IGCSE Mathematics 0580/23 October/November 2025 Past Paper with Answers

Question 1

The diagram shows an isosceles triangle. The two equal angles are each $25°$. Calculate the third angle.

1. $130°$
2. $155°$
3. $50°$
4. $75°$

Question 2

Work out $3 + 4 \times 3$.

1. $15$
2. $21$
3. $12$
4. $36$

Question 3

Find the value of $8^{\frac{1}{3}}$.

1. $2$
2. $4$
3. $3$
4. $\frac{8}{3}$

Question 4 · Answer A

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.

Question 5 · Answer A

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 $y = \frac{2}{5} \times 180 = 72$.

Question 6 · Answer A

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.

Question 7 · Answer A

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.

Question 8 · Answer A

Step 1: Count all points on the scatter diagram with $x > 70$. Step 2: There are $5$ such points.

Question 9 · Answer A

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

Question 10 · Answer A

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

Question 11 · Answer A

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.

Question 12 · Answer A

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

Question 13 · Answer A

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

Question 14 · Answer A

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

Question 15 · Answer A

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.

Question 16 · Answer A

Step 1: Interior angle $+$ exterior angle $= 180°$. Step 2: Exterior angle $= 180 - 123 = 57°$.

Question 17 · Answer A

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

Question 18 · Answer A

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

Question 19 · Answer A

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

Question 20 · Answer A

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

Question 21 · Answer A

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

Question 22 · Answer A

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 $x = 23°$.

Question 23 · Answer A

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 $x = -3$.

Question 24 · Answer A

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

Question 25 · Answer A

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: $x^2 - 7x + 12 = (x - 3)(x - 4)$.

Question 26 · Answer A

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: $(5 + 3n)(x + 2y)$.

Question 27 · Answer A

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 $5n + 3$.

Question 28 · Answer A

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 $\frac{n+2}{n+1}$.

Question 29 · Answer A

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 $2n - 3$.

Question 30 · Answer A

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

Question 31 · Answer A

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: $x = 360° - 45° = 315°$.

Question 32 · Answer A

Step 1: $A \cap B$ means the intersection of $A$ and $B$. Step 2: This is the region where both sets overlap.

Question 33 · Answer A

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

Question 34 · Answer A

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

Question 35 · Answer A

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

Question 36 · Answer A

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: $x = \frac{311}{990}$.

Question 37 · Answer A

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 $= 180 - 118 = 62°$.

Question 38 · Answer A

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 $y = 2x - 1$.

Question 39 · Answer A

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$^2$.

Question 40 · Answer A

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

Question 41 · Answer A

Step 1: $3\sqrt{8} \times \sqrt{8} = 3 \times (\sqrt{8})^2 = 3 \times 8 = 24$.

Question 42 · Answer A

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: $\frac{\sqrt{7} - 2}{3}$.

Question 43 · Answer A

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 $\frac{2}{5}$.

Question 44 · Answer A

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.

Question 45 · Answer A

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

Question 46 · Answer A

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: $\frac{11x - 29}{(2x - 5)(x - 3)}$.

Question 47 · Answer A

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 $x = 8$: $y = 3(8) - 2 = 22$.

Question 48 · Answer A

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 $(x - 7)$: $\frac{(2x + 3)(x - 7)}{(x + 7)(x - 7)} = \frac{2x + 3}{x + 7}$.

Question 49 · Answer A

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 $x$.