The correct option is C rational roots if y=0
Given : [1xy]⎡⎢⎣13102−1001⎤⎥⎦⎡⎢⎣1xy⎤⎥⎦=[0]
⇒[13+2x1−x+y]⎡⎢⎣1xy⎤⎥⎦=[0]
⇒ 2x2+y2+y+3x−xy+1=0
If y=0, 2x2+3x+1=0
Solving this, we get
x=−12,−1 (rational roots)
If y=−1, 2x2+4x+1=0
x=−4±√84=−2±√22 (irrational roots)