Answer
The curve does not intersect the XY-plane.
Work Step by Step
The space curve given by
$$ r(t) = \langle t, t^3, t^2+1 \rangle $$
intersects the xy-plane when $ z=0$, that is, we have the equation
$$ t^2+1=0\Longrightarrow t^2=-1$$
which has no solution in $ R $.
Hence, the curve does not intersect the xy-plane.
