Time to go over some of these.
For the last question, there was a much simpler way of doing it. Basically, the new circle on the base is going to be mathematically similar by a ratio of (216/360) or 1 : 0.6
0.6*15 = 9
Looking at 9 and 15 and knowing pythagoras, you shouldn't even need a calculator. It's a 345 triangle. The answer, of course, was 12cm.
For the hidden quadratics question, the fractions simplified to x²-6x+9=0. You then factorise it to two numbers which add to make -6 and multiply to make +9. Looking at the question there was only one answer, which threw me before I got to that point. But the factorisation comes to (x-3)(x-3), so the only answer is that x=3.
For the straight line, it's a lot simpler than it looks. If you draw two arcs from A and B, the line that connects to where they touch will be of equal distance from A and B.
For the mean of the numbers question, let us turn the question into algebra.
a+b+c+d+e+f+g+h/8 = 41
a+b/2= 29
Get rid of the denominators:
a+b+c+d+e+f+g+h=328
a+b=58
Then do a simultaneous equation:
a+b+c+d+e+f+g+h-a-b=328-58
c+d+e+f+g+h=270
270/6 = 45
As for how I did, I knew the answer to ever single question, so providing my working out is correct, I might even be able to think of 100%.