Sets — Paper 4 · IGCSE Mathematics - International 0607

8 The Venn diagram shows the sets M, E and T. U M E 8 T 4 U = {students at a school} M = {students who study mathematics} E = {students who study English} T = {students who study technology} n M + E + T = 8 ^ h n M , E , T l = 4 ^ h n M + E = 12 , n M + T = 14 and n E + T = 20 ^ h ^ h ^ h n M = 25 , n E = 30 , n T = 35 and n U = 56 ^ h ^ h ^ h ^ h (a) Complete the Venn diagram. [3] (b) Find (i) n M + E l , T l ^ ^ hh, … [1] (ii) n M + T l ^ h. … [1] (c) One of these students is chosen at random. Find the probability that this student studies English and mathematics but not technology. … [2] (d) Two of the 56 students are chosen at random. Find the probability that they both study technology. … [2] (e) A student who studies mathematics is chosen at random. Find the probability that this student also studies technology but not English. … [2] (f) Two students who study English are chosen at random. Find the probability that they both study mathematics but not technology. … [3]

14 marks

Mark scheme: 8(a) Correct values inside circles 3 B2 for 4 or 5 regions correct B1 for 2 or 3 regions correct M E 7 4 6 6 [8] 12 [4] 9 T 8(b)(i) 17 1 FT their diagram 8(b)(ii) 11 1 FT their diagram 8(c) 4 2 FT their 4 oe their 4 p 56 M1 for (k > their 4) or (p < 56) k 56 8(d) 1190 2 35 34 oe M1 for × 3080 56 55 8(e) 6 2 FT their 6 oe their 6 p 25 M1 for (k > their 6) or (p < 25) k 25 8(f) 12 3 their 4 (their 4) − 1 oe M2 for × (their 4 < 30) 870 30 29 a a − 1 or M1 for × (their a < 30) 30 29

This question in 0607/41 May/June 2017

9 In a survey, 40 students are asked if they like football, F, and if they like baseball, B. 22 like football, 19 like baseball and 6 do not like either football or baseball. (a) Complete the Venn diagram to show this information. U F B … … … 6 [2] (b) How many of these students (i) like both football and baseball, … [1] (ii) either like football or do not like baseball? … [1] (c) Find n (F + B l) . … [1] (d) Two of these students are chosen at random. Find the probability that they both like football. … [2] (e) (i) One of the 19 students who like baseball is chosen at random. Find the probability that this student also likes football. … [1] (ii) Two of the 19 students who like baseball are chosen at random. Find the probability that one likes football and one does not like football. … [3] (f) Another n students take part in the survey. They all like both baseball and football. A student is then chosen at random from the (40 + n) students. 5 The probability that a student likes both football and baseball is . 16 Find the value of n. n = … [3] (g) U F B On the Venn diagram, shade the region F l , B l. [1]

15 marks

Mark scheme: 9(a) 15, 7, 12 correctly placed 2 B1 for two correctly placed or M1 for 41 – (40 – 6) seen oe or correct equation 9(b)(i) 7 1 FT their Venn diagram 9(b)(ii) 28 1 FT their Venn diagram 9(c) 15 1 FT their Venn diagram 9(d) 462 2 22 21 oe M1 for × 1560 40 39 9(e)(i) 7 1 FT their Venn diagram 19 9(e)(ii) 168 3 their 7 their 12 their 12 their 7 oe M2 for × + × oe 342 19 18 19 18 or M1 for one of these products 9(f) 8 3 their 7 + n 5 M2 for = oe 40 + n 16 or M1 for at least two trials 9(g) 1

This question in 0607/43 May/June 2017

8 There are 100 students in a year group. Each student studies at least one of the languages, French (F), Italian (I) and Spanish (S). x students study all 3 languages. y students study French only. 18 students study Italian only. 4 students study French and Italian but not Spanish. 12 students study French and Spanish but not Italian. 2 students study Italian and Spanish but not French. 74 students study only one language. (a) Show this information on the Venn diagram. U F I x S [2] (b) Twice as many students study French as Italian. Find the number of students who study (i) all 3 subjects, x = … [2] (ii) French only, y = … [2] (iii) Spanish only. … [1]

7 marks

Mark scheme: 8(a) Correct Venn diagram 2 B1 for 2, 4, 12, 18 correct B1 for y and 56 – y correct oe 4 18 y 8 12 2 56 – y 8(b)(i) 8 2 M1 for 100 = 74 + 18 + x oe 8(b)(ii) 40 2 M1 for 16 + their ( x ) + y = 2(24 + their ( x )) oe 8(b)(iii) 16 1 FT 56 – their (b)(ii) (their (b)(ii) ⩽ 56)

This question in 0607/42 Oct/Nov 2019

8 A dance club has 90 members. Here is some information about types of dancing members like. 50 like Ballroom (B) 37 like Latin (L) 47 like Modern (M) 18 like Ballroom and Latin 15 like Ballroom and Modern 22 like Latin and Modern 8 like Ballroom, Latin and Modern (a) Complete the Venn diagram. U B L 10 8 M [2] (b) Write down the number of members who do not like any of these three types of dancing. … [1] (c) Two of the 90 members are chosen at random. Find the probability that they both like Ballroom and Latin but not Modern. … [2] (d) Two of the members who like Ballroom are chosen. Find the probability that one of these members likes Latin but not Modern and the other likes Modern but not Latin. … [3]

8 marks

Mark scheme: 8(a) Correct Diagram 2 Condone 3 omitted from outside region B1 for 3 subsets correct 25 (10) 5 7 (8) 14 3 18 8(b) 3 1 8(c) 90 2 10 9 oe M1 for × 8010 90 89 8(d) 140 3 10 their 7 their 7 10 M2 for × + × 2450 50 49 50 49 oe or M1 for one of these products

This question in 0607/43 Oct/Nov 2019

8 (a) When the weather is fine, the probability that Sara goes to the park is 0.9 . When the weather is not fine, the probability that Sara goes to the park is 0.2 . On any day, the probability that the weather is fine is 0.7 . (i) Complete the tree diagram. Weather Park Sara … goes Fine 0.7 … Sara does not go Sara … goes … Not fine … Sara does not go [3] (ii) Find the probability that, on any day, Sara goes to the park. … [3] (b) 30 students are asked if they like Mathematics (M) and if they like English (E). The Venn diagram shows the number of students in each subset. U M E 7 17 5 1 (i) Find n ( M , E l) . … [1] (ii) Two students are chosen at random. Find the probability that they both like Mathematics but not English. … [3]

10 marks

This question in 0607/41 May/June 2020

9 (a) The Venn diagram shows information about 115 people who play musical instruments. F = {people who play the flute} D = {people who play the drums} U F D 3x x + 2 4x + 1 8 (i) Calculate the number of people who play both the flute and the drums. … [3] (ii) On the Venn diagram, shade F l + D . [1] (iii) Briony plays both the flute and the drums. Use set notation to complete the statement. Briony … ( F + D ) [1] (b) Briony has 6 red socks, 4 green socks and 8 white socks. (i) She picks a sock at random. Find the probability that the sock is green. … [1] (ii) Briony replaces the sock. She now picks two socks at random, without replacement. Calculate the probability that the two socks are different colours. … [4]

10 marks

Mark scheme: 9(a)(i) 15 nfww 3 M2 for 8x = 104 or better or M1 for 3x + x + 2 + 4x + 1 + 8 [=115] oe If 0 scored, SC1 for 16 as final answer 9(a)(ii) correct shading 1 9(a)(iii) ∈ 1 9(b)(i) 2 1 oe 9 9(b)(ii) 104 4 M3 for oe 6 4 6 8 4 6 4 8 8 6 8 4 153 × + × + × + × + × + × 18 17 18 17 18 17 18 17 18 17 18 17 oe or M2 for 4 or 5 correct products added or M1 for 2 or 3 correct products added OR 6 12 4 14 8 10 M3 for × + × + × 18 17 18 17 18 17 or M2 for 2 correct products added or M1 for 1 correct product OR  6 5 4 3 8 7  M3 for 1 –  × + × + ×   18 17 18 17 18 17  or M2 for 1 – (two correct products added) or M1 for 1 – one correct product 52 If 0 scored SC1 for final answer oe 81

This question in 0607/42 Feb/March 2021

6 The Venn diagram shows the sets P, F and M. U P F M U = {integer values of x ; 2 G x G 12 } P = {prime numbers} F = {factors of 12} M = {multiples of 3} (a) List the elements of set P and the elements of set F. P = … F = … [2] (b) Write each element of U in the correct region of the Venn diagram. [2] (c) List the elements of (i) F , M , … [1] (ii) P l + M , … [1] (iii) ( P , F , M ) l. … [1] (d) Find n (( P + F ) l + M ) . … [1]

8 marks

Mark scheme: 6(a) [P =] 2, 3, 5, 7, 11 2 B1 for each [F =] 2, 3, 4, 6, 12 6(b) P F 2 FT their (a) B1 for at least 8 values correct 2 5 7 11 4 3 6 12 8 9 M 10 6(c)(i) 2, 3, 4, 6, 9, 12 1 FT their Venn diagram 6(c)(ii) 6, 9, 12 1 FT their Venn diagram 6(c)(iii) 8, 10 1 FT their Venn diagram 6(d) 3 1 FT their Venn diagram

This question in 0607/43 Oct/Nov 2021

10 (a) (i) In the Venn diagram, shade the region P , Q l. U P Q [1] (ii) Use set notation to describe the shaded region in the Venn diagram. U A B C … [1] (b) 20 students are asked if they like swimming (S ) and if they like tennis (T ). The Venn diagram shows the results. U S T 6 8 4 2 (i) How many students like swimming or tennis but not both? … [1] (ii) Find n ( S , T ) . … [1] (iii) One of the 20 students is chosen at random. Find the probability that this student likes swimming and tennis. … [1] (iv) Two of the 20 students are chosen at random. Find the probability that they both like tennis. … [2] (v) Two of the students who like swimming are chosen at random. Find the probability that (a) they both like tennis … [2] (b) one likes swimming only and one likes swimming and tennis. … [3]

12 marks

Mark scheme: 10(a)(i) 1 10(a)(ii)  A  B   C oe 1 10(b)(i) 10 1 10(b)(ii) 18 1 10(b)(iii) 8 1 oe 20 10(b)(iv) 33 2 n n  1 oe M1 for  , n  20 95 20 19 10(b)(v)(a) 4 2 n n  1 M1 for  , n  14 13 14 13 10(b)(v)(b) 48 3 6 8 M2 for 8 + 6 oe 91 14 13 14 13 n n or M1 for [2 × 14 oe n  14 ]14 13

This question in 0607/41 May/June 2023

9 (a) For each Venn diagram, shade the given set. U U A B P Q A + B P l , Q l [2] (b) There are 120 students in a year group. The Venn diagram below shows the number of students who study History (H), Geography (G) and Economics (E). U H G 25 9 32 4 7 5 17 x E (i) Find the value of x. … [1] (ii) One of the 120 students is chosen at random. Find the probability that this student studies both History and Geography. … [1] (iii) Two of the students who study Economics are chosen at random. Find the probability that one of these students also studies Geography but not History and the other student also studies History but not Geography. … [3] (iv) Three of the 120 students are chosen at random. Find the probability that two students study exactly two of the subjects and the other student studies all three subjects. … [3]

10 marks

Mark scheme: 9(a) U 2 B1 for each A B 9(b)(i) 21 1 9(b)(ii) 13 1 oe 120 9(b)(iii) 35 3 5 7 7 5 oe M2 for  +  oe 528 33 32 33 32 or M1 for either product correct 9(b)(iv) 3 3 21 20 4 oe M2 for k    oe 1003 120 119 118 k = 1, 3 or 6 p p − 1 4 or M1 for   oe 120 119 118

This question in 0607/43 Oct/Nov 2023

7 (a) Spinner A and spinner B are each fair 5-sided spinners. Spinner A is numbered 1, 2, 2, 3, 4. Spinner B is numbered 1, 2, 3, 4, 4. The two spinners are each spun once and the number on each spinner is recorded. Find the probability that (i) the number on spinner A is 6 … [1] (ii) the number on spinner B is not 4 … [1] (iii) the number on spinner A is the same as the number on spinner B … [3] (iv) the sum of the two numbers is 6. … [3] (b) (i) On the Venn diagram, shade A , B . U A B [1] (ii) Describe the shaded region using set notation. U A B … [1] (iii) The Venn diagram below shows the number of elements in each subset. U A B 3 8 4 1 6 9 7 5 C Find n ( A + B) + C l ` j. … [1]

11 marks

Mark scheme: 7(a)(i) 0 1 7(a)(ii) 3 1 oe 5 7(a)(iii) 6 3  1 1   2 1   1 1   1 2  oe M2 for                25  5 5   5 5   5 5   5 5  or correct sample space showing all 6 points or M1 for 2 correct products or correct sample space showing at least 3 points or list of all correct pairs 12 If 0 scored, SC1 for 25 7(a)(iv) 6 3  2 2   1 1   1 1  M2 for            25  5 5   5 5   5 5  or correct sample space showing all 6 points or M1 for 2 correct products or correct sample space showing at least 3 or for listing pairs that sum to 6 7(b)(i) 1 7(b)(ii) A  B / 1 7(b)(iii) 8 1

This question in 0607/41 May/June 2024

7 (a) There are 49 students in a year group. Each student studies at least one of the sciences, biology (B), chemistry (C) and physics (P). x students study all 3 sciences. y students study chemistry only. 12 students study physics only. 6 students study biology and chemistry but not physics. 11 students study biology and physics but not chemistry. 2 students study physics and chemistry but not biology. 25 students study only one science. (i) Show this information on the Venn diagram. U B x C P [2] (ii) Find the number of students who study all 3 sciences. … [2] (iii) The number of students that study biology is two times the number of students that study chemistry. Find the number of students who study (a) chemistry only … [2] (b) biology only. … [1] (b) A bag contains 7 red balls and 3 blue balls. In an experiment, three balls are chosen at random without replacement. Find the probability that at least two of the balls chosen are red. … [4]

11 marks

Mark scheme: 7(a)(i) 2 B1 for 4 correct regions 7(a)(ii) 5 2 M1 for 49 – 25 – 11 – 6 – 2 oe 7(a)(iii)(a) 3 2 M1 for either 30 + x − y or 2(8 + x + y ) oe 7(a)(iii)(b) 10 1 FT 13 – their 3, must be an integer 7(b) 49 4  7 6 5   7 6 3  oe M3 for     + 3      60  10 9 8   10 9 8   7 6 5   7 6 3  or M2 for     and      10 9 8   10 9 8   7 6 3  or 3       10 9 8   7 6 5   7 6 3  or M1 for     or      10 9 8   10 9 8 

This question in 0607/41 Oct/Nov 2024