GMAT Math : Coordinate Geometry

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Example Questions

Example Question #1 : Dsq: Graphing A Logarithm

Define a function $\displaystyle f$ as follows:

$f(x) = A \ln (Bx + C) +D$ $\displaystyle f(x) = A \ln (Bx + C) +D$

for nonzero real numbers A,B,C,D $\displaystyle A,B,C,D$.

Where is the vertical asymptote of the graph of $\displaystyle f$ in relation to the $\displaystyle y$-axis - is it to the left of it, to the right of it, or on it?

Statement 1: $\displaystyle A$ and $\displaystyle C$ are both positive.

Statement 2: $\displaystyle B$ and $\displaystyle D$ are of opposite sign.

Possible Answers:

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.

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:

Since only positive numbers have logarithms, the expression Bx+C $\displaystyle Bx+C$ must be positive, so

$\displaystyle Bx+C > 0$

Bx > -C $\displaystyle Bx > -C$

$x > - \frac{C}{B}$ $\displaystyle x > - \frac{C}{B}$

Therefore, the vertical asymptote must be the vertical line of the equation

$x = - \frac{C}{B}$ $\displaystyle x = - \frac{C}{B}$.

In order to determine which side of the $\displaystyle y$-axis the vertical asymptote falls, it is necessary to find the sign of $- \frac{C}{B}$ $\displaystyle - \frac{C}{B}$ ; if it is negative, it is on the left side, if it is positive, it is on the right side.

Assume both statements are true. By Statement 1, $\displaystyle C$ is positive. If $\displaystyle B$ is positive, then $- \frac{C}{B}$ $\displaystyle - \frac{C}{B}$ is negative, and vice versa. However, Statement 2, which mentions $\displaystyle B$, does not give its actual sign - just the fact that its sign is the opposite of that of $\displaystyle D$, which we are not given either. The two statements therefore give insufficient information.

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Example Question #2751 : Gmat Quantitative Reasoning

Define a function $\displaystyle f$ as follows:

$f(x) = A \ln (Bx + C) +D$ $\displaystyle f(x) = A \ln (Bx + C) +D$

for nonzero real numbers A,B,C,D $\displaystyle A,B,C,D$.

Give the equation of the vertical asymptote of the graph of $\displaystyle f$.

Statement 1: B = 7 $\displaystyle B = 7$

Statement 2: C = 10 $\displaystyle C = 10$

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.

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.

Correct answer:

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

Explanation:

Since a logarithm of a nonpositive number cannot be taken,

$\displaystyle Bx+C > 0$

Bx > -C $\displaystyle Bx > -C$

$x > - \frac{C}{B}$ $\displaystyle x > - \frac{C}{B}$

Therefore, the vertical asymptote must be the vertical line of the equation

$x = - \frac{C}{B}$ $\displaystyle x = - \frac{C}{B}$.

Each of Statement 1 and Statement 2 gives us only one of $\displaystyle B$ and $\displaystyle C$. However, the two together tell us that

$- \frac{C}{B} =- \frac{10}{7}$ $\displaystyle - \frac{C}{B} =- \frac{10}{7}$

making the vertical asymptote

$x=- \frac{10}{7}$ $\displaystyle x=- \frac{10}{7}$.

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Example Question #3 : Coordinate Geometry

Define a function $\displaystyle f$ as follows:

$f(x) = A \ln (Bx + C) +D$ $\displaystyle f(x) = A \ln (Bx + C) +D$

for nonzero real numbers A,B,C,D $\displaystyle A,B,C,D$.

Where is the vertical asymptote of the graph of $\displaystyle f$ in relation to the $\displaystyle y$-axis - is it to the left of it, to the right of it, or on it?

Statement 1: B+C = 5 $\displaystyle B+C = 5$

Statement 2: $\displaystyle BC = -36$

Possible Answers:

EITHER 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.

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.

Correct answer:

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

Explanation:

Only positive numbers have logarithms, so:

$\displaystyle Bx+C > 0$

Bx > -C $\displaystyle Bx > -C$

$x > - \frac{C}{B}$ $\displaystyle x > - \frac{C}{B}$

Therefore, the vertical asymptote must be the vertical line of the equation

$x = - \frac{C}{B}$ $\displaystyle x = - \frac{C}{B}$.

In order to determine which side of the $\displaystyle y$-axis the vertical asymptote falls, it is necessary to find out whether the signs of $\displaystyle B$ and $\displaystyle C$ are the same or different. If $\displaystyle B$ and $\displaystyle C$ are of the same sign, then their quotient $\frac{C}{B}$ $\displaystyle \frac{C}{B}$ is positive, and $- \frac{C}{B}$ $\displaystyle - \frac{C}{B}$ is negative, putting $x = - \frac{C}{B}$ $\displaystyle x = - \frac{C}{B}$ on the left side of the $\displaystyle x$-axis. If $\displaystyle B$ and $\displaystyle C$ are of different sign, then their quotient $\frac{C}{B}$ $\displaystyle \frac{C}{B}$ is negative, and $- \frac{C}{B}$ $\displaystyle - \frac{C}{B}$ is positive, putting $x = - \frac{C}{B}$ $\displaystyle x = - \frac{C}{B}$ on the right side of the $\displaystyle x$-axis.

Statement 1 alone does not give us enough information to determine whether $\displaystyle B$ and $\displaystyle C$ have different signs. $\displaystyle 2 + 3 = 5$, for example, but $\displaystyle 6 + (-1) = 5$, also.

From Statement 2, since the product of $\displaystyle B$ and $\displaystyle C$ is negative, they must be of different sign. Therefore, $- \frac{C}{B}$ $\displaystyle - \frac{C}{B}$ is positive, and $x = - \frac{C}{B}$ $\displaystyle x = - \frac{C}{B}$ falls to the right of the $\displaystyle y$-axis.

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Example Question #1 : Graphing

Define a function $\displaystyle f$ as follows:

$f(x) = A \ln (Bx + C) +D$ $\displaystyle f(x) = A \ln (Bx + C) +D$

for nonzero real numbers A,B,C,D $\displaystyle A,B,C,D$.

Give the equation of the vertical asymptote of the graph of $\displaystyle f$.

Statement 1: $\frac{B}{C} = \frac{2}{3}$ $\displaystyle \frac{B}{C} = \frac{2}{3}$

Statement 2: $\frac{A}{D} = 2$ $\displaystyle \frac{A}{D} = 2$

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:

Only positive numbers have logarithms, so:

$\displaystyle Bx+C > 0$

Bx > -C $\displaystyle Bx > -C$

$x > - \frac{C}{B}$ $\displaystyle x > - \frac{C}{B}$

Therefore, the vertical asymptote must be the vertical line of the equation

$x = - \frac{C}{B}$ $\displaystyle x = - \frac{C}{B}$.

Statement 1 alone gives that $\frac{B}{C} = \frac{2}{3}$ $\displaystyle \frac{B}{C} = \frac{2}{3}$. $\frac{C} {B}$ $\displaystyle \frac{C} {B}$ is the reciprocal of this, or $\frac{3}{2}$ $\displaystyle \frac{3}{2}$, and $- \frac{C}{B} = -\frac{3}{2}$ $\displaystyle - \frac{C}{B} = -\frac{3}{2}$, so the vertical asymptote is $x= -\frac{3}{2}$ $\displaystyle x= -\frac{3}{2}$.

Statement 2 alone gives no clue about either $\displaystyle B$, $\displaystyle C$, or their relationship.

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Example Question #2 : Graphing

Define a function $\displaystyle f$ as follows:

$f(x) = A \ln (Bx + C) +D$ $\displaystyle f(x) = A \ln (Bx + C) +D$

for nonzero real numbers A,B,C,D $\displaystyle A,B,C,D$.

Give the equation of the vertical asymptote of the graph of $\displaystyle f$.

Statement 1: B+C = 7 $\displaystyle B+C = 7$

Statement 2: BC = 12 $\displaystyle BC = 12$

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.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER 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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Since only positive numbers have logarithms,

$\displaystyle Bx+C > 0$

Bx > -C $\displaystyle Bx > -C$

$x > - \frac{C}{B}$ $\displaystyle x > - \frac{C}{B}$

Therefore, the vertical asymptote must be the vertical line of the equation

$x = - \frac{C}{B}$ $\displaystyle x = - \frac{C}{B}$.

Assume both statements to be true. We need two numbers $\displaystyle B$ and $\displaystyle C$ whose sum is 7 and whose product is 12; by trial and error, we can find these numbers to be 3 and 4. However, without further information, we have no way of determining which of $\displaystyle B$ and $\displaystyle C$ is 3 and which is 4, so the asymptote can be either $x = - \frac{3}{4}$ $\displaystyle x = - \frac{3}{4}$ or $x = - \frac{4}{3}$ $\displaystyle x = - \frac{4}{3}$.

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Example Question #3 : Graphing

Define a function $\displaystyle f$ as follows:

$f(x) = A \ln (Bx + C) +D$ $\displaystyle f(x) = A \ln (Bx + C) +D$

for nonzero real numbers A,B,C,D $\displaystyle A,B,C,D$.

Does the graph of $\displaystyle f$ have a $\displaystyle y$-intercept?

Statement 1: C = 8 $\displaystyle C = 8$.

Statement 2: D = 9 $\displaystyle D = 9$.

Possible Answers:

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.

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.

Correct answer:

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

Explanation:

The $\displaystyle y$-intercept of the graph of the function $f(x) = A \ln (Bx + C) +D$ $\displaystyle f(x) = A \ln (Bx + C) +D$, if there is one, occurs at the point with $\displaystyle x$-coordinate 0. Therefore, we find f(0) $\displaystyle f(0)$:

$f(0) = A \ln (B \cdot 0 + C) +D = A \ln C + D$ $\displaystyle f(0) = A \ln (B \cdot 0 + C) +D = A \ln C + D$

This expression is defined if and only if $\displaystyle C$ is a positive value. Statement 1 gives $\displaystyle C$ as positive, so it follows that the graph indeed has a $\displaystyle y$-intercept. Statement 2, which only gives $\displaystyle D$, is irrelevant.

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Example Question #7 : Coordinate Geometry

Define a function $\displaystyle f$ as follows:

$f(x) = A \ln (Bx + C) +D$ $\displaystyle f(x) = A \ln (Bx + C) +D$

for nonzero real numbers A,B,C,D $\displaystyle A,B,C,D$.

Where is the vertical asymptote of the graph of $\displaystyle f$ in relation to the $\displaystyle y$-axis - is it to the left of it, to the right of it, or on it?

Statement 1: $\displaystyle A$ and $\displaystyle D$ are both positive.

Statement 2: $\displaystyle B$ and $\displaystyle C$ are of opposite sign.

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.

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:

Since only positive numbers have logarithms,

$\displaystyle Bx+C > 0$

Bx > -C $\displaystyle Bx > -C$

$x > - \frac{C}{B}$ $\displaystyle x > - \frac{C}{B}$

Therefore, the vertical asymptote must be the vertical line of the equation

$x = - \frac{C}{B}$ $\displaystyle x = - \frac{C}{B}$.

Statement 1 gives irrelevant information. But Statement 2 alone gives sufficient information; since $\displaystyle B$ and $\displaystyle C$ are of opposite sign, their quotient $\frac{C}{B}$ $\displaystyle \frac{C}{B}$ is negative, and $- \frac{C}{B}$ $\displaystyle - \frac{C}{B}$ is positive. This locates the vertical asymptote on the right side of the $\displaystyle y$-axis.

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Example Question #2752 : Gmat Quantitative Reasoning

Define a function $\displaystyle f$ as follows:

$f(x) = A \ln (Bx + C) +D$ $\displaystyle f(x) = A \ln (Bx + C) +D$

for nonzero real numbers A,B,C,D $\displaystyle A,B,C,D$.

What is the equation of the vertical asymptote of the graph of $\displaystyle f$ ?

Statement 1: $\displaystyle A$ and $\displaystyle B$ are of opposite sign.

Statement 2: A+C = 0 $\displaystyle A+C = 0$

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:

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

Explanation:

Since only positive numbers have logarithms,

$\displaystyle Bx+C > 0$

Bx > -C $\displaystyle Bx > -C$

$x > - \frac{C}{B}$ $\displaystyle x > - \frac{C}{B}$

Therefore, the vertical asymptote must be the vertical line of the equation

$x = - \frac{C}{B}$ $\displaystyle x = - \frac{C}{B}$.

In order to determine which side of the $y\:$ $\displaystyle y\:$-axis the vertical asymptote falls, it is necessary to find the sign of $- \frac{C}{B}$ $\displaystyle - \frac{C}{B}$ ; if it is negative, it is on the left side, and if it is positive, it is on the right side.

Statement 1 alone only gives us that $\displaystyle B$ is a different sign from $\displaystyle A$; without any information about the sign of $\displaystyle C$, we cannot answer the question.

Statement 2 alone gives us that A+C = 0 $\displaystyle A+C = 0$, and, consequently, A = -C $\displaystyle A = -C$. This means that $\displaystyle A$ and $\displaystyle C$ are of opposite sign. But again, with no information about the sign of $\displaystyle B$, we cannot answer the question.

Assume both statements to be true. Since, from the two statements, both $\displaystyle B$ and $\displaystyle C$ are of the opposite sign from $\displaystyle A$, $\displaystyle B$ and $\displaystyle C$ are of the same sign. Their quotient $\frac{C}{B}$ $\displaystyle \frac{C}{B}$ is positive, and $- \frac{C}{B}$ $\displaystyle - \frac{C}{B}$ is negative, so the vertical asymptote $x = - \frac{C}{B}$ $\displaystyle x = - \frac{C}{B}$ is to the left of the $\displaystyle y$-axis.

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Example Question #9 : Coordinate Geometry

Define a function $\displaystyle f$ as follows:

$f(x) = A \ln (Bx + C) +D$ $\displaystyle f(x) = A \ln (Bx + C) +D$

for nonzero real numbers A,B,C,D $\displaystyle A,B,C,D$.

Does the graph of $\displaystyle f$ have a $\displaystyle y$-intercept?

Statement 1: A> D $\displaystyle A> D$.

Statement 2: $\displaystyle B$ and $\displaystyle C$ have different signs.

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.

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.

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 insufficient to answer the question.

Explanation:

$f(0) = A \ln (B \cdot 0 + C) +D = A \ln C + D$ $\displaystyle f(0) = A \ln (B \cdot 0 + C) +D = A \ln C + D$

This expression is defined if and only if $\displaystyle C$ is a positive value. However, the two statements together do not give this information; the values of $\displaystyle A$ and $\displaystyle D$ from Statement 1 are irrelevant, and Statement 2 does not reveal which of $\displaystyle B$ and $\displaystyle C$ is positive and which is negative.

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Example Question #1 : Dsq: Graphing Complex Numbers

Let $\displaystyle a$ and $\displaystyle b$ be real numbers.

What is the product of a + bi $\displaystyle a + bi$ and its complex conjugate?

Statement 1: $a^{2}+b^{2}= 71$ $\displaystyle a^{2}+b^{2}= 71$

Statement 2: $a^{2}-b^{2}= 39$ $\displaystyle a^{2}-b^{2}= 39$

Possible Answers:

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.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT 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 complex conjugate of an imaginary number a + bi $\displaystyle a + bi$ is a - bi $\displaystyle a - bi$, and

$(a + bi)(a - bi) = a^{2}+b^{2}$ $\displaystyle (a + bi)(a - bi) = a^{2}+b^{2}$.

Therefore, Statement 1 alone, which gives that $a^{2}+b^{2} = 71$ $\displaystyle a^{2}+b^{2} = 71$, provides sufficient information to answer the question, whereas Statement 2 provides unhelpful information.

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GMAT Math : Coordinate Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Dsq: Graphing A Logarithm

Example Question #2751 : Gmat Quantitative Reasoning

Example Question #3 : Coordinate Geometry

Example Question #1 : Graphing

Example Question #2 : Graphing

Example Question #3 : Graphing

Example Question #7 : Coordinate Geometry

Example Question #2752 : Gmat Quantitative Reasoning

Example Question #9 : Coordinate Geometry

Example Question #1 : Dsq: Graphing Complex Numbers

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