### All GMAT Math Resources

## Example Questions

### Example Question #1 : Dsq: Calculating The Slope Of A Line

Is the slope of the line positve, negative, zero, or undefined?

Statement 1:

Statement 2:

**Possible Answers:**

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.

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

, in slope-intercept form, is

Therefore, the sign of is the sign of the slope.

The first statement means that is positive - all that means is that both and are nonzero and of like sign. can be either positive or negative, and consequently, so can slope .

The second statement - that is positive - makes , the sign of the slope, negative.

### Example Question #1 : Dsq: Calculating The Slope Of A Line

Does a given line with intercepts have positive slope or negative slope?

Statement 1:

Statement 2:

**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 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 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.

The slope of a line through and is

From Statement 1 alone, we can tell that

,

so we know the sign of the slope.

From Statement 2 alone, we can tell that

But this can be positive or negative - for example:

but

### Example Question #2 : Dsq: Calculating The Slope Of A Line

Does a given line with intercepts have positive slope or negative slope?

Statement 1:

Statement 2:

**Possible Answers:**

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.

**Correct answer:**

The slope of a line through and is

If and have the same sign, then , making the slope negative; if and have the same sign, then , making the slope positive.

Statement 1 is not enough to determine the sign of .

Case 1:

Case 2:

So if we only know Statement 1, we do not know whether and have the same sign, and, subsequently, we do not know the sign of slope . A similar argument can be made that Statement 2 provides insufficient information.

If we know both statements, we can solve the system of equations as follows:

Therefore, we know and have unlike sign and .

### Example Question #4 : Dsq: Calculating The Slope Of A Line

Does a given line with intercepts have positive slope or negative slope?

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.

The slope of a line through and is

If and have the same sign, then , making the slope negative; if and have the same sign, then , making the slope positive.

If we know both statements, we try to solve the system of equations as follows:

This means that the system is dependent, and that the statements are essentially the same.

Case 1:

Then

Case 2:

Then

Thus from Statement 1 alone, it cannot be determined whether and have the same sign, and the sign of the slope cannot be determined. Since Statement 2 is equivalent to Statement 1, the same holds of this statement, as well as both statements together.

### Example Question #3 : Dsq: Calculating The Slope Of A Line

You are given two lines. Are they perpendicular?

Statement 1: The sum of their slopes is .

Statement 2: They have the same slope.

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

Statement 2 alone tells us that the lines are parallel, not perpendicular. Statement 1 alone is neither necessary nor helpful, as the sum of the slopes is irrelevant.

### Example Question #6 : Dsq: Calculating The Slope Of A Line

A line includes points and . Is the slope of the line positive, negative, zero, or undefined?

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:**

The slope of the line that includes points and is .

For the question of the sign of the slope to be answered, it must be known whether and are of the same sign or of different signs, or whether one of them is equal to zero.

Statement 1 alone does not answer this question, as it only states that the denominator is greater; it is possible for this to happen whether both are of like sign or unlike sign. Statement 2 only proves that - that is, that the denominator is positive.

If the two statements together are assumed, we know that . Since both the numerator and the denominator are positive, the slope of the line must be positive.

### Example Question #7 : Dsq: Calculating The Slope Of A Line

Is the slope of a line on the coordinate plane positive, zero, negative, or undefined?

Statement 1: The line is perpendicular to the -axis.

Statement 2: The line has no -intercept.

**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.

The -axis is horizontal, so any line perpendicular to it is vertical and has undefined slope. Statement 1 is sufficient.

A line on the coordinate plane with no -intercept does not intersect the -axis and therefore must be parallel to it - subsequently, it must be vertical and have undefined slope. This makes Statement 2 sufficient.

### Example Question #8 : Dsq: Calculating The Slope Of A Line

Is the slope of a line on the coordinate plane positive, zero, negative, or undefined?

Statement 1: It includes the origin.

Statement 2: It passes through Quadrant II.

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

Infinitely many lines pass through the origin, and infinitely many lines pass through each quadrant, so neither statement alone is sufficient to answer the question.

Suppose that both statements are known to be true. Since the line passes through quadrant II, it passes through a point , where are positive. It also passes through so its slope will be

which is a negative slope.

### Example Question #4 : Dsq: Calculating The Slope Of A Line

A line is on the coordinate plane. What is its slope?

Statement 1: The line is parallel to the line of the equation .

Statement 2: The line is perpendicular to the line of the equation .

**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.

If Statement 1 alone holds - that is, if it is known only that the line is parallel to the line of - then this equation can be rewritten in slope-intercept form. The slope can be deduced from this, and the two lines, being parallel, will have the same slope.

If Statement 2 alone holds - that is, if it is known only that the line is perpendicular to the line of the equation - then this equation can be rewritten in slope-intercept form. The slope can be deduced from this, and the first line, which is perpendicular to this one, will have the slope that is the opposite of the reciprocal of that.

Either statement alone will yield an answer.

### Example Question #10 : Dsq: Calculating The Slope Of A Line

Is the slope of a line on the coordinate plane positive, zero, negative, or undefined?

Statement 1: The line contains points in both Quadrant I and Quadrant II.

Statement 2: The line contains points in both Quadrant I and Quadrant III.

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

Examine the diagram below.

It can be seen from the red lines that no conclusions about the sign of the slope of a line can be drawn from Statement 1, since lines of positive, negative, and zero slope can contain points in both Quadrant I and Quadrant II.

If a line contains a point in Quadrant I and a point in Quadrant III, then it contains a point with positive coordinates and a point with negative coordinates ; its slope is

which is a positive slope.

Therefore, Statement 2 alone, but not Statement 1 alone, provides a definitive answer.

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