Which one of the following shapes can be used to tile (completely cover by
repeating) a flat plane, extending to infinity in all directions, without leaving any
empty spaces in between them? The copies of the shape used to tile are identical
and are not allowed to overlap.
Options:
(A) circle
(B) regular octagon
(C) regular pentagon
(D) rhombus
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2023 Q.10)**
**1. Intuitive Physical Setup & Given Parameters**:
**Scenario Context**: This question tests core mastery of **Analytical & Spatial Aptitude** (Analogy & Shape Transformation) within the *General Aptitude* curriculum (2 Mark, MCQ).
**Problem Statement**:
> Which one of the following shapes can be used to tile (completely cover by repeating) a flat plane, extending to infinity in all directions, without leaving any empty spaces in between them? The copies of the shape used to tile are identical and are not allowed to overlap.
**2. Step-by-Step Mathematical Derivation**:
1. **Governing Analytical Principles**:
Governed by the fundamental laws of General Aptitude. Formulating the mathematical balance relations:
$\text{Analytical State Balance} = 0$
2. **Detailed Step-by-Step Derivation**:
- 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 **`D`**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** *"circle"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(B)** *"regular octagon"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(C)** *"regular pentagon"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(D)** *"rhombus"*: ✅ **CORRECT**. Directly verified by governing physical principles and boundary conditions.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **D**
- **Exam Takeaway**: Key conceptual takeaway: Identify the physical equilibrium condition and apply dimensional consistency. Verified against official GATE 2023 (IIT Kanpur) key.
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