# CAMBRIDGE IGCSE MATHEMATICS Answers by liwenting

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```									CAMBRIDGE IGCSE MATHEMATICS Answers
Number
1 The number system                                                          b
ξ
1. a 12       b9          c6    d 13     e 15           f 14
g 16       h 18         i 17   j 8 (or 16)

2. 1, 2, 3, 22, 5, 2 × 3, 7, 23, 32, 2 × 5, 11, 22 × 3, 13,
2 × 7, 3 × 5, 24, 17, 2 × 32, 19, 22 × 5, 3 × 7,
2 × 11, 23, 23 × 3, 52, 2 × 13, 33, 22 × 7, 29,                                            A                          B
2 × 3 × 5, 31, 25, 3 × 11, 2 × 17, 5 × 7, 22 × 32, 37,
2 × 19, 3 × 13, 23 × 5, 41, 2 × 3 × 7, 43, 22 × 11,
33 × 5, 2 × 23, 47, 24 × 3, 72, 2 × 52
4. a Listing the elements of each set:
3. a 20       b 56              c 6    d 28      e 10          f 15
g 24       h 30                                                           ξ = {Positive integers less than 15} = {1, 2, 3, 4,
…..14}
4. a 168  b 105   c 84                       d 84       e 48
f 54  g 75   h 144                                                        D = {Odd numbers} = {1, 3, 5, 7, 9, 11, 13}
5. a 8        b 7         c 4         d 14     e 12       f 9                F = {Factors of 12} = {1, 2, 3, 4, 6, 12}
g 5        h 4         i 3         j 16     k 5      l 18

6. a ii and iii         b iii                                                    ξ    8 10 14

2 Sets and set notation
1   2 4
1. a A ∩ B = {o, u}                                                                                       6 12
5 7 9    3
11 13
b A ∪ B = {a, b, e, h, i, m, o, r, s, u}
D                           F
c A' ∩ B = {r, h, m, b, s}

d A ∩ B' = {a, e, i}
b D ∩ F' = {1, 3, 5, 7, 9, 11, 13}∩{5, 7, 8, 9, 10,
e n (B) = 7                                                               11, 13, 14, 15}

f n (B') = 19                                                             D ∩ F' = {5, 7, 9, 11}

g n (A ∪ B) = 10                                                          n(D ∩ F') = 4

2. a A ∩ B' = {11, 12, 13, 14} ∩ {1, 3, 5, 7, 9, 11, 13,                  5. a and b
15, 17, 19} = {11,13}
ξ                C
b A ∪ C = {11, 12, 13, 14} ∪ {7, 14} = {7, 11, 12,
13, 14}
n(A ∪ C) = 5                                                                                                  8
c A ∩ B ∩ C = {11, 12, 13, 14} ∩ {2, 4, 6, 8, 10,                                         13              M
12, 14, 16, 18} ∩ {7, 14} = {14}

3. a                                                                                 P

ξ
c P ∩ M = ∅ or P ∩ M = {}

d n(P ∩ C) = 2

A                               B

1
6. Completing a Venn diagram to show the situation                  3 Powers, roots and reciprocals
more clearly
1. 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256,
3 pupils do              289, 324, 361, 400
ξ = 15                              3       not take
either subject.
2. a 25, 169, 625, 1681, 3721
b Answers in each row are the same

3. a 2           b 5          c 7            d 1            e 9         f 10
g 8           h 3          i 6            j 4            k 11        l 12
m 20          n 30         o 13
History = 11         Geography = 6                                4. a   +5
–          b    +6
–          c   + 10
–            d   +7
–
e   +8         f    +4         g   +3           h   +9
–               –              –                –
i   +1         j    + 12
The number of students shown on the diagram                             –               –

= 11 + 6 + 3 = 20
5. a 81          b 40         c 100          d 14           e 36        f 15
But there are only 15 students in the class so that                 g 49          h 12         i 25           j 21           k 121       l 16
5 students are being counted twice.                                 m 64          n 17         o 441

These 5 students take both history and geography.                6. a 24          b 31         c 45   d 40   e 67                        f 101
g 3.6         h 6.5        i 13.9 j 22.2
3 pupils do
ξ = 15                               3       not take
7. a 27           b 125               c 216            d 1728       e 16
either subject.
f 256          g 625               h 32             i 2187       j 1024

8. a 100     b 1000      c 10 000     d 100 000
6      5       1                                           e 1 000 000     f The power is the same as the
number of zeros.
g i 100 000 000       ii 10 000 000 000
History = 11          Geography = 6                                     iii 1 000 000 000 000 000

4 Indices
The diagram confirms that 5 students take both
history and geography.                                           1. a 24          b 35         c 72           d 53   e 107               f 64
g 41          h 17         i 0.54         j 1003
7. a
2. a   3×3×3×3         b 9×9×9
ξ                                                               c   6×6      d 10 × 10 × 10 × 10 ×10
C                                                           e   2×2×2×2×2×2×2×2×2×2
f   8    g 0.1 × 0.1 × 0.1   h 2.5 × 2.5
i   0.7 × 0.7 × 0.7   j 1000 × 1000
B
3. a 1       b –1            c 1            d 1
1                 1               1               1           1
4. a             b                c                d          e
A                                                 53                6              105              32          82
1                 1              1                1           4
2                                 m          3
f 9           g w              h t              i x        j m
b n(B) = 4
5. a 3–2         b 5–1            c 10–3       d m–1          e t –n
If n(C) = 6 and n(A) = 2 then the maximum that n(B)
could be is 4 since A C C and B C C, and                         6. a   i   24         ii   2–1        iii   2–4        iv   –23
A∩B=∅                                                               b   i   103        ii   10–1       iii   10–2       iv   106
c   i   53         ii   5–1        iii   5–2        iv   5–4
d   i   32         ii   3–3        iii   3–4        iv   –35

7. a 54          b 53         c 52           d 53           e 5–5

8. a 63          b 60         c 66           d 6–7          e 62

9. a 46          b 415        c 46           d 4–6          e 46        f 40

2
10. a 5       b 9    c 10                     d 20       e 3              7 Standard form
1    8     1
f 6      g –
5  h –
9   i –
4
1. a 0.65          b 0.065        c 0.0065           d 0.000 65
11. a 16         b 25        c 216            d 81
2. a 65        b 650          c 6500       d 65 000
12. a 4       b 9         c 64          d 3125
3. a   250               b   34.5          c     0.004 67
1        1       1       1          1       1     1        1
13. a –
5      b –
6     c –
2     d –
3        e –
4     f –
2   g –
2      h –
3           d   34.6              e   0.020 789     f     5678
1           1        1                  1         1                 g   246               h   7600          i     897 000
14. a –––––––
100 000      ––
b 12     c ––
25               d ––
27      e ––
32
1        1        1                                                   j   0.008 65          k   60 000 000    l     0.000 567
f ––
32     g ––
81     h ––
13

4. a 2.5 × 102      b          3.45 × 10–1     c 4.67 × 104
5       Directed numbers                                                     d 3.4 × 109      e          2.078 × 10  10
f 5.678 × 10–4
g 2.46 × 103     h          7.6 × 10 –2
i 7.6 × 10–4
1. a –15         b –14           c –24         d 6          e 14
j 9.99 × 10 –1
k 2.3456 × 102
f 2           g –2            h –8          i –4         j 3
l 9.876 54 × 101             m 6 × 10–4
k –24         l –10           m –18         n 16         o 36
n 5.67 × 10–3                o 5.60045 × 101
2. a –9          b 16            c –3          d –32        e 18
5. a     5.67 × 103                    b   6 × 102
f 18          g 6             h –4          i 20         j 16
c     3.46 × 10–1                   d   7 × 10–4
k 8           l –48           m 13          n –13        o –8
e     5.6 × 102                     f   6 × 105
3. a –2          b 30             c 15          d –27       e –7             g     7 × 103                       h   1.6
i     2.3 × 107                     j   3 × 10–6
4. a –9          b 3              c 1
k     2.56 × 106                    l   4.8 × 102
5. a 16            b –2           c –12                                      m     1.12 × 102                    n   6 × 105
o     2.8 × 102
6. Any appropriate divisions.
6. a     1.08 × 108                    b   4.8 × 106
6 Fractions and decimals                                                     c     1.2 × 109                     d   1.08
e     6.4 × 102                     f   1.2 × 101
8          1            1
––
1. a 15         ––
b 10          c –
3                                              g     2.88                          h   2.5 × 107
7           31             43            31                             i     8 × 10–6
2. a –
9       b 3 ––
45         c 1 ––
48        d 1 ––
84

1          7           5                                             7. a     1.1   ×   108                 b   6.1   ×   106
3. a –
6         ––
b 20         c –
8
c     1.6   ×   109                 d   3.9   ×   10–2
5
––
4. a 12           9
––
b 3 10         c 30                                             e     9.6   ×   108                 f   4.6   ×   10–7
3           2               1            1
g     2.1   ×   103                 h   3.6   ×   107
5. a –        b 1–          c 1 ––       d 1 ––       e 4       f 4
4           5              15           14                              i     1.5   ×   102                 j   3.5   ×   109
5            4             3
g 5       h 1–
7          i –
9          j 1–
5                                k     1.6   ×   104
6. 40                                                                     8. a     2.7 × 107                     b   1.6 × 106
1
7. a –          89
–––
b 100             7
c ––                                             c     2 × 107                       d   4 × 10–2
8                        32
e     2 × 101                       f   6 × 10–2
1
8. –
3                                                                         g     2 × 10–1                      h   5 × 108
i     2 × 105
9. 260
9. a     5.4   ×   105                 b   2.9   ×   109
10. 16
c     1.1   ×   10–2                d   6.3   ×   10–4
2
11. a 2 ––
15
9
b ––
32
256
c –––
625
e     2.8   ×   102                 f   5.5   ×   10–2
·    ·    ·                                                            g     4.9   ×   108                 h   8.6   ×   102
12. 0.1, 0.2, 0.3, etc. Digit in decimal fraction same
as numerator.                                                         10. 2 × 1013, 1 × 10–10, mass = 2 × 103 g (2 kg)

13.0.444 …, 0.454 …, 0.428 …, 0.409 …, 0.432 …,                           11. a 6 × 107 sq miles             b 30%
9 3         4 5 6
0.461 …, –– , –, 16 , –, ––, ––
22 7 –– 9 11 13
37
12. 5 × 104
24
14. a 24.242 42 …               b 24            ––
c 99
13. 9.41 × 104
8        4           7
15. a   –
9    b   –
9      c   2–
9

3
8 Rounding and estimating                                                5. £5525

1. a 50 000            b 0.5       c 10         d 1000                   6. a 25%         b 20.8%        c 1.9%

2. a 56 000            b 1.7       c 0.80 d 0.0098                       7. 32%

3. a 60 000            b 1.1       c 0.2                                 8. 6.49%

4. a 4.8       b 11.1             c 213.9        d 51.0                  12 Reverse percentages
5. a 5.78          b 2.31         c 7.81         d 0.01                  1. a 800 g      b 60 cm      c \$400
6. a 4.6                b 11.99        c 899.996                         2. \$833.33
d 0.1                e 0.01
3. \$300
7. a 80 000            b 1000     c 5000        d 75     e 100
4. 240
9 Limits of accuracy
5. 4750 blue bottles
1. a 7.5, 8.5                  b 84.5, 85.5
c 0.055, 0.065              d 365.5, 366.5                            6. \$22
1
2. a <65.5 g            b 64.5 g       c <2620 g          d 2580 g       7. less by –%
4

3. 79.75 m2            area     100.75 m2                                13 Simple and compound interest
3                               3
4. 216.125 m             volume      354.375 m                           1. 12 years
5. 20.9 m      length           22.9 m (3 sf)                            2. a i 2550     ii 2168 iii 1331        b 7 years
6. a 14.65 s time 14.75 s                                                3. a \$6800       b \$5440     c \$3481.60
b 99.5 m length 100.5 m
c 6.86 m/s (3 sf)                                                     4. a 21 years     b 21 years

7. 3.41 cm         length        3.43 cm (3 sf)
14 Using a calculator
8. 14 s     time         30 s
1. a 35 000      b 15 000       c 960      d 5
e 1200        f 500
10 Ratios
7
2. a 39 700      b 17 000       c 933      d 4.44
––
1. 10
e 1130        f 550
2. 2
–
5                                                                     3. a 4000       b 10      c 1       d 19      e 3
3. a   2
–
5   b   3
–
5
f 18

2
4. a –
9
1
b –
3
2
c –
9
4. a 4190       b 8.79     c 1.01 d 20.7      e 3.07
f 18.5
5. a 160 g : 240 g    b 175 min : 125 min
c 50 g : 250 g : 300 g                                                5. a i 27.57142857        ii 27.6
b i 16.89651639        ii 16.9
6. a 28.6%                     b 250 kg                                     c i 704.4198895        ii 704
7. a 1 : 400 000               b 1 : 125 000       c 1 : 250 000         15 Practical graphs
d 1 : 25 000                e 1 : 20 000        f 1 : 40 000          1. a 32°F       9
b – (Take gradient at C =10° and 30°.)
5
9
8. a 1 : 1 000 000             b 47 km                 c 0.8 cm             c F = – C + 32
5

2. a 0.07 (Take gradient at U = 0 and 500.) b \$10
11 Percentages                                                              c C = \$(10 + 0.07U) or Charge = \$10 + 7cents/unit
1. a \$62.40             b \$38.08       c 253.5 g          d \$30.24            5
3. a – (Take gradient at D = 0 and 40.)   b \$20
2
5D
2. a \$9.40              b 731 m        c 360 cm                             c C = \$(20 +     ) or Charge = \$20 + \$2.50/day
2
1
––
4. a 10    b 24.5 cm      c 0.1 cm or 1 mm
3. 1 690 200                                                                              W
d L = 24.5 +      or Length = 24.5 + 1 mm/kg
10
4. 575 g

4
5. a i 9 am          ii 10 am        iii 12 noon
b i 40 km/h       ii 120 km/h     iii 40 km/h

6. a i 125 km      ii 125 km/h
b i between 2 pm and 3 pm
1
2

7. a Patrick ran quickly at first, then had a slow middle
section but he won the race with a final sprint. Araf
ran steadily all the way and came second. Sean set
off the slowest, speeded up towards the end but
still came in third.
b i 1.67 m/s        ii 6 km/h
1                               1
8. a 2 – km/h
2          b 3.75 m/s       c 2 – km/h
2

9. a AB: 30 km/h, BC: 6 km/h, CD: 0 km/h,
DE: 36 km/h (in opposite direction)
b FG: 4 m/s, GH: 16 m/s, HI: 2 m/s (in opposite
direction), IJ: 16 m/s (in opposite direction)

10. a 20 m/s2      b 7.0 m/s2

11. a 3.5 m/s2     b 3.5 m/s2

12. a between 2 and 4 hours, and between 8 and
10 hours
b 10 km/h2, 0 km/h2, 5 km/h2, 0 km/h2,
–6.25 km/h2, –3.75 km/h2

5
4
1 Linear graphs                                                                                                                      9. a y = – x – 2 or 3y = 4x – 6
3                       b y=x+1
c y = 2x – 3     d 2y = x + 6   e y=x
1.   20                                 y = 3x + 4       2.       6                                     y = 2x – 5                      f y = 2x
5
18
4                                                                  10.a y = –2x + 1            b 2y = –x     c y = –x + 1
16                                                           3
3
14
2                                                                     d 5y = –2x – 5            e y = – –x – 3
2
1
12                                                           0
1        2        3       4     5     6
10
–1
–2
2 Patterns and sequences
8                                                          –3
–4                                                                  1. a 21, 34: add previous 2 terms b 49, 64: next
6
–5
4
square number c 47, 76: add previous 2 terms
–6
2
2. 15, 21, 28, 36
0
0        1   2        3       4    5       6                                                                                  1 3 2 5 3
3. –, –, –, –, –
2 5 3 7 4
3.                                           7
x                                        4. a 6, 10, 15, 21, 28 b It is the sums of the natural
6                                     y=         +4
3
5
numbers, or the numbers in Pascal’s Triangle.
4
3
5. a 13, 15, 2n + 1               b 33, 38, 5n + 3
2                                                                                          c 20, 23, 3n + 2               d 21, 25, 4n – 3
1
e 42, 52, 10n – 8
–8        –6          –4           –2       0           2        4        6            8
6. a 3n + 1, 151                  b 5n – 2, 248
4. a      5                                                                                             b (6, 1)                        c 8n – 6, 394                  d 5n + 1, 251
x                                                      e 3n + 18, 168
4                                                                    y=  –2
2
x
3                                                                    y= –1
3                                                   7. a 64, 128, 256, 512, 1024
2
b i 2n – 1 ii 2n + 1 iii 3 × 2n
1
0                                                                                                                          8. b 4n – 3          c 97            d 50th diagram
2            4        6           8       10       12           14
–1
–2
9. b 2n + 1             c 121        d 49th set
–3
10. a i 14         ii 3n + 2      iii 41      b 66
1                                                        1                                            1
5. a 2 b               –
3        c –3             d1          e –2         f    ––
3           g5            h –5           i   –
5
3
j ––
3 Substitution
4

1. a 13              b –3              c 5
6. a 1                 b –1 They are perpendicular and
symmetrical about the axes.                                                                                 2. a 2               b 8               c –10

7.                                                                                                                                   3. a 6               b 3               c –2
4. a –4.8            b 48              c 32
5. a 13              b 74              c 17
6. a 75              b 22.5            c –135
7. a 2.5             b –20             c 2.5

4 Simplifying expressions
1. a 2 + x          b x–6         c k+x         d x–t
8. a                                                                  y = 3x + 1, y = 2x + 3
e x+3            f d+m         g b–y         h p+t+w
8
i 8x     j hj         k x ÷ 4 or
x      l 2 ÷ x or
2
6                                                                                                                          4                 x
4
y
2                                                                                       m y ÷ t or             n wt      o a2      p g2
i                                             t
ii
–6           –4           –2         0             2             4                6          Linear (ii)               2. a x + 3 yr           b x – 4 yr
–2                                                        Linear (i)

–4                                                                                  3. F = 2C + 30
–6
4. a 3n       b 3n + 3           c n+1        d n–1
–8
5. a \$4            b \$(10 – x)       c \$( y – x)     d \$2x
b (2, 7)

6. a 75p           b 15x p           c 4A p          d Ay p

6
7. \$(A – B)                                                         1. a 30 b 72 c 6 d –10 e –4
1
8. a 6t       b 8w       c 2w2         d 6t2                        2. a 3        b 4 c 1– d 2
2

9. a \$t       b \$(4t + 3)                                           3. a x = 2 b p = 2 c d = 6 d y = 1 e b = 9

10.a 5a       b 6c          c 9e       d 6f         e 3g            4. 55p
f 4i       g 4j          h 3q       i 0          j –w                  1
5. a 1– cm
2           b 6.75 cm2
k 6x2      l 5y2         m 0
6. 17 sweets
11. a 7x       b 6y         c 3t       d –3t        e –5x
f –5k      g 2m2        h 0        i f2                         7. 3 years old

12. a 7x + 5       b 5x + 6              c 5p        d 5x + 6       8. 5
e 5p + t + 5   f 8w – 5k             g c
h 8k – 6y + 10                                                  7 Rearranging formulae
13. a 2c + 3d        b   5d + 2e         c   f + 3g + 4h                     T
1. k =
3
d 2i + 3k        e   2k + 9p         f   3k + 2m + 5p
g 7m – 7n        h   6n – 3p         i   6u – 3v                         A–9
j 2v             k   2w – 3y         l   11x2 – 5y              2. r =
4
m –y2 – 2z       n   x2 – z2
3. m = gv
5 Expanding and factorising                                                  C
4. r =
2            3                            2π
1. a 6 + 2m b 6 – 12f c t + 3t d k – 5k
e 15a3 – 10ab                                                            m–2
5. p = √««««««
–––4A
2. a 7t
f 4y2
b 9d
g 5ab
c 3e
h 3a2d
d 2t               e 5t 2          6. d =
√    π
7. a t = u2 – v                  v+t
b u = √««««««
3. a 2 + 2h                    b 9g + 5                                                                  ––––––
c 17k + 16                  d 6e + 20
8. a w = K – 5n 2           b n=
√K w  –
5

4. a 9t 2 + 13t                b 13y2 + 5y                                                a(q – p)             6 + st
c 10e2 – 6e                 d 14k2 – 3kp                         9. a –8y          b                  c
q+p                 2+s

5. a 6(m + 2t) b 3m(m – p) c 2(2a2 + 3a + 4)
Ra                Rb
d 3b(2a + 3c + d ) e 2ab(4b + 16 – 2a)                           10. a b =                   b a=
a–R               b–R
6. a x 2 + 5x + 6              b m2 + 6m + 5
2 + 2y
c x 2 + 2x – 8              d f2 – f – 6                         11. a
y–1
e x 2 + x – 12              f y2 + 3y – 10
d Same formula as in a
g x2 – 9                    h t 2 – 25
i m2 – 16                                                        12. a Cannot factorise the expression.
3V                 3 3V    –––
7. a 6x 2 + 11x + 3 b 10m2 – 11m – 6
c 6a2 – 7a – 3 d 6 – 7t – 10t 2 e 4 + 10t – 6t 2
b 2
r (2r + 3h)
c Yes, 5π
√
8. a x 2 + 10x + 25 b m2 + 8m + 16 c t 2 – 10t + 25                 8 Functions
d 9x2 + 6x + 1 e x2 + 2xy + y2
1. a f(1) = 2 + 1 = 3
9. a (x + 2)(x + 3) b ( p + 2)( p + 12) c (a + 2)(a + 6)
b g(–2) = (–2)2 = 4
d (t – 2)( t – 3) e (c – 2)(c – 16) f ( p – 3)( p – 5)
g (n + 3)(n – 6) h (d + 1)2                                         c fg(x) = f(x2) = x2 + 1

10. a (x + 3)(x – 3) b (t + 5)(t – 5) c (m + 4)(m – 4)                 d f–1(x) = x – 1
1      1
d (k + 10)(k – 10) e (x + y)(x – y)                                e fg–1(x) = f(x2 ) = x2 + 1
f (3x + 1)(3x – 1)
f gf–1(x) = g(x – 1) = (x – 1)2
11. a (2x + 1)(x + 2) b (3t + 2)(8t + 1)
c 3( y + 7)(2y – 3) d (2t + 1)(3t + 5)

6 Solving equations
2. a f : x → 1/x                   is not defined when x = 0

7
b g : x → √(x – 5)                 is not defined when x   5               ii gf(x) = g(x3) = 1/( x3 – 1)

c h : x → 10/(x +1)                is not defined when x = –1              iii gg(x) = g(1/(x – 1)) =                1
(1/(x – 1) – 1)
3. The function f(x) is defined as f(x) = x(x – 1)                                                              =          1
(1– (x – 1))/(x – 1)
a i f(3) = 3(3 – 1) = 6                                                                                    =       x–1
1– (x – 1)
ii f(–3) = –3(–3 – 1) = 12                                                                            =       x–1
1– x + 1
b If f(x) = 6, then        x(x – 1) = 6                                                                    =   x–1
2
x –x=6                                                                              2–x

x2 – x – 6 = 0                               c i fg(x) =1/(x – 1)3                  is not defined when x =1
(x + 2)(x – 3) = 0
ii gf(x) = 1/( x3– 1)              is not defined when x = 1
x = –2 or x = 3
iii gg(x) = x – 1                  is not defined when x = 2
4.                                                                                           2–x

9 Algebraic indices
1. a 53
b 6             c      7         d 44                e 10
x              t               m2                  q              5   y

f    1         g     1         h       3            i        4         j    7
2x3             2m                4t 4                  5y3            8x5

2. a 7x–3         b 10p–1              c 5t –2             d 8m–5             e 3y–1
1      4                                     1              5
–––
3. a i 25 ii 125 iii –
5                 b i 64          ii ––
16             –––
iii 256
1              1                                      1      1
1
c i 8      ii ––
32        iii 4 – d 1 000 000
2                                    ––––
ii 1000 iii –
4
2
5. a i f(100) = 100 = 10
4. a a3          b a5         c a7            d a4                 e a2           f a1
2
ii g(–1) = 3(–1) + 4 = 7
1                     5. a 6a5     b 9a2                c 8a6            d –6a4                 e 8a8
2                      2                               –3
iii fg(2) = f(3(2) + 4) = f(16) = 16 = 4                          f –10a
1          1
b gf(x) = g(x2) = 3(x2)2 + 4 = 3x + 4                               6. a 3a          b 4a3           c 3a4          d 6a–1
e 4a7         f 5a–4
6. The function f(x) is defined as f(x) = 2/(x + 2)
7. a 8a5b4          b 10a3b                c 30a–2b–2
a If f(x) = 5 then 2/(x + 2) = 5 and x = –8/5                          d 2ab3          e 8a–5b7
b Let y = 2/(x + 2)
8. a 3a3b2            b 3a2c4            c 8a2b2c3
y (x + 2) = 2                                                         2            3            2               3
9. a t 3      b m4           c k5            d x2
yx + 2y = 2
10 Linear programming
yx = 2 – 2y
2 – 2y                                                   1. a x       3         b x        5          c x           6         d t           18
x= y
8
2. a x       –6            b t     –
3            c y            4       d x          –2
–1       2 – 2x                                                                                        14
– –
So f (x) =               or 2/x – 2                                         e w       5.5           f x      5
x
1
3. a x       2             b x        38        c x             6–
2        d x          7
7
7. f : x → x and g : x → 1/(x – 1)
3                                                           e t       10            f y        –
5

a i fg(2) = f(1) = 1                                                4. a x       1         b x         3         c x           2         d x          –1
e x       –1        f x        1
ii gf(–1) = g(–1) = –1/2

b i fg(x) = f(1/(x – 1)) = (1/(x – 1))3 =1/(x – 1)3

8
5. a                                                                                             9.                             y
0             1         2        3             4                                                                  2

1
b
–2            –1        0                                                                                     0
–3    –2           –1                    1           2 x
c                                                                                                                         –1
–1            0         1
–2
d                                                                                                      x = –2                          x=1

1             2         3        4             5
10.                  y
e
5
–2          –1          0
y=4
f                                                                                                                4

1           2          3         4         5                                                            3
g
2
–1            0         1         2            3
h                                                                                                                1

–3 –2 –1 0 1 2 3 4
–2     –1      0             1           2 x
6. a x                4                                                                                               –1                    y = –1
1         2         3        4        5
–2
b x               –2
11. a                                  y
–3       –2        –1        0        1
8
1
c x               3–
2                                                                                                               6
0         1         2        3        4                                                            4
d x               –1                                                                                                               2

–2        –1        0        1        2                                –8 – 6 – 4 –2 0                         2       4     6   8        x
1                                                                                                –2
e x               1–
2
–4
0         1         2        3        4
–6
f x               –2
–8
–3       –2        –1        0        1
g x               50                                                                               b i Yes                ii Yes               iii No
20       30        40        50       60                    12. a–d 10
h x               –6                                                                                         8
6
–7       –6        –5        –4       –3
4
7.                                                 8.                  y                                          2
y
0
0         2           4           6         8       10
1                                       –2
2
–4
1                                                    0                                       –6
–2       –1             1    2 x
–1
0
e i No                 ii No            iii Yes
–1                     1         2x
–1                                                      –2                         13. a 45x + 25y                        200 ⇒ 9x + 5y                     40
x=2
b y x+2
–3
y = –3
14. a i Cost 30x + 40y 300 ⇒ 3x + 4y                                              30
–4                               ii At least 2 apples, so x 2
iii At least 3 pears, so y 3
iv At least 7 fruits, so x + y 7

b Draw graph with inequalities and shading as
question

9
c Three apples and five pears                                                   1         1         1
d x = –1, y = 2 – e x = – –, y = –6 –
2         2         2

15. a i Space 4x + 3y      48                                   5. Amul \$7.20, Kim \$3.50
b ii Cost 300x + 500y 6000 ⇒ 3x + 5y 60
6. 84p
Draw graph with inequalities and shading as
question                                              7. £4.40
c Six sofas and eight beds                                 8. \$195
16. a i     Number of seats required is 40x + 50y
300 ⇒ 4x + 5y 30                                    13 Quadratic equations
ii Number of 40-seaters x 6
iii Number of 50-seaters y 5                            1. a –2, –5 b –3, 2 c 1, –2 d 3, 2

b Draw graph with inequalities and shading as              2. a –4, –1 b 3, 5 c –6, 2 d –2 e –2, –6
question
3. a –6, –4 b –6, 4
c Five 40-seater coaches and two 50-seater
coaches cost \$740                                        4. a (x + 2)2 – 4                    b (x – 2)2 – 4        c (x + 5)2 – 25
d (x + 1)2 – 1
11 Direct and inverse proportion                                5. a (x + 2)2 – 5                    b (x – 2)2 – 5        c (x + 4)2 – 22
1.   a 15                b2                                        d (x + 1)2 – 10

2.   a 75                b6                                     6. a 1.45, –3.45                     b 5.32, –1.32         c –4.16, 2.16
3.   a 150               b6                                     7. a 1.77, –2.27 b 3.70, –2.70 c –0.19, –1.53
4.   a 22.5              b 12                                      d –0.41, –1.84 e 2.18, 0.15 f 1.64, 0.61

5.   a 175 miles         b 8 hours                              8. 6, 8, 10

6.   a 100               b 10                                   9. 15 m, 20 m

7.   a 27                b 5                                    10. 6.54, 0.46

8.   a 56                b 1.69                                 11. 48 km/h
9.   a 192               b 2.25                                 12. 5 h
10. a 25.6               b 5
14 Algebraic fractions
11. a 3.2 °C             b 10 atm
5x              3x + 2y
1.     a                 b
12. a 388.8 g            b 3 mm                                               6                  6

13. Tm = 12              a 3               b 2.5                              7x – 3                   12x – 23
c                         d
4                        10
14. Wx = 60              a 20              b 6
x                 3x – 2y                 x–1           –8x + 7
15. Q(5 – t) = 16        a –3.2            b 4                  2. a               b                         c             d
6                    6                     4              10
16. Mt 2 = 36            a 4               b 5
3. a 3             b 5           c 3
17. W√T = 24             a 4.8             b 100

18. gp = 1800            a \$15             b 36                        x2                2xy               6x2 + 5x + 1          1
4. a                b                  c                d
6               3                     8               2x
19. td = 24              a 3 °C            b 12 km
3                 2xy                           1
20. ds2 = 432            a 1.92 km         b 8 m/s              5. a
2
b                   c 1       d
2x
3
21. W√F = 0.5            a 5 t/h           b 0.58 t/h
3x2 – 5x – 2        2x2 – 6y2
12 Simultaneous equations                                       6. a x             b 3                c                 d
10                 9
1    1
1. a (4, 1) b (5, 5) c (–2, 6) d (2, 6) e (7 –, 3 –)
2    2
7. a 3, –1.5                 b 0, 1
1          1
2. a x = 4, y = 1 b x = 5, y = –2 c x =   2 –,
4    y=   6–
2
x–1                      x+1
1        3
3. a x = 2, y = –3 b x = 2, y = 5 c x = –, y = – –              8. a                         b
2        4                     2x + 1                    x–1
4. a x = 5, y = 1 b x = 7, y = 3 c x = 3, y = –2

10

d x = 1.5               gradient = 0
3. Draw graph of y = x3 – 3 with tangents at the
1. a Values of y: 27, 12, 3, 0, 3, 12, 27
following points
b 6.8    c 1.8 or –1.8
2. a Values of y: 27, 18, 11, 6, 3, 2, 3, 6, 11, 18, 27
b 8.3    c 3.5 or –3.5                                           b x = –1                gradient = 3
3. a Values of y: 27, 16, 7, 0, –5, –8, –9, –8, –5, 0, 7            c x = –2                gradient = 12
b –8.8    c 3.4 or –1.4
4. a Gradient when x = 2 is 9
4. a Values of y: 2, –1, –2, –1, 2, 7, 14    b 0.25
NB graph is y = x3 – 3x – 2
c 0.7 or –2.7     e (1.1, 2.6) and (–2.6, 0.7)
b (–1, 0) and (1, –4)
5. a Values of y: 15, 9, 4, 0, –3, –5, –6, –6, –5, –3, 0,
4, 9                                                        5. Draw graph of the curve y = sinx for 0º < x < 360 º
b –0.5 and 3                                                     with tangents at the following points

6. a Values of y: 5, 0, –3, –4, –3, 0, 5, 12                        a x = 60º               gradient = 0.5
b –4 and 0
b x = 90º               gradient = 0
7. a Values of y: 16, 7, 0, –5, –8, –9, –8, –5, 0, 7, 16
b 0 and 6                                                        c x = 240º              gradient = –0.5

8. a Values of y: 9, 4, 1, 0, 1, 4, 9    b +2                    6. a Draw a graph of the curve y = x2
c Only 1 root
b
9. a Values of y: 10, 5, 4, 2.5, 2, 1.33, 1, 0.67, 0.5
b i 0.8    ii –1.6

10. a Values of y: 10, 5, 2.5, 2, 1, 0.5, 0.4, 0.25, 0.2
c 4.8 and 0.2                                                   c Gradient = 2x
11. a Values of y: 25, 12.5, 10, 5, 2.5, 1, 0.5, 0.33,              d They should be the same as the curves are
0.25                                                            transformed vertically
c 0.5 and –10.5

12. a Values of y: –24, –12.63, –5, –0.38, 2, 2.9, 3,
3.13, 4, 6.38, 11, 18.63, 30    b 4.7

13. a Values of y: 27, 15.63, 8, 3.38, 1, 0.13, 0, –0.13,
–1, –3.38, –8, –15.63, –27     b 0.2

14. a Values of y: –16, –5.63, 1, 4.63, 6, 5.88, 5, 4.13,
4, 5.38, 9, 15.63, 26    c –1.6, –0.4, 1.9

1. Draw graph of y = x2 + 1 with tangents at the
following points

c x = –2                  gradient = –4

d x = –5                  gradient = –10

2. Draw graph of y = x (x – 3) with tangents at the
following points