GATE AE 2017 • Q20 MCQ • 1 Mark Engineering Mathematics

Engineering Mathematics

Linear Algebra

Subtopic: Matrices & Determinants

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

Official Answer Key:

B

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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