GMAT Math : Functions/Series

Study concepts, example questions & explanations for GMAT Math

Create An Account

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

← Previous 1 2 3 4 5 6 7 Next →

Example Question #51 : Algebra

Define $f (x) = \left\{\begin{matrix} 0\textrm{ if } x< 0\\ x \textrm{ if } x \geq 0 \end{matrix}\right.$

Is f(A) greater than, less than, or equal to f (B) ?

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.

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:

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, f(A) = A > 0 = f(B) .

Report an Error

Example Question #22 : Dsq: Understanding Functions

Define $f (x) = \left\{\begin{matrix} 1\textrm{ if } x< 0\\ x \textrm{ if } x \geq 0 \end{matrix}\right.$

True or false: f(A) > f (B) .

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.

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 insufficient to answer the question.

Explanation:

Assume both statements. Then f(B) = 1 and f(A) = A . But this does not answer the question. For example,

If A = 2 , then

f( 2 ) = 2 > 1 = f (1)

making f(A) > f (B) true.

But if A = 0.5 , then

f( 0.5 ) = 0.5 < 1 = f (1 )

making f(A) > f (B) false.

Report an Error

Example Question #21 : Dsq: Understanding Functions

Define an operation $\bigstar$ on two real numbers as follows:

$a \bigstar b = a ^{2} + b$

Is $M \bigstar N$ positive, negative, or zero?

Statement 1: M > N

Statement 2: N > 0

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.

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:

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

Explanation:

Statement 1 alone does not provide a definite answer:

Case 1: If M = -1 and N = -2 , then M > N , and

$M \bigstar N = M^{2} + N = (-1 )^{2} + (-2) = 1-2 = -1$

Case 2: If M =1 and N = 0 , then M > N , and

$M \bigstar N = M^{2} + N = 1^{2} + 0 = 1 + 0 = 1$

However, if we assume Statement 2, then, since $M^{2} \geq 0$ , we can see:

$M \bigstar N = M^{2} + N > 0 + 0 > 0$ , and we know this is positive.

Report an Error

Example Question #24 : Dsq: Understanding Functions

Define an operation $\Delta$ on two real numbers as follows:

$a \Delta b = a - b ^{2}$

Is $M \Delta N$ positive, negative, or zero?

Statement 1: is negative.

Statement 2: is positive.

Possible Answers:

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.

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:

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

Explanation:

If we know Statement 1 only - that is negative - then, since $N ^{2}$ must be nonnegative,

$M \Delta N = M - N^{2}$ must be a negative number minus a nonnegative number. This makes $M \Delta N$ negative.

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

$2 \Delta \left ( 2 \right )= 2 - (2)^{2} = 2 - 4 = -2$

but

$6\Delta \left ( 2 \right )= 6 - (2)^{2} = 6 - 4 = 2$

Report an Error

Example Question #25 : Dsq: Understanding Functions

What is the measure of $\angle 1$ ?

Statement 1: $\angle 1$ is complementary to an angle that measures $62 ^{\circ }$ .

Statement 2: $\angle 1$ is supplementary to an angle that measures $152 ^{\circ }$ .

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 sufficient to answer the question, but NEITHER statement ALONE is 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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Complementary angles have degree measures that total $90 ^{\circ }$ , so the measure of an angle complementary to a $62 ^{\circ }$ angle would have measure $(90-62)^{\circ } = 28 ^{\circ }$ .

Supplementary angles have degree measures that total $180 ^{\circ }$ , so the measure of an angle supplementary to a $152 ^{\circ }$ angle would have measure $(180-152)^{\circ } = 28 ^{\circ }$ .

From either statement alone, we know that $m \angle 1 = 28 ^{\circ }$ .

Report an Error

Example Question #26 : Dsq: Understanding Functions

$\left \lfloor x \right \rfloor$ is defined as the greatest integer less than or equal to .

You are given that M>0,N>0 .

True or false: $\left \lfloor M \right \rfloor \cdot \left \lfloor N \right \rfloor < \left \lfloor MN \right \rfloor$

Statement 1: M + 0.5 is an integer.

Statement 2: N + 0.5 is an integer.

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.

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:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

If M>0 and M + 0.5 is an integer, then $M \in \left \{ 0.5, 1.5, 2.5, 3.5... \right \}$ ; 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: M = 0.5,N = 0.5

$\left \lfloor M \right \rfloor \cdot \left \lfloor N \right \rfloor = \left \lfloor 0.5 \right \rfloor \cdot \left \lfloor 0.5 \right \rfloor = 0 \cdot 0 = 0$

$\left \lfloor MN \right \rfloor = \left \lfloor 0.5\cdot 0.5\right \rfloor = \left \lfloor 0.25 \right \rfloor =0$

$\left \lfloor M \right \rfloor \cdot \left \lfloor N \right \rfloor < \left \lfloor MN \right \rfloor$ is a false statement.

Case 2: M = 1.5,N = 1.5

$\left \lfloor M \right \rfloor \cdot \left \lfloor N \right \rfloor = \left \lfloor 1.5 \right \rfloor \cdot \left \lfloor 1.5 \right \rfloor =1 \cdot 1 = 1$

$\left \lfloor MN \right \rfloor = \left \lfloor 1.5\cdot 1.5\right \rfloor = \left \lfloor 2.25 \right \rfloor =2$

$\left \lfloor M \right \rfloor \cdot \left \lfloor N \right \rfloor < \left \lfloor MN \right \rfloor$ is a true statement.

Report an Error

Example Question #27 : Dsq: Understanding Functions

$\left \lfloor x \right \rfloor$ is defined as the greatest integer less than or equal to .

You are given that M>0,N>0 .

True or false: $\left \lfloor M \right \rfloor \cdot \left \lfloor N \right \rfloor < \left \lfloor MN \right \rfloor$

Statement 1: is not 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.

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 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 both statements to be true. We can show at least one case where $\left \lfloor M \right \rfloor \cdot \left \lfloor N \right \rfloor < \left \lfloor MN \right \rfloor$ is true and one in which $\left \lfloor M \right \rfloor \cdot \left \lfloor N \right \rfloor < \left \lfloor MN \right \rfloor$ is false.

Case 1: M = 2.1, N = 2

Then $\left \lfloor 2.1 \right \rfloor \cdot \left \lfloor 2 \right \rfloor = 2 \cdot 2 = 4$ and $\left \lfloor 2.1 \cdot 2 \right \rfloor = \left \lfloor 4.2 \right \rfloor = 4$ .

This makes the two expressions equal and $\left \lfloor M \right \rfloor \cdot \left \lfloor N \right \rfloor < \left \lfloor MN \right \rfloor$ a false statement.

Case 2: M = 2.5, N = 2

Then $\left \lfloor 2.5 \right \rfloor \cdot \left \lfloor 2 \right \rfloor = 2 \cdot 2 = 4$ and $\left \lfloor 2.5 \cdot 2 \right \rfloor = \left \lfloor 5 \right \rfloor = 5$ .

This makes $\left \lfloor M \right \rfloor \cdot \left \lfloor N \right \rfloor < \left \lfloor MN \right \rfloor$ true.

Report an Error

Example Question #28 : 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.

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.

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:

From Statement 1 alone, it cannot be determined what the first two terms or the common ratio are; the terms, for example, could be 1, 4 or 2,3 , 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

$a , ar, ar^{2}, a r ^{3}$ .

a + ar = a (1 + r ) = 5

$ar^{2}+ a r ^{3} = ar^{2} (1+r)= 80$

Divide:

$\frac{ar^{2} (1+r) }{a(1+r)} =\frac{ 80}{5}$

$r ^{2} = 16$

r=4

a (1 + 4 ) = 5a = 5 , so a = 1 .

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.

Report an Error

Example Question #29 : Dsq: Understanding Functions

Given two functions and defined on the set of real numbers, which, if either, is the greater quantity, $\left ( f \circ g \right )(0 )$ or $\left ( g\circ f \right )(0 )$ ?

Statement 1: f (0 ) = g (0)

Statement 2: f (x)= g (x) + x for all real values of .

Possible Answers:

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.

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:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

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

Let N = f (0 ) = g (0) . Then

$\left ( f \circ g \right )(0 ) = f(g(0)) = f (N) = g(N)+N$

and

$\left (g \circ f \right )(0 ) = g(f(0)) = g (N)$ .

However, without knowing the sign of , we cannot determine whether f(N) or g(N) is the greater quantity.

Report an Error

Example Question #30 : Dsq: Understanding Functions

Define an operation $\Sigma$ on two real numbers as follows:

For all real numbers a,b ,

$a \Sigma b = 10 - a + b$ .

True or false: $M\Sigma N=N \Sigma M$

Statement 1: M - N = 1

Statement 2: M + N = 7

Possible Answers:

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.

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:

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

Explanation:

$M\Sigma N= 10 - M + N$

Similarly, $N\Sigma M= 10 - N + M$ .

Therefore, for $M\Sigma N=N \Sigma M$ , it must hold that

10 - M + N = 10 - N + M .

10 - M + N - 10 + M + N= 10 - N + M - 10 + M + N

2N= 2 M

N = M

M-N = 0

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

Report an Error

← Previous 1 2 3 4 5 6 7 Next →

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

ACT Tutors in Chicago, ACT Tutors in Seattle, Calculus Tutors in Los Angeles, SSAT Tutors in Miami, Reading Tutors in Houston, SSAT Tutors in San Francisco-Bay Area, SAT Tutors in Seattle, French Tutors in Los Angeles, Computer Science Tutors in Dallas Fort Worth, Chemistry Tutors in Boston

Popular Courses & Classes

Spanish Courses & Classes in Philadelphia, MCAT Courses & Classes in Miami, ISEE Courses & Classes in Phoenix, Spanish Courses & Classes in Seattle, Spanish Courses & Classes in San Diego, MCAT Courses & Classes in Boston, ACT Courses & Classes in Denver, GRE Courses & Classes in Boston, SSAT Courses & Classes in Washington DC, Spanish Courses & Classes in Phoenix

Popular Test Prep

GRE Test Prep in San Francisco-Bay Area, MCAT Test Prep in Chicago, ACT Test Prep in Denver, SAT Test Prep in Philadelphia, ISEE Test Prep in Phoenix, SAT Test Prep in New York City, GRE Test Prep in Phoenix, LSAT Test Prep in Denver, SSAT Test Prep in Atlanta, ISEE Test Prep in Denver

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

Email address:
Your name:
Feedback:

GMAT Math : Functions/Series

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #51 : Algebra

Example Question #22 : Dsq: Understanding Functions

Example Question #21 : Dsq: Understanding Functions

Example Question #24 : Dsq: Understanding Functions

Example Question #25 : Dsq: Understanding Functions

Example Question #26 : Dsq: Understanding Functions

Example Question #27 : Dsq: Understanding Functions

Example Question #28 : Dsq: Understanding Functions

Example Question #29 : Dsq: Understanding Functions

Example Question #30 : Dsq: Understanding Functions

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details