edexcel maths c1
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Edexcel Maths C1 Student BookTRANSCRIPT
Chapter 1 AnswersExercise 1A1 3 5 7 9 11 13 15 17 7x y 8m n 7p 6 x2 2 x2 6 x 8 6x2 12x 10 8x2 3x 13 3x2 14x 19 a 4 b 14c 20 6x 2 4 6 8 10 12 14 16 18 10t 2r 3a 2ac 4ab 2m2n 3mn2 9 x2 2 x 1 10c2d 8cd2 a2b 2a 8x2 9x 13 9d2 2c 13 r2
Exercise 1F1 a x5 d x3 1 g 3x 2 a 5 1 e 3 125 i 64 b x 2 e x5 h 5x b 9 1 f 125 j 9 4 c f i c g k
x412x0 6x 1 3 15 6
12 d h l1 16
2
664 49
Exercise 1G1 4 7 10 13 2 7 4 2 3 12 7 23 5 2 5 8 11 14 6 2 3 10 6 5 3 7 2 3 6 9 12 15 5 2 3 7 2 9 5 19 3
Exercise 1B1 4 7 10 13 16
x73x 2 5 x8 2p 7 27x8 32y6
2 5 8 11 14 17
6 x5 k5 p2 6a 9 24x11 4a6
3 6 9 12 15 18
2p2
y102a3 3a2b 2 63a12 6a12
Exercise 1H1 5 5 2 21 2
Exercise 1C1 3 5 7 9 11 13 15 17 19 9x 18 12y 9y2 3 x2 5 x 4 x2 5 x 10x2 8x 4x 1 3 x3 2 x2 5 x 10y2 14y3 6y4 11x 6 2x2 26x 2 4 6 8 10 12 14 16 18 20
2 4 6 8 101 4 1 3
x2 9 x xy 5x 20x2 5x 15y 6y3 3 x3 5 x2 2x 4 14y2 35y3 21y4 4x 10 7 x2 3 x 7 9x3 23x2
3 5 7 9 11 13
11 11 5 5
Exercise 1D1 3 5 7 9 11 13 15 17 19 21 23 4(x 2) 5(4x 3) 4(x2 5) x(x 7) x(3x 1) 5y(2y 1) x(x 2) 4x(x 3) 3xy(3y 4x) 5x(x 5y) 5y(3 4z2) xy(y x) 2 4 6 8 10 12 14 16 18 20 22 24 6(x 4) 2(x2 2) 6x(x 3) 2x(x 2) 2x(3x 1) 7x(5x 4) y(3y 2) 5y(y 4) 2ab(3 b) 4xy(3x 2y) 6(2x2 5) 4y(3y x) 15 17 19 21
13 13 1 3 2 3 7 2 5 3 2 5(2 5) 1 11(3 11) 2 14 187 3 1
2 1
5 5 2)(4 11 14) 5)
12 3 14 (3
16 5(4 18 5
21 2 35 1189 20 6
Mixed exercise 1I1 2 3 a a c a c a c e a a a a c
y8
b 6 x7
Exercise 1E1 3 5 7 9 11 13 15 17 19 21 23
x(x (x (x (x (x (2x (5x (2x (x (2x
4) 8)(x 3) 8)(x 5) 2)(x 3) 5)(x 2) 1)(x 2) 1)(x 3) 3)(x 5) 2)(x 2) 5)(2x 5) 4(3x 1)(3x 1) 2(3x 2)(x 1)
2 4 6 8 10 12 14 16 18 20 22 24
2x(x 3) (x 6)(x 2) (x 6)(x 2) (x 6)(x 4) (x 5)(x 4) (3x 2)(x 4) 2(3x 2)(x 2) 2(x2 3)(x2 4) (x 7)(x 7) (3x 5y)(3x 5y) 2(x 5)(x 5) 3(5x 1)(x 3)
4
5 6 7 8
15y 12 16x2 13x x(3x 4) x(x y y2) (x 1)(x 2) (x 7)(x 5) (5x 2)(x 3) b 2 3 x6 4 b 3375 9 4913 7 7 3 3 3 3 6 b 4 5
c b d b d b d f c
32x d 12b9 2 3 15x 25x 10x4 9 x3 3 x2 4 x 2y(2y 5) 2xy(4y 5x) 3x(x 2) (2x 3)(x 1) (1 x)(6 x) 1 6 x2 d 1x 3 2
b d
2 30
1 851 7
Chapter 2 AnswersExercise 2A1 5
y
y14 12 10 8 6 4 2 4 3 2 10 2 4 1 2 3 4 x 4 3 1 2
28 24 20 16 12 8 4 10 1 2 3 4 x
x
x
0
2
y22 18 14 10 6 2 4 3 0 2 10 1 2 3 4 x4 3 11 2 2 10 10 1 2 3 4 x 10 20
6
y30
x
x3
y8 6 4 2 4 3 030
2
10
1
2
3
4 x
7
y40
x
4
y4 3 2 10 4 8 12 164 33 4
20
1
2
3
4 x10
2
10 10
1
2
3
4 x
x
0
x
Chapter 2 Answers8 5 x 7 x 9 x 3 2 1 8 3 2 29 2 129 8 39 2 6 x 1 3 2 2
y20
8 No real roots 10 x 4 5 26 5
15
10
Exercise 2E5
1 2
3 2 3 2 3 5 2 5 1 3 2
5
, ,
0.38 or
2.62
17 3, 33
0.56 or 3.56 1.27 or 4.73 0.37
4
3
2
10 5
1
2
3
4 x
3 4
, 5.37 or ,
x9
1
y80 70 60 50 40 30 20 10 4 31 2
5 6
31
3.52 or 0.19
, 1.21 or 0.21 2 9 53 7 , 0.12 or 1.16 14 2 19 8 , 0.47 or 1.27 5 1 9 2 or 4 1 78 10 , 0.71 or 0.89 111 2 3 4 x
2
10
x
Exercise 2F1 a
y(0, 2)
Exercise 2B1 3 5 7 9 11 13 15 17 19
x x x x x x x x x x
21 x 23 x 25 x
0 or x 0 or x 1 or x 5 or x 3 or x 6 or x 1 2 or x 2 3 or x 1 or x 3 13 or x 5 3 1 11 3 1 2 or x
4 2 2 2 5 1 33 2
2 1
2 4 6 8 10 12 14 16 18 20
x x x x x x x x x x
0 or x 25 0 or x 6 1 or x 4 3 or x 2 4 or x 5 6 or x 2 1 3 3 or x 2 3 5 2 or x 2 3 or x 0 2 or x 2 3 137 6 11 6
( 2, 0)
( 1, 0) 0
x
b
y
22 x 24 x7 3
(0, 10)
1 or x 0 or x
26 x
0
x
Exercise 2C1 3 5 7 9 11 (x 2)2 4 (x 8)2 64 (x 7)2 49 3(x 4)2 48 5(x 2)2 20 3(x 3)2 247 2 2 4 6 8 10 12 (x 3)2 9 (x 1)2 1 2 4 2(x 4)2 32 2(x 1)2 2 2(x 5)2 285 4 3(x 1)2 112 6
c
y
( 5, 0)
0
(3, 0) x (0, 15)
Exercise 2D1 x 3 x 3 5 2 2 30 2 x 4 x 6 2 33 6
Chapter 2 Answersd
y
j
y
(0, 4) ( 3, 0) (0, 3) 0 ( 1 , 0) 2 ( 4, 0) (1 , 0) 2 0
x
x
e
y
2 3
4 4
(
3 2
, 0) 0 (0,
(1, 0)
x3)
Mixed exercise 2G1 a
y20 16
f
y
12 8 4
(0, 10) 0 ( 2 , 0) 3 ( 5 , 0) 2 6 5 3 4 3 2
xb
x
10 4
1
2 x
y12 8
g( 1, 0)
y0 ( 5 , 0) x3
4 3 2 10 4 8 1 2 3 x
(0,
5)
xh
3 4
12
y
2
a y c x
1 or1 5
2
b x d 5
2 3
or 7
5
or 3 17 , 0.44 or 0.65
2
(0, 0) 0
( 13, 0) x3
3
a b 2 c
5 2
4.56
7, 4.65 or 3 10 29 73 6
, 0.24 or
0.84 0.59
i
y4
d a
5
, 2.25 or
y
(0, 7) (0, 4)
( 1, 0) 0
(7, 0)
x( 4, 0)
( 1, 0) 0
x
Chapter 2 Answersb
y
d
y
(1, 0) (3 2,
0)
0 (0, 3)
x(0, 0) 0 (7 1 , 0) 2
x
c
y5 a p b 6 1 2 13 5 or x 4 3, q 7 3 2, r 7
(0, 6) ( 3, 0) 0 ( 1 , 0) 2
x7
x
Chapter 3 AnswersExercise 3A1 3 5
Mixed exercise 3F2 4 6
x x x
4, y 2 2, y 2 2 2 3, y
x x x
1, y 3 41, y 3 2 3, y 3
Exercise 3B1 3
x x
5, y 1, y
2 4
2 4
x x
51, y 2 13, y 4
61 4
1 2 3 4 5 6 7 8
Exercise 3C1 a b c d e f ( ( a b a 5, y 6 or x 6, y 5 3 0, y 1 or x 4, y 5 5 1, y 3 or x 1, y 3 41, y 41 or x 6, y 3 2 2 a 1, b 5 or a 3, b 1 u 11, v 4 or u 2, v 3 2 11, 15) and (3, 1) 11, 41) and (2, 5) 6 2 x 11, y 53 or x 3, y 1 2 4 x 3, y 1 or x 61, y 25 2 3 6 x 3 13, y 3 13 or x 3 13, y 3 13 b x 2 3 5, y 3 2 5 or x 2 3 5, y 3 2 5
x x x x
9 10 11 12 13
4, y 31 2 (3, 1) and ( 21, 13) 5 5 b x 4, y 3 and x x 11, y 21 and x 2 4 a x 101 2 3 x 4 a x 5, x 4 a x 21 2 b 1 x 5 2 c 1 x 21 2 2 k 31 5 x 0, x 1 a 1 13 a x 4, x 9 a 2x 2(x 5) 32 c 101 x 13 2
x
22, y 3 4, y b x b x
1 3 1 2
2, x 5, x
7 4
b x 1 13, x b y 3, y 3 b x(x 5) 104
1
13
2 3 4 5
Exercise 3D1 a d g j a d g j a d
2
3
21 2 No values
x x x x x x x x x
4 3 12 11 7 3 18 43 4
b x e x h x b x e x h x b 2 e x
7 11 1 1 3 7
c x f x i x c x f x i x 4 c 21 2
21 2 23 5 8 31 4 42 5 1 2
x4
x
3
Exercise 3E1 a c e g i k a c a c e a 8 2, x 5 1 x 7 2 1 x 11 2 2 3 x 3 x 0, x 5 5 x 2 1 x 1 2 2 x 4 1 x 0 4 5 x 3, x 2 k 6 3
x
x
2 3
4
4
b d f h j l b d b d f b
4
x4, x 2, x1 3,
3 3 21 2
x x x x11 2
x3
x 2 21, x 2 2 3 x 0 1, x 1 x1 4
x
3 No values 1 x 1, 2 8 p 0
x
3
Chapter 4 AnswersExercise 4A1 a h
y x
y6 1 0 1
10
2
3
x
i
y2
b
y xj
1 2
0
1 2
2
x
3 2
0
1 6
y
c
y6 3
0
1 2
x
3
2
10
x2 a
y x
d
y3 1 xb
1
0 1
1
3
10
y2
e
y24 2
0
0
2
3
4 xc
1
x
y2 1 0 2 x
f
y
10
2
xd
y
g
y x0 2 2 x
1 0
1
1
Chapter 4 Answerse b y
y
x(x
4)(x
1)
y
2
0
x4 10
x
f
y
c y
x(x
1)2
y
0 1
x1 0
x
g
y
d y
x(x
1)(3
x)
y3 0
3
1
x1 0 3 x
h
y3 0 1 3 x
e y
x2(x y
1)
0i
1
x
y
f
y
x(1 x)(1 y
x)
0
2
x10 1
x
j
y
g y
3x(2x
1)(2x
1)
y0
2
x10 2 1 2
x
3
a y
x(x
2)(x
1) h y
y
x(x 1)(x y
2)
2
0
1
x1
0
2
x
Chapter 4 Answersi
y
x(x
3)(x
3)
e
y
y
0 3 0 3
x
x
j
y
x2(x
9)
2
a
y
y27 0 9
x3 0
x
Exercise 4B1 a
y
b
y
0
x0 27
3
x
b
yc
y
0
x
1 0 1
x
c
y
d
y
0
x
8 0 2
x
d
y
e
y
0
x
1 8
0
1 2
x
Chapter 4 Answers
Exercise 4C1
Exercise 4Dy yx4
1
a i
y
y
x2
0
x yx2
1
0
1
x
y2
x(x2
1)iii x2
y yx2
ii 3 b i
x(x2
1)
y y x(x2)
0
x yx2
2
0
x y3
x
3
y y2
x
ii 1 c i
iii x(x
2)
3
x
0
x yx4
y
x2
y
y1 0 1
(x
1)(x
1)2
x
4
y yx8
ii 3 d i
iii x2
(x
1)(x
1)2
y y x2 (1 x)
0
x yx3
5
0
y
1
x yx2
y0
x
3
ii 2
iii x2(1
x)
2
x
x y8
e ix
y
0
4
x4)1
y
x
1
y
x(x
ii 1
iii x(x
4)
x
Chapter 4 Answersf i 2 a
y y x(x4)
y y
x2(x
4)
0 0 4
x
xx1
y
yb (0, 0); (4, 0); ( 1,
x(45)
x)
ii 3 g i
iii x(x
4)
1
x2)3
3 a
y
y y(x 1 2.5 2 0 4 0
x
x4)
y
x(1
x)3 y x(2x2)
5)
yii 1 h i
x(x
b (0, 0); (2, 18); ( 2, iii x(x 4) (x 2)3
4 a
y
y y y0x2
(x
1)(x
1)
y x1 0 1
(x
1)3
x
yii 2 i i iii
x32
x3
x
b (0, 5 a
1); (1, 0); (3, 8)
y y x2
y y x2
0
x y x3
0
x y27
x
ii 2 j i
iii
x3
x2b ( 3, 9) 6 a
y y y x22x
y2 0
x(x x
2)
0
2
3
x
yii 3 iii
x3x3 x(x2)
y
x(x
2)(x
3)
b (0, 0); (2, 0); (4, 8)
Chapter 4 Answers7 a 12 a
y
y
x
2
y y14x 2
y0 3
x2(x x
3)
y2 1b Only 2 intersections 8 a
(x2
1)(x
2)
0
1
2
x
y y3x(x 1)
b (0, 2); ( 3,
40); (5, 72)
13 a
y y2 0 2 (x 2)(x 2)2
10
1
x
y
(x
1)3
x y x28
b Only 1 intersection 9 a
y yx1
b (0,
8); (1,
9); ( 4,
24)
0
1
x
Exercise 4E1 a i
y
ii
y
iii
y
y
x(x
1)2 0
b Graphs do not intersect. 10 a
x
0
x
0
x
y
( 2, 0), (0, 4)
y
x(x
2)2
( 2, 0), (0, 8)
(0, 1 ), x 2
2
b i1 2
y
ii
y
iii
y
01 2
2
x4x2 0 (0, 2)
y
1
x(3
0
x0
x2
b 1, since graphs only cross once 11 a
2, 0), (0, 2) (1 2,
0), y
y y6xc i
y
ii
y
iii
y
1
0
4
x0
x
0
x
0
x
yb (0, 0); ( 2,
x3
3x2
4x
(0, 1), (1, 0) (0, 1), (1, 0) (0, 1), x 1
12); (5, 30)
Chapter 4 Answersd i
y
ii
y
iii
y
b
y
0 ( 1, 0), (0, 1), (1, 0)e i
x
0
x
0
x
1
0
y(0, 1), (1, 0) ii
x f(x
1)
(1, 0), y iii
1
c f(x
1)
x(x
1)2; (0, 0)
y
y0
y
4
a
y
y
f(x)
x
0 ( 3, 0), (0, 3), ( 3, 0)f
x
0
x0 2
x
(0, 3), (3 3, 0) ii
( 1 , 0), y 3 iii
3b
i
y
y
y
y2
y2
f(x)
2
0 (0, 9), (3, 0)
x(0,
0
x(0,
0
x yf(x 2)
2
0
x
27), (3, 0)
1 3 ),
x
3
c f(x
2)
(x
2)x2; (0, 0); ( 2, 0)
2
a
y y
f(x)
5
a
y
y
f(x)
2
0 2
1
x0 4
x
b i
y y4 1 0 f(x 2)b
y yf(x) 4
4ii
x
4 0
y y1 0 f(x) 2
2
2 f(x(x (x
x2)
y 4c f(x f(x) 2) 4
x2)(x 2); (2, 0); ( 2, 0) 2)2; (2, 0)
c f(x f(x) 3 a
2) 2
(x (x
1)(x 1)(x
4); (0, 4) 2) 2; (0, 0)
y y0 f(x) 1
Exercise 4F1 a i
y
f(2x) f(x)
ii
y
f(2x) f(x)
iii
yf(x)
x
0
x
0
xf(2x)
0
x
Chapter 4 Answersb i
y
ii
yf(x)
iii
yf(x) 0
j
i
y
f(x)
ii
y
f(x)
iii
yf(x)
f(x)
0
xf( x)
0
xf( x)
x0 f( x)
x1 f(x) 4
0
1 f(x) 4
x
0
x1 f(x) 4
c
i
y
f(x)
ii
y
f(x)
iii
y
f(1 x) 2
2
a
y
y
f(x)
0
f(1 x) 2
x
0
xf(1 x) 2 iii f(x) 0
0 f(x)
x2 0 4 2
x
d
i
y
f(4x) f(x)
ii
y
yf(x) 0b
y y
f(4x)
y y3f(x)
0
x
x
xf(4x)1 2
0 4
1 2
x2 0 2
f(4x) i
x
y
e
f(x)
ii
y
f(x)
iii
y
f( 1 x) 4
12
0
f( 1 x) 4
x
0
xf( 1 x) 4
0
f(x)
x
y yf( x)
y4
y
f(x)
f
i
y
2f(x) f(x)
ii
y
f(x)
iii
y
2 2f(x)
0 4
2
x
2
0
2
x
0
x
0 2f(x)
x
0
f(x)
x
F
a
y
y
f(x)
g i
y
f(x)
ii
yf(x)
iii
yf(x) 0 2 0 2
x
0 f(x)h
x
0
xf(x)
xf(x)b
y
y
f(1 x) 2
y
y
f(2x)
i
yf(x)
ii
yf(x) 0
iii
y
4f(x)
4
0
4
x
0 1
1
x
4f(x) 0i
xii
x4f(x)
0 f(x)
x y
i
y
f(x)
y
iii
y
f(x) 2 0 2
0
x1 f(x) 2
0 f(x)
1 f(x) 2
x
0
x1 f(x) 2
x
y
f(x)
Chapter 4 Answers4 a b
y
y
f(x)
y(4, 0)
0
3 x
0 (0, 2) (1, 4)
x(6, 4)
b
y y03 2
f(2x)
y
c
y(0, 4) ( 4, 2) (2, 0) ( 3, 0) 0
x0 3 x
x
y y
f(x)
d
y(2, 4) (0, 2) (3, 0) 0 (1 , 2 0)
x
3
0
xe
y(4, 12)
y
f( x) (0, 6)
5
a
y yf(x)f
(6, 0) 0 (1, 0)
x(8, 4)
2
0
1
2
x
y(0, 2)
b
y
y
f(2x)g
0 (2, 0)
(12, 0)
x
y(4, 2) (0, 1) 0 (1, 0) (6, 0)
1
0
1 2
1
x
x
y y0 f(1 x) 2h
( 4, 4)
y(0, 2)
4
2
4
x( 6, 0)
( 1, 0) 0
x
2
a y
4, x
1, (0, 2)
y
Exercise 4G1 a
y( 1, 2)
(3, 4)
y
4 2 0 1
x
(0, 0) 0 (5, 0)
x
Chapter 4 Answersb y 2, x 0, ( 1, 0) h y 2, x 1, (0, 0)
y
y
y
2 1 0
0
1
x
x
y
2
c y
4, x
1, (0, 0)
3
a A( 2,
6), B(0, 0), C(2,
3), D(6, 0)
y
y y4 ( 2, 0 1 6) 0 (2, 3) 6 x
xb A( 4, 0), B( 2, 6), C(0, 3), D(4, 6)
y( 2, 6)d y 0, x 1, (0, 2)
(4, 6) 3 4 06), B( 1, 0), C(0,
y
x3), D(2, 0)
0 2e y 2, x1 2,
1
x
c A( 2,
y1(0, 0)
0 3 ( 2, 6)
2
x
y y2d A( 8,
6), B( 6, 0), C( 4,
3), D(0, 0)
y01 2
x
6 0 ( 4, 3) 6) ( 8,e A( 4,
x
f
y
2, x
2, (0, 0)
y y2
3), B( 2, 3), C(0, 0), D(4, 3)
y( 2, 3) 0 3) (4, 3)
x
0
2
xf
( 4,A( 4,
18), B( 2, 0), C(0,
9), D(4, 0)
yg y 1, x 1, (0, 0)
y y1
2 0 9 4
x
0
1
x( 4, 18)
Chapter 4 Answersg A( 4, 2), B( 2, 0), C(0, 1), D(4, 0) iii x 0, y 0
y2 ( 4, 0 1 2) 4 x
y
0h A( 16, 6), B( 8, 0), C(0, 3), D(16, 0) iv x 2, y
x
y8 0 3 ( 16, 6) 2i A( 4, 6), B( 2, 0), C(0, 3), D(4, 0) 1, (0, 0)
16
x
y
0
x y1
( 4, 6)
y3
2
0
4
x
v x
2, y
0, (0, 1)
y3), D( 4, 0)
j
A(4,
6), B(2, 0), C(0,
y4 0 3 2 (4, 6)vi x
1 0 2
x
x
2, y
0, (0,
1)
y4 a i
x
2, y
0, (0, 2)
y2 2 0 2 0
1
x
xb f(x) 2
x
2
ii
x
1, y
0, (0, 1)
Mixed exercise 4H1 a
y
y
y
x2(x
2)
1 1 0
x
0
2
x
2xb x 0,
x23)
1, 2; points (0, 0), (2, 0), ( 1,
Chapter 4 Answers2 a 4 a x a 1 at A, x 3 at B
yB 1 A 0
y yx6
1
x
5
y yf(x 1)
x y x2(5, 0) 2x 5b
0 (0, 0)
x
b A( 3, c y 3 a
2) B(2, 3) 2x 5
y(4, 0)
x2
y y2
A( 3 , 4) 2
( 1, 0) (0, 3)
0
x
B(0, 0) 0b
x6 a
y
y y1 A(3, 2) (0, 0) 0 ( 2, 0)b
x(2, 0)
0 B(0, 0)c
x
yA(3, 2) 0 2)
y(0, 0) 0
(2, 0) x
x y yA(0, 4) 0 is asymptote7 a f b i 1
B(0,d
y
x2
4x
3
y
yB( 3, 0)e
2
0
xii
( 1, 0) 0 (0, 1)
(1, 0) x
yA(6, 4)
y
y0f
2 ( 1 , 0) 0 2 (1,
( 3 , 0) 2
B(3, 0)
x
x1)
y y3
A(3, 5)
B(0, 1) 0
x
Review Exercise 1 Answers1 a b c 2 a 3 a 4 5 6 7 8 9 a a a a a a b c a b c a a b c k a b c d
x(2x 1)(x 7) (3x 4)(3x 4) (x 1)(x 1)(x2 8) 9 b 27 2 b 1 42
c
1 27
10
11 12
13 14
b 4x 3 3 b 21 8 5 b 8 2 3 b 7 4 3 b 10 13 7 c 16 6 7 x 8 or x 9 7 x 0 or x 2 3 x or x 3 2 5 x 2.17 or 7.83 x 2.69 or x 0.186 x 2.82 or x 0.177 a 4, b 45 b x 4 3 5 (x 3)2 9 P is (0, 18), Q is (3, 9) x 3 4 2 6, x 1 (same root) a 5, b 11 discriminant < 0 so no real roots k 25 625 4 5 13 6 3 56 7
22 a Different real roots, determinant > 0, so k2 4k 12 > 0 b k < 2 or k > 6 23 0 < k < 8 9 24 a p2 8p 20 > 0 b p < 2 or p > 10 c x 25 a x(x b ( 3 2 2)(x 2) 13)
y
2
0
2
x
c
y
y
1
0 1
3
x
25
26
y(3, 2)
a
15 a a b
1, b
5 2
0
x
y0 2 4
x
3(2, 0) (4, 0) and (3, 2)
0
x
b
16 17 18 19
20 21
c discriminant 8 d 2 3k1 4 b x < 1 or x > 3 2 c 1 < x < 1 or x > 3 4 2 a 0 31 2
y
0
1 (1 1 , 2
2 2)
x
(1, 0) (2, 0) and (11, 2 27 a
2)
Review Exercise 1 Answers
y
30 a
y
0
3
x
2
0
3
4
x
(0, 0) and (3, 0) b
y
( 2, 0), (0, 0) and (4, 0) (0, 0) and (3, 0) b (0, 0), ( 1 (1 3 5), 10 3 5), 2 ( 1 (1 3 5), 10 3 5) 2
6
0
1
4
x
(1, 0) (4, 0) and (0, 6) c
y
3 0 2 8
x
(2, 0) (8, 0) and (0. 3) 28 a
y3 0
x
Asymptotes: y b (1 3,
3 and x
0
0)
29 a f(x) x(x2 8x 15) b f(x) x(x 3)(x 5) c y
3 0 3 5
x
(0, 0), (3, 0) and (5, 0)
Chapter 5 AnswersExercise 5A1 a e i e1 2
22 3
b f j b f 25 4 1 2
1
c 3 g k1 2
d 22 3
1 3
h 2 l d 0 h l 21 2 3 2
2 a 47 5
5
c g 2 k b d f h j l3 2
i 9 j 3 3 a 4x y 3 0 c 6x y 7 0 e 5x 3y 6 0 g 14x 7y 4 0 i 18x 3y 2 0 k 4x 6y 5 0 4 y 5x 3 5 2x 5y 20 0 1 6 y 7 2x 7 y 2x 3 8 (3, 0) 9 (5, 0) 3 10 (0, 5), ( 4, 0)
3x 4x 7x 27x 2x 6x
y
2
0
5y 30 0 3y 0 9y 2 0 6y 3 0 10y 5 0
2 3 4 5 6 7 8 9 10
( 3, 0) (0, 1) (0, 31) 2
y x y y y(3,
4 5
x5
y3 8
4 01 2 1 6
x
4x
x1)
13 2, y
x
1 3,
y
6x
23
Exercise 5E1 a c e g i k Perpendicular Neither Perpendicular Parallel Perpendicular Neither1 3
b d f h j l
Parallel Perpendicular Parallel Perpendicular Parallel Perpendicular
2 3 4 5 c g k p p2 3 5 1 2 1 3
y4x a c a c 3x 7x
x15 2x 01 2
y y y y y
Exercise 5B1 a e i1 2
12 3
b f j n
1 6 1 2
4 q q2
d 2 h 8 1 l 2
6 7
x 3 3x 11 2 2 3x 2y 5 0 4y 2 0
b d b d
y y y y
1 2 1 2
x x1 3 3 2
8
x x
13 3 17 2
m 1 2 3 4 5 6 7 8 7 12 41 3 21 41 4
q
p
Mixed exercise 5F1 2 3 4 5 6 7 8 9 10 a y 3x 14 b (0, 14) 1 1 3 a y 4 b y 2x 2x 2, (1, 1) 1 12 a y 7x 7, y x 12 b (9, 3) 11 5 a y b 22 6 12 x 3 3 a y 2x 2 b (3, 3) 11x 10y 19 0 1 a y 3 b y 1x 9 2x 4 4 3 a y 2x 2 b (4, 4) c 20 a 2x y 20 b y 1x 4 3 3 1 a 2 b 6 c 2x y 16 0 3 3 3 b y 3x 2 3 a 1 3 162 a 7x 5y 18 0 b 35 b y 1x 1 3 3 a y l2 l1 3 , 0) (2
26 5
Exercise 5C1 a c e g 2 y 3 y 4 2x 1 5 5 6 y 7 2x 8 8 5 9 y 10 6x
y y y y
2x
1
x 3 1 12 2x 2x0
b d f h
y y y y
3x 7 4x 11 2 5 3x 1 2b 2x
11 12 13 14
3x 6 2x 8 3y 242 5
x 3 3y 12 x15y 4 10
(0, 0) 00
x
4 3
(0,04 1
3)c 12x 3y b y 4x
Exercise 5D1 a c e g i
b (3, 3) 15 a x 2y 16 c ( 176, 674) b d f h j
17 0
0
y y y y y
4x 2x
4 4 4 9
x6 5
4x
x
y y y y y
x4x1 2 2 7
2
x x
23 1 8x 335 14
Chapter 6 AnswersExercise 6A1 24, 29, 34 Add 5 to previous term 2 2, 2, 2 Multiply previous term by 1 3 18, 15, 12 Subtract 3 from previous term 4 162, 486, 1458 Multiply previous term by 3 5 1, 1, 116 4 8 Multiply previous term by 1 2 6 41, 122, 365 Multiply previous term by 3 then 1 7 8, 13, 21 Add together the two previous terms 8 5, 161, 173 9 Add 1 to previous numerator, add 2 to previous denominator 9 2.0625, 2.031 25, 2.015 625 Divide previous term by 2 then 1 10 24, 35, 48 Add consecutive odd numbers to previous term 3 a b c d e f a a Uk 1 Uk Uk 1 Uk Uk 1 Uk Uk 1 Uk Uk 1 Uk Uk 1 U k 3k 2 4 2p 2, U1 1 3, U1 5 1, U1 3 1 1 2, U1 2k 1, U1 1 ( 1)k(2k 1), U1 b 3k2 2k 2 b 4 6p 1 c 130, c p
4 5
4 2
Exercise 6D1 2 Arithmetic sequences are a, b, c, h, l a 23, 2n 3 b 32, 3n 2 c 3, 27 3n d 35, 4n 5 e 10x, nx f a 9d, a (n 1)d a 5800 b (3800 200m) a 22 b 40 c 39 d 46 e 18 f n
3 4
Exercise 6E1 a c e g a d d a 24 78, 4n 2 23, 83 3n 27, 33 3n 39p, (2n 1)p 30 b 29 31 e 221 6 36, d 3, 14th term 5; 25, 20, 15 1 8 2, x b d f h c f 42, 2n 2 39, 2n 1 59, 3n 1 71x, (9 4n)x 32 77
Exercise 6B1 a b c d e f g h a e i Un Un 2) a a p U1 5 U1 7 U1 6 U1 4 2 U1 U1 1 3 1 U1 3 U1 1 14 6 4 4n2 4n (n 5)2 U2 8 U3 11 U10 32 U2 4 U3 1 U10 20 U2 9 U3 14 U10 105 U2 1 U3 0 U10 49 U2 4 U3 8 U10 1024 U2 1 U3 3 U10 5 2 5 6 3 U2 1 U3 U10 5 2 5 6 U2 0 U3 1 U10 512 b 9 c 11 d 9 f 9 g 8 h 14 j 5 4(n2 n) which is a multiple of 4 2 0 Un is smallest when n 5 (Un 2 3 4 5 6 7
2
x x
3 4 5 6 7
Exercise 6F1 a 820 d 294 g 1155 a 20 c 65 2550 i 222 500 1683, 3267 9.03, 141 days 1 d 5.5 2, a 6, d 2 b e h b d 450 c 1140 1440 f 1425 21(11x 1) 25 4 or 14 (2 answers)
12, b 22 1, b 3, c 0 1 51 2, q 2
2 3 4 5 6 7 8
Exercise 6C1 a c e g a b c 1, 4, 7, 10 b 3, 6, 12, 24 d 10, 5, 2.5, 1.25 f 3, 5, 13, 31 Uk 1 Uk 2, U1 3 Uk 1 Uk 3, U1 20 Uk 1 2Uk, U1 1 Uk , U1 100 d Uk 1 4 1 Uk, U1 1 e Uk 1 f Uk 1 2Uk 1, U1 3 g Uk 1 (Uk)2 1, U1 0 Uk1 2 1
ii 347 500
9, 4, 1, 6 2, 5, 11, 23 2, 3, 8, 63
2
Exercise 6G10 30
1
ar 1 11
(3r 4(11r 1
1) r)
br 1 16
(3r 6rr 1
1)
c 2 3 4
d
h Uk i j Uk Uk
2 2
, U1
26
Uk 1 Uk, U1 1, U2 1 2Uk 2( 1)k 1, U1 4
a 45 c 1010 19 49
b 210 d 70
Chapter 6 Answers
Mixed exercise 6H1 2 3 4 5, 8, 11 10 2, 9, 23, 51 a Add 6 to the previous term, i.e. Un (or Un 6n 1) b Add 3 to the previous term, i.e. Un (or Un 3n) c Multiply the previous term by 3, i.e. Un 1 3Un (or Un 3n 1) d Subtract 5 from the previous term, i.e. Un 1 Un 5 (or Un 15 5n) e The square numbers (Un n2) f Multiply the previous term by 1.2, i.e. Un 1 1.2Un (or Un (1.2)n 1) Arithmetic sequences are: a a 5, d 6 b a 3, d 3 d a 10, d 5 a 81 b 860 b 5050 32 a 13 780 c 42 198 a a 25, d 3 b 3810 a 26 733 b 53 467 a 5 b 45 a 4k 15 b 8k2 30k 30 1 c 4, 4 b 1500 m a U2 2k 4, U3 2k2 4k 4 b 5, 3 a 2450 b 59 000 c d 30 a d 5 b 59 11k 9 3
1
Un Un
6 3
1
5 6 7 8 9 10 11 12
13 15 16
17
18 b
c 1.5 d 415
Chapter 7 AnswersExercise 7A1 a i iv b 6 a i iv b 8 7 6.01 9 8.01 ii v ii v 6.5 h 6 8.5 8 h iii 6.1
Exercise 7G1 3 5 iii 8.1 7 1 10 2t 32 2
2 4 6
2 9.8 12 8t
12t 5r
2
2x
Exercise 7H1 a y c 3y 2 8 x71 4
3x 2x 12x
6 18 14 48
0 0 0 9
b d f b
4y
3x
4 22
0
Exercise 7B1 4 7 10 7 x61 3
34 3
4 x31 3
e y2 3
y y17y
x16x 2x 212 0
x
2 3
2 333 2
a 7y (12, 18) 9 9
x
54 6 3
x
654 3
x
3x 5x
8 11
4x1 3
9 12
2x1 2
4 5
y y
x, 4 y8x
x10, 8y
0; ( 3, 3)
x
145
0
x
x
13 2x 16 9x8
14 1 17 5x4
15 3x2 18 3x2
Exercise 7I1 4, 113, 1725 4 27 2 0, 2 2 3 ( 1, 0) and (12, 913) 3 27
Exercise 7C1 2 3 4 5 6 a 4x d 16x a 12 d 21 2 4, 0 ( 1, 1, 6, 1 4 8) 6 7 b x e 4 b 6 e 2 12 10x c 7 f 4 c 8x
4 2, 22 3 5 (2, 6 a 1 7 x 83 2
13) and ( 2, 15) 9
x220 2x 6x1 2 3 23 2 1 2 1 2 1 2 1
b
x
3
4, y
x
1 2
9 a
dy dx
3 2
x23 x)
b
(4, 16)
Exercise 7D1 2 3 a 4x a 0 a (21, 2 c (16, 61) 4 31)3
x x
(12 (41 2 3 2
x
2
b
x
3
c 4) and (2, 0)1 2,
x
3 2
b 111 2 b (4, 10 a x b 1 c 4116
x)1
x3 2
3 2 1 2
x1 2
d (1, 4) ( 2
4)
x
x
Exercise 7E1 a x d f i 24 3 1 31 2
11 6x2 b 6x 6x 3x 3 x23 4 2 1 2
1 2
x
1 2
2x
2
c
x
4
x
32 3 3
2x1 2 3 2
2
e21 2
x
1 2
12 6x2
10 2300 , 3 27 1, b1 3
14 a h 3 2 4 k 12x3 d 4 18x2
4, c 10x 5
5 ii y 2x 128 b1 5
x
x
g j2 2 9
15 a 3x2 b i 16 y 9x 17 a (4, 5
5x 2
x
2x c
7
iii
7 2
5
l 24x a 1
8
2x b
4 and 9y2 5)
x
Exercise 7F1 2 3 4 5 24x 159 2
3, 24 3x 2, 6x1 2
3 9 4
x
6x 3, 2, 302
x
3 2
18x 48x
4
30x 3x
16x 3, 6x
3
4
Chapter 8 AnswersExercise 8A1 y 3 y 5 y 7 y 9 y 11 y 13 y 15 y 17 y 19 y1 6
Exercise 8D2 y 4 y c c 6 y 8 y 10 y 12 y c c 14 y 16 y c 18 y 20 y10 3 9 2
x6c23 2
c
2 x5
c c c c c1 2
1
a c e
1 2 4 3 4 5 1 3 1 2 2 3
x4 x3 x5 2
x36 x2 2x 2 x28 33 3 2
c 9x c 4x 4x1 2
b c d
2x2 3
3
x1 2
c
x32x8 3 1 2
x3 5
1
x3
x2
3x
c
x2 7
5 3
x
x73
2
a c
x3 x2 x3 2 3 2
c c
b d f b d
1 3 3
x
2x5
1
x3
c c c
x61 2 1 2
c c
x4 3
x2
2 5
x21 2 1 2
4 3 4 3
x2 x3 2
2x 4x
10x
c 3
e
x
c7
a 2x c e3 5 1 4
4x 1
c c c3 2
2x 4x
x5
x31
c 3x c
3x12 9x 3 x21 3
2x
c c c
x3 x41 2 5 2
2
1
5x
x2 1 3 3x6 5 8 3
x21 3
x3 x2
x
2
3x
c g
c
x0.6
f h
4x8 5
x x1 2
5 2 3 2
c 2x 2 x31 2
x3
9x
c
x
c c j2 55 3
Exercise 8B1 a y b y c y d y e y f 2x2
x
1 21 2
4x
3 2 3 2
i c c c c
3x
2x
x2
3 x2
6x 2
c
5 x31 4
3x 3x 3x 4x1 3
2x 6x1
Exercise 8E1 a y c y
x4
x32 3 1 3 1 2
x23 2
21 3 12
b1 3
y y y
x46 x2 5
1
x21 25 2
3x
1 4 1
x44x 3x 4x5 3
x3
11 2
x
x
d2 3
x21 2
4x1 2
c2
e y 2 3 f(x)
x3
2 x2 1
4x1 2
f
x
6x
y
2 x51 3
x
c c
x42
g y h y 2 a f(x) b f(x) c f(x) d f(x) e f(x) f f(x)
3x1 2
4x 3x1 2
2 4
x3
y
1
x
5
2x 6x2
c c c 5 4 a f2(x) b f2(x) f4(x) 3
x x x33 4
3x
5x1 6
; f3(x)
x412 (n1 2
x61
x x2
61 2
xc22 3
xn51 2
1
1)
x25 x2 3x 3 3x1 3 3
x1 6
1; f3(x)
x
2
x
1;
4x 6x 2x
c c1 2 2 3 4 3
x3
x2
x
1
Mixed Exercise 8F1 2
2
x3 2
c c c
1 a 21 3
2 3
x33 2
3 2
x22
5x1 6
c
b
3 4
4
x35
3 2
2
x33
c
g f(x) h f(x)
x
3
x x2
1
x3 2
x35 2
x23 2
x
x
2
x
3 a 2 x4 44 5
2 x32 3
5x 6x1 2
c c 7
b
2x 2
4 3
x2
c
x
xt2
Exercise 8C11 4
5 x c c rx c t 3x c1
t3 2
3
t1 2
1; x c
x45
x2 x3 x31 2
2 44 3
2x3
1
3x1
c 4x c
6 2x 7 x
4x 121 3
3 2x 2 5 x4 72 3 3t
x2t2
4x 21
6 t3 8 c1 2
c1 2
8 a A 2x px t31 2
6. B1 2
93 2 3 2
b b
3 5
5
x33 2
9 2
4
x31 2
9x c
c
6t
x
2x qt2
c c
p 9 x5 5
2tx
p 10 t4 4
9 a
3 2
x
2x 8x1 2
2x
8x
10 a 5x
2 3
x
c
Review Exercise 2 (Chapters 5 to 8) Answers1 2 a Since P(3, 1), substitute values into y gives 1 5 6, so P on line. b x 2y 5 0 a AB 5 2 b 0 x 7y 9 c C is (0, 9 ) 7 a 0 x 3y 21 b P (3, 6) c 10.5 units2 a p 15, q 3 b 7x 5y 46 0 c x 11 4 7 1 a y 4 3x b C is (3, 3) c 15 units2 a P is (181, 13) 16 b 16241 units2 a d 3.5 b a 10 c 217.5 a 5 km, d 0.4 km a 3, 1, 1 b d 2 c n(n 4) a 750 b 14 500 c 155 a a2 4, a3 7 b 73 a a1 k, a2 3k 5 b a3 3a2 5 9k 20 c i 40k 90 ii 10(4k 9) a a5 16k 45 b k 4 c 81 a In general: Sn a (a d) (a (a (n 1)d) 2d) (a (n 5 2x d a5 2 2 e a100 3 1 dy 16 12x2 x 2 dx dy 17 a 4x 18x 4 dx b 2 x3 3 x 2 c 3 1 dy 18 a 6x 2x 2 dx 3 d2y b 6 x 2 2 dx c x3 8 x 2 c 3 dy d2y a i 15x2 7 ii 30x dx 3 dx 2 b x 2x 2 x 1 c 1 dy a 4 9 x 2 4x 2 dx b Substitute values, 8 8 c 3y x 20 d PQ 8 10 dy a 8x 5x 2, at P this is 3 dx b y 3x 5 5 c k 3 a At (3, 0), y 0 7x 21 b At P, y c Q (5, 15 1) 3 a P 2, Q 9, R 49 2 b 3x 2 2x 2 2x c When x 1, f(x) 5 1 , gradient of 2y 2 5 1 , so it is parallel with tangent. 21 1 3 3
3
4
5
19
6 7
20
8 9
21
10
11 12
22
23
13
11x
3 is
14
24 2)d)
1 3 3
x
2 x2 2x5 2
3x 4x1 2
13 3 0 for all
25 3x
Reversing the sum: Sn (a (n (a 1)d) d) (a a (n 2)d) (a (n 3)d)
Adding the two sums: 2Sn 2Sn Sn b c d e 15 a [2a (n 1)d] [2a [2a (n 1)d] n[2a (n 1)d] n [2a (n 1)d] 2 (n 1)d]
26 a 3x2 2 b 3x2 2 2 for all values of x since 3x2 values of x c y 1 x4 x2 7x 10 4 d 5y x 22 0 1 3 x 1 4 27 a y 3x 3 b (1, 2) and ( 1, 2 ) 3 28 a 2x3 5x2 12x b x(2x 3)(x 4) c y
109 n2 150n 5000 0 n 50 or 100 n 100 (gives a negative repayment) a2 4 2k a3 (4 2k)2 k(4 2k) 6k2 20k 16 b k 1 or k 7 3 2 c a2 3
3 2
0
4
x
(
3 2,
0), (0, 0) and (4, 0)
Review Exercise 2 Answers29 a PQ2 PQ b dy dx 12 132 170 3 x2 12 x 170 4x
2
dy dy 13, at Q, dx dx c x 13y 14 0 30 a x(x 3)(x 4) b y At P,
13
0
3
4
x
(0, 0), (3, 0) and (4, 0) c P (3 4 , 7 15) 7
Practice paper Answers1 a 4 b 64 2 2 x3 2 x 2 c 3 3 a 3, 5 b 36 4 a 27 10 2 b 20 2 5 x 3, y 3 and x 6 a x 2y 13 0 b y 2x c (23, 51) 5 5 7 a No intersections.3
8, y
2 3
y2 0 2
2
2
x
b
y(1 , 0) 2 2 0 3 5 2
x
c
y2 (0, 2) 0 2
2
2 1
x
x
8 a 670 b 5350 9 a i 2 ii c 4 b i x 5 ii x 7, x 3 iii x 7, 3 x 5 10 a P 9, Q 24, R 16 b 10 c x 10y 248 0
c 45 iii c 4
Examination style paper Answers1 a b 2 a b 3 a k 5 k 6 9 1 8x 3
y
2
x
b 4 a b 5 a 6 a b 7 a b c 8 a b c 9 a c
0 and y 2 3 420 0 k 2.4 a2 1, a3 4 24 3 24x2 3 x 2 2 5 9 48x 4 x 2 1 2 x4 6 x2 + 5 x c 0.4 (137, 4) 3 4 x3 3 x 2
x
y
( 2, 0)
(1, 0)
x
10 a y 9x 9 d 16 2 e 320 units2