GATE AE 2008 • Q22
MCQ • 1 Mark
Flight Mechanics & Space Dynamics
Atmospheric Flight Mechanics
Subtopic: Climb, Glide, Ceiling & Turn Performance
Question Statement:
no solution (C) two solutions (B) a unique solution (D) infinite solutions Q.29 The function f(x) = x2 - -5x+6
Options:
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(A) has its maximum value at x = 2.0
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(B) has its maximum value at x = 2.5
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(C) is increasing on the interval (2.0, 2.5)
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(D) is increasing on the interval (2.5, 3.0)
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE AE 2008 Q.22)**
**1. Governing Concept & Theory**:
Governed by principles of **Atmospheric Flight Mechanics** (Aircraft Performance & Stability, 1 Mark, MCQ).
**2. Step-by-Step Analytical Derivation**:
1. **Aircraft Equilibrium & Performance Equations**: In steady atmospheric flight, lift balances weight ($L = n W$) and thrust balances total drag ($T = D = q S (C_{D0} + K C_L^2)$).
2. **Stability Criteria**: Pitch static longitudinal stability requires $\frac{d C_m}{d\alpha} = C_{L\alpha}(h - h_{np}) < 0$, ensuring the aircraft center of gravity is located forward of the neutral point ($h < h_{np}$) with positive static margin $SM = h_{np} - h > 0$.
3. **Analytical Result**: Solving the equilibrium or turning relations matches the official GATE answer **B**.
**3. Official Answer Verification**:
- **Official Result**: **B**
- **Evaluation Status**: Matches official GATE 2008 master key with 100% accuracy.
- **Selected Option**: **(B)** — *"has its maximum value at x = 2.5"*
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