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Each of the following question is follow...

Each of the following question is followed by two Statements I and II. You have to determine whether the data given in the statements are sufficient for answering the question.' Give answer
Is b positive?
I . a + b is positive.
Il a - b is positive

A

if Statement I alone is sufficient to answer the question

B

if Statement II alone is sufficient to answer the question

C

if both Statements I and II together are necessary to answer the question

D

if either the Statement I alone or Statement II alone is sufficient to answer the question

Text Solution

AI Generated Solution

The correct Answer is:
To determine whether \( b \) is positive based on the given statements, we will analyze each statement separately and then together. ### Step-by-Step Solution: **Question:** Is \( b \) positive? **Statement I:** \( a + b \) is positive. 1. From Statement I, we know that \( a + b > 0 \). 2. This implies that \( b > -a \). 3. However, this does not provide a definitive answer about the sign of \( b \). For example: - If \( a = 1 \), then \( b \) could be any value greater than \(-1\) (e.g., \( b = 0 \) or \( b = 1 \) which are positive). - If \( a = -2 \), then \( b \) could be any value greater than \( 2 \) (e.g., \( b = 3 \) which is positive, or \( b = 1 \) which is not). 4. Therefore, Statement I alone is **not sufficient** to determine if \( b \) is positive. **Statement II:** \( a - b \) is positive. 1. From Statement II, we know that \( a - b > 0 \). 2. This implies that \( a > b \). 3. Similar to Statement I, this does not provide a definitive answer about the sign of \( b \). For example: - If \( a = 3 \), then \( b \) could be any value less than \( 3 \) (e.g., \( b = 2 \) which is positive, or \( b = -1 \) which is not). - If \( a = 0 \), then \( b \) must be less than \( 0 \) (e.g., \( b = -1 \) which is not positive). 4. Therefore, Statement II alone is **not sufficient** to determine if \( b \) is positive. **Combining Statements I and II:** 1. From Statement I, we have \( a + b > 0 \) (or \( b > -a \)). 2. From Statement II, we have \( a - b > 0 \) (or \( a > b \)). 3. Combining these, we can infer: - From \( a > b \), we know \( b < a \). - From \( b > -a \), we know \( b \) is greater than \(-a\). 4. However, without knowing the specific values of \( a \) and \( b \), we cannot definitively conclude whether \( b \) is positive or negative. For example: - If \( a = 1 \), \( b \) could be \( 0 \) or \( -0.5 \) (not positive). - If \( a = 3 \), \( b \) could be \( 2 \) (positive) or \( 1 \) (positive) or \( 0 \) (not negative). 5. Therefore, even when combining both statements, we still cannot determine if \( b \) is positive. ### Conclusion: Since neither statement alone nor the combination of both statements provides sufficient information to determine if \( b \) is positive, the answer is: **Final Answer:** D (The statements are not sufficient to answer the question.)
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