Active Maths 2 (Strands 1–5): Ch 1 Solutions
Chapter 1 Exercise 1.1
Q. 1. (i) Not well defined, as the meaning of the term ‘inexpensive’ is subjective.
(ii) Well defined.
(iii) Not well defined, as the meaning of the term ‘large‘ is subjective.
(iv) Well defined.
Q. 2. (i) A ∪ B = {0, 2, 3, 4, 5, 6, 7, 8}
(ii) A ∩ B = {2}
(iii) A’ = {0, 1, 4, 6, 8}
(iv) B’ = {1, 3, 5, 7}
(v) A \ B = {3, 5, 7}
(vi) B \ A = {0, 4, 6, 8}
(vii) (A ∪ B)’ = {1}
(viii) A’ ∪ B = {0, 1, 2, 4, 6, 8}
Q. 3. (a) U
A B
• 21
• 25
• 20 • 22 • 24
• 23
(b) (i) A ∪ B = {21, 23, 25}
(ii) A ∩ B = {23}
(iii) A’ = {20, 22, 24}
(iv) B’ = {20, 21, 22, 24, 25}
(v) (A ∩ B)’ = {20, 21, 22, 24, 25}
(vi) A’ ∪ B’ = {20, 21, 22, 24, 25}
(vii) B \ A = {} or ∅
(viii) (A \ B) ∪ (B \ A) = {21, 25}
(c) A \ B’ = {23}
B \ A’ = {23}
∴ A \ B’ = B \ A’
Answer is yes
(d) As B = A ∩ B
⇒ B ⊂ A
(e) Case 1 # (A ∪ B) = 0 ⇒ # (A ∩ B) = 0 In this case # (A ∪ B) = # (A ∩ B)
Case 2 # (A ∪ B) > 0 and # (A ∩ B) = 0 In this case # (A ∪ B) > # (A ∩ B)
Case 3 # (A ∪ B) > 0 and # (A ∩ B) > 0 As A ∩ B is a subset of A ∪ B,
# (A ∪ B) > # (A ∩ B)
Q. 4. (i) 23 = 8
(ii) {}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}
(iii) {x, y, z}
Q. 5. (i) A ∪ B (iii) (A ∩ B)’
(ii) A ∩ B (iv) B \ A
Q. 6. (i) U
X Y
(ii) U
X Y
(iii)
X Y
U
(iv) U
X Y
1
2 Active Maths 2 (Strands 1–5): Ch 1 Solutions
Q. 7. (a) U = {1, 2, 3, 4, 5, 6, 7, 8}
O = {1, 3, 5, 7}
P = {3, 5, 7}
(b)
• 4 • 6 • 8• 2
U
O P
• 3
• 5
• 7
• 1
(c) (i) 4
(ii) 3
(iii) 3
(iv) 1
(v) 4
(vi) 4
(d) = ; =
Q. 8. (i)
271
U(500)
I(103) S(146)
2083 126
83 + 20 + 126 = 229
500 − 229 = 271
(ii) 83 + 126 = 209
209 ____ 500 × 100 ____ 1 % = 41.8%
Q. 9. (i)
170
U(400)
G(150) V(100)
20130 80
150 + 100 + 170 = 420
420 − 400 = 20
(ii) 20
(iii) 80 ____ 400 × 100 ____ 1 % = 20%
(iv) 130 + 80 = 210
210 ____ 400 × 100 ____ 1 % = 52.5%
Q. 10. (i) {0} is the set containing the element 0 whereas ∅ is the set containing zero elements.
(ii) 0 is the element ‘zero’ whereas {0} is the set containing the element 0.
(iii) X and Y are disjoint (mutually exclusive) sets.
(iv) X and Y are both empty; X = Y = {}.
(v) X ∩ Y = X
(vi) X ∪ Y = Y
(vii) X = Y
Q. 11. (i) U
A B
Ans: A
(ii) U
A B
Ans: A ∩ B
(iii) U
A B
Ans: A ∪ B
(iv) U
A B
Ans: ∅
3Active Maths 2 (Strands 1–5): Ch 1 Solutions
Q. 12. (i) U
A B
(ii) U
A B
(iii) D
= (A \ B) ∪ (B \ A)
= (A ∪ B) \ (A ∩ B)
Q. 13. (i) A3 = {4k: k ∈ N} = {4, 8, 12, 16, 20, ……}
∴ First five elements are 4, 8, 12, 16, 20.
(ii) A1 = {2k: k ∈ N} = {2, 4, 6, 8, 10, 12, ……}
A2 = {3k: k ∈ N} = {3, 6, 9, 12, 15, 18, …….}
∴ A1 ∩ A2 = {6, 12, 18, 24, ….} = A5
Q. 14. (i) U(430)
H(250)
x
G(240)
3x250 – 3x 240 – 3x
(ii) Total Number = 250 + 240 − 3x + x
Total Number = 490 − 2x
(iii) 490 − 2x = 430
60 = 2x
x = 30
Exercise 1.2
Q. 1. (i)
B
U
C
A
(ii) B
U
C
A
(iii)
B
U
C
A
(iv)
B
U
C
A
(v)
B
U
C
A
4 Active Maths 2 (Strands 1–5): Ch 1 Solutions
(vi) B
U
C
A
Q. 2. (i) A ∩ B ∩ C
(ii) A \ (B ∪ C)
(iii) (B ∩ C) \ A
(iv) (A ∪ B ∪ C)’
Q. 3. (i) {4, 5}
(ii) {2, 8}
(iii) {8}
(iv) {2}
(v) {2, 3, 4, 5, 6, 8, 9}
(vi) {7}
Q. 4. (i) {0, 1, 2, 3, 5, 6, 8, 9}
(ii) {0}
(iii) {9}
(iv) {6}
(v) {0, 6, 9}
(vi) {4, 7}
Q. 5. (i) (X ∩ Y) \ Z
(ii) X \ Y
(iii) Z \ (X ∪ Y)
(iv) (X ∪ Y ∪ Z)’
Q. 6.
Y
U
Z
• 7
X
• 6
• 5• 1• 3
• 4• 2 • 0
Q. 7. (a) Q
U
R
• 8
P
• 7
• 5• 6
• 2• 1
• 3
• 4
(b) (i) {1}
(ii) {5, 6}
(iii) {1, 2, 3, 4, 5, 6, 7}
(iv) {6}
(v) {3, 4, 5, 7, 8}
(vi) {8}
(vii) {1}
(viii) ∅
(ix) {1, 7, 8}
(x) {1, 5, 6, 7}
(xi) {1, 6}
(xii) {1}
Q. 8. (i)
Y
U
Z
X
• f
• c• d
• b• a • e
(ii) {b, c}
(iii) {d}
(iv) 6
(v) (Y ∩ Z) \ X
Q. 9. (i)
B
U
C
A
• 1 • 9
• 3• 5• 7
• 2
• 4
• 6
• 8
5Active Maths 2 (Strands 1–5): Ch 1 Solutions
A = {2, 3, 5, 7}
B = {2, 4, 6, 8}
C = {1, 3, 5, 7, 9}
U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
(ii) {3, 5, 7}
(iii) {2}
(iv) # (A ∪ B ∪ C) = 9 = # U
(v) A \ (B ∪ C)
A ∩ B ∩ C
(B ∩ C) \ A
U \ (A ∪ B ∪ C)
Q. 10. (i) A = {2, 3}
B = {1, 2, 3, 4, 6, 12}
C = {1, 2, 4, 8}
(ii)
B
U
C
A
• 8
• 9
• 10
• 11
• 5
• 7
• 3
• 2 • 4• 1
• 6
• 12
(iii) {1, 2, 4}
(iv) {2}
(v) 7
(vi) A \ (B ∪ C)
(A ∩ C) \ B
Q. 11. (i) P ∩ Q ∩ R
(ii) P ∪ Q ∪ R
(iii) Q
(iv) Q \ (P ∪ R)
(v) P ∩ R
(vi) (P ∩ Q) \ R
(vii) R’
(viii) (P ∪ Q ∪ R)’
Q. 12. (i)
N
U
Z
P
• 0
• –2
• –10
• 1
2
• √3
• 9.0321
• 3
• 5• 2
• 23
• 1 • 4
(ii) N \ Z and P \ N
(iii) (a) P ⊂ N
(b) N ⊂ Z
Q. 13. A = {1, 2, 3, 4, 5, …., 18, 19, 20}
B = {2, 5, 8, 11, …….} (i.e. = {2, 5, 8, 11, 14, 17, 20, ….})
C = {3, 5, 7, 9, …….} (i.e. = {3, 5, 7, 9, 11, 13, 15, 17, 19,
….})
(a) {5, 11, 17}
(b) {2, 8, 14, 20}
(c) {3, 7, 9, 13, 15, 19}
Exercise 1.3
Q. 1. (i) 2
(ii) 6
(iii) 15
(iv) 5
(v) 19
(vi) 23
(vii) 21
Q. 2. (i) U(70)
B(30)
S(37)
M(35)
153
5710
1012 8
(ii) 3
6 Active Maths 2 (Strands 1–5): Ch 1 Solutions
Q. 3. (i) U(100)
S(48)
B(43)
R(55)
166
10
98
1422 15
(ii) 6
Q. 4. (i) U(30)
I(19)
G(12)
F(15)
21
5
23
105 2
(ii) 5
(iii) 2
(iv) 1 ___ 30
Q. 5. (i) L1(23)
L2(21)
L(31)
7
5
45
1015 6
(ii) 15
Q. 6. (i) 10 + 4 + 3 + 8 + 15 + 5 + 1 = 46
∴ x = 50 − 46
∴ x = 4
(ii) 7
Q. 7. (i) 7 + 10 + 6 + 9 + 2 = 34
45 − 34 = 11
∴ 3x + x + 5 − x = 11
⇒ 3x = 6
⇒ x = 2
(ii) # R = 9 + 6 + 5 = 20
Q. 8. (i)
T(16)
U(40)
C(18)
P(19)
86
2
26
11 – x 12 – xx
(ii) 18 + 12 + 11 − x + 6 = 40
47 − x = 40
x = 7
(iii) 4
(iv) 2 ___ 40 = 1 ___ 20 = 5%
(v) 26 ___ 40 = 13 ___ 20 = 65%
(vi) 17 ___ 40 = 42.5%
Q. 9. (i) S
U(100)
C(50)
A(41)
252
x16
5 2520 – x
9 – x
(ii) 41 + 2 + 25 + 9 − x = 77 − x
100 − (77 − x) = 25
23 + x = 25
x = 2
(iii) 46
(iv) 52
(v) 43
(vi) 93%
(vii) 57%
Q. 10. (i) B(18)
U = 39
C
P(23)
34
x
6 9 – x
11 – x 7
7Active Maths 2 (Strands 1–5): Ch 1 Solutions
(ii) 6 + (9 − x) + x + (11 − x) = 23
26 − x = 23
−x = 23 − 26
−x = −3
x = 3
(iii) 18 − [9 − x + x + 7] = 18 − 16
= 2
Two liked Brussels sprouts only.
(iv) 9 − 3 = 6
Six students liked Brussels sprouts and peas but not cauliflower.
Q. 11. (i) Tulips
U = 40
Daffodils
5
4
20
x 3
Roses
(28)
20 + 4 + 3 + x = 30
27 + x = 30
∴ x = 3
Tulips
U = 40
Daffodils
35
4
20 y1
3 3
Roses
(28)
28 + 3 + 3 + y + 5 = 40
39 + y = 40
∴ y = 1
(ii) P(roses or daffodils) = 34 ___ 40
= 17 ___ 20
(iii) P(roses and daffodils but not
tulips) = 3 ___ 40
Q. 12. (i)
B(13)
U = 35
SF(16)
55
7
1 49
3 1
NY(20)
(ii) P(None of the three cites) = 5 ___ 35
= 1 __ 7 (iii) P(New York only) = 9 ___ 35
(iv) P(Boston or New York) = 25 ___ 35
= 5 __ 7
Exercise 1.4
Q. 1. (i) B
U
C
• 1 • 6
• 8
• 7
• 4
• 5
• 0• 3
• 2 • 9
A
8 Active Maths 2 (Strands 1–5): Ch 1 Solutions
(ii) (A ∪ B) ∪ C = {0, 2, 3, 4, 5, 7, 8, 9} ∪ {1, 2, 6, 7, 8, 9} = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
A ∪ (B ∪ C) = {2, 3, 7, 8} ∪ {0, 1, 2, 4, 5, 6, 7, 8, 9} = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} ∴ (A ∪ B) ∪ C = A ∪ (B ∪ C)
(iii) (A ∩ B) ∩ C = {7, 8} ∩ {1, 2, 6, 7, 8, 9} = {7, 8}
A ∩ (B ∩ C) = {2, 3, 7, 8} ∩ {7, 8, 9} = {7, 8} ∴ (A ∩ B) ∩ C = A ∩ (B ∩ C)
(iv) (A \ B) \ C = {2, 3} \ {1, 2, 6, 7, 8, 9} = {3}
A \ (B \ C) = {2, 3, 7, 8} \ {0, 4, 5} = {2, 3, 7, 8} ∴ (A \ B) \ C ≠ A \ (B \ C)
Q. 2. (i)
B
U
C
• 10
• 2
• 9• 7
• 1• 0
• 5
• 4 • 8
• 3
• 6
A
(ii) A ∪ (B ∩ C) = {0, 2, 4, 5, 9} ∪ {2, 8} = {0, 2, 4, 5, 8, 9}
(A ∪ B) ∩ (A ∪ C) = {0, 1, 2, 4, 5, 7, 8, 9} ∩ {0, 2, 4, 5, 8, 9, 10} = {0, 2, 4, 5, 8, 9}
∴ A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
(iii) A ∩ (B ∪ C) = {0, 2, 4, 5, 9} ∩ {1, 2, 4, 7, 8, 9, 10} = {2, 4, 9}
(A ∩ B) ∪ (A ∩ C) = {2, 9} ∪ {2, 4} = {2, 4, 9}
∴ A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Q. 3. (i) (P ∪ Q) ∪ R = {b, c, d, e, f, g, h} ∪ {a, b, c, h}
= {a, b, c, d, e, f, g, h}
P ∪ (Q ∪ R) = {b, c, d, e, f} ∪ {a, b, c, e, f, g, h}
= {a, b, c, d, e, f, g, h}
∴ (P ∪ Q) ∪ R = P ∪ (Q ∪ R)
(ii) (P ∩ Q) ∩ R = {e, f} ∩ {a, b, c, h} = ∅
P ∩ (Q ∩ R) = {b, c, d, e, f} ∩ {h} = ∅
∴ (P ∩ Q) ∩ R = P ∩ (Q ∩ R)
9Active Maths 2 (Strands 1–5): Ch 1 Solutions
Q. 4. (i) X ∪ (Y ∩ Z) = {s, t, v, w} ∪ {u, z} = {s, t, u, v, w, z}
(X ∪ Y) ∩ (X ∪ Z) = {s, t, v, w, u, y, z} ∩ {s, t, v, w, u, x, z}
= {s, t, v, w, u, z}
∴ X ∪ (Y ∩ Z) = (X ∪ Y) ∩ (X ∪ Z)
(ii) X ∩ (Y ∪ Z) = {s, t, v, w} ∩ {u, w, y, z, s, x} = {s, w}
(X ∩ Y) ∪ (X ∩ Z) = {w} ∪ {s} = {s, w}
∴ X ∩ (Y ∪ Z) = (X ∩ Y) ∪ (X ∩ Z)
Q. 5. (X ∪ Z) ∪ Y = X ∪ (Z ∪ Y)
= X ∪ (Y ∪ Z)
= (X ∪ Y) ∪ Z
Q. 6. M ∪ (N ∩ O)
Q. 7. (T ∩ R) ∪ (T ∩ S)
Q. 8. (i) A’
(ii) B’
(iii) (A ∪ B)’ = Region 1
(iv) (A ∪ B)’ = A’ ∩ B’
(v) (A ∩ B)’
(vi) (A ∩ B)’ = A’ ∪ B’
Q. 9. B
U
C
• 6
• 7• 5
• 9• 3
• 2•8
•4•10• 1
A
(i) (A ∪ B)’ = {1, 6}
A’ ∩ B’ = {1, 4, 6, 8, 9, 10} ∩ {1, 2, 3, 6} = {1, 6}
∴ (A ∪ B)’ = A’ ∩ B’
(ii) (A ∩ C)’ = {1, 3, 4, 5, 6, 7, 8, 9, 10}
A’ ∪ C’ = {1, 4, 6, 8, 9, 10} ∪ {1, 3, 5, 7, 9} = {1, 3, 4, 5, 6, 7, 8, 9, 10}
∴ (A ∩ C)’ = A’ ∪ C’
(iii) A’ = {1, 4, 6, 8, 9, 10}
∴ (A’)’ = {2, 3, 5, 7} = A
(iv) A ∪ (B ∩ C) = {2, 3, 4, 5, 7, 8, 10}
(A ∪ B) ∩ (A ∪ C) = {2, 3, 4, 5, 7, 8, 9, 10} ∩ {2, 3, 4, 5, 6, 7, 8, 10}
= {2, 3, 4, 5, 7, 8, 10}
∴ A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
(v) A ∩ (B ∪ C) = {2, 5, 7}
(A ∩ B) ∪ (A ∩ C) = {5, 7} ∪ {2} = {2, 5, 7}
∴ A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
10 Active Maths 2 (Strands 1–5): Ch 1 Solutions
Q. 10. (i) (A’ ∪ B)’ = (A’)’ ∩ B’ = A ∩ B’ = {2, 3, 7, 8} ∩ {1, 2, 3, 6}
= {2, 3}
(ii) U
A B
• 7
• 8• 3
• 2
• 1 • 6
• 4
• 5
• 10
• 9
(A ∩ B’)’ = A’ ∪ (B’)’ = A’ ∪ B = {1, 4, 5, 6, 9, 10} ∪ {4, 5, 7, 8, 9, 10}
= {1, 4, 5, 6, 7, 8, 9, 10}
(A’ ∪ B)’ = {1, 4, 5, 6, 7, 8, 9, 10}’
= {2, 3}
(A ∩ B’)’ = {2, 3}’
= {1, 4, 5, 6, 7, 8, 9, 10}
∴ Answers verified
Revision Exercises
Q. 1. U = {1, 2, 3, 4, …, 19, 20}
A = {1, 2, 4, 5, 10, 20}
B = {2, 4, 6, 8, …, 18, 20}
(a) U
A B
• 2
• 10
• 20
• 4
• 5
• 1
• 3 • 7 • 9 • 11 • 13
• 15
• 17
• 19• 6
• 16
• 18
• 8• 12
• 14
(b) (i) {1, 2, 4, 5, 6, 8, 10, 12, 14, 16, 18, 20}
(ii) {2, 4, 10, 20}
(iii) {3, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19}
(iv) {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}
(v) {1, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19}
(vi) {1, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19}
(vii) {1, 5, 6, 8, 12, 14, 16, 18}
(viii) ∅
11Active Maths 2 (Strands 1–5): Ch 1 Solutions
Q. 2. (i) 4 (iv) 2
(ii) 4 (v) 6
(iii) 2 (vi) 2
Q. 3. (i) Y
Z
• v
• w• y• z
• s• t
• u• x
X
(ii) {w, y, z}
(iii) ∅
(iv) 8
(v) {X \ (Y ∪ Z)
(vi) {Y \ (X ∪ Z)
(X ∩ Y) \ Z
(X ∩ Z) \ Y
(Y ∩ Z) \ X
Q. 4. (i) T(1248)
U
J(1424)
480
1026
222
190 532
260116234
B(762)
(ii) 762 + 260 + 532 + 480 + 1026 = 3,060
(iii) 2034 _____ 3060 × 100 ____ 1 % = 66.47%
(iv) 856 _____ 3060 × 100 ____ 1 % = 27.97%
(v) 1554 _____ 3060 × 100 ____ 1 % = 51%
Q. 5. A = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30}
B = {3, 6, 9, 12, 15, 18, 21, 24, 27, 30}
C = {5, 10, 15, 20, 25, 30}
(i) B
U
C
• 7
• 17
• 19 • 23
• 29• 11
• 13
116
• 5 • 25
• 30
• 6• 2
• 4
• 8
• 14
• 16
• 22
• 26
• 28
• 12• 18• 24
• 10
• 20 • 15
A• 3
• 9
• 21
• 27
(ii) (A ∪ B) ∪ C = {2, 4, 8, 14, 16, 22, 26, 28, 10, 20, 6, 12, 18, 24, 30, 3, 9, 21, 27, 15}
∪ {10, 20, 30, 15, 25, 5}
= {2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30}
A ∪ (B ∪ C) = {2, 4, 8, 14, 16, 22, 26, 28, 10, 20, 6, 12, 18, 24, 30}
∪ {6, 12, 18, 24, 30, 3, 9, 21, 27, 15, 10, 20, 5, 25}
= {2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30}
∴ (A ∪ B) ∪ C = A ∪ (B ∪ C)
12 Active Maths 2 (Strands 1–5): Ch 1 Solutions
(iii) (A ∩ B) ∩ C = {6, 12, 18, 24, 30} ∩ {5, 10, 15, 20, 25, 30} = {30}
A ∩ (B ∩ C) = {2, 4, 6, 8, 10, 12, …, 28, 30} ∩ {15, 30} = {30}
∴ (A ∩ B) ∩ C = A ∩ (B ∩ C)
(iv) (A \ B) \ C = {2, 4, 8, 10, 14, 16, 20, 22, 26, 28} \ {5, 10, 15, 20, 25, 30}
= {2, 4, 8, 14, 16, 22, 26, 28}
A \ (B \ C) = {2, 4, 6, 8, …., 28, 30} \ {3, 6, 9, 12, 18, 21, 24, 27}
= {2, 4, 8, 10, 14, 16, 20, 22, 26, 28, 30}
∴ (A \ B) \ C ≠ A \ (B \ C)
(v) A ∪ (B ∩ C) = {2, 4, 6, …., 28, 30} ∪ {15, 30}
= {2, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30}
(A ∪ B) ∩ (A ∪ C) = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 3, 9, 15, 21, 27}
∩ {5, 10, 15, 20, 25, 30, 2, 4, 6, 8, 12, 14, 16, 18, 22, 24, 26, 28}
= {2, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30}
∴ A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
(vi) A ∩ (B ∪ C) = {2, 4, 6, 8, …., 28, 30} ∩ {5, 10, 15, 20, 25, 30, 3, 6, 9, 12, 18, 21, 24, 27}
= {6, 10, 12, 18, 20, 24, 30}
(A ∩ B) ∪ (A ∩ C) = {6, 12, 18, 24, 30} ∪ {10, 20, 30} = {6, 10, 12, 18, 20, 24, 30}
∴ A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Q. 6. (i) P(25)
U(49)
O(26)
6
5
8
3 9
17 – x 8 – xx
A(28)
(ii) 28 + 8 − x + 9 + 6 + 5 = 49
56 − x = 49
x = 7
(iii) 10 + 1 + 6 ___________ 49 = 17 ___ 49 × 100 ____ 1 %
≈ 35%
Q. 7. (i) U(30)
P(16) Q(6)
x16 – x 6 – x
y
To minimise y
maximise 16 + 6 − x = 22 − x
Let x = 0
⇒ Ans = 8
(ii) To maximise y, minimise 16 + 6 − x = 22 − x
Let x = 6
⇒ Ans = 14
(iii) U(u)
P(p) Q(q)
qp – q o
x
∴ u = p − q + q + o + x
∴ u = p + x
13Active Maths 2 (Strands 1–5): Ch 1 Solutions
Q. 8. (i)
B(25)
U
C(31)
0
116
7
8
15 – (x + y)1 + x – y
16 – x
y
x
F(23)
⇒
B(25)
U
C(31)0
116
7
8
210
5
2
11
F(23)
43 = 25 + 7 + x 23 + 17 − y = 3843 = 32 + x 40 − y = 38 x = 11 y = 2
(ii) 45 (iii) 26 ___ 45 × 100 ____ 1 % ≈ 58% (iv) 19 ___ 45 × 100 ____ 1 % ≈ 42%
Q. 9. (a) (i) {1, 2, 3, 4, 5, 6}
(ii) {5, 6}
(iii) {1, 2, 4, 5, 6}
(b) (P ∪ Q) ∩ (P ∪ R)
Q. 10. (a) 7 (b) One (c) Prime
Q. 11. (a) C
U(100)
D
24
116
8
4
2622
5
3
8
T(41)
(b) 41 + 22 + 5 ____________ 100 = 68 ____ 100 = 17 ___ 25 = 68%
(c) 3 ____ 100 = 3%
Q. 12. (i) B
C
A
• 6
• 1• 4
• 2
• 3• 5
(A \ B) ∪ (C ∩ B) = {1, 4, 3, 5}
= {1, 3, 4, 5}
(A ∪ B) ∩ (C \ B) = {1, 4}
(ii) U(20)
P
4
Q
x2x 5x
8x + 4 = 20
8x = 16
x = 2
# Q = 12
Q. 13. (a) U(110)
W(82)
15
S(57)
4438 13
82 + 57 = 139
139 + 15 = 154
154 − 110 = 44
82 − 44 = 38
57 − 44 = 13
14 Active Maths 2 (Strands 1–5): Ch 1 Solutions
(b) (i) U(u)
A(a)
y
B(b)
xa – x b – x
u = a + b − x + y
(ii) To minimise u, maximise x.
For maximum x, set b − x = 0 (as a > b)
⇒ b = x
⇒ u = a + (x) − x + y
⇒ u = a + y
⇒ u = y + a
