

r = \frac{10.5}{\tan(72^\circ)} = \frac{10.5}{3.07768...} \approx 3.4117

Answer

r = 3.41 \text{ cm}

Final answer

3.41

Detailed explanation

Walkthrough

First, recognise that $ABCD$ is a cyclic quadrilateral, so its opposite angles $\angle ABC$ and $\angle ADC$ add up to $180^\circ$ . This gives $\angle ADC = 54^\circ$ . Since $O$ lies on $AD$ , $\angle ODC$ is the same angle. Triangle $OCD$ is isosceles because $OC$ and $OD$ are both radii, so the base angles are equal ( $54^\circ$ each), leaving $\angle DOC = 72^\circ$ . Alternatively, use the angle at the centre theorem: the reflex angle $AOC$ is twice $\angle ABC$ ( $252^\circ$ ), so interior $\angle AOC = 108^\circ$ , and $\angle DOC = 180^\circ - 108^\circ = 72^\circ$ .

Next, apply the tangent-radius theorem: $DE$ is tangent at $D$ , so $\angle ODE = 90^\circ$ . Since $OCE$ is a straight line, $\angle DOE = \angle DOC = 72^\circ$ . In the right-angled triangle $ODE$ , the tangent ratio relates the opposite side $DE$ ( $10.5$ cm) to the adjacent side $OD$ (the radius $r$ ). Rearranging $\tan(72^\circ) = 10.5 / r$ gives $r = 10.5 / \tan(72^\circ) \approx 3.41$ cm.

Key Takeaways

Opposite angles in a cyclic quadrilateral sum to $180^\circ$ .
The angle at the centre is twice the angle at the circumference on the same arc.
A tangent to a circle is perpendicular to the radius at the point of tangency.
Trigonometric ratios in right-angled triangles can be used to find unknown lengths once angles are established via circle theorems.

Common Mistakes

Forgetting that opposite angles in a cyclic quadrilateral sum to $180^\circ$ instead of being equal.
Using the wrong angle in the tangent ratio (e.g., using $126^\circ$ or $54^\circ$ instead of $72^\circ$ ).
Forgetting that the tangent is perpendicular to the radius, leading to incorrect triangle setup.
Rounding the radius too early before calculating the final answer.

Things to Be Careful About

The answer must be given to 3 significant figures as is standard unless stated otherwise.
Ensure the angle used in the trigonometric ratio is the correct one inside the right-angled triangle ( $72^\circ$ , not $126^\circ$ or $54^\circ$ ).
Working must be shown to earn method marks; simply writing the answer scores no marks.

Techniques used

apply cyclic quadrilateral opposite angles propertyapply angle at centre theoremapply tangent perpendicular to radius propertyuse tangent ratio in right-angled triangle

(b)

Fig. 8

$P$ and $Q$ are points on a different circle, centre $O$ .
The angle of the minor sector $POQ$ is $82°$ .
The length of the minor arc $PQ$ is $7.3\,\text{cm}$ .

Calculate the area of the major sector.

______ $\text{cm}^2$

5M Similar

DifficultyMedium

Worked solution

Approach

Use the arc length formula to find the radius $r$ of the circle. Then calculate the area of the major sector using the reflex angle $(360^\circ - 82^\circ)$ and the radius found.

Working

The arc length formula is:

L = \frac{\theta}{360} \times 2\pi r

Substitute $L = 7.3$ and $\theta = 82^\circ$ :

7.3 = \frac{82}{360} \times 2\pi r

Solve for $r$ :

r = \frac{7.3 \times 360}{82 \times 2\pi} = \frac{2628}{164\pi} \approx 5.1007 \text{ cm}

The angle of the major sector is:

360^\circ - 82^\circ = 278^\circ

The area of the major sector is:

A = \frac{278}{360} \times \pi r^2 = \frac{278}{360} \times \pi \times (5.1007...)^2

A \approx 0.77222... \times 3.14159... \times 26.017... \approx 63.10 \text{ cm}^2

Answer

A = 63.1 \text{ cm}^2

Final answer

63.1

Detailed explanation

Walkthrough

Start with the arc length formula $L = \frac{\theta}{360} \times 2\pi r$ . Substitute the given minor arc length $L = 7.3$ cm and minor angle $\theta = 82^\circ$ to solve for the radius $r$ . This gives $r = \frac{7.3 \times 360}{82 \times 2\pi} \approx 5.1007$ cm. Keep this value unrounded in subsequent calculations to avoid rounding errors.

Next, find the angle of the major sector by subtracting the minor angle from $360^\circ$ : $360^\circ - 82^\circ = 278^\circ$ . Use the sector area formula $A = \frac{\theta}{360} \times \pi r^2$ with the major angle $278^\circ$ and the unrounded radius. This gives $A = \frac{278}{360} \times \pi \times (5.1007...)^2 \approx 63.1$ cm $^2$ .

Key Takeaways

The arc length formula relates arc length, central angle, and radius.
The major sector angle is $360^\circ$ minus the minor sector angle.
The sector area formula uses the central angle and radius squared.
Always keep intermediate values unrounded to ensure accuracy in the final answer.

Common Mistakes

Using the minor angle $82^\circ$ instead of the major angle $278^\circ$ when calculating the area of the major sector.
Rounding the radius too early (e.g., using $5.1$ instead of $5.1007...$ ), which can lead to a final answer slightly off.
Confusing the arc length formula with the sector area formula.
Forgetting to multiply by $2\pi$ in the arc length formula (using $\pi r$ instead of $2\pi r$ ).

Things to Be Careful About

The final answer must be given to 3 significant figures: $63.1$ cm $^2$ .
Use the unrounded radius value in the area calculation to avoid compounding rounding errors.
Ensure the angle used in the area formula is the reflex angle ( $278^\circ$ ), not the minor angle ( $82^\circ$ ).

Techniques used

use arc length formula to find radiuscalculate major sector angleuse sector area formula

Q10MediumAlgebra and GraphsNumber

The cost of apples is $x$ cents per kilogram.
Mina spends $9 on apples.

(a)

Write down an expression for the mass, in kilograms, of apples Mina receives.

______ $\text{kg}$

(b)

The cost of pears is 40 cents per kilogram more than the cost of apples.
Mina spends $9 on pears.
The mass of pears Mina receives is $0.75\,\text{kg}$ less than the mass of apples.

Form an equation in $x$ and show that it simplifies to $x^2 + 40x - 48\,000 = 0$ .

DifficultyMedium

Worked solution

Approach

Express the mass of pears in terms of $x$ , set up an equation showing that the mass of apples minus the mass of pears is $0.75\,\text{kg}$ , and clear algebraic denominators to simplify to the target quadratic equation.

Working

The cost of pears is $(x + 40)$ cents per kilogram.

Mina spends $9 (which is $900$ cents) on pears, so the mass of pears is:

\frac{900}{x + 40}

The mass of pears is $0.75\,\text{kg}$ less than the mass of apples:

\frac{900}{x} - \frac{900}{x + 40} = 0.75

Multiply the entire equation by $x(x + 40)$ to eliminate the fractions:

900(x + 40) - 900x = 0.75x(x + 40)

Expand both sides:

900x + 36\,000 - 900x = 0.75x^2 + 30x

36\,000 = 0.75x^2 + 30x

Divide every term by $0.75$ (or multiply by $\frac{4}{3}$ ):

x^2 + \frac{30}{0.75}x - \frac{36\,000}{0.75} = 0

x^2 + 40x - 48\,000 = 0

Answer

x^2 + 40x - 48\,000 = 0

Final answer

x^2 + 40x - 48000 = 0

Detailed explanation

Walkthrough

Find the mass of pears: pears cost 40 cents more per kilogram than apples, so their unit price is $(x + 40)$ cents/kg. Since Mina spends $9 ( $900$ cents), the mass of pears is $\frac{900}{x + 40}$ .
Form the equation: we are given that pears weigh $0.75\,\text{kg}$ less than apples. Therefore, $\text{mass of apples} - \text{mass of pears} = 0.75$ , giving:

\frac{900}{x} - \frac{900}{x + 40} = 0.75

Clear the denominators: multiply every term by the common denominator $x(x + 40)$ :

900(x + 40) - 900x = 0.75x(x + 40)

Expand and collect like terms:

36\,000 = 0.75x^2 + 30x

Rearrange to standard quadratic form and divide through by $0.75$ :

0.75x^2 + 30x - 36\,000 = 0 \implies x^2 + 40x - 48\,000 = 0

This precisely matches the required equation.

Key Takeaways

For "show that" questions, each step of algebraic manipulation must be clearly written out without skipping intermediate expansions.
Multiplying across by the common denominator is the most reliable way to clear fractions.

Common Mistakes

Subtracting in the wrong order, e.g. $\frac{900}{x + 40} - \frac{900}{x} = 0.75$ , which leads to incorrect signs.
Forgetting to multiply the right-hand side ( $0.75$ ) by $x(x + 40)$ .

Things to Be Careful About

All signs and division by $0.75$ must be shown clearly to gain full marks on a proof question.

Techniques used

form expressions for two different quantitiesset up a fractional equation from a worded differenceclear fractions using a common denominatorexpand and rearrange into standard quadratic form

(c)

Solve the equation $x^2 + 40x - 48\,000 = 0$ .
Show your working.

$x$ = ______ or $x$ = ______

DifficultyMedium-Easy

Worked solution

Approach

Factorise the quadratic expression $x^2 + 40x - 48\,000$ by finding two numbers that multiply to $-48\,000$ and add up to $40$ , or apply the quadratic formula.

Working

Find two numbers with a product of $-48\,000$ and a sum of $40$ . These numbers are $240$ and $-200$ .

Factorise the quadratic:

(x + 240)(x - 200) = 0

Set each factor equal to zero:

x + 240 = 0 \implies x = -240

x - 200 = 0 \implies x = 200

Alternatively, using the quadratic formula:

x = \frac{-40 \pm \sqrt{40^2 - 4(1)(-48\,000)}}{2(1)}

x = \frac{-40 \pm \sqrt{1600 + 192\,000}}{2} = \frac{-40 \pm \sqrt{193\,600}}{2} = \frac{-40 \pm 440}{2}

x = \frac{400}{2} = 200 \quad \text{or} \quad x = \frac{-480}{2} = -240

Answer

x = 200 \quad \text{or} \quad x = -240

Final answer

x = 200 or x = -240

Detailed explanation

Walkthrough

We need to solve $x^2 + 40x - 48\,000 = 0$ .

Using factorisation:
Look for two numbers that add to $+40$ and multiply to $-48\,000$ . Since $24 \times 20 = 480$ , we test $240$ and $-200$ :

240 + (-200) = 40 \quad \text{and} \quad 240 \times (-200) = -48\,000

This gives:

(x + 240)(x - 200) = 0

Solving gives $x = 200$ or $x = -240$ .

Both the quadratic formula and completing the square are equally valid alternative methods.

Key Takeaways

When solving pure quadratic equations in an exam, both positive and negative solutions must be stated unless the question asks specifically for a real-world quantity.

Common Mistakes

Discarding the negative root $-240$ in this part (the question asks to solve the algebraic equation, so both solutions are required).
Sign errors in factorisation, such as writing $(x - 240)(x + 200) = 0$ .

Things to Be Careful About

Double check the square root $\sqrt{193\,600} = 440$ if using the quadratic formula.

Techniques used

factorise a three-term quadraticapply the zero product property to find roots

(d)

Narinder buys $1.5\,\text{kg}$ of apples and $0.8\,\text{kg}$ of pears.

Work out the total amount he pays in dollars.

$ ______