**Given:** Statements: $A T = I \leq J \geq M$ **Conclusions:** I. $O \geq I$ II. $M \geq T$ III. $P M$ Find which of the conclusions is/are definitely true based on the statements.

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**Given:** Statements: $A < G < R > J \leq Q = Z \leq O; \ P > T = I \leq J \geq M$ **Conclusions:** I. $O \geq I$ II. $M \geq T$ III. $P > M$ Find which of the conclusions is/are definitely true based on the statements.

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**Solution:** 1. From the first statement: $A < G < R > J \leq Q = Z \leq O$ - Here, $O$ is greater than or equal to $Q$, which equals $Z$, and $J$ is less than or equal to $Q$. 2. From the second statement: $P > T = I \leq J \geq M$ - $P$ is greater than $T$, which equals $I$. - $I$ is less than or equal to $J$. - $J$ is greater than or equal to $M$. 3. Analyze conclusion I: $O \geq I$ - Since $O \geq Q = Z \geq J \geq I$ (because $I \leq J$), it follows that $O \geq I$. - So, conclusion I is true. 4. Analyze conclusion II: $M \geq T$ - From the statements, $J \geq M$ and $P > T = I \leq J$. - But no direct relation between $M$ and $T$ is established. - So, conclusion II cannot be confirmed. 5. Analyze conclusion III: $P > M$ - $P > T = I \leq J \geq M$ does not guarantee $P > M$ because $M$ could be equal to or greater than $T$ or $I$. - So, conclusion III cannot be confirmed. **Answer:** Only conclusion I is definitely true.

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