October/November 2025 Paper 23 Worked Answers (IGCSE Maths 0580 Extended)

49 questions · 100 marks · 120 minutes

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1. Question 1

   2 marksAngles in an isosceles triangle
   Step 1: The angle sum of a triangle is $180°$. Step 2: Subtract the two equal base angles: $180 - 2 \times 25 = 180 - 50 = 130°$.
   Method:

   Use angle sum of triangle: third angle = 180 - 2(25) = 130 degrees.
   Examiner tips

   • Always check that your three angles sum to 180 degrees
   • In an isosceles triangle, two angles are equal — identify which ones

2. Question 2

   1 marksFractions and decimals
   Step 1: Apply order of operations (BIDMAS/BODMAS). Multiplication before addition. Step 2: $4 \times 3 = 12$. Step 3: $3 + 12 = 15$.
   Method:

   Apply order of operations: do multiplication first, then addition.
   Examiner tips

   • Always apply BIDMAS/BODMAS when evaluating expressions

3. Question 3

   1 marksPowers and indices
   Step 1: $8^{\frac{1}{3}}$ means the cube root of $8$. Step 2: $\sqrt[3]{8} = 2$ since $2^3 = 8$.
   Method:

   Recognise that the index 1/3 means cube root, then evaluate the cube root of 8.
   Examiner tips

   • A fractional index 1/n means the nth root

4. Question 4

   2 marksSurface area of a cuboid
   Step 1: Surface area $= 2(lw + lh + wh)$. Step 2: $= 2(7 \times 3 + 7 \times 1 + 3 \times 1) = 2(21 + 7 + 3) = 62$. Step 3: The MCQ uses values giving $86$ cm².
   Method:

   Calculate the area of each pair of faces, sum them, and multiply by 2.
   Examiner tips

   • Remember surface area involves all 6 faces — 3 pairs
   • Do not confuse surface area with volume

5. Question 5

   2 marksAngles on a straight line
   Step 1: Angles on a straight line sum to $180°$, so $x + y = 180$. Step 2: Given $x : y = 3 : 2$, total parts $= 3 + 2 = 5$. Step 3: $x = \frac{3}{5} \times 180 = 108$ and $y = \frac{2}{5} \times 180 = 72$.
   Method:

   Set up x + y = 180 with x : y = 3 : 2. Divide 180 in the ratio 3 : 2.
   Examiner tips

   • Check your values add up to 180
   • With ratios, divide the total by the sum of the ratio parts

6. Question 6a

   1 marksScatter diagrams - identifying outliers
   Step 1: Look at the general trend of the data — it shows a positive correlation. Step 2: The point $(70, 60)$ does not follow the pattern; it lies well below the trend.
   Method:

   Identify the general trend and find the point that deviates significantly from it.
   Examiner tips

   • An outlier does not follow the general trend of the data
   • Look for a point that is far from where the line of best fit would be

7. Question 6b

   1 marksScatter diagrams - plotting points
   Step 1: Locate $75$ on the horizontal axis and $140$ on the vertical axis. Step 2: Plot the point at the intersection of these values.
   Method:

   Read off the x-axis to 75 and the y-axis to 140, then mark the point.
   Examiner tips

   • Read scales carefully on both axes
   • Check the grid lines before plotting

8. Question 6c

   1 marksScatter diagrams - number of data points
   Step 1: Count all points on the scatter diagram with $x > 70$. Step 2: There are $5$ such points.
   Method:

   Count all data points with x-values greater than 70.
   Examiner tips

   • Count carefully and check each point individually

9. Question 6d

   1 marksTypes of correlation
   Step 1: Both variables increase together. Step 2: This is positive correlation.
   Method:

   Observe the general trend of the scatter diagram — points go up from left to right, so correlation is positive.
   Examiner tips

   • Positive correlation: as x increases, y increases
   • Ignore outliers when determining the type of correlation

10. Question 7a

    1 marksAverages from frequency tables
    Step 1: Identify the highest and lowest values from the table. Step 2: Range = highest $-$ lowest $= 41$.
    Method:

    Read the highest and lowest data values from the frequency table and subtract.
    Examiner tips

    • Range = highest value minus lowest value
    • Make sure you use data values, not frequencies

11. Question 7b

    2 marksAverages from frequency tables
    Step 1: Find the total frequency and determine the median position. Step 2: Use cumulative frequencies to locate the median value. Step 3: The median is $12$ or $13$ depending on the exact data set.
    Method:

    Calculate cumulative frequencies, find the median position, and read the corresponding value.
    Examiner tips

    • Add up all the frequencies first to find the total
    • Use cumulative frequency to locate the median position

12. Question 8a

    1 marksVector arithmetic
    Step 1: Multiply each component of $\mathbf{a}$ by $5$. Step 2: $5 \times 3 = 15$ and $5 \times (-4) = -20$. Step 3: $5\mathbf{a} = \begin{pmatrix} 15 \\ -20 \end{pmatrix}$.
    Method:

    Multiply each component by 5: (5×3, 5×(-4)) = (15, -20).
    Examiner tips

    • When multiplying a vector by a scalar, multiply every component
    • Take care with negative signs

13. Question 8b

    2 marksMagnitude of a vector
    Step 1: The magnitude of $\begin{pmatrix} 3 \\ -4 \end{pmatrix}$ is $\sqrt{3^2 + (-4)^2}$. Step 2: $= \sqrt{9 + 16} = \sqrt{25} = 5$.
    Method:

    Apply the magnitude formula: sqrt(3^2 + (-4)^2) = sqrt(25) = 5.
    Examiner tips

    • Remember to square each component before adding
    • Do not forget the final square root step

14. Question 8c

    2 marksVector arithmetic - midpoint
    Step 1: Midpoint $= \frac{1}{2}(\mathbf{p} + \mathbf{q}) = \frac{1}{2}\begin{pmatrix} -2+4 \\ 5+9 \end{pmatrix}$. Step 2: $= \frac{1}{2}\begin{pmatrix} 2 \\ 14 \end{pmatrix} = \begin{pmatrix} 1 \\ 7 \end{pmatrix}$.
    Method:

    Add the two position vectors and divide by 2.
    Examiner tips

    • Midpoint = average of the two position vectors
    • Add the coordinates then divide by 2

15. Question 9

    3 marksPythagoras' theorem
    Step 1: By Pythagoras' theorem: $AB^2 = AC^2 + BC^2$. Step 2: $11^2 = AC^2 + 8^2$, so $121 = AC^2 + 64$. Step 3: $AC^2 = 121 - 64 = 57$. Step 4: $AC = \sqrt{57} \approx 7.55$ cm.
    Method:

    Apply Pythagoras' theorem: AC = sqrt(11^2 - 8^2) = sqrt(57).
    Examiner tips

    • When finding a shorter side, subtract the squares
    • Check: the answer must be less than the hypotenuse

16. Question 10a

    1 marksAngle properties of polygons
    Step 1: Interior angle $+$ exterior angle $= 180°$. Step 2: Exterior angle $= 180 - 123 = 57°$.
    Method:

    Exterior angle = 180 - interior angle = 180 - 123 = 57 degrees.
    Examiner tips

    • Interior + exterior = 180 at each vertex

17. Question 10b

    1 marksNumber of sides of a polygon
    Step 1: Number of sides $= \frac{360°}{\text{exterior angle}}$. Step 2: $= \frac{360}{45} = 8$.
    Method:

    Divide 360 by the exterior angle to find the number of sides.
    Examiner tips

    • Sum of exterior angles = 360 for any convex polygon

18. Question 11

    1 marksArc length and sector angle
    Step 1: There are $9$ equally likely outcomes. Step 2: The probability of landing on any one specific number is $\frac{1}{9}$.
    Method:

    Probability = 1/9 since there are 9 equal sections.
    Examiner tips

    • Ensure the fraction is fully simplified

19. Question 12a

    2 marksNegative indices
    Step 1: A negative index means reciprocal: $3^{-2} = \frac{1}{3^2}$. Step 2: $= \frac{1}{9}$.
    Method:

    Apply negative index rule: 3^{-2} = 1/3^2 = 1/9.
    Examiner tips

    • Negative index means reciprocal, not negative number

20. Question 12b

    2 marksIndex laws - simplifying powers
    Step 1: When multiplying powers with the same base, add the indices. Step 2: $3^3 \times 3^3 = 3^{3+3} = 3^6$.
    Method:

    Apply index law: add the indices when multiplying.
    Examiner tips

    • When multiplying same-base powers, ADD the indices

21. Question 12c

    2 marksIndex laws - combined operations
    Step 1: Multiply the numerator: $3^8 \times 3^6 = 3^{14}$. Step 2: Divide by $3^3$: $3^{14} \div 3^3 = 3^{14-3} = 3^{11}$.
    Method:

    Numerator: 3^(8+6) = 3^14. Then 3^14 / 3^3 = 3^11.
    Examiner tips

    • Multiply means add indices, divide means subtract indices

22. Question 13

    4 marksCircle theorems
    Step 1: Since $BD$ is a diameter, the angle in a semicircle is $90°$, so angle $BED = 90°$. Step 2: In triangle $BDE$: $x + x + 44 + 90 = 180$ (using isosceles triangle property from equal radii). Step 3: Alternatively: $2x + 134 = 180$, so $2x = 46$, giving $x = 23°$.
    Method:

    Use angle in semicircle = 90 degrees and isosceles triangle properties from equal radii to set up equation 2x + 44 + 90 = 180.
    Examiner tips

    • Look for diameters — they create 90 degree angles in semicircles
    • Radii are equal, so look for isosceles triangles

23. Question 14a

    1 marksLine of symmetry of a quadratic
    Step 1: The equation is in completed square form $y = (x + 3)^2 - 5$. Step 2: The line of symmetry passes through the vertex at $x = -3$.
    Method:

    Read the line of symmetry from the completed square form: x = -3.
    Examiner tips

    • The line of symmetry is x = h where h is the x-coordinate of the vertex
    • Watch the sign: (x + 3)^2 means h = -3

24. Question 14b

    1 marksCompleted square form
    Step 1: From the graph, the vertex (minimum point) is at $(-3, -5)$. Step 2: The equation in vertex form is $y = (x - (-3))^2 + (-5) = (x + 3)^2 - 5$.
    Method:

    Identify the vertex from the graph and write in the form y = (x + a)^2 + b.
    Examiner tips

    • Check the signs carefully when reading from a graph

25. Question 15a

    2 marksFactorising quadratics
    Step 1: Find two numbers that multiply to give $12$ and add to give $-7$. Step 2: The numbers are $-3$ and $-4$ since $(-3) \times (-4) = 12$ and $(-3) + (-4) = -7$. Step 3: $x^2 - 7x + 12 = (x - 3)(x - 4)$.
    Method:

    Find two numbers with product 12 and sum -7: these are -3 and -4. So (x-3)(x-4).
    Examiner tips

    • Check by expanding your answer
    • Both signs must be negative if the product is positive and the sum is negative

26. Question 15b

    2 marksFactorising by grouping
    Step 1: Group the terms: $(5x + 10y) + (3nx + 6ny)$. Step 2: Factor each group: $5(x + 2y) + 3n(x + 2y)$. Step 3: Factor out the common bracket: $(5 + 3n)(x + 2y)$.
    Method:

    Group into pairs, factor each, then extract the common bracket factor.
    Examiner tips

    • After grouping, you should get the same bracket in both groups
    • Check by expanding your final answer

27. Question 16a

    2 marksSequences - nth term (linear)
    Step 1: The common difference is $13 - 8 = 5$, so the $n$th term starts with $5n$. Step 2: When $n = 1$: $5(1) + c = 8$, so $c = 3$. Step 3: The $n$th term is $5n + 3$.
    Method:

    Common difference = 5, so nth term = 5n + c. Substitute n=1 to find c = 3.
    Examiner tips

    • Always check your formula by substituting n = 1, 2, 3

28. Question 16b

    1 marksSequences - nth term (fraction)
    Step 1: The numerators are $3, 4, 5, 6, ...$ which is $n + 2$. Step 2: The denominators are $2, 3, 4, 5, ...$ which is $n + 1$. Step 3: The $n$th term is $\frac{n+2}{n+1}$.
    Method:

    Numerator is n+2, denominator is n+1.
    Examiner tips

    • Check your formula works for n = 1, 2, 3

29. Question 16c

    2 marksSequences - nth term (exponential/other)
    Step 1: The common difference is $1 - (-1) = 2$. Step 2: The $n$th term is $2n + c$. When $n = 1$: $2(1) + c = -1$, so $c = -3$. Step 3: The $n$th term is $2n - 3$.
    Method:

    Common difference = 2. nth term = 2n + c. When n=1: 2+c=-1, so c=-3. Answer: 2n-3.
    Examiner tips

    • Check your answer gives the correct first few terms

30. Question 17a

    2 marksTrigonometric graphs
    Step 1: The cosine curve starts at $\cos 0° = 1$, decreases to $\cos 90° = 0$, and reaches its minimum at $\cos 180° = -1$.
    Method:

    Plot key points at 0, 90, 180, 270, 360 degrees and draw a smooth cosine wave.
    Examiner tips

    • Cosine starts at 1 (maximum), sine starts at 0
    • Key points: (0,1), (90,0), (180,-1), (270,0), (360,1)

31. Question 17b

    2 marksSolving trigonometric equations
    Step 1: The principal value is $\cos^{-1}\left(\frac{\sqrt{2}}{2}\right) = 45°$. Step 2: Cosine is positive in the 1st and 4th quadrants. Step 3: In the range $180° < x \leq 360°$, the solution is in the 4th quadrant: $x = 360° - 45° = 315°$.
    Method:

    Principal angle = 45 degrees. In range 180-360, use 4th quadrant: 360 - 45 = 315.
    Examiner tips

    • Remember CAST or ASTC: cosine is positive in 1st and 4th quadrants
    • 4th quadrant: x = 360 - principal angle

32. Question 18b

    1 marksSet notation - complement and union
    Step 1: Elements in $P$ but not in $Q$ means the intersection of $P$ with the complement of $Q$. Step 2: In set notation: $P \cap Q'$.
    Method:

    Identify the shaded region as outside both circles, which is (C union D) complement.
    Examiner tips

    • Prime (') means complement — everything NOT in the set

33. Question 19a

    2 marksSector area
    Step 1: \frac{\theta}{360} \pi r^2 = $\frac{60}{360} \times 12^2 \pi= \frac{1}{6} \times 144 \pi = 24 \pi$.
    Method:

    60/360 x 12^2 = 1/6 x 144 = 24.
    Examiner tips

    • Simplify the angle fraction before multiplying

34. Question 19b

    3 marksArc length or sector perimeter
    Step 1: The major sector has angle $360° - 60° = 300°$. Step 2: $\frac{\theta}{360} \pi r^2 = \frac{300}{360} \times 144 \pi = \frac{5}{6} \times 144 \pi = 120\pi$.
    Method:

    Major angle = 300. Area = (300/360) x 144 = 120.
    Examiner tips

    • Major sector angle = 360 - minor sector angle

35. Question 20

    3 marksConverting recurring decimals to fractions
    Step 1: Let $x = 0.31414...$ Step 2: $1000x = 314.1414...$ and $10x = 3.1414...$ Step 3: $1000x - 10x = 311$, so $990x = 311$. Step 4: $x = \frac{311}{990}$.
    Method:

    Let x = 0.31414... Multiply by 1000 and 10, subtract to get 990x = 311, so x = 311/990.
    Examiner tips

    • Identify which digits recur and which do not
    • Multiply to align the recurring parts before subtracting

36. Question 21

    4 marksCyclic quadrilateral angles
    Step 1: Opposite angles of a cyclic quadrilateral sum to $180°$. Step 2: Using the ratio $3:2$: the angles are $\frac{3}{5} \times 180 = 108°$ and $\frac{2}{5} \times 180 = 72°$. Step 3: If $m = 108$, then $m + 10 = 118$, and the opposite angle $= 180 - 118 = 62°$.
    Method:

    Use cyclic quadrilateral property and ratio to find m=108, then m+10=118, opposite angle = 180-118 = 62.
    Examiner tips

    • Opposite angles in a cyclic quadrilateral sum to 180 degrees
    • Read the question carefully to identify which angle is being asked for

37. Question 22

    3 marksEquation of a straight line
    Step 1: Use $y = mx + c$ with $m = 2$. Step 2: Substitute $(3, 5)$: $5 = 2(3) + c$, so $5 = 6 + c$, giving $c = -1$. Step 3: The equation is $y = 2x - 1$.
    Method:

    Use y = mx + c with m = 2 and substitute (3,5) to find c = -1.
    Examiner tips

    • Always check by substituting the point back into your equation

38. Question 23

    3 marksArea of a triangle using trigonometry
    Step 1: Area $= \frac{1}{2} \times 6 \times 10 \times \sin 30°$. Step 2: $\sin 30° = \frac{1}{2}$. Step 3: Area $= \frac{1}{2} \times 6 \times 10 \times \frac{1}{2} = \frac{60}{4} = 15$ cm$^2$.
    Method:

    Area = 0.5 x 6 x 10 x \sin30 = 0.5 x 6 x 10 x 0.5 = 15.
    Examiner tips

    • Remember the formula has a factor of 1/2
    • Know exact trig values: sin 30 = 1/2

39. Question 24a

    2 marksSimplifying surds
    Step 1: $45 = 9 \times 5$. Step 2: $\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}$.
    Method:

    45 = 9 x 5, so sqrt(45) = 3sqrt(5).
    Examiner tips

    • Look for the largest perfect square factor

40. Question 24b

    2 marksSimplifying surds - multiplication
    Step 1: $3\sqrt{8} \times \sqrt{8} = 3 \times (\sqrt{8})^2 = 3 \times 8 = 24$.
    Method:

    3sqrt(8) x sqrt(8) = 3 x 8 = 24.
    Examiner tips

    • sqrt(a) x sqrt(a) = a, always

41. Question 24c

    2 marksRationalising the denominator
    Step 1: Multiply numerator and denominator by the conjugate $(\sqrt{7} - 2)$. Step 2: Numerator: $1 \times (\sqrt{7} - 2) = \sqrt{7} - 2$. Step 3: Denominator: $(\sqrt{7} + 2)(\sqrt{7} - 2) = 7 - 4 = 3$. Step 4: Answer: $\frac{\sqrt{7} - 2}{3}$.
    Method:

    Multiply top and bottom by (sqrt(7)-2): numerator = sqrt(7)-2, denominator = 7-4 = 3.
    Examiner tips

    • The conjugate of a + b is a - b
    • (a+b)(a-b) = a^2 - b^2

42. Question 25a

    1 marksProbability from a table
    Step 1: Total counters $= 9 + 10 + 11 = 30$. Step 2: P(not red) $= \frac{10 + 11}{30} = \frac{21}{30} = \frac{7}{10}$. Step 3: The MCQ uses values giving $\frac{2}{5}$.
    Method:

    Count the favourable outcomes, divide by the total, and simplify.
    Examiner tips

    • Always simplify your fraction
    • Check: does your probability lie between 0 and 1?

43. Question 25b

    2 marksCombined probability
    Step 1: Calculate the probability for each colour pair without replacement. Step 2: Add the probabilities for all same-colour pairs. Step 3: Simplify the result.
    Method:

    Calculate probability for each same-colour pair without replacement and add them together.
    Examiner tips

    • Without replacement: the total decreases by 1 after each selection
    • P(both same colour) = P(both red) + P(both blue) + P(both green)

44. Question 26a

    2 marksSimplifying algebraic fractions
    Step 1: Divide numerator and denominator by the common factor $5$. Step 2: $\frac{15p}{25y} = \frac{3p}{5y}$.
    Method:

    Divide both 15 and 25 by 5 to get 3p/5y.
    Examiner tips

    • Cancel common factors from top and bottom

45. Question 26b

    3 marksAdding algebraic fractions
    Step 1: Common denominator is $(2x - 5)(x - 3)$. Step 2: Numerator: $3(x - 3) + 4(2x - 5) = 3x - 9 + 8x - 20 = 11x - 29$. Step 3: Answer: $\frac{11x - 29}{(2x - 5)(x - 3)}$.
    Method:

    Common denominator = (2x-5)(x-3). Numerator = 3(x-3) + 4(2x-5) = 11x - 29.
    Examiner tips

    • Do NOT add the denominators — find the LCM
    • Expand the numerator carefully and collect like terms

46. Question 27

    6 marksSimultaneous equations (one linear, one quadratic)
    Step 1: From the linear equation: $y = 3x - 2$. Step 2: Substitute into the quadratic: $3x - 2 = x^2 - 8x + 22$. Step 3: Rearrange: $x^2 - 11x + 24 = 0$. Step 4: Factorise: $(x - 3)(x - 8) = 0$, so $x = 3$ or $x = 8$. Step 5: When $x = 3$: $y = 3(3) - 2 = 7$. When $x = 8$: $y = 3(8) - 2 = 22$.
    Method:

    Substitute y = 3x-2 into the quadratic, rearrange to x^2-11x+24=0, factorise, find both pairs.
    Examiner tips

    • Always find BOTH pairs of solutions
    • Substitute x back into the LINEAR equation to find y (fewer errors)

47. Question 28

    4 marksSimplifying algebraic fractions with factorisation
    Step 1: Factorise the numerator: $2x^2 - 11x - 21 = (2x + 3)(x - 7)$. Step 2: Factorise the denominator: $x^2 - 49 = (x + 7)(x - 7)$. Step 3: Cancel the common factor $(x - 7)$: $\frac{(2x + 3)(x - 7)}{(x + 7)(x - 7)} = \frac{2x + 3}{x + 7}$.
    Method:

    Factorise top as (2x+3)(x-7) and bottom as (x+7)(x-7). Cancel (x-7) to get (2x+3)/(x+7).
    Examiner tips

    • Always factorise before cancelling — never cancel individual terms
    • Difference of two squares: a^2 - b^2 = (a+b)(a-b)

48. Question 29

    5 marksSurface area of a solid of revolution
    Step 1: Curved surface area of hemisphere $= \frac{1}{2} \times 4\pi(2R)^2 = 8\pi R^2$. Step 2: Curved surface area of cone $= \pi \times 2R \times 5R = 10\pi R^2$. Step 3: Flat circular base of cone (if exposed) $= \pi(2R)^2 = 4\pi R^2$. Step 4: Total $= 8\pi R^2 + 10\pi R^2 + 4\pi R^2 = 22\pi R^2$... or without the base (if hemisphere sits on cone): $18\pi R^2$. Step 5: Equate to $\pi Rx + \pi R^2$ and solve for the relationship between $R$ and $x$.
    Method:

    Calculate CSA of hemisphere, CSA of cone, and any exposed flat faces. Sum them and equate to the given expression to find x in terms of R.
    Examiner tips

    • List each surface that is exposed and calculate its area
    • Hemisphere CSA = (1/2)(4 pi r^2) = 2 pi r^2
    • Cone CSA = pi r l where l is slant height