GMAT Math : Data-Sufficiency Questions

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Example Questions

Example Question #5 : Dsq: Graphing Complex Numbers

Let and be real numbers.

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

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

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

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.

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.

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 #6 : Dsq: Graphing Complex Numbers

Let and be real numbers.

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

Statement 1: a + b = 12

Statement 2: 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.

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.

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 of and that 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 #7 : Dsq: Graphing Complex Numbers

Let and be real numbers.

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

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

Statement 2: 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 is a - bi , and

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

Therefore, it is necessary and sufficient to know in order to answer the question. Neither statement alone gives this information. However, the first statement can be rewritten by factoring out $a ^{2}- b^{2}$ 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 #8 : Dsq: Graphing Complex Numbers

Let be a positive integer.

Evaluate $i^{N}$ .

Statement 1: is a multiple of 16.

Statement 2: 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 $i^{N}$ , a positive integer, is equal to $i^{R}$ , where is the remainder of the division of by 4. Either statement alone is enough to prove that 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 and be real numbers.

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

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

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

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 is a - bi , and

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

Therefore, it is necessary and sufficient to know 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:

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

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

$=16 \sqrt{2}$

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

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Example Question #11 : Coordinate Geometry

Let and be real numbers.

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

Statement 1: a+b= 10

Statement 2: 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 is a - bi , and

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

a+b= 10

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

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

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Example Question #12 : Dsq: Graphing Complex Numbers

Let be a positive integer.

Evaluate $i^{N}$ .

Statement 1: is a multiple of 3.

Statement 2: is a multiple of 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 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 insufficient to answer the question.

Explanation:

Assume that both statements are true. The value of $i^{N}$ , a positive integer, is equal to $i^{R}$ , where is the remainder of the division of by 4, so we can use this fact to show that insufficient information is provided.

Case 1: N = 21 .

$21 \div 4 = 5\textup{ R }1$ , so

$i^{N} = i^{21} = i^{1}= i$

Case 2: N = 42 .

$42 \div 4 = 10 \textup{ R }2$ , so

$i^{N} = i^{42} = i^{2}= -1$

In both cases, both statements are true, but the value of $i^{N}$ differs.

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Example Question #651 : Data Sufficiency Questions

True or false: (A,B) and (C,D) are in the same quadrant of the rectangular coordinate plane.

Statement 1: and $D\;$ are of different sign.

Statement 2: and are of the same sign.

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:

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

Explanation:

Two points in the same quadrant have -coordinates of the same sign and -coordinates of the same sign.

It is possible for two points fitting the condition of Statement 1 to be in the same quadrant; (5,-6) and (6,-5) are two such points. However, it is also possible for two such points to be in different quadrants; (5,-6) and (-6,-5) are two such points. Therefore, Statement 1 alone gives insufficient information. By the same argument, Statement 2 alone gives insufficient information.

Assume both statements are true. and $D\;$ are of different sign by Statement 1. By Statement 2, and are of the same sign; therefore, they are both of the same sign as and the sign opposite that of $D\;$ , or vice versa. Therefore, in one ordered pair, both numbers are positive or both are negative, and in the other ordered pair, one number is positive and the other is negative. The two ordered pairs cannot represent points in the same quadrant.

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Example Question #652 : Data Sufficiency Questions

In which quadrant is the point (A,B) located: I, II, III, or IV?

Statement 1: 2A + B = 13

Statement 2: 3A - B = 32

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.

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.

Correct answer:

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

Explanation:

Assume Statement 1 alone. The set of points that satisfy the equation is the set of all points on the line of the equation

2x + y = 13 ,

which will pass through at least two quadrants on the coordinate plane. Therefore, Statement 1 provides insufficient information. By the same argument, Statement 2 is also insuffcient.

Now assume both statements to be true. The two statements together form a system of linear equations which can be solved using the elimination method:

2A + B = 13

$\underline{3A - B = 32}$

=45

$A = 45 \div 5 = 9$

Now, substitute back:

2 (9) + B = 13

18+ B = 13

B = 13-18 = -5

The point is (9,-5) , which has a positive -coordinate and a negative $y\,$ -coordinate and is consequently in Quadrant IV.

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Example Question #652 : Data Sufficiency Questions

In which quadrant is the point (A,B) located: I, II, III, or IV?

Statement 1: A > 0

Statement 2: A +B = 0

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.

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.

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:

Assume Statement 1 alone. The points (6,4) and (6,-4) each satisfy the condition of the statement; however, the former is in Quadrant I, having a positive -coordinate and a positive $y\,$ -coordinate; the latter is in Quadrant IV, having a positive -coordinate and a negative $y\,$ -coordinate.

Assume Statement 2 alone. The points (6,-6) and (-6,6) each satisfy the condition of the statement, since 6+ (-6)= -6+6 = 0 . However, the former is in Quadrant IV, having a positive -coordinate and a negative $y\,$ -coordinate; the latter is in Quadrant II, having a negative -coordinate and a positive $y\,$ -coordinate.

Assume both statements to be true. Statement 2 can be rewritten as B = -A ; since is positive from Statement 1, is negative. Since the point has a positive -coordinate and a negative $y\,$ -coordinate, it is in Quadrant IV.

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GMAT Math : Data-Sufficiency Questions

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #5 : Dsq: Graphing Complex Numbers

Example Question #6 : Dsq: Graphing Complex Numbers

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

Example Question #12 : Dsq: Graphing Complex Numbers

Example Question #651 : Data Sufficiency Questions

Example Question #652 : Data Sufficiency Questions

Example Question #652 : Data Sufficiency Questions

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