Question Statement:
Four hills H1, H^2, H^3, and H4 are present in an area. The following observations
are made about them:
i. Neither H2 nor H3 is the easternmost hill.
ii. Neither H2 nor H3 is the westernmost hill.
iii. Neither the easternmost hill nor the westernmost hill is the southernmost
hill.
iv. Two hills are located to the west of H^2.
v. The southernmost hill has at least two hills to its east.
The southernmost hill is ________.
Options:
-
(A) H1
-
(B) H2
-
(C) H3
-
(D) H4
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.9)**
**1. Intuitive Physical Setup & Given Parameters**:
Four hills $H_1, H_2, H_3, H_4$ are located in an area with cardinal directions (North, South, East, West).
We are given five constraints:
1. Neither $H_2$ nor $H_3$ is the easternmost hill.
2. Neither $H_2$ nor $H_3$ is the westernmost hill.
3. Neither the easternmost hill nor the westernmost hill is the southernmost hill.
4. Two hills are located to the west of $H_2$.
5. The southernmost hill has at least two hills to its east.
We need to identify which hill is the **southernmost hill**.
**2. Step-by-Step Analytical Deduction**:
1. **East-West Ordering**:
- By statements (i) and (ii), neither $H_2$ nor $H_3$ is easternmost or westernmost. Therefore, the easternmost and westernmost hills must be $\{H_1, H_4\}$.
- Statement (iv) says: *"Two hills are located to the west of $H_2$."*
Since there are 4 hills in total, having 2 hills to the west means $H_2$ is at rank 3 from the west (i.e., 2nd from the east).
- Thus, from West to East:
- Rank 1 (Westernmost): Either $H_1$ or $H_4$
- Rank 2: Must be $H_3$ (since $H_2$ is Rank 3 and $\{H_1, H_4\}$ are at the extremities)
- Rank 3: $H_2$
- Rank 4 (Easternmost): The other of $\{H_1, H_4\}$
- East-West ordering: $[\text{West: } H_1/H_4] < H_3 < H_2 < [\text{East: } H_4/H_1]$.
2. **Identify the Southernmost Hill**:
- By statement (iii): The southernmost hill is *neither* the easternmost nor the westernmost hill.
\text{Southernmost} \notin \{H_1, H_4\} \implies \text{Southernmost} \in \{H_2, H_3\}
- By statement (v): *"The southernmost hill has at least two hills to its east."*
- Look at the hills to the east:
- To the east of $H_2$, there is only 1 hill (the easternmost hill). This violates constraint (v)!
- To the east of $H_3$, there are 2 hills ($H_2$ and the easternmost hill). This satisfies constraint (v) perfectly!
- Therefore, the southernmost hill must be **$H_3$**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) H1**: ❌ **INCORRECT**. $H_1$ is an extremity (either easternmost or westernmost), ruled out by condition (iii).
- **(B) H2**: ❌ **INCORRECT**. $H_2$ has only one hill to its east, violating condition (v) which requires at least two hills to the east.
- **(C) H3**: ✅ **CORRECT**. $H_3$ is non-extremal and has exactly two hills ($H_2$ and the easternmost) to its east, satisfying all conditions.
- **(D) H4**: ❌ **INCORRECT**. $H_4$ is an extremity, ruled out by condition (iii).
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **C**
- **Exam Takeaway**: In multi-variable constraint elimination, project onto one 1D axis first (East-West ranking) before resolving the perpendicular axis (North-South constraints).