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Difficulty: HardStatement Independence Evaluation and Statement Combination

If kk and mm are non-zero real numbers, is k>mk > m?

(1) k2>m2k^2 > m^2
(2) k−mm2+1>0\frac{k - m}{m^2 + 1} > 0

  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.
The target question asks if k>mk > m. Evaluating Statement (1) alone: k2>m2k^2 > m^2 indicates ∣k∣>∣m∣|k| > |m|, which permits cases where k>mk > m (e.g., k=3,m=2k=3, m=2) and cases where k<mk < m (e.g., k=−5,m=2k=-5, m=2). Thus, Statement (1) is insufficient. Evaluating Statement (2) independently: k−mm2+1>0\frac{k - m}{m^2 + 1} > 0. For any real number mm, m2+1m^2 + 1 is strictly positive. Multiplying both sides of the inequality by (m2+1)(m^2 + 1) preserves the inequality sign, yielding k−m>0k - m > 0, which simplifies directly to k>mk > m. This yields a definitive 'Yes' answer. Therefore, Statement (2) alone is sufficient.

Step-by-Step Solution

1
Rephrase the target question
The target question asks whether the difference (k−m)(k - m) is strictly positive.
Determining if k>mk > m is algebraically equivalent to confirming k−m>0k - m > 0.
2
Evaluate Statement (1) independently
Statement (1) states k2>m2k^2 > m^2, which implies ∣k∣>∣m∣|k| > |m|. If k=3k = 3 and m=2m = 2, then 9>49 > 4 and 3>23 > 2 (Yes). However, if k=−5k = -5 and m=2m = 2, then 25>425 > 4 but −5<2-5 < 2 (No).
Because both 'Yes' and 'No' outcomes are possible, Statement (1) alone is not sufficient.
3
Evaluate Statement (2) independently without carrying over Statement (1)
Statement (2) states k−mm2+1>0\frac{k - m}{m^2 + 1} > 0. Since mm is a real number, m2≥0m^2 \ge 0, which guarantees m2+1≥1>0m^2 + 1 \ge 1 > 0. Multiplying the inequality by the strictly positive quantity (m2+1)(m^2 + 1) preserves the inequality direction, yielding k−m>0k - m > 0, or k>mk > m.
This yields a definitive 'Yes' answer, so Statement (2) alone is sufficient.

Key Concept

Statement Independence in Data Sufficiency
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Topics

Statement Independence Evaluation and Statement Combination

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