GMAT Question of the Day - Daily to your Mailbox; hard ones only

It is currently 16 Aug 2018, 12:46

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

If a and b are non-zero integers and a^8*b^4 - a^4*b^2 = 12. Which of

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 47946
If a and b are non-zero integers and a^8*b^4 - a^4*b^2 = 12. Which of  [#permalink]

Show Tags

New post 13 Sep 2017, 23:53
2
2
00:00
A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

65% (02:16) correct 35% (01:36) wrong based on 142 sessions

HideShow timer Statistics

If a and b are non-zero integers and \(a^8*b^4 - a^4*b^2 = 12\). Which of the following could be a^2 in terms of b?

I. 2/b
II. -2/b
III. 3/b

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II and III

_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Senior Manager
Senior Manager
User avatar
P
Joined: 02 Jul 2017
Posts: 294
GMAT 1: 730 Q50 V38
GMAT ToolKit User
Re: If a and b are non-zero integers and a^8*b^4 - a^4*b^2 = 12. Which of  [#permalink]

Show Tags

New post 14 Sep 2017, 00:10
=> a and b are non-zero integers
\(a^8∗b^4−a^4∗b^2=12\)
Find: a^2 in terms of b

\(a^8∗b^4−a^4∗b^2=12\)
=> \(a^4∗b^2(a^4∗b^2-1)=12\)
=> \(a^4∗b^2(a^2∗b-1)(a^2∗b+1)=12\)
here given a and b are integers => by putting value a= 1 and b=-2 or b=2 satisfy the equation
Now we have to find a^2 in terms of be... could be conditions

I. 2/b
II. -2/b
III. 3/b

So 1^2 = (2/2) => when a=1 and b=2 . I satisfy
So 1^2 = (-2/-2) => when a=1 and b=-2 . II satisfy
II doesnot satisfy for both b=-2 and b=2.

Answer : C
Senior Manager
Senior Manager
User avatar
G
Joined: 19 Apr 2016
Posts: 274
Location: India
GMAT 1: 570 Q48 V22
GMAT 2: 640 Q49 V28
GPA: 3.5
WE: Web Development (Computer Software)
Re: If a and b are non-zero integers and a^8*b^4 - a^4*b^2 = 12. Which of  [#permalink]

Show Tags

New post 14 Sep 2017, 00:30
2
Substitute the value for a^2 as 2/b, - 2/b and 3/b in given equation

LHS = RHS only for I and II
So option C is correct

Sent from my Redmi Note 4 using GMAT Club Forum mobile app
PS Forum Moderator
avatar
D
Joined: 25 Feb 2013
Posts: 1201
Location: India
GPA: 3.82
GMAT ToolKit User Premium Member Reviews Badge
Re: If a and b are non-zero integers and a^8*b^4 - a^4*b^2 = 12. Which of  [#permalink]

Show Tags

New post 14 Sep 2017, 08:36
2
1
Bunuel wrote:
If a and b are non-zero integers and \(a^8*b^4 - a^4*b^2 = 12\). Which of the following could be a^2 in terms of b?

I. 2/b
II. -2/b
III. 3/b

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II and III


Well working through options method has already been explained, so I'll go the Algebraic way :-)

Given \(a^8*b^4 - a^4*b^2 = 12\). it is evident that \(a^4b^2\) is common in LHS. So let's assume \(a^4b^2 = x\)

so our equation becomes \(x^2-x-12=0\). Now we can easily calculate the value of \(x\) and substitute

Solving the equation we get \(x= 4\) & \(-3\). \(-3\) is not possible as \(x\) is a square. Hence \(x= 4\).

or \(a^4b^2=4\). Taking square root of both the sides we get

\(a^2b =2\) or \(-2\)

So \(a^2 = \frac{2}{b}\) or \(\frac{-2}{b}\)

Option C
Manager
Manager
avatar
B
Joined: 04 May 2014
Posts: 162
Location: India
WE: Sales (Mutual Funds and Brokerage)
If a and b are non-zero integers and a^8*b^4 - a^4*b^2 = 12. Which of  [#permalink]

Show Tags

New post 14 Sep 2017, 22:08
Plugin
(a⁸xb⁴)-(a⁴xb²)=12
or
{(a²)⁴xb⁴}-{(a²)²xb²}=12

Let a²=\(\frac{-2}{b}\)

{(\(\frac{-2}{b}\))⁴*b⁴}-{(\(\frac{-2}{b}\))²*b²}=12

{(\(\frac{16}{b⁴}\)*b⁴)}-{(\(\frac{4}{b²}\)*b²)}=12

16-4=12
12=12
Now all the powers in the above equation are even hence positive \(\frac{2}{b}\) will also give the same result.
Answer I and II
Intern
Intern
avatar
B
Joined: 24 Jul 2017
Posts: 49
Location: India
WE: Information Technology (Computer Software)
Re: If a and b are non-zero integers and a^8*b^4 - a^4*b^2 = 12. Which of  [#permalink]

Show Tags

New post 15 Sep 2017, 03:00
Various methods to solve such problems.
1. Substitute appropriate nos to satisfy the equality
2. Algebraic method
3. Check for the option values.

Substituting a^2 as 2/b, -2/b and 3/b in equation , we can find that only I and II satisfies the equation.
Hence option C
Manager
Manager
avatar
B
Joined: 26 Jan 2016
Posts: 59
Location: India
Concentration: General Management, Statistics
GMAT 1: 670 Q49 V31
GMAT 2: 730 Q49 V41
GPA: 3
WE: Information Technology (Computer Software)
Re: If a and b are non-zero integers and a^8*b^4 - a^4*b^2 = 12. Which of  [#permalink]

Show Tags

New post 15 Sep 2017, 11:30
a^8*b^4-a^4*b^2=12
a^4*b^2(a^4*b^2-1)=12
lets say a^4*b^2 = x
therefore x(x-1) =12
That will give us
x^2-x-12=0
x=4 or x =-3

but any positive integer power of any integer cant be negative. so x = 4.
i.e a^4*b^2=4
since a and b are non zero integers a can be 1,-1 and b can be 2,-2.

now a^1 = 1 for -1 and 1.

So 1) 2/b will give us 1 since b can be 2.
2) -2/b can give us 1 if we take b as -2.
3) 3/b is not possible for any of the values of b.


Answer is C
Intern
Intern
avatar
B
Joined: 18 Aug 2017
Posts: 2
Re: If a and b are non-zero integers and a^8*b^4 - a^4*b^2 = 12. Which of  [#permalink]

Show Tags

New post 16 Sep 2017, 05:00
Can a^2 take a negative value? I figured it cannot which is why i eliminated C and chose A.
Senior Manager
Senior Manager
User avatar
P
Joined: 02 Jul 2017
Posts: 294
GMAT 1: 730 Q50 V38
GMAT ToolKit User
Re: If a and b are non-zero integers and a^8*b^4 - a^4*b^2 = 12. Which of  [#permalink]

Show Tags

New post 16 Sep 2017, 05:26
mehrotrayash wrote:
Can a^2 take a negative value? I figured it cannot which is why i eliminated C and chose A.



Yes a^2 cannot take a negative value

so for a^2 = -2/b will only be true if b is -ve.

=> a^2 = -2/-2 =1 which is positive.

So condition I: a^2 =2/b will be true when b is +ve and condition II: a^2 =-2/b will be true when b is -ve.

And questions is asking for possible options so as we have 2 options for b +ve and -ve condition I and II are possible.
Intern
Intern
avatar
B
Joined: 19 Sep 2016
Posts: 35
Re: If a and b are non-zero integers and a^8*b^4 - a^4*b^2 = 12. Which of  [#permalink]

Show Tags

New post 16 Sep 2017, 06:16
The equation states a^8b^4-a^4b^2=12. Now assuming a^4b^2=x so we can solve the equation by substituting the assumption and the equation now becomes x^2-x-12=0 and on solving the equation, either x=4 or -3. Since the value is non negative, -3 cannot be accepted. So the accepted value is 4. Now putting the same against the assumption, a^4b^2=4, so (a^2b)^2=2^2. So further solving the same becomes (a^2b)^2- 4=0. Which is also a^2-b^2 and can be expressed as (a-b)*(a+b). So using the same formula in the above equation, the simlified equation becomes (a^2b-2)*(a^2b+2)=0 which makes a^2= 2/b and a^2=-2/b. Hence answer is C.

Sent from my SM-N920G using GMAT Club Forum mobile app
Intern
Intern
avatar
B
Joined: 19 Sep 2016
Posts: 35
Re: If a and b are non-zero integers and a^8*b^4 - a^4*b^2 = 12. Which of  [#permalink]

Show Tags

New post 16 Sep 2017, 06:18
Hope this helps mehrotrayash to understand how it can take negative values. Not required to substitute values and check. Simplifying the equation to its lowest form as per the question also clears how one can arrive at the values. Hope this helps

Sent from my SM-N920G using GMAT Club Forum mobile app
Target Test Prep Representative
User avatar
G
Status: Head GMAT Instructor
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2727
Re: If a and b are non-zero integers and a^8*b^4 - a^4*b^2 = 12. Which of  [#permalink]

Show Tags

New post 16 Sep 2017, 07:29
Bunuel wrote:
If a and b are non-zero integers and \(a^8*b^4 - a^4*b^2 = 12\). Which of the following could be a^2 in terms of b?

I. 2/b
II. -2/b
III. 3/b

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II and III


Let’s factor (a^4)(b^2) from the left side:
(a^4)(b^2)[(a^4)(b^2) - 1] = 12

The arithmetic fact that 4 x 3 = 12 now comes into play for approaching the solution to this equation. We see that (a^4)(b^2) - 1 is exactly one less than (a^4)(b^2), so (a^4)(b^2) = 4 and (a^4)(b^2) - 1 = 3 OR (a^4)(b^2) = -3 and (a^4)(b^2) - 1 = -4. However, the latter can’t be true since (a^4)(b^2) is a nonnegative quantity. Thus, only (a^4)(b^2) = 4 and (a^4)(b^2) - 1 = 3 can be true.

Taking the square root of (a^4)(b^2) = 4, we have a^2 * b = 2 or a^2 * b = -2. Thus a^2 = 2/b or a^2 = -2/b.

Alternate Solution:

Let’s let (a^4)(b^2) = x. Then, the given equation becomes x^2 - x = 12, or x^2 - x - 12 = 0. We can factor this equation as (x - 4)(x + 3) = 0; therefore x = 4 or x = -3. Then, (a^4)(b^2) = 4 or (a^4)(b^2) = -3. Since (a^4)(b^2) is nonnegative, it can’t equal -3; thus, it must be true that (a^4)(b^2) = 4.

Taking the square root of (a^4)(b^2) = 4, we have a^2 * b = 2 or a^2 * b = -2. Thus, a^2 = 2/b or a^2 = -2/b.

Answer: C
_________________

Jeffery Miller
Head of GMAT Instruction

GMAT Quant Self-Study Course
500+ lessons 3000+ practice problems 800+ HD solutions

Senior Manager
Senior Manager
User avatar
G
Joined: 19 Oct 2012
Posts: 338
Location: India
Concentration: General Management, Operations
GMAT 1: 660 Q47 V35
GMAT 2: 710 Q50 V38
GPA: 3.81
WE: Information Technology (Computer Software)
Re: If a and b are non-zero integers and a^8*b^4 - a^4*b^2 = 12. Which of  [#permalink]

Show Tags

New post 16 Sep 2017, 16:02
The above equation can be simplified to:
\(a^4b^2 (a^2b+1) (a^2b-1) = 12\)
or \((a^2b)^2 (a^2b+1) (a^2b-1) =12\)

With some due diligence, we can find out that \(a^2b = 2\) or \(-2\).

Hence only option C implies. :-)
_________________

Citius, Altius, Fortius

Intern
Intern
avatar
B
Joined: 17 Mar 2017
Posts: 25
Concentration: Marketing, Operations
GMAT 1: 620 Q48 V28
GMAT 2: 660 Q44 V36
Re: If a and b are non-zero integers and a^8*b^4 - a^4*b^2 = 12. Which of  [#permalink]

Show Tags

New post 17 Sep 2017, 02:19
1
a^8*b^4 - a^4*b^2 = 12
(a^4*b^2) (a^4*b^2) - (a^4*b^2) = 12
(a^4*b^2) (a^4*b^2 - 1) = 12

This says that they are consecutive numbers... only solution possible is 4*3. So a^4*b^2 = 4

factoring will give us a^4 = 4/B^2.

so a^2 = + or - 2/b

Answer C.

Kudos doesn't hurt anyone.
Intern
Intern
avatar
Joined: 18 Sep 2017
Posts: 1
Re: If a and b are non-zero integers and a^8*b^4 - a^4*b^2 = 12. Which of  [#permalink]

Show Tags

New post 18 Sep 2017, 20:51
Let a^4b^2 = x

X^2-x-12=0

X=4 or -3

Which is

a^4*b^2 = 4 or -3
a^2 = 2/b or -2/b or i(3^0.5)

Hence answer is C


Sent from my iPhone using GMAT Club Forum mobile app
Senior Manager
Senior Manager
User avatar
S
Status: love the club...
Joined: 24 Mar 2015
Posts: 278
Re: If a and b are non-zero integers and a^8*b^4 - a^4*b^2 = 12. Which of  [#permalink]

Show Tags

New post 02 Oct 2017, 15:37
Bunuel wrote:
If a and b are non-zero integers and \(a^8*b^4 - a^4*b^2 = 12\). Which of the following could be a^2 in terms of b?

I. 2/b
II. -2/b
III. 3/b

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II and III



hi

the misdoings :-D that I did to this problem are as under

a^4 x b^2 ( a^4 x b^2 - 1 ) = 12
the important thing to notice here is

n(n-1) = 4 x 3 :-)

so, a^4 x b^2 = 4

a^2 = 2/b or -2/b

Answer: C

cheers through the kudos button if this helps
thanks
Re: If a and b are non-zero integers and a^8*b^4 - a^4*b^2 = 12. Which of &nbs [#permalink] 02 Oct 2017, 15:37
Display posts from previous: Sort by

If a and b are non-zero integers and a^8*b^4 - a^4*b^2 = 12. Which of

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  

Events & Promotions

PREV
NEXT


GMAT Club MBA Forum Home| About| Terms and Conditions and Privacy Policy| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.