A linear operator $T$ on a complex vector space $V$ has a characteristic polynomial $x^3(x-5)^2$ and minimal polynomial $x^3(x-5)$. Choose all correct options:
1) The Jordan canonical form of $T$ is uniquely determined by the given information.
2) There are exactly $2$ Jordan blocks in the Jordan decomposition of $T$.
3) The operator induced by $T$ on the quotient space $V/ \ \text{ker}(T-5I)$ is nilpotent, where $I$ is the identity operator.
4) The operator induced by $T$ on the quotient space $V/ \ \text{ker}(T)$ is a scalar multiple of the identity operator.
How can you argue that option 3 and 4 are incorrect?