Consider two real functions
U(x, y) = xy(x2 - y2) ,
V(x, y) = ax4 + by4 + cx2y2 + k ,
where k is a real constant and a, b, c are real coefficients. If U(x, y) + i V(x, y)
is analytic, then what is the value of a × b × c ?
Options:
(A) 1
8
(B) 3
28
(C) 5
36
(D) 3
32
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2023 Q.43)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *Section 3: Electromagnetic Theory* | **Topic**: *Electrostatics and Magnetostatics* (`Method of Images & Multipole Expansion`)
- **Question Type**: **MCQ** | **Marks**: **2 Marks**
- **Theoretical Framework**: Governed by foundational physical principles in **Electrostatics and Magnetostatics** (`Method of Images & Multipole Expansion`).
**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): `3
28`**.
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