If k is a real number, is k^3 > k? (1) k^2 - k 1

If k is a real number, is k^3 > k? (1) k^2 - k < 0 (2) k^2 > 1 — GMAT

If $k$ is a real number, is $k^3 > k$ ?

(1) $k^2 - k < 0$
(2) $k^2 > 1$

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

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

The option stating that Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient is correct because Statement (1) restricts

k

to the interval

(0, 1)

, where

k^3

is strictly less than

k

, giving a definitive 'No' to the stem question. A statement that yields a consistent 'No' is sufficient in GMAT Data Sufficiency. Statement (2) permits values both greater than 1 (where the answer is Yes) and less than -1 (where the answer is No), making it insufficient.

Step-by-Step Solution

Rephrase the question stem algebraically.

The question asks whether

k^3 - k > 0

, which factors into

k(k - 1)(k + 1) > 0

Simplifying the stem establishes the exact sign behavior of the expression across different intervals of

k

Evaluate Statement (1):

k^2 - k < 0

Factoring yields

k(k - 1) < 0

, which means

0 < k < 1

. For any

k

(0, 1)

k > 0

(k - 1) < 0

, and

(k + 1) > 0

. The product

k(k - 1)(k + 1)

is (positive) × (negative) × (positive) = negative. Thus,

k^3 < k

for all

k \in (0, 1)

. The answer to 'Is

k^3 > k

?' is a definitive NO.

In GMAT Data Sufficiency Yes/No structure logic, a statement that guarantees a definitive 'No' answer for all permissible values is SUFFICIENT.

Evaluate Statement (2):

k^2 > 1

Solving the inequality gives

k > 1

k < -1

. If

k = 2

k^3 = 8 > 2

(Answer: YES). If

k = -2

k^3 = -8

and

k = -2

, so

-8 > -2

is FALSE (Answer: NO). Because Statement (2) allows both YES and NO outcomes, it is INSUFFICIENT.

A statement that yields conflicting Yes and No answers depending on the chosen value within its domain is insufficient.

Key Concept

Value vs. Yes/No Data Sufficiency Decision Logic: Definitive 'No' Sufficiency

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