1. Let the frequency of the unknown fork be $f$.

2. Initially, the pair produces $4$ beats/sec with a fork of frequency $288\,\text{cps}$.

   Beat frequency is $|f-288|=4$ So, $f=292\,\text{cps} \quad \text{or} \quad f=284\,\text{cps}$

3. Now a little wax is placed on the unknown fork.

   Adding wax to a tuning fork decreases its frequency.

4. After adding wax, the beat frequency becomes $2$ beats/sec.

   We test the two possible initial frequencies:

   • If $f=292\,\text{cps}$, then after adding wax its frequency decreases toward $288$. Hence the difference from $288$ can reduce from $4$ to $2$, which is possible.

   • If $f=284\,\text{cps}$, then after adding wax its frequency decreases even further below $288$. Then the difference from $288$ would increase beyond $4$, not become $2$. This is not possible.

5. Therefore, the unknown fork must initially have frequency $f=292\,\text{cps}$

6. Checking options:

   • A: $286$ cps — incorrect
   • B: $292$ cps — correct
   • C: $294$ cps — incorrect
   • D: $288$ cps — incorrect

Therefore, the correct answer is Option B.