1).  4.3094 cm

  2).  5.4026 cm

  3).  6.0000 cm

     Notes

1). If O is the centre of the required circle, angle QOP = 60 degrees. So OQ = 1/sin 60, and OR = 1 + 1/sin 60. So the required diameter = 2 + 2/sin 60.

 2).Five coins may be arranged in similar fashion, each touching
     its two neighbours and the circumference of the required circle.
     Here angle QOP = 36 degrees. So required diameter = 2 + 2/sin 36.

3. If six circles are fitted into the required circle as in questions 1) and 2), there is room for a seventh in the middle, as shown.

  1. There were three nominations A, B and C for two places
    on a committee. The electors each voted for one pair of
    candidates and the result was A 10 votes, B 9 votes and
    C 7 votes. How many electors voted for both the two
    candidates A and B?

  2. There were four nominations for three places on another
    committee, each elector voting for three of them. If A
    got 30 votes, B got 34, C got 29 and D got 27, how many
    electors voted for the three candidates A, B and C?

  3. There were four nominations for two places on yet another
    committee, each elector voting for two of them. A got 31
    votes, B got 23, C got 29 and D got 27. The number of
    electors voting for A and B was the same as the number
    voting for B and D, and was one more than the number
    voting for B and C. How many voted for both A and B?