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We will show that for each sequence of quasinormable Fréchet spaces ${\left(E_n\right)}_{\mathbb{N}}$ there is a Köthe space λ such that $Ex{t}^1(\lambda \left(A\right),\lambda \left(A\right)=Ex{t}^1(\lambda \left(A\right),E_n)=0$ and there are exact sequences of the form $...\to \lambda \left(A\right)\to \lambda \left(A\right)\to \lambda \left(A\right)\to \lambda \left(A\right)\to E_n\to 0$. If, for a fixed ℕ, $E_n$ is nuclear or a Köthe sequence space, the resolution above may be reduced to a short exact sequence of the form $0\to \lambda \left(A\right)\to \lambda \left(A\right)\to E_n\to 0$. The result has some applications in the theory of the functor $Ex{t}^1$ in various categories of Fréchet spaces by providing a substitute for non-existing projective resolutions.