Examkin

Algebraic Equations and Systems in Data Sufficiency

42 soru

Soru 21Soru →

If xx and yy are real numbers such that x≠yx \neq y, what is the value of x+yx + y?

(1) x3−y3=7(x−y)x^3 - y^3 = 7(x - y)
(2) x2−y2=3(x−y)x^2 - y^2 = 3(x - y)

Cevabı ve açıklamayı göster

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

Cevap

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
The question stem explicitly states that x≠yx \neq y, establishing that x−y≠0x - y \neq 0. In Statement (2), factoring x2−y2x^2 - y^2 into (x−y)(x+y)=3(x−y)(x - y)(x + y) = 3(x - y) and dividing by non-zero (x−y)(x - y) directly yields the unique value x+y=3x + y = 3. In Statement (1), factoring gives x2+xy+y2=7x^2 + xy + y^2 = 7, which leaves x+yx + y undetermined because xyxy can vary. Consequently, Statement (2) alone is sufficient, while Statement (1) alone is not.

Adım Adım Çözüm

1
Analyze the question stem constraints and target expression.
We are given that xx and yy are real numbers with x≠yx \neq y, which implies x−y≠0x - y \neq 0. The goal is to determine a single unique value for x+yx + y.
Establishing x−y≠0x - y \neq 0 allows valid algebraic division by (x−y)(x - y) in the given statements.
2
Evaluate Statement (1) independently: x3−y3=7(x−y)x^3 - y^3 = 7(x - y).
Factor the left side using the difference of cubes identity: (x−y)(x2+xy+y2)=7(x−y)(x - y)(x^2 + xy + y^2) = 7(x - y). Since x−y≠0x - y \neq 0, divide both sides by (x−y)(x - y) to get x2+xy+y2=7x^2 + xy + y^2 = 7, which rewrites to (x+y)2−xy=7(x + y)^2 - xy = 7.
Because xyxy is unknown and variable, (x+y)(x + y) can take multiple values. For instance, (x,y)=(2,1)(x, y) = (2, 1) gives x+y=3x+y=3 and 4+2+1=74+2+1=7, whereas (x,y)=(7,−7)(x, y) = (\sqrt{7}, -\sqrt{7}) gives x+y=0x+y=0 and 7−7+7=77-7+7=7. Thus, Statement (1) alone is NOT sufficient.
3
Evaluate Statement (2) independently: x2−y2=3(x−y)x^2 - y^2 = 3(x - y).
Factor the left side using the difference of squares identity: (x−y)(x+y)=3(x−y)(x - y)(x + y) = 3(x - y). Since x−y≠0x - y \neq 0, divide both sides by (x−y)(x - y) to yield x+y=3x + y = 3.
This provides a single, unambiguous numerical value for x+yx + y. Therefore, Statement (2) alone IS sufficient.

Anahtar Kavram

Algebraic expression simplification using stem constraints in Data Sufficiency
Soru 22Soru →

If aa and bb are non-zero real numbers, what is the value of a2+9b2ab\frac{a^2 + 9b^2}{ab}?

(1) a2−3ab−18b2=0a^2 - 3ab - 18b^2 = 0
(2) a>0a > 0 and b>0b > 0

Cevabı ve açıklamayı göster

Cevap: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Cevap

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
The correct option is the choice stating that both statements together are sufficient, but neither statement alone is sufficient. Statement (1) produces two valid roots for the ratio of the variables, which lead to two different target values. Statement (2) specifies that both variables are positive, which eliminates the negative root and uniquely identifies the ratio and target value when combined with Statement (1).

Adım Adım Çözüm

1
Rephrase the target expression in terms of the ratio k=abk = \frac{a}{b}.
a2+9b2ab=a2ab+9b2ab=ab+9(ba)=k+9k\frac{a^2 + 9b^2}{ab} = \frac{a^2}{ab} + \frac{9b^2}{ab} = \frac{a}{b} + 9\left(\frac{b}{a}\right) = k + \frac{9}{k}. Finding the ratio k=abk = \frac{a}{b} is sufficient to answer the question.
Dividing the numerator by the denominator simplifies the expression into a function of a single ratio variable.
2
Evaluate Statement (1) independently.
Divide a2−3ab−18b2=0a^2 - 3ab - 18b^2 = 0 by b2b^2 to get (ab)2−3(ab)−18=0\left(\frac{a}{b}\right)^2 - 3\left(\frac{a}{b}\right) - 18 = 0, or k2−3k−18=0k^2 - 3k - 18 = 0. Factoring yields (k−6)(k+3)=0(k - 6)(k + 3) = 0, so k=6k = 6 or k=−3k = -3. If k=6k = 6, target value is 6+96=7.56 + \frac{9}{6} = 7.5. If k=−3k = -3, target value is −3+9−3=−6-3 + \frac{9}{-3} = -6.
Since two distinct numerical outcomes are possible for the target expression, Statement (1) alone is NOT sufficient.
3
Evaluate Statement (2) independently.
a>0a > 0 and b>0b > 0 implies k=ab>0k = \frac{a}{b} > 0, but gives no numerical constraint on kk.
Without an equation linking aa and bb, Statement (2) alone is NOT sufficient.
4
Evaluate Statement (1) and Statement (2) combined.
Statement (1) gives k=6k = 6 or k=−3k = -3. Statement (2) requires k>0k > 0, eliminating k=−3k = -3 and establishing k=6k = 6 uniquely. Thus, the target value is uniquely 7.57.5.
Combining both statements uniquely determines the ratio kk and therefore the target expression.

Anahtar Kavram

Rephrasing homogeneous algebraic expressions into single ratio variables and analyzing degree constraints in quadratic systems.
Soru 23Soru →

If xx and yy are real numbers, what is the value of x+yx + y?

(1) x2+y2=25−2xyx^2 + y^2 = 25 - 2xy
(2) x3+y3=125−3xy(x+y)x^3 + y^3 = 125 - 3xy(x+y)

Cevabı ve açıklamayı göster

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

Cevap

Statement (2) ALONE is sufficient to determine that x+y=5x + y = 5, but Statement (1) ALONE is not sufficient because it allows x+y=5x + y = 5 or x+y=−5x + y = -5.
Statement (2) alone allows us to factor the expression into (x+y)3=125(x+y)^3 = 125. Because real numbers have a unique real cube root, x+yx+y must equal 55. Thus, Statement (2) alone provides a single, definitive answer to the question stem. Statement (1) yields (x+y)2=25(x+y)^2 = 25, which leads to x+y=5x+y = 5 or x+y=−5x+y = -5, making it insufficient on its own.

Adım Adım Çözüm

1
Rephrase the target question
The target is to find a unique numerical value for the expression (x+y)(x + y). We do not need individual values for xx and yy.
Data Sufficiency targets involving expressions often do not require solving for individual variables.
2
Evaluate Statement (1): x2+y2=25−2xyx^2 + y^2 = 25 - 2xy
Rearranging terms gives x2+2xy+y2=25x^2 + 2xy + y^2 = 25, which factors as (x+y)2=25(x + y)^2 = 25. Taking the square root gives x+y=5x + y = 5 or x+y=−5x + y = -5.
Since there are two distinct real values for x+yx + y, Statement (1) alone is insufficient.
3
Evaluate Statement (2): x3+y3=125−3xy(x+y)x^3 + y^3 = 125 - 3xy(x+y)
Rearranging terms gives x3+3xy(x+y)+y3=125x^3 + 3xy(x+y) + y^3 = 125, which is the expanded form of (x+y)3=125(x + y)^3 = 125. Taking the cube root gives x+y=5x + y = 5.
For real numbers, every real number has exactly one real cube root. Thus, x+y=5x + y = 5 uniquely. Statement (2) alone is sufficient.

Anahtar Kavram

Algebraic Rephrasing and Degree of Real Polynomial Identities
Tahmini Süre:2m 0s
Soru 24Soru →

If xx and yy are real numbers, what is the value of x−yx - y?

(1) x2−y2=12(x+y)x^2 - y^2 = 12(x + y) and x+y≠0x + y \neq 0
(2) x2+y2=2xy+144x^2 + y^2 = 2xy + 144

Cevabı ve açıklamayı göster

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

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
The option stating that Statement (1) ALONE is sufficient while Statement (2) ALONE is not sufficient is correct. Factoring Statement (1) gives (x−y)(x+y)=12(x+y)(x - y)(x + y) = 12(x + y). Because x+y≠0x + y \neq 0, dividing both sides by (x+y)(x + y) establishes x−y=12x - y = 12 uniquely. In contrast, Statement (2) simplifies to (x−y)2=144(x - y)^2 = 144, yielding two possible values (1212 and −12-12), which makes Statement (2) insufficient on its own.

Adım Adım Çözüm

1
Evaluate Statement (1) independently
The statement gives x2−y2=12(x+y)x^2 - y^2 = 12(x + y). Factoring the difference of squares gives (x−y)(x+y)=12(x+y)(x - y)(x + y) = 12(x + y). Since x+y≠0x + y \neq 0, divide both sides by (x+y)(x + y) to get x−y=12x - y = 12.
Because a unique numerical value for x−yx - y is obtained, Statement (1) ALONE is sufficient.
2
Evaluate Statement (2) independently
Rearranging x2+y2=2xy+144x^2 + y^2 = 2xy + 144 yields x2−2xy+y2=144x^2 - 2xy + y^2 = 144, which contracts to (x−y)2=144(x - y)^2 = 144. Taking the square root gives x−y=12x - y = 12 or x−y=−12x - y = -12.
Because there are two distinct potential values for x−yx - y, Statement (2) ALONE is not sufficient.
3
Combine the evaluations
Statement (1) alone is sufficient, whereas Statement (2) alone is not sufficient.
No combination of statements is required since Statement (1) alone uniquely determines the target value.

Anahtar Kavram

Algebraic Stem Simplification and Degree Ambiguity in Data Sufficiency
Tahmini Süre:2m 0s
Soru 25Soru →

If xx and yy are real numbers such that x+y≠0x + y \neq 0, what is the value of the product xyxy?

(1) x3+y3=28(x+y)x^3 + y^3 = 28(x + y)
(2) x2+y2=20−xyx^2 + y^2 = 20 - xy

Cevabı ve açıklamayı göster

Cevap: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Cevap

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Combining both statements provides two independent linear equations in terms of (x+y)2(x+y)^2 and xyxy. Subtracting Statement (1) from Statement (2) eliminates the squared sum term (x+y)2(x+y)^2 and yields a unique value of −4-4 for the product xyxy, with real solutions existing for xx and yy. Therefore, both statements together are sufficient.

Adım Adım Çözüm

1
Analyze Statement (1) algebraically.
(x+y)2−3xy=28(x + y)^2 - 3xy = 28
Factor x3+y3x^3 + y^3 as (x+y)(x2−xy+y2)(x + y)(x^2 - xy + y^2). Since x+y≠0x + y \neq 0, divide both sides by (x+y)(x + y) to obtain x2−xy+y2=28x^2 - xy + y^2 = 28. Express x2+y2x^2 + y^2 as (x+y)2−2xy(x + y)^2 - 2xy, yielding (x+y)2−3xy=28(x + y)^2 - 3xy = 28. Because (x+y)2(x + y)^2 is unknown, xyxy can take multiple real values (e.g., if (x+y)2=16(x+y)^2 = 16, xy=−4xy = -4; if (x+y)2=36(x+y)^2 = 36, xy=8/3xy = 8/3). Thus, Statement (1) alone is insufficient.
2
Analyze Statement (2) algebraically.
(x+y)2−xy=20(x + y)^2 - xy = 20
Rearrange x2+y2=20−xyx^2 + y^2 = 20 - xy to x2+xy+y2=20x^2 + xy + y^2 = 20. Express x2+y2x^2 + y^2 as (x+y)2−2xy(x + y)^2 - 2xy, yielding (x+y)2−xy=20(x + y)^2 - xy = 20. Again, since (x+y)2(x + y)^2 is unknown, xyxy is not uniquely determined (e.g., if (x+y)2=16(x+y)^2 = 16, xy=−4xy = -4; if (x+y)2=4(x+y)^2 = 4, xy=−16xy = -16). Thus, Statement (2) alone is insufficient.
3
Combine both statements and solve the system of equations.
xy=−4xy = -4
Let u=(x+y)2u = (x + y)^2 and v=xyv = xy. Statement (1) gives u−3v=28u - 3v = 28 and Statement (2) gives u−v=20u - v = 20. Subtracting the first equation from the second yields (u−v)−(u−3v)=20−28  ⟹  2v=−8  ⟹  v=−4(u - v) - (u - 3v) = 20 - 28 \implies 2v = -8 \implies v = -4. Hence, xy=−4xy = -4 uniquely.
4
Verify existence of real solutions for xx and yy.
Real solutions exist because the discriminant is positive.
With u=16u = 16, we have x+y=4x + y = 4 or x+y=−4x + y = -4. For x+y=4x + y = 4 and xy=−4xy = -4, xx and yy are roots of t2−4t−4=0t^2 - 4t - 4 = 0, which has discriminant 16−4(1)(−4)=32>016 - 4(1)(-4) = 32 > 0. Thus real values of xx and yy exist.

Anahtar Kavram

System rephrasing in Data Sufficiency by substituting composite variables like (x+y)2(x+y)^2 and xyxy.
Soru 26Soru →

If rr and ss are non-zero real numbers, what is the value of r2+2s2rs\frac{r^2 + 2s^2}{rs}?

(1) r2−3rs+2s2=0r^2 - 3rs + 2s^2 = 0
(2) r+s=6r + s = 6

Cevabı ve açıklamayı göster

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

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
The correct option states that Statement (1) alone is sufficient while Statement (2) is not. Rephrasing the target expression r2+2s2rs\frac{r^2 + 2s^2}{rs} as rs+2(sr)\frac{r}{s} + 2\left(\frac{s}{r}\right) shows that the target value depends exclusively on the ratio t=rst = \frac{r}{s}. Statement (1) factors into (r−s)(r−2s)=0(r - s)(r - 2s) = 0, giving t=1t = 1 or t=2t = 2. Testing t=1t = 1 yields 1+2=31 + 2 = 3, and testing t=2t = 2 yields 2+1=32 + 1 = 3. Because both possible values of tt evaluate to the same constant result (33), Statement (1) alone uniquely determines the target value and is sufficient. Statement (2) allows infinitely many ratios for r/sr/s, giving different outcomes, and is therefore insufficient.

Adım Adım Çözüm

1
Rephrase the target expression algebraically.
Dividing each term in the numerator by the denominator gives r2+2s2rs=rs+2(sr)\frac{r^2 + 2s^2}{rs} = \frac{r}{s} + 2\left(\frac{s}{r}\right). Letting t=rst = \frac{r}{s}, the target expression is t+2tt + \frac{2}{t}.
Simplifying the question stem reveals that we only need to know the ratio rs\frac{r}{s}.
2
Evaluate Statement (1): r2−3rs+2s2=0r^2 - 3rs + 2s^2 = 0.
Divide the entire equation by s2s^2 (since s≠0s \neq 0) to get (rs)2−3(rs)+2=0\left(\frac{r}{s}\right)^2 - 3\left(\frac{r}{s}\right) + 2 = 0, or t2−3t+2=0t^2 - 3t + 2 = 0. Factoring yields (t−1)(t−2)=0(t - 1)(t - 2) = 0, so t=1t = 1 or t=2t = 2.
If t=1t = 1, t+2t=1+2=3t + \frac{2}{t} = 1 + 2 = 3.
If t=2t = 2, t+2t=2+22=3t + \frac{2}{t} = 2 + \frac{2}{2} = 3.
In both cases, the target expression equals 33.
Although there are two solutions for the ratio tt, both lead to the identical value of 33 for the target expression. Thus, Statement (1) alone is sufficient.
3
Evaluate Statement (2): r+s=6r + s = 6.
If r=2r = 2 and s=4s = 4, then rs=12\frac{r}{s} = \frac{1}{2}, giving a target value of 12+4=4.5\frac{1}{2} + 4 = 4.5.
If r=3r = 3 and s=3s = 3, then rs=1\frac{r}{s} = 1, giving a target value of 1+2=31 + 2 = 3.
Since different pairs of rr and ss yield different values for the expression, Statement (2) alone is not sufficient.
Knowing only the linear sum of two variables does not uniquely fix their ratio.

Anahtar Kavram

Question Stem Rephrasing and Symmetry in Homogeneous Quadratic Expressions
Soru 27Soru →

If aa and bb are real numbers such that a≠ba \neq b, what is the value of a+ba−b\frac{a + b}{a - b}?

(1) a2+b2=4aba^2 + b^2 = 4ab
(2) a>b>0a > b > 0

Cevabı ve açıklamayı göster

Cevap: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Cevap

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Statement (1) allows us to determine that the square of the target expression (a+b)/(a−b)(a+b)/(a-b) equals 3, which gives two possible values: 3\sqrt{3} and −3-\sqrt{3}. Statement (2) provides the condition a>b>0a > b > 0, ensuring that both a+ba+b and a−ba-b are positive, so their quotient must be positive. Combining both statements eliminates −3-\sqrt{3}, uniquely determining that the expression equals 3\sqrt{3}. Thus, both statements together are sufficient.

Adım Adım Çözüm

1
Express the square of the target expression in terms of basic algebraic components.
Let E=a+ba−bE = \frac{a + b}{a - b}. Squaring both sides yields E2=(a+b)2(a−b)2=a2+2ab+b2a2−2ab+b2E^2 = \frac{(a + b)^2}{(a - b)^2} = \frac{a^2 + 2ab + b^2}{a^2 - 2ab + b^2}.
Rewriting the ratio in squared form allows substitution of expressions involving a2+b2a^2 + b^2 and abab.
2
Evaluate Statement (1) independently.
Substitute a2+b2=4aba^2 + b^2 = 4ab into the squared ratio: E2=4ab+2ab4ab−2ab=6ab2ab=3E^2 = \frac{4ab + 2ab}{4ab - 2ab} = \frac{6ab}{2ab} = 3. Taking the square root gives E=3E = \sqrt{3} or E=−3E = -\sqrt{3}.
Because Statement (1) allows two distinct possible values for EE, Statement (1) alone is NOT sufficient.
3
Evaluate Statement (2) independently.
Statement (2) states a>b>0a > b > 0. This implies a+b>0a + b > 0 and a−b>0a - b > 0, so E>0E > 0, but no numerical value is specified.
Without quantitative equations, Statement (2) alone is NOT sufficient.
4
Evaluate Statements (1) and (2) combined.
From Statement (1), E=±3E = \pm\sqrt{3}. From Statement (2), E>0E > 0. Combining both rules out −3-\sqrt{3}, leaving uniquely E=3E = \sqrt{3}.
The two statements together establish a single, unique value for the target expression.

Anahtar Kavram

Evaluating algebraic ratios via squared identities and applying inequality sign constraints to eliminate redundant roots in Data Sufficiency.
Tahmini Süre:1m 30s
Soru 28Soru →

If uu and vv are non-zero real numbers, what is the value of u−2vu+v\frac{u - 2v}{u + v}?

(1) 3u2−5uv−2v2=03u^2 - 5uv - 2v^2 = 0
(2) u>v>0u > v > 0

Cevabı ve açıklamayı göster

Cevap: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Cevap

Both statements together are sufficient, but neither statement alone is sufficient.
The correct answer identifies that neither statement alone provides a unique value for the expression, but combining them eliminates the negative root case, leaving a unique value of 0 for the target expression.

Adım Adım Çözüm

1
Rephrase the target expression
Dividing numerator and denominator by vv, the target expression is uv−2uv+1\frac{\frac{u}{v} - 2}{\frac{u}{v} + 1}. Finding a unique value for uv\frac{u}{v} determines the target expression.
Simplifying the question target clarifies what information is necessary to achieve sufficiency.
2
Evaluate Statement (1) independently
Factor 3u2−5uv−2v2=03u^2 - 5uv - 2v^2 = 0 as (3u+v)(u−2v)=0(3u + v)(u - 2v) = 0. This gives 3u+v=0  ⟹  u=−v33u + v = 0 \implies u = -\frac{v}{3} or u−2v=0  ⟹  u=2vu - 2v = 0 \implies u = 2v. If u=2vu = 2v, then u−2vu+v=03v=0\frac{u - 2v}{u + v} = \frac{0}{3v} = 0. If u=−v3u = -\frac{v}{3}, then u−2vu+v=−v3−2v−v3+v=−73v23v=−72\frac{u - 2v}{u + v} = \frac{-\frac{v}{3} - 2v}{-\frac{v}{3} + v} = \frac{-\frac{7}{3}v}{\frac{2}{3}v} = -\frac{7}{2}. Two different values are possible.
Since Statement (1) produces two distinct numerical outcomes, it is not sufficient alone.
3
Evaluate Statement (2) independently
Statement (2) states u>v>0u > v > 0. This indicates that both uu and vv are positive, but gives no fixed algebraic equality for uv\frac{u}{v}.
Infinitely many positive pairs (u,v)(u, v) satisfy u>v>0u > v > 0 while yielding different values for the expression. Statement (2) is not sufficient alone.
4
Evaluate Statements (1) and (2) combined
From Statement (2), u>0u > 0 and v>0v > 0, so 3u+v>03u + v > 0. Therefore, the factor 3u+v=03u + v = 0 is impossible. This leaves u−2v=0u - 2v = 0 as the only valid relation, so u=2vu = 2v. Substituting u=2vu = 2v yields 2v−2v2v+v=0\frac{2v - 2v}{2v + v} = 0.
Combining the inequality constraint with the quadratic factorization uniquely determines the value of the target expression.

Anahtar Kavram

Algebraic Equations and Systems in Data Sufficiency
Tahmini Süre:2m 0s
Soru 29Soru →

If aa and bb are real numbers, what is the value of a2−b2a^2 - b^2?

(1) a+b=5a + b = 5
(2) (a−b)2=9(a - b)^2 = 9

Cevabı ve açıklamayı göster

Cevap: Statements (1) and (2) TOGETHER are NOT sufficient.

Cevap

The statements together are not sufficient to determine a unique value for a2−b2a^2 - b^2, because a2−b2a^2 - b^2 can equal either 1515 or −15-15.
The correct option states that both statements together are not sufficient. Factoring the target expression yields a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b). Statement (1) tells us a+b=5a+b = 5, and Statement (2) tells us a−b=±3a-b = \pm 3. Combining them yields two possible numerical results (1515 and −15-15). Since Data Sufficiency requires a single, unique numerical value, the information remains insufficient.

Adım Adım Çözüm

1
Rephrase the target expression using algebraic identities.
The target expression a2−b2a^2 - b^2 factors into (a+b)(a−b)(a + b)(a - b). To find a unique value, we need a unique value for the product (a+b)(a−b)(a + b)(a - b).
Factoring highlights the required components: the sum (a+b)(a + b) and the difference (a−b)(a - b).
2
Evaluate Statement (1) independently.
Statement (1) gives a+b=5a + b = 5. However, the value of a−ba - b is completely unknown.
Since (a−b)(a - b) can be any real number, a2−b2=5(a−b)a^2 - b^2 = 5(a - b) can take infinitely many values. Statement (1) is not sufficient.
3
Evaluate Statement (2) independently.
Statement (2) gives (a−b)2=9(a - b)^2 = 9, which implies a−b=3a - b = 3 or a−b=−3a - b = -3.
The sum a+ba + b is completely unknown, and a−ba - b has two potential values. Statement (2) is not sufficient.
4
Evaluate Statements (1) and (2) together.
From (1), a+b=5a + b = 5. From (2), a−b=3a - b = 3 or a−b=−3a - b = -3.
If a−b=3a - b = 3, then a2−b2=(5)(3)=15a^2 - b^2 = (5)(3) = 15.
If a−b=−3a - b = -3, then a2−b2=(5)(−3)=−15a^2 - b^2 = (5)(-3) = -15.
Because there are two distinct outcomes (1515 and −15-15), a unique value cannot be determined. Therefore, both statements together are not sufficient.

Anahtar Kavram

Quadratic non-linearity and root ambiguity in Data Sufficiency systems
Soru 30Soru →

If mm and nn are real numbers, what is the value of m+nm + n?

(1) m2−n2=0m^2 - n^2 = 0
(2) m2+n2=50m^2 + n^2 = 50

Cevabı ve açıklamayı göster

Cevap: Statements (1) and (2) TOGETHER are NOT sufficient.

Cevap

Statements (1) and (2) TOGETHER are NOT sufficient.
Evaluating both statements together yields four possible coordinate pairs: (5,5)(5, 5), (−5,−5)(-5, -5), (5,−5)(5, -5), and (−5,5)(-5, 5). The sum m+nm + n can equal 1010, −10-10, or 00. Because a Data Sufficiency question requires a single unique value to be deemed sufficient, having three distinct possible values means the statements together are insufficient.

Adım Adım Çözüm

1
Evaluate Statement (1) independently.
m2−n2=0  ⟹  (m−n)(m+n)=0  ⟹  m=nm^2 - n^2 = 0 \implies (m - n)(m + n) = 0 \implies m = n or m=−nm = -n.
If m=−nm = -n, then m+n=0m + n = 0. However, if m=nm = n, then m+n=2mm + n = 2m, which varies. Since we cannot determine a single numerical value for m+nm + n, Statement (1) alone is NOT sufficient.
2
Evaluate Statement (2) independently.
m2+n2=50m^2 + n^2 = 50.
Different pairs (m,n)(m, n) satisfy m2+n2=50m^2 + n^2 = 50. For example, if (m,n)=(5,5)(m, n) = (5, 5), then m+n=10m + n = 10. If (m,n)=(7,1)(m, n) = (7, 1), then m+n=8m + n = 8. Since m+nm + n is not uniquely determined, Statement (2) alone is NOT sufficient.
3
Evaluate Statements (1) and (2) combined.
From (1), m2=n2m^2 = n^2. Substitute into (2): n2+n2=50  ⟹  2n2=50  ⟹  n2=25  ⟹  n=5n^2 + n^2 = 50 \implies 2n^2 = 50 \implies n^2 = 25 \implies n = 5 or n=−5n = -5.
Since m2=25m^2 = 25, mm can also be 55 or −5-5. The possible ordered pairs (m,n)(m, n) are (5,5)(5, 5), (−5,−5)(-5, -5), (5,−5)(5, -5), and (−5,5)(-5, 5). Calculating m+nm + n for these pairs gives 1010, −10-10, and 00. Because multiple outcomes are possible, the combined statements are NOT sufficient.

Anahtar Kavram

Non-Linear Systems and Multiple Solution Ambiguity in Data Sufficiency
Soru 31Soru →

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

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Cevap: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is 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.
Soru 32Soru →

If xx and yy are real numbers, what is the value of x+yx + y?

(1) x3+y3=28x^3 + y^3 = 28
(2) x2−xy+y2=7x^2 - xy + y^2 = 7

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Cevap: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Cevap

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
The sum of cubes factors as x3+y3=(x+y)(x2−xy+y2)x^3 + y^3 = (x + y)(x^2 - xy + y^2). Neither statement alone provides enough information to determine x+yx + y, but combining both statements gives 28=(x+y)(7)28 = (x + y)(7), which uniquely determines x+y=4x + y = 4.

Adım Adım Çözüm

1
Rephrase the question target using the sum of cubes identity.
Recall x3+y3=(x+y)(x2−xy+y2)x^3 + y^3 = (x + y)(x^2 - xy + y^2). Thus, if x2−xy+y2≠0x^2 - xy + y^2 \neq 0, then x+y=x3+y3x2−xy+y2x + y = \frac{x^3 + y^3}{x^2 - xy + y^2}.
Factoring allows connecting the expression in Statement (1) directly to the expression in Statement (2).
2
Evaluate Statement (1) alone.
Statement (1) gives x3+y3=28x^3 + y^3 = 28. For example, if x=3x=3 and y=−1y=-1, x3+y3=27−1=26≠28x^3+y^3 = 27 - 1 = 26 \neq 28; if x=283x=\sqrt[3]{28} and y=0y=0, then x+y=283x+y=\sqrt[3]{28}. If x=3,y=1x=3, y=1, x3+y3=28x^3+y^3=28 and x+y=4x+y=4. Multiple values are possible.
A single cubic equation in two variables does not fix the sum x+yx+y.
3
Evaluate Statement (2) alone.
Statement (2) gives x2−xy+y2=7x^2 - xy + y^2 = 7. Multiple pairs such as (x,y)=(3,2)(x,y) = (3,2) give 9−6+4=79-6+4=7 (where x+y=5x+y=5) and (x,y)=(3,1)(x,y) = (3,1) give 9−3+1=79-3+1=7 (where x+y=4x+y=4).
A quadratic equation in two variables permits multiple sums for x+yx+y.
4
Evaluate both statements together.
Substitute Statement (1) and Statement (2) into the identity: 28=(x+y)(7)  ⟹  x+y=428 = (x + y)(7) \implies x + y = 4.
Dividing x3+y3x^3 + y^3 by x2−xy+y2x^2 - xy + y^2 yields a single unique value of 44 for x+yx + y.

Anahtar Kavram

Algebraic Expression Rephrasing and Factoring Identities in Data Sufficiency
Soru 33Soru →

If pp and qq are real numbers, what is the value of (p+q)2(p + q)^2?

(1) p2+q2=25p^2 + q^2 = 25
(2) pq=12pq = 12

Cevabı ve açıklamayı göster

Cevap: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Cevap

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Expanding the target expression gives (p+q)2=p2+2pq+q2(p + q)^2 = p^2 + 2pq + q^2. Statement (1) provides p2+q2=25p^2 + q^2 = 25 but leaves pqpq unknown, making it insufficient alone. Statement (2) provides pq=12pq = 12 but leaves p2+q2p^2 + q^2 unknown, making it insufficient alone. Combining both statements allows direct substitution into the identity: (p+q)2=25+2(12)=49(p + q)^2 = 25 + 2(12) = 49, which yields a single unique answer.

Adım Adım Çözüm

1
Rephrase the question stem algebraically.
(p+q)2=p2+2pq+q2(p + q)^2 = p^2 + 2pq + q^2
Expanding the target expression shows that calculating (p+q)2(p + q)^2 requires knowing the sum of squares (p2+q2)(p^2 + q^2) and the product pqpq.
2
Evaluate Statement (1) independently.
Insufficient
Given p2+q2=25p^2 + q^2 = 25, the term 2pq2pq remains unknown. For example, if p=5p = 5 and q=0q = 0, then (p+q)2=25(p+q)^2 = 25; if p=3p = 3 and q=4q = 4, then (p+q)2=49(p+q)^2 = 49. Since multiple outcomes exist, Statement (1) alone is insufficient.
3
Evaluate Statement (2) independently.
Insufficient
Given pq=12pq = 12, the term p2+q2p^2 + q^2 remains unknown. For example, if p=3p = 3 and q=4q = 4, then (p+q)2=49(p+q)^2 = 49; if p=1p = 1 and q=12q = 12, then (p+q)2=169(p+q)^2 = 169. Since multiple outcomes exist, Statement (2) alone is insufficient.
4
Evaluate Statements (1) and (2) together.
Sufficient
Substituting p2+q2=25p^2 + q^2 = 25 and pq=12pq = 12 into (p+q)2=(p2+q2)+2(pq)(p + q)^2 = (p^2 + q^2) + 2(pq) yields 25+2(12)=4925 + 2(12) = 49. This determines a unique value.

Anahtar Kavram

Algebraic Identity Expansion and Substitution in Data Sufficiency
Soru 34Soru →

If mm and nn are real numbers, what is the value of m2−n2m^2 - n^2?

(1) (m+n)2=36(m + n)^2 = 36
(2) m2+n2=20m^2 + n^2 = 20

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Cevap: Statements (1) and (2) TOGETHER are NOT sufficient.

Cevap

Statements (1) and (2) TOGETHER are NOT sufficient.
The correct response identifies that statements (1) and (2) together leave sign ambiguity in the difference of squares because mm and nn can swap values while maintaining the same sum and sum of squares.

Adım Adım Çözüm

1
Rephrase the target question
The target expression is m2−n2=(m+n)(m−n)m^2 - n^2 = (m + n)(m - n). To find a unique value, we need a unique numerical value for (m+n)(m−n)(m + n)(m - n).
Factoring the difference of squares clarifies what information is necessary.
2
Evaluate Statement (1) alone
(m+n)2=36  ⟹  m+n=6(m + n)^2 = 36 \implies m + n = 6 or m+n=−6m + n = -6. We have no information about m−nm - n.
Since m−nm - n can take infinitely many values, m2−n2m^2 - n^2 is not uniquely determined. Statement (1) is insufficient.
3
Evaluate Statement (2) alone
m2+n2=20m^2 + n^2 = 20.
Different pairs of numbers like (m,n)=(20,0)(m, n) = (\sqrt{20}, 0) give m2−n2=20m^2 - n^2 = 20, while (m,n)=(0,20)(m, n) = (0, \sqrt{20}) give m2−n2=−20m^2 - n^2 = -20. Statement (2) is insufficient.
4
Evaluate Statements (1) and (2) combined
Expand Statement (1): m2+2mn+n2=36m^2 + 2mn + n^2 = 36. Substitute Statement (2): 20+2mn=36  ⟹  2mn=16  ⟹  mn=820 + 2mn = 36 \implies 2mn = 16 \implies mn = 8.
Now test values satisfying m+n=6m + n = 6 and mn=8mn = 8:
Case 1: m=4,n=2  ⟹  m2−n2=42−22=16−4=12m = 4, n = 2 \implies m^2 - n^2 = 4^2 - 2^2 = 16 - 4 = 12.
Case 2: m=2,n=4  ⟹  m2−n2=22−42=4−16=−12m = 2, n = 4 \implies m^2 - n^2 = 2^2 - 4^2 = 4 - 16 = -12.
Since m2−n2m^2 - n^2 yields two distinct values (1212 and −12-12), the statements together are insufficient.
Symmetry between variables in a non-linear system creates multiple possible values for non-symmetric target expressions.

Anahtar Kavram

Symmetry and Degree Ambiguity in Algebraic Data Sufficiency
Soru 35Soru →

If xx and yy are positive real numbers, what is the value of x+yx + y?

(1) x2−y2=15x^2 - y^2 = 15
(2) x2y−xy2=12x^2 y - x y^2 = 12

Cevabı ve açıklamayı göster

Cevap: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Cevap

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Evaluating each statement individually reveals that neither provides enough constraints on its own to solve for the sum of the variables. When both statements are combined, substituting the expression for the sum into the product equation yields a quartic polynomial in terms of the difference of the variables. Because the function is strictly increasing for positive values, it possesses exactly one positive real root. This single valid root uniquely determines the sum of the variables, making both statements together sufficient.

Adım Adım Çözüm

1
Evaluate Statement (1) independently.
Statement (1) gives x2−y2=(x−y)(x+y)=15x^2 - y^2 = (x-y)(x+y) = 15. Multiple pairs of positive real numbers satisfy this (e.g., x=4,y=1x=4, y=1 gives x+y=5x+y=5; x=8,y=7x=8, y=7 gives x+y=15x+y=15). Thus, Statement (1) alone is NOT sufficient.
A single non-linear equation in two variables does not restrict x+yx+y to a unique value.
2
Evaluate Statement (2) independently.
Statement (2) gives x2y−xy2=xy(x−y)=12x^2 y - x y^2 = xy(x-y) = 12. Infinitely many positive real pairs satisfy this condition (e.g., x=4,y=1x=4, y=1 gives x+y=5x+y=5; x=3,y=1x=3, y=1 does not yield 12, but continuous choices of x,y>0x, y > 0 allow multiple sums). Thus, Statement (2) alone is NOT sufficient.
Knowing the product xy(x−y)=12xy(x-y) = 12 leaves the value of x+yx+y undetermined.
3
Combine Statement (1) and Statement (2).
Let u=x−yu = x-y and v=x+yv = x+y. Note that u>0u > 0 because x2−y2=15>0x^2 - y^2 = 15 > 0 and x,y>0x, y > 0.
From (1), v=15uv = \frac{15}{u}.
Also, xy=(x+y)2−(x−y)24=v2−u24xy = \frac{(x+y)^2 - (x-y)^2}{4} = \frac{v^2 - u^2}{4}.
Substituting into (2): (v2−u24)u=12  ⟹  u(v2−u2)=48\left(\frac{v^2 - u^2}{4}\right) u = 12 \implies u(v^2 - u^2) = 48.
Substitute v=15uv = \frac{15}{u} into the equation: u(225u2−u2)=48  ⟹  225u−u3=48  ⟹  u4+48u−225=0u\left(\frac{225}{u^2} - u^2\right) = 48 \implies \frac{225}{u} - u^3 = 48 \implies u^4 + 48u - 225 = 0.
Express the system in terms of the difference u=x−yu = x-y and sum v=x+yv = x+y to analyze uniqueness.
4
Determine the number of positive real roots for u4+48u−225=0u^4 + 48u - 225 = 0.
Let f(u)=u4+48u−225f(u) = u^4 + 48u - 225. For u>0u > 0, the derivative f′(u)=4u3+48>0f'(u) = 4u^3 + 48 > 0, meaning f(u)f(u) is strictly increasing for all positive uu. Since f(0)=−225<0f(0) = -225 < 0 and lim⁡u→∞f(u)=+∞\lim_{u \to \infty} f(u) = +\infty, there is exactly ONE positive real root for uu. Testing integer values shows f(3)=34+48(3)−225=81+144−225=0f(3) = 3^4 + 48(3) - 225 = 81 + 144 - 225 = 0, so u=3u = 3 uniquely. Substituting u=3u = 3 into v=15uv = \frac{15}{u} yields v=x+y=5v = x+y = 5 uniquely (with positive numbers x=4,y=1x = 4, y = 1).
A strictly increasing continuous function transitioning from negative to positive values has exactly one root.

Anahtar Kavram

Non-Linear Algebraic Systems and Root Uniqueness in Data Sufficiency
Soru 36Soru →

If xx and yy are real numbers such that x≠yx \neq y, what is the value of x+yx−y\frac{x+y}{x-y}?

(1) x2+y2=5xyx^2 + y^2 = 5xy
(2) x2−y2=12x^2 - y^2 = 12

Cevabı ve açıklamayı göster

Cevap: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Cevap

Both statements together are sufficient, but neither statement alone is sufficient.
Evaluating statement (1) reveals that (x+yx−y)2=73\left(\frac{x+y}{x-y}\right)^2 = \frac{7}{3}, leading to two possible values: 73\sqrt{\frac{7}{3}} and −73-\sqrt{\frac{7}{3}}. Thus, statement (1) alone is insufficient. Statement (2) states that (x+y)(x−y)=12(x+y)(x-y) = 12, which shows that (x+y)(x+y) and (x−y)(x-y) have the same sign, meaning their quotient must be positive. Combining both statements eliminates the negative value, yielding the unique result 73\sqrt{\frac{7}{3}}. Therefore, both statements together are sufficient.

Adım Adım Çözüm

1
Analyze Statement (1) algebraically.
From x2+y2=5xyx^2 + y^2 = 5xy, add 2xy2xy to both sides to obtain (x+y)2=7xy(x+y)^2 = 7xy, and subtract 2xy2xy from both sides to obtain (x−y)2=3xy(x-y)^2 = 3xy. Taking the ratio gives (x+yx−y)2=7xy3xy=73\left(\frac{x+y}{x-y}\right)^2 = \frac{7xy}{3xy} = \frac{7}{3}. Taking the square root yields x+yx−y=±73\frac{x+y}{x-y} = \pm \sqrt{\frac{7}{3}}.
Since two distinct numerical values are possible, Statement (1) alone is NOT sufficient.
2
Analyze Statement (2) algebraically.
Statement (2) gives x2−y2=12x^2 - y^2 = 12, which factors as (x+y)(x−y)=12(x+y)(x-y) = 12.
Knowing only the product of (x+y)(x+y) and (x−y)(x-y) does not provide enough information to determine the value of their quotient x+yx−y\frac{x+y}{x-y}. Statement (2) alone is NOT sufficient.
3
Combine Statement (1) and Statement (2).
Rewrite the target ratio using Statement (2): x+yx−y=(x+y)(x−y)(x−y)2=12(x−y)2\frac{x+y}{x-y} = \frac{(x+y)(x-y)}{(x-y)^2} = \frac{12}{(x-y)^2}. Because (x−y)2>0(x-y)^2 > 0 for x≠yx \neq y, the ratio must be strictly positive. Combining this with Statement (1), which requires x+yx−y=±73\frac{x+y}{x-y} = \pm \sqrt{\frac{7}{3}}, uniquely isolates the positive solution x+yx−y=73\frac{x+y}{x-y} = \sqrt{\frac{7}{3}}.
Together, the statements determine a single, unique value for the target expression.

Anahtar Kavram

Algebraic Rephrasing and Sign Constraints in Data Sufficiency
Tahmini Süre:2m 0s
Soru 37Soru →

If aa and bb are non-zero real numbers, what is the value of a2+b2ab\frac{a^2 + b^2}{ab}?

(1) a+b=3aba + b = 3ab
(2) a−b=aba - b = ab

Cevabı ve açıklamayı göster

Cevap: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Cevap

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Evaluating each statement independently shows that Statement (1) reduces the expression to 9ab−29ab - 2 and Statement (2) reduces it to ab+2ab + 2, both of which depend on the value of abab. Combining both statements yields a system of linear equations in aa and bb, giving a=1a = 1 and b=1/2b = 1/2. This uniquely determines the value of the target expression as 5/25/2. Therefore, both statements together are sufficient, but neither statement alone is sufficient.

Adım Adım Çözüm

1
Rephrase the target expression.
a2+b2ab=ab+ba\frac{a^2 + b^2}{ab} = \frac{a}{b} + \frac{b}{a}. Alternatively, squaring identities give a2+b2ab=(a+b)2−2abab=(a+b)2ab−2\frac{a^2+b^2}{ab} = \frac{(a+b)^2 - 2ab}{ab} = \frac{(a+b)^2}{ab} - 2.
Simplifying the target expression shows what combination of variables is required.
2
Evaluate Statement (1) alone: a+b=3aba + b = 3ab.
Substituting a+b=3aba+b = 3ab into the target expression yields (3ab)2−2abab=9ab−2\frac{(3ab)^2 - 2ab}{ab} = 9ab - 2.
Since the value depends on abab, and abab can take multiple non-zero values (e.g., if a=1,b=1/2a=1, b=1/2, ab=1/2ab=1/2 and the expression is 2.52.5; if a=2,b=2/5a=2, b=2/5, ab=4/5ab=4/5 and the expression is 5.25.2), Statement (1) alone is NOT sufficient.
3
Evaluate Statement (2) alone: a−b=aba - b = ab.
Expressing a2+b2=(a−b)2+2ab=(ab)2+2aba^2+b^2 = (a-b)^2 + 2ab = (ab)^2 + 2ab, the target expression becomes (ab)2+2abab=ab+2\frac{(ab)^2 + 2ab}{ab} = ab + 2.
Since the value depends on abab, which is not fixed by Statement (2) alone, Statement (2) alone is NOT sufficient.
4
Evaluate Statements (1) and (2) together.
Adding the two equations (a+b)+(a−b)=3ab+ab(a + b) + (a - b) = 3ab + ab gives 2a=4ab2a = 4ab. Since a≠0a \neq 0, dividing by 2a2a yields b=12b = \frac{1}{2}.
Substituting b=12b = \frac{1}{2} into a−b=aba - b = ab gives a−12=12a  ⟹  a=1a - \frac{1}{2} = \frac{1}{2}a \implies a = 1. With a=1a=1 and b=12b=\frac{1}{2}, ab=12ab = \frac{1}{2}, and a2+b2ab=1+1/41/2=52\frac{a^2+b^2}{ab} = \frac{1 + 1/4}{1/2} = \frac{5}{2}, giving a single unique value.

Anahtar Kavram

Algebraic System Reduction and Target Rephrasing in Data Sufficiency
Tahmini Süre:2m 0s
Soru 38Soru →

If uu and vv are real numbers, what is the value of u+2vu + 2v?

(1) 3u+6v=153u + 6v = 15
(2) u2−4v2=0u^2 - 4v^2 = 0

Cevabı ve açıklamayı göster

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

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Dividing Statement (1) by 3 directly gives u+2v=5u + 2v = 5, providing a unique numerical answer. Statement (2) permits multiple values for u+2vu + 2v because (u−2v)(u+2v)=0(u - 2v)(u + 2v) = 0 only guarantees that at least one factor is zero, not necessarily u+2v=0u + 2v = 0. Therefore, Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.

Adım Adım Çözüm

1
Analyze Statement (1) algebraically.
The equation is 3u+6v=153u + 6v = 15. Factoring out 3 gives 3(u+2v)=153(u + 2v) = 15. Dividing both sides by 3 yields u+2v=5u + 2v = 5.
This determines a unique, definitive numerical value for the target expression u+2vu + 2v. Therefore, Statement (1) alone is sufficient.
2
Analyze Statement (2) independently.
The equation u2−4v2=0u^2 - 4v^2 = 0 factors as (u−2v)(u+2v)=0(u - 2v)(u + 2v) = 0.
This statement implies that either u+2v=0u + 2v = 0 or u−2v=0u - 2v = 0. If u=4u = 4 and v=2v = 2, then u−2v=0u - 2v = 0 and u+2v=8u + 2v = 8. If u=0u = 0 and v=0v = 0, then u+2v=0u + 2v = 0. Since u+2vu + 2v can take multiple values, Statement (2) alone is not sufficient.

Anahtar Kavram

Algebraic Expression Simplification and Unique Determination in Data Sufficiency
Soru 39Soru →

If pp and qq are real numbers, what is the value of p2+4q2p^2 + 4q^2?

(1) p+2q=8p + 2q = 8
(2) pq=6pq = 6

Cevabı ve açıklamayı göster

Cevap: BOTH statements (1) and (2) TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Cevap

BOTH statements (1) and (2) TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
The target expression p2+4q2p^2 + 4q^2 can be related to the binomial square (p+2q)2=p2+4pq+4q2(p + 2q)^2 = p^2 + 4pq + 4q^2. Rearranging gives p2+4q2=(p+2q)2−4pqp^2 + 4q^2 = (p + 2q)^2 - 4pq. Neither statement alone provides both p+2qp + 2q and pqpq. However, combining Statement (1) (p+2q=8p + 2q = 8) and Statement (2) (pq=6pq = 6) allows direct substitution: p2+4q2=82−4(6)=40p^2 + 4q^2 = 8^2 - 4(6) = 40. Because this yields a single unique value, both statements together are sufficient.

Adım Adım Çözüm

1
Evaluate Statement (1) independently
Statement (1) states p+2q=8p + 2q = 8. Squaring both sides yields (p+2q)2=p2+4pq+4q2=64(p + 2q)^2 = p^2 + 4pq + 4q^2 = 64, so p2+4q2=64−4pqp^2 + 4q^2 = 64 - 4pq. Without the value of pqpq, a unique numerical value for p2+4q2p^2 + 4q^2 cannot be determined.
Statement (1) alone leaves one degree of freedom (the product pqpq is unknown).
2
Evaluate Statement (2) independently
Statement (2) states pq=6pq = 6. Knowing only the product of pp and qq allows infinitely many pairs (p,q)(p, q), resulting in infinitely many values for p2+4q2p^2 + 4q^2.
Statement (2) alone does not constrain the linear sum p+2qp + 2q.
3
Combine Statements (1) and (2)
From Statement (1), p2+4q2=64−4pqp^2 + 4q^2 = 64 - 4pq. Substituting pq=6pq = 6 from Statement (2) yields p2+4q2=64−4(6)=40p^2 + 4q^2 = 64 - 4(6) = 40. This provides a unique, definitive numerical answer.
Combining both statements eliminates all unknown parameters from the target expression p2+4q2p^2 + 4q^2.

Anahtar Kavram

Algebraic Identity Expansion and System Combination in Data Sufficiency
Tahmini Süre:1m 30s
Soru 40Soru →

If xx and yy are real numbers, what is the value of x−yx - y?

(1) (x−y)2=16(x - y)^2 = 16
(2) x2−y2=24x^2 - y^2 = 24

Cevabı ve açıklamayı göster

Cevap: Statements (1) and (2) TOGETHER are NOT sufficient.

Cevap

Statements (1) and (2) TOGETHER are NOT sufficient.
The option stating that statements (1) and (2) together are not sufficient is correct because combining both equations results in two valid candidate solution sets for (x,y)(x, y), specifically (5,1)(5, 1) and (−5,−1)(-5, -1). These produce two different values for the target expression x−yx - y (44 and −4-4), preventing a single unique determination.

Adım Adım Çözüm

1
Evaluate Statement (1) independently.
(x−y)2=16  ⟹  x−y=4(x - y)^2 = 16 \implies x - y = 4 or x−y=−4x - y = -4.
Taking the square root of both sides yields two possible values for x−yx - y, so Statement (1) alone is not sufficient.
2
Evaluate Statement (2) independently.
x2−y2=(x−y)(x+y)=24x^2 - y^2 = (x - y)(x + y) = 24.
Without knowing x+yx + y, the expression x−yx - y can take infinitely many numerical values, so Statement (2) alone is not sufficient.
3
Evaluate Statements (1) and (2) together.
Case 1: If x−y=4x - y = 4, then 4(x+y)=24  ⟹  x+y=64(x + y) = 24 \implies x + y = 6, giving (x,y)=(5,1)(x, y) = (5, 1) where x−y=4x - y = 4. Case 2: If x−y=−4x - y = -4, then −4(x+y)=24  ⟹  x+y=−6-4(x + y) = 24 \implies x + y = -6, giving (x,y)=(−5,−1)(x, y) = (-5, -1) where x−y=−4x - y = -4.
Both coordinate pairs (5,1)(5, 1) and (−5,−1)(-5, -1) satisfy both given statements, but yield two distinct values (44 and −4-4) for x−yx - y. Therefore, both statements combined remain insufficient.

Anahtar Kavram

Degree Ambiguity in Systems of Non-Linear Algebraic Equations
ÖncekiSayfa 2 / 3Sonraki
Algebraic Equations and Systems in Data Sufficiency Alıştırma Soruları — GMAT — Sayfa 2 | Examkin