Paper 12 · May/June 2026Mathematics 0580

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Paper 12, worked through. Cambridge IGCSE Mathematics, May/June 2026: worked solutions

Mathematics 0580/12 · May/June 2026 · 29 questions · 80 marks

Do the paper first. Then come back for the ones that got you.

Eighty marks in an hour and a half, so just over a minute a mark. There is no calculator, which means the numbers have been chosen to come out neatly: if you meet an ugly one, look back for a slip before pressing on. Most questions are worth one or two marks, so a wrong answer with the working shown can still earn something. The formula list is on page 2 of the paper.

  1. 1Finding a square root.[1]
    The question as printed, from page 3 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. A square root asks: which number, multiplied by itself, makes this one?
    2. 9 × 9 = 81, so the square root of 81 is 9.

    Answer9

    Know the square numbers up to 15 × 15 by heart. On a non-calculator paper they turn up again and again.

    Revise this: Squares, cubes and roots
  2. 2(a)Adding two two-digit numbers.[1]
    The question as printed, from page 3 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. Add the easy round number first: 63 + 30 = 93.
    2. You added 2 too many, so take 2 off: 93 − 2 = 91.

    Answer91

    Or set it out in columns: 3 + 8 = 11, write 1 and carry 1. Then 6 + 2 + 1 = 9.

  3. 2(b)Finding 50% of an amount.[1]
    The question as printed, from page 3 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. 50% means 50 out of 100, which is a half.
    2. Half of 526 is 263.

    Answer263

    Learn the easy ones: 50% is a half, 25% is a quarter, 10% is a tenth, 1% is a hundredth.

    Revise this: Percentages
  4. 3(a)Turning a percentage into a decimal.[1]
    The question as printed, from page 3 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. "Per cent" means "out of 100", so divide by 100.
    2. 35 ÷ 100 = 0.35. Each digit moves two places to the right.

    Answer0.35

    Revise this: Percentages
  5. 3(b)Turning a fraction into a percentage.[1]
    The question as printed, from page 3 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. A percentage is a fraction with 100 on the bottom. Multiply the top and the bottom of 4/5 by 20.
    2. 4/5 = 80/100, which is 80%.

    Answer80

    Or use one fifth = 20%, so four fifths = 4 × 20% = 80%.

    Revise this: Percentages
  6. 4Estimating the area of a curved shape drawn on a grid of 1 cm squares.[1]
    The question as printed, from page 3 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. Each square of the grid is 1 cm². Count the squares that lie completely inside the shape: about seven.
    2. Then deal with the part squares. Pair them up into wholes, or count each one that is more than half inside and ignore the rest. They add about six more.
    3. 7 + 6 = 13.

    AnswerAbout 13 cm². Any answer from 11 to 14 is accepted.

    It is an estimate, so a range of answers gets the mark. Tick each square on the diagram as you count it, so that none is counted twice.

    Revise this: Area and perimeter
  7. 5(a)Rounding a whole number to the nearest hundred.[1]
    The question as printed, from page 4 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. Find the hundreds digit in 65 432: it is the 4. The answer is either 65 400 or 65 500.
    2. Look at the digit after it, the 3. It is less than 5, so round down.

    Answer65 400

    Keep the zeros. They hold the other digits in their places: 654 is a completely different number.

    Revise this: Rounding, estimating and bounds
  8. 5(b)Rounding a decimal to three decimal places.[1]
    The question as printed, from page 4 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. Decimal places are counted from the decimal point. In 0.04857 the third one is the 8.
    2. The next digit is 5. Five or more rounds up, so the 8 becomes 9.

    Answer0.049

    Decimal places are not significant figures. To three significant figures this number would be 0.0486, which is a different answer.

    Revise this: Rounding, estimating and bounds
  9. 6(a)Placing one pair of brackets to make a calculation correct.[1]
    The question as printed, from page 4 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. Without brackets, the multiplications are done first: 7 × 3 = 21, 2 × 4 = 8, and 21 − 8 = 13. That is not 28.
    2. Brackets are worked out before anything else, so try them around the subtraction: 3 − 2 = 1.
    3. 7 × 1 × 4 = 28. It works.

    Answer7 × (3 − 2) × 4 = 28

    Only one pair is allowed. A second pair, even a harmless one, scores nothing.

    Revise this: Fractions, decimals and the order of operations
  10. 6(b)Placing one pair of brackets in a calculation with a division.[1]
    The question as printed, from page 4 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. Without brackets: 80 ÷ 8 = 10, then 9 − 10 − 6 = −7. Not 5.
    2. To reach 5 from 9 you need to take away 4, and 10 − 6 = 4. So put the brackets around everything after the first minus sign.
    3. 9 − (80 ÷ 8 − 6) = 9 − (10 − 6) = 9 − 4 = 5.

    Answer9 − (80 ÷ 8 − 6) = 5

    Inside the brackets the usual order still applies: divide before you subtract.

    Revise this: Fractions, decimals and the order of operations
  11. 7(a)Writing the shaded part of a rectangle as a fraction in its simplest form.[2]
    The question as printed, from page 4 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. Count all the small squares: 12 across and 4 down, so 12 × 4 = 48.
    2. Count the shaded ones, row by row: 6 + 6 + 5 + 5 = 22.
    3. The fraction shaded is 22/48. Divide the top and the bottom by 2: 11/24.
    4. 11 is a prime number and does not go into 24, so it cannot be simplified any further.

    Answer11/24

    22/48, or any other fraction equal to it, earns one of the two marks. Always check whether a fraction will cancel.

    Revise this: Fractions, decimals and the order of operations
  12. 7(b)Shading more squares so that a given fraction of the rectangle is left unshaded.[1]
    The question as printed, from page 4 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. One sixth of the rectangle is to stay white. 48 ÷ 6 = 8 squares.
    2. At the moment 48 − 22 = 26 squares are white.
    3. Shade 26 − 8 = 18 more, in any position.

    AnswerShade 18 more squares, so that 8 are left unshaded.

    Read the bold word. The fraction given is the part that is not shaded. Check by counting the white squares at the end: there must be exactly 8.

    Revise this: Fractions, decimals and the order of operations
  13. 8Finding the area of a triangle from its base and its height.[1]
    The question as printed, from page 5 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. Area of a triangle = 1/2 × base × height. The height is the one at right angles to the base.
    2. 1/2 × 10 × 6 = 30.

    Answer30 cm²

    The formula is at the front of the paper. 60 is the area of the rectangle around the triangle: do not forget to halve it.

    Revise this: Area and perimeter
  14. 9(a)Counting values in a range from a stem-and-leaf diagram.[1]
    The question as printed, from page 5 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. Use the key: the stem is the tens digit and each leaf is a units digit. So the row with stem 1 holds 11, 13, 15, 16 and 19.
    2. Times between 10 and 30 minutes are the rows with stems 1 and 2.
    3. Count the leaves: 5 in the first of those rows and 8 in the second. 5 + 8 = 13.

    Answer13

    Revise this: Charts and scatter diagrams
  15. 9(b)Finding the percentage of values below a given number.[1]
    The question as printed, from page 5 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. List the times below 18 minutes: 7, 8, 9, 11, 13, 15 and 16. That is 7 students.
    2. There are 20 students. 7/20 = 35/100 (multiply the top and the bottom by 5).

    Answer35

    19 is not less than 18, so it is not counted. Twenty students makes each one worth 5%.

    Revise this: Charts and scatter diagrams
  16. 9(c)Finding the gap between the largest and the smallest value.[1]
    The question as printed, from page 5 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. The slowest student is the last leaf in the diagram: 39 minutes.
    2. The quickest is the first: 7 minutes.
    3. 39 − 7 = 32.

    Answer32

    This difference is called the range. A stem-and-leaf diagram is already in order, so the first and last values are the smallest and largest.

    Revise this: Data, averages and range
  17. 10(a)Writing an equation from two expressions that are equal.[1]
    The question as printed, from page 6 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. Phil's number of cars and Ann's number of cars are the same, so put an equals sign between the two expressions.

    Answer7y − 52 = 78 − 6y

    An equation must have an equals sign. Write it exactly as given: there is no need to tidy it up in this part.

    Revise this: Solving linear equations
  18. 10(b)Solving an equation with the unknown on both sides, then using the solution.[3]
    The question as printed, from page 6 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. Get the y terms onto one side. Add 6y to both sides: 13y − 52 = 78.
    2. Get the numbers onto the other side. Add 52 to both sides: 13y = 130.
    3. Divide both sides by 13: y = 10.
    4. The question asks for the number of cars, so put y = 10 back into either expression: 7 × 10 − 52 = 18.
    5. Check with the other one: 78 − 6 × 10 = 18. They agree.

    Answer18

    Stopping at y = 10 earns two of the three marks. Read what the answer line is asking for. Using both expressions gives you a free check.

    Revise this: Solving linear equations
  19. 11(a)Changing kilograms to grams.[1]
    The question as printed, from page 6 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. "Kilo" means a thousand: 1 kg = 1000 g.
    2. 2 × 1000 = 2000.

    Answer2000

    Revise this: Units of measurement
  20. 11(b)Changing square centimetres to square millimetres.[1]
    The question as printed, from page 6 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. 1 cm = 10 mm. A square that is 1 cm by 1 cm is 10 mm by 10 mm, so 1 cm² = 10 × 10 = 100 mm².
    2. 7 × 100 = 700.

    Answer700

    For areas the conversion number is squared. 70 is the answer you get from multiplying by 10 only once.

    Revise this: Units of measurement
  21. 12(a)Naming the solid that a net folds into.[1]
    The question as printed, from page 7 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. The net is made of six rectangles, in three matching pairs: two that are 2 by 4, two that are 3 by 4 and two that are 2 by 3.
    2. Six rectangular faces in three pairs fold up into a box shape.

    AnswerCuboid

    A cube has six square faces. These faces are rectangles, so it is a cuboid.

    Revise this: Shapes and their names
  22. 12(b)Finding the corners of a net that come together when it is folded.[2]
    The question as printed, from page 7 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. Fold it in your head one edge at a time. Two edges that join must be the same length.
    2. X is at the end of a 2 cm edge: the top of the right-hand rectangle. That edge folds up against the 2 cm right-hand end of the rectangle just above it, the one 4 wide and 2 tall. So X lands on that rectangle's top right corner: 2 squares to the left of X and 2 squares up.
    3. Now the 4 cm top edge of that rectangle folds against the 4 cm right-hand side of the big rectangle at the top of the net. The same corner travels along it to the big rectangle's top right corner: 6 squares to the left of X and 6 squares up.

    AnswerThe top right corner of the 4 by 2 rectangle (2 left and 2 up from X), and the top right corner of the large rectangle at the top of the net (6 left and 6 up from X).

    Three faces meet at every corner of a cuboid, which is why there are exactly two other points. An extra X in a wrong place costs a mark.

    Revise this: Shapes and their names
  23. 13(a)Drawing a shape that is mathematically similar to a given trapezium.[1]
    The question as printed, from page 8 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. Similar means the same shape but a different size: every length is multiplied by the same number.
    2. Read T from the grid. Its parallel sides are 2 cm and 4 cm, they are 2 cm apart, and one of the other sides is at right angles to them.
    3. Double everything: parallel sides of 4 cm and 8 cm, 4 cm apart, with the right angles kept. The sloping side then goes 4 across and 4 down.

    AnswerFor example, a trapezium with parallel sides 4 cm and 8 cm, 4 cm apart, with right angles at one end. Halving T works too: sides 1 cm and 2 cm, 1 cm apart.

    It must not be the same size as T: that would be congruent, not "different". Multiply every length, including the height. Stretching it in one direction only does not give a similar shape.

    Revise this: Similar and congruent shapes
  24. 13(b)Drawing a rectangle with the same area as the trapezium.[2]
    The question as printed, from page 8 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. Find the area of T first. Area of a trapezium = 1/2 × (sum of the parallel sides) × the distance between them.
    2. 1/2 × (2 + 4) × 2 = 6 cm². You can check by counting: a 2 by 2 square (4) plus a triangle that is half of a 2 by 2 square (2).
    3. Any rectangle with an area of 6 cm² will do: 2 cm by 3 cm, or 1 cm by 6 cm.

    AnswerA rectangle 2 cm by 3 cm (or 1 cm by 6 cm). Its area is 6 cm².

    One mark is for a correct method for the area of the trapezium, so write that working down even though the answer is a drawing.

    Revise this: Area and perimeter
  25. 14Sharing a total in a given ratio.[2]
    The question as printed, from page 9 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. Add the parts of the ratio: 7 + 3 = 10 parts altogether.
    2. Find one part: 20 ÷ 10 = 2 pieces of fruit.
    3. Apples are 7 parts: 7 × 2 = 14.

    Answer14

    Check: oranges are 3 × 2 = 6, and 14 + 6 = 20. Make sure you give the one that was asked for.

    Revise this: Ratio and proportion
  26. 15Giving two geometrical properties of a parallelogram.[2]
    The question as printed, from page 9 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. A parallelogram is a four-sided shape with two pairs of parallel sides. Write down two facts that are true of every one.
    2. Sides: opposite sides are parallel, and opposite sides are equal in length.
    3. Angles and more: opposite angles are equal, the diagonals cut each other in half, and it has rotational symmetry of order 2.

    AnswerAny two, for example: opposite sides are parallel; opposite sides are equal in length.

    The fact has to be true for all parallelograms. "All four sides are equal" is only true of a rhombus, and "all angles are 90°" only of a rectangle.

    Revise this: Shapes and their names
  27. 16Finding the reciprocal of a decimal.[2]
    The question as printed, from page 9 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. The reciprocal of a number is 1 divided by it. For a fraction, that means turning it upside down.
    2. Write 0.8 as a fraction: 8/10, which simplifies to 4/5.
    3. Turn it over: 5/4. As a decimal that is 1.25.

    Answer1.25 (5/4 and 1 1/4 are also accepted).

    Check: a number times its reciprocal is always 1, and 0.8 × 1.25 = 1.

    Revise this: Fractions, decimals and the order of operations
  28. 17(a)Finding the third angle of a triangle.[1]
    The question as printed, from page 9 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. The three angles of a triangle add up to 180°.
    2. 180 − 100 − 30 = 50.

    Answer50°

    Revise this: Angles
  29. 17(b)Finding an interior angle of a regular ten-sided polygon.[2]
    The question as printed, from page 9 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. Start with the exterior angles: they always add up to 360°. A regular polygon has them all equal, so each one is 360 ÷ 10 = 36°.
    2. An interior angle and its exterior angle sit on a straight line, so they add up to 180°.
    3. Interior angle = 180 − 36 = 144°.

    Answer144°

    The other way: the interior angles add up to (10 − 2) × 180 = 1440°, and 1440 ÷ 10 = 144°. The exterior angle route has easier arithmetic with no calculator.

    Revise this: Angles in polygons
  30. 18(a)Collecting like terms.[2]
    The question as printed, from page 10 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. Like terms have exactly the same letter. Collect the x terms: 3x + 5x = 8x.
    2. Collect the y terms, keeping the sign in front of each: −2y + 7y = 5y.

    Answer8x + 5y

    The sign belongs to the term that follows it. An x term and a y term cannot be combined, so 13xy is wrong.

    Revise this: Expanding and factorising
  31. 18(b)Multiplying out two brackets and simplifying.[2]
    The question as printed, from page 10 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. Multiply each term in the first bracket by each term in the second. That gives four products.
    2. 3x × x = 3x². 3x × −7 = −21x. 5 × x = 5x. 5 × −7 = −35.
    3. Collect the two x terms: −21x + 5x = −16x.

    Answer3x² − 16x − 35

    One mark is for the four products with at least three correct, so write them out before simplifying. A positive times a negative is negative.

    Revise this: Expanding and factorising
  32. 19(a)Reading the gradient from the equation of a straight line.[1]
    The question as printed, from page 10 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. A straight line written as y = mx + c has gradient m, the number multiplying x.
    2. Here that number is 3.

    Answer3

    The 7 is where the line crosses the y-axis. Write the gradient as 3, not 3x.

    Revise this: Equations of straight lines
  33. 19(b)(i)Finding the y-coordinate of a point on the line.[1]
    The question as printed, from page 10 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. A point on the line fits its equation. P has x = 4, so put 4 in place of x.
    2. y = 3 × 4 + 7 = 19.

    Answerp = 19

    Revise this: Equations of straight lines
  34. 19(b)(ii)Finding the x-coordinate of a point on the line.[2]
    The question as printed, from page 10 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. This time the y-coordinate is known. Put −11 in place of y: −11 = 3w + 7.
    2. Subtract 7 from both sides: −18 = 3w.
    3. Divide both sides by 3: w = −6.

    Answerw = −6

    In a pair of coordinates the first number is x and the second is y. Check: 3 × −6 + 7 = −18 + 7 = −11.

    Revise this: Equations of straight lines
  35. 20Writing the inequality shown on a number line.[2]
    The question as printed, from page 10 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. The line covers the numbers from −4 up to 5, so x lies between them.
    2. The circle at −4 is empty: −4 is not included, so use <.
    3. The circle at 5 is filled in: 5 is included, so use ≤.

    Answer−4 < x ≤ 5

    Empty circle: not included (< or >). Filled circle: included (≤ or ≥). One mark is for each end.

    Revise this: Inequalities
  36. 21(a)Drawing the next diagram in a pattern.[1]
    The question as printed, from page 11 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. See how the pattern grows. The grey squares form one row, and it gets one square longer each time: diagram 5 has 5.
    2. White squares stand in columns above and below the grey row. In diagram 4 the columns are 3, 2, 1 and 0 squares tall, the same above and below.
    3. In diagram 5 every column is one taller: 4, 3, 2, 1 and 0 white squares above the five grey ones, and the same below.

    AnswerA row of 5 grey squares, with white columns of 4, 3, 2, 1 and 0 squares above it and the same below it.

    Shade the grey squares, or mark them in some way, so they can be told apart. The diagram is 5 squares wide and 9 tall, which just fits the grid.

    Revise this: Sequences
  37. 21(b)Completing a table for the fourth diagram and for the nth.[4]
    The question as printed, from page 11 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. Diagram 4: count from the picture. 4 grey squares, 12 white squares, 16 in total.
    2. Diagram n, grey squares: 1, 2, 3, 4 is the same as the diagram number, so n.
    3. Diagram n, total: 1, 4, 9, 16 are the square numbers, so n².
    4. Check with the white squares given in the table: n + n(n − 1) = n + n² − n = n². It fits.

    AnswerDiagram 4: 4, 12 and 16. Diagram n: n grey squares and n² in total.

    Two marks are for the three numbers and one each for the two expressions. Test an nth term on a diagram you know: for n = 3, n² = 9.

    Revise this: Sequences
  38. 21(c)Explaining why no diagram has a total of 120 squares.[1]
    The question as printed, from page 11 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. The total is always n², a square number.
    2. 10² = 100 and 11² = 121, so 120 falls between two square numbers that are next to each other. It is not a square number.

    Answer120 is not a square number.

    Give the reason, not just "it is not in the sequence". Showing that 120 lies between 10² and 11² makes it certain.

    Revise this: Sequences
  39. 22(a)Completing a tree diagram for picking a block twice, with the block put back.[2]
    The question as printed, from page 12 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. There are 7 + 5 = 12 blocks, so the probability of pink is 7/12 and of green is 5/12.
    2. The first block goes back in the bag, so the bag is exactly the same for the second pick. The probabilities do not change.
    3. First block, green: 5/12. Second block after pink: pink 7/12, green 5/12. Second block after green: green 5/12.

    Answer5/12 on the first green branch; 7/12 and 5/12 on the top pair of branches; 5/12 on the last green branch.

    Each pair of branches must add up to 1: 7/12 + 5/12 = 12/12. Use that to check every pair.

    Revise this: Probability of combined events
  40. 22(b)Using the tree diagram to find the probability of two greens.[2]
    The question as printed, from page 12 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. Follow the route green, then green. To find the probability of both happening, multiply along the branches.
    2. 5/12 × 5/12 = 25/144. Multiply the tops, and multiply the bottoms.

    Answer25/144

    Multiply along the branches. Adding gives 10/12, which is far too big for two greens in a row. 25/144 does not simplify.

    Revise this: Probability of combined events
  41. 23Finding the radius of a circle from its circumference.[2]
    The question as printed, from page 13 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. Circumference = 2 × π × radius, which is on the formula list.
    2. So 2 × π × r = 18π. Divide both sides by π: 2r = 18.
    3. Divide by 2: r = 9.

    Answer9 cm

    Leave π alone as a letter and it cancels. 18 is the diameter: halve it for the radius.

    Revise this: Circles, arcs and sectors
  42. 24Finding one value from two means.[3]
    The question as printed, from page 13 of the paper.The paper couldn’t be fetched just now, so the question isn’t shown. Open the whole paper
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    1. Mean = total ÷ how many, so total = mean × how many.
    2. The four children: 4 × 11 = 44 years altogether.
    3. All five children: 5 × 10 = 50 years altogether.
    4. The fifth child brought the difference: 50 − 44 = 6.

    Answer6 years

    Whenever a mean changes because someone joins or leaves, work with totals. The answer makes sense: the mean went down, so the new child is younger than 11.

    Revise this: Data, averages and range
  43. 25(a)Sorting the numbers 1 to 9 into a Venn diagram of primes and odd numbers.[2]
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    1. List each set. Primes from 1 to 9: 2, 3, 5 and 7. Odd numbers: 1, 3, 5, 7 and 9.
    2. Fill in the overlap first, with the numbers in both lists: 3, 5 and 7.
    3. The rest of P: 2. The rest of N: 1 and 9.
    4. Numbers in neither list go inside the rectangle but outside both circles: 4, 6 and 8.

    AnswerOverlap: 3, 5, 7. P only: 2. N only: 1, 9. Outside both circles: 4, 6, 8.

    1 is not a prime number, and 2 is the only even prime. Every one of the nine numbers must appear once: count them at the end.

    Revise this: Sets and Venn diagrams
  44. 25(b)Counting the members of the union of the two sets.[1]
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    1. P ∪ N means everything in P, or in N, or in both: all the numbers inside either circle.
    2. The n in front means "the number of members". Count them: 2, 3, 5, 7, 1 and 9.

    Answer6

    The answer is how many numbers there are, not a list of them. Do not count the overlap twice.

    Revise this: Sets and Venn diagrams
  45. 26(a)Writing a large number in standard form.[1]
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    1. Standard form is a number from 1 up to (but not including) 10, multiplied by a power of 10.
    2. Put the decimal point after the first digit: 6.32.
    3. To get from 6.32 back to 6320, the digits move 3 places, so multiply by 10³.

    Answer6.32 × 10³

    Revise this: Standard form
  46. 26(b)Writing a small number in standard form.[1]
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    1. Put the decimal point after the first digit that is not zero: 2.58.
    2. To get from 2.58 back to 0.0258, divide by 100, which is the same as multiplying by 10⁻².

    Answer2.58 × 10⁻²

    Numbers smaller than 1 have a negative power. The power counts how many places the digits move, not how many zeros there are.

    Revise this: Standard form
  47. 27Subtracting one mixed number from another, without a calculator.[3]
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    1. Turn each mixed number into a single fraction. Three and a third is (3 × 3 + 1) thirds = 10/3. One and three quarters is (1 × 4 + 3) quarters = 7/4.
    2. Give them the same denominator. 3 and 4 both go into 12: 10/3 = 40/12 and 7/4 = 21/12.
    3. Subtract the tops: 40/12 − 21/12 = 19/12.
    4. Turn it back into a mixed number: 12 goes into 19 once with 7 left over.

    Answer1 7/12

    Every step must be shown: decimals earn no method marks here. The answer has to be a mixed number in its simplest form, so 19/12 loses the last mark.

    Revise this: Fractions, decimals and the order of operations
  48. 28Rearranging a formula to make a different letter the subject.[2]
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    1. T has a minus sign in front of it. Add 4T to both sides to make it positive: Q + 4T = 3.
    2. Subtract Q from both sides: 4T = 3 − Q.
    3. Divide both sides by 4: T = (3 − Q) ÷ 4.

    AnswerT = (3 − Q)/4

    Divide the whole of 3 − Q by 4, not just one part of it. An equally good answer is (Q − 3)/−4, but the first is tidier.

    Revise this: Changing the subject of a formula
  49. 29(a)(i)Describing the single transformation that takes shape A to shape B.[2]
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    1. B is A flipped over, left to right, so it is a reflection. A reflection needs its mirror line.
    2. The mirror line is halfway between matching corners. The corner of A at (−3, 4) goes to the corner of B at (2, 4). Halfway between −3 and 2 is −0.5.
    3. The mirror line is vertical, so its equation is x = −0.5.

    AnswerReflection in the line x = −0.5

    Two marks: the word "reflection" and the equation of the line. A vertical line is x = a number. Check with another pair: (−4, 1) and (3, 1) are also the same distance from x = −0.5.

    Revise this: Transformations
  50. 29(a)(ii)Describing the single transformation that takes shape B to shape C.[3]
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    1. C is B turned, so it is a rotation. A rotation needs three things: the angle, the direction and the centre.
    2. Centre: the corner at (2, 4) belongs to both shapes. It has not moved, so it is the centre.
    3. Angle and direction: the top edge of B runs from (2, 4) to the right. On C the same edge runs from (2, 4) straight down. Right to down is a quarter turn clockwise: 90°.

    AnswerRotation, 90° clockwise, centre (2, 4)

    One mark for each of the three parts. Use tracing paper, which is allowed: trace B, hold your pencil point on (2, 4) and turn the paper. 270° anticlockwise is the same turn.

    Revise this: Transformations
  51. 29(b)Drawing the image of shape A after a translation.[2]
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    1. A translation slides the shape without turning it. The top number of the vector is the move across and the bottom number is the move up.
    2. Top number 1: move 1 to the right. Bottom number −3: move 3 down.
    3. Move each corner of A in turn: (−5, 4) → (−4, 1), (−3, 4) → (−2, 1), (−4, 1) → (−3, −2), (−6, 1) → (−5, −2) and (−4, 3) → (−3, 0).
    4. Join the new corners in the same order.

    AnswerA shape the same size and way up as A, with corners at (−4, 1), (−2, 1), (−3, −2), (−5, −2) and (−3, 0).

    Move one corner at a time and count squares. A shape moved the right way in only one direction earns one of the two marks.

    Revise this: Transformations

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