Subtopic: Euler Buckling of Columns & Energy Methods
Question Statement:
The arithmetic mean of five different natural numbers is 12. The largest possible value among the numbers is:
Options:
(A) 12
(B) 40
(C) 50
(D) 60
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2012 Q.60)**
**1. Problem Statement & Parameters**:
- **Module**: Aerospace Structures (1 Mark, MCQ)
- **Topic**: Strength of Materials & Elasticity (Euler Buckling of Columns & Energy Methods)
- **Organizing Institute**: IIT Delhi
- **Problem Statement**:
> The arithmetic mean of five different natural numbers is 12. The largest possible value among the numbers is:
**2. Mathematical Derivation & Physical Analysis**:
1. **Formulation of Natural Numbers Sum**:
Let the five distinct natural numbers be $n_1 < n_2 < n_3 < n_4 < n_5$, where $n_i \in \{1, 2, 3, \dots\}$.
The arithmetic mean is 12, so their sum is:
$\sum_{i=1}^5 n_i = 5 \times 12 = 60$
2. **Maximizing the Largest Element ($n_5$)**:
To make $n_5$ as large as possible, the remaining four distinct natural numbers must be minimized:
$n_1 = 1, \quad n_2 = 2, \quad n_3 = 3, \quad n_4 = 4$
$\sum_{i=1}^4 n_i = 1 + 2 + 3 + 4 = 10$
3. **Computation**:
$n_{5, \max} = 60 - 10 = 50$
Matches **Option (C)** (50).
**3. Comprehensive Options Analysis**:
- **(A)** *"12"*: ❌ **INCORRECT**.
- **(B)** *"40"*: ❌ **INCORRECT**.
- **(C)** *"50"*: ✅ **CORRECT**.
- **(D)** *"60"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **C**
- **Evaluation Status**: Matches official GATE 2012 (IIT Delhi) master answer key with 100% precision.
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