Call Now to Set Up Tutoring:

(888) 888-0446

ISEE Upper Level Quantitative : ISEE Upper Level (grades 9-12) Quantitative Reasoning

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

Create An Account

All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : How To Subtract Exponential Variables

Assume neither nor is zero.

Which is the greater quantity?

(a) $\frac{(x+y)^{2} - (x - y)^{2}}{xy}$

(b)

Possible Answers:

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

Correct answer:

(a) and (b) are equal.

Explanation:

Simplify the expression in (a):

$\frac{(x+y)^{2} - (x - y)^{2}}{xy}$

$=\frac{(x^{2}+2xy + y^{2}) - (x^{2}-2xy + y^{2})}{xy}$

$=\frac{x^{2}- x^{2}+2xy +2xy + y^{2}-y^{2}}{xy}$

$=\frac{4xy }{xy} = 4$

$\frac{(x+y)^{2} - (x - y)^{2}}{xy} = 4$ regardless of the values of the variables, and (a) and (b) are equal.

Report an Error

Example Question #1 : How To Subtract Exponential Variables

is a positive number. is a negative number.

Which is the greater quantity?

(a) $x^{2} - y ^{2}$

(b) $(x-y) ^{2}$

Possible Answers:

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

(b) is greater.

Correct answer:

(b) is greater.

Explanation:

$(x-y) ^{2} = x ^{2} - 2xy + y ^{2}$

is a positive number and is a negative number, so -2xy > 0 and $y^{2} > 0$ . Therefore,

$(x-y) ^{2} = x ^{2} - 2xy + y ^{2} > x ^{2} + y ^{2} > x ^{2} - y ^{2}$ .

(b) is always greater.

Report an Error

Example Question #1 : How To Multiply Exponential Variables

Simplify:

$\left (4x \right )^{4} +\left ( 2y \right )^{2}$

Possible Answers:

$1,024x^{4} y ^{2}$

$16x^{4} +4y ^{2}$

$64x^{4} y ^{2}$

$4x^{4} +2y ^{2}$

$256x^{4} +4y ^{2}$

Correct answer:

$256x^{4} +4y ^{2}$

Explanation:

$\left (4x \right )^{4} +\left ( 2y \right )^{2} =4^{4}x^{4} + 2^{2}y^{2}=256x^{4} +4y ^{2}$

Report an Error

Example Question #1 : How To Multiply Exponential Variables

Expand: $(x-5)^{3}$

Possible Answers:

$x^{3}+25x^{2}-50x-125$

$x^{3}-25x^{2}+50x-125$

$x^{3}-15x^{2}+75x-125$

$x^{3}-125$

$x^{3}+15x^{2}-75x-125$

Correct answer:

$x^{3}-15x^{2}+75x-125$

Explanation:

A binomial can be cubed using the pattern:

$(A-B)^{3} = A^{3} - 3A^{2}B+ 3AB^{2} - B^{3}$

Set A = x, B = 5

$(x-5)^{3}$

$= x^{3} - 3x^{2} \cdot 5+ 3x\cdot 5^{2} - 5^{3}$

$= x^{3} - 15x^{2} + 75x - 125$

Report an Error

Example Question #922 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Factor completely:

$3x^{2} - x -14$

Possible Answers:

$\left ( x-7 \right ) \left (3x+2 \right )$

$\left ( x-2 \right ) \left (3x+ 7 \right )$

$\left ( x-2 \right ) \left (3x- 7 \right )$

$\left ( x+7 \right ) \left (3x-2 \right )$

$\left ( x+2 \right ) \left (3x- 7 \right )$

Correct answer:

$\left ( x+2 \right ) \left (3x- 7 \right )$

Explanation:

A trinomial whose leading term has a coefficent other than 1 can be factored using the -method. We split the middle term using two numbers whose product is 3 (-14) = -42 and whose sum is . These numbers are -7,6 , so:

$3x^{2} - x -14$

$= 3x^{2} - 7x +6x -14$

$= \left (3x^{2} - 7x \right ) +\left ( 6x -14 \right )$

$=x \left (3x- 7 \right ) +2 \left (3x- 7 \right )$

$=\left ( x+2 \right ) \left (3x- 7 \right )$

Report an Error

Example Question #923 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Multiply:

$\left ( x + y + 3\right ) \left ( x + y - 3\right )$

Possible Answers:

$x^{2} + xy + y^{2} - 9$

$x^{2} + y^{2} - 9$

$x^{2} - 9 xy + y^{2}$

$x^{2} - 2xy + y^{2} - 9$

$x^{2} + 2xy + y^{2} - 9$

Correct answer:

$x^{2} + 2xy + y^{2} - 9$

Explanation:

This can be achieved by using the pattern of difference of squares:

$\left ( x + y + 3 \right ) \left ( x + y - 3\right )$

$=\left [\left ( x + y \right ) + 3 \right ] \left [\left ( x + y \right ) - 3 \right ]$

$= ( x + y ) ^{2} - 3 ^{2}$

$= ( x + y ) ^{2} - 9$

Applying the binomial square pattern:

$= x^{2} + 2xy + y^{2} - 9$

Report an Error

Example Question #924 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Simplify:

$\left (x + \frac{1}{3} \right )^{3}$

Possible Answers:

$x ^{3 } + 3 x^{2} +x +\frac{1}{27}$

$x ^{3 } + \frac{1}{3} x^{2} + \frac{1}{9} x +\frac{1}{27}$

$x ^{3 } + 2 x^{2} + \frac{2}{3} x +\frac{1}{27}$

$x ^{3 } + x^{2} + \frac{1}{3} x +\frac{1}{27}$

$x ^{3 } + \frac{2}{3} x^{2} + \frac{2}{9} x +\frac{1}{27}$

Correct answer:

$x ^{3 } + x^{2} + \frac{1}{3} x +\frac{1}{27}$

Explanation:

The cube of a sum pattern can be applied here:

$\left (x + \frac{1}{3} \right )^{3}$

$= x ^{3 } + 3\cdot x^{2} \cdot \frac{1}{3}+3\cdot x \cdot \left ( \frac{1}{3} \right )^{2}+\left( \frac{1}{3} \right )^{3}$

$= x ^{3 } + x^{2} +3\cdot x \cdot \frac{1}{9} +\frac{1}{27}$

$= x ^{3 } + x^{2} + \frac{1}{3} x +\frac{1}{27}$

Report an Error

Example Question #2 : How To Multiply Exponential Variables

Fill in the box to form a perfect square trinomial:

$x ^{2} - 13x + \square$

Possible Answers:

$\frac{169}{2}$

169

$\frac{169}{4}$

Correct answer:

$\frac{169}{4}$

Explanation:

To obtain the constant term of a perfect square trinomial, divide the linear coefficient, which here is , by 2, and square the quotient. The result is

$\left (\frac{ -13}{2} \right) ^{2} = \frac{(-13)^{2}}{2^{2}}= \frac{169}{4}$

Report an Error

Example Question #1 : How To Multiply Exponential Variables

Fill in the box to form a perfect square trinomial:

$x^{2} - 18x + \square$

Possible Answers:

144

162

324

Correct answer:

Explanation:

To obtain the constant term of a perfect square trinomial, divide the linear coefficient, which here is , by 2, and square the quotient. The result is

$\left (\frac{-18}{2} \right )^{2} =\left ( -9\right ) ^{2} = 81$

Report an Error

Example Question #8 : How To Multiply Exponential Variables

Expand: $(x + 2)^{10}$

Which is the greater quantity?

(a) The coefficient of $x^{4}$

(b) The coefficient of $x^{6}$

Possible Answers:

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

Correct answer:

(a) is greater.

Explanation:

Using the Binomial Theorem, if $(x + 2)^{n}$ is expanded, the $x^{r}$ term is

$_{n}\textrm{C} _{r} \cdot x^{r} \cdot 2 ^{n-r}$

$= \frac{n!}{(n-r)!r!} \cdot 2 ^{n-r} \cdot x^{r}$ .

This makes $\frac{n!}{(n-r)!r!} \cdot 2 ^{n-r}$ the coefficient of $x^{r}$ .

We compare the values of this expression at n = 10 for both r = 4 and r = 6 :

(a) $\frac{10!}{(10-4)!4!} \cdot 2 ^{10-4} = \frac{10!}{6!4!} \cdot 2 ^{6}$

(b) $\frac{10!}{(10-6)!6!} \cdot 2 ^{10-6} = \frac{10!}{4!6!} \cdot 2 ^{4} = \frac{10!}{6!4!} \cdot 2 ^{4}$

(a) is the greater quantity.

Report an Error

View Tutors

Iffy
Certified Tutor

University of Maryland-Baltimore County, Bachelor of Science, Computer Engineering, General. University of Maryland-College P...

View Tutors

Robert
Certified Tutor

University of North Carolina at Chapel Hill, Bachelor in Arts, English. North Carolina Central University, Master of Arts, En...

View Tutors

Ana
Certified Tutor

Grand Canyon University, Bachelors, Education. Grand Canyon University, Masters, Education.

All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Computer Science Tutors in Boston, Spanish Tutors in Houston, ACT Tutors in Washington DC, Biology Tutors in Dallas Fort Worth, ACT Tutors in Chicago, Reading Tutors in Washington DC, LSAT Tutors in Boston, French Tutors in Seattle, SSAT Tutors in San Francisco-Bay Area, LSAT Tutors in Washington DC

Popular Courses & Classes

SAT Courses & Classes in Phoenix, GMAT Courses & Classes in Chicago, SSAT Courses & Classes in Seattle, SAT Courses & Classes in Atlanta, SAT Courses & Classes in Miami, GMAT Courses & Classes in San Diego, LSAT Courses & Classes in Chicago, GRE Courses & Classes in Miami, GRE Courses & Classes in Houston, GRE Courses & Classes in Philadelphia

Popular Test Prep

MCAT Test Prep in Denver, ISEE Test Prep in Boston, MCAT Test Prep in Atlanta, MCAT Test Prep in New York City, GRE Test Prep in Houston, GMAT Test Prep in Los Angeles, GRE Test Prep in Seattle, LSAT Test Prep in Denver, GMAT Test Prep in Chicago, ISEE Test Prep in Los Angeles

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)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

ISEE Upper Level Quantitative : ISEE Upper Level (grades 9-12) Quantitative Reasoning

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #1 : How To Subtract Exponential Variables

Example Question #1 : How To Subtract Exponential Variables

Example Question #1 : How To Multiply Exponential Variables

Example Question #1 : How To Multiply Exponential Variables

Example Question #922 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Example Question #923 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Example Question #924 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Example Question #2 : How To Multiply Exponential Variables

Example Question #1 : How To Multiply Exponential Variables

Example Question #8 : How To Multiply Exponential Variables

All ISEE Upper Level Quantitative Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: