Question Statement:
For which of the following functions does the Laplacian vanish?
Options:
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(A) xey - yex
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(B) x cos(y) - y cos(x)
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(C) ex+iy
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(D) yx2 - y3
3 - xy
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2026 Q.27)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *Section 8: Electronics* | **Topic**: *Operational Amplifiers & Analog Circuits* (`Inverting and Non-inverting Op-Amp Configurations`)
- **Question Type**: **MCQ** | **Marks**: **1 Mark**
- **Theoretical Framework**: Governed by foundational physical principles in **Operational Amplifiers & Analog Circuits** (`Inverting and Non-inverting Op-Amp Configurations`).
**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 cos(y) - y cos(x)`**.
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