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Zorluk: OrtaAlgebraic Equations and Systems in Data Sufficiency

If rr and ss are non-zero real numbers, what is the value of r2+s2r^2 + s^2?

(1) r+s=6r + s = 6
(2) r3+s3=126r^3 + s^3 = 126

  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. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.Cevap
  4. D
    EACH statement ALONE is sufficient.
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Cevap

Both statements together are sufficient to determine the unique value of r2+s2=26r^2 + s^2 = 26, but neither statement alone is sufficient.
The correct answer specifies that both statements together are sufficient, but neither alone is sufficient. Statement (1) provides r+s=6r + s = 6, which leaves rsrs unknown. Statement (2) gives r3+s3=126r^3 + s^3 = 126, which also leaves rsrs undetermined. When combined, applying the identity r3+s3=(r+s)((r+s)2−3rs)r^3 + s^3 = (r + s)((r + s)^2 - 3rs) yields 126=6(36−3rs)126 = 6(36 - 3rs), solving uniquely for rs=5rs = 5. This allows exact calculation of r2+s2=(r+s)2−2rs=36−10=26r^2 + s^2 = (r+s)^2 - 2rs = 36 - 10 = 26.

Adım Adım Çözüm

1
Rephrase the target expression using fundamental algebraic identities
Note that r2+s2=(r+s)2−2rsr^2 + s^2 = (r + s)^2 - 2rs. To find r2+s2r^2 + s^2, we need both (r+s)(r + s) and the product rsrs.
Expressing the target in terms of sum and product simplifies the evaluation of statement sufficiency.
2
Evaluate Statement (1) alone
Statement (1) gives r+s=6r + s = 6. Without knowing rsrs or individual values of rr and ss, r2+s2=36−2rsr^2 + s^2 = 36 - 2rs can take infinitely many values. Statement (1) is NOT sufficient.
A single linear equation with two variables does not fix the value of rsrs.
3
Evaluate Statement (2) alone
Statement (2) gives r3+s3=126r^3 + s^3 = 126. Multiple pairs satisfy this (e.g., r=5,s=1r=5, s=1 gives 125+1=126125+1=126 where r2+s2=26r^2+s^2=26; r=1263,s=0r=\sqrt[3]{126}, s=0 is excluded since non-zero, but other non-integer pairs exist). Statement (2) is NOT sufficient.
A single cubic equation in two variables does not uniquely define r2+s2r^2 + s^2.
4
Evaluate Statement (1) and Statement (2) together
Use the sum of cubes identity r3+s3=(r+s)(r2−rs+s2)=(r+s)((r+s)2−3rs)r^3 + s^3 = (r + s)(r^2 - rs + s^2) = (r + s)((r + s)^2 - 3rs). Substitute r+s=6r + s = 6 and r3+s3=126r^3 + s^3 = 126: 126=6(62−3rs)  ⟹  21=36−3rs  ⟹  3rs=15  ⟹  rs=5126 = 6(6^2 - 3rs) \implies 21 = 36 - 3rs \implies 3rs = 15 \implies rs = 5. Thus, r2+s2=(r+s)2−2rs=62−2(5)=26r^2 + s^2 = (r + s)^2 - 2rs = 6^2 - 2(5) = 26. Both statements together are SUFFICIENT.
The system yields a unique numerical value for r2+s2r^2 + s^2.

Anahtar Kavram

Using polynomial identities (sum of cubes and square of a binomial) to evaluate sufficiency of non-linear symmetric systems.
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Konular

Algebraic Equations and Systems in Data Sufficiency

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