### All GMAT Math Resources

## Example Questions

### Example Question #21 : Graphing

Let be a positive integer.

Evaluate .

Statement 1: is a multiple of 3.

Statement 2: is a multiple of 7.

**Possible Answers:**

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.

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.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

Assume that both statements are true. The value of , a positive integer, is equal to , where is the remainder of the division of by 4, so we can use this fact to show that insufficient information is provided.

Case 1: .

, so

Case 2: .

, so

In both cases, both statements are true, but the value of differs.

### Example Question #22 : Graphing

True or false: and are in the same quadrant of the rectangular coordinate plane.

Statement 1: and are of different sign.

Statement 2: and are of the same sign.

**Possible Answers:**

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.

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

Two points in the same quadrant have -coordinates of the same sign and -coordinates of the same sign.

It is possible for two points fitting the condition of Statement 1 to be in the same quadrant; and are two such points. However, it is also possible for two such points to be in different quadrants; and are two such points. Therefore, Statement 1 alone gives insufficient information. By the same argument, Statement 2 alone gives insufficient information.

Assume both statements are true. and are of different sign by Statement 1. By Statement 2, and are of the same sign; therefore, they are both of the same sign as and the sign opposite that of , or vice versa. Therefore, in one ordered pair, both numbers are positive or both are negative, and in the other ordered pair, one number is positive and the other is negative. The two ordered pairs cannot represent points in the same quadrant.

### Example Question #23 : Graphing

In which quadrant is the point located: I, II, III, or IV?

Statement 1:

Statement 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 insufficient to answer the question.

**Correct answer:**

Assume Statement 1 alone. The set of points that satisfy the equation is the set of all points on the line of the equation

,

which will pass through at least two quadrants on the coordinate plane. Therefore, Statement 1 provides insufficient information. By the same argument, Statement 2 is also insuffcient.

Now assume both statements to be true. The two statements together form a system of linear equations which can be solved using the elimination method:

Now, substitute back:

The point is , which has a positive -coordinate and a negative -coordinate and is consequently in Quadrant IV.

### Example Question #24 : Graphing

In which quadrant is the point located: I, II, III, or IV?

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:**

Assume Statement 1 alone. The points and each satisfy the condition of the statement; however, the former is in Quadrant I, having a positive -coordinate and a positive -coordinate; the latter is in Quadrant IV, having a positive -coordinate and a negative -coordinate.

Assume Statement 2 alone. The points and each satisfy the condition of the statement, since . However, the former is in Quadrant IV, having a positive -coordinate and a negative -coordinate; the latter is in Quadrant II, having a negative -coordinate and a positive -coordinate.

Assume both statements to be true. Statement 2 can be rewritten as ; since is positive from Statement 1, is negative. Since the point has a positive -coordinate and a negative -coordinate, it is in Quadrant IV.

### Example Question #25 : Graphing

True or false: , , and are collinear points.

Statement 1:

Statement 2: and

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

Assume Statement 1 alone.

The proportion statement

can be rewritten by setting the reciprocals of the expressions equal:

The first expression is the slope of the line through and ; the second is the slope of the line through and . Since the slopes are equal, the three points are on the same line - collinear.

The three points cannot be assumed to be collinear from Statement 2 alone. For example, , , and collectively fit the condition of Statement 2, and all three points are easily seen to be on the line of the equation . However, , , and collectively fit the condition of Statement 2, and while the line through the first two points is again , is off that line, so the three points are noncollinear.

### Example Question #26 : Graphing

True or false: and are in the same quadrant of the rectangular coordinate plane.

Statement 1: and are of different sign.

Statement 2: and are of different sign.

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

Two points in the same quadrant have -coordinates of the same sign and -coordinates of the same sign; however, from Statement 1 alone, we find that the -coordinates of the points have different signs, and from Statement 2 alone, we find that this holds for the -coordinates. Therefore, from either statement alone, the points can be proved to be in different quadrants.

### Example Question #27 : Graphing

In which quadrant is the point located: I, II, III, or IV?

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:**

Assume Statement 1 alone. The set of points that satisfy the equation is the set of all points on the line of the equation

which will pass through at least two quadrants on the coordinate plane. Therefore, Statement 1 provides insufficient information.

Now assume Statement 2 alone. The set of points that satisfy the equation is the set of all points of the circle of the equation

This circle has as its center and as its radius. Since its center is , which is 5 units away from its closest axis, and the radius is less than 5 units, the circle never intersects an axis, so it is contained entirely within the same quadrant as its center. The center has negative - and -coordinates, placing it, and the entire circle, in Quadrant III.

### Example Question #28 : Graphing

In which quadrant is the point located: I, II, III, or IV?

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:**

BOTH statements TOGETHER are insufficient to answer the question.

Assume both statements. The points and each satisfy the conditions of both statements, since , , and . The former is in Quadrant I, having a positive -coordinate and a positive -coordinate; the latter is in Quadrant IV, having a positive -coordinate and a negative -coordinate.

### Example Question #29 : Graphing

True or false: , , and are collinear points.

Statement 1: and

Statement 2:

**Possible Answers:**

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.

Assume Statement 1 alone. The equations can be rewritten as follows:

The - and -coordinates of are the arithmetic means of those of and , so is the midpoint of the segment with those endpoints. Therefore, the three points are collinear.

Assume Statement 2 alone. The statement can be rewritten as follows:

The first expression is the slope of the line through and ; the second expression is the slope of the line through and . Since the slopes are equal, the three points are collinear.

### Example Question #30 : Graphing

What quadrant contains the point , where ?

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:**

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 alone tells you that and are of the same sign, so the point is in Quadrant I (both positive) or Quadrant III (both negative).

Statement 2 tells you that any of the following hold:

is positive and is negative - example:

is negative and is negative - example:

is positive and is positive - example:

This places the point in any quadrant except Quadrant II (where is negative and is positive).

The two statements together only eliminate two quadrants and leave both Quadrant I and Quadrant III as possibilities.