Geometry - GMAT Quantitative

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Question

Which of the following functions has as its graph a curve with -intercepts , , and ?

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Answer

A polynomial equation of degree 3 with solution set $\left \{ -2, 2, 3\right \}$ and leading term $x^{2}$ takes the form

We can rewrite this as follows:

$(x^{2}-2^{2})(x-3)= 0$

$(x^{2}-4)(x-3)= 0$

$x^{2}\cdot x- x^{2} \cdot 3 - 4\cdot x + 4 \cdot 3 = 0$

$x^{3} - 3 x^{2} - 4 x + 12 = 0$

The correct response is $f(x)= x^{3} - 3 x^{2} - 4 x + 12$ .

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Question

Which of the following functions has as its graph a curve with -intercepts , , and ?

Tap to reveal answer

Answer

A polynomial equation of degree 3 with solution set $\left \{ -2, 2, 3\right \}$ and leading term $x^{2}$ takes the form

We can rewrite this as follows:

$(x^{2}-2^{2})(x-3)= 0$

$(x^{2}-4)(x-3)= 0$

$x^{2}\cdot x- x^{2} \cdot 3 - 4\cdot x + 4 \cdot 3 = 0$

$x^{3} - 3 x^{2} - 4 x + 12 = 0$

The correct response is $f(x)= x^{3} - 3 x^{2} - 4 x + 12$ .

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Question

Which of the following functions has as its graph a curve with -intercepts , , and ?

Tap to reveal answer

Answer

A polynomial equation of degree 3 with solution set $\left \{ -2, 2, 3\right \}$ and leading term $x^{2}$ takes the form

We can rewrite this as follows:

$(x^{2}-2^{2})(x-3)= 0$

$(x^{2}-4)(x-3)= 0$

$x^{2}\cdot x- x^{2} \cdot 3 - 4\cdot x + 4 \cdot 3 = 0$

$x^{3} - 3 x^{2} - 4 x + 12 = 0$

The correct response is $f(x)= x^{3} - 3 x^{2} - 4 x + 12$ .

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Question

Which of the following functions has as its graph a curve with -intercepts , , and ?

Tap to reveal answer

Answer

A polynomial equation of degree 3 with solution set $\left \{ -2, 2, 3\right \}$ and leading term $x^{2}$ takes the form

We can rewrite this as follows:

$(x^{2}-2^{2})(x-3)= 0$

$(x^{2}-4)(x-3)= 0$

$x^{2}\cdot x- x^{2} \cdot 3 - 4\cdot x + 4 \cdot 3 = 0$

$x^{3} - 3 x^{2} - 4 x + 12 = 0$

The correct response is $f(x)= x^{3} - 3 x^{2} - 4 x + 12$ .

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Question

What is the median of the measures of the angles of a nonagon (a nine-sided polygon)?

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Answer

The sum of the measures of the nine angles of any nonagon is calculated as follows:

$180 \cdot \left ( 9-2\right ) =180 \cdot 7= 1,260^{\circ }$

The median of an odd quantity of numbers is the number that falls in the center position when they are arranged in ascending order; for nine numbers, it will be the fifth-highest number. We now need to show that we need to know the actual numbers in order to find the median.

Case 1: Each angle measures $140 ^{\circ }$ .

The set is $\left \{ 140,140,140,140,140,140,140,140,140\right \}$ and the median is 140.

Case 2: Eight of the angles measure $139 ^{\circ }$ and one of them measures $148 ^{\circ }$ .

The set is $\left \{}139, 139, 139, 139, 139, 139, 139, 139, 148 \right \}$ and the median is 139.

In both cases, the sum of the angle measures is 1,260, but the medians differ between the two.

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Question

Which of the following functions has as its graph a curve with -intercepts , , and ?

Tap to reveal answer

Answer

A polynomial equation of degree 3 with solution set $\left \{ -2, 2, 3\right \}$ and leading term $x^{2}$ takes the form

We can rewrite this as follows:

$(x^{2}-2^{2})(x-3)= 0$

$(x^{2}-4)(x-3)= 0$

$x^{2}\cdot x- x^{2} \cdot 3 - 4\cdot x + 4 \cdot 3 = 0$

$x^{3} - 3 x^{2} - 4 x + 12 = 0$

The correct response is $f(x)= x^{3} - 3 x^{2} - 4 x + 12$ .

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Question

A function is defined as

$f(x) = 2x^{4} + Bx^{3}+ Cx^{2}+ Dx +12$

where are integer coefficients whose values (which might be positive, negative, or zero) are not given. Which of the following cannot be an -intercept of the graph of no matter what the values of those three coefficients are?

Tap to reveal answer

Answer

Since the graph of a function has its -intercept at a point if and only if , finding possible -intercepts of the graph of is equivalent to finding a solution of . Since has integer coefficients, then by the Rational Zeroes Theorem, any rational solutions to the equation

$f(x) = 2x^{4} + Bx^{3}+ Cx^{2}+ Dx +12$

must be the quotient, or the (negative) opposite of the quotient, of a factor of constant coefficient 12 - that is, an element of $\left \{ 1, 2, 3, 4, 6, 12 \right \}$ - and a factor of leading coefficient 2 - that is, an element of $\left \{ 1,2 \right \}$ . Since all of the choices are positive, we will only look at possible positive solutions.

The quotients of an element of the first set and an element of the last are:

$\frac{1}{1} = 1$ ; $\frac{2}{1} = 2$ ; $\frac{3}{1} = 3$ ; $\frac{4}{1} = 4$ ; $\frac{6}{1} = 6$ ; $\frac{12}{1} = 12$ ;

$\frac{1}{2}$ ; $\frac{2}{2} = 1$ ; $\frac{3}{2}$ ; $\frac{4}{2} = 2$ ; $\frac{6}{2} = 3$ ; $\frac{12}{2} = 6$

Eliminating duplicates, the set of possible positive rational solutions to is

$\left \{ \frac{1}{2},1, \frac{3}{2} , 2,3,4,6,12 \right \}$ .

Of the five choices, only $\frac{1}{6}$ does not appear in the set of possible rational solutions of , so of the five choices, only $\left ( \frac{1}{6}, 0 \right )$ cannot be an -intercept of the graph.

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Question

What is the median of the measures of the angles of a nonagon (a nine-sided polygon)?

Tap to reveal answer

Answer

The sum of the measures of the nine angles of any nonagon is calculated as follows:

$180 \cdot \left ( 9-2\right ) =180 \cdot 7= 1,260^{\circ }$

The median of an odd quantity of numbers is the number that falls in the center position when they are arranged in ascending order; for nine numbers, it will be the fifth-highest number. We now need to show that we need to know the actual numbers in order to find the median.

Case 1: Each angle measures $140 ^{\circ }$ .

The set is $\left \{ 140,140,140,140,140,140,140,140,140\right \}$ and the median is 140.

Case 2: Eight of the angles measure $139 ^{\circ }$ and one of them measures $148 ^{\circ }$ .

The set is $\left \{}139, 139, 139, 139, 139, 139, 139, 139, 148 \right \}$ and the median is 139.

In both cases, the sum of the angle measures is 1,260, but the medians differ between the two.

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Question

Which of the following functions has as its graph a curve with -intercepts , , and ?

Tap to reveal answer

Answer

A polynomial equation of degree 3 with solution set $\left \{ -2, 2, 3\right \}$ and leading term $x^{2}$ takes the form

We can rewrite this as follows:

$(x^{2}-2^{2})(x-3)= 0$

$(x^{2}-4)(x-3)= 0$

$x^{2}\cdot x- x^{2} \cdot 3 - 4\cdot x + 4 \cdot 3 = 0$

$x^{3} - 3 x^{2} - 4 x + 12 = 0$

The correct response is $f(x)= x^{3} - 3 x^{2} - 4 x + 12$ .

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Question

A function is defined as

$f(x) = 2x^{4} + Bx^{3}+ Cx^{2}+ Dx +12$

where are integer coefficients whose values (which might be positive, negative, or zero) are not given. Which of the following cannot be an -intercept of the graph of no matter what the values of those three coefficients are?

Tap to reveal answer

Answer

Since the graph of a function has its -intercept at a point if and only if , finding possible -intercepts of the graph of is equivalent to finding a solution of . Since has integer coefficients, then by the Rational Zeroes Theorem, any rational solutions to the equation

$f(x) = 2x^{4} + Bx^{3}+ Cx^{2}+ Dx +12$

must be the quotient, or the (negative) opposite of the quotient, of a factor of constant coefficient 12 - that is, an element of $\left \{ 1, 2, 3, 4, 6, 12 \right \}$ - and a factor of leading coefficient 2 - that is, an element of $\left \{ 1,2 \right \}$ . Since all of the choices are positive, we will only look at possible positive solutions.

The quotients of an element of the first set and an element of the last are:

$\frac{1}{1} = 1$ ; $\frac{2}{1} = 2$ ; $\frac{3}{1} = 3$ ; $\frac{4}{1} = 4$ ; $\frac{6}{1} = 6$ ; $\frac{12}{1} = 12$ ;

$\frac{1}{2}$ ; $\frac{2}{2} = 1$ ; $\frac{3}{2}$ ; $\frac{4}{2} = 2$ ; $\frac{6}{2} = 3$ ; $\frac{12}{2} = 6$

Eliminating duplicates, the set of possible positive rational solutions to is

$\left \{ \frac{1}{2},1, \frac{3}{2} , 2,3,4,6,12 \right \}$ .

Of the five choices, only $\frac{1}{6}$ does not appear in the set of possible rational solutions of , so of the five choices, only $\left ( \frac{1}{6}, 0 \right )$ cannot be an -intercept of the graph.

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Question

What is the median of the measures of the angles of a nonagon (a nine-sided polygon)?

Tap to reveal answer

Answer

The sum of the measures of the nine angles of any nonagon is calculated as follows:

$180 \cdot \left ( 9-2\right ) =180 \cdot 7= 1,260^{\circ }$

The median of an odd quantity of numbers is the number that falls in the center position when they are arranged in ascending order; for nine numbers, it will be the fifth-highest number. We now need to show that we need to know the actual numbers in order to find the median.

Case 1: Each angle measures $140 ^{\circ }$ .

The set is $\left \{ 140,140,140,140,140,140,140,140,140\right \}$ and the median is 140.

Case 2: Eight of the angles measure $139 ^{\circ }$ and one of them measures $148 ^{\circ }$ .

The set is $\left \{}139, 139, 139, 139, 139, 139, 139, 139, 148 \right \}$ and the median is 139.

In both cases, the sum of the angle measures is 1,260, but the medians differ between the two.

← Didn't Know|Knew It →

Question

A function is defined as

$f(x) = 2x^{4} + Bx^{3}+ Cx^{2}+ Dx +12$

where are integer coefficients whose values (which might be positive, negative, or zero) are not given. Which of the following cannot be an -intercept of the graph of no matter what the values of those three coefficients are?

Tap to reveal answer

Answer

Since the graph of a function has its -intercept at a point if and only if , finding possible -intercepts of the graph of is equivalent to finding a solution of . Since has integer coefficients, then by the Rational Zeroes Theorem, any rational solutions to the equation

$f(x) = 2x^{4} + Bx^{3}+ Cx^{2}+ Dx +12$

must be the quotient, or the (negative) opposite of the quotient, of a factor of constant coefficient 12 - that is, an element of $\left \{ 1, 2, 3, 4, 6, 12 \right \}$ - and a factor of leading coefficient 2 - that is, an element of $\left \{ 1,2 \right \}$ . Since all of the choices are positive, we will only look at possible positive solutions.

The quotients of an element of the first set and an element of the last are:

$\frac{1}{1} = 1$ ; $\frac{2}{1} = 2$ ; $\frac{3}{1} = 3$ ; $\frac{4}{1} = 4$ ; $\frac{6}{1} = 6$ ; $\frac{12}{1} = 12$ ;

$\frac{1}{2}$ ; $\frac{2}{2} = 1$ ; $\frac{3}{2}$ ; $\frac{4}{2} = 2$ ; $\frac{6}{2} = 3$ ; $\frac{12}{2} = 6$

Eliminating duplicates, the set of possible positive rational solutions to is

$\left \{ \frac{1}{2},1, \frac{3}{2} , 2,3,4,6,12 \right \}$ .

Of the five choices, only $\frac{1}{6}$ does not appear in the set of possible rational solutions of , so of the five choices, only $\left ( \frac{1}{6}, 0 \right )$ cannot be an -intercept of the graph.

← Didn't Know|Knew It →

Question

What is the median of the measures of the angles of a nonagon (a nine-sided polygon)?

Tap to reveal answer

Answer

The sum of the measures of the nine angles of any nonagon is calculated as follows:

$180 \cdot \left ( 9-2\right ) =180 \cdot 7= 1,260^{\circ }$

The median of an odd quantity of numbers is the number that falls in the center position when they are arranged in ascending order; for nine numbers, it will be the fifth-highest number. We now need to show that we need to know the actual numbers in order to find the median.

Case 1: Each angle measures $140 ^{\circ }$ .

The set is $\left \{ 140,140,140,140,140,140,140,140,140\right \}$ and the median is 140.

Case 2: Eight of the angles measure $139 ^{\circ }$ and one of them measures $148 ^{\circ }$ .

The set is $\left \{}139, 139, 139, 139, 139, 139, 139, 139, 148 \right \}$ and the median is 139.

In both cases, the sum of the angle measures is 1,260, but the medians differ between the two.

← Didn't Know|Knew It →

Question

A function is defined as

$f(x) = 2x^{4} + Bx^{3}+ Cx^{2}+ Dx +12$

where are integer coefficients whose values (which might be positive, negative, or zero) are not given. Which of the following cannot be an -intercept of the graph of no matter what the values of those three coefficients are?

Tap to reveal answer

Answer

Since the graph of a function has its -intercept at a point if and only if , finding possible -intercepts of the graph of is equivalent to finding a solution of . Since has integer coefficients, then by the Rational Zeroes Theorem, any rational solutions to the equation

$f(x) = 2x^{4} + Bx^{3}+ Cx^{2}+ Dx +12$

must be the quotient, or the (negative) opposite of the quotient, of a factor of constant coefficient 12 - that is, an element of $\left \{ 1, 2, 3, 4, 6, 12 \right \}$ - and a factor of leading coefficient 2 - that is, an element of $\left \{ 1,2 \right \}$ . Since all of the choices are positive, we will only look at possible positive solutions.

The quotients of an element of the first set and an element of the last are:

$\frac{1}{1} = 1$ ; $\frac{2}{1} = 2$ ; $\frac{3}{1} = 3$ ; $\frac{4}{1} = 4$ ; $\frac{6}{1} = 6$ ; $\frac{12}{1} = 12$ ;

$\frac{1}{2}$ ; $\frac{2}{2} = 1$ ; $\frac{3}{2}$ ; $\frac{4}{2} = 2$ ; $\frac{6}{2} = 3$ ; $\frac{12}{2} = 6$

Eliminating duplicates, the set of possible positive rational solutions to is

$\left \{ \frac{1}{2},1, \frac{3}{2} , 2,3,4,6,12 \right \}$ .

Of the five choices, only $\frac{1}{6}$ does not appear in the set of possible rational solutions of , so of the five choices, only $\left ( \frac{1}{6}, 0 \right )$ cannot be an -intercept of the graph.

← Didn't Know|Knew It →

Question

What is the median of the measures of the angles of a nonagon (a nine-sided polygon)?

Tap to reveal answer

Answer

The sum of the measures of the nine angles of any nonagon is calculated as follows:

$180 \cdot \left ( 9-2\right ) =180 \cdot 7= 1,260^{\circ }$

The median of an odd quantity of numbers is the number that falls in the center position when they are arranged in ascending order; for nine numbers, it will be the fifth-highest number. We now need to show that we need to know the actual numbers in order to find the median.

Case 1: Each angle measures $140 ^{\circ }$ .

The set is $\left \{ 140,140,140,140,140,140,140,140,140\right \}$ and the median is 140.

Case 2: Eight of the angles measure $139 ^{\circ }$ and one of them measures $148 ^{\circ }$ .

The set is $\left \{}139, 139, 139, 139, 139, 139, 139, 139, 148 \right \}$ and the median is 139.

In both cases, the sum of the angle measures is 1,260, but the medians differ between the two.

← Didn't Know|Knew It →

Question

Which of the following functions has as its graph a curve with -intercepts , , and ?

Tap to reveal answer

Answer

A polynomial equation of degree 3 with solution set $\left \{ -2, 2, 3\right \}$ and leading term $x^{2}$ takes the form

We can rewrite this as follows:

$(x^{2}-2^{2})(x-3)= 0$

$(x^{2}-4)(x-3)= 0$

$x^{2}\cdot x- x^{2} \cdot 3 - 4\cdot x + 4 \cdot 3 = 0$

$x^{3} - 3 x^{2} - 4 x + 12 = 0$

The correct response is $f(x)= x^{3} - 3 x^{2} - 4 x + 12$ .

← Didn't Know|Knew It →

Question

A function is defined as

$f(x) = 2x^{4} + Bx^{3}+ Cx^{2}+ Dx +12$

where are integer coefficients whose values (which might be positive, negative, or zero) are not given. Which of the following cannot be an -intercept of the graph of no matter what the values of those three coefficients are?

Tap to reveal answer

Answer

Since the graph of a function has its -intercept at a point if and only if , finding possible -intercepts of the graph of is equivalent to finding a solution of . Since has integer coefficients, then by the Rational Zeroes Theorem, any rational solutions to the equation

$f(x) = 2x^{4} + Bx^{3}+ Cx^{2}+ Dx +12$

must be the quotient, or the (negative) opposite of the quotient, of a factor of constant coefficient 12 - that is, an element of $\left \{ 1, 2, 3, 4, 6, 12 \right \}$ - and a factor of leading coefficient 2 - that is, an element of $\left \{ 1,2 \right \}$ . Since all of the choices are positive, we will only look at possible positive solutions.

The quotients of an element of the first set and an element of the last are:

$\frac{1}{1} = 1$ ; $\frac{2}{1} = 2$ ; $\frac{3}{1} = 3$ ; $\frac{4}{1} = 4$ ; $\frac{6}{1} = 6$ ; $\frac{12}{1} = 12$ ;

$\frac{1}{2}$ ; $\frac{2}{2} = 1$ ; $\frac{3}{2}$ ; $\frac{4}{2} = 2$ ; $\frac{6}{2} = 3$ ; $\frac{12}{2} = 6$

Eliminating duplicates, the set of possible positive rational solutions to is

$\left \{ \frac{1}{2},1, \frac{3}{2} , 2,3,4,6,12 \right \}$ .

Of the five choices, only $\frac{1}{6}$ does not appear in the set of possible rational solutions of , so of the five choices, only $\left ( \frac{1}{6}, 0 \right )$ cannot be an -intercept of the graph.

← Didn't Know|Knew It →

Question

What is the median of the measures of the angles of a nonagon (a nine-sided polygon)?

Tap to reveal answer

Answer

The sum of the measures of the nine angles of any nonagon is calculated as follows:

$180 \cdot \left ( 9-2\right ) =180 \cdot 7= 1,260^{\circ }$

The median of an odd quantity of numbers is the number that falls in the center position when they are arranged in ascending order; for nine numbers, it will be the fifth-highest number. We now need to show that we need to know the actual numbers in order to find the median.

Case 1: Each angle measures $140 ^{\circ }$ .

The set is $\left \{ 140,140,140,140,140,140,140,140,140\right \}$ and the median is 140.

Case 2: Eight of the angles measure $139 ^{\circ }$ and one of them measures $148 ^{\circ }$ .

The set is $\left \{}139, 139, 139, 139, 139, 139, 139, 139, 148 \right \}$ and the median is 139.

In both cases, the sum of the angle measures is 1,260, but the medians differ between the two.

← Didn't Know|Knew It →

Question

A function is defined as

$f(x) = 2x^{4} + Bx^{3}+ Cx^{2}+ Dx +12$

where are integer coefficients whose values (which might be positive, negative, or zero) are not given. Which of the following cannot be an -intercept of the graph of no matter what the values of those three coefficients are?

Tap to reveal answer

Answer

Since the graph of a function has its -intercept at a point if and only if , finding possible -intercepts of the graph of is equivalent to finding a solution of . Since has integer coefficients, then by the Rational Zeroes Theorem, any rational solutions to the equation

$f(x) = 2x^{4} + Bx^{3}+ Cx^{2}+ Dx +12$

must be the quotient, or the (negative) opposite of the quotient, of a factor of constant coefficient 12 - that is, an element of $\left \{ 1, 2, 3, 4, 6, 12 \right \}$ - and a factor of leading coefficient 2 - that is, an element of $\left \{ 1,2 \right \}$ . Since all of the choices are positive, we will only look at possible positive solutions.

The quotients of an element of the first set and an element of the last are:

$\frac{1}{1} = 1$ ; $\frac{2}{1} = 2$ ; $\frac{3}{1} = 3$ ; $\frac{4}{1} = 4$ ; $\frac{6}{1} = 6$ ; $\frac{12}{1} = 12$ ;

$\frac{1}{2}$ ; $\frac{2}{2} = 1$ ; $\frac{3}{2}$ ; $\frac{4}{2} = 2$ ; $\frac{6}{2} = 3$ ; $\frac{12}{2} = 6$

Eliminating duplicates, the set of possible positive rational solutions to is

$\left \{ \frac{1}{2},1, \frac{3}{2} , 2,3,4,6,12 \right \}$ .

Of the five choices, only $\frac{1}{6}$ does not appear in the set of possible rational solutions of , so of the five choices, only $\left ( \frac{1}{6}, 0 \right )$ cannot be an -intercept of the graph.

← Didn't Know|Knew It →

Question

What is the median of the measures of the angles of a nonagon (a nine-sided polygon)?

Tap to reveal answer

Answer

The sum of the measures of the nine angles of any nonagon is calculated as follows:

$180 \cdot \left ( 9-2\right ) =180 \cdot 7= 1,260^{\circ }$

The median of an odd quantity of numbers is the number that falls in the center position when they are arranged in ascending order; for nine numbers, it will be the fifth-highest number. We now need to show that we need to know the actual numbers in order to find the median.

Case 1: Each angle measures $140 ^{\circ }$ .

The set is $\left \{ 140,140,140,140,140,140,140,140,140\right \}$ and the median is 140.

Case 2: Eight of the angles measure $139 ^{\circ }$ and one of them measures $148 ^{\circ }$ .

The set is $\left \{}139, 139, 139, 139, 139, 139, 139, 139, 148 \right \}$ and the median is 139.

In both cases, the sum of the angle measures is 1,260, but the medians differ between the two.

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