ABC is a triangle in a plane with vertices A(2,3,5), B(-1,3,2) and C(λ,5,μ). If the median through A is equally inclined to the coordinate axes, then the value of (λ³+μ³+5) is:

Let A = (2, 3, 5), B = (-1, 3, 2), C = (λ, 5, μ).
Let D be the midpoint of BC. Then D = ((λ-1)/2, 4, (μ+2)/2).
The direction ratios of AD are (λ-1)/2 - 2, 4 - 3, (μ+2)/2 - 5, i.e., (λ-5)/2, 1, (μ-8)/2.
Since the median AD is equally inclined to the coordinate axes, we have
(λ-5)/2 = 1 = (μ-8)/2
λ-5 = 2 => λ = 7
μ-8 = 2 => μ = 10
Therefore, λ³ + μ³ + 5 = 7³ + 10³ + 5 = 343 + 1000 + 5 = 1348