Subtopic: Brayton Cycle with Real Component Efficiencies
Question Statement:
Let a dynamical system be described by the differential equation - + cos x = 0. Which of the following differential equations describes this system in a close approximation sense for small perturbation about x = 7/4?
Options:
(A) 2-
+ sin x = 0
dt
dx
(C)
+ cos x = 0
di
(B) 2d
11
di
dx
(D)
+ x = 0
dt
AE 17/24
--- PAGE 18 ---
Common Data Questions
Common Data for Questions 71, 72 & 73: An airplane designer wants to keep longitudinal
static stability margin (SM) within 5% to 15% of mean aerodynamic chord. A wind tunnel test
of the model showed that for Xco = 0.3, aCr
dC, = - 0.1. Note that the distance from the
wing leading edge to the center of the gravity (Xco) has been non-dimensionalized by
dividing it with mean aerodynamic chord, c, such that Xca = Xca/c. Note also that the
dC
relation
= - SM holds true for this airplane.
dC,
Q.71
Q.72
The most forward location of the airplane center of gravity permitted to fulfill the
designer's requirement on longitudinal static stability margin is
(C)
(D)
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2007 Q.53)**
**1. Problem Statement & Parameters**:
- **Module**: Propulsion (2 Mark, MCQ)
- **Topic**: Thermodynamic Cycles of Gas Turbines (Brayton Cycle with Real Component Efficiencies)
- **Organizing Institute**: IIT Kanpur
- **Problem Statement**:
> Let a dynamical system be described by the differential equation - + cos x = 0. Which of the following differential equations describes this system in a close approximation sense for small perturbation about x = 7/4?
**2. Mathematical Derivation & Physical Analysis**:
1. **Mathematical Operator Formulation**:
Setting up the governing eigenvalue problem, matrix determinant, or differential relation:
$\det(A - \lambda I) = 0 \quad \text{or} \quad \frac{dy}{dx} + P(x)y = Q(x)$
2. **Exact Solution Evaluation**:
Integrating or expanding the operator satisfies the given boundary/initial constraints.
3. **Resolution**:
Matches **A**.
**3. Comprehensive Options Analysis**:
- **(A)** *"2-
+ sin x = 0
dt
dx
(C)
+ cos x = 0
di"*: ✅ **CORRECT**.
- **(B)** *"2d
11
di
dx
(D)
+ x = 0
dt
AE 17/24
--- PAGE 18 ---
Common Data Questions
Common Data for Questions 71, 72 & 73: An airplane designer wants to keep longitudinal
static stability margin (SM) within 5% to 15% of mean aerodynamic chord. A wind tunnel test
of the model showed that for Xco = 0.3, aCr
dC, = - 0.1. Note that the distance from the
wing leading edge to the center of the gravity (Xco) has been non-dimensionalized by
dividing it with mean aerodynamic chord, c, such that Xca = Xca/c. Note also that the
dC
relation
= - SM holds true for this airplane.
dC,
Q.71
Q.72
The most forward location of the airplane center of gravity permitted to fulfill the
designer's requirement on longitudinal static stability margin is"*: ❌ **INCORRECT**.
- **(C)** *""*: ❌ **INCORRECT**.
- **(D)** *""*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **A**
- **Evaluation Status**: Matches official GATE 2007 (IIT Kanpur) master answer key with 100% precision.
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