Cambridge IGCSE Mathematics 0580 — 2012 Oct/Nov Paper 4 · Variant 2

9 (a) = {25 students in a class} For Examiner's F = {students who study French} Use S = {students who study Spanish} 16 students study French and 18 students study Spanish. 2 students study neither of these. (i) Complete the Venn diagram to show this information. F S ..... ..... ..... ..... [2] (ii) Find n(F ' ). Answer(a)(ii) [1] (iii) Find n(F ∩ S)'. Answer(a)(iii) [1] (iv) One student is chosen at random. Find the probability that this student studies both French and Spanish. Answer(a)(iv) [1] (v) Two students are chosen at random without replacement. Find the probability that they both study only Spanish. Answer(a)(v) [2] (b) In another class the students all study at least one language from French, German and Spanish. For Examiner's Use No student studies all three languages. The set of students who study German is a proper subset of the set of students who study French. 4 students study both French and German. 12 students study Spanish but not French. 9 students study French but not Spanish. A total of 16 students study French. (i) Draw a Venn diagram to represent this information. [4] (ii) Find the total number of students in this class. Answer(b)(ii) [1]

Mark scheme: 9 (a) (i) 2 B1 for 2 outside of circles in diagram or all three of 5, 11, 7 correctly placed F S 5 11 7 2 (ii) 9 1ft ft their 2 + their 7 (iii) 14 1 11 1ft ft their 11 from diagram / 25 (iv) 25 42 7 (v) 600 oe =100 2ft isw incorrect cancelling ft their 7 from diagram for numerator their 7 their ( 7 − )1 M1 for × 25 24 their 7 their ( 7 ) After 0 scored, SC1 for × 25 25 IGCSE – October/November 2012 0580 42 (b) (i) F S 5 7 12 G 44 B1 for any correct diagram with blanks or F G zeros where needed and labelled 5 4 unambiguously 4 B1 for 4 in correct place 7 B1 for 12 in correct place B1 for 5 and 7 in correct place 12 S S F G 12 7 5 4 (ii) 28 1ft Correct or ft from their diagram

More questions on Sets

10 Consecutive integers are set out in rows in a grid. For Examiner's Use (a) This grid has 5 columns. 1 2 3 4 5 6 7 8 9 10 a b 11 12 13 14 15 n 16 17 18 19 20 c d 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 The shape drawn encloses five numbers 7, 9, 13, 17 and 19. This is the n = 13 shape. In this shape, a = 7, b = 9, c = 17 and d = 19. (i) Calculate bc O ad for the n = 13 shape. Answer(a)(i) [1] (ii) For the 5 column grid, a = n O 6. Write down b, c and d in terms of n for this grid. Answer(a)(ii) b = c = d = [2] (iii) Write down bc O ad in terms of n. Show clearly that it simplifies to 20. Answer(a)(iii) [2] (b) This grid has 6 columns. The shape is drawn for n = 10. For Examiner's Use 1 2 3 4 5 6 a b 7 8 9 10 11 12 n 13 14 15 16 17 18 c d 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 (i) Calculate the value of bc O ad for n = 10. Answer(b)(i) [1] (ii) Without simplifying, write down bc O ad in terms of n for this grid. Answer(b)(ii) [2] (c) This grid has 7 columns. 1 2 3 4 5 6 7 a b 8 9 10 11 12 13 14 n 15 16 17 18 19 20 21 c d 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Show clearly that bc O ad = 28 for n = 17. Answer(c) [1] Question 10 continues on the next page. (d) Write down the value of bc O ad when there are t columns in the grid. For Examiner's Use Answer(d) [1] (e) Find the values of c, d and bc O ad for this shape. 2 3 4 16 c d Answer (e) c = d = bc O ad = [2]

Mark scheme: 10 (a) (i) 20 1 (ii) n – 4 oe Accept unsimplified n + 4 oe n + 6 oe 2 B1 for two correct (iii) (n – 4)(n + 4) – (n – 6)(n + 6) M1 ft from their algebraic expressions can be implied by n2 – 4n + 4n – 16 – (n2 – 6n + 6n – 36) or n2 – 16 – (n2 – 36) n2 – 4n + 4n – 16 – (n2 – 6n + 6n Must have a line of algebra – 36) or better 20 E1 With no errors or omission of brackets (b) (i) 24 1 IGCSE – October/November 2012 0580 42 (ii) (n – 5)(n + 5) – (n – 7)(n + 7) 2 M1 for n – 5, n + 5, n – 7, n + 7 seen isw or n2 – 25 – (n2 – 49) isw or n2 – 25 – n2 + 49 isw (c) (11 × 23) – (9 × 25) Allow algebraic solution from 253 – 225 (n – 6)(n + 6) – (n – 8)(n + 8) [= 28] E1 (d) 4t oe 1 Accept unsimplified e.g. n2 – (t – 1)2 – [n2 – (t + 1)2] (e) c = 28 and d = 30 1 52 1

More questions on The four operations