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# Data Sufficiency Competition--Prizes can be won

Author Message
Manager
Joined: 12 Sep 2006
Posts: 91

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29 Sep 2006, 00:59
ok, did a little more thinking and i am changing my answer to E

i totally forgot about a, which could be negative.

so to continue my previous thread, i found a yes scenario when combining both statements, now i need to find a no scenario to spoil answer C--

to do this, let's look at

(x+1)(-x+2)

we have -x^2+x+2, where b+c is positive and bc is also positive but this time we have one negative root and one positive root
x=-1 or x=2

therefore INSUFF together

Cicerone, is this right??
Senior Manager
Joined: 28 Aug 2006
Posts: 304

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29 Sep 2006, 01:20
keeeeeekse, not convinced........
Try it again...............
Manager
Joined: 25 Sep 2006
Posts: 151

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29 Sep 2006, 02:32
Viperace wrote:
B

I) I just cant get
II) Sufficient

Question ask if both roots are positive
Only one condition for that is

b < -sqrt(b^2 - 4ac )
.
.
.
=> ac > 0

II) Product of two root > 0
which means ac>0 if u work it out

[-b + sqrt(b^2 - 4ac ) ]*[-b - sqrt(b^2 - 4ac ) ] >0
b^2 - (b^2 - 4ac )>0
=> ac >0

Woots, just found the mistake I made. However my answer doesnt changes

Question ask if both roots are positive
Only one condition for that is

b < -sqrt(b^2 - 4ac )
.
.
.
=> ac < 0
=> c <0 since given a>0

I) Sum of two root>0
This implies -b/a > 0
=> b <0
This is not sufficient to tell if c<0

II) Product of two root > 0
which means c/a>0 if u work it out

[-b + sqrt(b^2 - 4ac ) ]*[-b - sqrt(b^2 - 4ac ) ]/4a^2 >0
[b^2 - (b^2 - 4ac )]/4a^2>0
=> c/a >0
=> c > 0

Which is sufficient to answer the question. "both roots are positive?"NO
Condition from I) is not needed.
Manager
Joined: 25 Sep 2006
Posts: 151

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29 Sep 2006, 02:45
In fact,
b < 0 is a redundant information.

By knowing that
i) a > 0
ii) both root are positive

We can already deduce b < 0
Director
Joined: 13 Nov 2003
Posts: 789
Location: BULGARIA

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29 Sep 2006, 06:34
From A we know that -(b/a)>0 , given a>0 then b must be NEG
In order both roots to be positive , discriminant must be >0
or b^2-4*a*c>0 .
A) is not suff

From B) we get (4a*c)/4*(a^2) or c/a>0 provided a>0 then c is >0
Given a+b+c>0 and b<0 then a+c>b
Substitute in equation of discriminant b with a+c and get
(a+c)^2-4ac>0 we get a^2-2ac+c^2>0
this is (a-c)^2>0 which is always positive BUT we do not know if A is not EQUAL TO C which would make (a-c)^2=0
Director
Joined: 10 Oct 2005
Posts: 718

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29 Sep 2006, 10:15
4. The roots of a quadratic equatin ax^2+bx+c=0 are integers and a+b+c>0 and a>0. Are both the roots of the equation positive?

(1) Sum of the roots is positive.
(2) Product of the roots is positive.
IMHO C it is
1st alone
roots may be -2 and 4 sum is +ve
roots may be 2 and 4 and so on insuff
2)both +ve and both -ve insuff
both together -->from 2 we know that throots must be both +ve or -ve
but in order to have positive sum both should be +ve only
_________________

IE IMBA 2010

Senior Manager
Joined: 28 Aug 2006
Posts: 304

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04 Oct 2006, 08:58
GMAT Instructor
Joined: 04 Jul 2006
Posts: 1262

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04 Oct 2006, 17:06
cicerone wrote:
Here comes our next question

4. The roots of a quadratic equatin ax^2+bx+c=0 are integers and a+b+c>0 and a>0. Are both the roots of the equation positive?

(1) Sum of the roots is positive.
(2) Product of the roots is positive.

PRIZE: Vocabulary Practice Sotware (contains 3000 words)

ax^2+bx+c=0 => x^2 +(b/a)x +(c/a)=0 since a>0, a is not 0

If this quadratic equation has two integer roots m and n, then it can be written as
(x-m)(x-n)=0 where mn=c/a and -(m+n)=b/a

This is because (x-m)(x-n)=x^2-(m+n)x+mn

(1) We are told than m+n>0, which means that b/a<0 and so b<0
Also as m and n are integers m+n>=1, so |b/a|>=1 i.e |b|>=|a|=a

Summarizing, we know that a>0, b>0 and |b|=-b>=a

This means that b+a<=0 and since a+b+c>0, c>0 Thus ac>0 and so the roots have the same sign-

If the sum of the roots is positive, each must be >0 SUFFICIENT

(2) mn>0 means roots have the same sign. This doesn't tell me much.

Could they both be <0? Sure! (x+2)(x+1)=x^2+3x+2=0 has roots of -2 and -1 and a+b+c=6>0

Could they both be<0? Why not? (x-2)(x-6)=x^2-8x+12=0 has roots of 2 and 6 and a+b+c=5>0

NOT SUFFICIENT

Senior Manager
Joined: 28 Aug 2006
Posts: 304

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04 Oct 2006, 23:03
kevincan wrote:
cicerone wrote:
Here comes our next question

4. The roots of a quadratic equatin ax^2+bx+c=0 are integers and a+b+c>0 and a>0. Are both the roots of the equation positive?

(1) Sum of the roots is positive.
(2) Product of the roots is positive.

PRIZE: Vocabulary Practice Sotware (contains 3000 words)

ax^2+bx+c=0 => x^2 +(b/a)x +(c/a)=0 since a>0, a is not 0

If this quadratic equation has two integer roots m and n, then it can be written as
(x-m)(x-n)=0 where mn=c/a and -(m+n)=b/a

This is because (x-m)(x-n)=x^2-(m+n)x+mn

(1) We are told than m+n>0, which means that b/a<0 and so b<0
Also as m and n are integers m+n>=1, so |b/a|>=1 i.e |b|>=|a|=a

Summarizing, we know that a>0, b>0 and |b|=-b>=a

This means that b+a<=0 and since a+b+c>0, c>0 Thus ac>0 and so the roots have the same sign-

If the sum of the roots is positive, each must be >0 SUFFICIENT

(2) mn>0 means roots have the same sign. This doesn't tell me much.

Could they both be <0? Sure! (x+2)(x+1)=x^2+3x+2=0 has roots of -2 and -1 and a+b+c=6>0

Could they both be<0? Why not? (x-2)(x-6)=x^2-8x+12=0 has roots of 2 and 6 and a+b+c=5>0

NOT SUFFICIENT

Yes kevin, the answer is A.
I have a different approach...........

The roots of a quadratic equatin ax^2+bx+c=0 are integers and a+b+c>0 and a>0. Are both the roots of the equation positive?

Statement 1: Sum of the roots is positive.

ie -b/a = p where p>0
ie b= -(a x p).

Now consider the product of the roots.

Let c/a=q.

Clearly both p and q must be integers (since in the question it is given that both the roots are integers)

It is given that a+b+c>0
ie a-(axp)+(axq)>0
ie a(1-p+q)>0
It is given in the question that a>0
So (1-p+q)>0
ie q-p>-1
ie q-p>=0 (since p and q are integers)
ie q>=p
In statement 1 it is given that p>0
So clearly q>0

So from the first statement itself if sum is +ve we can conclude that product is also +ve.

Statement 2 alone is not sufficient..........

So A
Senior Manager
Joined: 28 Aug 2006
Posts: 304

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05 Oct 2006, 09:10
Here comes our next question

5. In triangle ABC angle A is the greatest angle. D is the foot of the
perpendicular dropped on to BC from A. Is triangle ABC right-angled?

Prize: A file on logical ability to solve the critical reasoning questions from GMAT.
Senior Manager
Joined: 28 Aug 2006
Posts: 304

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06 Oct 2006, 03:46
Hey folks, r u trying or not?
_________________
Senior Manager
Joined: 01 Oct 2006
Posts: 495

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06 Oct 2006, 08:08
St 1 is sufficient.
angle b = angle c
ab=ac
angle dac=45
thus angle bac =90

st 2 is not sufficient.

Senior Manager
Joined: 01 Oct 2006
Posts: 495

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06 Oct 2006, 08:09
St 1 is sufficient.
angle b = angle c
ab=ac
angle dac=45
thus angle bac =90

st 2 is not sufficient.

Senior Manager
Joined: 28 Aug 2006
Posts: 304

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06 Oct 2006, 10:50
_________________
Senior Manager
Joined: 28 Aug 2006
Posts: 304

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06 Oct 2006, 14:28
I am waiting..........
_________________
Senior Manager
Joined: 01 Oct 2006
Posts: 495

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07 Oct 2006, 05:54
Senior Manager
Joined: 31 May 2006
Posts: 368
Location: Phoenix AZ

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07 Oct 2006, 12:29
cicerone wrote:
Here comes our next question

5. In triangle ABC angle A is the greatest angle. D is the foot of the
perpendicular dropped on to BC from A. Is triangle ABC right-angled?

Prize: A file on logical ability to solve the critical reasoning questions from GMAT.

D.

If Angle A is right angle, then AD^2 = BD x DC
Senior Manager
Joined: 28 Aug 2006
Posts: 304

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08 Oct 2006, 05:56
Yes the answer must be D

In a right angled triangle

Clearly 1 says AD^2 = BD x DC
Sufficient

Clearly 2 says AD^2 !=BD x DC

Sufficient

So D

yogeshsheth, i have sent the link to the file.........
_________________
Senior Manager
Joined: 28 Aug 2006
Posts: 304

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08 Oct 2006, 12:54
Hey just relax with this question.................
No prize................
This is a simple one...........

6. How many sons does Mr. John have?

1. Mrs. John has 3 sons.
2. Peter is taller of the John's sons.

Don't relax too much............

A lot more to come
_________________
SVP
Joined: 05 Jul 2006
Posts: 1747

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08 Oct 2006, 14:02
How many sons does Mr. John have?

1. Mrs. John has 3 sons.
2. Peter is taller of the John's sons

ONE INSUFF

TWO

THEY ARE TWO

(TALLER )
08 Oct 2006, 14:02

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