Given the statements: 1.
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Given the statements:
- $T_1$ is twice as effective as $T_4$, but less effective than $T_5$ ($T_4 < T_1 < T_5$).
- $T_2$ is thrice as effective as $T_5$ ($T_2 = 3 T_5$).
- $T_3$ is equally effective as $T_4$ ($T_3 = T_4$).
Which of the following conclusions logically follow?
- Conclusion 1: $T_5$ has the highest efficacy.
- Conclusion 2: $T_4$ and $T_3$ have the least efficacy.
- Conclusion 3: $T_2$ has the highest efficacy.
Show answer & explanation
Based on the given relationships, T2 is the most effective because it is three times T5, which is already more effective than T1, T3, and T4. T3 and T4 are the least effective because they are equal and both are less than T1, T5, and T2.
Step-by-step Derivation:
Step 1: Analyze Statement 1: T1 is twice as effective as T4, and T1 is less effective than T5. This gives the inequality: T4 < T1 < T5.
Step 2: Analyze Statement 2: T2 is thrice as effective as T5. This means T2 = 3 * T5. Since T5 > 0, T2 > T5.
Step 3: Analyze Statement 3: T3 is equally effective as T4. This means T3 = T4.
Step 4: Combine all relationships into a single chain of efficacy: T3 = T4 < T1 < T5 < T2.
Step 5: Evaluate Conclusion 1: 'T5 has the highest efficacy.' This is False, as T2 > T5.
Step 6: Evaluate Conclusion 2: 'T4 and T3 have the least efficacy.' This is True, as T3 = T4 and both are the minimum values in the chain.
Step 7: Evaluate Conclusion 3: 'T2 has the highest efficacy.' This is True, as T2 is the maximum value in the chain.
Step 8: Conclusion 2 and Conclusion 3 are correct.