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number properties

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sud21 GMAT Destroyer! Default Avatar
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number properties Post Fri Jan 13, 2012 6:39 am
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  • Lap #[LAPCOUNT] ([LAPTIME])
    Is x>1?
    1). x^2<x
    2). x<rootx


    source search engines

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    sud21 GMAT Destroyer! Default Avatar
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    Post Sat Jan 14, 2012 11:59 am
    St.1 x^2<x, this is possible when x is a fraction, that is, a positive fraction less than 1 but greater than 0.

    If x=1/2 then x^2=1/4 which is less than x=1/2.

    If x=-1/2 then x^2=1/4 which is more than x=-1/2, thus not meeting the condition.

    Thus we can say that x can never be greater than 1 but it will be less than 1 but greater than 0.

    Since it is asked that whether x>1, we can say it as a no, so, sufficient.

    St.2 x<rootx, squaring both sides we get x<x^2 which is same as St.1 provided x is a positive number. Sufficient.

    Ans.D

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    Post Sat Jan 14, 2012 12:11 pm
    sud21 wrote:
    St.2 x<rootx, squaring both sides we get x<x^2 which is same as St.1 provided x is a positive number. Sufficient.
    Ans.D
    To add to the explanation, you cannot always square the terms of an equation without knowing the sign of the terms . x < y, DOESNOT always imply x^2 < y^2

    x<y, => x^2 < y^2, if x,y >0
    x<y, => x^2 > y^2, if x,y <0

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    Post Sat Jan 14, 2012 2:19 pm
    sud21 wrote:
    Is x>1?
    1). x^2<x
    2). x<rootx


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    Stmt 1 - can be reduced to
    x^2 - x < 0, which is x(x-1)<0,
    the two roots are 0 and 1. Check for x <0, 0<x<1 and x >1.
    We find that the condition holds only when 0<x<1, thus x > 1? Can be answered with an unequivocal NO. Hence sufficient.

    Stmt 2 - x < root (x), thus we know x >0 and x<1 since a negative number can not have a real root (GMAT does not deal with imaginary numbers) and only numbers < 1 are such that their roots are bigger than themselves. Thus sufficient.

    Hence D. Sorry solved the question right and forgot when both are sufficient independently we gotta choose D!! and not C Smile

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    Last edited by mj78ind on Sun Jan 15, 2012 7:01 am; edited 1 time in total

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    Post Sat Jan 14, 2012 10:14 pm
    Another approach to solve this kind of problem is interpretation of graphs and the approach is very quick and easy if you are familiar with graphs of standard functions.

    If we plot the graphs of x, x², and √x on the same axes, we will get the following diagram



    From the graph, it is very clear that x² is less than x, i.e. the yellow line is below the red one only for 0 < x < 1. And x is less than √x, i.e. the blue line is above the red one is only for 0 < x < 1.

    Hence, to satisfy any of the two statements x must be less than 1. Hence, both statements are individually sufficient to answer the question with NO.

    The correct answer is D.

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    Post Thu Mar 15, 2012 10:37 pm
    Is x>1?
    1). x^2<x
    2). x<rootx


    The answer is D. Got it in less than 2 mins...

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