Problem

The equation of a curve is such that $\frac{dy}{dx}=kx^3+\frac{2}{x^2}$, where $k$ is a constant. The curve passes through the point $S(2,20)$ and the gradient of the curve at $S$ is $\frac{65}{2}$.

(a) Find the value of $k$.

(b) The coordinates of a point $T$ on the curve are $(1,t)$.

Find the value of $t$.