All GMAT Math Resources
Example Questions
Example Question #10 : Graphing A Line
Which of these equations is represented by a line that does not intersect the graph of the equation ?
None of the other choices gives a correct answer.
We can find out whether the graphs of and intersect by first solving for in the first equation:
We then substitute in the second equation for :
Then we rewrite in standard form:
Since we are only trying to pdetermine whether at least one point of intersection exists, rather than actually find the point, all we need to do is to evaluate the discriminant; if it is nonnegative, at least one solution - and, consequently, one point of intersection - exists. In the general quadratic equation , this is , so here, the discriminant is
.
Therefore, the line of the equation intersects the parabola of the equation .
We do the same for the other three lines:
Then we rewrite in standard form:
.
The line of intersects the parabola.
The line of intersects the parabola.
Since the discriminant is negative, the system has no real solution. This means that the line of does not intersect the parabola of the equation , and it is the correct choice.
Example Question #41 : Graphing
A line with positive slope passes through the vertex and an -intercept of the parabola of the equation . What is the equation of the line?
The vertex of the parabola of an equation of the form has -coordinate . Here, we substitute , to obtain -coordinate
.
To find the -coordinate, substitute this for :
The vertex is .
To find the -intercepts of the parabola, substitute 0 for in the equation:
Either
, in which case ,
or
, in which case .
The -intercepts are and .
The line includes and either or , so we find the slope in each case using the slope formula.
If the line includes and :
.
If the line includes and :
We choose the first case, since the line has positive slope. The line through and has as its equation, using the point-slope form with and point :
Example Question #42 : Graphing
The graph of the equation shares its -intercept and one of its -intercepts with a line of positive slope. What is the equation of the line?
The -intercept of the line coincides with that of the graph of the quadratic equation, which is a horizontal parabola; to find the -intercept of the parabola, substitute 0 for in the quadratic equation:
The -intercept of the parabola, and of the line, is .
The -intercept of the line coincides with one of those of the parabola; to find the -intercepts of the parabola, substitute 0 for in the equation:
Either
, in which case ,
or
, in which case .
The -intercepts of the parabola are and , so the -intercept of the line is one of these. We will examine both possibilities.
If and be the - and -intercepts, respectively, of the line, then the slope of the line is . If the intercepts of the line are and , the slope of the line is ; if the intercepts are and , the slope is . We choose the latter, since we are looking for a line with positive slope; since its -intercept is , then we can substitute in the slope-intercept form of the line, , to get the correct equation, .
Example Question #1021 : Problem Solving Questions
Give the -coordinate of the point of intersection of the lines of the equations:
Round your answer to the nearest whole number, if applicable.
The lines of the equations do not intersect.
The point of intersection of the two lines has as its coordinates the values of and that make both of the given linear equations true. Therefore, we seek to find the solution of the system of equations:
We need only find , so multiply both sides of the two equations by 7 and 4, respectively. Then add:
,
making 9 the correct response.
Example Question #44 : Graphing
Which of the following equations can be graphed with a line perpendicular to the green line in the above figure, and with the same -intercept?
The - and -intercepts of the line are, respectively, and . If and are the - and -intercepts, respectively, of a line, the slope of the line is . This makes the slope of the green line .
Any line perpendicular to this line must have as its slope the opposite of the reciprocal of this, or . Since the desired line must also have -intercept , then the slope-intercept form of the line is
which can be rewritten in standard form:
Example Question #45 : Graphing
A triangle is formed by the -axis and the graphs of the equations
and
Give the area of the triangle.
The vertices of the triangle are the point of intersection of the graphs of the lines of the two equations, and the -intercepts of those lines.
The -intercept of the line of the equation can be found by setting and solving for :
The -intercept of the line of the equation can be found the same way:
The -intercepts are and ; these are two of the vertices of the triangle, and since the segment connecting them is horizontal, this will be taken as the base. The length of the base is the difference of the -coordinates:
The intersection of the lines of the equations and can be found by solving the system of linear equations, as follows:
The point of intersection, and the third vertex of the triangle, is .
Since we are taking the horiztonal segment to be the base, the height will be the vertical distance to this third point - namely, the -coordinate . The area is half the product of the base and the height:
Example Question #1022 : Problem Solving Questions
Which response comes closest to the area of the triangle on the rectangular coordinate plane whose sides are along the axes and the line of the equation ?
The intercepts of the line of the equation can be found by substituting 0 for each variable, in turn:
-intercept:
The -intercept is the point
-intercept:
The -intercept is the point .
This line and the axes together form a right triangle. The horizontal leg is the segment connecting the origin to , and its length is . The vertical leg is the segment connecting the origin to , and its length is . The area of a right triangle is half the product of these legs, which is
.
Of the five responses, 45 comes closest to the correct area.
Example Question #1 : Graphing An Ordered Pair
Which quadrants can include the point if nonzero numbers and have the same sign?
Either Quadrant I or Quadrant IV
Either Quadrant I or Quadrant II
Either Quadrant II or Quadrant III
Either Quadrant II or Quadrant IV
Either Quadrant I or Quadrant III
Either Quadrant I or Quadrant IV
If and have the same sign, then , and the point has a positive -coordinate. The sign of is the same as the common sign of and . Therefore, only the -coordinate restricts the possibilities.
The points witn positive -coordinates are in Quadrant I and Quadrant IV.
Example Question #102 : Coordinate Geometry
Which quadrant or axis can include the point ?
Quadrant IV
The -axis
Quadrant III
Quadrant II
Quadrant I
Quadrant I
The ordered pair consists of a positive -coordinate and a positive -coordinate. The only quadrant that consists entirely of positive - and -coordinates is Quadrant I.
Example Question #1031 : Gmat Quantitative Reasoning
Which quadrant or axis contains the point ?
The -axis
Quadrant III
Quadrant II
The -axis
Quadrant I
The -axis
The ordered pair consists of a positive -coordinate and a -coordinate of . The only part of the coordinate plane on which all points have a zero -coordinate is the -axis.