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
     ii about 12 – km/h
                 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
                                                                    cx=0                    gradient = –3

                                                                    d x = 1.5               gradient = 0
15 Quadratic and cubic graphs
                                                                 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
                                                                    ax=2                    gradient = 12
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

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

   ax=5                      gradient = 10

   bx=1                      gradient = 2

   c x = –2                  gradient = –4

   d x = –5                  gradient = –10

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

   ax=5                      gradient = 7

   bx=3                      gradient = 3


                                                            11

								
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