To solve the problem, we need to analyze the conditions under which a Wheatstone bridge remains balanced. A Wheatstone bridge is balanced when the ratio of the resistances in one branch is equal to the ratio of the resistances in the other branch, which can be expressed as:
\[
\frac{P}{Q} = \frac{R}{S}
\]
Where \(P\), \(Q\), \(R\), and \(S\) are the resistances in the bridge.
### Step-by-Step Solution:
1. **Condition of Balance**: In a balanced Wheatstone bridge, the current through the galvanometer is zero. This condition is maintained as long as the ratio of the resistances remains unchanged.
2. **Increasing the Battery EMF**:
- If we increase the EMF of the battery, it does not affect the ratio of the resistances \(P\), \(Q\), \(R\), and \(S\).
- Therefore, the balance condition \(\frac{P}{Q} = \frac{R}{S}\) still holds true.
- **Conclusion**: The current in the galvanometer remains zero.
3. **Increasing All Resistances by 10 Ohms**:
- If we increase each resistance by 10 ohms, the new resistances become \(P + 10\), \(Q + 10\), \(R + 10\), and \(S + 10\).
- The new balance condition becomes \(\frac{P + 10}{Q + 10} \neq \frac{R + 10}{S + 10}\) (this will not hold true for arbitrary values of \(P\), \(Q\), \(R\), and \(S\)).
- **Conclusion**: The current in the galvanometer will not remain zero.
4. **Making All Resistances Five Times**:
- If all resistances are multiplied by 5, we have new resistances as \(5P\), \(5Q\), \(5R\), and \(5S\).
- The balance condition becomes \(\frac{5P}{5Q} = \frac{5R}{5S}\), which simplifies to \(\frac{P}{Q} = \frac{R}{S}\).
- **Conclusion**: The current in the galvanometer remains zero.
5. **Interchanging the Battery and Galvanometer**:
- Interchanging the battery and the galvanometer does not affect the balance condition of the Wheatstone bridge.
- The ratios of the resistances remain unchanged.
- **Conclusion**: The current in the galvanometer remains zero.
### Final Results:
- The current in the galvanometer remains zero in the following cases:
- (1) When the battery EMF is increased.
- (3) When all resistances are made five times.
- (4) When the battery and the galvanometer are interchanged.
Thus, the correct options are **1, 3, and 4**.
