ANSWERS (Full Permission) PDF
ANSWERS (Full Permission) PDF
ANSWERS (Full Permission) PDF
2
A C 17 6 2 + 4 3 18 17 5 19 5 20 2
4 5 5
2 10 3 21 2 21 22 2 23 5 21 24 2 5 2 3
2 2 11
8 6 4 5 2 4 2
9
25 3 + 2 3 26
3
27 x 2 y 28 5 + 3 3
2
29 12 x 30 x + h + x 31 1 32 1
1 7 5 x a + 3 x + y
33 m 27
( 7 2 x ) ( m + 7 )
D
Exercise 1.3
17 a) {x 0 < x 1} b) {x 22 x < 3} 1 2 2 27 3 16 4 16
c) {x 1 < x 4} d) {x 22 x 0, 3 x 4} 5 8 6 8 7 4 8 3
9 4
e) {x 3 x 4} f) {x 22 x 0, 1 < x 4}
9 125 10 1 11 1 12 16
18 x > 210 19 x 23 20 x > 10 8 9 3
3
21 x > 2 3 22 3 x < 7 23 23 x < 1 13 264 14 x 2 y 6 15 2x 2 y 6 16 28x 3 y 9
8 2 4 27
24 False, x = 21 25 True 26 False, x = 0 32y 7 1 p2
17 18 19 20 28
27 False, x = 1 x
6 3
28 True 29 True 64m 3k
2 7
1
30 False, x = 21 31 False, x = 32 14.5 21 25 22 x 6 23 1 2 24 x
2 4a
33 9 34 8.2 35 π 2 3 3
26 (
x + 4y ) 2
36 11π 37 832 25 2 (a 2b) 27 p 2 + q 2 28 53x 21
2
3 77
38 25 x 3, closed, bounded
29 11 + 11 30 26 3n
9
( )
31 16 32 2 3x 2 y 4
39 210 < x 22, half-open, bounded x
6
x4
40 x 1, half-open, unbounded 33 1 + n
2
34 x
41 x < 4, open, unbounded
42 0 x < 2π, half-open, bounded Exercise 1.4
43 a x b, closed, bounded
2.54 × 1026 7.81 × 10 7.41 × 1023
2 23 6
1 2 3
44 ] 23, ∞ [ 45 ] − 4, 6 [ 46 ] 2∞,10 ] 4 1.04 × 10 5 4.98 6 1.99 × 10
47 [ 0,12 [ 48 ] 2∞, π [ 49 [ 23, 3 ] 7 1.49 × 10 8
8 8.99 × 10
25
9 1.50 × 10
8
9.11 × 10
231
10 11 0.0027 12 50 000 000
50 x 6 [ 6, ∞ [ 51 4 x < 10 [ 4,10 [
2.5 × 1018
3
13 0.000 000 090 35 14 4 180 000 000 000 15
52 x < 0 ] 2 ∞, 0 [ 53 0 < x < 25 ] 0, 25 [
2 × 10 8.2 × 10 ×
4 25
16 17 18 5.6 10
970
1.8 × 10 1 20 5 × 101 21 8.2 × 1025
( ) ( )
5
19
82 b) 21, 7 18 a) 533 b) 1, 11
22 5.56 × 10 17 a)
3 6 2
19 k = 1 or 9 20 k = 211 or 23
Exercise 1.5
1 x +2 x 2 20
2
2 6h 2 211h + 3 3 y 2 2 81
21 ( 5 )2 + ( 45 )2 = ( 50 )2 22 Sides are: 29, 29, 58
4 16x + 16x + 4 5 4n 2
210n + 25 23 Sides are: 45, 10, 45, 10 24 (5,1)
( )
2 2
6 4y 2 20y + 25 7 36a 2 49b 2
25 4, 1 26 (3,24) 27 (3.8,21.6)
2
8 4x + 12x + 9 2 y 2 9 a 3x 3 + 3a 2bx 2 + 3ab 2x + b 3 2
4 4 28 No solution 29 (21, 2) 30 (21, 3)
a x + 24a bx + 6a b x + 4ab x + b
3 3 2 2 2 3 4
10
(
11 4 12 3 31 23,28)
2 25x 8x2 212 32 Lines are coincident; solution set is all points on the line
13 x + y + z + 2xy + 2xz + 2yz 14 x + y
2 2
15 2m
9 2
16 x 2 2 2 x 2 + 1 + 2 y = 21 x 2 3
4 4
( ) ( )
17 12 (x + 2) (x 2 2)
(
18 2
x x 2 6) 33 20 , 40 34 1 , 3 35 (25,10)
19 (x + 4) (x 2 3) 20 2(m 21) (m + 7) 3 3 2
( )
( ) ( )
36 5,23 37 14.1, 10.4 38 11 ,2 18
21 (x 2 8) (x 2 2) 22 ( y + 1) ( y + 6) 19 19
23 3 (n 25) (n 2 2) 24 2x (x + 1) (x + 9)
25 (a + 4) (a 2 4)
26 (3y + 1) ( y 25)
Chapter 2
( )(
27 5n 2 + 2 5n 2 2 2
) 28 a (x + 3)
2
Exercise 2.1
29 (m + 1) (2n 21)
2
(
30 (x + 1) (x 21) x 2 + 1 ) 1 G 2 L 3 H 4 K
31 y (6 2 y )
(
32 2y 2 2y 2 25y 2 48
) 5 J 6 C
2
7 A
2
8 I
( )
3525
b) V =
2
14 a) 9.4
35 (n 2 2) (12n) m m22 1
3
36 37 P
3 x + 1 15 a) F = kx b) 6.25 c) 37.5 N
38 1 a +b
39 40 x + 2 16 { 26.2,21.5, 0.7, 3.2, 3.8} 17 r > 0
2n 5 18 19 20 t 3 21
41 21 42 4x 43 3x + 2 22 x ≠ ±3 23 21 x 1 and x ≠ 0
x + 1
3y 21 (2x 21) (x 21) 24 No, x = c is a vertical line
44 45 21 46
y + 2
x (x 2 2) 25 a) (i) 17 (ii) 7 (iii) 0
b) x < 4 c) Domain: x 4, range: h (x ) 0
49 22x + 5
12n 6 2 8x
47
n
48
2x 21 15 26 a) Domain {x : x ∈, x ≠ 5}, range {y : y ∈, y ≠ 0}
( )
b) y-intercept 0,2 1 , vertical asymptote x = 5,
52 x 2+ x + 3
2
b 2a 10 2 3x
50 51 5
(
ab x 2 3) x + 3x
2
horizontal asymptote y = 0
22
53 2x 54 55 6 y
x 2 2
2 2
2 y
x 6
56 2 57 1 58 2 5 x (x + 1) 4
7x 2 21 2
2
ab 2b
59
3y 210
60
( ) ( ) 61 x + 2
x 2 3 x + 2 2
( y + 2) ( y 25) 23x 2 2
x 2 2 2 0 2 4 6 8 10 x
10 25x 3 x + 2 xy + y x +h 2 x 2
62 63 64
4 2 3x
2
x2y h 4
6
Exercise 1.6
2 a = v + t
2
1 x = h 2 n 3 b1 = 2A 2b2 27 a) Domain {x : x < 23, x > 3}, range {y : y > 0}
m b h
b) Vertical asymptotes x = 23 and x = 3
2A gh
4 r = 5 k = 6 t = x y
θ f a +b 4
g
7 r = 3 3V 8 k = 9 y = 2 2 x 25
πh F (m1 + m2 ) 3
2
10 y = 24 11 y = 5 x + 6 12 x = 7
4 3
13 y = 24x + 11 5
14 y = 2 x 2 7
2 8 6 4 2 0 2 4 6 8 x
( )
15 a) 17 b) 0, 5 16 a) 40 b) (2, 3) 28 a) Domain {x : x ∈, x ≠ 22}, range {y :y ∈, y ≠ 2}
2
971
Answers
2
horizontal asymptote
( )
b) y-intercept 0,2 1 , vertical asymptote x = 22,
y=2
7
x ∈
(
f g ) (x ) = x, domain: x ∈; (g f ) (x ) = x, domain:
(
8 f g ) (x ) = x , domain: x ∈;
4
y
(
10 g f ) (x ) = 2x 4 + 4x 3 2 6x 2 + 4x , domain: x ∈
8 9 ( f g ) (x ) = 4x 2221 , domain: x ≠ 0, x ≠ ± 12 ;
6
( 4 2 x)
2
(g f ) (x ) = 4x 2 , domain: x ≠ 0, x ≠ 4
4
(
10 f g ) (x ) = 1 + x 2, domain: 21 x 1;
2
(
g f ) (x ) = 3 2x 6 + 4x 3 2 3 , domain: x ∈
10 8 6 4 2 0 6 x
(
2 4 11 f g ) (x ) = x, domain: x ≠ 23; (g f ) (x ) = x , domain:
2
x ≠ 23
2
4
12 ( f g ) (x ) = x 2 2 2 , domain: x ≠ ± 2;
x 21
6
(g f ) (x ) = 2x 212 , domain: x ≠ 1
8
(x 21)
{ 2 2
10
}
29 a) Domain x : 2 10 x 10 , range {y : 0 y 5}
13
a) (g h) (x ) = 9 2 x 2 , domain: 23 x 3, range: y 0
b) (h g ) (x ) = 2x + 11, domain: x 1, range: y 10
4 6
f
g
2 4
g
2 0 2 4 6 8 x 2
2
0 2 4 6 x
973
Answers
y
3
14 y f
6
2
4
f 2
1
f1
10 8 6 4 2 0 2 4 6 x
2
0 x
1 2 3
4
g 28 x > 0, f 21 (x ) = 4 2 2 x
6
y
8 2
f1
10
4 2 0 2 x
15 f 21 (x ) = 1 x + 3 , x ∈ 2
f
2 2
16 f 21 (x ) = 4x 2 7, x ∈
4
17 f 21 (x ) = x 2 , x 0
18 f 21 (x ) = 1 2 2, x ∈, x ≠ 0 29 x < 21, 21 x 1, x > 1 30 3
x
2
19 f 21 (x ) = 4 2 x , x 4 31 5 32 24 33 7
2
20 f 21 (x ) = x 2 + 5, x 0 34 g h = x 23
21 21 1 35 h g = x 2 3
21 21 1
2 2 2
21 f 21 (x ) = 1 x 2 b , x ∈
a a 36 ( ) 2 2
g h
21
= 1x+1 37
(
h g )21
= 2x + 2
22 f 21 (x ) = 1 + x + 1, x 21
38 ( a
_____
f (f (x)) = f )
2 b = _____
a
a
___________
2 b
23 f 21 (x ) = 1 + x ,21 x 1
x + b 2 b + b
12 x x + b
a 2 b = __ x + b
a . _____
24 f 21 (x ) = 3 x 21, x ∈ = _____ _____ 2 b = x + b 2 b = x
a
1 a
x+3 x + b
25 f 21 (x ) = x 2 2
Since f ( f (x )) = x, then the function f is its own inverse.
y
10
Exercise 2.4
8
1 y
6
4
8
f 2 6
10 8 6 4 2 0 2 4 6 8 10 x 4
2
2
4
f –1
6
4 2 0 2 4 x
8 2
10
4
26 x 2, f
21
(x ) = x +2 6
8
y
10 2 y
f 8
8
6 f1 6
4
4
2
2
0 2 4 6 8 10 x
27 x > 0, f 21 (x ) = 1 0 2 4 6 8 10 x
x
974
3 y 8 y
12 2
10
4 2 0 2 4 x
8 2
6 4
4 6
2 8
10
8 6 4 2 0 2 4 6 8 x
4 y
8 9 y
6 6
4
2 4
4 2 0 2 4 x
2 2
4
6
8 6 4 2 0 2 4 6 8 10 x
8
2
5 y
8
10 y
4
6
2
4
2 10 8 6 4 2 0 2 4 x
0
11 y
2 4 6 8 10 x 10
6 y
8
4
6
4
2
2
0 2 4 6 8 x
2 0 2 4 6 x
12 y
2
12
10
4
8
7 y 6
8 4
2
6
4 2 0 2 4 x
4
13 y
10
2 8
6
12 10 8 6 4 2 0 2 x 4
2
2
6 4 2 0 2 4 6 x
975
Answers
14 y d) y
10 4
8 3
6 2
4 1
2
5 4 3 2 1 0 1 2 3 4 5 x
1
4 2 0 2 4 x
2 2
4 3
6
8 e) y
4
10
3
2
15 y = 2x 2 + 5 16 y = 2x 1
17 y = 2 x + 1 18 y = 1 2 2
x 22 5 4 3 2 1 0 1 2 3 4 5 x
1
19 a) y 2
4 3
3
2
1 f) y
4
5 4 3 2 1 0 1 2 3 4 5 x 3
1
2 2
3 1
4
5 4 3 2 1 0 1 2 3 4 5 x
5 1
6 2
3
b) g) y
y 10
4 9
3 8
2 7
1 6
5
4 3 2 1 0 1 2 3 4 5 6 7 8 x
1 4
2 3
3 2
1
c) y
6 5 4 3 2 1 0 1 2 3 4 5 x
1
5
4
20 Horizontal translation 3 units right; vertical translation 5
3
units up (or reverse order).
2
21 Reflect over the x-axis; vertical translation 2 units up (or
1 reverse order).
5 4 3 2 1 0 1 2 3 4 5 x 22 Horizontal translation 4 units left; vertical shrink by factor
1
1 (or reverse order).
2 2
3 23 Horizontal translation 1 unit right; horizontal shrink by
4 6
2 4
2
2 0 2 4 6 8 10 12 14 x
2 6 4 2 0 2 4 6 x
2
4
4
6
6
8
8
b) 10
y
8
26 a) y
6 6
4 4
2 2
0 x
2 0 2 4 6 8 10 12 14 x 2 2
2
c) y 4
4
6
2
b) y
12 10 8 6 4 2 0 2 4 6 8 10 12 x 8
2
6
4
4
6
2
25 a) y x
2 0 2
2
c) y
8
6
4
2
4 2 0 2 4 6 x
2 0 2 x
Practice questions
2 1 a) a = 23, b = 1 b) range: y 0
2 a) 5 b) 3
b) y 3 a) g 21 (x ) = 23x + 4 b) x = 2
3
10 4 a) (g h) (x ) = 2x 2 3
b) 24
5 a) y
2
8
1
6
2 1 0 1 2 3 x
4
1
2
2
4 2 0 2 4 6 x
(2
)
maximum at 21,2 1 ; minimum at 0,2 3
2 ( )
977
Answers
1
2
8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 x
4 2 0 2 4 x
1
b) x = 4, x = 24 c) range: y 1
2
8 a) y
4
y
2
2
4 2 0 2 4 x
8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 x
2
1
4
2
b) h (x ) = 1 2 2
x+4 b) A′ (23,22)
( ) ( )
c) (i) x-intercept: 2 7 , 0 ; y-intercept: 0,2 7 14
2 4 y
(ii) Vertical asymptote: x = 24; horizontal asymptote: y1 = f (x) y2 = f (k)
y = 22
(iii) y m m+k
2 n n+k x
8 6 4 2 0 x
2
15 (
)
f g 21 (x ) = 3 x + 1
4
16 a) g (x ) = x b) 2
2x + 1 9
6 17 a) 2 2 x 2 ,x≠0
2 2
b) f (x ) 0
9 a) (i) 11 (ii) 7 (iii) 0
b) x < 23 18 f (x ) = x + 1 , x ≠ 2
21
c) (g f ) (x ) = x 2 2 x 22
19 a) 2 1 < A < 2 b) f 21 (x ) = 22x 21
10 a) 4
( )
b) g 21 h (x ) = 2x 2 + 6 c) x = ± 2 2
20
2
a) g (x ) = 3 x + 1
x 22
b) g (x ) = 3 x + 1
11 a) f 21 (x ) = 1 x + 1
3 3 21 a) S = {x : 2 3 < x < 3} b) f (x ) 3
b) ( f g ) (x ) = 12 21 x
3
x 22
4
c) ( f g ) (x ) = 12
21
978
Chapter 3 b) Horizontal translation 5 units right; vertical translation 7
units up.
Exercise 3.1 c) Minimum: (5, 7)
2
1
28; 28
2 0; 33 2 a) f (x ) = (x + 3) 21
3 29; 2375 4 0; 23c + 6
b) Horizontal translation 3 units left; vertical translation 1
5 k = 2 6 k = 2 unit down.
7 a) 216, 2, 2, 24, 24, 14, 62 c) Minimum: (23,21)
b) 3 a) f (x ) = 22 (x + 1) + 12
2
3
c) y
b) Horizontal translation 1 unit left; reflection over x-axis;
5 vertical stretch by factor 2; vertical translation 12 units up.
4 c) Maximum: (21,12)
( )
2
3 4 a) f (x ) = 4 x 2 1 + 8
2
2
1
b) Horizontal translation 2 unit right; vertical stretch by
1 factor 4; vertical translation 8 units up.
3 2 1 0 1 2 3 x
( )
c) Maximum: 1 , 8
2 2
1
5 a) f (x ) = 1 (x + 7) + 3
2 2 2
b) Horizontal translation 7 units left; vertical shrink by
1 3
3 factor 2 ; vertical translation 2 units up.
( )
4 c) Minimum: 27, 3
2
5 6 x = 2, x = 24 7 x = 5, x = 22
8 x= ,x=0 3 9 x = 6, x = 21
8 a) 52, 5, 0, 1, 24, 23, 40 2
b) 4 10 x = 3 11 x = 1 , x = 24
3
c) y 12 x = 3, x = 2 13 x = 2, x = 1
3 4
2
14 x = 22 ± 7 15 x = 5, x = 21
16 No real solution 17 x = 24 ± 13
1
18 x = 2, x = 24 19 x = 2 ± 22
2
0 3 x
3 2 1 1 2 20 a) x = 2 ± 5
1
b) Axis of symmetry: x = 2
2 c) Minimum value of f is 25
3 21 Two real solutions 22 No real solutions
4
23 Two real solutions 24 No real solutions
25 p = ±2 2 26 k < 4
5
27 k < 21, k > 1 28 m < 23, m > 3
6 29 k > 12
7 30 x 2 2 2 x ⇒ 2(x 2 x + 2) ⇒ 2 x 2 2 x + __
2 2
( ) 4
7
1 2 __
4
( )
2
⇒ 2 x 2 __ 1 2 __ 7 < 2 __ 7 for all x
2 4 4
9 a = 12 31 y = 22x 2 + 6x + 8 32 y = x 2 2 7 x 2 1
11 2 2
10 b = 2 3 33 21 < k < 15 34 m < 22 10 or m > 2 10
11 a) (i) (, ) (ii) (, )
( )
36 f (2) = 8
35 f x = 3x 2 + 5x 2 2
(iii) (, ) (iv) (, )
< >
(v) (, ) (vi) (, ) 37 x 1 or x 3
(vii) (, )
(viii) (, )
38
(2
)
∆ = (2 2t ) 2 4 (2) t 2 + 3 > 0 ⇒ 2 7t 2 2 4t 2 20 > 0 ;
b) I f leading term has positive coefficient and even because ∆1 = 2544 for 27t 2 2 4t 2 20 and leading
exponent, then (, ) . coefficient is negative, then graph of y = 27t 2 2 4t 2 20 is a
parabola opening down and always below x-axis; hence, ∆
If leading term has negative coefficient and even
exponent, then (, ) . for original equation is always negative; thus, no real roots
If leading term has positive coefficient and odd exponent, ( ) ( )
2 2a 2 21 ± 2a 2 21 2 4a (a) a 2 + 1 ± a 4 2 2a 2 + 1
2
= = ⇒ x = 2a
2a 2a 2a
Exercise 3.2 = a or x = 2 = 1
2a a
1 a) f (x ) = (x 25) + 7
2
979
Answers
5
40 a) sum 5 23, product 5 2 __
2 Exercise 3.4
1 y
b) sum 5 23, product 5 21
6
3
c) sum 5 0, product 5 2 __
2 4
d) sum 5 a, product 5 22a 2
e) sum 5 6, product 5 24 0
6 4 2 2 4 6 x
f) sum 5 __1 , product 5 2 __
2 2
3 3 4
41 4x 2 1 5x 1 4 5 0
6
1
42 a) __ b) __ 55
1 c) __
27
9 12
vertical asymptote: x = 22
43 a) 22 and 26 b) k 5 12 horizontal asymptote: y = 0
1
44 a) 2 __ b) 4x 2 1 x 1 1 5 0
4 2 y
45 a) x 2 2 19x 1 25 5 0 b) 25x 2 1 72x 2 5 5 0 6
4
Exercise 3.3 2
1 3x + 5x 25 = (x + 3) (3x 2 4) + 7
2
0
6 4 2 2 4 6 x
(
2 3x 4 2 8x 3 + 9x + 5 = (x 2 2) 3x 3 2 2x 2 2 4x + 1 + 7
) 2
(
3 x 3 25x 2 + 3x 2 7 = (x 2 4) x 2 2 x 21 211
) 4
4 9x 3 + 12x 2 25x + 1 = (3x 21) 3x 2 + 5x + 1
( ) 6
( )(
5 x 5 + x 4 2 8x 3 + x + 2 = x 2 + x 2 7 x 3 2 x + 1 + (27x + 9)
)
6 (x 2 7) (x 21) (2x 21) 7 (x 2 2) (2x + 1) (3x + 2) vertical asymptote: x = 2
horizontal asymptote: y = 0
8 (x 2 2) (x + 4) (3x + 2) 9 Q (x ) = x 2 2, R = 22
2
3 y
10 Q (x ) = x 2 + 2, R = 23 11 Q (x ) = 3, R (x ) = 20x + 5 12
12 Q (x ) = x 4 + x 3 + 4x 2 + 4x + 4, R = 22 10
13 P (2) = 5 14 P (21) = 217 8
()
6
15 P (27) = 2483 16 P 1 = 49
4 64 4
17 x = 2 + i or x = 2 2i 18 x = 1 + 5 or x = 12 5 2
2 2
19 k = 1 − x 3 or k = 2 1 − x 3 6 4 2 0 2 4 6 x
2
20 a = 5, b = 12
3 4
21 x 2 3x 2 2 6x + 8 22 x 4 2 3x 3 2 7x 2 + 15x + 18
23 x 3 2 6x 2 + 12x 2 8 24 x 3 2 x 2 + 2
25 x 4 + 2x 3 + x 2 + 18x 2 72 26 x 4 2 8x 3 + 27x 2 250x + 50
27 x = 2 + 3i, x = 3
4
( )
x-intercept: 1 , 0 , y -intercept: (0,1)
vertical asymptote: x = 1 horizontal asymptote: y = 4
28 a) a = 21, b = 22 b) 3x + 2 4 y
29 a = 4 , b = 1 4
3 3
30 x = 3, x = 21, x = 2 1 + 3 i, x = 2 1 2 3 i
4 4 4 4 2
31 a = 21, b = 24, c = 4 32 p = 25, q = 23, r = 251
33 a = 25 34 m = 22, n = 26
0 x
35 b = 18 36 b) R = 3 4 2 2 4
980
5 y
4 x- and y-intercept: (0, 0)
vertical asymptote: x = 1 oblique asymptote: y = x + 3
9 y
2
2
6 4 2 0 2 4 x
2 6 4 2 0 2 4 6 8 x
4 2
4 2 0 2 4 6 8 x
4 2 0 2 4 x
2
2
4
x-intercept: (2, 0) y-intercept: none
vertical asymptotes: x = 0 and x = 4
horizontal asymptote: y = 0
oblique asymptote: y = x vertical asymptote: x = 0
y 11 y
7
8
6
6
6 4 2 0 2 x 4
2
2
8 6 4 2 0 2 4 6 8 x
2
4
4
6
6
8
x- and y-intercept: (0, 0)
{ }
vertical asymptote: x = 22 horizontal asymptote: y = 0 domain {x : x ∈, x ≠ ± 2} range y : y 2 5 or y > 2
4
8 y 12 y
14 6
12 4
10 2
8 6 4 2 0 2 4 6 x
2
6
4
4
6
2
0 x domain {x : x ∈, x ≠ 24,1} range {y : y ∈, y ≠ 0}
6 4 2 2 4 6
2
4
6
981
Answers
13 y 17 y
2
4
4 2 0 2 4 x 10 8 6 4 2 0 2 4 6 8 10 x
domain {x : x ∈} range {y : 0 < y 1} x- and y-intercept: (0, 0)
14 y horizontal asymptote: y = 3
3
2 18 y
1 2
3 2 1 0 1 2 3 x
1
2 domain {x : x ∈, x ≠ 1}
3 range {y : y ∈, y ≠ 0} 0
–2 2 x
15 y
2
2
10 8 6 4 2 0 2 4 x
x-intercept: none y-intercept: 0, 1
vertical asymptotes: x = 22, x = 1 and x = 2
4 ( )
horizontal asymptote: y = 0
2
( ) ( )
19 a)
y
x-intercept: 5 , 0 y-intercept: 0, 5
2 18
vertical asymptotes: x = 26 and x = 3
y = 0 2
horizontal asymptote:
0
16 x
y
12
10
8
6 b) y
4
2
6 4 2 0 2 4 6 8 x
2
4
6
0 x
x-intercept: none
y-intercept: (0,21)
vertical asymptote: x = 1
oblique asymptote: y = x + 2
982
c) 33 1 > 2 ⇒ mn + 1 > 2n ⇒ mn 2 2n + 1 > 0; since
a) m + __
n
y
m > n ⇒ mn > n2 it follows that mn 2 2n + 1 > n2 2 2n + 1
and since n2 2 2n + 1 = (n 2 1)2 > 0 then mn 2 2n + 1 > 0
⇒ m + __ 1
n > 2
b) (m + n) (__ 1 + __ n1 ) > 4 ⇒ (m + n) (__
m 1 + __ n1 ) mn > 4mn ⇒
m
(m + n)(n + m) > 4mn ⇒ m2 + 2mn + n2 > 4mn ⇒
m2 2 2mn + n2 > 0 ⇒ (m 2 n)2 > 0 which is true for all
0 x
x and is equivalent to original inequality – thus,
(m + n) (__ 1 + __ n1 ) > 4 is true for all x.
m
34 x = 21 ± 13 , x = 1 or x = 22
2
35 (a + b + c)2 < 3 (a 2 + b 2 + c 2)
⇒ a 2 + b 2 + c 2 + 2ab + 2ac + 2bc < 3a 2 + 3b 2 + 3c 2
⇒ 0 < 2a 2 + 2b 2 + 2c 2 2 2ab 2 2ac 2 2bc
20 a) C ⇒ a 2 2 2ab + b 2 + b 2 2 2bc + c 2 + a2 2 2ac + c 2 > 0
8 ⇒ (a 2 b )2 + (b 2 c )2 + (c 2 a )2 > 0.
Since all the numbers are unequal, the squares of their
differences are strictly larger than zero therefore their sum
6 too is strictly larger than zero.
36 a) 1 < x < 3 b) x < 22, 21 < x < 1, x > 3
37 If a and b have the same sign, then a + b = a + b ; and if
4
a and b are of opposite sign, then a + b < a + b .
2 Practice questions
1 x = a or x = 3b 2 x 4
3 c = 5 4 a = 2 1 , b = 4, c = 22
2
0 5 10 15 t 5 ω = 22, p = 2, q = 28
b) At t = 2 minutes, concentration is 6.25 mg/l. 6 a) m > 22 b) 22 < m < 0
c) It continues to decrease and approaches zero as amount 7 a = 2, b = 21, c = 22
of time increases. 8 x < 5, x > 15
d) 50 minutes (49 minutes 55 seconds) 2
9 21 < k < 15
10 a) f (x ) = 2 2 3
Exercise 3.5 ( x + 2) + 1
2
1 =3 2 =9
x x b) (i) lim f (x ) = 2 (ii) lim f (x ) = 2
3 x = 5 or x = 22 4 x = 11 or x = 3 x→+ ∞ x→− ∞
5 x = 25 6 x = 1 or x = 22 c) (22, 21)
7 x = 2 or x = 22 8 x =1 11 k ∈ 12 a = 21
2
13 a = 7 , b = 21 14 a = 26
9 x =± 5 10 x = 27 or x = 2 125 4 4
8
15 a = 4 16 a = 22, b = 6
11 x = 3 or x = 2 12 x = 1 ± 41
4 17 a = 1 18 k = 6
13 x = 4 or x = 24 14 x = 15 or x = 9 19 k = 6 20 22.80 < k < 0.803 (3 s.f.)
3 2 21 k 4.5 22
15 No solution 16 x = 2 or x = 21 23 24 m 0
17 x = 2 or x = 1 18 =9
2 x 23 1 x 3 24 22.30 < x < 0 or 1 < x < 1.30
19 x = 25 20 =
±2 5
x or x = ±1 25 23 x 1 26 x < 21 or 4 < x 14
5 3
27 x 3 or x 27 28 x = 2 2i and x = 2
21 x = 1 + 41 22 = 49 x or x = 64 1
2 4
9 29 x <
3
23 2 2 < x < 2 24 < 22, x 3
x
3
25 210 x 6 26 x < 3 , x > 2
27 x > 17 2
28 24 x 21, 1 x 4
Chapter 4
2
29 x < 21, x > 2 30 x < 21, 2 2 < x < 3, x > 4 Exercise 4.1
3 1 21, 1, 3, 5, 7 2 21, 1, 5, 13, 29
31 a) p = 9 b) p < 9 c) p > 9
4 4 4 3 _32 , _34 , _38 , __
3 __ 3
16 , 32 4 5, 8, 11, 14, 17
32 x < 21, x > 1 5 1, 7, 25, 19, 229 6 3, 7, 13, 21, 31
3
7 21, 1, 3, 5, 7, 97
8 2, 6, 18, 54, 162, 4.786 3 1023
983
Answers
2 , 2 __
9 __ 6 , 2 __
2 , ___ 10 , ____
4 , ___ 50 2 Arithmetic, d 5 3, a10 5 27
3 3 11 9 27 1251
10 1, 2, 9, 64, 625, 1.776 3 1083 3 Geometric, r 5 2, b10 5 4096
11 3, 11, 27, 59, 123, 4.50 3 1015 4 Neither, not geometric, r 5 2, c10 5 21534
3 , ___ 39 , approx. 1
21 , ___ 5 Geometric, r 5 3, u10 5 78 732
12 0, 3, __ 6 Geometric, r 5 2.5, a10 5 7629.394 531 25
7 13 55
13 2, 6, 18, 54, 162, 4.786 3 1023 7 Geometric, r 5 22.5, a10 5 27629.394 531 25
14 21, 1, 3, 5, 7, 97 15 un = __ 1 un 2 1, u1 = __
1 8 Arithmetic, d 5 0.75, a10 5 8.75
4 3
16 un = ___
2
4a un 2 1, u1 = __ 1 a 17 un = un 2 1 + a 2 k, u1 = a 2 5k 9 Geometric, r 5 2 __ 2 , a10 5 2 ____ 1024
3 2 3 2187
10 Arithmetic, d 5 3 11 Geometric, r 5 23
18 un = n2 + 3 19 un = 3n 2 1
12 Geometric, r 5 2 13 Neither
2n 2 1
20 un = ______
21 2n 2 1
un = ______ 14 Neither 15 Arithmetic, d 5 1.3
n2 n + 3
89 16 a) 32 b) 23 1 5(n 2 1)
22 a) 1, 2, __ 3 , __
5 , __ 13 , ___
8 , ___ 34 , ___
21 , ___ 55 , ___
c) a1 5 23, an 5 an 2 1 1 5 for n . 1
2 3 5 8 13 21 34 55
23 a) 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 17 a) 29 b) 19 2 4(n 2 1)
c) a1 5 19, an 5 an 2 1 2 4 for n . 1
Exercise 4.2 18 a) 69 b) 28 1 11(n 2 1)
c) a1 5 28, an 5 an 2 1 1 11 for n . 1
19 , ___
1 3, ___ 23 , ___
27 , ___
31 , 7
19 a) 9.35 b) 10.05 2 0.1(n 2 1)
5 5 5 5
2 a) Arithmetic, d 5 2, a50 5 97 c) a1 5 10.05, an 5 an 2 1 2 0.1 for n . 1
b) Arithmetic, d 5 1, a50 5 52 20 a) 93 b) 100 2 (n 2 1)
c) Arithmetic, d 5 2, a50 5 97 c) a1 5 100, an 5 an 2 1 2 1 for n . 1
d) Not arithmetic, no common difference 21 a) 2 __
172 b) 2 2 1.5(n 2 1)
e) Not arithmetic, no common difference c) a1 5 2, an 5 an 2 1 2 1.5 for n . 1
f ) Arithmetic, d 5 27, a50 5 2341 22 a) 384 b) 3 3 2n 2 1
3 a) 26 c) a1 5 3, an 5 2an 2 1 for n . 1
b) an 5 22 1 4(n 2 1) 23 a) 8748 b) 4 3 3n 2 1
c) a1 5 22, an 5 an 2 1 1 4 for n . 1 c) a1 5 4, an 5 3an 2 1 for n . 1
4 a) 1 24 a) 25 b) 5 3 (21)n 2 1
b) an 5 29 2 4(n 2 1) c) a1 5 5, an 5 2an 2 1 for n . 1
c) a1 5 29, an 5 an 2 1 2 4 for n . 1 25 a) 2384 b) 3 3 (22)n 2 1
5 a) 57 c) a1 5 3, an 5 22an 2 1 for n . 1
b) an 5 26 1 9(n 2 1) 26 a) 2 _ 49 b) 972 3 (2 _ 13 )n 2 1
c) a1 5 26, an 5 an 2 1 1 9 for n . 1 c) a1 5 972, an 5 (2 _ 13 )an 2 1 for n . 1
6 a) 9.23 2187
27 a) ____
b) an 5 22 (2 _32 )n 2 1
b) an 5 10.07 2 0.12(n 2 1) 64
c) a1 5 22, an 5 2 _32 an 2 1, n . 1
c) a1 5 10.07, an 5 an 2 1 2 0.12 for n . 1
390 625
28 a) _______ b) an 5 35 (_ 57 )
n 2 1
7 a) 79 117 649
b) an 5 100 2 3(n 2 1) c) a1 5 35, an 5 _ 57 an 2 1, n . 1
b) an 5 26 (_ 12 )
c) a1 5 100, an 5 an 2 1 2 3 for n . 1 n 2 1
29 a) 2 __3
64
27 _
1
8 a) 2 ___ c) an 5 26, an 5 2 an 2 1, n . 1
4
b) an 5 2 2 _ 54 (n 2 1) 30 a) 1216 b) 9.5 3 2n21
c) a1 5 2, an 5 an 2 1 2 _ 54 for n . 1 c) a1 5 9.5, an 5 2an 2 1, n . 1
9 13, 7, 1, 25, 211, 217, 223 31 a) 69.833 729 609 375 5 __________ 893 871 739
12 800 000
( )
n 2 1
10 299, 299 _14 , 299 _12 , 299 _34 , 300 b) an 5 100 ___ 19
20
11 an 5 210 1 4(n 2 1) 5 4n 2 14
c) a1 5 100, an 5 __ 19
20 an 2 1, n . 1
12 an 5 2 ___ 11 (n 2 1) 5 251 1 ___
142 1 ___ 11 n
13 88
3 3
14 36
3 32 a) 0.002 085 685 73 5 ________ 2187
1 048 576
b) an 5 2 (_ 38 )
15 11 16 16 n 2 1
c) a1 5 2, an 5 _ 38 an 2 1, n . 1
17 11 18 9, 3, 23, 29, 215 33 6, 12, 24, 48 34 35, 175, 875
19 99.25, 99.50, 99.75 20 an 5 4n 2 1 35 36 36 21, 63, 189, 567
19n 2 277 ( )
n 2 1
21 an 5 _________
3
22 an 5 4n + 27 37 224, 24 1
38 1.5, an = 24 __
2
( )
n 2 1
23 Yes, 3271th term 24 Yes, 1385th term 39 a4 = ±3, r = ± __ 1 , a = 24 ± __ 1 40 ___ 49
25 No 2 n 2 3
41 10th term 42 Yes, 10th term
43 Yes, 10th term 44 2228.92
45 £945.23 46 €2968.79
Exercise 4.3
1 Geometric, r = 3a, g10 = 39a + 1 47 7745 thousands 48 ___ 98
9
984
49 10th term 50 €3714.87 Exercise 4.6
51 £2921.16 1 a) x 5 1 10x 4y 1 40x 3y 2 1 80x 2y 3 1 80xy 4 1 32y 5
b) a 4 2 4a 3b 1 6a 2b 2 2 4ab 3 1 b 4
Exercise 4.4 c) x 6 2 18x 5 1 135x 4 2 540x 3 1 1215x 2 2 1458x 1 729
1 11 280 10 5469
2 2_______ 3 0.7 d) 16 2 32x 3 1 24x 6 2 8x 9 1 x 12
1024 __ e) x 7 2 21bx 6 1 189b 2x 5 2 945b 3x 4 1 2835b 4x 3
10 16 1 4 √3 2 5103b 5x 2 1 5103b 6x 2 2187b 7
4 ___ 5 ________
7 39
449 f) 64n 6 1 192n 3 1 240 1 ___ 160 60 1 ___
1 ___ 12 1 ___
1
6 a) 52
___ ___
b) 7459
c) ____ n 3 n 6 n9 n 12
99 990 2475 81 2 _____
216__ 216
__
g) ___ 1 ___
x 2 96 √x 1 16x
2
7 13 026.135 (£13 026.14) x 4 x 2√ x
8 940 9 6578 2 a) 56 b) 0 c) 1225 d) 32 e) 0
n(7 + 3n) 3 a) x 7 1 14x 6y 1 84x 5y 2 1 280x 4y 3 1 560x 3y 4 1 672x 2y 5
10 42 625 11 ________
2 1 448xy 6 1 128y 7
12 17 terms 13 85 terms b) a 6 2 6a 5b 1 15a 4b 2 2 20a 3b 3 1 15a 2b 4 2 6ab 5 1 b 6
14 d = 4 15 a) 250, 125 250, b) 83 501 c) x 5 2 15x 4 1 90x 3 2 270x 2 1 405x 2 243
16 a = 1, d = 5 17 2890 d) x 18 2 12x 15 1 60x 12 2 160x 9 1 240x 6 2 192x 3 1 64
18 0.290 19 22.065 e) x 7 2 21bx 6 1 189b 2x 5 2 945b 3x 4 1 2835b 4x 3 2 5103b 5x 2
20 11 400 21 1.191 1 5103b 6x 2 2187b 7
22 49.2 __ 23 __ 6
5 f) 64n 6 1 192n 3 1 240 1 ___ 160 60 1 ___
1 ___ 12 1 ___
1
24
3 + √6
______
2
25 3, ___ 93 , ___
18 , ___
5 25 125 4
468 , ___
(
15 1 2 __
5 )
1n
g) ___ 216__
81 2 _____ 216
1 ___
n 3
__
x 2 96 √x 1 16x
n 6 n 9 n 12
2
3 , __ n x 4 x 2√ x
26 __ 1 , ___
1 , __ 1 ; ______
h) 112 i) 1792 √3
__
6 __4 10 3 __2n + 4 __ _____ __
27 2 2 1, √3 2 1, 1, √5 2 1; √n + 1
√ 2 1 j) 16 k) 223 1 10i √2
28 1.945, 152.42 29 127, 128 4 a) x 45 2 90x 43 1 3960x 41
30 819 , ___
___ 32 31 11 866 b) Does not exist as the powers of x decrease by 2’s starting
128 5 at 45. There is no chance for any expression to have zero
32 763 517 33 14 348 906
exponent.
34 <150
( )
45 x 2 (___ ( )
x )43 1 x )44 1 (___
45 x (___ ( )
x )45 5 2
43
c) 22 22 22 2
45 ___
43 44 43 x 41
( )
44 45
Exercise 4.5 1 45 ___ 2 2 ___ 2
44 x 43 x 45
1 a) 120 b) 120 c) 20 d) 336
2 a) 1 b) 1 c) 120 d) 120 d) ( )
45 x 24 (___
21
22 ( )
x )21 5 2 45 221x 3
21
3 a) 70 b) 70 c) 330 d) 330 n ) 5 _________
5 ( n! n!
5 _________ 5 ___________________
n!
4 a) 0 b) 39 916 800 c) 0 d) 10 k k!(n 2 k)! (n 2 k)!k! (n 2 k)!(n 2 (n 2 k))!
5 ( n
n 2 k )
5 a) F b) F c) T
6 24
6 (1 1 1)n 5 ( n ) 1 (
n ) 1 (
+ … 1 ( nn )
n
7 72 8 312 0 1 2)
9 16 777 216 10 262 144 2n 5 1 1 ( 1 n + … 1 ( nn ) ⇒ 2n 2 1 5 ( n1 ) 1 ( n2 )
n
1 ) (2 )
11 1 757 600 000 12 81 000
13 a) 40 320 b) 384 + … ( n
)
n
8 (_ 13 1 _ 23 ) 5 1
6
14 a) 40 320 b) 720 7 Answers vary
15 9 ( ) 10 ( _ 17 1 _ 67 ) 5 1
JANE, JAEN, JNAE, JNEA, JEAN, JENA, AJNE, AJEN, 8 n
_ 25 1 _ 35 5 1
ANJE, ANEJ, AEJN, AENJ, NJAE, NJEA, NEJA, NEAJ, 11 15 12 90 720
NAJE, NAEJ, EJAN, EJNA, EAJN, EANJ, ENJA, ENAJ 13 16 128 14 1.1045, 0.9045
16 Mag, Mga, Mai, …(60 of them) 15 Proof
17 a) 175 760 000 b) 174 790 000
7
16 a) __ 38
b) ___ 31 808
c) ______
18 a) 4080 b) 1680 c) 1050 d) 1980 9 110 9900
17 2145 152 18 35a 3 19 96 096
e) 3150
19 a) 296 b) 1460 c) 504 20 243n 5 2 810n 4m 1 1080n 3m 2 2 720n 2m 3 1 240nm 4 2 32m 5
20 a) 125 000 b) 117 600 c) 61 250 d) 176 400 21 7 838 208
21 768
22 a) 36 b) 256 Exercise 4.7
23 a) 5985 b) 2376 c) 2475 1 2 1 4 1 6 1 … 1 2n 5 n(n 1 1)
24 a) 2280 b) 748 c) 770 2–20 All proofs
25 a) 1 192 052 400 b) 4560, 0.000 38%
c) 265 004 096, 22.2% Practice questions
26 a) 74 613 b) 7560 1 D = 5, n = 20
27 151 200 2 €2098.63
985
Answers
√
√
2
11 a) __ 1 1 __ 1
5 ___
1
b) __
4 4 2 2
c) (i) __ 1 (ii) __ 1 1 (ii) 2
d) (i) ___
4 2 512
12 a) 1220 b) 36 920
13 a) Area A 5 1, Area B 5 _19 b) __1
81
c) 1 1 _ , 1 1 _ 1 (_ )
8 8 8 2 d) 0
9 9 9 (0, 1)
14 a) Neither, geometric converging, arithmetic, geometric
diverging
b) 6 0 x
15 a) (i) Kell: 18 400, 18 800; YBO: 18 190, 19 463.3
(ii) Kell: 198 000; YBO: 234 879.62
(iii) Kell: 21 600; YBO: 31 253.81 (ii) y
b) (i) After the second year
(ii) 4th year
16 a) 62 b) 936
17 a) 7000(1 1 0.0525)t b) 7 years
c) Yes, since 10 084.7 . 10 015.0
18 a) 11 b) 2 c) 15
19 15, 28 20 22, 27 21 10 300
22 Proof
23 a) an = 8n 2 3 b) 50 (0, 1)
24 2 099 520
25 6n 2 5 26 72 27 559
0 x
28 23, 3 29 9 30 62
36
31 2 ___
5
32 a) 4 b) 16(4n 2 1)
33 a) |x| < 1.5 b) 5
34 3168
n(3n + 1)
35 a) ________
b) 30
2
36 27
37 1275 ln 2
38 a) 4, 8, 16
b) (i) un = 2n (ii) proof
2
39 a) __ b) 9
3
40 2, 23 41 55 42 22, 4
986
2 y 5 y
100 10
90 8 1
p(x) =
80 6 2x 1
70 4
60 2
50 0
3 2 1 1 2 3 x
40 2
30 4
f (x) 3x 4
20 6
10 8
10
4 3 2 1 0 1 x
domain: x ∈ domain: x ∈, x ≠ 0 range: y < 21 or y > 0
range: y > 0
y-intercept: none
y-intercept: (0, 81) horizontal asymptotes: y = 0 and y = 21
horizontal asymptote: y = 0 (x-axis) 6
y
3 y
10 8
8 6 1
q(x) = 3 3
6 4 3x
g(x) 2x 8
4 2
2
2 1 0 1 2 3 4 5 x
2
3 2 1 0 1 2 3 4 5 x
2 4
4
6 domain: x ∈ range: y > 23
y-intercept: (0, 0) horizontal asymptote: y = 23
8
10 7 y
domain: x ∈ range: y < 8 4
y-intercept: (0, 7) horizontal asymptote: y = 8
k(x) = 2|x – 2| + 1
4 y
10
2
8
2 1 0 1 2 3 4 5 x
4
domain: x ∈ range: y > 1
( )
y-intercept: 0, 5 horizontal asymptote: y = 1
2 4
h(x) 4x 1 8 Domain: x ∈
range: if a > 0 ⇒ y > d , if a < 0 ⇒ y < d
( )
0 3 x y-intercept: 0, a (b) + d horizontal asymptote: y = d
2c
2 1 1 2
2
domain: x ∈ range: y > 21
y-intercept: (0, 0) horizontal asymptote: y = 21
987
Answers
( )
12t
16 a) A (t ) = 5000 1 + .09 2
12
b) A
50 000 1 0 1 2 x
a) Domain: x ∈, range: y > 0
( )
40 000
b) x-intercept: none, y-intercept: 0, 1
e
c) Horizontal asymptote: y = 0
30 000
2 y
10
20 000 g(x) ex + 1
8
10 000
6
0 5 10 15 20 25 t
c) A 4
50 000
2
40 000
1 0 1 2 3 x
30 000 (15.46, 20 000)
a) Domain: x ∈, range: y < 0
b) x-intercept: none, y-intercept: (0, e)
20 000
c) Horizontal asymptote: y = 0
10 000
0 5 10 15 20 25 t
minimum number of years is 16
988
3 3 2 1 0 1 2 6 y
x 20
18
2
16
14
4
12
10
6
h(x) 2ex 8
6
8
4
h(x) = eabs(x + 2) 1
2
10
y 6 5 4 3 2 1 0 1 2 x
a) Domain: x ∈, range: y < 0 a) Domain: x ∈, range: y > 0
b) x-intercept: none, y-intercept: (0,22)
b) x-intercept: (22, 0), y-intercept: (0, e 221)
c) Horizontal asymptote: y = 0 c) No asymptotes
4 y
( )
n
10 7 a) e = lim 1 + 1
x→∞ n
b) 0.366 032 3413, 0.367 861 0464, 0.367 879 2572
1 < 0.367 879 4412
c) 0.367 88; reciprocal of e, __
8 e
( )
x
5 4 3 2 1 0 1 2 3 4 x 1
1
2
1 0
h(x) = 1 e x
3 10 20 30 40 50 x
4 d) 20 days
13 a) 8 _ 12 % compounded semi-annually is the better investment.
a) Domain: x ∈, x 0, range: y < 0, y > 1 14 a) r ≈ 1.070 37 (6 s.f.) b) 7.037% (4 s.f.)
b) x-intercept: none, y-intercept: none 15 a) Less than 1 b) Less than 1
c) Horizontal asymptotes: y = 0 and y = 0 c) Greater than 1 d) Greater than 1
16 a) £1568.31, £2459.60
b) 15.4 years
c) 15.4 years
d) Same; doubling time is independent of initial amount
989
Answers
()
10 log 2 1024 = 10 11 log10 0.0001 = 24 12 log 4 1 = 2 1 26 x = 20 27 x = 104 28 x = 13
2 2 3 e
13 log 3 81 = 4 14 log10 1 = 0 15 ln 5 = x 29 x = 4 30 x = 98 31 x = ± e 16 ≈ ± 2980.96
16 log 2 0.125 = 23 17 ln y = 4
32 x = 2 or x = 4 33 x =9 34 x = 13
18 log10 y = x + 1 19 6 20 3 5
3 35 x = 3 36 x = 1 or x = 100
21 23 22 5 23 24 1
4 3 37 x > 5 1 38 x < 2 39 0 < x < ln 6
25 23 26 13 27 0 28 6 100
40 0.161 < x < 1.14 (approx. to 3 s.f.)
29 23 30 2 31 3 32 1
2
1
33 22 34 88 35 36 18 Practice questions
( )
2
37 1 38 π 39 1.6990 40 0.2386 1 a) (8, 0) b) (0, 2) c) 2 2 , 3
3
3
41 3.912 42 0.5493 43 1.398 44 0.2090 2 a) 183 g (3 s.f.) b) 154 years (3 s.f.)
45 4.605 46 13.82 47 x > 2 48 x ∈ n (n + 1)
49 x > 0 50 x < 8 51 22 x < 3 52 x < 0
3
( )
a) an = ln y n , Sn =
2
ln y
5 +
53 Domain {x : x > 0, x ≠ 1}, range {y : y ∈, y ≠ 0}
( )
b) an = ln xy n , Sn = n ln x +
n (n
2
1)
ln y
()
54 Domain {x : x > 1}, range {y : y 0}
4 x = 224e 5 y = 16 6 x = 0, ln 1 or 2 ln 2
55 Domain: x > 0, x ≠ 1, range: y < 0 2
7 x = e or e 2e
56 f (x) = log 4 x 57 f (x ) = log 2 x
8 a) x = 3 b) x = 6
58 f (x ) = log10 x 59 f (x ) = log 3 x a 2b 3 3
9 a) log b) ln ex
c
y
60 log 2 + log 2 m = 1 + log 2 m 61 log 9 2 log x 10 1900 years
1 2
62 ln x 63 log a + 3 log b 11 c = 22
5
log 10x + log (1 + r ) = log 10 + log x + t log (1 + r )
t
64 12 a) y
y = bx
65 3 lnm 2 ln n 66 log b p + log b q + log b r
log b p log b q
67 2 log b p + 3 log b q 2 log b r 68 +
4 4
log b q log b r log b p
69 + 2 70 log b p + 2 log b q 2 log b r
1
2 2 2
71 3 log b p + 3 log b q 2 12 log b r 72 log x (–1, b)
y4 p
73 log 3 72 74 ln 75 log b 4 76 log (0, 1)
4 qr
77 ( )
ln 36
e
78 9.97
79 25.32
80 2.06
0 x
81 20.179 82 4.32 83 1.86
log a b) y
84 log b a = a
= 1 85 log e = ln e = 1 y = b1 x
log a b log a b ln10 ln10
86 (
dB = 10log 216)
10
I
_____ 216
= 10(log I 2 log 10 ) = 10(log I + 16)
24
= 10log 10 + 160 = 10(24) + 160 = 120 decibels
() ()
0 x
ln 3 ln 4
2 3 13 a) k ≈ 0.000 4332 b) 17.7% (3 s.f.)
13 3 14 0 or 21 15 or
16 1 or 21
ln 6 ln 6 14 x ≈ 1.28
15 1.52 < x < 1.79 ∪ 17.6 < x < 19.1
17 a) $6248.58 b) 9 14 years
16 21 < x < 20.800 ∪ x > 1
18 12.9 years
17 a) x = 2 1 or x = 0
19 20 hours (≈ 19.93) 2
20 a) 24 years (≈ 23.45) b) 12 years (≈ 11.9) b) x = 1 or x =
log a e
c) 9 years (≈ 8.04) ln a 2 2 12 2 log a e
21 6 years
c) a = e 2
22 a) 99.7% b) 139 000 years 18 a = 22, b = 3
990
19 x = e , x = e
(iii) 81
(
324
0 81 ) (iv) 3
n
( n 1 1
3
0 3n
)
20 a) V = $265.33
5 25
b) 235 months ( )
13 11 , 8
3 3
14 (1, 24)
21 x = 5 3 or x = 5 3 15 5 16 (5, 1)
22 x = e 2 3 or x = 1 2 3
e
23 x = 22.50, 21.51 or 0.440 (3 s.f.)
Exercise 6.3
24 k = ln 2 ( 29
1 a)
27 ) b) M 5 29
27 2
1
(
)( )
( )
20 x x 4 3 4 3 3 5
25 a) f (x ) = ln x b) f 21 (x ) = 2 x2e or 2e x
x+2 e 21 12e c) ( 239
17
244
19
)
26 a) (i) Minimum value of f is 0.
(ii) From part (i) f (x) > 0 ⇒ e x 2 1 2 x > 0 ⇒ e x > 1 +
d) (i) N 5 2
1
3 5( 29
4 )(
27
3
)
(ii) N 5 214
211
27 26
( )
21
e) If AB 5 C then B 5 A C, while if BA 5 C, then B 5
x
d) n > e 100 CA21. Also, A21C CA21.
( )
0 0
2 _35
2 1
_ 9 __
11 2 _8 _1
5 5 5 2
Chapter 6 3 a) |A| 5 25 0 b) _ 65 _ 95 2 _75 c) 21
Exercise 6.1 and 6.2 1 1 21 _ 1
5
(
x 2 1
1 a) (i) x 2 3
y 1 3 y 1 1 ) ( 2x 2 7
(ii)
3y 2 7 11 2 y )
3x 1 3
__
√3
___
1
__ 3a 1 1
__ 21
b) x 5 23, y 5 5 c) x 5 3, y 5 23 4 a) 2 2__ b)
(
1 ___ √3
2x 2 2
d) AB 5
xy 2 x 1 y 1 11
xy 2 2x 1 6
23
)
;
2 __
2
5 x 5 2 or x 5 3
2
2a 2 2 a
(
22x 2 3y 1 1
)
2
BA 5
2 x 1 x 2 9
6 n 5 0.5
13
y 2 3y 2 6 4x 1 3y 2 6 1
__ 0 1 ___
2 12
2 a) x 5 2, y 5 210 b) p 5 2, q 5 24 7 a) X 5 b) Y 5
3
__ 7
__ 5
3 a) 0 1 0 0 1 2 0 b) 6 3 1 2 3 2 0 2
4 21 2 __
6 3
1 0 1 1 1 1 0 3 5 2 3 3 3 2 c) X Y 2 not commutative
0 1 0 2 0 0 2 1 2 9 1 3 1 0
5 24 3 4 25 28
0 1 2 0 1 0 0 2 3 1 6 1 2 4 8 a) PQ 5 33 5 21 , QP 5 8 0 24
1 1 0 1 0 1 0 3 3 3 1 4 3 0
2 23 2 7 10 8
2 1 0 0 1 0 0 2 3 1 2 3 6 0
0 0 2 0 0 0 0 0 2 0 4 0 0 4 _1
1 0 21 0 4 0
Matrix signifies the number of routes between each pair
b) P21 5 2 _75 _ 15 __
11 21
5 , Q 5 1 21 1
(
that go via one other city.
)
x 1 1
10 y 1 1
1 0 22 2 2 _74 1
4 a) A 1 C 5 0
2x 2 3
y 1 3
2x 1 y 1 7 x 2 3y 2x 1 2y 2 1 22 2 21
( 17m 1 2 26
) __
P21Q21 5 23 2 __
22 __
12
b)
4 2 9m
9
5 5 5
7m 2 2 217 __
15
24 4 22
c) Not possible d) x 5 3, y 5 1
e) Not possible f) m 5 3 2 __
7
__
1 __
11
20 20 20
5 a 5 23, b 5 3, c 5 2
Q21P21 5 __
17
5 2 _15 __
26
2 5
6 x 5 4, y 5 23
___
109
20 2 __
7
2 ___
157
7 m 5 2, n 5 3 20 20
8 Shop A: €18.77
( __
2 __ 1 __
11
)
7
9 2
a) 4 b) associative 20 20 20
22 12
(PQ)21 5 __
17 2 _15 2 __
26
( 222
c)
60 27
16
) d) associative 5
___
109
2 __
7 ___
157
5
20 20 2 20
10 AB 5 [88 142], which represents total profit.
11 r 5 3, s 5 22 22 2 21
12 a) (i) (
1
0 1 )
2 (
1
(ii)
0 1 )
3 (QP)21 5 __
23
5 2 __
22
5
__
12
5
( ) ( )
n n
1 4 1 __
15
(iii) (iv) 24 4 22
0 1 0 3n
(
b) (i) 9
0 9
18
) (ii) (
27 81
0 27 ) ( )
27
9 a)
22
3
27
b) ( )
3
22
991
Answers
10 x = 21 11 x = 1, y = 2
3 24 26
12 (0, 1) 13 (23, 229), (0, 1)
14 17x 2 8y + 37 = 0; y + 2 = 0; x + 5 = 0 15 165; 80; 136 11 a) 3 b) 0 22 23
c) 3
1
89
___ 129 ___ 0 0 2
16 x = or x = ; x = 24 or x = 22 or x = 23 ± √21 2
2 8
2 1 23 5
17 23; 3
0 1 2 216
18 a) 225 d) 21672 e) 0 0 36 2184 f) 21672
b) x 2 2 7x 2 25, constant = det(A) 209
0 0 0 2
c) 2 (a + d) 9
d) f (A) = 0
e) ad 2 bc ; x 2 2(a + d)x + (ad 2bc),
constant = det(A); f (A) = 0 Practice questions
19 a) 222 1 x 5 27 or x 5 1
b) x 3 2 x 2 2 22x + 22, constant = 2det(A)
c) Opposite of the sum of the main diagonal ( 2
a 1 4
2 a)
2a 2 2
2a 2 2
5 )
d) f (A) = 0
b) a 5 21; ( ( )
xy ) 5
1
21
Exercise 6.4 (
1
3 B 5 3
)
1 m = 2 or m = 3 4 12
2 a) a = 7, b = 2 b) (21, 2, 21) 28
4 a 5 ; b 5 ___
___ 59 ; c 5 ___20 ; d 5 ___
28
3 m = 2 33 33 33 33
( )
4 a) (21, 3, 2) b) (5, 8, 22) __1
19
__
2
19
( ) 5 a) A21 5
__
13 5 11 19 27 5
__
c) + t , + t ,t d) (27, 3, 22) 19 19
16 16 16 16
e) (21 + 2t, 2 2 3t, t) f) inconsistent b) (i) X 5 (C 2 B)A21 (ii) X 5 2
24
23
1 ( )
g) (22, 4, 3) h) (4, 22, 1) 6 a) A 1 B 5 (
a 1 1
c 1 d 1 1 c
b 1 2
)
5 a) k
≠ 21 ± 33
4
b) k = 1 b) AB 5 (
a 1 bd
c 1 d
2a 1 bc
3c
)
c)
1 0 0 22 23 1
0 1 0 3 3 21
0 0 1 22 24 1
7 (
0.1
a) 20.7
0.4
0.3
0.2
21.2 0.2 0.8
0.1
)
(
b) x 5 1.2, y 5 0.6, z 5 1.6
2
)
8 23
a) Q 5 1 ______
6 a) 71 ± i 251 b) k = 2 14 2 a
42 3
(
b) CD 5 214
24 1 4a
)
1 0 0 3 1 2 22 2 1 7a
5 5 5
1
c) D21 5 ______
5a 1 2 1 (
a
22
5 )
c) 0 1 0 2 4 23
5 5 5 9 a) (7, 2) b) (21, 2, 21)
0 0 1 3 6 21 1
5 5 0 0 1
10 a) B = A21C b) DA = 0 1 0 , B = 21
1 0 0 2 −16 −19
1 0 0 1 21 2 1 0 0 1 2
2 2 13 13
c) (1, 21, 2)
7 0 1 0 1 2 2 2 5 ; 0 1 0 1 −11 −9
2 3 6 13 13 11 a) Det = 0 b) = 5 c) (2 2 3t, 1 + t, t)
0 0 1 0 2 1 0 0 1 −1 12 11 12 No answer required 2 proof
3 3 13 13
B is the inverse of A
8 a) f (x) = 4x 2 2 6x 25
Chapter 7
b) f (x) = 1 (m 2 27) x 2 + 3 (17 2m) x + m, m ∈
2 2 Exercise 7.1
c) f (x) = 3x 3 2 2x 2 2 7x + 3
1 π 2 5π 3 2 3π 4 π
d) f (x) = 1 (42m) x 3 + 1 (42m) x 2 2 5 (42m) x + m, m ∈ 3 6 2 5
6 3 6 5 3π 6 5π 7 2 π 8 20π
4 18 4 9
−t 2 3 7t − 9
5 5 9 2 8π
9 m = 2, 19 10 m = 21, 3 3
−t 2 5 5 211t 10 135 2630
11 12 115 13 210
5t 5t 14 2143 15 300 16 115 17 89.95 ≈ 90
992
18 480 19 390°, 2330° 20 ___ 7p , 2 __
p 21 535°, 2185°
2 2 31 sin 17π = sin 5π = 1 ; cos 17π = cos 5π = 2 3
11 p
____ 13 p
____
22 , 2 11 p
23 , 2 __
____ p 6 6 2 6 6 2
6 6 3 3 3 2
24 3.25 + 2π ≈ 9.5, 3.25 2 2π ≈ 23.03 32 a) 2 b) 2 c) undefined
25 12.6 cm 26 14.7 cm 2 2
27 1.5 radians, or approx. 85.9 28 r ≈ 7.16 d) 2 e) 2 2 3
29 Area ≈ 13.96 ≈ 14.0 cm 2 30 Area ≈ 131 cm 2 3
31 a = 3 (radian measure), or a = 172 32 32 cm 33 a) 0.598 b) 2 3 c) 1 d) 1.04 e) 0
33 6.77 cm 3 2
34 a) 3π radians/second b) 11.9 km/hr 34 I, II 35 II
36 III 37 II
35 19.8 radians/second 36 v = ω 3r 38 I, IV 39 I
37 28.3 cm 38 20 944 sq metres 40 IV 41 II, IV
a) r ≈ 30.6 cm r ≈ 0.0771 cm π 2 2
( )
39 b)
40 150 3 cm
2
41 Area of circle = A
4π
Exercise 7.3
Exercise 7.2 1 y
1 a) t = π : 3 , 1 ; t = π : 1 , 3 2
6 2 2 3 2 2
1
2 0.6 3 1.0 4 0.5 5 0.5
6 2.7 7 0.1 8 0.3 9 1.6
π π2 0 π π 3π 2π 5π 3π x
2 2 2
10 a) I b) 3 , 1 1
2 2 2
1
11 a) IV b) ,2 3
2 2
2 y
2
12 a) IV b) ,2 2
2 2 π π2 0 π π 3π 2π 5π 3π x
b) (0,21)
2 2 2
13 a) Negative x-axis 1
14 a) II b) (20.416, 0.909) 2
15 a) I b) 2 , 2 3
2 2
16 a) IV b) (0.540, 0.841)
3 y
17 a) II b) 2 2 , 2
2 2 1
18 a) III b) (20.929,20.369)
19 sin π = 3 , cos π = 1 , tan π = 3
3 2 3 2 3 π π2 0 π
2 π 3π
2 2π 5π
2 3π x
20 sin 5π 1
= , cos 5π = 2 , tan 5π = 2 3
3
6 2 6 2 6 3
( ) ( ) ( )
1
21 sin 2 3π 2
= 2 , cos 2 3π = 2 , tan 2 3π = 1
2
4 2 4 2 4
22 π 1 π
sin = , cos = 0, tan π is undefined 4 y
2 2 2 2
23
( ) 3 2 ( )
sin 2 4π = 3 , cos 2 4π = 2 1 , tan 2 4π = 2 3
3 2 ( ) 3
1
( ) ( ) ( )
1
26 sin 2 7π 1
= , cos 2 7π = 2 3 , tan 2 7π = − 3
6 2 6 2 6 3
27 sin (1.25π) = 2 2 , cos (1.25π) = 2 2 , tan (1.25π) = 1 5 y
2 2
1
28 sin 13π π
= sin = ; cos1 13π π
= cos = 3
6 6 2 6 6 2
29 sin 10π = sin 4π 3
= 2 ; cos 10π = cos 4π = 21
3 3 2 3 3 2 π π2 0 π
2 π 3π
2 2π 5π
2 3π x
6 y
amplitude = 1 , period = 2π
4 2
b) Domain: x ∈ , range: 23.5 y 22.5
3
2 y
11 a)
1 4
3
π π2 0 π
2 π 3π
2 2π 5π
2 3π x
1 2
2 1
3
π 0 π 2π 3π 4π 5π x
4 1
2
y 3
7 1 4
π π2 0 π π 3π 2π 5π 3π x amplitude = 3, period = 2π
2 2 2 3
b) Domain: x ∈, range: 23.5 y 2.5
1 12 a) y
5
4
8 y
4 3
3 2
2 1
1
π 0 π 2π 3π 4π 5π x
1
2
3
4
994
y 31 x = 225 , 315 32 θ = π , 5π
6 6
8 33 t ≈ 1.5 hours
y = sec x 34 a) 80th day (March 21) and approximately 263rd day
6
(September 20)
4
b) 105th day (April 15) and approximately 238th day
2 (August 26)
0 c) 94 days – from 125th day to 218th day
π 2π x
2 35 x = π , 2π , 3π , 4π 36 θ = π , 7π , 11π
2 3 2 3 2 6 6
4 37 x = 245 , 63.4 38 x ≈ 21.87, 1.87
6 39 x ≈ 56.3 40 x = π , 3π
4 4
8 41 No solution 42 x ≈ 0 , 71.6 , 180 , 252
y Exercise 7.5
8 22 6 62 2
y = cot x 1 2 3 2 2 3
6 4 4
4 4 2 6 2 2 5 2 2 6 6 2 2 3
4 4
2
7 a) 6 + 2 b) 6+ 2+4
0 4 8
( ) (( ))
π 2π x
2 sin π 2 θ sin π cos θ 2 cos π sin θ cos θ
π 2 2 2
8 tan 2 2 θ = = = = cot θ
4 cos π 2 θ cos π cos θ + sin π sin θ sin θ
2 2 2
( )
6
9 sin π 2 θ = sin π cos θ 2 cos π sin θ = cos θ
8 2 2 2
( ) ( )
π 2θ = 1 = 1 = 1 = sec θ
10 csc
b) y = sec x, range: y 1, y 21; 2
sin π 2 θ sin π cos θ 2 cos π sin θ cos θ
2 2 2
y = csc x, range: y 1, y 21; y = cot x, range: y ∈ 4 7
11 a) b) c) 24
18 a) a = 2, b = 3, c = 21 b) 5π 5 25 25
π 18
19 a = 3, b = 2 , c = 21
4 12 a) 5 b) 2 4 5 c) 2 1
3 9 9
Exercise 7.4 13 sin 2θ = 2 4 5 , cos 2θ = 1 , tan 2θ = 24 5
1 x = π , 5π 2 x = 7π , 11π 9 9
3 3 6 6 14 sin 2θ = 24 , cos 2θ = 7 , tan 2θ = 24
3 x = , π 5π 4 x = , π 2π 25 25 7
4 4 3 3
π
15 sin 2θ = 4 , cos 2θ = 2 3 , tan 2θ = 2 4
5 x = , 3π , 5π , 7π 6 x = π , 5π , 7π , 11π 5 5 3
4 4 4 4 6 6 6 6
16 sin 2θ = 2 2 15 or sin 2θ = − 15 , cos 2θ = 2 7 , tan 2θ = 15
7 x = π , 3π , 5π , 7π 8 x = π , 2π , 4π , 5π 16 8 8 7
4 4 4 4 3 3 3 3 17 18 2cos x 19 tan x
2cos x
9 x = 0, 3π , π, 7π , 2π 10 x = 0, π , π, 3π , 2π
4 4 2 2 20 2sin x 21 1 + sin θ cos θ 22 1
π 5π π 3π 5π cos θ sin 3
θ
11 x = , 12 x = , , , 7π
3 3 4 4 4 4 23 sin θ + cos θ 24 1 + sin 2
θ 25 cos θ 3
sin 2θ
2
13 x ≈ 0.412, 2.73 14 x ≈ 1.91, 4.37 cos x
15 x ≈ 1.11, 4.25 16 x ≈ 5.64, 3.78, 2.50, 0.639 26 1 27 cos 2 θ 28 2 tan 2 θ
17 x ≈ 2.96, 5.32 18 x = π , 5π , 7π , 11π
2
29 2 sin a cos b 30 cos A 31 2 cos a cos b
6 6 6 6 32 1 33–46 No answers required (proofs)
19 x ≈ 5.85, 5.01, 2.71, 1.86 20 x ≈ 3.43, 0.291, 2.71,1.86
47 tan θ = 2 5x 48 x = π , π, 5π
21 5π , 3π , π , 2 π , 2 3π , 2 5π x + 14 3 3
2 2 2 2 2 2
π 5π
22 π ,2 11π 23 7π , 19π 49 x= , 50 x = 90 and 2 90
3 3
6 6 12 12
51 x ≈ 0.375, 2.77 52 x ≈ 0.615, 2.53, 3.76, 5.67
24 0, π , π , 3π , π, 5π , 3π , 7π , 2π
4 2 4 4 2 4 53 x= 3π , 7π 54 x = π , 2π
25 x = 5π , 3π 26 θ = 2 3π , π 4 4 3 3
6 2 4 4 55 x = 0, π , π, 5π 56 x = 0, π , 2π , π, 4π , 5π
27 x = 30 , 60 , 210 , 240 28 a = 2 π , π 4 4 3 3 3 3
6 6 57 x = 30 , 90 , 105 , 150 , 165 58 3 sin x 2 4 sin 3 x
29 θ = 2π , 4π 30 x = π , 5π
3 3 6 6 59 b) x = π , 3π , 5π , 7π
4 4 4 4
995
Answers
1 π 2 π 3 2 π 4 2π 1
2 4 3 3
5 0 6 2 π 7 π 8 3
3 3 2
9 12 10 Not possible 11 π 12 Not possible
4
13 3 14 24 15 Not possible 16 π
5 25 3
17 2 5 18 4 19 63
5 5 65
2 20 2 3 10 or 4 5 2 3 10
20 0 2 4 6 8 10 12 14 16 x
30 30
2
21 12 x
2
22 12 x 23 1 (iv) 3.98 m; sit in the 2nd row
x x 2
+1
( )
24 2
25 12 x
7 x cos π + 2.5
2x 12 x 9
1 + x θ =
( ) ( )( )
b) (ii) arctan 2
2x + x + 2x 12 x 2
3 x cos + 2.5 + 8.8 2 x sin π 1.8 2 x sin π
π
26 9 9 9
x2 + 1
27 ( 5 ) (
cos arcsin 4 + arcsin 5 = cos arccos 16
13 )
65
note: 20 = π
9
( ) ( ) ( ) ( )
(iii)
4
cos arcsin cos arcsin 5 2 sin arcsin 4 sin arcsin 5 = 16 θ
5 13 5 13 65
1
3 ⋅ 12 2 4 ⋅ 5 = 36 2 20 = 16 Q.E.D
5 13 5 13 65 65 65
28 ( 2 ) ()
sin arctan 1 + arcsin 1 = sin π
3 4
( ) ( ) ( ) ( )
sin arctan 1 cos arctan 1 + cos arctan 1 sin arctan 1 = 2
2 3 2 3 2
5 ⋅ 3 10 + 2 5 ⋅ 10 = 3 50 + 2 50 = 25 2 = 2 Q.E.D
5 10 5 10 50 50 50 2
x
29 x = 1 30 x ≈ 0.580, 2.56 0 2 4 6 8 10 12 14 16
2 (iv) 2.5 m; sit in the 3rd row
31 x ≈ 2.21 32 x ≈ 1.11, 4.25
33 x = , π 5π ; x ≈ 2.82, 5.96 34 x = π ; x ≈ 0.464
4 4 4
35 x ≈ 1.37, 4.91 Practice questions
36 x = π, 2π; x ≈ 0.912, 2.23, 4.05, 5.37 1 a) 135 cm b) 85 cm
37 x = 0, π; x ≈ 1.89, 5.03
θ
38 θ = arctan 2
d ()
2
c) t = 0.5 sec
x = 0, 2π
d) 1 sec
( )
0 2 4 6 8 10 12 14 16 d
2π x 2 9 + 4.2
( )
9 a) 1.6 sin
11 4
39 a) (ii) θ = arctan 2 7x
x + 15.84 b) Approximately 3.15 metres
c) Approximately 12:27 p.m. to 7:33 p.m.
10 x ≈ 0.785, 1.89
11 a) 15 cm
b) area ≈ 239 cm 2
996
12 k > 2.5, k < 22.5 13 k = 1, a = 22
14 sec θ = 2 3 c) θ ≈ 41.4 ; 48.6
2
60 , cos θ = 61 , tan θ = 2 915 ,
84 7 b) sin θ =
15 a) b) 2 13 c) 2 84 11 11 61
85 85 13 915 11 60
4 11 5 cot θ = , csc θ =
16 sin 2p = , sin 3p =
30 60
5 25 c) θ ≈ 44.8 ; 45.2
5 12
17 a) 2 b) c) 2 120 d) 119
13 13 169 169 8 b) sin θ = 9 181 , cos θ = 10 181 , cot θ = 10 , sec θ = 181 ,
1 181 181 9 10
18 tan θ = or 2 3
3 181
csc θ =
2(k + 1) tan a(k + 1) 9
19 tan x = tan a or tan x =
k 21 12k c) θ ≈ 42.0 ; 48.0
20 θ = ± 3π ,± π
8 8 9 b) sin θ = 7 65 , tan θ = 7 , cot θ = 4 , sec θ = 65 ,
65 4 7 4
21 b) x ≈ 0.412
c) cos (2) g (x ) 1 csc θ = 65
7
22 24.1 23 72 arccos 8 cm c) θ ≈ 60.3 ; 29.7
π 13
10 θ = 60 , π 11 θ = 45 , π
3 4
997
Answers
2 2
1
c) sin 330 = 2 , cos 330 =
3 , tan 330 = 2 1 , cot 330 = 22, sec 330 = 2 3 , csc 330 = 22
2 2 2 3
d) sin 270 = 21, cos 270 = 0, tan 270 = undef., cot 270 = 0, sec 270 = undef., csc 270 = 21
e) sin 240 = 2 3 , cos 240 = 2 1 , tan 240 = 3, cot 240 = 3 , sec 240 = 22, csc 240 = 2 2 3
2 2 3 3
f) sin 5π 2
= 2 , cos 5π 2
= 2 , tan 5π = 1, cot 5π = 1, sec 5π = 2 2, csc 5π =2 2
4 2 4 2 4 4 4 4
( ) 2 ( ) 6 2 ( ) 6 3 ( )
g) sin 2 π = 2 1 , cos 2 π = 3 , tan 2 π = 2 3 , cot 2 π = 2 3, sec 2 π = 2 3 , csc 2 π = 22
6 6 ( ) 6 3 ( )
6
( ) ( ) ( ) ( ) ( ) ( )
h) sin 7π = 3 , cos 7π 1
= 2 , tan 7π = 2 3, cot 7π 3
= 2 , sec 7π = 22, csc 7π =2 3
6 2 6 2 6 6 3 6 6 3
( )
2
3
( )
1
2
(
) (
)
i) sin 260 = 2 , cos 260 = , tan 260 = 2 3, cot 260 = 2 , sec 260 = 2, csc 260 = 2 2 3 3
3
(
) (
3
)
( )
j) sin 2 3π
2 ( )
= 1, cos 2 3π
2
= 0, tan 2( ) 3π
2
= undef., cot 2 ( )
3π
2
= 0, sec 2 ( )
3π
2 ( )
π
= undef., csc 2 = 1
6
( ) ( ) ( ) ( ) ( ) ( )
k) sin 5π = 1 , cos 5π = 2 3 , tan 5π = 2 3 , cot 5π = 2 3, sec 5π = 2 2 3 , csc 5π = 2
3 2 3 2 3 3 3 3 3 3
( ) 1
(
l) sin 2210 = 2 , cos 2210 = 2 , tan 2210 =
2
) 2
3
(
) 3
3
( ) ( )
, cot 2210 = 3, sec 2210 = 2 2 3 , csc 2210 = 22
3
( )
( ) π
m) sin 2 = 2 , cos 2 =
4 2
2
( ) π
4 2
2
( ) π
( )π
( )
π
, tan 2 = 21, cot 2 = 21, sec 2 = 2, csc 2 = 2 2
4 4 4 ( )
π
4
n) sin π = 0, cos π = 21, tan π = 0, cot π = undef., sec π = 21, csc π = undef.
o) sin 4.25π = 2 , cos 4.25π = 2 , tan 4.25π = 1, cot 4.25π = 1, sec 4.25π = 2, csc 4.25π = 2
2 2
6 sin θ = , tan θ = , cot θ = , sec θ = 17 , csc θ = 17
15 15 8
17 8 15 8 15
7 sin θ = 2 6 61 , cos θ = 5 61
61 61
8 cos θ = 21, tan θ = 0, cot θ = undef., sec θ = 21, csc θ = undef.
9 sin θ = 2 3 , cos θ = 1 , tan θ = 2 3, cot θ = 2 3 , csc θ = 2 2 3
2 2 3 3
20 c) 7.02 m
10 a) (i) 30 (ii) 85 21 1740 km
b) (i) 45 (ii) 7 1 ,0x < π
c) (i) 60 (ii) 20 22 a) sec θ =
12 x2 2
11 a) 6 3 b) 87.5 c) 675 2
12 y 1+ y 2
28.5 b) sin b =
13 a) 236 b) 97.4 1 + y2
14 a) 9.06 b) 119 23 cos = OA, tan = PB, cot = CP, sec = OB, csc = OC
15 ab sin θ
16 17
2hf cos θ
18 Verify
Exercise 8.3 and 8.4
x 3 h + f 1 Infinite triangles 2 One triangle 3 One triangle
19 a) A (x ) = 24 sin x b) 0 < x < 180 4 One triangle 5 Two triangles 6 One triangle
B = 115
y 7 BC ≈ 17.9, AC ≈ 27.0, AC
C = 65
8 AB ≈ 18.1, BC ≈ 22.5, BA
25 C = 111
9 AB ≈ 3.91, BC ≈ 1.56, AB
20 C = 43
10 AB ≈ 326, AC ≈ 149, BA
C ≈ 60.2 , AB C ≈ 48.8
15 11 AB ≈ 74.1, BA
10 12 BAC ≈ 75.5 , ABC ≈ 57.9 , ACB ≈ 46.6
C ≈ 60.6 , ACB ≈ 37.8
13 BA C ≈ 81.6 , AB
5
14 Two possible triangles:
C ≈ 55.9 , ACB ≈ 81.1 , AB ≈ 40.6
( )
0 (1) BA
0 30° 60° 90° 120°150°180° x c) 90 , 24
(2) BAC ≈ 124.1 , ACB ≈ 12.9 , AB ≈ 9.17
998
15 Two possible triangles:
C ≈ 72.2 , ACB ≈ 45.8 , AB ≈ 0.414
Chapter 9
(1) AB
Exercise 9.1 and 9.2
(2) ABC ≈ 107.8 , ACB ≈ 10.2 , AB ≈ 0.102
1
16 10.8 cm and 30.4 cm 17 51.3 , 51.3 , 77.4
18 71.6 or 22.4 19 Distance ≈ 743 metres
20 20.7
21 Area ≈ 151.2 cm 2 v 2u
22 a) BC = 5 sin 36 or BC 5
b) 5 sin 36 < BC < 5
c) BC < 5 sin 36
2uv
23 a) BC = 5 3 or BC 10 b) 5 3 < BC < 10
c) BC < 5 3 2u
24 x ≈ 64.9 m, y ≈ 56.9 m v
25 a) x = 5 c) 15 3
14
26 21 15 u
4
27 a) Obtuse triangle b) acute triangle v
28 21.1 uv
29 a) 14 b) cos θ = 3 , WY = 2 65 2u
5 u v
c) 2 5 d) 13.9
30 51.3 31–32 Verify
v2u
Exercise 8.5
2u v
1 a) tan 70 ≈ 2.75 b) y = x tan 70
2 (
)
a) tan 220 ≈ 20.364
(
b) y = x tan 220 ) ___
3 a) 1 b) y = 2x + 2 2 a) √41 b) u 5 (4, 25)
4
5
a) tan 22 ≈ 0.404
b) y = x tan 22 2 3
2
7 60.3 8 71.6 9 45
c) v 5 ____ (
4___
√
___ 41 41
√)
25
, ____
___ d) 1
45 6 33.7 3 a) √ 53 b) u 5 (7, 22)
10 a) y = 3 x
3
b) 56.6 c) v 5 ___ (
____
√
7
53 √ )
22
____
, ___
53
d) 1
11 AB ≈ 19.3 cm 4 a) 3 b) (23, 0) c) (21, 0) d) 1
12 O ≈ 71.8, SR
PR O ≈ 51.3, area ≈ 20.9 cm 2
5 a) 5 ___ b) (0, 5) c) (0, 1) d) 1
13 406.1 metres 14 2.70 metres › ___
6 ___ 5 (5, 26) b) √61
a) PQ d) (4, 25)
15 a) 1291.8 km b) 42.8 › ___
16 59.5 cm 7 a) PQ 5 (4, 6) b) 2 √13 d) (3, 7)
___› __
∆ABC = 72 cm , 2 ∆ABD = 24 3 ≈ 41.6 cm
17 2 2
, 8 a) PQ 5 (5, 5) b) 5 √2 d) (4, 6)
_ __› ___
∆BCD ≈ 34.6 cm ,
∆ACD ≈ 69.3 cm 2
9 a) PQ 5 (4, 6) b) 2 13 √ d) (3, 7)
DEF ≈ 41.9
18 19 43.0 metres 20 95.9 10 a, c
11 (1, 21) 12 (8, 21)
Practice questions 13 (4, 8) 14 (25, 25)
1 sin AO B = 24 2 sin 2θ = 21 , cos 2θ = 20 15 a) u 1 v 5 2i 1 2j, u 2 v 5 4i 2 4j, 2u 1 3v 5 3i 1 7j,
25 29 29 2u 2 3v 5 9i 2 11j __ __ ___
3 101.5 4 sin 2A = 120 b) |u 1 v| 5 2 √2 , |u 2 v| 5 4 √2 , |u| 1 |v| 5 2 √10 ,
169
|u| 2 |v| 5 0 ___
5 a) 29.1 m b) 41.9 m ____ ___
c) |2u 1 3v| 5 √58 , |2u 2 3v|5 √202 , 2|u| 1 3|v|5 5 √10 ,
C AB ≈ 86.4
6 ___
2|u| 2 3|v| 5 2√10
7 a) 38.2 b) 17.3 cm 2
8
a) ACB ≈ 116
b) 155 cm 2
L ≈ 31
16 ( 11
___
, 2 __
8 4 )
1
9 78.5 km 10 J K
17 u 5 _ 85 i 2 _ 75 j; v 5 2 _15 i 1 _ 45 j
11 a) 3.26 cm b) 7.07 cm 2 ___ ___
18 √
13 , √17
12 70.5
13 a) 91 m b) 1690 3 19 a) v 1 u b) v 1 0.5u c) v 2 u d) 0.5(v 2 u)
c) (ii) A2 = 26x (iii) x = 40 3 20 (6, 8)
d) (i) Supplementary angles have equal sines. 21 x = 3, y = 5
14 a) 2 2 + 4 b) 2 6 + 3 3 + 2 2 + 3 22 (6, 2)
15 Proof 23 5 (2, 3) 2 __
__ 1 (2, 1)
2 2
16 a) 0 < θ < 120 b) verify c) 60
2 24 r(1, 21) + (r 2 5)(21, 1)
17 a) 120 cm b) 2.16 c) 161 cm 2
25 2(2, 5) 2 5(3, 2)
18 Verify
19
cos θ = b
2a
26 ( )( ) (
x + y
( x y ) = _____
2
1
1 )( )
y 2 x
+ _____
2
21
1
999
Answers
( √
3 1
7 2 ___ ) (
, __
2 2
√
; ___ )
2 √2
, 2 ___
2 2
14
15
a) 5.6
√
440 2
__
b) ____
√
17
21 i 2 ___
8 ___ 28 j 16 a) 1 b) 0 21
c) _______
5 5 √
34
9 ± ____3___ (2i+ 3j) 17 No answer required – proof
__
√
13 48 ± 25 √3
18 _________
7
__
10 ± (4i + 3j) 39
5 19 a = 63.4°, b = 71.6°, = 45°
11 ± ____ 3___ (3i 2 2j) 20 No answer required – proof
√
_13
›
12 a) _ P _› 5 (840 cos 80°, 2840 sin 80°); Practice questions
W __› 5 (60 cos 30°, 260 sin 30°) 1 a) v 2 u
b) V 5 (840 cos 80° 1 60 cos 30°, 2840 sin 80° 2 60 sin 30°) b) ( _12 )(v 2 u)
5 (197.83, 2857.24)
c) ( _12 )(u 1 v)
c) Speed 5 879.77 km/h, bearing 167°
_› _› _› d) ( _ 32 )v 2 ( _ 12 )u
13 a) P 5 (520 cos 110)
_› i 1 (520 sin 110)
_› j 6___
_ _ 5 2177.85 i 1 488.64 j
2 a) (6, 21) b) ____
(6, 21)
› _› _› _› _› √
37
( )
__
5 (64 cos 160) i 1 (64 sin 160) j 5 260.14 i 1 21.89 j
W 25__
3 a) OR 5 15 b) 1__
c) ___ d) 75 √5
b) Speed 5 580.6 km/h, bearing 337.8° 5 √5 √
6
___› ___›
14 24.15, 6.47 11
4 a) MR 5 ( ) 23
, AC 5
4 6 ( )
15 200 m east of the initial point. b) 83.4°
16 Force 5 8176.152 N at an angle of 210.85° to the x-axis. _
__› _
__›
c) u 5 _ 21 MR
, v 5 2 _21 MR
⇒ u || v and |u| 5 |v |
17 Water 5 12.36, boat 5 38.04
63
5 m 5 ___ 37
, n 5 ___
18 T 5 35.89, S 5 41.57 46 46
19 35.9 km/h at N 12.88° W 6 a) 15 km/h, 19.7 km/h b)
6 ( ) ( )
4.5 ;
9
24
20 At N 11.54° W
c) 11.4 km d) At 8 a.m.
21 P = (10, 6)
e) 12.2 km f) 54 minutes
22 N 11.54° E, 293.9 km/h
23 a) (4, 6) b) (0, 22) and (20, 6) 7 a) y
24 No answer required – proof
I
25 No answer required – proof
26 No answer required – proof
27 a) 50 m b) 5 minutes
c) N 19.47° W, 5.3 minutes__
28 a) p = (220, 200 √3 ) b) speed = 410.37, N 32.42° E R
29 66.6 N, S 28.5° E (or N 151.5° E)
Exercise 9.4 __
1 a) 0, 90° b) 13, 54°
__
c) 11, 42°
__
d) 2 √3 , 30°
e) 4, 90° f) 3 √3 , 30° g) 212 √3 , 150° h) 216, 180° 0 x
2 a) 21 b) 21 c) (57, 238)
d) (212, 215) e) 26 f) 3
T
g) Scalar multiplication is distributive over addition of vectors.
Multiplication is not associative.
1000
( )
__ __
›__
5
b) IR 5
__ 13 p cis (0) 14 e cis (__
__ 2
) 15 _____
p
2 √3 __i ___
2
√3 i
+ , + __
2 2__ 2
2 25
6
__
√
3 i
1 2 ____ 2 √3 __i 21 √3 i
( )
8 a)
745
1000
b) 600 km/h c) at 1.5 hrs 16 1, __
__2
__2 __
17 _____
__
2
+ , 2i 18 2i, ____
__2 __ __ 2
+ ____
__ 2
√
6 + √2 ________ √6 2 √2 ____________ 9(2 6 + √2)
√ √
9( 6 + 2 ) √
( ) 19 _______
+ i , 2 i __________
d) 325 e) 451 km 2 2 8
__ 8
940 __
√
__ __ √
3 3 2 3 __________ i (3 3
+ 3)
2
9 2n 2 n 112 5 0 does not have real solutions, so it is not 20 23 √3 2 3 + i (3 √3 2 3), _______ 2
__ __ 4 4
possible. √
2 2 √
2
_____ (1 + i), ___
10 a = __ p 2 2 11 0 21
2
(1 + i)
__ 2
2
23 2 ____ 3 √3 i
22 6, ___
__4 4__ __ __ __ __ __ __
5 √6 2 15 √2
___________ 5 √6 + 15 √2 ____________ 25 √6 2 15 √2 ___________ 5 √6 2 15 √2
Chapter 10 23 2 i __________
, + i
48 48 __ 64 __ 64
__ __
3 √3 + 3 √
i(3 3 2 3)
Exercise 10.1 __
24 23 √3 + 3 + i(3 √3 + 3), _______
4
+ __________
4
1 5 + 2i 2 7 2 √7 i 3 26 + 0i 1 = __
25 z1 = 2 cis __ p , z2 = 4 cis ____ 2p , __ 1 cis 2 __ 1 = __
p , __ 1 cis __ p ,
4 27 + 0i 5 0 + 9i 5 i
6 0 2 __ 6 3 z1 2 6 z2 4 3
4 z1 __ 1 cis __
z1z2 = 8 cis ____ 2p , __ = p
7 21 2 i 8 25 + 9i 9 14 + 23i 6 z2 2 2 __ __
__ __ √ √
1 = ___ 2 p __ 3 p
10 22 + 7i 11 34 2 13i 12 5 2 i 26 √ p
z1 = 2 2 cis , z2 = 4 3 cis ____
__ √ 2p , __
z cis __ , z1 = ___ cis __
6 3 1 4 6 2 12 3
13 16 2 11 i 14 4 2 7 i 15 1 ,
29 29 13 13 __
__ z1 ___ √
6 __
25
16 ___ 17 2 1 2 18 i 18 8 2 i z1z2 = 8 √6 cis ____ 2p , __ = cis p
36 13 13 6 z2 6 2 __
__ √
2 3p
19 27 2 3i 20 4 + 10i 21 5 + 12 i 27 z1 = 8 cis , z2 = 3 2 cis , z1 = __
p
__ √ 23
_____ p __ 1 cis ____
2p 1 = ___
, __
z cis ___ ,
6 4 __ 1 8 6 2 6 4
13 13 __
27
_____ p z1 ____
__ 4 √2 11
____ p
z1z2 = 24 √2 cis , z = cis
22 48 + 36 i 23 2 + 9i 24 68 12 2 3 12 __
25 25 28
__
p __
z1 = 3 cis , z2 = 2 2 cis , z = ___
√ __ √ 22
_____ p 1
__ √
3 2p
cis ____ ,
25 8 2 63 i 26 7 + 4 i 27 5 + 12 i √
__
2 ___
2
__
3 1
z1 ___
3 __ 2
√
6
13 26 65 65 169 169 __1 2 p
z = cis , z1z2 = 2 √6 cis , z = cis _____
___ 2
____ p __ 25 p
2 8 3 6 2 ___ 4 6 __
28 12 + 8 i 29 498 + 553 i 30 2 33 2 56 i ___
p
__
__
p
__ 1 √
10 __ p 1 √
2 p
25 25 169 169 25 25 29 √
z1 = 10 cis , z2 = 2 2 cis √ , = cis , z = ___
__ 2 z1
__ ____ __ cis __
4 10 4 2 8 2
z
31 17 2 19 i 1 , y = 22; and x = 1, y = 1
__ √
32 x = 2 __ √ 3
___ p __1
___ 5
2
____ p
2 , z1z2 = 4 5 cis , z = cis
13 13 4 2 2 4 __
__ √
1 = __ 1 cis ____ 1 = ___ 3
33 a) 28 c) 248 30 z1 = 2 cis __ p , z2 = 2 √3 cis 0, __ 2p
__
,
cis 0,
34 a) 24i c) 246 3 __ z1 2 3 z2 6
__ z √
3
p , __ p
z1z2 = 4 √3 cis __ 1
= ___
cis __
35 x 2 + y 2 = 4 36 9 2 2 + 2 i 3 z2 3 3
3 3 31 b) (i)
37 x = 2 , y = 38 (1 + i )
2
___ 29
___ 1
65 65 2 1
39 5 + 12i 40 (x, y) = (2, 21) or (x, y) = (22, 1)
41 a) (x, y) = (1, 3) or (x, y) = (21, 23) 0.5
b) 2i, 21 2 i 0
3 ( 3 + i ) 3 (2 3 + i )
42 23i, , –0.5
2 2
43 1 2 2i, 3 44 f (x) = 2x 4 211x 3 + 15x 2 + 17x 211
2
45 f (x) = x 4 + 2x 3 + 8x + 16
46 5 2 2i, 23 47 1 + i 3,2 2 48 Verify (ii) arg(z1) = ____ 2p 25 p
, arg(z2) = _____
6 6
3 32 Verify__ __
49 a) k = 0 ±1 b) k = ± 3 ± 2 2 √
3 3i 22 √3 __
33 a) ___ 2 __
b) ______
c) √
3 i
50 z 1 = 1 + i, z 2 = 2 2i 51 z 1 = 7 2 4i , z 2 = 1 + 6i 2 2 3 __ __
3 3 34 |z1| = 4, arg(z1) = ____ 2p p , |z3| = 8 √2 ,
, |z2| = 2 √2 , arg(z2) = __
6 4
Exercise 10.2 arg(z3) = ___ p
__12
4 ) 6 ) ( ) 35 22 2 2 √3 18.5
__ __
1 2 √2 cis (__ p p
2 2 cis (__ 7p
3 2 √2 cis ___
4 36 a) {(x, y): x 2 + y 2 = 9}, the circle centre (0, 0) radius 3
( ) ( )
__ __
11p
4 2 √2 cis ____ 5p
5 4 cis ___ 3
6 3 √2 cis p
___ b) {(x, y): x = 0}, the y-axis
6 3 4 c) {(x, y): x = 4}, the line x = 4
7 4 cis ( __ ( ) 9 2 cis ( )
__
p ) 8 6 cis 7p
___ √ p
__
d) {(x, y): (x 2 3)2 1 y 2 = 4}, the circle centre (3, 0) radius 2
2 6 4
( )
__ e) {(x, y): 1 2 x 1 3 and y = 0}, the line segment between
10 15 cis p 1 cis (5.64)
11 __ 12 3 √2 cis ___ 3p (1, 0) and (3, 0)
5 4
1001
Answers
37 a) {(x, y): x 2 + y 2 < 9}, the disk centre (0, 0) radius 3 3 a) 2i c) 65 536
b) { (x, y): x 2} (y 1 3)2 2 4|, all points excluding the interior
of the disk centre (0, 3) radius 2
z √__ √__ 6 √__ √__
( )
4 a) z 1 = 2 cis 2 π ; z 2 = 2 cis 2 π ( ) 4
√
__ __
6 2 2
6 + 2 ________ 6 + √2
__
c) z = 1 _______ + i p = _______
; cos ___ ;
Exercise 10.3 2 __4 __ 4 12 4
√ √
6 2 2
1 22 2 2i 3 2 3 3 3i sin ___ p = ________
12 4
4 2 2 6 + i ( 2 + 6) 5 13 + 13i 3 z
3
3 3
π
2 2 5 1 = a 32 2i a 32
π
z 2b 2b
6 3e + 3ei 3
i i
7 2 2e 4 8 2e 6
2
2 2 6 z = 4 7 a = ___ 3
11 , b = __
π π
2i 2i i
3π
5 5
9 2 2e 6
10 4e 3 11 3 2e 4 8 b = 3 9 a = 0, b = 21
( )
7π
10 a) z 5 21 = (z 21) z 4 + z 3 + z 2 + z + 1
π i 3π
i i
12 4e 2 13 6e 6 14 3 2e 4
( ) ( )
1+i
π
15 πe 2πi or simply p 16 e 2 17 32i b) cis ± 2π ; cis ± 4π
5 5
( ) ( )
18 264 19 210 077 696 20 2262 144
c) z 2 2 cos 2 2π z + 1 z 2 2 2 cos 4π z + 1
21 1296 22 17 496(21 2 i) 1
23 ____ 5 5
1296
24 1
( 3 2i ) 11 a) 8i = 8 cis π
2
559 872
25 2128 3 2128i b) (i) z = 2 cis π
6
26 6 + i 2 , 2 6 2i 2 (ii) z = 3 + i
π 7π 13π
12 a) z = 1; arg (z ) = 2π
i i i
27 2e 9__; 2e 9 ;__2e 9
√ 3
2 √ 2
28 ± ___
±i ___
c) 3 + 3 3 i
2 __ 2 __ __ __ __ __ __ __
2 2
( √
6 √
29 2 ___
2
2 ___
4
√2 √ 6
+ i ___ 2 ___
4 4 ) ( √6 √
; 2 ___
4
2
+ ___
4
2 √
√
+ i 2 ___
4
2 ___
4 ) (
6
4 ) ( ) 13 25 2 12i
; 14 c) z 5 + 5z 3 + 10z + 10 + 53 + 15
2 z z z
__ __ __ __ __ __ __ __ 6
( √
6 √
___
2
2 ___
4
2 √
√
+ i ___
4
+ ___
4 ) (
6 √ 6 √
; ___
4
2
+ ___
4
√
6 √
+ i ___
4
2
2 ___
4 ) (
;
4 ) ( ) 15 p = 2 __ ; q = __
5 5
__ __ __ __ 16 (ii) z 12 = 4 cis 4π ; z 13 = 8 cis 6π ; z 14 = 16 cis π ; z 15 = 32
√2 √
2 √2 √
2 5 5 5
2 ___ + i ; 2 i
___ ___ ___
(iii)
2 2 2 2 Im
i
4π
i
10π
i
16π
i
22π
i
28π 6
30 5 18e 15 5
; 18e 15 5
; 18e 15 5
; 18e 15 5
; 18e 15
2
4
5
z1 2 z1 z1 Re
2π 4π 6π 8π
i i i i
31 2; 2e 5
; 2e 5
; 2e 5
; 2e 5
8 6 42 26 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
2
32 e
( );e
i −
π
16
i
3π
16
;e
i
7π
16
;e
i
11π
16
; ...;e
i
27π
16 z1
3 4 4
2 Z1 Z1
5
6 Re
5π 17π 29π 8–4
–8 –6 –2 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
33 2e
i
18
; 2e
i
18
; 2e
i
18 –2
3 10
–4
Z 1 12
–6
34 ±2, ±2i 14
–8 z1
4
16
i ( )
3π
35 8e 20 ; 8e 20 ; ...; 8e 20
i ( )
11π
i ( )
35π
–10
(iv) Enlargement scale factor of 2 with (0, 0) as centre, and a
–12
__ __ __ __ __ __ __ __
–14 2π . 4
( ) ( ) ( ) ( )
√
6 √2 2 √6
√ √6 √2 2 √6
√ rotation of
36 2 ___
2 ___ + i ___
2 ___ ; 2 ___
+ ___ + i 2 ___
2 ___ 17 b) (i) –16 5 1
2 2 2 2 2 2 2 2
;
__ __ __ __
2 ___
( ___ + ___
) + i (___ ) ; 2 2 + i 2 ; 2 2 i 2
√
6 2
√ √6 2 √ __ __ __ __
z2 1.5 z1
√ √ √ √
2 2 2 2
37 cos (4b) + i sin (4b) 38 cos (7b) + i sin (7b)
39 cos (3b) + i sin (3b) 40 cos (2b) + i sin (2b)
41 Proof 42–43 Verify (ii) 5π
6
45 b) 2 cos 2na = z n + 1n ; 2i sin 2na = z n 2 1n c) k = 4
z z 18 a = 3, b = 1
46 7 47 b) 1 2 i 48 b) 3 + 5 19 No answers required – proofs
2 20 a) z = 1 e iθ
49 524 288 3
50 __ 2 iθ
2
c) S∞ = e
Practice questions 12 1 e iθ
2
1 x = 2, y = 21 d) (i) S∞ = cos θ + i sin θ
2 a) 0 b) x 2 + y 2 2 xy 12 1 (cos θ + i sin θ)
2
1002
21 a = 8; b = 25; c = 26 50
Cumulative frequency
22 z = 2 + 4i 40
23 z 1 = 1 + 4i; z 2 = 7 2 1 i 30
2 2 z 14
24 a) z 1 = 2 + 2i; z 2 = 2 2 2i b) = 28i 20
z2
2 10
d) 0 e) n = 4k, where k
0
2.0 2.4 2.8 3.2
GPA
Exercise 11.1
Note: Some answers may differ from one person to another due 6 9
to different graph accuracies.
8
1 a) Student, all students in a community, random sample of
few students, qualitative 7
b) Exam, 10th-grade students in a country, a sample from a 6
Frequency
few schools, quantitative 5
c) Newborns, heights of newborns in a city, sample from a
4
few hospitals, quantitative
3
d) Children, eye colour of children in a city, sample of
children at schools, qualitative 2
e) Working persons, commuters in a city, sample of few 1
districts, quantitative 0
f) Country leaders, sample of few presidents, qualitative 60 70 80 90
Grades
g) Students, origin countries of a group of international
school students, qualitative
The grades appear to be divided into two groups, one with
2 Answers are not unique! mode around 65 and the other around 85. No outliers are
a) Skewed to the right as few players score very high detected.
b) Symmetric
c) Skewed to the right
7 a) 14
d) Unimodal, or bi-modal, symmetric or skewed, etc.
12
3 a) b) Quantitative
10
c) d) Qualitative
Frequency
8
4 a) Discrete b) Continuous
c) Continuous d) Discrete 6
e) Continuous f) Discrete (debatable!) 4
5 12 2
10 0
0 8 16 24 32
8 Months
Frequency
0
2.0 2.4 2.8 3.2 40
GPA
25 30
20 20
Percentage
15 10
10 0
8 0 8 16 24 32
5 C1
0 pparently, more than 35 out of the 50 will lose the
A
2.0 2.4 2.8 3.2 licence, about 70%.
GPA
1003
Answers
8 a) 35 b) c)
30 400
375
25
Percentage
20 300
15
Frequency
10 200
5
0 100
0.0 1.2 2.4 3.6 4.8
Time
0
60 75 90 105 120 130 135
b) 2
60 Speed
Cumulative frequency
50
25 = 6.25%.
d) As you see from the diagram, ___
40 400
30 12 a) Histogram of C1
20 100
10
0 80
Cumulative percentage
0.0 1.2 2.4 3.6 4.8
Time
60
pparently, about 15 customers have to wait more than
A
2 minutes.
40
9 a) Skewed to the right, there is a mode at about 7 days stay,
and a few extremes that stayed more than 20 days. A good
proportion stayed for about 3 days. 20
b) 6000
5000 0
4.95 5.00 5.05 5.10 5.15 5.20 5.25
4000 C1
3000 b) About 5% at the lower end and also about 5% at the
2000 upper end.
1000 13 a) b)
100
0 10 20 30 40
Days 80
c) Approximately 35% of the patients
Percentage
10 a) 40 minutes 60
b) Approximately 30%
c) 250 40
200
150 20
100
50 0
60 120 180 240 300 360 420
0 Time
18 20 22 24 26 28 30 32 34 36 38 40
c) As you see from the diagram, about 250 seconds.
Minutes
11 a)
Speed Frequency
60–75 20
75–90 70
90–105 110
105–120 150
120–135 40
135– … 10
1004
14 15
Histogram of mass
Histogram of time
18
12
16
16.1
10
14
Frequency density
Frequency density
8 12
10
6
8
4 6
2 4
2 2.25 0.48 0.27 0.04
0
5 10 20 30 45 60 0
0.5 200.5 400.5 600.5 800.5 1000.5 2000.5
Time
Mass
Exercise 11.2 c) 9
1 a) 6 b) 6 8
c) I t appears to be symmetric as the mean and median are 7
the same. A histogram supports this view.
6
2 a) 7.8 b) 7.5 c) 7 or 8
Frequency
5
3 Average 5 1.16, median 5 1. Median is more appropriate as
4
the data is skewed to the right.
4 Mean 5 7494.7, median 5 837.5. There are extreme values 3
and hence the median is more appropriate. 2
5 Mean 5 median 5 430. It appears to be symmetric and 1
hence either measure would be fine. 0
16 24 32 40 48
6 a) 49.56 b) 49.93 Grades
7 2.052
d) 110
8 a) 29.96
100
b)
90
Cumulative percentage
1 89 80
2 0223344 70
2 5666777
60
3 34
50
3 568
40
4 022
30
4 66
20
Median is 27 10
0
16 24 32 40 48
Grades
The median ≈ 27
1005
Answers
9 a) Chart of year c)
100
25 000
15 000 60
Count
10 000 40
5000 20
0
1970 1975 1980 1985 1990 1995 2000 2005 0
Year
15–19
20–24
25–29
30–34
35–39
40–44
45–49
50–54
55–59
60–64
65–69
70–74
75–79
There appears to be a decline in the total number of injuries.
b)
Pie chart of year Age
Percentage within all data.
Fatal* year Serious* year
From the graph, the median is approx. at 36.
11 Median ≈ 8 days; mean = 9.5 days
12 Median ≈ 28 minutes; mean = 28.7 minutes
13 Median ≈ 105; mean = 103 km/h
14 Median ≈ 5.075; mean = 5.09
15 Median ≈ 210; mean = 228.6
16 a) 41.6
Slight* year
b) 61.6
17 a) 61.4
Category
b) 63.8
1970
1990
Exercise 11.3 _____
Med. 71
Q3 79
Min 56
Max 80
100
80
Count
60 55 60 65 70 75 80
Rates
40
c) No outliers
20 2 a) Mean = 162.6, Sn – 1 = 23.35
0 b)
11 79
12 567
15–19
20–24
25–29
30–34
35–39
40–44
45–49
50–54
55–59
60–64
65–69
70–74
75–79
13 089
14 123679
Age 15 033445689
b) 37.6 16 02334568
17 1344789
18 02255779
19 8
20 9
21 08
Median = 162.5
1006
c) 100 4 a) 100
90
70
60 60
50
40 40
30
20 20
10
0 0
30–40
40–50
50–60
60–70
70–80
80–90
90–100
100–110
110–120
120 140 160 180 200 220
Passengers
Q1 ≈150, median ≈ 165, Q3 ≈ 182 Time
Percentage within all data.
b) Median = 63, IQR = 27
c) About 68
5 29.6
6 a) Mean = 72.1, Sn 2 1 = 6.1
b) New mean = 85.1, S will not change.
7 a)
x < 10 x < 20 x < 30 x < 40 x < 50
15 65 165 335 595
x < 60 x < 70 x < 80 x < 90 x < 100
120 140 160 180 200 220
815 905 950 980 1000
Passengers
800
3 a) and b) 700
100 600
500
Cumulative percentage of marks
400
80 300
200
100
60 0
0 10 20 30 40 50 60 70 80 90 100
Seats
40
c) (i) Around 50
(ii) Q1 5 40, Q3 5 60, IQR 5 20
20
(iii) About 170 days
0 (iv) Approximately 70 seats
0–9 10–19 20–29 30–39 40–49 50–60
Marks 8 a) 40
Percentage within all data.
Q1 ≈ 18, median ≈ 29, Q3 ≈ 39
30
Frequency
20
10
0
30–35
35–40
40–45
45–50
50–55
55–60
60–65
65–70
Time
1007
Answers
80 12 36.7
70 13 x = 6, y = 11
60
14 Mean = 11.12, variance = 24.6 (calculating σ2 = 23.6)
15 Standard deviation = 6.1, IQR 6
50
16 Standard deviation 4.5, IQR 6
40
17 Standard deviation 16.7, IQR 15
30 18 Standard deviation 0.056, IQR 0.05
20 19 Standard deviation 82.3, IQR 60
10
0 Practice questions _____
30–35
35–40
40–45
45–50
50–55
55–60
60–65
65–70
1 a) 12 b) √30.83
2 4
Time 3 a) Time
Percentage within all data. 1.6 2.1 2.6 3.1 3.6 4.1 4.6 5.1 5.6 6.1 6.6
Median = 53, IQR = 15 Frequency 2 2 6 4 11 10 5 5 3 2 0
c) Mean = 51.3 and Sn 2 1 = 34.8
14
9 a) Q1 5 165.1, median 5 167.64, Q3 5 177.8,
12
minimum 5 152, maximum 5 193
10
b)
Frequency 8
6
4
2
150 160 170 180 190 200
Height 0
1.6 2.1 2.6 3.1 3.6 4.1 4.6 5.1 5.6 6.1 6.6
20 Time
b) 86% c) approx. 4 d) 3.86, 1.1
e)
15 60
50
Percentage
10 40
Frequency
30
5 20
10
0 0
150 160 170 180 190 1.6 2.1 2.6 3.1 3.6 4.1 4.6 5.1 5.6 6.1 6.6
Height Time
f) M
inimum 5 1.6, Q1 5 3, median 5 4, Q3 5 4.5,
c) Mean 5 170.5, standard deviation 5 9.61
maximum 5 6.2
d) The heights are widely spread from very short to very tall
4 a) Median and IQR as the data is skewed with outliers.
players. Heights are slightly skewed to the right, bimodal
b) Mean 5 682.6, standard deviation 5 536.2
at 165 and 170, no apparent outliers. The heights between
c)
the first quartile and the median are closer together than 500
the rest of the data.
450
e)
140 400
120 350
100 300
80 250
60
200
40
150
20
100
0
173
152
155
157
160
163
165
168
170
175
178
180
183
185
188
191
193
50
Approximately 183 cm tall 0
100
300
500
700
900
1100
1300
1500
1700
1900
2100
2300
2500
f) 171.3
1008
d) Q1 5 300, median 5 500, Q3 5 800, IQR 5 500 b) 25
e) There are a few outliers on the right side. Outliers lie
20
above Q3 1 1.5IQR 5 1550.
Frequency
f) Data is skewed to the right, with several outliers from 15
1600 onwards. It is bimodal at 300–400.
10
5 a) Spain, Spain b) France
c) On average, it appears that France produces the more 5
expensive wines as 50% of its wines are more expensive 0
than most of the wines from the other countries. Italy’s 25 30 35 40 45 50 55
prices seem to be symmetric while France’s prices are Speed
skewed to the left. Spain has the widest range of prices. ata is relatively symmetric with possible outlier at 55.
D
6 a) Mean 5 52.65, standard deviation 5 7.66 The mode is approximately 37.
b) Median 5 51.34, IQR 5 2.65 Histogram created from table:
c) Apparently, the data is skewed to the right with a clear 35
outlier of 112.72! This outlier pulled the value of the mean 30
to the right and increased the spread of the data. The 25
Frequency
median and IQR are not influenced by the extreme value. 20
7 a) The distribution does not appear to be symmetric as the 15
mean is less than the median, the lower whisker is longer 10
than the upper one and the distance between Q1 and the 5
median is larger than the distance between the median 0
and Q3. Left skewed. 28.5 32.5 36.5 40.5 44.5 48.5 52.5
b) There are no outliers as Q1 2 1.5IQR 5 37 , 42 and Speed
Q3 1 1.5IQR 5 99 . 86. c) Mean 5 38.2, standard deviation 5 5.7
c) d) Speed Cu. frequency
26–30 8
42 86
31–34 23
60.25 70 75.75 35–38 54
40 50 60 70 80 90 39–42 78
d) See a) 43–46 88
8 a) 225
47–50 98
b) Q15205, Q3 5 255, 90th percentile5300,
10th percentile 5 190 51–54 100
c) IQR 5 50, since Q1 2 1.5IQR 5 130 . minimum and e) Median 5 37.6, Q1 5 34.5, Q3 5 41.3, IQR 5 6.8
Q3 1 1.5IQR 5 330 , 400 then there are outliers on f) There are outliers on the right since
both sides. Q3 1 1.5IQR 5 51.5 , maximum 5 54.
d)
227.5
e) T
he distribution has many outliers. Apparently skewed to 10 a) Mean 5 1846.9, median 5 1898.6,
the right with more outliers there. The middle 50% seem standard deviation 5 233.8,
to be very close together while the whiskers appear to be Q1 5 1711.8, Q3 5 2031.3, IQR 5 319.5
quite spread. b) Q1 2 1.5IQR 5 1232.55 . minimum, so there is an
outlier on the left.
9 a)
Speed Frequency c)
26–30 8
31–34 15
35–38 31
39–42 24 1000 1200 1400 1600 1800 2000 2200 2400
1009
Answers
1010
c) A
∪ B = all males or persons who drink; A ∩ C = all single
males; C ' = all non–single persons; A ∩ B ∩ C = all single c) 2426 0.306 d) 1045 0.924
males who drink; A ' ∩ B = all females who drink. 7917 1131
12 a) {(R, L, L, S), (L, R, L, R), …}, 81 10 005 269 265
21 a) 0.000 33 b) 0.000 35
b) {(R, R, R, R), (L, L, L, L), (S, S, S, S)} 29 900 492 777 412 792
c) {(R, R, L, L), (R, L, R, S), …} c) 777143527 0.9997 d) 85 266 221 0.1097
d) {(R, L, R, S), (S, S, R, L), …} 777 412 792 777 412 792
13 a) {(T, SY, O), (C, SN, O), …} 22 a) 3 0.107 b) 17 0.0895
b) {(T, SY, O), (T, SY, F), (B, SY, O), …} 28 190
c) {(C, SY, O), (C, SN, O), (C, SY, F), …} c) 153 0.805
d) C ∩ SY = {(C, SY, O), (C, SY, F)} 190
C ' = {(T, …, …), (B, …, …)} 23 7 0.4375 24 111 0.2775
16 400
C ∪ SY = all triplets containing C or SY.
25 a) 264 0.140 b) 166 0.00196
14 a) {(1, 1, 1), (1, 1, 0), (0, 1, 0), …}
b) X = {(1, 1, 0), (1, 0, 1), (0, 1, 1)} 1885 84 825
c) Y = {(1, 1, 1), (1, 1, 0), (1, 0, 1), (0, 1, 1)} c) 2584 0.0653
d) Z = {(1, 1, 1), (1, 1, 0), (1, 0, 1)} 395 85
e) Z ' = {(0, 1, 1), (0, 1, 0), (0, 0, 1), (0, 0, 0), (1, 0, 0)} 26 a) 0.096 b) 0.008 c) 0.512
X ∪ Z = {(1, 1, 0), (1, 0, 1), (0, 1, 1), (1, 1, 1)}
X ∩ Z = { (1, 1, 0), (1, 0, 1)} Exercise 12.4
Y ∪ Z = {(1, 1, 0), (1, 0, 1), (0, 1, 1), (1, 1, 1)} 1 __
7
20
Y ∩ Z = { (1, 1, 1), (1, 1, 0), (1, 0, 1)} 2 a) __5
b) __
4
c) __
2
d) __
1
e) _ 25
10 10 10 10
15 a) {1, 2, 31, 32, 41, 42, 51, 52, 341, 342, …, 3452} 1
b) {31, 32, 41, 42, 51, 52} 3 P(A ∩ B) = __
0 P(A)P(B)
9
c) All except {1, 2} 4 29
d) {1, 31, 41, 51, 341, 351, 431, 451, 531, 541, 3451, 4351, …} 35
5 0.90
6 a) 92%
Exercise 12.3 b) (i) 0.64% (ii) 15.36% (iii) 14.72%
1 a) __
3
10 b) _34 c) 48.68%
2 a) 0.63 b) 1 7 a) 10 000 b) __ 9
10 c) 0.3439 d) ___ 1000
3439
3 a) (i) __
1
(ii) __
7
(iii) __
4
(iv) __
10 8 a) __
15
16 b) _ 45 c) _ 15
52 26 13 13
b) (i) __
1
(ii) __
13 9 a) {(1, 1), (1, 2), …, (6, 6)}
51 17
b)
c) (i) __ 1
52 (ii) __
10
13
x 2 3 4 5 6 7 8 9 10 11 12
4 a) _45 b) __
11
30 c) 1 P(x) __1
36 __1 __1 _1 __ 5 _1 __ 5 _1 __ 1 __1 __1
18 12 9 36 6 36 9 12 18 36
5 a) _ 1 __
b) 1
2 12 c) __
11
36 d) __
11
12 e) _ 13 f) _ 23
6 a) _17 b) _47
10 a) __
7
15 b) __
11
75 c) __
9
35
7 a) (i) {(1, 1), (1, 2), …, (6, 6)}
(ii) _16 (iii) _ 29 (iv) _ 56 d) __
46
75 e) __
11
20
f) No: P(female) P(female/grade 12) 2 for example
b) (i) 0 (ii) _19 (iii) __5
36 (iv) 0
11 a) (i) 0.56 (ii) 0.15
8 a) 0.04 b) 0.55 c) 0.1548
d) 0.060 372 e) 0.104 022 b) __
15
56 c) no
9 a) Yes b) no c) no 12
10 a) 0.06 b) 0.42 c) 0.3364 d) 0.412 Conditions for
P(A) P(B) P(A B) P(A B) P(A|B)
11 a) 0.183 b) 0.69 events A and B
12 x > n 21 13 a) n = 20 b) n = 12 0.3 0.4 Mutually exclusive 0.00 0.7 0.00
5 2
14 a) b) 1 c) 1 0.3 0.4 Independent 0.12 0.58 0.30
18 3 4 0.1 0.5 Mutually exclusive 0.00 0.60 0.00
15 a) 1 b) 5 0.2 0.5 Independent 0.10 0.60 0.20
3 14
16 a) 3243 0.299 b) 143 0.0036 13 a) 0.30 b) yes
10 829 39 984 14 a) 65% b) 35% c) 52%
17 a) 144 0.0405 b) 943 0.417
3553 2261 15 a) 0.56 b) 0.10
18 a) 1 b) 4 16 a) 1 b) 91 c) 75
91 91 216 216 216
19 a) 78 0.308 b) 576 0.455 17 a) 0.21 b) 0.441 c) 0.657
253 1265
18 a) 23 b) 11 c) 15 or ___
144 48
( )
5 d) 9
20 a) 593 775 b) 608 0.230 144 144 23
2639 B = {(10, 5) , (10, 4) , ..., (10,1) , (1,10) , ..., (5,10)} ,
19 a) AA ∩ B
1011
Answers
1012
24 a) 1
4 L
7 W 18 1 19 4
8
3 L 3
4
3
1 5 L Exercise 13.2
8 W
1 f ′ (x ) = 22x 2 g ′ (x ) = 3x 2
2 L
3
3 h ′ (x ) = 1 4 r ′ (x ) = 2 23
b) ___
47
160 c) __
35
47 2 x x
5 (i) y
25 a) _13 b) __
7
12 c) _37
3
26 a) 0.9 Grows
2
0.4 Red 1
0.1 Does not grow
3 2 1 0 1 2 3 x
0.8 Grows 1
0.6 Yellow 2
0.2 Does not grow 3
b) (i) 0.36 (ii) 0.84 (iii) 0.429 4
27 a) _ 16 b) __
1
12 c) _29 5
28 a) (i) __ 8
21 (ii) _ 16 (iii) no, P(A B) P(A)P(B) 6
b) __ 10
17 c) ___
200
399 (ii) y
29 _1
3 30 _ 45 31 0.001 98 32 __
19
30
25
33 0.80 34 __
10
19 35 a) __
13 __11
20 b) 15 20
36 _ 25
()
2n 22 15
37 a) (i) ___ 5 25
(ii) ___ (iii) 1 5 10
36 216
6 6
b) No answer required – proof 5
c) __ 5
11 d) 0.432
3 2 1 0 1 2 3 x
38 a) 0.957 b) 0.301 5
39 _ 19
10
40 a) 0.25 b) 0.083
41 a) 0.80 b) 0.56 15
42 a) 0.732 b) __
11
61
20
43 a) _23 b) _ 2
9 c) _ 34
(iii) y
44 a) __1
10 b) proof 40
c) __11
90 d) __
3
11
_3
45 7 30
Chapter 13 20
Exercise 13.1
1 4 2 3x 2 3 2x 4 6 10
c) y
6 a) y ′ = 6x 2 4 b) 24
(4, 213) 250
7 a) y ′ = 22x 2 6 b) 0
200
150
8 a) y ′ = 2 64 b) 26 100
x
9 a) y ′ = 5x 2 3x 21 b) 1
4 2 50
10 a) y ′ = 2x 2 4 b) 0
7 6 5 4 3 2 1 0 1 2 3 4 5 6 x
11 a) y ′ = 2 2 2 1 + 9 b) 10 50
x x4 100
12 a) y ′ = 12 23 b) 3
x 150
(3, 130)
13 a = 25, b = 2 14 (0, 0)
15 (2, 8) and (22,28)
(
16 5 ,2 21
2
) 4 9 a) (0,25)
17 (1,22) b) Stationary point is neither a maximum nor minimum
18 a) Between A and B because 1st derivative is always positive.
b) Rate of change is positive at A, B and F; c) y
rate of change is negative at D and E; 4
rate of change is zero at C 2
c) Pair B and D, and pair E and F
19 a = 1, b = 5 20 a = 1 21 (3, 6) 4 3 2 1 0 1 2 3 4 x
22 a) 12.61 b) 12 23 f ′ (x ) = 2ax + b 2
–
24 a) 4. 6 degrees Celsius per hour 4
b) C ′ (t ) = 3 t 6 (0, 5)
c) t = 196 ≈ 2.42 hours 8
81
25–26 Proof 10
27 1 28 2 12 12
2 x x
5 5 1 10 a) (1, 4) , (3, 0)
30 −
29 or 2 b) (1, 4) maximum because 2nd derivative is negative at
(3 2 x )2 (x 2 3) 2 (x + 2)
3
x = 1
(3, 0) minimum because 2nd derivative is positive at
x = 3
Exercise 13.3 y
( )
c)
1 (1,27) 2 2 3 , 8 3 (3, 2) 8
2 7
4 a) y ′ = 2x 25 b) increasing for x > 5
2 6
c) decreasing for x < 5
2 5
5 a) y ′ = 26x 2 4 b) increasing for x < 2 2 4
(1, 4)
3
c) decreasing for x > 2 2 3
3
6 a) y ′ = x 2 21 b) increasing for x > 1, x < 21 2
c) decreasing for 21 < x < 1 1
(3, 0)
7 a) y ′ = 4x 3 212x 2 b) increasing for x > 3 0
2 1 1 2 3 4 5 x
c) decreasing for x < 0, 0 < x < 3 1
8 a) (3,2130) , (24, 213) 2
b) (3,2130) minimum because 2nd derivative is positive at
3
x = 3
(24, 213) maximum because 2nd derivative is negative at
4
5
x = 24
6
(
11 a) (21, 4) , (0, 6) , 5 ,2 279
2 16 )
b) (21, 4) minimum because 2nd derivative is positive at
x = 21
(0, 6) maximum because 2nd derivative is negative at
x = 0
1014
b) s
( )
Displacement function:
5 ,2 279 minimum because 2nd derivative is positive 1
s(t) t 3 4t 2 t
2 16
at 1 0 1 2 3 t
1
x=5 2
2
c) y
3
15
4
10 5
(0, 6)
5 6
(1, 4)
7
2 1 0 1 2 3 4 x v
5 12
10 10
15 8
20
( 5
2 , 279
16 ) 6
Velocity function:
v(t) 3t 2 8t 1
4
3 (
12 a) (21,14) , 7 ,2 122
27 ) 2
5
10
1 0 1 2 3 t
5
5
4 3 2 1 0 1 2 3 4 5 x
10
5 ( 7
3 , 122
27 )
( )
15
13 a) 1 ,2 1
4 4
( )
c) t ≈ 0.131, displacement ≈ 0.0646
1 1
b) ,2 minimum because 2nd derivative is positive at d) t = 1.3, displacement = 24.3
4 4
e) Object moves right at a decreasing velocity then turns left
x=1
4 with increasing velocity then slowing down and turning
c) y right with increasing velocity.
2 15 Relative maximum at (22,16); relative minimum at (2,16) ;
inflexion point at (0, 0)
16 Absolute minima at (22,24) and (2,24); relative maximum
1
at (0, 0) ; inflexion points at 2 2 3 ,2 20 and 2 3 ,2 20
3 9 3 9
17 Relative maximum at (22,24); relative minimum at (2, 4);
no inflexion points
3 3
1 0 1 2 3 x 18 Relative minimum at 2 4 , 3 2 ; inflexion point at (1, 0)
2 2
( 1
4 , 14 ) 19 Relative minimum at (21,22); relative maximum at (1, 2) ;
1
inflexion points at 2 2 ,2 7 2 , (0, 0) and 2 , 7 2
14 a) v (t ) = 3t 2 2 8t + 1; a (t ) = 6t 2 8
2 8 2 8
20 Relative minimum at (21, 0); absolute minimum at
(
2,227); relative maximum at (0, 5) ; inflexion points at
(
1.22,213.4) and (20.549, 2.32)
21 a) v (0) = 27 m s21, a (0) = 266 m s22
1015
Answers
(d)
d) y
b) v (3) = 45 m s21, a (3) = 78 m s22
c) t = 1 and t = 2 1 ; where displacement has a relative
2 4
maximum or minimum
d) t = 11 = 1.375 ; where acceleration is zero
8 0 x
22 x ≈ 5.77 tonnes; D ≈ 34.6 ($34 600); this cost is a minimum
because cost decreases to this value then increases
23 a 2 3, b = 4, c = 22
( )
24 Relative maximum at 22,2 15 , stationary inflexion point
4
at (1, 3) (e)
e)
y
f (x ) → x as x → ±∞
y
6
4 yx
2 x
5 4 3 2 1 0 1 2 3 4 5 x
2
4
26 a) Increasing on 1 < x < 5; decreasing on x < 1, x > 5
6
b) Minimum at x = 1; maximum at x = 5
8 27 a) Increasing on 0 x < 1, 3 < x < 5; decreasing on
1 < x < 3, x > 5
b) Minimum at x = 3; maximum at x = 1 and x = 5
25 a) y
28 x ≈ 0.5 and x ≈ 7.5
29 y
8
0 x 4
4 2 0 2 4 x
(b) 30 a) Right 1 < t < 4; left t < 1, t > 4
b) y b) v 0 = 224, a0 = 30
c) d max = 16 at t = 4, v max = 13.5 at t = 2.5
d) Velocity is maximum at t = 2.5
31 a) Maximum at x ≈ 6.50, minimum at x ≈ 20.215
b) Maximum is 7π + 1, minimum is π 21
4 4
0 x
Exercise 13.4
1 a) y = 24x 2 8 b) y = 4
27
c) y = 2x + 1 d) y = 22x + 4
(c)
2 a) y = 1 x + 19 b) x = 2 2
c) y 4 4 3
c) y = x + 1 d) y = 1 x + 11
2 4
3 At (0, 0) : y = 2x; at (1, 0) : y = 2x + 1; at (2, 0) : y = 2x 2 4
4 y = 22x
5 a) x = 1
0 x b) For y = x 2 2 6x + 20, eq. of tangent is y = 24x + 19
For y = x 3 2 3x 2 2 x , eq. of tangent is y = 24x + 1
( )
6 Normal: y = 1 x 2 7 ; intersection pt: 2 1 , 2 15
2 2 2 4
7 Eq. of tangent: y = 23x + 3; eq. of normal: y = 1 x 2 1
3 3
8 a = 4, b = 27
1016
9 a) y = 2x + 5
2
b) ( )
2 , 41
3 27
f ′ (3) = 10 > 0 (increasing) ∴ f (2) is a turning point
9
b) vertical asymptote: x = 0 (y-axis); oblique asymptote: y = 2x
( )
10 Eq. of tangent: y = 2 3 x + 1; eq. of normal: 1,3
4 5
y = 4 x 2 22 2
3 3 6 a = 1
11 y y = 12x + 4 7 a) y = 5x 2 7
35 y = 9x + 5 b) y = 2 1 x + 17
30 5 5
8 a) x = 1
25 b) 23 < x < 22, 1 < x < 3
20 c) x = 2 1
2
15 d) y
10 2
5
maximum at x 1
inflexion point at
0 x 1
4 2 2 4 6 x 12
–5
–10
3 2 1 0 1 2 3 x
–15
y = (1 + x)2 (5 x)
1
12 y = 11x 2 25 and y = 2x 21 minimum at x 2
13 y = (2 2 2 2) x and y = 2(2 2 + 2) x
2
14 a) y = 1 x + 4 b) 3 9 ≈ 2.08
12 3
15 y = 2 1 3 x + 3 3
2 a 2 a
16 xQ = 22x P , yQ = 28y P
9 b = 2, c = 3
10
function diagram
Practice questions f 1 d
1 a) Gradient = 3 b) y = 3x 2 9 f 2 e
4
c) y f 3 b
8 f 4 a
6
4
11 a) 2 b) 2 c) x ≈ 0.881
π
2
2 ( 3
2 , 9
4 ) 12 a) (i) x = 0
dy
(ii) y = 3
b) = 22
3 2 1 0 1 2 3 4 x dx x
2 c) Increasing for all x, except x = 0
dy
4 d) No stationary points because = 22 ≠ 0
dx x
13 Maximum at (21,1), minimum at (0, 0) , maximum at (1,1)
( ) ( )
d) Q 3 , 0 , R 0,2 9 14 a = 8 , b = 16
4 4 3 5
f) y = 2ax 2a 2 15 a) 10 m s21 b) 10 sec c) 50 metres
16 a) v = 14 2 9.8t
g) T
( )a
2
, 0(, U )
0,2a 2
b) t ≈ 1.43 sec
c) Velocity = 0, acceleration = 29.8 m s22
h) x-coord.: a + 0 = a ; y-coord.: a 2a = 0
2 2
17 (24,120)
2 2 2
2 A = 1, B = 2, C = 1
18 a) y = (2 3 ) x + π 3 2 2
3
3 a) 4x 215x 4 3 π
b) y = x 2 3 2 2
b) 2 12
3 9
x
( )
2
27 2r
4 a) x = 2 or 2 2; f ′ (1) = 26 < 0 (decreasing) and 19 a) h =
r
; V = πr 27 2r 2
b) r = 3
1017
Answers
37
20 a = 22, b = 8, c = 10 y
21 a) y = 27x + 1
y = f '(x)
b) y = x + 107
( )
7 7
22 a) Absolute minimum at 3 ,2 27
maximum
4 256
b) Domain: x ∈, range: y 2 27
256
c) Inflexion points at (0, 0) and ,2 1
1
2 16 ( ) y = f (x)
d) y a minimum b x
1 inflexion
points
inflexion
points
Chapter 14
Exercise 14.1
0 1 a) (_ 52 , 22, 0 )
1 1 2 x __
b) ( 3 , 2 √3 , 0 )
minimum
1 c) (21, 2, 22)
23 a) 2 5 b) 1 c) 3 d) 2 x + 2 d) (a, 24a, 2a)
3 4
24 a) f ′ (x ) = 3x 2 4 _12 , 23, 2 )
2 a) Q (2
2 x
b) P (_ 52 , 22, 0 )
b) f ′ (x ) = 3x 2 2 3 cos x
c) Q(0, 24a, 3a)
c) f ′ (x ) = 2 12 + 1
x 2 3 a) (x, y, z) 5 (t, t, 5 2 5t), or (x, y, z) 5 (1 1 t, 1 1 t, 2 5t)
d) f ′ (x ) = 2 14 91 b) (x, y, z) 5 (21 1 4t, 5t, 1 2 3t)
3x c) (x, y, z) 5 (2 2 4t, 3 2 6t, 4 1 t)
( )
25 3 solutions: 11 , 1105 , (2,215) , and (22, 5)
2 8
17 4 a) C(7, 28, 21)
26
2 b) C(21, __ 11 __ 29
2 , 3 )
27 ( )(
3
2, 2 , 22,2 2 )3
5
c) C(2 2 a, 4 2 2a, 2b 2 2)
a) (2 _13 , 1, _13 )
28 ( 21,22)
b) (1, 2 _53 , 21)
(
30 2, 20) , (4,16)
31 a) particle does not change direction for 0 < t < 2π ( a 1 b 1 c
c) ________
3
2a 1 2b 1 2c
, ___________
3
,
)
a 1 b 1 c
b) v = 1 + cost > 0 for 0 < t < 2π 6 a) D (21, 1, 26)
c) t = 0, π, 2π b) D (2
__ __
2 √2 , 2 √3 , 1 2 4 √5 )
__
d) Maximum value of s is 2π c) D ( _2 , 2 _3 , 24 )
5 2
y 7 m 5 5, n 5 1
6 8 a) v 5 _ 23 i 1 _ 23 j 2 _ 13 k
b) v 5 ____ 3___
i 2 ____ 2___ 1___
j 1 ____
k
√
14 √ 14 √ 14
4
c) v 5 _ 3 i 2 _ 3 j 2 _ 3 k
2 1 2
9 a) _ 23 (2i 1 2j 2 k)
2 2___
b) ____ (6i 2 4j + 2k)
√
14
5 (2i 2 j 2 2k)
c) __
0 2 4 6 x
3 ___
10 a) |u 1 v| 5 √29
___ __
32 a = 1 , b = 3 , c = 26, d = 2 5 ; y-coord. is 2 19 b) |u| 1 |v| 5 √14 1 ___ √5
__
4 4 2 2
( ) ( )
c) |23u| 1 |3v| 5 3 √14 1 3 √5
33 Absolute minimum points at 22,2 and 2,2 1
1
1 u 5 ____ i 1 ____ 3j
8 8 d) ___ ___ 2k
___ 2 _______
34 a) y = 2x + 2 b) y = 2x + π |u| √
14 √ 14 √ 14
2 1 u | 5 1
35 b) y = 2x, (1, 2) e) | ___
|u|
36 a) v = 50 2 20t b) s = 1062.5 m 11 a) (3, 4, 25) b) (0, 22, 5)
1018
__
12 a) (1, 2 _43 ) 6 ( 4i 1 2j 2 2k) c) 2 _23 i 1 _ 83 j 2 2k
b) √ 13 68.22
14 103.3°, 133.5°, 46.5°
13 0 14 ± 14 15 None 16 None
14 15 0
17 a) a = (8, 0, 0), b = (8, 8, 0), c = (0, 8, 0), d = (0, 0, 8), 16 k 5 2
e = (8, 0, 8), f = (8, 8, 8) 17 k = 0 or k = 4
b) l = (8, 4, 8), m = (4, 8, 8), n = (8, 8, 4) 18 x 5 220, y 5 214
c) proof 19 x 5 5
18
a) c = (8, 0, 12), d = (0, 10, 12)
b) f = (4, 5, 0), g = (4, 5, 12)
c) AG = (24, 5, 12) = FD
___›
_
0
20 117°, AC 5 6 ( )
, 33°
3
1
21 a) b = 2 2 b) b = 0 or b = _12
19 ± 6 c) b = _ 52 or b = 3 d) b = ±4
20
3
(a, b, µ) = 31 (
7
,2 15 , 6
7 7 )21 (a, b, µ) = (2,21, 3)
22 a) b 5 2 _12
23 (2140.8, 140.8, 18)
b) b 5 _12
24 t = 2
22 Not possible 23 Rectangle 25 t = 2 _ 12 26 t = 0 or t = _12
24
T1 = 125 ( 3 21) N;T2 = 175 3 2 2 6 N
2
27 90° or cos21 2
( ) 6
28 Proof
( )
9 a) 127° b) 63° c) 73°
a) o = 1 (ab) + (ac ) + (bc )
2 2 2
10 a) m 5 _ 13 b) m 5 2 _14 38
2
11 mA: r 5 (4, 22, 21) 1 m(21, 0, _32 ); b) a = 1 ab;b = 1 bc;c = 1 ac
mB: r 5 (3, 25, 21) 1 n( _12 , _92 , _32 ) 2 2 2
c) result obvious
mC: r 5 (3, 1, 2) 1 k( _12 , 2 _92 , 23); centroid ( __ 10
3 , 22, 0)
12 90, 90, 82, 74, 60, 54, 53, 52, 47, 43, 38, 37
1019
Answers
c) r 5 (2, 21) 1 t (7, 3)
x 2 1 7t
y 5 21 1 3t
(
30 16 , 35 , 13
11 11 11 ) (
11 11 11
) (
31 17 ,2 7 , 72 32 43 , 58 ,2 1
11 11 11
)
x 2t
d) r 5 (0, 2) 1 t (2, 24) y 5 2 2 4t
Exercise 14.5
1 B and C
9 a) t 5 _ 32 b) no c) m 5 _72 2 A
10 a) (i) (3, 24) (ii) (7, 24) (iii) 25 2 x
b) (i) (23, 1) (ii) (5, 212) (iii) 13 3 24 y = 26; 2x 2 4y + 3z 2 26 = 0
c) (i) (5, 22) (ii) (24, 27) (iii) 25 3 z
11 a) (296, 128) 13 , 2 ___
b) (___
2040 13 )
850
2 x
12 a) (24, 18) 4 0 y = 23; 2x + 3z + 3 = 0
b) r 5 (3, 2) 1 t (24, 18) 3 z
c) In 10 minutes 0 x 0 2 1
13 a) a 5 23, b 5 25 5 0 y = 3; 3z 2 3 = 0; r = 3 + t 21 + s 1
___
√21
3 z 1 0 0
b) 2____
6 5 x
___ ___ 6 1 y = 5; 5x + y 2 2z 2 5 = 0
√15 ____
√35
c) ____ , –2 z
6 2
14 a) 146.8° b) 3.87 0 x
c) (i) L1: r 5 (2, 21, 0) 1 t (0, 1, 2); L 2: r 5 (21, 1, 1) 7 1 y = 22; y 2 2z + 2 = 0
1 t (1, 23, 22) 22 z
1020
1 x 3 2 2 Practice
___
questions
___
› › ___› ___› ___ › ___
›
8 26 y = 23; r = 22 + 1 + m 0 1 a) OD 2 OC b) _ 12 (OD
2 OC ) c) _ 12 (OD
1 OC )
2 z 4 2 21 2 a) 5i 1 12j b) 10i 1 24j
___› ___› ___›
22 x
3 a) | OA | 5 | OB | 5 | OC | 5 6
9 2 y = 21; 22x + 2y + z = 21
( )
___› ___
1 z 21
b) AC 5
___ 1___
c) ____ d) 6 √11
√
11 √
12
18 x
10 23 = 5; 18x 2 3y 2 11z = 5 4 a) (10, 5) b) (23, 6); 90°
y
5 a 5 2, b 5 8
211 z
6 r 5 (3, 21) 1 t(4, 25)
p x ____
11 q = p 2 + q 2 + r 2; px + qy + rz = p 2 + q 2 + r 2 7 a) 39.4 b) (i) (9, 12), (18, 28) (ii) √481
y
c) 7 a.m. d) 24.4 km e) 54 minutes
r z
8 r 5 t(2i 1 3j)
4 x
12 4x 2 2y + 7z = 14; 22 y = 14; 9 b) (2, 3.25)
10 c) 90°
7 z
d) (i) 12x 2 5y 5 301 (ii) (28, 7)
1 2 4 11 117°
r = 2 + m 23 + n 1 12 2x 1 3y 5 5
2 22 22 13 a) (6, 20) b) (i) (6, 28) (ii) 10
8 x c) 4x 1 3y 5 84 d) collide at 15:00
13 8x + 17y 2 5z + 8 = 0; 17 y = 28; f) 26 km
25 z 14 72°
15 a) 3.94 m b) 1.22 m/s
2 1 23
r = 22 + s 1 + t 2
c) x 2 0.7y 5 2 29 , ___
d) (___
170 29 )
160
Chapter 15
d) 9x 2 15y + 4z 2 2 = 0 e) 15 Exercise 15.1
322
32 a) AB = 2i 2 3j + k; BC = i + j 1 a) y ′ = 12 (3x 2 8)
3
b) y ′ = 2 1
b) 2 i + j + 2k c) 6
2 12 x
d) 2x + y + 2z = 3 2 c) y ′ = cos 2 x 2 sin 2 x
()
2 2t d) y ′ = cos x e) y ′ = 2 4x 3
e) 21 + t f) 3 6
22
2
(
x2 + 4 )
26 + 2t f) y ′ =
(x 21)2
g) 1 (2i + j + 2k) h) E(24, 5, 6) 21 21
6 g) y ′ = or
2 (x + 2) (2x + 4) x + 2
3
33 Proof
34 a) P(4, 0, 23), Q(3, 3, 0), R(3, 1, 1), S(5, 2, 1) h) y ′ = 22 sin x cos x
b) 3x + 2y + 4z = 0
i) y ′ = 2x + 2 or
2x + 2
2 (12 x ) (2 2 2x ) 12 x
c) 0 3
35 a) 147° b) 2.29
2 21 + µ j) y ′ = 26x + 5 k) y ′ = 2
( ) 3 3 (2x + 5)
2
3x 2 25x + 7
2
c) (i) L1 : 21 + λ ; L2 : 12 3µ (ii) no solution
2λ 12 2µ
d) 9
(
l) y ′ = 2 (2x 21) 7x 4 2 2x 3 + 3
2
)
21 2 a) y = 212x 211 b) y = 9 x 2 2
36 a) (1, 21, 2) 5 5
b) 11i 2 7j 2 5k 1 1
c) v.u = 0 c) y = 2x 2 2π d) y = x +
2 2
1 6
(
3 a) v (t ) = 22t sin t 21 2
) b) velocity = 0
d) r = 21 + t 13
c) t = π + 1 ≈ 2.04, t = 1
2 25
37 a) (i) 25i + 3j + k (ii) 35 d) Accelerating to the right then slowing down, turning
b) (i) 25x + 3y + z = 5 2 around, accelerating to the left, slowing down, turning
y+2
(ii) x 25 = = z 21 around again, then accelerating to the right.
25 3
c) (0, 1, 2) d) 35
38
(
a) x 2 2 = y 25 = z + 1 b) 1 , 10 ,2 8
3 3
) 3
4 a) y = 212x + 38
b) y = 1 x + 7
12 4
(
c) A′ 2 , ,2
4
3 3
5
) 13
3
d) 654
3
5 a) y = 2 x + 5
3
6 a) y = 1 x + 1
3
b) y = 2 3 x + 6
2
b) y = 24x + 9
39 a) 3x 2 4y + z = 6 dy 4 4
d 2y 2
1 1 7 a) = 2 sin (2x ) ; = 4 cos (2x )
dx dx 2
( ) ( )
b) (ii) r = 2 + t 4 c) 53.7°
11 13 b) π , 0 and 3π , 0
a) (3µ 2 2, µ, 9 2 2µ) 4 4
( ) ( )
40
4 3 8 a) (i) (0, 0) and (4, 0) (ii) 4 , 256 (iii) 8 , 128
b) (i) r = 0 + 1 3 27 3 27
b) y
23 22 10 ( 4
3 , 256
27 )
_ __› 3µ 2 6 9
(ii) PM
= µ 8
12 2 2µ 7
c) (i) µ = 3 (ii) 3 6 6
d) 2x 2 4y + z = 5 e) verify
41 a) (1, 21, 2)
5
( 8
3 , 128
27 )
4
b) 2x 2 y + z = 5
3
c) (3, 1, 3) and (1, 2, 2)
2
42 a) (i) = µ
1
2 21
(ii) r = 1 + t 22 1 0 (0, 0) 1 2 3 (4, 0) 4 5 6 x
1 1
21
b) 3x 2 2y + z = 5 2
3
2 21
c) r = 1 + t 22
1 21
1022
9 c) f ′′ (3.8) = 0 and f ′′ (3) = 1 > 0, f ′′ (4) = 2 2 < 0, 2 a) y = 1 x + 3 3 2 π
3 625 2 6
therefore graph of f changes concavity from up to down
b) y = 2x + 1
at x = 3.8 verifying that graph of f does have an inflexion
point at x = 3.8 c) y = 16x + 4 2 2π
dy d 2y 24a 3 a) x = π , x = 5π
10 = 2a 2 ; 2 =
dx (x + a) dx (x + a)3 6 6
b) Maximum at π , minimum at 5π
d n y (21) n !
n +1
= n! 6 6
11 n +1
or 4 (0,21) is an absolute maximum
dx n
(x 21) (12 x)n + 1
( ) ( )
12 a) Max. at (0, 2); inflexion pts at or (22, 1) and (2, 1) 5 a) Maximum at π , 5 ; minimum at 3π ,23
2 2
( ) ( )
b) (i) None (ii) none (iii) all x ∈
3π 7π
c) (i) lim g (x ) = 0 (ii) lim g (x ) = 0 b) Minimum at ,21 and ,21
x→∞ x→− ∞ π 4 4
d) y 6 x =
2
7 a) f ′ (x ) = e x 2 3x 2 ; f ′′ (x ) = e x 2 6x
2 b) x 3.73 or x 0.910 or x 20.459
c) Decreasing on (2, 20.459) and (0.910, 3.73);
increasing on (20.459, 0.910) and (3.73, )
d) x 20.459 (minimum); x 0.910 (maximum);
x 3.73 (minimum)
e) x 0.204 or x 2.83
f) Concave up on (2, 0.204) and (2.83, ); concave down
on (0.204, 2.83)
6 4 2 0 2 4 6 x
8 The two functions intersect for all x such that
13 d (c ⋅ f (x )) = d (c ) ⋅ f (x ) + c ⋅ d ( f (x )) cos x = 1, i.e. x = k ⋅ 2π, k ∈. The derivatives for the
dx dx dx two functions are y ′ = 2e 2x and y ′ = 2e 2x (cos x + sin x ) .
= 0 ⋅ f (x ) + c ⋅ ( f (x )) = c ⋅ ( f (x ))
d d The derivatives are equal whenever x = k ⋅ 2π, k ∈ .
dx dx Therefore, the functions are tangent at all of the intersection
( )
14 y = x 2 x 2 2 6 = 0 when x = 0 and x = ± 6 ;
points.
()
9 a) 8 m s22 b) 2.09 m s21
y 1 = 2 23 < 0, so y < 0 for 0 < x < 1 10 y = ex
2 16
11 a) f ′ (x ) = 2x ln 2
dx
dy
( )
= 4x x 2 2 3 = 0 when x = 0, x = ± 3; when
b) y = x ln 2 + 1
dy dy
x = 1, = 2 11 < 0 , so < 0 for 0 < x < 1 c) f ′ (x ) = 2x ln 2 ≠ 0 for any x
2 dx 2 dx
( )
( ) 12 a) (21,22e ) and 3, 63
2
d y
= 12 x 2 21 = 0 when x = 0, x = ±1; when e
( )
2
dx
d 2y d 2y b) (21,22e ) is a minimum; 3, 63 is a maximum
x = 1, = 29 < 0, so 2 < 0 for 0 < x < 1
e
3 2 dx c) (i) lim h (x ) = 0
2
dx
d y x→∞
3
= 24x > 0 for 0 < x < 1 (ii) as x → 2∞, h (x ) increases without bound
dx
d) Horizontal asymptote y=0
Exercise 15.2 e) y
4
1 a) y ′ = x 2e x + 2xe x b) y ′ = 8x ln 8
3
c) y ′ = e sec e
x 2
( ) x
d) y ′ = cos x + x sin x2 + 1 2
(1 + cos x ) 1 ( 3, )6
x x e3
e) y ′ = xe 2e f) y ′ = 2 tan 3 (2x ) sec (2x ) ( 3, 0) ( 3, 0)
x 2 0 x
2 1 1 2 3 4
() ()
x 1
g) y ′ = 1 ln 1 h) y ′ = cos x
4 4 2
i) y ′ = 2xe + e 221 j) y ′ = 212 cos (3x ) sin (sin (3x ))
x x
3
(
e x 21
) 4
( )
3 3
k) y ′ = 2 ln 2 2x l) y ′ = cos x 2 sin x2 5
(cos x 2 sin x ) (1, 2e) 6
1023
Answers
13 a) a = π , b = π, c = 3π dy 2
23 y = x + 1 24 = 3x
(n) 2 2 x +1
( )
2 dx 3
33 0 34 y = 1 x 2 1 + 3
8 ln 2 ln 2
Exercise 15.3 35 Verify
dy dy 22xy 2 y 2 36 x = 1
1 = 2x 2 = 2 3
dx y
dx x + 2xy e2
37 a) g ′ (x ) = 12 ln x , g x = 23 + 2 ln x
3
dy
= cos y or
2 dy
= 1 ′′ ( )
dx dx 1 + x x2 x3
2
dy 22x + 3y 2 y 2 3
dy x 2 y + y 3 b) g ′ (x ) = 0 only at x = e ; g ′′ (e ) = 2 13 < 0 , ∴ abs. max.
4 = 5 = e
dx 26xy + 3xy 2 2y 2
dx x 3 + xy 2 at x = e, max. value of g is 1
dy 22xy 2 2y 2 xy 2
dy y 21 e
6 = 7 = dy dy
dx 2x 2
+ 2xy + xy dx cos y 2x 38 = 2 1 39 = 21
4x 3 2 2xy 3
dx x + 2x + 2 dx x + 1
( )
dy dy 2y
8 = 9 = dy 6 dy
= tan21 x + 2 x e x tan x
21
dx 3x 2 y 2 + 4y 3 dx x + e
y
40 = 41
dx x x −9
4 dx x +1
dy x + 2
10 =
42 f ′ (x ) = 0; the graph of f (x ) is horizontal
dx y + 3
dy dy 2
43 Verify
11 = 2sin 2 (x + y ) or = 2 2x
12
dx
dy 18x 2 xy 2 y
=
dx x + 1
( )
44 y = π + 4 x + π 2 4
2 4
dx x + 2 xy 45 a) For 0 x < π, f ′ (x ) = 21 , therefore f (x ) is linear
13 y = 2 x + 4 ; y = 5 x 2 24
7
b) y = 2x + π
5 5 7 7 2
14 y = 22x + 4 ; y = x + 1 3 46 10 ≈ 3.16 m
2 2
2 47 a) _14 m s21, __
1
20 m s
21
15 y = 2 x + π ; y = x + π 2 4
π 2
2 π 2π
352 32 23 b) 2 _ 14 m s22, 2 ___
13
800 m s
22
16 y =2 x 2 ; y = x 2 5655
23 23 352 176 c) T
he particle initially is moving very fast to the right and
dy x
17 x2 + y2 = r2 ⇒ = 2 x ; at point (x1, y1 ) , m = 2 1 ; then gradually slows down while continuing to move to
dx y y1
centre of circle is (0, 0) ; slope of line through (x1, y1 ) the right.
y
x y
d) lim s (t ) = π m
and (0, 0) is 1 ; because 2 1 × 1 = 21, the tangent to the t →∞ 2
x y x1
1 1
circle at (x1, y1 ) and the line through (x1, y1 ) and (0, 0) are Exercise 15.4
perpendicular 1 a) 218.1 cm/min b) 26.79 cm/min
a) ( 7, 0) , (2 7, 0) ;
dy 22x 2 y 2 a) 0.298 cm/sec b) 0.439 cm/sec
18 = , at both points
dx x + 2y 3 a) 2π cm/hr b) 8π cm/hr
dy
= 22
dx 4 dθ = 3 ≈ 0.0882 radians/min 5 26.4 m/sec
dt 34
b) 7 ,22 7 and 2 7 , 2 7 6 2 ft/sec 7 69.6 km/hr
3 3 3 3 dy 12
8 = ≈ 3.79 9 0.01 m/sec
7 7 7 7
c) 2 ,2 and 22 ,
3 3 3 dt 10
3 10 30 mm 3/sec 11 45 km/hr
( )
19 0, 0 __
8 √ 3
dy d 2 y 236y 2 216x 2 12 ____ < 4.62 cm/sec 13 1.5 units/sec
20 = 2 4x , 2 = 3
dx 9y 81y 3
dx 14 222.2̄ m/sec = 800 km/hr
2
dy 22 y d y 2y 2 4 15 a) 115 degrees/sec b) 57 degrees/sec
21 = , =
dx (x + 3)2 dx 2 (x + 3)2 16 2485 km/hr
dy 21 d 2 y
22 a) = 4 , 2 = 4 7
dx dx Exercise 15.5
3x 3 2 9x 3
dy y d y 4y 1 2 by 2
b) = 2 , 2 = 2 1 2
dx 3x dx 9x 2 13 3 cm by 6 23 cm
1024
3 5
dy
2 b) = 2e x cos (2x ) + e x sin (2x )
dx
4 b) S = 4x 2 + 3000 c) 7.21 cm 3 14.4 cm 3 9.61 cm
x dy
c) = 2x ln x + 2x ln 3 + x 2 1
5 x = 5 2π ≈ 12.5 cm 6 x ≈ 3.62 m
dx x
( )
7 Longest ladder ≈ 7.02 m 8 d ≈ 2.64 km
9 8 units 2 10 6 nautical miles 11 y = 2 x 2 , P (23, 0) , Q 0,2 3
1 3
2 2 2
5
R 2 12 a) x = 3; sign of h′′ (x ) changes from negative (concave
11 h = R 2 , r = 2
down) to positive (concave up) at x = 3
12 Distance of point P from point X is 2ac 2
b) x = 1; h′ (x ) changes from positive (h increasing) to
r 2c
13 x ≈ 51.3 cm, maximum volume ≈ 403 cm
3
negative (h decreasing) at x = 1
13 y = 5 x + 11
Practice questions 7 7
1 y 14 h = 8 cm, r = 4 cm
15 Maximum area is 32 square units; dimensions are 4 by 8
16 a) E b) A c) C
17 y =2 x + 1 32
5 5
18 a) y = 4x 2 4
b) y = 2 1 x + 1
4 4
19 a) Absolute minimum at 1 ,2 1
0 x 2e
e
b) Inflexion point at 1 ,2 3 3
3 2e
e
20 a) (i) a = 16 (ii) a = 54
2 a) (i) a = 24 (ii) b = 2
(i) f ′ (x ) = −3x 2 2 4x + 8 b) f ′ (x ) = 2x 2 a2 = 0 ⇒ x = 3 a ;
x 2
(ii) 22 + 2 7 , 22 2 2 7 3 a
3 3 f ′′ (x ) = 2 + 3 ⇒ f ′′ = 4 > 0; hence, f is concave
2a
2
(iii) f (1) = 5 x
up at any critical point, so it cannot be a maximum
c) (i) y = 8x (ii) x = 22
3 a) (i) v (0) = 0
(ii) v (10) ≈ 51.3 21 y = 22 x + 4
3
( )
b) (i) a (t ) = 0.99e 20.15t
(ii) a (0) = 0.99 22
y = π + 2 x 2 π ; y =
2
π π
c) (i) 66
(ii) 0
22
2 8 π + 2 x + 2π + 4 + 4
( )
(iii) As object falls it approaches terminal velocity
( )
4 a) 2 2 ,2 149 is a minimum, (24,13) is a maximum
23 a) Maximum at 0, 1 , inflexion points at 21, _____
2π √
(
1
___
2eπ )
(
3 27
b) 2 , ( 7
)101 is an inflexion point
and 1, )
1
_____
___
2eπ
√
3 27 b) lim f (x ) = 0; y = 0 (x-axis) is a horizontal asymptote
5 a) (i) g ′ (x ) = 2 33x
e x→ ± ∞ y
c)
(ii) e 3x > 0 for all x, hence 2 33x < 0 for all ( 0, )
1
2π
e 0.4
x ; therefore, f (x ) is decreasing for all x
b) (i) e + 2
(ii) g ′ (− 13 ) = 23e 0.3
c) y = 23ex + 2
(
1,
1
2eπ ) (1, 1
2eπ )
6 b) f ′ (3) = 0 and f ′′ (3) > 0 ⇒ stationary point at x = 3 and 0.2
graph of f is concave up at x = 3, so f (3) is a minimum
c) (4, 0) 0.1
7 a) 2 4
( 2x + 3)
3
dy 5 dy
= 25 x 2 (ln 2) 2 4x ln 2 + 2
2
25 x = 20 3 ≈ 34.6 km/hr 26 = or 44 a) (ii) f ′′ (x ) =
dx 6 dx 6
2x
dy dy
27 = 4 22x 2 28 = 2x ln x + x
dx 2x 2 2x + 1 dx b) (i) x = 2
ln 2
( )
29 sin x = 1 , sin x = 21 30 2 3 (ii) f ′′ 2 < 0; therefore, a maximum
2 4 ln 2
b) x = 1 + 17
31 a) f ′ (x ) = 2 2 + 2 ≈ 4.93, x = 2 2 2 ≈ 0.845
2x 21 4 c) x =
π ln 2 ln 2
32 x ≈ 20.586 33 c = 4 +
4 2
45 a) f ′ (t ) = 6 sec t tan t + 5 or f ′ (t ) = 6 sin3 t + 5
34 a) f ′ (x ) = π cos (πx ) e 1+ sin πx
b) x = 2n +1 cos t
n 2 b) (i) 3 + 5π (ii) 5
35 a) 1.5 dy 4
46 a) y = 21 b) =
dx 5
dy
1 47 a) = 3e sin (πx ) + πe cos (πx ) b) x ≈ 0.743
3x 3x
maximum dx
maximum
0.5 48 240 km/hr (
49 b) 2 ln b, a
c
1
) 2b
zero
50 a) p = 2 b) 2 4
7
1 0.5 0 0.5 1
51 x ≈ 0.460 52 1 radians/sec
10
minimum d 2y 24
minimum zero 53 = 54 y = 2 5 x + 13
dx 2 (2x 21)2 4 2
( )
1
55 a) f ′′ (x ) = 10 cos 5x 2 π
2
( )
1.5
b) f (x ) = 2 cos 5x 2 π + 7 2
2 5 2 5
b) (i) f ′ (x ) = 7x 2 3 1 , domain: 21.4 x 1.4, x ≠ ±1
domain: 21.4 < x < 1.4, x ≠ ±1
56 5 57 (20.803,22.08)
( )
3 x 2 21 3 4
58 a) k = ln 2 b) 510 bacteria per minute
(ii) Maximum at x = 3 , minimum at x = 2 3 20
7 7
c) x ≈ 1.1339
36 a = 24, b = 18
dy
37 a) = sec 2 x 2 8 cos x b) cos x = 1
dx 2 59 f (x ) = 2 1 x 3 + 12 x 2 2 3x + 2
5 5
38 a) y = 24x 2 8 b) (22, 0)
60 a) f ′ (x ) = 212 cos 2 (4x + 1) sin (4x + 1)
39 Proof
40 y = 2x + 2 b) x = π 2 2 , x = 3π 2 2 , x = π 21
8 8 4
41
a) (i) f ′ (x ) =
( )
2 x 2 21
dy 3x 2 (ln 3) 3
2 x+y
=
( ) 61
(ln 3) 3x + y 2 3
2
x2 + x + 1 dx
( )
(ii) A 1, __
3 ( ( ) )
1 , B(21, 3) or A(21, 3), B 1, __
1
3
62 a) f ′ (x ) = 3
3x + 1
b) y = 2 7 x + 14 + ln 7
3 3
b) (i) y 63 Verify
dy 12e
64 = 65 b) b = 6
dx e
dy
66 a) = 2 2k b) k = 2 67 3
dx 2k 21 2
1 0 1 x 68 a) 5 5 x + 4 + 5 (2 2 x ) minutes
2
c) (i) x = 1 (ii) 30 minutes
2
(iii) d T2 > 0 for x = 1; therefore, it’s a minimum
dx
2 (
69 a) P 2 1 ,2 1
)
2 2e
b) f ′′ (x ) = 4x + 4 = 0 at x = 21, and f ′′ (x ) changes sign at
(ii) x ≈ 20.347, 1.53, 1.88 x = 21
c) (i) Range of f : 1 , 3 (ii) range of f f : 1 , 7 c) (i) Concave up for x > 21
3
1 4 5 3 13 (ii) Concave down for x < 21
42 cm/s 43 y = x 2
2π 3 3
1026
_7 __
d) 5 u 5 2 u 4 1 c
5 ___
4x √x
6 _____
__
2 3 √x 1 c
y 7 3
7 23 cos 1 4 sin 1 c 8 t 3 1 2 cos t 1 c
__ __
4x 2√ 10x √x
x ______
9 ______ 2
1 c
10 3 sin 2 2 tan 1 c
5 3
11 __ 1
e 3t 2 1 1 c 12 2 ln|t| 1 c
3
13 __ 1 ln (3t 2 1 5) 1 c 14 e sin 1 c
1 x 6
1 2 (0, 0)
(2x 1 3)3 5x 4 1 ___
2x 3 1 cx 1 k
Q 15 ________
1 c
16 2 ___
6 4 3
P
17 2 __ x 5 1 __ x 4 1 __
x 2 1 2x 2 ___
11 4t
___
3
18 1 sin t 1 ct 1 k
5 4 2 20 3
e) Show true for n = 1: 19 3x 4 2 4x 2 1 7x 1 3 20 2 sin 1 __ 1 cos 2 1 c
2
f ' (x) = e 2x + 2xe 2x ( )
6
3x 2 + 7 1
= e 2x (1 + 2x) = (2x + 20) e 2x 21 +c 22 2 +c
36 ( )
3
Assume true for n = k, i.e. f (k) (x) 18 3x 2 + 5
( ) (2 x + 3)6 + c
5
= (2k x + k × 2k − 1) e 2x, k > 1 84 5x 3 + 2
d ( f (k) (x)) 23 +c 24
Consider n = k + 1, i.e. an attempt to find ___ 75 6
dx
( ) (2x + 3)6 + c
(k + 1) k 2x 2x k k − 1 3
f (x) = 2 e + 2e (2 x + k × 2 ) 2t 3 2 7
= (2k + 2(2k x + k × 2k −1)) e 2x 25 +c 26 2
9 18x 6
= (2 × 2k x + 2k + k × 2 × 2k − 1) e 2x cos (7x 2 3)
= (2k + 1 x + 2k + k × 2k) e 2x 27 2 +c 28 2 1 ln (cos (2θ 21) + 3) + c
7 2
= (2k + 1 x + (k + 1)2k) e 2x
P(n) is true for k ⇒ P(n) is true for k + 1, and since true for n = 29 (
1 tan 5θ 2 2 + c
) 30 1 sin (πx + 3) + c
+ 5 π
1, result proved by mathematical induction n 1 sec 2t + c 32 1 e x + 1 + c
2
31
70 72 arccos 8 cm 2 2
π 13
33 1 e 2t t + c 34 2 (ln θ) + c
3
71 a) y 3 3
36 2 1 3 25t 2 + c ( )
3
35 ln ln 2z + c
15
37 1 tan θ3 + c 38 2cos t + c
3
39 1 tan 6 2t + c 40 2 ln ( x + 2) + c
x 12
0 1 sec 5 2t + c 1
41 42 2 ln x 2 + 6x + 7 + c
10 3 3
43 2 k 4 a 2 2a 4 x 4 + c = 2 k 3 12a 2x 4 + c
2a 2a
2
44 ( 5
)
3x 2 2 x 2 2 x 21 + c 45 2 1 cot πt + c
π
b)
y
46 2 2 (1 + cos θ) + c
3
3
(
47 2 15t 3 2 3t 2 2 4t 2 8 12t + c
105
)
15
( 2
)
48 1 3r 2 + 2r − 13 2r 21 + c 49 1 ln e x + e −x + c ( 2 2
)
x
0 (
50 2 3t 2 + 20t + 230 t 25 + c
15
)
Exercise 16.2
1 2 __ 1 e2x + c
3
2 2e 2x (x2 + 2x + 2) + c
3
3 __ 2 2 1
x cos 3x 2 ___ sin 3x + __
x 2 sin 3x + c
9 27 3
Chapter 16 4 __ 1 (2 cos ax 2 a2x 2cos ax + 2ax sin ax) + c
a 3
Exercise 16.1 5 sin x(ln(sin x)21) + c 1 x2(ln x 2 2 1)+ c
6 __
2 2
x 1 2x 1 c
1 __ 2 t 3 2 t 21 t 1 c 7 __1 x3ln x 2 __ 1 x3 + c 8 2ex + x2ex 2 2xex 2 __ 1 x3 + c
2 3 9 3
4
x 2 ___ 3 2
3 __ x 1 c 2t 1 __
4 ___ t 2 3t 1 c 9 __ 12 (cos πx + πx sin πx) + c 10 ___ 3 cos 2t e3t + ___
2 e3t sin 2t + c
3 14 3 2 π 13 13
1027
Answers
______
11 √ 1 2 x2 + x arcsin x + c 12 ex(x3 2 3x2 + 6x 2 6) + c
()
1
13 2 e (cos 2x + sin 2x) + c
22x 2
( )
2
16 ln x + 1 2 2x + x ln x + x + c
kx
2
2
( )
41 1 arcsin (e x ) + e x 1 − e 2x + c 42 ln 1 e x + 1 e 2x + 9 + c
e (k sin x − cos x) + c 3 3
17 18 x tan x + ln cosx 43 2 x (ln x 2 2) + c
k2 + 1
19 2 sin 3 x 1 arctan x(1 + x2) 2 __
20 __ 1 x + c 2
3 2 2 44 12 ln (x + 2) + 8 + x 2 4x + c
x+2 2
21 2 x (ln x 2 2) + c 22 t tan t + ln cos x + c
2
23 Verification
2
( )
45 1 ln x 2 + 9 + c 1; x = 3 tan θ yields ln x + 9 + c 2 ; they
3
24 2x cos x + 4x sin x + 12x 2 cos x 2 24x sin x 2 24 cos x + c
4 3
differ by a constant
25 x 5 sin x + 5x 4 cos x 2 20x 3 sin x 2 60x 2 cos x + 120x sin x
+ 120 cos x + c 46 x 2 3 arctan (__ x ) + c1; x = 3 tan yields
3
(
26 e x x 4 2 4x 3 + 12x 2 2 24x + 24 + c
) 3(tan 2 ) + c2 = 3 (__ x 2 arctan __
3
x ) + c2
3
27 Proof 28 Proof 29 Proof
30 Proof 31 Proof
Exercise 16.4
1 24 2 40
Exercise 16.3 24
5 3 ___
3 4 0
1 1 cos 5t 2 1 cos 3t 2 1 cos t ; c cos t − cos t + c 25 __
6 80 48 8 5 3 176 √7 2 44
4 5 __________
6 0
2 cos t − cos t + c 5
6 4 7 2 8 2268
64
4
3 cos 3θ + c 9 ___
3
10 2
12
() () () ( )
__
1 3 1 2 5 1 1 7 1 11 ln ___ 11
44 2 8 √3
12 ___
4 3 cos t − 5 cos t + 7 cos t + c 3 3__
13 3 14 √p 1 1
5 sec x + cos x + c 6 1 tan 6 3x + c 15 a) 6 b) 6 c) 12 16 1
18
(
7 1 3 tan 4 θ 2 + 2 tan 6 θ 2 + c ) 17 4
π
18 0
24 __
19 π
20 __
8 2 sec 5 t − 2 sec 3 t + c 2 6
5 3 π
21 __ π
22 __
(
9 1 tan 3 5t 2 3 tan 5t + 15t + c
15
) 3
23 14 17 + 2 24 __
8
1
π
10 tant 2 sec t + c 11 csc t 2 cot t + c 3
12 2ln 1 − sin t + c 13 22x 2 3 ln sin x + cos x + c 25 In(2) 26 16 2 25 5
14 arctan (sec θ) + c 15 1 (arctan t ) + c
2 27 14 2 10 3
28 __
2
2
16 ln arctan t + c 17 arcsin (ln x ) + c π
29 π 2 2 3 2 1
3
30 __
( )
18 2cos x sin 2 x + 2 + c 19 2 (cos 2 x cos x 2 5 cos x ) + c
27 12 6
20
3
( 2cos
3
x
)
2 sin x + 4 + c
2
5
31 2 1 ln 37
2
( )
52
21
( sin (sin t )
3
)
cos 2 (sin t ) + 2 + c
( 4 2 4 ) (
32 2arctan 15 2 7 or 1 arcsin 1 − arcsin 3
4 () ( ))
22 ln sin θ + 2 sin θ + c
23 t sec t 2 ln sec t + tan t + c
2
33 __ 34 0
3
24 2ln (2 2 sin x ) + c
25 1 ln cos (e −2x ) + c 35 24 36 __ π
2 6
26 2 ln sec x + tan x + c 27 1 tan x + c π 3 2 3 3 arctan 3
1 4 3 2
2 37 arctan
9
38
(
28 1 arcsin 3x + 3x 12 9x 2 + c
6 x
) 6 18
1
39 __ 40 e 21
29 +c 6 2
4 x + 4
2
41 1 + e 42 2 cos(1) + 2
30 2 ln t + t 2 + 4 + 1 t t 2 + 4 + c 2
2 31
43 ___ 44 __ 2
π
( ) ( )
5
31 arctan e + c 3 1 t
32 1 arcsin 2 x + c 8
2 2 2 3 45 12 2 4 3 46 e 21
π
8
1 3 1 8e
33 ln x + 9x + 4 + c 34 ln 1 + sin 2x + sin x + c
2
47 π 48 sin x
3 2 2
6 ln 3 x
35 2 4 2 x + c 2
( )
36 1 ln x 2 + 16 + c
2
1028
28 25.36 29 m = 0.973 37
30 ___
12
sin t 2
49 2 t 50 22x sin 2x
x Exercise 16.7 __
2
51 2x sin 2x 52 cos t
127p
1 _____
64 √2 p
2 ______
1 + t
2
x 27 15
53 b 2a4 54 2csc θ 2 sec θ 70p
3 ____ 4 6p
5 + x 3
1
5 9p__ 6 2p
55 1 3 e x + 3x
( )
2
56 Yes √
3
7 ___
1 1 p 512p
8 _____
( )
( )
4x
4
2 15
1 3k +2 2 e 3 21
57 a) ln b) k = 9 Approx. 5.937p 32p
10 ____
3 2 3 3
58 Proof 59 2(12 x )
k +1
k (
1 + 12 x
+1 k + 2 ) 11 π ( 3 21)
12 23π
210
60 a) 0 b) 47 13 288π − 160π 5 14 64 π
3 15
( )
15 47
c) 1 1 16 1778 π
47 15 π 2 3
61 Proof 2 4 5
17 252 π 18 1419π
5
Exercise 16.5 _ _ _ _
19 9 π 20 a) 88 π b) 7 π
12 ((1 + 2 √2 ) ln |x 2 √2 | 1 (1 – 2 √2 ) ln|x 1 √2 | )
1 __ 8 15 6
2 3 ln|x 2 2| 2 2 ln|x| 1 c 21 40π 22 9π (2 2 2 )
3 12 ln|x 2 1 4x 1 3| 1 c
_
23 32 π 24 4 π (121 33 2 25 15 )
2 ln|x 1 1| 1 6 ln|x| 2 ____ 9 1 c
x 1 1 15 5
( ) ( )
4
5 ln|x 1 3| 1 3 ln|x 1 2| 2 2 ln|x| 1 c 25 2π ln 2 2 1 26 2π 11 11 2 2 2
4 3 3
ln|x 1 1| 1 3 ln|x| 1 _ 1x 1 c
( )
6
7 2 ln|x 1 2| 1 ln|x 2 1| 1 c 27 28 π ( 34 2 7 ) 28 π 1 2π 2 π + 2
3 2
8 3 ln|2x 2 1|
________ 2 2 ln|x 1 1| 1 c
29 284 π 30 2π
2 2 1 c 3
9 3 ln|x 1 2| 1 ____ x 1 2
31 256 π
10 ln|x 2 2| 2 4 ln|x 1 1| 1 3 ln|x| 1 _ 6x 1 c 15
11 2
2 ln|x 1 1| 1 2 ln|x| 1 c
_ __
12 ___
3
_
( )
3 arctan ___
√ 3x _______
√
3 2 ln|x 3 1 3|
2
2 ln|x|
1 ____ 3
1 c Exercise 16.8
1 __
70
3 m, 65 m 2 8.5 m to the left, 8.5
m
( )
__
23 arctan ___
√ ln|x 2 1 6| 4 2 m, 2 √2 m
13 ___ x_ 2 _______
6
ln|x|
1 ____ 3
1 c 3 1 m, 1 m
√
6 5 18 m, 28.67 m 6 __
p 4 m
4 m, __
p
_ _
14
√
___
( ) √2 x
22 arctan 4 2 __
___ 3 ln|x 2 1 8| 1 _
16 38 ln|x| 1 c 7 3t, 6 m, 6 m 2
8 t 2 4t 1 3, 0, 2.67 m
9 1 2 cos t, ___
2 ( ) (
3p 1 1 m, ___
3p 1 1 m
2 )
ln|x 2 5|
15 ______
3
2 ln|x 1 1|
1________ 2 ln|x| 1 c _____
3 10 4 2 2 √t 1 1
, 2.43 m, 2.91 m
1029
Answers
∫
ln 2
Exercise 16.9 1 2
c) v 5 p (1 1 e 2x)2 dx
0
x
1 y = ±10e x 2 y = ±e
( )
4
2
12 π 2 a5 + 2 a3
3 y = 2 2 4 y = 1
22x 32x 15 3
( )
( ) ( )
2 1 5
2
3
5 y = ln e
12ex
(
6 y = ln e x 2C ) 13 4
5 2
x +1 22 1 x +1 +c
2
3 2
(x + 1)2 2 1
7 y 3 =
3
2 2
8 y = 1
ln x + 1 + 1
14 a = 2 ___ 56
27
15 π e 2k 21
2
( )
16 k = 2 17 1800 m
9 2y 3 + 6y = 3x 2 + 6x + 72 10 y 2 = e x 21
2
x2 x2 18 2a by 2 a 2
11 arctan y = 2 + c 12 y + ln y = 2 2 x 11 3
1
y 21 19 a) ln x + 12k b) x > 1
= e (x − 1) + c
2
13 x + ln 14 2k e
x + Ce x + 1 y + 1 ( )
c) (ii) e k , 0
d) e
4
15 ( y + 1) ln y + 1 + 1 = (y + 1) (ln ln x ) + c
e) y = x 2e k f) Verify
16 1 + 2y 2 = c tan 4 x 17 arcsin y = 1 2 1 2 x 2 g) Common ratio = e
2
20 x 2 2 4y 2 = 4 21 v = v 02 + 4k
18 y = ln ln
(
e ex + 1 ) x3
19 y + ln y = 3 2 x 2 5
m
1 + e 22 a) y
2
4 ( )
20 cos y = 2 e x
+ 1 21 y = x e x 21
2
y = g(x)
22 2 ln y 2 y = e 2 2 23 y + ln sec y = 1 x 3 + x + c
2
2 x
3
Q R
( )
y + 1 = 3e (t 21) + c 25 e ( y + 1) = 2 13 sin 3 θ + c
2 3 t
24 2y
1
P
26 e 3y + 3y 2 = 3 (cos x + x sin x ) 2 2 A
y = f(x)
27 y = e x 2 x 2 + 2
0 p x
28 b) C = 78; m = 1 ln 8 ; 45.3 minutes 1
15 13
b) Proof c) 0.6937
( ( ))
p
∫ e 2 e 21 dx ≈ 0.467
2x 2 2x 2
d)
Practice questions 0
23 a) Verify
1 a) p 5 3 b) 3 square units
b) 2π ; 4π ; 6π
∫
ln 2
_x 2
2 a) (0, 1) b) V 5 ( e 2 ) dx 9 9 9
c) nπ (n + 1)
0
3 a 5 e 2
4 x
__
a) y 5 e 9
24 a) t = 0, 3, or 6
()
5 a) (i) 400 m (ii) v 5 100 2 8t, 60 m/s
b) (i) ∫ t sin π tdt
6
(iii) 8 s (iv) 1344 m (ii) 11.5 m
0 3
b) Distance needed 625
c) 2p cos x 2 __ x 2 1 c; 0.944 25 a) 0.435 b) 22t 2
6
7
b) 2.31
ln 3
2
2+t2 ( )
dy 2
8 a) (ii) (1.57, 0); (1.1, 0.55); (0, 0), (2, 21.66) 26 a) = 2x 2 + 2 1 + x 2
∫
__
p
p2
p2 22 dx 1+ x
b) x 5 __ c) (ii) x 2 cos x dx d) ___ b) Verify c) k = 0.918
2 0 2
9 a) 2p 27 6 m
b) Range: {y | 20.4 , y , 0.4} __
28 0.852
2 √3
c) (i) 23 sin 3 x 1 2 sin x (iii) ____
29 a) Verify
p 9
d) __ a) (i) A = 78; k = 1 ln 48 (ii) 45.3
2 15 78
( )
e) (i) _13 sin 3 x 1 c (ii) _ 13
__ 30 y = tan ln x
√
7 2
f) arccos ___ < 0.491
3
31
( x + 2)
2
2 6 (x + 2) + 12 ln x + 2 + 8 + c
∫
p
10 c) 3.696 72 d) (p 1 x cos x)dx 2 x+2
0
e) p 2 2 2 < 7.869 60
ln 5 < 0.805
11 a) (i) 10x 1 1 2 e 2x (ii) ___
2
21 ln (x 2 1)
________
b) (i) f (x) 5
2
1030
32 a) y 3 a) 0.26 b) 0.37 c) 0.77
6 d) 16.29 e) 8.126 f) 4.125; 2.013 25
g) E(aX + b) = aE(X) + b; V(aX + b) = a2V(X)
4 a) 0.969 b) 0.163 c) 3.5
4 5 k = __1
30
A x 12 14 16 18
2 P(X = x) 6k 7k 8k 9k
g(x)
6 a) k = __1
10 b) __
37
60 c) __
19
30
7
11 ; SD = __
d) E(X) = 16, SD = 7 e) E(Y) = ___
4 3 2 1 0 1 2 3 4 5 6 x 5 5
7 a) __
1
50
2 b) 0.35
0.30
4 0.25
0.20
6 0.15
f(x)
0.10
b) (i) x = 23; 0.05
(ii) x 2 int = e 2 2 3; y 2 int = ln 3 2 2
0.0
c) 21.34; 3.05 0 1 2 3 4
d) (ii) ∫
0
3.05
(4 2 (12 x ) 2 (ln (x + 3) 2 2)) dx
2
c) __
17
25
d) µ = 1.2; var = 0.9
(iii) 10.6
8 a) P(x = 18) = 0.2, P(x = 19) = 0.1, symmetric distribution.
e) 4.63 b) µ = 17, SD = 1.095
33 a) Verify 9 a) µ = 1.9, SD = 1.34
b) ln x 2 1 ln x 2 + 1 + c
2
( ) b) Between 0 and 5
θ
10 k = 0.667, E(X) = 5.44
c) y = 2e
e +1
2θ 11 a) k = 0.3 or 0.7
b) for k = 0.3: E(X) = 2.18; for k = 0.7: E(X) = 1.78
12 a)
Chapter 17 y 0 1 2 3
Exercise 17.1 P(Y = y) __
1 _2 _4 __
278
27 9 9
1 a) Discrete b) Continuous c) Continuous
d) Discrete e) Continuous f) Continuous b) 2
g) Discrete h) Continuous i) Continuous 13 a) k = __1
10 b) _ 12
j) Discrete k) Continuous l) Continuous 14 a) <See table below>
m) Discrete b) 0.85 c) 0.15 d) 48.87
2 a) 0.4 e) 2.057 f) 0.77
b) 0.5 x 45 46 47 48 49 50 51 52 53 54 55
CDF 0.05 0.13 0.25 0.4 0.65 0.85 0.9 0.94 0.97 0.99 1
0.4
15 a)
0.3 x 0 1 2 3 4 5 6
CDF 0.08 0.23 0.45 0.72 0.92 0.97 1
0.2
b) 0.72 c) 0.97
0.1 d) 2.63 e) 1.44
16 a) 0.90 b) 0.09 c) 0.009
0.0 d) Unacceptable, acceptable e) p(x) = (0.1x 2 1) 3 0.90
0 1 2 3 4 5
17 a) 0 b) 0.81 c) 0.162
c) 1.85, 1.19 e) 2.85, 1.19 d) Either acceptable or unacceptable, acceptable
f) E(X + b) = E(X) + b; V(X + b) = V(X) e) (x 2 1) (0.1x 2 2) 3 0.902, x > 1
18 n = 30
19 a) (i) _19 (ii) __1
81
b) (i) ___
73
648 (ii) ___
575
1296
1031
Answers
c) (ii) b)
x 1 2 3 4 5 6 Number of List the Write the Explain it, if Find the
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Exercise 17.4 16 a) __ 5 b) 0.1944 c) 0.1941 d) 0.6207
9
3
1 a) k = 2 __ b) 0.3125 c) 0.6875 17 b) 3, 3.1, 3.3 c) 0.475 d) 1 e) 0.64, no
2 10 15
___ ___
18 a) , c) 0.0803 d) 0.891
d) 0.375, 0.3473, 0.2437 3 4
1
2 a) __ 1
b) __ 1
c) __
6 8 2 e) (i) 0.987 (ii) 0.9999 (iii) 0.9996
7
__
d) , 0.697, 0.533 54
19 ___ 20 1.08
9 11
3 a) k = 2 b) 0.766 c) 0.234
d) 0.754, 0.765, 0.3127 Exercise 17.5
Note: some answers are rounded.
6
4 a) ___ 133
b) ___ 19
c) ___ 1 a) 0.5 b) 0.499 571 c) 0.158 655
37 148 74
50
___ d) 0.682 690 e) 0.022 750 f) 0
d) , 1.5, 0.528
37 2 a) 0.769 86 b) 0.161 514 c) 0.656 947
5 a) d) 0.999 944
y 3 a) 0.008 634 b) 0.982 732
0.8 4 1.28
0.6 5 1.96
0.4 6 a) 0.066 807 b) 0.682 69 c) 678.16
d) 134.898
0.2 7 a) 1.8% b) 509.975 c) 5.71
0 8 a) 0.9696 b) 0.546 746
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 x
9 a) 1 day b) 29 days c) 112 days
3
b) ___ c) 113 , 1.89, 0.757
___ d) 0.983 10 1.56 11 18.95
29 58 12 30.81 13 100.28
6 a) 24.7 hours b) 0.514 c) 0.264 14 29.95
7 a) 50 hours b) 50 hours c) 22.4 hours 15 µ = 21.037, = 4.252
d) 0.104 e) (i) 0.010 82 (ii) 0.9892 16 µ = 18.988, = 0.615
8 a) 17 µ = 121.936, = 34.39
y 18 a) µ = 6.966, = 0.324 b) 0.252
1.5 19 a) 0.655 422 b) 0.008 198 c) 82 bottles
20 a) 0.227 319 b) 0.55% c) 29.678
1.0
d) 229.182
21 a) Not likely: chance is 0.14% b) 15.87%
c) 68.27% d) 5396 e) 43 785
0.5 22 a) 6.817 b) 3.4315
c) µ = 64.14, = 7.545
23 7.3% 24 216.06 25 15.31
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 x 26 a) µ = 111.89, = 17.9 27 0.919
28 a) (i) = 1.355 (ii) µ = 110.37
7
b) __ c) 0.694 d) 134 barrels b) A = 108.63; B = 112.11
3
9 7 , 0.916
b) __
2 Practice questions
10 6 ; b = 5
b) a = ___ c) 1.25 1 a) 34.5% b) 0.416 c) 3325
125
2 a) (i) 0.393 (ii) 0.656 b) 50
11 1
a) k = ______ 3 a) 0.1 b) 10 d) 0.739
(b 2 a)
(a + b) (a 2 b)2
b) mean = median = ______ ; variance = _______
35
4 a) ___ 7
b) ___ 91
c) ___
2 12 128 32 128
12 a) (i) 0.378 (ii) 1.752 (iii) 1.892 5 a) a = 20.455, b = 0.682
b) 0.955 b) (i) 0.675
(ii) 0.428
13 a) __ 1
8 c) (ii) t = 62.6
0 0x <5 6 a) µ = 50 2 10(0.522 44) 44.8
b) HI: the mean speed has been affected by the campaign.
3 (x 2 7)2
b) f (x) = 5x 7 c) One-tailed test, as we are interested in a decrease in the
8
mean only (not also an increase).
0 x >7
7 a) 70.1% b) 0.002 26 c) p-value = 5.48%
c) 5.4126 d) 0.15 8 a) 0.0808 c) µ = 25.5, = 0.255 d) 12 500
14 a) k = 3 b) __ 4 c) 0.8409 9 a) (i) 0.345 (ii) 0.115 (iii) 0.540
5 b) 0.119 c) 737
15 b) 0.0183 c) π d) 0.8326 e) 0.641 f) 0.0769 10 a) 15.9% b) 210
2
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Answers
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