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Zorluk: Çok zorQuestion Stem Simplification and Target Rephrasing

If pp and qq are distinct non-zero real numbers, is p3−q3p−q>p2+q2\frac{p^3 - q^3}{p - q} > p^2 + q^2?

(1) pq+qp>2\frac{p}{q} + \frac{q}{p} > 2

(2) ∣p+q∣>∣p−q∣|p + q| > |p - q|

  1. A
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  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. EACH statement ALONE is sufficient.Cevap
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Cevap

EACH statement ALONE is sufficient.
Rephrasing the question stem is the key strategy. Factoring the numerator gives (p−q)(p2+pq+q2)p−q=p2+pq+q2\frac{(p - q)(p^2 + pq + q^2)}{p - q} = p^2 + pq + q^2. Subtracting p2+q2p^2 + q^2 from both sides simplifies the target question to 'Is pq>0pq > 0?'. Statement (1) simplifies to p2+q2pq>2\frac{p^2 + q^2}{pq} > 2; since p2+q2>0p^2 + q^2 > 0, this inequality requires pq>0pq > 0, answering 'Yes'. Statement (2) simplifies by squaring both sides to 4pq>0  ⟺  pq>04pq > 0 \iff pq > 0, also answering 'Yes'. Thus, each statement alone is sufficient.

Adım Adım Çözüm

1
Rephrase the question stem using algebraic factoring.
Since p≠qp \neq q, factor p3−q3=(p−q)(p2+pq+q2)p^3 - q^3 = (p - q)(p^2 + pq + q^2). The target inequality (p−q)(p2+pq+q2)p−q>p2+q2\frac{(p - q)(p^2 + pq + q^2)}{p - q} > p^2 + q^2 simplifies directly to p2+pq+q2>p2+q2p^2 + pq + q^2 > p^2 + q^2, which further simplifies to 'Is pq>0pq > 0?'
Simplifying the target question stem upfront converts a complex cubic rational expression into a simple condition about whether pp and qq have the same sign.
2
Evaluate Statement (1): pq+qp>2\frac{p}{q} + \frac{q}{p} > 2.
Combine fractions over a common denominator: p2+q2pq>2\frac{p^2 + q^2}{pq} > 2. Since pp and qq are distinct non-zero real numbers, (p−q)2>0  ⟹  p2+q2>2pq(p - q)^2 > 0 \implies p^2 + q^2 > 2pq. For p2+q2pq>2\frac{p^2 + q^2}{pq} > 2 to hold, pqpq must be positive (if pq<0pq < 0, the fraction would be negative). Thus, pq>0pq > 0 must be true.
Statement (1) yields a definitive 'Yes' to the rephrased target question 'Is pq>0pq > 0?'. Therefore, Statement (1) alone is sufficient.
3
Evaluate Statement (2): ∣p+q∣>∣p−q∣|p + q| > |p - q|.
Square both non-negative sides: (p+q)2>(p−q)2  ⟹  p2+2pq+q2>p2−2pq+q2  ⟹  4pq>0  ⟹  pq>0(p + q)^2 > (p - q)^2 \implies p^2 + 2pq + q^2 > p^2 - 2pq + q^2 \implies 4pq > 0 \implies pq > 0.
Statement (2) also yields a definitive 'Yes' to the target question 'Is pq>0pq > 0?'. Therefore, Statement (2) alone is sufficient.

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Question Stem Simplification and Target Rephrasing
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