In a class of 56 students, 13 study French (F), 12 study German (G) and 36 study neither. Find the number of students who study both French and German.
[2]
(b)
A student who studies French is chosen at random. Find the probability that this student also studies German.
[1]
(c)
Find n(F ∪ G′).
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
5
13 + 12 − x + 36 = 56 M1 5 A1
2
(b)
513
513B1
1
(c)
49
F ∪ G′ = everything except "G only" M1 49 A1
2
Question 2Structured · 9 marksA* challenge
An integer is chosen at random from 1 to 1000 inclusive.
(a)
Find the probability that it is divisible by 3 or by 5 but not by both.
[4]
(b)
Find the probability that it is divisible by exactly two of the numbers 2, 3 and 5.
[5]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
4011000
333 multiples of 3, 200 of 5 B1 66 multiples of 15 B1 333 + 200 − 2 × 66 M1 4011000A1
4
(b)
2331000
Venn diagram for 2, 3 and 5 M1 multiples of 30: 33 B1 6 not 5: 166 − 33 = 133; 10 not 3: 100 − 33 = 67; 15 not 2: 66 − 33 = 33 M1 233 A1 2331000A1
5
Question 3Structured · 3 marks
𝓔 = {x : x is an integer, 1 ≤ x ≤ 16} A = {factors of 16} B = {odd numbers}
(a)
List the elements of A ∩ B.
[1]
(b)
Find n(A ∪ B).
[1]
(c)
Use set notation to describe the set of even numbers that are not factors of 16.
[1]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
{1}
{1} B1
1
(b)
12
12 B1
1
(c)
A′ ∩ B′
A′ ∩ B′ B1
1
Question 4Structured · 5 marks
In a group of 32 students, 11 study French (F), 13 study Spanish (S) and 3 study both.
(a)
Complete a Venn diagram to show this information.
[2]
(b)
Find n(F ∪ S)′.
[1]
(c)
A student is chosen at random. Find the probability that the student studies only one of these languages.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
F only 8, both 3, S only 10, neither 11
two regions correct B1 all four correct B1
2
(b)
11
11 B1
1
(c)
916
18 seen M1 916A1
2
Question 5Structured · 5 marks
ℰ = {integers from 1 to 12} A = {multiples of 2} B = {multiples of 5}
Fig. 5.1
(a)
Complete the Venn diagram.
[2]
(b)
List the elements of A ∩ B.
[1]
(c)
Find n((A ∪ B)′).
[1]
(d)
A number is chosen at random from set A. Find the probability that it is also in set B.
[1]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
See diagram
All numbers in correct regions B2 B1 for at least 10 correct
2
(b)
{10}
10 B1
1
(c)
5
5 B1
1
(d)
16
16B1
1
Question 6Structured · 5 marksA* challenge
50 people were asked whether they owned a cat (C) or a dog (D). 29 own a cat, 20 own a dog and 4 own neither. x people own both.
(a)
Write an equation in x and solve it.
[3]
(b)
Two of the people are chosen at random. Find the probability that both own a dog only.
Riyaz Shaikh has mentored IGCSE, IB and A Level students for 15+ years, one-on-one, online and in Mumbai. The hardest questions on this site are the ones he uses to find a student's weak points.
Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in