GMAT Math : Geometry

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Example Questions

Example Question #581 : Geometry

Give the -intercept of the graph of the equation $f(x) = x^{2}+Ax+B$ .

Statement 1: A = 8

Statement 2: B = 10

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

The -intercept of the graph of an equation is the point at which x = 0 , so we evaluate f(0) :

$f(x) = x^{2}+Ax+B$

$f(0) = 0^{2}+A\cdot 0+B$

f(0) =B

The -intercept is simply the point (0,B) , so knowing is necessary, and knowing is neither necessary nor helpful.

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Example Question #582 : Geometry

Find the -intercept and y-intercept of the following straight line.

1. The line has a slope of 0.6.

2. The line passes through point (10,2)

Possible Answers:

Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

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

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

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

EACH statement ALONE is sufficient to answer the question asked.

Correct answer:

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

Explanation:

To find the - and y-intercepets, we need both statements of information.

Statement 1 is not enough information because we don't have a reference point for the slope. A slope by itself is not enough to define the line. There are an infinite amount of lines with a slope of 0.6.

Statement 2 is not enough information by itself since it only tells us about 1 point on the line. Again, there are an infinite number of lines that pass through the point (10,2).

Only by using both statements can we find the - and y-intercepts. Solving, we see the line is actually

y=0.6x-4

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Example Question #2 : Dsq: Calculating X Or Y Intercept

Give the -intercept of the graph of the function f(x) = | 3 x - A | + B

Statement 1: A = 8

Statement 2: B = 5

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

To find the -intercept of f(x) , evaluate f(0) :

f(x) = | 3 x - A | + B

$f(0) = | 3 \cdot 0 - A | + B$

f(0) = | - A | + B

Knowing both and is necessary and sufficient.

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Example Question #2 : X And Y Intercept

Give the -intercept of the graph of the function f(x) = | A x - 7| + B

Statement 1: A = 8

Statement 2: B = 5

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

To find the -intercept of f(x) , evaluate f(0) :

f(x) = | A x - 7| + B

$f(0) = | A \cdot 0 - 7| + B$

f(0) = | - 7 | + B

f(0) = 7 + B

Knowing is necessary and sufficient; the value of is irrelevant.

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Example Question #6 : X And Y Intercept

A line on the coordinate plane is neither horizontal nor vertical. Give its -intercept.

Statement 1: The line passes through (4,7) .

Statement 2: The line passes through (6,1) .

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Two points are necessary and sufficient to define a line. Therefore, neither statement alone is sufficient to determine the line, but both are sufficient. Once the line is defined, the -intercept - the point at which the line intersects the -axis - can be determined.

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Example Question #1 : X And Y Intercept

and are two distinct nonvertical lines on the coordinate plane.

True or false: and have the same -intercept.

Statement 1: and intersect at (10,0) .

Statement 2: and are perpendicular.

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

The question is equivalent to asking whether the lines intersect at a point on the -axis.

Assume Statement 1 alone. Since and are distinct lines, (10,0) , their common -intercept, is their sole point of intersection. They cannot intersect at a second point, so they cannot have the same -intercept.

Assume Statement 2 alone. Perpendicular lines are lines that meet at right angles; the question of their point of intersection is not answered by this statement.

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Example Question #7 : X And Y Intercept

A function f(x) is graphed on the coordinate plane. Give the -intercept of the graph.

Statement 1: f(5)= 0

Statement 2: f(0)= 7

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

The -intercept of the graph of is the point at which it intersects the -axis. Since this point has -coordinate 0, the -coordinate is f(0) . Statement 1 does not give us this value, but Statement 2 does.

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Example Question #8 : X And Y Intercept

and are two distinct nonvertical lines on the coordinate plane.

True or false: and have the same -intercept.

Statement 1: and have different -intercepts.

Statement 2: and both have slope $\frac{2}{3}$ .

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

The question is equivalent to asking whether the lines intersect at a point on the -axis.

Statement 1 only establishes that the lines pass through different points on the -axis; no clues are given about any of their other points.

Statement 2 establishes that the lines have the same slope, and, subsequently, are parallel - that is, they do not intersect at all. Therefore, they cannot have the same -intercept.

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Example Question #583 : Geometry

Continuous function g(x) has the set of all real numbers as its domain.

How many -intercepts does the graph g(x) have?

Statement 1: If M<N , then g(M)<g(N) .

Statement 2: g(0)= 1 .

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

The two statements together prvide insufficient information.

Assume both statements are true. By Statement 1, is a constantly increasing function, so it can intersect the -axis at most one time.

Now examine these two cases.

Case 1:

g(x) = x+1

g(0)= 0+1=1 .

Also, if M < N , then M+1 < N+1 , so g(M)<g(N) .

Since g(-1) = -1+1 = 0 , the function has exactly one -intercept.

Case 2:

$g(x)=2^{x}$

$g(0)=2^{0}= 1$

Also, if M < N , then $2^{M}< 2^{N}$ , so g(M)<g(N) .

However, 2 raised to any power must be positive, so there is no value for which g(N)= 0 . The function has no -intercepts.

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Example Question #11 : Dsq: Calculating X Or Y Intercept

Mandy's teacher challenged her to write numbers in the circle, the square, and the triangle in the pattern below in order to make an equation whose graph has -intercept (5,0) and -intercept (0, 1) .

$\square x+ \bigcirc y = \triangle$

Did Mandy succeed?

Statement 1: Mandy wrote a 5 in the triangle.

Statement 2: The number Mandy wrote in the square was five times the number she wrote in the circle.

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

Let A,B,C stand for the values Mandy wrote in the square, the circle, and the triangle, respectively. The equation becomes

A x+ B y =C .

From Statement 1 alone, we know that Mandy wrote a 5 in the triangle, but we do not know any of the others. The question of Mandy's success is unresolved.

Now assume Statement 2 alone is known. Then

A = 5B .

The equation can be rewritten as

5B x+ B y =C

and it can be rewritten in slope-intercept form as

B y =-5B x+C

$\frac{B y}{B} =\frac{-5B x+C}{B}$

$y= -5x + \frac{C}{B}$

Mandy wrote an equation whose line has slope .

However, the slope of a line through (5,0) and (0, 1) is

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{1-0}{0-5} = \frac{1}{-5}= -\frac{1}{5}$ .

Statement 2 alone is sufficient to establish that Mandy did not succeed.

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #581 : Geometry

Example Question #582 : Geometry

Example Question #2 : Dsq: Calculating X Or Y Intercept

Example Question #2 : X And Y Intercept

Example Question #6 : X And Y Intercept

Example Question #1 : X And Y Intercept

Example Question #7 : X And Y Intercept

Example Question #8 : X And Y Intercept

Example Question #583 : Geometry

Example Question #11 : Dsq: Calculating X Or Y Intercept

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