Step 1: Pin down the diagnosis first.
A gray membrane over the tonsil that extends onto the anterior pillar, plus fever and neck node swelling in a child, points straight to diphtheria. Trying to peel the membrane off usually makes it bleed underneath.
Step 2: Think about what rapid means in the lab.
The question asks for speed, not just accuracy. Two media are used for C. diphtheriae: Loeffler's serum slope for a quick look under the microscope, and tellurite agar for a slower but more specific culture.
Step 3: Work out why Loeffler's wins on speed.
Because Loeffler's medium is rich in coagulated serum, the bacillus multiplies fast and takes on its typical shape within hours. A stained smear at 6 to 8 hours already shows the granules that mark this organism, well before a tellurite plate would show colonies.
Step 4: Eliminate the distractors.
Nutrient agar carries no special nutrients for this fastidious organism. Blood agar helps with strep throat, not diphtheria speed. Lowenstein-Jensen medium is built for the tuberculosis bacillus and would not even grow C. diphtheriae well.
Step 5: Land on the answer.
The medium built for a fast presumptive diagnosis of C. diphtheriae is Loeffler's serum slope.
\[ \boxed{\text{Loeffler's serum slope}} \]