Chapter 1 Exercise 1 - Biology Leaving Cert. Sets 2.pdf · Daffodils 5 4 20 x 3 Roses (28) 20 + 4 +...

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


(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


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


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


(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


(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)


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)


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) ∅


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)


(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


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


(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

Chapter 1 Exercise 1 - Biology Leaving Cert. Sets 2.pdf · Daffodils 5 4 20 x 3 Roses (28) 20 + 4 +...

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Transcript of Chapter 1 Exercise 1 - Biology Leaving Cert. Sets 2.pdf · Daffodils 5 4 20 x 3 Roses (28) 20 + 4 +...