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                       Applied Mathematics Exam                                    

(2 Hours)

Question 1:

1) Solve the following equation and inequality:

   a) |5 - 3x| = |- 2x+ 7|              

   b) 6x² + x - 12 > 0

2) Determine the slope - intercept form of the linear equation which  passes through (2, - 4) and is parallel to the line 3x - 4y = 20.

3) Determine the inverse and the solution set for each of the following system of equations using the Gaussian Elimination Method:

       5x + 20y =25

       4x - 7y = - 26

Question 2:

1) A firm sells a product for $90 per unit, raw material costs $12 per unit, labor costs are $28 per unit, and annual fixed costs are $250,000.

   a) How many unites must be sold to break even?

   b) How many unites would have to be sold to earn an annual profit of $100,000.

2) The following is the Input-Output Table in certain economy (in million dollars);

a) Calculate the technological matrix A and interpret its elements.

b) Determine the equilibrium output levels for the two sectors required to satisfy a non industry demand of $1050 and $1575 in the two sectors respectively.

c) Solve the following LP problem using the corner-point method:

   Maximize    Z = 20 x1 + 10 x2

   Subject to:

              4 x1 + 3 x2  ≤ 48

              3 x1 + 5 x2 ≤ 60

                  x1            ≤ 9

              x1 , x2  ≥   0

Question 3:

1) Find each of the following:

   a)  Σ 2 i (i  -1)   ( where the summation sign is from i = 1 to 30)

   b) 12! / (4! 8!)

   c) 10C3  + 10C2

2) Given that:

   a)  f (x) = e ^(-x² + 2x + 10)   Find f' (x).

   b) Evaluate  ⌠ (1/ (2x + 10)) dx

   c) Evaluate   ⌠ ((3 x² + 4 x) / (Square Root of (2x³ + 4x² - 6))) dx

3) Given that E1 and E2 are any two events in sample space S such that:

                            _______

  P(E1) = 0.4,  P(E1 U E2) = 0.25,      Find P(E2).