Solution
= Solution
Attaching the cone collapses the circle $A=S^1\times\{x\}$ in the torus. The quotient has the homotopy type
$$
X'\simeq S^1\vee S^2.
$$
Equivalently, the exact sequence in part (b) kills one of the two generators of $H_1(T^2)$ and preserves the fundamental two-cycle. Therefore
$$
\boxed{H_i(X')\cong
\begin{cases}
\mathbb Z,&i=0,1,2,\\
0,&i>2.
\end{cases}}
$$