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M19-05

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M19-05  [#permalink]

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New post 16 Sep 2014, 01:05
1
5
00:00
A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

61% (01:30) correct 39% (01:41) wrong based on 170 sessions

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Re M19-05  [#permalink]

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New post 16 Sep 2014, 01:05
Official Solution:


Is \(1+(x+x^2)+(x^3+x^4)+...+(x^9+x^{10}) \gt 0\)?

(1) \(x \lt -1\). From this statement it follows that \(x+x^2 \gt 0\), \(x^3+x^4 \gt 0\), ..., \(x^9+x^{10} \gt 0\), so the sum is also more than zero. Sufficient.

(2) \(x^2 \gt 2\). This statement implies that \(x \lt -\sqrt{2}\) or \(x \gt \sqrt{2}\). Even if \(x\) itself is negative then still as above: \(x+x^2 \gt 0\), \(x^3+x^4 \gt 0\), ..., \(x^9+x^{10} \gt 0\), so the sum is also more than zero. Sufficient.


Answer: D
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Re: M19-05  [#permalink]

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New post 26 Mar 2017, 09:38
Bunuel wrote:
Is \(1 + x + x^2 + ... + x^{10}\) positive?


(1) \(x \lt -1\)

(2) \(x^2 \gt 2\)




So essentially we need to figure whether x is between 0-1 or not. is that correct? Thank you.
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Re: M19-05  [#permalink]

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New post 02 Apr 2017, 16:20
Strange question...
Wouldn't the answer be positive regardless what the statement is?
two statements add no value to the answer to this question...
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Re: M19-05  [#permalink]

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New post 04 Apr 2017, 08:35
1
Bunuel wrote:
Is \(1 + x + x^2 + ... + x^{10}\) positive?


(1) \(x \lt -1\)

(2) \(x^2 \gt 2\)


Hi,

The Q in the present format may not require any statement.

\(1 + x + x^2 + ... + x^{10}\)

1) If x is POSITIVE, it will always be positive.

2) If x is NEGATIVE and x is between 0 and -1.
\(1 + x + x^2 + ... + x^{10}= (1+x)+(x^2+x^3)+....(x^8+x^9)+x^{10}\)
Here x is between 0 and 1..
So 1+x will be positive.
\(x^2+x^3\) will have x^2 as positive and x^3 as negative but the numeric value of lower powers will be greater. Example (1/2)^2=1/4 whereas (1/2)^3 is 1/8....
Similarly all other brackets too would be positive and thus total equation will be positive..
3) If x is NEGATIVE and <-1..
\(1 + x + x^2 + ... + x^{10}= 1+(x+x^2)+(x^3+x^4)+.....+(x^9+x^{10})\)..
Here each bracket will be POSITIVE as higher power mean higher value and each bracket has higher EVEN power.
Again overall POSITIVE.

Bunuel, the Q will be correct if we remove 1 from equation and Q becomes
Is \(x + x^2 + ... + x^{10}\) positive?
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Re: M19-05  [#permalink]

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New post 04 Apr 2017, 08:43
chetan2u wrote:
Bunuel wrote:
Is \(1 + x + x^2 + ... + x^{10}\) positive?


(1) \(x \lt -1\)

(2) \(x^2 \gt 2\)


Hi,

The Q in the present format may not require any statement.

\(1 + x + x^2 + ... + x^{10}\)

1) If x is POSITIVE, it will always be positive.

2) If x is NEGATIVE and x is between 0 and -1.
\(1 + x + x^2 + ... + x^{10}= (1+x)+(x^2+x^3)+....(x^8+x^9)+x^{10}\)
Here x is between 0 and 1..
So 1+x will be positive.
\(x^2+x^3\) will have x^2 as positive and x^3 as negative but the numeric value of lower powers will be greater. Example (1/2)^2=1/4 whereas (1/2)^3 is 1/8....
Similarly all other brackets too would be positive and thus total equation will be positive..
3) If x is NEGATIVE and <-1..
\(1 + x + x^2 + ... + x^{10}= 1+(x+x^2)+(x^3+x^4)+.....+(x^9+x^{10})\)..
Here each bracket will be POSITIVE as higher power mean higher value and each bracket has higher EVEN power.
Again overall POSITIVE.

Bunuel, the Q will be correct if we remove 1 from equation and Q becomes
Is \(x + x^2 + ... + x^{10}\) positive?


Thank you. I'm revising the question.
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Re: M19-05  [#permalink]

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New post 06 Apr 2017, 09:23
If I'm understanding this correctly, you just need to know if x is between 0 and -1.

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Re: M19-05  [#permalink]

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New post 31 Jul 2017, 04:46
Bunuel wrote:
chetan2u wrote:
Bunuel wrote:
Is \(1 + x + x^2 + ... + x^{10}\) positive?


(1) \(x \lt -1\)

(2) \(x^2 \gt 2\)


Hi,

The Q in the present format may not require any statement.

\(1 + x + x^2 + ... + x^{10}\)

1) If x is POSITIVE, it will always be positive.

2) If x is NEGATIVE and x is between 0 and -1.
\(1 + x + x^2 + ... + x^{10}= (1+x)+(x^2+x^3)+....(x^8+x^9)+x^{10}\)
Here x is between 0 and 1..
So 1+x will be positive.
\(x^2+x^3\) will have x^2 as positive and x^3 as negative but the numeric value of lower powers will be greater. Example (1/2)^2=1/4 whereas (1/2)^3 is 1/8....
Similarly all other brackets too would be positive and thus total equation will be positive..
3) If x is NEGATIVE and <-1..
\(1 + x + x^2 + ... + x^{10}= 1+(x+x^2)+(x^3+x^4)+.....+(x^9+x^{10})\)..
Here each bracket will be POSITIVE as higher power mean higher value and each bracket has higher EVEN power.
Again overall POSITIVE.

Bunuel, the Q will be correct if we remove 1 from equation and Q becomes
Is \(x + x^2 + ... + x^{10}\) positive?


Thank you. I'm revising the question.


The question is still not reversed; please remove 1...thanks
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Re: M19-05  [#permalink]

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New post 20 Aug 2017, 00:12
it is a GP
1+X^2+X^3....+X^10= (X^11-1)/(X-1)
RHS is always positive if |X|>1

1- Statement: 1..says X<-1, means |X|>1...therefore RHS is positive always: Statement is sufficient
2- Statement: 2..says X^2>2, means |X|>1...therefore RHS is positive always: Statement is sufficient

hence Answer is D
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Re: M19-05  [#permalink]

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New post 04 Nov 2018, 07:22
buan15 wrote:

The question is still not reversed; please remove 1...thanks


+1
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Re: M19-05  [#permalink]

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New post 04 Nov 2018, 09:07
barryseal wrote:
buan15 wrote:

The question is still not reversed; please remove 1...thanks


+1



what does that mean?
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Re: M19-05  [#permalink]

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New post 04 Nov 2018, 10:05
prabsahi wrote:
barryseal wrote:
buan15 wrote:

The question is still not reversed; please remove 1...thanks


+1



what does that mean?


that question is still not adjusted
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Re: M19-05  [#permalink]

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New post 14 Apr 2019, 06:31
Hi!!

The question is: 1+x+x2+...+x10 positive?

Why do you assume that the set is 1+x+x2+x3+x4+x5+x6+x7+x8+x9+x10?

Since the question stem dosen't say anything about the nature of the set, you can form many different sets such as
1+x+x2+x2+x3+x3+x3+......+x10
So, we can have several answers
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Re: M19-05   [#permalink] 14 Apr 2019, 06:31
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