2x^2-5x+k=0 for the roots to be real, the discriminant (of the quadratic equation) must be greater than or equal to zero.
(25-8k)≥0
25≥8k
k≤25/8
k≤3.125
That is for it to be real...
For it to be rational...
(25-8k)^(1/2) must be rational...
The only one from your choice that satisfy that condition is zero.