### All GMAT Math Resources

## Example Questions

### Example Question #1 : Coordinate Geometry

Define a function as follows:

for nonzero real numbers .

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:**

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.

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.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

Since only positive numbers have logarithms, the expression must be positive, so

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

.

In order to determine which side of the -axis the vertical asymptote falls, it is necessary to find the sign of ; if it is negative, it is on the left side, if it is positive, it is on the right side.

Assume both statements are true. By Statement 1, is positive. If is positive, then is negative, and vice versa. However, Statement 2, which mentions , does not give its actual sign - just the fact that its sign is the opposite of that of , which we are not given either. The two statements therefore give insufficient information.

### Example Question #1 : Coordinate Geometry

Define a function as follows:

for nonzero real numbers .

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

Statement 1:

Statement 2:

**Possible Answers:**

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.

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.

**Correct answer:**

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

Since a logarithm of a nonpositive number cannot be taken,

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

.

Each of Statement 1 and Statement 2 gives us only one of and . However, the two together tell us that

making the vertical asymptote

.

### Example Question #3 : Coordinate Geometry

Define a function as follows:

for nonzero real numbers .

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:

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

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.

**Correct answer:**

Only positive numbers have logarithms, so:

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

.

In order to determine which side of the -axis the vertical asymptote falls, it is necessary to find out whether the signs of and are the same or different. If and are of the same sign, then their quotient is positive, and is negative, putting on the left side of the -axis. If and are of different sign, then their quotient is negative, and is positive, putting on the right side of the -axis.

Statement 1 alone does not give us enough information to determine whether and have different signs. , for example, but , also.

From Statement 2, since the product of and is negative, they must be of different sign. Therefore, is positive, and falls to the right of the -axis.

### Example Question #1 : Coordinate Geometry

Define a function as follows:

for nonzero real numbers .

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

Statement 1:

Statement 2:

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

Only positive numbers have logarithms, so:

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

.

Statement 1 alone gives that . is the reciprocal of this, or , and , so the vertical asymptote is .

Statement 2 alone gives no clue about either , , or their relationship.

### Example Question #5 : Coordinate Geometry

Define a function as follows:

for nonzero real numbers .

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

Statement 1:

Statement 2:

**Possible Answers:**

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.

Since only positive numbers have logarithms,

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

.

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 or .

### Example Question #6 : Coordinate Geometry

Define a function as follows:

for nonzero real numbers .

Does the graph of have a -intercept?

Statement 1: .

Statement 2: .

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

The -intercept of the graph of the function , if there is one, occurs at the point with -coordinate 0. Therefore, we find :

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.

### Example Question #7 : Coordinate Geometry

Define a function as follows:

for nonzero real numbers .

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.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

Since only positive numbers have logarithms,

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

.

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

### Example Question #8 : Coordinate Geometry

Define a function as follows:

for nonzero real numbers .

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

Statement 1: and are of opposite sign.

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

Since only positive numbers have logarithms,

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

.

In order to determine which side of the -axis the vertical asymptote falls, it is necessary to find the sign of ; 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 , and, consequently, . 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 is positive, and is negative, so the vertical asymptote is to the left of the -axis.

### Example Question #9 : Coordinate Geometry

Define a function as follows:

for nonzero real numbers .

Does the graph of have a -intercept?

Statement 1: .

Statement 2: and have different signs.

**Possible Answers:**

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.

The -intercept of the graph of the function , if there is one, occurs at the point with -coordinate 0. Therefore, we find :

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.

### Example Question #641 : Data Sufficiency Questions

Let and be real numbers.

What is the product of and its complex conjugate?

Statement 1:

Statement 2:

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

The complex conjugate of an imaginary number is , and

.

Therefore, Statement 1 alone, which gives that , provides sufficient information to answer the question, whereas Statement 2 provides unhelpful information.