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Difficulty: Very hardValue vs. Yes/No Data Sufficiency Decision Logic

If aa and bb are real numbers, is a2+b2<4a^2 + b^2 < 4?

(1) (a−2)2+(b−2)2=18(a - 2)^2 + (b - 2)^2 = 18
(2) a2−6a+b2−8b=−24a^2 - 6a + b^2 - 8b = -24

  1. A
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.Answer
  3. C
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
  4. D
    EACH statement ALONE is sufficient.
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Statement (1) permits values of a2+b2a^2 + b^2 ranging from 2 to 50, which can be either less than 4 (yielding a 'Yes') or greater than or equal to 4 (yielding a 'No'), making Statement (1) insufficient. Statement (2) defines a geometric locus of points where a2+b2≥16a^2 + b^2 \ge 16, which means a2+b2<4a^2 + b^2 < 4 is conclusively false for all possible values. Because Statement (2) yields a definitive 'No' answer, it is sufficient on its own. Consequently, Statement (2) alone is sufficient, but Statement (1) alone is not sufficient.

Step-by-Step Solution

1
Understand the Data Sufficiency decision logic for Yes/No questions.
A statement is sufficient if it yields a definitive 'Yes' (always true) OR a definitive 'No' (always false). It is insufficient only if it allows both 'Yes' and 'No'.
Data Sufficiency requires determinism regarding the truth value of the target question.
2
Evaluate Statement (1): (a−2)2+(b−2)2=18(a - 2)^2 + (b - 2)^2 = 18.
This equation represents a circle centered at (2,2)(2, 2) with radius r=18=32r = \sqrt{18} = 3\sqrt{2}. The distance from the origin to the center is 22+22=22\sqrt{2^2 + 2^2} = 2\sqrt{2}. The distance d=a2+b2d = \sqrt{a^2 + b^2} of any point on the circle from the origin ranges from ∣22−32∣=2|2\sqrt{2} - 3\sqrt{2}| = \sqrt{2} to 22+32=522\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}. Thus, a2+b2=d2a^2 + b^2 = d^2 ranges from (2)2=2(\sqrt{2})^2 = 2 to (52)2=50(5\sqrt{2})^2 = 50.
Since a2+b2a^2 + b^2 can be 22 (which is <4< 4, giving 'Yes') or 5050 (which is ≥4\ge 4, giving 'No'), Statement (1) cannot yield a single definitive Yes/No answer and is INSUFFICIENT.
3
Evaluate Statement (2): a2−6a+b2−8b=−24a^2 - 6a + b^2 - 8b = -24.
Completing the square gives (a−3)2−9+(b−4)2−16=−24  ⟹  (a−3)2+(b−4)2=1(a - 3)^2 - 9 + (b - 4)^2 - 16 = -24 \implies (a - 3)^2 + (b - 4)^2 = 1. This is a circle centered at (3,4)(3, 4) with radius 11. The distance from the origin to the center (3,4)(3,4) is 32+42=5\sqrt{3^2 + 4^2} = 5. The minimum distance from the origin to any point on this circle is 5−1=45 - 1 = 4, so d≥4d \ge 4. Therefore, a2+b2=d2≥42=16a^2 + b^2 = d^2 \ge 4^2 = 16.
Because a2+b2≥16a^2 + b^2 \ge 16 for all points satisfying Statement (2), the inequality a2+b2<4a^2 + b^2 < 4 is NEVER true. Statement (2) yields a definitive 'No' answer, which makes it SUFFICIENT.
4
Conclude the final Data Sufficiency choice.
Statement (1) alone is insufficient, while Statement (2) alone is sufficient.
A definitive 'No' from Statement (2) satisfies sufficiency under standard Data Sufficiency decision logic.

Key Concept

Value vs. Yes/No Data Sufficiency Decision Logic
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Topics

Value vs. Yes/No Data Sufficiency Decision Logic

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