# Edexcel IGCSE Further Pure Maths answers

3 Edexcel IGCSE Further Pure Mathematics 2.3 Exercise 2C 1 x 5 0 or x 5 4 2 x 5 0 or x 5 25 3 x 5 0 or x 5 2 4 x √5 0 or...

4

a

625

b

9

c

7

d

2

5

a

1.30

b

0.602

c

3.85

d

20.105

1.1 Exercise 1A

6

a

1.04

b

1.55

c

20.523 d

2

3

a e i m q a d g a

x7 k5 2a3 27x8 4a6 x5 x3 1 _ 3x 2 65

b f j n r b e h b

6x5 y10 2p27 24x11 6a12 x22 x5 5x 69

e

6 _3 1

f

6 ___ 64

j

125

i

c g k o

2p2 5x8 6a29 63a12

3x22 p2 3a2b22 32y6

d h l p

c f i c

x4 12x0 = 12 6 x21 3 d

1 ___ 125

g

1

h

66

9 _ 4

k

5 _

l

64 __

6

__

1 __ 16

9 7√2

__

12 9√5

2

__

13 23√5

14 2

√5 16 ___ 5__ √ 5 19 ___ 5___ √ 13 22 ____ 13

√ 11 17 ____ 11 1 20 __ 2 1 23 __ 3

__

__

11 23√7

15 19√3 __

___

√2 18 ___ 2 1 21 __ 4

2

3

a c a c e a e

log4 256 5 4 log10 1 000 000 5 6 241 = 16 _ 92 = 3 105 = 100 000 3 b 2 1 6 f _2

3 4

d a c e

e

log10 120 b log6 36 = 2 1 d log8 2 = _3

a b c

3 loga x 1 4 loga y 1 loga z 5 loga x 2 2 loga y 2 1 2 loga x

d

loga x 1 _2 loga y 2 loga z

e

1 1 _ 1 _ log

log2 8 = 3 log12 144 = 2 log10 10 = 1

a d a a a d a

c g

2

a

2

2.460 0.458 1.27 1 _ 2 , 512 6.23 1.66 1 _ 2 , 512

3.465 0.774 2.09 1 _ 1 __ 16 , 4 2.10

c

b

1 _ 1 __ ,

c 2.52

4.248

c 0.721 c 0.431

16 4

y x

x

y=6

y=4

x

(b)

1

d h

b e b e b

y = ( 41)

log3 (_9 ) 5 22 log11 11 5 1 52 = 25 521 = 0.2 7 21

log5 80

1

1

b d b d

c

1.6 Exercise 1F

1.3 Exercise 1C 1

log2 9

1

8 6√5

10 12√7

b

68 log6 (__ 81 )

__

7 √3

__

log2 21

1.5 Exercise 1E

√2 3 5__ 6 √3

__

a

__

2 6√___ 2 5 3√10 __

3

49

1 2√__ 7 4 4√2 __

1 2

1.2 Exercise 1B __

1.4 Exercise 1D

Edexcel IGCSE Further Pure Mathematics

1

3.00

(a) 1 (c)

1 10

(4  )

1 NB __

x

2x

=4

x

2x

so (c) is y = 4

1

2

x

y = ( 31 )

7

y y=3

(c)

x

So x = log580

10 _______ 10

y = log3x

i.e. x = 2.72270 …

(b)

= 2.72 3sf

x

1

Edexcel IGCSE Further Pure Mathematics

(=  loglog 805 )

(a) 1

7x = 123

b ⇒

x = log7123

(=  loglog 123 7 ) 10 ________

x

10

NB y = log3x is a reflection of y = 3 in the line y = x. x 1 y = __ is y = 32x 3

i.e. x = 2.47297 …

(  ) y

3

8

y = log4x

y = log3x

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

y = lx

1

x

1 (b)

y = 1x = 1 y = log3x = 1 ⇒ x = 31 = 3 So coordinates of intersection are (3, 1)

Exercise 1G a

y8

2

a

3x6

3

a

4

a

1

5

a

6

a

b

1 _ ,9

2 4 6 8 10 12 14 16 18 20 22 24

2x(x 1 3) (x 1 6)(x 1 2) (x 2 8)(x 2 2) (x 2 6)(x 1 4) (x 1 5)(x 2 4) (3x 2 2)(x 1 4) 2(3x 1 2)(x 2 2) 2(x2 1 3)(x2 1 4) (x 1 7)(x 2 7) (3x 1 5y)(3x 2 5y) 2(x 1 5)(x 2 5) 3(5x 2 1)(x 1 3)

2 4 6 8 10

(x 2 3)2 2 9 1 (x 1 _2 )2 2 _1 4 2(x 1 4)2 2 32 2(x 2 1)2 2 2 5 25 2(x 2 _4 )2 2 __

9

2.1 Exercise 2A

y

a c

c

Chapter 2

(a)

4

= 2.47 3sf b 12

9

3

x

(b)

a

1 9 _3 , 9 10 2_1 , 22

y = log6x

1

(a)

2

5x = 80

a

6x7

b 62 3375 4 __ b _____ 9__ 4913 __ √7 ___ b 4√5 7__ √3 15 ___ __ b ___ 3 √5 2 logd p + logd q

c

32x

d

c

6x2

d

12b9 1 _31 __ x 2

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

2.2 Exercise 2B 1 3 5 7 9

(x 1 2)2 2 4 (x 2 8)2 2 64 (x 2 7)2 2 49 3(x 2 4)2 2 48 5(x 1 2)2 2 20

11 3(x 1 _2 )2 2 __ 4 3

b

loga p = 4, logd q = 1

27

8 1

12 3(x 2 _6 )2 2 __ 12 1

1 3 5 7 9 11

x 5 0 or x 5 4 x 5 0 or x 5 2 x 5 21 or x 5 22 x 5 25 or x 5 22 x 5 3 or x 5 5 x 5 6 or x 5 21

2 4 6 8 10 12

x 5 0 or x 5 25 x 5 0 or x 5 6 x 5 21 or x 5 24 x 5 3 or x 5 22 x 5 4 or x 5 5 x 5 6 or x 5 22

1 13 x 5 2 _2 or x 5 23 3 2 15 x 5 2 _ or x 5 _

3 1 14 x 5 2 _3 or x 5 _2 3 5 16 x 5 _ or x 5 _

17 x 5 _3 or x 5 22

18 x 5 3 or x 5 0

19 x 5 13 or x 5 1

20 x 5 2 or x 5 22

√5 21 x 5 6 ___

22

3

2

1

__

3

___

√ 11 23 x 5 1 6 ____ 3

2

7

__

27 x 5 23 6 2√2 ___

26 x 5 0 or x 5 2 __ 62 11

___

28 x = 26 ± √33 __

29 x 5 5 6 √30

30 x = 22 ± √6

3 √29 31 x 5 __ 6 ____ 2 2

3 __ 32 x = 1 ± __√2 2

1 √129 33 x 5 __ 6 _____ 8 8

34 No real roots

3 √39 35 x 5 2 __ 6 ____

4 √ 26 36 x = 2 __ ± ____

___

____

___

2

___

5

5

1

x2 2 2x + 1 = 0 ⇒ (x 2 1)2 = 0 2

__

___

+3 6 √17 10 _________ , 20.56 or 3.56 2 __

11 23 6 √3 , 21.27 or 24.73 ___

5 6 √ 33 12 ________ , 5.37 or 20.37 2 ___ √ 31 13 5 6 ____, 23.52 or 0.19 3 __

1 6 √2 14 _______ , 1.21 or 20.21 2

14___ 22 6 √19 16 __________ , 0.47 or 21.27 5 ___

21 6 √78 18 __________ , 0.71 or 20.89 11

so equal roots x51

so two real roots __ __ √ 2 ± 8 x 5 ______ = 1 ± √ 2 or 2.41, 20.414 3sf 2 3 b2 2 4ac = (23)2 2 4(22) 5 17 so two real roots

2.5 Exercise 2E x2 + 5x + 2 = 0 a + b = 25 ab = 2

1

___

4

2

___

√ 53 15 9 6 ____, 20.12 or 21.16

1

b2 2 4ac = (22)2 2 4(21) 5 8

3 ± √17 x 5 ________

2x2 = x 1 4 = 0 b2 2 4ac = (21)2 2 4 3 (2) 3 (24) = 33 so two real roots ___ √ 1 ± 33 x = ________ 4 x = 1.69 or 21.19 3sf

17 2 or 2 _4

2.4 Exercise 2D b2 2 4ac = (22)2 2 4 3 1 5 0

2x2 2 x 2 4

8

√5 9 23 6 ___, 20.38 or 22.62 2

___

x 5 3 6 √13

7

1

3x2 + x 2 7

2

24 x 5 1 or x 5 2 _6

25 x 5 2 _2 or x 5 _3

2

3x2 = 7 2 x = 0 b2 2 4ac = 12 2 4 3 3 3 (27) = 1 + 84 = 85 so two real___ roots 21 ± √85 _________ x = 6 x = 1.37 or 21.70 3sf

7

Edexcel IGCSE Further Pure Mathematics

2.3 Exercise 2C

= 3.56 or 20.562 3sf

b2 2 4ac = (23)2 2 4 3 4 5 9 2 16 = 27 so no real roots

a

2a + 1 + 2b + 1 = 2(a + b) + 2 = 210 + 2 = 28 (2a + 1)(2b + 1) = 4ab + 2(a + b) + 1 = 8 2 10 + 1 = 21  new equation is x2 1 8x 2 1 5 0

b

ab + a2b2 = ab(1 + ab) = 2(1 + 2) = 6 (ab)(a2b2) = (ab)3 = 23 = 8  new equation is x2 2 6x 1 8 5 0

5 b2 2 4ac = (1)2 2 4 3 2 3 (22) 5 17 so two real roots ___

21 ± √17 x 5 _________ = 0.781, 21.28 3sf 4 6 b2 2 4ac = (21)2 2 4 3 3 3 3 5 235 so no real roots

3

___

x2 + 6x + 1

2

=0 a + b = 26 ab = 1

a

Edexcel IGCSE Further Pure Mathematics

b

4

(a + 3) + (b + 3) = (a + b) + 6 = 0 (a + 3)(b + 3) = ab + 3(a + b) + 9 = 1 2 18 + 9 = 28 2  new equation is x 2 8 5 0 b a2 + b2 (a + b)2 2 2ab a __ __ + = _______ = ______________ b a ab ab 36 2 2 _______ = = 34 1 b a __ __ 3 =1 b a  new equation is x2 2 34x 1 1 5 0 x2 2 x + 3

3

=0 a+b=1 ab = 3

a

b

4

a

b

a

25 ± √17 _________ , 20.44 or 24.56

b

2 ± √ 7 , 4.65 or 20.65

c

23 ± √29 _________ , 0.24 or 20.84

d

2

__

___

10 ___ 5 ± √73 ________ , 2.25 or 20.59 6

a

6

64

7

a

ab 5 t, a2 1 b2 5 2t(2t 2 1)

b

√ 577 t 5 1 1 _____ 2

c

x2 2 2√577 x 1 1 5 0

____

____

2x2 2 7x 1 3 = 0

8

x2 2 _2 x + _2 = 0 7

7

ab = _2 3

a2 + b2 = (a + b)2 2 2ab = 12 2 6 = 25 a2 3 b2= (ab)2 = 32 = 9  new equation is x2 1 5x 1 9 5 0 x2 1 x 2 1 = 0 a + b = 21 ab = 21 1 b + a 21 1 __ __ + = _____ = ____ = 1 a b ab 21 1 __ 1 ___ 1 ____ 1 __ 3 = = = 21 a b ab 21  new equation is x2 2 x 2 1 5 0 b a+b a _____ ___ + = _____ = 1 ab a + b a + b

3

a

_ __ a2 + b2 = (a + b)2 2 2ab = __ 4 2232= 4

b

a 2 b = √(a 2 b)2 = √ a2 + b2 2 2ab

49

________

4

37

3

_____________

___

________

_ __ _ = √__ 4 2 2. 2 = √ 4 = 2 37

c

3

25

5

a3 2 b3 = (a 2 b)(a2 + b2 + ab) _ _ __ ___ = _2 ( __ 4 + 2) = 2 3 4 = 8 5 37

3

5

43

215

x2 2 2tx + t = 0

9 a

a + b = 2t ab = t a2 + b2 = (a + b)2 2 2ab = 4t2 2 2t = 2t(2t 2 1)

b

a + b = 2t a 2 b = 24 ⇒ 2a = 2t + 24 a = t + 12, b = t 2 12  ab = t ⇒ (t + 12)(t 2 12) = t

a c a c e

x(3x 1 4) x(x 1 y 1 y2) (x 1 1)(x + 2) (x 2 7)(x 1 5) (5x 1 2)(x 2 3)

b d b d f

2y(2y 1 5) 2xy(4y 1 5x) 3x(x 1 2) (2x 2 3)(x + 1) (1 2 x)(6 + x)

a

y 5 21 or 22

b

x 5 _3 or 25

c

1 x 5 2 _ or 3

d

√7 5 6 ___

5

3

a + b = _2

(a + 2) + (b + 2) = (a + b) + 4 = 1 + 4 = 5 (a + 2)(b + 2) = ab + 2(a + b) + 4 =312+4=9  new equation is x2 2 5x + 9 5 0

Mixed Exercise 2F 2

b

p 5 3, q 5 2, r 5 27

b ab a 21 _____ 3 _____ = _______ = ____ = 21 a + b a + b (a + b)2 1 2  new equation is also x 2 x 2 1 5 0

1

__

√7 22 6 ___ 3

5

2

__

2

i.e. t2 2 144 = t or 0 = t2 2 t 2 144 ________

____

1 + √ 577 1 ± √ 1 + 576  t = ____________  t = _________ (t > 0) 2 2 c

2

2

b a +b 2t(2t 2 1) a __ __ + = _______ = _________ = 2(2t 2 1)

b a ab t b a __ 3 __ = 1 b a  equation is x2 2 2(2t 2 1) x + 1 = 0 ____

or x2 2 2√ 577 x + 1 = 0

Chapter 3

3.3 Exercise 3C

1

2

a c e a c e

x2 1 5x 1 3 x2 2 3x 1 7 x2 2 3x – 2 6x2 1 3x 1 2 2x2 2 2x 2 7 25x2 1 3x 1 5

b d

x2 1 x2 9 x2 2 3x 1 2

b d

3x2 1 2x2 2 23x2 1 5x 2 7 + 6x2

2x3

3

a

b

2x3 + 5x2 2 5x + 1

2x2

=

x2 + 3x 2 1

3x3 + 2x2 2 3x 2 2

=

(3x 1 2)(x2 2 1)

=

x2 2 1

=

(3x 2 1)(2x2 1 x 2 2)

=

2x2 1 x 2 2

3

2

(3x + 2)

6x3 + x2 2 7x 1 2 3

2

6x + x 2 7x 1 2  ________________ 3x 2 1

d

4x3 + 4x2 1 5x 1 12

2x2

 ___________________ = 2x 1 3 2x3 + 7x2 1 7x 1 2

=

2x3 + 7x2 1 7x 1 2  _________________ = 2x 2 1

3.2 Exercise 3B 1 2 3 4 5

6

7 8 9

1

+3x2

(x 2 1)(x 1 3)(x 1 4) (x 1 1)(x 1 7)(x 2 5) (x 2 5)(x 2 4)(x 1 2) (x 2 2)(2x 2 1)(x 1 4) a (x 1 1)(x 2 5)(x 2 6) b (x 2 2)(x 1 1)(x 1 2) c (x 2 5)(x 1 3)(x 2 2) a (x 2 1)(x 1 3)(2x 1 1) b (x 2 3)(x 2 5)(2x 2 1) c (x 1 1)(x 1 2)(3x 2 1) d (x 1 2)(2x 2 1)(3x 1 1) e (x 2 2)(2x 2 5)(2x 1 3) 2 216 p 5 3, q 5 7

c

26

d

0

1

a

x 5 5, y 5 6 or x 5 6, y 5 5

b

x 5 0, y 5 1 or x 5 _5 , y 5 _5

c

x 5 21, y 5 23 or x 5 1, y 5 3

d

x 5 4_2 , y 5 4_2 or x 5 6, y 5 3

e

a 5 1, b 5 5 or a 5 3, b 5 21

4

1

3

1

2

f u 5 1_2 , v 5 4 or u 5 2, v 5 3 (211, 215) and (3, 21)

16x2

3

(21_6 , 24_2 ) and (2, 5)

2x2 2 x 1 4

4

a

x = 21_2 , y = 5_4 or x = 3, y = 21

b

x = 3, y = _2 or x = 6_3 , y = 22_6

a

x 5 3 1 √ 13 , y 5 23 1 √13 or x 5 3 2 √13 , ___ y 5 23 2 √__13 __ __ x 5 2 2 3√__ 5 , y 5 3 1 2√5 or x 5 2 1 3√5 , y 5 3 2 2√5

22x2

=

a 27 b 27 18 30 29 8 8__ 27 a = 5, b = 28 p 5 8, q 5 3

3.4 Exercise 3D

22x2

4x3 + 4x2 1 5x 1 12

e

(2x 2 1)(x2 + 3x 2 1)

2x3 + 5x2 2 5x + 1  _________________ 2x 2 1

3x + 2x 2 3x 2 2  _________________ c

=

1 2 3 4 6 7 8 9

Edexcel IGCSE Further Pure Mathematics

3.1 Exercise 3A

(2x 1 3)(2x2 2 x 1 4)

1

16x2

(2x 1 1)(x2 1 3x 1 2) 1x2

5

x2 1 3x 1 2

1

b

1

3

1

1

5

1

___

___

___

3.5 Exercise 3E 1

2

3

a

x,4

b

x>7

c

x . 2_2

d g j a d g j a d

x < 23 x . 212 8 x>3 x , 18 x,4 3 x > _4 1 x . 2_2 No values

e h k b e h

x , 11 x,1 1 x . 1_7 x,1 x.3 x . 27

f i

x , 2_5 x < ??

c f i

x < 23_4 2 x > 4_5 1 x < 2 _2

b e

2,x,4 x54

c

2_2 , x , 3

1

3

1

1

5

3.6 Exercise 3F

2

a c e

3,x,8 x , 22, x . 5 1 2 _2 , x , 7

b d f

24 , x , 3 x < 24, x > 23 1 x , 22, x . 2_2

g

1 1 _ , x , 1_

h

x , _3 , x . 2

j l b d

1 2 x , 22_, x . _

i k a c

2

2

23 , x , 3 x , 0, x . 5 25 , x , 2 1 _ 2,x,1

1

2

3

21_2 , x , 0 x , 21, x . 1 1 23 , x , _4 1

13 a Let a = no. of adults, and a + c < 14

c = no. of children. (no more than 14 passengers) (money raised must cover cost of £72) (more children than adults) (at least 2 adults)

12a + 8c > 72 c .a

3.7 Exercise 3G 1 2 3 4 5

23 < x , 4 y , 2 or y > 5 2y 1 x > 10 or 2y 1 x < 4 22 < 2x 2 y < 2 4x 1 3y < 12, y > 0 and y , 2x 1 4 3x 3x 2 3 6 y . < ___ 2 3, y < 0 and y > 2___ 4 2 3x 2x 7 x > 0, y > 0, y , 2 ______ and y < 2 ______ 219 316 8 y > 0, y < x + 2, y < 2x 2 2 and y < 18 2 2x 9

a >2 c a=2

14 P

12 10

+

Edexcel IGCSE Further Pure Mathematics

1

12

a=c

8 6

Q M

4

N

2

4

+

2

6

8

3a + 2c = 18

10

11

6

10

12

14

d

a + c = 14

NB 12a + 8c > 72 requires line 3a + 2c = 18 b To find smallest sized group you need to consider points close to M and N M(2, 6) is 2 adults and 6 children Points close to N are (3, 5) and (4, 5) So the smallest sized group is 8: 2 adults and 6 children or 3 adults and 5 children. c To find the maximum amount of money that can be made you need to consider points close to P and Q P(2, 12) raises 2 3 12 1 12 3 8 = £120 Q(7, 7) is not in the region ( c . a) but (6, 8) is on d (6, 8) raises 6 3 12 1 8 3 8 = £136 So the maximum amount available for refreshments is £64 from taking 6 adults and 8 children

y

28

28

24

24

20

20

+

b

16

R

16

3b + 2a = 40 12 R

S

+

12

8

+

8

4

10a + 14b = 140

28

T

d

4

16

20

24

28

32

x

To find maximum profit drag the profit line towards the edges R, S, T. It will first cross at T, then R and finally S. T (16, 0) gives a profit of £192 R (0, 18) gives a profit of £270 S is not a point giving whole numbers for x and y but the nearby points are (6, 14) and (7, 13) (6, 14) gives a profit of 6 3 12 + 14 3 15 = £282 (7, 13) gives a profit of 7 3 12 + 13 3 15 = £279 So maxmimum profit is £282 from making 6 ornament A and 14 ornament B.

a = no. of machine A b = no. of machine B 4a + 5b 5

a 2i 2 11j 73.9° a – 15 200 1 a r 5 _2 , r 5 23 46.5°, 133.5°

b

13 2 __ i2_j

b

12 791

b

10

5

5

1 _ 2

a a a c e 37 a b c d e f 38 – 39 a c

40 a b 41 a d 42 a 43 a c

2y 1 x 5 25 b (25, 0) c (10, 0) 1 4 m/s2 b 25_3 m 4y 5 x 1 23 b y 5 24x 1 26 16 (23, 38) d 6__ 17 28 12__ 51 22p 1 q 5 28, 3p 1 q 5 18 22, 24 (x 1 2)(x 2 3)(x 2 4) – 12 2 , 2 __ 5

7__ 12 7

2y 5 3x 2 18 156 x2 x x3 1 1 ___ 2 ____2 1 _____3 2p 8p 16p p 5 6 _2 – b – e – b 1 2 2x 2 4x2 0.087%

b d

3y 5 22x 1 51 216p

1

c

– 408 4000 b d

4.76

2.76132 a 5 1, b 5 25, c 5 8

1

44 45 46 47

3 _ 8

65 66 67

3

112

51 52 53 54 55

56 57 58 59

60 61 62 63

O

68

x

4 m/s2 b 90 m 1 1 4 _ _ b 2 a b 2b 2 _3 a c – 2 3 dy a ___ 5 10x cos 3x 2 15x2 sin 3x dx dy 3e3x(x2 1 3) 2 2x e3x b ___ 5 ___________________ dx (x2 1 3)2 0.212 m/s a 1.39 b 28.7° p , 25, p . 2 a 1, 3.75, 5.89, 6.92 b – c 0.79 d 2.1 7 7 9 9 _ _ a A 5 2 2, B 5 2 4 b 2 _4 , x 5 _2 c (1, 4), (7,10) d (2, 0), (5, 0) e – f 24 (22, 1), (21, 3) p2 a i __ 1 6 ii 9 b p 5 64 4 c x2 2 10x 1 9 5 0 9 a 2 __ b 2 11 , 5 c 4 d 16 380 dy a ___ 5 10x e2x 1 2(5x2 2 2)e2x dx dy 2x3 2 x4 1 4x 2 2 b ___ 5 _________________ dx (x 2 x2)2 ln 4 91.1° 2 23__5 x2 x x2 x b 1 1 ___ 2 ___ a 1 1 ___ 2 ____ 12 144 12 72 x x2 c |x| , 4 d 1 1 __ 2 ___ 6 72 e 0.308

48 a 49 a 50

1

cos 2A 5 2 cos2 A 2 1 sin 2A 5 2 sin A cos A – 17.7°, 102.3°, 137.7° __ 3√ 3 ____ e 8 (6, 21), (1,4) a p5 1 5p4qx 1 10p3q2x 1 10p2q3x 1 5pq4x4 1 q5x5 6 12 b p 5 _5 , q 5 __ 5 or p 5 22, q 5 4 a 3 b q 5 20 c a 5 2, b 5 1 d 9 ___ ___ ___ a i √20 ii √40 iii √20 b A 5 90°, B 5 C 5 45° ___ c (5, 5) d √10 660° 67.4° a 2 b log p c r 5 n 2 1, s 5 n d – 12 12 2 a 2x 2 5x 1 2 5 0 b x2 2 ___ x 1 ___ 5 0 p p 8 3 c _3 d _2 24 , p , 3

64 a b c d

69 70 71 72 73

Practice examination papers

Edexcel IGCSE Further Pure Mathematics

e |x| , _6 2y 5 x 2 2 p[_14 e8 1 4e4 1 27_34 ] a – b c 15, 75, 105, 165 d a i y52 ii x 5 21 3 b (0, 3), (2 _2 , 0) c y

Paper 1 1 2 3 4 5

80.4° or 99.6° a – b p 5 210, q 5 33 20 cm2/s x 5 2 y 5 3, x 5 3 y 5 2 4 a p 5 6, q 5 24 b 5i 1 _3 j

6 a 7 a 8 a b 9 a 10 a c 11 a d

__

( _13, 2√_53 )

b 7r __ 3 ___ 1

25 __ p 3

c 3_2 d 11 2 2 a6 1 6a5bx 1 15a4b2x2 1 20a3b3x3 1 15a2b4x4 1 6ab5x5 1 b6x6 4 4 a 5 2 b 5 _3 , a 5 22 b 5 2 _3 5 b 28 c 9, 3 (2, 4) b y 5 4x 2 4 y54 b 8 units2 11.0 cm b 11.9 cm c 40.1° 101.4° e 61.9° 5

b

1

29

Paper 2 1 2e2x sin 3x 1 3e2x cos 3x 2 a 37.0° b 17.2 cm2 2 3 a i y53 ii x 5 2 b i (2_3 , 0) c y

ii (0, 4)

4

Edexcel IGCSE Further Pure Mathematics

3

30

O

4 a

x

0

y

1

2

0.5

x

2 23

1.0

1.5

2.0

2.5

graph drawn 0, 3, 4 1 6 _2

7 a 8 a 9 a d

(ln 3, 36), (0, 4) b 32.02 c 82.2 units2 – b 2.71 c – d 138 – ___ b 2280 c 37 46√37 e 9x2 1 280 1 3 5 0

c i 1.9 – a 5 120x

b b

__

b

√3 1 1 __ i _______ √3 2 1

c

2 tan u tan 2u 5 _________ 1 2 tan2 u

d

√2

e

20 __

__

29

21

3.5

4.0

0.649 21.28 24.52 28.61 212.8 215.9 215.9 29.40

b 5 a 6 a

10 a

3.0

ii 1.3 c 11.8 m c 4

__

√3 2 1 __ ii _______ √3 1 1