Question

Is $a^2 > 3a-b^4$?
1. $3a-b^4 = -5$
2. a $>$ 5 and b $>$ 0

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

Hint:

For inequalities involving quadratics like $a^2 - 3a$, remember the properties of parabolas. Knowing the vertex and direction of opening can help you find the minimum or maximum value of the expression over a given range, which is often the key to solving the inequality.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This is a "Yes/No" data sufficiency question involving an inequality. We need to determine if \(a^2\) is greater than the expression \(3a-b^4\). A statement is sufficient if we can definitively answer "Yes" or "No".

Step 2: Key Formula or Approach:
We can analyze the inequality by either substituting values from the statements or by rearranging the inequality to a more telling form, such as \(a^2 - 3a + b^4 > 0\). We also need to remember key properties of real numbers, such as \(x^2 \ge 0\) for any real number \(x\).

Step 3: Detailed Explanation:
Analyzing Statement (1): $3a-b^4 = -5$
This statement gives us the exact value of the right side of the inequality.
The question "Is \(a^2 > 3a-b^4\)? " can be rewritten by substituting the value from this statement: \[ \text{Is } a^2 > -5? \] The square of any real number \(a\) is always non-negative, meaning \(a^2 \ge 0\).
Since \(0\) is greater than \(-5\), it must be true that \(a^2 > -5\).
The answer to the question is always "Yes".
Therefore, Statement (1) ALONE is sufficient.
Analyzing Statement (2): a $>$ 5 and b $>$ 0
This statement provides constraints on the values of \(a\) and \(b\).
Let's rearrange the original inequality: \[ \text{Is } a^2 - 3a + b^4 > 0? \] Let's analyze the expression \(a^2 - 3a\). This is a quadratic in \(a\). The function \(f(a) = a^2 - 3a\) is an upward-opening parabola. Its value increases as \(a\) moves away from the vertex, which is at \(a = 1.5\).
Since we are given that \(a > 5\), we are on the increasing part of the parabola, far from the vertex. The minimum value of \(a^2-3a\) for \(a>5\) will be greater than its value at \(a=5\). \[ \text{At } a=5, a^2 - 3a = 5^2 - 3(5) = 25 - 15 = 10. \] So, for \(a > 5\), we have \(a^2 - 3a > 10\).
Now let's consider the term \(b^4\). We are given that \(b > 0\). This means \(b^4\) must be a positive number, so \(b^4 > 0\).
Now combine the parts: \[ a^2 - 3a + b^4 > 10 + 0 \] \[ a^2 - 3a + b^4 > 10 \] Since \(10 > 0\), the expression \(a^2 - 3a + b^4\) is always greater than 0.
The answer to the question is always "Yes".
Therefore, Statement (2) ALONE is sufficient.

Step 4: Final Answer:
Both Statement (1) and Statement (2) are independently sufficient to answer the question. The correct option is (E).