### All GMAT Math Resources

## Example Questions

### Example Question #21 : Functions/Series

Define

Is greater than, less than, or equal to ?

Statement 1: is a positive number.

Statement 2: is a negative number.

**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:**

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

One must have information about both and to answer the question, so neither statement is sufficient by itself.

Now assume both statements to be true. Then, being positive and being negative, .

### Example Question #22 : Dsq: Understanding Functions

Define

True or false: .

Statement 1: is a positive number

Statement 2: is a negative number

**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.

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 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.

Assume both statements. Then and . But this does not answer the question. For example,

If , then

making true.

But if , then

making false.

### Example Question #21 : Dsq: Understanding Functions

Define an operation on two real numbers as follows:

Is positive, negative, or zero?

Statement 1:

Statement 2:

**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.

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 does not provide a definite answer:

Case 1: If and , then , and

Case 2: If and , then , and

However, if we assume Statement 2, then, since , we can see:

, and we know this is positive.

### Example Question #21 : Dsq: Understanding Functions

Define an operation on two real numbers as follows:

Is positive, negative, or zero?

Statement 1: is negative.

Statement 2: is positive.

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

If we know Statement 1 only - that is negative - then, since must be nonnegative,

must be a negative number minus a nonnegative number. This makes negative.

If we know Statement 2 only - that is positive - then we do not have a definite answer. For example,

but

### Example Question #24 : Dsq: Understanding Functions

What is the measure of ?

Statement 1: is complementary to an angle that measures .

Statement 2: is supplementary to an angle that measures .

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

Complementary angles have degree measures that total , so the measure of an angle complementary to a angle would have measure .

Supplementary angles have degree measures that total , so the measure of an angle supplementary to a angle would have measure .

From either statement alone, we know that .

### Example Question #2961 : Gmat Quantitative Reasoning

is defined as the greatest integer less than or equal to .

You are given that .

True or false:

Statement 1: is an integer.

Statement 2: is an integer.

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

If and is an integer, then ; likewise for .

We show that the two statements together are not enough to draw a conclusion by assuming both statements are true and taking two cases:

Case 1:

is a false statement.

Case 2:

is a true statement.

### Example Question #27 : Dsq: Understanding Functions

is defined as the greatest integer less than or equal to .

You are given that .

True or false:

Statement 1: is not an integer.

Statement 2: is an integer.

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

Assume both statements to be true. We can show at least one case where is true and one in which is false.

Case 1:

Then and .

This makes the two expressions equal and a false statement.

Case 2:

Then and .

This makes true.

### Example Question #22 : Dsq: Understanding Functions

Gary is writing out a geometric sequence, in order. How many terms does he need to write out before he writes a term greater than or equal to 1,000?

Statement 1: The sum of the first two terms is 5.

Statement 2: The sum of the third and fourth terms is 80.

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

From Statement 1 alone, it cannot be determined what the first two terms or the common ratio are; the terms, for example, could be or , giving a different common ratio in each case. By a smiliar argument Statement 2 alone is also insufficient.

If both statements are assumed, though, the following can be deduced. If is the common ratio and is the first term, then the first four terms are

.

Divide:

, so .

Knowing the common ratio and the first term allows us to determine all of the numbers of the sequence and to find the first one greater than or at least 1,000.

### Example Question #25 : Dsq: Understanding Functions

Given two functions and defined on the set of real numbers, which, if either, is the greater quantity, or ?

Statement 1:

Statement 2: for all real values of .

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 is a consequence of Statement 2, so we need only show that Statement 2 provides insufficient information.

Let . Then

and

.

However, without knowing the sign of , we cannot determine whether or is the greater quantity.

### Example Question #22 : Functions/Series

Define an operation on two real numbers as follows:

For all real numbers ,

.

True or false:

Statement 1:

Statement 2:

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

Similarly, .

Therefore, for , it must hold that

.

Statement 1 contradicts this; Statement 2 neither confirms nor contradict this.

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