Question

a $<$ b. Is a positive?
1. b = 0.
2. \( \sqrt{a} < a \)

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question ask
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Hint:

The behavior of functions like \( \sqrt{x} \) and \( x^2 \) changes at key points like 0 and 1. Remember these rules for a positive number \( x \): - If \( 0 < x < 1 \), then \( x^2 < x < \sqrt{x} \). (e.g., for \(x=1/4\), \(1/16 < 1/4 < 1/2\)). - If \( x > 1 \), then \( \sqrt{x} < x < x^2 \). (e.g., for \(x=4\), \(2 < 4 < 16\)). The inequality \( \sqrt{a} < a \) immediately tells you that \( a > 1 \).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We are given the condition \( a < b \) and asked a 'Yes/No' question: "Is \( a > 0 \)?". A statement will be sufficient if it forces the answer to be always 'Yes' or always 'No'.

Step 2: Detailed Explanation:
Analyzing Statement (1): b = 0.
We combine this with the given information \( a < b \).
Substituting \( b = 0 \) into the inequality, we get:
\[ a < 0 \] This definitively tells us that \( a \) is negative. Therefore, the answer to the question "Is a positive?" is a definite 'No'.
Since we have a conclusive answer, statement (1) is sufficient.
Analyzing Statement (2): \( \sqrt{a} < a \)
First, for \( \sqrt{a} \) to be a real number, we must have \( a \geq 0 \). This already eliminates the possibility of \( a \) being negative.
Now let's analyze the inequality \( \sqrt{a} < a \). We can square both sides since both sides are non-negative:
\[ a < a^2 \] \[ a^2 - a > 0 \] \[ a(a-1) > 0 \] This inequality is true when both factors are positive or both are negative.
- Case (i): \( a > 0 \) and \( a-1 > 0 \), which means \( a > 1 \).
- Case (ii): \( a < 0 \) and \( a-1 < 0 \), which means \( a < 0 \). However, this contradicts our initial condition that \( a \geq 0 \) for the square root to be real.
So, the only solution is \( a > 1 \).
If \( a > 1 \), then \( a \) is definitely positive. The answer to the question "Is a positive?" is a definite 'Yes'.
Thus, statement (2) is sufficient.

Step 3: Final Answer:
Each statement alone provides enough information to definitively answer the question.