GATE AE 2020 • Q9 MCQ • 2 Marks General Aptitude

General Aptitude

Quantitative Aptitude

Subtopic: Arithmetic & Numerical Estimation

Question Statement:

$a, b, c$ are real numbers. The quadratic equation $a x^2 - b x + c = 0$ has equal roots, which is $\beta$. Then:

Options:

  • (A) $\beta = b/a$
  • (B) $\beta^2 = ac$
  • (C) $\beta^3 = \frac{bc}{2a^2}$
  • (D) $b^2 \ne 4ac$

Official Answer Key:

C

Step-by-Step Solution & Explanation

### **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)** *"$\beta = b/a$"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws. - **(B)** *"$\beta^2 = ac$"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws. - **(C)** *"$\beta^3 = \frac{bc}{2a^2}$"*: ✅ **CORRECT**. Directly verified by governing physical principles and boundary conditions. - **(D)** *"$b^2 \ne 4ac$"*: ❌ **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.
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