Test: Coordinate Geometry - GMAT MCQ

Statement (1): Line L has a positive slope.

A positive slope indicates that the line is increasing as you move from left to right in the xy-plane. However, it doesn't provide specific information about whether the line passes through Quadrant II or not. There are lines with positive slopes that do not cross into Quadrant II. Therefore, Statement (1) alone is not sufficient to answer the question.

Statement (2): At its closest approach, Line L is never closer than 3 units to the origin.

This statement provides information about the distance between Line L and the origin but doesn't directly tell us if Line L passes through Quadrant II. It only gives a constraint on the distance, but the line can still have various orientations and pass through different quadrants. Therefore, Statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we still don't have enough information to determine if Line L passes through Quadrant II. While Statement (1) tells us that the slope is positive, it doesn't provide information about the intercepts or the specific direction of the line. Statement (2) provides a constraint on the distance to the origin but doesn't give a conclusive answer about the quadrant. Therefore, Statements (1) and (2) together are not sufficient to answer the question.

In conclusion, neither statement alone nor both statements together provide enough information to determine if Line L passes through Quadrant II. Therefore, the answer is E: Statements (1) and (2) together are not sufficient to answer the question asked, and additional data are needed.

To determine the value of b in the equation ax + by + c = 0, where abc ≠ 0, given that the line has a slope of -3, let's analyze the given statements:

Statement (1): a = 2

In the equation ax + by + c = 0, the coefficient of x is a, and the coefficient of y is b. Statement (1) tells us the value of a is 2, but it doesn't directly provide information about the value of b. However, we can determine the slope of the line using the equation by rearranging it into the slope-intercept form, y = mx + b, where m is the slope.

Rearranging the equation ax + by + c = 0, we get:
by = -ax - c
y = (-a/b)x - c/b

Comparing this equation with the slope-intercept form, we can see that the slope of the line is -a/b. Since we know the slope is -3, we can substitute -3 for -a/b and solve for b:
-3 = -a/b
-3 = -2/b
b = 2/3

Therefore, Statement (1) alone is sufficient to determine the value of b.

Statement (2): c = 5/2

Statement (2) provides information about the value of c in the equation, but it doesn't directly provide information about the value of b or the slope of the line. Therefore, Statement (2) alone is not sufficient to answer the question.

In conclusion, Statement (1) alone is sufficient to determine the value of b, while Statement (2) alone is not sufficient. Thus, the answer is A: Statement (1) alone is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement (1): Line L has a positive X-intercept.

The X-intercept is the point where the line intersects the X-axis, which occurs when the value of y is zero. If Line L has a positive X-intercept, it means the line crosses the X-axis in the positive X region. However, this statement doesn't provide information about the behavior of the line in other quadrants. Line L could still pass through Quadrant II, but it could also pass through other quadrants. Therefore, Statement (1) alone is not sufficient to answer the question.

Statement (2): Line L has a negative Y-intercept.

The Y-intercept is the point where the line intersects the Y-axis, which occurs when the value of x is zero. If Line L has a negative Y-intercept, it means the line crosses the Y-axis in the negative Y region. However, similar to Statement (1), this statement alone doesn't provide enough information about the behavior of the line in other quadrants. Line L could still pass through Quadrant II, but it could also pass through other quadrants. Therefore, Statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we still don't have enough information to determine if Line L passes through Quadrant II. While Statement (1) provides information about the positive X-intercept and Statement (2) provides information about the negative Y-intercept, we don't know the behavior of the line outside those intercepts. Line L could still pass through Quadrant II, but it could also pass through other quadrants or have different orientations. Therefore, Statements (1) and (2) together are not sufficient to answer the question.

In conclusion, neither statement alone nor both statements together provide enough information to determine if Line L passes through Quadrant II. Therefore, the answer is C: Both statements (1) and (2) together are sufficient to answer the question asked, but neither statement alone is sufficient.

To determine if the y-intercept of line L is smaller than that of line M, let's analyze the given statements:

Statement (1): The slope of line L is negative, and the slope of line M is positive.

Knowing the slopes of the lines doesn't provide direct information about their y-intercepts or their relative positions along the y-axis. Therefore, Statement (1) alone is not sufficient to answer the question.

Statement (2): Line L has a positive x-intercept.

This statement provides information about the x-intercept of line L but doesn't provide direct information about the y-intercept of either line or their relative positions along the y-axis. Therefore, Statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we still don't have enough information to determine if the y-intercept of line L is smaller than that of line M. The relationship between the slopes and the x-intercepts doesn't provide conclusive information about the y-intercepts or their relative positions. Therefore, Statements (1) and (2) together are not sufficient to answer the question.

In conclusion, neither statement alone nor both statements together provide enough information to determine if the y-intercept of line L is smaller than that of line M. Therefore, the answer is E: Statements (1) and (2) together are not sufficient to answer the question asked, and additional data are needed.

Statement (1): |m| = |n| and |r| = |t|

This statement tells us that the absolute values of the x-coordinates and y-coordinates of both points are equal. However, it doesn't provide any information about the signs or magnitudes of m, n, r, or t. Therefore, we cannot determine if points A and B are equidistant from the origin based on this statement alone.

Statement (2): |m| = |r| and |n| = |t|

This statement tells us that the absolute values of the x-coordinates and the absolute values of the y-coordinates of both points are equal. However, similar to Statement (1), it doesn't provide any information about the signs or magnitudes of m, n, r, or t. Therefore, we cannot determine if points A and B are equidistant from the origin based on this statement alone.

When we consider both statements together, we still cannot determine if points A and B are equidistant from the origin. Although both statements provide information about the equality of absolute values, they do not provide any information about the signs or magnitudes of the coordinates. Therefore, Statements (1) and (2) together are not sufficient to answer the question.

In conclusion, Statement (2) alone is sufficient to answer the question, while Statement (1) alone is not sufficient. Thus, the answer is B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Statement (1): The line L doesn't have an X-intercept.

If the line L doesn't have an x-intercept, it means the line does not intersect or cross the x-axis. This information alone does not provide any direct information about whether the line passes through Quadrant III or not. It is possible for a line to not have an x-intercept and still pass through Quadrant III. Therefore, Statement (1) alone is not sufficient to answer the question.

Statement (2): The y-intercept of line L is positive.

The y-intercept is the point where the line intersects the y-axis, which occurs when the value of x is zero. If the y-intercept of line L is positive, it means the line crosses the y-axis above the origin. However, this statement alone does not provide any information about whether the line passes through Quadrant III or not. A line can have a positive y-intercept and still pass through Quadrant III if it slopes downward. Therefore, Statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we have the following information: line L doesn't have an x-intercept (from Statement 1) and the y-intercept of line L is positive (from Statement 2). These two statements together do not provide enough information to determine if the line L passes through Quadrant III. The line could have a negative slope and pass through Quadrant III, or it could have a positive slope and not pass through Quadrant III.

Therefore, both statements (1) and (2) together are sufficient to answer the question asked, but neither statement alone is sufficient. The answer is C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1): The slope of line m is 2.

The slope-intercept form of a linear equation is y = mx + b, where m is the slope. Since we are given that the slope of line m is 2, we have the form y = 2x + b. However, we don't have enough information to determine the value of the y-intercept, b. Therefore, Statement (1) alone is not sufficient to answer the question.

Statement (2): The x-intercept of line m is 3.

The x-intercept is the point where the line intersects the x-axis, which occurs when the value of y is zero. If the x-intercept of line m is 3, it means the line crosses the x-axis at the point (3, 0). However, we still don't have information about the slope or the y-intercept of the line, so we can't determine the equation of line m. Therefore, Statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we know the slope of line m is 2 (from Statement 1) and the x-intercept is 3 (from Statement 2). With this information, we can determine the equation of line m. The slope-intercept form of the equation becomes y = 2x + b. Substituting the x-intercept value of 3, we have 0 = 2(3) + b, which simplifies to 0 = 6 + b. Solving for b, we find that b = -6. Therefore, the equation of line m is y = 2x - 6.

Thus, both statements (1) and (2) together are sufficient to answer the question asked, but neither statement alone is sufficient. The answer is C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1): Line L2 is perpendicular to line L1.

If line L2 is perpendicular to line L1, it means their slopes are negative reciprocals of each other. If the slope of line L1 is positive, then the slope of line L2 would be negative. Therefore, Statement (1) alone is sufficient to determine that the slope of line L2 is negative.

Statement (2): Line L1 and L2 intersect in the I st quadrant.

This statement provides information about the intersection point of lines L1 and L2 but doesn't provide direct information about the slope of line L2. Knowing the intersection point alone doesn't determine the slope of line L2. Therefore, Statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we know that line L2 is perpendicular to line L1 (from Statement 1), and they intersect in the I st quadrant (from Statement 2). If line L1 passes through points in each quadrant except the IIIrd quadrant, and line L2 is perpendicular to line L1, then line L2 would pass through points in each quadrant except the II nd quadrant.

Since we know the slope of line L1 is positive, and line L2 is perpendicular to line L1, we can determine that the slope of line L2 is negative.

Therefore, Statement (1) alone is sufficient to answer the question, while Statement (2) alone is not sufficient. Thus, the answer is A: Statement (1) alone is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement (1): The product of the slopes of line l and line k is -1.

If the product of the slopes of two lines is -1, it means that the lines are perpendicular to each other. In this case, line l and line k intersect at a point, so line k must be the perpendicular bisector of line l. Therefore, the slope of line l must be the negative reciprocal of the slope of line k.

Statement (2): Line k passes through the origin.

Knowing that line k passes through the origin doesn't provide direct information about the slope of line l.

When we consider both statements together, we can conclude that line k is perpendicular to line l (from Statement 1) and passes through the origin (from Statement 2). The slope of line k can be determined since we know it passes through the origin, but the slope of line l cannot be determined solely from this information.

Therefore, both statements together are sufficient to determine that line k is perpendicular to line l, but we still don't have enough information to determine the exact slope of line l. Thus, the answer is C: Both statements (1) and (2) together are sufficient to answer the question asked, but neither statement alone is sufficient.

Statement (1): The road distance from Exit J to Exit L is 21 kilometers.

This statement provides information about the distance between Exit J and Exit L, but it doesn't directly tell us the distance from Exit K to Exit L. Therefore, Statement (1) alone is not sufficient to answer the question.

Statement (2): The road distance from Exit K to Exit M is 26 kilometers.

This statement gives information about the distance between Exit K and Exit M, which is unrelated to the distance from Exit K to Exit L. Therefore, Statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we still cannot determine the road distance from Exit K to Exit L. The relationship between the distances from Exit J to Exit L and from Exit K to Exit L is not provided, nor is the relationship between the distances from Exit K to Exit M and from Exit K to Exit L. Therefore, Statements (1) and (2) together are not sufficient to answer the question.

In conclusion, neither statement alone nor both statements together provide enough information to determine the road distance from Exit K to Exit L. Therefore, the answer is E: Statements (1) and (2) together are not sufficient to answer the question asked, and additional data are needed.