GATE PH 2024 • Q30
MSQ • 1 Mark
Section 1: Mathematical Physics
Differential equations
Subtopic: First and Second Order ODEs
Question Statement:
Consider a volume integral
I = ∫ ∇2 (1
r) dV
V
over a volume V, where r = √x2 + y2 + z2. Which of the following statement
is/are correct?
Options:
-
(A) I = -4𝜋, if r = 0 is inside the volume V
-
(B) Integrand vanishes for r ≠ 0
-
(C) I = 0, if r = 0 is not inside the volume V
-
(D) Integrand diverges as r → ∞
Official Answer Key:
A; B; C
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2024 Q.30)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *Section 1: Mathematical Physics* | **Topic**: *Linear Algebra & Vector Calculus* (`Matrices, Eigenvalues & Eigenvectors`)
- **Question Type**: **MCQ** | **Marks**: **1 Mark**
- **Theoretical Framework**: Governed by foundational physical principles in **Linear Algebra & Vector Calculus** (`Matrices, Eigenvalues & Eigenvectors`).
**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): `Integrand vanishes for r ≠ 0`**.
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.
Launch CBT Mock Test