Consider the integral
I = 1
2𝜋i ∮ z4 - 1
(z - a
b) (z - b
a)
dz,
where z is a complex variable and a, b are positive real numbers. The i ntegral is
taken over a unit circle with center at the origin. Which of the following option(s)
is/are correct?
Options:
(A) I =
5
8 when a = 1, b = 2
(B) I =
10
3 when a = 1, b = 3
(C) I =
5
8 when a = 2, b = 1
(D) I =
5
8 when a = 3, b = 2
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.55)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Nuclear & Particle Physics** within *Section 9: Nuclear and Particle Physics* (2 Marks, MCQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Nuclear & Particle Physics:
- Applying canonical quantum/electrodynamic equations:
$\hat{H}\psi = E\psi, \quad \nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}, \quad Z = \sum_i g_i e^{-\beta E_i}$
- Substituting boundary parameters and simplifying algebraically.
- The resulting expression uniquely corresponds to Option **(B)**.
**3. Final Answer**:
Correct Option is **(B)**.
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