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# M24 #12 - formatting error

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Director
Joined: 12 Jul 2008
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M24 #12 - formatting error [#permalink]

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15 Aug 2008, 07:51
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Do lines $$y = ax^2 + b$$ and $$y = cx^2 + d$$ cross?

1. $$a = -c$$
2. $$b \gt d$$

[Reveal] Spoiler: OA
E

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What am I supposed to do w/ #2?
Math Expert
Joined: 02 Sep 2009
Posts: 39751
Re: M24 #12 - formatting error [#permalink]

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09 Feb 2010, 08:51
9
KUDOS
Expert's post
zoinnk wrote:
Do lines $$y = ax^2 + b$$ and $$y = cx^2 + d$$ cross?

1. $$a = -c$$
2. $$b \gt d$$

[Reveal] Spoiler: OA
E

Source: GMAT Club Tests - hardest GMAT questions

What am I supposed to do w/ #2?

I'd like to comment on this one:

First of all: equations given ARE NOT linear equations. We know that if two different lines do not cross each other they are parallel. How can we tell, based on the equations, whether the lines are parallel? We can check the slopes of these lines: parallel lines will have the same slopes. NOT that the slopes of lines must be negative reciprocals of each other (as it was mentioned in the earlier posts): in this case they are perpendicular not parallel.

Second of all: we have quadratic equations. These equations when drawn give parabolas. The question is: do they cross? This CANNOT be transformed to the question: "are they parallel?" In the wast majority of cases the word "parallel" is used for lines. Well, we can say that concentric circles are parallel, BUT GMAT, as far as I know, uses this word ONLY about the lines (tutors may correct me if I'm wrong). Next, the word "parallel" when used for curves (lines, ...) means that these curves remain a constant distance apart. So strictly speaking two parabolas to be parallel they need not only not to intersect but also to remain constant distance apart. In this case, I must say that this cannot happen. If a curve is parallel (as we defined) to the parabola it won't be quadratic: so curve parallel to a parabola is not a parabola. So I think that at this point we can stop considering this concept in regard to the original question.

So in which cases parabolas do not cross? There are number of possibilities: We can shift the vertex: the parabolas $$y=x^2$$ and $$y=x^2+1$$ will never intersect (note that they won't be exactly parallel but they will never intersect). We can consider downward and upward parabolas and in some cases they also never intersect... Of course there can be other cases as well.

As for the solution. We can follow the way dzyubam proposed (and I think it's the fastest way, provided we can identify correct examples) and consider two cases. First case: $$y=-x^2+1$$ and $$y=x^2+0$$ (upward and downward parabolas), which satisfies both statements, and see that in this case answer is YES, they cross each other; and the second case: $$y=x^2+1$$ and $$y=-x^2+0$$ (also upward and downward parabolas), which also satisfies both statements, and see that in this case answer is NO, they do not cross each other. Two different answers to the question, hence not sufficient.

We can solve the question algebraically as well:

Do lines $$y = ax^2 + b$$ and $$y = cx^2 + d$$ cross?

(1) $$a = -c$$ --> $$y_1= ax^2 + b$$ and $$y_2=-ax^2 + d$$, now if they 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 $$\geq0$$ --> $$d=0-8a(b-d)\geq0$$? --> $$d=-8a(b-d)\geq0$$? Now can we determine whether this is true? We know nothing about $$a$$, $$b$$, and $$d$$, hence no. Not sufficient.

(2) $$b>d$$ --> 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 $$\geq0$$ --> $$d=0-4(a-c)(b-d)\geq0$$? --> $$d=-4(a-c)(b-d)\geq0$$? Now can we determine whether this is true? We know that $$b-d>0$$ but what about $$a-c$$? Hence no. Not sufficient.

(1)+(2) $$a=-c$$ and $$b>d$$ --> $$y_1= ax^2 + b$$ and $$y_2=-ax^2 + d$$ --> same steps as above --> $$2ax^2+(b-d)=0$$ --> and the same question remains: is $$d=-8a(b-d)\geq0$$ true? $$b-d>0$$ but what about $$a$$? Not sufficient.

Hope it helps.
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Last edited by Bunuel on 09 Feb 2010, 09:13, edited 3 times in total.
CIO
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Posts: 1218
Re: M24 #12 - formatting error [#permalink]

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06 Sep 2008, 12:47
2
KUDOS
You guys might have forgotten that we are dealing with quadratic equations here. The upward and downward parabolas may not cross even if the "slope" is different for the two equations. See the OE for more info.

The right answer is E.

jallenmorris wrote:
Do the lines cross is another way of asking if they are parallel.

Statement 1 is sufficient. We know that if a = 1 and -c = a, then c = -1. These are not parallel. Also keep in mind the forumula for a line. Is y = [slope]x + [y-intercept] (or minus y-intercept).

Statement 2 is telling us information about the y-intercept. It is saying that no matter what d is, b is greater than d. This doesn't tell us any information about a or c, which are the most important because these two variables determine the slope of the line. Without knowing the slope of the lines, we cannot know if the lines cross.

Answer is A, and the anwer with the info in A is "Yes, the lines do cross."

zoinnk wrote:
Do lines $$y = ax^2 + b$$ and $$y = cx^2 + d$$ cross?

1. $$a = -c$$
2. $$b \gt d$$

What am I supposed to do w/ #2?

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Director
Joined: 29 Aug 2005
Posts: 860
Re: M24 #12 [#permalink]

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07 Dec 2008, 06:20
2
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Answer should be E. We know (given S1) that one of them is upward parabola, the other is inverted. However, we don't know which one is inverted and which one is upward. I.e. - we dont know the sign of "a", it can be a=2 or a=-2
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Joined: 30 Apr 2008
Posts: 1874
Location: Oklahoma City
Schools: Hard Knocks
Re: m24 #12 [#permalink]

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05 Mar 2009, 21:23
2
KUDOS
This question is a bit different than a normal line equation. A normal line equation is y = [slope]x + [y intercept], but here the X is actually squared. This will lead to a curve similar to a parabola rather than a straight light. I don't think this is avery good question, but that's just me. I have not seen this before becuase I only did the first 12 or 13 math tests before my GMAT.

As for your questions prior to the DS question. Yes, generally, all you need to know for lines that cross is the slope. Remember that a Cartesian plane extends to infinity, so if the slopes are not IDENTICAL, then the lines will cross at some point. The y-intercept doesn't matter because at some point, the lines will cross unless they are EXACTLY parallel. To determine perpendicular lines, you are correct, the opposite sign (positive or negative) and reciprocal will be the test for perpendicular.

If slope is $$-\frac{1}{C}$$ then the perpendicular slope will be $$C$$ because we take the reciprocal so $$\frac{C}{1}$$ and it's positive because the other was negative.

dczuchta wrote:
Hello, would someone pls answer my question here, thank you.

I believe it was several times in past test answers that you only need slopes of lines to determine whether or not they cross, the angle they form together or whether or not they are perpendicular. Assuming that S1 stated "a=-1/c" would we then know that two lines cross are perpendicular and therefore cross?
_________________________________
Do lines ax^2 + b and cx^2 + d cross?

1) a= -c
2) b > d

Would someone pls explain this question's answer further: What do we really need to know if two lines cross? The answer just gives examples. Also, if we have both slopes and know that one is the negative reciprocal of the other, do we then know they are perpendicular? Thankyou

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J Allen Morris
**I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a$$. GMAT Club Premium Membership - big benefits and savings Manager Joined: 11 Sep 2009 Posts: 100 Re: M24 #12 [#permalink] ### Show Tags 06 Nov 2009, 23:56 2 This post received KUDOS if cross ax2 + b = cx2 + d x2(a - c) = d – b x2 = (d-b)/(a-c) Thus, three criteria occur: (1) a != c (2) if a > c, d > b (3) if a < c, d < b 1. a = -c, we do not know about b and d  insufficient 2. b > d, what about a and c?  insufficient Both are still insufficient because a can be either more or less than c SVP Joined: 30 Apr 2008 Posts: 1874 Location: Oklahoma City Schools: Hard Knocks Re: M24 #12 - formatting error [#permalink] ### Show Tags 20 Aug 2008, 09:45 1 This post received KUDOS Do the lines cross is another way of asking if they are parallel. Statement 1 is sufficient. We know that if a = 1 and -c = a, then c = -1. These are not parallel. Also keep in mind the forumula for a line. Is y = [slope]x + [y-intercept] (or minus y-intercept). Statement 2 is telling us information about the y-intercept. It is saying that no matter what d is, b is greater than d. This doesn't tell us any information about a or c, which are the most important because these two variables determine the slope of the line. Without knowing the slope of the lines, we cannot know if the lines cross. Answer is A, and the anwer with the info in A is "Yes, the lines do cross." zoinnk wrote: Do lines $$y = ax^2 + b$$ and $$y = cx^2 + d$$ cross? 1. $$a = -c$$ 2. $$b \gt d$$ What am I supposed to do w/ #2? _________________ ------------------------------------ J Allen Morris **I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a$$.

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Intern
Joined: 07 Feb 2009
Posts: 49
Re: M24 #12 [#permalink]

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11 Mar 2009, 23:16
1
KUDOS
Both statements required to answer this question.

Explanation:
Lets equate these two equations first.
We have no idea about the signs of a,b,c or d. so we cant say anything about the nature of the graph whether it is concave or convex.
After equating these equation we can get x^2 = (d-b)/(a-c).
For getting real values of x, the above result must satisfy that a!=c!=0.
Also, (d-b)/(a-c) must always be positive.
To find this we must have both the conditions. i.e, if a=-c and b>d then x^2 will be equal to a negative value so we can't have real values of x. Thus we can say that these two lines are not crossing each other.
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Joined: 02 Oct 2007
Posts: 1218
Re: M24 #12 [#permalink]

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30 Mar 2009, 08:36
1
KUDOS
The OA is E.

Quote:
Consider $$y = -x^2 + 1$$ , $$y = x^2 + 0$$ (the answer is YES) and $$y = x^2 + 1$$ , $$y = -x^2 + 0$$ (the answer is NO).

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Re: M24 #12 - formatting error [#permalink]

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27 Feb 2012, 23:21
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Expert's post
devinawilliam83 wrote:
Hi I am getting c. Please see the attached image.. please expl this solution is not why this solution is not correct

Please read the solutions above with algebraic as well as number plugging approaches showing that answer is E, not C.

The problem with your reasoning is that the case you've given is not the only possible one, there are the cases when the graphs do not intersect.

Consider the graphs y=x^2+1 and y=-x^2 to get answer NO:
Attachment:

1.gif [ 4.4 KiB | Viewed 1764 times ]

Consider the graphs y=-x^2+1 and y=x^2 to get answer YES:
Attachment:

2.gif [ 5.18 KiB | Viewed 1764 times ]

Hope it's clear.
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Re: M24 #12 - formatting error [#permalink]

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05 Sep 2008, 12:44
jallenmorris wrote:
Do the lines cross is another way of asking if they are parallel.

Statement 1 is sufficient. We know that if a = 1 and -c = a, then c = -1. These are not parallel. Also keep in mind the forumula for a line. Is y = [slope]x + [y-intercept] (or minus y-intercept).

Statement 2 is telling us information about the y-intercept. It is saying that no matter what d is, b is greater than d. This doesn't tell us any information about a or c, which are the most important because these two variables determine the slope of the line. Without knowing the slope of the lines, we cannot know if the lines cross.

Answer is A, and the anwer with the info in A is "Yes, the lines do cross."

zoinnk wrote:
Do lines $$y = ax^2 + b$$ and $$y = cx^2 + d$$ cross?

1. $$a = -c$$
2. $$b \gt d$$

What am I supposed to do w/ #2?

what if a and c are both 0??? then we can't confidently say that the lines will cross. I think it should be E.
Manager
Joined: 18 Nov 2008
Posts: 116
Re: M24 #12 [#permalink]

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29 Nov 2008, 02:52
A

We can rewrite the question as 'are two lines y=ax^2+b and y=cx^2+d NOT parralel?'
They can be parralel only if have the same slope.

1) Slopes are different, so lines must cross, Suff.
2) there is no info about slopes, Insuff.
Intern
Joined: 10 Sep 2008
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03 Mar 2009, 14:42
Hello, would someone pls answer my question here, thank you.

I believe it was several times in past test answers that you only need slopes of lines to determine whether or not they cross, the angle they form together or whether or not they are perpendicular. Assuming that S1 stated "a=-1/c" would we then know that two lines cross are perpendicular and therefore cross?
_________________________________
Do lines ax^2 + b and cx^2 + d cross?

1) a= -c
2) b > d

Would someone pls explain this question's answer further: What do we really need to know if two lines cross? The answer just gives examples. Also, if we have both slopes and know that one is the negative reciprocal of the other, do we then know they are perpendicular? Thankyou
VP
Joined: 05 Jul 2008
Posts: 1408
Re: M24 #12 [#permalink]

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11 Mar 2009, 22:38
Can some one explain this parabola and how to figure these out similar to normal lines?

I will feel good if I see such Q on the GMAT
Intern
Joined: 07 Feb 2009
Posts: 49
Re: M24 #12 [#permalink]

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13 Mar 2009, 02:54
Hi icandy

If we equate the two given equations, we get x^2=(d-b)/(a-c).
We need to equate these two equations, because if they cross each other, they must satisfy(must have equal values of y) for a specific value of x. we get this value of x from x^2=(d-b)/(a-c).

Or try this way:
y=ax^2+b
and
y=cx^2+d

equating these equations will give:
ax^2+b = cx^2+d
ax^2 - cx^2=d-b
x^2(a-c)=(d-b)
x^2=(d-b)/(a-c)

Hope this will help.
Manager
Joined: 07 Jul 2007
Posts: 137
Re: M24 #12 [#permalink]

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16 Mar 2009, 08:44
1. a = -c
2. b > d

1) If a = -c then first equation will become as follow,

1) -cx2 –y + b = 0
2) Cx2 –y + b = 0
As per rule two lines are parallel (if a1/a2 = b1/b2) but here a1/a2 = -1 and b1/b2 = 1 and so they are not equal and so line must intercept.

2) b > d, does not give enough information about a and c and so it is not sufficient.

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Re: M24 #12 [#permalink]

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05 Apr 2009, 02:43
Hi abhishekik,
If we dont have real values then how do we say that lines dont intersect !! Imaginary numbers can also be plotted on plane and they can also be an intersection point??!!
http://en.wikipedia.org/wiki/Imaginary_number

Or my understanding is not correct? Please explain.

abhishekik wrote:
Both statements required to answer this question.

Explanation:
Lets equate these two equations first.
We have no idea about the signs of a,b,c or d. so we cant say anything about the nature of the graph whether it is concave or convex.
After equating these equation we can get x^2 = (d-b)/(a-c).
For getting real values of x, the above result must satisfy that a!=c!=0.
Also, (d-b)/(a-c) must always be positive.
To find this we must have both the conditions. i.e, if a=-c and b>d then x^2 will be equal to a negative value so we can't have real values of x. Thus we can say that these two lines are not crossing each other.
Manager
Joined: 02 Aug 2009
Posts: 111
Re: M24 #12 - formatting error [#permalink]

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08 Feb 2010, 11:14
Statement 1: It states that each parabola has a different sign (or they both equal 0). If one slope is positive with an intercept that is positive and one is negative with an intercept that is negative the lines will never cross. (There is nothing defining which line has a positive or negative slope).

Statement 2: Obviously not sufficient says nothing about the slopes.

Together: Since we don't know which line is positive and which is negative or rather if there both not zero than the y intercept can either mean they cross or don't so still not definitive.

E
Manager
Joined: 27 Aug 2009
Posts: 137
Re: M24 #12 - formatting error [#permalink]

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08 Feb 2010, 13:46
I get A

if product of two slops =-1 then the lines are perpendicular to each other so they cross. Not sure why OA is E can anyone explain
Manager
Joined: 27 Aug 2009
Posts: 137
Re: M24 #12 - formatting error [#permalink]

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08 Feb 2010, 13:49
I get E

only if product of two slops =-1 then the lines are perpendicular to each other so they cross.
Re: M24 #12 - formatting error   [#permalink] 08 Feb 2010, 13:49

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# M24 #12 - formatting error

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