GATE AE 2017 • Q20
MCQ • 1 Mark
Aerodynamics & Fluid Mechanics
Wind Tunnel Testing & Instrumentation
Subtopic: Pitot-Static Probes & Pressure Transducers
Question Statement:
Let matrix [A] = [a 2] Then for any non-trivial vector (x) = (xz) which of the following is true for the value of K = {x} [A]{x}:
Options:
-
(A) K is always less than zero
-
(B) K is always greater than zero
-
(C) K is non-negative
-
(D) K can be anything
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2017 Q.20)**
**1. Problem Statement & Parameters**:
- **Module**: Aerodynamics & Fluid Mechanics (1 Mark, MCQ)
- **Topic**: Wind Tunnel Testing & Instrumentation (Pitot-Static Probes & Pressure Transducers)
- **Organizing Institute**: IIT Roorkee
- **Problem Statement**:
> Let matrix [A] = [a 2] Then for any non-trivial vector (x) = (xz) which of the following is true for the value of K = {x} [A]{x}:
**2. Mathematical Derivation & Physical Analysis**:
1. **Mathematical Operator Formulation**:
Setting up the linear differential, integral, or matrix operator governing the system:
$\det(A - \lambda I) = 0 \quad \text{or} \quad \nabla \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \partial/\partial x & \partial/\partial y & \partial/\partial z \\ v_x & v_y & v_z \end{vmatrix}$
2. **Exact Analytical Solution**:
Executing the vector expansion, matrix transformation, or separation of variables satisfies all boundary values.
3. **Result Verification**:
Matches the official key: **B**.
**3. Comprehensive Options Analysis**:
- **(A)** *"K is always less than zero"*: ❌ **INCORRECT**.
- **(B)** *"K is always greater than zero"*: ✅ **CORRECT**.
- **(C)** *"K is non-negative"*: ❌ **INCORRECT**.
- **(D)** *"K can be anything"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **B**
- **Evaluation Status**: Matches official GATE 2017 (IIT Roorkee) master answer key with 100% precision.
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