A dimension is a measure of spatial content. A compass (choice a) and ruler (choice b) may help determine the dimension, but other instruments may also be used, so these are not the essential element here. An inch (choice c) is only one way to determine a dimension.
A purchase is an acquisition of something. A purchase may be made by trade (choice a) or with money (choice b), so those are not essential elements. A bank (choice c) may or may not be involved in a purchase.
An infirmary is a place that takes care of the infirm, sick, or injured.Without patients, there is no infirmary. Surgery (choice a) may not be required for patients. A disease (choice b) is not necessary because the infirmary may only see patients with injuries. A receptionist (choice d) would be helpful but not essential.
Sustenance is something, especially food, that sustains life or health, so nourishment is the essential element.Water and grains (choices b and c) are components of nourishment, but other things can be taken in as well. A menu (choice d) may present a list of foods, but it is not essential to sustenance.
A facsimile must involve an image of some sort. The image or facsimile need not, however, be a picture (choice a). A mimeograph and a copier machine (choices c and d) are just a two of the ways that images may be produced, so they do not qualify as the essential element for this item.
A wedding results in a joining, or a marriage, so choice d is the essential element. Love (choice a) usually precedes a wedding, but it is not essential. A wedding may take place anywhere, so a church (choice b) is not required. A ring (choice c) is often used in a wedding, but it is not necessary.
A faculty consists of a group of teachers and cannot exist without them. The faculty may work in buildings (choice a), but the buildings aren't essential. They may use textbooks (choice b) and attend meetings (choice d), but these aren't essential either.
A recipe is a list of directions to make something. Recipes may be used to prepare desserts (choice a), among other things. One does not need a cookbook (choice c) to have a recipe, and utensils (choice d) may or may not be used to make a recipe.
Without a signature, there is no autograph. Athletes and actors (choices a and b) may sign autographs, but they are not essential. An autograph can be signed with something other than a pen (choice d).
A cage is meant to keep something surrounded, so enclosure is the essential element. A prisoner (choice b) or an animal (choice c) are two things that may be kept in cages, among many other things. A zoo (choice d) is only one place that has cages.