Solutions to Unit 3A Specialist 3A Specialist Mathematics by A.J. Sadler Prepared by: Glen Prideaux 2009 ii Preface The answers in the Sadler text book sometimes are not enough. For

  • View
    213

  • Download
    0

Embed Size (px)

Text of Solutions to Unit 3A Specialist 3A Specialist Mathematics by A.J. Sadler Prepared by: Glen Prideaux...

  • Solutions to

    Unit 3A Specialist Mathematics

    by A.J. Sadler

    Prepared by:Glen Prideaux

    2009

  • ii

  • Preface

    The answers in the Sadler text book sometimes are not enough. For those times when your really need to seea fully worked solution, look here.

    It is essential that you use this sparingly!You should not look here until you have given your best effort to a problem. Understand the problem here,

    then go away and do it on your own.

    Contents

    Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Exercise 1A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Exercise 1B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Miscellaneous Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Exercise 2A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Exercise 2B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Exercise 2C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Exercise 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Exercise 2E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Miscellaneous Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

    Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Exercise 3A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Exercise 3B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Exercise 3C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Exercise 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Miscellaneous Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

    Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Exercise 4A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Exercise 4B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Exercise 4C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Miscellaneous Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

    Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Exercise 5A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Exercise 5B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Exercise 5C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Miscellaneous Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

    Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Exercise 6A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Exercise 6B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Exercise 6C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Exercise 6D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Exercise 6E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Miscellaneous Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

    Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Exercise 7A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Exercise 7B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Miscellaneous Exercise 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

    Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Exercise 8A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Exercise 8B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Exercise 8C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Exercise 8D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Miscellaneous Exercise 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

    Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Exercise 9A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Miscellaneous Exercise 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

    iii

  • iv

  • CHAPTER 1

    Chapter 1

    Exercise 1A

    1.-2 -1 0 1 2 3 4 5 6 7 8 9 10

    2 2

    x = 3 or x = 7

    2.-2 -1 0 1 2 3 4 5 6 7 8 9 10

    1 1

    x = 5 or x = 7

    3.-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2

    2 2

    x = 6 or x = 2

    4.

    -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3

    5 5

    x = 8 or x = 2

    5.-2 -1 0 1 2 3 4 5 6 7 8 9 10

    3 3

    x = 4

    6.

    -3 -2 -1 0 1 2 3 4 5 6 7 8 9

    5 5

    x = 3

    7. x+ 3 = 7x = 4

    or x+ 3 = 7x = 10

    8. x 3 = 5x = 8

    or x 3 = 5x = 2

    9. No solution (absolute value can never be nega-tive).

    10. x 2 = 11x = 13

    or x 2 = 11x = 9

    11. 2x+ 3 = 72x = 4x = 2

    or 2x+ 3 = 72x = 10x = 5

    12. 5x 8 = 75x = 15x = 3

    or 5x 8 = 75x = 1

    x =15

    13. Find the appropriate intersection and read thex-coordinate.

    (a) Intersections at (3,4) and (7,4) so x = 3 orx = 7.

    (b) Intersections at (-2,4) and (6,4) so x = 2or x = 6.

    (c) Intersections at (4,2) and (8,6) so x = 4 orx = 8.

    14.

    x

    y

    -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8

    -2-1

    123456789

    10

    y = 5

    y = |x|

    y = 3 0.5x

    y = |2x+ 3|

    (a) Intersections at (-4,5) and (1,5) so x = 4or x = 1.

    (b) Intersections at (-6,6) and (2,2) so x = 6or x = 2.

    (c) Intersections at (-4,5) and (0,3) so x = 4or x = 0.

    (d) Intersections at (-3,3) and (-1,1) so x = 3or x = 1.

    15.-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2

    1 1

    x = 7 or x = 5

    16. No solution (absolute value can never be nega-tive).

    17.-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2

    2.52.5

    x = 5.5

    18.

    -12 -10 -8 -6 -4 -2 0 2 4 6

    88

    x = 3

    19.0 1 2 3 4 5 6 7 8 9 10 11 12

    22

    x = 8

    20.

    -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14

    88

    x = 2

    21.-6 -5 -4 -3 -2 -1 0

    1.5 1.5

    x = 3.5

    1

  • Exercise 1A

    22.

    -1 0 1 2 3 4 5 6 7 8 9 10 11

    5 5

    x = 5

    23. x+ 5 = 2x 14x = 19

    |19 + 5| = |2 19 14||24| = |24| X

    or

    x+ 5 = (2x 14)x+ 5 = 2x+ 14

    3x = 9x = 3

    |3 + 5| = |2 3 14||8| = | 8| X

    24. 3x 1 = x+ 92x = 10x = 5

    |3 5 1| = |5 + 9||14| = |14| X

    or

    (3x 1) = x+ 93x+ 1 = x+ 94x = 8x = 2

    |32 1| = | 2 + 9|| 7| = |7| X

    25. 4x 3 = 3x 11x = 8

    |48 3| = |38 11|| 35| = | 35| X

    or

    4x 3 = (3x 11)4x 3 = 3x+ 11

    7x = 14x = 2

    |4 2 3| = |3 2 11||5| = | 5| X

    26. 5x 11 = 5 3x8x = 16x = 2

    |5 2 11| = |5 3 2|| 1| = | 1| X

    or

    (5x 11) = 5 3x5x+ 11 = 5 3x

    6 = 2xx = 3

    |5 3 11| = |5 3 3||4| = | 4| X

    27. x 2 = 2x 6x = 4x = 4

    |4 2| = 2 4 6|2| = 2 X

    or

    (x 2) = 2x 6x+ 2 = 2x 63x = 8

    x =83

    |83 2| = 2 8

    3 6

    |23| 6= 2

    3The second solution is not valid. The only so-lution is x = 4.

    28. x 3 = 2xx = 3

    | 3 3| = 23| 6| 6= 6

    or

    (x 3) = 2xx+ 3 = 2x3x = 3x = 1

    |1 3| = 2 1| 2| = 2 X

    The first solution is not valid. The only solutionis x = 1.

    29. x 2 = 0.5x+ 10.5x = 3x = 6

    |6| 2 = 0.5 6 + 14 = 4 X

    or

    2

  • Exercise 1A

    x 2 = 0.5x+ 11.5x = 3

    x = 2

    | 2| 2 = 0.52 + 10 = 0 X.

    30. x+ 2 = 3x+ 64x = 4x = 1

    |1 + 2| = 3 1 + 6|3| = 3 X

    or

    (x+ 2) = 3x+ 6x 2 = 3x+ 6

    2x = 8x = 4

    |4 + 2| = 3 4 + 6|6| 6= 3 6

    The second solution is invalid. The only solutionis x = 1.

    31. x 1:

    x+ 5 + x 1 = 72x+ 4 = 7

    2x = 3x = 1.5 X

    5 x 1:

    x+ 5 (x 1) = 7x+ 5 x+ 1 = 7

    6 6= 7 = no soln

    x 5:

    (x+ 5) (x 1) = 7x 5 x+ 1 = 7

    2x 4 = 72x = 11x = 5.5 X

    32. x 4:

    x+ 3 + x 4 = 22x 1 = 2

    2x = 3x = 1.5

    = no soln (out of domain)

    3 x 4:

    x+ 3 (x 4) = 2x+ 3 x+ 4 = 2

    7 6= 2 = no soln

    x 3:

    (x+ 3) (x 4) = 2x 3 x+ 4 = 2

    2x+ 1 = 22x = 1x = 0.5

    = no soln (out of domain)

    The equation has no solution.

    33. x 3:

    x+ 5 + x 3 = 82x+ 2 = 8

    2x = 6x = 3 X

    5 x 3: