The question given at Hornblower's Lieutenant examination, that of what to do in response to a "full a-back", relates to the sails of a ship suddenly flattening against the mast due to a change in wind. The correct answer (which Hornblower clearly does not know) is to put the helm over, haul up the mainstay and spanker sails, and then shift the helm in the opposite direction in order to fill the after-sails with wind in order to regain headway. Once the wind has been recovered, the ship should then brace up the sails and bring the ship's course directly in front of the wind. The maneuver to regain wind after a fall a-back is known as "chapelling", or "Chapel Ship", and would have been something which Hornblower absolutely should have known.
The very first question asked at the Lieutenant's examination - to define a "Rhumb Line", relates to a course on the Earth's surface cutting all meridians at the same angle, used as the standard method of plotting a ship's course on a chart. In contrast to the confused looks of Hornblower and the others present, this is actually a basic question of navigation which any of the Midshipman at the examination should have known.