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4024/21 May/Jun 2023, Question 5: Three-Set Venn Diagrams

Cambridge O Level Mathematics 4024 · Paper 21 (Calculator) · Calculator · 12 marks

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The question, in short

Using sets P, Q and R inside ℰ = {1,...,12}, complete a Venn diagram and answer set questions; then use prime factor forms of M and N to find x, y, a square factor and the HCF.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (ai) · 3 marks

Place every element of ℰ into the correct region of the three-circle Venn diagram, keeping the given 1 and 2.

  1. 1P = {3, 6, 9, 12}, Q = {1, 3, 5, 7, 9, 11}, R = {1, 2, 3, 4, 6, 8, 12}
  2. 2P ∩ Q ∩ R: 3; P ∩ R only: 6, 12; P ∩ Q only: 9; P only: none
  3. 3Q ∩ R only: 1; Q only: 5, 7, 11; R only: 2, 4, 8
  4. 4Outside the circles: 10

AnswerP only: none Q only: 5, 7, 11 R only: 2, 4, 8 P ∩ Q only: 9 P ∩ R only: 6, 12 Q ∩ R only: 1 P ∩ Q ∩ R: 3 Outside the circles: 10

Part (aii) · 1 mark

State how many elements of ℰ lie outside set R.

  1. 1n(R) = 7
  2. 2n(R′) = 12 − 7 = 5

Answer5

Part (aiii) · 1 mark

Write down the elements common to Q and to the union of P with R.

  1. 1P ∪ R = {1, 2, 3, 4, 6, 8, 9, 12}
  2. 2Odd ones: (P ∪ R) ∩ Q = {1, 3, 9}

Answer1, 3, 9

Part (aiv) · 1 mark

In words, name the kind of number found in P and R but not in Q.

  1. 1P ∩ R ∩ Q′ = {6, 12}: even multiples of 3
  2. 2Multiples of 6

AnswerMultiples of 6

Part (av) · 2 marks

A member of R is picked at random; work out the probability it is divisible by 3.

  1. 1R = factors of 24 in ℰ = {1, 2, 3, 4, 6, 8, 12}, so 7 numbers.
  2. 2Multiples of 3 in R: 3, 6, 12 — that is 3 numbers.
  3. 3P(multiple of 3) = 3/7

Answer3/7

Part (bi) · 2 marks

Using the given LCM and prime factor forms, determine the values of x and y.

  1. 1LCM power of 2: 2x = 8, so x = 4
  2. 2LCM power of 3 is 6 and M has 3⁴, so x − y = 6
  3. 34 − y = 6, so y = −2
  4. 4x = 4, y = −2

Answerx = 4, y = −2

Part (bii) · 1 mark

Identify the greatest perfect square that divides M.

  1. 1M = 2⁸ × 3⁴ × 5 × 7
  2. 2Largest square factor = 2⁸ × 3⁴ = 256 × 81 = 20 736

Answer20736

Part (biii) · 1 mark

Express the HCF of M and N as a product of primes.

  1. 1N = 2³ × 3⁶ × 5²
  2. 2HCF takes the lower power of each prime: 2³, 3⁴, 5¹
  3. 3HCF = 2³ × 3⁴ × 5

Answer2³ × 3⁴ × 5

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Also searched as: 4024/21/M/J/23 · 4024_s23_qp_21 · 4024 s23 21 · question 5.
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