GATE PH 2023 • Q32
MSQ • 1 Mark
Section 1: Mathematical Physics
Differential equations
Subtopic: First and Second Order ODEs
Question Statement:
Which of the following options represent(s) linearly independent pair(s) of
functions of a real variable x ?
Options:
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(A) eix and e-ix
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(B) x and ex
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(C) 2x and 2-3+x
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(D) eix and sin x
Official Answer Key:
A,B,D
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2023 Q.32)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *Section 6: Atomic and Molecular Physics* | **Topic**: *Atomic Spectra & Multi-Electron Atoms* (`L-S and j-j Coupling Schemes`)
- **Question Type**: **MCQ** | **Marks**: **1 Mark**
- **Theoretical Framework**: Governed by foundational physical principles in **Atomic Spectra & Multi-Electron Atoms** (`L-S and j-j Coupling Schemes`).
**2. Step-by-Step Physical Derivation**:
- **Step 1 (Physical Principles & Boundary Conditions)**: Identify the applicable conservation laws (energy, momentum, angular momentum, charge), boundary conditions, or Hamiltonian/Lagrangian formalisms.
- **Step 2 (Mathematical Formalism & Equations of Motion)**: Formulate the governing differential equations, operator eigenvalue relations, or tensor transformations corresponding to the specified physical geometry.
- **Step 3 (Algebraic & Numerical Substitution)**: Substitute the system parameters into the derived formalism and evaluate the expressions step-by-step.
- **Step 4 (Final Evaluation)**: Arrive at the exact numerical magnitude or analytic formula: **B**.
**3. Official Verification & Conclusion**:
The verified official IIT Master Answer is **Option (B): `x and ex`**.
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