Subtopic: Boolean Algebra, De Morgan's Laws & Karnaugh Maps
Question Statement:
P, Q, R and S are four types of dangerous microbes recently found in a human habitat. The area of each circle with its diameter printed in brackets represents the growth of a single microbe surviving human immunity system within 24 hours of entering the body. The danger to human beings varies proportionately with the toxicity, potency and growth attributed to a microbe shown in the figure (Toxicity: P=800, Q=600, R=300, S=200; Potency: P=0.4, Q=0.5, R=0.6, S=0.8; Diameter: P=50 mm, Q=40 mm, R=30 mm, S=20 mm).
A pharmaceutical company is contemplating the development of a vaccine against the most dangerous microbe. Which microbe should the company target in its first attempt?
(A) P (B) Q (C) R (D) S
Options:
(A) P
(B) Q
(C) R
(D) S
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2011 Q.63)**
**1. Quantitative Aptitude (Proportionality & Danger Index)**:
- Danger index $D \propto \text{Toxicity} \times \text{Potency} \times \text{Growth area}$.
- Growth area $\propto d^2$ (square of diameter):
- For $P$: $D_P \propto 800 \times 0.4 \times (50)^2 = 320 \times 2500 = 800,000$.
- For $Q$: $D_Q \propto 600 \times 0.5 \times (40)^2 = 300 \times 1600 = 480,000$.
- For $R$: $D_R \propto 300 \times 0.6 \times (30)^2 = 180 \times 900 = 162,000$.
- For $S$: $D_S \propto 200 \times 0.8 \times (20)^2 = 160 \times 400 = 64,000$.
**2. Comparison**:
- Microbe $P$ has by far the highest composite danger index ($800,000$).
- Therefore, the pharmaceutical company should prioritize developing a vaccine against **P**.
**3. Conclusion**:
- **Official Master Answer**: **A**
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