### All GMAT Math Resources

## Example Questions

### Example Question #31 : Dsq: Understanding Functions

Define an operation on two real numbers as follows:

Is positive, negative, or zero?

Statement 1:

Statement 2:

**Possible Answers:**

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.

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.

is positive or zero, so if Statement 1 is assumed, and is negative, then , the product of a negative number and a nonnegative number, must be nonpositive. It can be negative or zero, however.

If Statement 2 is assumed, the expression can be of any sign, since, although must be positive, no information is given about the sign of .

If both statements are assumed, then since is nonzero, is positive; also, is negative. is the product of two numbers of unlike sign, and therefore, it can be determined that it is a negative number.

### Example Question #61 : Algebra

John is writing out an arithmetic sequence. How many terms does he need to write out before he writes a term greater than or equal to 1,000?

Statement 1: The fourth term is 50.

Statement 2: The twentieth term is 418.

**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.

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:**

Knowing one term of a sequence will not help you find any other terms, so neither statement alone will answer the question. But knowing two terms and knowing that the sequence is arithmetic will allow you to find the common difference .

The twentieth term is greater than the fourth term, so take the difference and divide by 16:

Now solve this inequality for to find the minimum number of terms needed to exceed 1,000:

### Example Question #32 : Dsq: Understanding Functions

Define an operation on the real numbers as follows:

Is positive, negative, or zero?

Statement 1:

Statement 2:

**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.

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.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

, and . Therefore, since is a positive number, the sign of is the sign of . This makes Statement 1 neither necessary nor helpful. We need to know whether is greater than, equal to, or less than 0, or, equivalently, whether is greater than, equal to, or less than 10. Statement 2 does not tell us this either.

### Example Question #31 : Dsq: Understanding Functions

What is ?

(1)

(2)

**Possible Answers:**

A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient

B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient

C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient

D: EACH statement ALONE is sufficient

E: Statements (1) and (2) TOGETHER are not sufficient

**Correct answer:**

C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient

Statement (1) does not give us any information about g(x), so it is not sufficient.

Statement (2) alone gives us the relationship between the functions f and g but does not give us any information about f(x), so it is not sufficient.

Both statement together allow us to get an expression for both f and g:

With an expression for both functions we can estimate f(g(1)):

and

So the correct answer is C.

### Example Question #67 : Algebra

Does a function have an inverse?

Statement 1:

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

A necessary and sufficient condition for not to have an inverse is for for distinct in the domain of .

If , then , meaning that pairs at least two domain values, 4 and , with the same range value. Therefore, from Statement 1 alone, it follows that does not have an inverse.

If , then , meaning pairs two different domain values with two different range values. This is nt a contradiction of the conditions for an inverse to not exist, but this does not prove that an inverse does exist. Statement 2 is unhelpful either way.

### Example Question #31 : Dsq: Understanding Functions

Does a function have an inverse?

Statement 1: Every vertical line passes through the graph of exactly once.

Statement 2: Every horizontal line passes through the graph of exactly once.

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

Statement 1 is simply an assertion that the relation passes the vertical line test and is therefore a function. Since we know already that the relation is a function, this statement is unhelpful.

If Statement 2 is assumed, it follows that the graph passes the horizontal line test for the existence of an inverse. Statement 2 alone answers the question.

### Example Question #34 : Dsq: Understanding Functions

A relation comprises ten ordered pairs. Is it a function?

Statement 1: The domain of the relation is .

Statement 2: The range of the relation is .

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

The relation comprises ten ordered pairs. If Statement 1 alone is known, then the domain comprises ten elements, each of which must appear in exactly one ordered pair. Therefore, no domain element is matched with more than one range element, and the relation is a function.

If Statement 2 alone is known, then the range comprises ten elements, each of which must appear in exactly one ordered pair. But nothing is known about the domain. If no domain element is repeated among the ordered pairs, the relation is a function; otherwise, the relation is not a function.

### Example Question #62 : Algebra

True or false:

Statement 1: for all real values of .

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

. So, for to be negative, and must be of unlike sign.

From Statement 1 it can be determined that is negative, but no information about the sign of can be determined.

From Statement 2, it can be determined that and is therefore negative, but no information about the sign of can be determined.

From the two statements together, both and can be proved negative, so their product, , is positive.

### Example Question #32 : Dsq: Understanding Functions

Let and be functions, the domain of both of which is the set of real numbers. Let .

True or false:

Statement 1: The range of is the set

Statement 2: The range of is the set

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

, so we need to determine whether .

From Statement 1 alone, since the range of is - that is, the set of all positive numbers, then regardless of the value of ,

.

Therefore, Statement 1 alone yields an affirmative answer to the question.

From Statement 2 alone, regardless of the value or , , but we do not know the value or range of values of . Statement 2 alone is unhelpful.

### Example Question #71 : Algebra

Is a given relation a function?

Statement 1: The domain of the relation is .

Statement 2: The range of the relation is .

**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.

The two statements together do not prove or disprove the relation to be a function.

The relations defined by the sets of points

.

and

have the domain and range given in the statements, but the former is a function, since each domain element is matched with exaclty one element, and the latter is not a function, since domain element 1 is matched with two different range elements.

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