Graphing - GMAT Quantitative
Card 0 of 400
Find the graph of 
.
I) 
is a linear equation which passes through the point 
.
II) 
crosses the y-axis at 1300.
Find the graph of
I)
II)
Find the graph of 
.
I) 
is a linear equation which passes through the point 
.
II) 
crosses the y-axis at 1300.
To graph a linear equation, we need some combination of slope, y-intercept, or two points.
Statement I tells us 
is linear and gives us one point.
Statement II gives us the y-intercept of 
.
We can use Statement I and Statement II to find the slope of 
. Then, we can plot the given points and continue the line in either direction to get our graph.
Slope:
Plugging in the provided value of 
, 1300, we have the equation of the line 
:
Find the graph of
I)
II)
To graph a linear equation, we need some combination of slope, y-intercept, or two points.
Statement I tells us
Statement II gives us the y-intercept of
We can use Statement I and Statement II to find the slope of
Slope:
Plugging in the provided value of
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Graph a line, if possible.
Statement 1: The slope is 4.
Statement 2: The y-intercept is 4.
Graph a line, if possible.
Statement 1: The slope is 4.
Statement 2: The y-intercept is 4.
Statement 1): The slope is 4.
Write the slope-intercept form, and substitute the slope.
The point and the y-intercept are unknown. Either of these will be needed to solve for the graph of this line.
Statement 1) by itself is not sufficient to graph a line.
Statement 2): The y-intercept is 4.
Substitute the y-intercept into the incomplete formula.
The function 
can then be graphed on the x-y coordinate plane.
Therefore:
Statement 1): The slope is 4.
Write the slope-intercept form, and substitute the slope.
The point and the y-intercept are unknown. Either of these will be needed to solve for the graph of this line.
Statement 1) by itself is not sufficient to graph a line.
Statement 2): The y-intercept is 4.
Substitute the y-intercept into the incomplete formula.
The function
Therefore:
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Let 
be a positive integer.
True or false: 
Statement 1: 
is a prime number.
Statement 2: 
is a two-digit number ending in a 7.
Let
True or false:
Statement 1:
Statement 2:
If 
is a positive integer, then 
if and only if 
is a multiple of 4.
It follows that if 
, 
cannot be a prime number. Also, every multiple of 4 is even, so as an even number, 
cannot end in 7. Contrapositively, if Statement 1 is true and 
is prime, or if Statement 2 is true and if 
ends in 7, it follows that 
is not a multiple of 4, and 
.
If
It follows that if
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Let 
and 
be real numbers.
What is the product of 
and its complex conjugate?
Statement 1: 
Statement 2: 
Let
What is the product of
Statement 1:
Statement 2:
The complex conjugate of an imaginary number 
is 
, and
.
Therefore, Statement 1 alone, which gives that 
, provides sufficient information to answer the question, whereas Statement 2 provides unhelpful information.
The complex conjugate of an imaginary number
Therefore, Statement 1 alone, which gives that
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Let 
and 
be real numbers.
What is the product of 
and its complex conjugate?
Statement 1: 
Statement 2: 
Let
What is the product of
Statement 1:
Statement 2:
The complex conjugate of an imaginary number 
is 
, and
.
Therefore, it is necessary and sufficient to know the values of both 
and 
in order to answer the problem. Each statement alone gives only one of these values, so each statement alone provides insufficient information; the two together give both, so the two statements together provide sufficient information.
The complex conjugate of an imaginary number
Therefore, it is necessary and sufficient to know the values of both
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Let 
be a positive integer.
Evaluate 
.
Statement 1: 
is a multiple of 16.
Statement 2: 
is a multiple of 20.
Let
Evaluate
Statement 1:
Statement 2:
The value of 
, 
a positive integer, is equal to 
, where 
is the remainder of the division of 
by 4. Either statement alone is enough to prove that 
is divisible by 4, since, if a number is divisible by a given number (16 or 20 in these statements), it is divisible by any factor of that number (with 4 being a factor of both).
The value of
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Let 
be a positive integer.
Evaluate 
.
Statement 1: 
is a multiple of 3.
Statement 2: 
is a multiple of 7.
Let
Evaluate
Statement 1:
Statement 2:
Assume that both statements are true. The value of 
, 
a positive integer, is equal to 
, where 
is the remainder of the division of 
by 4, so we can use this fact to show that insufficient information is provided.
Case 1: 
.
, so
Case 2: 
.
, so
In both cases, both statements are true, but the value of 
differs.
Assume that both statements are true. The value of
Case 1:
Case 2:
In both cases, both statements are true, but the value of
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Let 
and 
be real numbers.
What is the product of 
and its complex conjugate?
Statement 1: 
Statement 2: 
Let
What is the product of
Statement 1:
Statement 2:
The complex conjugate of an imaginary number 
is 
, and
.
Statement 1 alone provides insufficient information, as seen in these two scenarios, both of which feature values of 
and 
that add up to 12:
Case 1: 
Then 
, and the product of this number and its complex conjugate is 
.
Case 2: 
Then 
, and the product of this number and its complex conjugate is 
.
The two cases result in different products.
For a similar reason, Statement 2 alone provides insufficient information.
If both statements are assumed to be true, they form a system of equations that can be solved as follows:
Backsolve:
Since we know that 
and 
, then we know that the desired product is 
.
The complex conjugate of an imaginary number
Statement 1 alone provides insufficient information, as seen in these two scenarios, both of which feature values of
Case 1:
Then
Case 2:
Then
The two cases result in different products.
For a similar reason, Statement 2 alone provides insufficient information.
If both statements are assumed to be true, they form a system of equations that can be solved as follows:
Backsolve:
Since we know that
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Let 
and 
be real numbers.
From the number 
, subtract its complex conjugate. What is the result?
Statement 1: 
Statement 2: 
Let
From the number
Statement 1:
Statement 2:
The complex conjugate of an imaginary number 
is 
, and
.
Therefore, it is necessary and sufficient to know 
in order to answer the question. Neither statement alone gives this information. However, the first statement can be rewritten by factoring out 
as a difference of squares:
Since 
, then by substitution,
A system of linear equations has now been formed; subtract both sides of the equations as follows:
We need go no further; since 
, the desired difference is 
.
The complex conjugate of an imaginary number
Therefore, it is necessary and sufficient to know
Since
A system of linear equations has now been formed; subtract both sides of the equations as follows:
We need go no further; since
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Let 
and 
be real numbers.
From the number 
, subtract its complex conjugate. What is the result?
Statement 1: 
Statement 2: 
Let
From the number
Statement 1:
Statement 2:
The complex conjugate of an imaginary number 
is 
, and
.
Therefore, it is necessary and sufficient to know 
in order to answer the question. Neither statement alone gives this information. However, the two statements together form a linear system that can be solved as follows:
We need go no further; since 
, the desired sum is 
.
The complex conjugate of an imaginary number
Therefore, it is necessary and sufficient to know
We need go no further; since
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Let 
and 
be real numbers.
What is the sum of 
and its complex conjugate?
Statement 1: 
Statement 2: 
Let
What is the sum of
Statement 1:
Statement 2:
The complex conjugate of an imaginary number 
is 
, and
.
Therefore, it is necessary and sufficient to know 
in order to answer the question. Neither statement alone gives this information. However, the two statements together form a linear system that can be solved as follows:
We need go no further; since 
, this is the desired sum.
The complex conjugate of an imaginary number
Therefore, it is necessary and sufficient to know
We need go no further; since
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Let 
and 
be real numbers.
What is the sum of 
and its complex conjugate?
Statement 1: 
Statement 2: 
Let
What is the sum of
Statement 1:
Statement 2:
The complex conjugate of an imaginary number 
is 
, and
.
We show, however, that the two statements are insufficient to determine the sum by examining two scenarios.
Case 1: 
.
, and since 
, 
. The conditions of both statements are satisfied.
The sum of the numbers is 
.
Case 2: 
.
, and since 
, 
. The conditions of both statements are satisfied.
The sum of the numbers is 
.
In both cases, the conditions of both statements are satisfied, but the sum of the number and its complex conjugate differs between the two.
The complex conjugate of an imaginary number
We show, however, that the two statements are insufficient to determine the sum by examining two scenarios.
Case 1:
The sum of the numbers is
Case 2:
The sum of the numbers is
In both cases, the conditions of both statements are satisfied, but the sum of the number and its complex conjugate differs between the two.
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Let 
and 
be real numbers.
From the number 
, subtract its complex conjugate. What is the result?
Statement 1: 
Statement 2: 
Let
From the number
Statement 1:
Statement 2:
The complex conjugate of an imaginary number is , and
.
Therefore, it is necessary and sufficient to know in order to answer the question. Statement 1 does not give this value, and is unhelpful here; Statement 2 does give this value.
The complex conjugate of an imaginary number
Therefore, it is necessary and sufficient to know
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Let 
and 
be real numbers.
What is the sum of 
and its complex conjugate?
Statement 1: 
Statement 2: 
Let
What is the sum of
Statement 1:
Statement 2:
The complex conjugate of an imaginary number 
is 
, and the sum of the two numbers is
.
Therefore, it is necessary and sufficient to know 
in order to answer the question. Statement 1 alone gives this information; Statement 2 does not, and it is unhelpful.
The complex conjugate of an imaginary number
Therefore, it is necessary and sufficient to know
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Graph the exponential function 
.
I) 
is a monomial.
II) 
has a base of 4.
Graph the exponential function
I)
II)
An exponential function follows the general form of
Statement I tells us that there is only one term, so the 
part of the equation isn't needed for this exponential function.
Statement II tells us that in this case, 
.
However, we could have nearly anything as our exponent. We are unable to make an accurate graph of this function, so more information is needed.
An exponential function follows the general form of
Statement I tells us that there is only one term, so the
Statement II tells us that in this case,
However, we could have nearly anything as our exponent. We are unable to make an accurate graph of this function, so more information is needed.
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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:
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
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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:
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
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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:
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
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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:
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.
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True or false: , , and 
are collinear points.
Statement 1: 
and 
Statement 2: 
True or false:
Statement 1:
Statement 2:
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
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