The work done by [tex]\vec F[/tex] is
[tex]\displaystyle\int_C\vec F\cdot\mathrm d\vec r[/tex]
where [tex]C[/tex] is the given curve and [tex]\vec r(t)[/tex] is the given parameterization of [tex]C[/tex]. We have
[tex]\mathrm d\vec r=\dfrac{\mathrm d\vec r}{\mathrm dt}\mathrm dt=k(1-2t)\,\vec\imath+\vec\jmath[/tex]
Then the work done by [tex]\vec F[/tex] is
[tex]\displaystyle\int_0^1((t-kt(1-t))\,\vec\imath+kt^2(1-t)\,\vec\jmath)\cdot(k(1-2t)\,\vec\imath+\vec\jmath)\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^1((k-k^2)t-(k-3k^2)t^2-(k+2k^2)t^3)\,\mathrm dt=-\frac k{12}[/tex]
In order for the work to be 1, we need to have [tex]\boxed{k=-12}[/tex].
