1. Give the dual of following Linear Programming problem:
Minimize z = 3x_1 + 2x_2 - 5x_3 + x_4
Subject to	
    	 x_1 + x_2 + x_3 + x_4 = 1
      -x_1 + x_2 - 3x_3 - x_4 <= 2
                          x_2 >= 0
      x_1 + 2x_2 - x_3 + x_4  >=-2
		          x_4 <= 0
	
2. Show that
	Dual of a dual is the primal.

3. Consider following linear programming problem 
	
	Maximize z = 5x_1 + 10x_2
	Subject to
       	         x_1 + 3x_2 <= 50
   		4x_1 + 2x_2 <=60
                        x_1 <=5
	           x_1, x_2 >= 0
   a. State the dual of the preceding LPP
   b. Given that (5,15) is an optimal solution to this (P), use the principle of complementary 
		slackness to find the optimal solution to the (DP).

4. The Game of Morra. 
	Two players simultaneously throw out one or two fingers and call out their guess as to what the total sum of 
	the outstretched fingers will be. If a player guesses right, but his opponent does not, he receives payment 
	equal to his guess. In all other cases, it is a draw. 
(a) List the pure strategies for this game. 
(b) Write down the payoff matrix for this game.
(c) Formulate the row player's problem as a linear programming problem. (Hint: Recall that the row player's problem 
	is to minimize the maximum expected payout.) 
(d) What is the value of this game?

5.  Write dual of the following LPP

	a)	Maximize z = c`x
		Subject to
			Ax = b,
			x are unrestricted in signs.
	b)	Manimize z = c'|x|
		Subject to
			Ax = b,
			x are unrestricted in signs.