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
<100
(floor area is 100m2)
2a + 3b
<40
(no. of operators is 40)
Profit would be P = 100a + 140b
(this is parallel to 10a + 14b)
The maximum profit wil run along RQ
Nearest to Point R (0, 13) will raise £1820
Point (1, 12) will raise £1780
Point Q (20, 0) will raise £2000
So maximum profit is from 20 Machine A
Let
12
4a + 5b = 100
Let
15
8
+
24
+
16
+
12
+
8
+
4
+
4
Q
20
x = no. of ornament A
y = no. of ornament B
3x + 2y
<48
(machine time is 48 h)
1.5x + 2.5y <45
(craftsman’s time is 45 h)
Profit P = 12x + 15y (e.g. 12x + 15y = 120 is drawn)
Edexcel IGCSE Further Pure Mathematics
+
14
Mixed Exercise 3H
1 x 5 4, y 5 3_2
3
1
2 (3, 1) and (22_5 , 21_5 )
1
3 x 5 21_2 , y 2 2_4 and x 5 4, y 5 2 _2
1
4 a x . 10_2
b x , 22, x . 7
1
1
1
5 3,x,4
1
6 a x , 2_2
1
b _2 , x , 5
c _2 , x , 2_2
x , 0, x . 1
A = 2, B = 4, C = 25
p 5 1, q 5 3
(x 2 2)(x + 4)(2x 2 1)
7
1
7_4
a p 5 1, q 5 215
a r 5 3, s 5 0
a (x 2 1)(x 1 5)(2x 1 1)
22
218
1
7
8
9
10
11
12
13
14
15
16
17
1
b
b
b
(x 1 3)(2x 2 5)
13
1__
27
1
25, 2 _2 , 1
7
__
3 √5
18 2, 2 __ ± ___
2
2
1
19 _, 3
i
2
y
2
20 A: y . 2x + 2, 4y + 3x + 12 . 0
B: y , 2x + 2, 4y + 3x + 12 . 0
C: y , 2x + 2, 4y + 3x + 12 , 0
D: y . 2x + 2, 4y + 3x + 12 , 0
21 {y > 3 and 2x 1 y < 6}
22
Edexcel IGCSE Further Pure Mathematics
j
y
1
2
2
0
x
2
1
2
a
0
3
b
y
x
1
2
y
2
0
1
x
1
1
c
0
2
d
y
x
1
y
2
Chapter 4
e
4.1
Exercise 4A
1
a
b
y
1
f
y
2 x
0
2
y
x
0
2
y
2 x
0
1
x
0 1
6
0
3 2
10
2
c
x
3
6
6
y
i
3
3
e
x
10
2
y
10
3
f
h
y
3
3
0
3
d
y
y
g
x
1
x
1
0
y
j
y
3 x
1
1 x
0
y
2
x
0
x
2
24
3
0
g
2
8
h
y
1 0
10
4 x
3
1
x
2
0 1
x
b y = x(x + 4)(x 1 1)
y
y
x
2
y
1
a y = x(x + 2)(x 2 1)
0
1
x
4
10
x
y
x
0
1
e
d y = x(x + 1)(3 2 x)
y
1
e y = x2(x 2 1)
y
x
0
3 x
0
f y = x(1 2 x)(1 1 x)
y
y
2
y
a
b
y
27
0
x
1
10
x
1
x
0
3
0
g y = 3x(2x 2 1)(2x 1 1) h y = x(x + 1)(x 2 2)
27
y
y
c
10
2
0
1
0
3
1
9
x
0
b
x
0
d
x
2
x
x
1
2
y
x
y
0
y
1
8
y
0
x
1
e
4.3
Exercise 4C
1
c
y
8
0
0
x
4.2
Exercise 4B
0
d
x
y
y
a
2
j y = x2(x 2 9)
i y = x(x 2 3)(x 1 3)
1
y
x
1
2
3
x
3
Edexcel IGCSE Further Pure Mathematics
c y = x(x + 1)2
y
y
y
0
x
4
x
x
0
y
2
x
9
2
b
y
x
x
0
Edexcel IGCSE Further Pure Mathematics
ii
2
x
c
y
3
x
1
3
iii x(x 1 2) 5 2 __
x
i
y
2
y
x
y
4
y
x
0
1
x
1
y
y
8
x
d
y
x
0
y
3
y
5
ii 3
iii x2 5 (x 1 1)(x 2 1)2
i
x
0
y
x2 (1
x)
x
1
y
3
y
ii
x
x
0
y
e
8
x
2
2
iii x2(1 2 x) 5 2 __
x
i
y
4.4
Exercise 4D
1 a
0
y
i
y
y
x2
ii
1
0
y = x(x2 2 1)
ii 3
iii x2 5 x(x2 2 1)
10
2)
x
0
4
x(x
x
x
y
y
0
2
2
y
y
y
2
y
3
i
1
x
1
1
x
1
iii x(x 2 4) 5 __
x
4
y
x( x
x
4)
2
x
(x
1)(x
1)2
y
i
j
y
x( x
i
y
4)
y
0
g
x
y
3
1
iii x(x 2 4) 5 2 __
x
i
2 a
y
4
y
y
y
2)3
(x
x2(x
4)
x
0
2
0
x3
ii 3
iii 2x3 5 2x(x 1 2)
y
h
x
0
2
x( x
x
y
4)
b
3 a
ii 1
iii x(x 2 4) 5 (x 2 2)3
i
x(4
x)
(0, 0); (4, 0); (21, 25)
y
y
1
y
2
y
x
x)3
y
x(2x
y
x
0
i
x(1
x3
b
4 a
x
0
2.5
ii
2)
1
y
ii
x
4
x(x
Edexcel IGCSE Further Pure Mathematics
f
5)
(0, 0); (2, 18); (22, 22)
y
2
2
iii 2x3 5 2 __
x
i
y
y
(x
1)(x
1)
y
y
x2
1 0
(x
1)3
x
1
x
0
y
x3
b
5 a
(0, 21); (1, 0); (3, 8)
y
y
ii 2
iii 2x3 5 x2
x2
x
0
y
b
27
x
(23, 9)
11
y
6 a
y
x2
11 a
0
Edexcel IGCSE Further Pure Mathematics
y
b
7 a
x(x
2
2)(x
0
1
y
b
12 a
2
0
x2(x
x3
3x2
y
3)
1
Only 2 intersections
y
(x
y
0
3x(x
1)
b
13 a
1
(x2
1)(x
2)
x
2
(0, 2); (23, 240); (5, 72)
y
x
1
y
2
1)3
(x
2)(x
x
0 2
y
Only 1 intersection
x2
y
9 a
1
y
0
x
x(x
1)2
Graphs do not intersect
y
1
2
y
01
x(x
2
2
y
1
(0, 28); (1, 29); (24, 224)
x
1
y
b
2
y
10
b
10 a
14x
x
3
y
b
4x
y
2
b
8 a
x
(0, 0); (22, 212); (5, 30)
x
y
6x
4
y
3)
(0, 0); (2, 0); (4, 8)
y
y
x
3
b
12
y
2x
2)2
x
4x2
1, since graphs only cross once
4.5
Exercise 4E
1
y
O
4
x
8
2)2
y
2
Transformation f(x + 1) gives
y
x
1
1
y = _____
x+1
x
1 O
Finally transformation f(x) + 3 gives
y
y
y=3
1
y = 3 + _____
x+1
x
1 12
x
O
Asymptotes x = 21, y = 3
x = 1
y
4
1
1
8 y = _____ 2 1 Start with y = __
x22
x
y
x
Transformation f(x 2 2) gives
y
x
O
1 12
2
5
x
1
y = _____
x22
y
Finally f(x) 2 1 gives
5
y
2
O
150
x
330
1
y = _____ 2 1
x22
x
1
Asymptotes x = 2, y = 21
1
9 y = 2 + _____
Vertical asymptote is x = 1
(put denominator = 0)
Horizontal asymptote is y 5 2 (let x ⇒ ∞)
x21
5
6
Edexcel IGCSE Further Pure Mathematics
3
y
y
x = 0, y = 1
2
1
3
O
1
x
Asymptotes x 5 1 and
y52
3 + 2x 2 (1 + x) + 1
1
10 y = ______ = ___________ = 2 + _____
1+x
1+x
1+x
y
1
1
7 y = 3 + _____ Start with y = __
x+1
x
x
y
x
1
3
2
O
x
x = 0, y = 3
Asymptotes x = 21, y = 2
13
3
11 y = 2 + _____
12x
2
Vertical asymptote x = 1
Horizontal asymptote y = 2
Let x = 0 and y = 2 + 3 = 5
y
NB This is a series of
2
3
transformations of y = 2 __
O 1
x
y
1)
y
0
1
x
(0, 0)
x
(1, 0)
y
c
1
0
y
y=5
x
O
3
y
ln(2x)
x = 3
3
13 y = _____ 2 4
22x
d
Let x = 0 and y = _2 2 4 = 2 2 _3
3
2
This is a series of transformations
3
of y = 2 __
x
x
( 12 , 0)
Vertical asymptote x = 2
Horizontal asymptote y = 24
y = 3ln (x 2 2)
y
x>2
y
2
x
O
Use transformation f(x 2 2)
4.6
Exercise 4F
a
b
y
y
e
x
1
y
Now use transformation 3f(x)
y
y
4e
2x
y = 3ln (x 2 2)
(0, 4)
1
y
e
x
x
y
x)
ln(4
d
y
2ex
y
ln(4)
x
c
y
3
4
(3, 0)
x
4
(0, 3)
y
y
y
e
y
1
10e 2 x
6
16
y
f
x
110
(0, 3
y
100e
x
10
x
3
ln(x
x
ln2)
10
2
6
y
2
3
f
y
x
4
x
e
x
1)
(0,
2
x
3
y
4
1
x
1
Start with y = lnx
y
4
(0, 2)
14
2 ln(x)
x
2
ln(x
x
x
Vertical asymptote x = 23
Horizontal asymptote y = 5
Let x = 0 and y = 5 2 _3 = 4_3
Again this is a series of
2
transformations of y = 2 __
x
y
b
+
Edexcel IGCSE Further Pure Mathematics
2
12 y = 5 2 _____
3+x
y
a
(e
3
2, 0)
x
2)
2
3
x
⇒ _3 ex = 2 2 2x
1
4.7
Exercise 4G
1
a
b
 2 + _3 ex = 4 – 2x so draw y = 4 2 2x
1
x 5 1, y 5 4.21; x 5 5, y 5 3.16
3
y
a
b
and intersection at  0.65 or 0.7 to 1sf
x 5 1, y 5 21ln1 5 2; x 5 4, y 5 3.39
y
+
5
5
y=4
+
y=x
4
+
3
4
5
6
x
0
4
x
+
+
y = 2.5
+
d
+
+
)
y=x
lnx = 0.5
2
y
⇒ 2 + lnx = 2.5 so draw y = 2.5
Intersection
 1.60
1
4
x = ex 2 2
⇒3 lnx =0 x 2 2
1
2
3
4
⇒ 2 + lnx = x so draw y =x
1
2
Intersection  3.1 to 1sf
x5
30, y 5 2.60; x 5 75, y 5 1.98
1
+
c
x
+
+
y
4
+
9
a
b
y
0
4
7
1
3
y=2
10
20
30
40
50
60
70
80
0
+
8
y=6
+
2
2
5
90 x
y=2
+
+
6
+
0
20
30
40
50
60
70
80
90 x
0
+
+
1
10
+
+
+
1
4
3
4
+
1
3
3
+
b
2
+
( 
1
+
a
e – = 0.5 ⇒ 3 + 2e – = 3 + 2 3 0.5 = 4
Draw y = 4 and intersection is at  1.35
x22
x = 22 ln _____
2
x
_
x22
⇒ e– 2 = _____
2
x
1
_
_
⇒ 2e– 2 = x 2 2  3 + 2e– 2 x = 1 + x
Draw y 5 1 1x and intersection at x  2.55
1
x 5 0, y 5 2 1 _3 = 2.33; x = 2.5, y = 6.06
2
y = 4 – 2x
+
+
2
1
_
2x
+
d
1
_
2x
+
3
+
2
+
1
+
1
5
+
2
y
1
c
y = 2.5
+
2
+
y=1+x
Edexcel IGCSE Further Pure Mathematics
+
+
3
+
+
+
+
4
2
c
1
–1
1
2
3
4
5
6
c
ex = 12 ⇒ 2 + _3 ex = 2 + _3 3 12 = 6
d
Draw line y = 6 and intersection is at  2.45
x = ln (6 2 6x)
⇒ ex = 6 2 6x
1
1
x
2 + 2 cosx = 5 sin 2x
⇒ 2 = 5 sin 2x 2 2 cosx
Draw y = 2
and intersections at  25.1 and 74.6
or 25 and 75 to 2sf
15
Mixed Exercise 4H
a
y
x2(x
y
y
1.0
2)
y= x
4
+
1
b
0.8
0.6
x
0.4
+
2
0
0.2
y
1
y
y
x2
a
c
2x
5
d
A(23, 22) B(2, 3)
y = x2 + 2x 2 5
y = x2 2 2x 2 3
= (x 2 3)(x + 1)
7
x
ln(x 2 1) = 0
⇒ 1 2 ln(x 2 1) = 1
Intersection at x = 2
so draw y = 1
x
_
x = 1 + e1 + 4
ln(x 2 1) = 1 + _4
x
x
_
x
3
1
⇒ x 2 1 = e1 2 4
x
x
x
⇒ 1 2 ln(x 2 1) = 1 2 (1 1 _4 ) = _4 so draw y = _4
and intersection at ≈ 2.5
3
y
b
y = x2 2 2x 1 4
= (x 2 1)2 + 3
Chapter 5
4
3
x
1
5.1
Exercise 5A
y
4
a
y = 2(x2 2 4x + 3)
= (3 2 x)(x 2 1)
1
x
3
1
2
3
y
b
5
a
y = 2(x2 2 4x 1 5)
= 2 [ (x 2 2)2 +1 ]
= 21 2 (x 2 2)2
y = _2 ex 1 4
1
2
x
1
3
4
5
y
4.5
x
b
6
a
1
y
y = ln(x 1 1) 1 2
2
(x = 0, y = 2)
y = 0 ⇒ ln(x + 1) = 22 0.86
⇒x+1
= e22
 x
= e22 2 1 ≈ 20.86
x=5
y = 1 2 ln4 = 20.39
x=6
y = 1 2 ln5 = 20.61
Arithmetic sequences are a, b, c, h, l
a 23, 2n + 3
b 32, 3n + 2
c 23, 27 2 3n d 35, 4n 2 5
e 10x, nx
f a + 9d, a + (n 2 1)d
a £5800
b £(3800 1 200m)
a 22
b 40
c 39
d 46
e 18
f n
5.2
Exercise 5B
y=4
(x = 0, y = 4.5)
16
6
0.8
6
x
x
0
5
0.6
x
1
y
3
4
0.4
y
B
b
c
3
+
x 5 0, 21, 2; points (0, 0), (2, 0), (21, 23)
A
2
+
b
a
1
0.2
+
2
x2
+
Edexcel IGCSE Further Pure Mathematics
2x
x
2
3
4
5
6
7
a 78, 4n 2 2
b
c 23, 83 2 3n
d
e 227, 33 2 3n
f
g 39p, (2n 2 1)p
h
a 30
b 29
d 31
e 221
d56
a = 36, d = 23, 14th term
24
x = 5; 25, 20, 15
1
7 3 _2 , x 5 8
42, 2n 1 2
39, 2n 2 1
59, 3n 2 1
271x, (9 2 4n)x
c 32
f 77
1
3
4
5
6
a 820
b 450
d 2294
e 1440
g 21155
h 21(11x 1 1)
a 20
b 25
d 4 or 14 (2 answers)
2550
i £222 500
ii £347 500
1683, 32674
£9.03, 141 days
7
8
d 5 2 _2 , 25.5
a = 6, d = 22
2
c
f
21140
1425
c
65
1
10
a
2
3
4
5
6
7
8
9
30
∑ (3r + 1)
b
r =1
2
3
4
16
∑ 4(11 − r )
d
r =1
a 45
c 1010
19
49
b
d
5.8
Exercise 5H
∑ 6r
r =1
210
112
2
3
9
2
6_3
Doesn’t exist
Doesn’t exist
1
_
1 2 r if |r|,|
a
c
Geometric r 5 2
Not geometric
b
d
Not geometric
Geometric r 5 3
3 2 _3
4 20
e
Geometric r 5 _2
f
Geometric r 5 21
g
a
c
e
Geometric r 5 1
135, 405, 1215
7.5, 3.75, 1.875
p3, p4, p5
h
b
d
f
Geometric r 5 _4
232, 64, 2128
1 ___
1 ____
1
__
64 , 256 , 1024
28x4, 16x5, 232x6
5
6
7
8
9
a
3√3
b
9√3
1
__
1
5.6
Exercise 5F
1
10
__
1 a
c
e
g
i
2
_
2 3
5.5
Exercise 5E
1
a 255
b 63.938 (3 dp)
2
c 2728
d 546_3
e 5460
f 19 680
g 5.994 (3 dp)
h 44.938 (3 dp)
9
5
_
_
,
2
4
4
264 2 1 5 1.84 x 1019
a £49 945.41
b £123 876.81
a 2.401
b 48.8234
19 terms
22 terms
26 days, 98.5 miles on the 25th day
25 years
r =1
11
c
∑ (3r − 1)
10, 6250
a 5 1, r 5 2
1
±_8
26 (from x 5 0), 4 (from x 5 10)
5.7
Exercise 5G
1
5.4
Exercise 5D
1
2
3
4
5
a
b
c
d
486, 39 366, 2 3 3n21
100
25 ___
25 _____
__
8 , 128 , n 2 1
2
232, 2512, (22)n21
1.610 51, 2.357 95, (1.1)n21
__
b
d
f
h
j
Doesn’t exist
Doesn’t exist
1
4_2
90
1
1
_
_
1 1 2x if |x|, 2
Edexcel IGCSE Further Pure Mathematics
5.3
Exercise 5C
2
40
1
__
5 13_
3
3
23
__
99
4
40 m
1
r , 0 because
S∞ , S3, a 5 12, r 5 2 _2
__
2
10 r = ± __
3
√
Mixed Exercise 5I
1 a
b
c
Add 6 to the previous term, i.e. Un11 5 Un 1 6
(or Un 5 6n 2 1)
Add 3 to the previous term, i.e. Un11 5 Un 1 3
(or Un 5 3n)
Multiply the previous term by 3, i.e. Un11 5 3Un
(or Un 5 3n21)
17
d
Edexcel IGCSE Further Pure Mathematics
e
f
2
3
4
5
6
7
8
9
10
11
12
a
b
c
a
32
a
a
a
a
a
b
a
b
d
a
b
c
d
a
c
Subtract 5 from the previous term,
i.e. Un11 5 Un 2 5 (or Un 5 15 2 5n)
The square numbers (Un 5 n2)
Multiply the previous term by 1.2, i.e. Un11 5 1.2Un
(or Un 5 (1.2)n21)
Arithmetic sequences are:
a 5 5, d 5 6
a 5 3, d 5 3
a 5 10, d 5 25
81
b 860
£13 780
b £42 198
a 5 25, d 5 23
b 23810
26 733
b 53 467
5
b 45
d=5
b 59
9
_
c 1.5
d 415
11k 2 3
Not geometric
b Geometric r = 1.5
1
Geometric r = _2
c Geometric r = 22
Not geometric
e Geometric r = 1
0.8235 (4 dp), 10x (0.7)n21
640, 5 3 2n21
24, 4 3 (21)n21
3
1 n21
___
_
128 , 3 3 (2 2 )
4092
b 19.98 (2 dp)
50
d 3.33 (2 dp)
13 a
9
b
c
14 b
d
Doesn’t converge
60.72
3.16
d
c
15 b
16 a
c
17 a
18 a
c
200
76, 60.8
367
1
1
1, _3 , 2 _9
0.8
50
c
19 a
2 _2
b
1
333_3
1
3
_
, 22
4
Chapter 6
6.1
Exercise 6A
18
1
2
1 1 8x 1 28x2 1 56x3
1 2 12x 1 60x2 2 160x3
3
2
3
1 1 5x 1 __
4 x 1 15x
4
1 2 15x 1 90x2 2 270x3
45
8
_
3
16
__
3
182.25
8.95 3 1024
b
d
d
0.876
380
b
d
10
0.189 (3sf)
c
14
5
6
7
8
a p55
b 210
c 280
1 2 0.6x 1 0.15x2 2 0.02x3, 0.94148, accurate to 5 dp
a 220x3
b 120x3
c 1140x3
b = 22
6.2
Exercise 6B
1
2
3
4
6
a
b
1 1 6x 1 12x2 1 8x3, valid for all x
1 1 x 1 x2 1 x3, |x|, 1
c
d
3
1 1 _2 x 2 _8 x2 1 __
16 x , |x| , 1
1
1 2 6x 1 24x2 2 80x3, |x| , _2
e
1 2 x 2 x2 2 _3 x3, |x| , _3
f
___ 3
__
2
1 2 15x 1 __
2 x 1 2 x , |x| , 10
g
3
1 2 x 1 _8 x2 2 __
16 x , |x| , 4
h
√2
2x2 1 ..., |x| , ___
1
1
1
5
1
75
5
125
1
5
__
12
2
1
|x| < _
2
3x 9
27 3
1 1 ___ 2 _8 x2 1 __
16 x , 10.148 891 88, accurate to 6 d.p.
2
a = ±8, ±160x3
9x 27x2 27x3
1 2 ___ + _____ + _____, x = 0.01, 955.339(1875)
2
8
16
Mixed Exercise 6C
1
a
p 5 16
b
2
a
b
1 2 20x 1 180x2 2 960x3
0.817 04, x = 0.01
3
a
n58
4
a
b
c
d
1 + 24x + 264x2 + 1760x3
1.268 16
1.268 241 795
0.006 45% (3 sf)
5
a
1 2 12x 1 48x2 2 64x3, all x
b
1 1 2x 1 4x2 1 8x3, |x| , _2
b
7
8
9
35
__
8
1
1 2 2x 1 6x2 2 18x3, |x| , _3
x3
x x2
1 2 __ 2 ___ 2 ____
4 32 128
27 4
3
135 6
___
1 2 _2 x2 1 __
8 x 2 16 x
1
c
6
c
270
x x2 x3 1145
1 + __ 2 ___ + ___, _____
2
8 16 512
a
b
c
n 5 22, a 5 3
2108
1
|x| , _3
21890
Chapter 7
7.3
Exercise 7C
7.1
Exercise 7A
c
1
5
1
_
a 1 _b
2
1
1
2 _a 2 _b + c
3
3
4
____
c
e
 23.9
d2a
a1b2d
b
d
a1b1c
a1b1c2d
2a 1 2b
1
b 2 _2 a
3
_
2a 2 b
Yes (l 2)
Yes (l 21)
l = _12 , m = 23
b
b
d
b
e
b
a1b
c
b 2 3a
2a 2 b
Yes (l 4)
c
Yes (l 23)
f
l = 22, m = 1
l = _14 , m = 5
l = 4, m = 8_12
d
l = 22, m = 21
5
1
5
6
2 _6 a 1 (l 2 _6 )b
2ma 1 (m 2 l)b
l 5 _12 , m 5 _16
7
2
i2a+b
ii _3 a 2 _4 b
7
3
2
(_3 l 2 m)a + (_4 2 _4 l 1 m)b = 0
f
13
__
a
2a 1 b
b
1
1
_
a 1 _b
c
3 _
3
_
a b
e
a
1
_
a
d
f
2 _8 a 1 _8 b
5 : 3, k 5 35
5
1
2
a
b
c
d
e
f
3i 2 j, 4i 1 5j, 22i 1 6j
i 1 6j
25i
1 7j ___
___
√___
40 = 2√10
√37
___
√ 74
3
a
14
__
4
2a 1 kb
d
e
2b
1
1
2 _4 a + _4 b
4
4
2
2
5
3
4
3
d
v 5 _3 , w 5 _3
a
AC = x + y; BE = _3 y 2 x
2
2 _4 a + _4 b
1:3
___›
___›
1
2 _3 a + b, 2a + 3b, AG = 3EB ⇒ parallel
5
)
(  )
1
___›
i
iii
12
9
c
b
1 a 1 _b
2__
4
1
( 
5
13 212
1
1
____
___
√ 10 23
(  )
(  )
(  )
(  ) (  )
ii
b
c
1
___
  )
(221
229
___›
XM 21
3
___›
210
XZ 5
6
7
v3
8 1 w 210
0
6
b
3
3
1
_
a 1 _b
d
a
13
8
c
b
3
13
8
(  )
(  )
53
1 27
c ___
25 24
27 or 223
m 5 3, n 5 1
m 5 22, n 5 5
7
6
__
a + __b
10
b
1
2
6
__
13
 16 )
(21
  3)
(12
Mixed Exercise 7E
b 2 a, _6 (b 2 a), _6 a 1 _6 b
b
c
d
a
b
d
e
f
g
h
___›
a
No
No
a
8
___›
OC 5 22a 1 2b, OD 5 3a 1 2b, OE 5 22a 1 b
1
b2a
5
7
2
Edexcel IGCSE Further Pure Mathematics
25
a
a
c
a
d
a
2
___›
7.4
Exercise 7D
7.2
Exercise 7B
1
2
6
b
√ 569
a
c
6
d
a
2
3
4
1
__›
__›
1
BF = v(_3 y – x)
1
___›
__›
AF = x + BF = x + v(_3 y – x)
1
v = _4
3
(  )
(  )
(  )
___
v 1 w 5 4 , √41
5
5 , √___
2v – w 5 22
29
___
1 , √ 50
v – 2w 5 27
19
6
7
a
b
Edexcel IGCSE Further Pure Mathematics
(  )
(  )
(  )
5
Chloe (7
  ); Leo (54  ); Max (23  )
___
p 1 q = 5 , √41
4
9 , √___
3p 1 q = 2
85
2 7 ____
p 2 3q = 216 , √ 305
Chloe: 74 km, 2.9 km/h
Leo: 41 km, 2.1 km/h
Max: 13 km, 1.2 km/h
3
12
4
4_3
5
2_4
6
1
_
7
26
8
25
1
2
3
4
5
6
7
8
22
e
2
_
3
1
_
2
i
a
e
i
4
7
_
5
29
b
21
f
5
_
4
1
_
2
j
b
f
j
25
2
23
a 4x 2 y 1 3 5 0
c 6x 1 y 2 7 5 0
e 5x 2 3y 1 6 5 0
g 14x 2 7y 2 4 5 0
i 18x 1 3y 1 2 5 0
k 4x2 6y 1 5 5 0
y 1 5x 1 3
2x 1 5y 1 20 5 0
1
y 1 2 _2 x 1 7
2
y 5 _3 x
(3, 0)
y 5 2x 1 1
y 5 2x 2 3
1
y 5 _2 x 1 12
b
d
f
g
y 5 2x
h
y 52 _2 x 1 2b
a Perpendicular
c Neither
e Perpendicular
g Parallel
i Perpendicular
k Neither
1
y 5 2 _3 x
4x 2 y 1 15 5 0
1
a y 5 22x 1 _2
b
d
f
h
j
l
Parallel
Perpendicular
Parallel
Perpendicular
Parallel
Perpendicular
b
y 5 _2 x
c
y 5 2x 2 3
d
y 5 _2 x 2 8
5
a
y 5 3x 1 11
b
d
6
7
c y 5 _3 x 1 2
3x 1 2y 2 5 5 0
7x 2 4y 1 2 5 0
y 5 2 _3 x 1 __
3
c
3
d
1
_
2
y 5 3x 2 6
g
1
_
2
h
2
3
y 5 2x 1 8
22
2
2 _3
2
l
d
h
l
3
_
2
4
5
2x 2 3y + 24 = 0
1
2 _5
k
c
g
k
b
d
f
h
j
l
3
_
2
3
0
22
1
2 _2
3x 2 y 2 2 5 0
4x 2 5y 2 30 5 0
7x 2 3y 5 0
27x 1 9y 2 2 5 0
2x 1 6y 2 3 5 0
6x 2 10y 1 5 5 0
6
7
y = _5 x 1 3
2x 1 3y 2 12 5 0
8
9
8
_
2
3
4
1
a
2
20
6
c
3
_
d
2
g
1
_
h
8
l
1
_
5
e
21
f
1
_
i
2
_
j
24
k _3
q2 2 p2
_______
5q1p
q2p
3
m 1
2
b
1
_
7
n
2
2
1
2
1
2
5
y 5 _3 x 24
4
8.4
Exercise 8D
1
8.2
Exercise 8B
y 5 3x 1 7
y 5 24x 2 11
2
y 5 _3 x 2 5
10 6x 1 15y 2 10 = 0
5
9 (_3 , 0)
10 (0, 5), (24, 0)
1
_
4
a
c
e
8.1
Exercise 8A
a
1
8.3
Exercise 8C
Chapter 8
1
1
2
1
1
1
13
y 5 2 _2 x 1 __
2
3
17
1
4
7
10
11 a
2
5
8
11
10
__
√5
____
√ 113
5c
3
6
9
12
13
___
2√10
___
a√53
___
d√ 61
5
____
b
√ 106
__
3b√5
__
2e√5
a
b
c
d
2
a
c
(0, 6); (4, 10) is ratio 3 : 1
3 3 10 1 1 3 6
3 3 4 1 1 3 0 ______________
_____________
,
= (3, 9)
4
4
(1, 5); (22, 8) is ratio 1 : 2
(2 3 5 1 1 3 8)
2 3 1 1 1 3 (22) ______________
________________
,
= (0, 6)
3
3
(3, 27); (22, 8) is ratio 3 : 2
2 3 (27) 1 3 3 8
2 3 3 1 3x 3 (22) ________________
_________________
,
= (0, 2)
5
5
(22, 5); (5, 2) is ratio 4 : 3
4 3 5 1 3 3 (22) _____________
4321335
2
________________
,
= (2, 3_7 )
7
7
416 218
(4, 2); (6, 8)
midpoint ______, ______
2
2
= (5, 5)
( 
( 
( 
( 
)
( 
)
( 
0 1 12 6 1 2
midpoint _______, ______
2
2
= (6, 4)
(2, 2); (24, 6)
d
(26, 4); (6, 24)
( 
224 216
midpoint ______, ______
2
2
= (21, 4)
( 
)
b
(0, 14)
2 a
y = 2 _2 x + 4
b
y 5 _7 x 1 __
7 , y 5 2x 1 12
y 5 2 _2 x + _2 , (1, 1)
b
(9, 3)
4 a
y
b
222
5 a
y
b
(3, 3)
12
3
1
1
8 a
y = _2 x 2 2
9 a
2x 1 y 5 20
3
b (4, 4)
b 6
b
y 5 _4 x 1 _4
c
20
b
y 5 _3 x 1 _3
c
2x 1 y 2 16 = 0
1
1
__
____________________
12 a
________
__
AB = √ (4 2 21)2 + (11 2 1)2 = √ 52 + 102 = 5√ 5
__
__
1
1
.... Area of Δ ABD = __ AB.CD = __ 3 √ 5 3 5√5
2
2
25
___
=
2
1
1
_
_
y = 3x 1 3
2 8x7
3 4x3
2
__
1
4 _3 x2 3
_
1
5 _4 x2 3
2
_
1
6 _3 x2 3
7 23x24
8 24x25
9 22x23
10 25x26
11 2 _3 x2 3
4
_
12 2 _2 x2 2
13 22x23
14 1
15 3x2
16 9x8
17 5x4
18 3x2
4
1
1
3
_
9.2
Exercise 9B
6 11x 2 10y + 19 = 0
y 5 2 _2 x 5 3
_______
CD = √ 22 + 12 = √ 5
)
y 5 23x 1 14
5
11
__
= 2 __
12 x + 6
3
3
5 _2 x 2 _2
ΔABD
1 7x6
1 a
2
d
9.1
Exercise 9A
)
26 1 6 4 2 4
midpoint _______, ______
2
2
= (0, 0)
1
x
L
Chapter 9
Mixed Exercise 8G
1
_
C
A
)
c
10 a
B
)
(0, 6); (12, 2)
7 a
or x 1 2y 2 16 = 0
x = 0 ⇒ y = 8  D is (0, 8)
)
1
)
y
b
3 a
( 
2y 2 14 = 2x + 2
8.6
Exercise 8F
1
(21, 1); (4, 11) in ratio 3 : 2
2 3 (21) + 3 3 4 2 3 1 + 3 3 11
is ________________, ______________
5
5
so c = (2, 7)
11 2 1 10
1
MAB = _______ = ___ = 2  mi = 2 __
4 2 21 5
2
1
Equation of l is y 2 7 = 2 _2 (x 2 2)
Edexcel IGCSE Further Pure Mathematics
8.5
Exercise 8E
9
4
a
4x3 2 x22
b
2x23
2
a
0
b
3
a
(2_2 , 26_4 )
11_2
b
(4, 24) and (2, 0)
c
(16, 231)
d
(_2 , 4), (2 _2 , 24)
1
1
c
3
_
2x2 2
1
1
1
1
21
4
Edexcel IGCSE Further Pure Mathematics
5
1
_
a
x 22
b
26x23
d
4 3
_
x 2 2x2
e
22
26x24 +__
2x
f
1 2 _3
1
_
x 2 _x22
g
23x22
i
5x 2 + _2 x2 2
j
3x2 2 2x + 2
k
12x3 1 18x2
l
24x 2 8 + 2x22
a
1
3
2
3
2
3
_
3
1
_
b
2
_
9
1
_
1
c
p
p
= 28 sin __4 [ sin __2 = sin 90° = 1 ]
= 28
2x24
c
h 3 + 6x22
24
d
9.4
Exercise 9D
4
1
c
9.3
Exercise 9C
1
a
b
c
d
e
f
2
3
4
d
dy
y = e2x ⇒ ___ = 2e2x
dx
dy
26x
y=e
⇒ ___ = 26e26x
dx
dy
x
2
y = e + 3x ⇒ ___ = ex + 6x
dx
dy
___
y = sin 2x ⇒ = 2 cos 2x
dx
dy
y = cos 3x ⇒ ___ = 23 sin 3x
dx
dy
y = 3 sin 4x + 4 cos 3x ⇒ ___ =
dx
12 cos 4x 212 sin 3x
dy
y = sin 5x ⇒ ___ = 5 cos 5x
dx
dy
1
1
1
1
_
b y = 2 sin 2 x ⇒ ___ = 2 3 _2 cos _2 x = cos _2 x
dx
dy
c y = sin 8x ⇒ ___ = 8 cos 8x
dx
dy
2
2
2
2
d y = 6 sin _3 x ⇒ ___ = 6 3 _3 cos _3 x = 4 cos _3 x
dx
dy
e y = 2 cos x ⇒ ___ = 22 sin x
dx
dy
5
5
5
5
_
f y = 6 cos 6 x ⇒ ___ = 6 3 _6 sin _6 x = 25 sin _6 x
dx
dy
g y = cos 4x ⇒ ___ = 24 sin 4x
dx
dy
x
x
x
1
h y = 4 cos ( _2 ) ⇒ ___ = 24 3 _2 sin _2 = 22 sin _2
dx
dy
y = 2e2x ⇒ ___ = 22e2x @ (0, 2)
dx
dy
p
y = 3 sin x ⇒ ___ = 3 cos x @ ( __3 = x )
dx
2
3
p
22
2
2
y = 4e3x ⇒ y′ = 4e3x 3 6x = 24x e3x
8(1 1 2x)3
b
y = 9e32x ⇒ y′ = 9e3 2 x 3 21 = 29e3 2 x
c
y = e26x ⇒ y′ = e26x 3 26 = 26e26x
d
y = ex
a
c
d
4
a
b
c
5
3
p
p
1
 m = 3 cos ( __3 ) = _2 [ cos __3 = cos 60° = _2 ]
 m = 28 sin ( 2 3 __4 )
2
a
b
a
dy
p
y = 4 cos 2x ⇒ ___ = 28 sin 2x @ ( x = __4 )
dx
y = (1 + 2x)4 ⇒ y′ = 4(1 + 2x)3 3 2 = 8(1 + 2x)3
y = (1 + x2)3 ⇒ y′ = 3(1 + x2)2 3 2x = 6x(1 1 x2)2
1
1
_
_
2
1
_______
y = (3 +4x) 2 ⇒ y′ = _2 (3 + 4x)2 2 3 4 = ________
√ 3 1 4x
y = (x2 + 2x)3 ⇒ y′ = 3(x2 + 2x)2 3 (2x + 2)
= 6(x 1 1)(x2 1 2x)2
2
+ 2x
⇒ y′ = ex
2
+ 2x
3 (2x + 2)
x2 + 2x
Remember x is in radians
5
a
b
a
b
c
6
a
b
= 2(x + 1) e
y = sin(2x + 1) ⇒ y′ = cos(2x + 1) 3 2 =
2 cos(2x 1 1)
y = cos(2x2 + 4) ⇒ y′ = 2sin(2x2 + 4) 3 4x ⇒
24x sin(2x2 1 4)
y = sin3x ⇒ y′ = 3 sin2 3 cos x
3 sin2 x cos x
y = cos2 2x ⇒ y′ = 2 cos 2x 3 (2sin 2x) 3 2
= 24 sin 2x cos 2x
y = x (1 + 3x)5 ⇒ y′ = (1 + 3x)5 + x.5(1 + 3x)4 3 3
= (1 + 3x)4 [1 + 3x + 15x]
= (1 + 3x)4 (1 + 18x)
2
3
y = 2x (1 + 3x ) ⇒ y′ = 2(1 + 3x)3
+ 2x 3 3(1 + 3x2)3 3 6x
= 2(1 + 3x2)2 [1 + 18x2]
3
y = x (2x + 6)4 ⇒ y′ = 3x2 (2x + 6)4
+ x3 3 4(2x + 6)3 3 2
= x2 (2x + 6)3 [6x + 18 + 8x]
= 2x2 (2x + 6)3 (7x + 9)
y = xe2x ⇒ y′ = e2x + x.2 e2x = e2x(1 1 2x)
y = (x2 + 3) e2x ⇒ y′ = 2x 3 e2x + (x2 + 3)(2e2x)
= e2x(2x 2 x2 2 3)
2
2
x2
y = (3x 2 5) e ⇒ y′ = 3 3 ex + (3x 2 5)ex 3 2x
2
= ex (6x2 2 10x 1 3)
y = x sin x ⇒ y′ = sin x + x cos x
y = sin2x cos x
⇒ y′ = 2 sin x cos x 3 cos x + sin2x (2sin x)
y′ = sin x (2 cos2x 2 sin2x)
7
a
b
y = ex cos x ⇒ y′ = ex cos x 2 ex sin x
+ x2 3 3 (3x 2 1)2 3 3 when x = 1
= ex (cos x 2 sin x)
(x + 1) 3 5 25x 3 1
5x
= __________________
y = _____ ⇒ y′
x+1
(x + 1)2
5x + 5 2 5x
= ___________
(x + 1)2
5
= ________2
(x 1 1)
(3x 2 2) 3 2 2 2x 3 3
2x
y = _______ ⇒ y′ = ____________________
3x 2 2
(3x 2 2)2
m = 2 3 23 + 1 3 9 3 22 = 16 + 36 = 52
b
m = 6e0 + 2e0 = 8
c
d
24
= _________2
(3x 2 2)
3x2
y = _________ ⇒ y′
(2x 2 1)2
(2x 2 1)2 3 6x 2 3x2 3 2(2x 2 1) 3 2
= __________________________________
(2x 2 1)4
6x (2x 2 1)[2x 2 1 2 2x]
= ______________________
(2x 2 1)4
26x
= _________3
(2x 2 1)
8
9
e2x 3
1 2 2x
12x3
= _________________ = _______
4x
e
e2x
x
y = ___ ⇒ y′
e2x
b
(x + 1)ex 2 ex 3 1
ex
xex
y = _____ ⇒ y′ = ________________ = _______
x+1
(x + 1)2
(x + 1)2
c
e
y = ___ ⇒ y′
a
sin x
y = _____ ⇒ y′
x
b
cos x 3 ex + ex sin x
ex
y = _____ ⇒ y′ = _________________
cos x
cos2x
x
x2
x2
x2
2
x2
7
ex(sin x 1 cos x)
= ______________
cos2 x
c
sin2 x
y _____ ⇒ y′
e2x
1
x 3 2xe 2 e 3 1 e (2x 2 1)
= _________________ = ___________
x2
x cos x 2 sin x 3 1
= ________________
x2
x cos x 2 sin x
= ____________
x2
e2x 3 2 sin x cos x 2 sin2 x 3
p
y = 3 sin2x ⇒ y′ = 6 sin x cos x when x = __
4
1__
___
1
___
m = 6 3 __ 3 √ = 3
2
√2
p
y = x cos x ⇒ y′ = cos x 2 x sin x when x = __
2
p __
p
p
p
__
__
__
m = cos 2 sin = 0 2 3 1
2 2
2
2
p
__
=2
2
9.5
Exercise 9E
2
3
4
5
6
2e2x
a
x2
y = (2x 1 3)e2x ⇒ y′ = 2(2x + 3)e2x + 2 3 e2x
when x = 0
6x 2 4 2 6x
= ___________
(3x 2 2)2
c
y = x2 (3x 2 1)3 ⇒ y′ = 2x (3x 2 1)3
a y 1 3x 2 6 5 0
b 4y 2 3x 2 4 5 0
c y5x
a 7y 1 x 2 48 5 0
b 17y 1 2x 2 212 5 0
y 5 28x 1 10, 8y 2 x 2 145 5 0
e
y 5 2ex 2 __
2
1
_
y 5 3e
y = x sin x ⇒ y′ = sin x + x cos x @ (p, 0)
 m = sin p + p cos p = 2p
 equation of tangent:
y 2 0 = 2p (x 2 p)
or
y = p2 2 xp
p
y = 2 cos2 x ⇒ y′ = 4 cos x (2sin x) @ (__4 , 1)
1
1__ ___
. __ = 22

m = 24.____
√2 √2
1
normal has gradient = __
2
 equation of normal is:
p
1
y 2 1 = __ x 2 __
2
4
or
8y 2 8 = 4x 2 p
( 
2e2x
= _____________________________
e4x
2 sin xe2x (cos x 2 sin x)
= _____________________
e4x
2 sin x(cos x 2 sin x)
= __________________
e2x
or
Edexcel IGCSE Further Pure Mathematics
c
10 a
)
8y 2 4x = 8 2 p
9.6
Exercise 9F
1
1 _4 x4 1 x2 1 c
2 22x21 + 3x + c
23
5
_
3 2x 2 2 x3 1 c
Edexcel IGCSE Further Pure Mathematics
4
5
6
7
_
4 _2
_
x 2 4x 2 1 4x 1 c
3
1
3
x4 1 x23 1 rx 1 c
t3 1 t21 1 c
1
2 3
2 _2
_
1t1c
3 t 1 6t
1
1
_
_
1
8 _2 x2 1 2x 2 2 2x2 2 1 c
p
9 ____5 1 2tx 2 3x21 1 c
5x
p
10 _4 t4 1 q2t 1 px3t 1 c
11 a
b
c
1 4
_
x 1 x3 1 c
2
3
2x 2 __ 1 c
x
4 3
_
2
3 x 1 6x 1 9x 1 c
d
2 3
1
_
x 1 _ x2 23x 1 c
e
_
4 _2
_
x 1 2x 2 1 c
12 a
3
5
2
3
5
∫2 sin 3x dx
b
∫3e4x dx
c
∫2 cos 3x dx
d
∫2e2x dx
13 a
b
c
d
2
= 2 __ cos 3x 1 c
3
3
= __ e4x 1 c
4
2
= __ sin 3x 1 c
3
= 22e2x 1 c
x4
5ex 1 4 cos x 1 ___ 1 c
2
22 cos x 2 2 sin x 1 x2 1 C
2
5ex 1 4 sin x 1 __ 1 c
x
ex 2 cos x 1 sin x 1 C
9.7
Exercise 9G
1
2
3
4
5
6
24
3t2 1 8t 2 5
b 6t 1 8
v 5 6 m/s, a 5 14 m/s2
8
v 5 3t2 2 4t 1 3
b a 5 6t 2 4
9
a 5 2t 1 10
b a 5 14 m/s2
2
v 5 t + 10t + 5
t3
2
⇒ s 5 __
3 + 5t + 5t + d; s = 0 when t = 0 ⇒ d = 0
8
2
 when t = 2 s = _3 + 20 + 10 5 32_3
10 a 6 2 2t
b 2 m/s
c
v 5 24 + 6t 2 t2
t3
⇒
s 5 24t + 3t2 2 __ + d
3
27
s = 100, t = 3 ⇒ 100 = 24 3 3 + 3 3 9 2 ___ + d
3
i.e. 100 5 72 + 27 2 9 + d
 d = 10
t3

s 5 10 + 24t + 3t2 2 __
3
7
10t
48 2 32t
a 40 1 10t
b 70 m/s
a 30 2 1t
b 0 m/s
a a 5 32
b
v 5 32t 1 100
⇒ s 5 16t2 1 100t 1 d; when t = 0, s = 0 ⇒ d = 0
 s 5 16t2 1 100t
a a 5 232
b
v 5 160 2 32t
⇒ s 5 160t 2 16t2 1 d;
t = 0, s = 384 ⇒ d = 384
 s 5 384 + 160t 2 16t2
c
s 5 0 ⇒ 16(t2 2 10t 2 24) = 0
i.e. 16(t 2 12)(t 1 2) = 0
 passes through origin when t = 12
a
c
a
a
c
9.8
Exercise 9H
1
2
3
5
228
10
9
3
(2 _4 , 2 _4 )
c
(2 _3 , 1 __
27 ), (1, 0)
5
d
(3, 218), (2 _3 , __
27 )
e
(1, 2), (21,22)
f
(3, 27)
1
b
b
c
c
2 _5
12.25
a
a
a
217
4
1
1
b (_2 , 9 _4 )
1
1 14
7p ___
3p ___
  8 , 2 12 e ) minimum
  8 , 12 e ) maximum, (___
(___
__
√
3p
__
__
4
7p
__
4
√
9.9
Exercise 9I
8
2
19_3
b
d
9_4
21
e
8__
12
a
b
i 2 ln2
8
i _3
ii
ii
64
__
p
3
a
A(1, 3), B(3, 3)
b
4
6_3
1_3
5
a
(2, 12)
b
6
3_8
13_3
1
2
a
c
3
5
2p
15
1
2
3
1
9.10
Exercise 9J
1
2
3
4
8
_
p
9
6p
15e2
dy
y = 5x4 ⇒ ___
dx
3
 δy ≈ 20x δx
= 20x3
x
= 20x3 3 ____
200
x
[ 0.5% of x = ____
200 ]
x4
= ___
10
δy
x4
= ___ × 100 = _________4 × 100 = 2%
y
10 3 5x
 % change in y
dy
y = 3x2 ⇒ ___
dx
 δy ≈ 6x δx
5
= 6x
6x2
x
= 6x 3 ____ = ____
100 100
δy
6x2
= ___ × 100 = __________ × 100
y
3x2 3 100
= 2%
 % change in y
6
dv
4
For a sphere: V = __ pr3 = __ = 4pr2
3
dr
δr ≈ 0.02 cm
δv ≈ 4pr2 dr
Use r = 1 ⇒
δv ≈ 4p 3 0.02
= 0.25 cm3
c
9 b A is (21, 0); B is (_3 , 9__
27 )
10 3x2 cos 3x + 2x sin 3x
x cos x 2 sin x
12 _____________
x2
13 b y = 2x + 1
14 a 2(x3 2 2x)ex + (3x2 2 2)ex
56p
15 ____
5
5
20 m 3 40 m; 800 m2
2000p cm2
40 cm
800
_____
cm2
4+p
27 216 mm2
4
5
10.1
Exercise 10A
1
2
3
4
g
1 a
3 a
b
7__
32
31
j
m
5
p
a
d
9°
90°
225°
26.4°
99.2°
200°
0.479
0.909
2p
__
45
p
__
6
5p
__
12
2p
__
3
4p
__
3
11p
___
6
0.873
2.79
b
e
h
b
e
12°
140°
270°
57.3°
143.2°
c
f
i
c
f
75°
210°
540°
65.0°
179.9°
b
e
b
0.156
20.897
c
1.74
c
p
__
f
p
__
i
5p
__
l
10p
___
o
7p
__
c
f
1.75
5.59
p
__
n
18
p
__
4
4p
__
9
3p
__
4
3p
__
2
b
e
1.31
4.01
e
h
k
8
3
8
9
4
3
x
___
1 2 2_2
2 2x 2 __
3
3
x
2
x
21
c f9(x) 5 _____ . 0 for all values of x
x
4 (1, 4)
x
px
5 a y 5 1 2 __ 2 ___
2
4
2
______
c
m2 (0.280 m2)
41p
d2v
10
6 b __
c ____2 , 0  maximum
3
dx
2300p
2
e 22_9 %
d _______
27
250
7 a ____ 2 2x
x2
b (5, 125)
( 
8 b
a
d
g
a
d
g
a
d
a
d
Mixed Exercise 9L
x 5 4, y 5 20
d2y 15
b ____2 5 ___ . 0  minimum
8
dx
7
5
2 (1, 211) and (_3 , 212__
27 )
13
Chapter 10
Exercise 9K
1
2
3
__
OP = 3; f ″(x) > 0 so minimum when x = ±2√2
(maximum when x = 0
Edexcel IGCSE Further Pure Mathematics
20x4
i.e. δy = _____
200
__
)
x = ±2√2 , or x = 0
10.2
Exercise 10B
1 a
b
c
2
3
4
5
6
7
8
9
10
i 2.7
ii 2.025
iii 7.5p (23.6)
2
i 16_3
1
i 1_
ii 1.8
iii 3.6
ii 0.8
iii 2
10p
___
cm
3
3
2p__
5√ 2 cm
a 10.4 cm
7.5
0.8
p
a __
3
6.8 cm
a (R 2 r) cm
b
1_4
b
4p
cm
(6  + ___
3 )
c
2.43
1
25
Edexcel IGCSE Further Pure Mathematics
10.3
Exercise 10C
1 a 19.2 cm2
c 1.296p cm2
1
e 5_3 p cm2
2 a 4.47
c 1.98
3 12 cm2
4 b 120 cm2
2
5 40_3 cm
6 a 12
7 8.88 cm2
8 a 1.75 cm2
9
10
11
12
b
d
f
b
x 5 23.2, y 5 2.06
__
4 a 3.19 cm
b 1.73 cm (√ 3 cm)
c 9.85 cm
d 4.31 cm
e 6.84 cm (isosceles) f 9.80 cm
5 a 108(.2)°
b 90°
c 60°
d 52.6°
e 137°
f 72.2°
6 a 23.7 cm3
b 4.31 cm3
c 20.2 cm3
7 a 155°
b 13.7 cm
8 a x = 49.5, area = 1.37 cm2
b x = 55.2, area = 10.6 cm2
c x = 117, area = 6.66 cm2
9 6.50 cm2
10 a 36.1 cm3
b 12.0 cm3
6.75p cm2
38.3 cm2
5 cm2
3.96
c 1.48 cm2
b 25.9 cm2
c 25.9 cm2
4.5 cm2
b 28 cm
78.4 cm
b 28 cm
10.4
Exercise 10D
__
__
1
a
√2
___
2__
√
3
d ___
2
1
g __
2 __
√2
j 2 ___
2
__
√
3
m ___
3
b
e
h
k
n
√3
2 ___
2
__
√
3
___
2 __
√2
2 ___
2
21
__
2 √3
c
1
2 __
2
f
2 _2
1
__
i
2 √3
_____
l
21
o
√3
2
__
10.5
Exercise 10E
1 a
b
c
d
e
f
2 a
d
3 a
b
c
26
x 5 84, y 5 6.32
x 5 13.5, y 5 16.6
x 5 85, y 5 13.9
x 5 80, y 5 6.22 (Isosceles )
x 5 6.27, y 5 7.16
x 5 4.49, y 5 7.49 (right-angled
48.1
b 45.6
48.7
e 86.5
x 5 74.6, y 5 65.4
x 5 105, y 5 34.6
x 5 59.8, y 5 48.4
x 5 120, y 5 27.3
x 5 56.8, y 5 4.37
c
f
14.8
77.4
10.6
Exercise 10F
1 a 11.7 cm
b 14.2 cm
d 63.4°
2 a 18.6 cm
b 28.1 cm
3 a 14.1 cm
b 17.3 cm
4 a 28.3 cm
b 34.6 cm
c 19.5°
5 a 4.47 m
b 4.58 m
d 12.6°
e 26.6°
6 a 407 m
b 402 m
c 13.3°
7 a 43.3 cm
b 68.7 cm
8 a 28.9 cm
b 75.7 cm
9 a 16.2 cm
b 67.9°
d 71.6°
10 a 26.5 cm
b 61.8°
11 a 30.3°
b 31.6°
12 a 36.9°
b 828 cm2
13 a 15 m
b 47.7°
14 a 66.4°
b 32.9°
15 46.5 m
16 a OW = 4290 m, OS = 2760 m
b 36.0°
c 197 km/h
c
34.4°
c
c
c
48.6°
35.4°
35.1°
c
29.2°
c
8.57°
c
c
c
81.2 cm
22.4°
55.3 cm2
c
c
1530 cm2
68.9°
c
€91 300
10.7
Exercise 10G
1
a
2
d
g
a
2
sin u
_____
b 5
c 2cos2 A
2
cos u
e tan x0
f tan 3A
4
h sin2 u
i 1
LHS 5 sin2 θ + cos2 θ + 2 sin θ cos θ
= 1 + 2 sin θ cos θ
= RHS
c
d
e
f
g
3
4
a
d
g
j
a
d
g
5
j
a
b
c
d
e
LHS
1 2 cos2 θ sin2 θ
sin θ
5 _________ = _____ = sin θ 3 _____
cos θ
cos θ
cos θ
LHS
= sin θ tan θ = RHS
sin x° cos x° sin2 x° + cos2x°
5 ______ + ______ = ______________
cos x° sin x°
sin x° cos x°
1
___________
=
= RHS
sin x° cos x°
LHS
5 cos2 A 2 (1 2 cos2 A) = 2 cos2 A 2 1
= 2 (1 2 sin2 A) 2 1 = 1 2 2 sin2 A = RHS
LHS 5(4 sin2θ 2 4 sin θ cos θ + cos2 θ)
+ (sin2 θ + 4 sin θ cos θ + 4 cos2 θ)
= 5 (sin2 θ + cos2 θ) = 5 = RHS
LHS 5 2 2 (sin2 θ 2 2 sin θ cos θ + cos2 θ)
= 2 (sin2 θ + cos2 θ)
2 (sin2 θ 2 2 sin θ cos θ + cos2 θ)
= sin2 θ + 2 sin θ cos θ + cos2 θ
= (sin θ + cos θ)2 = RHS
LHS 5 sin2 x(1 2 sin2y) 2 (1 2 sin2x) sin2y
= sin2x 2 sin2y = RHS
sin 35°
b sin 35°
c cos 210°
tan 31°
e cos u
f cos 7u
sin 3u
h tan 5u
i sin A
cos 3x
__
√3
___
1
b 0
c
2
__
__
√
√
2
2
1
___
___
e
f 2 _2
2
2
__
√3
__
p
p
5 sin __ cos u 1 cos __ sin u
6
6
p
5 sin __ 1 u 5 R.H.S.
6
( 
10.8
Exercise 10H
1
2
3
h
3
i
LHS 5 sin A cos 60° 1 cos A sin 60°
1 sin A cos 60° 2 cos A sin 60°
5 2 sin A cos 60°
1
5 2 sin A (_2 ) 5 sin A 5 R.H.S.
cos A cos B 2 sin A sin B cos (A 1 B)
LHS 5 _____________________ 5 __________
sin B cos B
sin B cos B
5 R.H.S.
sin x cos y 1 cos x sin y
LHS 5 ____________________
cos x cos y
cos x sin y
sin x cos y _________
_________
1
5
cos x cos y cos x cos y
5 tan x 1 tan y 5 R.H.S.
cos x cos y 2 sin x sin y
LHS 5 ____________________ 1 1
sin x sin y
5 cot x cot y 2 1 1 1 5 cot x cot y 5 R.H.S.
__
p
p
LHS 5 cos u cos __ 2 sin u sin __ 1 √3 sin u
3 __
3
__
√3
1
5 _2 cos u 2 ___ sin u 1 √3 sin u
2__
√3
1
_
___
sin u
5 2 cos u 2
2
270°
60°, 300°
140°, 220°
90°, 270°
45.6°, 134.4°
2120, 260, 240, 300
2144, 144
150, 330, 510, 690
b
d
f
h
j
b
d
f
a
2p, 0, p, 2p
b
4
a
b
c
d
e
g
i
5
a
1
√2
a
c
e
g
i
a
c
e
c
__
√3
___
)
60°, 240°
15°, 165°
135°, 315°
230°, 310°
135°, 225°
2171, 28.63
2327, 232.9
251, 431
4p 2p 2p
2 ___, 2 ___, ___
3
3 3
7p 5p p 3p
2 ___, 2 ___, __, ___
d 20.14, 3.00, 6.14
4
4 4 4
0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°, 360°
60°, 180°, 300°
1
1
1
1
22_2 °, 112_2 °, 202_2 °, 292_2 °
30°, 150°, 210°, 330°
300°
f 225°, 315°
90°, 270°
h 50°, 170°
165°, 345°
7p
p
2 ___, 2 ___
12
12
b
Edexcel IGCSE Further Pure Mathematics
b
1.48, 5.85
10.9
Exercise 10I
1
2
3
4
a
b
a
b
a
b
c
d
e
f
g
a
b
c
30°, 210°
135°, 315°
p, 2p
0.59, 3.73
60°, 120°, 240°, 300°
0°, 180°, 199°, 341°, 360°
60°, 300°
30°, 60°, 120°, 150°, 210°, 240°, 300°, 330°
270°
0°, 18.4°, 180°, 198°, 360°
194°, 270°, 346°
17p ____
23p
5p ____
11p ____
___
,
,
,
12 12 12 12
2p 4p
0.841, ___, ___, 5.44
3 3
4.01, 5.41
27
Mixed Exercise 10J
Edexcel IGCSE Further Pure Mathematics
1
cos2 u sin2 u
sin4 3u
1
a
b
c
2
a
3
a
4
b
a
b
5
a
6
7
8
4 + tan x
tan y = __________
b
1
2 tan x 2 3
2 sin 2u 5 cos 2u ⇒ 2 sin 2u \ cos 2u 5 1
⇒ 2 than 2u 5 1 ⇒ than 2u 5 0.5
13.3, 103.3, 193.3, 283.3
225, 345
22.2, 67.8, 202.2, 247.8
23p
2p 5p 5p 11p
11p ____
____
,
b ___, ___, ___, ____
12 12
3 6 3
6
0°, 131.8°, 228.2°
0, p, 2p
a Max 1, u = 100°; Min = 21, u = 280°
9
b
Max 1, u = 330°; Min = 21, u = 150°
a
1
i _2
b
23.8°, 203.8°
27
28
29
30
31
32
33
34
35
36
iii √3 \3
Review exercise
5
A 5 5, B 5 2 _2 , C 5
113
5
_
fmin 5 2 ___
4 ,x52
1 a
b
2
228_4
1
y
4
3
2
O
2
3 a 3
4 814
5 a –
6 a
b
b
1
i _4 b 2 a
2
_
c
5
2_2
1
3
c
x
4
125_2
1
P 5 126.8°, Q 5 R 5 26.6°
2
_
1
1
ii _3 a 1 _6 b
5
1
iii _3 a 2 _6 b
b
3
a –
b 54.5°, 234.5°
1
4_2 m
21.8°, 38.2°,120°
a 3.18, 6.69, 13.04
b –
c 2.3
d 0.6 (0.5 acceptable)
6
4
2
11 x 2 18x 1 135x
12 23 , p , 2
13 –
7
8
9
10
28
b
c
e
1 _3
__
6√ 3
109.5°
c
y 5 2 _5 (x 2 1)
b
1.11, 2.68
b
2, 64
b
–
1
6
2
23 449_5 p
24 a
c
25 a
26 a
__
1
ii _2
14 a (3.14), (5, 24)
15 a –
b 12 cm
d 54.7°
1
16 _2 x%
5
5
17 a–b (23, 2 _3 ), (2, _2 )
d –
18 a 0.253, 2.89
c 1.91, 2.30
19 a 625
c x 5 2, y 5 3
20 –
21 82.8°
22 a 8i 2 j
cos 2u 5 2 cos2 u 2 1
b sin 2u 5 2 sin u cos u
–
d 0.767, 1.33, 2.86
e 4
26
b 50
c 17
(2, 4), (5, 16)
b x < 2, x > 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
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Edexcel IGCSE Further Pure Maths answers