O-LevelMathematicsNumber4024/21 · May/June 2023 · Q512 Marks

= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} P = {x : x is a multiple of 3} Q = {x: x is an odd number} R = {x:x is a factor of 24}

(a)(i)Complete the Venn diagram. [Figure 5.1][3]
(a)(ii)Find n(R').[1]
(a)(iii)List the elements of (P ∪ R) ∩ Q.[1]
(a)(iv)Describe, in words, the type of number represented by P' ∩ R ∩ Q'.[1]
(a)(v)A number, m, is chosen at random from the elements of R. Find the probability that m is a multiple of 3.[2]
(b)(i)M = 2²ˣ ×3⁴×5×7 N = 2³ × 3ˣ⁻ʸ × 5² The lowest common multiple (LCM) of M and N is 2⁸×3⁶×5²×7. Find the value of x and the value of y.[2]
(b)(ii)Find the largest square number that is a factor of M.[1]
(b)(iii)Find the highest common factor (HCF) of M and N. Give your answer as a product of its prime factors.[1]

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Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

[3] (a)(i) P: {3, 6, 9, 12}, Q: {1, 3, 5, 7, 9, 11}, R: {1, 2, 3, 4, 6, 8, 12} Common elements: P∩Q={3,9}, P∩R={3,6,12}, Q∩R={1,3}. P∩Q∩R={3}. Venn diagram with numbers in correct regions. B2 for Venn diagram with 1 error, omission or repeat or B1 for Venn diagram with 2 or 3 errors, omissions or repeats
[1] (a)(ii) 5
[1] (a)(iii) 1, 3, 9 FT their Venn diagram
[1] (a)(iv) Multiples of 6 oe
[2] (a)(v) 3/7 oe B1FT for answer k/their7 or their3/k
[2] (b)(i) x = 4 y = -2 B1 for x = 4 or B1FT for y = (their x) – 6 their x must be a positive integer
[1] (b)(ii) 20736
[1] (b)(iii) 2³ × 3⁴ × 5 cao

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