If x and y are real numbers, is x^2 + y^2 < 25? (1) x + y = 7 (2) (x - 3)^2 + (y - 4)^2 = … — GMAT

If $x$ and $y$ are real numbers, is $x^2 + y^2 < 25$ ?

(1) $x + y = 7$
(2) $(x - 3)^2 + (y - 4)^2 = 0$

A
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.Cevap
C
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D
EACH statement ALONE is sufficient.
E
Statements (1) and (2) TOGETHER are NOT sufficient.

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

Statement (1) allows multiple outcomes for

x^2 + y^2

(both values less than 25 and values greater than or equal to 25), so it is not sufficient. Statement (2) forces

x=3

and

y=4

because the sum of non-negative real squares can only be zero when each square term is zero. Substituting

x=3

and

y=4

yields

x^2 + y^2 = 25

, which conclusively answers 'No' to the question stem (

25 < 25

is false). Hence, Statement (2) alone is sufficient.

Evaluate Statement (1) independently

Statement (1) gives

x + y = 7

. If

x = 3.5

and

y = 3.5

, then

x^2 + y^2 = 12.25 + 12.25 = 24.5 < 25

(Yes). If

x = 7

and

y = 0

, then

x^2 + y^2 = 49 + 0 = 49 \not< 25

(No).

Since both 'Yes' and 'No' are possible, Statement (1) alone is insufficient.

Evaluate Statement (2) independently without carrying over Statement (1)

Statement (2) states

(x - 3)^2 + (y - 4)^2 = 0

. Since the square of a real number is non-negative, the sum of two squared terms can equal zero if and only if each term is zero:

x - 3 = 0 \implies x = 3

and

y - 4 = 0 \implies y = 4

This uniquely fixes the values of

x

and

y

Calculate the target expression using values from Statement (2)

Substitute

x = 3

and

y = 4

into

x^2 + y^2

3^2 + 4^2 = 9 + 16 = 25

. The question asks if

x^2 + y^2 < 25

. Since

25 < 25

is false, the answer is a definitive 'No'.

A definitive 'No' answer means the statement provides enough information to answer the question, so Statement (2) alone is sufficient.

Statement Independence in Data Sufficiency and Definitive Yes/No Decision Rules

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