Let `L` denots value of `cos^(2)(alpha - beta)` if `sin2alpha + sin2beta = (1)/(2), cos2alpha + cos2beta = (sqrt(3))/(2)` and `M` denotes value of `(1)/(log_(xy)xyz) + (1)/(log_(yz)xyz) + (1)/(log_(zx)xyz)` then `16L^(2) + M^(2)` is

Answer

5

Explanation

`1 + 1 + 2cos2(alpha - beta) = (1)/(4) + (3)/(4) = 1`
`2 + 2(2cos^(2)(alpha - beta) - 1) = 1`
`cos^(2)(alpha - beta) = (1)/(4) = L rArr L = (1)/(4)`
`m = log_(xyz)xy + log_(xyz)yz + log_(xyz)(xyz)^(2) = 2`
`:. 16L^(2) + m^(2) = 1 + 4 = 5`