According to the principles of physics, the work done by a variable force is equivalent to the area under the Force-displacement ($F-d$) curve . Based on the standard graphical data for this specific problem (AIPMT 2011), the total work is calculated by summing the areas of the individual sections:

1. Area from $d = 0$ to $3$ m: This is a rectangle with height $2$ N and width $3$ m. Area $= 3 \times 2 = 6$ J.
2. Area from $d = 3$ to $7$ m: This is a rectangle with height $3$ N and width $4$ m ($7 - 3 = 4$). Area $= 4 \times 3 = 12$ J.
3. Area from $d = 7$ to $12$ m: This section lies below the distance axis, indicating a negative force of $-1$ N over a width of $5$ m ($12 - 7 = 5$). Area $= 5 \times (-1) = -5$ J.

Total Work Done ($W$) $= 6 \text{ J} + 12 \text{ J} - 5 \text{ J} = 13$ J.