Edexcel AS and A Level Modular Mathematics: Core Mathematics 1 [1 ed.] 0435519107, 9780435519100

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Chapter 1 Answers Exercise 1A 1 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 1B 1 4 7 10 13 16

x7 3x2 5 x8 2p7 27x8 32y6

2 5 8 11 14 17

6 x5 k5 p2 6a9 24x11 4a6

Exercise 1C 1 3 5 7 9 11 13 15 17 19

9x  18 12y  9y2 3x2  5x 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

Exercise 1D 1 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

x(x  4) (x  8)(x  3) (x  8)(x  5) (x  2)(x  3) (x  5)(x  2) (2x  1)(x  2) (5x  1)(x  3) (2x  3)(x  5) (x  2)(x  2) (2x  5)(2x  5) 4(3x  1)(3x  1) 2(3x  2)(x  1)

2p2

y10 2a3 3a2b2 63a12 6a12

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

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)

1

2

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)

a d g a e i

x5 x3 1  3x 2 5 13 12 5  64

b e h b f j

x2 x5 5x

1 4 7 10 13

27  42  3  127  235 

9

2 5 8 11 14

x4 12x0  12 6x1 3 d 116 1 h 6 5  l 6449 6

62  310  65  37  2

Exercise 1H 5  1  5 2  3  2 5

c f i c g k

1 125 9  4

Exercise 1G

3 6 9 12 15

52  3  72  95  193 

1  1 2  11 5  4  5

1  2

1  3 7  13 1  3  9  2 3  7  11  2 5   3  13  2 5(2  5 ) 15  1 11(3  11 ) 17  2 14  187  19  3

6

1  4

8

1 3

2  5  10  1 12 3  5  (3  2 )(4  5 ) 14  11 16 5(4  14 ) 5  21  18  2 35  1189  20  6

21 1

Mixed exercise 1I 1 2 3

Exercise 1E 1 3 5 7 9 11 13 15 17 19 21 23

3 6 9 12 15 18

Exercise 1F

4

5 6 7 8

a a c a c a c e a a

y8

b 6 x7

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 37 5   b 3 9 4913

7  a  7 3  a  3 c 33 6

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 12 x 3

b 45  b 2 1 30  851  d  7

Chapter 2 Answers Exercise 2A 1

5

y

y

28

14

24

12

20

10

16

8

12

6

8

4

4

2 4 3 2 1 0 2

x0

2

1

2

3

4 3 2 1 0

4 x

1

2

3

4 x

x1

4

y 22

6

y

18

30

14 10

20

6 2 4 3 2 1 0

10

1

2

3

4 x 4 3 2 1 0

x0

x  112 3

1

2

3

4 x

1

2

3

4 x

10

y 8 6 4 2 4 3 2 1 0

1

2

3

4 x

7

y 40

x0

30

4

y 4 3 2 1 0 4

20

1

2

3

4 x 10

8 12 16

x0

4 3 2 1 0

x   34

10

Chapter 2 Answers 8

3 29  5 x     2 2 1 1  29 7 x     8 8 3 39  9 x     2 2

y 20

15

3 6 x  1  2  2 8 No real roots 4 26  10 x     5 5

10

Exercise 2E 5

4 3 2 1 0

1

2

3

4 x

5

y 80 70 60 50 40 30 20 10 4 3 2 1 0

x

1

2

3

x  0 or x  4 x  0 or x  2 x  1 or x  2 x  5 or x  2 x  3 or x  5 x  6 or x  1 x  12 or x  3 x  23 or x  32 x  13 or x  2 x  13 or x  1



5 21 x    3 1  11  23 x   3 25 x  12 or x  73

(x  2)2  4 (x  8)2  64 (x  7)2  49 3(x  4)2  48 5(x  2)2  20 3(x  32)2  247

Exercise 2F 1

2 4 6 8 10 12 14 16 18 20

x  0 or x  25 x  0 or x  6 x  1 or x  4 x  3 or x  2 x  4 or x  5 x  6 or x  2 x  13 or x  32 x  32 or x  52 x  3 or x  0 x  2 or x  2

a

y (0, 2)

(2, 0)

x

(1, 0) 0

b

22 x  3  13 

y

(0, 10)

24 x  1 or x  76

2 4 6 8 10 12

(x  3)2  9 (x  12)2  14 2(x  4)2  32 2(x  1)2  2 2(x  54)2  285 3(x  16)2  112

x

0

26 x  0 or x  161

Exercise 2C 1 3 5 7 9 11

4 x

1 2

Exercise 2B 1 3 5 7 9 11 13 15 17 19

3 3  3 , 1.27 or 4.73 5  33  4 , 5.37 or 0.37 2 5  31  5 , 3.52 or 0.19 3 1  2  6 , 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 11

x  1 9

3  5  1 , 0.38 or 2.62 2 3  17  2 , 0.56 or 3.56 2

c

y

(5, 0)

0

(3, 0) x

Exercise 2D 1 x  3  22  3 x  5  30 

2 x  6  33  4 x  2  6 

(0, 15)

Chapter 2 Answers d

j

y

y

(0, 4) (0, 3) 0 ( 12 , 0)

(3, 0)

e

x

x

0

y

( 32 , 0)

(12 , 0)

(4, 0)

2

4

3

4

(1, 0)

x

0

Mixed exercise 2G 1

(0, 3)

a

y 20 16

f

y

12 8 4

(0, 10) 0 ( 23 , 0)

( 52 , 0)

6 5 4 3 2 1 0 4 x  3

x b

1

2 x

y 12

g

y (1, 0)

0

8 4

( 5 , 0) x 3

3 2 1 0 4

1

2

3 x

8 (0, 5)

h

x  34 2

y

(0, 0) 0

( 13, 3

3

0) x

12

a y  1 or 2

b x  23 or 5

c x  15 or 3

5  7  d  2

5  17  a , 0.44 or 4.56 2 b 2  7 , 4.65 or 0.65 3  29  c , 0.24 or 0.84 10

i

5  73  d , 2.25 or 0.59 6

y 4

a

y

(0, 7)

(1, 0)

(0, 4)

(7, 0) 0

x

(1, 0) (4, 0)

0

x

Chapter 2 Answers b

d

y

y

(1, 0) ( 32 , 0)

x

0

(0, 0) 0

(0, 3) c

y 5

( 12 , 0)

(3, 0) 0

a p  3, q  2, r  7 b 2 

(0, 6)

(7 12 , 0)

3 7

6

1  13 

7

x  5 or x  4

x

x

Chapter 3 Answers Exercise 3A 1 3 5

Mixed exercise 3F

x  4, y  2 x  2, y  2 x  23, y  2

2 4 6

x  1, y  3 x  412, y  3 x  3, y  3

2 4

x  512, y  6 x  134, y  14

Exercise 3B 1 3

x  5, y  2 x  1, y  4

Exercise 3C 1

2 3 4 5

a x  5, y  6 or x  6, y  5 b x  0, y  1 or x  45, y  35 c x  1, y  3 or x  1, y  3 d x  412, y  412 or x  6, y  3 e a  1, b  5 or a  3, b  1 f u  112, v  4 or u  2, v  3 (11, 15) and (3, 1) (116, 412) and (2, 5) a x  112, y  534 or x  3, y  1 b x  3, y  12 or x  613, y  256 a x  3  13 , y  3  13  or x  3  13 , y  3  13  b x  2  35 , y  3  25  or x  2  35 , y  3  25 

Exercise 3D 1

2

3

a d g j a d g j a d

x4 x 3 x  12 x  117 x3 x  18 x4 x  34 x  212 No values

b x7 e x  11 h x1

c x  212 f x  235 i x 8

b x1 e x3 h x  7

c x 314 f x  425 i x 12

b 2x4 e x4

c 212  x  3

Exercise 3E 1

2 3

4

a c e g i k a c a c e a

3x8 x  2, x  5 12  x  7 1 1   2 x 12 3  x  3 x  0, x  5 5  x  2 1   x  1 2 2x4 14  x  0 5  x  3, x  4 2  k  6

b d f h j l b d b d f b

4  x  3 x 4, x  3 x  2, x  212 x  13, x  2 x  212, x  23 112 x 0 x  1, x  1 3  x  14 x3 No values 1  x  1, 2  x  3 8  p  0

1 2 3 4 5 6 7 8

9 10 11 12 13

x  4, y  312 (3, 1) and (215, 135) b x  4, y  3 and x  223, y  13 x  112, y  214 and x  4, y  12 a x  1012 b x  2, x  7 3x4 a x  5, x  4 b x  5, x  4 a x  212 b 12  x  5 c 12  x  212 k 315 x  0, x  1 a 1 13  b x  1  13 , x  1  13  a x  4, x  9 b y  3, y  3 a 2x  2(x  5)  32 b x(x 5)  104 c 1012  x  13

Chapter 4 Answers Exercise 4A 1

a

h

y

y

1 0

2

x

3

i

x

0 1

1

6

y 2

b

y

12 0

0

3 2

x

1

j

6

c

y

y

0

3

6

2

a

y

0

1

y 3 1 0

3

1

x

1

1 x b

e

x

1 2

x

3 2 1 0

d

x

2

1 2

y

y

2

24

0

2

3

c f

0

2

4 x

1

y

y 2

1 0

1

x

2

d g

0

2 x

0

2 x

y

y

1 0

1

x

1 2

x

Chapter 4 Answers e

b y  x(x  4)(x  1)

y

y

x

0

2

1 0

4 f

x

c y  x(x  1)2

y

y

x

0 1

g

y

x

0

1

d y  x(x  1)(3  x)

y 3 0

3

x

1

h

y

3 x

0

1

e y  x2(x  1)

y 3 0

3 x

1

0 i

y

f

0

2

y  x(1  x)(1  x) y

x 1 0

j

x

1

y

x

1

g y  3x(2x  1)(2x  1)

y 0

2

x 0  12

3

a y  x(x  2)(x  1)

0

x

h y  x(x  1)(x  2)

y

2

1 2

y

1

x

1

0

2

x

Chapter 4 Answers i

y  x(x  3)(x  3)

e

y

y

x

0 0

3

j

3

y  x2(x  9)

x

2

a

y

y 27 0

9

x x

0

3

Exercise 4B 1

a

y

b

y

x

0

0

x

3

27 b

y c

x

0

y

1 0

c

y

d

x

0

y

8 0

d

y

e

0

x

x

1

x

2

y

1 8

0

1 2

x

Chapter 4 Answers

Exercise 4C 1

Exercise 4D y

1

y

a i

y

x

x

0

y

y  x2

4

1

2

0

x

1

x

y  x(x2  1) 2

y

iii x2  x(x2  1)

ii 3 b i

y  x2

y y  x(x  2)

x

0

y   x2

x

0

2

3

y  x 3

y y   x2

3 iii x(x  2)  

ii 1

x

0

x

c i y  x2

y

y   x4

4

y  (x  1)(x  1)2 0

1

y  x8

iii x2  (x  1)(x  1)2

ii 3 d i

y

x

0

y  x2 (1  x)

y  x3

5

x

1

y

0

y

y

y   x3 0

x

1

2 iii x2(1  x)  

ii 2

x

2

x

x

e i

y

y   x8

0

y ii 1

1

x

4

x

y  x(x  4) 1 iii x(x  4)  

x

Chapter 4 Answers f

i

2 a

y

y

y  x2(x  4)

y  x(x  4) x

0 0

x

4

y  x(4  x)

y   x1

b (0, 0); (4, 0); (1, 5) 1

iii x(x  4)  

ii 3

3 a

x

g i

y

y y  (x  2)3

1

2.5 2 0

4

x

0

y  x(1  x)3

x

y  x(2x  5) y  x(x  4)

b (0, 0); (2, 18); (2, 2)

iii x(x  4)  (x  2)

ii 1 h i

3

4 a

y

y y  (x  1)(x  1) y  x2

y  (x  1)3 x

0

1 0

x

1

y  x3 2 iii x3  

ii 2 i

b (0, 1); (1, 0); (3, 8)

x

i

y

5 a

y

y y  x2

x2

x

0

x

0

y

y  x3

27

x

iii x3  x2

ii 2

b (3, 9) j

i

y 6 a

y  x(x  2) 2

0

x

y  x3 ii 3

iii x3  x(x  2)

y y  x2  2x 0

y  x(x  2)(x  3) b (0, 0); (2, 0); (4, 8)

2

3

x

Chapter 4 Answers 7 a

12 a

y

y

y  x2

y  14x  2

y  x2(x  3) 0

x

3

y  (x2  1)(x  2) 2 1

0

1

x

2

b Only 2 intersections 8 a

y

b (0, 2); (3, 40); (5, 72)

y  3x(x  1)

13 a

1 0

y

x

1

y  (x  2)(x  2)2 2

y  (x  1)3

x

0 2

y   x2  8 b Only 1 intersection 9 a

y y

0

1

b (0, 8); (1, 9); (4, 24)

1

x

Exercise 4E

x

1

y

a i

y

ii

y

iii

y  x(x  1)2

10 a

0

x

y  x(x 

b i

01 2

2

(0, 12 ), x 2

(2, 0), (0, 8)

2)2

y

ii

y

x

0

(2, 0), (0, 4)

y

 12

x

0

b Graphs do not intersect.

y

iii

x 0

y  1  4x2

x

0

x x

0

(0, 2)

(3

b 1, since graphs only cross once

2, 0), (0, 2) ( 12 , 0), y  2

11 a

y

1

y  6x

0

4

c i

y

b (0, 0); (2, 12); (5, 30)

y

iii

y

x 0

y  x3  3x2  4x

ii

x

0

x

0

(0, 1), (1, 0) (0, 1), (1, 0)

(0, 1), x  1

x

Chapter 4 Answers d i

y

0

x

iii

x

0

1

x y  f(x  1)

0

c f(x  1)  x(x  1)2; (0, 0)

y

4

a

y  f(x)

y

x

0

x

0

y

x

(1, 0), y  1

y

ii

b

0

(0, 1), (1, 0)

y

y

iii

0

x

(1, 0), (0, 1), (1, 0) e i

y

ii

0 ( 3, 0), (0, 3), ( 3, 0)

(0, 3), (3 3, 0)

x

2

( 13 , 0), y  3 b

f

i

y

y

ii

iii

y

y

y  f(x)  2

2

x

0

x

0

0

(0,  13 ), x  3

(0, 27), (3, 0) a

y y  f(x)

2

0 2

2

0

c f(x  2)  (x  2)x2; (0, 0); (2, 0) 5

a

y

y  f(x)

x

1

0 b i

y  f(x  2)

b

x

4

y y

4 1 0

4 ii

y  f(x)  4 x

4

y

0

2

y  f(x)  2 1

x

2

y  f(x  2)

(0, 9), (3, 0)

2

x

y  f(x  2) 4

x

0

x

2

c f(x  2)  (x  2)(x  2); (2, 0); (2, 0) f(x)  4  (x  2)2; (2, 0)

c f(x  2)  (x  1)(x  4); (0, 4) f(x)  2  (x  1)(x  2)  2; (0, 0) 3

a

Exercise 4F

y

1

a i

y

1

ii

y

f(2x)

iii

y f(x)

y  f(x) 0

f(2x) f(x)

x

0

x

f(x) 0

0

x f(2x)

x

Chapter 4 Answers b i

y

y

ii

iii

j

y

f(x)

x

0

x 0 f(x)  f(x)

y

i

0

f(x)

f(x)

x

0

x

y

iii

f(12 x)

a

0

2

f(x)

iii

y

x

x

0

1 f(x) 4

1 f(x) 4

y  f(x)

y

x

0

f(12 x)

f(12 x)

2

0

1 f(x) 4

f(x)

f(x)

f(x)

x

0

y

ii

y

ii

f(x)

x

0

f(x) c

y

i

f(x)

2

x

4 d

y

i

f(4x)

y

ii

iii

f(x)

f(x)

f(x) 0

y

0

x

x

0

y

i

0

i

f

y

f(x)

f(4x)

 12

f( 14 x)

2f(x)

f(x)

y

iii

x

0

x

0

f( 14 x)

ii

0

1 2

y

f(x)

iii

y

y  3f(x)

x 2

4

y

ii

y

x

f(4x)

e

y y  f(4x)

b

f( 14 x)

2

x

12

y

x

f(x)

0

y y  f(x)

2 2f(x)

0

2

x

y  f(x)

4

2

0

2

x

4

f(x) 0

x

0

x

0

x

f(x)

F

a

y

y  f(x)

2f(x) g i

y

ii

f(x)

y

iii

y

0

x

0

2

f(x)

f(x)

x

2

y

y  f(12 x)

0

4

x

2

x

x

0

f(x)

b

f(x)

f(x)

x

0

y

y  f(2x)

h

y

i

ii

y

iii

f(x) 4f(x) 0 i

i

y

0

0

1 f(x) 2

x

f(x)

y

y

0 f(x)

x

0

4f(x) ii

x

4

4f(x)

f(x)

x

f(x)

y

iii

x

1 f(x) 2

y

0

f(x)

x 1 f(x) 2

2

0

y  f(x)

0 1

1

x

Chapter 4 Answers 4

a

y

b

y  f(x)

y (4, 0)

3 x

0

(0, 2)

x

0

(6, 4)

(1, 4) b

y y  f(2x)

c

y

y (0, 4) (4, 2)

0

(2, 0)

x

3 2

x

(3, 0) 0 3 x

0

d

y  f(x)

y (2, 4)

y

(0, 2) (3, 0) 0

3

x

0

e

( 12 ,

x

0)

y (4, 12)

y  f(x) (0, 6) 5

a

y

(6, 0)

y  f(x) f

0

2

1

x

0 (1, 0)

x

2

y

(8, 4)

(0, 2) b

y

0 (2, 0)

y  f(2x) g

1

0

1 2

1

(12, 0)

y (4, 2)

x

(6, 0)

(0, 1)

x

0 (1, 0)

y h

y  f(12 x) 0

4

2

4

y

(4, 4)

(0, 2)

x

(1, 0) 0

(6, 0)

2

x

a y  4, x  1, (0, 2)

y

Exercise 4G 1

a

y

y4

(3, 4)

2 (1, 2) (0, 0) 0 (5, 0)

0

x

1

x

x

Chapter 4 Answers b y  2, x  0, (1, 0)

h y  2, x  1, (0, 0)

y

y

1

y  2

x

0

x

1

0

y2

3

c y  4, x  1, (0, 0)

a A(2, 6), B(0, 0), C(2, 3), D(6, 0)

y

y 0

y4

6 x

(2, 3)

(2, 6) 1

0

x b A(4, 0), B(2, 6), C(0, 3), D(4, 6)

y (2, 6) (4, 6)

d y  0, x  1, (0, 2)

y

3 4

0

x

1

x

0

c A(2, 6), B(1, 0), C(0, 3), D(2, 0)

y

2

1

e y  2, x  12, (0, 0)

0

x

2

3

y (2, 6)

y2

d A(8, 6), B(6, 0), C(4, 3), D(0, 0)

y 0

x

1 2

6

x

0 (4, 3) f

(8, 6)

y  2, x  2, (0, 0)

e A(4, 3), B(2, 3), C(0, 0), D(4, 3)

y

y (2, 3)

y2

2

0

(4, 3)

x

0 (4, 3)

x f

A(4, 18), B(2, 0), C(0, 9), D(4, 0)

y g y  1, x  1, (0, 0)

2

y

0

y1

9 0

1

x (4, 18)

4

x

Chapter 4 Answers g A(4, 2), B(2, 0), C(0, 1), D(4, 0)

iii x  0, y  0

y

y

2

4 x

0 1 (4, 2)

x

0 h A(16, 6), B(8, 0), C(0, 3), D(16, 0)

y iv x  2, y  1, (0, 0)

8 16

0 3

x

y

(16, 6) 2 i

(4, 6)

x

0

A(4, 6), B(2, 0), C(0, 3), D(4, 0)

y  1

y 3

2

j

4

0

v x  2, y  0, (0, 1)

x

y

A(4, 6), B(2, 0), C(0, 3), D(4, 0)

1

y

0 4

0 3

x

2

x

2 (4, 6)

vi x  2, y  0, (0, 1)

y 4

a i

x  2, y  0, (0, 2)

y 2

2 2

1

0

x

x

0

2 b f(x)   x2

x  1, y  0, (0, 1)

ii

Mixed exercise 4H 1

y

a

y

1 1 0

x

0

y  x2(x  2)

2

x

2x  x2 b x  0, 1, 2; points (0, 0), (2, 0), (1, 3)

Chapter 4 Answers 2

a

y y1x

B

A

1

y  x6

0

x

4

a x  1 at A, x  3 at B

5

a

y y  f(x  1) (5, 0)

y  x2  2x  5 b A(3, 2) B(2, 3)

b

y

c y  x2  2 x  5 3

a

(4, 0)

y

A( 32 , 4)

x

0

(1, 0)

y2

(0, 3)

x

B(0, 0) 0

6 b

x

0 (0, 0)

a

y

y A(3, 2)

y1

b

c

A(3, 2)

d

(2, 0)

y

y

0 B(0, 2)

x

(0, 0) 0 (2, 0)

x

0 B(0, 0)

(0, 0) 0

(2, 0) x

x y  0 is asymptote

7

y

a f  1 b i

y  x2  4 x  3

y

A(0, 4)

y2 B(3, 0) e

(1, 0) 0 (0, 1)

x

0

ii

y

(1, 0) x

y

A(6, 4) ( 32 , 0)

y2 0 f

B(3, 0)

y

x

A(3, 5)

y3 B(0, 1) 0

x

( 12 , 0) 0

x (1, 1)

Review Exercise 1 Answers 1 a b c 2 a 3 a 4 5 6 7 8 9

10

11 12

13 14

x(2x  1)(x  7) (3x  4)(3x  4) (x  1)(x  1)(x2  8) 9 b 27 2 b 14

c

2 

1 27

a 625 b 43 x 3 a 45  b 21  8 5  a 13 b 8  2 3  a 63  b 7  4 3  a 567  b 10  13 7  c 16  6 7  a x   8 or x  9 b x  0 or x  72 c x  32 or x  35 a x  2.17 or 7.83 b x  2.69 or x  0.186 c x  2.82 or x  0.177 a a  4, b  45 b x  4  35  a (x  3)2  9 b P is (0, 18), Q is (3, 9) c x  3  4 2  k  6, x  1 (same root) a a  5, b  11 b discriminant < 0 so no real roots c k  25 d y

25

5 0 15 a a  1, b  2 b y

22 a Different real roots, determinant > 0, so k2  4k  12 > 0 b k < 2 or k > 6 23 0 < k < 89 24 a p2  8p  20 > 0 b p < 2 or p > 10 ) (3 13 c x   2 25 a x(x  2)(x  2) b

y

2

0

y

c

1

26

x

2

0 1

3

x

a

y

x

(3, 2)

0

2

4

x

3 (2, 0) (4, 0) and (3, 2)

16 17 18 19

20 21

x 0 c discriminant  8  k < 23  d 23 y  4, x  2 or y  2, x  4 a x2  4 x  8  0 , y  6  23  b x  2  23 x  2, y  1 or x  13, y  137 a x > 14 b x < 12 or x > 3 c 14 < x < 12 or x > 3 a 0 312

b

y

0

1

2 (1 12 , 2)

(1, 0) (2, 0) and (112, 2) 27 a

x

Review Exercise 1 Answers 30 a

y

0

3

x

2

(0, 0) and (3, 0) b

6

1

x

4

(1, 0) (4, 0) and (0, 6) c

y

3 0

2

8

x

(2, 0) (8, 0) and (0. 3) 28 a

y 3

x

0

Asymptotes: y  3 and x  0 b (13, 0) 29 a f(x)  x(x2  8x  15) b f(x)  x(x  3)(x  5) c y

3 0

0

3

(2, 0), (0, 0) and (4, 0) (0, 0) and (3, 0) b (0, 0), ), 10  35 ), (12(1  35 ), 10  35 ) (12(1  35

y

0

y

3

(0, 0), (3, 0) and (5, 0)

5

x

4

x

Chapter 5 Answers Exercise 5A 1 a 2 e 23 i

1  2

2 a 4 e 3

4 5 6 7 8 9 10

7  5

b 1 f j

c 3

5  4 1  2

g

1  2

k 2 23

b 5

c

f

g 2

2

i 9 j 3 a 4x  y  3  0 c 6xy70 e 5x  3y  6  0 g 14x  7y  4  0 i 18x  3y  2  0 k 4x  6y  5  0 y  5x  3 2x  5y  20  0 y  12 x  7 y  23 x (3, 0) (53, 0) (0, 5), (4, 0)

k b d f h j l

d

1  3

h 2 l

32

d 0 h 2

3  2

l 12 3x  y  2  0 4x  5y  30  0 7 x  3y  0 27x  9y  2  0 2x  6y  3  0 6x  10y  5  0

m 1 2 3 4 5 6 7 8

7 12 413 214 1 4

26 5

Exercise 5C

1 a y  2x  1 c y  x  3 e y  12 x  12 g y  2x 2 y  3x  6 3 y  2x  8 4 2x  3y  24  0 5 15 6 y  25 x  3 7 2x 3y  12  0 8 85 9 y  43 x  4 10 6x  15y  10  0

b d f h

y  3x  7 y  4x  11 y  23 x  5 y  12 x  2b

y  4x  4 y  2x  4 yx4 y  4x  9 y  65 x

d 2 h 8 l 12

6 7

a Perpendicular c Neither e Perpendicular g Parallel i Perpendicular k Neither y  13 x 4x  y  15  0 a y  2x  12 c y  x  3 a y  3x  11 c y  23 x  2 3x  2y  5  0 7x  4y  2  0

b d f h j l

Parallel Perpendicular Parallel Perpendicular Parallel Perpendicular

b d b d

y  12 x y  12 x  8 y  13 x  133 y  32 x  127

Mixed exercise 5F 1 2 3 4 5 6 7 8 9 10 11 12 13 14

a y  3x  14 b a y  12 x  4 b a y  17 x  172, y  x  12 b a y  152 x  161 b a y  32 x  32 b 11x  10y  19  0 a y  12 x  3 b a y  32 x  2 b (4, 4) c a 2x  y  20 b a 12 b 6 c 3  3   b a   3 1  3  a 7x  5y  18  0 b b y  13 x  13 a y l2 l1 3 ( 2 , 0)





(0, 0) 0

(0, 14) y  12 x  32, (1, 1) (9, 3) 22 (3, 3)

y  14 x  94 20 y  13 x  43 2x  y  16  0

y  3 x  23  1 2 6 35

x

(0, 3) 4

1

b (3, 3) 15 a x  2y  16  0 c (176, 674)

Exercise 5D 1 a c e g i

1

5

b 16 c 35 1 f 2 g 12 j 4 k 13 2 2 q p n   q  p qp

(3, 0) (0, 1) (0, 312) y  45 x  4 xy50 y   38 x  12 y  4x  13 y  x  2, y  16 x  13, y  6x  23 (3, 1)

Exercise 5E

2 3 4

Exercise 5B 1 a 12 e 1 i 23

2 3 4 5 6 7 8 9 10

b d f h j

yx2 y  4x  23 y  12 x  1 y  8x  33 y  27 x  154

c 12x  3y  17  0 b y  4x

Chapter 6 Answers Exercise 6A

3

1 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 14, 18, 116 Multiply previous term by 12 6 41, 122, 365 Multiply previous term by 3 then 1 7 8, 13, 21 Add together the two previous terms 8 59, 161, 173 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

4 5

1 2

3 4

2

3 4 5 6 7

Exercise 6C 1

2

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 Uk1  Uk  2, U1  3 Uk1  Uk  3, U1  20 Uk1  2Uk, U1  1 U d Uk1  k , U1  100 4 e Uk1  1  Uk, U1  1 f Uk1  2Uk  1, U1  3 g Uk1  (Uk)2  1, U1  0

9, 4, 1, 6 2, 5, 11, 23 2, 3, 8, 63

Uk  2 h Uk1  , U1  26 2 i Uk2  Uk1  Uk, U1  1, U2  1 j Uk1  2Uk  2(1)k1, U1  4

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

Exercise 6E

Exercise 6B

a U1  5 U2  8 U3  11 U10  32 b U1  7 U2  4 U3  1 U10  20 U2  9 U3  14 U10  105 c U1  6 U2  1 U3  0 U10  49 d U1  4 U2  4 U3  8 U10  1024 e U1  2 f U1  13 U2  12 U3  35 U10  56 g U1  13 U2  12 U3  35 U10  56 h U1  1 U2  0 U3  1 U10  512 a 14 b 9 c 11 d 9 e 6 f 9 g 8 h 14 i 4 j 5 Un 4n2  4n  4(n2  n) which is a multiple of 4 Un  (n  5)2  2  0 Un is smallest when n  5 (Un  2) a  12, b  22 a  1, b  3, c  0 p  12, q  512

Uk1  Uk  2, U1  1 Uk1  Uk  3, U1  5 Uk1  Uk  1, U1  3 Uk1  Uk  12, U1  1 Uk1  Uk  2k  1, U1  1 Uk1  Uk  (1)k(2k  1), U1  1 3k  2 b 3k2  2k  2 c 130, 4 4  2p b 4  6p c p  2

Exercise 6D

1 1

a b c d e f a a

2 3 4 5 6 7

a 78, 4n  2 c 23, 83  3n e 27, 33  3n g 39p, (2n  1)p a 30 b 29 d 31 e 221 d6 a  36, d  3, 14th term 24 x  5; 25, 20, 15 x  12, x  8

b d f h c f

42, 2n  2 39, 2n  1 59, 3n  1 71x, (9  4n)x 32 77

Exercise 6F 1

2 3 4 5 6 7 8

a 820 d 294 g 1155 a 20 c 65 2550 i £222 500 1683, 3267 £9.03, 141 days d  12, 5.5 a  6, d  2

b e h b d

450 c 1140 1440 f 1425 21(11x  1) 25 4 or 14 (2 answers)

ii £347 500

Exercise 6G 10

1

a

 (3r  1) r1

c

 4(11  r) r1

30

b

 (3r  1) r1

d

 6r r1

11

2 3 4

a 45 c 1010 19 49

16

b 210 d 70

Chapter 6 Answers

Mixed exercise 6H 1 2 3 4

5 6 7 8 9 10 11 12

13 15 16

17

5, 8, 11 10 2, 9, 23, 51 a Add 6 to the previous term, i.e. Un1  Un  6 (or Un  6n  1) b Add 3 to the previous term, i.e. Un1  Un  3 (or Un  3n) c Multiply the previous term by 3, i.e. Un1  3Un (or Un  3n1) d Subtract 5 from the previous term, i.e. Un1  Un  5 (or Un  15  5n) e The square numbers (Un  n2) f Multiply the previous term by 1.2, i.e. Un1  1.2Un (or Un  (1.2)n1) 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 c 14, 4 b 1500 m a U2  2k  4, U3  2k2  4k  4 b 5, 3 a £2450 b £59 000 c d  30 a d5 b 59

11k  9 18 b  3 c 1.5 d 415

Chapter 7 Answers Exercise 7A 1

2

a i iv b 6 a i iv b 8

Exercise 7G

7 6.01

ii v

9 8.01

6.5 h6

ii v

iii 6.1

8.5 8h

iii 8.1

1

2t  3

2

2

3

12t2

4

9.8

5

1  5r2

6

12  8t

7

10  2x

Exercise 7H

a y  3x  6  0

b

4y  3x  4  0

c 3y  2x  18  0

d

e y  12x  14

f

2

a 7y  x  48  0

b

yx y  16x  22 17y  2x  212  0

3

(129, 189)

4

y  x, 4y  x  9  0; (3, 3) y  8x  10, 8y  x  145  0

1

Exercise 7B 1

7 x6 

4

1 3

7

3x4

x

8 x7

2 2  3



4  3

5

1 4

8

4x5

x

10 5x6

11 13 x

13 2x3 16 9x8

14 1 17 5x4



4  3

3

4 x3

6

1 3

9

2x3

2

x

 3

12 12 x



3  2

5

Exercise 7I

15 3x2 18 3x2

1 4, 1134, 172257 2 0, 22  3 (1, 0) and (123, 91237)

Exercise 7C 1

a 4x  6

b x  12

d 16x  7

e 4  10x

2

a 12

b 6

c 7

d 212

e 2

f

3

4, 0

4

(1, 8)

5

1, 1

6

6, 4

5 (2, 13) and (2, 15) 9 6 a 1  2

b

x

4

x  3

7 x  4, y  20 8

Exercise 7D 1

4 2, 223

c 8x

a 4x3  x2

b x3

c x

3

3  2

1

x

 2

 2x

1 1 dy   9 a   6x 2  32 x 2 b dx 1    12 x 2 (12  3x)

 2

 32 x

1112

2

a 0

b

3

a (212, 614)

b (4, 4) and (2, 0)

c (16, 31)

d (12, 4) (12, 4)

3

 2

3  2

1

 2

10 a x  x  x 1  2

3  2

(4  x) 1

 2 1  2

(4, 16)

b 1 x  x

1 3

 2

c 4116

Exercise 7E 1

1

a x

 2

b 6x3

d

4  3

f

 3 1  3

x  2x 3

2

x

2

4

1  2

 x

 x2 g 3x2

5x  32 x



1  2

h 3  6x2

3 x2  2 x  2

j

l 24x  8  2x2 a 1 b 29

c 4

Exercise 7F 1

24x  3, 24

2

15  3x2, 6x3

3

9  2

4

30x  2, 30

5

3x2  16x3, 6x3  48x4

1

x

 2

1

 2

1  2

3  2

i

e 6x

2

c x4

 6x3, 94 x

3

 2

 18x4

k 12x3  18x2 d 4

11 6x2  12 x

1

 2

 2x2

10 2300 12 ,  3 27 14 a  1, b  4, c  5 15 a 3x2  10x  5 b i

1  3

ii y  2x  7

16 y  9x  4 and 9y  x  128 17 a (45, 25)

b

1  5

iii 725 

Chapter 8 Answers Exercise 8A

Exercise 8D

1  6

1 y  x6  c

2 y  2 x5  c

3 yx c

4 yx

3

5 y  2x

3  5

c

2

1

1

c

5  3

6 y x c

3 

7 y  83 x 2  c

8 y  27 x7  c

9 y  12 x6  c

10 y  x3  c

1 

11 y  2x 2  c 13 y  4x

1

 2

12 y  10x

1

 2

2

c

4 

14 y  92 x 3  c

c

15 y  3x12  c

3

16 y  2x7  c

1  3

17 y  9x  c

18 y  5x  c

19 y  3x  c

20 y  130 x0.6  c

2

1

a y  2x  x

1

3  2

 4x  c

b y  5x3  3x2  2x

3

 2

1

c y  14 x4  3x  6x1  c

5 

h y  x  2x 2

1

 3x

a f(x)  6x  3x 2

1

 2

1  2

c f(x)  x  x

1

 2

d f(x)  5x  4x 2

2  3

f(x)  3x  2x 3

3

c

1  6

4



c

2

2

5

c

 x c 3 

h f(x)  x2  x2  43 x  c

4

x x c 2

5  2

2 2x 4  3

1

 3x  c

4

5 x4  x3  rx  c

6 t3  t 1  c

2 t3 3

1

 2

 6t

a

1  3

x3  2 x2  4 x  c

b

1  3 3

c

1  2

x2  83 x 2  4x  c

d

2  5

e

2  3

x 2  4x 2  c

f

2x 2  43 x 2  c

b

4 x 2  x3  c

3 

5 

3 

3 

1 

3 1  a 2x 2    c

x

5 

3 

1 

3 

x 2  43 x 2  c 1 

2

5  3

1

x  2x    c x

1

c

3 5

1

f

4x 2  65 x 2  c

1 

5 

5  2

3  2

g

1 3

x3  3 x2  9 x  c

j

2 5

x 2  3 x2  6 x 2  c

1  2

h

8  5

x  83 x  2x  c

i

3 x  2 x 2  2 x3  c

1 

3 

5 

a y  x3  x2  2

1

d

y  x4  2  3x  1 x y  6x  12 x2  4

f

y  25 x 2  6x 2  1

b

2  3

5 

1 

1 f(x)  12 x4    12

x

2

3

y  1     x x x3 x4 a f2(x)  ; f3(x)   3

12

f2(x)  x  1; f3(x)  21 x2  x  1; f4(x)  61 x3  21 x2  x  1

2  3

x3  32 x2  5x  c

2 

5 

3 

5 

4 

3 

1 

3  4

x 3  32 x 3  c

b

2x 2  43 x 2  c

b

3  5

b

2x 2  8x 2  c

2

1  3

4  5

4 

b

5 

1 

3 

x 2  23 x 2  6 x 2  c

tc

p 9 x5  2tx  3x1  c 5

8

1 2

x  4x  4x  c 1 

x2  2 x 2  2 x

1 

3 

6 2x 2  4x 2  c

1  2

3  2

3 2 x  x3  c

7

x 2  2x 2  c

x x3  12 x2  3x  c

5 x  t3  t2  t  1; x  7

Exercise 8C 1

4  5

x3  32 x2    16 x 3 a 2 x4  2 x3  5 x  c 2

3  2

4

e

1 a

g f(x)  13 x3  x1  23 x 2  c

1 4

2  3

Mixed Exercise 8F 1  2

1 2

d

3 4 5 … (n  1)

c

 3

x3  6 x2  9 x  c

xn1 b 

c

2

e f(x)  3x  6x

4

 5x  c

b f(x)  x6  x6  x

f

2

 3 x  4 x2  c  2

4  3

e y  x3  2 x2  4 x 

y  3x 3  2x5  12 x2  c

5

c

1  3

e y  4x  4x3  4x  c



3 2x    c

c y  23 x 2  112x3  13

1  2

g y  4x

b

3 

1 

d y  x4  3x 3  x1  c

1  3

x4  x3  c

Exercise 8E

c

1  2

f

1  2

x  2  c d 2    3x  c x x x 1 1 3 4 3 2 e 4 x  3 x  2 x  3x  c

Exercise 8B 2

a

1

 2

7 x  1213 8 a A  6. B  9

c

p 10 t4  q2t  px3t  c 4

9 a

3 2

1

x

 2

 2x 1 

3

 2 3 

10 a 5x  8x 2  23 x 2  c

x 3  92 x 3  9x  c

Review Exercise 2 (Chapters 5 to 8) Answers 1

2

3

4

5

6 7

8 9

10

11 12

13

14

a Since P(3, 1), substitute values into y  5  2x 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, 97) 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  1147 a y  13 x  4 b C is (3, 3) c 15 units2 a P is (181, 1136) 12 1 2 b  64 units a d  3.5 b a  10 c 217.5 a  5 km, d  0.4 km a 3, 1, 1 b d2 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 k4 c 81 a In general: Sn  a  (a  d)  (a  2d)  …  (a  (n  2)d)  (a  (n  1)d) Reversing the sum: Sn  (a  (n  1)d)  (a  (n  2)d)  (a  (n  3)d)  …  (a  d)  a Adding the two sums: 2Sn  [2a  (n  1)d]  [2a  (n  1)d]  …  [2a  (n  1)d]

b c d e 15 a b c

2Sn  n[2a  (n  1)d] n Sn   [2a  (n  1)d] 2 £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 k  1 or k  73 a2  23

d a5  2 e a100  23 1 dy  16   12x2  x 2 dx dy 17 a   4x  18x4 dx b 23 x3  3x2  c 1 dy  18 a   6x  2x 2 dx 3 d2y  b 2  6  x 2 dx 3 

19

20

21

22

23

c x3  83 x 2  c dy d2y a i   15x2  7 ii 2  30x dx 3 dx  b x  2x 2  x1  c 1 dy  a   4  92 x 2  4x dx b Substitute values, 8  8 c 3y  x  20  d PQ  8 10 dy a   8x  5x2, at P this is 3 dx b y  3x  5 c k  53 a At (3, 0), y  0 b At P, y  7x  21 c Q  (5, 15 13) a P  2, Q  9, R  4 1 

1 

3 

b 3x 2  29x 2  2x 2 c When x  1, f’(x)  512, gradient of 2y  11x  3 is 512, so it is parallel with tangent. 24

1  3 3

x  2x2 3x  13 5 

1 

25 3x  2x 2  4x 2  3 26 a 3x2  2 b 3x2  2  2 for all values of x since 3x2  0 for all values of x c y  14 x4  x2  7x  10 d 5y  x  22  0 27 a y  13 x3  x1  43 b (1, 2) and (1, 23) 28 a 2x3 5x2  12x b x(2x  3)(x  4) c y

 32

0

(32, 0), (0, 0) and (4, 0)

4

x

Review Exercise 2 Answers 29 a PQ2  12  132  170 PQ  170  dy b   3x2  12 x 4x2 dx dy dy At P,   13, at Q,   13 dx dx c x  13y  14  0 30 a x(x  3)(x  4) b y

0

3

(0, 0), (3, 0) and (4, 0) c P  (347, 157)

4

x

Practice paper Answers 1 a 4 b 64 3 

2 2x3  23 x 2  c 3 a 3, 5 b 36  4 a 27  102 b 202  5 x  3, y  3 and x  8, y  23 6 a x  2y  13  0 b y  2x c (235, 515) 7 a No intersections.

y 2 2

0

2

x

2

b

y (12 , 0) 2

0

2

x

3 5

c

y 2 (0, 2) 2

0

2

x

2

x  1 8 a 670 b 5350 9 a i 2 ii c  4 b i x5 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 Answers 1

a b 2 a b 3 a

k5 k6 9 1 8x3

y

x

2

b 4 a b 5 a 6 a b 7 a b c 8 a b c 9 a c

x  0 and y  2 3 420 0  k  2.4 a2  1, a3  4 24 3 24x2  32 x2 5 9    48x  4 x 2 1 2x4  6x2 + 5x  c 0.4 (137, 43) 4 x3  3 x  2 y

(2, 0)

10 a y  9x  9  d 16 2 e 320 units2

(1, 0)

x