### **Detailed Step-by-Step Solution (GATE AE 2025 Q.25)**
**1. Intuitive Physical Setup & Given Parameters**:
- **Physical Concept**: Longitudinal static pitch stability of an aircraft requires that any disturbance in pitch produces a restoring pitching moment to return the aircraft to its equilibrium trim state.
- **Governing Variable**: Static margin $K_n = \frac{x_{np} - x_{cg}}{\bar{c}}$, where $x_{np}$ is the longitudinal position of the aerodynamic neutral point and $x_{cg}$ is the center of gravity location.
**2. Step-by-Step Mathematical Derivation**:
1. **Pitching Moment Stability Derivative**:
For an aircraft disturbed by an incremental angle of attack $\Delta \alpha > 0$, static stability requires a pitch-down (restoring) moment:
$C_{m\alpha} = \frac{\partial C_m}{\partial \alpha} < 0$
2. **Neutral Point and Static Margin Formulation**:
The pitching moment about the center of gravity relates to the neutral point by:
$C_{m\alpha} = -C_{L\alpha} \left( \frac{x_{np} - x_{cg}}{\bar{c}} \right)$
Since the lift curve slope of the complete aircraft $C_{L\alpha} > 0$:
$C_{m\alpha} < 0 \iff x_{np} - x_{cg} > 0 \implies x_{cg} < x_{np}$
Therefore, the center of gravity $x_{cg}$ must lie forward (ahead) of the neutral point $x_{np}$, yielding a positive static margin:
$\text{Static Margin} = \frac{x_{np} - x_{cg}}{\bar{c}} > 0$
**3. Comprehensive Options & Distractor Analysis**:
- **Option (A)** (aft of neutral point): ❌ **Incorrect**. If $x_{cg} > x_{np}$, the static margin is negative and $C_{m\alpha} > 0$, making the aircraft statically unstable in pitch (pitch-up divergence).
- **Option (B)** (forward of neutral point): ✅ **CORRECT**. $x_{cg} < x_{np}$ ensures a positive static margin and a negative restoring pitching moment slope $C_{m\alpha} < 0$.
- **Option (C)** ($C_{m\alpha} > 0$): ❌ **Incorrect**. Positive $C_{m\alpha}$ means pitch divergence, not stability.
- **Option (D)** (negative static margin): ❌ **Incorrect**. Negative static margin indicates instability.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **B**
- **Exam Rule**: For longitudinal static stability: $x_{cg} < x_{np}$ (cg forward of neutral point), static margin
gt; 0$, and $\frac{\partial C_m}{\partial \alpha} < 0$.