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Edge-02 takes over when Edge-01 goes down because it is the only router in the HSRP group.

Changing the priority, preemption, tracking, or timers has no effect because there is only one router in the group.

The only way this question makes sense is if we assume that tracking on R1 to the Core caused Edge-01 to decrement its  priority, but that is not what the question is stating.

You are correct, Edge-02 will automatically promote itself to active state when Edge-01 goes down. No option in this question is correct, but if we have to choose best answer, it would be standby preempt. The standby preempt command enables the HSRP router to take over the role of active router, if it has the highest priority.