Algebraic Equations and Systems in Data Sufficiency Practice Questions — GMAT

Question 1Question →

If $x$ and $y$ are real numbers such that $x \neq 3y$ , what is the value of $x + 3y$ ?

(1) $x^2 + 6xy + 9y^2 = 49$
(2) $x^2 - 9y^2 = 12(x - 3y)$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

Statement (1) simplifies to

(x + 3y)^2 = 49

, which means

x + 3y

could be

7

or

-7

, so it is not sufficient. Statement (2) factors as

(x - 3y)(x + 3y) = 12(x - 3y)

. Given

x \neq 3y

, dividing both sides by

(x - 3y)

uniquely determines

x + 3y = 12

. Thus, Statement (2) alone is sufficient while Statement (1) alone is not.

Step-by-Step Solution

1

Analyze Statement (1) algebraically

The expression

x^2 + 6xy + 9y^2

is a perfect square trinomial equal to

(x + 3y)^2

. Therefore,

(x + 3y)^2 = 49

, which gives

x + 3y = 7

or

x + 3y = -7

.

Because there are two distinct valid values for

x + 3y

, Statement (1) alone does not provide a unique value.

2

Analyze Statement (2) algebraically

The left side

x^2 - 9y^2

factors as

(x - 3y)(x + 3y)

. The equation becomes

(x - 3y)(x + 3y) = 12(x - 3y)

.

Since the stem specifies

x \neq 3y

, we know

x - 3y \neq 0

, allowing us to divide both sides by

(x - 3y)

without dividing by zero.

3

Solve the simplified equation from Statement (2)

Dividing gives

x + 3y = 12

.

This yields a single, unique value for the target expression

x + 3y

, making Statement (2) alone sufficient.

Key Concept

Algebraic Stem Rephrasing and Quadratic Degree Evaluation in Data Sufficiency

Estimated Time:1m 30s

Question 2Question →

If $x$ and $y$ are real numbers, what is the value of $\frac{x}{y}$ ?

(1) $x^2 y - x y^2 = 12$
(2) $x - y = 3$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

Combining Statement (1) and Statement (2) allows substituting

x - y = 3

into

xy(x - y) = 12

, yielding

xy = 4

. Solving the system

x - y = 3

and

xy = 4

gives two valid real solution pairs:

(4, 1)

and

(-1, -4)

. The quotient

\frac{x}{y}

evaluates to

4

in the first case and

\frac{1}{4}

in the second case. Because a unique numerical value for

\frac{x}{y}

cannot be determined, the statements together are not sufficient.

Step-by-Step Solution

1

Evaluate Statement (1) independently.

Factor Statement (1):

xy(x - y) = 12

. Many pairs of real numbers

(x, y)

satisfy this, such as

(4, 1)

which gives

\frac{x}{y} = 4

, or

(3, 1)

which gives

3(1)(2) = 6 \neq 12

. For

(4, 1)

,

\frac{x}{y} = 4

. For

(x, y) = (2, -1)

,

2(-1)(3) = -6 \neq 12

. However, choosing

x = \frac{3 + \sqrt{17}}{2}

and

y = \frac{-3 + \sqrt{17}}{2}

gives a different ratio. Since

\frac{x}{y}

can take infinitely many values, Statement (1) is not sufficient.

A single equation with two variables generally allows infinitely many solutions.

2

Evaluate Statement (2) independently.

Statement (2) gives

x - y = 3

. If

x = 4

and

y = 1

, then

\frac{x}{y} = 4

. If

x = 5

and

y = 2

, then

\frac{x}{y} = 2.5

. Since multiple values for

\frac{x}{y}

exist, Statement (2) is not sufficient.

A linear difference condition alone does not fix the ratio of two unknown quantities.

3

Evaluate Statements (1) and (2) combined.

Substitute

x - y = 3

into

xy(x - y) = 12

to get

3xy = 12 \implies xy = 4

. Express

y

as

x - 3

and substitute into

xy = 4

:

x(x - 3) = 4 \implies x^2 - 3x - 4 = 0 \implies (x - 4)(x + 1) = 0

. This yields two real solutions for

x

:

x = 4

or

x = -1

. If

x = 4

, then

y = 1

and

\frac{x}{y} = \frac{4}{1} = 4

. If

x = -1

, then

y = -4

and

\frac{x}{y} = \frac{-1}{-4} = \frac{1}{4}

.

Because two distinct values for

\frac{x}{y}

(

4

and

\frac{1}{4}

) are possible, combined sufficiency fails.

Key Concept

Degree of non-linear algebraic systems and non-unique solutions in Data Sufficiency

Estimated Time:2m 0s

Question 3Question →

If $x$ and $y$ are real numbers, what is the value of $x + y$ ?

(1) $3x + 3y = 12$
(2) $x - y = 2$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

The correct option correctly identifies that Statement (1) alone is sufficient because dividing

3x + 3y = 12

by 3 yields

x + y = 4

, directly answering the question without needing to solve for

x

and

y

individually. Statement (2) alone leaves

x + y

undetermined.

Step-by-Step Solution

1

Analyze the target expression in the question stem.

The target to find is a single value for

x + y

.

Rephrasing the question stem clarifies what exact numerical value or algebraic combination is required.

2

Evaluate Statement (1) independently:

3x + 3y = 12

.

Factor out 3 to get

3(x + y) = 12

, then divide by 3 to obtain

x + y = 4

.

Statement (1) gives a unique value for the target expression

x + y

, so Statement (1) alone is sufficient.

3

Evaluate Statement (2) independently:

x - y = 2

.

If

x = 3

and

y = 1

, then

x - y = 2

and

x + y = 4

. If

x = 4

and

y = 2

, then

x - y = 2

and

x + y = 6

.

Multiple values for

x + y

are possible, so Statement (2) alone is not sufficient.

Key Concept

Question Stem Rephrasing and Linear Expression Combination

Estimated Time:1m 0s

Question 4Question →

If $x$ and $y$ are real numbers, what is the value of $x^3 - y^3$ ?

(1) $x - y = 2$
(2) $x^2 + y^2 = 10$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

Combining Statement (1) and Statement (2) allows us to determine the exact value of

xy = 3

. Substituting

x - y = 2

,

x^2 + y^2 = 10

, and

xy = 3

into the factored form

x^3 - y^3 = (x - y)(x^2 + xy + y^2)

yields

2 \times (10 + 3) = 26

. Because this value is uniquely determined, both statements together are sufficient while neither statement alone is sufficient.

Step-by-Step Solution

1

Rephrase the target expression using algebraic identities.

The target expression

x^3 - y^3

factors into

(x - y)(x^2 + xy + y^2)

.

To find

x^3 - y^3

, we need the values of

(x - y)

and

(x^2 + xy + y^2)

, or equivalently,

(x - y)

,

(x^2 + y^2)

, and

xy

.

2

Evaluate Statement (1) alone.

Statement (1) gives

x - y = 2

.

Without knowing

x^2 + y^2

or

xy

,

x^3 - y^3

can take infinitely many values (for example, if

x=2, y=0

,

x^3-y^3=8

; if

x=3, y=1

,

x^3-y^3=26

). Thus, Statement (1) alone is NOT sufficient.

3

Evaluate Statement (2) alone.

Statement (2) gives

x^2 + y^2 = 10

.

Without knowing

x - y

or

xy

,

x^3 - y^3

can take multiple values (for example, if

x=\sqrt{10}, y=0

,

x^3-y^3=10\sqrt{10}

; if

x=3, y=1

,

x^3-y^3=26

). Thus, Statement (2) alone is NOT sufficient.

4

Combine Statement (1) and Statement (2).

Square Statement (1):

(x - y)^2 = 2^2 \implies x^2 - 2xy + y^2 = 4

. Substitute

x^2 + y^2 = 10

:

10 - 2xy = 4 \implies 2xy = 6 \implies xy = 3

.

By squaring

x - y

and using

x^2 + y^2

, we isolate a unique value for the product term

xy

.

5

Substitute known values into the target expression.

x^3 - y^3 = (x - y)(x^2 + y^2 + xy) = 2 \times (10 + 3) = 26

.

Since the expression evaluates to a single, unique numerical value (

26

), the two statements together are sufficient.

Key Concept

Algebraic Expression Rephrasing and Solution Set Invariance in Data Sufficiency

Estimated Time:2m 0s

Question 5Question →

If $a$ and $b$ are real numbers, what is the value of $a^2 - b^2$ ?

(1) $a + b = 7$
(2) $a - b = 3$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

Both statements together are sufficient, but neither statement alone is sufficient.

Factoring the target expression yields

a^2 - b^2 = (a + b)(a - b)

. Neither statement alone gives both factors, so neither statement alone is sufficient. Combining Statement (1), which gives

a + b = 7

, and Statement (2), which gives

a - b = 3

, allows us to calculate

a^2 - b^2 = 7 \times 3 = 21

. Therefore, both statements together are sufficient.

Step-by-Step Solution

1

Rephrase the target question stem algebraically.

Recognize that

a^2 - b^2 = (a + b)(a - b)

. The objective is to find the value of the product

(a + b)(a - b)

.

Simplifying the target expression reduces the problem to finding the sum and difference of the variables.

2

Evaluate Statement (1) independently.

Statement (1) states

a + b = 7

. Knowing only

a + b

leaves

a - b

unknown, so

a^2 - b^2 = 7(a - b)

can take infinitely many values. Statement (1) is insufficient.

A single linear equation with two variables cannot determine the value of a two-factor product unless the second factor is fixed.

3

Evaluate Statement (2) independently.

Statement (2) states

a - b = 3

. Knowing only

a - b

leaves

a + b

unknown, so

a^2 - b^2 = 3(a + b)

can take infinitely many values. Statement (2) is insufficient.

A single difference equation does not uniquely determine the sum of two variables.

4

Evaluate Statements (1) and (2) together.

Combining both statements gives

a + b = 7

and

a - b = 3

. Substituting these values into the rephrased target expression yields

a^2 - b^2 = (7)(3) = 21

. A unique numerical value is obtained. Statements combined are sufficient.

Having both component factors allows direct computation of their product without needing to solve for individual values of

a

and

b

first.

Key Concept

Algebraic Rephrasing and Difference of Squares in Data Sufficiency

Estimated Time:1m 0s

Question 6Question →

What is the value of the real number $k$ ?

(1) $4k - 8 = 12$
(2) $k^2 = 25$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

Solving Statement (1) gives

4k = 20

, which simplifies to

k = 5

. Because this is a single, unique value, Statement (1) alone is sufficient. Statement (2) gives

k^2 = 25

, which has two valid solutions (

k = 5

or

k = -5

). Without additional constraints, Statement (2) alone is not sufficient to determine a single value for

k

. Therefore, Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.

Step-by-Step Solution

1

Evaluate Statement (1) independently

4k - 8 = 12 \implies 4k = 20 \implies k = 5

Solving the linear equation gives a single, unique value for

k

. Thus, Statement (1) alone is sufficient.

2

Evaluate Statement (2) independently

k^2 = 25 \implies k = 5

or

k = -5

Taking the square root yields two distinct possible values for

k

. Because

k

cannot be uniquely determined, Statement (2) alone is not sufficient.

3

Select the Data Sufficiency decision key

Statement (1) is sufficient, but Statement (2) is not.

This corresponds to the standard option where Statement (1) alone is sufficient.

Key Concept

Linear vs. Quadratic Degree Sufficiency in Data Sufficiency

Question 7Question →

If $x$ is a real number, what is the value of $(x + 3)^2$ ?

(1) $x^2 + 6x = 7$
(2) $x = 1$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

Answer: EACH statement ALONE is sufficient.

Answer

Each statement alone is sufficient.

Expanding the target expression

(x + 3)^2

gives

x^2 + 6x + 9

. Statement (1) provides

x^2 + 6x = 7

, which can be directly substituted into the expanded expression to get

7 + 9 = 16

, yielding a unique value. Statement (2) explicitly provides

x = 1

, which gives

(1 + 3)^2 = 16

. Because each statement independently provides enough information to determine a single value for the expression, the option stating that each statement alone is sufficient is correct.

Step-by-Step Solution

1

Rephrase the target expression using algebraic expansion.

(x + 3)^2 = x^2 + 6x + 9

Expanding the binomial allows direct comparison with given algebraic statements.

2

Evaluate Statement (1) using the expanded target expression.

Since

x^2 + 6x = 7

, substituting this into

x^2 + 6x + 9

gives

7 + 9 = 16

. Alternatively, solving

x^2 + 6x - 7 = 0

yields

x = 1

or

x = -7

. If

x = 1

,

(1 + 3)^2 = 16

. If

x = -7

,

(-7 + 3)^2 = (-4)^2 = 16

. Both roots yield the exact same value of

16

.

Statement (1) alone uniquely determines the target value.

3

Evaluate Statement (2) independently.

Given

x = 1

, substituting directly into the target expression gives

(1 + 3)^2 = 4^2 = 16

.

Statement (2) alone uniquely determines the target value.

4

Conclude sufficiency based on independent evaluations.

Both Statement (1) alone and Statement (2) alone are sufficient.

Since each statement independently determines a single value of 16, EACH statement ALONE is sufficient.

Key Concept

Algebraic Expression Expansion and Target Rephrasing in Data Sufficiency

Estimated Time:1m 0s

Question 8Question →

If $p$ and $q$ are numbers, what is the value of the product $pq$ ?

(1) $(p + q)^2 = 49$
(2) $p^2 + q^2 = 25$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

Both statements together are sufficient to determine that

pq = 12

, but neither statement alone is sufficient.

The correct option states that both statements together are sufficient, but neither alone is sufficient. Expanding

(p+q)^2 = p^2 + 2pq + q^2

allows substituting

(p+q)^2 = 49

from statement (1) and

p^2 + q^2 = 25

from statement (2), yielding

49 = 25 + 2pq

, which uniquely determines

pq = 12

. Neither statement alone isolates

pq

.

Step-by-Step Solution

1

Rephrase the target question using algebraic identities

Recall the identity

(p + q)^2 = p^2 + 2pq + q^2

. Rearranging gives

2pq = (p + q)^2 - (p^2 + q^2)

. Thus, knowing both

(p + q)^2

and

(p^2 + q^2)

will determine

pq

.

Target rephrasing simplifies evaluating statement sufficiency.

2

Evaluate Statement (1) independently

Statement (1) gives

(p + q)^2 = 49

. If

p = 7

and

q = 0

, then

pq = 0

. If

p = 4

and

q = 3

, then

(4+3)^2 = 49

and

pq = 12

. Multiple values for

pq

exist.

A statement is sufficient only if it yields one unique value for the target expression.

3

Evaluate Statement (2) independently

Statement (2) gives

p^2 + q^2 = 25

. If

p = 5

and

q = 0

, then

pq = 0

. If

p = 4

and

q = 3

, then

4^2 + 3^2 = 25

and

pq = 12

. Multiple values for

pq

exist.

Statement (2) alone does not yield a unique product.

4

Combine Statement (1) and Statement (2)

Substitute

(p + q)^2 = 49

and

p^2 + q^2 = 25

into

(p + q)^2 = p^2 + 2pq + q^2

to get

49 = 25 + 2pq 2pq = 24 pq = 12

. A single, unique value is found.

Combining the statements provides enough information to determine the value of the target expression

pq

uniquely.

Key Concept

Algebraic Expression Manipulation via Quadratic Identities

Question 9Question →

If $a$ and $b$ are non-zero real numbers, what is the value of $\frac{a}{b}$ ?

(1) $5a - 3b = 0$
(2) $a + b = 8$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

Statement (1) ALONE is sufficient to determine the value of

\frac{a}{b}

, but statement (2) alone is not sufficient.

Statement (1) can be rearranged algebraically to

5a = 3b

, which directly gives

\frac{a}{b} = \frac{3}{5}

, providing a single definitive value. Statement (2),

a + b = 8

, allows infinitely many pairs of

(a, b)

yielding different ratios for

\frac{a}{b}

. Therefore, Statement (1) alone is sufficient, while Statement (2) alone is not.

Step-by-Step Solution

1

Rephrase the question stem target

The target is to find a numerical value for the ratio

\frac{a}{b}

.

Simplifying the target helps determine if an equation directly yields the ratio without needing individual variable values.

2

Evaluate Statement (1) independently:

5a - 3b = 0

Rearranging gives

5a = 3b

. Dividing both sides by

5b

(since

b \neq 0

) yields

\frac{a}{b} = \frac{3}{5}

.

This yields a single, unique numerical value for the target expression. Thus, Statement (1) alone is sufficient.

3

Evaluate Statement (2) independently:

a + b = 8

If

a = 4

and

b = 4

, then

\frac{a}{b} = 1

. If

a = 2

and

b = 6

, then

\frac{a}{b} = \frac{1}{3}

.

Multiple values of the ratio

\frac{a}{b}

are possible. Thus, Statement (2) alone is not sufficient.

Key Concept

Determining ratios from homogeneous linear equations in Data Sufficiency

Question 10Question →

If $m$ and $n$ are non-zero real numbers, what is the value of $\frac{m^2 + n^2}{mn}$ ?

(1) $m^2 n + m n^2 = 6(m + n)$
(2) $m^2 - n^2 = 3(m - n)$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

Both statements together are sufficient, but neither statement alone is sufficient.

Evaluating each statement alone yields multiple possible numerical outcomes because factoring produces independent cases (

m+n=0

vs

mn=6

for the first statement, and

m-n=0

vs

m+n=3

for the second). However, when combining both statements, three of the four sub-cases fail: two violate non-zero or consistency conditions, and the sub-case where

mn=6

and

m+n=3

has a negative discriminant (

3^2 - 4(1)(6) = -15 < 0

), yielding no real solutions. This leaves only the sub-case

mn=6

and

m-n=0

, which uniquely determines the value of the target expression to be 2. Therefore, both statements together are sufficient.

Step-by-Step Solution

1

Rephrase the target expression.

The target expression

\frac{m^2 + n^2}{mn}

can be rewritten as

\frac{m}{n} + \frac{n}{m}

. We need a unique numerical value for this expression.

Simplifying the target helps evaluate what parameters or relationships are required.

2

Evaluate Statement (1) independently.

Rearrange Statement (1):

mn(m + n) - 6(m + n) = 0 \implies (mn - 6)(m + n) = 0

. This gives two cases: Case 1:

m + n = 0 \implies n = -m

. Since

m, n \neq 0

,

\frac{m^2 + (-m)^2}{m(-m)} = \frac{2m^2}{-m^2} = -2

. Case 2:

mn = 6

. For instance, if

m = 2, n = 3

, then

\frac{4+9}{6} = \frac{13}{6}

. Multiple values exist (

-2

and

\frac{13}{6}

), so Statement (1) alone is NOT sufficient.

Factoring instead of dividing by

(m+n)

preserves the root

m+n=0

.

3

Evaluate Statement (2) independently.

Rearrange Statement (2):

(m - n)(m + n) - 3(m - n) = 0 \implies (m - n)(m + n - 3) = 0

. This gives two cases: Case 1:

m - n = 0 \implies m = n

. Since

m, n \neq 0

,

\frac{m^2 + m^2}{m^2} = 2

. Case 2:

m + n = 3

. For instance, if

m = 1, n = 2

, then

\frac{1+4}{2} = \frac{5}{2}

. Multiple values exist (

2

and

\frac{5}{2}

), so Statement (2) alone is NOT sufficient.

Factoring preserves the root

m-n=0

.

4

Combine Statement (1) and Statement (2).

We test the four combinations of cases:
- Case A (

m + n = 0

) & Case X (

m - n = 0

): System gives

m = 0, n = 0

. Contradicts the condition that

m, n

are non-zero.
- Case A (

m + n = 0

) & Case Y (

m + n = 3

):

0 = 3

, impossible.
- Case B (

mn = 6

) & Case X (

m - n = 0

):

m = n \implies m^2 = 6 \implies m = n = \pm\sqrt{6}

. Here

\frac{m^2+n^2}{mn} = \frac{6+6}{6} = 2

.
- Case B (

mn = 6

) & Case Y (

m + n = 3

):

n = 3 - m \implies m(3-m) = 6 \implies m^2 - 3m + 6 = 0

. The discriminant is

b^2 - 4ac = 9 - 24 = -15 < 0

, which yields no real solutions for

m

and

n

.

Only Case B & Case X yields valid real non-zero solutions, providing a unique value of 2.

Checking real constraints eliminates non-real systems and isolates a single valid numerical outcome.

Key Concept

Factoring non-linear algebraic systems without illegal variable division, and verifying real-number constraints using discriminants in Data Sufficiency.

Estimated Time:2m 30s

Question 11Question →

If $x$ and $y$ are real numbers such that $x + y \neq 0$ , what is the value of $\frac{x^3 + y^3}{x + y}$ ?

(1) $x^2 - xy + y^2 = 12$
(2) $x^2 + y^2 = 20$ and $xy = 8$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

Answer: EACH statement ALONE is sufficient.

Answer

EACH statement ALONE is sufficient.

Factoring the numerator using the sum of cubes identity

x^3 + y^3 = (x + y)(x^2 - xy + y^2)

allows canceling

(x + y)

, since

x + y \neq 0

. Thus, the question asks for the value of

x^2 - xy + y^2

. Statement (1) directly states that

x^2 - xy + y^2 = 12

, which is sufficient. Statement (2) gives

x^2 + y^2 = 20

and

xy = 8

, so

x^2 - xy + y^2 = (x^2 + y^2) - xy = 20 - 8 = 12

, which is also sufficient. Since each statement independently yields a unique value, the correct choice states that each statement alone is sufficient.

Step-by-Step Solution

1

Simplify and rephrase the target expression in the question stem.

Since

x + y \neq 0

, factor the numerator using the sum of cubes formula

x^3 + y^3 = (x + y)(x^2 - xy + y^2)

. The expression simplifies to

\frac{(x + y)(x^2 - xy + y^2)}{x + y} = x^2 - xy + y^2

. The question target is equivalent to finding the value of

x^2 - xy + y^2

.

Simplifying the target expression before analyzing the statements eliminates unnecessary variables and reveals the exact algebraic value needed.

2

Evaluate Statement (1) independently.

Statement (1) gives

x^2 - xy + y^2 = 12

. Since this matches the simplified question target directly, the value is uniquely determined as 12.

Statement (1) provides the exact value of the rephrased target expression.

3

Evaluate Statement (2) independently.

Statement (2) provides

x^2 + y^2 = 20

and

xy = 8

. Substituting these into the target expression

x^2 - xy + y^2 = (x^2 + y^2) - xy

gives

20 - 8 = 12

. The value is uniquely determined as 12.

Statement (2) supplies component values that combine to form the target expression uniquely.

4

Determine the final Data Sufficiency decision.

Because Statement (1) alone is sufficient and Statement (2) alone is sufficient, the correct choice is that EACH statement ALONE is sufficient.

Both statements independently yield a single, consistent answer to the rephrased question.

Key Concept

Question Stem Rephrasing and Algebraic Identity Simplification in Data Sufficiency

Question 12Question →

If $x$ and $y$ are real numbers, what is the value of $x + y$ ?

(1) $x^2 y + xy^2 = 30$
(2) $x^2 + y^2 = 19 - xy$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

Statements (1) and (2) TOGETHER are NOT sufficient to determine a unique value for

x + y

.

Evaluating both statements together produces the polynomial relation

(x + y)^3 - 19(x + y) - 30 = 0

. Factoring gives

(x + y - 5)(x + y + 2)(x + y + 3) = 0

. Each of the three roots (

5

,

-2

,

-3

) produces real numbers

x

and

y

satisfying both original equations. Because

x + y

can take three different values, the two statements together are not sufficient to determine a unique value.

Step-by-Step Solution

1

Analyze Statement (1) independently

Factor Statement (1):

x^2 y + xy^2 = xy(x + y) = 30

.

If

(x, y) = (2, 3)

, then

xy(x + y) = 6(5) = 30

, giving

x + y = 5

. If

(x, y) = (1, 5)

, then

xy(x + y) = 5(6) = 30

, giving

x + y = 6

. Multiple values exist, so Statement (1) alone is insufficient.

2

Analyze Statement (2) independently

Rearrange Statement (2):

x^2 + y^2 + xy = 19

, which equals

(x + y)^2 - xy = 19

.

If

(x, y) = (3, 2)

, then

3^2 + 2^2 + 6 = 19

, giving

x + y = 5

. If

(x, y) = (-5, 3)

, then

(-5)^2 + 3^2 + (-15) = 19

, giving

x + y = -2

. Multiple values exist, so Statement (2) alone is insufficient.

3

Evaluate Statements (1) and (2) together

Express

xy

in terms of

S = x + y

: from Statement (1),

xy = \frac{30}{S}

; from Statement (2),

xy = S^2 - 19

.

Equating the two expressions for

xy

yields

\frac{30}{S} = S^2 - 19 \implies S^3 - 19S - 30 = 0

.

4

Solve the cubic polynomial for

S = x + y

and check for real roots

(x, y)

S^3 - 19S - 30 = (S - 5)(S + 2)(S + 3) = 0

, giving

S = 5

,

S = -2

, or

S = -3

.

For

S = 5

,

xy = 6

, giving real solutions

\{x, y\} = \{2, 3\}

. For

S = -2

,

xy = -15

, giving real solutions

\{x, y\} = \{-5, 3\}

. For

S = -3

,

xy = -10

, giving real solutions

\{x, y\} = \{-5, 2\}

. Since three distinct valid real sums exist, the combined statements are not sufficient.

Key Concept

Non-linear systems of equations may yield multiple distinct real solutions. Two equations in two variables do not guarantee a unique solution in Data Sufficiency unless the system is linear or constrained to a single root.

Question 13Question →

If $u$ and $v$ are real numbers such that $u + v \neq 0$ , what is the value of $\frac{u - v}{u + v}$ ?

(1) $u^2 - 3uv + 2v^2 = 0$
(2) $2u^2 - 5uv + 2v^2 = 0$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

The correct answer is the option stating that both statements together are sufficient, but neither alone is sufficient. Statement (1) yields two possible ratios (

u=v

giving 0, or

u=2v

giving

1/3

), so it is not sufficient. Statement (2) also yields two possible ratios (

u=v/2

giving

-1/3

, or

u=2v

giving

1/3

), so it is not sufficient. When combined,

u=v

and

u=v/2

would require

u=v=0

, which is forbidden by the condition

u+v \neq 0

. Therefore,

u=2v

is the only valid relation, producing a unique value of

1/3

.

Step-by-Step Solution

1

Rephrase the target question stem.

Dividing the numerator and denominator of

\frac{u - v}{u + v}

by

v

(assuming

v \neq 0

) shows that knowing the ratio

\frac{u}{v}

uniquely determines the value of the expression.

Simplifying the target expression reduces the problem to determining whether a unique ratio

\frac{u}{v}

exists.

2

Evaluate Statement (1) independently.

Factor

u^2 - 3uv + 2v^2 = 0

as

(u - v)(u - 2v) = 0

. This implies

u = v

or

u = 2v

. If

u = v

, then

\frac{u - v}{u + v} = \frac{0}{2v} = 0

. If

u = 2v

, then

\frac{u - v}{u + v} = \frac{2v - v}{2v + v} = \frac{v}{3v} = \frac{1}{3}

.

Since Statement (1) allows two distinct numerical values (

0

and

\frac{1}{3}

), Statement (1) alone is NOT sufficient.

3

Evaluate Statement (2) independently.

Factor

2u^2 - 5uv + 2v^2 = 0

as

(2u - v)(u - 2v) = 0

. This implies

u = \frac{1}{2}v

or

u = 2v

. If

u = \frac{1}{2}v

, then

\frac{u - v}{u + v} = \frac{\frac{1}{2}v - v}{\frac{1}{2}v + v} = \frac{-\frac{1}{2}v}{\frac{3}{2}v} = -\frac{1}{3}

. If

u = 2v

, then

\frac{u - v}{u + v} = \frac{1}{3}

.

Since Statement (2) allows two distinct numerical values (

-\frac{1}{3}

and

\frac{1}{3}

), Statement (2) alone is NOT sufficient.

4

Evaluate Statement (1) and Statement (2) combined.

Combine the conditions: Statement (1) requires

u = v

or

u = 2v

; Statement (2) requires

u = \frac{1}{2}v

or

u = 2v

. Testing

u = v

in Statement (2) yields

2v^2 - 5v^2 + 2v^2 = -v^2 = 0 \implies v = 0

, which implies

u = 0

, but

u + v \neq 0

rules out

u = v = 0

. Testing

u = \frac{1}{2}v

in Statement (1) similarly leads to

u = v = 0

. Thus, the only non-zero solution satisfying both equations is

u = 2v

, which gives a unique value of

\frac{1}{3}

.

Combining both statements eliminates the ambiguous cases that violate the stem constraint, establishing a unique value.

Key Concept

Homogeneous Quadratic Factoring and Constraint Evaluation in Data Sufficiency

Question 14Question →

If $u$ and $v$ are real numbers such that $u \neq v$ , what is the value of $\frac{u + v}{u - v}$ ?

(1) $u^2 - v^2 = 4(u - v)^2$
(2) $u^2 + v^2 = 5uv$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

Statement (1) allows factoring the difference of squares into

(u - v)(u + v) = 4(u - v)^2

. Because

u \neq v

, we can divide by

(u - v)

to determine directly that

\frac{u + v}{u - v} = 4

, which provides a unique value. Statement (2) gives

\left(\frac{u + v}{u - v}\right)^2 = \frac{7}{3}

, which yields two possible values (

\pm\sqrt{\frac{7}{3}}

) and is therefore not sufficient. Thus, Statement (1) alone is sufficient, but Statement (2) alone is not.

Step-by-Step Solution

1

Rephrase the target question and examine constraints.

The target expression is

\frac{u + v}{u - v}

. The stem specifies

u \neq v

, so

u - v \neq 0

, making the fraction well-defined.

Establishing non-zero denominator conditions allows division by

(u - v)

when evaluating statements.

2

Evaluate Statement (1) independently.

Rewrite the left side of

u^2 - v^2 = 4(u - v)^2

as

(u - v)(u + v) = 4(u - v)^2

. Since

u - v \neq 0

, divide both sides by

(u - v)

to get

u + v = 4(u - v)

. Divide both sides by

(u - v)

again to obtain

\frac{u + v}{u - v} = 4

.

This yields a single, definitive numerical value for the target expression. Statement (1) is sufficient.

3

Evaluate Statement (2) independently.

Expand

(u + v)^2 = u^2 + v^2 + 2uv

and

(u - v)^2 = u^2 + v^2 - 2uv

. Substituting

u^2 + v^2 = 5uv

gives

(u + v)^2 = 7uv

and

(u - v)^2 = 3uv

. Taking the ratio gives

\left(\frac{u + v}{u - v}\right)^2 = \frac{7uv}{3uv} = \frac{7}{3}

(since

uv \neq 0

). Taking the square root yields

\frac{u + v}{u - v} = \sqrt{\frac{7}{3}}

or

-\sqrt{\frac{7}{3}}

.

Because two distinct numerical values are possible, Statement (2) alone does not yield a unique answer. Statement (2) is not sufficient.

Key Concept

Factoring difference of squares and avoiding degree miscounts in quadratic ratios

Question 15Question →

If $x$ and $y$ are real numbers, what is the value of $x + y$ ?

(1) $x^2 + 2xy + y^2 - 5x - 5y + 6 = 0$
(2) $xy = 2$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

The correct response identifies that neither statement alone establishes a single value for

x + y

, but combining them eliminates

x + y = 2

because no real numbers

x

and

y

have a sum of

2

and a product of

2

(discriminant

2^2 - 4(2) = -4 < 0

). Only

x + y = 3

yields real values for

x

and

y

(discriminant

3^2 - 4(2) = 1 \ge 0

). Thus, both statements together are sufficient to determine that

x + y = 3

.

Step-by-Step Solution

1

Analyze Statement (1) by grouping terms in terms of

(x + y)

.

Rewrite

x^2 + 2xy + y^2 - 5(x + y) + 6 = 0

as

(x + y)^2 - 5(x + y) + 6 = 0

, which factors into

((x + y) - 2)((x + y) - 3) = 0

. Hence,

x + y = 2

or

x + y = 3

. Since there are two possible values, Statement (1) alone is NOT sufficient.

Determining whether Statement (1) provides a unique value for the target expression.

2

Analyze Statement (2) independently.

Statement (2) gives

xy = 2

. Multiple real pairs satisfy this (e.g.,

(2, 1) \Rightarrow x + y = 3

or

(1, 2) \Rightarrow x + y = 3

vs.

(\sqrt{2}, \sqrt{2}) \Rightarrow x + y = 2\sqrt{2}

). Thus, Statement (2) alone is NOT sufficient.

Testing whether

xy = 2

alone restricts

x + y

to a single numerical value.

3

Combine Statement (1) and Statement (2) under the real number constraint.

Let

S = x + y

and

P = xy = 2

. For

x

and

y

to be real numbers, they must be the real roots of

t^2 - St + P = 0

, requiring discriminant

\Delta = S^2 - 4P \ge 0

.
- Case 1: If

S = 2

, then

\Delta = 2^2 - 4(2) = -4 < 0

. No real solutions exist for

x

and

y

.
- Case 2: If

S = 3

, then

\Delta = 3^2 - 4(2) = 1 \ge 0

. Real solutions exist (e.g.,

x = 2, y = 1

).

Applying the real-number constraint to eliminate algebraically impossible cases.

4

Conclude total sufficiency.

Only

x + y = 3

produces real values for

x

and

y

. Therefore, both statements together uniquely determine

x + y = 3

.

Completing the Data Sufficiency evaluation.

Key Concept

Quadratic expression rephrasing and discriminant constraints for real variable existence in Data Sufficiency

Estimated Time:2m 0s

Question 16Question →

If $x$ and $y$ are real numbers, what is the value of $x^2 - y^2$ ?

(1) $x^3 - y^3 = 26$ and $x^2 + xy + y^2 = 13$
(2) $x + y = 4$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

Factoring the target expression gives

x^2 - y^2 = (x + y)(x - y)

. Statement (1) simplifies via the difference of cubes formula

x^3 - y^3 = (x - y)(x^2 + xy + y^2)

to

x - y = 2

and

xy = 3

. However, solving for

x + y

yields

(x + y)^2 = 16

, which gives two solutions (

x + y = 4

or

x + y = -4

) and thus two values for

x^2 - y^2

(

8

or

-8

). Thus, Statement (1) alone is insufficient. Statement (2) gives

x + y = 4

without constraining

x - y

, making it insufficient alone. Combining both statements provides

x + y = 4

and

x - y = 2

, giving a unique product of

8

. Hence, both statements together are sufficient.

Step-by-Step Solution

1

Rephrase the target expression using algebraic identities

x^2 - y^2 = (x + y)(x - y)

Determining

x^2 - y^2

requires knowing either the product

(x + y)(x - y)

or the individual values of

x + y

and

x - y

.

2

Evaluate Statement (1) independently

Using the difference of cubes identity

x^3 - y^3 = (x - y)(x^2 + xy + y^2)

, substitute the given values:

26 = (x - y)(13) \implies x - y = 2

. Next, expand

(x - y)^2 = x^2 - 2xy + y^2 = 4

. Subtracting this from

x^2 + xy + y^2 = 13

gives

3xy = 9 \implies xy = 3

. Then

(x + y)^2 = (x - y)^2 + 4xy = 4 + 4(3) = 16

, which implies

x + y = 4

or

x + y = -4

. Thus

x^2 - y^2 = (4)(2) = 8

or

(-4)(2) = -8

.

Because Statement (1) produces two distinct possible values (

8

and

-8

), Statement (1) ALONE is NOT sufficient.

3

Evaluate Statement (2) independently

x + y = 4

Without any information about

x - y

,

x^2 - y^2 = 4(x - y)

can take infinitely many values. Statement (2) ALONE is NOT sufficient.

4

Evaluate Statements (1) and (2) combined

From Statement (1),

x - y = 2

. From Statement (2),

x + y = 4

. Multiplying these equations yields

x^2 - y^2 = (x + y)(x - y) = (4)(2) = 8

.

A single, unique numerical value is determined. Therefore, BOTH statements TOGETHER are sufficient.

Key Concept

Algebraic Factoring and Non-Linear System Ambiguity in Data Sufficiency

Estimated Time:2m 0s

Question 17Question →

If $a$ and $b$ are real numbers, what is the value of $(a + b)^2$ ?

(1) $a^2 + b^2 + 4a - 6b = -13$
(2) $ab = -6$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

Statement (1) can be rewritten by completing the square as

(a + 2)^2 + (b - 3)^2 = 0

. Since

a

and

b

are constrained to be real numbers,

(a + 2)^2 \geq 0

and

(b - 3)^2 \geq 0

. The only way their sum can equal

0

is if

a = -2

and

b = 3

. This uniquely determines

(a + b)^2 = (-2 + 3)^2 = 1

, making Statement (1) alone sufficient. Statement (2) gives

ab = -6

, which allows multiple values for

(a + b)^2

(e.g.,

1

or

25

), making it insufficient. Hence, the option stating that Statement (1) alone is sufficient while Statement (2) alone is not is the correct choice.

Step-by-Step Solution

1

Analyze Statement (1):

a^2 + b^2 + 4a - 6b = -13

Rearrange and group terms:

(a^2 + 4a) + (b^2 - 6b) = -13

. Complete the square for both variables by adding

4

and

9

to both sides:

(a^2 + 4a + 4) + (b^2 - 6b + 9) = -13 + 4 + 9

, which simplifies to

(a + 2)^2 + (b - 3)^2 = 0

.

Grouping and completing the square reveals the sum-of-squares structure.

2

Evaluate the real number constraint on Statement (1)

Because

a

and

b

are real numbers,

(a + 2)^2 \geq 0

and

(b - 3)^2 \geq 0

. The sum of two non-negative terms can equal zero if and only if both terms are independently zero:

a + 2 = 0 \implies a = -2

and

b - 3 = 0 \implies b = 3

.

The sum of non-negative real squares equaling zero forces each squared term to be zero.

3

Calculate the target expression using values from Statement (1)

(a + b)^2 = (-2 + 3)^2 = (1)^2 = 1

. Since this yields a single unique value, Statement (1) ALONE is sufficient.

A unique value for the target expression establishes sufficiency.

4

Analyze Statement (2):

ab = -6

If

a = 2

and

b = -3

, then

ab = -6

and

(a + b)^2 = (2 - 3)^2 = 1

. If

a = 1

and

b = -6

, then

ab = -6

and

(a + b)^2 = (1 - 6)^2 = 25

. Multiple values are possible.

Testing specific valid cases proves that Statement (2) does not yield a unique result.

5

Conclude Data Sufficiency determination

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

Statement (1) produces a unique answer, whereas Statement (2) does not.

Key Concept

Completing the Square and Trivial Inequality for Real Squares in Data Sufficiency

Question 18Question →

If $x$ and $y$ are non-zero real numbers, what is the value of $\frac{x}{y}$ ?

(1) $x^2 y - x y^2 = 2xy$
(2) $x^2 + y^2 = 5xy$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

Statements (1) and (2) together are NOT sufficient.

The statement that both statements together are not sufficient is correct because combining Statement (1) (

x = y + 2

) and Statement (2) (

x^2 + y^2 = 5xy

) leads to a quadratic equation

3y^2 + 6y - 4 = 0

with two distinct real roots for

y

. Evaluating

\frac{x}{y} = 1 + \frac{2}{y}

for both roots yields two distinct possible values,

\frac{5 + \sqrt{21}}{2}

and

\frac{5 - \sqrt{21}}{2}

. Because a unique numerical value cannot be determined, the information provided is not sufficient.

Step-by-Step Solution

1

Evaluate Statement (1) independently.

Simplifies to

x - y = 2 \implies x = y + 2

.

Since

x, y \neq 0

,

xy \neq 0

. Dividing

x^2 y - x y^2 = 2xy

by

xy

yields

x - y = 2

. Thus

\frac{x}{y} = \frac{y+2}{y} = 1 + \frac{2}{y}

, which varies with

y

. Not sufficient.

2

Evaluate Statement (2) independently.

Yields

(\frac{x}{y})^2 - 5(\frac{x}{y}) + 1 = 0

.

Dividing

x^2 + y^2 = 5xy

by

y^2

gives a quadratic in

k = \frac{x}{y}

:

k^2 - 5k + 1 = 0

. Solving for

k

gives

k = \frac{5 \pm \sqrt{21}}{2}

. Because there are two distinct real solutions for the ratio, Statement (2) alone is not sufficient.

3

Evaluate Statements (1) and (2) combined.

Two distinct pairs of real numbers satisfy both conditions, giving two distinct values for

\frac{x}{y}

.

Substitute

x = y + 2

into

x^2 + y^2 = 5xy

:

(y+2)^2 + y^2 = 5y(y+2) \implies 3y^2 + 6y - 4 = 0

. This gives two real roots for

y

:

y = \frac{-3 \pm \sqrt{21}}{3}

. Substituting each

y

back into

\frac{x}{y} = 1 + \frac{2}{y}

yields

\frac{x}{y} = \frac{5 + \sqrt{21}}{2}

and

\frac{x}{y} = \frac{5 - \sqrt{21}}{2}

. Since two distinct ratios remain possible, combined statements are not sufficient.

Key Concept

Non-linear systems in Data Sufficiency often result in multiple valid solutions, requiring explicit verification of solution uniqueness rather than assuming two equations with two variables yield a single solution.

Estimated Time:2m 15s

Question 19Question →

If $p$ and $q$ are real numbers, what is the value of $p + q$ ?

(1) $p^2 q + p q^2 = 36$ and $p^2 + 2pq + q^2 = 9$
(2) $p^2 - q^2 = 15$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

Answer: Statement (1) ALONE is sufficient, but statement (2) alone is 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

yields two potential values for the sum:

p+q = 3

or

p+q = -3

. Combining each sum with

pq(p+q) = 36

gives corresponding products

pq = 12

and

pq = -12

. For real numbers

p

and

q

, the discriminant

S^2 - 4P

must be non-negative. The sum

p+q = 3

gives a negative discriminant

3^2 - 4(12) = -39

, which yields no real solutions for

p

and

q

. The sum

p+q = -3

gives a positive discriminant

(-3)^2 - 4(-12) = 57

, which yields valid real numbers. Thus,

p+q

uniquely equals

-3

.

Statement (2) alone is not sufficient because

p^2 - q^2 = (p-q)(p+q) = 15

allows infinitely many different values for

p+q

depending on

p-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

p^2 + 2pq + q^2 = 9

, we have

(p + q)^2 = 9

, which implies

p + q = 3

or

p + q = -3

. Factor the second equation:

p^2 q + p q^2 = pq(p + q) = 36

.

Factoring exposes the sum

S = p + q

and product

P = pq

structure of the algebraic system.

2

Evaluate the real number constraint for Case 1 (

p + q = 3

) in Statement (1).

If

p + q = 3

, then

pq(3) = 36 \implies pq = 12

. For real numbers

p

and

q

with sum

3

and product

12

, they must be real roots of

t^2 - 3t + 12 = 0

. The discriminant is

\Delta = (-3)^2 - 4(1)(12) = 9 - 48 = -39 < 0

.

A negative discriminant means no real numbers

p

and

q

exist for which

p + q = 3

and

pq = 12

. Thus,

p + q = 3

is impossible.

3

Evaluate the real number constraint for Case 2 (

p + q = -3

) in Statement (1).

If

p + q = -3

, then

pq(-3) = 36 \implies pq = -12

. The corresponding quadratic equation for

p

and

q

is

t^2 + 3t - 12 = 0

, with discriminant

\Delta = 3^2 - 4(1)(-12) = 57 > 0

. Real solutions exist.

Since a positive discriminant guarantees real solutions for

p

and

q

,

p + q

must uniquely equal

-3

. Therefore, Statement (1) alone is sufficient.

4

Evaluate Statement (2) independently.

p^2 - q^2 = 15 \implies (p - q)(p + q) = 15

. For example, if

p = 4, q = 1

, then

p + q = 5

. If

p = 8, q = 7

, then

p + q = 15

.

Without knowing

p - q

, infinitely many pairs of real numbers satisfy

p^2 - q^2 = 15

with different sums

p + q

. Statement (2) alone is not sufficient.

Key Concept

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

Question 20Question →

If $a$ and $b$ are positive real numbers, what is the value of $a^2 + b^2$ ?

(1) $a^4 - b^4 = 5(a^2 - b^2)$
(2) $ab = 6$ and $a + b = 5$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

Statement (1) simplifies to

(a^2 - b^2)(a^2 + b^2 - 5) = 0

, which allows

a^2 + b^2 = 5

when

a \neq b

, but allows infinitely many values for

a^2 + b^2

when

a = b

. Hence, Statement (1) alone is insufficient. Statement (2) provides

a + b = 5

and

ab = 6

, allowing us to use

(a + b)^2 = a^2 + b^2 + 2ab

to find

a^2 + b^2 = 25 - 12 = 13

uniquely. Therefore, Statement (2) alone is sufficient.

Step-by-Step Solution

1

Analyze Statement (1) algebraically.

Factor the left side as

(a^2 - b^2)(a^2 + b^2) = 5(a^2 - b^2)

. Rearranging yields

(a^2 - b^2)(a^2 + b^2 - 5) = 0

.

Dividing by

(a^2 - b^2)

is valid only if

a \neq b

.

2

Evaluate sufficiency of Statement (1).

If

a \neq b

, then

a^2 + b^2 = 5

. However, if

a = b > 0

, then

a^2 - b^2 = 0

, which makes the equation true for any positive value of

a

, meaning

a^2 + b^2 = 2a^2

can take infinitely many values (e.g., if

a=b=1

,

a^2+b^2=2

; if

a=b=3

,

a^2+b^2=18

). Thus, a unique value cannot be determined.

Multiple possible values mean Statement (1) is not sufficient.

3

Evaluate sufficiency of Statement (2).

Use the algebraic identity

(a + b)^2 = a^2 + 2ab + b^2

. Substituting

a + b = 5

and

ab = 6

gives

5^2 = a^2 + 2(6) + b^2

, so

25 = a^2 + b^2 + 12 \implies a^2 + b^2 = 13

.

This determines a single, unique numerical value for

a^2 + b^2

.

4

Conclude overall sufficiency.

Statement (1) is NOT sufficient, but Statement (2) IS sufficient.

Statement (2) alone resolves the target question.

Key Concept

Algebraic Equations and Systems in Data Sufficiency

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