DSQ: Graphing a point - GMAT Quantitative
Card 0 of 64
In which quadrant is the point 
located: I, II, III, or IV?
Statement 1: 
Statement 2: 
In which quadrant is the point
Statement 1:
Statement 2:
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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.
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
In which quadrant is the point 
located: I, II, III, or IV?
Statement 1: 
Statement 2: 
In which quadrant is the point
Statement 1:
Statement 2:
Tap to see back →
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.
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
In which quadrant is the point 
located: I, II, III, or IV?
Statement 1: 
Statement 2: 
In which quadrant is the point
Statement 1:
Statement 2:
Tap to see back →
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.
Assume Statement 1 alone. The points
Assume Statement 2 alone. The points
Assume both statements to be true. Statement 2 can be rewritten as
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.
True or false:
Statement 1:
Statement 2:
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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.
Two points in the same quadrant have
It is possible for two points fitting the condition of Statement 1 to be in the same quadrant;
Assume both statements are true.
True or false: , , and 
are collinear points.
Statement 1: 
and 
Statement 2: 
True or false:
Statement 1:
Statement 2:
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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.
Assume Statement 1 alone. The equations can be rewritten as follows:
The
Assume Statement 2 alone. The statement can be rewritten as follows:
The first expression is the slope of the line through
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.
True or false:
Statement 1:
Statement 2:
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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.
Two points in the same quadrant have
In which quadrant is the point 
located: I, II, III, or IV?
Statement 1: 
Statement 2: 
In which quadrant is the point
Statement 1:
Statement 2:
Tap to see back →
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.
Assume both statements. The points
True or false: , , and are collinear points.
Statement 1: 
Statement 2: 
and 
True or false:
Statement 1:
Statement 2:
Tap to see back →
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.
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
The three points cannot be assumed to be collinear from Statement 2 alone. For example,
In which quadrant is the point located: I, II, III, or IV?
Statement 1:
Statement 2:
In which quadrant is the point
Statement 1:
Statement 2:
Tap to see back →
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.
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
In which quadrant is the point located: I, II, III, or IV?
Statement 1:
Statement 2:
In which quadrant is the point
Statement 1:
Statement 2:
Tap to see back →
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.
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
In which quadrant is the point located: I, II, III, or IV?
Statement 1:
Statement 2:
In which quadrant is the point
Statement 1:
Statement 2:
Tap to see back →
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.
Assume Statement 1 alone. The points
Assume Statement 2 alone. The points
Assume both statements to be true. Statement 2 can be rewritten as
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.
True or false:
Statement 1:
Statement 2:
Tap to see back →
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.
Two points in the same quadrant have
It is possible for two points fitting the condition of Statement 1 to be in the same quadrant;
Assume both statements are true.
True or false: , , and are collinear points.
Statement 1: and
Statement 2:
True or false:
Statement 1:
Statement 2:
Tap to see back →
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.
Assume Statement 1 alone. The equations can be rewritten as follows:
The
Assume Statement 2 alone. The statement can be rewritten as follows:
The first expression is the slope of the line through
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.
True or false:
Statement 1:
Statement 2:
Tap to see back →
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.
Two points in the same quadrant have
In which quadrant is the point located: I, II, III, or IV?
Statement 1:
Statement 2:
In which quadrant is the point
Statement 1:
Statement 2:
Tap to see back →
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.
Assume both statements. The points
True or false: , , and are collinear points.
Statement 1:
Statement 2: and
True or false:
Statement 1:
Statement 2:
Tap to see back →
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.
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
The three points cannot be assumed to be collinear from Statement 2 alone. For example,
In which quadrant is the point located: I, II, III, or IV?
Statement 1:
Statement 2:
In which quadrant is the point
Statement 1:
Statement 2:
Tap to see back →
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.
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
In which quadrant is the point located: I, II, III, or IV?
Statement 1:
Statement 2:
In which quadrant is the point
Statement 1:
Statement 2:
Tap to see back →
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.
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
In which quadrant is the point located: I, II, III, or IV?
Statement 1:
Statement 2:
In which quadrant is the point
Statement 1:
Statement 2:
Tap to see back →
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.
Assume Statement 1 alone. The points
Assume Statement 2 alone. The points
Assume both statements to be true. Statement 2 can be rewritten as
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.
True or false:
Statement 1:
Statement 2:
Tap to see back →
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.
Two points in the same quadrant have
It is possible for two points fitting the condition of Statement 1 to be in the same quadrant;
Assume both statements are true.