Question Statement:
In the given figure, the numbers associated with the rectangle, triangle, and ellipse
are 1, 2, and 3, respectively. W hich one among the given options is the most
appropriate combination of P, Q, and R ?
Options:
-
(A) P = 6; Q = 5; R = 3
-
(B) P = 5; Q = 6; R = 3
-
(C) P = 3; Q = 6; R = 6
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(D) P = 5; Q = 3; R = 6
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.4)**
**1. Intuitive Physical Setup & Given Parameters**:
The figure depicts overlapping geometric shapes:
- Numbers associated with basic shapes:
- Rectangle $= 1$
- Triangle $= 2$
- Ellipse $= 3$
The numbers inside each intersection region represent the sum (or product) of the indices of the shapes containing that region.
We need to find the combination of values for regions $P$, $Q$, and $R$.
**2. Step-by-Step Geometric & Combinatorial Derivation**:
1. **Map Shape Containment for Each Region**:
- Region $P$ lies at the common intersection of all three shapes: Rectangle ($1$), Triangle ($2$), and Ellipse ($3$). The combined composite index or sum:
P = 1 + 2 + 3 = 6
- Region $Q$ lies in the intersection of Triangle ($2$) and Ellipse ($3$):
Q = 2 + 3 = 5
- Region $R$ lies solely within Ellipse ($3$):
R = 3
2. Combining the values:
P = 6, \quad Q = 5, \quad R = 3
**3. Comprehensive Options & Distractor Analysis**:
- **(A) P = 6; Q = 5; R = 3**: ✅ **CORRECT**. Accurately matches the multi-shape membership summation.
- **(B) P = 5; Q = 6; R = 3**: ❌ **INCORRECT**. Swaps $P$ and $Q$.
- **(C) P = 3; Q = 6; R = 6**: ❌ **INCORRECT**. Incorrect shape assignments.
- **(D) P = 5; Q = 3; R = 6**: ❌ **INCORRECT**. Inconsistent with region topology.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A**
- **Exam Takeaway**: Identify the shape boundary set enclosing each designated region and apply the additive combination rule.