GMAT Math : Graphing

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Example Questions

Example Question #1 : Dsq: Graphing Complex Numbers

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

From the number a + bi $\displaystyle a + bi$, subtract its complex conjugate. What is the result?

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

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

Possible Answers:

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 insufficient 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:

The complex conjugate of an imaginary number a + bi $\displaystyle a + bi$ is a - bi $\displaystyle a - bi$, and

$\displaystyle (a + bi)- (a - bi) = a -a +bi+bi = 2bi$.

Therefore, it is necessary and sufficient to know $\displaystyle b$ in order to answer the question. Statement 1 does not give this value, and is unhelpful here; Statement 2 does give this value.

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Example Question #2 : Dsq: Graphing Complex Numbers

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

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

Statement 1: a = 12 $\displaystyle a = 12$

Statement 2: b= 18 $\displaystyle b= 18$

Possible Answers:

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.

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 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 the sum of the two numbers is

$\displaystyle (a + bi)+ (a - bi) = 2a$.

Therefore, it is necessary and sufficient to know $\displaystyle a$ in order to answer the question. Statement 1 alone gives this information; Statement 2 does not, and it is unhelpful.

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Example Question #531 : Geometry

Let $\displaystyle N$ be a positive integer.

True or false: $i^{N} = 1$ $\displaystyle i^{N} = 1$

Statement 1: $\displaystyle N$ is a prime number.

Statement 2: $\displaystyle N$ is a two-digit number ending in a 7.

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 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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

If $\displaystyle N$ is a positive integer, then $i^{N} = 1$ $\displaystyle i^{N} = 1$ if and only if $\displaystyle N$ is a multiple of 4.

It follows that if $i^{N} = 1$ $\displaystyle i^{N} = 1$, $\displaystyle N$ cannot be a prime number. Also, every multiple of 4 is even, so as an even number, $\displaystyle N$ cannot end in 7. Contrapositively, if Statement 1 is true and $\displaystyle N$ is prime, or if Statement 2 is true and if $\displaystyle N$ ends in 7, it follows that $\displaystyle N$ is not a multiple of 4, and $i^{N} \ne 1$ $\displaystyle i^{N} \ne 1$.

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Example Question #4 : 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 = 12 $\displaystyle a = 12$

Statement 2: $b = \sqrt{2}$ $\displaystyle b = \sqrt{2}$

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:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is 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, it is necessary and sufficient to know the values of both $\displaystyle a$ and $\displaystyle b$ in order to answer the problem. Each statement alone gives only one of these values, so each statement alone provides insufficient information; the two together give both, so the two statements together provide sufficient information.

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Example Question #532 : Geometry

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

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

Statement 1: $a ^{2}+ b^{2}= 50$ $\displaystyle a ^{2}+ b^{2}= 50$

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

Possible Answers:

EITHER 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.

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:

The complex conjugate of an imaginary number a + bi $\displaystyle a + bi$ is a - bi $\displaystyle a - bi$, and

$\displaystyle (a + bi)+ (a - bi) = 2a$.

We show, however, that the two statements are insufficient to determine the sum by examining two scenarios.

Case 1: $\displaystyle a = b= 5$.

$a ^{2}+ b^{2}=5^{2}+ 5^{2} = 25+25 = 50$ $\displaystyle a ^{2}+ b^{2}=5^{2}+ 5^{2} = 25+25 = 50$, and since $\displaystyle a = b= 5$, $a ^{2} = b^{2}=5 ^{2} =25$ $\displaystyle a ^{2} = b^{2}=5 ^{2} =25$. The conditions of both statements are satisfied.

The sum of the numbers is $2a = 2 \cdot 5 = 10$ $\displaystyle 2a = 2 \cdot 5 = 10$.

Case 2: $\displaystyle a = b= -5$.

$a ^{2}+ b^{2}=(-5)^{2}+ (-5)^{2} = 25+25 = 50$ $\displaystyle a ^{2}+ b^{2}=(-5)^{2}+ (-5)^{2} = 25+25 = 50$, and since $\displaystyle a = b= -5$, $a ^{2} = b^{2}=(-5) ^{2}= 25$ $\displaystyle a ^{2} = b^{2}=(-5) ^{2}= 25$. The conditions of both statements are satisfied.

The sum of the numbers is $2a = 2 \cdot (-5) = - 10$ $\displaystyle 2a = 2 \cdot (-5) = - 10$.

In both cases, the conditions of both statements are satisfied, but the sum of the number and its complex conjugate differs between the two.

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Example Question #11 : Coordinate Geometry

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: $\displaystyle a + b = 12$

Statement 2: $\displaystyle a - b = 16$

Possible Answers:

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.

BOTH statements TOGETHER are insufficient 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:

The complex conjugate of an imaginary number a + bi $\displaystyle a + bi$ is a - bi $\displaystyle a - bi$, and

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

Statement 1 alone provides insufficient information, as seen in these two scenarios, both of which feature values of $\displaystyle a$ and $\displaystyle b$ that add up to 12:

Case 1: $\displaystyle a = b= 6$

Then $\displaystyle a + bi = 6+6i$, and the product of this number and its complex conjugate is $a^{2}+b^{2} = 6^{2}+6^{2} =36+36 = 72$ $\displaystyle a^{2}+b^{2} = 6^{2}+6^{2} =36+36 = 72$.

Case 2: $\displaystyle a =5, b = 7$

Then $\displaystyle a + bi = 5+7i$, and the product of this number and its complex conjugate is $a^{2}+b^{2} = 5^{2}+7^{2} =25+49= 74$ $\displaystyle a^{2}+b^{2} = 5^{2}+7^{2} =25+49= 74$.

The two cases result in different products.

For a similar reason, Statement 2 alone provides insufficient information.

If both statements are assumed to be true, they form a system of equations that can be solved as follows:

$\displaystyle a + b = 12$

$\underline{a - b = 16}$ $\displaystyle \underline{a - b = 16}$

$\displaystyle 2a$ =28 $\displaystyle =28$

a = 14 $\displaystyle a = 14$

Backsolve:

$\displaystyle 14 + b = 12$

b=-2 $\displaystyle b=-2$

Since we know that a = 14 $\displaystyle a = 14$ and b=-2 $\displaystyle b=-2$, then we know that the desired product is $a^{2}+b^{2} =14 ^{2}+(-2)^{2} =196+4 = 200$ $\displaystyle a^{2}+b^{2} =14 ^{2}+(-2)^{2} =196+4 = 200$.

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Example Question #7 : Dsq: Graphing Complex Numbers

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

From the number a + bi $\displaystyle a + bi$, subtract its complex conjugate. What is the result?

Statement 1: $a ^{2}- b^{2}= 36$ $\displaystyle a ^{2}- b^{2}= 36$

Statement 2: $\displaystyle a +b = 12$

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.

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.

Correct answer:

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

$\displaystyle (a + bi)- (a - bi) = a -a +bi+bi = 2bi$.

Therefore, it is necessary and sufficient to know $\displaystyle b$ in order to answer the question. Neither statement alone gives this information. However, the first statement can be rewritten by factoring out $a ^{2}- b^{2}$ $\displaystyle a ^{2}- b^{2}$ as a difference of squares:

$a ^{2}- b^{2}= 36$ $\displaystyle a ^{2}- b^{2}= 36$

$\displaystyle (a+b)(a-b)= 36$

Since $\displaystyle a +b = 12$, then by substitution,

$\displaystyle 12(a-b)= 36$

$\frac{12(a-b)}{12}= \frac{36}{12}$ $\displaystyle \frac{12(a-b)}{12}= \frac{36}{12}$

a -b=3 $\displaystyle a -b=3$

A system of linear equations has now been formed; subtract both sides of the equations as follows:

$\displaystyle a +b = 12$

$-\underline{(a -b=3)}$ $\displaystyle -\underline{(a -b=3)}$

$\displaystyle 2b$ = 9 $\displaystyle = 9$

We need go no further; since 2b=9 $\displaystyle 2b=9$, the desired difference is $\displaystyle (2b)i = 9i$.

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Example Question #8 : Dsq: Graphing Complex Numbers

Let $\displaystyle N$ be a positive integer.

Evaluate $i^{N}$ $\displaystyle i^{N}$.

Statement 1: $\displaystyle N$ is a multiple of 16.

Statement 2: $\displaystyle N$ is a multiple of 20.

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.

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The value of $i^{N}$ $\displaystyle i^{N}$, $\displaystyle N$ a positive integer, is equal to $i^{R}$ $\displaystyle i^{R}$, where $\displaystyle R$ is the remainder of the division of $\displaystyle N$ by 4. Either statement alone is enough to prove that $\displaystyle N$ is divisible by 4, since, if a number is divisible by a given number (16 or 20 in these statements), it is divisible by any factor of that number (with 4 being a factor of both).

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Example Question #9 : Dsq: Graphing Complex Numbers

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

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

Statement 1: $a+ b = 12\sqrt{2}$ $\displaystyle a+ b = 12\sqrt{2}$

Statement 2: $a - b = 4\sqrt{2}$ $\displaystyle a - b = 4\sqrt{2}$

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:

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

$\displaystyle (a + bi)+ (a - bi) = 2a$.

Therefore, it is necessary and sufficient to know $\displaystyle a$ in order to answer the question. Neither statement alone gives this information. However, the two statements together form a linear system that can be solved as follows:

$a+ b = 12\sqrt{2}$ $\displaystyle a+ b = 12\sqrt{2}$

$+( \underline{a - b = 4\sqrt{2}})$ $\displaystyle +( \underline{a - b = 4\sqrt{2}})$

$\displaystyle 2a$ $=16 \sqrt{2}$ $\displaystyle =16 \sqrt{2}$

We need go no further; since $2a=16 \sqrt{2}$ $\displaystyle 2a=16 \sqrt{2}$, this is the desired sum.

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Example Question #11 : Coordinate Geometry

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

From the number a + bi $\displaystyle a + bi$, subtract its complex conjugate. What is the result?

Statement 1: a+b= 10 $\displaystyle a+b= 10$

Statement 2: $\displaystyle a -b = 4$

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.

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:

The complex conjugate of an imaginary number a + bi $\displaystyle a + bi$ is a - bi $\displaystyle a - bi$, and

$\displaystyle (a + bi)- (a - bi) = a -a +bi+bi = 2bi$.

Therefore, it is necessary and sufficient to know $\displaystyle b$ in order to answer the question. Neither statement alone gives this information. However, the two statements together form a linear system that can be solved as follows:

a+b= 10 $\displaystyle a+b= 10$

$- \underline{(a -b = 4)}$ $\displaystyle - \underline{(a -b = 4)}$

$\displaystyle 2b$ $\displaystyle =6$

We need go no further; since 2b=6 $\displaystyle 2b=6$, the desired sum is $\displaystyle (2b)i = 6i$.

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GMAT Math : Graphing

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Dsq: Graphing Complex Numbers

Example Question #2 : Dsq: Graphing Complex Numbers

Example Question #531 : Geometry

Example Question #4 : Dsq: Graphing Complex Numbers

Example Question #532 : Geometry

Example Question #11 : Coordinate Geometry

Example Question #7 : Dsq: Graphing Complex Numbers

Example Question #8 : Dsq: Graphing Complex Numbers

Example Question #9 : Dsq: Graphing Complex Numbers

Example Question #11 : Coordinate Geometry

All GMAT Math Resources

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