Maximise Z = 3x + 4y subject to x + y ≤ 4, x ≥ 0 and y ≥ 0.

Minimise Z = 5x + 10y subject to x + 2y ≤ 120, x + y ≥ 60, x − 2y ≥ 0 and x, y ≥ 0.

Define a convex feasible region and state why the optimum of a linear objective function occurs at a corner point.

Define the feasible region and the objective function of a linear programming problem, and state where the optimum occurs.

A factory has two machines. Machine A makes 60% of the output with 2% defective, and machine B makes 40% with 5% defective. An item chosen at random is defective. Find the probability that it came from machine A.

A rectangle is to be made with a perimeter of 40 cm. Find the dimensions that give the greatest area.

Write the smallest equivalence relation from the set
cos x
A to A, where A = {1, 2, 3}. (i) Evaluate: ∫ dx
3cos x - 5
 0 1 -2 
 OR