Find the shortest distance between the lines

and .

Find the particular solution of the differential equation , given that y = π/4.

OR

Find the particular solution of the differential equation , given that y = 0 when x = π/3.

Let , and Find a vector which is perpendicular to both and and

Suppose a girl throws a die. If she gets 1 or 2 she tosses a coin three times and notes the number of tails. If she gets 3,4,5or 6, she tosses a coin once and notes whether a ‘head’ or ‘tail’ is obtained. If she obtained exactly one ‘tail’, what is the probability that she threw 3,4,5 or 6 with the ride ?

Two numbers are selected at random (without replacement) from the first five positive integers. Let X denote the larger of the two numbers obtained. Find the mean and variance of X.