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

 It is currently 18 Oct 2019, 14:16 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

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.  For all integers n, n* = n(n – 1). What is the value of x* w

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

Hide Tags

Manager  Affiliations: PMP
Joined: 13 Oct 2009
Posts: 210
For all integers n, n* = n(n – 1). What is the value of x* w  [#permalink]

Show Tags

1
6 00:00

Difficulty:   75% (hard)

Question Stats: 52% (01:52) correct 48% (01:59) wrong based on 151 sessions

HideShow timer Statistics

For all integers n, n* = n(n – 1). What is the value of x* when x is an integer?

(1) x* = x

(2) (x – 1) * = x – 2

_________________
Thanks, Sri
-------------------------------
keep uppp...ing the tempo...

Press +1 Kudos, if you think my post gave u a tiny tip
Math Expert V
Joined: 02 Sep 2009
Posts: 58453
Re: what is the value of x*  [#permalink]

Show Tags

3
2
srini123 wrote:
quick question, when (x-1)(x-2) = (x - 2) why cant we cross out (x-2) and arrive at x-1=1 and x=2 ?

Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We can not divide by zero.

Never multiply (or reduce) inequality by variable (or expression with variable) if you don't know the sign of it or are not certain that variable (or expression with variable) doesn't equal to zero.

You cannot reduce by x-2 as x may be equal to 2, and in this case you are dividing by 0.
_________________
General Discussion
Manager  Joined: 18 Aug 2009
Posts: 246
Re: what is the value of x*  [#permalink]

Show Tags

IMO E.

1) x* = x(x-1) = x
x = 2 or 0

2) ( x – 1) * = (x-1)(x–2) = x-2
x=2 or 0

combining 1 and 2, not possible to answer if x is 0 or 2.
Manager  Joined: 29 Oct 2009
Posts: 177
GMAT 1: 750 Q50 V42 Re: what is the value of x*  [#permalink]

Show Tags

2
1
Question Stem : n* = n(n-1) ; x = ?

St. (1) : x* = x(x-1) = x
$$x^2 - x = x$$
$$x^2 -2x = 0$$
$$x(x-2) = 0$$
Thus x can be equal to 0 or 2.
Insufficient.

St. (2) : (x-1)* = (x-1)(x-2) = x - 2
$$x^2 - 3x + 2 = x - 2$$
$$(x-2)(x-2) = 0$$
Thus x can only be equal to 2.
Hence Sufficient.

_________________
Click below to check out some great tips and tricks to help you deal with problems on Remainders!
http://gmatclub.com/forum/compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html#p651942

Word Problems Made Easy!
1) Translating the English to Math : http://gmatclub.com/forum/word-problems-made-easy-87346.html
2) 'Work' Problems Made Easy : http://gmatclub.com/forum/work-word-problems-made-easy-87357.html
3) 'Distance/Speed/Time' Word Problems Made Easy : http://gmatclub.com/forum/distance-speed-time-word-problems-made-easy-87481.html
Manager  Affiliations: PMP
Joined: 13 Oct 2009
Posts: 210
Re: what is the value of x*  [#permalink]

Show Tags

sriharimurthy wrote:
Question Stem : n* = n(n-1) ; x = ?

St. (1) : x* = x(x-1) = x
$$x^2 - x = x$$
$$x^2 -2x = 0$$
$$x(x-2) = 0$$
Thus x can be equal to 0 or 2.
Insufficient.

St. (2) : (x-1)* = (x-1)(x-2) = x - 2
$$x^2 - 3x + 2 = x - 2$$
$$(x-2)(x-2) = 0$$
Thus x can only be equal to 2.
Hence Sufficient.

quick question, when (x-1)(x-2) = (x - 2) why cant we cross out (x-2) and arrive at x-1=1 and x=2 ?
_________________
Thanks, Sri
-------------------------------
keep uppp...ing the tempo...

Press +1 Kudos, if you think my post gave u a tiny tip
Manager  Joined: 29 Oct 2009
Posts: 177
GMAT 1: 750 Q50 V42 Re: what is the value of x*  [#permalink]

Show Tags

1
1
srini123 wrote:

quick question, when (x-1)(x-2) = (x - 2) why cant we cross out (x-2) and arrive at x-1=1 and x=2 ?

As a general rule, never cross multiply variables on two sides of an equation unless we can be sure that it cannot be equal to 0.

In this case it turned out that the answer would have come the same. However, there is no way of knowing that in advance.
Take the following case :
(x-2)(x-2) = (x-2) ----> if we cross out (x-2) on both sides, we will get x = 3. However, x = 2 also satisfies this equation.
(x-3)(x-2) = (x-2) ----> again if we cross out (x-2) on both sides we will get x = 4. However, x = 2 also satisfies this equation.

One more thing to note is that an algebraic expression, the highest power of x will determine the number of roots of x. Therefore a quadratic equation in x will always have 2 values that satisfy it. It can be that the two values are the same as they are in our case. However, they will still be treated as two values. Thus we will say that the roots of the equation are 2 and 2.
_________________
Click below to check out some great tips and tricks to help you deal with problems on Remainders!
http://gmatclub.com/forum/compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html#p651942

Word Problems Made Easy!
1) Translating the English to Math : http://gmatclub.com/forum/word-problems-made-easy-87346.html
2) 'Work' Problems Made Easy : http://gmatclub.com/forum/work-word-problems-made-easy-87357.html
3) 'Distance/Speed/Time' Word Problems Made Easy : http://gmatclub.com/forum/distance-speed-time-word-problems-made-easy-87481.html

Originally posted by sriharimurthy on 20 Nov 2009, 17:17.
Last edited by sriharimurthy on 20 Nov 2009, 17:20, edited 1 time in total.
Math Expert V
Joined: 02 Sep 2009
Posts: 58453
Re: what is the value of x*  [#permalink]

Show Tags

sriharimurthy wrote:
One more thing to note is that an algebraic expression, the highest power of x will determine the number of roots of x. Therefore a quadratic equation in x will always have 2 values that satisfy it. It can be that the two values are the same as they are in our case. However, they will still be treated as two values. Thus we will say that the roots of the equation are 2 and 2.

One little thing: in quadratic equation highest power of x will determine maximum # of roots. Quadratic equation can have 0, 1, or 2 roots.

When discriminant more than 0 you'll have 2 roots;
When discriminant is zero you'll have 1 root (and not two treated as one);
When discriminant is less than 0 you won't have any real roots.

x^2-4x+3=0 has 2 roots;
x^2-4x+4=0 has 1 root;
x^2-4x+5=0 has 0 real roots.
_________________
Manager  Joined: 29 Oct 2009
Posts: 177
GMAT 1: 750 Q50 V42 Re: what is the value of x*  [#permalink]

Show Tags

Bunuel wrote:

One little thing: in quadratic equation highest power of x will determine maximum # of roots. Quadratic equation can have 0, 1, or 2 roots.

When discriminant more than 0 you'll have 2 roots;
When discriminant is zero you'll have 1 root (and not two treated as one);
When discriminant is less than 0 you won't have any real roots.

x^2-4x+3=0 has 2 roots;
x^2-4x+4=0 has 1 root;
x^2-4x+5=0 has 0 real roots.

In cases when the discriminant is more than 0 or less than 0, you will always have two distinct roots. In one case real and in the other a pair of complex conjugates.

What you have stated is perfectly valid provided we specify that we are only concerned with the real and distinct roots of an equation.

When the discriminant is 0, you are right in saying that it will have only one distinct root. However, the term root by itself does not imply distinct or real. Thus when the discriminant is 0 it actually has a double root (which is to account for its multiplicity).

So, unless we are asked to find the number of distinct real solutions of a quadratic equation am I not right in saying that it will always have two roots?
_________________
Click below to check out some great tips and tricks to help you deal with problems on Remainders!
http://gmatclub.com/forum/compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html#p651942

Word Problems Made Easy!
1) Translating the English to Math : http://gmatclub.com/forum/word-problems-made-easy-87346.html
2) 'Work' Problems Made Easy : http://gmatclub.com/forum/work-word-problems-made-easy-87357.html
3) 'Distance/Speed/Time' Word Problems Made Easy : http://gmatclub.com/forum/distance-speed-time-word-problems-made-easy-87481.html
Math Expert V
Joined: 02 Sep 2009
Posts: 58453
Re: what is the value of x*  [#permalink]

Show Tags

sriharimurthy wrote:
In cases when the discriminant is more than 0 or less than 0, you will always have two distinct roots. In one case real and in the other a pair of complex conjugates.

What you have stated is perfectly valid provided we specify that we are only concerned with the real and distinct roots of an equation.

When the discriminant is 0, you are right in saying that it will have only one distinct root. However, the term root by itself does not imply distinct or real. Thus when the discriminant is 0 it actually has a double root (which is to account for its multiplicity).

So, unless we are asked to find the number of distinct real solutions of a quadratic equation am I not right in saying that it will always have two roots?

When the discriminant is negative quadratic equation has no real roots. As I stated above.

GMAT is dealing ONLY with real numbers. No need to complicate this.

When the quadratic equation has one root it's rarely called "double root". More common to say that it has 1 root. So usually when we say root of equation we think about the distinct root, even if we don't specify it.

So we can say:
Discriminant positive - 2 roots;
Discriminant 0 - 1 root, even not specifying that it's distinct;
Discriminant negative - no real roots, for GMAT no root at all.

Though you are right in saying that quadratic equation always has two roots if:
A. We consider complex roots, which is not the case for GMAT;
B. We consider the concept of "double root", though I've never seen GMAT even mentioning the double root.

From my experience in GMAT, we can say that it's considering quadratic equation as parabola and the roots as its intersection with X-axis. One tangent point - one root, two intersections - 2 roots, no intersection - no root.

Again as I said no need to complicate.
_________________
SVP  Joined: 29 Aug 2007
Posts: 1942
Re: What is the value of x*?  [#permalink]

Show Tags

apoorvasrivastva wrote:
For all integers n, n* = n(n-1). What is the value of x*?

(1) X* = X
(2) (X - 1)* = (X - 2)

OA after some discussion !! (1) X* = X
x* = x (x-1)

x (x-1) = x
x^2 - 2x = 0
x (x - 2) = 0
x = 0 or 2 ....NSF.

(2) (X - 1)* = (X - 2)
(x - 1)* = (x - 1) (x - 2)
(x - 1) (x - 2) = (x - 2)
(x - 1) (x - 2) - (x - 2)=0
(x - 2) [x - 1) -1]= 0
(x - 2) (x-2) = 0
x = 2. Suff

B.

Updated for overlooking....
_________________
Gmat: http://gmatclub.com/forum/everything-you-need-to-prepare-for-the-gmat-revised-77983.html

GT
SVP  Joined: 29 Aug 2007
Posts: 1942
Re: What is the value of x*?  [#permalink]

Show Tags

apoorvasrivastva wrote:
GMAT TIGER wrote:
apoorvasrivastva wrote:
For all integers n, n* = n(n-1). What is the value of x*?

(1) X* = X
(2) (X - 1)* = (X - 2)

OA after some discussion !! (1) X* = X
x* = x (x-1)

x (x-1) = x
x^2 - 2x = 0
x (x - 2) = 0
x = 0 or 2 ....NSF.

(2) (X - 1)* = (X - 2)
(x - 1)* = (x - 1) (x - 2)
(x - 1) (x - 2) = (x - 2)
(x - 1) (x - 2) - (x - 2)=0
(x - 2) [x - 1) -1]= 0 -------> (x-2)(x-2)=0
(x - 2) (x- 2) = 0 --------------> therefore x is necessarily 2 so st.2 is sufficient Thanks... I updated...
_________________
Gmat: http://gmatclub.com/forum/everything-you-need-to-prepare-for-the-gmat-revised-77983.html

GT
Manager  Joined: 29 Oct 2009
Posts: 177
GMAT 1: 750 Q50 V42 Re: what is the value of x*  [#permalink]

Show Tags

Quote:
Again as I said no need to complicate.

For all practical purposes, your statements regarding the number of roots of an equation are perfectly valid Bunuel.

I completely agree with you when you say that it is getting unnecessarily complicated for a GMAT discussion.

The concept of double root and the relevance of its multiplicity is way out of GMAT scope.

Still, it was interesting to discuss.

Cheers.
_________________
Click below to check out some great tips and tricks to help you deal with problems on Remainders!
http://gmatclub.com/forum/compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html#p651942

Word Problems Made Easy!
1) Translating the English to Math : http://gmatclub.com/forum/word-problems-made-easy-87346.html
2) 'Work' Problems Made Easy : http://gmatclub.com/forum/work-word-problems-made-easy-87357.html
3) 'Distance/Speed/Time' Word Problems Made Easy : http://gmatclub.com/forum/distance-speed-time-word-problems-made-easy-87481.html
SVP  Joined: 29 Aug 2007
Posts: 1942
Re: what is the value of x*  [#permalink]

Show Tags

srini123 wrote:
sriharimurthy wrote:
Question Stem : n* = n(n-1) ; x = ?

St. (1) : x* = x(x-1) = x
$$x^2 - x = x$$
$$x^2 -2x = 0$$
$$x(x-2) = 0$$
Thus x can be equal to 0 or 2.
Insufficient.

St. (2) : (x-1)* = (x-1)(x-2) = x - 2
$$x^2 - 3x + 2 = x - 2$$
$$(x-2)(x-2) = 0$$
Thus x can only be equal to 2.
Hence Sufficient.

quick question, when (x-1)(x-2) = (x - 2) why cant we cross out (x-2) and arrive at x-1=1 and x=2 ?

However it is already explained, never do that in GMAT math. You can do that only if both variables are +ve.
_________________
Gmat: http://gmatclub.com/forum/everything-you-need-to-prepare-for-the-gmat-revised-77983.html

GT
Manager  Joined: 19 Nov 2007
Posts: 137
Re: what is the value of x*  [#permalink]

Show Tags

St1: solving the equation x^2-2x=0; x=0 or 2;INSUFF
St2: solving the equation x^2-4x+4=0; x=2;SUFF

Hence answer is B
Manager  Joined: 12 Sep 2014
Posts: 142
GMAT 1: 740 Q49 V41 GPA: 3.94
Re: For all integers n, n* = n(n – 1). What is the value of x* w  [#permalink]

Show Tags

Statement 1: x*=x
So, x = x(x-1)
Don't cross-out/eliminate anything--just solve for x: x=0, 2 INS

Statement 2: (x-1)*=(x-2)==(x-1)(x-2)
Again, don't eliminate anything--just solve for x: x=2
SUF! Choose B
Non-Human User Joined: 09 Sep 2013
Posts: 13260
Re: For all integers n, n* = n(n – 1). What is the value of x* w  [#permalink]

Show Tags

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________ Re: For all integers n, n* = n(n – 1). What is the value of x* w   [#permalink] 13 Apr 2019, 10:00
Display posts from previous: Sort by

For all integers n, n* = n(n – 1). What is the value of x* w

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

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