In `Delta ABC`, circumrdius is 3 inradius is 1.5 units. The value of a `acot^(2)A+b^(2)cot^(3)B+c^(3)cot^(4)C` is

Options

A. `13 sqrt(3)`
B. `11sqrt(6)`
C. 21
D. none of these

Answer

A

Explanation

Given `R:r=3:1.5=2`
`rArr Delta ABC` must be equilateral.
So `a=b=c=2R sin.(pi)/(3)=R sqrt(3)` (By sine rule)
Now `a cot^(2)A+b^(2)cot^(3)B+c^(3)cot^(4)C`
`=R sqrt(3)((1)/(sqrt(3)))^(2)+(R sqrt(3))^(2)((1)/(sqrt(3)))^(3)+(R sqrt(3))^(3)((1)/(sqrt(3)))^(4)`
`=(R )/(sqrt(3))+(R^(2))/(sqrt(3))+(R^(3))/(sqrt(3))=(3+3^(2)+3^(3))/(sqrt(3))=(39)/(sqrt(3))=13sqrt(3)`