Find the shortest distance between the lines and .

Solve the differential equation (x² – y²) dx + 2xydy = 0.

OR

Find the particular solution of the differential equation , given that y = 0 when x = 1.

Find x such that the four points A (4, 4, 4), B (5, x, 8),

C (5, 4, 1) and D (7, 7, 2) are coplanar.

Two groups are competing for the positions of the Board of Directors of a corporation. The probabilities that the first and second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins. Find the probability that the new product introduced was by the second group.

From a lot of 20 bulbs which include 5 defectives, a sample of 3 bulbs is drawn at random, one by one with replacement. Find the probability distribution of the number of defective bulbs. Also, find the mean of the distribution.