Sets, Vectors and Functions [CIE Extended Mathematics 0580 (2017-2018) Sets, Vectors and Functions]

1	2017/m/22/q17 (a) A total of 20 trucks were tested at a checkpoint. • 6 trucks failed the test for brakes (B) • 7 trucks failed the test for lights (L) • 9 trucks passed the tests for both brakes and lights. (i) Complete the Venn diagram. [2] (ii) Find \( n(B' \cap L') \). [1] (b) In the Venn diagram below, shade the region \((P \cup Q) \cap Q'\). [1]
2	2017/s/22/q23 (a) \(\mathcal{E} = \{ \text{students in a class} \}\)      \(P = \{ \text{students who study physics} \}\)      \(C = \{ \text{students who study chemistry} \}\)  The Venn diagram shows numbers of students. (i) Find the number of students who study physics or chemistry. [1] (ii) Find \( n(P \cap C') \). [1] (iii) A student who does not study chemistry is chosen at random. Find the probability that this student does not study physics. [1] (b) On the Venn diagram below, shade the region \( D \cup E' \). [1]
3	2018/m/22/q19 (a) Use set notation to complete the statements for the Venn diagram above. (i) \( c \ \ldots\ldots\ X \) [1] (ii) \( \ldots\ldots\ = \{a, m, e\} \) [1] (iii) \( Y \cap Z = \ldots\ldots \) [1] (b) List the elements of \((X \cup Y \cup Z)'\). [1] (c) Find \( n(X' \cap Z) \). [1]
4	2018/s/22/q23 The Venn diagram shows information about the number of elements in sets A, B and $\mathcal{E}$ (a) $n(A \cup B) = 23$ Find the value of x.  [1] (b) An element is chosen at random from $\mathcal{E}$ Find the probability that this element is in ${A \cup B}'$ [2]
5	2018/w/22/q7  x = {x: is an integer and 5 < x < 12} , D={5,10} (a) Put a ring around the correct statement from the list below. $D = ∅\quad \quad C \cap D={10} \quad \quad 6 ∈ D \quad \quad D ⊂ C$ [1] (b) Find $n(C \cup D)$.  [1]
6	2018/s/23/q12 \( n(\mathcal{E}) = 20,\ n(P) = 10,\ n(Q) = 13 \) and \( n(P \cup Q') = 5 \)  Work out \( n(P \cap Q) \). You may use the Venn diagram to help you. [2]
7	2017/s/21/q17 (a) In this Venn diagram, shade the region \( F \cup G' \). [1] (b) \(\mathcal{E} = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}\) \(A = \{x: x \text{ is an odd number}\}\) \(B = \{x: x \text{ is a square number}\}\) \(C = \{x: x \text{ is a multiple of } 3\}\) (i) Write all the elements of \(\mathcal{E}\) in the Venn diagram below. [2] (ii) Another number is included in the set \(\mathcal{E}\). This number is in the region \( A \cap B \cap C \). Write down a possible value for this number. [1]
8	2017/w/21/q15 (a) \(Q = \{1, 2, 3, 4, 5, 6\}\) Write down a set \( P \) where \( P \subseteq Q \). [1] (b) Shade these regions in the Venn diagrams. [2]
	to be continued....

Answer Key

1. (a)(i) Figure (ii) 9 (b) Figure
2. (a)(i) 24 (ii) 5 (iii) \( \frac{7}{12} \) (b) Figure
3. (a)(i) \( \in \) (ii) \( X \cap Y \) (iii) \( \varnothing \) (b) \( u, v, w \) (c) 5
4. (a) 5 (b) \( \frac{7}{30} \)
5. (a) \( C \cap D=\{10\} \) (b) 7
6. 8
7. (a) diagram (b)(i) diagram (ii) Any even square number that is also a multiple of 3
8. (a)Fewer than 6 elements from \( \{1,2,3,4,5,6\} \) or \( \varnothing \) (b) Diagram
9. (a) Diagram (b)(i) Diagram (b)(ii)3 (b)(iii) \( \varnothing \) or \(\{ \} \)

10. (a) diagram (b) (i) \& (ii) \( \| \) (c) \( 21,23,24,29 \)
(d)(i) 5 (ii) \( 9(\mathrm{e}) \subset \) or \( \subseteq \)
11. (a) \( \frac{94}{200} \) (b) 14.1 or 14.07
12. (a) 27 (b) 3 (c) \( \frac{x-7}{3} \)
13. (a) 10 (b) \( \frac{x+7}{6} \) (c) -2
14. (a) 2 (b) \( 30-4 x \) (c) \( 15-4 x^{2} \)
15. (a)(i) \( 5 x^{3}+2 \)
(ii) \( \frac{x-2}{5} \)
(b) 5
16. (a) \( 3+12 x \) (b) \( 24 x+31 \)
17. \( 25 x^{2}-8 \)
18. (a) -7
(b) \( \frac{4}{64} \) (c) \( \frac{3-x_{i}}{2} \)
(d) \( 4^{3-2 x}, \frac{64}{16^{x}}(\mathrm{e}) \) 1.5
19. (a) -11
(b) 7 (c) 25
(d) \( 9 x^{2}-8 x+2(\mathrm{e}) \frac{x+2}{3} \)
20. (a) 3 (b) \( -\frac{2}{5} \) (c) \( -2 x-7 \) (d) 26 (e) \( \frac{1-x}{2} \) (f) \( p=-20, q=26 \)
21. (a)(i) 1
(ii) 8
(iii) \( \frac{10-x}{x} \)
(iv) 5 (b) \( \frac{-3 x^{2}+5 x+18}{x+1} \)
22. (a)(i)26 (a)(ii) \( x^{2}+4 x+5 \) (a)(iii)5 (a)(iv) \( \frac{x+3}{2} \) (b)(i) \( [0] .70 \) (b)(ii) 4
23. (a) 0 (b) 5
(c) \( \frac{x+1}{2} \)
(d) \( 9 x+16 \)
(e) \( 9 x^{2}+ 24 x+16(\mathrm{f}) 27 \)
24. (a) \( (9,-4) \) (b) -5
25.
(a) Point at \( (3,5) \)
(b) \( \binom{1}{-3} \)
(c) \( \binom{0}{4} \) or \( \binom{0}{-4} \)
26. (a)(i) \( \binom{30}{-20} \)
(ii) \( \binom{-6}{4} \)
(b) -4
27. (a)(i) \( \binom{-19}{-2} \)
(ii) 3.61 or 3.605 to 3.606
(iii) \( [n=]-\frac{1}{2} \) or \( [n=] 2 \frac{1}{2} \) (b)(i)(a) \( -\vec{a}+2 \vec{c} \)
(b) \( \frac{3}{8}(-\vec{a}+2 \vec{c}) \)
(c) \( \frac{1}{2}(5 \vec{a}-2 \vec{c}) \)
(d) \( \frac{1}{8}(5 \vec{a}-2 \vec{c}) \)
(ii) 4
28. (a) \( 2 \vec{a}+\vec{b} \) (b) \( D \) (c) \( \overrightarrow{C F} \) and \( \overrightarrow{B G} \)
29. \( \frac{1}{3} \vec{a}+\frac{2}{3} \vec{b} \)
30. \( 2 \vec{q}+\vec{p} \)
31. (a) \( 6 \vec{a}-2 \vec{b} \) or \( 2(3 \vec{a}-\vec{b}) \)
(b) \( 5 \vec{a}-\vec{b} \)
32. \( \frac{2}{7} \vec{p}+\frac{5}{7} \vec{q} \)
33. \( \frac{2}{3} \vec{p}+\frac{1}{3} \vec{q} \)

34. (a) \( \vec{c}+\frac{2}{3} \vec{a} \) (b)(i) \( \frac{2}{5} \vec{a}+\frac{3}{5} \vec{c} \) (b)(ii) \( 3: 2 \)