### All GMAT Math Resources

## Example Questions

### Example Question #81 : Functions/Series

Give the common difference of an arithmetic sequence .

Statement 1:

Statement 2:

**Possible Answers:**

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

**Correct answer:**

EITHER STATEMENT ALONE provides sufficient information to answer the question.

The common difference of an arithmetic sequence is the difference of two consecutive terms, . It can be calculated from the difference of any two terms and from the formula

By Statement 1 alone, setting and ,

.

By Statement 2 alone, setting and ,

.

### Example Question #81 : Functions/Series

How many values of make

a true statement?

**Possible Answers:**

Infinitely many

Two

One

None

Four

**Correct answer:**

One

, so the problem is equivalent to solving

.

It can be seen that ; since the expressions within the two sets of absolute value bars are opposites, it follows that

,

and the equation is equivalent to

This is the only solution, so the correct choice is one.

### Example Question #82 : Functions/Series

How many values of make

a true statement?

**Possible Answers:**

One

Infinitely many

Two

None

Four

**Correct answer:**

Infinitely many

, so the problem is equivalent to solving

.

It can be seen that ; since the expressions within the two sets of absolute value bars are opposites, it follows that

,

and the equation is equivalent to

Therefore, the statement is true for all real values of , and the correct response is infinitely many.

### Example Question #83 : Functions/Series

The domain and the range of a function are both the set . Also, exists.

Which of the following tables *cannot* show the values of for each of the given domain elements?

**Possible Answers:**

None of the other responses is correct.

**Correct answer:**

None of the other responses is correct.

A function has an inverse if and only if, if are in the domain of , then implies that , or, contrapositively, if , then . In each of the four choices, no two values of are matched with the same value of , so implies that in all four choices. Since the entire domain is given to be , all of the given functions have inverses.

### Example Question #83 : Functions/Series

Which of the following is an example of a relation which is *not* a function?

Pairing each positive integer with...

**Possible Answers:**

...the word "blue" if there exists a positive number with absolute value , and the word "purple" if there exists a negative number with absolute value .

...the word "brown" if and 20 are relatively prime, and the word "gray" if is divisible by either 4, 5, or both.

...the word "yellow" if has four or more factors, and the word "black" if has fewer than four factors.

...with the word "green" if is a prime integer, the word "red" if is a composite integer, and the word "white" if .

...the word "orange" if is odd, and the word "turquoise" if is divisible by 2.

**Correct answer:**

...the word "blue" if there exists a positive number with absolute value , and the word "purple" if there exists a negative number with absolute value .

A relation is a function if and only if, for each value in the domain, there is one and only one value in the range that can be paired with . Let us examine all of the relations.

*Pairing each positive integer with the word "orange" if is odd, and the word "turquoise" if is is divisible by 2:*

Every positive integer is either odd (not divisible by 2) or even (divisible by 2); no integer is both.

*Pairing each positive integer with the word "brown" if and 20 are relatively prime, and the word "gray" if divisible by either 2, 5, or both:*

If is relatively prime to 20, then it shares only one factor with 20, which is 1; otherwise, shares at least one prime factor with 20 - 2, 5, or both.

*Pairing each positive integer with the word "yellow" if has four or more factors, and the word "black" if has fewer than four factors:*

This is simply saying that either * *does fall in a set or does not fall in the same set.

*Pairing each positive integer with with the word "green" if is a prime integer, the word "red" if is a composite integer, and the word "white" if .*

Every positive integer except 1 is either prime or composite, and no number is both; 1 is considered neither.

In each of the above cases, each positive integer belongs in exactly one of the described sets, so is paired with only one word. Each of these relations is a function.

Now consider this relation:

*Pairing each positive integer with the word "blue" if there exists a positive number with absolute value , and the word "purple" if there exists a negative number with absolute value .*

For each positive integer , there exists one positive integer and one negative integer with absolute value ; for example, if , then

.

Therefore, each positive integer is paired with *two* values, "blue" and "purple", and the relation is not a function.

### Example Question #1311 : Problem Solving Questions

For any values , , define the operation as follows:

Which of the following expressions is equal to ?

**Possible Answers:**

**Correct answer:**

Substitute and for and in the expression for :