The universal set is E = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. A = {1, 2, 3, 4, 5}. B = {4, 5, 6, 7}. Write down the elements of A ∩ B.
(Total for Question 1 is 1 mark)
2
Using the sets given in Question 1, write down the elements of A ∪ B.
(Total for Question 2 is 1 mark)
3
Using the sets in Question 1, find n(A ∪ B).
(Total for Question 3 is 1 mark)
4
Using the sets in Question 1, write down the elements of A' (the complement of A within E).
(Total for Question 4 is 1 mark)
5
Priya says that 8 is a member of set A (using the sets in Question 1). Is she correct? Give a reason for your answer.
(Total for Question 5 is 1 mark)
6
Ms Nowak's form group has 20 pupils. 9 of the pupils play a musical instrument (M) and 11 pupils are in a sports team (T). 4 pupils do both. Find n(M ∪ T).
(Total for Question 6 is 2 marks)
7
Using the information in Question 6, find the number of pupils in the form group who do neither.
(Total for Question 7 is 1 mark)
8
On a two-set Venn diagram showing sets A and B inside E, describe in words the region that represents A ∩ B'.
(Total for Question 8 is 1 mark)
9
Describe in words the region that represents A' ∩ B'.
(Total for Question 9 is 1 mark)
10
A class of 24 students was asked whether they study French (F) or Spanish (S). The Venn diagram shows: French only = 6, both French and Spanish = 4, Spanish only = 9, and neither = 5. Write down the number of students who study both French and Spanish.
(Total for Question 10 is 1 mark)
11
Using the diagram in Question 10, write down n(F).
(Total for Question 11 is 1 mark)
12
Using the diagram in Question 10, find the probability that a student, chosen at random, studies French only.
(Total for Question 12 is 1 mark)
13
Using the diagram in Question 10, find P(F ∩ S).
(Total for Question 13 is 1 mark)
14
Using the diagram in Question 10, find P(F').
(Total for Question 14 is 1 mark)
15
A survey of 50 Year 11 students at a school in Leeds asked whether they study Drama (D) and/or Music (M). 28 students study Drama, 19 students study Music, and 8 students study neither.
(a)Find the number of students who study both Drama and Music.(3)
(b)Find the probability that a randomly selected student studies Drama only.(2)
(Total for Question 15 is 5 marks)
16
The universal set is E = {1, 2, 3, ..., 20}. A = {factors of 20}. B = {multiples of 5}.
(a)List the elements of set A.(1)
(b)List the elements of set B.(1)
(c)Find A ∩ B.(1)
(Total for Question 16 is 3 marks)
17
Chidinma organises a fun run in Bristol with 90 entrants. Every entrant runs the 5 km race (R), the 10 km race (F), or both; nobody runs neither. n(R) = 63 and n(F) = 47.
(a)Find n(R ∩ F).(2)
(b)Find the probability that a randomly chosen entrant ran only the 10 km race.(2)
(Total for Question 17 is 4 marks)
18
In a class of 32 students, x study French only, 2x study Spanish only, x study both French and Spanish, and 4 students study neither.
(a)Form an equation in x and solve it to find the value of x.(3)
(b)Hence write down the number of students who study French.(1)
(Total for Question 18 is 4 marks)
19
Dr Kaur, a vet in Cardiff, surveys 60 pet owners. 35 own a dog (D), 24 own a cat (C), and 10 own both a dog and a cat. One of the dog owners is chosen at random. Find the probability that this dog owner also owns a cat.
(Total for Question 19 is 3 marks)
20
The Venn diagram shows solar panels (S) and heat pumps (H) for the 48 houses on a street in Manchester. S only = 3y, H only = y, both S and H = 2y, and 12 houses have neither.
(a)Form an equation in y and solve it.(3)
(b)A house is selected at random. Find P(S).(1)
(Total for Question 20 is 4 marks)
21
60 students were surveyed about which sports they play: football (F), basketball (B) and rugby (R). The three-circle Venn diagram shows: F only = 14, B only = 9, R only = 7, F and B only = 6, F and R only = 5, B and R only = 4, all three sports = 3, and none of the three = 12.
(a)Find the number of students who play at least two of the three sports.(2)
(b)Find the probability that a randomly selected student plays exactly one sport.(2)
(Total for Question 21 is 4 marks)
22
A gym in Glasgow, run by manager Callum, has 44 members in total. Of these, x members use the pool only, x2 members use both the pool and the gym floor, and (2x + 1) members use the gym floor only. The remaining 3 members use neither facility.
(a)Show that x2 + 3x - 40 = 0.(2)
(b)Solve the equation to find the value of x, explaining why the other solution is not valid.(2)
(c)Hence find the probability that a randomly selected member uses the pool (including those who use both facilities).(1)
(Total for Question 22 is 5 marks)
23
A and B are sets such that A is a subset of B (written A ⊆ B, meaning every element of A is also in B). State, in terms of A and B, what A ∩ B and A ∪ B simplify to, and justify your answer using a numerical example.
(Total for Question 23 is 3 marks)
Mark scheme · 5.23D Venn Diagrams: Fluency and Exam Drill
Question 1
B1 {4, 5} oe (any order)
Answer: A ∩ B = {4, 5}
Question 2
B1 {1, 2, 3, 4, 5, 6, 7} cao
Answer: A ∪ B = {1, 2, 3, 4, 5, 6, 7}
Question 3
B1 7 cao
Answer: 7
Question 4
B1 {6, 7, 8, 9, 10} oe
Answer: A' = {6, 7, 8, 9, 10}
Question 5
B1 No, with correct reason, e.g. A = {1,2,3,4,5} does not contain 8, so 8 is not a member of A
Answer: No; 8 is not a member of A
Question 6
M1 9 + 11 - 4 oe (uses n(M) + n(T) - n(M ∩ T))
A1 16 cao
Answer: 16
Question 7
B1 4 cao, ft from Question 6
Answer: 4
Question 8
B1 correct description, e.g. the elements in A but not in B (the 'A only' region)
Answer: The elements that are in A but not in B (the A only region).
Question 9
B1 correct description, e.g. the elements outside both circles, in neither A nor B
Answer: The elements outside both circles, i.e. in neither A nor B.
Question 10
B1 4 cao
Answer: 4
Question 11
B1 10 cao
Answer: 10
Question 12
B1 6/24 oe (= 1/4)
Answer: 6/24 = 1/4
Question 13
B1 4/24 oe (= 1/6)
Answer: 4/24 = 1/6
Question 14
B1 14/24 oe (= 7/12), ft from n(F)
Answer: 14/24 = 7/12
Question 15
(a) M1 50 - 8 (= 42) used for n(D ∪ M)
(a) M1 28 + 19 - 42 (oe) to find the overlap
(a) A1 5 cao
(a) Answer: 5
(b) M1 28 - 5 (= 23) oe, ft from part (a)
(b) A1 23/50 cao
(b) Answer: 23/50
Question 16
(a) B1 {1, 2, 4, 5, 10, 20} cao (any order)
(a) Answer: A = {1, 2, 4, 5, 10, 20}
(b) B1 {5, 10, 15, 20} cao
(b) Answer: B = {5, 10, 15, 20}
(c) B1 {5, 10, 20} cao, ft from parts (a) and (b)
(c) Answer: A ∩ B = {5, 10, 20}
Question 17
(a) M1 63 + 47 - 90 oe
(a) A1 20 cao
(a) Answer: 20
(b) M1 47 - 20 (= 27) oe, ft from part (a)
(b) A1 27/90 oe (= 3/10)
(b) Answer: 27/90 = 3/10
Question 18
(a) M1 x + 2x + x + 4 = 32 oe (correct equation formed)
(a) M1 4x = 28 (oe, correct simplification)
(a) A1 x = 7 cao
(a) Answer: x = 7
(b) B1 14 cao, ft from part (a)
(b) Answer: 14
Question 19
M1 identifies 35 (the dog owners) as the reduced sample space
M1 10 identified as the number who own both a dog and a cat