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Difficulty: HardAlgebraic Equations and Systems in Data Sufficiency

If pp and qq are real numbers, what is the value of p+qp + q?

(1) p2q+pq2=36p^2 q + p q^2 = 36 and p2+2pq+q2=9p^2 + 2pq + q^2 = 9
(2) p2−q2=15p^2 - q^2 = 15

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

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (1) alone is sufficient because simplifying (p+q)2=9(p+q)^2 = 9 yields two potential values for the sum: p+q=3p+q = 3 or p+q=−3p+q = -3. Combining each sum with pq(p+q)=36pq(p+q) = 36 gives corresponding products pq=12pq = 12 and pq=−12pq = -12. For real numbers pp and qq, the discriminant S2−4PS^2 - 4P must be non-negative. The sum p+q=3p+q = 3 gives a negative discriminant 32−4(12)=−393^2 - 4(12) = -39, which yields no real solutions for pp and qq. The sum p+q=−3p+q = -3 gives a positive discriminant (−3)2−4(−12)=57(-3)^2 - 4(-12) = 57, which yields valid real numbers. Thus, p+qp+q uniquely equals −3-3.

Statement (2) alone is not sufficient because p2−q2=(p−q)(p+q)=15p^2 - q^2 = (p-q)(p+q) = 15 allows infinitely many different values for p+qp+q depending on p−qp-q. Hence, the correct statement selection is that Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.

Step-by-Step Solution

1
Rephrase and analyze Statement (1) algebraically.
From p2+2pq+q2=9p^2 + 2pq + q^2 = 9, we have (p+q)2=9(p + q)^2 = 9, which implies p+q=3p + q = 3 or p+q=−3p + q = -3. Factor the second equation: p2q+pq2=pq(p+q)=36p^2 q + p q^2 = pq(p + q) = 36.
Factoring exposes the sum S=p+qS = p + q and product P=pqP = pq structure of the algebraic system.
2
Evaluate the real number constraint for Case 1 (p+q=3p + q = 3) in Statement (1).
If p+q=3p + q = 3, then pq(3)=36  ⟹  pq=12pq(3) = 36 \implies pq = 12. For real numbers pp and qq with sum 33 and product 1212, they must be real roots of t2−3t+12=0t^2 - 3t + 12 = 0. The discriminant is Δ=(−3)2−4(1)(12)=9−48=−39<0\Delta = (-3)^2 - 4(1)(12) = 9 - 48 = -39 < 0.
A negative discriminant means no real numbers pp and qq exist for which p+q=3p + q = 3 and pq=12pq = 12. Thus, p+q=3p + q = 3 is impossible.
3
Evaluate the real number constraint for Case 2 (p+q=−3p + q = -3) in Statement (1).
If p+q=−3p + q = -3, then pq(−3)=36  ⟹  pq=−12pq(-3) = 36 \implies pq = -12. The corresponding quadratic equation for pp and qq is t2+3t−12=0t^2 + 3t - 12 = 0, with discriminant Δ=32−4(1)(−12)=57>0\Delta = 3^2 - 4(1)(-12) = 57 > 0. Real solutions exist.
Since a positive discriminant guarantees real solutions for pp and qq, p+qp + q must uniquely equal −3-3. Therefore, Statement (1) alone is sufficient.
4
Evaluate Statement (2) independently.
p2−q2=15  ⟹  (p−q)(p+q)=15p^2 - q^2 = 15 \implies (p - q)(p + q) = 15. For example, if p=4,q=1p = 4, q = 1, then p+q=5p + q = 5. If p=8,q=7p = 8, q = 7, then p+q=15p + q = 15.
Without knowing p−qp - q, infinitely many pairs of real numbers satisfy p2−q2=15p^2 - q^2 = 15 with different sums p+qp + q. Statement (2) alone is not sufficient.

Key Concept

Evaluating non-linear algebraic systems in Data Sufficiency using real variable discriminant constraints
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Topics

Algebraic Equations and Systems in Data Sufficiency

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