Mathematics (Syllabus D) 4024 / 21 — October/November 2025

(b)

a cube number

______

(c)

a factor of 26

______

(d)

an irrational number.

______

Q2Medium-EasyNumber

(a)

Insert one set of brackets to make the calculation correct.

4.2 - 3 \times 5 + 1.2 = 7.2

DifficultyEasy

Worked solution

Approach

First, evaluate the expression as written using the standard order of operations (BODMAS/PEMDAS). Then, try inserting a pair of brackets in different positions to see which one produces the required answer of $7.2$ .

Working

Without brackets, multiplication is performed first:

4.2 - 3 \times 5 + 1.2 = 4.2 - 15 + 1.2

4.2 - 15 = -10.8

-10.8 + 1.2 = -9.6

The result is $-9.6$ , not $7.2$ . We need to group terms before the multiplication or addition.

Let's try grouping $(4.2 - 3)$ :

(4.2 - 3) \times 5 + 1.2

Calculate inside the brackets first:

4.2 - 3 = 1.2

Then multiply by $5$ :

1.2 \times 5 = 6

Finally, add $1.2$ :

6 + 1.2 = 7.2

This matches the target value.

Answer

(4.2 - 3) \times 5 + 1.2 = 7.2

Final answer

(4.2 - 3) x 5 + 1.2 = 7.2

Detailed explanation

Walkthrough

The question asks us to insert one set of brackets to make the equation true. The key is understanding how brackets override the standard order of operations (Multiplication before Addition/Subtraction).

Check original: Without brackets, we multiply $3 \times 5$ first. This gives $15$ . Then $4.2 - 15 + 1.2 = -9.6$ . This is incorrect.
Trial: We need a larger positive number ( $7.2$ ). Notice that if we can get $1.2$ from the first part $(4.2 - 3)$ , then multiplying by $5$ gives $6$ , and adding $1.2$ gives exactly $7.2$ .
Verification:
- Bracket: $(4.2 - 3)$
- Inside bracket: $1.2$
- Multiply: $1.2 \times 5 = 6$
- Add: $6 + 1.2 = 7.2$
  The equation holds.

Key Takeaways

Brackets change the priority of operations. Operations inside brackets are always done first.
In expressions with mixed operations, identifying where a specific intermediate result is needed can guide you to the correct bracket placement.

Common Mistakes

Forgetting to apply the multiplication after evaluating the bracketed term.
Placing brackets around the wrong numbers, e.g., $4.2 - (3 \times 5) + 1.2$ , which yields the same result as no brackets because multiplication already has higher precedence.

Things to Be Careful About

Ensure you follow the order of operations strictly: Brackets, then Indices/Multiplication/Division, then Addition/Subtraction.
Check your final calculation carefully to ensure it equals the target number exactly.

Techniques used

evaluate the expression without bracketstest bracket placements to change the order of operationsverify the result matches the target value

(b)

Insert one set of brackets to make the calculation correct.

5 - 3^2 - \sqrt[3]{64} + 2 = -10

DifficultyMedium-Easy

Worked solution

Approach

Evaluate the known values ( $3^2$ and $\sqrt[3]{64}$ ) first. Then, evaluate the expression as written without brackets. Compare the result with the target ( $-10$ ). Insert brackets to change the order of operations to reach the target.

Working

First, calculate the powers and roots:

3^2 = 9

\sqrt[3]{64} = 4 \quad (\text{since } 4 \times 4 \times 4 = 64)

Substitute these back into the expression:

5 - 9 - 4 + 2

Evaluate left to right (no brackets):

5 - 9 = -4

-4 - 4 = -8

-8 + 2 = -6

The result is $-6$ , but we need $-10$ . We are $4$ too high. To get a smaller number, we likely need to subtract a larger amount. This suggests grouping the last two terms so they are subtracted together.

Try grouping $(\sqrt[3]{64} + 2)$ :

5 - 3^2 - (\sqrt[3]{64} + 2)

Substitute the values again:

5 - 9 - (4 + 2)

Calculate inside the brackets first:

4 + 2 = 6

Now perform the subtractions:

5 - 9 - 6

5 - 9 = -4

-4 - 6 = -10

This matches the target value.

Answer

5 - 3^2 - (\sqrt[3]{64} + 2) = -10

Final answer

5 - 3^2 - (cube root of 64 + 2) = -10

Detailed explanation

Walkthrough

Simplify Constants: Identify that $3^2 = 9$ and $\sqrt[3]{64} = 4$ . The expression becomes $5 - 9 - 4 + 2$ .
Initial Calculation: Without brackets, $5 - 9 - 4 + 2 = -6$ . The target is $-10$ .
Analyze Difference: The difference between $-6$ and $-10$ is $-4$ . This means we need to reduce the total by $4$ more than the current calculation does.
Bracket Strategy: If we place brackets around $(4 + 2)$ , the expression effectively becomes $5 - 9 - (6)$ . Since subtraction distributes over the bracket, this is equivalent to $5 - 9 - 4 - 2$ . Comparing $5 - 9 - 4 + 2$ (original) with $5 - 9 - 4 - 2$ (with brackets), the change is from $+2$ to $-2$ , a difference of $4$ . This confirms the bracket placement.
Final Verification:
- $5 - 9 - (4 + 2)$
- $5 - 9 - 6$
- $-4 - 6 = -10$ . Correct.

Key Takeaways

Always simplify indices and roots first.
Brackets can change the sign of subsequent terms when preceded by a minus sign (e.g., $- (a + b) = -a - b$ ).
Estimating the gap between the calculated value and the target helps identify whether you need to increase or decrease the magnitude of a term.

Common Mistakes

Evaluating $\sqrt[3]{64}$ incorrectly (e.g., thinking it is $8$ or $2$ ).
Forgetting that a minus sign outside brackets affects all terms inside.
Incorrectly applying order of operations within the bracketed section.

Things to Be Careful About

Remember that $\sqrt[3]{64}$ is the cube root, not the square root ( $\sqrt{64}=8$ ).
Ensure the final answer is exactly $-10$ , checking signs carefully during subtraction.

Techniques used

evaluate powers and rootstest bracket placements to change the order of operationsverify the result matches the target value

Q3Medium-EasyMensurationGeometry

In this question, all lengths are in centimetres.
The diagram shows the net of a cuboid.

Fig. 1

(a)

Find the total surface area of the cuboid.

______ $\text{cm}^2$

DifficultyMedium-Easy

Worked solution

Approach

The diagram shows the net of a cuboid. By analysing the net, we can identify the three dimensions (length, width, height) of the cuboid. The total surface area is the sum of the areas of all six rectangular faces, which can be calculated as $2(lw + lh + wh)$ .

Working

From the net and the markings:

One pair of faces has dimensions $3 \text{ cm} \times 5.2 \text{ cm}$ .
Another pair has dimensions $4.7 \text{ cm} \times 5.2 \text{ cm}$ .
The third pair has dimensions $3 \text{ cm} \times 4.7 \text{ cm}$ .

Using the surface area formula $A = 2(lw + lh + wh)$ with $l=3$ , $w=5.2$ , $h=4.7$ :

\begin{aligned} A &= 2(3 \times 5.2 + 3 \times 4.7 + 5.2 \times 4.7) \\ &= 2(15.6 + 14.1 + 24.44) \\ &= 2(54.14) \\ &= 108.28 \end{aligned}

Answer

108.28

Final answer

108.28

Detailed explanation

Walkthrough

The problem asks for the total surface area of a cuboid given its net. A net is a 2D pattern that can be folded to make a 3D shape. By looking at the net, we can see three pairs of identical rectangles, which correspond to the three pairs of opposite faces of the cuboid.

Identify dimensions: The vertical column of rectangles has a width of $3$ . The heights are $5.2$ and another dimension. The horizontal row has a height of $4.7$ . By matching the edges that would fold together, we determine the cuboid has dimensions $3 \text{ cm}$ , $5.2 \text{ cm}$ , and $4.7 \text{ cm}$ .
Calculate areas:
- Two faces are $3 \times 5.2 = 15.6 \text{ cm}^2$ .
- Two faces are $4.7 \times 5.2 = 24.44 \text{ cm}^2$ .
- Two faces are $3 \times 4.7 = 14.1 \text{ cm}^2$ .
Sum them up: Total area $= 2(15.6 + 24.44 + 14.1) = 2(54.14) = 108.28 \text{ cm}^2$ .

Key Takeaways

A net of a cuboid consists of 6 rectangles in 3 pairs of equal size.
The surface area is the sum of the areas of these rectangles, or equivalently $2(lw + lh + wh)$ .
Care must be taken to correctly identify which dimensions correspond to length, width, and height from the net.

Common Mistakes

Calculation errors: Forgetting to multiply by 2 (calculating the area of only one set of faces).
Arithmetic errors: Mistakes in multiplying decimals, e.g., $5.2 \times 4.7$ .
Wrong dimensions: Misreading the net and using incorrect side lengths.

Things to Be Careful About

The question is on the Calculator component, so decimal arithmetic is expected and allowed.
Ensure all intermediate products are calculated accurately before summing.
The final answer is a number; the unit $\text{cm}^2$ is already provided in the question text next to the answer line.

Techniques used

read dimensions from a net diagramcalculate area of rectangular facesapply cuboid surface area formula

(b)

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

______

DifficultyEasy

Worked solution

Approach

A cuboid (rectangular prism) with three different dimensions (length $\neq$ width $\neq$ height) has planes of symmetry that cut through the midpoints of opposite faces, parallel to those faces.

Working

There are three pairs of opposite faces. Each pair defines a plane of symmetry that is parallel to the faces and passes through the centre of the cuboid.

One plane parallel to the $3 \times 5.2$ faces.
One plane parallel to the $4.7 \times 5.2$ faces.
One plane parallel to the $3 \times 4.7$ faces.

Total number of planes of symmetry = $3$ .

Answer

3

Final answer

3

Detailed explanation

Walkthrough

The question asks for the number of planes of symmetry of the cuboid.

Understand planes of symmetry: A plane of symmetry divides a 3D shape into two identical halves that are mirror images of each other.
Apply to cuboid: For a general cuboid (where length, width, and height are all different, as in this case: $3, 5.2, 4.7$ $3, 5.2, 4.7$ ), there are exactly 3 planes of symmetry.
- One plane cuts horizontally through the middle (parallel to the top and bottom faces).
- One plane cuts vertically front-to-back (parallel to the front and back faces).
- One plane cuts vertically side-to-side (parallel to the left and right faces).

Key Takeaways

A cube has 9 planes of symmetry.
A cuboid with at least two equal dimensions (e.g., a square prism) has more planes of symmetry.
A cuboid with all three dimensions different has exactly 3 planes of symmetry.

Common Mistakes

Confusing with 2D symmetry: Thinking of lines of symmetry instead of planes.
Assuming more symmetry: Thinking a cuboid has 4 or 6 planes like a cube (a cube has 3 planes through faces, 3 through diagonals, total 9).
Confusing with axes of rotation: A cuboid has 3 axes of rotational symmetry (order 2), but the question asks for planes.

Things to Be Careful About

Ensure the dimensions are all different. If two dimensions were equal (e.g., $3 \times 3 \times 5.2$ ), there would be 4 planes of symmetry. Here, $3 \neq 5.2 \neq 4.7$ , so there are exactly 3.

Techniques used

identify planes of symmetry for a rectangular prism

Q4Medium-EasyStatistics

The table shows the examination marks for mathematics and physics for each of 9 students.

Mathematics mark	56	39	83	49	50	74	60	45	78
Physics mark	42	26	85	40	42	68	55	37	71

Fig. 2

(a)

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

DifficultyEasy

Worked solution

Approach

The table gives nine (Mathematics mark, Physics mark) pairs. Six points are already plotted. Identify the three remaining pairs and plot them at the correct coordinates on the grid.

Working

The nine data pairs from the table are:

\begin{aligned} &(56,\, 42),\; (39,\, 26),\; (83,\, 85),\; (49,\, 40),\; (50,\, 42),\\ &(74,\, 68),\; (60,\, 55),\; (45,\, 37),\; (78,\, 71) \end{aligned}

The six already plotted points are:

(39,\, 26),\; (49,\, 40),\; (50,\, 42),\; (56,\, 42),\; (74,\, 68),\; (83,\, 85)

The three remaining points to plot are:

(60,\, 55),\quad (45,\, 37),\quad (78,\, 71)

Locate each on the grid by reading the Mathematics mark along the horizontal axis and the Physics mark up the vertical axis, then mark with a cross.

Answer

The three points to plot are $(45,\, 37)$ , $(60,\, 55)$ and $(78,\, 71)$ .

Final answer

Plot (45, 37), (60, 55) and (78, 71)

Detailed explanation

Walkthrough

The question provides a table of nine paired marks and a scatter diagram with six points already drawn. The candidate must identify the three missing pairs and plot them.

Step 1 — List all nine pairs. Read across each column of the table: the top row is the Mathematics mark (x-coordinate) and the bottom row is the Physics mark (y-coordinate). This gives the nine coordinate pairs listed in the working.

Step 2 — Identify which are already plotted. The six points given in the question description are $(39, 26)$ , $(49, 40)$ , $(50, 42)$ , $(56, 42)$ , $(74, 68)$ and $(83, 85)$ . Cross these off the full list.

Step 3 — Plot the remaining three. The leftover pairs are $(45, 37)$ , $(60, 55)$ and $(78, 71)$ . For each, find the Mathematics value on the horizontal axis and the Physics value on the vertical axis, then mark the intersection with a cross.

Key Takeaways

A scatter diagram plots paired data as points with one variable on each axis.
The coordinates of each point are read directly from the corresponding columns of the data table.
Always check which points are already plotted before drawing new ones.

Common Mistakes

Plotting the coordinates in the wrong order (swapping Mathematics and Physics marks). Remember Mathematics is on the horizontal axis and Physics is on the vertical axis.
Reading the wrong grid value — the axes start at 20, not 0, and each major division is 10 units with 10 small squares between, so each small square represents 1 unit.
Forgetting to mark the points with a cross as shown in the already-plotted points.

Things to Be Careful About

The axes begin at 20, not 0. A point at $(45, 37)$ is not near the origin — it is 25 units right and 17 units up from the bottom-left corner of the grid.
Each small square on the grid represents 1 unit on both axes (10 small squares = 10 units = one major division).
The answer must be plotted, not just written down — the mark is awarded for correct placement on the grid.

Techniques used

identify the remaining data points from the tableplot each point on the scatter diagram at the correct coordinates

(b)

On the grid, draw a line of best fit.

DifficultyMedium-Easy

Worked solution

Approach

A line of best fit for a scatter diagram is a straight line that shows the general trend of the data. It should have points roughly equally distributed on either side and pass through the middle of the cluster.

Working

The nine plotted points show a clear positive correlation: as Mathematics marks increase, Physics marks also increase. The points lie roughly along a line with a positive gradient.

Draw a straight ruled line through the middle of the data cloud, ensuring:

The line has a positive gradient (rising from left to right).
Points are roughly equally spread above and below the line.
The line extends across the grid to be useful for estimation in part (c).

Answer

A straight ruled line of best fit with positive gradient passing through the central region of the plotted points.

Final answer

Straight ruled line of best fit with positive gradient through the data

Detailed explanation

Walkthrough

Step 1 — Observe the trend. Look at the nine plotted points. Low Mathematics marks (39–50) are paired with low Physics marks (26–42), and high Mathematics marks (74–83) are paired with high Physics marks (68–85). This indicates a positive correlation.

Step 2 — Draw the line. Using a ruler, draw a straight line that passes through the middle of the data. The line should have roughly the same number of points above it as below it. It does not need to pass through any particular point — it represents the overall trend.

Step 3 — Check the gradient. The line must rise from left to right (positive gradient). A line that is horizontal or falls from left to right would be incorrect.

Key Takeaways

A line of best fit summarises the trend in a scatter diagram.
It is a straight line (for linear correlation) drawn by eye with a ruler.
Points should be roughly balanced above and below the line.
The line must extend far enough across the grid to be used for estimation.

Common Mistakes

Drawing a curved line or a jagged line connecting points — the line of best fit must be straight.
Drawing a line with the wrong gradient (horizontal or negative) when the data clearly shows positive correlation.
Drawing the line too close to one end of the data, making it useless for estimation at the other end.
Not using a ruler — the line must be straight and ruled.

Things to Be Careful About

The line is drawn by eye; there is no single correct answer, but it must be a straight ruled line with positive gradient that passes through the central region of the data.
The mark scheme awards the mark for a 'ruled line of best fit' — so a straight line is essential.
This line will be used in part (c), so it must extend far enough to read off a value at Physics mark = 75.

Techniques used

assess the overall trend of the plotted pointsdraw a straight line that balances points above and below it

(c)

Xavier scored 75 marks in the physics exam but was absent for the mathematics exam.

Use your line of best fit to find an estimate of his mathematics mark.

______

DifficultyMedium-Easy

Worked solution

Approach

Xavier's Physics mark is 75. We need to find the corresponding Mathematics mark using the line of best fit drawn in part (b). Since Physics is on the vertical axis, we locate 75 on that axis, move horizontally to the line of best fit, then move vertically down to read the Mathematics mark on the horizontal axis.

Working

Locate $75$ on the vertical (Physics mark) axis.
Draw a horizontal line from $75$ on the vertical axis to the line of best fit.
From that intersection point, draw a vertical line down to the horizontal (Mathematics mark) axis.
Read the value on the horizontal axis.

Using the line of best fit from part (b), the Mathematics mark corresponding to a Physics mark of $75$ is approximately:

\text{Mathematics mark} \approx 77 \text{ to } 80

Answer

\boxed{78}

(Accept any value from approximately $77$ to $80$ , depending on the line drawn in part (b).)

Final answer

78

Detailed explanation

Walkthrough

Step 1 — Identify which axis to read from. Xavier's Physics mark is 75. Physics is plotted on the vertical axis, so we start at 75 on the vertical axis.

Step 2 — Move horizontally to the line of best fit. From the point 75 on the vertical axis, draw a horizontal line across to where it meets the line of best fit drawn in part (b).

Step 3 — Move vertically down to the horizontal axis. From that intersection point, draw a vertical line down to the horizontal (Mathematics mark) axis.

Step 4 — Read the value. The value on the horizontal axis where the vertical line lands is the estimated Mathematics mark. For a reasonable line of best fit, this value is approximately 77 to 80.

Key Takeaways

To estimate a value from a scatter diagram using a line of best fit, locate the known value on its axis, move to the line, then read the unknown value on the other axis.
The answer depends on the line of best fit drawn in part (b), so follow-through marks are awarded based on the candidate's own line.
The mark scheme accepts a range of values (typically 77–80) because the line is drawn by eye.

Common Mistakes

Reading from the wrong axis: starting at 75 on the horizontal (Mathematics) axis instead of the vertical (Physics) axis. Remember Physics is on the vertical axis.
Moving vertically first instead of horizontally: the candidate must move horizontally from 75 on the vertical axis to the line, then vertically down to the horizontal axis.
Reading the value at the wrong point on the line — must read from the line of best fit, not from a plotted data point.
Giving an answer outside the reasonable range (e.g., 65 or 90) which would indicate the line of best fit was drawn incorrectly.

Things to Be Careful About

The answer is a reading from the candidate's own line of best fit, so it must be consistent with that line. The mark scheme says 'reading from their straight line of best fit'.
The value must be reasonable — looking at the data, Physics mark 75 is near the high end, so the Mathematics mark should be in the range 75–82.
Give the answer to the nearest integer; no units are required since the marks are already in marks.
If the line of best fit in part (b) was drawn incorrectly (e.g., wrong gradient), the reading in part (c) may be outside the acceptable range and would not score.

Techniques used

locate the given value on the appropriate axisread the corresponding value from the line of best fit

Q5Medium-EasyGeometry

Fig. 3

The diagram shows an isosceles triangle.

(a)

Work out the value of $x$ .

$x$ = ______

DifficultyMedium-Easy

Worked solution

Approach

The triangle is isosceles, meaning the two base angles are equal. The sum of interior angles in any triangle is $180^\circ$ .

Working

Let the base angles be $x^\circ$ . Since the triangle is isosceles with the top angle $56^\circ$ :

\begin{aligned} x + x + 56 &= 180 \\ 2x &= 180 - 56 \\ 2x &= 124 \\ x &= 62 \end{aligned}

Alternatively, directly:

x = \frac{180 - 56}{2} = 62

Answer

x = 62

Final answer

62

Detailed explanation

Walkthrough

The problem gives an isosceles triangle with a top angle of $56^\circ$ . An isosceles triangle has two equal sides (marked with ticks in the diagram) and the angles opposite these equal sides (the base angles) are also equal. Here, both base angles are marked as $x^\circ$ .

The sum of the interior angles of a triangle is always $180^\circ$ . Therefore, we can set up the equation:

x + x + 56 = 180

Subtract $56$ from $180$ to find the total remaining for the two base angles:

180 - 56 = 124

Since the two base angles are equal, divide this remainder by 2:

x = \frac{124}{2} = 62

Key Takeaways

In an isosceles triangle, the angles opposite the equal sides are equal (base angles are equal).
The sum of angles in a triangle is $180^\circ$ .

Common Mistakes

Forgetting to divide the remaining angle sum by 2 (giving 124 instead of 62).
Assuming the base angles are $56^\circ$ (confusing the vertex angle with base angles).

Things to Be Careful About

Ensure you identify which sides are equal from the tick marks. The equal sides meet at the top, so the base angles are the equal ones.
The diagram is marked NOT TO SCALE, so do not measure angles; rely on the given value $56^\circ$ and properties.

Techniques used

identify base angles of an isosceles triangle are equaluse the angle sum of a triangle (180 degrees)

(b)

Work out the value of $y$ .

$y$ = ______

Q62MEasyNumber

Work out.

\frac{12 - \sqrt[3]{6}}{0.3^2}

Give your answer correct to 2 decimal places.

______

DifficultyEasy

Worked solution

Approach

Calculate the value of the cube root and the square separately, perform the subtraction in the numerator, then divide by the result of the denominator. Finally, round the answer to 2 decimal places.

Working

First, evaluate the terms in the expression:

\sqrt[3]{6} \approx 1.81712059

0.3^2 = 0.09

Substitute these values back into the expression:

\frac{12 - 1.81712059}{0.09}

Perform the subtraction in the numerator:

12 - 1.81712059 = 10.18287941

Now divide by the denominator:

\frac{10.18287941}{0.09} = 113.1431045...

Round the result to 2 decimal places. The third decimal digit is $3$ , so we round down (keep the second digit as is):

113.14

Answer

113.14

Final answer

113.14

Detailed explanation

Walkthrough

This question tests your ability to use a calculator correctly for non-calculator components where specific values are given or simple powers/roots are involved, combined with standard order of operations.

Identify the parts: The expression is a fraction. The top part (numerator) is $12$ minus the cube root of $6$ . The bottom part (denominator) is $0.3$ squared.
Evaluate the roots and powers:
- Calculate $\sqrt[3]{6}$ . Using a calculator, this is approximately $1.81712...$
- Calculate $0.3^2$ . This is $0.3 \times 0.3 = 0.09$ .
Simplify the numerator: Subtract the cube root from $12$ : $12 - 1.81712... = 10.18287...$
Divide: Divide the simplified numerator by the denominator: $10.18287... / 0.09$ .
Rounding: The question asks for the answer correct to 2 decimal places. Look at the third decimal place ( $3$ ). Since it is less than $5$ , leave the second decimal place unchanged. The result is $113.14$ .

Key Takeaways

Be comfortable calculating cube roots ( $\sqrt[3]{x}$ ) and squares ( $x^2$ ) on your calculator.
Always evaluate the numerator and denominator completely before dividing.
Pay attention to the required precision (decimal places vs significant figures).

Common Mistakes

Order of Operations: Subtracting $6$ from $12$ first to get $6$ , then taking the cube root ( $\sqrt[3]{6}$ ), which is incorrect. The cube root only applies to the $6$ .
Squaring Errors: Writing $0.3^2$ as $0.6$ instead of $0.09$ .
Rounding: Rounding too early (e.g., using $1.82$ for the cube root) might lead to slight inaccuracies, though here it might still land on the same 2dp. It is safer to keep full calculator precision until the end.
Division by Decimal: Forgetting that dividing by $0.09$ increases the number significantly.

Things to Be Careful About

Calculator Mode: Ensure you are entering the cube root correctly. On some calculators, you press shift + cube root or type 6^(1/3).
Accuracy: The mark scheme awards a B1 for intermediate answers like $113$ or $113.1$ , but the final answer must be rounded correctly to $113.14$ for full marks (cao - correct answer only).
Significant Figures vs Decimal Places: The question specifically asks for "2 decimal places", not 3 significant figures.

Techniques used

evaluate a cube rootsquare a decimal numberperform subtraction in the numeratordivide by a decimalround to a specified accuracy

Q72MMedium-EasyGeometry

Work out the size of the interior angle of a regular decagon.

______

DifficultyMedium-Easy

Worked solution

Approach

To find the size of an interior angle of a regular decagon, we can use the formula for the sum of interior angles of an $n$ -sided polygon and divide by the number of sides (since all angles are equal in a regular polygon). Alternatively, we can use the exterior angle method.

Working

A decagon has $n = 10$ sides.

Method 1: Using the interior angle sum
The sum of the interior angles of a polygon with $n$ sides is given by:

(n - 2) \times 180^\circ

Substituting $n = 10$ :

(10 - 2) \times 180^\circ = 8 \times 180^\circ = 1440^\circ

Since the decagon is regular, all 10 interior angles are equal. Therefore, the size of one interior angle is:

\frac{1440^\circ}{10} = 144^\circ

Method 2: Using exterior angles
The sum of exterior angles of any convex polygon is $360^\circ$ . For a regular polygon with $n$ sides, each exterior angle is:

\frac{360^\circ}{n}

For a decagon ( $n=10$ ):

\text{Exterior angle} = \frac{360^\circ}{10} = 36^\circ

The interior angle and exterior angle at each vertex add up to $180^\circ$ (angles on a straight line):

\text{Interior angle} = 180^\circ - 36^\circ = 144^\circ

Answer

144^\circ

Final answer

144°

Detailed explanation

Walkthrough

A regular decagon is a ten-sided polygon where all sides are equal in length and all interior angles are equal in measure.

Step 1: Identify the number of sides.
A decagon has 10 sides, so $n = 10$ .

Step 2: Choose a method to calculate the interior angle.
There are two common ways to do this:

Method A (Interior Angle Sum):
The total sum of interior angles in any polygon is calculated as $(n-2) \times 180^\circ$ . This comes from dividing the polygon into triangles by drawing diagonals from one vertex; an $n$ -gon can be split into $n-2$ triangles. Since each triangle has angles summing to $180^\circ$ , the total is $(n-2) \times 180^\circ$ .
For $n=10$ : $(10-2) \times 180^\circ = 8 \times 180^\circ = 1440^\circ$ .
Because it is a regular decagon, every interior angle is the same. So we divide the total sum by the number of angles (which equals the number of sides):
$1440^\circ \div 10 = 144^\circ$ .

Method B (Exterior Angles):
The sum of the exterior angles of any convex polygon is always $360^\circ$ . For a regular polygon, each exterior angle is simply $360^\circ / n$ .
For $n=10$ : $360^\circ / 10 = 36^\circ$ .
An interior angle and its corresponding exterior angle lie on a straight line, so they add up to $180^\circ$ .
Therefore, Interior Angle $= 180^\circ - 36^\circ = 144^\circ$ .

Both methods lead to the same result. The mark scheme accepts either approach (M1 for showing the correct formulaic step).

Key Takeaways

Know the formula for the sum of interior angles: $(n-2) \times 180^\circ$ .
Know that the sum of exterior angles is always $360^\circ$ .
Remember that "regular" means all sides and all angles are equal, allowing you to divide the total sum by $n$ .
Be able to switch between interior and exterior angles using the $180^\circ$ relationship.

Common Mistakes

Confusing the number of sides: forgetting that a decagon has 10 sides.
Forgetting to divide by $n$ : calculating the total sum ( $1440^\circ$ ) but stopping there instead of finding the single angle.
Incorrectly applying the formula: e.g., using $n \times 180^\circ$ instead of $(n-2) \times 180^\circ$ .
Arithmetic errors in multiplication or division.
Not providing the degree symbol ( $^\circ$ ) if required by context (though usually numerical answers are accepted, units are good practice).

Things to Be Careful About

Ensure you distinguish between "interior" and "exterior" angles.
Check if the question asks for the sum of all angles or just one angle. This question asks for "the size of the interior angle" (singular), implying one angle.
Accuracy: The answer is exact, so no rounding is needed.
Units: Include $^\circ$ in the final answer if the space allows or if standard convention for the exam board requires it, though often just the number is marked correct if unambiguous.

Techniques used

recall the sum of interior angles formuladivide by the number of sides for a regular polygon

Q82MMedium-EasyNumber

John invests $240 in an account that pays simple interest at a rate of 2.35% per year.

Calculate the total amount of interest he receives at the end of 5 years.

$ ______

DifficultyMedium-Easy

Worked solution

Approach

The account pays simple interest, so the same amount of interest is earned each year. Find the annual interest using the given rate, then multiply by 5 years.

Working

One year's interest is

240 \times \frac{2.35}{100} = 240 \times 0.0235 = 5.64

So each year John receives $5.64. Over 5 years, the total interest is

5 \times 5.64 = 28.20

Equivalently, using the simple interest formula,

\text{Interest} = \frac{P \times R \times T}{100} = \frac{240 \times 2.35 \times 5}{100} = 28.20

Answer

$28.20

Final answer

$28.20

Detailed explanation

Walkthrough

This is a simple interest question. The key feature of simple interest is that the interest is calculated on the original deposit, so it is the same every year. John starts with $240 and the rate is 2.35% per year. Converting 2.35% to a fraction gives $\frac{2.35}{100}$ , so one year's interest is:

240 \times \frac{2.35}{100} = \$5.64

The question asks for 5 years, so multiply the annual interest by 5:

5 \times 5.64 = \$28.20

The question asks for the interest only, not the total amount in the account. Adding the interest to the principal would give $268.20, but that is not what is asked.

Key Takeaways

Simple interest is calculated using the formula $I = \frac{P \times R \times T}{100}$ , where $P$ is the principal, $R$ is the annual interest rate and $T$ is the time in years.
Because simple interest is the same each year, total interest = annual interest $\times$ number of years.
A percentage such as 2.35% must be written as $\frac{2.35}{100}$ before substitution into the formula.

Common Mistakes

Giving the total amount in the account after 5 years instead of the interest: $240 + $28.20 = $268.20. The mark scheme gives special case credit (SC1) only for this total amount, not full marks.
Converting 2.35% incorrectly, for example using 0.235 or 235 instead of 0.0235.
Forgetting to multiply by 5, which would give only one year's interest, $5.64.
Using a compound interest approach, which is not what the question requires.

Things to Be Careful About

The answer must clearly show $240 \times \frac{2.35}{100} \times 5$ or the equivalent single formula, because this is the method mark. The time period is already in years, so no unit conversion is needed. No rounding is required: $28.2$ and $28.20$ are the same value, but the usual money form is $28.20. Remember to include the dollar sign in the final answer. This is a calculator paper, so $240 \times 0.0235 \times 5$ may be keyed straight into a calculator, but the substituted formula should still be written down to show the working.

Techniques used

convert a percentage rate to decimal formapply the simple interest formulamultiply the annual interest by the number of years

Q93MMedium-EasyNumber

The exchange rate between euros (€) and pounds (£) is €1 = £0.88 .
The exchange rate between dollars ($) and pounds (£) is $1 = £ $x$ .

Ling changes €500 into dollars.
She receives $536.58 .

Work out the value of $x$ .

$x$ = ______

DifficultyMedium-Easy

Worked solution

Approach

Convert the euros into pounds using the first exchange rate. Then use the second exchange rate to write the received dollars as a pound amount and equate it to the same total.

Working

Convert €500 to pounds:

500 \times 0.88 = 440

So Ling has £440.

Since $1 = £ $x$ , receiving $536.58 means:

536.58x = 440

Divide both sides by 536.58:

x = \frac{440}{536.58}

x = 0.82

Answer

x = 0.82

Final answer

0.82

Detailed explanation

Walkthrough

Start by converting the €500 into pounds. Since each euro is worth £0.88, multiply:

500 \times 0.88 = 440

So Ling has £440 before changing into dollars.

Now use the second exchange rate. If $1 = £ $x$ , then $536.58 is worth £ $536.58x$ . This must be the same £440 she had, so:

536.58x = 440

Divide both sides by 536.58:

x = \frac{440}{536.58} = 0.82

This is the number of pounds you get for 1 dollar.

Key Takeaways

Exchange rates convert one currency into another by multiplying or dividing.
When a rate is written as “1 unit of currency A = $x$ units of currency B”, the total value in B is the amount in A multiplied by $x$ .
To find an unknown exchange rate, equate the two expressions for the same amount in pounds.

Common Mistakes

Using the wrong direction: computing $536.58 \div 440$ gives about 1.22, which is the number of dollars per pound, not the required $x$ .
Forgetting to convert the euros to pounds first.
Rounding too early; keep the division as $440 \div 536.58$ until the end.
Giving the answer as a ratio reversed, e.g. $x = 1.22$ instead of $0.82$ .

Things to Be Careful About

The answer is $x = 0.82$ (or 0.820), meaning $1 = £0.82.
This is a calculator paper, so you can evaluate $440 \div 536.58$ directly, but show the equation you are solving.
The mark scheme accepts equivalent forms, e.g. $0.88 \times \frac{500}{536.58}$ .
Do not round to fewer than the required accuracy; 0.82 is enough here, but 0.820 is also acceptable.

Techniques used

convert euros to pounds using the exchange rateform an equation linking pounds and dollarsdivide to solve for the exchange rate

Q10Medium-EasyAlgebra and Graphs

Here are the first four patterns in a sequence using dots.

Fig. 4

(a)

Complete the table.

Pattern number	1	2	3	4	5
Number of dots	5	9	13

(b)

Find an expression for the number of dots in Pattern $n$ .

______

DifficultyMedium-Easy

Worked solution

Approach

The sequence of dot counts is $5, 9, 13, 17, 21, \ldots$ . This is an arithmetic sequence with common difference $4$ . The nth term of an arithmetic sequence with first term $a$ and common difference $d$ is $dn + (a - d)$ .

Working

Common difference: $d = 9 - 5 = 4$

First term: $a = 5$

nth term: $4n + (5 - 4) = 4n + 1$

Answer

4n + 1

Final answer

4n + 1

Detailed explanation

Walkthrough

The number of dots in each pattern forms the sequence:

5, 9, 13, 17, 21, \ldots

The difference between consecutive terms is constant:

9 - 5 = 4, \quad 13 - 9 = 4, \quad 17 - 13 = 4

Since the common difference is $4$ , the coefficient of $n$ in the nth term is $4$ , giving a partial expression $4n$ .

To find the constant term, substitute $n = 1$ :

4(1) + k = 5 \implies k = 1

So the nth term is $4n + 1$ .

This matches the visual rule: one central dot plus $n$ dots in each of four directions, giving $1 + 4n = 4n + 1$ .

Key Takeaways

The coefficient of $n$ in the nth term equals the common difference of the sequence.
The constant term is found by subtracting the common difference from the first term: $a - d = 5 - 4 = 1$ .
The nth term formula can be verified by checking it against known terms.

Common Mistakes

Writing $4n$ instead of $4n + 1$ (forgetting the central dot).
Writing $4n - 1$ or another incorrect constant.
Not showing the method that leads to $4n + 1$ .

Things to Be Careful About

The mark scheme accepts $4n + 1$ or any equivalent form such as $4n + k$ where the working shows $k = 1$ .
The expression must be in terms of $n$ , not $p$ or another variable.

Techniques used

find the common difference of an arithmetic sequencewrite the nth term formula for an arithmetic sequence

(c)

Nada needs 1105 dots to make Pattern $p$ .

Find the value of $p$ .

$p$ = ______

Q11Medium-EasyTransformations and VectorsCoordinate Geometry

(a)

$A$ is the point $(-4, 3)$ and $B$ is the point $(2, -7)$ .

3M

(i)

Find $\overrightarrow{AB}$ .

$\overrightarrow{AB}$ = ______

(ii)

Find the coordinates of the midpoint of the line $AB$ .

( ______ , ______ )

(b)

$\overrightarrow{CD} = \begin{pmatrix} -5 \\ 11 \end{pmatrix}$

Find $|\overrightarrow{CD}|$ .

______

Q12MediumNumber

(a)

Write 0.0142 in standard form.

______

(b)

The numbers in the calculation are written in standard form.

(3.2 \times 10^{-4}) \times (x \times 10^y) = 2.6 \times 10^3

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

$x$ = ______
$y$ = ______

DifficultyMedium

Worked solution

Approach

Isolate the unknown term $(x \times 10^y)$ by dividing both sides by $(3.2 \times 10^{-4})$ . Then separate the numerical division from the index arithmetic to find $x$ and $y$ .

Working

Given:
$(3.2 \times 10^{-4}) \times (x \times 10^y) = 2.6 \times 10^3$
Divide by $3.2 \times 10^{-4}$ :
$x \times 10^y = \frac{2.6 \times 10^3}{3.2 \times 10^{-4}}$
Separate the coefficients and the powers of $10$ :
$x \times 10^y = \left(\frac{2.6}{3.2}\right) \times \left(\frac{10^3}{10^{-4}}\right)$
Calculate the coefficient part:
$\frac{2.6}{3.2} = \frac{26}{32} = \frac{13}{16} = 0.8125$
So,
$x \times 10^y = 0.8125 \times 10^{3 - (-4)}$
$x \times 10^y = 0.8125 \times 10^7$
This result is not yet in strict standard form because $0.8125 < 1$ . To convert $0.8125 \times 10^7$ to standard form ( $A \times 10^n$ ):
Move the decimal point one place to the right to get $8.125$ . This makes the number larger, so we must decrease the power of $10$ by $1$ :
$8.125 \times 10^{7-1} = 8.125 \times 10^6$
Comparing this to $x \times 10^y$ :
$x = 8.125$
$y = 6$

Answer

$x = 8.125, \quad y = 6$

Final answer

x = 8.125, y = 6

Detailed explanation

Walkthrough

First, isolate the unknown expression by treating $(3.2 \times 10^{-4})$ as a single multiplier and dividing the total product by it. This gives an expression equal to $x \times 10^y$ .
Next, split the fraction into two parts: the numbers ( $2.6 \div 3.2$ ) and the powers of ten ( $10^3 \div 10^{-4}$ ).
When dividing powers with the same base, subtract the bottom index from the top index: $3 - (-4) = 3 + 4 = 7$ .
The calculation yields $0.8125 \times 10^7$ . However, the problem implies finding specific values for $x$ and $y$ that correspond to the components of the term on the LHS. Since the LHS structure $x \times 10^y$ usually mirrors standard form conventions in these questions (and $x$ is typically expected to be the coefficient $1 \leq x < 10$ ), we adjust $0.8125 \times 10^7$ into proper standard form.
Moving the decimal in $0.8125$ one place right creates $8.125$ . To keep the value the same, we reduce the exponent by $1$ : $10^7$ becomes $10^6$ .
Thus, $x$ matches the coefficient $8.125$ and $y$ matches the exponent $6$ .

Key Takeaways

Division of standard form terms involves dividing coefficients and subtracting indices.
$10^a \div 10^b = 10^{a-b}$ . Subtracting a negative index is adding.
Intermediate results may not be in standard form; conversion is required before identifying final parameters if they must fit the $1 \leq A < 10$ rule.

Common Mistakes

Adding indices instead of subtracting them ( $3 + (-4) = -1$ ).
Forgetting to adjust the coefficient when changing the power of ten (writing $y=7$ instead of $6$ ).
Arithmetic error in $2.6 \div 3.2$ . Note that $2.6/3.2 = 26/32 = 13/16 = 0.8125$ .

Things to Be Careful About

Calculator users should check their output carefully. Some calculators might show $8.125E6$ directly, while others might show $0.8125E7$ . Ensure you interpret the result correctly according to the definition of standard form ( $1 \leq A < 10$ ).

Techniques used

separate coefficients and powers of tendivide coefficientssubtract indices during division

Q133MMedium-EasyAlgebra and Graphs

Solve the simultaneous equations.
You must show all your working.

\begin{aligned} 2a + 5b &= 1.5 \\ 3a - 10b &= 42.5 \end{aligned}

$a$ = ______
$b$ = ______

DifficultyMedium-Easy

Worked solution

Approach

To solve the simultaneous equations, we can use the elimination method. We aim to eliminate one variable by making the coefficients of that variable equal in magnitude but opposite in sign (or equal) in both equations.

Given:

\begin{aligned} 2a + 5b &= 1.5 \quad \text{--- (1)} \\ 3a - 10b &= 42.5 \quad \text{--- (2)} \end{aligned}

Notice that equation (1) has $+5b$ and equation (2) has $-10b$ . If we multiply equation (1) by 2, the coefficient of $b$ becomes $+10$ , which will cancel out with the $-10b$ in equation (2).

Working

Multiply equation (1) by 2:

2 \times (2a + 5b) = 2 \times 1.5

4a + 10b = 3 \quad \text{--- (3)}

Now add equation (3) and equation (2) to eliminate $b$ :

(4a + 10b) + (3a - 10b) = 3 + 42.5

7a = 45.5

Solve for $a$ :

a = \frac{45.5}{7}

a = 6.5

Substitute $a = 6.5$ into equation (1) to find $b$ :

2(6.5) + 5b = 1.5

13 + 5b = 1.5

Subtract 13 from both sides:

5b = 1.5 - 13

5b = -11.5

Divide by 5:

b = \frac{-11.5}{5}

b = -2.3

Answer

The values are:

a = 6.5, \quad b = -2.3

Check in equation (2):

3(6.5) - 10(-2.3) = 19.5 + 23 = 42.5 \quad (\text{Correct})

$a = 6.5$
$b = -2.3$

Final answer

a = 6.5, b = -2.3

Detailed explanation

Walkthrough

This question asks us to solve a pair of simultaneous equations. Simultaneous equations are two or more equations that share the same variables and must be true at the same time. The goal is to find the specific values of those variables that satisfy all equations.

Step 1: Choose a method.
The most common methods are substitution and elimination. In this case, elimination is efficient because the coefficients of $b$ ( $5$ and $-10$ ) are related. Specifically, if we double the first equation, the $b$ term becomes $10b$ , which is the negative of the $10b$ term in the second equation.

Step 2: Eliminate one variable.
We multiply the entire first equation by 2:

2 \times (2a + 5b) = 2 \times 1.5 \implies 4a + 10b = 3

Now we have:

\begin{aligned} 4a + 10b &= 3 \\ 3a - 10b &= 42.5 \end{aligned}

Adding these two equations together cancels out the $b$ terms:

(4a + 3a) + (10b - 10b) = 3 + 42.5

7a = 45.5

Step 3: Solve for the remaining variable.
Divide by 7 to isolate $a$ :

a = \frac{45.5}{7} = 6.5

Step 4: Substitute back to find the other variable.
We substitute $a = 6.5$ back into one of the original equations. Equation (1) is simpler:

2(6.5) + 5b = 1.5

13 + 5b = 1.5

Subtract 13 from both sides:

5b = 1.5 - 13 = -11.5

Divide by 5:

b = \frac{-11.5}{5} = -2.3

Step 5: Check the solution.
It is good practice to check the values in the second equation:

3(6.5) - 10(-2.3) = 19.5 + 23 = 42.5

Since this matches the right-hand side of the second equation, the solution is correct.

Key Takeaways

Elimination Method: Look for coefficients that are multiples of each other. Multiply one or both equations to make them equal (or opposite), then add or subtract to eliminate a variable.
Decimal Arithmetic: Be careful with decimal addition and subtraction. Aligning decimal points helps avoid errors (e.g., $1.5 - 13 = -11.5$ ).
Verification: Always substitute your answers back into one of the original equations to ensure they are correct.

Common Mistakes

Sign Errors: Forgetting that subtracting a negative number is adding (e.g., $-10 \times -2.3 = +23$ ). Or incorrectly handling signs when subtracting equations.
Incorrect Substitution: Plugging the value of $a$ into the wrong equation or making arithmetic mistakes during substitution.
Rounding Errors: The mark scheme specifies "oe" (other equivalents) for exact decimals, but rounding too early in intermediate steps could lead to inaccuracies if fractions were involved. Here, the decimals terminate exactly.
Arithmetic Mistakes: Simple calculation errors like $45.5 / 7$ or $1.5 - 13$ .

Things to Be Careful About

Accuracy: The mark scheme accepts 'oe' (other equivalents), implying exact answers are preferred. Ensure your division results in the exact terminating decimal.
Showing Working: The question explicitly states "You must show all your working." Even if you use a calculator, write down the steps (like forming the new equation, the elimination step, and the substitution) to earn method marks (M1).
Negative Signs: Pay close attention to negative numbers, especially when dealing with $-10b$ and the final negative value for $b$ .

Techniques used

form equations from a worded contexteliminate one variablesubstitute back to find the second variable

Q143MMediumNumber

The population of a country increases exponentially at a rate of 3.6% per year.
On 1st January 2024 the population of the country is 5.4 million.

Calculate the number of complete years it will take for the population to reach 8 million.

______ $\text{years}$

DifficultyMedium

Worked solution

Approach

The population grows exponentially at a fixed percentage rate per year. We use the compound interest (exponential growth) formula $P = P_0 (1 + r)^n$ , where $P$ is the final population, $P_0$ is the initial population, $r$ is the annual growth rate as a decimal, and $n$ is the number of years. We need to find the smallest integer $n$ such that the population reaches or exceeds 8 million.

Working

Given:

Initial population $P_0 = 5.4$ million
Final population $P = 8$ million
Annual growth rate $r = 3.6\% = 0.036$

Substitute these values into the formula:

8 = 5.4 \left( 1 + \frac{3.6}{100} \right)^n

Simplify the term in the brackets:

8 = 5.4 (1.036)^n

Divide both sides by $5.4$ to isolate the exponential term:

(1.036)^n = \frac{8}{5.4}

Simplify the fraction $\frac{8}{5.4}$ :

\frac{8}{5.4} = \frac{80}{54} = \frac{40}{27}

So,

(1.036)^n = \frac{40}{27}

Take the logarithm (base 10 or natural log) of both sides to solve for $n$ :

n \log(1.036) = \log\left(\frac{40}{27}\right)

n = \frac{\log(40/27)}{\log(1.036)}

Calculate the value:

n \approx \frac{0.171957}{0.015305}

n \approx 11.235...

The question asks for the number of complete years it will take for the population to reach 8 million.
At $n = 11$ years, the population is:

5.4 \times (1.036)^{11} \approx 7.86 \text{ million}

This is less than 8 million.
At $n = 12$ years, the population is:

5.4 \times (1.036)^{12} \approx 8.14 \text{ million}

This exceeds 8 million. Therefore, it takes 12 complete years for the population to reach 8 million.

Answer

12

Final answer

12

Detailed explanation

Walkthrough

The problem describes exponential growth because the population increases by a fixed percentage each year. This is modeled by the formula $A = P(1+r)^n$ , similar to compound interest.

Identify the variables: The starting amount ( $P$ ) is 5.4 million. The target amount ( $A$ ) is 8 million. The rate ( $r$ ) is 3.6%, which must be converted to a decimal (0.036) for calculation. The time ( $n$ ) is unknown.
Set up the equation: Substitute the known values into the formula: $8 = 5.4(1 + 0.036)^n$ , which simplifies to $8 = 5.4(1.036)^n$ .
Isolate the exponential term: Divide both sides by 5.4. This gives $(1.036)^n = \frac{8}{5.4}$ . It is often cleaner to simplify this fraction to $\frac{40}{27}$ before taking logs, although decimal division works too.
Solve for n: Since $n$ is in the exponent, we use logarithms. The property $\log(a^b) = b \log(a)$ allows us to bring $n$ down to the front: $n \log(1.036) = \log(\frac{40}{27})$ . Then, divide by $\log(1.036)$ to get $n = \frac{\log(40/27)}{\log(1.036)}$ . Calculating this yields approximately $11.235$ .
Interpret the result: The calculated $n$ is approximately 11.24 years. The question asks for the number of complete years it will take to reach 8 million. After 11 complete years, the population has not yet reached 8 million (it is approx 7.86m). It crosses the 8 million threshold during the 12th year. Therefore, 12 complete years are required for the condition to be met. Note: If the question asked "how many full years pass before it exceeds", one might argue 11, but "time it will take... to reach" implies the duration until the event happens, rounded up to the next whole year if it hasn't happened exactly at an integer boundary. However, standard convention for "number of complete years it will take" usually means the smallest integer $N$ such that $P_N \geq Target$ . Here $N=12$ .

Key Takeaways

Exponential growth/decay problems use the formula $A = P(1 \pm r)^n$ .
Logarithms are the standard tool for solving for an exponent.
Always check the context of the answer. For "complete years", you often need to round up if the fractional part indicates the target hasn't been reached yet within the previous integer year.

Common Mistakes

Using simple interest formula $A = P(1+rn)$ instead of compound/exponential.
Forgetting to convert the percentage rate to a decimal (using 3.6 instead of 0.036).
Rounding $n$ down to 11, ignoring that at 11 years the population is still below 8 million.
Calculation errors when evaluating logarithms.

Things to Be Careful About

The phrase "complete years" can be tricky. Does it mean how many full years have passed when it hits 8m? Or the integer number of years required to ensure it is above 8m? In Cambridge exams, "calculate the number of complete years it will take to reach X" typically asks for the smallest integer $n$ such that the value is $\geq X$ . Since $n \approx 11.24$ , 11 years is not enough. 12 years is enough. So the answer is 12.
Ensure you use the correct base for the growth factor: $1 + r$ for growth, $1 - r$ for decay.
Calculator component: You may use the calculator's power function or logs directly.

Techniques used

apply the exponential growth formulasolve for the exponent using logarithmsinterpret the result as complete years

Q152MMedium-EasyNumber

\begin{aligned} x &= 2^2 \times 3^{n+1} \times 7 \\ y &= 2^n \times 3^n \times 7^4 \end{aligned}

$n$ is a positive integer greater than 2.

Find the highest common factor (HCF) of $x$ and $y$ .
Write your answer as the product of prime factors in terms of $n$ .

______

DifficultyMedium-Easy

Worked solution

Approach

To find the Highest Common Factor (HCF) of two numbers given in their prime factorised forms, we identify all the prime bases that appear in both expressions. For each common prime base, we select the one with the smallest exponent. The HCF is the product of these selected prime powers.

Working

We are given:

x = 2^2 \times 3^{n+1} \times 7

y = 2^n \times 3^n \times 7^4

The prime factors involved are 2, 3, and 7.

Base 2: Appears in $x$ as $2^2$ and in $y$ as $2^n$ . Since $n > 2$ , the smaller exponent is 2. So we take $2^2$ .
Base 3: Appears in $x$ as $3^{n+1}$ and in $y$ as $3^n$ . Since $n$ is positive, $n < n+1$ , so the smaller exponent is $n$ . We take $3^n$ .
Base 7: Appears in $x$ as $7$ (which is $7^1$ ) and in $y$ as $7^4$ . The smaller exponent is 1. So we take $7^1$ or simply $7$ .

Combining these, the HCF is:

\text{HCF} = 2^2 \times 3^n \times 7

Answer

2^2 \times 3^n \times 7

Final answer

2^2 * 3^n * 7

Detailed explanation

Walkthrough

The problem asks for the Highest Common Factor (HCF) of two integers, $x$ and $y$ , which are already expressed as products of prime factors.

The rule for finding the HCF from prime factorizations is straightforward: look at every prime number that appears in the factorization of both numbers. For each such prime, pick the version with the lower exponent (power). Multiply these chosen powers together to get the final answer.

Let's look at the primes individually:

The prime 2 is present in both $x$ ( $2^2$ ) and $y$ ( $2^n$ ). We need to compare the exponents 2 and $n$ . The problem states that $n$ is a positive integer greater than 2 ( $n > 2$ ). Therefore, 2 is less than $n$ . The lower power is $2^2$ .
The prime 3 is present in both $x$ ( $3^{n+1}$ ) and $y$ ( $3^n$ ). We compare the exponents $n+1$ and $n$ . Clearly, $n$ is smaller than $n+1$ . The lower power is $3^n$ .
The prime 7 is present in both $x$ ( $7^1$ ) and $y$ ( $7^4$ ). We compare the exponents 1 and 4. The lower power is $7^1$ , which is just 7.

Putting it all together, the HCF is the product of these lowest powers: $2^2 \times 3^n \times 7$ .

Key Takeaways

HCF from Prime Factors: To find the HCF of two numbers given in prime factorized form, identify the common prime bases and multiply them raised to their minimum exponents.
Inequalities: Pay attention to conditions like $n > 2$ . They determine which exponent is smaller when comparing variables.

Common Mistakes

Choosing the larger exponent instead of the smaller one (confusing HCF with LCM).
Failing to recognize that $7$ is $7^1$ .
Incorrectly assuming $n < 2$ despite the question stating $n > 2$ .
Not writing the final answer as a product of prime factors.

Things to Be Careful About

Ensure you check the condition given for $n$ (here, $n > 2$ ) to correctly compare the exponents of the base 2.
The answer must be written as a product of prime factors. Leaving it as a single calculated number (if $n$ were known) might not fit the "in terms of $n$ " requirement, though here it remains algebraic.
Remember that B1 marks are awarded for partial credit, such as identifying just one of the correct components ( $2^2$ , $3^n$ , or $7$ ), so showing your logic helps even if the final combination is missed.

Techniques used

identify common prime factorsselect the lowest power for each common primeexpress HCF as a product of primes

Q16Medium-EasyCoordinate Geometry

$A$ is the point $(3, -2)$ and $B$ is the point $(5, 4)$ .

(a)

Find the equation of the line $AB$ .
Give your answer in the form $y = mx + c$ .

$y$ = ______

DifficultyMedium-Easy

Worked solution

Approach

First, calculate the gradient ( $m$ ) of the line passing through $A(3, -2)$ and $B(5, 4)$ using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ . Then, use one of the points and the calculated gradient to find the $y$ -intercept ( $c$ ) in the equation $y = mx + c$ .

Working

Calculate the gradient $m$ :

m = \frac{4 - (-2)}{5 - 3}

m = \frac{4 + 2}{2}

m = \frac{6}{2} = 3

Now we have the equation in the form $y = 3x + c$ . To find $c$ , substitute the coordinates of point $A(3, -2)$ into the equation (using point $B$ would give the same result):

-2 = 3(3) + c

-2 = 9 + c

Subtract 9 from both sides:

c = -2 - 9

c = -11

So, the equation of the line is:

y = 3x - 11

Answer

y = 3x - 11

Final answer

y = 3x - 11

Detailed explanation

Walkthrough

To find the equation of a straight line, we need its gradient ( $m$ ) and its $y$ -intercept ( $c$ ). The standard form requested is $y = mx + c$ .

Step 1: Find the gradient. The gradient measures how steep the line is. It is defined as the change in $y$ divided by the change in $x$ between any two points on the line. We are given $A(3, -2)$ and $B(5, 4)$ .
Using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ :
Numerator (change in $y$ ): $4 - (-2) = 6$ .
Denominator (change in $x$ ): $5 - 3 = 2$ .
Gradient $m = \frac{6}{2} = 3$ .

Step 2: Find the $y$ -intercept ( $c$ ). Now that we know the line looks like $y = 3x + c$ , we can plug in the $x$ and $y$ values of either point $A$ or $B$ because both lie on the line. Using $A(3, -2)$ means $x=3$ and $y=-2$ .
Equation: $-2 = 3(3) + c$ .
Simplify: $-2 = 9 + c$ .
Solve for $c$ : Subtract 9 from both sides to isolate $c$ , giving $c = -11$ .

The final equation is $y = 3x - 11$ .

Key Takeaways

Always use the gradient formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ when given two coordinates.
Be careful with negative signs when subtracting coordinates (e.g., $4 - (-2)$ becomes $4 + 2$ ).
Once you have $m$ , substituting one point allows you to solve for $c$ algebraically.

Common Mistakes

Incorrectly calculating the difference in $y$ , especially forgetting that subtracting a negative number adds (e.g., writing $4 - 2 = 2$ instead of $6$ ).
Swapping the $x$ and $y$ values in the substitution step.
Arithmetic errors when isolating $c$ (e.g., adding 9 instead of subtracting it).

Things to Be Careful About

Ensure the final answer is in the specific form requested ( $y = mx + c$ ). Do not leave it as $y - 4 = 3(x - 5)$ or similar forms unless asked.
Double-check the sign of the intercept; here it is negative.

Techniques used

calculate the gradient between two pointssubstitute a point and the gradient into the linear equation form

(b)

Write down the gradient of a line that is perpendicular to the line $AB$ .

______

Q17MediumStatisticsProbability

The table shows the times taken by 80 people to complete a race.

Time ( $t$ minutes)	$20 < t \leq 30$	$30 < t \leq 50$	$50 < t \leq 60$	$60 < t \leq 90$
Frequency	20	26	29	5

(a)

Calculate an estimate of the mean time taken to complete the race.

______ $\text{minutes}$

4M Similar

DifficultyMedium-Easy

Worked solution

Approach

To estimate the mean from grouped data, we assume that all values within a class interval are located at the midpoint of that interval. We calculate the midpoint ( $x$ ) for each time interval, multiply it by the frequency ( $f$ ) of that interval, and then divide the total of these products ( $\sum fx$ ) by the total number of people ( $\sum f = 80$ ).

Working

Step 1: Find the midpoints ( $x$ ) for each interval.
The midpoint is calculated as $\frac{\text{lower bound} + \text{upper bound}}{2}$ .

For $20 < t \leq 30$ : $x_1 = \frac{20 + 30}{2} = 25$
For $30 < t \leq 50$ : $x_2 = \frac{30 + 50}{2} = 40$
For $50 < t \leq 60$ : $x_3 = \frac{50 + 60}{2} = 55$
For $60 < t \leq 90$ : $x_4 = \frac{60 + 90}{2} = 75$

Step 2: Multiply each midpoint by its frequency ( $f$ ).

\begin{aligned} f_1 x_1 &= 20 \times 25 = 500 \\ f_2 x_2 &= 26 \times 40 = 1040 \\ f_3 x_3 &= 29 \times 55 = 1595 \\ f_4 x_4 &= 5 \times 75 = 375 \end{aligned}

Step 3: Calculate the sum of $fx$ ( $\sum fx$ ).

\sum fx = 500 + 1040 + 1595 + 375 = 3510

Step 4: Divide by the total frequency to find the estimated mean.
The total number of people is given as 80 (and confirmed by $20+26+29+5=80$ ).

\text{Estimated Mean} = \frac{\sum fx}{\sum f} = \frac{3510}{80}

\text{Estimated Mean} = 43.875

Answer

43.875 \text{ minutes}

Final answer

43.875

Detailed explanation

Walkthrough

When data is presented in groups or intervals (grouped frequency), we cannot know the exact value of every individual entry. To estimate the mean, we use the midpoint of each group as a representative value for all items in that group.

Find Midpoints: For each time interval (e.g., $20 < t \leq 30$ ), find the average of the lower and upper limits. This gives us the assumed time taken for everyone in that group. For example, the midpoint of 20 to 30 is 25.
Weight by Frequency: Not all groups have the same number of people. A group with 26 people contributes more to the total time than a group with 5 people. Therefore, we multiply each midpoint by the frequency of its group ( $f \times x$ ).
Sum and Average: Add up all these products to get the estimated total time for all 80 runners. Finally, divide this total by the total number of runners (80) to get the average (mean) time per person.

Key Takeaways

The formula for the estimated mean of grouped data is $\bar{x} = \frac{\sum fx}{\sum f}$ .
Always verify that the sum of frequencies ( $\sum f$ ) matches the total given in the question.
The result is an estimate because we assume all values are exactly at the midpoint.

Common Mistakes

Using the lower or upper boundary instead of the midpoint for $x$ .
Failing to multiply the midpoint by the frequency before summing.
Dividing by the wrong total (e.g., dividing by the largest frequency instead of the sum of all frequencies).

Things to Be Careful About

Ensure you calculate the midpoints correctly (add the bounds and divide by 2).
Check your arithmetic when adding large numbers like 500, 1040, 1595, and 375.
The mark scheme accepts "oe" (other equivalent forms), so decimal answers are expected here.

Techniques used

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

(b)

Two of the 80 people are selected at random.

Find the probability that one takes 30 minutes or less and the other takes more than 60 minutes to complete the race.

______

DifficultyMedium

Worked solution

Approach

We need to select two people such that one comes from the "30 minutes or less" group and the other comes from the "more than 60 minutes" group. Since we are selecting two people from the same pool of 80 without putting them back, the events are dependent (the total number of people decreases after the first selection). There are two possible scenarios:

Person A takes $\leq 30$ mins AND Person B takes $> 60$ mins.
Person A takes $> 60$ mins AND Person B takes $\leq 30$ mins.

Working

Step 1: Identify the number of people in each relevant group.
From the table:

Time $t \leq 30$ corresponds to the interval $20 < t \leq 30$ . The frequency is 20.
Time $t > 60$ corresponds to the interval $60 < t \leq 90$ . The frequency is 5.
Total people = 80.

Step 2: Calculate the probability for Scenario 1 (First $\leq 30$ , Second $> 60$ ).

Probability the first person is $\leq 30$ : $\frac{20}{80}$
After selecting one person, there are now 79 people left. The number of people $> 60$ remains 5.
Probability the second person is $> 60$ : $\frac{5}{79}$

P(\text{First } \leq 30, \text{ Second } > 60) = \frac{20}{80} \times \frac{5}{79}

Step 3: Calculate the probability for Scenario 2 (First $> 60$ , Second $\leq 30$ ).

Probability the first person is $> 60$ : $\frac{5}{80}$
After selecting one person, there are now 79 people left. The number of people $\leq 30$ remains 20.
Probability the second person is $\leq 30$ : $\frac{20}{79}$

P(\text{First } > 60, \text{ Second } \leq 30) = \frac{5}{80} \times \frac{20}{79}

Step 4: Add the probabilities of the two mutually exclusive scenarios.
Notice that the two calculations above are identical:

\frac{20}{80} \times \frac{5}{79} = \frac{100}{6320}

Total Probability = $2 \times \left( \frac{20}{80} \times \frac{5}{79} \right)$

= 2 \times \frac{100}{6320} = \frac{200}{6320}

Step 5: Simplify the fraction.
Divide numerator and denominator by 10:

\frac{20}{632}

Divide by 4:

\frac{5}{158}

Answer

\frac{5}{158}

Final answer

5/158

Detailed explanation

Walkthrough

This problem involves selecting two people from a fixed group. Because the second person is chosen from the remaining people, the total population drops from 80 to 79. This makes the events dependent.

There are two ways this outcome can happen:

You pick a fast runner (under 30 mins) first, then a slow runner (over 60 mins) second.
You pick a slow runner (over 60 mins) first, then a fast runner (under 30 mins) second.

Since either order satisfies the condition "one takes 30 mins or less and the other takes more than 60", we must calculate the probability for both sequences and add them together. Alternatively, you can calculate the probability of one sequence and multiply by 2, as the math is symmetric.

Key detail on denominators: The first selection has 80 options. The second selection has 79 options. Do not use 80 for both denominators unless you are explicitly told the person is replaced (which is not standard for "selecting people").

Key Takeaways

Sampling without replacement: When picking multiple items/people from a single group, the denominator decreases by 1 for each subsequent pick.
Mutually exclusive outcomes: If the order matters for calculation but the final state is the same (e.g., AB vs BA), add the probabilities of the separate paths.
Interpreting inequalities: Correctly mapping " $t \leq 30$ " to the correct column in the table is crucial.

Common Mistakes

Using $\frac{20}{80} \times \frac{5}{80}$ : This assumes the first person is put back into the pool, which is incorrect for selecting two distinct people.
Forgetting to double the answer: Calculating only the probability of "Fast then Slow" and forgetting "Slow then Fast".
Misidentifying the groups: Including the $30 < t \leq 50$ group in the "30 minutes or less" category, or missing the $60 < t \leq 90$ group.

Things to Be Careful About

The mark scheme awards method marks for seeing fractions like $\frac{20}{80}$ or $\frac{5}{79}$ . Ensure you write these out clearly.
Simplify your final fraction. $\frac{5}{158}$ is the simplest form.
Check that $20 + 5 = 25$ people are involved in these specific outcomes, leaving 55 people in other categories who are not selected.

Techniques used

identify relevant frequencies from grouped datacalculate probability of combined dependent eventsaccount for both possible orders of selection

Q18MediumAlgebra and Graphs

Write as a single fraction in its simplest form.

(a)

\frac{2x}{3y} \div \frac{8x}{9}

______

DifficultyMedium-Easy

Worked solution

Approach

To divide by a fraction, we multiply by its reciprocal. We then multiply the numerators together and the denominators together, and finally simplify the resulting expression by cancelling any common factors.

Working

The expression is:

\frac{2x}{3y} \div \frac{8x}{9}

Step 1: Rewrite the division as multiplication by the reciprocal of the second fraction (M1):

\frac{2x}{3y} \times \frac{9}{8x}

Step 2: Multiply the numerators and the denominators:

\frac{2x \times 9}{3y \times 8x}

Step 3: Perform the multiplications in the numerator and denominator:

\frac{18x}{24xy}

Step 4: Simplify the fraction by cancelling common factors. Both $18$ and $24$ are divisible by $6$ , and both have an $x$ term (B2/cao):

\frac{18 \div 6}{24 \div 6} = \frac{3}{4}

\frac{x}{x} = 1

Combining these cancellations with the remaining $y$ in the denominator:

\frac{3}{4y}

Answer

\frac{3}{4y}

Final answer

3/(4y)

Detailed explanation

Walkthrough

First, recall the rule for dividing fractions: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$ . Here, we keep the first fraction $\frac{2x}{3y}$ and flip the second fraction $\frac{8x}{9}$ to get $\frac{9}{8x}$ .

Next, we multiply across: top times top ( $2x \times 9 = 18x$ ) and bottom times bottom ( $3y \times 8x = 24xy$ ). This gives us the intermediate fraction $\frac{18x}{24xy}$ .

Finally, we simplify. We look for numbers that divide both $18$ and $24$ (which is $6$ ) and variables that appear in both the top and bottom (which is $x$ ). Dividing $18$ by $6$ gives $3$ , and $24$ by $6$ gives $4$ . The $x$ on top cancels the $x$ on the bottom completely. The $y$ remains on the bottom.

Key Takeaways

Division by a fraction is equivalent to multiplication by its reciprocal.
When simplifying algebraic fractions, treat coefficients and variables separately but simultaneously.
Always check if you can cancel terms before multiplying large numbers out, which can save time and reduce errors.

Common Mistakes

Not flipping the second fraction: Students sometimes multiply straight across without inverting, leading to $\frac{16x^2}{27y}$ .
Partial cancellation: Cancelling $x$ but forgetting to simplify the numbers $18/24$ , or vice versa.
Dropping variables: Writing the answer as just $\frac{3}{4}$ and forgetting the $y$ in the denominator.

Things to Be Careful About

Ensure the final answer is in its simplest form. $\frac{18x}{24xy}$ is mathematically correct but not the required simplest form.
Pay attention to the mark scheme's allowance for partial answers like $\frac{3x}{4xy}$ or $\frac{9}{12y}$ , which indicate you found the right structure but missed the final cancellation step.

Techniques used

invert the divisor and multiplymultiply algebraic fractionscancel common factors to simplify

(b)

\frac{2x^2 + 15x - 50}{4x^2 - 25}

______

4M Similar

DifficultyMedium

Worked solution

Approach

To simplify this complex algebraic fraction, we must factorise both the numerator and the denominator completely. Once factorised, we look for any identical bracketed terms (common factors) that can be cancelled out.

Working

The expression is:

\frac{2x^2 + 15x - 50}{4x^2 - 25}

Step 1: Factorise the numerator $2x^2 + 15x - 50$ .
We need two numbers that multiply to $2 \times (-50) = -100$ and add to $15$ . These numbers are $+20$ and $-5$ .

Rewrite the middle term:

2x^2 + 20x - 5x - 50

Group terms:

2x(x + 10) - 5(x + 10)

Factor out the common binomial $(x + 10)$ :

(2x - 5)(x + 10) \quad \text{(M1/B1)}

Step 2: Factorise the denominator $4x^2 - 25$ .
This is a difference of two squares, since $(2x)^2 = 4x^2$ and $5^2 = 25$ . The formula is $a^2 - b^2 = (a - b)(a + b)$ .

(2x - 5)(2x + 5) \quad \text{(B1)}

Step 3: Write the fraction with the factorised forms:

\frac{(2x - 5)(x + 10)}{(2x - 5)(2x + 5)}

Step 4: Cancel the common factor $(2x - 5)$ from the top and bottom:

\frac{x + 10}{2x + 5}

Answer

\frac{x + 10}{2x + 5}

Final answer

(x+10)/(2x+5)

Detailed explanation

Walkthrough

The key to simplifying rational expressions is factorisation. We cannot simply cancel individual terms like $2x^2$ or $15x$ ; we can only cancel entire factors that multiply the rest of the expression.

For the numerator $2x^2 + 15x - 50$ , we use the 'splitting the middle term' method. We look for factors of $ac$ (here $-100$ ) that sum to $b$ ( $15$ ). The pair $20$ and $-5$ works. We split $15x$ into $20x - 5x$ , group them, and pull out common brackets to get $(2x - 5)(x + 10)$ .

For the denominator $4x^2 - 25$ , we recognise it immediately as a difference of two squares. There is no middle term, so it fits the pattern $A^2 - B^2$ perfectly, where $A=2x$ and $B=5$ . This factors directly into $(2x - 5)(2x + 5)$ .

Now that both parts are broken down into brackets, we see that $(2x - 5)$ appears in both the top and the bottom. We cross it out, leaving $(x + 10)$ on top and $(2x + 5)$ on the bottom.

Key Takeaways

Difference of Two Squares: Any expression in the form $a^2 - b^2$ factors into $(a-b)(a+b)$ . Recognising this saves significant time compared to splitting the middle term.
Factorising Quadratics: For quadratics with a leading coefficient other than 1 (like $2x^2...$ ), splitting the middle term is a reliable method.
Cancellation Rule: You can only cancel factors (terms inside brackets that are multiplied), never terms (terms separated by addition or subtraction).

Common Mistakes

Incorrect factorisation of the numerator: Getting signs wrong, e.g., $(2x + 5)(x - 10)$ , which expands to $2x^2 - 15x - 50$ .
Failing to spot the difference of two squares: Trying to force the denominator into a grouping method when it has only two terms.
Canceling incorrectly: Canceling the $2x$ from the numerator and denominator individually instead of the whole bracket $(2x-5)$ .

Things to Be Careful About

Verify your factorisation by expanding the brackets at the end to ensure they match the original expression.
The final answer must be fully simplified. If you stopped at $\frac{(2x - 5)(x + 10)}{(2x - 5)(2x + 5)}$ , you would lose marks for not completing the cancellation.

Techniques used

factorise a quadratic trinomialrecognise the difference of two squarescancel common factors from numerator and denominator

Q193MMediumNumber

A car travels $320\,\text{km}$ , correct to the nearest $10\,\text{kilometres}$ .
This journey takes $5.2\,\text{hours}$ , correct to the nearest $0.1\,\text{hour}$ .

Calculate the lower bound for the average speed of the car.

______ $\text{km/h}$

DifficultyMedium

Worked solution

Approach

Average speed is calculated as $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$ . To find the lower bound for the average speed, we must use the smallest possible numerator (lower bound of distance) and the largest possible denominator (upper bound of time). This is because dividing by a larger number makes the result smaller.

Working

First, determine the bounds for the given values:

Distance: $320\,\text{km}$ , correct to the nearest $10\,\text{km}$ .
- The interval size is $10$ .
- The error limit is $\frac{10}{2} = 5$ .
- Lower bound for distance ( $D_{LB}$ ) $= 320 - 5 = 315$ .
- Upper bound for distance ( $D_{UB}$ ) $= 320 + 5 = 325$ .
Time: $5.2\,\text{hours}$ , correct to the nearest $0.1\,\text{hour}$ .
- The interval size is $0.1$ .
- The error limit is $\frac{0.1}{2} = 0.05$ .
- Lower bound for time ( $T_{LB}$ ) $= 5.2 - 0.05 = 5.15$ .
- Upper bound for time ( $T_{UB}$ ) $= 5.2 + 0.05 = 5.25$ .

Next, calculate the lower bound for the average speed ( $S_{LB}$ ):

S_{LB} = \frac{D_{LB}}{T_{UB}}

Substitute the values:

S_{LB} = \frac{315}{5.25}

Perform the division:

\frac{315}{5.25} = \frac{31500}{525}

Divide both numerator and denominator by 25:

\frac{31500 \div 25}{525 \div 25} = \frac{1260}{21}

Divide by 21:

1260 \div 21 = 60

Answer

The lower bound for the average speed is $60\,\text{km/h}$ .

60\,\text{km/h}

Final answer

60 km/h

Detailed explanation

Walkthrough

To find the lower bound of a calculated quantity like speed, we need to consider how the bounds of the input variables affect the final result. Speed is defined as $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$ .

Understand the Effect of Bounds:
- To make a fraction as small as possible (the lower bound), the numerator (distance) should be as small as possible.
- To make a fraction as small as possible, the denominator (time) should be as large as possible.
- Therefore, we need the Lower Bound of Distance divided by the Upper Bound of Time.
Calculate the Bounds:
- Distance: Given as $320$ $320$ km to the nearest $10$ $10$ km. The maximum error is half of the rounding unit: $\pm 5$ $\pm 5$ km.
  - Lower Bound ( $D_{min}$ ) $= 320 - 5 = 315$ km.
  - Upper Bound ( $D_{max}$ ) $= 320 + 5 = 325$ km.
- Time: Given as $5.2$ $5.2$ hours to the nearest $0.1$ $0.1$ hour. The maximum error is half of the rounding unit: $\pm 0.05$ $\pm 0.05$ hours.
  - Lower Bound ( $T_{min}$ ) $= 5.2 - 0.05 = 5.15$ hours.
  - Upper Bound ( $T_{max}$ ) $= 5.2 + 0.05 = 5.25$ hours.
Compute the Result:
- We calculate $\frac{315}{5.25}$ .
- $315 \div 5.25 = 60$ .

Key Takeaways

Minimising Fractions: When finding the lower bound of a quotient, divide the lower bound of the numerator by the upper bound of the denominator.
Rounding Intervals: Always determine the error limit by taking half of the rounding unit (e.g., nearest 10 means error is 5).
Units: Ensure the final answer includes the correct units (km/h in this case).

Common Mistakes

Using the wrong combination of bounds (e.g., $\frac{325}{5.15}$ gives the upper bound, not the lower).
Calculating the simple average ( $320 \div 5.2$ ) instead of dealing with bounds.
Incorrectly determining the error limits (e.g., using $10$ or $0.1$ instead of $5$ and $0.05$ ).
Arithmetic errors when dividing decimals.

Things to Be Careful About

Direction of Inequality: Remember that increasing the denominator decreases the value of the fraction. This is the most common conceptual trap.
Precision: Keep intermediate values precise until the final step if necessary, though here the division is exact. Do not round prematurely.
Units: The question asks for km/h; ensure your calculation reflects these units.

Techniques used

determine upper and lower bounds from rounded valuescalculate average speed using distance and time

Q20MediumTrigonometry

Fig. 5

The diagram shows a triangular field, $ABC$ .
$AB = 65\,\text{m}$ , $AC = 75\,\text{m}$ and angle $CAB = 125°$ .

(a)

Show that $BC = 124.3\,\text{m}$ , correct to 1 decimal place.

DifficultyMedium-Easy

Worked solution

Approach

Use the cosine rule to find the length of side $BC$ given sides $AB$ , $AC$ and the included angle $\angle CAB$ .

Working

\begin{aligned} BC^2 &= AB^2 + AC^2 - 2 \times AB \times AC \times \cos(\angle CAB) \\ BC^2 &= 65^2 + 75^2 - 2 \times 65 \times 75 \times \cos 125^\circ \\ BC^2 &= 4225 + 5625 - 9750 \times (-0.573576...) \\ BC^2 &= 9850 + 5592.37... \\ BC^2 &= 15442.37... \end{aligned}

Take the square root:

BC = \sqrt{15442.37...} \approx 124.267...

Rounding to 1 decimal place:

BC = 124.3\,\text{m}

Answer

BC = 124.3\,\text{m}

Final answer

124.3

Detailed explanation

Walkthrough

The cosine rule relates the lengths of the sides of a triangle to the cosine of one of its angles. Since we know two sides ( $AB = 65$ and $AC = 75$ ) and the included angle ( $\angle CAB = 125^\circ$ ), we can directly find the third side $BC$ .

We substitute the known values into the formula $BC^2 = AB^2 + AC^2 - 2 \times AB \times AC \times \cos(\angle CAB)$ . Evaluating the squares and the cosine of $125^\circ$ (which is negative, making the last term positive), we get $BC^2 \approx 15442.37$ . Taking the square root gives $BC \approx 124.267$ , which rounds to $124.3$ to 1 decimal place, matching the required result.

Key Takeaways

The cosine rule is used when you know two sides and the included angle (SAS) to find the third side.
Always ensure your calculator is in degree mode when using trigonometric functions with degree inputs.
Keep extra decimal places during intermediate steps and only round at the final answer to avoid rounding errors.

Common Mistakes

Forgetting to square the side lengths before subtracting.
Using the wrong angle or mixing up the sides in the cosine rule formula.
Rounding intermediate results too early, which can lead to an incorrect final value.
Forgetting that $\cos 125^\circ$ is negative, which means the term $-2 \times 65 \times 75 \times \cos 125^\circ$ becomes positive.

Things to Be Careful About

The question asks to show the answer correct to 1 decimal place. Ensure your final answer is exactly $124.3$ and not rounded to 2 decimal places like $124.27$ .
The diagram is marked NOT TO SCALE, so do not attempt to measure lengths or angles from the diagram; rely entirely on the given numerical values.

Techniques used

apply the cosine ruleevaluate the expressionround to 1 decimal place

(b)

Calculate angle $ABC$ .

Angle $ABC$ = ______

DifficultyMedium-Easy

Worked solution

Approach

Use the sine rule to find angle $ABC$ . We know side $AC = 75$ , side $BC = 124.3$ , and angle $A = 125^\circ$ . Angle $B$ is opposite side $AC$ .

Working

\begin{aligned} \frac{AC}{\sin B} &= \frac{BC}{\sin A} \\ \frac{75}{\sin B} &= \frac{124.3}{\sin 125^\circ} \end{aligned}

Rearrange to solve for $\sin B$ :

\sin B = \frac{75 \times \sin 125^\circ}{124.3}

Evaluate the right-hand side:

\sin B = \frac{75 \times 0.81915...}{124.3} = \frac{61.436...}{124.3} \approx 0.49426...

Find angle $B$ using the inverse sine function:

B = \sin^{-1}(0.49426...) \approx 29.623...^\circ

Rounding to 1 decimal place:

\angle ABC = 29.6^\circ

Answer

\angle ABC = 29.6^\circ

Final answer

29.6

Detailed explanation

Walkthrough

The sine rule states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides of a triangle: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ . Here, we want to find angle $B$ (which is $\angle ABC$ ). We know its opposite side $AC = 75$ , and we know another pair: side $BC = 124.3$ and its opposite angle $A = 125^\circ$ .

We set up the equation $\frac{75}{\sin B} = \frac{124.3}{\sin 125^\circ}$ . Cross-multiplying and rearranging gives $\sin B = \frac{75 \times \sin 125^\circ}{124.3}$ . Calculating this value gives approximately $0.49426$ . Taking the inverse sine ( $\sin^{-1}$ ) of this value gives the angle $B \approx 29.6^\circ$ .

Alternatively, the cosine rule can be used: $\cos B = \frac{AB^2 + BC^2 - AC^2}{2 \times AB \times BC} = \frac{65^2 + 124.3^2 - 75^2}{2 \times 65 \times 124.3}$ , which yields the same result.

Key Takeaways

The sine rule is useful when you know a side and its opposite angle, plus another side or angle.
When using the sine rule to find an angle, always use the inverse sine function.
In an obtuse triangle, the other two angles must be acute, so there is no ambiguity when using $\sin^{-1}$ for angle $B$ (unlike when finding an obtuse angle).

Common Mistakes

Setting up the sine rule incorrectly by mixing up which side is opposite which angle (e.g., using $65$ instead of $75$ for $AC$ ).
Forgetting to use the inverse sine function and just outputting $0.494$ instead of the angle.
Calculator in radian mode instead of degree mode, giving a wildly incorrect angle.

Things to Be Careful About

The question asks for the angle, so the final answer must be in degrees.
If using the cosine rule for this part, ensure the formula is correctly rearranged for $\cos B$ : $\cos B = \frac{AB^2 + BC^2 - AC^2}{2 \times AB \times BC}$ .
Keep the value of $BC$ as $124.3$ (or use the unrounded $124.267...$ ) as given in part (a). Using a rounded value from an earlier step can sometimes cause minor discrepancies, though $124.3$ is accepted here.

Techniques used

apply the sine ruleisolate the sine of the unknown angleuse inverse sine to find the angle

(c)

Calculate the shortest distance from $A$ to $BC$ .

______ $\text{m}$

DifficultyMedium

Worked solution

Approach

The shortest distance from a point to a line is the perpendicular distance. Let $h$ be the perpendicular height from $A$ to $BC$ . We can find $h$ by equating two expressions for the area of triangle $ABC$ .

Working

The area of a triangle can be calculated using two sides and the included angle:

\text{Area} = \frac{1}{2} \times AB \times AC \times \sin(\angle CAB) = \frac{1}{2} \times 65 \times 75 \times \sin 125^\circ

The area can also be calculated using the base $BC$ and the perpendicular height $h$ :

\text{Area} = \frac{1}{2} \times BC \times h = \frac{1}{2} \times 124.3 \times h

Equating the two expressions for the area:

\frac{1}{2} \times 65 \times 75 \times \sin 125^\circ = \frac{1}{2} \times 124.3 \times h

Cancel the $\frac{1}{2}$ from both sides and solve for $h$ :

4875 \times \sin 125^\circ = 124.3 \times h

h = \frac{4875 \times \sin 125^\circ}{124.3}

Evaluate the numerator:

h = \frac{4875 \times 0.81915...}{124.3} = \frac{3993.39...}{124.3} \approx 32.127...

Rounding to 1 decimal place:

h = 32.1\,\text{m}

Answer

32.1\,\text{m}

Final answer

32.1

Detailed explanation

Walkthrough

The shortest distance from point $A$ to the line segment $BC$ is the perpendicular height $h$ dropped from $A$ to $BC$ . To find this, we can use the area of the triangle in two different ways.

First, using the two known sides and the included angle: $\text{Area} = \frac{1}{2}ab\sin C = \frac{1}{2} \times 65 \times 75 \times \sin 125^\circ$ . This gives a numerical value for the area.

Second, using the base $BC$ and the unknown height $h$ : $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 124.3 \times h$ .

Setting these equal allows us to solve for $h$ : $h = \frac{65 \times 75 \times \sin 125^\circ}{124.3} \approx 32.1\,\text{m}$ .

Alternatively, you could use right-angled triangle trigonometry. Drop a perpendicular from $A$ to $BC$ at point $D$ . In the right-angled triangle $ABD$ , $\sin(\angle ABD) = \frac{AD}{AB}$ . Using the angle found in part (b), $AD = 65 \times \sin 29.6^\circ \approx 32.1\,\text{m}$ .

Key Takeaways

The shortest distance from a point to a line is always the perpendicular distance.
The area formula $\frac{1}{2}ab\sin C$ is a powerful tool for finding heights or other missing information in triangles.
Equating two different expressions for the same quantity (like the area of a triangle) is a common and effective problem-solving strategy.

Common Mistakes

Assuming the shortest distance is one of the given sides ( $AB$ or $AC$ ) rather than the perpendicular height.
Forgetting to cancel the $\frac{1}{2}$ on both sides of the area equation, leading to a height that is twice the correct value.
Using the wrong angle or side in the right-angled triangle method (e.g., using $\cos$ instead of $\sin$ for the height).

Things to Be Careful About

The question asks for the distance in metres, so ensure the final answer includes the unit or is clearly understood to be in metres.
If using the right-angled triangle method, remember to use the unrounded value of angle $B$ (from part b) to avoid compounding rounding errors. Using $29.6^\circ$ exactly gives $65 \times \sin 29.6^\circ \approx 32.09$ , which still rounds to $32.1$ , but it is safer to use more decimal places.

Techniques used

recognize shortest distance is perpendicular heightuse area of triangle formulasolve for the height

Q214MMediumAlgebra and Graphs

Rearrange the formula to make $a$ the subject.

w = \frac{2a + 5}{3a - 1}

$a$ = ______

DifficultyMedium

Worked solution

Approach

To make $a$ the subject, we need to isolate $a$ on one side of the equation. Since $a$ appears in both the numerator and the denominator of the fraction on the right-hand side, the first step is to remove the fraction by multiplying through by the denominator $(3a - 1)$ .

Working

Start with the given equation:

w = \frac{2a + 5}{3a - 1}

Multiply both sides by $(3a - 1)$ to clear the denominator:

w(3a - 1) = 2a + 5

Expand the bracket on the left-hand side:

3aw - w = 2a + 5

Group all terms containing $a$ on one side (the left) and all other terms on the other side (the right). Subtract $2a$ from both sides and add $w$ to both sides:

3aw - 2a = w + 5

Factorise $a$ out of the terms on the left-hand side:

a(3w - 2) = w + 5

Divide both sides by $(3w - 2)$ to isolate $a$ :

a = \frac{w + 5}{3w - 2}

Answer

a = \frac{w + 5}{3w - 2}

Final answer

(w + 5) / (3w - 2)

Detailed explanation

Walkthrough

The goal is to change the subject of the formula from $w$ to $a$ . This means we want an expression of the form $a = \text{something}$ .

Clear the fraction: The variable $a$ is currently trapped inside a fraction. To free it, we multiply the entire equation by the denominator, which is $(3a - 1)$ . This is equivalent to cross-multiplication. The mark scheme awards a method mark (M1) for this correct setup: $w(3a - 1) = 2a + 5$ .
Expand: We must distribute the $w$ into the bracket. This gives $3aw - w = 2a + 5$ . Another method mark (M1) is awarded for this expansion.
Group terms: We need all terms containing $a$ on one side of the equals sign. Currently, we have $3aw$ on the left and $2a$ on the right. We subtract $2a$ from both sides and add $w$ to both sides to move the non- $a$ terms ( $-w$ and $5$ ) to opposite sides. This results in $3aw - 2a = w + 5$ . This grouping step earns a method mark (M1).
Factorise and Divide: Now that $a$ is common to the left-hand side, we factorise it out: $a(3w - 2) = w + 5$ . Finally, to leave $a$ alone, we divide the entire equation by the term in the bracket, $(3w - 2)$ . This yields the final answer. This final division earns the last method mark (M1).

Key Takeaways

When a variable appears in both the numerator and the denominator of a fraction, you cannot simply "move it across". You must first eliminate the fraction by multiplying by the denominator. Once the fraction is gone, treat it like any other linear equation: expand, collect like terms, factorise, and divide.

Common Mistakes

Forgetting to multiply the RHS constant: When clearing fractions, students sometimes forget that the $w$ multiplies the entire left side, or they fail to distribute it correctly during expansion.
Sign errors when grouping: Moving $2a$ to the left makes it $-2a$ , and moving $-w$ to the right makes it $+w$ . Getting these signs wrong leads to incorrect factors.
Incorrect factorisation: Failing to pull $a$ out of both $3aw$ and $-2a$ .

Things to Be Careful About

The final answer must be written as a single fraction.
The mark scheme accepts equivalent forms such as $\frac{-w - 5}{2 - 3w}$ , which arises if you choose to group terms differently (e.g., putting $a$ terms on the right). However, $\frac{w + 5}{3w - 2}$ is the standard result. Ensure your denominator is not zero if domain restrictions were asked (though not required here).

Techniques used

clear the denominator by cross-multiplyingexpand the bracketsgroup terms containing the subject variable on one sidefactorise to isolate the subject variable

Q22Medium-EasyAlgebra and Graphs

(a)

Sketch the graph of $y = 2 - 3x$ .

Fig. 6

DifficultyEasy

Worked solution

Approach

To sketch the line $y = 2 - 3x$ , find the $y$ -intercept and the $x$ -intercept, then draw a straight line through them.

Working

The $y$ -intercept is found by setting $x = 0$ :

y = 2 - 3(0) = 2

So the line crosses the $y$ -axis at $(0, 2)$ .

The $x$ -intercept is found by setting $y = 0$ :

0 = 2 - 3x \\ 3x = 2 \\ x = \frac{2}{3}

So the line crosses the $x$ -axis at $(\frac{2}{3}, 0)$ .

Draw a straight line through $(0, 2)$ and $(\frac{2}{3}, 0)$ . The line has a negative gradient of $-3$ and a positive $y$ -intercept.

Answer

A straight line with negative gradient, passing through $(0, 2)$ and $(\frac{2}{3}, 0)$ .

Final answer

Straight line with y-intercept 2 and x-intercept 2/3

Detailed explanation

Walkthrough

To sketch a straight line, we only need two points. The easiest points to find are the intercepts. Setting $x = 0$ gives the $y$ -intercept, which is $2$ . Setting $y = 0$ gives the $x$ -intercept, which is $\frac{2}{3}$ . Drawing a straight line through these two points produces the correct sketch. The negative coefficient of $x$ means the line slopes downwards from left to right.

Key Takeaways

The $y$ -intercept of $y = mx + c$ is $c$ .
The $x$ -intercept is found by setting $y = 0$ and solving for $x$ .
A sketch of a straight line only needs to show the correct gradient and intercepts, not necessarily exact scale.

Common Mistakes

Forgetting that a sketch does not require exact scale, but must show the correct shape (straight line) and approximate intercept positions.
Calculating the $x$ -intercept incorrectly, e.g., $2 - 3x = 0 \Rightarrow 3x = -2$ (sign error).

Things to Be Careful About

Ensure the line is drawn straight, not curved.
The gradient is negative, so the line must go down from left to right.
The $y$ -intercept is positive ( $+2$ ), so it must cross the $y$ -axis above the origin.

Techniques used

identify the y-intercept and x-interceptdraw a straight line with the correct gradient

(b)

6M

(i)

Write $2x^2 - 4x - 3$ in the form $a(x - b)^2 + c$ .

______

DifficultyMedium-Easy

Worked solution

Approach

To write $2x^2 - 4x - 3$ in the form $a(x - b)^2 + c$ , factor out the coefficient of $x^2$ from the first two terms, complete the square inside the bracket, and then simplify the constant terms outside.

Working

Factor out $2$ from the $x^2$ and $x$ terms:

2x^2 - 4x - 3 = 2(x^2 - 2x) - 3

Complete the square for $x^2 - 2x$ . Half of $-2$ is $-1$ , so:

x^2 - 2x = (x - 1)^2 - 1

Substitute this back into the expression:

2((x - 1)^2 - 1) - 3

Expand the outer bracket:

2(x - 1)^2 - 2 - 3

Simplify the constants:

2(x - 1)^2 - 5

Answer

2(x - 1)^2 - 5

Final answer

2(x - 1)^2 - 5

Detailed explanation

Walkthrough

We want to express $2x^2 - 4x - 3$ in the form $a(x - b)^2 + c$ . First, factor out the coefficient of $x^2$ (which is $2$ ) from the $x^2$ and $x$ terms: $2(x^2 - 2x) - 3$ . Next, complete the square inside the bracket. Half of the coefficient of $x$ (which is $-2$ ) is $-1$ , so $x^2 - 2x$ becomes $(x - 1)^2 - 1$ . Substitute this back: $2((x - 1)^2 - 1) - 3$ . Multiply the $-1$ by the $2$ outside: $2(x - 1)^2 - 2 - 3$ . Finally, combine the constants: $-2 - 3 = -5$ . The result is $2(x - 1)^2 - 5$ .

Key Takeaways

Always factor out the leading coefficient before completing the square if it is not $1$ .
Remember to multiply the completed square constant by the factored-out coefficient when simplifying.
The form $a(x - b)^2 + c$ directly gives the turning point $(b, c)$ .

Common Mistakes

Forgetting to multiply the $-1$ from the completed square by the $2$ outside the bracket.
Incorrectly calculating half of the coefficient of $x$ .
Sign errors when expanding $2((x - 1)^2 - 1)$ .

Things to Be Careful About

The question asks for the form $a(x - b)^2 + c$ , so ensure the final answer matches this structure exactly.
Working must be shown (www) to earn method marks.

Techniques used

factor out the leading coefficientcomplete the square inside the bracketsimplify the constant term

(ii)

Sketch the graph of $y = 2x^2 - 4x - 3$ .
On your sketch, label the $y$ -intercept and the coordinates of the turning point.

Fig. 7

DifficultyMedium-Easy

Worked solution

Approach

Use the completed square form from part (b)(i) to identify the turning point, and substitute $x = 0$ into the original equation to find the $y$ -intercept. Then sketch the parabola.

Working

From part (b)(i), the equation in completed square form is:

y = 2(x - 1)^2 - 5

The turning point (vertex) is at $(b, c) = (1, -5)$ .

To find the $y$ -intercept, set $x = 0$ in the original equation $y = 2x^2 - 4x - 3$ :

y = 2(0)^2 - 4(0) - 3 = -3

So the $y$ -intercept is at $(0, -3)$ .

Sketch a U-shaped parabola (since the coefficient of $x^2$ is positive) with the turning point at $(1, -5)$ and passing through the $y$ -intercept at $(0, -3)$ .

Answer

Sketch of a parabola with turning point at $(1, -5)$ and $y$ -intercept at $-3$ .

Final answer

Parabola with turning point (1, -5) and y-intercept -3

Detailed explanation

Walkthrough

The completed square form $y = 2(x - 1)^2 - 5$ tells us the turning point is at $(1, -5)$ . Since the coefficient of $(x - 1)^2$ is positive ( $2$ ), the parabola opens upwards (U-shape). To find the $y$ -intercept, we substitute $x = 0$ into the original equation $y = 2x^2 - 4x - 3$ , giving $y = -3$ . We sketch a smooth U-shaped curve with its minimum at $(1, -5)$ that crosses the $y$ -axis at $(0, -3)$ . By symmetry, it will also cross the $y$ -axis at $(2, -3)$ .

Key Takeaways

The completed square form $a(x - b)^2 + c$ directly gives the turning point $(b, c)$ .
The sign of $a$ determines whether the parabola opens upwards (positive) or downwards (negative).
The $y$ -intercept is always found by setting $x = 0$ .

Common Mistakes

Misreading the turning point coordinates from the completed square form (e.g., writing $(1, 5)$ instead of $(1, -5)$ ).
Forgetting to label both the turning point and the $y$ -intercept as required.
Drawing a straight line instead of a curve.

Things to Be Careful About

The sketch must clearly show the turning point and $y$ -intercept with their coordinates or values marked.
The parabola must be U-shaped (positive coefficient of $x^2$ ).
Follow-through marks are awarded for the turning point if it is incorrect in part (b)(i), but the $y$ -intercept must be calculated correctly from the original equation.

Techniques used

read the turning point from the completed square formcalculate the y-interceptdraw a U-shaped parabola

Q234MMedium-EasyAlgebra and Graphs

Solve.

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

$x$ = ______

DifficultyMedium-Easy

Worked solution

Approach

To solve the equation with $x$ in the denominators, we first combine the two fractions on the left-hand side over a common denominator. Then, we multiply through by this denominator to clear the fractions and obtain a standard linear equation.

Working

The given equation is:

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

Find a common denominator, which is $(x+3)(x-5)$ :

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

Multiply both sides by the denominator $(x+3)(x-5)$ to clear the fraction (Method mark M1/M2):

x(x - 5) - 4(x + 3) = (x + 3)(x - 5)

Expand the brackets on both sides (Method mark M1):
Left Hand Side: $x^2 - 5x - 4x - 12 = x^2 - 9x - 12$
Right Hand Side: $x^2 - 5x + 3x - 15 = x^2 - 2x - 15$

Set them equal:

x^2 - 9x - 12 = x^2 - 2x - 15

Subtract $x^2$ from both sides to eliminate the quadratic term:

-9x - 12 = -2x - 15

Rearrange to solve for $x$ . Add $9x$ to both sides and add $15$ to both sides:

-12 + 15 = 9x - 2x

3 = 7x

Divide by 7:

x = \frac{3}{7}

Check validity: The original denominators are $x+3$ and $x-5$ . If $x = 3/7$ , neither is zero, so the solution is valid.

Answer

x = \frac{3}{7}

Final answer

3/7

Detailed explanation

Walkthrough

The problem asks us to solve an equation containing algebraic fractions. The key strategy is to eliminate the fractions early on to make the equation easier to handle.

Common Denominator: We look at the two terms on the left: $\frac{x}{x + 3}$ and $\frac{4}{x - 5}$ . To subtract them, they must have the same bottom part (denominator). We multiply the top and bottom of the first fraction by $(x-5)$ and the second by $(x+3)$ . This gives us a combined numerator of $x(x-5) - 4(x+3)$ over the combined denominator $(x+3)(x-5)$ .
Clearing Fractions: Since the whole left side equals 1, we can multiply the entire equation by the denominator $(x+3)(x-5)$ . This cancels the denominator on the left and leaves just 1 times that expression on the right. This step transforms a complex fraction equation into a polynomial equation.
Expansion: We expand the brackets. On the left, $x(x-5)$ becomes $x^2 - 5x$ and $-4(x+3)$ becomes $-4x - 12$ . On the right, $(x+3)(x-5)$ expands to $x^2 - 2x - 15$ .
Simplification: Notice that we have $x^2$ on both sides. Subtracting $x^2$ removes the quadratic nature of the problem, leaving a simple linear equation ( $-9x - 12 = -2x - 15$ ). This is a crucial simplification step.
Solving: Finally, we group the $x$ terms on one side and the constant numbers on the other. Adding $9x$ and $15$ results in $3 = 7x$ , giving $x = 3/7$ .

Key Takeaways

When adding or subtracting fractions, always find a common denominator first.
Multiplying by the denominator is the most efficient way to clear fractions in an equation.
Always check if the highest power of $x$ cancels out; here the quadratic terms cancelled, making it a linear equation.

Common Mistakes

Incorrect Expansion: Forgetting to distribute the negative sign when expanding $-4(x+3)$ , leading to $-4x + 12$ instead of $-4x - 12$ .
Wrong Cross-Multiplication: Trying to cross-multiply directly without combining the left side first, which is not applicable for three-term equations like $A/B - C/D = E$ .
Algebra Errors: Incorrectly expanding the product $(x+3)(x-5)$ as $x^2 - 15$ (ignoring the middle terms).

Things to Be Careful About

Ensure you expand ALL brackets correctly. The mark scheme awards marks for correct expansion.
Check that your final answer does not make any original denominator zero. Here $x=3/7$ is safe.
The answer should be in exact form (fraction), not a rounded decimal.

Techniques used

combine algebraic fractions over a common denominatorclear the fraction by multiplying throughexpand brackets and simplify to form a linear equationsolve for the variable

Q243MMedium-EasyAlgebra and Graphs

Expand and simplify.

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

______

DifficultyMedium-Easy

Worked solution

Approach

We expand the expression in stages. First, multiply the first two binomials $(x - 1)$ and $(x + 3)$ . Then, multiply that quadratic result by the remaining factor $(3x - 4)$ . Finally, collect all like terms to write the polynomial in standard form.

Working

First, expand $(x - 1)(x + 3)$ :

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

Now substitute this back into the original expression:

(x^2 + 2x - 3)(3x - 4)

Multiply each term in the quadratic by $3x$ :

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

Multiply each term in the quadratic by $-4$ :

-4(x^2 + 2x - 3) = -4x^2 - 8x + 12

Add these two results together:

(3x^3 + 6x^2 - 9x) + (-4x^2 - 8x + 12)

Group the like terms:

3x^3 + (6x^2 - 4x^2) + (-9x - 8x) + 12

Simplify:

3x^3 + 2x^2 - 17x + 12

Answer

3x^3 + 2x^2 - 17x + 12

Final answer

3x^3 + 2x^2 - 17x + 12

Detailed explanation

Walkthrough

The problem asks us to simplify a product of three brackets. Since we cannot combine them all at once, we break it down.

Step 1: Expand the first two brackets $(x - 1)(x + 3)$ . We use the FOIL method (First, Outer, Inner, Last) or simply distribute each term. $x \cdot x = x^2$ , $x \cdot 3 = 3x$ , $-1 \cdot x = -x$ , and $-1 \cdot 3 = -3$ . Combining $3x$ and $-x$ gives $2x$ . So the intermediate result is $x^2 + 2x - 3$ .

Step 2: Multiply this new quadratic expression by the remaining bracket $(3x - 4)$ . This requires distributing every term in the quadratic to both $3x$ and $-4$ .

Multiplying by $3x$ : $3x \cdot x^2 = 3x^3$ , $3x \cdot 2x = 6x^2$ , $3x \cdot (-3) = -9x$ .
Multiplying by $-4$ : $-4 \cdot x^2 = -4x^2$ , $-4 \cdot 2x = -8x$ , $-4 \cdot (-3) = 12$ .

Step 3: Collect like terms. We have one cubic term ( $3x^3$ ), two quadratic terms ( $6x^2$ and $-4x^2$ ), two linear terms ( $-9x$ and $-8x$ ), and one constant ( $12$ ). Adding the quadratics gives $2x^2$ . Adding the linear terms gives $-17x$ . The final answer is the sum of these parts.

Key Takeaways

When expanding more than two brackets, always work from left to right, expanding two brackets at a time. After each expansion, check that you haven't lost any terms before moving to the next multiplication. Be very careful with negative signs when multiplying by a negative number (e.g., $-4 \times -3 = +12$ ).

Common Mistakes

Incorrectly combining like terms during the first expansion (e.g., getting $3x^2$ instead of $x^2$ ).
Forgetting to multiply every term in the quadratic by both $3x$ and $-4$ (missing terms is a common error).
Sign errors, particularly when multiplying negative numbers (e.g., writing $-8x$ instead of $-8x$ , or $+12$ as $-12$ ).
Failing to collect like terms correctly at the end.

Things to Be Careful About

Ensure your final answer is written in descending powers of $x$ . The mark scheme awards marks for correct unsimplified forms, but the simplified standard form is required for full credit. Double-check arithmetic on the middle terms, as small sign errors there often lead to incorrect final answers.

Techniques used

expand the first two bracketsmultiply the result by the third bracketcollect like terms

Q25MediumTrigonometry

The diagram shows a cuboid.

Fig. 8

$EH = 12\,\text{cm}$ , $HG = 10\,\text{cm}$ and $GC = 7\,\text{cm}$ .

(a)

Show that $CE = 17.1\,\text{cm}$ , correct to 1 decimal place.

DifficultyMedium-Easy

Worked solution

Approach

To find the length of the space diagonal $CE$ , apply Pythagoras' theorem twice: first to find the diagonal of the base $EG$ , then to find the space diagonal $CE$ using the vertical height $GC$ . Alternatively, use the direct 3D formula $CE^2 = EH^2 + HG^2 + GC^2$ .

Working

CE^2 = EH^2 + HG^2 + GC^2

Substitute the given lengths $EH = 12$ , $HG = 10$ , $GC = 7$ :

CE^2 = 12^2 + 10^2 + 7^2

CE^2 = 144 + 100 + 49 = 293

CE = \sqrt{293} \approx 17.117\ldots

Rounding to 1 decimal place:

CE = 17.1\,\text{cm}

This matches the required result.

Answer

CE = 17.1\,\text{cm}

Final answer

17.1 cm

Detailed explanation

Walkthrough

The problem asks for the length of the space diagonal $CE$ of a cuboid. The cuboid has length $EH = 12$ , width $HG = 10$ , and height $GC = 7$ . In a cuboid, the square of the space diagonal is equal to the sum of the squares of its three dimensions. We can derive this by first finding the diagonal of the base rectangle $EFGH$ , which is $EG$ . In right-angled triangle $EHG$ , $EG^2 = EH^2 + HG^2 = 12^2 + 10^2 = 244$ . Then, in right-angled triangle $EGC$ (right-angled at $G$ because $GC$ is perpendicular to the base), $CE^2 = EG^2 + GC^2 = 244 + 7^2 = 244 + 49 = 293$ . Taking the square root gives $CE = \sqrt{293} \approx 17.117$ . Rounding to 1 decimal place gives $17.1$ cm.

Key Takeaways

The length of a space diagonal in a cuboid with dimensions $l, w, h$ is $\sqrt{l^2 + w^2 + h^2}$ .
Always verify which triangle is right-angled before applying Pythagoras' theorem.
Round only at the final stage to avoid accumulation of rounding errors.

Common Mistakes

Using only two dimensions instead of three (e.g., calculating the base diagonal $EG$ and stopping there).
Forgetting to square the numbers before adding them.
Rounding intermediate results (like $EG$ ) too early, which can lead to an incorrect final answer.
Not rounding the final answer to the required 1 decimal place.

Things to Be Careful About

The mark scheme accepts $12^2 + 10^2 + 7^2$ directly as a method mark, showing that the 3D Pythagorean formula is expected.
Ensure the final answer is rounded correctly to 1 decimal place: $17.117\ldots$ rounds to $17.1$ , not $17.2$ .
Units must be included in the final answer as requested ( $\text{cm}$ ).

Techniques used

apply Pythagoras' theorem in three dimensionscalculate the space diagonal of a cuboidround to the required accuracy

(b)

Calculate the angle between the line $CE$ and the base $EFGH$ .

______