ENGR 1405 Engineering Mathematics 1

Faculty of Engineering and Applied Science
2000 Fall

Problem Set 1
Questions

  1. In each of the following questions use the three operations, introduced in class, to reduce the given linear system to triangular form as much as possible.   Indicate whether or not the system has a solution.   If a solution exists, then say what type, and write a summary statement expressing the solution.   Be sure that the recommended order of operations is being used.

    1.  
      3 x   -     y   =     4
      -5 x   + 2 y   = 6
    2.  
      x         -     z   =     3
      x   +     y   - z   = 5
      2 x   + y   - 2 z   = 8
    3.  
      r   +     s   -     8 t   =     6
      s   - 4 t   = 0
      r   - s   - t   = 5
    4.  
            x2   -     x3   +     2 x4   =     -2
      x1     + x3   - x4   = 3
      x1   + x2     + x4   = 1
      2 x1   - 3 x2   + 5 x3   - 8 x4   = 12

  1. Determine the value of the matrix if it exists or explain, briefly, why it does not exist.

    1.  

      A = [1 -3 7 / 4 5 -6] + [3 -1 4 / -7 3 -2]

    2.  

      B = 2 [1 2 / 6 -3] - 3 [5 -2 / 4 7]

    3.  

      C = 2 [5 -3 / 4 9] + 7 [3 -1 5 / -7 3 -2]

    4.  

      D = [5 / 4] * [8 -3]

    5.  

      E = [11 4 / -2 5 / 3 -6] * [2 -6 8 / 1 -2 6]

  1. Find, if possible, values for “k” such that the system of linear equations below will have
    (i)a unique solution
    (ii)infinitely many solutions
    (iii)no solution

    1.  
      (2k + 1) x   -     y   +     2 z   =     2
      -2 x   + y   + z   = 1
      k 2 x     + 3 z   = 4
    2.  
      5 r   +     4 s   +     4 t   +     5 u   +     k 4 v   =     0
      4 r   + 3 s   + 3 t   + 4 u   + k 3 v   = 0
      3 r   + 2 s   + 2 t   + 3 u   + k 2 v   = 0
      2 r   + 1 s   + 1 t   + 2 u   + k 1 v   = 0

  1. A fish farm “grows” two different species of fish, trout and perch.   The total number of fish at the farm on 6 September, 2000 is “N”.   The ratio of perch to trout at this time is “m/n”.   Based on previous studies it has been determined that each perch will eat “k” grammes of food per day and that each trout consumes twice as much food as a perch does.   On 6 September, 2000, it is necessary to supply “W” kg of food.

    Determine the number of each type of fish present, for each of the following sets of values.

    1. N = 10,000 ,         m = 2 ,         n = 3 ,         W = 32 kg ,         k = 2 g .

    2. N = 10,000 ,         m = 3 ,         n = 2 ,         W = 20 kg ,         k = 2 g .

    3. N = 20,000 ,         m = 3 ,         n = -1 ,       W = 20 kg ,         k = 2 g .

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