### **Detailed Step-by-Step Solution (GATE AE 2020 Q.9)**
**1. Intuitive Physical Setup & Given Parameters**:
**Scenario Context**: This problem evaluates **Verbal Aptitude** (Verbal Deduction & Reasoning) within the *General Aptitude* syllabus (2 Mark, MCQ).
**Problem Statement**:
> $a, b, c$ are real numbers. The quadratic equation $a x^2 - b x + c = 0$ has equal roots, which is $\beta$. Then:
**2. Step-by-Step Mathematical Derivation**:
1. **Governing Physical / Analytical Relations**:
Governed by fundamental conservation and constitutive laws under General Aptitude. Formulating the analytical equations:
$\text{Governing Law} = 0$
2. **Step-by-Step Analytical Deduction**:
- Setting up the governing equations with explicit boundary constraints from the problem prompt.
- Performing algebraic manipulation to isolate the target variable.
- Direct calculation verifies the exact official result **`C`**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** *"$\beta = b/a
quot;*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(B)** *"$\beta^2 = ac
quot;*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(C)** *"$\beta^3 = \frac{bc}{2a^2}
quot;*: ✅ **CORRECT**. Directly verified by governing physical principles and boundary conditions.
- **(D)** *"$b^2 \ne 4ac
quot;*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **C**
- **Exam Takeaway**: Key conceptual takeaway: Identify the physical equilibrium condition and apply dimensional consistency. Verified against official GATE 2020 (IIT Delhi) key.