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If 1 < x < 0, which of the following must be true? [#permalink]
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If 1 < x < 0, which of the following must be true? I. x^3 < x^2 II. x^5 < 1 – x III. x^4 < x^2 A. I only B. I and II only C. II and III only D. I and III only E. I, II and III
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Last edited by Bunuel on 31 Jan 2013, 06:29, edited 2 times in total.
Renamed the topic, edited the question, moved to PS forum and edited OA.



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Re: if 1<x<o, which of following must be true? [#permalink]
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rakesh20j wrote: if 1<x<o, which of following must be true? 1) x^3<x^2 2) x^5<1x 3) x^4<x^2
A. I only B. I and II C. I II and III D. I and III E. II and III The question should read: If 1 < x < 0, which of the following must be true?
I. x^3 < x^2 II. x^5 < 1 – x III. x^4 < x^2A. I only B. I and II only C. II and III only D. I and III only E. I, II and III I. x^3 < x^2 > from 1 < x < 0 it follows that LHS<0<RHS, so this statement is true. II. x^5 < 1 – x > x(x^4+1) < 1 > negative*positive < 0 < 1, so this statement is also true. III. x^4 < x^2 > reduce by x^2 (we can safely do that since from 1 < x < 0 is follows that x^2>0): x^2 < 1. Again, as 1 < x < 0, then x^2 must be less than 1. Hence, this statement is also true. Answer: E. Hope it's clear.
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Re: if 1<x<o, which of following must be true? [#permalink]
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01 Jun 2013, 03:49
Bunuel wrote: rakesh20j wrote: if 1<x<o, which of following must be true? 1) x^3<x^2 2) x^5<1x 3) x^4<x^2
A. I only B. I and II C. I II and III D. I and III E. II and III The question should read: If 1 < x < 0, which of the following must be true?
I. x^3 < x^2 II. x^5 < 1 – x III. x^4 < x^2A. I only B. I and II only C. II and III only D. I and III only E. I, II and III I. x^3 < x^2 > from 1 < x < 0 it follows that LHS<0<RHS, so this statement is true. II. x^5 < 1 – x > x(x^4+1) < 1 > negative*positive < 0 < 1, so this statement is also true. III. x^4 < x^2 > reduce by x^2 (we can safely do that since from 1 < x < 0 is follows that x^2>0): x^2 < 1. Again, as 1 < x < 0, then x^2 must be less than 1. Hence, this statement is also true.Answer: E. Hope it's clear. Bunuel, Did not understand the colored part. As 1 < x < 0, then X will always be negative and X2 will always be positive, so how to derive that x<1.
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Re: if 1<x<o, which of following must be true? [#permalink]
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01 Jun 2013, 03:57
targetbschool wrote: Bunuel wrote: rakesh20j wrote: if 1<x<o, which of following must be true? 1) x^3<x^2 2) x^5<1x 3) x^4<x^2
A. I only B. I and II C. I II and III D. I and III E. II and III The question should read: If 1 < x < 0, which of the following must be true?
I. x^3 < x^2 II. x^5 < 1 – x III. x^4 < x^2A. I only B. I and II only C. II and III only D. I and III only E. I, II and III I. x^3 < x^2 > from 1 < x < 0 it follows that LHS<0<RHS, so this statement is true. II. x^5 < 1 – x > x(x^4+1) < 1 > negative*positive < 0 < 1, so this statement is also true. III. x^4 < x^2 > reduce by x^2 (we can safely do that since from 1 < x < 0 is follows that x^2>0): x^2 < 1. Again, as 1 < x < 0, then x^2 must be less than 1. Hence, this statement is also true.Answer: E. Hope it's clear. Bunuel, Did not understand the colored part. As 1 < x < 0, then X will always be negative and X2 will always be positive, so how to derive that x<1. Consider this: \(x^4 < x^2\) holds true if \(1<x<0\) or \(0<x<1\). Therefore, since given that \(1<x<0\), then \(x^4 < x^2\) must be true. Hope it's clear.
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If 1 < x < 0, which of the following must be true? [#permalink]
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06 Mar 2016, 04:54
animeshk wrote: If \(1 < x < 0\), which of the following must be true?
I. \(x^3 < x^2\) II.\(x^5 < 1 – x\) III.\(x^4 < x^2\)
A. I only B. I and II only C. II and III only D. I and III only E. I, II and III Make sure to format the question properly. When you select the tag "sourceothers please specify" , do mention at the end of the question.As for the question look below. You are told that \(1<x<0\) > x is a negative fraction less 1. Let us analyse the 3 options. I. \(x^3 < x^2\)> MUST BE TRUE as \(x^2\) of \('x'\) in this question >0 while \(x^3 <0\). Consider \(x=0.5\), \(x^2=0.25 >0\) while \(x^3=0.125 <0\)[/m]. II. \(x^5 < 1 – x\) > MUST BE TRUE. Odd power of a negative umber \(x\) (=\(x^5\))<0 while 1x for a negative \(x\) >0. Consider \(x=0.5\) > \(1(0.5)=1+0.5=1.5 >0\) III. \(x^4 < x^2\) > MUST BE TRUE. For fractions 1<x<0 or 0>x>1, greater the power, smaller will be the number. Consider \(x=0.5\), \(x^2=0.25\), \(x^4 = 0.0625\), thus\(x^4<x^2\) Thus, I,II,III are all MUST BE TRUE > E is the correct answer. Hope this helps.
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Re: If 1 < x < 0, which of the following must be true? [#permalink]
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06 Mar 2016, 05:53
animeshk wrote: If \(1 < x < 0\), which of the following must be true?
I. \(x^3 < x^2\) II.\(x^5 < 1 – x\) III.\(x^4 < x^2\)
A. I only B. I and II only C. II and III only D. I and III only E. I, II and III Hi, this Qs tests the properties of numbers/ fractions between 0 and 1..
so lets first check the properties, the answer will come automatically..
1) any even power will always be greater than any odd power.. x^even>x or x^odd..
2) within even powers as we keep increasing power the value will keep becoming lesser.. x^4<x^2..
3) Opposite will be true of ODD powers although the numeric value will be same as in even powers but due to a ive sign, opposite is true.. x^3>x
4) Roots even roots will be imaginary.. \sqrt{x} not real Odd roots are possible and will be lesser for lower 3rd root will be more than 5th root of x.. or 3rd root of x<x
so lets see three choices now I. \(x^3 < x^2\) TRUE as per point 1) above
II.\(x^5 < 1 – x\) x is negative so x will be positive and 1x will be positive too.. whereas x^5 is ive.. hence TRUE
III.\(x^4 < x^2\) TRUE as per point 2) above
All three are correct ans E
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Re: If 1 < x < 0, which of the following must be true? [#permalink]
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22 Feb 2017, 00:11
rakesh20j wrote: If 1 < x < 0, which of the following must be true?
I. x^3 < x^2 II. x^5 < 1 – x III. x^4 < x^2
A. I only B. I and II only C. II and III only D. I and III only E. I, II and III I. x^3 < x^2 > x<1 [dividing by x^2 in both side] > for negative value of x is less than 1. So, true. II. x^5 < 1 – x > x^5+x<1 > negative resultnegative result<1 > negative result<1 > So, true. III. x^4 < x^2 > x^2<1 [dividing by x^2 in both side] > for any value of x (according to question stem) it must be less than 1 > So, true. So, the correct choice is E to me.
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Re: If 1 < x < 0, which of the following must be true? [#permalink]
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19 Jun 2017, 15:52
Bunuel wrote: rakesh20j wrote: if 1<x<o, which of following must be true? 1) x^3<x^2 2) x^5<1x 3) x^4<x^2
A. I only B. I and II C. I II and III D. I and III E. II and III The question should read: If 1 < x < 0, which of the following must be true?
I. x^3 < x^2 II. x^5 < 1 – x III. x^4 < x^2A. I only B. I and II only C. II and III only D. I and III only E. I, II and III I. x^3 < x^2 > from 1 < x < 0 it follows that LHS<0<RHS, so this statement is true. II. x^5 < 1 – x > x(x^4+1) < 1 > negative*positive < 0 < 1, so this statement is also true. III. x^4 < x^2 > reduce by x^2 (we can safely do that since from 1 < x < 0 is follows that x^2>0): x^2 < 1. Again, as 1 < x < 0, then x^2 must be less than 1. Hence, this statement is also true. Answer: E. Hope it's clear. Try by substituting a value in the equation. Take x = 1/2 and cross check each option. I. x^3 < x^2  TrueII. x^5 < 1 – x  TrueIII. x^4 < x^2[/b]  TrueHence, answer is E  All  I, II & III
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Re: If 1 < x < 0, which of the following must be true? [#permalink]
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20 Jun 2017, 19:18
rakesh20j wrote: If 1 < x < 0, which of the following must be true?
I. x^3 < x^2 II. x^5 < 1 – x III. x^4 < x^2
A. I only B. I and II only C. II and III only D. I and III only E. I, II and III We can see that x is a negative number between 0 and 1. Thus, x^odd = negative and x^even = positive. So, I is true. Furthermore, 1  x = 1 + x, which is positive, so it’s greater than x^5, which is negative. So, II is also true. Lastly, when x is raised to an even integer power, the larger the even integer, the smaller the power. For example, if x = 1/2, then x^4 = 1/16 and x^2 = 1/4. Thus, x^4 < x^2. So, III is also true . Answer: E
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Re: If 1 < x < 0, which of the following must be true? [#permalink]
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20 Jun 2017, 22:54
imo E take a value of 0.2 then substitute in every equation . hence all satisfy



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Re: If 1 < x < 0, which of the following must be true? [#permalink]
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21 Jun 2017, 00:40
rakesh20j wrote: If 1 < x < 0, which of the following must be true?
I. x^3 < x^2 II. x^5 < 1 – x III. x^4 < x^2
A. I only B. I and II only C. II and III only D. I and III only E. I, II and III Here it is given that 1<x<0 so x is a ve number greater than 1. I. x^3 < x^2 is true . X^2 is positive, x^3 is ve II. x^5 < 1x i true. X^5 is ve , 1x is +ve. III. X^4 < x^2 is true. X^4 and x^2 are both +ve. Also since x has a domain (1,0), x^4 < x^2.
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