Sample paper · GCSE Foundation · grades 1–5
GCSE Foundation sample Paper 2 (calculator)
The real Paper 2 is 1.5 hour 30 minutes and 80 marks, calculator, and all three papers carry equal weight. This sample is 20 original questions in the same content proportions as the Foundation qualification — number 25%, algebra 20%, ratio, proportion and rates of change 25%, geometry and measures 15%, probability 7.5%, statistics 7.5% — with a calculator allowed. AO1 / AO2 / AO3 at this tier: 50% / 25% / 25%.
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Answer key: GCSE Foundation sample Paper 2 (calculator)
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- (c) 5/8 — Method: write the decimal over 1000 using its three decimal places, then simplify. Working: 0.625 = 625/1000 = 5/8 (dividing both numerator and denominator by 125). Answer: 5/8. 25/4 comes from writing the decimal over 100 instead of 1000, as if there were only two decimal places. 31/50 comes from rounding 0.625 to 0.62 before converting. 8/5 comes from simplifying correctly to 5/8 and then writing the fraction upside down.
- (d) 2x + 4 — Method: collect the like terms, which here are the two terms in x, and leave the number term on its own because a number and a term in x are unlike. Working: x + x is one lot of x added to one more lot of x, which is 2x; the 4 has nothing like it to join with, so it is written after the 2x. Answer: 2x + 4. The distractors: x² + 4 comes from multiplying the two x terms instead of adding them; x + 4 comes from treating x + x as a single x, as though the repeated letter counted only once; 6x comes from collecting the unlike terms together, adding 1 + 1 + 4 and attaching the letter to that total.
- (c) 3 hours — Time = distance ÷ speed, so 210 ÷ 70 = 3 hours. Working out 210 − 70 = 140 subtracts the speed from the distance instead of dividing. Working out 210 + 70 = 280 adds the two values together, which does not give a time. Working out 210 × 70 = 14700 multiplies distance and speed together instead of dividing. The journey takes 3 hours.
- (c) a square — A cube has six identical square faces, and looking at it from directly above, directly from the front, or directly from the side each shows one of these square faces face-on, undistorted — so all three views are squares of the same size. "a triangle" would be the plan or elevation of a solid such as a pyramid or cone, not a cube. "a circle" belongs to a sphere or a cylinder viewed along its axis, not a cube. "a rectangle that is not a square" would appear if the cube's edges were not all equal, which is not true of a cube.
- (d) 15 — Method: each of Spinner A's outcomes can be paired with each of Spinner B's outcomes, so the two counts are combined by multiplying, not adding. Working: Spinner A has 3 outcomes and Spinner B has 5, so there are 3 × 5 = 15 equally likely outcomes in total. Answer: 15. Watch out: adding the two counts, 3 + 5 = 8, undercounts drastically — every one of Spinner A's 3 outcomes pairs with all 5 of Spinner B's outcomes, not just one each. Writing down 3 counts Spinner A alone and leaves Spinner B out completely. And writing down 25 comes from squaring Spinner B's own outcome count, 5 × 5, and forgetting Spinner A altogether.
- (b) A person's shoe size and their favourite colour — A person's shoe size is not linked to which colour they prefer, so these two show no correlation. The other three pairs are all genuinely correlated: distance travelled and fuel used rise together, which is positive correlation; hours of revision and test score generally rise together, which is also positive correlation; and as outdoor temperature rises, fewer woolly hats are sold, which is negative correlation. Negative correlation is still a real relationship between two variables — it is not the same thing as no relationship at all, so the temperature and hats pair is not the answer to this question.
- (a) 1/4 — First find the fraction of the people who are children: the pool is 3 + 5 = 8 equal shares and the children take 5, so the children are 5/8 of the people. The boys are 2/5 of that, and 'of' means multiply: 2/5 × 5/8 = 10/40 = 1/4. Answering 2/5 gives the boys as a fraction of the children only, 3/20 multiplies by the adults' fraction 3/8 instead of the children's, and 3/5 is the fraction of the children who are girls.
- (b) 6 — Method: for any point that lies on y = k/x, the value of k is found by multiplying the x-coordinate and the y-coordinate together, since k = x × y. Working: k = 2 × 3 = 6. Answer: k = 6. Distractor refutation: 1.5 comes from dividing the y-coordinate by the x-coordinate instead of multiplying them. 5 comes from adding the two coordinates instead of multiplying them. 9 comes from misreading the point's x-coordinate as 3 instead of 2, then multiplying 3 × 3.
- (c) 180 pages — The gradient is (9 − 24) ÷ (300 − 0) = −15 ÷ 300 = −0.05, so the cartridge uses 0.05 ml of ink per page. At p = 300 there are 9 ml left. The extra pages before the cartridge is empty is 9 ÷ 0.05 = 180 pages. Giving 9 as the answer confuses the millilitres of ink remaining with the number of pages remaining — they are different quantities with different units. Multiplying instead of dividing, 9 × 0.05 = 0.45, does not undo the rate correctly. Working out the total number of pages a full cartridge lasts, 24 ÷ 0.05 = 480 pages, answers how many pages the cartridge prints in total from full, not how many more pages it can print from the 300-page point.
- (a) the lens-shaped region where the two circles overlap — The region within 30 km of the ship is a circle of radius 30 km around the ship; the region within 20 km of the lighthouse is a circle of radius 20 km around the lighthouse. Since 30 + 20 = 50 km is more than the 40 km between them, and 30 − 20 = 10 km is less than 40 km, the two circles genuinely overlap, in a lens-shaped region where both conditions hold at once. "the whole smaller circle, since it fits inside the larger one" would only be true if the 40 km distance between the centres were small enough for one circle to swallow the other, which it is not here. "no overlap — 30 km and 20 km don't add to 40 km" wrongly assumes the two radii must add up to exactly the distance between the centres for an overlap to exist; any total greater than that distance is enough. "a straight strip, exactly halfway between the two" confuses this with an equidistant locus, which is a different problem from the one asked here.
- (b) 1/12 — Method: the coin does not affect the dice, so the two events are independent and the probability that both happen is the product of their probabilities. Working: heads has probability 1/2 and a 6 on an ordinary dice has probability 1/6. Multiplying gives 1 on the top and 2 × 6 = 12 on the bottom. Answer: the probability is 1/12. The distractors: 2/3 comes from adding 1/2 and 1/6 instead of multiplying them; 1/6 comes from using the dice alone and ignoring the condition on the coin; 1/8 comes from counting the possible results as 6 + 2 = 8 and treating the winning result as one of those eight.
- (c) −2 — On a number line, negative numbers get smaller as their size (ignoring the sign) gets bigger, so −2 is closest to zero and is the largest of the four. A candidate who ignores the negative signs and orders the numbers as if they were positive, largest digit first, would pick −9 as the 'largest'. Continuing that same reversed ranking, the next number in digit order is −7. The number one step further along that same reversed ranking is −4, still short of the true largest value, −2.
- (d) y = (x + 3)(x − 5) — A root at x = −3 means the matching bracket must be zero when x = −3, so the bracket is (x − (−3)) = (x + 3). A root at x = 5 means the other bracket is (x − 5). So the equation is y = (x + 3)(x − 5); checking the y-intercept, (0 + 3)(0 − 5) = 3 × (−5) = −15, which matches the given value. The option (x − 3)(x + 5) swaps the signs of both roots. The option (x + 3)(x + 5) keeps the correct sign for the first root but gets the second wrong. The option (x − 3)(x − 5) gets the first root's sign wrong.
- (d) 4:1 — Write the ratio online : in-store using the numbers in the question: 180 : 45. Divide both parts by their highest common factor, 45, to give 4 : 1. (1:4 comes from writing the ratio the wrong way round, in-store : online. 3:1 comes from subtracting the in-store orders from the online orders, 180 − 45 = 135, and comparing that to the in-store orders, 135:45, instead of dividing. 4:5 comes from comparing the online orders to the total number of orders, 180:225.)
- (c) The scale factor is 2, not 1, so the sides are not equal — Congruent shapes must be exactly the same size as well as the same shape, which means a scale factor of 1. Here the scale factor between the triangles is 2, so the sides are different lengths and the triangles cannot be congruent, even though they are similar. 'Similar triangles are never congruent' is too strong — a scale factor of exactly 1 would make them both similar and congruent. 'The angles are not necessarily equal' is wrong, since similar triangles always have equal matching angles. 'Congruent triangles must have a right angle' is an unrelated, false fact about congruence.
- (c) 600 — Method: round each number to the nearest 100, then add the rounded values. Working: 113 is nearer to 100 than to 200, so it rounds to 100; 491 is nearer to 500 than to 400, so it rounds to 500; adding those gives the estimate. Answer: 600. The distractors: 500 comes from rounding each number down to the hundred below instead of to the nearest hundred, giving 100 + 400; 700 comes from rounding each number up to the hundred above, giving 200 + 500; 604 is the exact total, worked out in full when the question asks for an estimate.
- (c) 4x − 3 — Subtract each term inside the bracket: 5x − (x + 3) = 5x − x − 3 = 4x − 3. Writing 4x + 3 comes from subtracting the x but not the 3, keeping its sign positive. Writing 6x − 3 comes from adding the x term, 5x + x = 6x, instead of subtracting it. Writing 5x − 3 comes from ignoring the x term inside the bracket and only subtracting the 3.
- (b) 2 : 5 — 40% as a fraction is 40/100 = 2/5, so a = (2/5)b, giving a : b = 2 : 5. Swapping the two numbers gives 5 : 2, which is the ratio b : a instead. Treating 40% as the fraction of the total (a + b) rather than of b alone gives a : (a + b) = 2 : 5, which rearranges to a : b = 2 : 3 — a different statement from the one in the question. Inverting that same mistaken ratio gives 3 : 2.
- (b) 12 — Method: round the number of pupils to the nearest 10, divide by the number of passengers each minibus can carry, then round up because a part-full minibus still needs a whole vehicle. Working: 179 rounds to 180 (nearest 10); 180 ÷ 16 = 11.25; 11 minibuses only carry 176 passengers, so a 12th minibus is needed for the rest. Answer: 12. 11 comes from rounding 11.25 to the nearest whole number in the usual way, without checking that the leftover pupils still need transporting. 10 comes from rounding 179 down to 170 instead of to the nearest 10, 180. 180 comes from stopping after rounding the number of pupils, without dividing by the number of passengers each minibus carries at all.
- (b) −1% — Method: write each change as a multiplier and multiply them. A 10% fall is × 0.9 and a 10% rise is × 1.1. Working: 0.9 × 1.1 = 0.99, so the final price is 99% of the original, which is 1% less. Answer: an overall change of −1%. The distractors: 0% comes from assuming a 10% fall and a 10% rise cancel — they do not, because the rise is 10% of a smaller amount; +1% has the size right but the sign wrong, from reading the multiplier 0.99 as 1% above 1 instead of 1% below it; −2% comes from finding the 1% fall and then counting it once for each of the two changes.
How the 20 questions are shared out
- Number — 5 questions (25% of the qualification)
- Algebra — 4 questions (20% of the qualification)
- Ratio, proportion and rates of change — 5 questions (25% of the qualification)
- Geometry and measures — 3 questions (15% of the qualification)
- Probability — 2 questions (7.5% of the qualification)
- Statistics — 1 question (7.5% of the qualification)
Where an area has fewer printable questions than its share, the shortfall is filled from the other areas. These are original questions, not past papers.