NIMCET 2010 — Reasoning PYQ

The question specifies numbers between $1$ and $1000$, with both excluded.

Therefore, we are looking for:

2. Test Integer Values for $k$:

• For $k=1$:

  $$1^6=1$$

  (Excluded as per the question)

• For $k=2$:

  $$2^6=64$$

  (This is within the range 1 < 64 < 1000)

  Note: $64=8^2$ and $64=4^3$.

• For $k=3$:

  $$3^6=729$$

  (This is within the range 1 < 729 < 1000)

  Note: $729=27^2$ and $729=9^3$.

• For $k=4$:

  $$4^6=(2^2{)}^6=2^{12}=4096$$

  (This is greater than 1000, so it is out of range)

3. Conclusion:

The numbers that satisfy the condition are $64$ and $729$.

Total count = $2$.

Final Answer:

There are $2$ such numbers.

The correct option is (c).