# GMAT Math : Coordinate Geometry

## Example Questions

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### 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.

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 insufficient 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.

Explanation:

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 #2 : 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:

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.

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 sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

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:

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.

EITHER statement ALONE is 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.

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

Explanation:

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 #4 : 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:

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.

EITHER statement ALONE is 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.

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

Explanation:

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:

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.

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.

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.

Explanation:

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:  .

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.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 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.

Explanation:

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.

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.

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.

Explanation:

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:

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.

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.

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,

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.

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.

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.

Explanation:

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 #1 : Dsq: Graphing Complex Numbers

Let  and  be real numbers.

What is the product of  and its complex conjugate?

Statement 1:

Statement 2:

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.

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.

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  is , and

.

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

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