GATE BT 2021 • Q51
NAT • 4 Marks
Section 2: General Biotechnology
Biomolecules, enzyme kinetics
Subtopic: Fundamentals of Biomolecules, enzyme kinetics
Question Statement:
to 0.43
2
49
5
NAT
2 to 2
2
50
5
NAT
6 to 6
2
0
51
5
NAT
3.70 to 3.80
2
52
5
NAT
23.0 to 24.0
2
0
53
5
NAT
1.37 to 1.38
2
54
5
NAT
2 to 2
2
0
55
5
NAT
1.90 to 2.00
2
Official Answer Key:
0 to 0 OR 4 to
Step-by-Step Solution & Explanation
### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2021 Q.51)**
**1. Governing Concept & Definition**:
- **Topic**: Coordinate Geometry & Area of Triangle
- The area of a triangle formed by vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is:
$\text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$
**2. Detailed Step-by-Step Calculation**:
- **Given Vertices**:
- $(x_1, y_1) = (k, 0)$
- $(x_2, y_2) = (2, 0)$
- $(x_3, y_3) = (0, -2)$
- **Area Formulation**:
$\text{Area} = \frac{1}{2} |k(0 - (-2)) + 2(-2 - 0) + 0(0 - 0)| = \frac{1}{2} |2k - 4| = |k - 2|$
- **Given Area $= 2$ square units**:
$|k - 2| = 2$
- Case 1: $k - 2 = 2 \implies k = 4$
- Case 2: $k - 2 = -2 \implies k = 0$
- Therefore, $k$ can be either $0$ or $4$.
**3. Final Answer**:
- **Calculated Value**: **0 or 4**
- **Official GATE Accepted Range**: **0 to 0 OR 4 to 4**
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