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IGCSE Maths · Edexcel 4XMA1 · Unit 1 Higher · Number

Sets, Notation and Venn Diagrams — Practice Questions

12 exam-style questions (40 marks), building from warm-up to grade 8–9. Show full working: Edexcel gives method marks even when the final answer is wrong.

Section A — Warm-up (grades 4–5)
  1. (2 marks)ξ = {1, 2, 3, …, 12}. A = {multiples of 3}. B = {factors of 12}.
    (a) List the members of A ∩ B.
    (b) Find n(A ∪ B).
  2. (2 marks)Using the sets in Question 1:
    (a) List the members of A′ ∩ B.
    (b) Is it true that 9 ∈ B? Give a reason.
  3. (2 marks)Use set notation to describe the shaded region in each diagram.
    ξAB
    (a)
    ξAB
    (b)
  4. (4 marks)ξ = {x : x is an integer, 1 ≤ x ≤ 15}. P = {prime numbers}. E = {even numbers}.
    (a) List the members of P ∩ E.
    (b) Find n(P′ ∩ E′).
    (c) A number is chosen at random from ξ. Find the probability that it is in P ∪ E.
Section B — Core (grades 5–7)
  1. (4 marks)50 people were asked whether they like coffee or tea. 32 like coffee, 21 like tea and 9 like neither.
    (a) Draw a Venn diagram to show this information.
    (b) One of the people is chosen at random. Find the probability that they like tea but not coffee.
    HintFind how many like at least one drink first.
  2. (4 marks)n(ξ) = 60. The Venn diagram shows the number of elements in each region.
    ξAB2xx + 33x − 110
    (a) Find the value of x.
    (b) Find n(B).
  3. (5 marks)100 students study at least one of French (F), German (G) and Spanish (S), or none of them.
    5 study all three. 12 study French and German. 15 study French and Spanish. 10 study German and Spanish.
    45 study French, 38 study German and 40 study Spanish.
    (a) Complete a three-set Venn diagram.
    (b) How many students study none of the three languages?
    (c) Find the probability that a student chosen at random studies exactly two languages.
    HintStart with the 5 in the centre, then subtract it from each pair.
  4. (3 marks)A = {x : x is an integer, x² < 10}. B = {x : x is an integer, 0 < x < 4}.
    (a) List the members of A.
    (b) Explain why B ⊂ A.
  5. (4 marks)The Venn diagram shows the number of students in a group of 40 who have a cat (C) or a dog (D).
    ξCD1461010
    A student is chosen at random. Find
    (a) P(C)
    (b) P(C ∩ D′)
    (c) the probability that the student has a dog, given that they have a cat.
Section C — Stretch (grades 8–9)
  1. (3 marks)ξ = {x : x is an integer, 1 ≤ x ≤ 30}. A = {square numbers}. B = {multiples of 4}.
    Find n((A ∪ B)′).
  2. (3 marks)A and B are any two sets. For each statement, write always true, sometimes true or never true.
    (a) A ∩ B ⊂ A ∪ B
    (b) n(A ∪ B) = n(A) + n(B)
    (c) A ∪ A′ = ∅
  3. (4 marks)n(ξ) = 48, n(A) = 25, n(B) = 20.
    n(A ∩ B) = x and n(A′ ∩ B′) = 2x.
    (a) Find the value of x.
    (b) An element is chosen at random from ξ. Find the probability that it is in B but not in A.
    HintWrite n(A ∪ B) in terms of x first.
When you’re doneRevise the method in the notes, then mark your answers with the worked solutions.
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