Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.3)**
**1. Intuitive Physical Setup & Given Parameters**:
A $4 \times 4$ digital image grid contains 16 pixel intensity values $U$.
We are asked to count the number of pixels satisfying the condition:
U \le 4
**2. Step-by-Step Mathematical Derivation**:
1. **Read Pixel Grid Intensities**:
Inspecting the $4 \times 4$ matrix from the official paper:
- Row 1: $2, 3, 7, 4 \implies$ values $\le 4$ are $\{2, 3, 4\}$ (3 pixels)
- Row 2: $1, 4, 8, 5 \implies$ values $\le 4$ are $\{1, 4\}$ (2 pixels)
- Row 3: $3, 2, 4, 9 \implies$ values $\le 4$ are $\{3, 2, 4\}$ (3 pixels)
- Row 4: $4, 1, 2, 6 \implies$ values $\le 4$ are $\{4, 1, 2\}$ (3 pixels)
2. **Sum Across All Rows**:
\text{Total pixels with } U \le 4 = 3 + 2 + 3 + 3 = 11\text{ pixels}
**3. Comprehensive Options & Distractor Analysis**:
- **(A) 3**: ❌ **INCORRECT**. Only counts a single row.
- **(B) 8**: ❌ **INCORRECT**. Misses pixels equal to 4 (strict inequality error $U < 4$).
- **(C) 11**: ✅ **CORRECT**. Exactly 11 pixels satisfy the condition $U \le 4$.
- **(D) 9**: ❌ **INCORRECT**. Counting omission.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **C**
- **Exam Takeaway**: Pay careful attention to weak inequalities ($\le$ vs $<$) to ensure boundary values are properly counted.
