GMAT Math : Graphing

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.

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.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

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.

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.

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

Explanation:

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:

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 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.

Explanation:

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:

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.

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:

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

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.

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

Explanation:

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.

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.

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.

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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:

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.

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.

Explanation:

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:

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.

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 insufficient to answer the question.

Explanation:

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:

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.

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.

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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:

BOTH statements TOGETHER are insufficient 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 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.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

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.

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