A particle moves in $x-y$ plane and at time $t$ is at the point $\left(t^2,t^3-2t\right)$, then which of the following is/are correct?

$x=t^2 \Rightarrow v_x=\frac{dx}{dt}=2t$
$\Rightarrow a_x=\frac{d{v}_x}{dt}=2$
$\Rightarrow y=t^3-2t$
$\Rightarrow v_y=\frac{dy}{dt}=3t^2-2$
$\Rightarrow a_y=\frac{d{v}_y}{dt}=6t$
At $t=0, v_x=0, v_y=-2, a_x=2$ and $a_y=0$ 

$\therefore v=-2\hat{j}$ and $a=2\hat{i}$ or $v{\perp }_a$ 

At $t=\sqrt{\frac{2}{3}}$, $v_y=0,v_x\ne 0$
Hence, the particle is moving parallel to $x$-axis.