Math AA HL

A nswers 443

Chapter 8 Prior Knowledge ⎛ 11⎞ ⎛ 2⎞
15 a ⎜⎜⎜⎝−1175⎟⎟⎠⎟ b ⎜⎜⎝⎜ −90⎟⎟⎠⎟
1 2x + 5y = 23
16 a a + 4 b b a + 1 b
20 3 2

3 x = λ, y = 13 − 7λ, z = 35 − 19λ

Exercise 8A 17 a − 3 a + b b − 1 b + 1 a
2 2 2
⎛⎜⎝23⎠⎞⎟, ⎛3⎞
1 a = b = ⎜⎝1⎠⎟ 18 a 3 a − b b − 4 b + 1 a
2 3 2

2 a = ⎜⎝⎛−31⎞⎠⎟, b = ⎛ 2⎞ 19 a p = 3, q = 15 b p = 4, q = 16
⎜⎝−3⎟⎠

3 a = ⎛⎜⎝−41⎠⎞⎟, b = ⎛−3⎞ 20 a p = −6, q = 3 b p = 2, q = −8
⎜⎝ 2⎠⎟
21 a p = 4, q = 1 b p = 45, q = −1
a ⎛⎜⎝−−21⎞⎠⎟, b ⎛−1⎞
4 = = ⎝⎜ −3⎟⎠ 22 a p = −3, q = −18 b p = 4, q = −2

5 a i + 2j + 3k b 4i + 5j + 6k 23 a p = −2, q = −10 b p = −3, q = −3
6 a −3i + j b 2i − 2j

7 a 3i b −5j ⎛1/3⎞ ⎛3/5⎞
b a = ⎜⎜⎜⎝40/5⎠⎟⎟⎟
⎛ 2⎞ ⎛ 3⎞ 24 a a = ⎜ 2/3⎟⎟
8 a ⎝⎜⎜⎜ −31⎠⎟⎟⎟ b ⎝⎜⎜⎜ −41⎟⎟⎟⎠ ⎜
⎝⎜ 2/3⎟⎠

⎜⎛−1/ 6⎞ ⎛ 1/ 14 ⎞
⎛1⎞ ⎛ 0⎞ ⎜ 1/ 6 ⎟ ⎜ ⎟
9 a ⎜⎜⎝⎜ 03⎟⎟⎠⎟ b ⎜⎜⎜⎝−21⎟⎟⎟⎠ 25 a a = ⎟ b a = ⎜ −2/ 14 ⎟

⎝⎜ 2/ 6 ⎠⎟ ⎜⎝ 3/ 14 ⎟⎠

⎛ −1⎞ ⎛−3⎞ 26 a a= 3 i + 1 j − 1 k
10 a ⎝⎜⎜⎜ −42⎟⎟⎟⎠ 11 11 11
⎜ 01⎟⎟⎟⎠
11 a a + b b ⎜⎝⎜ b a= 1 i − 2 j + 1 k
12 a −a − b − c 6 6 6

⎛−3⎞ b b+c 27 a a= 1 i − 4 j b a= 2 j − 3 k
13 a ⎜⎜⎝⎜ −73⎟⎟⎠⎟ b −b − c 17 17 13 13

⎛ 3⎞ 28 a a + 1 b b 1 b − 1 a c 2a − b
b ⎜⎜⎝⎜ −73⎟⎟⎟⎠ 2 2 2

29 a a − b b 1 a + 2 b c − 1 a + 2 b
⎛ 8⎞ 3 3 6 3

⎛ 11⎞ ⎛ 22⎞ 30 a ⎝⎜⎜⎜ −190⎟⎟⎠⎟ b 146
14 a ⎜⎜⎜⎝−241⎟⎠⎟⎟ b ⎜⎜⎜⎝−3261⎠⎟⎟⎟

444 A nswers

31 ±2 11 10 a p = −15, q = 22 b p = 1, q = −3
11 a p = 10, q = 11
32 1 ± 41 12 a p = 3, q = −1 b p = 0, q = −3
10 1
b p = − 3 , q = −6

2⎛ ⎞ 3⎛ ⎞ 1⎛− ⎞
033 ⎜ 1⎜ ⎟
⎟ 1⎜ ⎟

3/4⎜ ⎟ 13 a ⎝⎜ − ⎟⎠ 4b ⎜ ⎟
2⎜ ⎟
⎝⎜ − ⎠⎟
⎜⎝ ⎟⎠
34 −2 8⎛ ⎞
2b ⎜ − ⎟ 1 3
35 a 2 i + 2 j − 1 k 14 a 2i − 2j − k b 2 i − 2 j + 2k
3 3 3 4 2⎜ ⎟
4 1 7 1 7
36 −3 ⎜⎝ ⎟⎠ 15 a 2 j + 2 k b −i + 2 j + 2 k

37 −2 6⎛− ⎞ 3
3b ⎜ ⎟ 16 2
⎛4 2⎞
2⎜ ⎟ 1⎜ ⎟ 17 b 1 : 3

38 ⎜ − ⎟ ⎝⎜ − ⎟⎠ 18 a 30 b 3 i − j − 3 k
2 2
2⎜ ⎟ 6⎛ ⎞
3b ⎜ − ⎟
⎝⎠
1⎜ ⎟
39 3, −5 1⎛ ⎞
3 ⎜⎝ ⎠⎟ 2⎜ ⎟
65 19 a 2⎜ ⎟ b 227
40 3 8⎛ − ⎞
0b ⎜ ⎟ ⎝⎜ − ⎟⎠

41 27 + 6 2 10⎜ ⎟ 20 3i − 4j

Exercise 8B ⎜⎝− ⎟⎠ 2⎛ ⎞
1321 ⎜ ⎟
2⎛ ⎞ b 26
31 a ⎜ ⎟ 1⎜ ⎟
b2
9⎜ ⎟ ⎜⎝− ⎟⎠
b 53
⎝⎜ ⎟⎠ 22 a 1⎛ ⎞ b No
b Yes, no 2⎜ ⎟
8⎛ ⎞ b No 2⎜ ⎟
02 a ⎜ ⎟ b Yes, yes
⎝⎜ ⎟⎠
10⎜ ⎟
23 a (13, 4, −6) b (6, 3, 1)
⎝⎜ ⎠⎟
24 a BC = c − b, MN = 1 c − 1 b
2⎛− ⎞ 2 2
33 a ⎜ − ⎟ 1
b Parallel, MN = 2 BC
9⎜ ⎟
26 a p = 6, q = 5 b 3:2
⎜⎝− ⎠⎟
27 11 i − j − 12 k
4 a 53 5 5

5a 2 28 −2, 23
− 15
6 a 94
29 a 3 i + 3 j − 2k b ⎛⎝21 ,123 , 0⎞
7 a Yes, no 2 2 ⎠
8 a No
9 a Yes, yes

A nswers 445

30 a 1 a + 1 b + 1 c c 2:1 30 6 3
2 3 6 4
31 −
3 2 3 1 2 13 b
31 a d = 5 b + 5 c, e = 2 a − 2 c, f = 3 a + 0, 3
2
32

Exercise 8C 33 2

1 a 35 b 20 34 a 1.6 b 68.7°, 21.3°, 90°
2 a 67.3 b 9.64 c 61.8
b b2− a2
3 a −54.0 b −36.9 36 a a + b, b − a c 45
b −56
4 a 16 37 b 2
b 10
5 a −16 b9 Exercise 8D

6a9 b 67° ⎛2⎞ ⎛−1⎞
b 101° ⎝⎜⎜⎜ 51⎟⎟⎟⎠ + ⎜ 22⎟⎟⎠⎟, yes
7 a 64° b 65° 1 a r = t ⎜⎝⎜
8 a 108° b 111°
9 a 61° ⎛−1⎞ ⎛4⎞
10 a 96° b 58.5 ⎜ 03⎟⎟⎟⎠ + t ⎜⎜⎝⎜ 51⎟⎟⎟⎠, yes
b r = ⎜⎜⎝
11 a 25.5
⎛4⎞ ⎛4⎞
12 a −4.5 b −3 2 a r = ⎜⎜⎜⎝03⎟⎟⎟⎠ + t ⎜⎜⎝⎜ 03⎠⎟⎟⎟, no

13 a 156.5 b 615.5 ⎛−1⎞ ⎛0⎞
⎜ 51⎟⎟⎟⎠ + t ⎜⎝⎜⎜ 70⎠⎟⎟⎟, no
14 a 51.5 b 420.5 b r = ⎜⎜⎝

15 a 2 b − 1
7 2
3 a r = ⎝⎜⎛41⎞⎠⎟+ λ ⎛⎝⎜ −31⎞⎟⎠, no
16 a 3 b2

17 a 9 b 16 b r = ⎜⎝⎛72⎞⎠⎟+ λ ⎛⎝⎜ −92⎟⎠⎞ , yes
7
4 b −12
18 a 5

19 a 19 b7 c 32 ⎛1⎞ ⎛ 1⎞
4 a r = ⎝⎜⎜⎜ 05⎠⎟⎟⎟+ λ ⎜⎜⎝⎜ −33⎟⎟⎟⎠
20 141°
21 40.0°
2
22 3 ⎛−1⎞ ⎛ 3⎞

23 48.2° b r = ⎜ 51⎟⎠⎟⎟+ λ ⎜⎜⎝⎜ −22⎟⎟⎠⎟
⎝⎜⎜
24 98.0°

26 3 ⎛0⎞ ⎛ 1⎞
5 a r = ⎜⎜⎜⎝23⎠⎟⎟⎟+ λ ⎜⎜⎝⎜ −03⎟⎟⎠⎟
27 61.0°, 74.5°, 44.5°
28 94.3°, 54.2°, 31.5°
29 b 41.8°, 48.2°

c 161

446 Answers

4 2⎛ ⎞ ⎛ ⎞ 5 2⎛− ⎞ ⎛⎞

3 3b r = ⎜ − ⎟+ λ ⎜ ⎟ 2 7b r = ⎜ ⎟+ λ ⎜ ⎟
0 1⎜ ⎟ ⎜⎟ 2 1⎜ ⎟ ⎜⎟
⎝⎜ − ⎠⎟ ⎝⎜ ⎠⎟
⎜⎝ ⎠⎟ ⎝⎜ − ⎠⎟

41 416 ⎞ ⎛3/2⎞ 1⎛ ⎞
a r = ⎛ ⎞⎛ ⎠⎟ 4 2r ⎜ ⎟
⎜⎝− ⎟⎠ + λ ⎝⎜ ⎜ ⎟+
20 a = λ ⎜ − ⎟
1 3⎜ ⎜ ⎟
41 23b
⎝
r = ⎛ ⎞⎛ ⎞ ⎟ ⎜⎝ ⎠⎟
⎜⎝ ⎟⎠ + λ ⎝⎜ − ⎟⎠ ⎠

7 a x = 3 + 8λ, y = 5 + 2λ, z = 2 + 4λ ⎛−1/3⎞ ⎛4/3⎞
b x = 4 − 2λ, y = 1− + 3λ, z = 2 + 5λ 3 1b
r = ⎜ ⎟⎜ ⎟
⎜ ⎟+ λ ⎜ ⎟
4 2⎜ ⎟
⎟ ⎜ − ⎠
⎝ ⎠ ⎝
8 a x = 2 + λ, y = 3, z = 4− λ
b x = 2λ, y = 3 + λ, z = −1 2 3⎛ ⎞ ⎛ ⎞ 1 3⎛ − ⎞
⎛⎞

9 a x = −1, y = 3 + 5λ b x = 4 − 3λ, y = 2λ 21 a 1 5r = ⎜ − ⎟+ λ ⎜ ⎟ 1 4b r = ⎜ ⎟+ λ ⎜ − ⎟
4 0⎜ ⎟ ⎜⎟ 2 0⎜ ⎟ ⎜⎟
10 a x − 3y = 17 b 3x + 4 y = −2 ⎝⎜ ⎠⎟ ⎝⎜ ⎠⎟
⎜⎝ ⎟⎠ ⎝⎜ − ⎠⎟

11 a 2x + 3y = 6 b 5x − 3y = −12 3 0⎛ ⎞ ⎛ ⎞ 0 1⎛ ⎞ ⎛ ⎞
4 0r = ⎜ − ⎟+ λ ⎜ ⎟ 2 0b r = ⎜ ⎟+ λ ⎜ ⎟
12 a y = 4 b x=2 22 a 0 1⎜ ⎟ ⎜⎟ 2 0⎜ ⎟ ⎜⎟
⎝⎜ ⎠⎟ ⎜⎝ ⎟⎠
x −1 y −7 z − 2 ⎜⎝ ⎠⎟ ⎝⎜ − ⎟⎠
−1 1
13 a = = 2 23 a 31.4° b 31.8°

b x −3 = y +1 z 24 a 38.1° b 80.0°
2 4= 5
25 a 83.7° b 7.13°
−
2x −1 y 2 4 − 3z
14 a 4 = + = 6
3 4⎛ ⎞
3 ⎛− ⎞

b 2 − 3x y − 2 2z +1 26 a 1 2r = ⎜ − ⎟+ λ ⎜ ⎟ b No
3 = = 6 5 3⎜ ⎟ ⎜⎟
1 ⎜⎝ − ⎠⎟
⎜⎝ ⎟⎠

15 a 2x − 2 = 3y = 2 − z 27 x + 1 = y −1 z−2
1 1 1= −3
3 2
−
x 3 2 − 2y 3z 21
b − = 1 = + 28 81.8°

4 y −5 2 30 No
−2
16 a x = −1, = z 31 a 2.55ms−1
2 b r = (12i − 5j + 11k) + t(0.5i + 2j + 1.5k)
y
b x = 1, 3 = z−3 c No
−1
32 16.8 m
17 a x + 1 = 3 − z , y = 2 b 76.4°
33 a p = 12, q = 5
5 y 2 1
x−2 34 b (0, 3, 0)
b −3 + , z −3
= = 7 4⎛ ⎞
1 ⎛− ⎞
18 a y = 1, z = 5 1 2r = ⎜ ⎟+ λ ⎜ − ⎟
b x = 4, z = −3 35 a 2 3⎜ ⎟ ⎜⎟ b (−5, −5 −11)
⎝⎜ ⎟⎠
⎜⎝ ⎟⎠

2 5⎛ ⎞ ⎛ ⎞ 2 2⎛ ⎞ ⎛ ⎞
1 3r = ⎜ − ⎟+ λ ⎜ − ⎟ 1 3r = ⎜ ⎟+ λ ⎜ − ⎟
19 a 3 1⎜ ⎟ ⎜⎟ 36 a 4 6⎜ ⎟ ⎜⎟ b7
⎝⎜ ⎟⎠ ⎜⎝ ⎠⎟
⎝⎜ ⎠⎟ ⎜⎝ ⎠⎟

A nswers 447

c (−8, 16, −26), (12, −14, 34) 9 (4, 3, 3)

37 a x − 1 4 − y z + 1 b ⎛⎞ 10 Skew 2z − 3, y = 2z − 9;x = 11 − 3z , y = z − 27
11 5
3 = 2 = 3 1 3⎜ ⎟ a x= 5
22 2⎜ ⎟
b (1, −5, 2)
3⎜⎝ ⎠⎟

131/3 40/23 1338 a⎛ ⎞ ⎛− ⎞ 12 (3, −2, 1)
⎟+ λ ⎜
r=⎜ − ⎟⎜ ⎟ b 14 a (0, −22, 0)
⎠⎟ ⎝⎜ ⎟
⎜ ⎟⎠ 15 a (8, 7, 1)
⎝⎜ −
16 3; (3, 7, 2)

⎛1/2⎞ ⎛3/2⎞ 17 b No
7 0r
39 a = ⎜ ⎟⎜ ⎟ b 69.4° 18 a 54, 3 b No
⎜ ⎟+ λ ⎜ ⎟
2 4⎜ 19 b 1.58 m
⎟ ⎜− ⎟
⎝⎠ ⎝ ⎠
20 a 0.1 b 47.2 km/h
40 13.2°
21 a 7λ + 5μ = 11 b3

41 ⎝⎛694 , 4 , 199⎞⎠ Exercise 8F
9

42 a 3⎛ ⎞ b 3.32 ms−1 c 14.9 m 1 a 60.6 b 34.6
1⎜ − ⎟ 2 a 251
4⎜ ⎟ 3 a 64.3 b 11.5

⎜⎝ ⎟⎠ 2⎛− ⎞
64 a ⎜ ⎟
43 a r1 = 3i + t (−2i + 5j), r2 = 5j + t (4i + j) b 25.8
b 52t2 − 76t + 34 1⎜ ⎟
1⎛ − ⎞
c 2.50 m ⎜⎝ − ⎟⎠ 10b⎜ ⎟
⎜ −
7⎝⎜ ⎟
⎠⎟

2⎛ ⎞ ⎛ 596⎞ 9⎛ − ⎞ ⎛−23⎞
0 59644 r = ⎜ ⎟+ t ⎜ − ⎟
0 298⎜ ⎟ ⎜ ⎟ 195 a ⎜ − ⎟ 1b ⎜ ⎟
⎟⎠ ⎜ ⎟
⎜⎝ ⎟⎠ ⎜⎝ 2⎝⎜ ⎠⎟ 8⎜ ⎟

45 a (9, −5, 8) c (3, 4, −3) ⎜⎝ ⎟⎠

11 11 (= 6.08) 6 a −5i − 11j − 2k b 12i + 6j + 9k
6
46 b 48.5° d 7 a 50 b 30

e 4.55 8 a 120 b 80

47 3 9 a 10 b 30

48 6 10 a 70 b 90
11
153
Exercise 8E 11 a 2 b 33

1 a (10, −7, −2) b (−1, 1, 6) 12 a 15 3 9
2 a (0.5, 0, 1) b (4.5, 0, 0) 2 b2
3 a (3, 3, 1) b (7, 2, 4)
6 a Parallel b Parallel 13 a 446 b 3 66
7 a Not parallel b Not parallel 2 2
8 a Same line b Same line
14 17.5

15 0.775

16 0.630

17 5 6

448 A nswers

18 −8i − 5j + k 6⎛ ⎞

0⎛ ⎞ 0⎛ ⎞ 1 19b ir ⎜ − ⎟= ii 6x − y + 2z = 19
1⎜ − ⎟
1⎜ ⎟ •
2⎜ ⎟
19 a ⎜⎝ ⎠⎟ b ⎜⎜⎜⎝−11// 2⎟
2⎟ ⎜⎝ ⎠⎟

⎠⎟ 5 a i r • (3i − 2j + 5k) = 5
ii 3x − 2 y + 5z = 5
⎛ 3/14⎞
1/14⎟⎟
⎜2⎝⎜ − /14⎟⎠ b i r • (4i + j − 2k) = −2
ii 4x + y − 2z = −2
20 ⎜

21 b 13a × b 3⎛ ⎞

22 b 0 6ai 1 9r ⎜ − ⎟= − ii 3x − y = −9

•
0⎜ ⎟
⎛ 18⎞ ⎛−18⎞
23 a ⎜ −12⎟, ⎜ 12⎟ ⎝⎜ ⎠⎟

72 72⎜ ⎟ ⎜ ⎟ b p = −q 4⎛ ⎞
0 10b i r • ⎜ ⎟= −
⎝⎜ ⎟⎠ ⎜⎝− ⎟⎠ ii 4x − 5z = −10
5⎜ ⎟
25 b 42.6
⎝⎜ − ⎠⎟
19
26 2 7 a 10x + 13y − 12z = 38
b 3x + y + z = 1
27 a (11, −2, 0) b 21.9
8 a x + 5y = 22
30 a C(5,4,0), F(5,0,2), G(5,4,2), H(0,4,2) b x + 20y + 7z = 152

b 11.9 9 a x + y + z = 10
b 40x + 5y + 8z = 580
Exercise 8G
⎛12⎞ 1⎛ ⎞ 3⎛ ⎞
3 2 1⎛ ⎞ 4 0 8r = ⎜ ⎟+ λ ⎜ ⎟+ μ ⎜ − ⎟
⎛− ⎞ ⎛⎞ 10 a

4 5 21 a r = ⎜ ⎟+ λ ⎜ ⎟+ μ ⎜ − ⎟, yes 10 5 10⎜ ⎟

⎜⎝ ⎟⎠
2 5 2⎜ ⎟ ⎜⎟ ⎜⎟ ⎜⎟
⎜⎝ ⎟⎠ ⎝⎜ − ⎠⎟ ⎜⎝− ⎟⎠
⎝⎜ − ⎠⎟
⎜⎟ 1 1 1⎛ ⎞
⎝⎜ ⎠⎟

4 3 3⎛ ⎞ ⎛− ⎞ ⎛− ⎞
⎛− ⎞ ⎛⎞ 0 1 0b r = ⎜ ⎟+ λ ⎜ ⎟+ μ ⎜ ⎟

1 1 1b r = ⎜ − ⎟+ λ ⎜ ⎟+ μ ⎜ − ⎟, yes 0 0 1⎜ ⎟

⎝⎜ ⎟⎠
2 1 0⎜ ⎟ ⎜⎟ ⎜⎟
⎜⎝ ⎟⎠ ⎝⎜ ⎟⎠
⎝⎜ ⎠⎟
⎜⎟ ⎜⎟
⎜⎝ ⎠⎟ ⎝⎜ ⎠⎟

3 2 1⎛ ⎞
1 1 1⎛ ⎞ ⎛− ⎞ ⎛⎞

⎛− ⎞ ⎛⎞ 11 a 1 2 0r = ⎜ − ⎟+ λ ⎜ ⎟+ μ ⎜ ⎟

0 5 22 a r = ⎜ ⎟+ λ ⎜ ⎟+ μ ⎜ − ⎟, no 3 1 1⎜ ⎟

⎜⎝ ⎟⎠
2 2 3⎜ ⎟ ⎜⎟ ⎜⎟
⎝⎜ − ⎟⎠ ⎜⎝− ⎠⎟
⎜⎝ ⎠⎟
⎜⎟ ⎜⎟ 1 5 6⎛− ⎞
⎝⎜ ⎠⎟ ⎜⎝ ⎟⎠

0 0 5⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛⎞ ⎛− ⎞

2 4 3b r = ⎜ ⎟+ λ ⎜ ⎟+ μ ⎜ ⎟, no 1 2 2b r = ⎜ − ⎟+ λ ⎜ ⎟+ μ ⎜ ⎟
5 3 4⎜ ⎟

⎜⎝ ⎠⎟
0 1 0⎜ ⎟ ⎜⎟ ⎜⎟ ⎜⎟
⎜⎝ ⎠⎟ ⎝⎜ − ⎠⎟ ⎜⎝− ⎠⎟
⎝⎜ ⎠⎟
⎜⎟ 9 11 8⎛ ⎞
⎜⎝− ⎠⎟

3 a r = (j + k) + λ (3i + j − 3k) + μ(i − 3j), no ⎛− ⎞ ⎛− ⎞
0 1 1r = ⎜ ⎟+ λ ⎜
b r = (i − 6j + 2k) + λ (5i − 6j) + iμ(− + 3j − k), 12 a ⎟+ μ ⎜ − ⎟
yes 0 0 2⎜ ⎟ ⎜
⎟⎜⎟
⎝⎜ ⎠⎟ ⎜⎝ ⎠⎟ ⎜⎝ ⎟⎠

3⎛ ⎞ ⎛ 11⎞ ⎛−10⎞ ⎛−16⎞
5 4r • ⎜ − ⎟= − ii 3x − 5y + 2z = −4 7 21 17b r = ⎜ − ⎟+ λ ⎜
4ai ⎟+ μ ⎜ ⎟
2⎜ ⎟
3 1 3⎜ ⎟
⎝⎜ ⎠⎟
⎝⎜ ⎠⎟
⎜⎟ ⎜⎟
⎜⎝ − ⎠⎟ ⎜⎝ − ⎠⎟

A nswers 449

13 Yes 0 3⎛ ⎞ ⎛ ⎞ 0 9⎛ ⎞ ⎛ ⎞

3 0 1⎛ ⎞ 11 a 8 5r = ⎜ ⎟+ λ ⎜ − ⎟ 3 5b r = ⎜ ⎟+ λ ⎜ − ⎟
⎛⎞ ⎛− ⎞ 1 11⎜ ⎟

⎝⎜ ⎠⎟
14 a 1 2 1r = ⎜ − ⎟+ λ ⎜ ⎟+ μ ⎜ ⎟ 1 1⎜ ⎟ ⎜⎟ ⎜⎟
2 1 1⎜ ⎟ ⎝⎜ − ⎠⎟ ⎜⎝− ⎟⎠
⎝⎜ − ⎟⎠
⎝⎜ ⎟⎠
⎜⎟ ⎜⎟ 3 1⎛ ⎞ ⎛ ⎞ 1 2⎛ ⎞ ⎛ ⎞
⎝⎜ ⎠⎟ ⎝⎜ ⎠⎟

b No b No 12 a 0 1r = ⎜ ⎟+ λ ⎜ ⎟ 0 1b r = ⎜ ⎟+ λ ⎜ ⎟
1 0⎜ ⎟ ⎜⎟ 0 0⎜ ⎟ ⎜⎟
15 a 4x − y + 7z = 39 ⎜⎝ ⎠⎟ ⎝⎜ ⎟⎠
⎜⎝ ⎟⎠ ⎝⎜ ⎠⎟
16 a 5i + j − 4k b 14
13 a ( −11, 64, 88) b ( − 2, 1, 0)
1
17 a 3 10 (i + 5j − 8k) b p = 30, q = 1 14 a Prism b Prism
2⎛ ⎞
15 a ∏1 and ∏2 are parallel
3b ⎜ − ⎟
18 a (1,−3, 14) 1⎜ ⎟ b ∏1 and ∏3 are parallel

⎝⎜ ⎠⎟ 2 1⎛ ⎞ 1 2⎛ ⎞ ⎛ ⎞
c 2x − 3y + z = 25 ⎛− ⎞
19 a (10,11,−6) 7⎛ ⎞ 1 1r = ⎜ ⎟+ λ ⎜ ⎟ 1 1b r = ⎜ − ⎟+ λ ⎜ ⎟
9b ⎜ − ⎟ 16 a 0 1⎜ ⎟ ⎜⎟ 0 1⎜ ⎟ ⎜⎟
c 7x − 9y − 5z = 1 ⎜⎝ ⎠⎟ ⎝⎜ ⎠⎟
5⎜ ⎟ ⎝⎜ ⎠⎟ ⎝⎜ ⎟⎠

⎜⎝ − ⎠⎟ 1 3⎛ ⎞ 6 2⎛ ⎞

⎛− ⎞ ⎛− ⎞
1 4r = ⎜ − ⎟+ λ ⎜ ⎟ 0 1b r = ⎜ ⎟+ λ ⎜ ⎟
⎛11⎞ 6⎛ ⎞ 2⎛ ⎞ 17 a 0 1⎜ ⎟ ⎜⎟ 1 1⎜ ⎟ ⎜⎟
⎝⎜ ⎟⎠ ⎜⎝ − ⎠⎟
12 3 1520 r = ⎜ ⎟+ λ ⎜ − ⎟+ μ ⎜ ⎟ ⎝⎜ ⎠⎟ ⎜⎝− ⎠⎟

⎜⎟ ⎜⎟
⎜⎝ ⎠⎟ ⎝⎜ ⎟⎠
13 1 6⎜ ⎟ 3 1⎛ ⎞ ⎛ ⎞ 1 2⎛ ⎞ ⎛ ⎞
0 1r = ⎜ ⎟+ λ ⎜ ⎟ 0 1b r = ⎜ ⎟+ λ ⎜ ⎟
⎝⎜ ⎠⎟

1⎛ ⎞ 18 a 1 0⎜ ⎟ ⎜⎟ 0 0⎜ ⎟ ⎜⎟
0⎜ ⎟ ⎜⎝ ⎟⎠ ⎝⎜ ⎠⎟
21 b 2⎜ ⎟ c x + 2z = 8 ⎜⎝ ⎟⎠ ⎝⎜ ⎠⎟

⎝⎜ ⎠⎟ 3⎛ ⎞
1⎜ − ⎟
Exercise 8H 19 a 1⎜ ⎟ b 8°

⎝⎜ ⎠⎟

1 a 46.4° b 10.8° 2⎛ ⎞ 3 2⎛− ⎞
2 a 17.5° b 51.0° 2⎜ ⎟ ⎛⎞
3 a 75.8° b 47.6° 1⎜ ⎟
4 a 17.7° b 51.9° 20 a 3 2b r = ⎜ − ⎟+ λ ⎜ ⎟
5 a 48.2° b 51.9° ⎝⎜ − ⎠⎟ 4 1⎜ ⎟ ⎜⎟
⎜⎝− ⎠⎟
⎜⎝ ⎟⎠

c (3, 3, 1) d9

6 a 60° b 60° 21 a 8i − 16j + 24k b x − 2y + 3z = 9
7 a (7, 3, 11) b (5, 3, 6)
8 a (1, 3, 5) b (10, 4, −3) c 69.1°

22 a (9, 6, 7) b 7.42° c 12.1

23 a 32.5° b (1, 1, 5) c 3.39

9 a (−7, 0, 3) b (−7, 1, 5) 4 1⎛ ⎞ ⎛ ⎞

0 1⎛ ⎞ ⎛ ⎞ 1 3⎛ ⎞ 24 a 1 3r = ⎜ ⎟+ λ ⎜ − ⎟ b (3, 4, −3)
⎛− ⎞ 2 5⎜ ⎟ ⎜⎟
1 3r = ⎜ ⎟+ λ ⎜ − ⎟ 1 4b r = ⎜ − ⎟+ λ ⎜ ⎟ ⎜⎝ ⎟⎠
10 a 2 5⎜ ⎟ ⎜⎟ 0 1⎜ ⎟ ⎜⎟ ⎜⎝ ⎠⎟
⎜⎝− ⎠⎟ ⎜⎝ ⎟⎠
⎝⎜ ⎟⎠ ⎝⎜ ⎠⎟ c (2, 7, − 8)

450 Answers

0⎛ ⎞ b 2x − y + z = 0 Chapter 8 Mixed Practice
5⎜ ⎟
25 a 5⎜ ⎟ 1 b a 1 a 3 b
2 2 4
⎝⎜ ⎟⎠ 1 a − b +

3 0⎛ ⎞ ⎛ ⎞ 2 a −14 b 19
4 1d r = ⎜ ⎟+ λ ⎜ ⎟ 16
e (3, 4, 0)
2 1⎜ ⎟ ⎜⎟ 5π
⎜⎝ ⎟⎠ 3 12
⎝⎜ − ⎟⎠

f 47.1° 4 a 6i − 5j − 2k b i + 2j − 2k
c 65
26 a 63 b 83.3 seconds
b Intersect along a line 5 a 128ms−1

27 a Prism

⎛ 0.4⎞ 1⎛ ⎞ 2⎛ ⎞
06 a ⎜
0 1b a = −2, r = ⎜ ⎟ c (3, 6, 1)
⎟+ λ ⎜ ⎟
0.8 0⎜ ⎟ ⎜⎟ 7⎝⎜⎜ k − ⎟
⎝⎜ ⎟⎠ ⎠⎟
⎝⎜ − ⎟⎠

3 1⎛ ⎞ d 1
⎛− ⎞ 130
0 1r = ⎜ ⎟+ λ ⎜ ⎟
28 a 4 4⎜ ⎟⎜⎟ c Prism 1 1⎛ ⎞ ⎛ ⎞
⎝⎜ ⎟⎠
⎝⎜ ⎟⎠ ⎛ 20⎞ 3 57 a r = ⎜ − ⎟+ λ ⎜ ⎟ b 88.5°
b ⎜ −15⎟ d 23.6
29 a c = 1 2 1⎜ ⎟ ⎜⎟
5⎜ ⎟ ⎝⎜ − ⎟⎠
⎜⎝ ⎠⎟
⎝⎜ ⎟⎠
c 24
y−4
c x +3 = −3 = z −1 2⎛− ⎞
4 1 78 a ⎜ ⎟
b (3, 3, 8)
2⎛ − ⎞ 3⎜ ⎟
30 a ⎜ −12⎟ b 31.6
62⎜ ⎟ ⎜⎝ − ⎟⎠
d 2x − 7y + 3z = 9
⎝⎜ ⎟⎠ 2⎛− ⎞
7c ⎜ ⎟
c x + 6y − 31z = −86 e (−4.75, 3.49, 3.30)
3⎜ ⎟
f 248 b 5
3 ⎜⎝ − ⎠⎟ 4
2 2⎛ ⎞ ⎛ ⎞
9 a 3x + y − z = 6

8 0 3c r = ⎜ 1 1
31 b 15 ⎟+ λ ⎜ − ⎟ 10 a i 2 (c − a) bi 3 (a − b)
8/15 5⎜ ⎟ ⎜ ⎟
d 132 6 6
⎜⎝ ⎟⎠ ⎝⎜ ⎠⎟ 2 2
32 b 4i − 4j − 16k
x −1 y +1 z−5 11 6i − j + k
1 −1 −4
d = = 19
3
b k = 1, c = −4 12

33 a k ≠ 1 13 b 107°, 73.2° c 5
4
c k = 1, c ≠ −4

15 133

16 p = 3 , q = 1
8 8

A nswers 451

17 a 2(1 − cosα), 2(1 + cosα) 28 355

b 72.9° 29 c x − 2 = y − 2 = z − 3
18 a Perpendicular
b No 3 7

19 a 2⎛ ⎞ b ii (1,−3, 14) 30 b 1⎛− ⎞ c 30 d5
3⎜ − ⎟ 5 5r • ⎜ ⎟= − d 91.2
1⎜ ⎟
2⎜ ⎟
⎝⎜ ⎟⎠
⎝⎜ ⎠⎟

c 2x − 3y + z = 25 31 5

32 a 51.7° b (1, 1, −1)

3⎛ ⎞ 33 a sin3α − 1 b π c 0; Parallel
2 1⎜ − ⎟ ⎛ ⎞ 2
3 4r ⎜ ⎟ ⎜ ⎟ ⎛−33 .28, 12 , 45⎞ 6
⎜ ⎟+ ⎜ 7 18⎠
20 a = ⎠⎟ λ − ⎟ b ⎝ 34 b 1
c 0 5⎜⎝ ⎠⎟ c
⎝⎜ x 1 1⎛ ⎞ ⎛ ⎞ ⎛ ⎞

2.63 y 1 1k = −1, ⎜ ⎟= ⎜ ⎟+ λ ⎜ − ⎟

0 4⎛ ⎞ ⎛ ⎞ ⎝⎛1272 8 197⎞⎠ z 0 1⎜ ⎟ ⎜ ⎟ ⎜⎟
1 3r = ⎜ − ⎟+ λ ⎜ − ⎟ , 17 , ⎜⎝ ⎟⎠
⎝⎜ ⎠⎟ ⎜⎝ ⎠⎟

21 a 2 3⎜ ⎟ ⎜⎟ b − − 1 0 0⎛ ⎞ ⎛ ⎞ ⎛ ⎞
c ⎝⎜ ⎠⎟ ⎜ 2⎟, ⎜ 1⎟, ⎜ 2⎟
⎝⎜ ⎠⎟ 2 2 1⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎛−1⎞

1.62 35 a ⎝⎜ ⎠⎟ ⎝⎜ ⎟⎠ ⎝⎜ ⎠⎟ b ⎜ 1⎟
ii ⎜ 1⎟⎠⎟
2 1⎛− ⎞ ⎜⎝
(1, 1, 2)
⎛⎞ 1 3 −x + y + z = 3
2 2
22 a 4 1r = ⎜ ⎟+ λ ⎜ ⎟ b ci ,

2 0⎜ ⎟ ⎜⎟ ⎛1 , 5 , 5⎞
⎝⎜ ⎠⎟ ⎝3 3 3⎠
⎝⎜ ⎟⎠

c6 d3 d

3⎛ ⎞ c 3x + 10y + 15z = 21 36 a (4, 1,−2) c d 5 26
⎜ 10⎟ 2
23 b 15⎜ ⎟
1337 t = , d = 14
⎝⎜ ⎠⎟ 3

⎛1/2⎞ 2⎛ ⎞ 38 8

24 a 2 3r= ⎜ − ⎟ λ ⎜ ⎟ 0 1⎛ ⎞ ⎛ ⎞
⎜ ⎟+ ⎜ ⎟
4/3 2⎜

⎝
⎟ ⎝⎜ − ⎠⎟ 39 a 11 3r = ⎜ ⎟+ λ ⎜ − ⎟ b 3x + y = 11
⎠
9 2⎜ ⎟ ⎜⎟
b Yes ⎜⎛at ⎛11 , 0, 0⎞⎞ ⎝⎜ − ⎠⎟
6⎜ ⎟⎟ ⎜⎝ ⎠⎟
⎝
⎝ ⎠⎠ 40 b (5, −7, 6), (−1, 5, −6)

c 61.0°

25 a x −3 y −1 z+4 b (0, 2, −3) 2 3⎛ ⎞ ⎛ ⎞
3 = 1= −1 1 1c r = ⎜ − ⎟+ λ ⎜ − ⎟
e 32
− 0 4⎜ ⎟ ⎜⎟
⎝⎜ ⎠⎟
c (−3, 3, −2) ⎜⎝ ⎠⎟

26 a 15i − 5j + 10k b 3.74 m s−1 c No [second possible solution]:

3⎛ t ⎞ 2 1⎛ ⎞ ⎛ ⎞
1 3r = ⎜ − ⎟+ λ ⎜ ⎟
27 a ⎜ 5 − 4t⎟ b 30 km 0 0⎜ ⎟ ⎜⎟
⎜ ⎟ ⎝⎜ ⎠⎟
⎜⎝ t ⎟⎠ ⎜⎝ ⎠⎟

452 A nswers

41 a ⎛64 , 4 ,199⎞⎠ b 29 7a late?
⎝9 9 3
transport
c ⎛65 , − 10 ,197⎠⎞ L
⎝9 9 0.6
0.4 0.3
42 a 1 a × b b c cosθ B
2 b 0.78
8a 0.7
d 1 e 2 L
3 99 size

f B ⎛⎝h = 2203⎠⎞ 164 L
184 L
43 c (4, −1, 3) d 3x − 2y + z = 10 0.1
T
⎛1⎞ ⎛ 1 ⎞
e r = ⎜⎜⎜⎝21⎟⎠⎟⎟+ t ⎜⎜⎜⎝−11⎟⎟⎟⎠ 0.9

44 b 76 c 95 d 7 30 c 0.538
18 2

45 b 1 broken?

c − 8 B
5
44 0.1
46 a r = p + λn d 26 S

47 11:52 0.9
B
Chapter 9 Prior Knowledge

1 a 19 b 1 0.2 B
39 3 L B

2 a 0.4 b 2.5 0.8 c 0.5

3a2 b e23

Exercise 9A b 0.157 c 0.273
9 a 0.16 b 0.5
1 a 0.243 b 0.913
2 a 0.462 b 0.226 10 a 0.523 b 8
3 a 0.667 b 0.875 17
4 a 0.136 b 0.238 11 a 0.3
b 0.3
12 a 0.87
5 a 0.182 b 0.205 13 a 0.914 b 0.724
6 a 0.0714 b 0.286 b 0.482

A nswers 453

14 a late? Exercise 9B

transport 1a1 b1

W 0.05 L 0.2 × 0.05 = 0.01 2 a 2.5 b 14.5

3a1 b1

0.2 0.95 L 0.2 × 0.95 = 0 .19 4 a 0.388 b 0.595

0.4 0.1 L 0.4 × 0.1 = 0.04 5 a E(Y) = 11, Var(Y) = 28
b E(Y) = 47, Var(Y) = 80

C 0.9 L 0.4 × 0.9 = 0.36 6 a E(Y) = 1, Var(Y) = 28
0.2 L 0.4 × 0.2 = 0.08 b E(Y) = 41, Var(Y) = 80
0.4 2
7 a E (Y ) = 4, Var (Y ) = 9

D 0.8 L 0.4 × 0.8 = 0.32 b E (Y ) = 4, Var (Y ) = 3
5
8 a E(Y) = −30.5, Var(Y) = 33.3
b 0.13 c 0.0769 b E(Y) = −55, Var(Y) = 180

15 a fruit ripe? 9 a E(Y) = 4.4, Var(Y) = 0.2
b E(Y) = 2.8, Var(Y) = 0.3

S 0.6 R 13 ×0.6 = 51 10 a E(Y) = 24, Var(Y) = 18
0.4 R 31 ×0.4 = 125 b E(Y) = 30, Var(Y) = 12
13 0.9 R 13 ×0.9 = 130
31 0.1 R 13 ×0.1 = 310 11 a E (Y ) = 2, Var (Y ) = 1
0.5 R 13 ×0.5 = 16 6
B 0.5 R 31 ×0.5 = 61 2
b E (Y ) = 3, Var (Y ) = 5
31 c 0.3
12 a 0.3 b 1.9 c 2.69
P b 0.435
13 a 0.4 b 1.17

14 a 4.2, 4.96 b 13.6, 44.64
c 13.8
15 a 8 b 34.25, 85.9
15
7 10 16
b 0.667 16 a 25
16 0.471 8

17 0.451 17 a 4.1, 3.09

2 by 25
7
18 P(Y = y) 0.2 0.3 0.4 0.1

3 18 a 6, 4
19 7
bw −3 −1 1 3
20 2.86%
P(W = w) 0.1 0.2 0.3 0.4

21 27.7% 19 E(B) = 3.1, Var(B) = 0.3
20 E(V ) = −6, Var(V ) = 2.56
22 0.460

23 a m(m −1) + n(n −1)
(m + n) (m + n −1)

454 A nswers

21 a y 01234 b 0.469 6x
x
P(Y = y) 0.1 0.2 0.2 0.3 0.2 f(x)

b E(Y) = 2.3, Var(Y) = 1.61 0
c E(X) = 233, Var(Y) = 16100
5 a 0.583
22 a x 012 3
b 0.865
P(X = x) 1 3 3 1
8 8 8 8

b 3 c 50 cents
2

23 Mean 3.67, s.d. 0.972

24 b a + b = 0.3 c 1

25 0.85 25 b 50 − c, 2500 4
26 6 6 f(x)
a 5,
0
27 14 f(x)
28 1.5

Exercise 9C

1 a 4 , 0.355 b 3 , 0.599
65 14

2 a 11 , 1 b 1 , 1
60 3 12 6

3 a 2,116 b 3 3 , 19
7 56

4 a 0.62

f(x)

0 x 0x
5 7.5

A nswers 455

6 a 0.449 15 a 4 b5

f(x) 16 a 136,392 b 133,191

17 a 0.571, 0.142 b 0.386, 0.0391

18 a 12 ,1265 b 3 , 1
5 20 400

19 a 4.79, 3.08 b 3.83, 1.97

20 a 5, 265 b 4, 8
3

21 a 1.14, 0.197 b 0.750, 0.111

22 a 0.172 b 0.467

2 58 x 23 a 11 b 20
32 27
0
24 a 20.2 b 0.129 c 3.97
b 0.729 1 9 c 0.709
25 a 9 b 4
f(x)

26 0.293

27 a 3 b2
4 b 2.25
28 a 3.87 d 2.47
b 0.467
c 3.63 b1
b3
29 a 0

30 a 0.821

30 a ln3

32 a 1 b 8
3 15
2 x
0 4 33 a 22 minutes b 8

34 a i 0.237 ii 0.881

7a1 b8 b 4.63 minutes
8 a 2.67
b 0.167 35 b k c 0.9k
9 a 0, π 2
b π , 3π
2 2 36 a −0.418 b 0.0820
b 0.836
10 a 5.66 b 4.47 37 a 2.95

11 a 1 ln 2 b 1 ln 2 38 0.833
3 2
1 1
12 a 0.572 be 39 a 100 b 8

13 a 1.19 b 0.742 c 15.5 d 16.7

14 a 2.73 b 3.82

456 A nswers

40 a f(x) 45 a f(t)

k

a x 5 t
b0 10

b a+b b 2
2 3
2π
41 a f(x) 46 a π+4

b f(x)

k

0 x
2x
b2 c 36
25 012

c 1.07

42 b 5 Chapter 9 Mixed Practice
3

43 0.984 1 a E(X ) = 3.2, Var(X ) = 1.56
b E(Y) = −7.6, Var(Y) = 14.04
44 a 0.901

b 37.50 2 a 0.1
c 2
b E(V ) = 4.2, SD(V ) = 2.14
c E(W ) = 5.8, SD(W ) = 2.14

3 b 20 c3
27

A nswers 457

4 a f(x) 20 a 1
25

b f(m)

0 4 x

b 8 c 2.46 m
3 0 10
b 6.47, 6.06
5 a 1.87, 0.379 c 0.436 d 5.65, 1.69 e 4%

6 0.667 21 0.782

7a x 1234 11
22 32
P(X = x) 1 5 4 11
13 26 13 26
23 a i
b 40 d 23 A 0.02 F 0.6 × 0.02 = 0 .012
13 0.6 0.98 F 0.6 × 0.98 = 0 .588

8a h 0123

P(H = h) 1 3 3 1
8 8 8 8
0.4 0.01 F 0.4 × 0.01 = 0 .004
b 1.5 c No d 6.75 B 0.99 F 0.4 × 0.99 = 0 .396

9 1.61 b a = 0.2, b = 0.3
10 a 4
7 ii 0.016 iii 0.75
11 a = 0.12, b = 0.26 8
b bi 12
12 a 0.48 35
b 0.538
13 a 0.6
ii x 0123
6
14 23 b 0.567 P(X = x) 4 18 12 1
35 35 35 35
15 a 0.0355

c3 d $1.50 9
16 0.288 7
iii

17 a 20, 20 b 0.156 24 a 56
c 0.132 d Student’s 45

18 a 1.018 b 23 months c 0.269 b 26
d i 0.619 ii 0.919 3

6 c 8 − 2 10
19 19

458 A nswers

25 a i 0.407 ii 0.275 Exercise 10A
b 0.0676
c 0.0340 1 a  b3

26 0.00258 2 a 1.5 b 0.5

27 b 2 c 1 3a6 b 
2
28 a f(x) 4a3 b5
1 1
5 a 2 ln 2 b ln4
1
6ae b 2 e4

7 a a = 2, b = −1 b a = 3, b = −2

8 a a = 5, b = 4 b a = 11, b = 6

9 a a = 2, b = 2 b a = 3, b = 3

10 a a = 1, b = 1 b a = e2, b = −e2
1
11 a a = e , b = 0 b a = 4e−2, b = e− −2
b a = 4,b = −3
12 a a = 2, b = 1

13 a 3 b2

x 14 a 1 b 1.5
2
0  15 a 0 b0
16 a 0.5
1 dπ 17 a 0.5 b1
2 g 0.375 18 a 1
b e 3.40 19 a Divergent b −4

f 3.87 h 1 b 
3
b Divergent
Chapter 10 Prior Knowledge 20 a Convergent b Convergent

1 a 14(2x + 1)6 21 a Divergent b Divergent

b x2 (3ln x + 1) 22 a Convergent b Convergent

cos x − sin x 23 a 2 b0
ex
c 24 a 3x2 b 4x3

2 (2 + 8x + 4x2)e2x 25 a 2x b2

3   26 a 2x + 2 b 3x2 + 1
b 120x3
4 3 ln x 10 x 3 c 27 a 60x2 b 180x2 − 48x
2 3 2 b 8e2x
+ + 28 a 24x + 12
b sin x
5 1 29 a 2
5
x3
30 a − cos x
6 5 + 5sec x − 2sec2 x

7 11 32 a i 1.5
1 2x − + x + ii 0
iii 1

A nswers y Exercise 10B 459
y = 1.5
c y = 1x
1 2y = 1x4 x
1a4 b2

2 a −2 b 2.5

3a1 b −2
4a1 1
5a2 b 2

b4

6a0 b0

7a∞ b∞

8a∞ b∞

ln 2 b a = 1 or 3 x 91 b0
33 2 10 0 1
1 b 4
34 a Divergent 11 2
12 a 
cy
13 a ∞
1 14 25 b0
2 15 1
y
a=1 1
y 16 4

17 a 1

xc

1

4 2

1 19 0
2
20 0
35 a = −0.5, b = −1
1
21 2

x Exercise 10C
3
a=3 1 a 4 b −6

2 a −1 b − e
2

460 A nswers

3 a −e2 b e 24 ⎛3 8 , 3 8 ⎞ and (2,−2)
−2 3 3 ⎠
4 a −3 ⎝

b 1 y25 a
−4

5 a y (2x + y) b y
− x (x + 2 y) −x

6 a x2y2 − x2y + y3
xy2 − x3
y
b 2x2y + 4xy2 + x + 2y3 x

2 b 2x2 + 4xy − 1
x +2y +2
xy7 a − 2

8 a y 2(1 − e(x+ y) ) b xe(x2)
y 2e(x+ y) + 1 = − ye(y2)

9 a 1− y cos(xy) sin(x + y) c (1,−1)
x cos(xy) −1 b − sin(x + y) +1
c y = 2x Exercise 10D
10 b 1
−2 1 a 
b4
11 a (0, 1) b y = x +1
2 a 12 b 7e

12 a − ln 2 b 1 3 a − 3 per second b   
13 y = 2x − 5 −2 e 4

y x14 = 4 − 4 a    b 3 per hour
−2
15 b 2 − 3 5 −2
16 y = 1 − 1.5x and y = −1 − 1.5x
6 −21

17 a (0,0),(0,1),(0,−1) 7 15
b     16

18 a   b x = 1, x = −1 8 20 cm2 s−1

19 a 0 or 5 9 22.6 mm2 per day

y 2x 2 y 23x 2 10 1.92 cm s−1
5 5 5 5
b = − or = + 11 a 10 cm2 s−1

y sin y b 0.8 cm s−1
20 yey − xy cos y − 1
12 5 cm s−1
sin x + x cosx 9π
21 sin y + y cos y
13 7.65 cm
922 − y3
14 2.68 m s−1

23 (− 2, 12) and ( 2, − 12) 15 Increasing

16 0.2 ≈ 0.0894 m s−1
5

A nswers 461

Exercise 10E 12 a −15 cot x⎛ ⎞ + c b 12 tan x⎛ ⎞ + c
3⎝ ⎠ 4⎝ ⎠

1a0 b8 32x c 25x c
2 ln 3 5 ln 2
2 a −6 b −7 13 a + b +

e3 a − −1 b −2e−2 14 a 1 arctan (2 x) c b 1 arctan (4 x ) c
4 Max: 2 Min: −0.25 2 4
+ +

5 Max: 1 Min: 2 15 a arcsin(3x) + c b 2arcsin (5x) + c
e e2

6 4.5 16 a arctan(x + 2) + c b arctan(x − 3) + c

7 0.25

825 17 a 1 arctan ⎛ x + 1⎞ c b 1 arctan ⎛ x + 2⎞ c
9 a x (10 − 2x)2 0  x  5 3 ⎝ + 4 ⎝ +
3 ⎠ 4 ⎠

b 5 c 2000 cm3 d0 18 a 1 arctan ⎛ x 3+ ⎞ + c
3 27 2 ⎝ 2⎠

10 96 cm2 b 43.2 s b 1 arctan ⎛x −55⎠⎞ + c
5
11 a 0.654 cm3 s−1 ⎝

12 1.12 arcsin ⎛ x + 2⎞ c arcsin ⎛ x − 1⎞ c
⎝ ⎝
13 0.254 19 a 4 ⎠ + b 3 ⎠ +

15 a 2 2 m b 5 5m 20 a arcsin ⎛x 2− ⎞ + c b arcsin ⎛ x 1+ ⎞ + c
5⎠ ⎝ 3⎠
⎝

Exercise 10F 21 a ln x + 1 + ln x + 3 + c
b ln x + 3 + ln x − 2 + c
1a7 b 10
3 22 a ln x − 3 − ln x + 1 + c
b ln x + 2 − ln x + 3 + c
2 a 2−8 3 b −7 2

3 a 324ln3 b − 3 ln 2 23 a ln 2x − 1 − ln x + 1 + c
64 b ln 3x + 1 − ln x + 1 + c
1 1 π⎛ ⎞
4 a 2 ln 3 b ln5
y 4 8 x24 − = ⎜ − ⎟
3 2−4 3 19 4⎝ ⎠
2 5
5 a b y 3 81 x π625 −
⎛⎞
6 a 2 tan x + 3sec x + c =− ⎜ − ⎟
b −5cot x − 2 cosec x + c
⎝⎠

26 2

7 a 2 tan x + 3sec x + c 27 6
b −3cot x + cosec x + c 4 + x2

2x 3x π ,2 3π , 228 ⎛ ⎞
ln 2 c ln3 c ⎛ ⎞⎟and ⎜ −⎟
8 a + b + ⎜4 4⎝
⎠⎝ ⎠

9 a 3arcsin x + c b 4arctan x + c 29 (log3 4, 4)

10 a 2arcsin x + x + c b 2x − 3arcsin x + c 30 3
2
11 a −4cot(2x − 1) + b 3tan(3x + 1) + c 7
ln 2
31

462 A nswers

1 4 a ex − ln(ex + 1) + c
232 ± 1
33 y = 3tan(πx) + 2 b ex + 2 e2x + ln (ex − 1) + c

35 a 11 b ln x − 2 + c 5 a 2(1 + sin x) − 1 (1 + sin x) 2 + c
2 1x − − x + x + 1 2
1
36 a 21 b ln 9⎛ ⎞ b 2 (1 + cos x)2 − 2 (1 + cos x) + c
2 2x + − x − 2⎝ ⎠

43 352 ml 6 a 256 b 10
15 21
1
44 a 2 x − x2 bπ 7a ln ⎛16⎞ b 2 ln 5⎛ ⎞
9⎜ ⎟ 3 2⎜ ⎟
2 ln 2
3 ⎝⎠ ⎝⎠

45 a tan x b 8a 2 b π
2
46 π 1 4
6 4
1 x +1 1 ⎛x 3⎞ 9a (π − 2) bπ
4 x +5 3
47 a ln + c b arctan ⎝ + ⎠ + c 7
10
3 10

c ln (x2 + 6x + 18) − 2 arctan ⎛x + 3⎞ + c 4 (11 2 − 4)
105
⎝ 2 ⎠ 11

49 a (2x − 2)2 + 25 12 6 + 8 32

b 1 arctan ⎛2 x − 2⎞ + c
10
⎝ 5 ⎠

50 a arcsin x + x 13 2e x + c

1− x2 14 14
b x arcsin x + 1 − x2 + c 3

51 1 15 a 11 b ln ⎛ ex ⎞ c
u − u+1 e⎜ x
52 ln 2 + 1⎟+
⎝ ⎠

Exercise 10G 16 2 + ln 2

17 1 (ln x)3 +c
3
2 (x 1)5/2 2 (x 1)23 c
1 a 5 + − 3 + + tan x 1 tan3 x +c
3
18 +

b 2 (x − 2)52 + 4 (x − 2)23 + c 19 arcsec(ex) + c
5 3
1⎛ ⎞
2 a 1 (2x + 1)7 − 1 (2x + 1)6 + c 20 arctan 6⎜ ⎟
28 24
⎝⎠

b 811(3x − 2)9 + 1 (3x − 2)8 + c 21 x − ln 1 + ex + c
36
22 1 arcsin ⎛ 5x ⎞ c
2 x)23 x)12 5 2 2⎜ ⎟+
3 a 3 (2 4(2 c
⎝⎠

+ − + + π
4
2 1)23 2)21 23
3
b (x − + 2(x − + c

A nswers 463

y 1 xHint: If you use technology to sketch = ± − 2 13 5e6 +1
36
you might see a familiar shape which helps to explain
this answer. 15 8ln 2 − 4

24 a eu = x + x2 + 4 b ln ⎛1 + 5⎞ 16 e− −x (x2 + 2x + 2) + c
2 ⎟
⎜ 2 ⎠
⎝

17 π2 − 4 1
2
Exercise 10H 18 x (x + 1) ln x − x 2 − x + c

1 a 1 (2 x sin 2 x + cos 2 x) + c 19 a 1 tan 2 x + c
4 2
1 1 1
b 9 (3x sin 3x + cos 3x) + c b 2 x tan 2 x − 4 ln (sec 2 x) + c

2 a 2 x⎛ cos x⎛ ⎞ 2 sin x⎛ ⎞⎞ c 20 b 1 (x 2 + 1) arctan x − 1 x + c
2⎜ ⎟+ 2⎜ ⎟⎟+ 2 2
⎜−
⎝ ⎝⎠ ⎝ ⎠⎠

b 3⎛ x cos⎛⎜⎝3x⎠⎟⎞+ 3 sin x⎛ ⎞⎞ c 21 a tan x b 1− 2 (2 + ln 2)
⎜− 3⎜ ⎟⎟+ 4
⎝ 22 a J = ex sin x − I
⎝ ⎠⎠

3 a − 1 e−2 x (2 x + 1) + c b − 1 e−3 x (3 x + 1) + c 1 ex (sin x cos x) c
4 9 2
b + +

4 a 1 x 2 (2 ln x − 1) + c b 1 x 3 (3 ln x − 1) + c 23 x ln x − x + c
4 9
26 1 e3x (3 sin 2 x 2 cos 2 x) c
1 1 13 − +
x 4x
5 a − (ln x + 1) + c b − 2 (2 ln x + 1) + c 1
10
27 e− x (3 sin 3x − cos3x) + c

6 a 2 x x (3ln x − 2) + c b 2 x (ln x − 2) + c 28 b 6 − 2e
9

7 a 1 e3x (9x 2 − 6 x + 2) + c Exercise 10I
27
1
b − 4 e−2 x (2 x 2 + 2 x + 1) + c 1 a 26 b 45
3 4
1
8 a 4 (−2 x 2 cos 2 x + 2 x sin 2 x + cos 2 x) + c 2 a 1.83 b 0.848

b 1 (−9 x 2 cos 3x + 6 x sin 3x + 2 cos 3x) + c 3 a 5.10 b 8.77
27
4 a 2.48 b 0.527
9 a 3⎛⎜ x 2 sin x⎛ ⎞ 6x cos x⎛ ⎞ 18sin x⎛ ⎞⎞ c 5 a 1370 b 230
⎝ 3⎜ ⎟+ 3⎜ ⎟− 3⎜ ⎟⎟+ 6 a 91.7 b 512
7 a 11.8 b 33.0
⎝⎠ ⎝⎠ ⎝ ⎠⎠ 8 a 3.14 b 7.07
9 a 101 b 134
b 2 x⎛ 2 sin x⎛ ⎞ 4 x cos x⎛ ⎞ 8 sin x⎛ ⎞⎞ c 10 a 12.6 b 45.7
2⎜ ⎟+ 2⎜ ⎟− 2⎜ ⎟⎟+ 11 a 3.59 b 0.771
⎜ 12 a 93.2 b 48.7
⎝ ⎝⎠ ⎝⎠ ⎝ ⎠⎠

10 1 (1 + e2)
4

11 π − 1
2

12 − 2 xe−3x − 2 e−3x +C
3 9

464 A nswers

13 a ln 5 b 4 π 33 1 : 3
5
14 3 34 a y = ln(x − 2) b π (e2 − 8e − 10)
2
15 2 35 a y

16 a −3 b 18π
b 107 (3s.f.)
17 a 32 b 7.75
3
ii 30.1
18 a 2.43

19 b 5.25 1 y = cosx
c i 44.1   x

20 2π 1

21 3646

22 π2
2
y
23 a

y =x c 3π2 b sinθ = 1 − a2
36 a 1 − sinθ

Chapter 10 Mixed Practice

123

23
3
0 x
3 a sec x(sec2 x + tan2 x)

b 127 c 153 4 10x (ln10)3

24 a A (0, 2), B (2, 0) 5 19.0

b 4.39 6 8
5
c 32.7
20 7 2arctan(3x) + c
3 π
25 8 259.2cm3 s−1

26 32 π 9 −16.3
5
10 181
27 a a − 1
11 b 10 c 600 cm2
28 a hx + ry = rh
x y r29 a 2 + 2 = 2 12 a 12
2 1x − − x +
30 a (0, 0) and (1, 1) 3π 13 y = x
b 10 1 1⎞
3 3e⎠⎟
31 π2 − 2π 14 a e3x + 3x e3x c ⎛ ,
4 ⎜−
2 2 ⎝
3 3e2
32 a (0, 0) and (2, 4) b 8π d ⎛ ,− ⎞
3 ⎜− ⎟
⎝ ⎠

A nswers 465

e y 40 a a2 arcsin x⎛ ⎞ xa 1− x2 c
2 a⎝ ⎠ 2 a2
+ +

y = xex b 3 m, θ = π
2
41 In(42++ 135) 6

42 0.243 m s−1

43 0.721

44 0

0 x 45 −1
inection minimum 47 (3,e6)
48 a (−2,4) , (2, 4− )

c (−2,4) is a max, (2,−4) is a min

19 a 4 b No 2349 b m, θ = π
9
6

20 ln3 50 a y = − a h b(x − a)

1 −
221 −
y51 a

23 a ln 11 b 1.52 c 5.33

24 a (1.73, 0) b0

25 π (e4 − 1) r2 y = r2  2
r y =  r2  x2
π2
26 4

27 π

29 1 (1 + 3e4)
16

30 1 + e2 b 1 − 2  rx
t 1 e c31 a − ( + ) −t + e

32 1 (x 2 + 1)52 − 1 (x 2 + 1)23 +c 4
5 3 3
b
33 b 4(e2 − 1)
52 a y = 0.000545x3 − 0.0582x2 +1.69x +10
35 a = −2e−1, b = 4e−1
b 74.4 l
1
36 4 π (e2 − 1) 153 a + k2

37 a 1 c k arctan k − 1 ln (1 + k 2 )
1− x2 2

b x arcsin x + 1− x2 + c 54 a 1, 12 ,13 , 1 ,51, 1 , 1 , 81 , 91 ,110
4 6 7
4 938 a (x − )2 +
b 2 when n is even, −2 when n is odd
b ln x2 − 8x + 25 + 5arctan⎛x − 4⎞ + c
⎝ c 18
3 ⎠
55 (−3, 0), (3, 0), (−1, −4), (−1, 4)

466 Answers

56 a The total area under the curve must be 1. 13 a 2.49 b 1.4

b X = μ + σZ, so the mean is increased by μ and 14 a 2.98 b 0.655
the variance is multiplied by σ2.
15 a 1.36 b 1.46
Link: You learnt about linear transformations
of random variables in Section B. 16 a 4.56 b 3.82

Chapter 11 Prior Knowledge 17 a y = A + Bx − 2e2x b y = 3 + 2x − 2e2x

18 a i 0.9 ii 3.8

1 sin x2 + c b i 0.8 ii 3.6
2
y xc = 2, b ii is furthest away

2 3ex − y 19 a 31.4
2y + x
b Use a smaller step length

3 −8 20 183 m
4 x2
5 1035 21 177 m
6 90
22 y

Exercise 11A 1
123456 x
1 a 1st order linear
1
b 1st order linear 2
3
2 a 2nd order linear 4
5
b 2nd order linear

3 a 2nd order linear

b 3rd order linear

4 a 1st order non-linear

b 1st order non-linear

5 a 1st order non-linear 23 y
2
b 2nd order non-linear

6 a 2nd order non-linear

b 1st order non-linear

7 a dB = kB b dh k
dt dt = h
1.5
d2s k d2s k t
8 a dt2 = s2 b dt2 =

dI kI (N I) dR kR (N − R) 1
dt dt = t
9 a = − b

10 a y = 1 + x3 b y = 3 − cos x 0.5
3

11 a y = 1+ − e2x b y = 5 − arctan x 1 2 3 4x
2
1 b 1.1
12 a y = 1 + x − ln x b y = x + 3x

A nswers 467

24 a 1.46 m b 3 seconds 19 a 1
b 15 minutes 4
25 a dr = −0.0328
dt b 24 000

26 0.889 V20 b = 6000t + 90000

27 b 3.96 21 a v = 100 − 100e−0.1t b 40.8 m
22 y = − 9 − 4e−2x
28 (9.46, 3.71) 23 y = − ln(c − ex)

Exercise 11B 1
2
1 a y = Ae2x b y = Ae−x 24 y = ln(4ex − 3)
2 a y = Aex − 1 b y = Ae−x + 1
25 A = 2, B = −2

3 a y = − c 3 b y= 2 26 y = 102 − 2cos x
c − x4 27 a sin x + cos y = 1
x+ 3
π
4 a y = Ax b y = x2 +c b ±
2

5 a y = cx2 − x b y = c + x 28 y = Ax3 − x
x
2 dy x y y⎛ ⎞
dx y x f x⎜⎝ ⎠⎟
6 a x c + 2ln x b y= c x2 29 a = + =
x2 + 2
x (ln A x)2 b y2 = 2x2 ln(Ax)
7 a y = 4 b y = x c + 2ln x
(2x + y)2 2
−2 30 a 4x2 = 1+ y + 1 ⎛ y = f(x)
x3 + x ⎞
4 x⎝⎜ ⎠⎟
8 y = − c

9 y = 3 3sin x + c b y = x tan ⎛1 ln x⎞
2 ⎝4 ⎠

10 y = ln(x2 + c) 31 y = x ln ⎛ Ax ⎞
)2 ⎟
11 y = 4etanx x⎜ y− ⎟⎠

⎜⎝(

12 2 y 2 = 3x3 + 18 32 a y = x + c
x

13 y = Ax b y = x + 4

1 x
x2 +
14 y = − c 33 y x 3e2− 4
x
=

15 y = 3 9 (x2 + 2) 34 y = 3tan(3ln⎢ Ax⎥)
2 35 y = sin(ln(1 + x2))

= + (x +2)2
y 1 Ae16 a
b y = 1+ ex2+4x
36 y = 2(1 − x)

18 a k = 1 37 y = arcsin(tan x + c)
5
t 600e35t
b m = 25e− 5 38 a N = 1 + 2e35t

c 3.47 seconds

468 Answers

b Increases towards the limit of 300 Exercise 11C

N 1 a y = ce− x + ex b y = cex + xex
400 2
350 c x2 c x3
300 2 a y = x + 3 b y = x + 4
250
200 3 a y = cx + x2 b y = c + x2
150 x2 4
100
50 4 a y = ccos x − cos x ln(cos2 x)
b y = ccos x + x cos x
e5 a 3x b y = 1 ex + ce−3x
6 y = −ex + ce2x 4

8t e7 a x2 b y = (x + c)e−x2 + 4

1234567 8 y = (5x + c) e−2x2 + 3

40 b y = 1 9 b y = 2x3 ln x + cx3
2ex22
1 + 10 a x2 3x2 2
b x2
y Ax 1 y= +

41 b = − x + 1 11 y = (2x + c) ecosx

42 (2x − 3y + 3)2 = 6x − 2x2 12 y = (3x + 4)e−5x2

43 a dv = 4u − v 13 y = c − 1
du 2u + v x x2

b (y − x − 4)3 (y + 4x + 1)2 = c 15 b y = (x + 4 − π4)sec2 x

44 y = arctan(x + c) − x e16 a sin x
b y = 1 + ce−sinx
3 1
45 A = 4 , B = −2 17 y ln x+ 2
x
=

M 1 Ae46 b ⎛α 3 ⎛ 3 c 3 y 2x3 + 3x2 − 12 x + c
3− β ⎞ ⎛α ⎞ 6(x + 2)
= ⎞ − ⎝⎜ β⎠⎟ 18 =
⎟⎠
⎜⎝β⎠⎟ ⎜⎝

dM 19 y = x2(x2 + 2)
2(x2 + 1)

( )3 20 y = 2 + c
x2

21 y = 3x2 + c
x −1
1 ⎛1
22 y = 2 sin x + ⎝2 x + c⎞sec x
⎠

10 20 30 40 50 t e(e − 1)
23 4
24 y = (3sin x + c)sec2 x

A nswers 469

25 y = −cot x + 2 cosec x Exercise 11D

26 y = x2 ln(x − 3) + cx2 1 a x − x3 + x5 b 1 − x2 + x4

27 y = (x + 2)cos x 6 120 2 24

28 y = x2 + c 12 a + x + x2 b x − x2 + x3
2(x2 −1)
2 3

29 a i 5 + (v0 − 5)e−2t ii 5 3 a 1 + x − x2 b x − x2 + 3x3
8
vt0 −5 2 8 2
+1
bi 5(t + 1) + 4 a x + x3 + 3x5 2 6b π − x − x3
40
dy 6 x+c21x+82 1−
dx
30 a 2 xy 2 + 2 x 2 y 1 x x2 x3 b1 − x x3

5 a − 2 + 8 − 48 b 3 162

31 b y = 3 1 (ex + c) 9x4 9x6
x 2 2
6 a 1 + 3x2 + +
1
32 b y =− 2 x 2 − x 4x9
3
y33 = x2 + cx b 1 + 2x3 + 2x6 +

dz 7 a 1 − 9x4 + 27x8 b 1 − 2x6 + 2x12
dx 2 8 3
34 a + z tan x = 2 cos2 x 64x3 9x2
4 x 8x2 3 −3x 2 9x3
8 a − + b − −

35 b z = x3 + cx2 39 a x + x2 + x3 x3
1
c y = 3 x3 + cx2 + d b 1 + x − 3

P e e36 a i x2 7x3 1 x 11x2
= α ( t−γ − −βt) 10 a x + − 24 b + − 18
2 3
β −γ

1 lnii ⎛β⎞ 11 a x − 3x2 + 11x3 b x − x2 + 23x3
β −γ ⎜⎝γ ⎠⎟ 2 6 24

iii Diagram of growth that rises and falls, with 12 a f′(x) = sec2 x, f′′(x) = 2sec2 x tan x, f′′′(x)
t-value at maximum labelled
= 4sec2 x tan2 x + 2sec4 x

P t cb = (α + )e−βt b x + x3
3
37 a [Bi] = [Bi]0 e−k1t
c 0.849%

b Rate of change of Po governed by generation 13 a f′(x) = sec x tan x, f′′(x) = sec x tan2 x + sec3 x,
from bismuth decay and loss due to Po decay f′′′(x) = sec x tan3 x + 5sec3 x tan x

c [Po] = kk12[−Bik]01 (e−k1t − e−k2t) b 1 + x2

[Pb] k1kk22−[Bki1]0 ⎛1 e−k2t 1 e− k1t⎞⎠⎟+ [Bi]0 2
k⎜⎝ 2 k1
d = − c 1.02, 0.0332%

e [Bi]0 14 x − 9 x3 + 27 x5
2 8

2 315 a −x − x2 − x3 b 79
− 750

470 A nswers

16 a 1+ x − x2 + x3 b 6e3 + 6e2 − 3e + 2 33 a x − x3 + x5 b 1
6e3 3
e 2e2 3e3 1 2 24
234 −
c 0.270%

17 x + x2 − 2x3 x2 x4 x6 x8
3
2 35 a − 2 + 3 − 4 b2

18 2x + 2x2 − 4x3 1− 3x 11x2
3 2 +8
36
5x2 5x3 3x2 7x3
19 a 1+ 2x + 2 + 3 b 1+ x + 2 + 6 ln 2 x x2 x3

37 a + 2 − 8 + 24

20 1− x2 b 2 ln 2 x 7x2 11x3
4 12
4 − − +

21 a −3x − 9x2 − 9x3 b 3 38 a x2 − x4 + x6 b0
2 −2
6 120

22 2 x2 x3 x4 x5 1
2
23 a x x3 2x5 b x x2 5x3 x4 39 a − 2 + 4 − 4 b
15 6
+ 3 + + + + 2

24 a x − 2x3 e40 a xln2 b 1 + (ln 2) x + (ln 2)2 x2
3 2

b Use sin x cos x = 1 sin 2x c ln 2
2 ln3

25 a x + x3 b 364 41 a f''(x) = sec2 x, f'''(x) = 2sec2 x tan x, f(4)(x)
3 375
= 2sec4 x + 4sec2 x tan2 x
x2 x4 1
26 a 2 x − 4x3 b 6 b x + + c −3
3 5 2 12

1 , x (1 − x2 )− 3 , (1 − 2 )− 3 3x2 (1 − x2 )− 5 42 a x − x3 + x5
1− 2 2 2
27 a x2 x + 6 120

x3 2 4x3 2 43 a i ln(1 + x) ≈ x − x2 + x3 − x4 + x5
3 3
b x + 6 c x − d 2 3 4 5

1+ x x2 x3 ii arctan x ≈ x − x3 + x5

28 a e − 2e2 + 3e3 3 5

1 229 a − x2 + x4 b 23 b x − x2 + x4 − 11x5
30 60
2 4

c 2.66% 44 a 1(− )n b 1(− )n
(2n + 1)! (2n + 1)!
d 2.42 × 10−6% – approximation is much better
at small values of x
2 1n (− )n
5630 a x − x2 + x3 45 b n! − (2n)!

b 3 c 1 n x 46 a (−1)n−1 n (for n 1)
8 2

31 a 1 + 1 x2 b x + x3 47 y = − 1 x + 3
2 4
6

32 b 2 48 (0, 2), local maximum
3

A nswers 471

49 a 1 b 1 18 x ∑ (−1)r 2252 …(3n − 1)2 x3n+1
2 30 (3r + 1)!
+ r=1

50 b x + x2 + x3 + x4 + x5 (−1)r (−2)r r!
2r r! (2r + 1)!
2 6 24 A∑ x2r B∑ x 2 r +1

Exercise 11E 19 r=0 + r=0

1 a (k 1) aakk++11 −ak2 Hint: You might not have found the answer in exactly
b (k 1)
+ = ak this form. Can you see why?
+ =
11
2 a (k + 2)(k + 11))aakk++22 = 3−a4kak 2r r! = 2 × 4 × 6… × 2r
b (k + 2)(k + =
20 b i x ii 1 (3x 2 − 1)
2
3 a (k + 2)(k + 1) ak+2 +(k + 1)ak+1 = ak 2(r − k)
21 a ar+2 = (r + 2)(r + 1) ar
b (k + 2)(k + 1)ak+2 −(k + 1)ak+1 = −2ak
3x2 y x 2 x3
4 a 1 + 3x + 2 b k is an odd positive integer; y = x and = − 3

b 1+ 2x + 3x2 22 b 2x2 − 1, 4x3 − 3x

5 a 2 + 4x + 8x2 b 3 − 9x + 27x2 B
x2 x
6 a 1+ 2x − 3x2 b −1 + 2x + 23 b y=A x+
2
x2
7 1 + x +
2
Chapter 11 Mixed Practice
8 a 1+ 2x + 2x2 b5

9 a 1 + 1 x3 − 1 x6 b c + x + 1 x4 − 1 x7 1 a y = Ae23sin2x b y = 5e23sin2x
2 8 8 56

c 0.10001 2 y = Aex3−2x

10 1+ 2x + 2x2 − x3 e3 a −x3 b y = 3ex3 − 2

6 4 b y = (x2 + 1)(arctan x + c)

11 a 1 − x3 b 0.979 5 1.44

6

12 1− x − x2 + x3 6 x − 7x3 + 7x5
6 40
2 3
x2
13 y = −2 + 3x − 13 x 2 + 91 x 3 7 a 1 + x +
2 6 2

14 a ∑ x k +1 b c + ∑ xk +2 8a 2x − 8x3 + 32x5 b 8
k! k !(k 2) 3 5 3
k=0 k=0 + x2 x4
A Bx 2 12
c 1.17 9 a 1 + x2 b + + +

1 + (1 − e) x e2 − e 2 10 a 2.23

15 + 2 x b (ii) y = cos x (sin x + 2)

x2n 11 a R = R0e−kt b ln 2

17 a ∑ 1 k 1 1
2 4 4
n=0 2n n ! c The time taken to go from to and from
1 etc will be the same.
b e x2 to 8
2

472 A nswers

y12 a 26 a 1 1 x2 , x , 1 + 2x2
(1 −
10 − (1 − x2)23 x2)52
9
8 b x + x3 c 1
7 3
6 6
5
4 27 2 + 4x − x3
3
2 2
1
28 1+ 2x + 1 − e x2
1 2 3 4 5 6 7 8 9 10 x
2

29 a y = kx d 50

30 a ii y = ± sin 2x + c

iii c = 1

b 0.6 b i y = sin 2x + 3 , y ∈ ⎣⎡ 2, 2⎤⎦

13 a  ii 2.99
b − ln(e−x + e − 1); 0.0181
iii 13.0

31 y = 1+ x
1− x
c Decrease the step length

14 a  32 a dy = v − 1 ,where v = y
e 1b x2 (x − ) ; 0.0392 dx ln v x
b y (ln(xy) − 1) = x ln(ex)
15 y = 1 + 2e−tanx
17 y = (x + 4)e2x2 12
b y = 2e3− 3 c 1.92%
18 a 3.92 x −−

19 a y = arctan(ln x − 1) 33 3333 a v + + v −

b 2.39% b dy = v + 9 + v , where v = y
dx 1 + v x

N 2e20 = 0.2⎛t−12 cos⎛⎝π6t⎞⎠⎞⎠ 34 16ey −x = (2x − 3y − 3)
⎝π

21 a dy + (2x) y = 2x (1 + x2) 35 y = (x − 1)2
dx 4x
4(cos x − sin x)
b y = x2 + ce−x2 37 a cos x − sin x , (sin x −2 , (sin x + cos x)3
sin x + cos x + cos x)2
dy y
22 a dx = 3v + 2v2 where v = x x2 x4

y x3 38 a cos xesinx b 1+ x + 2 − 8
− x2
b = 6 1
6
c

23 2x2 39 a 5 c 3 terms
9 −4

2 324 x − x2 − x3 40 1 × 5 × 9…(4n + 1) x 2 n +1
(2n + 1)!
x4 ∑

25 6 n=0

A nswers 473

Analysis and approaches 6 a 21
1 1x − − x +

HL: Practice Paper 1 b ln ⎛27⎞
⎝16⎠

1 y − π = − 2 ⎛x − 1⎞ 7 a 2t b π , 5π c −2
3 3⎠ 1− t2 8
4 ⎝ 8

1 9 (0, 0), (−1, 1)
3
2 a 10 a 3

b 5 b ii −3
16
5 7⎛− ⎞
⎛− ⎞
10 4iii r = ⎜ ⎟+ λ ⎜ ⎟
(4 + 3i) 0 1⎜ ⎟ ⎜⎟
3 a 5 ⎜⎝ ⎠⎟
⎜⎝ ⎠⎟

b p = 2 ,q = 1 c i 46 ii 3
5 −5 2 23
11 b e−t
4a y
0 for t  0
c i F(t) = ⎪⎧ 0
e− −t for t >
3 1⎨
(4, 2)
⎪⎩

d i e−3 (ii) ln 2

ei C 1 1 ii 1 − 6e−5
e− −5 1 − e−5

12 a i 2 sin ⎛x + π⎞

x ⎝ 4⎠

16 ii x = 3π , 7π
4 4

iii f′(x) = (sin x 1 x)2 >0
+ cos

by iv y

31 34  74 2 x

1 (4,2) 6 x

5 ln 9 S Cb i + = π ii π − 1 ln 2
4
4 8

474 Answers

Analysis and approaches Analysis and approaches

HL: Practice Paper 2 HL: Practice Paper 3

1 a un+1 = 1.04un + 100 b 32 c 33.5% 1 a i and ii
y
2 a 2 + x + 2y b x
y y = x2  2x + 1
y=0
3 a 21, −4 b 11 1

4 15.7 ⎛ 4⎞ y
b ⎜⎝⎜⎜ 1100⎟⎠⎟⎟
5 1 y = 3x
2 y = x2  2x + 1

6 a 3 km per minute. Two solutions

c Goes through the point at t = 3 for A and t = 6 iii If a = 0 or a = −4 there is one solution. x
x
for B. If −4 < a < 0 there is no solution.

d No – closest distance is 6.44 km Otherwise there are two solutions.

7 a x > 1.76
1.76
b x > k

8 a 86 400
1
bi 5

ii 1
15

10 a σ = 2.57, μ = 11.3

b Expectation: 2.4, Standard deviation: 2.11

c 0.64 d 0.684 e 0.6

11 a ii y = 1+ ex
1+ x

bi ex − 2 ⎝⎜⎛ddxy ⎞⎟⎠2 − 2 d2 y − 2y d2 y
dx dx2
2

ii −3, 16, −65

ii y ≈ 2 − 3x + 8x 2 − 113 x 3
6
c 1.47

12 a iii −cos6θ + 6cos4θ − 15cos2θ + 10

b ii 1 + 3x2 + x4

5nπ 2
8
c

A nswers y iii 475
y
iv y = x2  2x + 1

y = 4x 1
( 1, 4) x

They are tangential. x
bi y
iv ⎛ b3 ,1 − 2b b⎞
1 ⎝ 3
3⎠
ii y 3
v 34
1
ci y

x

y = x

y= lnx

1x

ii y

x

y = lnx
1x
y= x

476 A nswers
iii
y Be the Examiner answers

2.1 Solution 2

3.1 Solution 2

3.2 Solution 2

4.1 Solution 1

y = lnx 4.2 Solution 3
5.1 Didn’t prove true for n = 1

y = 0.2x 1 Should have said ‘Assume true for n = k’

x Can’t use result for n = k + 1 in final step

6.1 Solution 3

6.2 Solution 2

7.1 Solution 3

iv y − ln p 1 (x p) 7.2 Solution 2
p
= − 8.1 Solution 3

ve 8.2 Solution 2

1vi k < 0 or k = 8.3 Solution 3
ed 0.1286
8.4 Solution 2

2 a ii 5 9.1 Solution 3

b ii Bn = (n + 1)!An, B3 = 120 9.2 Solution 2
e B3 × 43 = 7680
10.1 Solution 2

10.2 Solution 3

11.1 Solution 1

Glossary

Argand diagram Another term for the complex plane Euler form A way of writing a complex number, z, in

Argument (of a complex number) The angle a terms of its modulus, r, and argument, θ: z = reiθ

number in the complex plane makes with the real axis, Euler’s method An iterative method that approximates
the solution of a differential equation
measured anticlockwise
Even function A function such that f(−x) = f(x) for all x
Base vectors The vectors i, j and k, which are of
magnitude 1 and parallel to the x, y and z axes in the domain of f

respectively First-order differential equation An equation with a
yBoundary conditions Values d first derivative term but no higher derivatives
xof x (other than x = 0)of y or d at other values

Cartesian equation (of plane) The form General solution (of differential equation) The
solution containing an unknown constant

⎛ n1 ⎞ General solution (of system of equations) A form of
⎜ ⎟ solution in which the variables are expressed in terms
n1x + n2y + n3z = d, where n = ⎜ n2 ⎟is a normal to of parameter(s)
⎜ ⎟
⎜⎝ n3 ⎠⎟ Homogenous differential equation A differential

the plane and d = a • n for a point in the plane with ( )equation dy f xy
position vector a that can be written in the form dx =

Cartesian form (of complex number) A way of Imaginary part If z = x + iy, then the imaginary part of z
is the real number y
writing a complex number, z, in terms of its real and
Inconsistent (system of equations) A set of
imaginary parts: z = x + iy, where x, y ∈ simultaneous equations that do not have a solution

Combination An arrangement of items where the order Inductive step A step in proof by induction that
does not matter

Complex conjugate If z = x + iy, then the complex establishes the result for the next integer by building

conj ugate of z is z* = x − iy on the result for the previous integer

Complex number A number that can be written in the yxInitial The y d at x = 0
conditions value of or d
 i 1form x + iy, where x, y ∈ and = −
Integrating factor A function that can be multiplied
Complex plane A Cartesian plane where the x-axis
through a first order linear differential equation so that
represents the real part of a complex number and the d
the LHS can be expressed as dx (f(x, y ))
y-axis the imaginary part

Components (of a vector) The number of units in the Integration by parts A method for integrating a
direction of the coordinate axes

Consistent (system of equation) A set of product of two functions: ∫ u dv dx = uv − ∫v du dx
simultaneous equations that have solution(s) dx dx

Continuous function A function whose graph can be Linear differential equation A differential equation
drawn without taking pen from paper
where neither y nor any of its derivatives are multiplied

Continuous random variable A variable that can take together, or have any non-linear function applied to
any real value in a given interval (which may be finite
or infinite) them

L’Hôpital’s rule A rule for finding limits of the form 0 or ∞
0 ∞
Maclaurin series An infinite series in positive integer
Counterexample A particular case that disproves a powers of x that represents a function
statement

Cross product Another term for vector product Modulus (of a complex number) The distance of a
number from the origin in the complex plane
Degree (of a polynomial) The highest power of x in a
Modulus–argument (polar) form A way of writing
polynomial
a complex number, z, in terms of its modulus, r, and
Direction vector (of line) A vector parallel to a given
line argument, θ: z = r(cos θ+ i sin θ)

Displacement vector A vector from one point to Oblique asymptote An asymptote that is neither
another point horizontal nor vertical

Dot product Another term for scalar product Odd function A function such that f(−x) = −f(x) for all x
in the domain of f

478 Glossary

Order (of a polynomial) Another term for degree Scalar product A scalar value given by | a | | b | cos θ,
where θis the angle between a and b
Parametric form (of equation of line) A form of the
Scalar product form (of the equation of a plane)
equation where x, y and z are expressed in terms of a
The form r • n = a • n, where a is a point in the plane
parameter
and n is a normal to the plane
Partial fractions Two or more rational functions that
sum to give a more complicated rational function Self-inverse function A function f such that f −1(x) = f(x)
for all x in the domain of f
Particular solution A solution where the values of any
constants have been found Skew Straight lines that do not intersect and are not
parallel
Permutation An arrangement of items where the order
matters Solid of revolution A 3D shape formed by rotating part

Polynomial An expression that can be written as a sum of a curve 360° around the x-axis (or y-axis)
of terms involving only non-negative integer powers
Unit vector A vector of magnitude 1
of x
Variance (of random variable) A measure of spread:
Position vector A vector from the origin to a point
Var(X ) = E(X 2) − [E(X )]2
Probability density function A function, f, that gives
Vector A quantity that has both magnitude and
∫the variable: direction

P(a Vector equation (of plane) The form r = a + λd1 + μd2,
distribution ofb a continuous random
where a is a point in the plane and d1 and d2 are two
< X < b) = a f(x ) dx vectors that lie in the plane

Proof by contradiction An indirect method of proof Vector product A vector perpendicular to the two given
that starts by assuming the statement is false and
shows that this leads to an impossible or contradictory vectors with magnitude | a | | b | sin θ, where θis the
conclusion
angle between a and b
Real part If z = x + iy, then the real part of z is the real
number x Volume of revolution The volume of a solid of
revolution

Recurrence relation A formula that defines the next
term of a sequence from previous term(s)

Index

3D shapes 57, 317 higher derivatives 302–3 representing 56, 63, 68
implicit differentiation 309–13 roots 74–7, 79, 80–3
Abel, Niels 121 integration by parts 327–30 two-dimensional coordinates of
acceleration 303 integration by substitution 324–5
algebra 16 L’Hôpital’s rule 305–8, 371 61–2
limits 299–300, 302, 305–8
algebraic expressions 41 optimization 314–15 see also coefcients
binomial theorem 17–24 partial fractions 321
fundamental theorem of 77 radians 302 complex plane 61–2, 474
partial fractions 22–4 related rates of change 312–13 components, of a vector 178–80, 200,
properties of a vector product 229 solid of revolution 333, 475
properties of scalar product 202–3 volumes of revolution 333–5, 475 474
representation of a function 133 compound angle identities 44–6,
systems of linear equations 25–7 see also trigonomic functions
48–9, 302
see also calculus Cantor, Georg 103 conditional probability 263–4
Cardano, Girolamo 84 conjugate pairs 77
Al-Samawal 84 Cardano’s formula 121 consistent system of equation 474
AND rule 4–8 Cartesian constant coefficients 359
angles continuity 297–8
equation of line 210–13 continuous functions 297–8, 474
nding with vectors 200–2, 213–14, equation of plane 236–9, 474 continuous random variables 274–81,
242–9 form (of complex number) 58–61,
474
see also triangles 66, 69–71, 80, 474 contradiction, proof by 95, 102–3, 475
Cauchy–Riemann conditions 88 convergent functions 299–300
approximation 262, 276, 307, 370 cosecant (cosec) 37
arbitrary constant 362 choosing r items 6–7 cosine (cos) 36–40, 44–5, 48–9, 87–8,
arccos 40–2
arcsin 39–40 chord, gradient of 300 99, 163–5, 200, 302
arctan 41, 320 circular reasoning 79 cotangent (cot) 37–8
area coefficients 6, 22, 74–7 counterexample, disproof by 95,

bounded by a curve 332–3 comparing 374 104–5, 474
triangles 229–30 constant 359 counting principles
vector products 229–31 polynomials 110, 118–20
Argand diagram 61–2, 65–8, 80–3, 474 quadratic equations 122–5 basic techniques 2–8
argument (of a complex number) real 76 problem solving 11–12
65–6, 474 coincident lines 223 cross product 226, 237, 474
assumptions 102–3 collinear points 193–4, 207 cubic inequalities 139–40
see also proof column vector 178 cubic polynomial graphs 110
combination 474 curves 311
base vectors 179–80, 190, 474 gradient of 310
Bayes, Thomas 265 combinations of n items 6–7
Bayes’ theorem 264–7 degree, of a polynomial 110, 474
binomial complex conjugates 59–90, 67, De Moivre’s theorem 79–80, 87, 99
74–7, 474 dependent variables 352
coefcients 6, 22 derivative of the function 300
expansion 6, 18–20, 22–4, 366 complex exponentials 88 derivatives 302–3, 317, 319
theorem 17–24, 87 complex numbers 474
boundary conditions 352, 474 differential equations 352
Brahmagupta 84 Argand diagram 61–2, 65–8, determinant 28
80–3, 474 differential equations 352–5, 357–9,
calculus 295
applied 317–21 calculations with 59–90, 68 362–4, 374–6, 474
area bounded by a curve 332–3 Cartesian form 58–61, 66, 69–71,
with complex numbers 88 see also calculus
continuous functions 297–8 80, 474
derivatives 302–3, 317, 319 denition 474 differentiation 297–8
differentiation 300–1, 309–13, 320 De Moivre’s theorem 79–80, 87 from rst principles 300–1
fundamentals of 297 Euler form 69–71, 79, 474 implicit 309–13
geometric interpretation of exponential form 69–70, 79
integrals 332 factorizing polynomials 74–7 diffusion 221
indices 70 direction vectors 209, 474
modulus-argument (polar) form Dirichlet function 298
discrete random variables 270–1
65–8, 474 displacement 303
powers 70 displacement vectors 190–1,
quadratic equations 74–7
216–17, 474

480 Index

disproof by counterexample 104–5, graphs of the functions y 143–55 i, number 57–61
474 imaginary parts 59–60, 68, 474
modulus 143–7 implicit differentiation 309–13
see also proof odd and even 163–5, 474 inconsistent system of equations 28,
properties of 163–5
divergent functions 299–300 rational functions of the form 134–7 474
divisibility self-inverse 166–7, 475 independent variables 352
solving inequalities 139–41 induction, proof by 95–100
and mathematical induction 98 inductive reasoning 100
and polynomials 118–20, 136–7 see also trigonomic functions inductive step 96, 474
domain restriction 165–6 inequalities, solving 139–41, 146–7
dot product 199, 474 fundamental theorem of algebra 77 infinite polynomials 17–18
double angle identities 47 infinite series 366–7
Gabriel’s Horn, 335 infinity 299, 335
equations Galois, Evariste 121 initial conditions 352, 474
complex numbers 58–60, 74–7 generalization 6, 94 integrating factors 362–4, 474
consistent 474 integration
determinant 28 and validity 102
differential 352–5, 357–9, 362–4, by parts 327–30, 474
374–6 see also binomial theorem by substitution 324–5
inconsistent 28, 474 intersection of lines 221–4
of a line 207–17 general solution inverse function 165–6
linear 25–6 of differential equation 352–3, 474 inverse trigonomic functions 39–42
modulus function 146–7 of system of equations 27, 246, 474 irrational numbers 102–3
of a plane 234–9, 475 isoperimetric problem 317
quadratic 58, 74–6, 122–5 geometric
simultaneous 25 interpretation of integrals 332 kinematics 216–17, 303
solving 25–7 representations 57
systems of 25–7 sequences 18 Ladder problem 317
law of the excluded middle 103
see also trigonomic functions Gödel, Kurt 104 L’Hôpital’s rule 305–8, 371
Gödel’s incompleteness theorem 104 linear differential equations 352, 474
equivalence 49 gradient linear equations 17, 25–6
Euclid 102 lines
Euler, Leonard 355 of the chord 300
Euler form 69–71, 79, 474 of a curve 310 coincident 223
Euler’s method 353–5, 474 of the tangent 300 equations of 207–17
even functions 163–5, 474 graphs intersection of 221–4
excluded middle, law of 103 continuous functions 297–8, 474 parallel 223–4
expanding brackets 17 cubic inequalities 139–40 skew 221–3, 475
expanding expressions 48–9 horizontal asymptote 134–7, 140, Liu Hui 84
exponential form 69–70, 79, 88, 307 logarithms 363
exponents, laws of 79, 363 151–5 Lorentz force 227
modulus function 143–7
factorization 19 oblique asymptote 135–6 Maclaurin series 366–71, 374–6, 474
factorizing polynomials 74–7, 113, odd and even functions 163–5 magnitude, of a vector 185, 191–2,
polynomials 110–13
118–20 rational functions of the form 226–9
Fermat’s little theorem 98 mathematical induction 96–100
first-order differential equation 352, 134–7
symmetries 164–5 see also proof
474 symmetries of trigonometric matrices 28
first principles, differentiation from Maurolico, Francesco 96
graphs 48–9 mean 280–1
300–1 transformations 151–5 median 277–8
formulas, as generalizations 6 vertical asymptote 134–7, 140, mode 277
fractals 57 modelling real life situations
fractions, partial 22–4, 321, 475 151–5
functions optimization 314–15
x-intercept 112–13, 134–7, 140, 151–5 using probability 263, 265
convergent 299–300 y-intercept 112–13, 134–7, 140, using trigonomic functions 35
cubic inequalities 139–40 using vectors 176–8
derivative of 300 151–5 modular arithmetic 98
divergent 299–300
domain restriction 165–6 heat equation 352
nding inverse function 165–6 higher derivatives 302–3
higher order polynomials 125
Hilbert’s Hotel 8
homogeneous differential equations

357–9, 474
horizontal asymptote 134–7, 140, 151–5
hyperbola 134

Index 481

modulus (of a complex number) number of roots of 77 real coefficients 76
65–6, 474 order of 110, 474 real parts 59, 61, 87, 475
quadratic 110, 122–5 recurrence relation 374–5, 475
modulus-argument (polar) form quartic 110 remainder theorem 118–20
65–8, 474 quotient 118–20 rhombus 192
real coefcients 76 right-angled triangle 36
modulus equations 146–7 remainder theorem 118–20 right hand screw rule 227
modulus function 143–7 position vectors 190–5, 207–8, 475 roots
Monty Hall problem 267 positive integers 17–18
Moving Sofa problem 317 mathematical proof 96 complex numbers 74–7, 79, 80–3
multiple angle identities 48 powers 18, 79 quadratic equations 122–5
powers of unity 80–1
nC 6–8, 18 complex numbers 70 rotations 68
negative integers 79
negative integers 18 non-negative integers 110 see also volumes of revolution
powers 79 positive integers 17–18, 79
probability scalar product 199–204, 214, 475
negative numbers Bayes’ theorem 264–7 scalar product form 236–7, 475
debts as 84 conditional 263–4 scalar quantities 177
history of 84 continuous random variables secant (sec) 36–9
square root of 58 self-inverse functions 166–7, 475
274–81 sequences 95
neural networks 17 density function 274–5, 475
discrete random variables 270–1 see also series
nP 7–8, 12 linear transformations 270–1 series
random variables 263, 270–1
number i 57–61 tree diagram 265–6 differential equations 352–5,
number line 57, 63 variance 270–1, 280–1, 475 374–6
problem solving
complex numbers 61–2 counting techniques 11–12 Euler’s method 353–5
equivalence 49 integrating factors 362–4
oblique asymptote 135–6, 474 using vectors 176–8 Maclaurin series 366–71, 374–6, 474
Occam’s razor 112 and mathematical induction 97
odd functions 163–5, 474 see also trigonomic functions recurrence relation 374–5
optimization 314–15 simultaneous equations 25
order, of a polynomial 110, 474 proof 94 sine (sin) 36–42, 44–5, 48–9, 87–8, 99,
OR rule 4–8 compound angle identities 44–6 163–5, 229, 302
by contradiction 95, 102–3, 475 skew lines 221–3, 475
parallel lines 223–4 deductive 95 solid of revolution 333, 475
parallelogram 184, 192, 229–30 differentiation 302 space 112, 176, 221
parallel vectors 183–5, 192, 203–4, disproof by counterexample 95, speed 177
104–5 square numbers 97
223–4, 229 and generalization 102 square root
parametric form 210, 475 by induction 95–100, 474 negative numbers 58
partial differential equations 352 validity of 102
partial fractions 22–4, 321, 475 see also roots
particular solution 352–3, 375, 475 Pythagoras’ theorem 185
patterns, as representations 56, 63 substitution 324–5
quadratic denominator 321
permutations of n items 4–7, 11–12, quadratic equations 58, 74–6, 122–5 tangent (tan) 36–41, 46–7, 163–5, 300
quadratic polynomial graphs 110 Thomae function 298
475 quartic polynomial graphs 110 three dimensions
perpendicular vectors 203–4, 214, 229,
radians 302 intersection of lines 221–4
237–8 Ramsey theory 8
physics, and vectors 227 random variables 263, 270–1 see also 3D shapes
polynomials 109–10, 475 rational functions of the form 134–7
rational numbers 18 Torricelli’s trumpet 335
complex coefcients 75–7 ratios transformations 151–5
cubic 110 tree diagram 265–6
degree of 110, 474 calculus 306 triangles
division 118–20, 136–7 trigonomic functions 36
expressions of higher powers areas of 229–30
radians 302
109–10 right-angled 36
factorizing 74–7, 113, 118–20
factors 112 see also calculus; trigonomic
function of a factor 112
graphs 110–13 functions
higher order 125 trigonometry 34, 57
innite 17–18

482 Index

trigonomic functions variance 270–1, 475 position 190–5, 207–8, 475
combining 35 continuous random variables representing 178–82
complex numbers 87–8 280–1 right hand screw rule 227
compound angle identities 44–6, scalar multiplication 183–5
48–9, 302 vectors 68, 475 scalar product 199–204, 214, 475
differentiation 302 addition 180–1 scalar product form 236–7, 475
double angle identities 47 algebraic properties of a vector subtraction 181
expanding expressions 48–9 product 229 unit vectors 185, 190, 475
identities for powers 87–8 angles 200–2, 213–14, 242–9 vector product 226–31, 475
inverse trigonomic 39–42 areas 229–31 zero vector 181
multiple angle identities 48, 87–8 Cartesian form 210–13, 236–9 velocity 177, 216–17, 303
proving identities 44–8 collinear points 193–4, 207 vertical asymptote 134–7, 140, 151–5
range and domain of 37–42 components 178–80, 200, 474 volumes of revolution 333–5, 475
reciprocal 36–8 cross product 226, 237, 474
symmetries of trigonometric dening 176–8 wave equation 352
graphs 48–9 direction 209, 474
uses of 35 displacement 190–1, 216–17, 474 x-intercept 112–13, 134–7, 140, 151–5
distances 191–2
see also calculus equation of a line 207–17 y-intercept 112–13, 134–7, 140, 151–5
equation of a plane 234–9, 475
two dimensions, intersection of and geometry 190–5, 227 zero
lines 221 intersection of lines 221–4 denominator as 135, 301
kinematics 216–17 divided by zero 301, 305–8
unit vectors 185, 190, 475 Lorentz force 227 factorizing polynomials 112
magnitude of 185, 191–2, 226–9 gradient as 315
validity, and generalization 102 parallel 183–5, 192, 203–4, 223–4, numerator as 301
variables 474 229 remainder theorem 119
parametric form 210 vector 181
continuous random 280–1, 474 perpendicular 203–4, 214, 229,
dependent 352 237–8
discrete random 270–1 and physics 227
independent 352
separation of 357–9

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real world. mathematics and in applications of
mathematics, including economics,
n Engage and excite students with examples and epidemiology, linguistics,
philosophy and natural sciences.
photos of maths in the real world, plus
inquisitive starter activities to encourage their Between them they have
problem-solving skills considerable experience of teaching
IB Diploma Mathematics at
n Build mathematical thinking with our ‘Toolkit’ Standard and Higher Level, and two
of them currently teach at the
and mathematical exploration chapter, along University of Cambridge.
with our new toolkit feature of questions,
investigations and activities

n Develop understanding with key concepts and

applications integrated throughout, along
with TOK links for every topic

n Prepare your students for assessment with

worked examples, and extended essay support

n Check understanding with review exercise at

the end of the coursebook

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