Relation between AM,GM,HM for 2 Numbers

Q.

If the harmonic mean between two positive numbers is to their geometric mean as 12:13. The numbers are in the ratio

1. 12:13
   

2. $\frac{1}{12}:\frac{1}{13}$
   

3. 4:9
   

4. $\frac{1}{4}:\frac{1}{9}$

Q. Let a, b and c be positive real numbers such that $a+b+c=6$ Then range of $a{b}^2{c}^3$ is

1. (0, 1)
2. (0, 108]
3. (6, 108]
4. (0, ∞)

Q. Interval of r for which x2 + y2 = r2 and x2 +y2 -6x-8y +9 0 cut at two distinct points is

Q.

If the length of a rectangle is increased by 5 meters and breadth decreased by 5 square metres. If the length is increased by 3 meters and breadth increased by 2 meters , the area would increased by 50 square metres. What are the length and breadth?