A unary operator â—Š is defined as â—Š x = x2
- 4x + 3. If ◊β = ◊(3 - 2β), what is the value of β?
A) 1 only
B) - 1 only
C) ± 1
D) None of these
Help plz!!
This topic has expert replies
-
- GMAT Instructor
- Posts: 2630
- Joined: Wed Sep 12, 2012 3:32 pm
- Location: East Bay all the way
- Thanked: 625 times
- Followed by:119 members
- GMAT Score:780
Let's change the diamond to a # so it's easier to type.
#x, for any value of x, is defined as x² - 4x + 3. So #2 = 2² - 4*2 + 3, #10 = 10² - 4*10 + 3, etc.
From there, we can say that
#ß = ß² - 4ß + 3
and
#(3 - 2ß) = (3 - 2ß)² - 4*(3 - 2ß) + 3
Since we're told these are equal, we've got
ß² - 4ß + 3 = (3 - 2ß)² - 4*(3 - 2ß) + 3
or
ß² - 4ß + 3 = 9 - 12ß + 4ß² - 12 + 8ß + 3
or
0 = 3ß² - 3
or
3 = 3ß²
or
1 = ß²
so ß = ±1
#x, for any value of x, is defined as x² - 4x + 3. So #2 = 2² - 4*2 + 3, #10 = 10² - 4*10 + 3, etc.
From there, we can say that
#ß = ß² - 4ß + 3
and
#(3 - 2ß) = (3 - 2ß)² - 4*(3 - 2ß) + 3
Since we're told these are equal, we've got
ß² - 4ß + 3 = (3 - 2ß)² - 4*(3 - 2ß) + 3
or
ß² - 4ß + 3 = 9 - 12ß + 4ß² - 12 + 8ß + 3
or
0 = 3ß² - 3
or
3 = 3ß²
or
1 = ß²
so ß = ±1
-
- GMAT Instructor
- Posts: 2630
- Joined: Wed Sep 12, 2012 3:32 pm
- Location: East Bay all the way
- Thanked: 625 times
- Followed by:119 members
- GMAT Score:780
You could also cheat a bit here if you have no time or no idea what to do. Since the function tells us that #ß = #(3 - 2ß), it's pretty reasonable to guess that ß = 3 - 2ß, so ß = 1 is probably a solution.
Since we're squaring terms inside the function, there's also a decent chance that -1 is a solution too, so 1 and -1 seem like good guesses.
Since we're squaring terms inside the function, there's also a decent chance that -1 is a solution too, so 1 and -1 seem like good guesses.
-
- GMAT Instructor
- Posts: 2630
- Joined: Wed Sep 12, 2012 3:32 pm
- Location: East Bay all the way
- Thanked: 625 times
- Followed by:119 members
- GMAT Score:780
We could also save time by trying the answers. Since all of them (except D) involve 1 and -1, let's try both.
First case, ß = 1:
#1 = #(3 - 2*1)
#1 = #1
Well, duh!
Second case, ß = -1:
#(-1) = #(3 - 2*(-1))
#(-1) = #5
(-1)² - 4*(-1) + 3 = 5² - 4*5 + 3
1 + 4 = 5² - 20
Also true! So ß = -1 is another solution.
First case, ß = 1:
#1 = #(3 - 2*1)
#1 = #1
Well, duh!
Second case, ß = -1:
#(-1) = #(3 - 2*(-1))
#(-1) = #5
(-1)² - 4*(-1) + 3 = 5² - 4*5 + 3
1 + 4 = 5² - 20
Also true! So ß = -1 is another solution.
-
- GMAT Instructor
- Posts: 2630
- Joined: Wed Sep 12, 2012 3:32 pm
- Location: East Bay all the way
- Thanked: 625 times
- Followed by:119 members
- GMAT Score:780
It's also reasonable to wonder how we can exclude D, since we could have lots of solutions, right?
When we go from this step
#(3 - 2ß) = (3 - 2ß)² - 4*(3 - 2ß) + 3
to this step
ß² - 4ß + 3 = (3 - 2ß)² - 4*(3 - 2ß) + 3
We can see that our equation is in one variable (ß) and its highest power is ². So we invoke the Fundamental Theorem of Algebra - which is as mighty as it sounds! - to tell us that there are AT MOST two solutions to this problem.
When we go from this step
#(3 - 2ß) = (3 - 2ß)² - 4*(3 - 2ß) + 3
to this step
ß² - 4ß + 3 = (3 - 2ß)² - 4*(3 - 2ß) + 3
We can see that our equation is in one variable (ß) and its highest power is ². So we invoke the Fundamental Theorem of Algebra - which is as mighty as it sounds! - to tell us that there are AT MOST two solutions to this problem.
GMAT/MBA Expert
- Jay@ManhattanReview
- GMAT Instructor
- Posts: 3008
- Joined: Mon Aug 22, 2016 6:19 am
- Location: Grand Central / New York
- Thanked: 470 times
- Followed by:34 members
Hi Hmna,
This is not a GMAT question. It helps to prepare GMAT type questions. Pl. post only GMAT questions.
We have â—Š x = x^2- 4x + 3;
We are given that ◊β = ◊(3 - 2β)
â—ŠB = B^2 - 4B + 3;
â—Š(3 - 2B) = (3 - 2B)^2 - 4(3 - 2B) + 3 = 9 - 12B + 4B^2 - 12 + 8B + 3 = 4B^2 - 4B
=> B^2 - 4B + 3 = 4B^2 - 4B
=> 3B^2 = 3
=> B^2 = 1
=> [spoiler]B = +/-1[/spoiler]
The correct answer: C
Hope this helps!
Relevant book: Manhattan Review GMAT Number Properties Guide
-Jay
_________________
Manhattan Review GMAT Prep
Locations: New York | Beijing | Auckland | Milan | and many more...
Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.
-
- GMAT Instructor
- Posts: 2630
- Joined: Wed Sep 12, 2012 3:32 pm
- Location: East Bay all the way
- Thanked: 625 times
- Followed by:119 members
- GMAT Score:780
It does test GMAT concepts, though, so I'd say it's fair game and not bad practice. Its only non-GMATy features are the use of the term 'unary' and the four answer choices.Jay@ManhattanReview wrote:
This is not a GMAT question.
GMAT/MBA Expert
- Jay@ManhattanReview
- GMAT Instructor
- Posts: 3008
- Joined: Mon Aug 22, 2016 6:19 am
- Location: Grand Central / New York
- Thanked: 470 times
- Followed by:34 members
I agree with you, Matt; however, we must advise the new poster that the source being referred to is probably not the GMAT source. And maybe he/she is practicing few non-GMAT type questions. As I see that Hmna has posted four questions and each have only four options.Matt@VeritasPrep wrote:It does test GMAT concepts, though, so I'd say it's fair game and not bad practice. Its only non-GMATy features are the use of the term 'unary' and the four answer choices.Jay@ManhattanReview wrote:
This is not a GMAT question.