Topic 2 Answers

Cambridge IGCSE™ International Mathematics
2 Algebra
All answers were written by the authors. In examination, the way marks would be awarded to
answers like these may be different.
Exercise 2.1 page 96

1 a 4x – 12
b 10p – 20
c –42x + 24y
d 6a – 9b – 12c
e –14m + 21n
f –16x + 6y
2 a 3x2 – 9xy
b a2 + ab + ac
c 8m2 – 4mn
d –15a2 + 20ab
e 4x2 – 4xy
f 24p2 – 8pq
3 a –2x2 + 3y2
b a–b
c 7p – 2q
d 3x – 4y + 2z
3
e 3x – y
2
f 2x2 – 3xy
4 a 12r3 – 15rs + 6rt
b a3 + a2b + a2c
c 6a3 – 9a2b
d p2q + pq2 – p2q2
e m3 – m2n + m3n
f a6 + a5b
Cambridge IGCSE International Mathematics Online Teacher’s Guide 1

© Hodder & Stoughton Ltd 2018
Cambridge IGCSE International Mathematics answers

1 a –a – 8
b 4x – 20
c 3p – 16
d 21m – 6n
e 3
f –p – 3p2
2 a 8m2 + 28m + 2
b x–4
c 2p + 22
d m – 12
e a2 + 6a + 2
f 7ab – 16ac + 3c
3 a 4x + 4
3
b 5x – y
2
9 5
c x– y
4 2
9 1
d x– y
2 2
e 7x – 4y
f 0

1 a 2(2x – 3)
b 6(3 – 2p)
c 3(2y – 1)
d 2(2a + 3b)
e 3(p – q)
f 4(2m + 3n + 4r)
2 a a(3b + 4c – 5d)
b 2p(4q + 3r – 2s)
c a(a – b)
d 2x(2x – 3y)
e ab(c + d + f)
f 3m(m + 3)

3 a 3pq(r – 3s)
b 5m(m – 2n)
c 4xy(2x – y)
d b2(2a2 – 3c2)
e 12(p – 3)
f 6(7x – 9)
4 a 6(3 + 2y)
b 7(2a – 3b)
c 11x(1 + y)
d 4(s – 4t + 5r)
e 5q(p – 2r + 3s)
f 4y(x + 2y)
5 a m(m + n)
b 3p(p – 2q)
c qr(p + s)
d ab(1 + a + b)
e p3(3 – 4p)
f b2c(7b + c)
6 a m(m2 – mn + n2)
b 2r2(2r – 3 + 4s)
c 28xy(2x – y)
d 18mn(4m + 2n – mn)

1 a 0
b 30
c 14
d 20
e –13
f –4
2 a –3
b –30
c 20
d –16
e –40
f 42

3 a –160
b –23
c 42
d –17
e –189
f 113
4 a 48
b –8
c 15
d 16
e –5
f 9
5 a 12
b –5
c –5
d 7
e 7
f 36

1 a n=r–m
b m=p–n
c n = 3p – 2m
d q = 3x – 2p
cd
e a=
b
ab
f d=
c
4m
2 a x=
3y
7 pq
b r=
5
c
c x=
3
y−7
d x=
3

3r + 9
e y=
5
5y − 9
f x=
3
2a − 5
3 a b=
6
6b + 5
b a=
2
3x − 7 y
c z=
4
3x − 7 y
d x=
4
3x − 4 z
e y=
7
8+q
f p=
2r
4 a p = 4r
4
b p=
3r
n
c p=
10
d n = 10p
2t
e p=
q+r
2t
f q= −r
p
3m − n
5 a r=
t ( p + q)
3m − n
b t=
t ( p + q)
rt ( p + q ) + n
c m=
3
d n = 3m – rt(p + q)
3m − n
e p= −q
rt
3m − n
f q= −p
rt

ab
6 a d=
ce
cde
b a=
b
ab
c c=
de
d a = cd – b
a
e b= d−
c
a
f c=
d −b

1 a x2 + 5x + 6
b x2 + 7x + 12
c x2 + 7x + 10
d x2 + 7x + 6
e x2 + x – 6
f x2 + 5x – 24
2 a x2 + 2x – 24
b x2 – 3x – 28
c x2 – 2x – 35
d x2 – 2x – 15
e x2 – 2x – 3
f x2 + 2x – 63
3 a x2 – 5x + 6
b x2 – 7x + 10
c x2 – 12x + 32
d x2 + 6x + 9
e x2 – 6x + 9
f x2 – 12x + 35
4 a x2 – 9
b x2 – 49
c x2 – 64
d x2 – y2
e a2 – b2
f p2 – q2

5 a 2y2 + 7y + 6
b 3y2 + 25y + 28
c 2y2 + 17y + 8
d 4y2 + 6y + 2
e 6y2 + 23y + 20
f 18y2 + 15y + 3
6 a 2p2 + 13p – 24
b 4p2 + 13p – 35
c 6p2 + p – 12
d 12p2 + 13p – 35
e 18p2 – 2
f 28p2 + 44p – 24
7 a 4x2 – 4x + 1
b 9x2 + 6x + 1
c 16x2 – 16x + 4
d 25x2 – 40x + 16
e 4x2 + 24x + 36
f 4x2 – 9
8 a –4x2 + 9
b 16x2 – 9
c –16x2 + 9
d –25y2 + 49
e 8y2 – 18
f 25y2 – 70y + 49

1 a (x + y)(a + b)
b (x – y)(a + b)
c (3 + x)(m + n)
d (m + n)(4 + x)
e (m – n)(3 + x)
f (x + z)(6 + y)
2 a (p + q)(r – s)
b (p + 3)(q – 4)
c (q – 4)(p + 3)

d (r + 2t)(s + t)
e (s + t)(r – 2t)
f b + c)(a – 4c)
3 a (y + x)(x + 4)
b (x – 2)(x – y)
c (a – 7)(b + 3)
d (b – 1)(a – 1)
e (p – 4)(q – 4)
f (m – 5)(n – 5)
4 a (m – 3)(n – 2)
b (m – 3r)(n – 2r)
c (p – 4q)(r – 4)
d (a – c)(b – 1)
e (x – 2y)(x – 2z)
f (2a + b)(a + b)
Exercise 2.8
1 a (a – b)(a + b)
b (m – n)(m + n)
c (x – 5)(x + 5)
d (m – 7)(m + 7)
e (9 – x)(9 + x)
f (10 – y)(10 + y)
2 a (12 – y)(12 + y)
b (q – 13)(q + 13)
c (m – 1)(m + 1)
d (1 – t)(1 + t)
e (2x – y)(2x + y)
f (5p – 8q)(5p + 8q)
3 a (3x – 2y)(3x + 2y)
b (4p – 6q)(4p + 6q)
c (8x – y)(8x + y)
d (x – 10y)(x + 10y)
e (pq – 2p)(pq + 2p)
f (ab – cd)(ab + cd)

4 a (mn – 3y)(mn + 3y)

1 1  1 1 
b  2 x − 3 y  2 x + 3 y 
  
c (p2 – q2)(p2 + q2)
d 4(m2 – 3y2)(m2 + 3y2)
e (4x2 – 9y2)(4x2 + 9y2)
f (2x – 9y2)(2x + 9y2)
Exercise 2.9
1 a 60
b 240
c 2400
d 280
e 7600
f 9200
2 a 2000
b 9800
c 200
d 3200
e 998 000
f 161
3 a 68
b 86
c 1780
d 70
e 55
f 5
4 a 72.4
b 0.8
c 65
d 15
e 1 222 000
f 231


1 a (x + 4)(x + 3)
b (x + 6)(x + 2)
c (x + 12)(x + 1)
d (x – 3)(x – 4)
e (x – 6)(x – 2)
f (x – 12)(x – 1)
2 a (x + 5)(x + 1)
b (x + 4)(x + 2)
c (x + 3)2
d (x + 5)2
e (x + 11)2
f (x – 6)(x – 7)
3 a (x + 12)(x + 2)
b (x + 8)(x + 3)
c (x – 6)(x – 4)
d (x + 12)(x + 3)
e (x + 18)(x + 2)
f (x – 6)2
4 a (x + 5)(x – 3)
b (x – 5)(x + 3)
c (x + 4)(x – 3)
d (x – 4)(x + 3)
e (x + 6)(x – 2)
f (x – 12)(x – 3)
5 a (x – 4)(x + 2)
b (x – 5)(x + 4)
c (x + 6)(x – 5)
d (x + 6)(x – 7)
e (x + 7)(x – 9)
f (x + 9)(x – 6)
6 a (2x + 2)(x + 1)
b (2x + 3)(x +2)
c (2x – 3)(x + 2)
d (2x – 3)(x – 2)
e (3x + 2)(x + 2)
f (3x – 1)(x + 4)

g (2x + 3)2
h (3x – 1)2
i (3x + 1)(2x – 1)

P
1 a x=
2m
T
b x=
3
y2
c x=
m
d x =  p2 − q2 − y 2
e x =  y 2 − n 2 − m2
p2 − q2 + y2
f x=
4
P
2 a x=
Qr
P
b x=
Qr
Pr
c x=
Q
n
d x=
m
wst
e x=
r
wr
f x=
p+q
3 a x = (rp)2
2
 mn 
b x= 
 p 
k
c x=
g2
r2g
d x=
4 2

4m2 r
e x=
p2
4m2 r
f x=
p2

v−u
1 a t=
a
b u =  v 2 − 2as
v2 − u 2
c s=
2a
1
s − at 2
d u= 2
t
2 ( s − ut )
e a=
t2
2( s − ut )
t=
f a
A
2 a r=
 s2 + t 2
A 2
b h= ( ) − r2
r
vf
c U=
v− f
fu
d v=
u− f
2
 t 
e l =  g
 2 
 2 
2
f g = l 
 t 


xp
1 a
yq
q
b
y
p
c
r
d
d
c
bd
e
c2
f p
2 a m2
b r5
c x6
d xy2
e abc3
r3
f
pq
2x
3 a
y
b 4q2
c 5n
d 3x3y
e 3p
2
f
3mn
2a
4 a
3b
2y
b
x
c 2
d 3xy
e 3
f 6xy
8y
5 a
3

5p
b
2
p3
c
s
ef
d
c2
4qm
e
9p
x3
f
yz

4
1 a
7
a+b
b
7
11
c
13
c+d
d
13
x+ y+z
e
3
p2 + q2
f
5
3
2 a
11
c−d
b
11
4
c
a
2a − 5b
d
3
2x − 3y
e
7
1
f −
2x

1
3 a
2
3
b
2a
5
c
3c
7
d
2x
3
e
2p
1
f −
2w
3p − q
4 a
12
x − 2y
b
4
3m − n
c
9
x − 2y
d
12
5r + m
e
10
5s − t
f
15
7x
5 a
12
9x − 2 y
b
15
m
c
2
m
d
p
x
e
2y
4r
f
7s


5
1 a
6
8
b
15
11
c
28
11
d
15
29
e
36
24
f
35
3a + 2b
2 a
6
5a + 3b
b
15
7 p + 4q
c
28
6a + 5b
d
15
9 x + 20 y
e
36
10 x + 14 y
f
35
a
3 a
6
2a
b
15
11 p
c
28
11a
d
15
29 x
e
36
24 x
f
35

m
4 a
10
r
b
10
x
c −
4
29 x
d
28
23 x
e
6
p
f
6
p
5 a
2
2c
b
3
4x
c
5
m
d
3
q
e
5
w
f
4
3m
6 a
2
7m
b 3
m
c −
2
5m
d
2
p
e
3
36q
f
7
pr − p
7 a
r
x + xy
b
y

mn + m
c
n
a + ab
d
b
2xy − x
e
y
2 pq − 3 p
f
q

3x + 4
1 a
2 x2
10 − x2
b
2 x4
7x + 4
c
x ( x + 1)
(3x + 4)
2 a ( x + 1)( x + 2)
m−7
b
( m + 2 )( m − 1)
3p − 7
c
( 3)( p − 2 )
p −
w + 11
d
( w − 1) ( w + 3)
3y
e
( y + 4 )( y + 1)
12 − m
f
( m − 2 )( m + 3)
x
3 a
x+2
y
b
y+4
m+2
c
m−3
p
d
p−5

m
e
m+4
m +1
f
m+2
x
4 a
x+3
x
b
x+4
y
c
y −1
x
d
x+3
x
e
x+2
x
f
x +1
x
5 a
x +1
x
b
x+3
x
c
x−3
x
d
x+2
x
e
x−3
x
f
x+7

1 a x = –4
b y=5
c y = –5
d p = –4
e y=8
f x = –5.5

2 a x = 41
3
b x=5
c x=6
d y = –8
e y=4
f m = 10
3 a m=1
b p=3
c k = –1
d x = –21
e x=2
f y=3
4 a x=6
b y = 14
c x=4
d m = 12
e x = 35
f p=8
5 a x = 15
b x = –5
c x = 7.5
d x=8
e x = 2.5
f x = 10
6 a x=5
b x = 14
c x = 22
d x=5
e x=8
f x=2
7 a y = 10
b x = 17
c x = 13
d y = –9
e x=4
f x = 6.5


1 a i 3x + 60 = 180
ii x = 40
iii 40°, 60°, 80°
b i 3x = 180
ii x = 60
iii 20°, 80°, 80°
c i 18x = 180
ii x = 10
iii 20°, 50°, 110°
d i 6x = 180
ii x = 30
iii 30°, 60°, 90°
e i 7x – 30 = 180
ii x = 30
iii 10°, 40°, 130°
f i 9x – 45 = 180
ii x = 25
iii 25°, 55°, 100°
2 a i 12x = 360
ii x = 30
iii 90°, 120°, 150°
b i 11x + 30 = 360
ii x = 30
iii 90°, 135°, 135°
c i 12x + 60 = 360
ii x = 25
iii 35°, 80°, 90°, 155°
d i 10x + 30 = 360
ii x = 33
iii 33°, 94°, 114°, 119°
3 a i 11x – 80 = 360
ii x = 40
iii 40°, 80°, 80°, 160°
b i 10x + 60 = 360
ii x = 30
iii 45°, 90°, 95°, 130°

c i 16x + 8 = 360
ii x = 22
iii 44°, 96°, 100°, 120°
d i 9x + 45 = 360
ii x = 35
iii 35°, 55°, 120°, 150°
e i 6x + 18 = 180
ii x = 27
iii 50°, 50°, 130°, 130°
4 a 20
b 25
c 14
d 25
e 31
f 40
5 a 50
b 13
c 40
d 40
6 a 5
b 2
c 7
d 1.1
e 25
f 15

1 a 132.7 cm2 (1 d.p.)
A
b r=

c 12.6 cm (1 d.p.)
2 a 1206 cm3 (4 s.f.)
b 5.4 cm (1 d.p.)
3 a 37.8 °C (1 d.p.)
9
b F= C + 32
5
c 50 °F

4 a 125 m
2s
b t=
a
c 25.8 s (1 d.p.)
1
5 a A= (a + b) h
2
b 20 cm2
c 5 cm
6 a C = 2.5n + 5
C −5
b n=
2 12
c 30 km
7 a $9.40
b C = 10 – 0.2x
xy
c C = 10 −
100
8 a T = xe
b T = xe + yl
c C = 20 – (xe + yl + zc)
9 a C = 15.5n + 20
C − 20
b i n=
15.5
ii 23.5 metres
10 a 50 – 2x
b 30 – 2x
c x
d x(50 – 2x)(30 – 2x)
e 1872 cm2

1 a x = 4, y = 2
b x = 6, y = 5
c x = 6, y = –1
d x = 5, y = 2
e x = 5, y = 2
f x = 4, y = 9

2 a x = 3, y = 2
b x = 7, y = 4
c x = 1, y = 1
d x = 1, y = 5
e x = 1, y = 10
f x = 8, y = 2
3 a x = 5, y = 4
b x = 4, y = 3
c x = 10, y = 5
d x = 6, y = 4
e x = 4, y = 4
f x = 10, y = –2
4 a x = 5, y = 4
b x = 4, y = 2
c x = 5, y = 3
d x = 5, y = –2
e x = 1, y = 5
f x = –3, y = –3
5 a x = –5, y = –2
b x = –3, y = –4
2
c x = 4, y = 3
3
d x = 2, y = 7
e x = 1, y = 1
f x = 2, y = 9
6 a x = 2, y = 3
b x = 5, y = 10
c x = 4, y = 6
d x = 4, y = 4
7 a x = 1, y = –1
2
b x = 11 , y = 8
3
c x = 4, y = 0
d x = 3, y = 4
e x = 2, y = 8
f x = 1, y = 1


1 a x = 2, y = 3
b x = 1, y = 4
c x = 5, y = 2
d x = 3, y = 3
e x = 4, y = 2
f x = 6, y = 1
2 a x = 1, y = 4
b x = 5, y = 2
c x = 3, y = 3
d x = 6, y = 1
e x = 2, y = 3
f x = 2, y = 3
3 a x = 0, y = 3
b x = 5, y = 2
c x = 1, y = 7
d x = 6, y = 4
e x = 2, y = 5
f x = 3, y = 0
4 a x = 1, y = 0.5
b x = 2.5, y = 4
1
c x= ,y=4
5
3 1
d x= ,y=
4 2
1
e x = 5, y =
3
1
f x= ,y = 1
2


1 10 and 7
2 16 and 9
3 x = 1, y = 4
4 x = 5, y = 2
5 60 and 20 years old
6 60 and 6 years old

1 a 7
b 3
c 6
d 7
e 10
2
16
f 3
2 a 7
b 6
c 6
d 10
e 3
3 a 8
b 5
c 8
d 6
e 6
4 a 6, 18, 26
b 160, 214, 246, 246
c 50°, 80°, 100°, 140°, 170°
d 80°, 80°, 80°, 160°, 160°, 160°
e 150°, 150°, 150°, 150°, 120°, 120°, 120°, 120°


1 a –4 and –3
b –2 and –6
c –1 and –12
d 2 and 5
e 2 and 3
f 2 and 4
2 a –5 and 2
b –2 and 5
c –7 and 2
d –2 and 7
e –5 and 3
f –3 and 5
3 a –3 and –2
b –3
c –8 and –3
d 4 and 6
e –4 and 3
f –2 and 6
4 a –2 and 4
b –4 and 5
c –6 and 5
d –6 and 7
e –7 and 9
f –9 and 6

1 a –3 and 3
b –4 and 4
c –5 and 5
d –11 and 11
e –12 and 12
f –15 and 15

2 a –2.5 and 2.5

b –2 and 2
c –1.6 and 1.6
1 and 1
d −
2 2
1 and 1
e −
3 3
1 and 1
f −
20 20
3 a –4 and –1
b –5 and –2
c –4 and –2
d 2 and 4
e 2 and 5
f –4 and 2
4 a –2 and 5
b –5 and 2
c –3 and 6
d –6 and 3
e –4 and 6
f –6 and 8
5 a –4 and 3
b –6 and –2
c –9 and 4
d –1
e –2
f –9 and –8
6 a 0 and 8
b 0 and 7
c –3 and 0
d –4 and 0
e 0 and 9
f 0 and 4
7 a –1.5 and –1
b –1 and 2.5
1
c –1 and
3
d –5 and − 1
2

e 1.5 and 5
1 1
f −1 and
3 2
8 a –12 and 0
b –9 and –3
c –8 and 4
d –7 and 2
e –6 and 6
f –10 and 10

1 –4 and 3
2 –6 and 7
3 2
4 4
5 Height = 3 cm, base length = 12 cm
6 Height = 20 cm, base length = 2 cm
7 Base length = 6 cm, height = 5 cm
8 a 9x + 14 = 50
b x=4
c 11 m × 6 m

1 a –3.14 and 4.14
b –5.87 and 1.87
c –6.14 and 1.14
d –4.73 and –1.27
e –6.89 and 1.89
f 3.38 and 5.62
2 a –5.30 and –1.70
b –5.92 and 5.92
c –3.79 and 0.79
d –1.14 and 6.14
e –4.77 and 3.77
f –2.83 and 2.83

3 a –0.73 and 2.73

b –1.87 and 5.87
c –1.79 and 2.79
d –3.83 and 1.83
e 0.38 and 2.62
f 0.39 and 7.61
4 a –0.85 and 2.35
b –1.40 and 0.90
c 0.14 and 1.46
d –0.5 and –2
e –0.39 and 1.72
f –1.54 and 1.39

1 a (3, 2)
b (5, 2)
c (2, 1)
d (2, 1)
e (–4, 1)
f (4, −2)
2 a (3, −2)
b (−l, −1)
c (–2, 3)
d (−3, −3)
e Infinite solutions
f No solution
3 The lines in part (e) are the same line. The lines in part (f) are parallel.


1 a i
ii (1, 0) and (2, 0)

b i
ii (–6, 0) and (2, 0)

c i
ii (3, 0) and (5, 0)

d i
ii No roots
e i
ii No roots
f i
ii (3, 0)
2 a i
ii (–2, 0) and (3, 0)

b i
ii (4, 0) and (6, 0)

c i
ii (–2, 0) and (8, 0)

d i
ii (–5, 0)


1 a x<4
b x>1
c x≤3
d x≥7
e x<2
f x>2
2 a x<7
b x ≥ –2
c x > –3
d x ≥ –12
e x > –24
f x ≥ –3
3 a x<2
b x ≤ 12
c x≥2

d x ≥ –2
e x > –0.5
f x≥2

1 a 2<x≤4
b 1≤x<5
c 3.5 ≤ x < 5
d 2 ≤ x < 4.2
2 a 1<x≤5
b –1 ≤ x < 1
c 2<x<3
d No solution


1 a


2 a



1


1 a 33
b 25
c 42
d 64
e 86
f 51
2 a 23 × 32
b 45 × 52
c 32 × 43 × 52
d 2 × 74
e 62
f 33 × 42 × 65
3 a 4×4
b 5×5×5×5×5×5×5
c 3×3×3×3×3
d 4×4×4×6×6×6
e 7×7×2×2×2×2×2×2×2
f 3×3×4×4×4×2×2×2×2
4 a 32
b 81
c 64
d 216
e 1 000 000
f 256
g 72
h 125 000


1 a 36
b 87
c 59
d 410
e 24
f 35 × 66
g 48 × 59 × 62
h 24 × 510 × 68
2 a 44
b 53
c 2
d 63
e 63
f 8
g 43
h 37
3 a 54
b 412
c 1010
d 315
e 68
f 86
4 a 23
b 3
c 53
d 45
e 28
f 64 × 85
g 43 × 52
h 4 × 67

5 a c8
b m2
c b9
d m3n6
e 2a4b
f 3x3y2
uv 3
g
2
x2 y3 z 2
h
3
6 a 12a5
b 8a5b3
c 8p6
d 16m4n6
e 200p13
f 32m5n11
g 24xy3
h a(d+e)b(d+e)

1 a 8
b 25
c 1
d 1
e 1
f 0.25
1
2 a
4
1
b
9
3
c
50
1
d
200
e 1
1
f
1000

3 a 1
b 2
c 4
1
d
2
1
e
6
f 10
4 a 12
b 32
c 225
d 80
e 7
f 64

1 a 2
b 4
c 3
d 3
e 4
f 0
2 a 4
b 1
3
c
2
d –1
e 2
f 3
3 a –3
b –4
c –5
2
d −
3
e –2
1
f −
2

4 a –3
b –7
c –3
d 2
e 8
f 5

1 a 7 cm
b 0 cm
c 5 hours
d
1
e Approx. 5 hours
2
2 a Approx. 2.5
1
b −
2
3 a Approx. 4.3
b Approx. 3.3
c Approx. 2.3
4 a Approx. 0.3
b Approx. 2.4


1 a 4
b 5
c 10
d 3
e 9
f 10
2 a 2
b 3
c 2
d 2
e 6
f 4
3 a 8
b 32
c 27
d 64
e 1
f 9
4 a 25
b 8
c 32
d 100
e 32
f 27
1
5 a
2
1
b
3
1
c
2
1
d
3
1
e
2
1
f
6

1
6 a
3
1
b
2
1
c
4
1
d
3
1
e
6
1
f
3

1 a 1
b 7
c 2
d 1
e 81
f 6
2 a 25
b 2
c 2
d 27
e 4
f 2
3 a 4
b 2
c 64
d 9
e 3
f 27


1 a i 3n + 2
ii 32
b i 4n − 4
ii 36
c i n − 0.5
ii 9.5
d i −3n + 9
ii −21
e i 3n – 10
ii 20
f i −4n − 5
ii −45
2 a
Position 1 2 5 12 50 n
Term 1 5 17 45 197 4n – 3
b
Term 5 11 29 59 449 6n – 1
c
Position 1 3 8 50 100 n
Term 2 0 –5 –47 –97 –n + 3
d
Position 1 2 3 10 100 n
Term 3 0 –3 –24 –294 –3n + 6
e
Term 1 10 16 25 145 3n – 5
f
Term –5.5 –7 –11.5 –34 –79 –1.5n – 4

3 a i +4
ii 4n + 1
iii 201
b i +1
ii n−1
iii 49
c i +3
ii 3n – 13
iii 137
d i +0.5
ii 0.5n + 5.5
iii 30.5
e i +4
ii 4n − 62
iii 138
f i −3
ii −3n + 75
iii −75

1 un = n2 + 1
2 un = n2 – 1
3 un = n2 + 5
4 un = n2 + 8
5 un = n2 – 3
6 un = 2n2 + 2
7 un = 2n2 – 2
8 un = 3n2 + 2
9 un = 4n2 – 4
10 un = 5n2 – 4


1 un = n3 + 10
2 un = n3 – n
3 un = n3 – 5
4 un = n3 + n2
5 un = n3 + n2 + 5n
6 un = n3 + 3n2 + 5n – 2
7 un = n3 + n2 – n
8 un = n3 + 5n + 7

1 a Geometric
b Geometric
c Not geometric
d Geometric
e Not geometric
f Not geometric
2 a i 3
ii 162, 486
iii un = 2(3)n−1
1
b i
5
1 1
ii ,
25 125
n−1
1
iii un = 25  
5
d i −3
ii −243, 729
iii un = (−3)n
3 a −6, −12, −24
b 8
4 a −4
1
b
4
c −65 536
5 €38 203.20

6 a Because 0.85 ≠ 0
b $3276.80
c Because 0.8n ≠ 0 where n is the number of years

1 a 3
b 21
c 27
d 3
e 10
2 a 0.5
b 8
c 24.5
d 8
e 16
3 a 24
3
b
8
1
c
9
d 1
4 a 0.25
b 25
c 4
d 1
16

1 a i y ∝ x3
ii y = kx3
1
b i y∝
x3
k
ii y=
x3
c i t∝P
ii t = kP

1
d i s∝
t
k
ii s=
t
e i A ∝ r2
ii A = kr2
1
f i T∝
g
k
ii T=
g
2 10.5
1
3 a
2
b 2
4 32
1
5 a
8
b 0.4
6 75

1 a h = kt2
b k=5
c 45 m
d 6 s
2 a v=k e
b 3
c 49 J
1
3 a l = km 3
b 3
c l = 6 cm
4 a P = kI2
b 5 amps

House of cards page 174

1 155
2 5612
3 The formula can be proved using the method of differences explained in Topic 2 as follows:
Height of house 1 2 3 4 5
Number of cards 2 7 15 26 40
1st difference 5 8 11 14
2nd difference 3 3 3
Comparing with the algebraic table below:

Position 1 2 3 4 5
16a + 4b + c
25a + 5b + c
Term
4a + 2b + c
9a + 3b + c
a+b+c
1st difference
3a + b
5a + b
9a + b
7a + b
2nd difference 2a 2a 2a
It can be deduced that:

3
2a = 3 therefore a =
2
1
3a + b = 5 therefore b =
2
a+b+c=2 therefore c = 0.
3 1 1
This produces the rule c = n2 + n which factorises to c = n(3n + 1).
2 2 2
Chequered boards page 174

1 Student’s own diagrams and results, presented in a logical table.
mn
2 Where either m or n is even, the number of black and white squares is given by . Where
2
both m and n are odd, the number of black and white squares differ by one. The number of
mn – 1 mn + 1
black squares is and the number of white squares is , assuming that the bottom
2 2
right-hand corner is white.

Modelling: Linear programming page 175

1

ICT Activity 1 page 175

1

ICT Activity 2 page 176

1 x ≈ 2.7
2 x ≈ 2.6
3 x ≈ 2.1
4 x ≈ 0.6
Student assessment 1 page 177

1 a 6x – 9y + 15z
b 8pm – 28p
c –8m2n + 4mn2
d 20p3q – 8p2q2 – 8p3
e –2x – 2
f 22x2 – 14x
g 2
5 2
h x −x
2
2 a 8(2p – q)
b p(p – 6q)
c 5pq(p – 2q)
d 3pq(3 – 2p + 4q)
3 a 0
b –7
c 29
d 7
e 7
f 35
4 a n = p – 4m
4x − 5z
b y=
3
10 px
c y=
3
3w
d y= −x
m
pqt
e r=
4mn
f q = r(m – n) – p


1 a (q + r)(p – 3r)
b (1 – t2)(1 + t2)
c 750 000
d 50
2 a x2 – 2x – 8
b x2 – 16x + 64
c x2 + 2xy + y2
d x2 – 121
e 6x2 – 13x + 6
f 9x2 – 30x + 25
3 a (x – 11)(x + 7)
b (x – 3)(x – 3)
c (x – 12)(x + 12)
d 3(x – 2)(x + 3)
e (2x – 3)(x + 4)
f (2x – 5)2
p
4 a f =
m
m
b t=
5
2
 A
c p =  −q
r 
ty
d x=
y −t
5 a x4
b nq
c y3
4
d
q
2m
6 a
11
3p
b −
16
3x
c
4y

6m + 13n
d
30 p
9p
7 a
20
m
b
10
−p
c
12
7 x − 23
8 a
( x − 5 )( x − 2 )
a−b
b
a+b
1
c
x+3

1 a 0.53 m3 (2 d.p.)
V
b r=
h
c 5.05 cm
2 a 66 °C (2 s.f.)
b –11 °C (2 s.f.)
9
c F= C + 32
5
d 320 °F
3 a 15 hours
b H = 1200(T – k)
c 5000 m
4 a 524 cm3 (3 s.f.)
3V
b r= 3
4
c r = 8.42 m (3 s.f.)
5 a £251.50
x − 1.5
b n=
0.05
c n = 470


1 a –6
b 6
c 4
d 2.4
2 a 0.5
b 4
c 9.5
d 5
3 a 6
b 15
c 22
d 6
4 a 8.5
1
b 4
3
c 8.5
d 12
5 a x = 7, y = 4
b p = 1, q = 2
c x = 7, y = 1
d m = 3, n = 5

1 a 9x – 90 = 360
b x = 50
c 50°, 60°, 100°, 150°
2 2
3 135°, 85°, 95°, 95°, 130°
4 –4 and 5
5 2.77 and –1.27 (3 s.f.)
6 3<x≤5
7 All values other than 0


1 a x – y = 30, x + y + 40 = 180
b x = 85, y = 55
2 a
b 5x + 100
c 92°
d
e 92° + 82° + 72° + 62° + 52° = 360°

3 a
b 120 – 4x2
c 4x2 – 64 = 0 x = 4

4 b 0.44 and 4.56 (2 d.p.)

5 a
c 10 cm, 8 cm, 6 cm
6 a x≤7
b y≤9
7 a 1≤p<4
b 4<x≤7

1 a 22 × 35
b 214
2 a 6×6×6×6×6
1
b
2 2 2 2 2
3 a 27 000
b 125
4 a 27
b 77 × 312
c 26
d 33
e 4–1
f 22
5 a 5
b 16
c 49
d 48
6 a 2.5
b –0.5
c 0
d 2
9


1 a 2
b 81
1
c
3
d 64
e 3
f 2
g 6
h 32
2 a 15
b 4
c 3
d 3125
e 16
f 1
g 27
h 10
3 a
b Approx. –2.6

1 a i un = 4n – 3
ii u10 = 37
b i un = –3n + 4
ii u10 = –26

2 a u5 = 27, u100 = 597
b u5 = 3 , u100 = –46
2
3 a
Term 17 14 11 –10 –55 –3n + 20
b
Position 2 6 10 80 n 2
1
Term –4 –2 0 35 n–5 –4
2
4 a $515.46
b 3 years
5 a $253.50
b $8.08
1
6 a r= −
3
b u1 = 243
c n = 10
7 un = n3 + 3n2 + 4
8 un = n3 + n2 + 3n + 5

1 a 1.5
b 15
c 3
d 12
2 a 10
b 2.5
c 1
d 20
1
3 a
3
b 72
1
c
3
d 12

4 a 5
5
b
4
1
c
2
d 1
1
5 a
3
4
b
3
c ±2
d ±1

1 a
x 1 2 4 8 16 32
y 32 16 8 4 2 1
b 1.6
2 a
x 1 2 4 5 10
y 5 10 20 25 50
b
x 1 2 4 5 10
y 20 10 5 4 2
c
x 4 16 25 36 64
y 4 8 10 12 16
3 a 0.8
b 0.8
4 a 1.6 (1 d.p.)
b 2


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Cambridge IGCSE™ International Mathematics

Exercise 2.1 page 96

Cambridge IGCSE International Mathematics Online Teacher’s Guide 1

Exercise 2.2 page 96

Exercise 2.3 page 97

Cambridge IGCSE International Mathematics Online Teacher’s Guide 2

Exercise 2.4 page 99

Cambridge IGCSE International Mathematics Online Teacher’s Guide 3

Exercise 2.5 page 99

Cambridge IGCSE International Mathematics Online Teacher’s Guide 4

Cambridge IGCSE International Mathematics Online Teacher’s Guide 5

Exercise 2.6 page 100

Cambridge IGCSE International Mathematics Online Teacher’s Guide 6

Exercise 2.7 page 101

Cambridge IGCSE International Mathematics Online Teacher’s Guide 7

Cambridge IGCSE International Mathematics Online Teacher’s Guide 8

4 a (mn – 3y)(mn + 3y)

Cambridge IGCSE International Mathematics Online Teacher’s Guide 9

Exercise 2.10 page 104

Cambridge IGCSE International Mathematics Online Teacher’s Guide 10

Exercise 2.11 page 105

Cambridge IGCSE International Mathematics Online Teacher’s Guide 11

Exercise 2.12 page 106

Cambridge IGCSE International Mathematics Online Teacher’s Guide 12

Exercise 2.13 page 106

Cambridge IGCSE International Mathematics Online Teacher’s Guide 13

Exercise 2.14 page 107

Cambridge IGCSE International Mathematics Online Teacher’s Guide 14

Cambridge IGCSE International Mathematics Online Teacher’s Guide 15

Exercise 2.15 page 108

Cambridge IGCSE International Mathematics Online Teacher’s Guide 16

Cambridge IGCSE International Mathematics Online Teacher’s Guide 17

Exercise 2.16 page 109

Cambridge IGCSE International Mathematics Online Teacher’s Guide 18

Exercise 2.17 page 110

Cambridge IGCSE International Mathematics Online Teacher’s Guide 19

Cambridge IGCSE International Mathematics Online Teacher’s Guide 20

Exercise 2.18 page 112

Cambridge IGCSE International Mathematics Online Teacher’s Guide 21

Exercise 2.19 page 117

Cambridge IGCSE International Mathematics Online Teacher’s Guide 22

Exercise 2.20 page 125

Cambridge IGCSE International Mathematics Online Teacher’s Guide 23

Cambridge IGCSE International Mathematics Online Teacher’s Guide 24

Exercise 2.21 page 126

Cambridge IGCSE International Mathematics Online Teacher’s Guide 25

Exercise 2.22 page 128

Exercise 2.23 page 129

Cambridge IGCSE International Mathematics Online Teacher’s Guide 26

Exercise 2.24 page 133

Exercise 2.25 page 133

Cambridge IGCSE International Mathematics Online Teacher’s Guide 27

2 a –2.5 and 2.5

Cambridge IGCSE International Mathematics Online Teacher’s Guide 28

Exercise 2.26 page 133

Exercise 2.27 page 136

Cambridge IGCSE International Mathematics Online Teacher’s Guide 29

3 a –0.73 and 2.73

Exercise 2.28 page 140

Cambridge IGCSE International Mathematics Online Teacher’s Guide 30