Define an equivalence relation. Give an example of an equivalence relation on the set of integers.
A factory manufactures two products, P and Q. Each unit of P requires 3 hours in the manufacturing department and 2 hours in the packaging department. Each unit of Q requires 4 hours in manufacturing and 3 hours in packaging. The total available hours in the manufacturing and packaging departments are 120 and 90 hours, respectively. Formulate this problem as a linear programming problem to maximize profit.
A bag contains 4 white, 5 red, and 6 black balls. If three balls are drawn at random, what is the probability that they are all of the same color?
Find the equation of a plane passing through the point (1, -2, 3) and perpendicular to the line with direction ratios 2, -1, and 4.
If two vectors a and b satisfy a . b = 0, show that the vectors are perpendicular to each other.
A 10-meter ladder is leaning against a vertical wall. The bottom of the ladder is pulled along the ground away from the wall at a speed of 1 meter per second. How fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 meters from the wall?
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