Question

Is 'b' positive?
(I) a + b is positive.
(II) a - b is positive.

• A. Statement I alone is sufficient to answer the question.
• B. Statement II alone is sufficient to answer the question.
• C. Both statement I and II together are necessary to answer the question.
• D. Both statements I and II together are not sufficient to answer the question.

Hint:

Pick numbers to test both statements together; b's sign is not fixed even then.

Answer 1

The Correct Option is D

Instead of picking numbers first, this can be reasoned out algebraically by combining the two inequalities directly.

1. From statement (I): $a + b > 0$, which means $a > -b$.
2. From statement (II): $a - b > 0$, which means $a > b$.
3. Adding the two original inequalities: $(a+b) + (a-b) > 0 \Rightarrow 2a > 0 \Rightarrow a > 0$. So together, the statements only guarantee that $a$ is positive, not that $b$ is.
4. What both statements really say is $-a < b < a$ with $a > 0$, meaning $b$ lies strictly between $-a$ and $a$. This range includes both positive and negative values of $b$ (for example $b = a/2$ or $b = -a/2$ both satisfy it).
5. Since $b$ can be positive or negative even after combining both statements, the data given is not sufficient.

\[\boxed{\text{Data in both statements together is not sufficient (option 4)}}\]