Two identical conducting rods are first connected independently to two vessels, one containing water at `100^@C` and the other containing ice at `0^@C`. In the second case, the rods are joined end to end and connected to the same vessels. Let `q_1 and q_2` gram per second be the rate of melting of ice in the two cases respectively. The ratio `q_1/q_2` is
(a) `1/2` (b)`2/1` (c)`4/1` (d)`1/4`

Explanation

To solve the problem, we need to analyze the heat transfer through the conducting rods in both scenarios and determine the ratio of the rates of melting of ice, \( q_1 \) and \( q_2 \).

### Step-by-Step Solution:

1. **Understanding the Setup**:
- In the first case, we have two identical conducting rods connected independently to two vessels: one at \( 100^\circ C \) and the other at \( 0^\circ C \) (ice).
- In the second case, the rods are joined end to end and connected to the same vessels.

2. **Heat Transfer in the First Case**:
- The heat transfer \( Q_1 \) through each rod can be expressed using Fourier's law of heat conduction:
\[ Q_1 = k \cdot A \cdot \frac{\Delta T}{L} \]
where:
- \( k \) is the thermal conductivity of the rods,
- \( A \) is the cross-sectional area of the rods,
- \( \Delta T \) is the temperature difference (which is \( 100^\circ C - 0^\circ C = 100^\circ C \)),
- \( L \) is the length of each rod.

- Since there are two rods, the total heat transfer rate \( q_1 \) will be:
\[ q_1 = 2 \cdot k \cdot A \cdot \frac{100}{L} \]

3. **Heat Transfer in the Second Case**:
- In this case, the two rods are connected end to end, effectively doubling the length. The heat transfer \( Q_2 \) can be expressed as:
\[ Q_2 = k \cdot A \cdot \frac{\Delta T}{2L} \]
- The temperature difference remains the same at \( 100^\circ C \) and \( 0^\circ C \), so:
\[ q_2 = k \cdot A \cdot \frac{100}{2L} \]

4. **Finding the Ratio \( \frac{q_1}{q_2} \)**:
- Now we can find the ratio of the rates of melting of ice:
\[ \frac{q_1}{q_2} = \frac{2 \cdot k \cdot A \cdot \frac{100}{L}}{k \cdot A \cdot \frac{100}{2L}} \]
- Simplifying this gives:
\[ \frac{q_1}{q_2} = \frac{2 \cdot 100}{\frac{100}{2}} = \frac{2 \cdot 100 \cdot 2}{100} = 4 \]

5. **Conclusion**:
- Thus, the ratio \( \frac{q_1}{q_2} \) is \( 4 \), which means the correct answer is:
\[ \frac{q_1}{q_2} = 4 \quad \text{(Option C)} \]