IGCSE Mathematics Paper 2 — Unit Assessment Mock 2 — Sets, Percentages, Time, Money

Information for the whole question

ξ = {x : x is an integer and 20 ≤ x ≤ 35}
A = {x : x is a multiple of 4}
B = {x : x divides 120 exactly}

(a)(i) [1]

List the elements of A ∩ B.

Model Answer — (a)(i)

A = {20, 24, 28, 32} and B = {20, 24, 30}

B1 20, 24

⚠ If you missed marks here: the two elements must be in both lists. 30 divides 120 exactly but is not a multiple of 4; 28 and 32 are multiples of 4 but do not divide 120 exactly (120 ÷ 28 and 120 ÷ 32 are not whole numbers). Write both sets out in full first — guessing at the intersection is where the marks go.

B1 — 20 and 24, and nothing else

(a)(ii) [2]

Find n((A ∪ B)′).

Model Answer — (a)(ii)

A ∪ B = {20, 24, 28, 30, 32}

M1 n(A ∪ B) = 5, and n(ξ) = 16 because ξ runs from 20 to 35 inclusive

A1 11

⚠ If you missed marks here: two traps here. Answering 16 gives n(ξ), not the complement. And n(ξ) is 16, not 15 or 25 — counting inclusively from 20 to 35 gives 35 − 20 + 1 numbers. Elements in both A and B must be counted once only in the union.

M1 — n(A ∪ B) = 5, or the union listed, with n(ξ) = 16 A1 — 11

Information for the whole question

52 people were asked which of three apps they use: music (M), news (N) and photos (P). In the Venn diagram below, every one of the eight regions is labelled with how many people it contains. The letter x stands for a whole number.

(b)(i) [3]

Altogether 52 people were asked. Write down an equation in x for the total, and solve your equation to find the value of x.

Model Answer — (b)(i)

M1 adds all eight regions: 9 + 7 + 6 + x + 4 + 2x + 3 + 5

M1 34 + 3x = 52, so 3x = 18

A1 x = 6

Check: the eight regions become 9, 7, 6, 6, 4, 12, 3, 5 and they total 52 ✓

⚠ If you missed marks here: the two x terms are not the same size — one region holds x and another holds 2x, so they collect to 3x, not 2x. Missing the 5 outside all three circles is the other common slip: a person who uses none of the apps is still one of the 52.

M1 — all eight regions added M1 — 34 + 3x = 52 or 3x = 18 A1 — x = 6

(b)(ii) [2]

Find n(N ∩ M′).

Model Answer — (b)(ii)

This is the part of N that lies outside M: the news-only region and the news-and-photos-only region.

M1 7 + 2x = 7 + 12

A1 19

For comparison, n(N) = 7 + 6 + 12 + 3 = 28, which is a different question.

⚠ If you missed marks here: answering 7 is reading one number off the diagram. answering 28 is giving n(N) and forgetting the ∩ M′ part. Shade the region on the diagram before you add anything — M′ removes every region that touches circle M, including the centre.

M1 — the two correct regions identified (7 and 2x) A1 — 19

(b)(iii) [2]

One of the 52 people is chosen at random. Find the probability that this person uses exactly two of the three apps. Give your answer as a fraction in its simplest form.

Model Answer — (b)(iii)

M1 exactly two means the three overlap-only regions: x + 4 + 2x = 6 + 4 + 12 = 22

A1 11/26

22 out of 52, and both divide by 2.

⚠ If you missed marks here: the centre region, 3, is not included — those people use all three, not exactly two. Including it gives 25/52, which will not simplify at all, and that is your warning sign. Always cancel the fraction: an uncancelled 22/52 loses the accuracy mark.

M1 — 22 seen, or the three overlap-only regions identified A1 — 11/26

(a) [3]

A charity’s annual income fell by 12.5% last year. Last year its income was £294 000. Calculate its income in the year before that.

Model Answer — (a)

M1 £294 000 is 100% − 12.5% = 87.5% of the earlier income

M1 294 000 ÷ 0.875, and by hand 87.5% = ⁷⁄₈, so this is 294 000 × 8 ÷ 7

A1 £336 000

Check forwards: 12.5% of 336 000 = 42 000, and 336 000 − 42 000 = 294 000 ✓

⚠ If you missed marks here: adding 12.5% of 294 000 gives 330 750, which is wrong — the 12.5% was a share of the earlier, larger income. Notice 12.5% = ¹⁄₈, so 87.5% = ⁷⁄₈: dividing by ⁷⁄₈ is multiplying by ⁸⁄₇, which makes this a non-calculator question.

M1 — 87.5% (or 0.875, or ⁷⁄₈) linked to £294 000 M1 — divides rather than adding 12.5% on A1 — £336 000

Information for the whole question

A shop records its takings for three weeks.

Week 1: £2400 · Week 2: £2760 · Week 3: £2415

(b)(i) [2]

Show that the takings fell by 12.5% from week 2 to week 3.

Model Answer — (b)(i)

M1 2760 − 2415 = 345, then 345 ÷ 2760

A1 12.5% — since 2760 = 8 × 345, the fraction is ¹⁄₈ = 0.125, shown as required

⚠ If you missed marks here: a "show that" question is marked on the working, not on the answer — writing 12.5% on its own scores nothing. Divide by 2760, the week-2 figure, because that is what the fall is a fall from.

M1 — 345 ÷ 2760 seen A1 — reaches 0.125 or 12.5% with the working shown

(b)(ii) [2]

Calculate the overall percentage change in the takings from week 1 to week 3. State clearly whether it is an increase or a decrease.

Model Answer — (b)(ii)

M1 2415 − 2400 = 15, then 15 ÷ 2400 × 100

A1 0.625% increase

By hand: 15 ÷ 2400 = 1 ÷ 160 = 0.00625

⚠ If you missed marks here: week 1 to week 2 is a 15% rise and week 2 to week 3 is a 12.5% fall, but the answer is not 15 − 12.5 = 2.5%. Those two percentages are shares of different amounts, so they cannot be subtracted. Compare week 3 with week 1 directly, or multiply the multipliers: 1.15 × 0.875 = 1.00625.

M1 — compares 2415 with 2400 (or multiplies 1.15 × 0.875) A1 — 0.625%, stated as an increase

A shop prices a camera at £252. At that price the shop would make a profit of 5% on the amount it paid for the camera. The camera does not sell, so the shop sells it for £216. Calculate the percentage loss on the amount the shop paid.

Model Answer — (c)

M1 the amount paid: 252 ÷ 1.05 = £240

M1 loss = 240 − 216 = £24, then 24 ÷ 240 × 100

A1 10%

By hand: 252 ÷ 1.05 = 25 200 ÷ 105 = 240 ✓

⚠ If you missed marks here: the tempting move is to compare £216 with £252 and answer 14.3%. But profit and loss are always measured against what the shop paid, and that is £240, which you have to find first. Two steps, and step one is a reverse percentage.

M1 — 252 ÷ 1.05 to reach the £240 cost price M1 — the loss divided by £240 (not by £252 or £216) A1 — 10%

Information for the whole question

A lorry drives 120 km at an average speed of 60 km/h. It then stops for 40 minutes. It then drives a further 90 km at an average speed of 45 km/h.

(a)(i) [2]

Show that the total time for the journey, including the stop, is 4 hours 40 minutes.

Model Answer — (a)(i)

M1 120 ÷ 60 = 2 hours and 90 ÷ 45 = 2 hours

A1 4 hours 40 minutes — 2 h + 2 h + 40 min, as required

⚠ If you missed marks here: both driving times must be shown for the method mark; a "show that" answer with no working scores nothing. Note that time = distance ÷ speed, so the speed goes on the bottom.

M1 — both driving times found (2 h and 2 h) A1 — adds the 40-minute stop and reaches 4 h 40 min

(a)(ii) [2]

The lorry leaves at 08:40. Find the time at which it arrives.

Model Answer — (a)(ii)

M1 08:40 + 4 hours = 12:40, then + 40 minutes

A1 13:20

Check backwards: 13:20 − 4 h 40 min = 08:40 ✓

⚠ If you missed marks here: adding the hours and the minutes in separate columns gives 12:80, which is not a time. Forty minutes past 12:40 crosses the hour: 20 minutes takes you to 13:00, and 20 more gives 13:20.

M1 — 4 h 40 min added to 08:40 A1 — 13:20

(a)(iii) [3]

Calculate the average speed for the whole journey, including the stop. Give your answer in km/h.

Model Answer — (a)(iii)

M1 total distance = 120 + 90 = 210 km

M1 total time = 4 h 40 min = 4²⁄₃ hours = ¹⁴⁄₃ hours, then 210 ÷ ¹⁴⁄₃ = 210 × 3 ÷ 14

A1 45 km/h

Check: 45 × ¹⁴⁄₃ = 15 × 14 = 210 km ✓

⚠ If you missed marks here: averaging the two speeds gives (60 + 45) ÷ 2 = 52.5 km/h and is wrong, even though the two driving times happen to be equal — because the 40-minute stop is still part of the journey. Average speed is always total distance ÷ total time, and every minute counts as time, moving or not. The other slip is writing 4 h 40 min as 4.4 hours; it is 4²⁄₃, or 4.666... hours.

M1 — total distance 210 km M1 — divides by the total time as a number of hours (¹⁴⁄₃, not 4.4) A1 — 45 km/h

(b) [3]

A late-running airport shuttle leaves at 11:40 pm. The journey takes 2 hours 45 minutes. Write down the departure time using the 24-hour clock, and find the arrival time using the 24-hour clock. State the day on which the shuttle arrives.

Model Answer — (b)

B1 departure is 23:40 in the 24-hour clock

M1 23:40 + 20 minutes = 00:00, then 2 hours 25 minutes more

A1 arrives 02:25 on the next day

Total check: 23:40 plus 2 h 45 min is 26:25, and 26:25 − 24:00 = 02:25 the next day ✓

⚠ If you missed marks here: writing 11:40 pm as 11:40 turns a departure just before midnight into one just before noon — pm times from 1 pm onwards need 12 added, so 11:40 pm is 23:40. The other error is answering 25:85, or 02:25 without saying it is the next day. When a time passes 24:00, subtract 24 hours and move the date on.

B1 — 23:40 M1 — adds 2 h 45 min, handling the crossing of midnight A1 — 02:25, stated as the next day

(a) [4]

A trader buys 40 identical chairs. Each chair costs £22, and the delivery charge for the whole order is £120. He sells all 40 chairs and makes a profit of 25% on his total cost. Calculate the selling price of one chair.

Model Answer — (a)

M1 total cost = 40 × 22 + 120 = 880 + 120 = £1000

M1 total income = 1000 × 1.25 = £1250

M1 1250 ÷ 40

A1 £31.25

Check: 40 × £31.25 = £1250, a profit of £250 on £1000, which is 25% ✓

⚠ If you missed marks here: adding 25% to the £22 chair price gives £27.50 and leaves the delivery out completely. The 25% is on the whole cost, delivery included, so work out the total cost first, add the profit to the total, and only divide by 40 at the very end.

M1 — total cost £1000, delivery included M1 — increases the total cost by 25% M1 — divides the total income by 40 A1 — £31.25

(b) [4]

Paint is sold in three tins.

    Tin A: 2.5 litres for £18
    Tin B: 4 litres for £32, with 25% off
    Tin C: 750 ml for £6

Work out which tin gives the best value for money. You must show your working.

Model Answer — (b)

M1 a correct cost per litre for any one tin

M1 applies the discount: £32 × 0.75 = £24, and converts 750 ml to 0.75 litres

A1 Tin A £7.20/litre, Tin B £6.00/litre, Tin C £8.00/litre

A1 Tin B is the best value

By hand: 18 ÷ 2.5 = 7.20; 24 ÷ 4 = 6.00; 6 ÷ 0.75 = 8.00.

⚠ If you missed marks here: two traps sit in this one question. 750 ml is 0.75 litres, not 7.5 — there are 1000 ml in a litre. And Tin B is only better once the 25% is taken off; at the full £32 it costs £8.00 per litre, the same as Tin C. Put every tin into the same units and the same quantity before comparing anything.

M1 — one correct cost per litre M1 — the 25% discount applied and 750 ml converted to 0.75 litres A1 — all three rates correct: £7.20, £6.00, £8.00 per litre A1 — states Tin B, supported by the comparison

A restaurant bill is £84. A service charge of 12.5% is added to the bill. The total is then shared equally between 6 people. Calculate how much each person pays.

Model Answer — (c)

M1 12.5% of 84 = £10.50, so the total is £94.50 (or 84 × 1.125)

A1 £15.75

By hand: 10% of 84 = 8.40, and 2.5% is a quarter of that, 2.10, so the charge is £10.50. Then 94.50 ÷ 6 = 15.75 ✓

⚠ If you missed marks here: dividing £84 by 6 first and then adding 12.5% gives the same answer here, but sharing before adding a charge will not always work — add the charge to the bill, then share. The other slip is reading 12.5% as £12.50.

M1 — the total bill £94.50 found (or 12.5% of 84 = £10.50) A1 — £15.75

IGCSE Mathematics Paper 2 (Extended) — non-calculator

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