Test: Coordinate Geometry - GMAT MCQ

From statement (1), we know that the distance between point P and point X is 4.5. Since the radius of the circular area is 4, point X lies outside the circular area. However, we don't have any information about the position of point Y relative to the circular area based on this statement alone.

From statement (2), we know that the distance between point X and point Y is 9. This statement alone doesn't provide any information about the position of any of the points relative to the circular area.

Considering both statements together, we can deduce the following: If point X lies outside the circular area (as indicated by statement 1), and the distance between point X and point Y is 9 (as indicated by statement 2), then point Y must also lie outside the circular area. Thus, together, the statements are sufficient to conclude that point Y lies outside the circular area.

Therefore, 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) tells us that the angle between Line 1 and Line 2 is 40º. However, this information alone does not provide any direct information about the slope of Line 1. It is possible for Line 1 to have a positive, negative, or zero slope depending on the specific configuration of the lines. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2) states that Line 2 has a slope of 1. This information alone does not provide any information about the slope of Line 1. The slopes of two lines can be completely unrelated, so statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we can deduce the following: If the angle between Line 1 and Line 2 is 40º (as stated in statement 1), and Line 2 has a slope of 1 (as stated in statement 2), then Line 1 must have a positive slope. This is because when two lines intersect with a 40º angle and one of the lines has a positive slope, the other line must also have a positive slope to form the given angle. Thus, together, the statements are sufficient to conclude that the slope of Line 1 is positive.

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 whether line m is perpendicular to line n, we need to analyze the given statements.

Statement (1) states that if the slope of line m is represented by y and the slope of line n is represented by z, then –yz = 1. From this statement, we can conclude that the product of the slopes of lines m and n is equal to -1. In the rectangular coordinate system, two lines are perpendicular if and only if the product of their slopes is -1. Therefore, statement (1) alone is sufficient to answer the question, and the answer cannot be E or C.

Statement (2) tells us that line m has a slope of -1, and line n passes through the point (-x, -x). While this information provides the slope of line m, it does not provide any direct information about the slope of line n. Without the slope of line n, we cannot definitively determine whether line m is perpendicular to line n. Thus, statement (2) alone is not sufficient to answer the question.

Considering the statements together, we have the slope of line m (-1) from statement (2) and the relationship between the slopes of lines m and n (-yz = 1) from statement (1). Combining these two pieces of information, we can conclude that line m is perpendicular to line n. Therefore, each statement alone is sufficient to answer the question.

Hence, the answer is D: EACH statement ALONE is sufficient to answer the question asked.

Statement (1) tells us that line L passes through the point (4, -8). However, this information alone does not provide any direct information about the intersection with the graph of y = x². It is possible for a line to pass through a point on the graph of y = x² or be tangent to it at that point. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2) states that line L passes through the point (-4, 16). Similarly to statement (1), this information alone does not provide any direct information about the intersection with the graph of y = x². The line can intersect, be tangent to, or simply pass through the point on the graph. Therefore, statement (2) alone is not sufficient to answer the question.

Considering both statements together, we have the information that line L passes through the points (4, -8) and (-4, 16). However, even with this information, we still cannot determine whether line L intersects the graph of y = x². It is possible for the line to intersect or be tangent to the graph at some other point or segment. Therefore, together, the statements are still not sufficient to answer the question.

Since neither statement alone is sufficient to answer the question, and the combination of both statements is also insufficient, the answer is B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Statement (1) tells us that a = 4. This implies that point A has coordinates (4, 0) in the rectangular plane. However, we still don't have enough information to determine the measure of angle BAO based on this statement alone. We need the value of b or some relationship between a and b to calculate the angle. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2) states that b = α√3. This gives us the value of b in terms of α. However, we still don't have any information about the value of α, which is necessary to determine the measure of angle BAO. Hence, statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we know that a = 4 and b = α√3. However, we still don't have any direct information about the relationship between α and the angle BAO. Without knowing the specific value or relationship of α, we cannot determine the measure of the angle. Therefore, together, the statements are still not sufficient to answer the question.

Since statement (2) alone is sufficient to provide the value of b, but statement (1) alone is not sufficient, the answer is B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Statement (1) states that ab < 0. This means that the product of a and b is negative. If the product of a and b is negative, it implies that a and b have opposite signs. When the x-intercept of a line is greater than 0, it means that the line crosses the x-axis to the right of the origin. Given that a and b have opposite signs, it indicates that the line y = ax + b crosses the x-axis to the right of the origin. Therefore, statement (1) alone is sufficient to answer the question.

Statement (2) tells us that a + b < 0. This statement provides a relationship between the sum of a and b and 0, but it doesn't give us direct information about the x-intercept or its position. Knowing that a + b is less than 0 doesn't provide conclusive information about the position of the x-intercept. Therefore, statement (2) alone is not sufficient to answer the question.

Considering both statements together, we have the information that ab < 0 from statement (1) and a + b < 0 from statement (2). From this combined information, we can deduce that a and b have opposite signs and their sum is negative. This indicates that the line y = ax + b crosses the x-axis to the right of the origin, and thus, the x-intercept is greater than 0. Therefore, together, the statements are sufficient to answer the question.

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

Statement (1) tells us that the x-intercept of the line is a such that a < x. However, this information alone does not provide any direct information about the slope of the line. Knowing the x-intercept does not necessarily determine whether the slope is positive or negative.

Statement (2) tells us that both x and y are less than 0. This information alone also does not directly provide any information about the slope of the line passing through the point (x, y). Again, knowing the signs of x and y does not determine the slope of the line.

By considering both statements together, we still cannot determine the slope of the line definitively. The x-intercept from statement (1) and the negative values of x and y from statement (2) do not give us enough information to determine whether the slope is negative or positive. Therefore, the correct answer is (C): BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

From the equation of line j, bx + ay = 5, we can rearrange it as ay = -bx + 5 and rewrite it as y = (-b/a)x + 5/a.

Similarly, from the equation of line k, 2bx - 3ay = -5, we can rearrange it as -3ay = -2bx - 5 and rewrite it as y = (-2b/-3a)x - 5/-3a, which simplifies to y = (2b/3a)x + 5/3a.

Comparing the equations of the lines, we can see that the slopes (-b/a and 2b/3a) must be equal for the lines to intersect at one point.

Now let's analyze the statements:

Statement (1) tells us that the lines j and k intersect at the point (1, -3). This means that the coordinates (x, y) = (1, -3) satisfy both equations. By substituting these values into the equations, we can form a system of equations:

a + b = 5 (from line j)
2b/3a - 3a = -5 (from line k)

From this system, we can solve for the values of a and b. Therefore, Statement (1) alone is sufficient to answer the question.

Statement (2) tells us that a - b = -3. While this provides a relationship between a and b, it does not give us enough information to determine their specific values. Therefore, Statement (2) alone is not sufficient to answer the question.

By combining both statements, we have the equations from Statement (1) and the relationship from Statement (2):

a + b = 5
a - b = -3

By solving this system of equations, we can find the values of a and b, and hence determine the value of a + b. Therefore, each statement alone is sufficient to answer the question.

Hence, the answer is (D): EACH statement ALONE is sufficient to answer the question asked.

Statement (1) tells us that Line C is perpendicular to a line with the equation y = -2x + 5. Perpendicular lines have negative reciprocal slopes. The given line has a slope of -2. Therefore, the slope of Line C must be the negative reciprocal of -2, which is 1/2. Hence, from statement (1) alone, we can determine the value of a as 1/2.

Statement (2) tells us that Line C is parallel to a line with the equation y = (1 - a)x + b + 2. Parallel lines have the same slope. In this case, the slope of Line C is equal to the slope of the given line, which is 1 - a. Therefore, from statement (2) alone, we can determine the value of a as 1 - a.

Both statements individually provide enough information to determine the value of a. Statement (1) allows us to directly determine the value of a as 1/2, and statement (2) allows us to determine the value of a as 1 - a. Therefore, each statement alone is sufficient to answer the question.

Hence, the answer is (D): EACH statement ALONE is sufficient to answer the question asked.

Statement (1) tells us that a = 2. However, it does not provide any information about the value of b. Without knowing the value of b, we cannot determine if the point (a, b) lies above the line y = x. Statement (1) alone is not sufficient to answer the question.

Statement (2) tells us that b = a + 2. With this information, we can substitute b = a + 2 into the equation y = x to obtain a new equation: a + 2 = a. This equation simplifies to 2 = 0, which is not possible. Therefore, statement (2) leads to a contradiction, and it is not possible to find consistent values for a and b that satisfy the condition. Hence, statement (2) alone is sufficient to answer the question, as it indicates that the point (a, b) cannot lie above the line y = x.

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