Answers Chapter 1
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