GCSE Maths 2018 Edexcel Foundation Paper 2
🖩 Calculator Paper
Calculator button sequences are shown for tricky calculations. You may use a calculator for all questions.
Mark Scheme Legend
- M1 = Method mark (correct method applied)
- P1 = Process mark (completing a process)
- A1 = Accuracy mark (correct answer)
- B1 = Independent mark (correct statement or value)
- C1 = Communication mark (clear explanation)
Table of Contents
- Question 1 (Fractions/Percentages)
- Question 2 (Rounding)
- Question 3 (Indices)
- Question 4 (Place Value)
- Question 5 (Unit Conversions)
- Question 6 (Number Properties)
- Question 7 (Systematic Listing)
- Question 8 (Money Problem)
- Question 9 (Speed/Distance)
- Question 10 (Prime Numbers)
- Question 11 (Solving Equations)
- Question 12 (Pie Chart)
- Question 13 (Probability)
- Question 14 (Frequency Tree)
- Question 15 (Geometric Reasoning)
- Question 16 (Frequency Table)
- Question 17 (Ratio/Recipe)
- Question 18 (Transformations)
- Question 19 (Perimeter/Cost)
- Question 20 (Algebra/Indices)
- Question 21 (LCM/HCF)
- Question 22 (Equation of Line)
- Question 23 (Ratio/Percentages)
- Question 24 (Quadratic Graph)
- Question 25 (Pressure/Force/Area)
- Question 26 (Volume of Prism)
Question 1 (1 mark)
Write \( \frac{4}{50} \) as a percentage.
Step 1: Understanding Percentages
💡 Why we do this:
Percent means “out of 100”. To turn a fraction into a percentage, we want the denominator (bottom number) to be 100.
✏ Working:
We have \( \frac{4}{50} \).
To get from 50 to 100, we multiply by 2.
Whatever we do to the bottom, we must do to the top.
\[ \frac{4 \times 2}{50 \times 2} = \frac{8}{100} \]\( \frac{8}{100} \) is 8%.
Final Answer: 8%
✓ (B1) for 8
Question 2 (1 mark)
Write \( 1.59 \) correct to 1 decimal place.
Step 1: Identifying the decimal place
💡 Why we do this:
The 1st decimal place is the digit immediately after the decimal point. We need to decide whether to keep it as it is or round it up based on the next digit.
✏ Working:
Number: \( 1.59 \)
First decimal digit: 5
Next digit (decider): 9
Since 9 is 5 or greater, we round the 5 up to 6.
\[ 1.59 \rightarrow 1.6 \]Final Answer: 1.6
✓ (B1) for 1.6
Question 3 (1 mark)
Work out the value of \( 3^5 \).
Step 1: Calculate the power
💡 Why we do this:
\( 3^5 \) means 3 multiplied by itself 5 times. You can use your calculator for this.
✏ Working:
\[ 3 \times 3 \times 3 \times 3 \times 3 \]Calculator: \( 3^5 = 243 \)
Final Answer: 243
✓ (B1) for 243
Question 4 (1 mark)
Write down a 6 digit number that has 4 as its thousands digit.
You can only use the digit 4 once.
Step 1: Place value
💡 Why we do this:
A 6-digit number has place values: Hundred-thousands, Ten-thousands, Thousands, Hundreds, Tens, Units.
We need the ‘Thousands’ column to be 4.
✏ Working:
Place values: HTh TTh Th H T U
We put 4 in the Th column: _ _ 4 _ _ _
We can fill the other spots with any digit except 4 (since we can only use it once).
Example: 1 2 4 5 6 7
Example: 5 6 4 0 0 0
Final Answer: Any 6-digit number like 564,000 where the 4th digit from the right is 4 and no other digit is 4.
✓ (B1) for a suitable 6 digit number
Question 5 (3 marks)
(a) Change 35 cm to mm.
(b) Change 7700 millilitres to litres.
(c) Change 0.32 kilograms to grams.
Part (a): cm to mm
💡 Conversion Fact: 1 cm = 10 mm. To go from cm to mm, we multiply by 10.
✏ Working:
\[ 35 \times 10 = 350 \]✓ (B1) 350
Part (b): ml to litres
💡 Conversion Fact: 1 litre = 1000 millilitres. To go from ml to litres, we divide by 1000.
✏ Working:
\[ 7700 \div 1000 = 7.7 \]✓ (B1) 7.7
Part (c): kg to grams
💡 Conversion Fact: 1 kg = 1000 grams. To go from kg to grams, we multiply by 1000.
✏ Working:
\[ 0.32 \times 1000 = 320 \]✓ (B1) 320
Final Answers: (a) 350 mm, (b) 7.7 litres, (c) 320 grams
Question 6 (3 marks)
Margaret is thinking of a number.
She says,
“My number is odd. It is a factor of 36 and a multiple of 3”
There are two possible numbers Margaret can be thinking of.
Write down these two numbers.
Step 1: List factors of 36
💡 Why we do this:
Factors are numbers that divide into 36 exactly.
✏ Working:
1, 2, 3, 4, 6, 9, 12, 18, 36
✓ (P1) for starting to list factors
Step 2: Filter for Odd numbers
✏ Working:
From our list: 1, 3, 9
(All others are even)
Step 3: Filter for Multiples of 3
✏ Working:
From 1, 3, 9:
- 1 is not a multiple of 3
- 3 is a multiple of 3 (3 × 1)
- 9 is a multiple of 3 (3 × 3)
So the numbers are 3 and 9.
Final Answer: 3 and 9
✓ (A2) for both correct (A1 for one correct)
Question 7 (2 marks)
Mohsin, Yusuf and Luke are going to play a game.
At the end of the game, one of them will be in First place, one of them will be in Second place and one of them will be in Third place.
Use the table below to list all the possible outcomes of the game.
| First place | Second place | Third place |
|---|---|---|
Step 1: Systematic Listing
💡 Strategy: Pick one person to be First, then swap the other two for Second and Third. Repeat for each person.
✏ Working:
Let M = Mohsin, Y = Yusuf, L = Luke.
If Mohsin is First:
- M, Y, L
- M, L, Y
If Yusuf is First:
- Y, M, L
- Y, L, M
If Luke is First:
- L, M, Y
- L, Y, M
✓ (M1) for at least 3 correct combinations
Final Answer: All 6 combinations listed above.
✓ (A1) fully correct list
Question 8 (5 marks)
Neil buys 30 pens, 30 pencils, 30 rulers and 30 pencil cases.
Price list
| pens | 6 for 82p |
| pencils | 15 for 45p |
| rulers | 10 for £1.25 |
| pencil cases | 37p each |
What is the total amount of money Neil spends?
Step 1: Calculate cost of 30 pens
💡 Why we do this: Pens are sold in packs of 6. We need to find how many packs make 30 pens.
✏ Working:
Packs needed: \( 30 \div 6 = 5 \) packs.
Cost: \( 5 \times 82p = 410p \) (or £4.10)
✓ (P1) process to find cost of 30 pens
Step 2: Calculate cost of 30 pencils
✏ Working:
Packs needed: \( 30 \div 15 = 2 \) packs.
Cost: \( 2 \times 45p = 90p \)
✓ (P1) process to find cost of 30 pencils
Step 3: Calculate cost of 30 rulers
✏ Working:
Packs needed: \( 30 \div 10 = 3 \) packs.
Price per pack: £1.25 = 125p.
Cost: \( 3 \times 125p = 375p \) (or £3.75)
✓ (P1) process to find cost of 30 rulers
Step 4: Calculate cost of 30 pencil cases
✏ Working:
These are sold individually.
Cost: \( 30 \times 37p = 1110p \) (or £11.10)
Step 5: Total cost
💡 Important: Make sure all units are the same (all pence or all pounds) before adding.
✏ Working:
Total in pence: \( 410 + 90 + 375 + 1110 = 1985p \)
Convert to pounds: \( 1985 \div 100 = £19.85 \)
✓ (P1) adding at least 3 costs
Final Answer: £19.85
✓ (A1) cao
Question 9 (4 marks)
Emily drives 186 miles in 3 hours.
(a) What is her average speed?
Sarah drives at an average speed of 58 mph for 4 hours.
(b) How many miles does Sarah drive?
Part (a): Speed
💡 Formula: \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \)
✏ Working:
\[ \text{Speed} = \frac{186}{3} = 62 \text{ mph} \]✓ (M1) for 186 ÷ 3
✓ (A1) 62
Part (b): Distance
💡 Formula: \( \text{Distance} = \text{Speed} \times \text{Time} \)
✏ Working:
\[ \text{Distance} = 58 \times 4 = 232 \text{ miles} \]✓ (M1) for 58 × 4
✓ (A1) 232
Question 10 (3 marks)
(a) Write down all the prime numbers between 20 and 30.
Catherine says,
“2 is the only even prime number.”
(b) Is Catherine right?
You must give a reason for your answer.
Part (a): Prime Numbers
💡 Why we do this: A prime number has exactly two factors: 1 and itself.
✏ Working:
Let’s check numbers between 20 and 30:
- 21 (3 × 7) – No
- 22 (Even) – No
- 23 – Prime
- 24 (Even) – No
- 25 (5 × 5) – No
- 26 (Even) – No
- 27 (3 × 9) – No
- 28 (Even) – No
- 29 – Prime
✓ (B2) 23 and 29
Part (b): Even Primes
✏ Working:
Yes, Catherine is right.
Reason: Any other even number can be divided by 2, so it would have more than two factors (1, 2, itself).
✓ (C1) Yes with explanation
Question 11 (3 marks)
(a) Solve \( x + x + x = 51 \)
(b) Solve \( \frac{y}{4} = 3 \)
(c) Solve \( 2f + 7 = 18 \)
Part (a)
✏ Working:
\( 3x = 51 \)
\( x = \frac{51}{3} = 17 \)
✓ (B1) 17
Part (b)
✏ Working:
Multiply both sides by 4:
\( y = 3 \times 4 = 12 \)
✓ (B1) 12
Part (c)
✏ Working:
Subtract 7 from both sides:
\( 2f = 18 – 7 \)
\( 2f = 11 \)
Divide by 2:
\( f = \frac{11}{2} = 5.5 \)
✓ (B1) 5.5
Question 12 (3 marks)
A group of football fans were asked what their half time snack was.
The table below gives information about their answers.
| Snack | Number of fans |
|---|---|
| burger | 11 |
| pie | 17 |
| hot dog | 8 |
Draw an accurate pie chart for this information.
Step 1: Calculate Total
✏ Working:
Total fans = \( 11 + 17 + 8 = 36 \)
Step 2: Calculate Degrees per person
✏ Working:
A circle is 360°.
Degrees per fan = \( 360 \div 36 = 10^\circ \)
Step 3: Calculate Angles
✏ Working:
Burger: \( 11 \times 10^\circ = 110^\circ \)
Pie: \( 17 \times 10^\circ = 170^\circ \)
Hot dog: \( 8 \times 10^\circ = 80^\circ \)
Check: \( 110 + 170 + 80 = 360^\circ \)
✓ (M1) method to find angles
✓ (A1) correct angles calculated
Step 4: Draw Chart
✓ (A1) fully correct labelled pie chart
Question 13 (2 marks)
A scout group has a raffle to raise money for charity.
There is 1 prize to be won in the raffle.
Laura buys 12 raffle tickets.
A total of 350 raffle tickets are sold.
Find the probability that Laura does not win the prize.
Step 1: Probability of winning
✏ Working:
Laura has 12 tickets out of 350.
P(Win) = \( \frac{12}{350} \)
Step 2: Probability of NOT winning
💡 Strategy: The total probability is 1. So P(Not Win) = 1 – P(Win).
✏ Working:
\( 1 – \frac{12}{350} = \frac{350}{350} – \frac{12}{350} = \frac{338}{350} \)
✓ (M1) for 350 – 12
Final Answer: \( \frac{338}{350} \) (or simplified \( \frac{169}{175} \))
✓ (A1) oe
Question 14 (3 marks)
Each worker in a factory is either left-handed or right-handed.
22 of the 45 workers are male.
16 of the 34 right-handed workers are female.
Complete the frequency tree for this information.
Step 1: Fill in Males and Females
✏ Working:
Total workers = 45.
Male workers = 22.
Female workers = 45 – 22 = 23.
Step 2: Right-handed logic
✏ Working:
Total right-handed = 34.
Right-handed females = 16.
Right-handed males = Total RH – Female RH = 34 – 16 = 18.
Step 3: Left-handed logic
✏ Working:
Left-handed males = Total Males – RH Males = 22 – 18 = 4.
Left-handed females = Total Females – RH Females = 23 – 16 = 7.
✓ (C3) for complete correct tree
Completed Tree Values:
- Male: 22
- Female: 23
- Male Left: 4
- Male Right: 18
- Female Left: 7
- Female Right: 16
Question 15 (2 marks)
Mary needs to work out the size of angle \( x \) in this diagram.
She writes:
\( x = 63^\circ \) because base angles of an isosceles triangle are equal.
Mary is wrong.
(a) Explain why.
William needs to work out the size of angle \( y \) in this diagram.
William writes:
| Working | Reason |
| angle \( EGH = 57^\circ \) | because corresponding angles are equal |
| \( y = 180^\circ – 57^\circ \) \( y = 123^\circ \) |
because angles on a straight line add up to \( 180^\circ \) |
One of William’s reasons is wrong.
(b) Write down the correct reason.
Part (a): Mary’s Error
💡 Geometric Reasoning: In an isosceles triangle, the angles opposite the equal sides are equal. Here, the marks show AB = AC, so the angles at B and C are NOT the equal base angles. The equal angles are at B and C only if AB = AC. Wait, let’s look at the marks.
The marks are on AC and AB. This means angle B = angle C. Wait, the marks are on sides AB and AC. That makes A the vertex angle, and B and C the base angles. So x should equal 63.
Wait, looking closely at the diagram in the PDF: The ticks are on AC and AB. This means the angles opposite them (Angle B and Angle C) are equal. So x SHOULD be 63. Why is Mary wrong?
Ah, check diagram carefully. The ticks are on sides AC and AB? No, looking at P11 crop. The ticks are on AB and AC. So triangle is isosceles with AB=AC. This makes angle ABC = angle ACB. So x should be 63.
Wait… re-read question. “She writes x = 63 because base angles…”.
Let’s re-examine the diagram crop. The tick is on AC. Where is the other tick? It’s on AB. So AB=AC. The angles opposite these sides are angle C (which is x) and angle B (which is 63). So x = 63.
Is it possible the angle labelled 63 is NOT a base angle? If the ticks were on AB and BC, then A and C would be base angles.
Let’s look at the mark scheme. “x is not a base angle or states x = 54”. This implies x is the VERTEX angle. This means the 63s are the base angles. For that to be true, the ticks must be on AB and AC, meaning angle B = angle C. But if x is 54, then 180 – 63 – 63 = 54. This means B and C are 63.
Hold on. If x is 54, then the equal angles are 63 and 63. That means B is 63, and the other base angle (let’s call it A?) is 63.
Let’s look at the SVG I drew based on the image. The arc for x is at C. The arc for 63 is at B. The ticks are on AB and BC? No, it looks like AB and AC.
Actually, looking at the crop very closely: The angle at B is 63. Angle at C is x. The tick marks are on sides AC and AB. This implies Angle B = Angle C. So x = 63.
Why does the mark scheme say “x is not a base angle”? This implies the diagram shows something different. Maybe the angle 63 is at A? No, it’s clearly at B. Maybe the ticks are on AB and BC? If ticks on AB and BC, then base angles are A and C. Then x = A. B is the vertex.
Let’s check the modification notes in the Mark Scheme. “Lines have been made longer… AC = BC”. AHA! The original diagram likely has AC = BC. If AC = BC, then the angles opposite are A and B. So A = B = 63. And x is at C (the vertex).
So, assuming AC = BC (ticks on AC and BC): Base angles are at A and B. Angle A = 63. Angle B = 63. Angle C (x) = 180 – 63 – 63 = 54.
So Mary is wrong because x is the vertex angle, not a base angle.
✏ Correct Answer:
The diagram shows the sides AC and BC are equal (indicated by dash marks). Therefore, the base angles are at A and B.
Angle A = Angle B = 63°.
Angle \( x \) is the third angle, not a base angle.
✓ (C1) Explanation stating x is not a base angle
Part (b): William’s Error
💡 Geometric Reasoning: William used “corresponding angles”. Let’s check the position.
The angle 57 is at E (inside parallel lines). Angle EGH is at G (inside parallel lines, same side). These are Allied (or Co-interior) angles.
Or if he calculated EGH, maybe he meant the angle alternate to 57? The alternate angle is at G, but on the left (FGE). That would be 57.
He says “Angle EGH = 57”. EGH and the 57 angle are Alternate angles (Z-angles). Are they? 57 is acute. EGH is obtuse. Wait.
The 57 is marked at vertex E. The transversal goes down to G.
Angle DEG = 57. Angle FGE (marked y) is on the left. EGH is on the right.
Alternate angles are equal. Angle FGE = Angle DEG = 57 (Alternate/Z-angles). So y should be 57.
William calculated y = 123. This implies he thinks they add to 180 (Allied). But he justified EGH = 57 using “corresponding”.
Actually, Alternate angles are equal. So y should be 57. William is calculating y as 123.
Let’s look at the reason “because corresponding angles are equal”. The angle 57 and angle EGH are Alternate angles, not corresponding. (Z shape). Corresponding angles make an F shape.
✓ (C1) “Alternate angles are equal”
Question 16 (3 marks)
Marla buys some bags of buttons.
There are 19 buttons or 20 buttons or 21 buttons or 22 buttons in each bag.
The table gives some information about the number of buttons in each bag.
| Number of buttons | Frequency |
|---|---|
| 19 | … |
| 20 | 7 |
| 21 | 3 |
| 22 | 1 |
The total number of buttons is 320.
Complete the table.
Step 1: Calculate buttons from known bags
✏ Working:
20 buttons \(\times\) 7 bags = 140 buttons
21 buttons \(\times\) 3 bags = 63 buttons
22 buttons \(\times\) 1 bag = 22 buttons
Subtotal = \( 140 + 63 + 22 = 225 \) buttons.
✓ (P1) for start of process
Step 2: Find remaining buttons
✏ Working:
Total buttons = 320.
Remaining buttons = \( 320 – 225 = 95 \) buttons.
Step 3: Calculate missing frequency
✏ Working:
These 95 buttons are in bags of 19.
Number of bags = \( 95 \div 19 = 5 \).
✓ (P1) complete process
✓ (A1) 5
Final Answer: The missing frequency is 5.
Question 17 (3 marks)
Here is the list of ingredients for making 30 biscuits.
Ingredients for 30 biscuits
- 225 g butter
- 110 g caster sugar
- 275 g plain flour
- 75 g chocolate chips
Lucas has the following ingredients:
- 900 g butter
- 1000 g caster sugar
- 1000 g plain flour
- 225 g chocolate chips
What is the greatest number of biscuits Lucas can make?
You must show your working.
Step 1: Calculate “batches” for each ingredient
💡 Strategy: We need to find out how many times we can make the recipe with each ingredient. The lowest number will be our limit (limiting reactant).
✏ Working:
Butter: \( 900 \div 225 = 4 \) batches
Sugar: \( 1000 \div 110 = 9.09… \) batches
Flour: \( 1000 \div 275 = 3.63… \) batches
Choc chips: \( 225 \div 75 = 3 \) batches
✓ (P1) process to find number of batches for at least 2 ingredients
Step 2: Identify the limiting ingredient
What this tells us: The chocolate chips run out first (we can only make 3 batches). Even though we have enough butter for 4 batches, we can’t make them without chips.
✓ (P1) identifying limiting factor (3 batches)
Step 3: Calculate total biscuits
✏ Working:
1 batch = 30 biscuits
3 batches = \( 3 \times 30 = 90 \) biscuits
✓ (A1) 90
Final Answer: 90
Question 18 (2 marks)
Describe fully the single transformation that maps shape A onto shape B.
Step 1: Identify Transformation Type
💡 Observation: Shape B is a mirror image of Shape A. It hasn’t been rotated or resized. This is a Reflection.
Step 2: Find the Mirror Line
✏ Working:
Shape A is on the right, Shape B is on the left.
They are equidistant from the y-axis (the vertical line where x = 0).
e.g., A point is at x=3, B point is at x=-3.
✓ (B2) Reflection in the y-axis (or x=0)
Final Answer: Reflection in the y-axis (or x = 0)
Question 19 (5 marks)
A farmer has a field in the shape of a semicircle of diameter 50 m.
The farmer asks Jim to build a fence around the edge of the field.
Jim tells him how much it will cost.
Jim takes three days to build the fence.
Work out the total cost.
Step 1: Calculate Curved Length (Arc)
💡 Formula: Circumference = \( \pi \times d \). Semicircle arc = \( \frac{1}{2} \times \pi \times d \).
✏ Working:
\[ \text{Arc} = 0.5 \times \pi \times 50 = 25\pi \approx 78.54 \text{ m} \]✓ (P1) process to find circumference
Step 2: Calculate Total Perimeter
✏ Working:
Perimeter = Arc + Diameter (straight edge)
\[ 78.54 + 50 = 128.54 \text{ m} \]✓ (P1) process to find total perimeter
Step 3: Calculate Fence Cost
✏ Working:
\[ 128.54 \times £29.86 = £3838.20 \]✓ (P1) process to find cost of fence
Step 4: Calculate Labour Cost
✏ Working:
\[ 3 \text{ days} \times £180 = £540 \]Step 5: Total Cost
✏ Working:
\[ £3838.20 + £540 = £4378.20 \]✓ (P1) adding costs
✓ (A1) answer in range 4377 – 4392
Final Answer: £4378.20
Question 20 (5 marks)
(a) Simplify \( m^3 \times m^4 \)
(b) Simplify \( (5np^3)^3 \)
(c) Simplify \( \frac{32q^9 r^4}{4q^3 r} \)
Part (a)
💡 Rule: When multiplying with same base, add powers. \( a^x \times a^y = a^{x+y} \)
✏ Working:
\[ 3 + 4 = 7 \implies m^7 \]✓ (B1) m^7
Part (b)
💡 Rule: Cube everything inside the bracket. \( (ab)^n = a^n b^n \).
✏ Working:
- \( 5^3 = 125 \)
- \( n^3 \)
- \( (p^3)^3 = p^{3 \times 3} = p^9 \)
Combine: \( 125n^3p^9 \)
✓ (B2) 125n^3p^9
Part (c)
💡 Rule: Divide numbers, subtract powers. \( \frac{a^x}{a^y} = a^{x-y} \)
✏ Working:
- Numbers: \( 32 \div 4 = 8 \)
- q: \( q^9 \div q^3 = q^{9-3} = q^6 \)
- r: \( r^4 \div r^1 = r^{4-1} = r^3 \)
Combine: \( 8q^6r^3 \)
✓ (B2) 8q^6r^3
Question 21 (3 marks)
(a) Find the lowest common multiple (LCM) of 40 and 56.
\( A = 2^3 \times 3 \times 5 \)
\( B = 2^2 \times 3 \times 5^2 \)
(b) Write down the highest common factor (HCF) of A and B.
Part (a): LCM
✏ Method 1: Listing Multiples
40: 40, 80, 120, 160, 200, 240, 280…
56: 56, 112, 168, 224, 280…
LCM = 280
✏ Method 2: Prime Factors
\( 40 = 2^3 \times 5 \)
\( 56 = 2^3 \times 7 \)
LCM = \( 2^3 \times 5 \times 7 = 8 \times 5 \times 7 = 280 \)
✓ (M1) for listing or prime factors
✓ (A1) 280
Part (b): HCF
💡 Rule: For HCF from prime factors, take the lowest power of each common prime.
✏ Working:
Common primes: 2, 3, 5
Lowest power of 2: \( 2^2 \)
Lowest power of 3: \( 3^1 \)
Lowest power of 5: \( 5^1 \)
HCF = \( 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60 \)
✓ (B1) 60
Question 22 (3 marks)
The line L is shown on the grid.
Find an equation for L.
Step 1: Formula y = mx + c
💡 Key Concept: \( m \) is the gradient (slope), \( c \) is the y-intercept (where line crosses y-axis).
Step 2: Find the y-intercept (c)
✏ Working:
The line crosses the vertical y-axis at \( -6 \).
So, \( c = -6 \).
Step 3: Find the gradient (m)
✏ Working:
Pick two clear points on the line: \( (0, -6) \) and \( (2, 0) \).
Gradient = \( \frac{\text{Change in y}}{\text{Change in x}} \)
Gradient = \( \frac{0 – (-6)}{2 – 0} = \frac{6}{2} = 3 \)
So, \( m = 3 \).
✓ (M1) method to find gradient
Step 4: Write Equation
✏ Working:
\( y = 3x – 6 \)
✓ (A1) y = 3x – 6
Final Answer: \( y = 3x – 6 \)
Question 23 (5 marks)
Raya buys a van for £8500 plus VAT at 20%.
Raya pays a deposit for the van.
She then pays the rest of the cost in 12 equal payments of £531.25 each month.
Find the ratio of the deposit Raya pays to the total of the 12 equal payments.
Give your answer in its simplest form.
Step 1: Calculate Total Cost (with VAT)
✏ Working:
Base price = £8500
VAT = 20% of 8500 = \( 0.2 \times 8500 = 1700 \)
Total Cost = \( 8500 + 1700 = £10,200 \)
✓ (P1) process to find total cost
Step 2: Calculate Total Payments
✏ Working:
12 payments of £531.25
\( 12 \times 531.25 = £6,375 \)
✓ (P1) process to find total payments
Step 3: Calculate Deposit
✏ Working:
Deposit = Total Cost – Total Payments
\( 10,200 – 6,375 = £3,825 \)
✓ (P1) process to find deposit
Step 4: Form Ratio
✏ Working:
Ratio = Deposit : Payments
\( 3825 : 6375 \)
Step 5: Simplify Ratio
✏ Working:
Divide by 25: \( 153 : 255 \)
Divide by 51: \( 3 : 5 \)
(Or use calculator fraction button \( \frac{3825}{6375} = \frac{3}{5} \))
✓ (A1) 3 : 5
Final Answer: 3 : 5
Question 24 (6 marks)
(a) Complete the table of values for \( y = x^2 – x – 6 \)
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| y | 6 | -4 | -6 |
(b) On the grid, draw the graph of \( y = x^2 – x – 6 \) for values of x from -3 to 3
(c) Use your graph to find estimates of the solutions to the equation \( x^2 – x – 6 = -2 \)
Part (a): Table
✏ Working:
When x = -2: \( (-2)^2 – (-2) – 6 = 4 + 2 – 6 = 0 \)
When x = 1: \( 1^2 – 1 – 6 = 1 – 1 – 6 = -6 \)
When x = 2: \( 2^2 – 2 – 6 = 4 – 2 – 6 = -4 \)
When x = 3: \( 3^2 – 3 – 6 = 9 – 3 – 6 = 0 \)
Values: 0, -6, -4, 0
✓ (B2) Correct table
Part (b): Graph
💡 Strategy: Plot the points from the table and join them with a smooth curve.
✓ (A1) Fully correct curve
Part (c): Solve Equation
💡 Strategy: Draw the line \( y = -2 \) (horizontal line). Find the x-values where this line crosses your curve.
✏ Working:
The line y = -2 crosses the curve at approx x = -1.6 and x = 2.6.
✓ (A1) Values in range -1.5 to -1.7 AND 2.5 to 2.7
Final Answer: approx -1.6 and 2.6
Question 25 (3 marks)
A force of 70 newtons acts on an area of 20 cm².
\( \text{pressure} = \frac{\text{force}}{\text{area}} \)
The force is increased by 10 newtons.
The area is increased by 10 cm².
Helen says,
“The pressure decreases by less than 20%”
Is Helen correct?
You must show how you get your answer.
Step 1: Calculate Initial Pressure
✏ Working:
\[ P_{\text{old}} = \frac{70}{20} = 3.5 \text{ N/cm}^2 \]Step 2: Calculate New Pressure
✏ Working:
New Force = 70 + 10 = 80
New Area = 20 + 10 = 30
\[ P_{\text{new}} = \frac{80}{30} = 2.66… \text{ (or } 2.67 \text{)} \]✓ (P1) process to find new pressure
Step 3: Calculate Percentage Decrease
💡 Formula: \( \frac{\text{Change}}{\text{Original}} \times 100 \)
✏ Working:
Decrease = \( 3.5 – 2.66… = 0.833… \)
Percentage = \( \frac{0.833…}{3.5} \times 100 \approx 23.8\% \)
✓ (P1) process to compare
Step 4: Conclusion
✏ Working:
The decrease is ~24%.
Helen said “less than 20%”.
24% is MORE than 20%.
So Helen is No.
✓ (A1) No, supported by correct figures
Question 26 (5 marks)
Here is a triangular prism.
Wait, let’s verify dimensions from diagram text.
Diagram shows height 7.2 cm. The slanted side is labeled? No, looking at P22… 8.4 is the hypotenuse? No. The dashed line implies 8.4 is the base of the triangle? Actually, looking at the right angle symbol… it’s between the vertical side (7.2?) and the base?
Let’s look at the diagram again. Vertical side: 7.2. Hypotenuse: 8.4. So base is unknown. Length: 18.
Work out the volume of the prism.
Give your answer correct to 3 significant figures.
Step 1: Calculate the missing base of the triangle
💡 Pythagoras Theorem: \( a^2 + b^2 = c^2 \). Here, \( c = 8.4 \) (hypotenuse) and \( a = 7.2 \).
✏ Working:
\( \text{base}^2 = 8.4^2 – 7.2^2 \)
\( \text{base}^2 = 70.56 – 51.84 = 18.72 \)
\( \text{base} = \sqrt{18.72} \approx 4.326… \text{ cm} \)
✓ (P1) Pythagoras to find missing side
Step 2: Calculate Area of Cross-section (Triangle)
✏ Working:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] \[ \text{Area} = 0.5 \times 4.326… \times 7.2 \approx 15.57… \text{ cm}^2 \]✓ (P1) process to find area
Step 3: Calculate Volume
✏ Working:
\[ \text{Volume} = \text{Area} \times \text{Length} \] \[ \text{Volume} = 15.57… \times 18 \approx 280.33… \text{ cm}^3 \]✓ (P1) process to find volume
Step 4: Rounding
✏ Working:
Round 280.33… to 3 significant figures.
First 3 digits are 2, 8, 0.
So, 280.
✓ (A1) 280
Final Answer: 280 cm³
