What occurs if a t cell binds to an antigen and the t cell does not receive a co-stimulatory signal?

The correct answer is that "the T cell enters a state of anergy".

The activation of T cells requires two signals: (1) antigen specific signal presented by an antigen presenting cell (either a macrophage or a dendritic cell) that activates t cell receptors and (2) co-stimulatory signals that is not antigen specific but rather found in the plasma membrane of the antigen presenting cell (i. e. CD28). In the absence of a co-stimulatory signal, the t cell will enter a state of anergy or the inability to produce an immune response toward an offending antigen.