GMAT Math : Data-Sufficiency Questions

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Example Questions

Example Question #5 : Dsq: Graphing A Logarithm

Define a function as follows:

$f(x) = A \ln (Bx + C) +D$

for nonzero real numbers A,B,C,D .

Give the equation of the vertical asymptote of the graph of .

Statement 1: B+C = 7

Statement 2: BC = 12

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Since only positive numbers have logarithms,

Bx+C > 0

Bx > -C

$x > - \frac{C}{B}$

Therefore, the vertical asymptote must be the vertical line of the equation

$x = - \frac{C}{B}$ .

Assume both statements to be true. We need two numbers and whose sum is 7 and whose product is 12; by trial and error, we can find these numbers to be 3 and 4. However, without further information, we have no way of determining which of and is 3 and which is 4, so the asymptote can be either $x = - \frac{3}{4}$ or $x = - \frac{4}{3}$ .

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Example Question #6 : Dsq: Graphing A Logarithm

Define a function as follows:

$f(x) = A \ln (Bx + C) +D$

for nonzero real numbers A,B,C,D .

Does the graph of have a -intercept?

Statement 1: C = 8 .

Statement 2: D = 9 .

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

The -intercept of the graph of the function $f(x) = A \ln (Bx + C) +D$ , if there is one, occurs at the point with -coordinate 0. Therefore, we find f(0) :

$f(0) = A \ln (B \cdot 0 + C) +D = A \ln C + D$

This expression is defined if and only if is a positive value. Statement 1 gives as positive, so it follows that the graph indeed has a -intercept. Statement 2, which only gives , is irrelevant.

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Example Question #7 : Dsq: Graphing A Logarithm

Define a function as follows:

$f(x) = A \ln (Bx + C) +D$

for nonzero real numbers A,B,C,D .

Where is the vertical asymptote of the graph of in relation to the -axis - is it to the left of it, to the right of it, or on it?

Statement 1: and are both positive.

Statement 2: and are of opposite sign.

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

Since only positive numbers have logarithms,

Bx+C > 0

Bx > -C

$x > - \frac{C}{B}$

Therefore, the vertical asymptote must be the vertical line of the equation

$x = - \frac{C}{B}$ .

Statement 1 gives irrelevant information. But Statement 2 alone gives sufficient information; since and are of opposite sign, their quotient $\frac{C}{B}$ is negative, and $- \frac{C}{B}$ is positive. This locates the vertical asymptote on the right side of the -axis.

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Example Question #2 : Graphing

Define a function as follows:

$f(x) = A \ln (Bx + C) +D$

for nonzero real numbers A,B,C,D .

What is the equation of the vertical asymptote of the graph of ?

Statement 1: and are of opposite sign.

Statement 2: A+C = 0

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Since only positive numbers have logarithms,

Bx+C > 0

Bx > -C

$x > - \frac{C}{B}$

Therefore, the vertical asymptote must be the vertical line of the equation

$x = - \frac{C}{B}$ .

In order to determine which side of the $y\:$ -axis the vertical asymptote falls, it is necessary to find the sign of $- \frac{C}{B}$ ; if it is negative, it is on the left side, and if it is positive, it is on the right side.

Statement 1 alone only gives us that is a different sign from ; without any information about the sign of , we cannot answer the question.

Statement 2 alone gives us that A+C = 0 , and, consequently, A = -C . This means that and are of opposite sign. But again, with no information about the sign of , we cannot answer the question.

Assume both statements to be true. Since, from the two statements, both and are of the opposite sign from , and are of the same sign. Their quotient $\frac{C}{B}$ is positive, and $- \frac{C}{B}$ is negative, so the vertical asymptote $x = - \frac{C}{B}$ is to the left of the -axis.

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Example Question #3 : Coordinate Geometry

Define a function as follows:

$f(x) = A \ln (Bx + C) +D$

for nonzero real numbers A,B,C,D .

Does the graph of have a -intercept?

Statement 1: A> D .

Statement 2: and have different signs.

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

The -intercept of the graph of the function $f(x) = A \ln (Bx + C) +D$ , if there is one, occurs at the point with -coordinate 0. Therefore, we find f(0) :

$f(0) = A \ln (B \cdot 0 + C) +D = A \ln C + D$

This expression is defined if and only if is a positive value. However, the two statements together do not give this information; the values of and from Statement 1 are irrelevant, and Statement 2 does not reveal which of and is positive and which is negative.

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Example Question #2 : Graphing

Let and be real numbers.

What is the product of a + bi and its complex conjugate?

Statement 1: $a^{2}+b^{2}= 71$

Statement 2: $a^{2}-b^{2}= 39$

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

The complex conjugate of an imaginary number a + bi is a - bi , and

$(a + bi)(a - bi) = a^{2}+b^{2}$ .

Therefore, Statement 1 alone, which gives that $a^{2}+b^{2} = 71$ , provides sufficient information to answer the question, whereas Statement 2 provides unhelpful information.

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Example Question #1 : Dsq: Graphing Complex Numbers

Let and be real numbers.

From the number a + bi , subtract its complex conjugate. What is the result?

Statement 1: a = 7

Statement 2: b = 12

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

The complex conjugate of an imaginary number a + bi is a - bi , and

(a + bi)- (a - bi) = a -a +bi+bi = 2bi .

Therefore, it is necessary and sufficient to know in order to answer the question. Statement 1 does not give this value, and is unhelpful here; Statement 2 does give this value.

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Example Question #2 : Dsq: Graphing Complex Numbers

Let and be real numbers.

What is the sum of a + bi and its complex conjugate?

Statement 1: a = 12

Statement 2: b= 18

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

The complex conjugate of an imaginary number a + bi is a - bi , and the sum of the two numbers is

(a + bi)+ (a - bi) = 2a .

Therefore, it is necessary and sufficient to know in order to answer the question. Statement 1 alone gives this information; Statement 2 does not, and it is unhelpful.

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Example Question #3 : Dsq: Graphing Complex Numbers

Let be a positive integer.

True or false: $i^{N} = 1$

Statement 1: is a prime number.

Statement 2: is a two-digit number ending in a 7.

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

If is a positive integer, then $i^{N} = 1$ if and only if is a multiple of 4.

It follows that if $i^{N} = 1$ , cannot be a prime number. Also, every multiple of 4 is even, so as an even number, cannot end in 7. Contrapositively, if Statement 1 is true and is prime, or if Statement 2 is true and if ends in 7, it follows that is not a multiple of 4, and $i^{N} \ne 1$ .

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Example Question #4 : Dsq: Graphing Complex Numbers

Let and be real numbers.

What is the product of a + bi and its complex conjugate?

Statement 1: a = 12

Statement 2: $b = \sqrt{2}$

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

The complex conjugate of an imaginary number a + bi is a - bi , and

$(a + bi)(a - bi) = a^{2}+b^{2}$ .

Therefore, it is necessary and sufficient to know the values of both and in order to answer the problem. Each statement alone gives only one of these values, so each statement alone provides insufficient information; the two together give both, so the two statements together provide sufficient information.

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GMAT Math : Data-Sufficiency Questions

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #5 : Dsq: Graphing A Logarithm

Example Question #6 : Dsq: Graphing A Logarithm

Example Question #7 : Dsq: Graphing A Logarithm

Example Question #2 : Graphing

Example Question #3 : Coordinate Geometry

Example Question #2 : Graphing

Example Question #1 : Dsq: Graphing Complex Numbers

Example Question #2 : Dsq: Graphing Complex Numbers

Example Question #3 : Dsq: Graphing Complex Numbers

Example Question #4 : Dsq: Graphing Complex Numbers

All GMAT Math Resources

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