Test: Coordinate Geometry - GMAT MCQ

To determine if the point (a, b) lies on the line l with the equation 5x + 6y = 90, we can substitute the values of a and b into the equation and see if it satisfies the equation.

Statement (1): (a + b - 16)(5a + 6b - 90) = 0

If (a + b - 16) = 0, then a + b = 16. Substituting a + b = 16 into the equation 5x + 6y = 90, we get 5a + 6b = 80.

If (5a + 6b - 90) = 0, then 5a + 6b = 90.

We have two equations: a + b = 16 5a + 6b = 90

Solving these two equations simultaneously, we can find the values of a and b. If the values of a and b satisfy both equations, then the point (a, b) lies on the line.

However, statement (1) alone does not provide enough information to determine the values of a and b, and therefore we cannot conclude whether the point (a, b) lies on the line or not.

Statement (2): (b - a - 4)(5a + 6b - 90) = 0

If (b - a - 4) = 0, then b - a = 4. Substituting b - a = 4 into the equation 5x + 6y = 90, we get 5a + 6b = 106.

If (5a + 6b - 90) = 0, then 5a + 6b = 90.

We have two equations: b - a = 4 5a + 6b = 90

Solving these two equations simultaneously, we can find the values of a and b. If the values of a and b satisfy both equations, then the point (a, b) lies on the line.

Similar to statement (1), statement (2) alone does not provide enough information to determine the values of a and b, and therefore we cannot conclude whether the point (a, b) lies on the line or not.

When we consider both statements together, we have the following equations:

a + b = 16 5a + 6b = 90

b - a = 4 5a + 6b = 90

Combining these equations, we can solve for the values of a and b. If the values of a and b satisfy both sets of equations, then the point (a, b) lies on the line.

Since both statements together are sufficient to determine the values of a and b and determine if the point lies on the line, the correct answer is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

To determine the number of points at which the two graphs intersect, we need to analyze each statement individually and then consider them together.

Statement (1): The equation of the first graph is x² + y² = 4.
This equation represents a circle centered at the origin (0, 0) with a radius of 2. The graph of this equation is a circle with its center at the origin and a radius of 2. However, this statement does not provide any information about the second graph, so it is not sufficient to determine the number of intersection points.

Statement (2): The second graph has equation y = ax² + 4, where a is a constant.
This equation represents a parabola that opens upward. The constant "a" determines the shape and orientation of the parabola. However, the statement does not provide any specific value for "a," so we cannot determine the exact shape of the parabola or its intersection points with the first graph. Therefore, statement (2) alone is not sufficient.

Considering both statements together:
When we combine the equations from both statements, we have a system of equations:
x² + y² = 4 (from statement 1)
y = ax² + 4 (from statement 2)

However, this system of equations is not enough to determine the exact values of x and y or the number of intersection points. We have two equations with three variables (x, y, and a). Without additional information about the value of "a" or any other constraints, we cannot uniquely determine the number of intersection points.

Since neither statement alone nor the two statements together are sufficient to answer the question, the correct answer is (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

To determine if point C lies between point A and point B, we need to compare the distances between each pair of points.

Let's consider the statements one by one:

Statement (1): The distance between point A and point B is 25.
This statement alone does not provide any information about the position of point C. It only gives us the distance between point A and point B. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): Point A lies to the left of point B.
This statement tells us the relative position of points A and B. Since the number line is a linear continuum, if point A is to the left of point B, it means that point B is located further to the right than point A. However, we still don't have enough information to determine the position of point C relative to points A and B. Therefore, statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we can combine the information provided:

From statement (1), the distance between point A and point B is 25.

From statement (2), point A lies to the left of point B.

We know that the distance between point A and point C is 5, and the distance between point B and point C is 20. Since the total distance between A and B is 25, and the distance between A and C is 5, it follows that the distance between C and B must be 25 - 5 = 20. This matches the information provided in statement (2), which states that the distance between B and C is 20.

Based on this, we can conclude that point C lies between points A and B. Therefore, both statements (1) and (2) together are sufficient to answer the question.

Hence, the correct solution is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Given information:

• The four corners of the quadrilateral are (a,5), (b,5), (a,0), and (b,0).
• a + c = 12.
• a < b and both a and b are positive values.

We need to find the area of the quadrilateral.

Statement 1: b + c = 6
Statement 1 alone is sufficient to answer the question.

Explanation:
The coordinates of the four corners of the quadrilateral are (a,5), (b,5), (a,0), and (b,0). From the given information, we can deduce that the quadrilateral has two sides of length b - a and two sides of length 5.

Since the opposite sides of a rectangle are equal in length, if the quadrilateral is a rectangle, then b - a = 5.

From statement 1, we have b + c = 6. Since a + c = 12, subtracting the equations gives b - a = 6 - 12 = -6. However, it is given that both a and b are positive values, so the quadrilateral cannot be a rectangle.

Therefore, statement 1 alone is sufficient to determine that the quadrilateral is not a rectangle, which means the quadrilateral is not necessarily a parallelogram. Without knowing the specific shape of the quadrilateral, we cannot determine its area.

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

The given question asks whether the slope of line m is greater than the slope of line n, both passing through the point (1,2). We need to determine if we can answer this question based on the information provided in statements (1) and (2) individually or together.

Statement (1): The x-intercept of m is greater than 1, and that of n is less than 1.

This statement provides information about the x-intercepts of the lines. The x-intercept is the point at which a line crosses the x-axis. If the x-intercept is greater than 1 for line m and less than 1 for line n, it means that line m is steeper (has a greater slope) than line n. The reason for this is that the greater the x-intercept, the larger the change in x for a given change in y, indicating a steeper slope. Therefore, statement (1) alone is sufficient to determine that the slope of line m is greater than the slope of line n.

Statement (2): The y-intercept of m = 4, and that of n = -2.

This statement provides information about the y-intercepts of the lines. The y-intercept is the point at which a line crosses the y-axis. However, the y-intercept alone does not provide any information about the slope of the lines. Therefore, statement (2) alone is not sufficient to determine the relationship between the slopes of lines m and n.

When we consider both statements together, we know that line m has a greater x-intercept and line n has a smaller x-intercept, as stated in statement (1). Additionally, statement (2) does not provide any relevant information to compare the slopes of the lines. Therefore, when combined, the statements still do not provide enough information to determine the relationship between the slopes of lines m and n.

Based on the analysis of the individual statements and their combination, the answer is (D) EACH statement ALONE is sufficient to answer the question asked.

To analyze the question and determine the slopes of lines L and K, let's examine each statement individually and then combine them.

Statement (1): L passes through (5, 0), and K passes through (-5, 0).
This statement provides the x-intercepts for both lines but does not provide any information about their slopes. The fact that both lines pass through the x-axis at different points does not allow us to compare their slopes. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): L and K intersect with each other in the 2nd quadrant.
This statement tells us that lines L and K intersect in the second quadrant, but it still does not provide enough information to determine the slopes of the lines. Two lines can intersect in different ways, resulting in different slopes. Thus, statement (2) alone is not sufficient to answer the question.

Combining both statements:
Although combining the statements provides additional information about the positions of the lines, it still does not yield enough data to determine the slopes. We don't have any information about the points of intersection or any other relevant details. Thus, even when considering both statements together, we cannot determine the slopes of lines L and K. Therefore, the answer is option (E): Statements (1) and (2) together are not sufficient to answer the question, and additional data are needed.

In summary, both statements individually and together are insufficient to determine the slopes of lines L and K.

Statement (1): The X-Intercept of line k is greater than 1.

The X-intercept of a line represents the point at which the line intersects the X-axis. If the X-intercept of line K is greater than 1, it means that the line crosses the X-axis at a point to the right of x = 1. However, this statement alone does not provide enough information to determine whether line K intersects circle C. It is possible for the line to intersect or not intersect the circle, depending on its slope and position.

Statement (2): The slope of line k is -1/10.

The slope of a line determines its direction and steepness. A negative slope indicates a line that is decreasing as it moves from left to right. While this information tells us about the direction of line K, it still does not provide enough information to determine whether the line intersects circle C. The position of the line in relation to the circle is crucial in making that determination.

Combining the two statements does not provide any additional information that would allow us to definitively determine whether line K intersects circle C. Therefore, both statements together are not sufficient to answer the question.

Hence, the correct answer is (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Statement (1): In the coordinate plane, the point (-1, p) lies inside the square S. The sides of the square are 6 each, and its diagonals intersect at the origin.

From this statement, we know that the point (-1, p) lies inside the square S, and the sides of the square are 6 units each. We also know that the diagonals of the square intersect at the origin.

Since the diagonals of a square are equal in length and intersect at the midpoint, we can conclude that the length of each diagonal is equal to the square root of 2 times the side length of the square. In this case, the length of each diagonal is 6√2.

Since the diagonals intersect at the origin, the distance from the origin to the point (-1, p) is equal to half the length of a diagonal, which is 6√2 / 2 = 3√2.

Now, the question asks if p < 3. To answer this, we need to compare the value of p with 3.

However, statement (1) alone does not provide any information about the value of p. We know the distance from the origin to (-1, p) is 3√2, but we don't have any direct information about the value of p. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): In the coordinate plane, the point (p, -1) lies inside the circle with equation x^2 + y^2 = 25.

From this statement, we know that the point (p, -1) lies inside the circle with equation x^2 + y^2 = 25, which is a circle with a radius of 5 centered at the origin.

Since the y-coordinate of the point (p, -1) is -1, we can conclude that the distance from the origin to (p, -1) is less than 5.

However, this information alone does not provide any direct information about the value of p. We cannot determine if p is less than 3 based on statement (2) alone. Therefore, statement (2) alone is not sufficient to answer the question.

Combining the statements:

Together, we know from statement (1) that the distance from the origin to (-1, p) is 3√2, and from statement (2) that the distance from the origin to (p, -1) is less than 5.

However, even when combining the statements, we still do not have enough information to determine if p is less than 3. The two statements provide information about different points and distances in the coordinate plane, and there is no direct relationship between them that allows us to determine the value of p.

Therefore, both statements together are not sufficient to answer the question.

In conclusion, the correct answer is (A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient to answer the question asked.

The question asks for the minimum distance between a point on line L and a point on line M in the x-y coordinate plane.

Statement (1): The absolute value of the difference between the y-intercepts of the two lines is 4.
This statement provides information about the y-intercepts of the two lines. However, it doesn't give us any information about the slopes of the lines or the position of the lines in the coordinate plane. Without any additional information, we cannot determine the minimum distance between the two lines. Statement (1) alone is not sufficient.

Statement (2): The absolute values of the slopes of the two lines are both equal to 2.
This statement provides information about the slopes of the two lines. However, it doesn't give us any information about the y-intercepts or the position of the lines in the coordinate plane. Without any additional information, we cannot determine the minimum distance between the two lines. Statement (2) alone is not sufficient.

Combining both statements:
By combining both statements, we still don't have enough information to determine the minimum distance between the two lines. We need additional information about the positions of the lines in the coordinate plane (e.g., their equations, relative positions, etc.) to calculate the minimum distance. Therefore, the combined statements (1) and (2) are also not sufficient.

Since neither statement alone, nor the combined statements, provide enough information to determine the minimum distance between the two lines, the answer is (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

We have a line 'l' passing through the origin and the point (m, n), where mn ≠ 0. Our goal is to determine if n > 0.

Statement (1): The line 'l' has a negative slope.
This statement tells us that the line has a negative slope, but it doesn't provide any direct information about the coordinates (m, n). The slope alone doesn't determine whether n is greater than 0 or not. For example, the line could have a negative slope and pass through a point below the x-axis, resulting in n being negative. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): m < n
This statement provides a direct comparison between m and n. If m is less than n, it implies that the point (m, n) is above the line y = x (the diagonal line passing through the origin with a slope of 1). Since the line 'l' passes through the origin and (m, n), and mn ≠ 0, we can conclude that n must be positive. Therefore, statement (2) alone is sufficient to answer the question.

By combining both statements, we have the following information: the line 'l' has a negative slope, and m < n. This means that the line 'l' is below the line y = x, and the point (m, n) is above it. Since the line passes through the origin, the point (m, n) must be in the first quadrant. Hence, n > 0.

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