Mathematics (Syllabus D) 4024 / 23 — May/June 2025

(b)

Write $34159$ correct to 2 significant figures.

______

DifficultyEasy

Worked solution

Approach

To round $34159$ to 2 significant figures, we identify the first two non-zero digits and look at the third digit to determine rounding.

Working

The number is $34159$ .

The first significant figure is $3$ .
The second significant figure is $4$ .
The third significant figure is $1$ .

Since the third digit ( $1$ ) is less than $5$ , we do not round up the second digit. We keep $34$ and replace the following digits with zeros to maintain the magnitude of the number.

So, $34159$ becomes $34\,000$ .

Answer

34\,000

Final answer

34 000

Detailed explanation

Walkthrough

We need to round $34159$ to 2 significant figures. Significant figures start counting from the first non-zero digit on the left.

The first significant digit is $3$ .
The second significant digit is $4$ .
The next digit (the one that decides whether to round up) is $1$ .

Because $1 < 5$ , we round down, meaning the second digit ( $4$ ) remains unchanged. All digits after the second significant figure are replaced by zeros to keep the number's scale correct. Thus, $34159$ becomes $34000$ .

Key Takeaways

Significant figures count from the first non-zero digit.
Zeros between non-zero digits are significant.
When rounding to fewer significant figures than the total digits, trailing zeros are placeholders.

Common Mistakes

Starting from the wrong end: Counting from the right instead of the left.
Ignoring the placeholder zeros: Writing $34$ instead of $34000$ .
Rounding incorrectly: Seeing the $1$ and thinking it rounds up because of the subsequent $5$ and $9$ (rounding is determined solely by the immediate next digit).

Things to Be Careful About

Distinguish clearly between decimal places (counting from the decimal point) and significant figures (counting from the first non-zero digit).
The mark scheme expects $34\,000$ cao.

Techniques used

identify the first two significant digitslook at the third significant digit to decide roundingreplace remaining digits with zeros

Q22MEasyTransformations and Vectors

A rectangle has dimensions $2.4\,\text{cm}$ by $5.6\,\text{cm}$ .
The rectangle is enlarged by a scale factor of $3.25$ .

Work out the dimensions of the enlarged rectangle.

______ $\text{cm}$ by ______ $\text{cm}$

DifficultyEasy

Worked solution

Approach

To find the dimensions of an enlarged shape, multiply each original dimension by the scale factor. The problem states the rectangle is enlarged by a scale factor of $3.25$ .

Working

The original dimensions are $2.4\,\text{cm}$ and $5.6\,\text{cm}$ . The scale factor is $3.25$ .

Calculate the first new dimension:

2.4 \times 3.25

Break this down for clarity (or use calculator):

2.4 \times 3 = 7.2

2.4 \times 0.25 = 2.4 \div 4 = 0.6

7.2 + 0.6 = 7.8

So the first dimension is $7.8\,\text{cm}$ .

Calculate the second new dimension:

5.6 \times 3.25

Break this down:

5.6 \times 3 = 16.8

5.6 \times 0.25 = 5.6 \div 4 = 1.4

16.8 + 1.4 = 18.2

So the second dimension is $18.2\,\text{cm}$ .

Alternatively, using fractions:
$3.25 = 3\frac{1}{4} = \frac{13}{4}$

2.4 \times \frac{13}{4} = 0.6 \times 13 = 7.8

5.6 \times \frac{13}{4} = 1.4 \times 13 = 18.2

Answer

7.8 \text{ cm by } 18.2 \text{ cm}

Final answer

7.8 cm by 18.2 cm

Detailed explanation

Walkthrough

An enlargement changes the size of a shape while keeping its angles the same. The ratio of any length in the new shape to the corresponding length in the original shape is the scale factor.

Here, we are given:

Original width = $2.4\,\text{cm}$
Original length = $5.6\,\text{cm}$
Scale factor = $3.25$

To get the new dimensions, we simply multiply each original dimension by the scale factor.

First dimension: $2.4 \times 3.25$ . Since $3.25$ is the same as $3 + 0.25$ (or $3 \frac{1}{4}$ ), we can think of this as finding three times $2.4$ and adding one quarter of $2.4$ .
$3 \times 2.4 = 7.2$
$2.4 \div 4 = 0.6$
$7.2 + 0.6 = 7.8$

Second dimension: $5.6 \times 3.25$ . Similarly, take three times $5.6$ and add one quarter of $5.6$ .
$3 \times 5.6 = 16.8$
$5.6 \div 4 = 1.4$
$16.8 + 1.4 = 18.2$

The order does not matter; the rectangle is $7.8\,\text{cm}$ by $18.2\,\text{cm}$ .

Key Takeaways

In an enlargement with scale factor $k$ , every linear dimension is multiplied by $k$ .
Area scales by $k^2$ and volume by $k^3$ , but here we only need lengths.
Decimal multiplication can be simplified by breaking the multiplier into integer and fractional parts (e.g., $0.25$ is $\frac{1}{4}$ ).

Common Mistakes

Forgetting to multiply both dimensions (only multiplying one side).
Multiplying by the wrong number (e.g., adding instead of multiplying).
Calculation errors with decimals.

Things to Be Careful About

Ensure you include the units ( $\text{cm}$ ) in your final answer if required by the context or blanks provided.
Check that the new dimensions are larger than the original ones since the scale factor is greater than $1$ .

Techniques used

multiply dimensions by the scale factorcalculate decimal products

Q32MMedium-EasyNumber

Ang and Bou share $104 in the ratio $7 : 6$ .

Calculate the amount they each receive.

Ang $ ______
Bou $ ______

DifficultyMedium-Easy

Worked solution

Approach

Add the two parts of the ratio to find the total number of parts. Divide the total amount by this sum to find the value of one part, then multiply by each person's ratio number.

Working

The ratio $7 : 6$ has $7 + 6 = 13$ parts.

\frac{104}{13} = 8

So one part is worth $8.

Ang receives $7$ parts:

7 \times 8 = 56

Bou receives $6$ parts:

6 \times 8 = 48

Check: $56 + 48 = 104$ .

Answer

Ang receives $56 and Bou receives $48.

Final answer

Ang receives $56, Bou receives $48

Detailed explanation

Walkthrough

The ratio $7 : 6$ means the $104 is split into $7 + 6 = 13$ equal parts. Ang takes 7 of those parts and Bou takes 6. So first find the value of one part by dividing the total by 13. Then multiply by 7 and 6. This is the standard method for sharing in a ratio.

Key Takeaways

A ratio $a : b$ divides a total into $a + b$ equal parts.
The value of one part is the total divided by the sum of the ratio parts.
Multiply the value of one part by each ratio number to find each share.
Check by adding the shares back to the total.

Common Mistakes

Adding the ratio parts incorrectly, for example using $7 + 6 = 12$ .
Swapping the amounts: Ang should get the larger share because $7 > 6$ .
Dividing by only one of the ratio numbers instead of by the sum $7 + 6$ .
Forgetting to show the method: the mark scheme awards M1 for $\frac{104}{7 + 6} \times k$ where $k = 1$ , $6$ or $7$ . Unsupported answers may not receive full credit.

Things to Be Careful About

The total is $104, not $13 or $7.
The answer must include both amounts, with units.
Since this is a calculator paper, decimal division is acceptable, but here $104 \div 13 = 8$ is exact.
The ratio order matters: Ang's share uses 7, Bou's share uses 6.

Techniques used

add the parts of the ratiodivide the total by the sum of the ratio partsmultiply the unit share by each ratio numbercheck the shares sum to the total

Q4MediumGeometry

(a)

Shade one small square so the diagram has rotational symmetry of order 2.

Fig. 4a

DifficultyMedium-Easy

Worked solution

Approach

Rotational symmetry of order 2 means the shape looks the same after a 180-degree rotation. For a 4x4 grid, the centre of rotation is the central point where the four middle squares meet. We can map each shaded square to its 180-degree rotated position using the rule: a square at row $r$ , column $c$ maps to row $5-r$ , column $5-c$ .

Working

List the currently shaded cells (row, column):

(1,1), (1,2), (1,4), (2,2), (2,4), (3,3), (4,1), (4,3), (4,4)

Find the 180-degree rotation of each shaded cell:

$(1,1) \to (4,4)$ — both shaded
$(1,2) \to (4,3)$ — both shaded
$(1,4) \to (4,1)$ — both shaded
$(2,2) \to (3,3)$ — both shaded
$(2,4) \to (3,1)$ — $(2,4)$ is shaded, but $(3,1)$ is white
$(3,3) \to (2,2)$ — both shaded
$(4,1) \to (1,4)$ — both shaded
$(4,3) \to (1,2)$ — both shaded
$(4,4) \to (1,1)$ — both shaded

The only shaded cell without a matching partner is $(2,4)$ . Its rotational partner is $(3,1)$ , which is currently unshaded. Shading cell $(3,1)$ completes the rotational symmetry of order 2.

Answer

Shade the square in row 3, column 1.

Final answer

Row 3, column 1

Detailed explanation

Walkthrough

Rotational symmetry of order 2 means that if you rotate the diagram 180 degrees around its centre, it looks exactly the same. For a 4x4 grid, the centre is the point where the four central squares meet. To check for this symmetry, we can pair up squares that are opposite each other through the centre. The rule for a 4x4 grid is that a square at position $(r, c)$ maps to $(5-r, 5-c)$ .

We list all the shaded squares in Fig. 4a and find their partners:

$(1,1)$ and $(4,4)$ are both shaded.
$(1,2)$ and $(4,3)$ are both shaded.
$(1,4)$ and $(4,1)$ are both shaded.
$(2,2)$ and $(3,3)$ are both shaded.
$(2,4)$ is shaded, but its partner $(3,1)$ is white.

Since only one square can be shaded, we must shade $(3,1)$ to match $(2,4)$ . This gives the completed diagram shown in .

Key Takeaways

Rotational symmetry of order 2 is equivalent to 180-degree rotational symmetry.
For an even-sized square grid, the centre of rotation is the central intersection point, not a square itself.
You can systematically check symmetry by mapping each element to its rotated position using coordinate rules.

Common Mistakes

Choosing a square that creates reflectional symmetry instead of rotational symmetry.
Miscounting the rows or columns, leading to the wrong partner cell.
Forgetting that the centre of rotation for a 4x4 grid is between squares, not on a square.

Things to Be Careful About

The question asks for rotational symmetry of order 2, not reflectional symmetry or rotational symmetry of order 4.
Only one square may be shaded; ensure no other cells are altered.
Coordinate mapping must be precise: row $r$ maps to row $5-r$ , and column $c$ maps to column $5-c$ .

Techniques used

identify the centre of rotationmap each shaded cell to its 180-degree rotated positionfind the missing cell to complete the symmetry

(b)

Shade one small square so the diagram has 1 line of symmetry.

Fig. 4b

DifficultyMedium

Worked solution

Approach

We need to shade exactly one square so that the resulting diagram has exactly one line of symmetry. We test the possible lines of symmetry for a 4x4 grid: vertical, horizontal, and the two diagonals.

Working

The shaded cells in Fig. 4b are:

(1,1), (1,3), (2,2), (2,4), (3,2), (3,4), (4,1), (4,3), (4,4)

Test the main diagonal (top-left to bottom-right), where $(r,c) \to (c,r)$ :

$(1,1) \to (1,1)$ ✓
$(1,3) \to (3,1)$ ✗
$(2,2) \to (2,2)$ ✓
$(2,4) \to (4,2)$ ✗
$(3,2) \to (2,3)$ ✗
$(3,4) \to (4,3)$ ✓
$(4,1) \to (1,4)$ ✗
$(4,3) \to (3,4)$ ✓
$(4,4) \to (4,4)$ ✓
Multiple cells are missing, so the main diagonal won't work with just one shading.

Test the anti-diagonal (top-right to bottom-left), where $(r,c) \to (5-c, 5-r)$ :

$(1,1) \to (4,4)$ ✓
$(1,3) \to (2,4)$ ✓
$(2,2) \to (3,3)$ ✗
$(2,4) \to (1,3)$ ✓
$(3,2) \to (3,2)$ ✓
$(3,4) \to (1,2)$ ✗
$(4,1) \to (4,1)$ ✓
$(4,3) \to (2,1)$ ✗
$(4,4) \to (1,1)$ ✓
Three cells are missing: $(3,3)$ , $(1,2)$ , $(2,1)$ . Not one cell.

Test the vertical line between columns 2 and 3, where $(r,c) \to (r, 5-c)$ :

$(1,1) \to (1,4)$ ✗
Multiple missing.

Test the horizontal line between rows 2 and 3, where $(r,c) \to (5-r, c)$ :

$(1,1) \to (4,1)$ ✓
$(1,3) \to (4,3)$ ✓
$(1,4) \to (4,4)$ $(1, 4) \to (4, 4)$ — if we shade $(1,4)$ $(1, 4)$ :
- $(1,4) \to (4,4)$ ✓
$(2,2) \to (3,2)$ ✓
$(2,4) \to (3,4)$ ✓
$(3,2) \to (2,2)$ ✓
$(3,4) \to (2,4)$ ✓
$(4,1) \to (1,1)$ ✓
$(4,3) \to (1,3)$ ✓
$(4,4) \to (1,4)$ ✓

Shading cell $(1,4)$ completes the horizontal line of symmetry between rows 2 and 3. Checking other lines: the vertical and diagonal symmetries are not present, so there is exactly one line of symmetry.

Answer

Shade the square in row 1, column 4.

Final answer

Row 1, column 4

Detailed explanation

Walkthrough

We need to add exactly one shaded square to Fig. 4b to create a diagram with exactly one line of symmetry. We systematically test the possible lines of symmetry for a square grid.

The shaded cells are at: $(1,1), (1,3), (2,2), (2,4), (3,2), (3,4), (4,1), (4,3), (4,4)$ .

Testing the horizontal line between rows 2 and 3 (reflection maps row $r$ to row $5-r$ ):

Row 1 and Row 4: $(1,1)\leftrightarrow(4,1)$ , $(1,3)\leftrightarrow(4,3)$ . Row 4 has $(4,4)$ but Row 1 is missing $(1,4)$ .
Row 2 and Row 3: $(2,2)\leftrightarrow(3,2)$ , $(2,4)\leftrightarrow(3,4)$ . These pairs match perfectly.

By shading $(1,4)$ , the top row becomes a mirror of the bottom row. The completed diagram is shown in .

We verify no other lines of symmetry exist: the vertical line fails because $(1,1)$ has no partner $(1,4)$ until we shade it, but then $(1,3)$ needs $(1,2)$ which is not shaded. The diagonals require multiple additional shadings. Thus, exactly one line of symmetry is achieved.

Key Takeaways

Testing lines of symmetry systematically is key: vertical, horizontal, and both diagonals.
When reflecting across a horizontal line in a 4-row grid, row $r$ maps to row $5-r$ .
Ensure the final diagram has exactly the required number of symmetries, not more.

Common Mistakes

Shading a cell that creates multiple lines of symmetry when only one is required.
Misidentifying the reflection rule (e.g., using $(r, 5-c)$ for a horizontal line).
Assuming the diagonal will work without checking all pairs.

Things to Be Careful About

The question asks for exactly 1 line of symmetry. Adding a cell that creates rotational symmetry or additional reflectional lines will lose marks.
Only one square may be shaded.
Coordinate mapping must be precise: for horizontal reflection, row $r$ maps to row $5-r$ ; for vertical, column $c$ maps to column $5-c$ .

Techniques used

test possible lines of symmetryreflect shaded cells across the candidate lineidentify the single cell that completes the symmetry

Q5EasyMensuration

(a)

Write $6300\,\text{cm}$ in metres.

______ $\text{m}$

(b)

Write $450\,\text{cm}^3$ in litres.

______ $\text{litres}$

Q6EasyAlgebra and Graphs

Simplify.

(a)

$5a + 3b + 2a - 7b$

______

(b)

$c^{12} \div c^4$

______

Q72MEasyAlgebra and Graphs

Write down all the integer values of $x$ that satisfy the inequality.

-3 \leq x < 1

______

DifficultyEasy

Worked solution

Approach

Interpret the inequality symbols to determine the allowable range, then list all integers that fall strictly within or on the boundaries of this range.

Working

The inequality is given as:

-3 \leq x < 1

This compound inequality combines two conditions:

$x \geq -3$ : The value of $x$ can be equal to $-3$ because of the $\leq$ symbol.
$x < 1$ : The value of $x$ must be strictly less than $1$ because of the $<$ symbol.

We list the integers starting from $-3$ and counting upwards, stopping immediately before reaching $1$ :

x = -3, -2, -1, 0

Answer

-3, -2, -1, 0

Final answer

-3, -2, -1, 0

Detailed explanation

Walkthrough

The problem asks for all integer values of $x$ that satisfy $-3 \leq x < 1$ . First, examine the left-hand boundary: $-3 \leq x$ . The symbol $\leq$ means 'less than or equal to', which explicitly includes $-3$ in the solution set. Next, examine the right-hand boundary: $x < 1$ . The symbol $<$ means 'strictly less than', which explicitly excludes $1$ . Since we are looking for integers (whole numbers), we simply count up by ones from the lower bound until we reach the upper bound, writing down every whole number we encounter. This yields $-3, -2, -1,$ and $0$ .

Key Takeaways

Understanding the distinction between strict inequalities ( $<$ and $>$ ) and inclusive inequalities ( $\leq$ and $\geq$ ) is essential for correctly handling boundary values. When a question specifies 'integer values', only whole numbers are accepted; decimals and fractions must be excluded even if they lie within the range.

Common Mistakes

Including the upper bound ( $1$ ) when the inequality is strict ( $<$ ), or incorrectly excluding the lower bound ( $-3$ ) when it is inclusive ( $\leq$ ). Listing non-integers such as $-2.5$ or $0.75$ . Forgetting negative integers altogether or miscounting the sequence.

Things to Be Careful About

Always verify the direction and type of each inequality sign before listing values. Mark schemes often award partial credit (e.g., B1 for 3 correct integers without extras, or 4 correct with one extra), so precision matters. Ensure your final answer clearly separates each integer with commas or spaces, and double-check that no decimal approximations have been written.

Techniques used

interpret inequality symbolsidentify integer constraintslist valid values within bounds

Q8Medium-EasyProbability

A bag contains 4 black tiles and 6 white tiles.
A tile is chosen at random from the bag and is then replaced.
A second tile is then chosen at random.

(a)

Complete the tree diagram.

Fig. 8a

DifficultyEasy

Worked solution

Approach

Calculate the probability of selecting a black tile and a white tile from the bag. Since the tile is replaced, the probabilities for the second selection are identical to the first.

Working

Total number of tiles = $4 + 6 = 10$ .

Probability of selecting a black tile:

P(\text{Black}) = \frac{4}{10} = \frac{2}{5}

Probability of selecting a white tile:

P(\text{White}) = \frac{6}{10} = \frac{3}{5}

Because the tile is replaced, the probabilities for the second selection remain the same:

After Black: $P(\text{Black}) = \frac{2}{5}$ , $P(\text{White}) = \frac{3}{5}$
After White: $P(\text{Black}) = \frac{2}{5}$ , $P(\text{White}) = \frac{3}{5}$

The completed tree diagram has $\frac{4}{10}$ (or $\frac{2}{5}$ ) on the Black branches and $\frac{6}{10}$ (or $\frac{3}{5}$ ) on the White branches.

Answer

Tree diagram filled with $\frac{4}{10}$ (or $\frac{2}{5}$ ) for Black and $\frac{6}{10}$ (or $\frac{3}{5}$ ) for White on all relevant branches.

Final answer

Tree diagram with 2/5 (or 4/10) for Black and 3/5 (or 6/10) for White on all branches

Detailed explanation

Walkthrough

First, find the total number of tiles in the bag: 4 black + 6 white = 10 tiles.
The probability of picking a black tile is the number of black tiles divided by the total: $\frac{4}{10}$ , which simplifies to $\frac{2}{5}$ .
The probability of picking a white tile is $\frac{6}{10}$ , which simplifies to $\frac{3}{5}$ .
The problem states the tile is "replaced", meaning it is put back into the bag before the second draw. This is crucial because it means the total number of tiles and the number of each colour remain the same for the second selection.
Therefore, every branch on the tree diagram gets the same probability values: $\frac{2}{5}$ (or $\frac{4}{10}$ ) for Black and $\frac{3}{5}$ (or $\frac{6}{10}$ ) for White.

Key Takeaways

Probability of an event = number of favourable outcomes / total number of outcomes.
"With replacement" means the probabilities do not change between selections.
A probability tree diagram must have probabilities on every branch, and the branches from any single node must sum to 1.

Common Mistakes

Forgetting that the probabilities sum to 1 on each branch (e.g., writing $\frac{4}{10}$ and $\frac{5}{10}$ ).
Changing the probabilities for the second set of branches (calculating without replacement instead of with replacement).
Not simplifying fractions (though $\frac{4}{10}$ and $\frac{6}{10}$ are usually accepted as per the mark scheme "oe").

Things to Be Careful About

Check whether the question says "replaced" or "not replaced". Here, "replaced" means independent events with constant probabilities.
The mark scheme accepts unsimplified fractions like $\frac{4}{10}$ and $\frac{6}{10}$ ("oe").

Techniques used

calculate single event probabilitiesfill in a probability tree diagram

(b)

Work out the probability that both tiles are white.

______

DifficultyMedium-Easy

Worked solution

Approach

Use the probability tree diagram from part (a). To find the probability that both tiles are white, multiply the probability of selecting a white tile on the first draw by the probability of selecting a white tile on the second draw.

Working

From part (a), $P(\text{White}) = \frac{6}{10}$ .

Since the tile is replaced, the events are independent:

P(\text{Both White}) = P(\text{White}) \times P(\text{White})

P(\text{Both White}) = \frac{6}{10} \times \frac{6}{10}

P(\text{Both White}) = \frac{36}{100}

This can be simplified to $\frac{9}{25}$ , but $\frac{36}{100}$ is accepted.

Answer

\frac{36}{100} \text{ or } \frac{9}{25}

Final answer

36/100 (or 9/25)

Detailed explanation

Walkthrough

The question asks for the probability that both tiles are white. On the tree diagram, this corresponds to the path: First tile is White AND Second tile is White.
To find the probability of both events happening, we multiply the probabilities along that path.
From part (a), the probability of the first tile being white is $\frac{6}{10}$ .
Because the tile is replaced, the probability of the second tile being white is also $\frac{6}{10}$ .
Multiply these two probabilities: $\frac{6}{10} \times \frac{6}{10} = \frac{36}{100}$ .

Key Takeaways

To find the probability of combined events along a path in a tree diagram, multiply the probabilities of the individual branches.
$P(A \text{ and } B) = P(A) \times P(B)$ for independent events.

Common Mistakes

Adding the probabilities instead of multiplying them ( $\frac{6}{10} + \frac{6}{10}$ ).
Using the wrong probabilities from the tree diagram (e.g., using the probability for Black instead of White).
Forgetting to simplify the final fraction (though unsimplified answers are often accepted if correct).

Things to Be Careful About

Ensure you are multiplying the correct branches: White then White, not Black then White or any other combination.
The mark scheme accepts $\frac{36}{100}$ as an answer ("oe"), so simplification to $\frac{9}{25}$ is not strictly required but good practice.

Techniques used

multiply probabilities along branches for combined events

Q9Medium-EasyNumber

(a)

Ameerah invests $480 in a savings account.
The account pays simple interest at a rate of $3.6\%$ per year.

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

$ ______

DifficultyMedium-Easy

Worked solution

Approach

Simple interest is paid only on the original $480, so the interest is the same each year. First find the interest for one year by taking $3.6\%$ of $480, multiply by 5, then add that total interest to the original amount.

Working

Interest for one year:

480 \times \frac{3.6}{100} = 480 \times 0.036 = 17.28

Interest for 5 years:

17.28 \times 5 = 86.40

Value of the investment after 5 years:

480 + 86.40 = 566.40

Answer

The value of the investment is $566.40.

Final answer

$566.40

Detailed explanation

Walkthrough

Since the account pays simple interest, interest is calculated only on the original $480 each year. Convert $3.6\%$ to a decimal: $3.6 \div 100 = 0.036$ . The interest for one year is therefore $480 \times 0.036 = 17.28$ . Because simple interest gives the same amount every year, the total interest over 5 years is $17.28 \times 5 = 86.40$ . The final value of the investment is the original amount plus the interest, so $480 + 86.40 = 566.40$ .

Key Takeaways

Simple interest is calculated on the initial principal only.
The annual interest can be found by multiplying the principal by the decimal version of the percentage rate.
To find the total value, always add the total interest to the original principal.

Common Mistakes

Writing $86.40 as the final answer instead of $566.40. The question asks for the value of the investment, which includes the original $480.
Using a compound interest method when the question clearly says simple interest.
Converting $3.6\%$ incorrectly to $0.36$ rather than $0.036$ .

Things to Be Careful About

The rate is $3.6\%$ , not $36\%$ , so the decimal must be $0.036$ .
The final answer is a currency value, so it should be written to 2 decimal places: $566.40.
The mark scheme gives partial credit for the interest step, but the final answer must include the addition of the original $480.

Techniques used

convert the annual percentage rate into a decimalmultiply the principal by the annual rate and the number of yearsadd the total simple interest to the original investment

(b)

Ben invests $600 in a different savings account.
The account pays compound interest at a rate of $2.7\%$ per year.

Calculate the total amount of interest paid to Ben at the end of 4 years.

$ ______

DifficultyMedium-Easy

Worked solution

Approach

Compound interest means the balance is multiplied by $1+\frac{2.7}{100}=1.027$ at the end of each year. After 4 years the amount is $600 \times 1.027^4$ . To find the interest, subtract the original $600 from this amount.

Working

The compound growth factor is

1 + \frac{2.7}{100} = 1.027

Amount after 4 years:

600 \times 1.027^4 = 600 \times 1.11245326 = 667.471958\ldots

Total interest paid:

667.471958\ldots - 600 = 67.471958\ldots = 67.47 \quad (\text{to 2 d.p.})

Answer

The total amount of interest paid to Ben is $67.47.

Final answer

$67.47

Detailed explanation

Walkthrough

For compound interest, the balance is multiplied by the same growth factor at the end of each year. Since the account earns $2.7\%$ per year, the factor is $1 + \frac{2.7}{100} = 1.027$ . After 4 years, the balance is $600 \times 1.027^4$ . Multiplying gives $667.47$ to 2 decimal places. This is the total amount in the account, not the interest. To find the amount of interest actually paid, subtract the original $600: $667.47 - 600 = 67.47$ .

Key Takeaways

For compound interest, use the multiplier $1 + \frac{r}{100}$ , where $r$ is the percentage rate.
The multiplier is used as many times as there are years.
When asked for interest, sub-the original amount after finding the final balance.

Common Mistakes

Writing $667.47 as the final answer. That is the balance, but the question asks for interest only.
Using simple interest: $600 \times 0.027 \times 4 = 64.80$ , which is incorrect.
Rounding $1.027^4$ too early, e.g. using $1.11$ , and then getting an inaccurate final answer.
Forgetting to subtract the original $600 from the balance.

Things to Be Careful About

The rate is $2.7\%$ , so the growth factor is $1.027$ , not $1.27$ .
The period is 4 years, so the exponent is 4.
The final answer is a monetary amount, so write it as $67.47.
The mark scheme gives full credit for the correctly evaluated compound amount minus the principal, so clearly show the subtraction step in your working.

Techniques used

use the compound growth factor $1+\frac{r}{100}$raise the growth factor to the number of compounding yearssubtract the original amount to find the total interest

Q10MediumAlgebra and Graphs

$\text{f}(x) = 3x - 5 \qquad \text{g}(x) = 2 - 6x$

(a)

Find $\text{f}(2)$ .

______

(b)

Solve $\text{g}(x) = 18$ .

$x$ = ______

(c)

Find $\text{fg}(4)$ .

______

DifficultyMedium

Worked solution

Approach

Find $\text{fg}(4)$ by first evaluating the inner function $\text{g}(4)$ , and then substituting that result into the outer function $\text{f}$ .

Working

First, find $\text{g}(4)$ :

\begin{aligned} \text{g}(4) &= 2 - 6(4) \\ &= 2 - 24 \\ &= -22 \end{aligned}

Now substitute this result into $\text{f}(x)$ :

\begin{aligned} \text{fg}(4) = \text{f}(-22) &= 3(-22) - 5 \\ &= -66 - 5 \\ &= -71 \end{aligned}

Answer

-71

Final answer

-71

Detailed explanation

Walkthrough

The notation $\text{fg}(4)$ means $\text{f}(\text{g}(4))$ . This is a two-step process. First, we apply the rule for $\text{g}$ to the number $4$ . $\text{g}(4) = 2 - 6(4) = 2 - 24 = -22$ . Second, we take this result, $-22$ , and apply the rule for $\text{f}$ to it. $\text{f}(-22) = 3(-22) - 5 = -66 - 5 = -71$ .

Alternatively, one could find the general formula for $\text{fg}(x)$ by substiting $(2-6x)$ into $\text{f}$ : $\text{fg}(x) = 3(2-6x) - 5 = 6 - 18x - 5 = 1 - 18x$ . Then $\text{fg}(4) = 1 - 18(4) = 1 - 72 = -71$ . Both methods yield the same result.

Key Takeaways

Composite functions are evaluated from the inside out.
$\text{fg}(x)$ is not the same as $\text{gf}(x)$ ; order matters.

Common Mistakes

Evaluating $\text{f}(4)$ first and then putting that into $\text{g}$ (calculating $\text{gf}(4)$ instead).
Arithmetic errors with negative numbers (e.g., $3 \times -22 = -66$ , then forgetting to subtract $5$ again).

Things to Be Careful About

Watch the signs carefully. Subtracting $5$ from a large negative number makes it more negative.

Techniques used

evaluate inner functionevaluate outer function using inner resultsubstitute expressions for composite functions

(d)

The domain of $\text{g}(x)$ is $x > -8$ .

Find the range of $\text{g}(x)$ .

______

DifficultyMedium-Easy

Worked solution

Approach

Start with the given domain inequality for $x$ and manipulate it algebraically to form the expression for $\text{g}(x)$ . The resulting inequality will define the range.

Working

Given the domain:

x > -8

The function is $\text{g}(x) = 2 - 6x$ . We need to transform $x > -8$ into an inequality involving $2 - 6x$ .

Step 1: Multiply by $-6$ . Remember that multiplying an inequality by a negative number reverses the direction of the inequality sign.

-6x < -6(-8)

-6x < 48

Step 2: Add $2$ to all parts of the inequality.

2 - 6x < 48 + 2

2 - 6x < 50

Since $\text{g}(x) = 2 - 6x$ , we have:

\text{g}(x) < 50

Answer

\text{g}(x) < 50

Final answer

g(x) < 50

Detailed explanation

Walkthrough

The domain tells us what values $x$ can take ( $x > -8$ ). The range is the set of output values $\text{g}(x)$ . Since $\text{g}(x)$ is a linear function with a negative gradient ( $-6$ ), it is strictly decreasing. As $x$ increases, $\text{g}(x)$ decreases. Therefore, the boundary at $x = -8$ will give the upper bound for the range.

Algebraically, we start with $x > -8$ . To get $-6x$ , we multiply by $-6$ . Crucially, because we multiplied by a negative number, the $>$ sign flips to $<$ , giving $-6x < 48$ . Then we add $2$ to both sides to match the structure of $\text{g}(x) = 2 - 6x$ , resulting in $2 - 6x < 50$ . Thus, $\text{g}(x) < 50$ .

Key Takeaways

When manipulating inequalities, multiplying or dividing by a negative number reverses the inequality symbol.
The range of a linear function over an open interval is also an open interval.

Common Mistakes

Forgetting to reverse the inequality sign when multiplying by $-6$ .
Calculating the value at the boundary ( $50$ ) but writing it as an equality ( $\text{g}(x) = 50$ ) instead of an inequality ( $\text{g}(x) < 50$ ).

Things to Be Careful About

The domain is strict ( $>$ ), so the range must also be strict ( $<$ ). Do not include equality.

Techniques used

determine range from domain using inequality propertiesapply operations to inequalities

Q11Medium-EasyNumber

$\mathscr{E} = \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$
$A = \{x : x \text{ is a prime number}\}$
$B = \{x : x \text{ is a factor of } 36\}$

(a)

Complete the Venn diagram.

Fig. 11a

DifficultyMedium-Easy

Worked solution

Approach

List the elements of sets $A$ and $B$ from the universal set $\mathscr{E}$ , then place them into the appropriate regions of the Venn diagram: $A$ only, $A \cap B$ , $B$ only, and outside both.

Working

The universal set is:

\mathscr{E} = \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}

Set $A$ contains the prime numbers in $\mathscr{E}$ :

A = \{2, 3, 5, 7, 11\}

Set $B$ contains the factors of $36$ that are in $\mathscr{E}$ . Since $36 = 2^2 \times 3^2$ , its factors are $1, 2, 3, 4, 6, 9, 12, 18, 36$ . The ones in $\mathscr{E}$ are:

B = \{2, 3, 4, 6, 9, 12\}

Find the intersection $A \cap B$ (elements in both $A$ and $B$ ):

A \cap B = \{2, 3\}

Find elements in $A$ only ( $A \setminus B$ ):

\{5, 7, 11\}

Find elements in $B$ only ( $B \setminus A$ ):

\{4, 6, 9, 12\}

Find elements in $\mathscr{E}$ but not in $A \cup B$ (outside both sets):

\mathscr{E} \setminus (A \cup B) = \{8, 10\}

Answer

Venn diagram with $A$ containing $5, 7, 11$ ; intersection containing $2, 3$ ; $B$ containing $4, 6, 9, 12$ ; outside containing $8, 10$ .

Final answer

Venn diagram with A containing 5, 7, 11; intersection containing 2, 3; B containing 4, 6, 9, 12; outside containing 8, 10

Detailed explanation

Walkthrough

First, determine the explicit elements of sets $A$ and $B$ by checking each number in the universal set $\mathscr{E}$ . For $A$ , identify which numbers are prime: $2, 3, 5, 7, 11$ . For $B$ , identify which numbers divide $36$ without remainder: $2, 3, 4, 6, 9, 12$ (note that $1$ and $18, 36$ are not in $\mathscr{E}$ ). Next, find the intersection $A \cap B$ by looking for numbers common to both lists, which are $2$ and $3$ . Place these in the overlapping region. Then place the remaining primes ( $5, 7, 11$ ) in the part of $A$ that does not overlap $B$ , and the remaining factors ( $4, 6, 9, 12$ ) in the part of $B$ that does not overlap $A$ . Finally, any numbers in $\mathscr{E}$ that are neither in $A$ nor $B$ ( $8$ and $10$ ) go outside both circles but inside the universal set box.

Key Takeaways

When completing a Venn diagram from set definitions, always start by listing the explicit elements of each set within the universal set. Find the intersection first, then distribute the remaining elements to their respective unique regions, and finally place any leftover universal set elements outside the defined sets.

Common Mistakes

Forgetting that $1$ is not a prime number (so $1$ is not in $A$ , though it is not in $\mathscr{E}$ anyway).
Forgetting that $1$ is not a prime number (so $1$ is not in $A$ , though it is not in $\mathscr{E}$ anyway).
Including numbers like $18$ or $36$ in set $B$ even if they are not in the universal set $\mathscr{E}$ .
Placing elements in the wrong region, such as putting $4$ in $A$ because it is even, or forgetting to put $8$ and $10$ outside both sets.
Listing elements in the wrong order or missing elements in part (b).

Things to Be Careful About

Always check that every element in the universal set $\mathscr{E}$ is placed somewhere in the Venn diagram. None should be left out.
Remember that $0$ is not prime and $1$ is not prime.
In part (b), the complement $B'$ includes everything outside $B$ , which means elements in $A$ only and elements outside both sets.
Elements must be listed in ascending order and separated by commas.

Techniques used

identify prime numbers in the universal setidentify factors of 36 in the universal setplace elements in correct Venn diagram regions

(b)

List the elements of $A \cup B'$ .

______

DifficultyMedium-Easy

Worked solution

Approach

Find the complement of $B$ (elements in $\mathscr{E}$ not in $B$ ), then find the union of $A$ and $B'$ by combining their elements.

Working

From part (a), $B = \{2, 3, 4, 6, 9, 12\}$ . The complement $B'$ contains all elements in $\mathscr{E}$ that are not in $B$ :

B' = \{5, 7, 8, 10, 11\}

Set $A = \{2, 3, 5, 7, 11\}$ . The union $A \cup B'$ contains all elements that are in $A$ or in $B'$ (or both):

A \cup B' = \{2, 3, 5, 7, 8, 10, 11\}

Alternatively, from the Venn diagram, $A \cup B'$ consists of all elements in $A$ (the left circle) plus all elements outside $B$ (everything except the right circle). This gives $\{2, 3, 5, 7, 11\} \cup \{5, 7, 8, 10, 11\} = \{2, 3, 5, 7, 8, 10, 11\}$ .

Answer

2, 3, 5, 7, 8, 10, 11

Final answer

2, 3, 5, 7, 8, 10, 11

Detailed explanation

Walkthrough

The complement $B'$ is the set of all elements in the universal set $\mathscr{E}$ that are not in $B$ . From the universal set $\mathscr{E} = \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$ and $B = \{2, 3, 4, 6, 9, 12\}$ , removing the elements of $B$ leaves $B' = \{5, 7, 8, 10, 11\}$ . The union $A \cup B'$ is the set of all elements that belong to $A$ or to $B'$ (or both). Combining $A = \{2, 3, 5, 7, 11\}$ and $B' = \{5, 7, 8, 10, 11\}$ gives $\{2, 3, 5, 7, 8, 10, 11\}$ . Listing them in ascending order gives the final answer.

Key Takeaways

The complement $B'$ includes everything outside set $B$ , which means elements in $A$ only and elements outside both sets. The union $A \cup B'$ combines all elements from both sets, removing duplicates.

Common Mistakes

Forgetting that elements in $A$ only are also part of $B'$ , so they must be included in $A \cup B'$ .
Missing elements like $8$ or $10$ which are outside both $A$ and $B$ but inside $\mathscr{E}$ , so they are in $B'$ and thus in $A \cup B'$ .
Listing elements out of order or with missing commas.

Things to Be Careful About

Ensure all elements are listed in ascending order.
Do not include elements that are in $B$ (such as $4, 6, 9, 12$ ) unless they are also in $A$ , but here none are.
Follow-through marks are awarded for using their own Venn diagram from part (a), so consistency is key.

Techniques used

find the complement of Bfind the union of A and B'list elements in ascending order

(c)

Find $\text{n}(A \cap B')$ .

______

Q12Medium-EasyAlgebra and Graphs

These are the first four terms of a sequence.

$16 \qquad 13 \qquad 10 \qquad 7$

(a)

Write down the next term of the sequence.

______

(b)

Find an expression for the $n\text{th}$ term of the sequence.

______

DifficultyMedium-Easy

Worked solution

Approach

For an arithmetic sequence, the $n$ th term is given by the form $an + b$ , where $a$ is the common difference and $b$ is determined using the first term.

Working

From part (a), we know the common difference is $-3$ . This means the coefficient of $n$ is $-3$ . So the expression starts as:
$-3n$
We need to find the constant term $b$ such that $-3n + b$ gives the correct values. Let's test for $n=1$ (the first term):
$-3(1) + b = 16$
$-3 + b = 16$
Add 3 to both sides:
$b = 19$
So the expression is $-3n + 19$ or $19 - 3n$ .

Let's verify with the second term ( $n=2$ ):
$19 - 3(2) = 19 - 6 = 13$
This matches the given sequence. Let's verify with the fourth term ( $n=4$ ):
$19 - 3(4) = 19 - 12 = 7$
This also matches.

Thus, the expression for the $n$ th term is $19 - 3n$ .

Answer

$19 - 3n$

Final answer

19 - 3n

Detailed explanation

Walkthrough

To find the $n$ th term of an arithmetic sequence, we use the linear formula $un = dn + c$ , where $d$ is the common difference.

Step 1: Identify the common difference ( $d$ ). From part (a), the difference is $-3$ . This becomes the number multiplying $n$ . So we have $-3n$ .

Step 2: Find the constant ( $c$ ). We compare $-3n$ to the actual sequence terms.
For $n=1$ , $-3(1) = -3$ . The actual first term is $16$ . To get from $-3$ to $16$ , we must add $19$ . Therefore, the constant is $+19$ .
Alternatively, you can think of the "zeroth term" (term 0). If the first term is $16$ and we go back by $3$ , the term at position $0$ would be $19$ . Thus the formula is $19 - 3n$ .

Step 3: Write the final expression. Combining these parts gives $19 - 3n$ .

Key Takeaways

The coefficient of $n$ in the $n$ th term formula is always equal to the common difference of the arithmetic sequence.
The constant term adjusts the scaled $n$ value to match the starting point of the sequence.

Common Mistakes

Using the positive value of the difference ( $3n$ ) instead of the negative one ( $-3n$ ).
Forgetting to add the constant adjustment (e.g., writing just $-3n$ ).
Confusing the position $n$ with the term value itself.

Things to Be Careful About

The mark scheme accepts equivalent forms like $-3n + 19$ . Both are correct.
Always check your answer against at least two terms in the original sequence to ensure accuracy.

Techniques used

determine the coefficient of n from the common differencecalculate the zeroth term to find the constant offsetcombine into the linear expression un + an

Q13MediumNumber

(a)

Write $1.23 \times 10^{-4}$ as an ordinary number.

______

DifficultyEasy

Worked solution

Approach

To convert $1.23 \times 10^{-4}$ to an ordinary number, we evaluate the power of ten and multiply it by $1.23$ . A negative exponent indicates that the decimal point moves to the left.

Working

The exponent is $-4$ , so we move the decimal point in $1.23$ four places to the left:

1.23 \rightarrow 0.123 \rightarrow 0.0123 \rightarrow 0.00123 \rightarrow 0.000123

Alternatively, using fraction notation:

1.23 \times 10^{-4} = 1.23 \times \frac{1}{10^4} = 1.23 \times \frac{1}{10000}

\frac{1.23}{10000} = 0.000123

Answer

0.000123

Final answer

0.000123

Detailed explanation

Walkthrough

Standard form (or scientific notation) expresses numbers as $A \times 10^n$ , where $1 \leq A < 10$ and $n$ is an integer. To reverse this process and find the ordinary number, we simply calculate the value of $10^n$ and multiply it by $A$ .

In this case, $n = -4$ . A negative power means the number is less than 1. Specifically, $10^{-4}$ is equal to $0.0001$ . Multiplying $1.23$ by $0.0001$ shifts the decimal point 4 places to the left. Starting at $1.23$ , one place left is $0.123$ , two places is $0.0123$ , three places is $0.00123$ , and four places is $0.000123$ .

Key Takeaways

Positive powers of 10 ( $10^n$ ) shift the decimal point to the right, making the number larger.
Negative powers of 10 ( $10^{-n}$ ) shift the decimal point to the left, making the number smaller.
The magnitude of the exponent tells you exactly how many places to move the decimal point.

Common Mistakes

Moving the decimal point in the wrong direction (right instead of left for negative exponents).
Miscounting the number of zeros or places to shift.
Forgetting to add leading zeros before the first non-zero digit when shifting past the start of the number.

Things to Be Careful About

Ensure the final answer is written in standard decimal notation, not in standard form. The mark scheme accepts 'cao' (correct answer only), meaning any equivalent correct decimal representation is fine, but $1.23 \times 10^{-4}$ itself is not the final answer.

Techniques used

convert from standard form to an ordinary number

(b)

$(8.2 \times 10^4) + (x \times 10^y) = 9.1 \times 10^5$

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

$x$ = ______
$y$ = ______

DifficultyMedium

Worked solution

Approach

The equation involves adding two numbers in standard form. To add them directly, they must have the same power of 10. We can either convert everything to ordinary numbers, or convert both terms to the same power of 10 (e.g., $10^5$ ) before adding. Once added, we compare the result to $9.1 \times 10^5$ to identify $x$ and $y$ .

Working

Given:

(8.2 \times 10^4) + (x \times 10^y) = 9.1 \times 10^5

Let's convert the known terms to ordinary numbers to see the total sum clearly:

8.2 \times 10^4 = 82,000

9.1 \times 10^5 = 910,000

So the equation becomes:

82,000 + (x \times 10^y) = 910,000

Subtract $82,000$ from both sides to isolate the second term:

x \times 10^y = 910,000 - 82,000

x \times 10^y = 828,000

Now we need to express $828,000$ in the form $x \times 10^y$ , where typically $1 \leq x < 10$ for standard form.

Convert $828,000$ to standard form:
Move the decimal point 5 places to the left to get a number between 1 and 10:

828,000 = 8.28 \times 10^5

Comparing this to $(x \times 10^y)$ :

x = 8.28

y = 5

Answer

x = 8.28, \quad y = 5

Final answer

x = 8.28, y = 5

Detailed explanation

Walkthrough

When adding or subtracting numbers in standard form, the powers of 10 must match. You cannot simply add the coefficients if the exponents are different.

Method 1: Convert to Ordinary Numbers
This is often the safest way to avoid errors. Convert $8.2 \times 10^4$ to $82,000$ and $9.1 \times 10^5$ to $910,000$ . Subtract $82,000$ from $910,000$ to find the missing value: $828,000$ . Finally, convert $828,000$ back into standard form $x \times 10^y$ . Since there are 6 digits, the power is $10^5$ , leaving $8.28$ as the coefficient.

Method 2: Match Powers of 10
Rewrite $8.2 \times 10^4$ to have a power of $10^5$ . To increase the exponent by 1, divide the coefficient by 10:

8.2 \times 10^4 = 0.82 \times 10^5

Now the equation is:

0.82 \times 10^5 + x \times 10^y = 9.1 \times 10^5

Assuming $y=5$ (since the result is $10^5$ and the other term was adjusted to $10^5$ ), we can add the coefficients:

0.82 + x = 9.1

x = 9.1 - 0.82 = 8.28

Thus $x=8.28$ and $y=5$ .

Both methods yield the same result. The mark scheme explicitly accepts $x=8.28$ and $y=5$ , or the intermediate ordinary number $828,000$ .

Key Takeaways

Standard form requires the coefficient $A$ to be between 1 and 10.
Addition/Subtraction: Align powers of 10 first, then operate on coefficients.
Conversion: Moving the decimal point left increases the positive exponent; moving right decreases it (or increases the negative exponent).

Common Mistakes

Adding coefficients directly without matching powers ( $8.2 + 9.1 = 17.3$ ), which is incorrect because $10^4 \neq 10^5$ .
Incorrectly counting the number of places to move the decimal point when converting back to standard form.
Confusing $x$ and $y$ . Remember $x$ is the coefficient and $y$ is the exponent.
Arithmetic errors in subtraction ( $910,000 - 82,000$ ).

Things to Be Careful About

The question asks for $x$ and $y$ . Ensure you provide both values.
Check if the resulting $x$ needs to be in a specific form. Here, $8.28$ is the natural standard form coefficient. The mark scheme allows $828,000$ or $8.28 \times 10^5$ as evidence, but $x$ and $y$ are distinct variables.

Techniques used

convert standard form terms to ordinary numbersperform addition with aligned powers of tenexpress the result back in standard form

Q14Medium-EasyAlgebra and Graphs

Factorise.

(a)

$5x^2 + 15xy$

______

DifficultyEasy

Worked solution

Approach

Identify the Highest Common Factor (HCF) of the terms $5x^2$ and $15xy$ , then divide each term by this HCF to find the remaining expression inside the brackets.

Working

The two terms are $5x^2$ and $15xy$ .

First, look at the numbers: $5$ and $15$ . The HCF is $5$ .
Next, look at the variables: $x^2$ and $xy$ . The HCF is $x$ .
Therefore, the overall HCF is $5x$ .

Divide each term by $5x$ :

\frac{5x^2}{5x} = x

\frac{15xy}{5x} = 3y

Write the HCF outside the bracket and the results inside:

5x(x + 3y)

Check by expanding: $5x \cdot x = 5x^2$ and $5x \cdot 3y = 15xy$ .

Answer

5x(x + 3y)

Final answer

5x(x + 3y)

Detailed explanation

Walkthrough

To factorise an algebraic expression means to write it as a product of brackets. We start by finding the largest factor that divides into every term in the expression.

Numbers: The coefficients are $5$ and $15$ . Since $5$ goes into both ( $5 \div 5 = 1$ and $15 \div 5 = 3$ ), we take $5$ out.
Variables: The first term has $x^2$ (which is $x \times x$ ) and the second has $xy$ (which is $x \times y$ ). Both contain at least one $x$ , so we take $x$ out.
Combine: The total factor to pull out is $5x$ .
Simplify: Divide the original terms by $5x$ to see what remains inside the brackets. $5x^2 \div 5x = x$ and $15xy \div 5x = 3y$ . This gives $(x + 3y)$ .

Key Takeaways

Always check for a common numerical factor AND a common variable factor. The final answer must expand back exactly to the original expression.

Common Mistakes

Forgetting the variable part of the HCF (e.g., writing $5(x^2 + 3xy)$ ). The mark scheme awards partial credit (B1) for this, but full marks require the full HCF $5x$ .
Dividing incorrectly (e.g., thinking $15xy \div 5x = 15y$ ).

Things to Be Careful About

Make sure you don't leave a '1' behind if the entire term was factored out (not applicable here, but common in other questions like $3x - 3$ ). Ensure all terms are accounted for inside the bracket.

Techniques used

identify the highest common factor of numerical coefficientsidentify the highest common factor of variable termsfactorise by extracting the HCF

(b)

$2ax + 4bx - 3ay - 6by$

______

DifficultyMedium-Easy

Worked solution

Approach

This expression has four terms. We use the method of grouping: split the expression into two pairs, factorise each pair separately, and then look for a common bracket.

Working

Expression: $2ax + 4bx - 3ay - 6by$

Step 1: Group the terms.
Group the first two terms and the last two terms:

(2ax + 4bx) - (3ay + 6by)

Note: When grouping the negative terms, be careful with signs. It is often easier to factor out a positive number from the second group by keeping track of the minus sign outside.

Alternatively, just factorise normally:
From the first group $(2ax + 4bx)$ , the common factor is $2x$ :

2x(a + 2b)

From the second group $(-3ay - 6by)$ , the common factor is $-3y$ :

-3y(a + 2b)

Step 2: Combine the groups.
Now we have:

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

Step 3: Factorise the whole expression.
Both parts contain the same bracket $(a + 2b)$ . Pull this out:

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

Check by expanding: $a(2x) + a(-3y) + 2b(2x) + 2b(-3y) = 2ax - 3ay + 4bx - 6by$ . This matches the original.

Answer

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

Final answer

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

Detailed explanation

Walkthrough

When there are four terms, 'grouping' is the standard technique.

Pair them up: Look at the first two terms ( $2ax$ and $4bx$ ) and the last two ( $-3ay$ and $-6by$ ).
Factorise Pair 1: What do $2ax$ and $4bx$ share? They share $2$ and $x$ . So, $2x(a + 2b)$ .
Factorise Pair 2: What do $-3ay$ and $-6by$ share? They share $3$ and $y$ . Because they are both negative, we factor out $-3y$ to make the inside positive. $-3y(a + 2b)$ .
Alternative: You could factor out $+3y$ to get $-3y(a+2b)$ or keep the minus inside $3y(-a-2b)$ . The key is that the content inside the bracket must match the other pair.
Spot the Match: Both lines now end with $(a + 2b)$ . This is the common binomial factor.
Final Bracket: Write $(a + 2b)$ once, and put whatever is left ( $2x$ and $-3y$ ) in the second bracket: $(2x - 3y)$ .

Key Takeaways

Factorising by grouping relies on the 'hidden' common bracket appearing after the first step. Always verify your answer by expanding it fully to ensure it equals the original question.

Common Mistakes

Sign errors: When factoring the second group, forgetting that the terms were negative leads to $(2x + 3y)(a + 2b)$ which expands to $2ax + 4bx + 3ay + 6by$ (wrong signs).
Mismatched brackets: If the brackets don't match, the grouping order might be wrong (e.g., pairing $2ax$ with $-3ay$ ). Try different pairings if the first attempt fails.

Things to Be Careful About

The mark scheme accepts equivalent forms such as $(3y - 2x)(-a - 2b)$ . However, the standard positive form $(2x - 3y)(a + 2b)$ is safest. Ensure you distribute the negative sign correctly when factoring out $-3y$ from $-3ay - 6by$ .

Techniques used

group terms into pairsfactorise each pair individuallyextract the common binomial factor

Q154MMedium-EasyAlgebra and Graphs

Fig. 15

Find the three inequalities that define the unshaded region, $R$ .

______
______
______

DifficultyMedium-Easy

Worked solution

Approach

Identify the equation of each of the three boundary lines bounding region $R$ . Then determine whether $R$ lies above or below each line, and use the line style (solid or dashed) to decide whether the inequality is strict or non-strict.

Working

The region $R$ is bounded by three lines:

A horizontal dashed line passing through $y = 2$ .
The equation is $y = 2$ . Since the line is dashed and the region $R$ lies below it, the inequality is:

y < 2

A solid line passing through $(0, 0)$ and $(4, 2)$ .
The gradient is $m = \frac{2 - 0}{4 - 0} = \frac{1}{2}$ and the y-intercept is $0$ .
The equation is $y = \frac{1}{2}x$ . Since the line is solid and the region $R$ lies above it, the inequality is:

y \geq \frac{1}{2}x

A solid line passing through $(0, 2)$ and $(2, 0)$ .
The gradient is $m = \frac{0 - 2}{2 - 0} = -1$ and the y-intercept is $2$ .
The equation is $y = 2 - x$ . Since the line is solid and the region $R$ lies above it, the inequality is:

y \geq 2 - x

We can verify with a test point inside $R$ , such as $(1.5, 1.5)$ :

$1.5 < 2$ (true)
$1.5 \geq \frac{1}{2}(1.5) = 0.75$ (true)
$1.5 \geq 2 - 1.5 = 0.5$ (true)

Answer

y < 2, \quad y \geq \frac{1}{2}x, \quad y \geq 2 - x

Final answer

y < 2, y >= 1/2 x, y >= 2 - x

Detailed explanation

Walkthrough

The question asks for three inequalities that define the unshaded triangular region $R$ . We do this by finding the equation of each boundary line and then deciding which side of the line contains $R$ .

Step 1: The horizontal boundary.
The top boundary is a dashed horizontal line crossing the y-axis at $y = 2$ . Its equation is simply $y = 2$ . Because the line is dashed, the boundary is not included in the region, meaning we use a strict inequality ( $<$ or $>$ ). Since $R$ is below this line, the y-values in $R$ are less than $2$ , giving $y < 2$ .

Step 2: The line with positive gradient.
The bottom-right boundary is a solid line passing through the origin $(0, 0)$ and the point $(4, 2)$ . We calculate its gradient: $m = \frac{2 - 0}{4 - 0} = \frac{1}{2}$ . With a y-intercept of $0$ , the equation is $y = \frac{1}{2}x$ . Because the line is solid, the boundary is included, so we use $\geq$ or $\leq$ . Region $R$ lies above this line (for a given $x$ , the y-values in $R$ are greater than those on the line), giving $y \geq \frac{1}{2}x$ .

Step 3: The line with negative gradient.
The bottom-left boundary is a solid line passing through $(0, 2)$ and $(2, 0)$ . We calculate its gradient: $m = \frac{0 - 2}{2 - 0} = -1$ . With a y-intercept of $2$ , the equation is $y = 2 - x$ . Because the line is solid, we use $\geq$ or $\leq$ . Region $R$ lies above this line, giving $y \geq 2 - x$ .

Verification:
We can confirm by picking a point clearly inside $R$ , like $(1.5, 1.5)$ , and checking it against all three inequalities. All three hold true, confirming our directions are correct.

Key Takeaways

To represent a region with inequalities, find the equation of each boundary line in the form $y = mx + c$ .
A dashed boundary line means the region does not include the line itself, requiring a strict inequality ( $<$ or $>$ ).
A solid boundary line means the region includes the line, requiring a non-strict inequality ( $\leq$ or $\geq$ ).
Use a test point inside the region to verify the correct inequality direction.

Common Mistakes

Using $=$ instead of $<$ or $>$ for the dashed line (e.g., writing $y = 2$ instead of $y < 2$ ).
Confusing the direction of the inequality (e.g., writing $y \leq \frac{1}{2}x$ when the region is above the line).
Failing to notice that one line is dashed while the others are solid, leading to incorrect inequality symbols.
Writing the equations of the lines incorrectly, such as misreading the y-intercept or gradient from the grid.

Things to Be Careful About

Always check the line style: dashed means strict inequality ( $<$ or $>$ ), solid means non-strict ( $\leq$ or $\geq$ ).
Ensure the inequality symbols are correct for the region's position relative to each line. A quick test point inside the region can prevent sign errors.
The mark scheme accepts equivalent forms, such as $y \geq \frac{1}{2}x$ or $2y \geq x$ , but the standard $y = mx + c$ form is expected.
Do not include the shaded regions in your final answer; only the three inequalities defining $R$ are required.

Techniques used

find the equation of each boundary linedetermine the inequality direction using a test pointdistinguish between strict and non-strict inequalities from line style

Q16MediumStatisticsNumber

The table gives information about the ages of the 80 members of a gym.

Age ( $a$ years)	$16 < a \leq 18$	$18 < a \leq 24$	$24 < a \leq 40$	$40 < a \leq 100$
Frequency	5	31	19	25

(a)

Calculate an estimate of the mean age.

______ $\text{years}$

DifficultyMedium

Worked solution

Approach

To estimate the mean from a grouped frequency table where classes have different widths, we assume that all values within a class are concentrated at the class midpoint ( $x$ ). We calculate $fx$ for each row, sum these products ( $\sum fx$ ), and divide by the total frequency ( $\sum f$ ).

Working

Step 1: Find the midpoints ( $x$ ) for each age group.
The midpoint is calculated as $\frac{\text{Lower Bound} + \text{Upper Bound}}{2}$ .

For $16 < a \leq 18$ : $x = \frac{16 + 18}{2} = 17$
For $18 < a \leq 24$ : $x = \frac{18 + 24}{2} = 21$
For $24 < a \leq 40$ : $x = \frac{24 + 40}{2} = 32$
For $40 < a \leq 100$ : $x = \frac{40 + 100}{2} = 70$

Step 2: Calculate the product $fx$ for each row.
Multiply the frequency ( $f$ ) by the midpoint ( $x$ ).

\begin{aligned} &5 \times 17 = 85 \\ &31 \times 21 = 651 \\ &19 \times 32 = 608 \\ &25 \times 70 = 1750 \end{aligned}

Step 3: Sum the frequencies and the $fx$ values.
Total Frequency ( $\sum f$ ):

5 + 31 + 19 + 25 = 80

Sum of Products ( $\sum fx$ ):

85 + 651 + 608 + 1750 = 3094

Step 4: Calculate the estimated mean.

\text{Estimated Mean} = \frac{\sum fx}{\sum f} = \frac{3094}{80}

\frac{3094}{80} = 38.675

Answer

38.675 \text{ years}

Final answer

38.675

Detailed explanation

Walkthrough

When data is presented in groups (intervals) rather than as individual numbers, we cannot find the exact mean. Instead, we estimate it by assuming every person's age falls exactly on the middle of their age group (the midpoint).

Find Midpoints: For each column, add the lower and upper age limits and divide by 2. Note that for the last group ( $40 < a \leq 100$ ), the midpoint is $\frac{40+100}{2} = 70$ . This assumes the distribution is roughly even across that wide range.
Weighted Values: Multiply the number of people in each group (frequency) by that group's representative age (midpoint). This gives the estimated total age contribution of that group.
- Group 1: 5 people aged approx 17 $ightarrow$ 85 years total.
- Group 2: 31 people aged approx 21 $ightarrow$ 651 years total.
- Group 3: 19 people aged approx 32 $ightarrow$ 608 years total.
- Group 4: 25 people aged approx 70 $ightarrow$ 1750 years total.
Total Sum: Add up all the estimated total ages ( $3094$ ) and divide by the total number of gym members ( $80$ ).

Key Takeaways

The formula for the estimated mean of grouped data is $\bar{x} = \frac{\sum fx}{\sum f}$ .
The variable $x$ represents the class midpoint.
Class midpoints must be calculated carefully, especially for non-uniform class widths.

Common Mistakes

Using the lower or upper bound instead of the midpoint for $x$ .
Failing to multiply the frequency by the midpoint (calculating just $x$ ).
Incorrectly calculating the midpoint for the final interval ( $40$ to $100$ ).
Arithmetic errors when summing large numbers like $1750 + 608 + 651 + 85$ .

Things to Be Careful About

Ensure you use the correct boundaries. The notation $16 < a \leq 18$ means the continuous interval runs from 16 to 18.
The question asks for an estimate, so using the midpoint is the standard accepted method (M1 for correct midpoints soi).
Check your division; $3094 \div 80$ results in a terminating decimal here.

Techniques used

determine midpoints of grouped intervalscalculate the sum of frequency times midpointdivide by total frequency to estimate the mean

(b)

Work out the percentage of the members of the gym who are more than 24 years old.

______ $\%$

DifficultyMedium-Easy

Worked solution

Approach

We need to find the proportion of members who are strictly older than 24 years and express this as a percentage of the total membership.

Working

Step 1: Identify the relevant groups.
The condition is "more than 24 years old" ( $a > 24$ ).
Looking at the table:

$16 < a \leq 18$ : No (all are $\leq 18$ )
$18 < a \leq 24$ : No (all are $\leq 24$ )
$24 < a \leq 40$ : Yes (all are $> 24$ )
$40 < a \leq 100$ : Yes (all are $> 24$ )

So we sum the frequencies for these two groups:

19 + 25 = 44

Step 2: Calculate the percentage.
The total number of members is $80$ .

\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100\%

\frac{44}{80} \times 100\%

Simplifying the fraction:

\frac{44}{80} = \frac{11}{20} = 0.55

Converting to percentage:

0.55 \times 100\% = 55\%

Answer

55\%

Final answer

55

Detailed explanation

Walkthrough

First, look at the age intervals to see which ones contain only ages greater than 24. The interval $18 < a \leq 24$ includes people who are exactly 24, so they do not count as "more than 24". The next interval starts with $24 < a$ , meaning everyone in that group and the one after is strictly older than 24.

Add the frequencies for the groups $24 < a \leq 40$ (19 members) and $40 < a \leq 100$ (25 members) to get 44 members.

Finally, divide this count by the total number of members (80) and multiply by 100 to get the percentage.

Key Takeaways

Pay attention to inequality symbols. " $\leq 24$ " means 24 is included in that group, so those people are not "more than 24".
A percentage is simply a fraction out of 100.

Common Mistakes

Including the group $18 < a \leq 24$ in the count (getting $31+19+25=75$ ).
Calculating the percentage of the wrong base (e.g., dividing by 100 instead of 80).
Arithmetic errors in the final division/multiplication.

Things to Be Careful About

The phrase "more than 24" excludes 24 itself. If the question said "24 or older", the second group would be included.

Techniques used

identify subset of data satisfying conditionform fraction of part over wholeconvert fraction to percentage

Q17MediumGeometry

Fig. 17

Points $B$ , $D$ , $E$ and $F$ lie on a circle.
$AC$ is a tangent to the circle at $B$ .

Angle $BEF = 50^\circ$ and $EF = BF$ .

(a)

Find the value of $x$ .
Give a geometrical reason for your answer.

$x$ = ______ because ______

DifficultyEasy

Worked solution

Approach

The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. Apply this theorem directly to find $x$ .

Working

$AC$ is a tangent to the circle at $B$ , and $BF$ is a chord. The angle between the tangent and the chord is $x^\circ$ (i.e., $\angle FBC$ ). The angle subtended by chord $BF$ in the alternate segment is $\angle BEF$ .

x = \angle BEF

Given that $\angle BEF = 50^\circ$ :

x = 50

Answer

$x = 50$ because alternate segment theorem.

Final answer

x = 50

Detailed explanation

Walkthrough

The alternate segment theorem states that the angle between a tangent to a circle and a chord drawn from the point of contact is equal to the angle subtended by that chord in the alternate segment of the circle. Here, $AC$ is the tangent at $B$ , and $BF$ is the chord. The angle between them is $\angle FBC = x^\circ$ . The angle in the alternate segment (the segment on the other side of $BF$ from $C$ ) is $\angle BEF$ , which is given as $50^\circ$ . Therefore, $x$ is immediately $50$ .

Key Takeaways

The alternate segment theorem is a powerful tool for finding angles involving tangents and chords. Always identify the tangent, the chord, and the angle in the alternate segment clearly.

Common Mistakes

Confusing which angle is in the alternate segment. The alternate segment is the one on the opposite side of the chord from the angle between the tangent and chord.
Forgetting to state the theorem clearly as the geometrical reason; simply writing "alternate segment" is sufficient, but vague reasons like "circle theorem" may not score.

Things to Be Careful About

The mark scheme awards one mark for the correct value ( $50$ ) and one mark for the reason. The reason must specifically name the "alternate segment theorem". Do not write unsupported answers or vague circle theorems.

Techniques used

apply the alternate segment theorem

(b)

Find the value of $y$ .
Give a geometrical reason for each step of your working.

$y$ = ______

DifficultyMedium

Worked solution

Approach

To find $y$ , first use the fact that $\triangle BEF$ is isosceles to find $\angle BFE$ . Then, use the property that opposite angles in a cyclic quadrilateral sum to $180^\circ$ to find $y$ .

Working

In $\triangle BEF$ , we are given $EF = BF$ . This means $\triangle BEF$ is an isosceles triangle, so the base angles are equal:

\angle EBF = \angle BEF = 50^\circ

The sum of angles in a triangle is $180^\circ$ , so we can find $\angle BFE$ :

\angle BFE = 180^\circ - 50^\circ - 50^\circ = 80^\circ

Points $B$ , $D$ , $E$ , and $F$ lie on the circle, forming a cyclic quadrilateral $BDEF$ . A key property of cyclic quadrilaterals is that opposite angles sum to $180^\circ$ . The angle opposite to $\angle BFE$ is $\angle BDE$ (which is $y^\circ$ ):

\angle BDE + \angle BFE = 180^\circ

y + 80^\circ = 180^\circ

y = 100

Answer

$y = 100$ .

Final answer

y = 100

Detailed explanation

Walkthrough

The question asks for $y$ , which is the angle $\angle BDE$ at vertex $D$ in the cyclic quadrilateral $BDEF$ . To find this, we first need the opposite angle $\angle BFE$ .

Step 1: Look at $\triangle BEF$ . We are told $EF = BF$ , which makes it an isosceles triangle. The angles opposite the equal sides are equal, so $\angle EBF = \angle BEF = 50^\circ$ . Using the triangle angle sum ( $180^\circ$ ), we calculate the remaining angle: $\angle BFE = 180^\circ - 50^\circ - 50^\circ = 80^\circ$ .

Step 2: Now consider the cyclic quadrilateral $BDEF$ . The vertices $B, D, E, F$ all lie on the circle. Opposite angles in a cyclic quadrilateral always add up to $180^\circ$ . The angle opposite $\angle BFE$ is $\angle BDE$ (labeled $y^\circ$ in the diagram). Therefore, $y + 80 = 180$ , giving $y = 100$ .

Key Takeaways

Isosceles triangles have two equal base angles. Always use this when two sides of a triangle are marked equal.
Opposite angles in a cyclic quadrilateral sum to $180^\circ$ . Identify the four points on the circle and match up the correct opposite pairs.

Common Mistakes

associating the reasons with the wrong angles. The mark scheme explicitly states: "Reasons must not be associated with an incorrect angle". For example, claiming $\angle BDE = \angle BFE$ or misidentifying which angles are opposite in the cyclic quadrilateral.
forgetting to calculate $\angle BFE$ first and trying to jump straight to $y$ .
not showing the intermediate step for $\angle BFE = 80^\circ$ ; the marks are awarded for the correct reasons applied to the correct angles.

Things to Be Careful About

The mark scheme awards 1 mark for the final answer ( $100$ ) and 1 mark each for two correct reasons (isosceles triangle, opposite angles of cyclic quadrilateral sum to $180^\circ$ ). Max 2 marks if the answer is not fully correct, meaning working must be shown and reasons must be valid.
Ensure you write $180^\circ$ not just $180$ when stating the circle theorem reason, as precision matters in geometry proofs.

Techniques used

identify isosceles triangle propertiescalculate the third angle in a triangleapply the cyclic quadrilateral opposite angles theorem

Q18MediumAlgebra and GraphsMensuration

The diagram shows a cuboid.

Fig. 18

The length of the cuboid is $x\,\text{cm}$ .
The height of the cuboid is 3 times its length.
The width of the cuboid is $4\,\text{cm}$ less than its length.

(a)

Write down expressions, in terms of $x$ , for the height of the cuboid and the width of the cuboid.

$\text{height} = \_\_\_\_\_\_\,\text{cm}$
$\text{width} = \_\_\_\_\_\_\,\text{cm}$

(b)

The surface area of the cuboid is $200\,\text{cm}^2$ .

Form an equation in $x$ and show that it simplifies to $7x^2 - 16x - 100 = 0$ .

DifficultyMedium

Worked solution

Approach

The surface area of a cuboid is the sum of the areas of its six rectangular faces. Write this sum using the expressions for length, height, and width from part (a), set it equal to 200, and simplify to the required quadratic equation.

Working

The three pairs of opposite faces have areas $x \times 3x$ , $x \times (x - 4)$ , and $3x \times (x - 4)$ . The total surface area is:

2x(3x) + 2x(x - 4) + 2(3x)(x - 4) = 200

Expand each bracket:

6x^2 + 2x^2 - 8x + 6x^2 - 24x = 200

Combine like terms on the left side:

14x^2 - 32x = 200

Subtract 200 from both sides to form a quadratic equation equal to zero:

14x^2 - 32x - 200 = 0

Divide the entire equation by 2:

7x^2 - 16x - 100 = 0

This matches the required form.

Answer

7x^2 - 16x - 100 = 0

Final answer

7x^2 - 16x - 100 = 0

Detailed explanation

Walkthrough

A cuboid has three pairs of identical rectangular faces. The dimensions are length $x$ , height $3x$ , and width $x - 4$ . The areas of the three distinct faces are $x(3x) = 3x^2$ , $x(x - 4) = x^2 - 4x$ , and $3x(x - 4) = 3x^2 - 12x$ . Multiplying each by 2 for the pairs gives $6x^2$ , $2x^2 - 8x$ , and $6x^2 - 24x$ . Adding these together gives $14x^2 - 32x$ . Setting this equal to the given surface area of 200 cm $^2$ yields $14x^2 - 32x = 200$ . Rearranging gives $14x^2 - 32x - 200 = 0$ , and dividing by 2 simplifies it to $7x^2 - 16x - 100 = 0$ .

Key Takeaways

The surface area of a cuboid with dimensions $l$ , $w$ , $h$ is $2(lw + lh + wh)$ . When dimensions are algebraic expressions, careful expansion and collecting of like terms is required. Always check that the final equation matches the required form exactly.

Common Mistakes

Forgetting to multiply by 2 for the pairs of faces.
Expansion errors, such as $2x(x - 4) = 2x^2 - 4x$ (missing the 8x) or $2(3x)(x - 4) = 6x^2 - 12x$ (missing the 12x).
Failing to divide by 2 at the end, leaving $14x^2 - 32x - 200 = 0$ instead of the required $7x^2 - 16x - 100 = 0$ .
Not setting the equation equal to zero.

Things to Be Careful About

The mark scheme requires 'no errors or omissions' for the final A1 mark. Show all expansion steps clearly. The final answer must be exactly $7x^2 - 16x - 100 = 0$ . Remember that 'show that' questions require starting from the given information and working towards the target, not the other way around.

Techniques used

write the surface area formula for a cuboidsubstitute algebraic expressions for dimensionsexpand and collect like termsform a quadratic equation equal to zero

(c)

Solve the equation $7x^2 - 16x - 100 = 0$ .
You must show all your working and give your answers correct to 2 decimal places.

$x$ = ______ or $x$ = ______

DifficultyMedium-Easy

Worked solution

Approach

Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a = 7$ , $b = -16$ , and $c = -100$ to find the roots.

Working

Substitute the values into the quadratic formula:

x = \frac{-(-16) \pm \sqrt{(-16)^2 - 4 \times 7 \times -100}}{2 \times 7}

Simplify the numerator and denominator:

x = \frac{16 \pm \sqrt{256 + 2800}}{14}

x = \frac{16 \pm \sqrt{3056}}{14}

Calculate $\sqrt{3056} \approx 55.281098$ :

x = \frac{16 \pm 55.281098}{14}

Find the two values:

x = \frac{16 + 55.281098}{14} = \frac{71.281098}{14} \approx 5.0915

x = \frac{16 - 55.281098}{14} = \frac{-39.281098}{14} \approx -2.8058

Round to 2 decimal places:

x = 5.09 \quad \text{or} \quad x = -2.81

Answer

x = 5.09 \quad \text{or} \quad x = -2.81

Final answer

x = 5.09 or x = -2.81

Detailed explanation

Walkthrough

The equation is $7x^2 - 16x - 100 = 0$ . Identify $a = 7$ , $b = -16$ , $c = -100$ . The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . Substituting gives $x = \frac{16 \pm \sqrt{256 - 4(7)(-100)}}{14}$ . The discriminant is $256 + 2800 = 3056$ . The square root of 3056 is approximately 55.2811. Adding and subtracting this from 16 and dividing by 14 gives the two roots: $71.2811 / 14 \approx 5.09$ and $-39.2811 / 14 \approx -2.81$ . Both must be rounded to 2 decimal places as instructed.

Key Takeaways

The quadratic formula is a reliable method for solving any quadratic equation that does not factorise easily. Always double-check the signs of $a$ , $b$ , and $c$ , especially when $c$ is negative, as $-4ac$ becomes positive.

Common Mistakes

Sign errors in the discriminant, such as calculating $256 - 2800 = -2544$ instead of $256 + 2800 = 3056$ .
Forgetting the $\pm$ symbol and only giving one answer.
Rounding incorrectly (e.g., $-2.8058$ rounds to $-2.81$ , not $-2.80$ ).
Not showing the substitution into the formula, which loses method marks.

Things to Be Careful About

The question explicitly asks for answers 'correct to 2 decimal places'. Do not give more or fewer decimal places. The mark scheme awards B1 for the final answers, so both must be correct and properly rounded. Showing the quadratic formula substitution is essential for the B2 mark.

Techniques used

apply the quadratic formulaevaluate the discriminantcalculate the two roots to 2 decimal places

(d)

Find the height of the cuboid.

______ $\text{cm}$

DifficultyEasy

Worked solution

Approach

The height of the cuboid is given by the expression $3x$ from part (a). Substitute the positive value of $x$ found in part (c) to calculate the height.

Working

From part (c), the positive root is $x \approx 5.0915...$ (use the unrounded value for accuracy).
Height $= 3x$ :

\text{height} = 3 \times 5.0915... = 15.2745...

Round to 3 significant figures (or 1 decimal place, as the mark scheme accepts 15.27 to 15.3):

\text{height} \approx 15.3 \, \text{cm}

Answer

15.3 \, \text{cm}

Final answer

15.3

Detailed explanation

Walkthrough

Part (a) established that the height is $3x$ cm. Part (c) gave two values for $x$ : $5.09$ and $-2.81$ . Since $x$ represents a physical length, it must be positive, so we use $x = 5.0915...$ (using more decimal places from the calculator to avoid rounding errors). Multiplying by 3 gives $15.2745...$ cm. The mark scheme accepts any answer from 15.27 to 15.3, so 15.3 cm is appropriate.

Key Takeaways

When a physical quantity like length must be positive, discard negative roots. Always use the unrounded value from the previous step in calculations to avoid compounding rounding errors.

Common Mistakes

Using the rounded value $5.09$ exactly: $3 \times 5.09 = 15.27$ , which is acceptable, but using $5.0915...$ gives $15.2745...$ , which rounds to $15.3$ . Both are typically accepted, but carrying more decimals is safer.
Forgetting that length cannot be negative and attempting to use $x = -2.81$ .
Not including the unit 'cm' in the final answer if required (though the blank has 'cm' next to it, it's good practice to be aware).

Things to Be Careful About

The mark scheme states 'FT $3 \times$ their positive root from (c)'. This means if a candidate made an error in (c) but correctly substituted their (incorrect) positive root into $3x$ , they would still get the mark for (d). Always use the positive root for physical dimensions.

Techniques used

substitute the positive root into the height expressionround to appropriate accuracy

Q19MediumTrigonometry

The diagram shows triangle $ABC$ and triangle $ABD$ .

Fig. 19

$AB = 12\,\text{cm}$ , $BD = 7\,\text{cm}$ and $AD = 6\,\text{cm}$ .
Angle $DAC = 32^\circ$ and angle $BCA = 48^\circ$ .

(a)

Calculate angle $BAD$ .

$\text{Angle } BAD$ = ______

DifficultyMedium-Easy

Worked solution

Approach

In triangle ABD, all three side lengths are known (AB = 12, AD = 6, BD = 7). Use the cosine rule to find the angle opposite to BD, which is angle BAD.

Working

\begin{aligned} BD^2 &= AB^2 + AD^2 - 2(AB)(AD)\cos(\angle BAD) \\ 7^2 &= 12^2 + 6^2 - 2(12)(6)\cos(\angle BAD) \\ 49 &= 144 + 36 - 144\cos(\angle BAD) \\ 49 &= 180 - 144\cos(\angle BAD) \\ 144\cos(\angle BAD) &= 131 \\ \cos(\angle BAD) &= \frac{131}{144} \end{aligned}

Now find the angle:

\angle BAD = \cos^{-1}\left(\frac{131}{144}\right) \approx 24.53^\circ

Answer

\angle BAD = 24.5^\circ

Final answer

24.5

Detailed explanation

Walkthrough

We are given the three side lengths of triangle ABD: AB = 12, AD = 6, and BD = 7. To find angle BAD, we use the cosine rule, which relates the sides of a triangle to the cosine of one of its angles. The form used is $b^2 = a^2 + c^2 - 2ac\cos B$ , adapted here as $BD^2 = AB^2 + AD^2 - 2(AB)(AD)\cos(\angle BAD)$ . Substituting the values gives $49 = 180 - 144\cos(\angle BAD)$ , which simplifies to $\cos(\angle BAD) = 131/144$ . Taking the inverse cosine yields approximately $24.53^\circ$ , which rounds to $24.5^\circ$ to three significant figures.

Key Takeaways

The cosine rule can be used to find an angle when all three sides of a triangle are known. Always ensure the correct side is squared on the left-hand side (the side opposite the angle you want to find).

Common Mistakes

Forgetting to square the side lengths before substituting into the cosine rule.
Mixing up the sides in the formula $2ac\cos B$ ; the two sides forming the angle must be used here (AB and AD), not the side opposite (BD).
Rounding the angle to fewer than three significant figures when not specified.

Things to Be Careful About

The diagram is marked NOT TO SCALE, so do not attempt to measure angles or lengths with a ruler.
Keep extra decimal places during intermediate calculations (e.g., use $24.53^\circ$ rather than $24.5^\circ$ ) to avoid rounding errors in part (b).
The final answer should be given to three significant figures as is standard for 4024 unless otherwise stated.

Techniques used

apply the cosine rule to find an angleevaluate the inverse cosine

(b)

Calculate $AC$ .

$AC$ = ______ $\text{cm}$

DifficultyMedium

Worked solution

Approach

First, find angle BAC by adding angle BAD (from part a) and angle DAC (given). Then use the angle sum of triangle ABC to find angle ABC. Finally, apply the sine rule to find AC.

Working

Angle BAC is the sum of angle BAD and angle DAC:

\angle BAC = 24.53^\circ + 32^\circ = 56.53^\circ

In triangle ABC, the angles sum to $180^\circ$ . We can find angle ABC:

\angle ABC = 180^\circ - \angle BAC - \angle BCA = 180^\circ - 56.53^\circ - 48^\circ = 75.47^\circ

Alternatively, note that $\angle ABC = 180^\circ - (32^\circ + 48^\circ) - \angle BAD = 100^\circ - 24.53^\circ = 75.47^\circ$ .

Now apply the sine rule in triangle ABC to find AC:

\frac{AC}{\sin(\angle ABC)} = \frac{AB}{\sin(\angle BCA)}

\frac{AC}{\sin 75.47^\circ} = \frac{12}{\sin 48^\circ}

AC = \frac{12 \sin 75.47^\circ}{\sin 48^\circ}

AC \approx \frac{12 \times 0.9682}{0.7431} \approx 15.63

Answer

AC = 15.6 \text{ cm}

Final answer

15.6

Detailed explanation

Walkthrough

To find AC, we work in the larger triangle ABC. We already know AB = 12 and angle BCA = 48°. We need one more angle. Angle BAC is made up of angle BAD (found in part a as $24.53^\circ$ ) and angle DAC (given as $32^\circ$ ), so angle BAC = $56.53^\circ$ . The sum of angles in triangle ABC is $180^\circ$ , so angle ABC = $180^\circ - 56.53^\circ - 48^\circ = 75.47^\circ$ . With two angles and a side opposite one of them, we use the sine rule: $AC / \sin(\angle ABC) = AB / \sin(\angle BCA)$ . Substituting the values gives $AC = (12 \times \sin 75.47^\circ) / \sin 48^\circ \approx 15.6$ cm.

Key Takeaways

When a triangle is split into smaller triangles, angles can be combined. The sine rule is useful when you know a side and its opposite angle, plus another angle or side.

Common Mistakes

Using angle BAD directly in the sine rule for triangle ABC instead of angle BAC.
Forgetting to add angle DAC to angle BAD to get the full angle BAC.
Rounding angle ABC too early (e.g., using $75.5^\circ$ instead of $75.47^\circ$ ) can lead to a slightly incorrect final answer.
Inverting the sine rule fraction (e.g., $12 \sin 48^\circ / \sin 75.47^\circ$ ).

Things to Be Careful About

Always use unrounded intermediate values (like $24.53^\circ$ ) in subsequent parts to avoid compounding rounding errors.
The sine rule requires matching each side with the sine of its opposite angle: AC is opposite angle ABC, and AB is opposite angle BCA.

Techniques used

find the third angle of a triangle using angle sumapply the sine rule to find a side length

Q204MMediumAlgebra and Graphs

Solve.

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

$x$ = ______

DifficultyMedium

Worked solution

Approach

The equation contains two rational expressions with different denominators. We combine them over a common denominator, clear the fractions by multiplying both sides by that denominator, and then solve the resulting linear equation.

Working

Given:

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

Find a common denominator, which is $(x - 1)(x + 2)$ :

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

Combine the numerators:

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

Multiply both sides by $(x - 1)(x + 2)$ to clear the fraction:

3x(x + 2) + 4(x - 1) = 3(x - 1)(x + 2)

Expand all brackets:
Left Hand Side (LHS):

3x^2 + 6x + 4x - 4 = 3x^2 + 10x - 4

Right Hand Side (RHS):

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

Set LHS equal to RHS:

3x^2 + 10x - 4 = 3x^2 + 3x - 6

Subtract $3x^2$ from both sides:

10x - 4 = 3x - 6

Rearrange to solve for $x$ . Subtract $3x$ from both sides and add $6$ to both sides:

10x - 3x = -6 + 4

7x = -2

Divide by 7:

x = -\frac{2}{7}

Check validity: The original denominators are $x-1$ and $x+2$ . If $x = -\frac{2}{7}$ , neither denominator is zero ( $-\frac{9}{7} \neq 0$ and $\frac{12}{7} \neq 0$ ). Thus, the solution is valid.

Answer

x = -\frac{2}{7}

Final answer

-2/7

Detailed explanation

Walkthrough

The problem asks us to solve an equation involving algebraic fractions. The key strategy is to eliminate the denominators so we can work with a standard polynomial equation.

Combine Fractions: The two terms on the left have denominators $(x-1)$ and $(x+2)$ . To add them, we need a common denominator, which is the product $(x-1)(x+2)$ . We multiply the numerator and denominator of the first term by $(x+2)$ and the second term by $(x-1)$ . This gives us a single fraction: $\frac{3x(x+2) + 4(x-1)}{(x-1)(x+2)}$ .
Clear Denominators: Since the entire left side equals 3, we can multiply both sides of the equation by the common denominator $(x-1)(x+2)$ . This removes the fraction entirely. Note: We must assume $x \neq 1$ and $x \neq -2$ for the original expression to be defined. The final answer should be checked against these restrictions.
$3x(x+2) + 4(x-1) = 3(x-1)(x+2)$
Expand and Simplify: Expand the brackets on both sides.
On the left: $3x^2 + 6x + 4x - 4 = 3x^2 + 10x - 4$ .
On the right: $3(x^2 + x - 2) = 3x^2 + 3x - 6$ .
Equating them: $3x^2 + 10x - 4 = 3x^2 + 3x - 6$ .
Solve Linear Equation: Notice that the $3x^2$ terms appear on both sides. Subtracting $3x^2$ cancels them out, leaving a simple linear equation: $10x - 4 = 3x - 6$ . Rearranging terms to isolate $x$ gives $7x = -2$ , so $x = -2/7$ .
Check Restrictions: The values $x=1$ and $x=-2$ would make the original denominators zero. Our result $-2/7$ is neither of these, so it is a valid solution.

Key Takeaways

When solving equations with algebraic fractions, finding a common denominator or multiplying through by the least common multiple of the denominators is the most efficient method.
Be careful when expanding brackets, especially with negative signs (e.g., $4(x-1)$ becomes $4x-4$ ).
Quadratic terms often cancel out in these types of problems, reducing the complexity to a linear equation.
Always check that the solution does not violate any domain restrictions (denominators cannot be zero).

Common Mistakes

Incorrect Expansion: Failing to expand $3x(x+2)$ correctly as $3x^2+6x$ or missing the constant term in $4(x-1)$ . The mark scheme awards M1 for correct expansion of all brackets.
Sign Errors: When moving terms across the equals sign, students often forget to change the sign. For example, turning $3x-6$ into $+3x+6$ .
Ignoring Restrictions: Not checking if the solution makes a denominator zero. While rare in such problems, it's a critical step.
Algebraic Manipulation Errors: Making arithmetic errors when combining like terms (e.g., $10x - 3x = 7x$ ).

Things to Be Careful About

The mark scheme accepts 'oe' (or equivalent), so simplified forms like $-2/7$ are required. Do not leave it as $2/-7$ without noting equivalence, though usually simplest form is expected.
Ensure you show the expansion step clearly, as marks are allocated for it (M1 for correct expansion).
Remember that 'caution' applies to the domain: $x \neq 1, -2$ . Although this specific question doesn't yield those values, it's good practice to verify.
In the final answer, write the fraction clearly. Decimal approximations might not be accepted unless specified, and exact form is preferred in algebra.

Techniques used

find a common denominator for algebraic fractionsclear the denominators by multiplying throughexpand and simplify the resulting equationsolve the linear equation for x

Q21MediumNumber

The width of a rectangle is $5.4\,\text{cm}$ , correct to the nearest $0.1\,\text{cm}$ .

(a)

Write down the upper bound of the width of the rectangle.

______ $\text{cm}$

(b)

The perimeter of the rectangle is $26.4\,\text{cm}$ , correct to the nearest $0.1\,\text{cm}$ .

Calculate the lower bound of the length of the rectangle.

______ $\text{cm}$

DifficultyMedium

Worked solution

Approach

To find the lower bound of the length, we must use the lower bound of the perimeter and the upper bound of the width. The perimeter $P$ of a rectangle is given by $P = 2(\text{length} + \text{width})$ . We need to rearrange this to solve for length: $\text{length} = \frac{P}{2} - \text{width}$ . To minimize the length, we use the smallest possible perimeter (lower bound) and subtract the largest possible width (upper bound).

Working

First, find the lower bound of the perimeter.
The perimeter is $26.4\,\text{cm}$ , correct to the nearest $0.1\,\text{cm}$ .

\text{Lower Bound of Perimeter} = 26.4 - 0.05 = 26.35\,\text{cm}

From part (a), the upper bound of the width is $5.45\,\text{cm}$ .

Using the perimeter formula:

2(L + W) = P

L + W = \frac{P}{2}

L = \frac{P}{2} - W

Substitute the lower bound of $P$ and the upper bound of $W$ :

L_{\text{min}} = \frac{26.35}{2} - 5.45

L_{\text{min}} = 13.175 - 5.45

L_{\text{min}} = 7.725

Answer

7.725 \,\text{cm}

Final answer

7.725

Detailed explanation

Walkthrough

The problem asks for the lower bound of the length of a rectangle. We know the dimensions are constrained by their rounding.

Understand the relationship: Perimeter $P = 2(l + w)$ , so $l = \frac{P}{2} - w$ .
Determine bounds for knowns:
- The perimeter $P$ is $26.4$ cm (nearest $0.1$ ). Its lower bound is $26.4 - 0.05 = 26.35$ cm.
- The width $w$ is $5.4$ cm (nearest $0.1$ ). From part (a), its upper bound is $5.4 + 0.05 = 5.45$ cm.
Apply logic for extremes: To get the smallest possible length ( $l_{\text{min}}$ ), we need the smallest total perimeter ( $P_{\text{min}}$ ) and we must subtract the largest possible width ( $w_{\text{max}}$ ). If we used the minimum width, the calculated length would be larger than necessary. If we used the maximum perimeter, the length would also be larger. So, $l_{\text{min}} = \frac{P_{\text{min}}}{2} - w_{\text{max}}$ .
Calculate:
$l_{\text{min}} = \frac{26.35}{2} - 5.45 = 13.175 - 5.45 = 7.725$

Key Takeaways

When calculating bounds of derived quantities, choose the combination of bounds that pushes the result in the desired direction (minimum or maximum).
For subtraction ( $A - B$ ), the minimum result comes from the minimum of A minus the maximum of B.

Common Mistakes

Using the lower bound of the width instead of the upper bound. This is incorrect because subtracting a smaller number yields a larger result.
Using the upper bound of the perimeter. This would yield the upper bound of the length, not the lower bound.
Arithmetic errors when dividing $26.35$ by $2$ or subtracting $5.45$ .

Things to Be Careful About

Pay close attention to whether the question asks for the upper or lower bound. The choice of inputs ( $P_{\text{min}}$ vs $P_{\text{max}}$ , $w_{\text{min}}$ vs $w_{\text{max}}$ ) changes accordingly.
Ensure exact values are used in intermediate steps to avoid rounding errors before the final answer.

Techniques used

calculate the lower bound of the perimeteruse the perimeter formula to isolate the lengthcalculate the lower bound of the length

Q22MediumGeometryTrigonometryNumberMensuration

The diagram shows a pyramid $ABCV$ .

Fig. 22

The base of the pyramid is an equilateral triangle, $ABC$ , with sides of length $34\,\text{cm}$ .
The height $VO$ is perpendicular to the base of the pyramid.
$VA = VB = VC = 82\,\text{cm}$ .

(a)

Write down the number of planes of symmetry of the pyramid.

______

(b)

$M$ is the midpoint of $AC$ .
The ratio $MO : OB$ is $1 : 2$ .

10M

(i)

Show that $OB = 19.6\,\text{cm}$ correct to 1 decimal place.

DifficultyMedium

Worked solution

Approach

First calculate the length of the altitude $MB$ of the equilateral triangle $ABC$ using Pythagoras' theorem in the right-angled triangle $AMB$ . Then use the ratio $MO : OB = 1 : 2$ to find the length of $OB$ .

Working

Since $M$ is the midpoint of $AC$ and $AC = 34\,\text{cm}$ :

AM = \frac{34}{2} = 17\,\text{cm}

In the right-angled triangle $AMB$ , by Pythagoras' theorem:

MB = \sqrt{AB^2 - AM^2} = \sqrt{34^2 - 17^2}

MB = \sqrt{1156 - 289} = \sqrt{867} \approx 29.44486\,\text{cm}

Given the ratio $MO : OB = 1 : 2$ , the total number of parts along $MB$ is $1 + 2 = 3$ . Therefore, $OB$ represents $\frac{2}{3}$ of the total length $MB$ :

OB = \frac{2}{3} \times \sqrt{867} \approx \frac{2}{3} \times 29.44486 = 19.6299\dots\,\text{cm}

Rounding to $1$ decimal place:

OB = 19.6\,\text{cm}

Answer

OB = 19.6\,\text{cm} \quad (\text{correct to 1 decimal place})

Final answer

19.6 cm

Detailed explanation

Walkthrough

The base $ABC$ is an equilateral triangle with side length $34\,\text{cm}$ . The line segment $MB$ connects vertex $B$ to the midpoint $M$ of the opposite side $AC$ , making $MB$ perpendicular to $AC$ .
In right-angled triangle $AMB$ , the hypotenuse is $AB = 34\,\text{cm}$ and the base is $AM = 17\,\text{cm}$ . Using Pythagoras' theorem gives $MB = \sqrt{34^2 - 17^2} = \sqrt{867} \approx 29.445\,\text{cm}$ .
The point $O$ divides $MB$ in the ratio $MO : OB = 1 : 2$ . Thus, $OB$ is $\frac{2}{1 + 2} = \frac{2}{3}$ of the entire length $MB$ .
Multiplying gives $OB = \frac{2}{3} \times 29.445\dots \approx 19.6299\,\text{cm}$ , which rounds to $19.6\,\text{cm}$ correct to $1$ decimal place.

Key Takeaways

In an equilateral triangle of side $s$ , the altitude is $h = s\sin 60° = \frac{\sqrt{3}}{2}s$ .
The centroid divides each median in a $1 : 2$ ratio from the base to the vertex.

Common Mistakes

Using $\frac{1}{2}$ instead of $\frac{2}{3}$ for the ratio fraction.
Using the entire side length $34$ instead of half ( $17$ ) for $AM$ .
Rounding prematurely before multiplying by $\frac{2}{3}$ .

Things to Be Careful About

For a 'Show that' question, you must show the unrounded value (e.g. $19.62\dots$ ) before stating the final rounded value $19.6$ .

Techniques used

apply Pythagoras' theorem to find triangle altitudedivide a line segment in a given ratio

(ii)

Calculate the angle between the edge $VB$ and the base of the pyramid.

______

DifficultyMedium-Easy

Worked solution

Approach

The angle between the edge $VB$ and the base of the pyramid is the angle between $VB$ and its projection onto the base, which is the line segment $OB$ . Since $VO$ is perpendicular to the base, triangle $VOB$ is right-angled at $O$ .

Working

In the right-angled triangle $VOB$ :

\cos(\angle VBO) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{OB}{VB}

Substitute $OB = 19.6299\dots\,\text{cm}$ (or $19.6\,\text{cm}$ ) and $VB = 82\,\text{cm}$ :

\cos(\angle VBO) = \frac{19.6299}{82} \approx 0.239389

\angle VBO = \cos^{-1}(0.239389) \approx 76.151°

Rounding to $1$ decimal place (or $3$ significant figures):

\angle VBO = 76.2° \quad (\text{or } 76.1° \text{ using } 19.6)

Answer

76.2°

Final answer

76.2°

Detailed explanation

Walkthrough

The angle between a line and a plane is defined as the angle between the line and its orthogonal projection on that plane.
Since $VO$ is perpendicular to the base, the projection of $VB$ onto the base is the segment $OB$ .
Therefore, the required angle is $\angle VBO$ in the right-angled triangle $VOB$ where $\angle VOB = 90°$ .
In triangle $VOB$ , the hypotenuse is $VB = 82\,\text{cm}$ and the adjacent side is $OB \approx 19.63\,\text{cm}$ .
Applying the cosine ratio: $\cos(\angle VBO) = \frac{OB}{VB} = \frac{19.63}{82}$ .
Taking the inverse cosine gives $\angle VBO \approx 76.15°$ , which rounds to $76.2°$ (or $76.1°$ if using the rounded value $19.6$ ).

Key Takeaways

In 3D trigonometry, always locate the right angle where the perpendicular height meets the base plane.
Choose the correct trigonometric ratio based on the known sides (adjacent and hypotenuse $\rightarrow$ cosine).

Common Mistakes

Using $\sin$ or $\tan$ instead of $\cos$ .
Choosing the wrong angle, such as $\angle BVO$ at the apex rather than $\angle VBO$ at the base.

Things to Be Careful About

Retain accuracy by using the more precise value of $OB$ from part (b)(i) to avoid rounding discrepancies.

Techniques used

identify the angle between a line and a plane in 3Duse basic cosine ratio to find an angle

(iii)

Calculate the volume of the pyramid.

______ $\text{cm}^3$

5M Similar

DifficultyMedium

Worked solution

Approach

Find the perpendicular height $VO$ of the pyramid using Pythagoras' theorem in the right-angled triangle $VOB$ .
Find the area of the equilateral triangle base $ABC$ .
Calculate the volume of the pyramid using the formula $V = \frac{1}{3} \times \text{Base Area} \times \text{Height}$ .

Working

First, find the height $VO$ from triangle $VOB$ :

VO^2 + OB^2 = VB^2

VO = \sqrt{VB^2 - OB^2} = \sqrt{82^2 - 19.6299^2} = \sqrt{6724 - 385.333} = \sqrt{6338.667} \approx 79.6157\,\text{cm}

Next, calculate the area of the base equilateral triangle $ABC$ :

\text{Area of base} = \frac{1}{2} \times AC \times MB = \frac{1}{2} \times 34 \times \sqrt{867} \approx 500.5627\,\text{cm}^2

(\text{or } \text{Area} = \frac{1}{2} \times 34 \times 34 \times \sin 60° = 578 \times \frac{\sqrt{3}}{2} \approx 500.5627\,\text{cm}^2)

Now calculate the volume of the pyramid:

\text{Volume} = \frac{1}{3} \times \text{Base Area} \times VO

\text{Volume} = \frac{1}{3} \times 500.5627 \times 79.6157 \approx 13283.7\,\text{cm}^3

Rounding to $3$ significant figures:

\text{Volume} = 13300\,\text{cm}^3

Answer

13300\,\text{cm}^3 \quad (\text{or } 13280\text{ to } 13290)

Final answer

13300 cm^3

Detailed explanation

Walkthrough

Find the vertical height $VO$ : In the vertical right-angled triangle $VOB$ , the hypotenuse is $VB = 82\,\text{cm}$ and the base is $OB \approx 19.63\,\text{cm}$ . Using Pythagoras' theorem: $VO = \sqrt{82^2 - 19.63^2} = \sqrt{6724 - 385.3} \approx 79.62\,\text{cm}$
Find the area of the equilateral base $ABC$ : Using the triangle area formula $\frac{1}{2}ab\sin C$ with $a = b = 34\,\text{cm}$ and $C = 60°$ : $\text{Base Area} = \frac{1}{2} \times 34 \times 34 \times \sin(60°) \approx 500.56\,\text{cm}^2$
Calculate the volume: Apply the formula for the volume of any pyramid, $V = \frac{1}{3} \times \text{Base Area} \times h$ : $V = \frac{1}{3} \times 500.56 \times 79.62 \approx 13284\,\text{cm}^3$ Rounding to $3$ significant figures yields $13300\,\text{cm}^3$ .

Key Takeaways

The volume of a pyramid is always $\frac{1}{3} \times \text{base area} \times \text{perpendicular height}$ .
The perpendicular height is found from the vertical right-angled triangle formed by the apex, the centre of the base, and a base vertex.

Common Mistakes

Forgetting the $\frac{1}{3}$ factor in the pyramid volume formula.
Using the slant height ( $82\,\text{cm}$ ) instead of the perpendicular height ( $VO \approx 79.62\,\text{cm}$ ).
Using an incorrect formula for the area of an equilateral triangle.

Things to Be Careful About

Intermediate rounding: carrying full precision throughout ensures the final answer falls cleanly into the accepted range ( $13281$ to $13286$ , or $13300$ to $3$ s.f.).

Techniques used

calculate height using Pythagoras' theorem in 3Dcalculate area of an equilateral trianglecalculate volume of a pyramid

Q234MMediumProbability

A box contains 13 pencils.
There are 4 red pencils, 7 green pencils and 2 yellow pencils in the box.
Two pencils are chosen at random from the box without replacement.

Work out the probability that the two pencils are different colours.

______

DifficultyMedium

Worked solution

Approach

There are two main methods to solve this: finding the probability of each specific case where the colours differ and adding them, or finding the probability that they are the same colour and subtracting from 1. We will use the direct method.

The total number of pencils is $13$ . There are $4$ red ( $R$ ), $7$ green ( $G$ ), and $2$ yellow ( $Y$ ).
Since the pencils are chosen without replacement, the total number of pencils decreases by one after the first pick.

The possible combinations for different colours are:

Red then Green
Green then Red
Red then Yellow
Yellow then Red
Green then Yellow
Yellow then Green

Alternatively, we can group these into three pairs based on the two colours involved:

One Red and one Green (in either order)
One Red and one Yellow (in either order)
One Green and one Yellow (in either order)

We will calculate the probability for each pair.

Working

Case 1: One Red and one Green
This can happen as Red then Green ( $RG$ ) or Green then Red ( $GR$ ).
$P(RG) = \frac{4}{13} \times \frac{7}{12} = \frac{28}{156}$
$P(GR) = \frac{7}{13} \times \frac{4}{12} = \frac{28}{156}$
Total probability for Red and Green:
$P(\text{Red and Green}) = \frac{28}{156} + \frac{28}{156} = \frac{56}{156}$

Case 2: One Red and one Yellow
This can happen as Red then Yellow ( $RY$ ) or Yellow then Red ( $YR$ ).
$P(RY) = \frac{4}{13} \times \frac{2}{12} = \frac{8}{156}$
$P(YR) = \frac{2}{13} \times \frac{4}{12} = \frac{8}{156}$
Total probability for Red and Yellow:
$P(\text{Red and Yellow}) = \frac{8}{156} + \frac{8}{156} = \frac{16}{156}$

Case 3: One Green and one Yellow
This can happen as Green then Yellow ( $GY$ ) or Yellow then Green ( $YG$ ).
$P(GY) = \frac{7}{13} \times \frac{2}{12} = \frac{14}{156}$
$P(YG) = \frac{2}{13} \times \frac{7}{12} = \frac{14}{156}$
Total probability for Green and Yellow:
$P(\text{Green and Yellow}) = \frac{14}{156} + \frac{14}{156} = \frac{28}{156}$

Total Probability
Add the probabilities from all three cases:
$P(\text{different colours}) = \frac{56}{156} + \frac{16}{156} + \frac{28}{156}$
$P(\text{different colours}) = \frac{56 + 16 + 28}{156} = \frac{100}{156}$

Simplification
Both numerator and denominator are divisible by 4:
$\frac{100 \div 4}{156 \div 4} = \frac{25}{39}$

Answer

$\frac{25}{39}$

Final answer

25/39

Detailed explanation

Walkthrough

The problem asks for the probability of choosing two pencils of different colours from a box containing 13 pencils (4 Red, 7 Green, 2 Yellow) without replacement. "Without replacement" means that after the first pencil is picked, there are only 12 pencils left in the box.

There are two primary ways to approach this:

Method 1: Direct Calculation (Summing Mutually Exclusive Cases)
The event "different colours" covers three scenarios involving pairs of colours:

Red and Green: The first could be Red and the second Green, OR the first Green and the second Red.
- $P(R \text{ then } G) = \frac{4}{13} \times \frac{7}{12}$
- $P(G \text{ then } R) = \frac{7}{13} \times \frac{4}{12}$
- Sum = $\frac{28}{156} + \frac{28}{156} = \frac{56}{156}$
Red and Yellow: Similar logic applies.
- $P(R \text{ then } Y) = \frac{4}{13} \times \frac{2}{12}$
- $P(Y \text{ then } R) = \frac{2}{13} \times \frac{4}{12}$
- Sum = $\frac{8}{156} + \frac{8}{156} = \frac{16}{156}$
Green and Yellow:
- $P(G \text{ then } Y) = \frac{7}{13} \times \frac{2}{12}$
- $P(Y \text{ then } G) = \frac{2}{13} \times \frac{7}{12}$
- Sum = $\frac{14}{156} + \frac{14}{156} = \frac{28}{156}$

Finally, add these sums together: $\frac{56 + 16 + 28}{156} = \frac{100}{156}$ . Simplify by dividing top and bottom by 4 to get $\frac{25}{39}$ .

Method 2: Complement (Subtracting Same Colours from 1)
It is often easier to calculate the probability of the opposite event (picking two pencils of the same colour) and subtracting that from 1.

$P(\text{Same}) = P(RR) + P(GG) + P(YY)$
$P(RR) = \frac{4}{13} \times \frac{3}{12} = \frac{12}{156}$
$P(GG) = \frac{7}{13} \times \frac{6}{12} = \frac{42}{156}$
$P(YY) = \frac{2}{13} \times \frac{1}{12} = \frac{2}{156}$
Total $P(\text{Same}) = \frac{12 + 42 + 2}{156} = \frac{56}{156}$

Then:
$P(\text{Different}) = 1 - P(\text{Same}) = 1 - \frac{56}{156} = \frac{156 - 56}{156} = \frac{100}{156}$
Simplify to $\frac{25}{39}$ .

Key Takeaways

When dealing with "without replacement", always remember the denominator decreases by 1 for the second event.
For "different" conditions, it is often safer to list all mutually exclusive outcomes (RR, GG, YY for same; RG, GR, RY, YR, GY, YG for different) rather than guessing.
Using the complement ( $1 - P(\text{same})$ ) is usually faster because there are fewer terms to add (3 terms vs 6 terms). Both methods must yield the same result.
Always simplify your final fraction.

Common Mistakes

Ignoring "without replacement": Students often write $\frac{4}{13} \times \frac{7}{13}$ instead of $\frac{4}{13} \times \frac{7}{12}$ . This leads to the incorrect answer $\frac{100}{169}$ (which gets SC1).
Order errors: Calculating only $P(RG)$ and forgetting $P(GR)$ , effectively halving the probability for that pair.
Arithmetic errors: Adding fractions incorrectly or failing to simplify $\frac{100}{156}$ correctly.
Misinterpreting the question: Calculating the probability of picking specific colours (e.g., just Red and Green) instead of any two different colours.

Things to Be Careful About

Ensure you simplify the final fraction. The mark scheme accepts equivalent forms like $\frac{100}{156}$ but standard practice requires simplest form ( $\frac{25}{39}$ ).
Watch out for the "SC1" trap: if you treat the picks as independent (replacement), you get denominators of $13 \times 13 = 169$ . The question explicitly states "without replacement", so the denominator must be $13 \times 12 = 156$ .

Techniques used

calculate probabilities of combined events without replacementsum probabilities of mutually exclusive outcomessimplify fractions to lowest terms