M6.3 · Scatter graphs and correlation
State — 4 marks
A student investigates the relationship between the thickness of insulation material (in cm) and the rate of heat loss (in W) from a hot water tank. The student collects data from 12 different tanks with varying insulation thicknesses and plots the results on a scatter graph. The graph shows points that follow a clear downward trend from left to right, with most points lying close to an imaginary straight line, though three points deviate noticeably from this pattern.
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(a) State the type of correlation shown by the scatter graph.
[1 mark]
-
(b) State the strength of the correlation shown by the scatter graph, and justify your answer with reference to the distribution of points.
[2 marks]
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(c) State what the three points that deviate from the main trend suggest about the reliability of using this scatter graph to make predictions about heat loss from insulation thickness. Explain your reasoning.
[1 mark]
Show mark scheme
- (a) Negative correlation (accept 'inverse correlation')
- (b) Strong correlation (1 mark). Most points lie close to / follow a clear linear trend / the points are clustered tightly around an imaginary line (1 mark)
- (c) The outliers/anomalous results suggest predictions may be unreliable / there may be other variables affecting heat loss that are not accounted for (1 mark)
M5.1 · Basic probability and sample space
Show — 4 marks
A game is played using two fair spinners, each numbered 1, 2 and 3. The spinners are spun once each and the player's score is the product of the two numbers.
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(a) Complete the sample space diagram to show all possible scores.
| | 1 | 2 | 3 |
|-----|---|---|---|
| 1 | | | |
| 2 | | | |
| 3 | | | |
[2 marks]
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(b) Show that the probability of scoring an even number is 5/9.
[2 marks]
Show mark scheme
- (a) All 9 products correctly filled in sample space diagram (or list of all 9 outcomes with products)
- (b) Identifies 5 even outcomes from their sample space
- (b) Shows working leading to 5/9 with correct conclusion stated
M4.3 · Mensuration (area, volume)
Describe — 3 marks
A manufacturer produces two cuboid storage containers. Container P has dimensions 10 cm × 6 cm × 5 cm. Container Q is mathematically similar to Container P, but each dimension is twice as long.
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(a) Describe how to calculate the volume of Container P.
[1 mark]
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(b) Describe how the volume of Container Q compares to the volume of Container P.
[2 marks]
Show mark scheme
- (a) Multiply the three dimensions together (10 × 6 × 5) or reference to length × width × height
- (b) State that the volume of Q is 8 times the volume of P (or equivalent)
- (b) Explain that doubling all three dimensions means 2 × 2 × 2 = 8 (or 2³)
M3.3 · Direct and inverse proportion
Calculate — 2 marks
A student is investigating how the resistance of a wire changes with its length. They use a wire made of constantan and measure the resistance at different lengths. The wire has a constant cross-sectional area.
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(a) When the wire is 50 cm long, its resistance is 10 Ω. Calculate the resistance of the same wire when it is 100 cm long.
[1 mark]
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(b) Explain why the resistance changes in the way shown in your answer to part (a).
[1 mark]
Show mark scheme
- (a) Correct answer of 20 Ω (accept 20)
- (a) Working must show recognition that resistance is directly proportional to length, e.g. R ∝ L or 10/50 = R/100
- (b) States that resistance is directly proportional to length
- (b) OR states that when length doubles, resistance doubles
- (b) OR explains that longer wire has more material for charge carriers to collide with
M2.3 · Sequences
Calculate — 2 marks
A florist is creating a display with rows of flowers. The number of flowers in each row follows a sequence pattern. Row 1 has 4 flowers, Row 2 has 7 flowers, Row 3 has 10 flowers, and Row 4 has 13 flowers.
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(a) Calculate the number of flowers needed for Row 5.
[1 mark]
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(b) Calculate the number of flowers needed for Row 10.
[1 mark]
Show mark scheme
M5.2 · Combined events and tree diagrams
Show — 3 marks
A bag contains 4 green counters and 6 yellow counters. Mia takes a counter at random from the bag, notes its colour, and replaces it. She then takes a second counter at random. The tree diagram shows the possible outcomes.
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(a) Write down the probability that Mia takes a green counter on her first draw.
[1 mark]
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(b) Show that the probability of Mia taking two counters of different colours is 12/25.
[2 marks]
Show mark scheme
- (a) 4/10 oe
- (b) Correct method: (4/10 × 6/10) + (6/10 × 4/10) oe
- (b) Complete working shown leading to 12/25 oe (e.g., 24/100 + 24/100 = 48/100 = 12/25)
M3.2 · Percentages and percentage change
Compare — 3 marks
A shop sells two different models of headphones. Model A originally costs £80 and Model B originally costs £120. During a sale, Model A is reduced by 15% and Model B is reduced by £18.
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(a) Calculate the sale price of Model A.
[1 mark]
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(b) Calculate the sale price of Model B.
[1 mark]
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(c) Compare which headphone has the greater percentage reduction. Show your working.
[1 mark]
Show mark scheme
- (a) £68
- (b) £102
- (c) Model A: 15% reduction
- (c) Model B: 15% reduction
- (c) Conclusion that both have the same percentage reduction (accept equivalent working showing both equal 15%)
M5.3 · Venn diagrams and set notation
Explain — 2 marks
A teacher surveys 30 students about their favourite subjects. The Venn diagram shows the sets: M = students who like Mathematics and E = students who like English. The numbers in each region represent the number of students.
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(a) One student is chosen at random from the class. Explain what the notation P(M ∪ E) represents in this context.
[1 mark]
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(b) There are 12 students who like Mathematics, 15 who like English, and 7 who like both. Explain why the number of students who like neither subject is 10.
[1 mark]
Show mark scheme
- (a) The probability of selecting a student who likes Mathematics OR English (or both)
- (b) 12 + 15 − 7 = 20 students like at least one subject, so 30 − 20 = 10 like neither
M5.1 · Basic probability and sample space
Explain — 2 marks
A fair spinner is divided into 8 equal sectors numbered 1 to 8.
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(a) Write down all the possible outcomes when the spinner is spun once.
[1 mark]
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(b) Explain why the probability of landing on a number greater than 6 is 1/4.
[1 mark]
Show mark scheme
- (a) 1, 2, 3, 4, 5, 6, 7, 8 (accept any order)
- (b) There are 2 numbers greater than 6 (7 and 8) out of 8 possible outcomes
- (b) 2/8 = 1/4
M2.1 · Algebraic manipulation
Explain — 4 marks
A student is investigating the motion of a toy car rolling down a ramp. They measure the distance travelled and the time taken. To find the average speed, they need to rearrange and use the equation: distance = speed × time, or d = s × t
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(a) The equation for distance is d = s × t, where d is distance in metres, s is speed in m/s, and t is time in seconds. Rearrange this equation to make speed (s) the subject.
[1 mark]
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(b) The toy car travels 2.4 metres in 3 seconds down the ramp. Use your rearranged equation from part (a) to calculate the average speed of the car. Show your working.
[2 marks]
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(c) Explain why it is important to rearrange the equation before substituting numerical values into it.
[1 mark]
Show mark scheme
- (a) Correctly rearranges to s = d / t or s = d ÷ t (1 mark)
- (b) Correctly substitutes values: s = 2.4 / 3 or s = 2.4 ÷ 3 (1 mark)
- (b) Correct final answer: s = 0.8 m/s (1 mark)
- (c) Explains that rearranging ensures the unknown quantity is isolated / makes the calculation clearer / avoids confusion about which value to divide or multiply (1 mark)
M5.2 · Combined events and tree diagrams
Explain — 2 marks
A fair spinner has 4 equal sections numbered 1, 2, 3 and 4. The spinner is spun twice. The tree diagram below shows all possible outcomes.
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(a) Write down the probability of getting a 3 on the first spin.
[1 mark]
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(b) Explain why there are 16 possible outcomes shown on the tree diagram.
[1 mark]
Show mark scheme
- (a) 1/4 or 0.25
- (b) Because there are 4 outcomes for the first spin AND 4 outcomes for the second spin, and 4 × 4 = 16 (or equivalent)
M1.1 · Operations with integers, decimals, fractions
Describe — 4 marks
A physics student is investigating the density of different materials. They measure the mass of a copper block as 178.5 g and calculate its volume using water displacement. The volume reading changes from 15.2 cm³ to 35.7 cm³ when the block is submerged.
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(a) Calculate the volume of the copper block in cm³. Show your working.
[1 mark]
-
(b) The student needs to convert the mass to kilograms. Describe the process of converting 178.5 g to kilograms as a decimal.
[1 mark]
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(c) Using your answers from parts (a) and (b), describe the steps you would use to calculate the density of copper in kg/cm³. You do not need to give a final numerical answer.
[2 marks]
Show mark scheme
- (a) Subtract 15.2 from 35.7 to get 20.5 cm³ (or equivalent correct subtraction of decimals)
- (b) Divide 178.5 by 1000 (or move the decimal point three places to the left) to convert grams to kilograms, giving 0.1785 kg
- (c) Divide the mass in kilograms (0.1785) by the volume in cm³ (20.5)
- (c) Use the formula density = mass ÷ volume (or equivalent statement showing the correct operation with the values from parts a and b)
GCSE Perfect