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Zorluk: OrtaQuestion Stem Simplification and Target Rephrasing

If mm and nn are non-zero real numbers such that m2≠n2m^2 \neq n^2, is m3−m2n+mn2−n3m2−n2<0\frac{m^3 - m^2 n + m n^2 - n^3}{m^2 - n^2} < 0?

(1) m+n=−5m + n = -5
(2) m−n=3m - n = 3

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

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Simplifying the target expression by factoring gives (m−n)(m2+n2)(m−n)(m+n)=m2+n2m+n\frac{(m - n)(m^2 + n^2)}{(m - n)(m + n)} = \frac{m^2 + n^2}{m + n}. Because m2+n2>0m^2 + n^2 > 0 for all non-zero real numbers, the quotient is negative if and only if m+n<0m + n < 0. Statement (1) gives m+n=−5<0m + n = -5 < 0, providing a definitive 'Yes' answer. Statement (2) gives m−n=3m - n = 3, which allows m+nm + n to be either positive or negative depending on the specific values of mm and nn. Thus, Statement (1) alone is sufficient.

Adım Adım Çözüm

1
Rephrase the target question stem algebraically
The target expression m3−m2n+mn2−n3m2−n2\frac{m^3 - m^2 n + m n^2 - n^3}{m^2 - n^2} simplifies to m2+n2m+n\frac{m^2 + n^2}{m + n}.
Factor the numerator by grouping: m2(m−n)+n2(m−n)=(m−n)(m2+n2)m^2(m - n) + n^2(m - n) = (m - n)(m^2 + n^2). Factor the denominator: m2−n2=(m−n)(m+n)m^2 - n^2 = (m - n)(m + n). Since m2≠n2m^2 \neq n^2, m−n≠0m - n \neq 0, so cancel (m−n)(m - n).
2
Determine the condition for the simplified expression to be negative
The rephrased target question is 'Is m+n<0m + n < 0?'
Since mm and nn are non-zero real numbers, m2+n2>0m^2 + n^2 > 0 always. Therefore, the sign of m2+n2m+n\frac{m^2 + n^2}{m + n} depends solely on the denominator m+nm + n.
3
Evaluate Statement (1): m+n=−5m + n = -5
Statement (1) is SUFFICIENT.
Statement (1) directly tells us m+n=−5m + n = -5, which is less than 00. This yields a definitive 'Yes' to the rephrased question 'Is m+n<0m + n < 0?'.
4
Evaluate Statement (2): m−n=3m - n = 3
Statement (2) is NOT SUFFICIENT.
Knowing m−n=3m - n = 3 gives m=n+3m = n + 3, so m+n=2n+3m + n = 2n + 3. If n=0.5n = 0.5, m+n=4>0m + n = 4 > 0 ('No'). If n=−5n = -5, m+n=−7<0m + n = -7 < 0 ('Yes'). Since m+nm + n can be positive or negative, Statement (2) is insufficient.

Anahtar Kavram

Question Stem Simplification in Data Sufficiency
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