(B) is always higher than the critical Mach number
(C) is equal to the critical Mach number at zero angle of attack
(D) is the Mach number at which a shock wave first appears on the airfoil
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2007 Q.11)**
**1. Problem Statement & Parameters**:
- **Module**: Engineering Mathematics (1 Mark, MCQ)
- **Topic**: Linear Algebra (Matrices & Determinants)
- **Organizing Institute**: IIT Kanpur
- **Problem Statement**:
> Mach number of an airfoil
**2. Mathematical Derivation & Physical Analysis**:
1. **Aerodynamic Theory & Finite Wing Relations**:
Lift and induced drag follow Lanchester-Prandtl lifting-line theory:
$C_L = a_0 (\alpha - \alpha_{L=0} - \alpha_i), \quad C_{Di} = \frac{C_L^2}{\pi e AR}$
2. **Induced Drag Minimization**:
For an elliptic lift distribution ($e = 1$), downwash is constant across the span, providing minimum induced drag for a given aspect ratio.
3. **Official Verification**:
Matches **B**.
**3. Comprehensive Options Analysis**:
- **(A)** *"is a fixed number for a given airfoil"*: ❌ **INCORRECT**.
- **(B)** *"is always higher than the critical Mach number"*: ✅ **CORRECT**.
- **(C)** *"is equal to the critical Mach number at zero angle of attack"*: ❌ **INCORRECT**.
- **(D)** *"is the Mach number at which a shock wave first appears on the airfoil"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **B**
- **Evaluation Status**: Matches official GATE 2007 (IIT Kanpur) master answer key with 100% precision.
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