These are the first four patterns in a sequence made from black beads and from white beads. (a) Complete the table for Pattern 5. Pattern number 1 2 3 4 5 --- --- --- --- --- --- Number of white beads 5 7 9 11 Total number of beads 6 10 15 21 (b) Find an expression for (i) the number of white beads in Pattern $n$ \_\_\_\_\_\_ (ii) the total number of beads in Pattern $n$. \_\_\_\_\_\_ (c) Bill has only 88 white beads but lots of black beads. Pattern $p$ is the largest…

(a) White beads: 13, Total: 28 · (b)(i) 2n + 3 · (b)(ii) (1/2)n^2 + (5/2)n + 3 · (c) 42

(a) Complete the table of values for $y = x^3 - 5x + 4$. $x$ -3 -2 -1 0 1 2 3 --- --- --- --- --- --- --- --- $y$ 6 8 4 0 2 16 (b) Draw the graph of $y = x^3 - 5x + 4$ for $-3 \leq x \leq 3$. (c) By drawing a suitable straight line on the grid, find the solutions of the equation $x^3 - 5x - 3 = 0$. $x$ = \_\_\_\_\_\_ , $x$ = \_\_\_\_\_\_ , $x$ = \_\_\_\_\_\_

(a) -8 · (b) Smooth cubic curve through (-3,-8), (-2,6), (-1,8), (0,4), (1,0), (2,2), (3,16) · (c) x ≈ -1.8, x ≈ -0.7, x ≈ 2.5

$ABCD$ and $EFGD$ are rectangles. (i) Show that $CD = x - 1$. (ii) The area of rectangle $ABCD$ is $\frac{1}{5}$ of the area of rectangle $EFGD$. Form an equation in $x$ and show that it simplifies to $x^2 - 6x + 3 = 0$. (i) Use the quadratic formula to solve the equation $x^2 - 6x + 3 = 0$. You must show all your working and give your answers correct to 2 decimal places. $x$ = \_\_\_\_\_\_ or $x$ = \_\_\_\_\_\_ (ii) Calculate the shaded area. \_\_\_\_\_\_ $\text{units}^2$

(a)(i) x - 1 · (a)(ii) x^2 - 6x + 3 = 0 · (b)(i) 5.45 or 0.55 · (b)(ii) 53.4

4024/22 May/June 2025 — Mathematics (Syllabus D) Solved Past Paper

Q11MMedium-EasyGeometry

Fig. 1

The diagram shows part of a pattern with rotational symmetry of order 4.

Complete the diagram.

DifficultyMedium-Easy

Worked solution

Approach

The diagram must have rotational symmetry of order 4 about the centre of the 6 by 6 grid. The centre of a 6 by 6 grid is the intersection of the grid lines at coordinates $(3.5, 3.5)$ if we use $(x, y)$ with $x$ as the column from the left and $y$ as the row from the top. To complete the pattern, we rotate every existing shaded square and line segment by $90^\circ$ clockwise (and consequently $180^\circ$ and $270^\circ$ ) about this centre.

Working

1. Identify the centre of rotation:
The grid is $6 \times 6$ . The centre is at $(3.5, 3.5)$ .

2. Rotate the shaded squares:

Square 1: Top-left at $(2, 1)$ , bottom-right at $(2, 2)$ .
Rotating $90^\circ$ clockwise about $(3.5, 3.5)$ :
$\Delta x = 2 - 3.5 = -1.5$ , $\Delta y = 2 - 3.5 = -1.5$ .
New position: $(3.5 + (-1.5), 3.5 - (-1.5)) = (2, 5)$ .
This corresponds to the square at row 5, column 2.
Rotating again gives the square at row 5, column 5 (bottom-right).
Rotating again gives the square at row 2, column 5 (top-right, already present).
Square 2: Top-left at $(5, 2)$ .
Rotating $90^\circ$ clockwise gives the square at row 3, column 5.
Rotating again gives the square at row 5, column 4 (bottom-right, already present).
Rotating again gives the square at row 4, column 2.

3. Rotate the line segments:
The existing lines form a partial zig-zag from $(2, 2)$ through $(3, 3)$ to $(5, 2)$ and $(5, 5)$ . Rotating these by $90^\circ$ , $180^\circ$ , and $270^\circ$ about $(3.5, 3.5)$ completes the four-fold symmetric pattern, connecting the newly shaded squares in a similar zig-zag fashion around the centre.

Answer

The completed diagram has four shaded squares at $(2, 1)$ , $(5, 2)$ , $(4, 5)$ , and $(2, 5)$ (using column, row from top-left), plus their $180^\circ$ and $270^\circ$ rotations, with line segments connecting their corners to form a fully symmetric pattern about the grid centre $(3.5, 3.5)$ .

\text{Shaded squares at: } (2,1), (5,2), (4,5), (2,5), (5,5), (3,4), (4,2), (1,3) \text{ (and symmetric equivalents)}

Final answer

Grid completed with 8 shaded squares and connecting lines forming 4-fold rotational symmetry about the centre (3.5, 3.5).

Detailed explanation

Walkthrough

The question asks to complete a pattern with rotational symmetry of order 4. Order 4 means the pattern looks the same after rotations of $90^\circ$ , $180^\circ$ , $270^\circ$ , and $360^\circ$ .

Find the centre: For a $6 \times 6$ grid, the centre of rotation is the exact middle, which is the grid intersection at $(3.5, 3.5)$ .
Rotate the elements: Take each shaded square and line segment. Rotate them $90^\circ$ clockwise around the centre. For example, the top-left shaded square at column 2, row 1 rotates to column 2, row 5. Repeat this to get the $180^\circ$ and $270^\circ$ positions.
Draw the lines: Connect the corners of the newly shaded squares in the same zig-zag pattern relative to the centre as the original lines.

Key Takeaways

Rotational symmetry of order $n$ means the pattern repeats $n$ times around a central point, with each repetition separated by $\frac{360^\circ}{n}$ .
For order 4, the angle of rotation is $90^\circ$ .
The centre of rotation for a square grid is its geometric centre.

Common Mistakes

Using the wrong centre of rotation (e.g., a corner or edge midpoint instead of the grid centre).
Confusing rotational symmetry with line (reflection) symmetry.
Failing to rotate all elements (both shaded squares and line segments).
Drawing the pattern with order 2 symmetry instead of order 4 (as seen in the mismatched marking scheme text provided).

Things to Be Careful About

Ensure the grid is read correctly ( $6 \times 6$ in the question image, though the provided marking scheme text describes a $5 \times 5$ grid with order 2 symmetry; the solution follows the question text and image which specify order 4 and a $6 \times 6$ grid).
The final answer must be a completed diagram; no numerical calculation is required other than identifying the correct positions for the rotated elements.
All four quadrants must be identical under $90^\circ$ rotation.

Techniques used

identify the centre of rotationrotate given elements by 90 degreescomplete the pattern to achieve the required order of rotational symmetry

Similar questions

Q2EasyNumber

(a)

Write eighteen thousand and twelve in figures.

______

1M Similar

(b)

Write down a prime number between 20 and 30.

______

1M Similar

(c)

Find the reciprocal of $\frac{1}{9}$ .

______

1M Similar

Q3Medium-EasyNumberMensuration

(a)

Write the ratio $175\,\text{ml} : 2.5\,\text{litres}$ in its simplest form.

______ : ______

2M Similar

DifficultyMedium-Easy

Worked solution

Approach

Convert both quantities to the same unit, then simplify the ratio by dividing both terms by their highest common factor.

Working

2.5\ \text{litres} = 2.5 \times 1000 = 2500\ \text{ml}

So the ratio is

175 : 2500

Divide both terms by $25$ :

175 \div 25 = 7, \qquad 2500 \div 25 = 100

Therefore

175 : 2500 = 7 : 100

Answer

7 : 100

Final answer

7 : 100

Detailed explanation

Walkthrough

A ratio compares quantities in the same unit, so before simplifying $175\,\text{ml} : 2.5\,\text{litres}$ we must convert both to millilitres. Since $1$ litre $= 1000$ ml, $2.5$ litres $= 2.5 \times 1000 = 2500$ ml. The ratio is therefore $175 : 2500$ . To simplify, divide both terms by their highest common factor. $175$ and $2500$ are both divisible by $25$ : $175 \div 25 = 7$ and $2500 \div 25 = 100$ . No further common factor remains, so the simplest form is $7 : 100$ .

Key Takeaways

A ratio is only meaningful when both quantities are in the same unit.
Simplifying a ratio means dividing both terms by the same factor until no common factor remains.
$1$ litre $= 1000$ ml, so litres are converted to millilitres by multiplying by $1000$ .

Common Mistakes

Simplifying $175 : 2.5$ without converting, which incorrectly gives $70 : 1$ .
Dividing only one term of the ratio by $25$ .
Reversing the order and writing $100 : 7$ instead of $7 : 100$ .
The mark scheme accepts only $7 : 100$ as the final answer (cao), but gives M1 for the converted form $175 : 2500$ or $0.175 : 2.5$ .

Things to Be Careful About

The two quantities are in different units: ml and litres, so the conversion must be done first.
The final ratio has no units.
$175 : 2500$ is not the final answer because it is not in simplest form.

Techniques used

convert litres to millilitresrewrite the ratio in the same unitdivide both terms by a common factor

(b)

Dan, Erika and Fatik share $540 in the ratio $4 : 5 : 3$ .

Find the amount that Erika receives.

$ ______

2M Similar

Q42MMedium-EasyGeometry

Fig. 2

Find the value of $x$ and the value of $y$ .

$x$ = ______
$y$ = ______

DifficultyMedium-Easy

Worked solution

Approach

Identify the relationships between the angles using the properties of parallel lines. The three horizontal lines are parallel to each other, and the two transversal lines are parallel to each other. Use co-interior angles to find $x$ and corresponding angles to find $y$ .

Working

To find $x$ :
The angle $128°$ and angle $x°$ are co-interior (same-side interior) angles between the parallel horizontal lines, formed by the right transversal. Co-interior angles sum to $180°$ .

\begin{aligned} x + 128 &= 180 \\ x &= 180 - 128 \\ x &= 52 \end{aligned}

To find $y$ :
The angle $y°$ is below the top horizontal line and to the right of the left transversal. Because the two transversal lines are parallel, the angle below the top horizontal line and to the right of the right transversal is equal to $y°$ (corresponding angles). Let us call this angle $z°$ , so $z = y$ .

Now, because the top and middle horizontal lines are parallel and cut by the right transversal, angle $z°$ (below the top horizontal line, right of right transversal) and the angle $128°$ (below the middle horizontal line, right of right transversal) are corresponding angles. Corresponding angles are equal.

y = 128

Answer

x = 52, \quad y = 128

Final answer

x = 52, y = 128

Detailed explanation

Walkthrough

The diagram contains two sets of parallel lines: three horizontal lines and two transversal lines. We need to find $x$ and $y$ by applying angle properties.

Finding $x$ :
Look at the right transversal intersecting the middle and bottom horizontal lines. The angle marked $128°$ is below the middle horizontal line and to the right of the transversal. The angle $x°$ is above the bottom horizontal line and to the right of the transversal. These two angles are on the same side of the transversal and between the parallel horizontal lines, making them co-interior angles (also called same-side interior angles). Co-interior angles always add up to $180°$ .

x + 128 = 180

Subtracting $128$ from $180$ gives $x = 52$ .

Finding $y$ :
Look at the top horizontal line intersecting the two parallel transversals. The angle $y°$ is below the top horizontal line and to the right of the left transversal. Because the two transversals are parallel, the angle in the same relative position on the right transversal (below the top horizontal line, to the right of the right transversal) must be equal to $y°$ . This is the corresponding angles property.

Now consider the top and middle horizontal lines, which are parallel, cut by the right transversal. The angle we just identified (below the top horizontal line, right of the right transversal) and the angle $128°$ (below the middle horizontal line, right of the right transversal) are in the same relative position at each intersection. These are corresponding angles, so they are equal.

Therefore, $y = 128$ .

Key Takeaways

Co-interior angles between parallel lines sum to $180°$ .
Corresponding angles are equal when parallel lines are cut by a transversal.
When two sets of parallel lines intersect, you can chain corresponding angle arguments across one set of parallels to relate angles on the other set.

Common Mistakes

Assuming $x$ and $128°$ are corresponding angles and setting $x = 128$ . They are co-interior, so they must sum to $180°$ .
Forgetting that both the horizontal lines AND the transversals are parallel. If a student only uses the horizontal parallels, they might struggle to link $y$ to $128°$ without using the transversal parallel property first.
Giving $y$ as $52°$ by incorrectly applying co-interior angles to the wrong pair of lines.

Things to Be Careful About

Always check which lines are marked as parallel (arrows indicate this). Here, both the horizontal lines and the transversals have parallel markings.
Co-interior angles are sometimes confused with alternate interior angles (which are equal). Co-interior angles are on the same side of the transversal and sum to $180°$ .
The diagram is marked NOT TO SCALE, so do not estimate angles visually; rely only on the parallel line properties and the given $128°$ .
Final answers must be exact integer values as derived from the angle sum properties.

Techniques used

identify co-interior angles between parallel linesidentify corresponding angles across parallel transversals

Similar questions

Q51MEasyMensuration

Convert $6.1\,\text{m}^2$ to $\text{cm}^2$ .

______ $\text{cm}^2$

DifficultyEasy

Worked solution

Approach

To convert an area from square metres ( $\text{m}^2$ ) to square centimetres ( $\text{cm}^2$ ), we must use the relationship between the linear units. Since $1 \text{ m} = 100 \text{ cm}$ , squaring both sides gives the conversion factor for area.

Working

The relationship between metres and centimetres is:

1 \text{ m} = 100 \text{ cm}

To find the conversion for area (square units), we square this relationship:

1 \text{ m}^2 = (100 \text{ cm})^2 = 100 \times 100 \text{ cm}^2 = 10,000 \text{ cm}^2

Now, multiply the given value in $\text{m}^2$ by $10,000$ to get the value in $\text{cm}^2$ :

6.1 \text{ m}^2 = 6.1 \times 10,000 \text{ cm}^2

Performing the multiplication:

6.1 \times 10,000 = 61,000

Answer

61,000 \text{ cm}^2

Final answer

61000

Detailed explanation

Walkthrough

This question requires converting a measurement of area from one unit to another. The key step is determining the correct conversion factor.

Identify the linear conversion: We know that there are $100$ centimetres in $1$ metre ( $1 \text{ m} = 100 \text{ cm}$ ).
Determine the area conversion: Area is measured in square units. To convert square metres to square centimetres, we must square the linear conversion factor. This means multiplying $100$ by itself: $100 \times 100 = 10,000$ . Therefore, $1 \text{ m}^2 = 10,000 \text{ cm}^2$ .
Calculate the final value: Multiply the given number of square metres ( $6.1$ ) by the conversion factor ( $10,000$ ). Moving the decimal point four places to the right converts $6.1$ into $61,000$ .

Key Takeaways

Always distinguish between linear conversions (length) and area conversions. For length, $1 \text{ m} = 100 \text{ cm}$ . For area, you must square the factor: $1 \text{ m}^2 = 10,000 \text{ cm}^2$ .
A common error is to simply multiply by $100$ when dealing with squares or cubes; remember that the exponent on the unit dictates the power to which the conversion factor must be raised.

Common Mistakes

Multiplying by $100$ instead of $10,000$ . This happens if the student treats the area conversion like a linear length conversion.
Dropping zeros incorrectly during multiplication.

Things to Be Careful About

Ensure you read the units correctly. The question asks for $\text{cm}^2$ , not $\text{cm}$ . If it asked for $\text{mm}^2$ , the factor would be $1,000,000$ ( $1000^2$ ).

Techniques used

convert square metres to square centimetres using the area conversion factor

Similar questions

Q6Medium-EasyProbability

In a survey, students are asked to choose their favourite type of movie.
The table shows the results.

Type of movie	Romance	Thriller	Science Fiction	Other
Relative frequency	0.1		0.35	0.3

(a)

Calculate the relative frequency of a student choosing Thriller.

______

2M Similar

DifficultyMedium-Easy

Worked solution

Approach

The sum of all relative frequencies in a complete distribution must equal 1. We can find the missing relative frequency for Thriller by subtracting the known relative frequencies from 1.

Working

Let $R$ be Romance, $T$ be Thriller, $S$ be Science Fiction, and $O$ be Other.
We are given:

P(R) = 0.1

P(S) = 0.35

P(O) = 0.3

The sum of relative frequencies is:

P(R) + P(T) + P(S) + P(O) = 1

Substitute the known values:

0.1 + P(T) + 0.35 + 0.3 = 1

Sum the known values:

0.75 + P(T) = 1

Solve for $P(T)$ :

P(T) = 1 - 0.75

P(T) = 0.25

Answer

0.25 \text{ or } \frac{1}{4}

Final answer

0.25

Detailed explanation

Walkthrough

Relative frequency represents the proportion of times an outcome occurs out of the total number of observations. Since every student must choose exactly one category (Romance, Thriller, Science Fiction, or Other), the sum of the relative frequencies for all categories must add up to 1 (representing 100% of the students).

To find the missing value for Thriller, we first add up the relative frequencies we already know: Romance ( $0.1$ ), Science Fiction ( $0.35$ ), and Other ( $0.3$ ).
$0.1 + 0.35 + 0.3 = 0.75$
Since the total must be 1, we subtract this sum from 1 to find the remaining portion assigned to Thriller:
$1 - 0.75 = 0.25$
This means the relative frequency for Thriller is $0.25$ . This can also be written as the fraction $\frac{1}{4}$ or the percentage $25\%$ .

Key Takeaways

The sum of all relative frequencies (or probabilities) in a sample space is always 1.
If you have a table of relative frequencies with one missing entry, you can find it by subtracting the sum of the others from 1.

Common Mistakes

Adding the numbers instead of subtracting them from 1.
Miscalculating the sum of the known frequencies (e.g., adding $0.1 + 0.35 + 0.3$ incorrectly).
Forgetting that relative frequencies are proportions between 0 and 1.

Things to Be Careful About

Ensure you include all given categories in your sum before subtracting from 1.
The answer can be expressed as a decimal, fraction, or percentage unless specified otherwise. $0.25$ , $\frac{1}{4}$ , and $25\%$ are all correct.

Techniques used

use the property that relative frequencies sum to 1subtract known values from 1

(b)

500 students take part in the survey.

Calculate the number of students who choose Science Fiction.

______

1M Similar

Q73MMediumNumber

Chris wants to exchange $350 for euros (€) at the bank.
The bank only gives euros in multiples of €5.
The exchange rate is $1 = €0.92.

Calculate the number of euros he receives and his change from $350.

Chris receives € ______
His change is $ ______

DifficultyMedium

Worked solution

Approach

Convert the dollars to euros using the exchange rate, then round down to the nearest €5 because the bank only issues multiples of €5. Work out the dollar value of the euros actually received, and subtract from $350 to find the change.

Working

Convert $350 to euros:

350 \times 0.92 = 322

So the full conversion gives €322. The bank only gives multiples of €5, so the largest multiple of €5 not exceeding 322 is:

320

Chris receives €320.

Now find the dollar value of €320:

\frac{320}{0.92} = 347.826\ldots

His change is therefore:

350 - 347.826\ldots = 2.1739\ldots

Rounded to 2 decimal places:

2.17

Alternatively, using the mark scheme expression:

\frac{350 \times 0.92 - 320}{0.92} = \frac{322 - 320}{0.92} = 2.1739\ldots

Answer

Chris receives €320 and his change is $2.17.

Final answer

Chris receives €320 and his change is $2.17

Detailed explanation

Walkthrough

Start by converting the $350 into euros. Since each dollar is worth €0.92, multiply:

350 \times 0.92 = 322

So without any restriction Chris would receive €322. However, the bank only gives euros in multiples of €5, so he cannot receive €322. The largest multiple of €5 that is not more than 322 is €320, so that is what he receives.

To find his change in dollars, work out how many dollars €320 is worth. Because €0.92 = $1, divide by 0.92:

\frac{320}{0.92} = 347.826\ldots

This means the €320 he receives is worth about $347.83. He paid $350, so his change is:

350 - 347.826\ldots = 2.1739\ldots

Rounded to 2 decimal places, this is $2.17.

Key Takeaways

To convert from dollars to euros, multiply by the exchange rate.
To convert from euros back to dollars, divide by the exchange rate.
When a bank only gives notes in certain multiples, round the converted amount down to the largest allowed multiple.
Change is the original amount paid minus the value of what was received.

Common Mistakes

Giving €322 as the answer and forgetting that the bank only gives multiples of €5.
Subtracting €320 directly from $350 to get $30; this mixes different currencies and is wrong.
Rounding €322 up to €325 instead of down to €320.
Giving the change as €2 instead of $2.17, or failing to round to 2 decimal places.

Things to Be Careful About

Keep the units separate: dollars and euros are not interchangeable.
The exchange rate direction matters. €0.92 per dollar means multiply dollars by 0.92 to get euros, and divide euros by 0.92 to get dollars.
The mark scheme accepts the expression $\frac{350 \times 0.92 - 320}{0.92}$ as a single method mark, so showing the conversion and the change calculation clearly is important.
This is a calculator paper, so decimal values such as 347.826... are acceptable, but the final money answer should be given to 2 decimal places.

Techniques used

convert dollars to euros using the exchange rateround down to a multiple of 5convert euros back to dollarssubtract to find the change

Similar questions

Q82MMedium-EasyGeometry

Calculate the interior angle of a regular octagon.

______

DifficultyMedium-Easy

Worked solution

Approach

To find the interior angle of a regular octagon, we can use two common methods: either finding the exterior angle first or using the sum of interior angles. Both lead to the same result.

Working

Method 1: Using the Exterior Angle
The sum of the exterior angles of any convex polygon is always $360°$ .
An octagon has $8$ sides (and therefore $8$ vertices).
Since the octagon is regular, all its exterior angles are equal.

\text{Exterior angle} = \frac{360°}{8}

\text{Exterior angle} = 45°

The interior angle and exterior angle at each vertex lie on a straight line, so they add up to $180°$ .

\text{Interior angle} = 180° - 45°

\text{Interior angle} = 135°

Method 2: Using the Sum of Interior Angles
The sum of the interior angles of an $n$ -sided polygon is given by:

(n - 2) \times 180°

For an octagon, $n = 8$ :

\text{Sum} = (8 - 2) \times 180°

\text{Sum} = 6 \times 180°

\text{Sum} = 1080°

Since the octagon is regular, all $8$ interior angles are equal. We divide the total sum by $8$ :

\text{Interior angle} = \frac{1080°}{8}

\text{Interior angle} = 135°

Answer

135°

Final answer

135°

Detailed explanation

Walkthrough

The problem asks for the measure of one interior angle of a regular octagon. A regular polygon has equal sides and equal interior angles. An octagon is defined as having 8 sides.

There are two standard ways to solve this:

Exterior Angle Method: It is a fundamental property that the sum of the exterior angles of any convex polygon is $360°$ . By dividing this sum by the number of sides ( $8$ ), we find the size of one exterior angle. Since the interior and exterior angles form a linear pair (they add up to $180°$ ), subtracting the exterior angle from $180°$ gives the interior angle.
$\text{Interior} = 180° - \frac{360°}{8}$
Interior Sum Method: The sum of the interior angles of an $n$ -sided polygon is calculated using the formula $(n-2) \times 180°$ . For an octagon, this is $(8-2) \times 180° = 1080°$ . Because the polygon is regular, we divide this total sum by the number of angles ( $8$ ) to get the individual angle size.
$\text{Interior} = \frac{(8-2) \times 180°}{8}$

Both methods are mathematically equivalent and yield $135°$ .

Key Takeaways

The sum of exterior angles of any polygon is always $360°$ .
The sum of interior angles of an $n$ -sided polygon is $(n-2) \times 180°$ .
For a regular polygon, divide the respective sum by the number of sides/angles to find the individual angle measure.
Interior and exterior angles at a vertex are supplementary (add to $180°$ ).

Common Mistakes

Forgetting that the polygon must be regular to simply divide the sum; if it were irregular, the angles could vary.
Confusing the formula for the sum of interior angles with the formula for the exterior angle sum ( $360°$ ).
Arithmetic errors when calculating $1080 \div 8$ or $360 \div 8$ .
Providing the answer without the degree symbol ( $°$ ) where appropriate, though sometimes accepted depending on specific mark scheme leniency, it is best practice to include units.

Things to Be Careful About

Ensure you identify the number of sides correctly: an octagon has 8, not 6 (hexagon) or 10 (decagon).
Remember to subtract from 180 if using the exterior angle method.
On calculator papers, ensure the final division is performed accurately. $1080 / 8 = 135$ exactly.

Techniques used

apply the interior angle sum formula for a polygondivide by the number of sides to find the regular interior angle

Similar questions

Q9Medium-EasyNumber

(a)

On the Venn diagram, shade the region represented by $(A \cap B)'$ .

Fig. 3

1M Similar

DifficultyEasy

Worked solution

Approach

The notation $(A \cap B)'$ represents the complement of the intersection of sets $A$ and $B$ . The intersection $A \cap B$ is the region where the two circles overlap. The complement means we shade everything EXCEPT that overlapping region.

Working

Identify the intersection $A \cap B$ : This is the central lens-shaped region where circle $A$ and circle $B$ overlap.
Identify the complement $(A \cap B)'$ : This is the entire universal set $\xi$ minus the intersection.
Shade the regions: Shade the part of $A$ that does not overlap with $B$ , the part of $B$ that does not overlap with $A$ , and the region outside both circles but inside the rectangle $\xi$ .
Leave unshaded: Only the central overlapping region $A \cap B$ remains white.

Answer

All regions shaded except the central intersection $A \cap B$ .

Final answer

Shade all regions except the intersection $A \cap B$

Detailed explanation

Walkthrough

The symbol $\cap$ denotes intersection, which is the set of elements belonging to both $A$ and $B$ . On a Venn diagram, this is the overlapping area of the two circles. The apostrophe $'$ denotes the complement, meaning everything in the universal set $\xi$ that is NOT in the specified set. Therefore, $(A \cap B)'$ includes everything outside the intersection: the part of $A$ only, the part of $B$ only, and the region outside both circles. We shade all these three regions and leave only the intersection unshaded.

Key Takeaways

$A \cap B$ is the overlap between two sets.
The complement $X'$ means everything in the universal set except $X$ .
$(A \cap B)'$ is shaded everywhere except the central overlap.

Common Mistakes

Shading only the region outside both circles (this is $(A \cup B)'$ , not $(A \cap B)'$ ).
Shading the intersection instead of leaving it unshaded.
Forgetting to shade the region outside the circles but inside the universal set rectangle.

Things to Be Careful About

Ensure the shading is solid and clearly covers all three required regions.
The intersection must be completely white (unshaded).
The universal set boundary (the rectangle) must be respected; do not shade outside it.

Techniques used

interpret set intersection notationidentify the complement regionshade the appropriate area on a Venn diagram

(b)

In a class of 22 students:

12 play the piano
9 play the guitar
6 do not play the piano or the guitar.

Find the number of students who play both the piano and the guitar.
You may use the Venn diagram to help you.

Fig. 4

______

2M Similar

DifficultyMedium-Easy

Worked solution

Approach

Let $x$ be the number of students who play both the piano and the guitar. We can express the number of students in each disjoint region of the Venn diagram in terms of $x$ and the given totals, then use the overall total to form an equation.

Working

Let $x$ be the number of students who play both instruments ( $A \cap B$ ).

Students who play only the piano: $12 - x$
Students who play only the guitar: $9 - x$
Students who play neither: $6$

The sum of all disjoint regions equals the total number of students:

(12 - x) + x + (9 - x) + 6 = 22

Simplify the left side:

12 + 9 + 6 - x = 22

27 - x = 22

Solve for $x$ :

x = 27 - 22 = 5

Alternatively, using the union formula:
Number who play at least one instrument = $22 - 6 = 16$ .

|P \cup G| = |P| + |G| - |P \cap G|

16 = 12 + 9 - x

16 = 21 - x \implies x = 5

Answer

5

Final answer

5

Detailed explanation

Walkthrough

We are given the total number of students (22), the number who play piano (12), the number who play guitar (9), and the number who play neither (6). We need to find the number who play both.

First, find the number of students who play at least one instrument. Since 6 play neither, this is $22 - 6 = 16$ .

Let $x$ be the number who play both. Then:

Piano only = $12 - x$
Guitar only = $9 - x$

The total who play at least one is the sum of these three disjoint groups:
$(12 - x) + x + (9 - x) = 16$

Simplify:
$21 - x = 16$
$x = 5$

So 5 students play both instruments.

Key Takeaways

The total number of elements is the sum of all disjoint regions in a Venn diagram.
If $n(A)$ and $n(B)$ are given, the intersection can be found using $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ .
Always verify that calculated regions are non-negative.

Common Mistakes

Adding $12 + 9 + 6 = 27$ and forgetting to subtract the total $22$ to find the overlap.
Setting up the equation incorrectly, such as $12 + 9 + x + 6 = 22$ , which double-counts the intersection.
Forgetting that the 6 students who play neither are outside both circles.

Things to Be Careful About

Ensure all calculated regions (only piano, only guitar, both, neither) are non-negative.
The final answer is just the number, no units required unless specified.
Working must be shown to earn method marks.

Techniques used

define a variable for the intersectionexpress disjoint regions in terms of the variableform an equation using the total number of elementssolve the linear equation

Q102MEasyAlgebra and Graphs

Simplify.

\frac{7x^2y^5}{x^3y^2}

______

DifficultyEasy

Worked solution

Approach

To simplify the fraction $\frac{7x^2y^5}{x^3y^2}$ , we apply the quotient rule for indices: $\frac{a^m}{a^n} = a^{m-n}$ . We handle the constant, the $x$ terms, and the $y$ terms separately.

Working

Step 1: Separate the components.

\frac{7 \cdot x^2 \cdot y^5}{x^3 \cdot y^2}

Step 2: Simplify the $x$ terms.
The variable $x$ appears as $x^2$ in the numerator and $x^3$ in the denominator. Subtracting the powers:

\frac{x^2}{x^3} = x^{2-3} = x^{-1} = \frac{1}{x}

Alternatively, cancelling two $x$ 's from top and bottom leaves one $x$ in the denominator.

Step 3: Simplify the $y$ terms.
The variable $y$ appears as $y^5$ in the numerator and $y^2$ in the denominator. Subtracting the powers:

\frac{y^5}{y^2} = y^{5-2} = y^3

Step 4: Combine all parts.
Combine the constant $7$ , the simplified $x$ part ( $\frac{1}{x}$ ), and the simplified $y$ part ( $y^3$ ):

7 \cdot \frac{1}{x} \cdot y^3 = \frac{7y^3}{x}

Answer

\frac{7y^3}{x}

Final answer

7y^3/x

Detailed explanation

Walkthrough

The problem asks us to simplify an algebraic fraction. The key rule here is the quotient rule of indices, which states that when you divide two terms with the same base, you subtract their exponents: $\frac{a^m}{a^n} = a^{m-n}$ .

Identify the bases: We have three 'bases' here: the number $7$ , the letter $x$ , and the letter $y$ .
Handle the numbers: The number $7$ is only in the numerator, so it remains $7$ .
Handle the $x$ 's: We have $x^2$ on top and $x^3$ on the bottom. Since the larger power is at the bottom, the result will have $x$ in the denominator. Specifically, $x^{2-3} = x^{-1}$ , which is equivalent to $\frac{1}{x}$ . Alternatively, think of it as cancelling out two $x$ 's from both top and bottom, leaving one $x$ left over at the bottom.
Handle the $y$ 's: We have $y^5$ on top and $y^2$ on the bottom. Since the larger power is on top, the result has $y$ in the numerator. Specifically, $y^{5-2} = y^3$ .
Final Assembly: Put these pieces together: $7$ times $y^3$ divided by $x$ .

Key Takeaways

Quotient Rule: Always subtract the exponent of the denominator from the exponent of the numerator ( $m - n$ ).
Negative Indices: A negative exponent indicates the term belongs in the opposite part of the fraction (numerator becomes denominator, or vice versa). For example, $x^{-1} = \frac{1}{x}$ .
Simplicity: The standard simplified form usually prefers positive indices and places variables in the numerator where possible.

Common Mistakes

Adding instead of subtracting: Students often add the exponents (e.g., $2+3=5$ ) instead of subtracting them. This is the rule for multiplication, not division.
Incorrect cancellation: Cancelling variables incorrectly (e.g., cancelling $x^2$ and $x^3$ to get $x$ or $1$ without tracking which side they remain on).
Handling constants: Sometimes students try to cancel the $7$ with nothing, or forget it entirely.

Things to Be Careful About

Order of subtraction: Ensure you subtract the bottom power from the top power ( $Top - Bottom$ ). If the result is negative, move the variable to the other side of the fraction line to make the exponent positive.
Final Form: The mark scheme accepts $\frac{7y^3}{x}$ or $7y^3x^{-1}$ . However, $\frac{7y^3}{x}$ is the most conventional simplified form.

Techniques used

apply index laws to divide terms with the same basesubtract exponents for common variableswrite the final expression in simplest form

Similar questions

Q11Medium-EasyNumber

(a)

Zaya invests $4500 in a savings account.
The account pays simple interest at a rate of 3.2% per year.

Calculate the value of the investment at the end of 5 years.

$ ______

3M Similar

DifficultyMedium-Easy

Worked solution

Approach

To calculate the total value of the investment, first find the simple interest earned over $5$ years using the formula $I = \frac{P \times R \times T}{100}$ , where $P$ is the principal, $R$ is the annual interest rate, and $T$ is the time in years. Then, add the interest to the original principal to find the final amount.

Working

Calculate the simple interest earned:

\text{Interest} = \frac{4500 \times 3.2 \times 5}{100}

\text{Interest} = \frac{72000}{100} = 720

Calculate the total value of the investment:

\text{Total value} = 4500 + 720 = 5220

Answer

5220

Final answer

5220

Detailed explanation

Walkthrough

Identify the given values for simple interest:
- Principal ( $P$ ) = $4500
- Rate ( $R$ ) = $3.2\%$
- Time ( $T$ ) = $5$ years
Compute the simple interest earned over the $5$ -year period using the formula:

I = \frac{P \times R \times T}{100} = \frac{4500 \times 3.2 \times 5}{100} = 720

The question asks for the value of the investment at the end of $5$ years, which means the total balance in the account (principal plus interest), not just the interest earned:

\text{Total Value} = 4500 + 720 = 5220

Key Takeaways

Simple interest is calculated only on the initial principal amount.
Distinguish carefully between the "interest earned" and the "total value of the investment" (total amount = principal + interest).

Common Mistakes

Stopping after calculating the interest ($720) and forgetting to add the original principal ($4500).
Confusing simple interest with compound interest.

Things to Be Careful About

Ensure all arithmetic is checked carefully.
The currency symbol $ is already provided outside the answer line, so only the numerical value $5220$ is required.

Techniques used

calculate simple interest using formulacalculate total investment value by adding interest to principal

(b)

Aisha invests $2750 in a different savings account.
This account pays compound interest at a rate of 2.1% per year.

Calculate the total amount of interest she receives at the end of 3 years.

$ ______

3M Similar

DifficultyMedium-Easy

Worked solution

Approach

Use the compound interest formula $A = P \left(1 + \frac{r}{100}\right)^n$ to find the total amount in the account at the end of $3$ years. Then, subtract the initial principal from this total amount to find the total interest earned.

Working

Calculate the total amount in the account after $3$ years:

A = 2750 \times \left(1 + \frac{2.1}{100}\right)^3

A = 2750 \times 1.021^3

A = 2750 \times 1.064332261 = 2926.913718

Calculate the total interest received by subtracting the principal:

\text{Interest} = 2926.913718 - 2750 = 176.913718

Rounding to $2$ decimal places for currency:

\text{Interest} = 176.91

Answer

176.91

Final answer

176.91

Detailed explanation

Walkthrough

Identify the given values for compound interest:
- Principal ( $P$ ) = $2750
- Annual rate ( $r$ ) = $2.1\%$
- Time ( $n$ ) = $3$ years
The formula for the total accrued amount under compound interest is:

A = P\left(1 + \frac{r}{100}\right)^n = 2750 \times (1.021)^3

Evaluate this expression:

A = 2750 \times 1.064332261 = 2926.913718

The question specifically asks for the total amount of interest received, not the total investment value. Therefore, subtract the principal from the total amount:

\text{Interest} = 2926.913718 - 2750 = 176.913718

Since monetary amounts are given to $2$ decimal places (cents), the answer is $176.91$ .

Key Takeaways

Compound interest formula $A = P(1 + r)^n$ calculates the total value, not the interest.
To find the compound interest alone: $\text{Interest} = A - P$ .

Common Mistakes

Giving the total final amount ( $2926.91$ ) instead of just the interest earned ( $176.91$ ).
Incorrectly rounding intermediate values during computation rather than rounding the final answer to $2$ decimal places.
Using the simple interest formula instead of the compound interest formula.

Things to Be Careful About

Keep full calculator precision until the final step to avoid rounding errors.
Pay close attention to what is asked: part (a) asked for the total value, while part (b) asked for the interest.

Techniques used

calculate total amount using compound interest formulafind compound interest by subtracting principal from total amount

Q122MMedium-EasyAlgebra and Graphs

Factorise.

7hx + 6fy - 21fx - 2hy

______

DifficultyMedium-Easy

Worked solution

Approach

Group the four terms into two pairs so that each pair has a common factor. Factor each pair, then factor out the common binomial factor that appears.

Working

Group the terms with common factors:

7hx + 6fy - 21fx - 2hy = 7hx - 21fx + 6fy - 2hy

Factor the first pair:

7hx - 21fx = 7x(h - 3f)

Factor the second pair, taking care with the sign:

6fy - 2hy = 2y(3f - h) = -2y(h - 3f)

So the whole expression becomes:

7x(h - 3f) - 2y(h - 3f)

Factor out the common binomial $(h - 3f)$ :

(7x - 2y)(h - 3f)

Answer

(7x - 2y)(h - 3f)

An equivalent accepted form is $(2y - 7x)(3f - h)$ .

Final answer

(7x - 2y)(h - 3f)

Detailed explanation

Walkthrough

We have four terms and no factor is common to all of them, so we factor by grouping. Look for pairs of terms that share a factor: $7hx$ and $-21fx$ both contain $7x$ ; $6fy$ and $-2hy$ both contain $2y$ . Factor these pairs to get $7x(h - 3f)$ and $2y(3f - h)$ . Since $3f - h$ is the negative of $h - 3f$ , rewrite the second as $-2y(h - 3f)$ . Now both terms have the binomial factor $h - 3f$ , so take it out: $(7x - 2y)(h - 3f)$ .

Key Takeaways

Four-term expressions with no common factor can often be factorised by grouping into pairs.
After factoring each pair, look for a common binomial factor.
A difference such as $a - b$ can be written as $-(b - a)$ ; this is essential for matching signs.
The order of factors does not matter, so $(7x - 2y)(h - 3f)$ and $(h - 3f)(7x - 2y)$ are the same.

Common Mistakes

Stopping after the first grouping: $7x(h - 3f) + 2y(3f - h)$ is not fully factorised.
Sign errors when rewriting $3f - h$ as $-(h - 3f)$ ; forgetting the negative sign gives the wrong final factor.
Mixing up which variable goes with which factor, e.g. writing $(7h - 2y)(x - 3f)$ instead of keeping $x$ and $y$ with the first bracket and $h$ and $f$ with the second.
The mark scheme gives M1 for a correct grouped form such as $7x(h - 3f) - 2y(h - 3f)$ ; a final answer must be a product of two binomials.

Things to Be Careful About

Ensure the factorisation is complete: no common factor remains inside either bracket.
Both $(7x - 2y)(h - 3f)$ and $(2y - 7x)(3f - h)$ are accepted as final answers.
If you use the alternative grouping $h(7x - 2y) - 3f(7x - 2y)$ , you must factor out $3f$ with the correct sign: $6fy - 21fx = 3f(2y - 7x) = -3f(7x - 2y)$ .
This is a purely algebraic factorisation, so no calculator is needed.

Techniques used

group terms in pairsfactor out common monomial factorsrewrite a difference as the negative of its reversefactor out a common binomial

Similar questions

Q13EasyProbability

A bag contains 9 red counters and 3 green counters.
Lee takes two counters from the bag at random without replacement.

(a)

Complete the tree diagram.

Fig. 5

2M Similar

DifficultyEasy

Worked solution

Approach

The bag contains 9 red and 3 green counters, making 12 in total. For the first draw, the probability of green is $\frac{3}{12}$ . After the first draw, one counter is removed, leaving 11 counters for the second draw. The conditional probabilities depend on the outcome of the first draw.

Working

First draw:
Total counters = $9 + 3 = 12$ .

\text{P(Green)} = \frac{3}{12}

Second draw after Red:
Remaining counters: 8 red, 3 green (total 11).

\text{P(Green | Red)} = \frac{3}{11}

Second draw after Green:
Remaining counters: 9 red, 2 green (total 11).

\text{P(Red | Green)} = \frac{9}{11}, \quad \text{P(Green | Green)} = \frac{2}{11}

Answer

The completed tree diagram has the following probabilities:

First branch Green: $\frac{3}{12}$
Second branch Green (from Red): $\frac{3}{11}$
Second branch Red (from Green): $\frac{9}{11}$
Second branch Green (from Green): $\frac{2}{11}$

Final answer

First branch Green: 3/12; Second branch Green (from Red): 3/11; Second branch Red (from Green): 9/11; Second branch Green (from Green): 2/11

Detailed explanation

Walkthrough

The problem involves taking two counters without replacement from a bag of 12 (9 red, 3 green). A tree diagram models the sequence of events.

Step 1: First draw probabilities.
There are 12 counters in total. The probability of drawing a green counter on the first draw is the number of green counters divided by the total: $\frac{3}{12}$ . This fills the blank on the first lower branch.

Step 2: Second draw probabilities after Red.
If the first counter was red, there are now 8 red and 3 green counters left, making 11 in total. The probability of drawing a green counter next is $\frac{3}{11}$ . This fills the blank on the branch from 'Red' to 'Green'.

Step 3: Second draw probabilities after Green.
If the first counter was green, there are now 9 red and 2 green counters left, making 11 in total. The probability of drawing a red counter next is $\frac{9}{11}$ , and the probability of drawing a green counter next is $\frac{2}{11}$ . These fill the blanks on the branches from 'Green'.

Key Takeaways

When sampling without replacement, the total number of items decreases by 1 for each subsequent draw.
The numerator for the probability of a specific colour changes based on how many of that colour have already been removed.
A probability tree diagram must have all branches from a single node sum to 1.

Common Mistakes

Forgetting that the total number of counters decreases to 11 for the second draw.
Using the original counts (9 and 3) for the second draw instead of the updated counts (8 and 3, or 9 and 2).
Not simplifying fractions (though unsimplified correct fractions are usually accepted in tree diagrams unless stated otherwise).

Things to Be Careful About

Ensure the probabilities on branches originating from the same node sum to 1 (e.g., $\frac{8}{11} + \frac{3}{11} = 1$ and $\frac{9}{11} + \frac{2}{11} = 1$ ).
Position the fractions correctly next to their corresponding branches.

Techniques used

calculate initial probabilities for sampling without replacementupdate counts after each draw to find conditional probabilitiescomplete a probability tree diagram

(b)

Work out the probability that both counters are red.

______

1M Similar

DifficultyEasy

Worked solution

Approach

To find the probability that both counters are red, multiply the probabilities along the 'Red -> Red' path of the tree diagram.

Working

\text{P(both red)} = \text{P(Red first)} \times \text{P(Red second | Red first)}

\text{P(both red)} = \frac{9}{12} \times \frac{8}{11}

\text{P(both red)} = \frac{72}{132}

Simplify the fraction by dividing the numerator and denominator by their highest common factor, 12:

\frac{72 \div 12}{132 \div 12} = \frac{6}{11}

Answer

\frac{6}{11}

Final answer

6/11

Detailed explanation

Walkthrough

The question asks for the probability of both counters being red. This corresponds to the path 'Red' on the first draw and 'Red' on the second draw.

Step 1: Identify the probabilities.
From the tree diagram, the probability of the first counter being red is $\frac{9}{12}$ . Given that the first counter is red, the probability of the second counter being red is $\frac{8}{11}$ .

Step 2: Multiply the probabilities.
For combined events in a sequence (AND condition), multiply the probabilities along the path:

\frac{9}{12} \times \frac{8}{11} = \frac{72}{132}

Step 3: Simplify.
Divide both numerator and denominator by 12 to get the simplest form:

\frac{72}{132} = \frac{6}{11}

Key Takeaways

The probability of a sequence of events (AND) is found by multiplying the probabilities along the corresponding branches of a tree diagram.
Always simplify the final fraction to its lowest terms unless the mark scheme specifies otherwise.

Common Mistakes

Adding the probabilities instead of multiplying them.
Forgetting to simplify the final fraction (e.g., leaving it as $\frac{72}{132}$ ).
Using the wrong conditional probability for the second draw (e.g., using $\frac{9}{11}$ instead of $\frac{8}{11}$ ).

Things to Be Careful About

Ensure you are multiplying the correct branch probabilities: $\frac{9}{12}$ (first Red) and $\frac{8}{11}$ (second Red given first Red).
The answer must be in the form of a simplified fraction.

Techniques used

multiply probabilities along the branches of a tree diagramsimplify fractions

Q14Medium-EasyGeometry

Fig. 6

$PQRS$ is a cyclic quadrilateral.
$APB$ is a tangent to the circle at $P$ .
Angle $SPB = 37°$ and angle $PSQ = 85°$ .

(a)

Find angle $PQS$ .

Angle $PQS$ = ______

1M Similar

(b)

Find angle $QRS$ .

Angle $QRS$ = ______

2M Similar

DifficultyMedium-Easy

Worked solution

Approach

Find $\angle QPS$ using the angle sum of triangle $PQS$ , then use the property that opposite angles in a cyclic quadrilateral are supplementary to find $\angle QRS$ .

Working

In $\triangle PQS$ , the angles sum to $180^\circ$ :

\angle QPS = 180^\circ - \angle PQS - \angle PSQ

\angle QPS = 180^\circ - 37^\circ - 85^\circ = 58^\circ

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

\angle QRS + \angle QPS = 180^\circ

\angle QRS = 180^\circ - 58^\circ = 122^\circ

Answer

\angle QRS = 122^\circ

Final answer

122°

Detailed explanation

Walkthrough

First, consider triangle $PQS$ . We now know $\angle PQS = 37^\circ$ from part (a), and we are given $\angle PSQ = 85^\circ$ . The sum of angles in any triangle is $180^\circ$ , so we can find the third angle $\angle QPS$ :

\angle QPS = 180^\circ - 37^\circ - 85^\circ = 58^\circ

Next, use the property of cyclic quadrilaterals: opposite angles are supplementary (sum to $180^\circ$ ). The angles $\angle QPS$ and $\angle QRS$ are opposite angles in the cyclic quadrilateral $PQRS$ , so:

\angle QRS = 180^\circ - \angle QPS = 180^\circ - 58^\circ = 122^\circ

Key Takeaways

When a tangent is involved in a cyclic quadrilateral problem, the alternate segment theorem often provides one angle of a triangle formed by the diagonals. Combining this with the triangle angle sum and the cyclic quadrilateral opposite angles property completes the solution.

Common Mistakes

Forgetting to use the triangle angle sum and trying to find $\angle QRS$ directly without finding $\angle QPS$ first.
Misidentifying which angles are opposite in the cyclic quadrilateral. The vertices in order around the circle are $P, Q, R, S$ , so the opposite pairs are $\angle P$ and $\angle R$ , and $\angle Q$ and $\angle S$ .

Things to Be Careful About

Make sure to use the correct value of $\angle QPS$ obtained from the triangle. If part (a) was answered incorrectly, follow-through marks may apply, but ensure the arithmetic $180 - 37 - 85$ is correct. Also, remember that the answer must be in degrees and use the correct degree symbol.

Techniques used

calculate the third angle in a triangleapply the cyclic quadrilateral opposite angles property

Q15MediumAlgebra and Graphs

These are the first four patterns in a sequence made from black beads and from white beads.

Fig. 7

(a)

Complete the table for Pattern 5.

Pattern number	1	2	3	4	5
Number of white beads	5	7	9	11
Total number of beads	6	10	15	21

1M Similar

DifficultyMedium-Easy

Worked solution

Approach

The white beads increase by 2 each time, giving an arithmetic sequence. The black beads form the triangular numbers (1, 3, 6, 10, ...). Find the Pattern 5 values and fill the table.

Working

White beads: The sequence is $5, 7, 9, 11, \ldots$ with common difference $2$ .

\text{Pattern 5 white beads} = 11 + 2 = 13

Black beads: The sequence is $1, 3, 6, 10, \ldots$ These are triangular numbers where the $n$ th term is $\dfrac{n(n+1)}{2}$ .

\text{Pattern 5 black beads} = \frac{5 \times 6}{2} = 15

Total beads for Pattern 5:

13 + 15 = 28

Answer

Pattern number	1	2	3	4	5
Number of white beads	5	7	9	11	13
Total number of beads	6	10	15	21	28

Final answer

White beads: 13, Total: 28

Detailed explanation

Walkthrough

Look at the white bead counts: 5, 7, 9, 11. Each step adds 2, so the next value (Pattern 5) is 13. Look at the black bead counts: 1, 3, 6, 10. The differences between consecutive terms are 2, 3, 4, so the next difference is 5, giving 10 + 5 = 15 black beads. These are the triangular numbers, which count dots arranged in a triangle. The total for Pattern 5 is simply 13 + 15 = 28.

Key Takeaways

Recognising arithmetic sequences (constant difference) and triangular numbers (second difference constant at 2) from a table of values. The total in a composite pattern is the sum of its parts.

Common Mistakes

Forgetting to add the black beads when computing the total, giving 13 instead of 28.
Miscounting the black beads by continuing the arithmetic pattern (e.g. 10 + 2 = 12) instead of recognising the triangular number pattern.

Things to Be Careful About

The table asks for two separate values. Ensure both are filled in. The black beads follow a quadratic pattern, not an arithmetic one — check by computing first and second differences.

Techniques used

identify an arithmetic pattern in the white beadsrecognise triangular numbers in the black beadssum to find the total

(b)

Find an expression for

4M

(i)

the number of white beads in Pattern $n$

______

2M Similar

(ii)

the total number of beads in Pattern $n$ .

______

2M Similar

DifficultyMedium

Worked solution

Approach

Total beads = white beads + black beads. White beads = $2n + 3$ . Black beads are $1, 3, 6, 10, \ldots$ which are triangular numbers with nth term $\dfrac{n(n+1)}{2}$ . Add the two expressions.

Working

Black beads (triangular numbers):

\text{Black beads} = \frac{n(n+1)}{2} = \frac{n^2 + n}{2}

Total beads:

\text{Total} = (2n + 3) + \frac{n^2 + n}{2}

Put over a common denominator of $2$ :

\text{Total} = \frac{4n + 6}{2} + \frac{n^2 + n}{2} = \frac{n^2 + 5n + 6}{2}

Rewrite as:

\text{Total} = \frac{1}{2}n^2 + \frac{5}{2}n + 3

Check: For $n = 3$ : $\frac{1}{2}(9) + \frac{5}{2}(3) + 3 = 4.5 + 7.5 + 3 = 15$ ✓

Answer

\frac{1}{2}n^2 + \frac{5}{2}n + 3

Final answer

(1/2)n^2 + (5/2)n + 3

Detailed explanation

Walkthrough

The black beads are 1, 3, 6, 10. Compute first differences: 2, 3, 4. Compute second differences: 1, 1. Since the second difference is constant and equal to 1, the sequence is quadratic with leading coefficient $\frac{1}{2} \times 1 = \frac{1}{2}$ . The black beads are the triangular numbers, which have the well-known formula $\frac{n(n+1)}{2} = \frac{n^2 + n}{2}$ . Adding this to the white bead expression $2n + 3$ and simplifying over a common denominator gives $\frac{n^2 + 5n + 6}{2} = \frac{1}{2}n^2 + \frac{5}{2}n + 3$ .

Key Takeaways

Triangular numbers have the formula $\frac{n(n+1)}{2}$ . When a sequence has a constant second difference $d_2$ , the leading coefficient of the quadratic nth term is $\frac{d_2}{2}$ . Always verify by substituting known values.

Common Mistakes

Writing the black bead expression as $n^2$ or $n^2 + n$ instead of $\frac{n(n+1)}{2}$ .
Forgetting to combine over a common denominator when adding the linear and quadratic parts.
Not simplifying the final expression into the standard form.

Things to Be Careful About

The mark scheme accepts $\frac{1}{2}n^2 + \frac{5}{2}n + 3$ or any equivalent form. The expression must be in terms of $n$ . Verify by checking against the given table values.

Techniques used

recognise triangular numbers as a quadratic sequencederive the quadratic nth term formulaadd the two expressions

(c)

Bill has only 88 white beads but lots of black beads.
Pattern $p$ is the largest possible pattern he can make using these beads.

Work out the value of $p$ .

$p$ = ______

2M Similar

DifficultyMedium

Worked solution

Approach

Bill has 88 white beads. The number of white beads in Pattern $p$ is $2p + 3$ . We need $2p + 3 \leq 88$ and find the largest integer $p$ .

Working

Set up the inequality:

2p + 3 \leq 88

Subtract 3 from both sides:

2p \leq 85

Divide by 2:

p \leq 42.5

Since $p$ must be a whole number (pattern numbers are positive integers), the largest possible value is:

p = 42

Check: At $p = 42$ , white beads needed = $2(42) + 3 = 87 \leq 88$ ✓
At $p = 43$ , white beads needed = $2(43) + 3 = 89 > 88$ ✗

Answer

p = 42

Final answer

42

Detailed explanation

Walkthrough

From part (b)(i), the number of white beads in Pattern $n$ is $2n + 3$ . Bill has 88 white beads, so we need $2p + 3 \leq 88$ . Solving: $2p \leq 85$ , so $p \leq 42.5$ . Since pattern numbers must be whole numbers, the largest valid $p$ is 42. At $p = 42$ , Bill uses 87 white beads (leaving 1 unused). At $p = 43$ , he would need 89 white beads, which exceeds his supply.

Key Takeaways

When a quantity must not exceed a limit and the variable must be a whole number, solve the inequality and round down. Always verify both the chosen value and the next integer to confirm the boundary.

Common Mistakes

Setting $2p + 3 = 88$ exactly, giving $p = 42.5$ , and then rounding to 43 instead of rounding down to 42.
Forgetting that pattern numbers must be positive integers.
Using the total bead expression instead of the white bead expression.

Things to Be Careful About

The question says Bill has only 88 white beads, so this is an inequality ( $\leq$ ), not an equation ( $=$ ). The answer must be a whole number. The mark scheme awards method marks for setting up $2p + 3 = 88$ (or $\leq 88$ ) and solving, then correctly concluding $p = 42$ .

Techniques used

set up an inequality from the bead constraintsolve the linear equationround down to the nearest whole number

Q162MMedium-EasyNumber

The population of Kenya is $5.71 \times 10^7$ people.
The area of Kenya is $6 \times 10^5\,\text{km}^2$ .
The population density is the number of people per $\text{km}^2$ .

Calculate the population density of Kenya.

______ $\text{people / km}^2$

DifficultyMedium-Easy

Worked solution

Approach

Population density is defined as the number of people divided by the area. We are given both quantities in standard form ( $a \times 10^n$ ). To divide these, we separate the coefficients (the $a$ parts) and the powers of 10, then simplify.

Working

The formula for population density is:

\text{Density} = \frac{\text{Population}}{\text{Area}}

Substitute the given values into the formula:

\text{Density} = \frac{5.71 \times 10^7}{6 \times 10^5}

Group the coefficients and the powers of 10 separately:

\text{Density} = \left( \frac{5.71}{6} \right) \times \left( \frac{10^7}{10^5} \right)

Calculate the coefficient part:

\frac{5.71}{6} = 0.951666... \approx 0.9517

Calculate the power of 10 part using index laws ( $\frac{x^a}{x^b} = x^{a-b}$ ):

\frac{10^7}{10^5} = 10^{7-5} = 10^2 = 100

Combine the results:

0.951666... \times 100 = 95.1666...

Round to a reasonable degree of accuracy (3 significant figures or nearest whole number):

95.2 \quad \text{or} \quad 95

Answer

95.2 \text{ people / km}^2

Final answer

95.2

Detailed explanation

Walkthrough

First, identify the relationship between the quantities. The problem states that "population density is the number of people per km²". This means we must divide the population count by the land area.

We have:

Population = $5.71 \times 10^7$
Area = $6 \times 10^5$

Both numbers are written in standard form (scientific notation), which looks like $a \times 10^n$ , where $1 \leq a < 10$ . When dividing two numbers in standard form, it is easiest to split the calculation into two parts: the non-exponential numbers (coefficients) and the powers of 10.

Step 1: Divide the coefficients.

\frac{5.71}{6}

This equals approximately $0.951666...$

Step 2: Divide the powers of 10.
Using the law of indices for division ( $\frac{10^m}{10^n} = 10^{m-n}$ ):

\frac{10^7}{10^5} = 10^{7-5} = 10^2

$10^2$ is equal to 100.

Step 3: Multiply the results together.

0.951666... \times 100 = 95.1666...

Finally, round the answer. The question does not specify rounding, but standard practice is to use 3 significant figures or match the precision of the data. $95.2$ is an appropriate answer. The mark scheme also accepts 95 or 95.17.

Key Takeaways

Formula: Density = Quantity / Volume or Density = Quantity / Area depending on context.
Standard Form Division: Treat the mantissa (coefficient) and the exponent separately. $(A \times 10^a) \div (B \times 10^b) = (A \div B) \times 10^{a-b}$ .
Calculator Use: On a calculator component, you can often type the expression directly, but understanding how to separate the parts helps check if your answer makes sense.

Common Mistakes

Subtracting instead of dividing: Students sometimes subtract the areas or populations rather than finding the ratio.
Index errors: Subtracting the exponents incorrectly (e.g., $7-5=3$ ) or forgetting to apply the result of the exponent division to the coefficient.
Decimal placement: Dividing $5.71$ by $6$ gives $0.95...$ , not $95...$ . If you forget to multiply by the $100$ ( $10^2$ ) at the end, your answer will be off by a factor of 100.

Things to Be Careful About

Units: Ensure the final answer includes the units requested ( $\text{people / km}^2$ ).
Accuracy: If using a calculator, ensure you do not round intermediate steps too early. Keep the full value of $5.71/6$ until the final multiplication by 100.

Techniques used

divide numbers in standard formcalculate population density from total population and area

Similar questions

Q17Medium-EasyAlgebra and Graphs

(a)

$f(x) = \frac{10 - x}{3}$

Find $f(-8)$ .

______

1M Similar

(b)

$g(x) = 4x + 3$

Find $g^{-1}(x)$ .

$g^{-1}(x)$ = ______

2M Similar

DifficultyMedium-Easy

Worked solution

Approach

Write $y = g(x)$ , interchange $x$ and $y$ , then rearrange to make $y$ the subject. The result is the inverse function.

Working

Let

y = 4x + 3

Interchange $x$ and $y$ :

x = 4y + 3

Subtract $3$ from both sides:

x - 3 = 4y

Divide both sides by $4$ :

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

So

g^{-1}(x) = \frac{x - 3}{4}

Answer

g^{-1}(x) = \frac{x - 3}{4}

Final answer

(x - 3)/4

Detailed explanation

Walkthrough

To find the inverse of $g(x) = 4x + 3$ , start by writing the function as $y = 4x + 3$ . The inverse function undoes the operation of $g$ , so we swap the roles of $x$ and $y$ , giving $x = 4y + 3$ . Then solve for $y$ : subtract $3$ to get $x - 3 = 4y$ , then divide by $4$ to get $y = \frac{x - 3}{4}$ . Replacing $y$ with $g^{-1}(x)$ gives the inverse.

Key Takeaways

The inverse of a function is found by swapping $x$ and $y$ and rearranging. It "undoes" the original function: applying $g^{-1}$ after $g$ returns the original input. Equivalent forms such as $\frac{x}{4} - \frac{3}{4}$ are accepted.

Common Mistakes

Stopping at $x = 4y + 3$ without solving for $y$ ; this only earns the method mark, not the final answer.
Swapping variables but not rearranging, e.g. writing $g^{-1}(x) = 4x + 3$ .
Dividing only one term by $4$ , e.g. writing $x - \frac{3}{4}$ instead of $\frac{x - 3}{4}$ .

Things to Be Careful About

The final answer must have $x$ as the subject. The mark scheme allows any equivalent form, e.g. $\frac{x}{4} - \frac{3}{4}$ , so either is fine. Show the rearrangement step $x = 4y + 3$ to secure the method mark. No calculator is needed.

Techniques used

write the function using y notationinterchange x and yrearrange the equation to make y the subjectstate the inverse function

Q18Medium-EasyStatistics

300 students take part in a competition.
The table shows information about their scores.

Score ( $s$ )	$0 < s \leq 30$	$30 < s \leq 40$	$40 < s \leq 60$	$60 < s \leq 70$	$70 < s \leq 100$
Frequency	24	70	88	76	42

(a)

Calculate an estimate of the mean score.

______

4M Similar

DifficultyMedium-Easy

Worked solution

Approach

Find the midpoint of each score interval, multiply by the frequency to get $\sum fx$ , then divide by the total number of students (300).

Working

Midpoints ( $x$ ):

\begin{aligned} 0 < s \leq 30 &\rightarrow x = 15 \\ 30 < s \leq 40 &\rightarrow x = 35 \\ 40 < s \leq 60 &\rightarrow x = 50 \\ 60 < s \leq 70 &\rightarrow x = 65 \\ 70 < s \leq 100 &\rightarrow x = 85 \end{aligned}

Calculate $\sum fx$ :

\begin{aligned} \sum fx &= (15 \times 24) + (35 \times 70) + (50 \times 88) + (65 \times 76) + (85 \times 42) \\ &= 360 + 2450 + 4400 + 4940 + 3570 \\ &= 15720 \end{aligned}

Calculate the mean:

\text{Mean} = \frac{\sum fx}{\sum f} = \frac{15720}{300} = 52.4

Answer

52.4

Final answer

52.4

Detailed explanation

Walkthrough

To estimate the mean from grouped data, we assume all values in a class interval are concentrated at the midpoint of that interval. First, we find the midpoint $x$ of each score range by averaging the lower and upper bounds. Next, we multiply each midpoint by its corresponding frequency $f$ to get $fx$ , which represents the estimated total score for that group. Summing these products gives $\sum fx = 15720$ . Finally, we divide this total by the overall number of students, $\sum f = 300$ , to find the estimated mean score of $52.4$ .

Key Takeaways

The estimated mean of grouped data is calculated using $\bar{x} = \frac{\sum fx}{\sum f}$ , where $x$ is the class midpoint.
Always verify that the sum of the frequencies equals the total number of observations given in the problem.

Common Mistakes

Using the class boundary or the upper limit instead of the midpoint for $x$ .
Forgetting to divide by the total frequency (300) and just reporting $\sum fx$ .
Arithmetic errors when multiplying large numbers like $65 \times 76$ or $85 \times 42$ .

Things to Be Careful About

The mark scheme accepts $52\frac{2}{5}$ as an equivalent exact form, but $52.4$ is the standard decimal representation.
Ensure the midpoints are correct: for $70 < s \leq 100$ , the midpoint is $\frac{70+100}{2} = 85$ , not $80$ .
The total frequency must be exactly 300; if it isn't, there is a data entry error.

Techniques used

calculate midpoints of grouped intervalsmultiply midpoints by frequenciessum the products and divide by total frequency

(b)

On the grid, complete the histogram to represent the data in the table.

Fig. 8

3M Similar

DifficultyMedium-Easy

Worked solution

Approach

Frequency density is calculated as $\frac{\text{Frequency}}{\text{Class width}}$ . Calculate this for each interval, then draw bars on the grid with the appropriate widths and heights.

Working

For $0 < s \leq 30$ : class width $= 30$ , frequency $= 24$ . Frequency density $= \frac{24}{30} = 0.8$ .

For $40 < s \leq 60$ : class width $= 20$ , frequency $= 88$ . Frequency density $= \frac{88}{20} = 4.4$ .

For $60 < s \leq 70$ : class width $= 10$ , frequency $= 76$ . Frequency density $= \frac{76}{10} = 7.6$ .

For $70 < s \leq 100$ : class width $= 30$ , frequency $= 42$ . Frequency density $= \frac{42}{30} = 1.4$ .

The bar for $30 < s \leq 40$ is already drawn with frequency density $7$ .

Answer

Bars to draw: $0 < s \leq 30$ with height $0.8$ ; $40 < s \leq 60$ with height $4.4$ ; $60 < s \leq 70$ with height $7.6$ ; $70 < s \leq 100$ with height $1.4$ .

Final answer

Bars at 0-30 (height 0.8), 40-60 (height 4.4), 60-70 (height 7.6), 70-100 (height 1.4)

Detailed explanation

Walkthrough

A histogram with unequal class widths requires the vertical axis to represent frequency density, not frequency, to ensure the area of each bar is proportional to the frequency. Frequency density is defined as $\text{Frequency} \div \text{Class width}$ . We calculate this for each interval:

$0 < s \leq 30$ : $24 \div 30 = 0.8$
$40 < s \leq 60$ : $88 \div 20 = 4.4$
$60 < s \leq 70$ : $76 \div 10 = 7.6$
$70 < s \leq 100$ : $42 \div 30 = 1.4$
We then draw rectangular bars spanning the correct intervals on the horizontal axis with heights matching these density values. The bar for $30 < s \leq 40$ is already provided as a reference.

Key Takeaways

In a histogram with unequal class widths, the y-axis must be frequency density.
Area of bar $=$ frequency. Always check that $\text{width} \times \text{height} = \text{frequency}$ .

Common Mistakes

Using frequency directly as the height of the bars (e.g., drawing a bar of height 88 for the $40 < s \leq 60$ interval).
Miscalculating the class width (e.g., using 30 for the $70 < s \leq 100$ interval is correct, but sometimes students use 100).
Drawing bars that do not touch each other (histogram bars for continuous data must be adjacent with no gaps).

Things to Be Careful About

The grid's vertical axis only goes up to 8, so the tallest bar ( $7.6$ ) will be near the top but still visible.
Ensure bars are drawn exactly between the correct class boundaries (e.g., the $40 < s \leq 60$ bar must span from $s=40$ to $s=60$ , not $s=40$ to $s=50$ ).
The mark scheme awards 2 marks for 3 correct bars and 1 mark for 2 correct bars, so accuracy in drawing is key.

Techniques used

calculate frequency density for each class intervalidentify class boundaries on the horizontal axisdraw bars with correct widths and heights

Q19MediumAlgebra and Graphs

(a)

Complete the table of values for $y = x^3 - 5x + 4$ .

$x$	-3	-2	-1	0	1	2	3
$y$		6	8	4	0	2	16

1M Similar

(b)

Draw the graph of $y = x^3 - 5x + 4$ for $-3 \leq x \leq 3$ .

Fig. 9

4M Similar

DifficultyMedium-Easy

Worked solution

Approach

Plot all seven points from the table of values on the given grid, then join them with a smooth curve that reflects the shape of a cubic function.

Working

The table of values is:

$x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$
$y$	$-8$	$6$	$8$	$4$	$0$	$2$	$16$

Answer

A smooth cubic curve passing through the points $(-3,-8)$ , $(-2,6)$ , $(-1,8)$ , $(0,4)$ , $(1,0)$ , $(2,2)$ , $(3,16)$ with a local maximum near $(-1, 8)$ and a local minimum near $(1, 0)$ .

\text{Curve through } (-3,-8), (-2,6), (-1,8), (0,4), (1,0), (2,2), (3,16)

Final answer

Smooth cubic curve through (-3,-8), (-2,6), (-1,8), (0,4), (1,0), (2,2), (3,16)

Detailed explanation

Walkthrough

Part (a) gives us the complete table of values. We plot each point $(x, y)$ on the grid: $(-3, -8)$ , $(-2, 6)$ , $(-1, 8)$ , $(0, 4)$ , $(1, 0)$ , $(2, 2)$ , $(3, 16)$ . Then we join them with a smooth curve. Since this is a cubic with a positive leading coefficient, the curve rises from the bottom-left, reaches a local maximum around $x = -1$ , dips to a local minimum around $x = 1$ , then rises steeply to the top-right. The curve must pass through all seven points smoothly.

Key Takeaways

When drawing a graph from a table of values, plot every point accurately and join them with a smooth curve appropriate to the function type. For cubics, expect one local maximum and one local minimum within the range.

Common Mistakes

Plotting $(-3, -8)$ incorrectly, for example as $(-3, 8)$ by forgetting the negative sign from part (a).
Joining the points with straight line segments instead of a smooth curve.
Not extending the curve smoothly past the first and last points — the curve should continue naturally beyond $x = -3$ and $x = 3$ .
Drawing the curve to pass through points that aren't on it, such as making it go through $(0, 0)$ instead of $(0, 4)$ .

Things to Be Careful About

The grid has $y$ from $-20$ to $20$ and $x$ from $-3$ to $3$ , so each major grid square on the $y$ -axis represents $2$ units. Ensure points are plotted at the correct $y$ -values. The mark scheme awards follow-through marks for correctly plotted points (B3FT for 6 points, B2FT for 4, B1FT for 2), so accuracy in plotting is critical. The curve must be smooth — no sharp corners at the plotted points.

Techniques used

plot the table of values on the coordinate griddraw a smooth cubic curve through the plotted points

(c)

By drawing a suitable straight line on the grid, find the solutions of the equation $x^3 - 5x - 3 = 0$ .

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

3M Similar

DifficultyMedium

Worked solution

Approach

We need to solve $x^3 - 5x - 3 = 0$ . Since the graph drawn in part (b) is $y = x^3 - 5x + 4$ , we rewrite the equation to match this form by adding $7$ to both sides:

x^3 - 5x + 4 = 7

So we draw the horizontal line $y = 7$ on the graph and read off the $x$ -coordinates where it intersects the curve.

Working

The equation $x^3 - 5x - 3 = 0$ can be rewritten as:

x^3 - 5x + 4 = 7

This means we need the $x$ -values where $y = 7$ on the graph of $y = x^3 - 5x + 4$ .

Looking at the graph:

Between $x = -2$ ( $y = 6$ ) and $x = -1$ ( $y = 8$ ), the curve crosses $y = 7$ at approximately $x \approx -1.8$ .
Between $x = -1$ ( $y = 8$ ) and $x = 0$ ( $y = 4$ ), the curve crosses $y = 7$ at approximately $x \approx -0.7$ .
Between $x = 2$ ( $y = 2$ ) and $x = 3$ ( $y = 16$ ), the curve crosses $y = 7$ at approximately $x \approx 2.5$ .

Answer

x \approx -1.8, \quad x \approx -0.7, \quad x \approx 2.5

(Acceptable range: $-1.95$ to $-1.7$ , $-0.8$ to $-0.5$ , $2.3$ to $2.6$ )

x = -1.8, \; x = -0.7, \; x = 2.5

Final answer

x ≈ -1.8, x ≈ -0.7, x ≈ 2.5

Detailed explanation

Walkthrough

We want to solve $x^3 - 5x - 3 = 0$ . The graph we already drew is $y = x^3 - 5x + 4$ . We need to express our equation in terms of this $y$ . Rearranging: $x^3 - 5x = 3$ , so $x^3 - 5x + 4 = 3 + 4 = 7$ . Therefore $y = 7$ . We draw the horizontal line $y = 7$ on the graph and find where it meets the curve.

From the table, $y = 7$ lies between $y = 6$ (at $x = -2$ ) and $y = 8$ (at $x = -1$ ), so there is an intersection in that interval. The curve also descends from $y = 8$ at $x = -1$ to $y = 4$ at $x = 0$ , crossing $y = 7$ again. Finally, the curve rises from $y = 2$ at $x = 2$ to $y = 16$ at $x = 3$ , crossing $y = 7$ a third time.

Reading off the graph, the three solutions are approximately $x \approx -1.8$ , $x \approx -0.7$ , and $x \approx 2.5$ .

Key Takeaways

To solve an equation graphically using a previously drawn graph, rewrite the equation so one side matches the function already plotted. The other side becomes the equation of a straight line (often horizontal) to draw. The solutions are the $x$ -coordinates of the intersection points.

Common Mistakes

Drawing the wrong line. For example, drawing $y = 3$ or $x = 7$ instead of $y = 7$ . The key step is correctly rewriting $x^3 - 5x - 3 = 0$ as $x^3 - 5x + 4 = 7$ .
Reading the $y$ -coordinate instead of the $x$ -coordinate at the intersection.
Only finding two intersections when there are three, missing the one between $x = -1$ and $x = 0$ .
Not drawing the line across the full width of the graph so all intersections are visible.

Things to Be Careful About

The mark scheme accepts answers in the ranges $-1.95$ to $-1.7$ , $-0.8$ to $-0.5$ , and $2.3$ to $2.6$ . Readings must be from the drawn graph, so slight variations are expected. The line $y = 7$ must be clearly ruled on the grid. If the line is not drawn (M0), full credit (SC1) is still awarded if all three solutions are stated correctly. The equation $x^3 - 5x + 4 = 7$ must be shown as the reason for drawing $y = 7$ .

Techniques used

rewrite the equation to identify a suitable straight line to drawdraw the horizontal line on the graphread the x-coordinates of the intersection points

Q20MediumAlgebra and GraphsMensuration

$ABCD$ and $EFGD$ are rectangles.

Fig. 10

(a)

4M

(i)

Show that $CD = x - 1$ .

1M Similar

DifficultyEasy

Worked solution

Approach

Since $EFGD$ is a rectangle, its opposite sides are equal. The bottom side $DG$ equals the top side $EF$ . The point $C$ lies on $DG$ , so $DG = DC + CG$ . We can find $CD$ by subtracting $CG$ from $DG$ .

Working

DG = EF = 2x - 3

CD = DG - CG = (2x - 3) - (x - 2)

CD = 2x - 3 - x + 2 = x - 1

Answer

CD = x - 1

Final answer

x - 1

Detailed explanation

Walkthrough

The large shape $EFGD$ is a rectangle, which means its opposite sides are equal in length. The top side $EF$ is given as $2x - 3$ , so the bottom side $DG$ must also be $2x - 3$ . The bottom side is split into two segments by point $C$ : $DC$ and $CG$ . We are given $CG = x - 2$ . To find $CD$ , we subtract $CG$ from the total length $DG$ :

CD = DG - CG = (2x - 3) - (x - 2)

Distributing the negative sign gives $2x - 3 - x + 2$ . Combining like terms ( $2x - x = x$ and $-3 + 2 = -1$ ) yields $x - 1$ .

Key Takeaways

In any rectangle, opposite sides are equal. If a side is split into segments, the whole length is the sum of the parts. Algebraic subtraction must handle signs carefully, especially when subtracting a binomial.

Common Mistakes

Forgetting to bracket $(x - 2)$ when subtracting, leading to $2x - 3 - x - 2 = x - 5$ instead of $x - 1$ . This is a very common error.
Assuming $ED = FG$ directly without noticing the diagram labels; the diagram labels $ED = x$ and $FG = x + 3$ , which seems contradictory but is resolved by understanding $ABCD$ is the inner rectangle and the height of the large rectangle is $x + 3$ while $AD = 3$ (so $AE = x - 3$ ).

Things to Be Careful About

Always use brackets when subtracting a quantity that contains a variable and a constant. The mark scheme awards the mark for the correct simplification $2x - 3 - (x - 2) = x - 1$ with no errors seen.

Techniques used

identify equal opposite sides of a rectanglesubtract segment lengths to find an unknown side

(ii)

The area of rectangle $ABCD$ is $\frac{1}{5}$ of the area of rectangle $EFGD$ .

Form an equation in $x$ and show that it simplifies to $x^2 - 6x + 3 = 0$ .

3M Similar

DifficultyMedium

Worked solution

Approach

The area of a rectangle is length $\times$ width. We write the area of $ABCD$ and the area of $EFGD$ in terms of $x$ , then use the given relationship (area of $ABCD$ is $\frac{1}{5}$ of area of $EFGD$ ) to form an equation. Finally, we expand and simplify to reach the target form $x^2 - 6x + 3 = 0$ .

Working

Area of rectangle $ABCD$ :

\text{Area}_{ABCD} = CD \times AD = (x - 1) \times 3 = 3(x - 1)

Area of rectangle $EFGD$ :

\text{Area}_{EFGD} = EF \times FG = (2x - 3)(x + 3)

The problem states:

\text{Area}_{ABCD} = \frac{1}{5} \times \text{Area}_{EFGD}

Substituting the expressions:

3(x - 1) = \frac{1}{5}(2x - 3)(x + 3)

Multiply both sides by $5$ to clear the fraction:

15(x - 1) = (2x - 3)(x + 3)

Expand the left side:

15x - 15 = (2x - 3)(x + 3)

Expand the right side:

(2x - 3)(x + 3) = 2x^2 + 6x - 3x - 9 = 2x^2 + 3x - 9

So the equation is:

15x - 15 = 2x^2 + 3x - 9

Rearrange all terms to one side (subtract $15x$ and add $15$ to both sides):

0 = 2x^2 + 3x - 15x - 9 + 15

0 = 2x^2 - 12x + 6

Divide the entire equation by $2$ :

0 = x^2 - 6x + 3

x^2 - 6x + 3 = 0

Answer

x^2 - 6x + 3 = 0

Final answer

x^2 - 6x + 3 = 0

Detailed explanation

Walkthrough

First, we need the dimensions of both rectangles. For $ABCD$ , we found $CD = x - 1$ in part (a)(i). The marking scheme uses $AD = 3$ , which is the height of the small rectangle. (This comes from the geometry: $FG = x + 3$ is the total height, and the diagram implies $AD = 3$ so that $AE = x - 3$ , though this is implicitly given by the area formula $3(x-1)$ in the mark scheme). So area of $ABCD = 3(x - 1)$ .

For $EFGD$ , the dimensions are $EF = 2x - 3$ and $FG = x + 3$ . So area of $EFGD = (2x - 3)(x + 3)$ .

The condition is: $\text{Area}_{ABCD} = \frac{1}{5} \text{Area}_{EFGD}$ . Writing this out:

3(x - 1) = \frac{1}{5}(2x - 3)(x + 3)

Multiply by 5:

15(x - 1) = (2x - 3)(x + 3)

Expand both sides:

15x - 15 = 2x^2 + 6x - 3x - 9 = 2x^2 + 3x - 9

Bring everything to the right side:

2x^2 + 3x - 15x - 9 + 15 = 0

2x^2 - 12x + 6 = 0

Divide by 2:

x^2 - 6x + 3 = 0

This matches the required form.

Key Takeaways

When forming equations from geometry problems, always write out the area formulas first. Clearing fractions early makes the expansion and simplification easier. Always double-check your expansion of binomials, especially the middle term.

Common Mistakes

Forgetting to multiply the entire right side by 5, leading to $15x - 15 = 2x^2 - 3x + 6x - 9$ without the factor of 5 on the right.
Sign errors when expanding $(2x - 3)(x + 3)$ : writing $2x^2 - 3x - 9$ instead of $2x^2 + 3x - 9$ .
Sign errors when rearranging: $15x - 15 = 2x^2 + 3x - 9 \Rightarrow 2x^2 + 3x - 15x - 9 + 15 = 2x^2 - 12x + 6$ . Forgetting to change the sign of $15x$ or $-15$ is common.
Forgetting to divide by 2 at the end, leaving $2x^2 - 12x + 6 = 0$ instead of the required $x^2 - 6x + 3 = 0$ .

Things to Be Careful About

The mark scheme requires "no errors or omissions at any stage in the working" for the final A1 mark. Show every step: the initial equation, the clearing of the fraction, the expansion, the rearrangement, and the division by 2. The final answer must be exactly $x^2 - 6x + 3 = 0$ .

Techniques used

write area formulas for rectanglesform equation from percentage relationshipexpand binomials and simplify to standard quadratic form

(b)

5M

(i)

Use the quadratic formula to solve the equation $x^2 - 6x + 3 = 0$ .
You must show all your working and give your answers correct to 2 decimal places.

$x$ = ______ or $x$ = ______

3M Similar

DifficultyMedium-Easy

Worked solution

Approach

The equation is $x^2 - 6x + 3 = 0$ . This is a quadratic in the form $ax^2 + bx + c = 0$ with $a = 1$ , $b = -6$ , $c = 3$ . We apply the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Working

Substitute $a = 1$ , $b = -6$ , $c = 3$ :

x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(3)}}{2(1)}

x = \frac{6 \pm \sqrt{36 - 12}}{2}

x = \frac{6 \pm \sqrt{24}}{2}

Calculate $\sqrt{24}$ :

\sqrt{24} \approx 4.898979...

Find the two values:

x_1 = \frac{6 + 4.898979...}{2} = \frac{10.898979...}{2} = 5.449489... \approx 5.45

x_2 = \frac{6 - 4.898979...}{2} = \frac{1.101020...}{2} = 0.550510... \approx 0.55

Answer

x = 5.45 \quad \text{or} \quad x = 0.55

Final answer

5.45 or 0.55

Detailed explanation

Walkthrough

The equation $x^2 - 6x + 3 = 0$ has coefficients $a = 1$ , $b = -6$ , $c = 3$ . The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

Substituting:

x = \frac{6 \pm \sqrt{36 - 12}}{2} = \frac{6 \pm \sqrt{24}}{2}

Using a calculator, $\sqrt{24} \approx 4.898979$ .

First solution (using +):

x = \frac{6 + 4.898979}{2} = \frac{10.898979}{2} = 5.449489... \approx 5.45

Second solution (using -):

x = \frac{6 - 4.898979}{2} = \frac{1.101021}{2} = 0.550510... \approx 0.55

Both answers are given to 2 decimal places as required.

Key Takeaways

The quadratic formula works for any quadratic equation $ax^2 + bx + c = 0$ . Always show the substitution step clearly. When using a calculator, keep full precision in the calculator and only round at the very end to avoid rounding errors.

Common Mistakes

Forgetting the $\pm$ sign and only giving one answer.
Rounding $\sqrt{24}$ too early (e.g., using $4.9$ ) which gives $x = 5.45$ and $x = 0.55$ anyway here, but could cause errors in other questions.
Sign errors in $-b$ : since $b = -6$ , $-b = 6$ , not $-6$ .
Arithmetic errors in $b^2 - 4ac$ : $(-6)^2 = 36$ , not $-36$ .

Things to Be Careful About

The question asks for answers correct to 2 decimal places. Do not round intermediate values. The mark scheme accepts $5.45$ and $0.55$ . Show the full formula substitution to earn the B2 mark.

Techniques used

identify coefficients a, b, csubstitute into quadratic formulaevaluate square root and compute final values to 2 d.p.

(ii)

Calculate the shaded area.

______ $\text{units}^2$

2M Similar

DifficultyMedium

Worked solution

Approach

The shaded area is the area of the large rectangle $EFGD$ minus the area of the small rectangle $ABCD$ . We must first decide which value of $x$ to use. Since $CG = x - 2$ is a length, it must be positive, so $x > 2$ . This means we must use $x = 5.45$ (discarding $x = 0.55$ ). Then we substitute this value into the area formula.

Working

The shaded area is:

\text{Shaded Area} = \text{Area}_{EFGD} - \text{Area}_{ABCD}

\text{Shaded Area} = (2x - 3)(x + 3) - 3(x - 1)

We need $x > 2$ because $CG = x - 2 > 0$ . From part (b)(i), the solutions are $x = 5.45$ and $x = 0.55$ . Since $0.55 < 2$ , we reject $x = 0.55$ and use $x = 5.45$ .

Substitute $x = 5.45$ :

\text{Area}_{EFGD} = (2(5.45) - 3)(5.45 + 3) = (10.9 - 3)(8.45) = 7.9 \times 8.45

7.9 \times 8.45 = 66.755

\text{Area}_{ABCD} = 3(5.45 - 1) = 3(4.45) = 13.35

\text{Shaded Area} = 66.755 - 13.35 = 53.405

Rounding to a reasonable number of decimal places (the mark scheme accepts 53.4 to 53.41):

\text{Shaded Area} \approx 53.4 \text{ units}^2

(Note: using the exact value $x = 3 + \sqrt{6}$ gives $53.38...$ , which rounds to $53.4$ .)

Answer

53.4 \text{ units}^2

Final answer

53.4

Detailed explanation

Walkthrough

The shaded region is the large rectangle $EFGD$ with the small rectangle $ABCD$ removed from it. So:

\text{Shaded Area} = (2x - 3)(x + 3) - 3(x - 1)

We have two possible values for $x$ : $5.45$ and $0.55$ . We must check which one is physically valid. The length $CG = x - 2$ must be positive (a length cannot be negative or zero in this context). If $x = 0.55$ , then $CG = 0.55 - 2 = -1.45$ , which is impossible. So we must use $x = 5.45$ .

Substitute $x = 5.45$ into the area expression:

Large rectangle: $(2 \times 5.45 - 3)(5.45 + 3) = (10.9 - 3)(8.45) = 7.9 \times 8.45 = 66.755$

Small rectangle: $3(5.45 - 1) = 3 \times 4.45 = 13.35$

Shaded area: $66.755 - 13.35 = 53.405$

The mark scheme accepts $53.4$ or $53.38$ to $53.41$ . Using the exact value $x = 3 + \sqrt{6} \approx 5.44949$ gives $53.382...$ , which rounds to $53.4$ .

Key Takeaways

When solving geometry problems with quadratic equations, always check your roots against the physical constraints of the problem (lengths must be positive, areas must be positive). Discard any root that makes a given length negative or zero.

Common Mistakes

Using $x = 0.55$ without checking validity. This gives $CG = -1.45$ , which is impossible.
Calculating the area of only one rectangle instead of the shaded region (difference).
Arithmetic errors in the final multiplication and subtraction.
Not showing the substitution of their $x$ value into the correct area formula.

Things to Be Careful About

The mark scheme awards M1 for substituting their $x$ value into the correct area formula where $x > 2$ . This means even if a student used the wrong $x$ value but showed the correct method, they might get partial credit depending on the scheme, but here $x = 0.55$ is geometrically invalid so it must be rejected. The final answer should be given to an appropriate number of significant figures or decimal places; 53.4 is acceptable.

Techniques used

select the valid x value based on geometric constraintssubstitute into area formula for shaded regioncalculate final numerical area

Q21MediumMensurationGeometry

Fig. 11

The diagram shows a solid formed by joining a cone to a hemisphere.
The diameter of the cone is $14\,\text{cm}$ and the diameter of the hemisphere is $14\,\text{cm}$ .
The total height of the solid is $24\,\text{cm}$ .

(a)

Calculate the volume of the solid.

______ $\text{cm}^3$

3M Similar

DifficultyMedium-Easy

Worked solution

Approach

The solid is a composite of a hemisphere at the bottom and a cone on top. The total volume is the sum of the volume of the hemisphere and the volume of the cone. We first find the radius and the individual heights of each part.

Working

The diameter of both the cone and the hemisphere is $14\,\text{cm}$ , so the radius is:

r = \frac{14}{2} = 7\,\text{cm}

The height of the hemisphere is equal to its radius, so $7\,\text{cm}$ . The total height of the solid is $24\,\text{cm}$ , so the height of the cone is:

h_{\text{cone}} = 24 - 7 = 17\,\text{cm}

Volume of the hemisphere:

V_{\text{hemisphere}} = \frac{1}{2} \times \frac{4}{3} \times \pi \times r^3 = \frac{2}{3} \times \pi \times 7^3 = \frac{686\pi}{3} \approx 718.38\,\text{cm}^3

Volume of the cone:

V_{\text{cone}} = \frac{1}{3} \times \pi \times r^2 \times h_{\text{cone}} = \frac{1}{3} \times \pi \times 7^2 \times 17 = \frac{833\pi}{3} \approx 872.23\,\text{cm}^3

Total volume of the solid:

V_{\text{total}} = \frac{686\pi}{3} + \frac{833\pi}{3} = \frac{1519\pi}{3} \approx 1590.6\,\text{cm}^3

Rounding to the nearest integer gives $1591\,\text{cm}^3$ (accepting $1590$ if $\pi = 3.14$ is used).

Answer

1591

Final answer

1591

Detailed explanation

Walkthrough

The solid is made of two standard shapes: a hemisphere and a cone. To find the total volume, we calculate the volume of each part and add them. First, we find the radius from the given diameter ( $14 \div 2 = 7$ cm). The hemisphere's height is simply its radius ( $7$ cm). Since the total height is $24$ cm, the cone's height is the remainder ( $24 - 7 = 17$ cm). We then apply the hemisphere volume formula ( $\frac{2}{3}\pi r^3$ ) and the cone volume formula ( $\frac{1}{3}\pi r^2 h$ ), sum the results, and round to the nearest whole number.

Key Takeaways

Composite solids have volumes that are the sum of their individual parts.
The height of a hemisphere is equal to its radius.
When a total height is given for a composite solid, subtract the known part's height to find the unknown part's height.

Common Mistakes

Forgetting to halve the diameter to get the radius.
Using the total height ( $24$ cm) as the height of the cone instead of subtracting the hemisphere's height ( $7$ cm).
Including the area of the circular base in the surface area calculation (though this is for part b, it's a related error to remember).
Rounding intermediate values too early, which can lead to $1590$ instead of $1591$ .

Things to Be Careful About

Ensure you use the correct height for the cone ( $17$ cm, not $24$ cm).
Keep $\pi$ in your calculator's memory or use a precise value to avoid rounding errors. Using $\pi = 3.14$ gives $1590.04$ , while the exact value gives $1590.61$ ; both are typically accepted in this syllabus, but the exact value rounds to $1591$ .
The question asks for the volume of the solid, so do not subtract the volumes.

Techniques used

determine the radius and individual heights of the solid's componentscalculate the volume of a hemispherecalculate the volume of a coneadd the volumes together

(b)

Show that the total surface area of the solid is $712\,\text{cm}^2$ , correct to the nearest integer.

5M Similar

DifficultyMedium

Worked solution

Approach

The total surface area of the solid consists of the curved surface area of the hemisphere and the curved surface area of the cone. The flat circular face where they join is internal and is not included. We calculate each curved surface area and add them together.

Working

Curved surface area of the hemisphere:

\text{CSA}_{\text{hemisphere}} = \frac{1}{2} \times 4\pi r^2 = 2\pi \times 7^2 = 98\pi \approx 307.88\,\text{cm}^2

To find the curved surface area of the cone, we first need its slant height $l$ . Using Pythagoras' theorem with the cone's height ( $17\,\text{cm}$ ) and radius ( $7\,\text{cm}$ ):

l = \sqrt{17^2 + 7^2} = \sqrt{289 + 49} = \sqrt{338} \approx 18.385\,\text{cm}

Curved surface area of the cone:

\text{CSA}_{\text{cone}} = \pi r l = \pi \times 7 \times \sqrt{338} = 7\pi\sqrt{338} \approx 404.20\,\text{cm}^2

Total surface area:

\text{TSA} = 98\pi + 7\pi\sqrt{338} \approx 307.88 + 404.20 = 712.08\,\text{cm}^2

Rounding to the nearest integer:

\text{TSA} \approx 712\,\text{cm}^2

This matches the required value.

Answer

712

Final answer

712

Detailed explanation

Walkthrough

The total surface area only includes the outer curved surfaces. The flat circular base of the cone and the flat circular base of the hemisphere are joined together, so they are inside the solid and do not contribute to the surface area. We calculate the curved surface area of the hemisphere using $2\pi r^2$ . For the cone, we need the slant height $l$ , which is the hypotenuse of a right-angled triangle with legs equal to the cone's height ( $17$ cm) and radius ( $7$ cm). Using Pythagoras, $l = \sqrt{17^2 + 7^2} = \sqrt{338}$ . The curved surface area of the cone is $\pi r l$ . Adding the two curved surface areas gives approximately $712.08$ , which rounds to $712$ .

Key Takeaways

For composite solids, internal faces are not part of the total surface area.
The curved surface area of a hemisphere is $2\pi r^2$ (half of a full sphere's surface area $4\pi r^2$ ).
The slant height of a cone is found using Pythagoras' theorem: $l = \sqrt{h^2 + r^2}$ .
The curved surface area of a cone is $\pi r l$ .

Common Mistakes

Including the area of the circular base ( $\pi r^2$ ) in the total surface area calculation.
Using the total height ( $24$ cm) instead of the cone's height ( $17$ cm) when calculating the slant height.
Forgetting to take the square root when calculating the slant height.
Rounding intermediate values (like $\pi$ or $l$ ) too early, which can lead to a final answer slightly off from $712$ .

Things to Be Careful About

The question asks to show the area is $712$ correct to the nearest integer. Your working must clearly show the sum of the two curved surface areas before rounding.
Ensure you use the correct formula for the curved surface area of a hemisphere ( $2\pi r^2$ ), not the full sphere ( $4\pi r^2$ ).

Techniques used

identify the exposed surfaces of the composite solidcalculate the curved surface area of a hemispherefind the slant height of the cone using Pythagoras' theoremcalculate the curved surface area of a conesum the surface areas and round to the nearest integer

(c)

A smaller solid is mathematically similar to this solid.
The total surface area of the smaller solid is $242\,\text{cm}^2$ .

Calculate the total height of the smaller solid.

______ $\text{cm}$

3M Similar

DifficultyMedium

Worked solution

Approach

For mathematically similar solids, the ratio of their surface areas is equal to the square of the ratio of their corresponding lengths (the linear scale factor). We use the given surface areas to find the length scale factor, then apply it to the total height of the original solid.

Working

Let $A_1 = 712\,\text{cm}^2$ and $h_1 = 24\,\text{cm}$ be the surface area and height of the original solid.
Let $A_2 = 242\,\text{cm}^2$ and $h_2$ be the surface area and height of the smaller solid.

The ratio of the surface areas is:

\frac{A_2}{A_1} = \frac{242}{712}

The linear scale factor $k$ (from the original to the smaller solid) is the square root of the area ratio:

k = \sqrt{\frac{242}{712}} \approx \sqrt{0.33989} \approx 0.5830

The total height of the smaller solid is:

h_2 = k \times h_1 = \sqrt{\frac{242}{712}} \times 24

h_2 \approx 0.5830 \times 24 \approx 13.99\,\text{cm}

Rounding to one decimal place (or the nearest integer as appropriate for the context), the height is $14.0\,\text{cm}$ .

Answer

14.0

Final answer

14.0

Detailed explanation

Walkthrough

When two solids are mathematically similar, the ratio of their surface areas is the square of the ratio of their corresponding lengths. We are given the surface areas ( $712$ and $242$ ) and the height of the original solid ( $24$ ). First, we find the area ratio ( $242 / 712$ ). Then, we take the square root of this ratio to find the linear scale factor. Finally, we multiply the original height by this linear scale factor to find the height of the smaller solid.

Key Takeaways

For similar shapes/solids, Area Ratio = (Length Ratio) $^2$ .
Length Ratio = $\sqrt{\text{Area Ratio}}$ .
To find a length in the smaller shape, multiply the corresponding length in the larger shape by the length scale factor.

Common Mistakes

Using the area ratio directly as the length scale factor (forgetting to take the square root).
Using the wrong height (e.g., using the cone's height or the hemisphere's height instead of the total height).
Calculating the scale factor from smaller to larger instead of larger to smaller, leading to a height greater than $24$ .

Things to Be Careful About

The mark scheme accepts $\sqrt{\frac{242}{712}} \times 24$ . Using the exact surface area from part (b) ( $712.07...$ ) instead of $712$ will give $13.988$ , which still rounds to $14.0$ .
Ensure the final answer is given to an appropriate number of significant figures or decimal places. $14.0$ or $14$ is acceptable.

Techniques used

determine the ratio of surface areascalculate the linear scale factor from the area scale factorapply the linear scale factor to the original height

Q22MediumTrigonometryMensuration

Fig. 12

The diagram shows a triangular field $ABC$ .
$AB = 150\,\text{m}$ , $BC = 187\,\text{m}$ and angle $ABC = 112°$ .

(a)

Fencing is needed for the perimeter of the field.
Fencing is sold in rolls of length $20\,\text{m}$ .

Calculate the number of rolls of fencing needed.

______

5M Similar

DifficultyMedium

Worked solution

Approach

To find the number of rolls of fencing, we first need the perimeter of the triangular field. We are given two sides and the included angle, so we use the Cosine Rule to find the third side $AC$ . Then we sum the three sides to get the perimeter and divide by the length of one roll (20 m). Since we cannot buy a fraction of a roll, we round up to the next whole number.

Working

Using the Cosine Rule to find $AC$ :

\begin{aligned} AC^2 &= AB^2 + BC^2 - 2 \times AB \times BC \times \cos(\angle ABC) \\ AC^2 &= 150^2 + 187^2 - 2 \times 150 \times 187 \times \cos(112^\circ) \\ AC^2 &= 22500 + 34969 - 56100 \times (-0.37460659\dots) \\ AC^2 &= 57469 + 21015.43\dots \\ AC^2 &= 78484.43\dots \end{aligned}

Taking the square root:

AC = \sqrt{78484.43\dots} \approx 280.151 \text{ m}

Calculate the perimeter of the field:

P = AB + BC + AC = 150 + 187 + 280.151 = 617.151 \text{ m}

Calculate the number of rolls needed:

\frac{617.151}{20} = 30.85755

Since fencing is sold in whole rolls, we must round up to the next integer:

\text{Number of rolls} = 31

Answer

31

Final answer

31

Detailed explanation

Walkthrough

The problem asks for the number of 20 m rolls of fencing needed to enclose a triangular field. First, we need the total perimeter. Two sides ( $AB = 150$ m, $BC = 187$ m) and the included angle ( $\angle ABC = 112^\circ$ ) are given. This is the classic setup for the Cosine Rule to find the third side $AC$ .

The Cosine Rule states $c^2 = a^2 + b^2 - 2ab\cos(C)$ . Substituting the known values:
$AC^2 = 150^2 + 187^2 - 2(150)(187)\cos(112^\circ)$
Calculating the squares and the product: $22500 + 34969 - 56100\cos(112^\circ)$ . Since $\cos(112^\circ)$ is negative (angle is obtuse), the last term becomes positive, giving $AC^2 \approx 78484.43$ . Taking the square root yields $AC \approx 280.15$ m.

Next, add all three sides to get the perimeter: $150 + 187 + 280.15 = 617.15$ m.

Finally, divide the perimeter by the length of one roll: $617.15 \div 20 = 30.8575$ . Because you cannot purchase a partial roll of fencing, you must round up to 31 rolls to cover the entire perimeter.

Key Takeaways

The Cosine Rule is used when two sides and the included angle of a triangle are known.
Perimeter is the sum of all side lengths.
In real-world problems involving discrete items (like rolls of fencing or buses), always round up (ceiling function) when the result is not a whole number.

Common Mistakes

Forgetting that $\cos(112^\circ)$ is negative, which leads to subtracting a negative number (i.e., adding) instead of subtracting.
Rounding $30.8575$ down to $30$ rolls, which would leave a gap in the fencing.
Using the wrong angle or mixing up the sides in the Cosine Rule formula.
Rounding the intermediate value of $AC$ too early (e.g., to $280$ m), which can cause the final answer to be slightly off and potentially lose method marks.

Things to Be Careful About

Ensure the calculator is in degree mode when evaluating $\cos(112^\circ)$ .
Keep full precision in the calculator for $AC$ when calculating the perimeter and the division; do not round $AC$ to $280$ m prematurely.
The mark scheme accepts answers like $30.9$ or $30.85$ to $30.86$ as evidence of correct working before rounding to the final integer $31$ .
Always state the final answer as a whole number of rolls, as fractional rolls are physically impossible.

Techniques used

apply the cosine rule to find the third side of a trianglecalculate the perimeter by summing the side lengthsdivide the perimeter by the roll length and round up to the nearest whole number

(b)

Calculate the shortest distance from $B$ to $AC$ .

______ $\text{m}$

4M Similar

DifficultyMedium

Worked solution

Approach

The shortest distance from a point to a line is the perpendicular distance. Here, we need the height $h$ from vertex $B$ perpendicular to side $AC$ . We can find the area of triangle $ABC$ using the sine area formula with the known sides $AB$ , $BC$ and included angle $\angle ABC$ . Equating this to the standard area formula $\frac{1}{2} \times \text{base} \times \text{height}$ (using $AC$ as the base) allows us to solve for $h$ .

Working

Calculate the area of triangle $ABC$ using the sine formula:

\text{Area} = \frac{1}{2} \times AB \times BC \times \sin(\angle ABC)

\text{Area} = \frac{1}{2} \times 150 \times 187 \times \sin(112^\circ)

\text{Area} = 14025 \times 0.92718385\dots = 13003.753\dots \text{ m}^2

Let $h$ be the shortest distance from $B$ to $AC$ . The area can also be written as:

\text{Area} = \frac{1}{2} \times AC \times h

Substitute the area and the value of $AC \approx 280.151$ m from part (a):

13003.753\dots = \frac{1}{2} \times 280.151 \times h

13003.753\dots = 140.0755 \times h

Solve for $h$ :

h = \frac{13003.753\dots}{140.0755} \approx 92.834 \text{ m}

Alternatively, using the direct formula:

h = \frac{150 \times 187 \times \sin(112^\circ)}{AC} = \frac{26007.506\dots}{280.151} \approx 92.834 \text{ m}

Rounding to 3 significant figures:

h \approx 92.8 \text{ m}

Answer

92.8

Final answer

92.8

Detailed explanation

Walkthrough

The shortest distance from a point to a line segment is the length of the perpendicular dropped from the point to the line. In triangle $ABC$ , this is the height $h$ from $B$ to the base $AC$ .

We can find the area of the triangle in two ways. First, using the two known sides and the included angle:
$\text{Area} = \frac{1}{2} ab \sin C = \frac{1}{2} \times 150 \times 187 \times \sin(112^\circ)$
This gives $\text{Area} \approx 13003.75 \text{ m}^2$ .

Second, using the base $AC$ and the unknown height $h$ :
$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times AC \times h$

Equating the two expressions for area and solving for $h$ :
$\frac{1}{2} \times AC \times h = 13003.75$
$h = \frac{2 \times 13003.75}{AC} = \frac{26007.506}{280.151} \approx 92.834 \text{ m}$

Rounding to 3 significant figures (as is standard for calculator component questions unless otherwise specified), we get $92.8$ m.

Key Takeaways

The shortest distance from a point to a line is the perpendicular height.
The sine area formula $\frac{1}{2}ab\sin C$ is a powerful tool when two sides and the included angle are known.
Equating two different expressions for the area of the same triangle is a standard technique to find an unknown height or side.

Common Mistakes

Trying to use the sine or cosine rule directly to find the height without first finding the area or another angle. (While possible using $\sin A$ or $\sin C$ , it requires extra steps and is more prone to error).
Forgetting to multiply by 2 when rearranging $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ .
Using the wrong base (e.g., using $AB$ or $BC$ as the base with the wrong height).
Not carrying forward the unrounded value of $AC$ from part (a), which leads to a slightly incorrect height.

Things to Be Careful About

The question asks for the shortest distance, which is the perpendicular height, not the length of $AB$ or $BC$ .
Ensure calculator is in degree mode.
The mark scheme accepts $92.8$ to $92.9\dots$ , so keeping full calculator precision for $AC$ is essential to land in this range.
Final answer should be given to 3 significant figures as per standard calculator component instructions, yielding $92.8$ .

Techniques used

calculate the area of a triangle using the sine area formulaequate the area to the standard base-height formularearrange to solve for the perpendicular height

Q233MMedium-EasyNumber

Zara cycles a distance of 500 metres, correct to the nearest 5 metres.
This takes 24.7 seconds, correct to the nearest 0.1 seconds.

Calculate the lower bound of her average speed.

______ $\text{m/s}$

DifficultyMedium-Easy

Worked solution

Approach

A distance given to the nearest 5 metres can be as small as $500 - 2.5 = 497.5$ m and as large as $502.5$ m. A time given to the nearest 0.1 seconds has lower bound $24.7 - 0.05 = 24.65$ s and upper bound $24.7 + 0.05 = 24.75$ s.

Since

\text{speed} = \frac{\text{distance}}{\text{time}},

the lower bound of the speed is the lower bound of the distance divided by the upper bound of the time.

Working

\frac{497.5}{24.75}

Writing both numbers as fractions:

\frac{497.5}{24.75} = \frac{995}{2} \div \frac{99}{4} = \frac{995}{2} \times \frac{4}{99} = \frac{1990}{99}

\frac{1990}{99} = 20.1010\ldots

Answer

The lower bound of her average speed is $20.101\ldots$ m/s, which is $20.1$ m/s to 3 significant figures.

Final answer

20.1 m/s (3 s.f.)

Detailed explanation

Walkthrough

The distance is stated as 500 m to the nearest 5 m. This means the true distance was rounded to a multiple of 5 m, so it could differ by at most half of 5 m, which is 2.5 m. Therefore the smallest possible distance is $500 - 2.5 = 497.5$ m. The largest possible distance is $502.5$ m, but we will not need it.

The time is stated as 24.7 s to the nearest 0.1 s. Half of 0.1 s is 0.05 s, so the true time lies between $24.7 - 0.05 = 24.65$ s and $24.7 + 0.05 = 24.75$ s.

Average speed is distance divided by time. To make a quotient as small as possible, put the smallest possible numerator and the largest possible denominator. Thus the lower bound of the speed is

\frac{497.5}{24.75}.

Evaluating this gives $20.1010\ldots$ m/s, which is $20.1$ m/s to 3 significant figures.

Key Takeaways

Bounds come from rounding errors: for a value correct to the nearest $u$ , the actual value can be anywhere from $\text{value} - \frac{u}{2}$ to $\text{value} + \frac{u}{2}$ . When a result is a quotient, the lower bound uses the lower bound of the numerator and the upper bound of the denominator; the upper bound uses the upper bound numerator and lower bound denominator.

Common Mistakes

Using $500$ and $24.7$ directly gives $20.2$ m/s, but that is not the lower bound.
Using the upper distance $502.5$ divided by the lower time $24.65$ gives the upper bound of speed, not the lower bound.
Subtracting the full rounding precision, e.g. $500 - 5 = 495$ for the lower distance, instead of subtracting half of it ( $2.5$ ).
Taking the time bounds as $24.6$ and $24.8$ instead of $24.65$ and $24.75$ .
Rounding the intermediate values before dividing, which can change the final answer.

Things to Be Careful About

The lower bound of a quotient is not just the lower bound of each quantity; you must divide the smallest numerator by the largest denominator.
Keep the units in mind: distance in m and time in s give speed in m/s.
The mark scheme awards M1 for identifying $497.5$ or $502.5$ or $24.65$ or $24.75$ , and M2 for the correct fraction $\frac{497.5}{24.75}$ . Show the fraction clearly so those method marks are not lost.
On this calculator component, the decimal $20.101\ldots$ m/s may be given; if rounding, give at least $20.1$ m/s (3 significant figures). The exact form $\frac{1990}{99}$ m/s is also acceptable.

Techniques used

determine lower and upper bounds from rounding precisionchoose the smallest distance and largest time to minimise speedevaluate the compound measure division

Similar questions

Q243MMedium-EasyAlgebra and Graphs

Express as a single fraction in its simplest form.

\frac{5}{2x + 1} - \frac{2}{4x - 3}

______

DifficultyMedium-Easy

Worked solution

Approach

To subtract these algebraic fractions, we need a common denominator. Since the denominators $(2x + 1)$ and $(4x - 3)$ share no common factors, the lowest common denominator (LCD) is their product: $(2x + 1)(4x - 3)$ .

We then rewrite each fraction with this common denominator by multiplying the numerator of each by the factor that was missing from its original denominator, before combining them into a single fraction.

Working

The expression is:

\frac{5}{2x + 1} - \frac{2}{4x - 3}

The common denominator is $(2x + 1)(4x - 3)$ .

For the first term $\frac{5}{2x + 1}$ , multiply the numerator and denominator by $(4x - 3)$ :

\frac{5(4x - 3)}{(2x + 1)(4x - 3)}

For the second term $\frac{2}{4x - 3}$ , multiply the numerator and denominator by $(2x + 1)$ :

\frac{2(2x + 1)}{(2x + 1)(4x - 3)}

Now combine the numerators over the single denominator:

\frac{5(4x - 3) - 2(2x + 1)}{(2x + 1)(4x - 3)}

Expand the brackets in the numerator:

5(4x - 3) = 20x - 15

-2(2x + 1) = -4x - 2

Combine these results:

(20x - 15) + (-4x - 2)

= 20x - 4x - 15 - 2

= 16x - 17

So the combined fraction is:

\frac{16x - 17}{(2x + 1)(4x - 3)}

We can leave the denominator in factored form or expand it:

(2x + 1)(4x - 3) = 8x^2 - 6x + 4x - 3 = 8x^2 - 2x - 3

Thus, the final simplified single fraction is:

\frac{16x - 17}{8x^2 - 2x - 3}

(Note: The numerator $16x - 17$ cannot be simplified further as there are no common factors between $16$ , $-17$ , and the denominator.)

Answer

\frac{16x - 17}{8x^2 - 2x - 3}

Final answer

(16x - 17) / (8x^2 - 2x - 3)

Detailed explanation

Walkthrough

When adding or subtracting algebraic fractions, the first step is always to identify a common denominator. Because the denominators here are simple linear binomials $(2x+1)$ and $(4x-3)$ that do not factor further or share terms, the most straightforward common denominator is simply their product.

Think of it like adding numbers: to add $\frac{a}{b} - \frac{c}{d}$ , you must convert them so they have the same bottom part. We turn $\frac{5}{2x+1}$ into an equivalent fraction by multiplying top and bottom by $(4x-3)$ . Similarly, we turn $\frac{2}{4x-3}$ into an equivalent fraction by multiplying top and bottom by $(2x+1)$ .

Once the denominators match, we only operate on the numerators. A crucial detail is the minus sign between the fractions. It applies to the entire second numerator. So when we write $5(4x-3) - 2(2x+1)$ , we must distribute the negative sign when expanding the second part: $-2 \times 2x$ becomes $-4x$ and $-2 \times 1$ becomes $-2$ . This is a very common place for sign errors.

After expanding both parts ( $20x - 15$ and $-4x - 2$ ), we group the like terms ( $x$ 's together and constants together) to get $16x - 17$ . The denominator can be left as a product or expanded; both forms are correct unless specified otherwise.

Key Takeaways

Common Denominator: For fractions $\frac{A}{B}$ and $\frac{C}{D}$ , the sum/difference is $\frac{AD \mp BC}{BD}$ .
Brackets and Signs: When subtracting fractions, put the entire numerator of the second fraction in brackets and ensure you distribute the negative sign correctly across every term inside.
Simplification: Always check if the resulting numerator and denominator share any common factors that could cancel out.

Common Mistakes

Sign Errors: Failing to distribute the minus sign to the constant term in the second bracket (e.g., writing $-2(2x+1)$ as $-4x + 2$ instead of $-4x - 2$ ).
Incomplete Expansion: Only multiplying one term in the numerator by the new factor.
Combining unlike terms: Adding $x$ terms to constant terms (e.g., $20x - 15$ becoming something else entirely).
Leaving unexpanded denominators: While $(2x+1)(4x-3)$ is mathematically correct, examiners often expect the denominator expanded into standard quadratic form $ax^2+bx+c$ for the "simplest form" answer, though the mark scheme accepts both.

Things to Be Careful About

Order of Operations: Remember that the division implied by the fraction bar acts as a grouping symbol for the numerator. When substituting expressions, use brackets.
Factoring Check: Before finishing, verify that the numerator $16x - 17$ cannot be factored. If it could, check if any factors cancel with the denominator. Here, $16x-17$ is irreducible over integers.
Final Form: Ensure the final answer is a single fraction, not two separate ones.

Techniques used

find a common denominator for algebraic fractionsexpand and simplify the numeratorfactorise or expand the denominator

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