GMAT Math : Geometry

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Example Questions

Example Question #12 : X And Y Intercept

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

Statement 1: The line has slope $\frac{1}{4}$ .

Statement 1: The line passes through the origin.

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.

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

Statement 1 provides insufficient information, since the slope of the line alone is not enough from which to deduce the -intercept. Statement 2 alone tells us that the line passes through the point (0,0) ; since this is on the -axis, this is the -intercept.

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Example Question #13 : X And Y Intercept

Stanley's teacher challenged him 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 (4,0) .

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

Did Stanley succeed?

Statement 1: Stanley wrote a 4 in the square.

Statement 2: Stanley wrote a 16 in the triangle.

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:

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

Explanation:

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

A x+ B y =C .

To find the -coordinate of the -intercept, set y = 0 and solve for :

$Ax+ B \cdot 0 =C$

Ax=C

$x=\frac{C}{A}$

It is therefore necessary and sufficient to know the values of and - the values Ralph wrote in the square and the triangle - in order to determine the -intercept of Ralph's equation. Each statement alone give only one of those numbers; the two together give both.

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

A function f(x) is graphed on the coordinate plane. It has exactly one -intercept. What is it?

Statement 1: f(5)= 0

Statement 2: f(0)= 7

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.

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

EITHER statement ALONE is sufficient 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 -intercept of the graph of is the point at which it intersects the -axis. Since this point has -coordinate 0, the -coordinate is the value for which f(x)= 0 . Statement 1 gives us this value; Statement 2 does not.

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Example Question #12 : Dsq: Calculating X Or Y Intercept

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)= 0 .

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 sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

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.

Correct answer:

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

Explanation:

Statement 1 alone establishes that is always increasing. Its graph cannot have more than one -intercept; if it does, then the graph of the function must have a vertex between two intercepts, violating this statement. But it does not answer the question as to how many intercepts has, as seen in these two cases:

Case 1:

g(x) = x+1

This is a linear function that is always increasing—it is in slope-intercept form, and its slope is 1, a positive number. The graph of has exactly one -intercept.

Case 2:

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

An exponential function with a base greater than 1, such as this, is an increasing function; however, 2 raised to any power must be positive, so there is no value for which g(N)= 0 . The graph of has no -intercepts.

Statement 2 alone establishes that at least one -intercept exists - since g(0)= 0 , (0,0) is an -intercept. It does not, however, rule out the possibility of more -intercepts.

Now assume both statements are true. Since Statement 1 establishes that there is at most one -intercept, and Statement establishes that there is at least one -intercept, the two statements together establish that there is exactly one.

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Example Question #13 : Dsq: Calculating X Or Y Intercept

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

$y = \square x+ \bigcirc$

Did Ray succeed?

Statement 1: Ray wrote a 5 in the square.

Statement 2: Ray wrote a 7 in the circle.

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

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.

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:

If we let stand for the number Ray wrote in the square and stand for the number he wrote in the circle, the equation becomes

y = mx+b ,

the slope-intercept form of the equation of a line. In this form, the -intercept is solely determined by the value of —namely, the value that Ray wrote in the circle. Statement 1 is therefore irrelevant, and Statement 2 alone establishes that Ray did not succeed.

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Example Question #17 : X And Y Intercept

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 0<M<N , then g(M)<g(N) .

Statement 2: If M<N < 0 , then g(M)>g(N) .

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.

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements are true.

By Statement 1, is decreasing on the domain interval $\left ( -\infty, 0\right )$ ; by Statement 2, is increasing on the domain interval $(0, \infty)$ . Therefore, must have its minimum value when x= 0 .

This does not, however, tell us the number of -intercepts. For example, the graph of $g(x)=x^{2}$ has as its minimum point (0,0) , and, subsequently, exactly one -intercept. The graph of $g(x)=x^{2} + 1$ has as its minimum point (0,1) and, subsequently, no -intercepts.

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Example Question #18 : X And Y Intercept

Ralph's teacher challenged him 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 (0, 4) .

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

Did Ralph succeed?

Statement 1: Ralph wrote a in the square.

Statement 2: Ralph wrote a in the triangle.

Possible Answers:

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

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

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

A x+ B y =C .

To find the -coordinate of the -intercept, set x = 0 and solve for :

$A \cdot 0 + B y =C$

B y =C

$y =\frac{C}{B}$

It is therefore necessary and sufficient to know the values of and - the values Ralph wrote in the circle and the triangle - in order to determine the -intercept of Ralph's equation. Statement 1 is irrelevant, and Statement 2 only provides the value Ralph wrote in the triangle.

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Example Question #19 : X And Y Intercept

Gloria's teacher challenged her to write numbers in the circle and the square in the pattern below in order to make an equation whose graph has -intercept (6,0) .

$y = \square x+ \bigcirc$

Did Gloria succeed?

Statement 1: Gloria wrote a in the square.

Statement 2: Gloria wrote a in the circle.

Possible Answers:

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.

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:

If we let stand for the number Gloria wrote in the square and stand for the number she wrote in the circle, the equation becomes

y = mx+b ,

the slope-intercept form of the equation of a line. We can find the -coordinate of the -intercept by setting y = 0 and solving for :

mx+b = 0

mx+b - b= 0 - b

mx = -b

$\frac{mx }{m}= \frac{-b}{m}$

$x = -\frac{b}{m}$

Therefore, it is necessary and sufficient to know both and - both of the numbers Gloria wrote - to determine the -intercept of the equation she made. Neither statement provides both values, but both statements together do.

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Example Question #591 : Geometry

In the -plane, the equation of line is

ax+by+c=0 $\left ( abc\neq 0 \right )$ .

The slope of line is 2. What is the value of ?

(1) a+b=4

(2) c=-3

Possible Answers:

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

EACH statement ALONE is sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Correct answer:

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Explanation:

The slope of line is $-\frac{a}{b}=2$

From statement (1) we get another function of and . Therefore, we can calculate the values of and .

From $-\frac{a}{b}=2$

we can get a=-2b .

Plug a=-2b into a+b=4 , then we can get

b=-4

and

$a=-2\cdot -4=8$

Statement (2) only tells us the value of , which is useless to get the value of , because we have three unknown numbers with only two equations given.

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Example Question #592 : Geometry

What is the equation of a vertical parabola on the coordinate plane?

Statement 1: The vertex of the parabola is (2,4) .

Statement 2: The parabola passes through the point (4,7) .

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:

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

Explanation:

Each statement alone gives one point of the parabola, which, even it is known to be the vertex, is not enough to determine its equation.

Assume both statements are true. The equation of a vertical parabola with vertex (h,k) can be written as

f(x) = a(x-h)+k

Statement 1 gives the values of and as 2 and 4, respectively, so the equation of the parabola is

$f(x) = a(x-2)^{2}+3$

for some .

From Statement 2, we can substitute 4 and 7 for and f(x) , respectively, and solve

$7= a(4-2)^{2}+3$

7=4a+3

for , yielding the complete equation:

4a = 4

a=1

This makes the equation $f(x) = (x-2)^{2}+3$ .

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #12 : X And Y Intercept

Example Question #13 : X And Y Intercept

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

Example Question #12 : Dsq: Calculating X Or Y Intercept

Example Question #13 : Dsq: Calculating X Or Y Intercept

Example Question #17 : X And Y Intercept

Example Question #18 : X And Y Intercept

Example Question #19 : X And Y Intercept

Example Question #591 : Geometry

Example Question #592 : Geometry

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