PROBLEM 1 Let's say that someone is drawing a circle with a compass. As that circle is being drawn a second compass is attached to the first such that the needle leg is attached to the pencil of the first. Only instead of a needle it is a small wheel. As the first compass inscribes its circle the second compass is tracing out the circumference of the first and drawing a circle at twice the rate of rotation of the first. An analogy would be someone at the edge of a merry-go-round that is rotating above a piece of paper. The rider endeavors to trace out a circle on the paper as the merry-go-round rotates at twice the rate of rotation of the merry-go-round. Is there an exponential relation between the relative rates of rotation and the area incribed under the curve created by the second compass? PROBLEM 2 on 7 Feb. a star ( R.A 5h9m44s) is in transit at Sidney (longitude 155degree 12minute 23 second east direction) when time by observer watch which should keep local times is 8hour 0 minutes 33 seconds . given that mean sun R.A AT mean noon at green which on 7 Feb.is 21 h 8 m 1 s and that 1 hour of S.T IS equivalent to 59m 50.2 s of mean time . find to nearest second how much the watch is slow or fast ? PROBLEM 3 I ask people at random if they have two children and also if one is a boy born on a tuesday. After a long search I finally find someone who answers yes. What is the probability that this person has two boys? Assume an equal chance of giving birth to either sex and an equal chance to giving birth on any day. PROBLEM 4 You have twelve coins. You know that one is fake. The only thing that distinguishes the fake coin from the real coins is that its weight is imperceptibly different. You have a perfectly balanced scale. The scale only tells you which side weighs more than the other side. What is the smallest number of times you must use the scale in order to always find the fake coin? PROBLEM 5 A CIRCULAR CHIMNEY OF BRICK MASONRY HAS UNIFORM SECTION THROUGHOUT ITS HEIGHT HAVING AN INNER DIAMETER OF 2 METER AND OUTER DIAMETER OF 3 METER . FIND THE MAXIMUM HEIGHT OF CHIMNEY IF NO TENSION IS TO DEVELOP AND IF ALLOWABLE COMPRESSIVE STRESS IN MASONRY IS 1.5 N/MM^2 . TAKE A UNIFORM WIND PRESSURE INTENSITY OF 1.5 KN/M^2 TAKE UNIT WEIGHT OF BRICK MASONRY AS 22 KN/M^2