GMAT Math : DSQ: Graphing complex numbers

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Example Question #10 :Coordinate Geometry

Letandbe real numbers.

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

声明1: $a^{2}+b^{2}= 71$

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

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.

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.

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 is a - bi , and

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

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

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Example Question #11 :Coordinate Geometry

Letandbe real numbers.

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

声明1: a = 7

Statement 2: b = 12

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.

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.

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 is a - bi , and

(a + bi)- (a - bi) = a -a +bi+bi = 2bi .

Therefore, it is necessary and sufficient to knowin 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 #12 :Coordinate Geometry

Letandbe real numbers.

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

声明1: a = 12

Statement 2: b= 18

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:

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 is a - bi , and the sum of the two numbers is

(a + bi)+ (a - bi) = 2a .

Therefore, it is necessary and sufficient to knowin 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 #13 :Coordinate Geometry

Letbe a positive integer.

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

声明1:is a prime number.

Statement 2:is a two-digit number ending in a 7.

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 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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Ifis a positive integer, then $i^{N} = 1$ if and only ifis a multiple of 4.

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

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Example Question #1 :Dsq: Graphing Complex Numbers

Letandbe real numbers.

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

声明1: a = 12

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

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 insufficient 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:

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

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

Therefore, it is necessary and sufficient to know the values of bothandin 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 #15 :Coordinate Geometry

Letandbe real numbers.

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

声明1: $a ^{2}+ b^{2}= 50$

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

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 insufficient to answer the question.

Explanation:

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

(a + bi)+ (a - bi) = 2a .

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

Case 1: a = b= 5 .

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

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

Case 2: a = b= -5 .

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

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

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 #16 :Coordinate Geometry

Letandbe real numbers.

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

声明1: a + b = 12

Statement 2: a - b = 16

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:

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 is a - bi , and

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

Statement 1 alone provides insufficient information, as seen in these two scenarios, both of which feature values ofandthat add up to 12:

Case 1: a = b= 6

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

Case 2: a =5, b = 7

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

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:

a + b = 12

$\underline{a - b = 16}$

=28

a = 14

Backsolve:

14 + b = 12

b=-2

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

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Example Question #17 :Coordinate Geometry

Letandbe real numbers.

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

声明1: $a ^{2}- b^{2}= 36$

Statement 2: a +b = 12

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 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.

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 is a - bi , and

(a + bi)- (a - bi) = a -a +bi+bi = 2bi .

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

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

(a+b)(a-b)= 36

Since a +b = 12 , then by substitution,

12(a-b)= 36

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

a -b=3

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

a +b = 12

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

= 9

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

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Example Question #18 :Coordinate Geometry

Letbe a positive integer.

Evaluate $i^{N}$ .

声明1:is a multiple of 16.

Statement 2:is a multiple of 20.

Possible Answers:

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.

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

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The value of $i^{N}$ ,a positive integer, is equal to $i^{R}$ , whereis the remainder of the division ofby 4. Either statement alone is enough to prove that能被4整除,因为,如果一个数字是可分的吗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 #19 :Coordinate Geometry

Letandbe real numbers.

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

声明1: $a+ b = 12\sqrt{2}$

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

Possible Answers:

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.

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 is a - bi , and

(a + bi)+ (a - bi) = 2a .

Therefore, it is necessary and sufficient to knowin 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:

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

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

$=16 \sqrt{2}$

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

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GMAT Math : DSQ: Graphing complex numbers

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #10 :Coordinate Geometry

Example Question #11 :Coordinate Geometry

Example Question #12 :Coordinate Geometry

Example Question #13 :Coordinate Geometry

Example Question #1 :Dsq: Graphing Complex Numbers

Example Question #15 :Coordinate Geometry

Example Question #16 :Coordinate Geometry

Example Question #17 :Coordinate Geometry

Example Question #18 :Coordinate Geometry

Example Question #19 :Coordinate Geometry

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