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Active Maths 2 (Strands 1–5): Ch 22 Solutions
Chapter 22 Exercise 22.1
1
Q. 1. (i) 2
–2
–4
–6
–8
–10
–12
0
–2 2
x
y
4 60
(ii)
–1
2
4
6
8
10
12
14
16
18
1 2 3 4 5 6 7 8 9 10 11 12 13 140
0 x
y
(iii)
–9 –8 –7 –6 –5 –4 –3 –2
2
–2
–4
–6
–8
–10
–12
–14
–16
–1
0
0 1
x
y (iv)
–2
2
4
6
–1 1 2–2 0
0 x
y
2 Active Maths 2 (Strands 1–5): Ch 22 Solutions
Q. 2. (i)
x
y
–4
–2
0
0–1
2
1 2–2
(ii)
x
y
–3 –2 –1 10
0
–2
2
4
6
(iii)
0
02
1 2 3 4 5
4
6
x
y
(iv)
0
5
10
15
20
25
30
35
0
1 2 3 4 5 6 7 8 9 10 11 12
x
y
Q. 3.
–3 –2 –1
–5
5
10
15
–10
–15
1 2 3 4 500 x
y
(i) f(2.5) = 7 (iii) x = −1
(ii) x = 9 __ 4 (iv) x ≥ 1
Q. 4.
654321–1
2
4
6
8
10
12
14
16
0
0 x
y
(i) v = 8.1 m/s (iii) v = 15 m/s
(ii) t = 5 __ 3 s (iv) t = 5 s
Q. 5. x x2 + 3x − 4 y (x,y)−4 (−4)2 + 3(−4) − 4 0 (−4,0)−3 (−3)2 + 3(−3) − 4 −4 (−3,−4)−2 (−2)2 + 3(−2) − 4 −6 (−2,−6)−1 (−1)2 + 3(−1) − 4 −6 (−1,−6)0 (0)2 + 3(0) − 4 −4 (0,4)1 (1)2 + 3(1) − 4 0 (1,0)
x0–1
–1
1
2
3
4
5
6
7
8
–2
–3
–4
–5
–6
–7
1 2 3–2–3–4–5–6
y
(i) −6.25
(ii) x = −1 and x = −2
(iii) x2 + 3x − 4 = −6
x2 + 3x + 2 = 0
(x + 1)(x + 2) = 0
⇒ x = −1 OR x = −2
3Active Maths 2 (Strands 1–5): Ch 22 Solutions
Q.
6. (i)
x
y
4321–1
2
4
6
8
10
12
–2–3 0
0
(ii)
x
y
43210
0
–1
–2
2
4
–4
–6
–8
–2–3–4
(iii)
–10
–8
–6
–4
–2
–1 1 2 3
x
y
0–2–3
2
4
6
8
10
0
(iv)
–2 –1
5
10
15
20
25
30
1 2 3 40
0 x
y
(v)
–2–2
2
4
6
8
10
–1 1 2 30
0 x
y
Q. 7. (i) x 6x − x2 y (x,y)0 6(0) − (0)2 0 (0,0)1 6(1) − (1)2 5 (1,5)2 6(2) − (2)2 8 (2,8)3 6(3) − (3)2 9 (3,9)4 6(4) − (4)2 8 (4,8)5 6(5) − (5)2 5 (5,5)6 6(6) − (6)2 0 (6,0)
(ii)
00
1
1
2
3
4
5
6
7
8
9
10
2 3 4 5 6
x
y
(iii) 9 m (iv) 0.35 m OR 5.6 m
4 Active Maths 2 (Strands 1–5): Ch 22 Solutions
Q. 8.
–1
–1
1
2
3
4
5
6
7
8
–2
–3
–4
–5
–2 1 2 3 4
0
0
x
y
(i) f(1.5) = −3.25
(ii) x = −1.6 OR x = 2.6
(iii) −1.6 ≤ x ≤ 2.6
(iv) −4.25
Q. 9. (i)
x
y
161412108642–2
10
20
30
40
50
60
0
0
(ii) f(0.7) = 5.01 m
(iii) 51.25 m
(iv) x = 14.7 seconds
Q. 10.
–8 –7 –6 –5 –4 –3 –2 –1–2
2
4
6
8
10
12
14
–4
–6
–8
–10
–12
1 20
0 x
y
(i) f(−4.5) = −9.8
(ii) x = 0.5 OR x = −6.5
(iii) f(x) = 2 x = 0.74 OR x = −6.74
(iv) x = −7.1 OR x = 1.1
(v) −12
Q. 11. (i)
–2
1 2 3 4 5
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
00
x
y
5Active Maths 2 (Strands 1–5): Ch 22 Solutions
(ii) t = 0.2 OR t = 4.1
(iii) Max [h(t)] = 31.3 m
Max [g(t)] = 20 m
(iv) h(t) ball t = 2.5 s
g(t) ball t = 5 s
(v) 4 seconds
Q. 12. (i) f(7) = 0.25(7) + 4
= 5.75
(ii) x = 0.7 OR x = 6.1
(iii) x ≤ 0.7 OR x ≥ 6.1
(iv) 0.7 ≤ x ≤ 6.1
Q. 13.
x
y
0
0
–1–1
1
2
3
4
5
6
–2
–3
–4
–5
–6
–7
–8
–9
–10
1 2 3–2–3
(i) f(−1.5) = 5.75
(ii) f(x) = 2
x = −3 OR x = 1
(iii) Max (f(x)) = 6
(iv) −3 ≤ x ≤ 1
Q. 14.
–4 –3 –2 –1–1
1
2
3
4
5
6
7
8
9
10
0
–2
–3
1 2 30
y
x
(i) x = −2.6 and x = 0.6
(ii) x ≤ −2.6 OR x ≥ 0.6
Q. 15. (i)
0
0
1
1
2
3
4
5
6
7
8
9
2 3 4 5 6
x
y
(ii) t = 2 s OR t = 4 s
6 Active Maths 2 (Strands 1–5): Ch 22 Solutions
(iii)
0
0
1
1
2
3
4
5
6
7
8
9
2 3 4 5 6
x
y
(iv) Missiles collide 0.2 seconds after take-off
(v) j (0.2) = 1.2(0.2) + 1
= 1.24 m above the ground
Q. 16. (i)
x0
–2
2
4
6
8
10
12
14
16
18
20
22
24
26
0
–1 1 2 3 4 5 6 7 8 9 10 11
y
(ii) The optimal quantity to produce is 500 units
(iii) €2,500
(iv) 400 units OR 600 units
(v) It is more lucrative to produce 725 units than 250 units
as P(725) = €2,000 while P(250) = €,1875
Exercise 22.2
Q. 1.
x
y
–1
1
2
3
4
5
6
7
8
9
–1 1 2 30
0
–2
Q. 2.
–2
2
4
6
8
10
12
14
16
18
20
22
24
26
28
0
–2 –1 1 2 3 40
x
y
Q. 3.
x
y
–2 –1
5
10
15
20
25
1 2 3
–5
0
0
7Active Maths 2 (Strands 1–5): Ch 22 Solutions
Q. 4.
x
y
–1 1 2 30
5
10
15
20
25
30
–2
Q. 5.
x–2 –1
5
10
15
20
25
30
35
40
45
50
55
1 2 30
0
y
Q. 6.
x
–2 –1
10
20
30
40
50
60
70
80
90
100
110
1 2 30
0
y
Q. 7.
–2 –1
2
4
6
8
10
12
14
16
1 2 30
0 x
y
Q. 8.
–2 –1 1 2 30
02
–2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
x
y
Q. 9. f(x) = a . 2x
f(0) = a . 20
= a
From the graph f(0) = 4
⇒ a = 4
8 Active Maths 2 (Strands 1–5): Ch 22 Solutions
Q. 10. f(x) = a .2x
f(0) = a . 20 = 2
From the graph f(0) = 2
⇒ a = 2
Q. 11. (i) x < −2.1 OR x > 0.6
(ii) −2.1 ≤ x ≤ 0.6
Q. 12. (i) −1.8 < x < 0
(ii) x ≤ −1.8 OR x ≥ 0
Q. 13.
x
y
–2
2
4
6
8
10
12
–1 1 20
0
–2–3–4–5–6–7–8–9–10–11–12–13
x < −11.9 OR x > −0.8
Q. 14.
x
y
–1
2
4
6
8
10
12
14
16
18
0
1 2 3 4 50–2–3–4–5
x ≥ −0.2
Q. 15. (i) 26 = 64
(ii)
0
0
1
20
40
60
80
100
120
140
2 3 4 5 6 7
x
y
(iii) 16
(iv) 6 days
Q. 16. (i) & (ii)
x
y
00
5
10
15
20
25
30
35
40
45
50
55
60
65
1 2 3 4 5 6
(iii) 5 days and 18 hours
9Active Maths 2 (Strands 1–5): Ch 22 Solutions
Exercise 22.3
Q. 1. (i) g(x) → Red graph
(ii) h(x) → Blue graph
Q. 2. (i) f(x) → Blue graph
(ii) g(x) → Red graph
Q. 3. (i) g(x) → Red graph
(ii) f(x) → Blue graph
Q. 4.
3
x
y
y
g (x)
f (x)
h (x)
21–1–2
1
2
3
4
5
6
7
8
9
10
11
12
–3 00
Q. 5. f(x) = 3x − 3
Q. 6. g(x) = 3 + 2x + 2
Revision ExercisesQ. 1. (i)
21–1–2
–4
–6
–8
–10
–12
–14
–16
2
4
6
8
10
12
–20
0 x
y
(ii)
1–1–2
–4
2
4
6
8
10
12
14
–2–3 00 x
y
Q. 2. (i) V = 5x
(ii)
02000–2000 6000 10000 14000 18000 22000 26000 30000 34000 38000 42000 46000 50000
x
y
0
20000
40000
60000
80000
100000
120000
140000
160000
180000
200000
220000
240000
260000
280000
300000
10 Active Maths 2 (Strands 1–5): Ch 22 Solutions
(iii) f = 45,000
(iv)
02000–2000 6000 10000 14000 18000 22000 26000 30000 34000 38000
x
y
0
20000
40000
60000
80000
100000
120000
140000
160000
180000
200000
220000
240000
260000
280000
300000
(v) T = 45,000 + 5x
(vi)
02000–2000 6000 10000 14000 18000 22000 26000 30000 34000 38000
x
y
0
20000
40000
60000
80000
100000
120000
140000
160000
180000
200000
220000
240000
260000
280000
300000
(vii) S(x) = 6x
(viii)
02000–2000 6000 10000 14000 18000 22000 26000 30000 34000 38000
x
y
0
20000
40000
60000
80000
100000
120000
140000
160000
180000
200000
220000
240000
260000
280000
300000
11Active Maths 2 (Strands 1–5): Ch 22 Solutions
(ix) 45,000 units
(x) T = 45,000 + 5x
S = 6x
Solve the simultaneous equations:
y = 45,000 + 5x 6x = 45,000 + 5x
S = 6x x = 45,000 units
Q. 3. (i)
21–1–1
–2
–3
–4
–5
–6
1
2
3
4
5
6
–2–3–4 00 x
y
(ii)
3
x
y
21–1–2
–4
–2
2
4
6
8
10
12
14
16
18
20
22
24
–3 00
(iii)
00
–2
2
4
6
8
–4
–6
1–1–2–3 2
y
x
(iv) y
x
323028262422201816141210
8642
0 1 2–1–2–3–4
0
Q. 4. 4
3
2
1
00
–1–1–2 1 2 3 4
–2
–3
–4
–5
y
x
(i) f(1.5) = −4.75
(ii) x = −1.2 OR x = 3.2
(iii) x2 − 2x − 4 ≤ −1
f(x) ≤ − 1
Answer: −1 ≤ x ≤ 3
(iv) −5
12 Active Maths 2 (Strands 1–5): Ch 22 Solutions
Q. 5. (i) y
x
21–1–2 00
2
4
6
8
10
14
12
18
20
22
24
26
28
16
(ii) y
x
21–1–2 00
1
2
3
4
5
7
6
9
10
11
12
8
(iii) y
x
21–1–2 00
2
4
6
8
10
14
12
16
(iv) y
x
21–1–2 00
2
4
6
8
10
14
12
18
16
(v) y
x21–1–2 0
0
5
10
15
20
25
35
30
45
40
13Active Maths 2 (Strands 1–5): Ch 22 Solutions
Q. 6. (i) A horizontal translation, two units to the right
OR (0,1) → (2,1)
(ii) g(x) = 2x − 2
Q. 7. (i) Week Amount
1 1
2 2
3 4
4 8
5 16
6 32
7 64
8 128
(ii)
10
20
30
40
50
60
70
80
90
100
110
120
130
0
1 2 3 4 5 6 7 80
y
x
(iii) This is an exponential function.
(iv) Yes. The pattern could also be
1, 2, 4, 7, 11, 16, ...
Q. 8. (i) f(x) → Black curve
(ii) g(x) → Blue curve
Q. 9.
0
1
–1
–2
2
3
4
5
6
7
8
9
10
0
1–1 2 3 4 5 6
x
y
Range of values of x for which f(x) ≥ g(x):
0 ≤ x ≤ 1Q. 10.
3
x
y
21–1–2
–6
–4
–2
2
4
6
8
10
12
14
16
18
20
–3 00
0.3 < x < 3
Q. 11. (a) x2 + x − 2 = 0 (x + 2)(x − 1) = 0 x = −2 OR x = 1
(b) He should have drawn the line y = 2x + 7
i.e. he subtracted 7 rather than adding 7.
(c) y = 2x + 7
14 Active Maths 2 (Strands 1–5): Ch 22 Solutions
(d) x = −2 OR x = 1
21–1
–2
2
4
6
8
10
12
14
–2–3–4 0
0 x
y
Q. 12. (a) x = −4 and x = 2
(b) (3)2 + 2(3) + k = 2
9 + 6 + k = 2
k = 2 − 15
k = −13
(c) x2 + 2x + 1 = 0
(x + 1)(x + 1) = 0
⇒ x = −1
k = 1
(d)
–5
–4
–3
–2
–1
0
1
2
3
4
5
6
7
–6
–7
–8
–9
–5 –4 –3 –2 –1 0 1 2 3
x
y
(e) x = −5 OR x = 3
∴ (x + 5)(x − 3) = 0
x2 + 2x − 15 = 0
k = −15
Q. 13. f(x) = x2 − 6x + 22
0
2
–2
4
6
8
10
12
14
16
18
20
22
0
1–1 2 3 4
x
y
(i) f(10) = (1)2 − 6(1) + 22
= 17 hours
(ii) f(2) = (2)2 − 6(2) + 22
= 14 hours
(iii) 13 hours
(iv) Yes, as the graph increases for x > 3.
Q. 14. (a) A = 1 __ 2 (3x)(x + 2) − x2
= 3 __ 2 x2 + 3x − x2
= 1 __ 2 x2 + 3x
(b) 1 __ 2 x2 + 3x = 3.5
x2 + 6x = 7
x2 + 6x − 7 = 0
15Active Maths 2 (Strands 1–5): Ch 22 Solutions
(c)
15
10
5
–5
–10
–15
–20
20
–8 –7 –6 –5 –4 –3 –2 –1 00
1
x
y
(d) x = 1
(e) Base = 3 cm
Height = 3 cm
(f) x2 + 6x − 7 = 0
(x + 7)(x − 1) = 0
x = −7 OR x = 1
Q. 15. (a)
0
20
40
60
80
100
120
140
160
180
200
220
240
260
0
1 2 3 4 5 6 7 8
x
y
(b) Sarah
(c) H = 4
(d) 25 + 30H = 35 + 27.5H
2.5H = 10
H = 10 ___ 2.5
H = 4
Q. 16. (−2,−9)