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# M24-12

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Math Expert
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M24-12 [#permalink]

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16 Sep 2014, 00:21
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Difficulty:

55% (hard)

Question Stats:

46% (01:32) correct 54% (01:05) wrong based on 28 sessions

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If $$ac \ne 0$$, do graphs $$y=ax^2+b$$ and $$y=cx^2+d$$ intersect?

(1) $$a = -c$$

(2) $$b \gt d$$
[Reveal] Spoiler: OA

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Re M24-12 [#permalink]

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16 Sep 2014, 00:21
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Official Solution:

Notice that graphs of $$y=ax^2+b$$ and $$y=cx^2+d$$ are parabolas.

Algebraic approach:

(1) $$a = -c$$. Given: $$y_1= ax^2 + b$$ and $$y_2=-ax^2 + d$$. Now, if these two parabolas cross, then for some $$x$$, $$ax^2+b=-ax^2 + d$$ should be true, which means that equation $$2ax^2+(b-d)=0$$ must have a solution, some real root(s), or in other words discriminant of this quadratic equation must be $$\ge 0$$. So, the question becomes: is $$discriminant=0-8a(b-d) \ge 0$$? Or, is $$discriminant=-8a(b-d) \ge 0$$? Now can we determine whether this is true? We know nothing about $$a$$, $$b$$, and $$d$$, hence no. Not sufficient.

(2) $$b \gt d$$. The same steps: if $$y_1= ax^2 + b$$ and $$y_2= cx^2 + d$$ cross, then for some $$x$$, $$ax^2 +b=cx^2+d$$ should be true, which means that equation $$(a-c)x^2+(b-d)=0$$ must have a solution or in other words discriminant of this quadratic equation must be $$\ge 0$$. So, the question becomes: is $$discriminant=0-4(a-c)(b-d) \ge 0$$? Or, is $$discriminant=-4(a-c)(b-d) \ge 0$$? Now can we determine whether this is true? We know that $$b-d \gt 0$$ but what about $$a-c$$? Hence no. Not sufficient.

(1)+(2) We have that $$a=-c$$ and $$b \gt d$$, so $$y_1= ax^2 + b$$ and $$y_2=-ax^2 + d$$. The same steps as above: $$2ax^2+(b-d)=0$$ and the same question remains: is $$discriminant=-8a(b-d) \ge 0$$ true? $$b-d \gt 0$$ but what about $$a$$? Not sufficient.

Else consider two cases.

First case: $$y=-x^2+1$$ and $$y=x^2+0$$ (upward and downward parabolas). Notice that these parabolas satisfy both statements and they cross each other;

Second case: $$y=x^2+1$$ and $$y=-x^2+0$$ (also upward and downward parabolas). Notice that these parabolas satisfy both statements and they do not cross each other.

Answer: E
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Re: M24-12 [#permalink]

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17 Aug 2016, 22:11
Is Parabola concept tested in Gmat?..if yes then how often?

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Re: M24-12 [#permalink]

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18 Aug 2016, 06:50
subhajit1 wrote:
Is Parabola concept tested in Gmat?..if yes then how often?

Yes but not very often.
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Re M24-12 [#permalink]

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23 Aug 2016, 04:36
I think this is a high-quality question and I agree with explanation.

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Re: M24-12 [#permalink]

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28 Aug 2016, 05:38
Good question. Could be solver quickly through graphical approach.

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Re: M24-12 [#permalink]

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20 Jul 2017, 01:57
hi brunel

i struggle with parabola questions..may be because of lack of clarity in the concepts.
In the algebra approach i understood why both the equations have been equated but how you break to get the dicriminants i.e from 2ax2+(b−d)=0 to discriminant=0−8a(b−d)≥0 ?

also, can you help via diagram as i could not understand second method

thanks in advance

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Re: M24-12 [#permalink]

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26 Sep 2017, 05:21
Bunnuel, just to check if my graphical solution is right:

If they would say that a is positive, and c is negative, would the answer be (C) ? (Given that one parabola opens upward and the ther one downwards?

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Re: M24-12   [#permalink] 26 Sep 2017, 05:21
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# M24-12

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