Set language, notation & Venn diagrams – IGCSE Maths exam questions with mark scheme

1Grade 4–5HigherCalculator3 marks

$\xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$

$A = \{\text{multiples of } 3\}$ and $B = \{\text{even numbers}\}$

(a) List the members of $A \cap B$. [1]

(b) Find $n(A \cup B)$. [1]

(a) $\{6, 12\}$ (order irrelevant; no extras)B1
(b) $8$B1
(c) $\{1, 5, 7, 11\}$ (order irrelevant; no extras)B1
Answer: (a) 6, 12 (b) 8 (c) 1, 5, 7, 11
Examiner tip: In (b) n( ) means 'how many', so the answer is a number, not a list — and 6 and 12 are counted only once.

2Grade 4–5HigherCalculator4 marks

The Venn diagram shows the number of elements in each region of sets $A$ and $B$.

(a) Find $n(A)$. [1]

(b) Find $n(A \cap B')$. [1]

An element is chosen at random from set $B$.

(d) Find the probability that this element is also in set $A$. [1]

(a) $13$B1
(b) $9$B1
(c) $5$B1
(d) $\dfrac{4}{11}$ (oe, e.g. 0.364 or better)B1
Answer: (a) 13 (b) 9 (c) 5 (d) $\frac{4}{11}$
Examiner tip: In (d) the element is already known to be in B, so the denominator is n(B) = 11, not the total of 25.

3Grade 6–7HigherCalculator4 marks

There are 60 students in a year group.

32 of the students study French.
25 of the students study German.
9 of the students study neither French nor German.

(a) Work out how many students study both French and German. [3]

One of the students who study French is chosen at random.

(b) Find the probability that this student also studies German. [1]

for $60 - 9\ (= 51)$ or a Venn diagram with 9 outside both setsM1
for $32 + 25 - 51$ oe, or a Venn diagram with $32 - x$, $x$, $25 - x$ and $9$ and the equation $32 - x + x + 25 - x + 9 = 60$M1
$6$A1
(b) $\dfrac{6}{32}$ oe, e.g. $\dfrac{3}{16}$ (ft their answer to (a) over 32)B1
Answer: (a) 6 (b) $\frac{6}{32} = \frac{3}{16}$
Examiner tip: Students in both subjects are counted twice when you add 32 and 25 — subtract the 51 who study at least one language to find them.

4Grade 6–7HigherCalculator4 marks

$\mathcal{E}$, $A$, $B$ and $C$ are sets.

$n(\mathcal{E}) = 50 \qquad n(B) = 20 \qquad n(A) = 8 \qquad n(C) = 12$

$A \subset B \qquad B \cap C = \varnothing$

(a) Find $n(B \cap A')$. [1]

(b) Find $n(A \cup C)$. [1]

(a) $12$B1
(b) $20$B1
(c) for $50 - (20 + 12)$ oe, or a correct Venn diagram showing $8$ in $A$, $12$ in $B$ outside $A$, $12$ in $C$ with $C$ separate from $B$M1
$18$A1
Answer: (a) 12 (b) 20 (c) 18
Examiner tip: $A \subset B$ means every element of A is inside B, so don't add n(A) again when counting the elements outside all three sets.

5Grade 6–7HigherCalculator5 marks

40 people were asked which of the drinks tea ($T$), coffee ($C$) and juice ($J$) they like.

The Venn diagram shows the number of people in each region, in terms of $x$.

(a) Find the value of $x$. [2]

One of the people who like coffee is chosen at random.

(b) Find the probability that this person also likes tea. [3]

(a) for $2x + (x + 3) + 5 + 4 + x + 3 + 2 + 7 = 40$ oe, e.g. $4x + 24 = 40$M1
$x = 4$A1
(b) for $n(C) = (x + 3) + 4 + 2 + 3 = 16$ (ft their $x$)M1
for $n(C \cap T) = 4 + 2 = 6$ used as the numeratorM1
$\dfrac{6}{16}$ oe, e.g. $\dfrac{3}{8}$A1
Answer: (a) $x = 4$ (b) $\frac{6}{16} = \frac{3}{8}$
Examiner tip: A 'given' probability uses only the people in the given set — here the 16 who like coffee, not all 40 people.

6Grade 4–5HigherCalculator3 marks

$\xi = \{x : x \text{ is an integer and } 1 \leqslant x \leqslant 20\}$

$P = \{x : x \text{ is a prime number}\}$

$Q = \{x : x^2 < 30\}$

(a) List the members of $P \cap Q$. [1]

(b) Find $n((P \cup Q)')$. [2]

(a) $\{2, 3, 5\}$B1
(b) for listing $P = \{2, 3, 5, 7, 11, 13, 17, 19\}$ and $Q = \{1, 2, 3, 4, 5\}$, or $n(P \cup Q) = 10$M1
$10$A1
Answer: (a) 2, 3, 5 (b) 10
Examiner tip: 1 is not a prime number, but it is in Q because 1² = 1 < 30 — check the set rule for every candidate.

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