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.