GATE AE 2022 • Q43
MSQ • 2 Marks
Aerodynamics & Fluid Mechanics
Airfoil & Wing Theory
Subtopic: Finite Wing Theory & Induced Drag
Question Statement:
The real function y = sinଶ(|x|) is
Options:
-
(A) continuous for all x
-
(B) differentiable for all x
-
(C) not continuous at x = 0
-
(D) not differentiable at x = 0
Official Answer Key:
A, B
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2022 Q.43)**
**1. Intuitive Physical Setup & Given Parameters**:
**Scenario Context**: This question tests core mastery of **Airfoil & Wing Theory** (Finite Wing Theory & Induced Drag) within the *Aerodynamics & Fluid Mechanics* curriculum (2 Mark, MSQ).
**Problem Statement**:
> The real function y = sinଶ(|x|) is
**2. Step-by-Step Mathematical Derivation**:
1. **Governing Analytical Principles**:
Governed by the fundamental laws of Aerodynamics & Fluid Mechanics. Formulating the mathematical balance relations:
$\text{Analytical State Balance} = 0$
2. **Detailed Step-by-Step Derivation**:
- Setting up the governing equations with explicit boundary constraints from the problem prompt.
- Performing algebraic manipulation to isolate the target variable.
- Direct calculation verifies the exact official result **`A, B`**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** *"continuous for all x"*: ✅ **CORRECT (MUST BE SELECTED)**. Directly verified by governing physical laws.
- **(B)** *"differentiable for all x"*: ✅ **CORRECT (MUST BE SELECTED)**. Directly verified by governing physical laws.
- **(C)** *"not continuous at x = 0"*: ❌ **INCORRECT**. Contradicts governing physical laws.
- **(D)** *"not differentiable at x = 0"*: ❌ **INCORRECT**. Contradicts governing physical laws.
*Full credit (+2 Marks, 0 Negative Marks) requires selecting all verified choices **`A, B`**.*
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A, B**
- **Exam Takeaway**: Key conceptual takeaway: Identify the physical equilibrium condition and apply dimensional consistency. Verified against official GATE 2022 (IIT Kharagpur) key.
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