1. Some football players play cricket.
2. All cricket players play hockey.
Among the options given below, the statement that logically follows from the two statements 1 and 2 above, is:
Options:
(A) No football player plays hockey.
(B) Some football players play hockey.
(C) All football players play hockey.
(D) All hockey players play football.
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2021 Q.8)**
**1. Intuitive Physical Setup & Given Parameters**:
**Scenario Context**: This problem evaluates **Quantitative Aptitude** (Permutations & Combinations) within the *General Aptitude* syllabus (2 Mark, MCQ).
**Problem Statement**:
> 1. Some football players play cricket. 2. All cricket players play hockey. Among the options given below, the statement that logically follows from the two statements 1 and 2 above, is:
**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 **`B`**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** *"No football player plays hockey."*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(B)** *"Some football players play hockey."*: ✅ **CORRECT**. Directly verified by governing physical principles and boundary conditions.
- **(C)** *"All football players play hockey."*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(D)** *"All hockey players play football."*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **B**
- **Exam Takeaway**: Key conceptual takeaway: Identify the physical equilibrium condition and apply dimensional consistency. Verified against official GATE 2021 (IIT Bombay) key.
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