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Algebra
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Source: Beat The GMAT — Data Sufficiency |
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theCodeToGMAT
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theCodeToGMAT
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According to me, answer is [spoiler]{E}[/spoiler].. what mistake i made? 
To find: A is +ve?
Statement 1:
x^2-2x+A is positive for all x
(x)(x) - 2(x) + A ---> Is positive
Assume "x" as 10 & A = -1
(10)(10) - 2(10) - 1 ==> 100 - 20 - 1 ==> Positive; NO "A" is NEGATIVE
Assume "x" as 10 & A = 1
(10)(10) - 2(10) + 1 ==> 100 - 20 + 1 ==> Positive; Yes "A" is Positive
INSUFFICIENT
Statement 2:
Ax^2+1
A (x) (x) + 1 --> Is positive
if "x" = 1 & A = -1/2, then
(-0.5) (1) (1) + 1 ==> 0.5 --> Positive; NO "A" is NEGATIVE
if "x" = 1 & A = 1/2, then
(0.5) (1) (1) + 1 ==> 1.5 --> Positive; YES "A" is POSITIVE
INSUFFICIENT
Combining...
we cannot deduce anything from A & X combination
[spoiler]{E}[/spoiler]
To find: A is +ve?
Statement 1:
x^2-2x+A is positive for all x
(x)(x) - 2(x) + A ---> Is positive
Assume "x" as 10 & A = -1
(10)(10) - 2(10) - 1 ==> 100 - 20 - 1 ==> Positive; NO "A" is NEGATIVE
Assume "x" as 10 & A = 1
(10)(10) - 2(10) + 1 ==> 100 - 20 + 1 ==> Positive; Yes "A" is Positive
INSUFFICIENT
Statement 2:
Ax^2+1
A (x) (x) + 1 --> Is positive
if "x" = 1 & A = -1/2, then
(-0.5) (1) (1) + 1 ==> 0.5 --> Positive; NO "A" is NEGATIVE
if "x" = 1 & A = 1/2, then
(0.5) (1) (1) + 1 ==> 1.5 --> Positive; YES "A" is POSITIVE
INSUFFICIENT
Combining...
we cannot deduce anything from A & X combination
[spoiler]{E}[/spoiler]
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One a\way to solve is:
s A>0?
(1) x^2-2x+A is positive for all x:
Quadratic expression x^2-2x+A is a function of of upward parabola (it's upward as coefficient of x^2 is positive). We are told that this expression is positive for all x --> x^2-2x+A>0, which means that this parabola is "above" X-axis OR in other words parabola has no intersections with X-axis OR equation x^2-2x+A=0 has no real roots.
Quadratic equation to has no real roots discriminant must be negative --> D=2^2-4A=4-4A<0 --> 1-A<0 --> A>1.
Sufficient.
(2) Ax^2+1 is positive for all x:
Ax^2+1>0 --> when A\geq0 this expression is positive for all x. So A can be zero too.
Not sufficient.
s A>0?
(1) x^2-2x+A is positive for all x:
Quadratic expression x^2-2x+A is a function of of upward parabola (it's upward as coefficient of x^2 is positive). We are told that this expression is positive for all x --> x^2-2x+A>0, which means that this parabola is "above" X-axis OR in other words parabola has no intersections with X-axis OR equation x^2-2x+A=0 has no real roots.
Quadratic equation to has no real roots discriminant must be negative --> D=2^2-4A=4-4A<0 --> 1-A<0 --> A>1.
Sufficient.
(2) Ax^2+1 is positive for all x:
Ax^2+1>0 --> when A\geq0 this expression is positive for all x. So A can be zero too.
Not sufficient.
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Could I request the experts to explain, thanks!
[email protected] wrote:One a\way to solve is:
s A>0?
(1) x^2-2x+A is positive for all x:
Quadratic expression x^2-2x+A is a function of of upward parabola (it's upward as coefficient of x^2 is positive). We are told that this expression is positive for all x --> x^2-2x+A>0, which means that this parabola is "above" X-axis OR in other words parabola has no intersections with X-axis OR equation x^2-2x+A=0 has no real roots.
Quadratic equation to has no real roots discriminant must be negative --> D=2^2-4A=4-4A<0 --> 1-A<0 --> A>1.
Sufficient.
(2) Ax^2+1 is positive for all x:
Ax^2+1>0 --> when A\geq0 this expression is positive for all x. So A can be zero too.
Not sufficient.
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GMATGuruNY
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Statement 1:
In other words, the graph of y = x² - 2x + A lies entirely above the x-axis, so that the value of y is always positive.
A parabola of the form y = ax² + bx + c, where a>0, opens UPWARD.
The result is a U-shaped graph that looks like this:
U.
The DISCRIMINANT of the parabola is equal to b² - 4ac.
The U-shaped graph will lie entirely above the x-axis -- and thus will yield only positive values for y -- if its discriminant is negative.
Implication:
Since y = x² - 2x + A must yield only positive values for y, its discriminant must be negative.
In y = x² - 2x + A, a=1, b=-2, and c=A.
Since b² - 4ac < 0, we get:
(-2)² - 4*1*A < 0
4 - 4A < 0
-4A < -4
A > 1.
SUFFICIENT.
Statement 2:
In other words, the graph of y = Ax² + 1 lies entirely above the x-axis, so that the value of y is always positive.
Case 1: A=0, so that y = Ax² + 1 becomes y=1.
Here, the graph is a horizontal line that lie entirely above the x-axis.
Case 2: A=1, so that y = Ax² + 1 becomes y = x² + 1
Here, since x² cannot be negative, every value for y will be positive, yielding a graph that lies entirely above the x-axis.
Since A=0 in Case 1, but A>0 in Case 2, INSUFFICIENT.
The correct answer is A.
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theCodeToGMAT
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Hi Mitch,
Can you please suggest what mistake i made in understanding the problem?
Can you please suggest what mistake i made in understanding the problem?
theCodeToGMAT wrote:According to me, answer is [spoiler]{E}[/spoiler].. what mistake i made?
To find: A is +ve?
Statement 1:
x^2-2x+A is positive for all x
(x)(x) - 2(x) + A ---> Is positive
Assume "x" as 10 & A = -1
(10)(10) - 2(10) - 1 ==> 100 - 20 - 1 ==> Positive; NO "A" is NEGATIVE
Assume "x" as 10 & A = 1
(10)(10) - 2(10) + 1 ==> 100 - 20 + 1 ==> Positive; Yes "A" is Positive
INSUFFICIENT
Statement 2:
Ax^2+1
A (x) (x) + 1 --> Is positive
if "x" = 1 & A = -1/2, then
(-0.5) (1) (1) + 1 ==> 0.5 --> Positive; NO "A" is NEGATIVE
if "x" = 1 & A = 1/2, then
(0.5) (1) (1) + 1 ==> 1.5 --> Positive; YES "A" is POSITIVE
INSUFFICIENT
Combining...
we cannot deduce anything from A & X combination
[spoiler]{E}[/spoiler]
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GMATGuruNY
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The constraint in statement 1 is that x²-2x+A is positive FOR ALL VALUES OF X, not just x=10.theCodeToGMAT wrote:According to me, answer is [spoiler]{E}[/spoiler].. what mistake i made?
To find: A is +ve?
Statement 1:
x^2-2x+A is positive for all x
(x)(x) - 2(x) + A ---> Is positive
Assume "x" as 10 & A = -1
If x=0 and A=-1, then x²-2x+A < 0, which does not satisfy the constraint that x²-2x+A is positive for all values of x (including x=0).
To guarantee that x²-2x+A is positive FOR ALL VALUES OF X, A must be greater than 1, as shown in my solution above.
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theCodeToGMAT
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Thanks Mitch!..GMATGuruNY wrote:The constraint in statement 1 is that x²-2x+A is positive FOR ALL VALUES OF X, not just x=10.theCodeToGMAT wrote:According to me, answer is [spoiler]{E}[/spoiler].. what mistake i made?
To find: A is +ve?
Statement 1:
x^2-2x+A is positive for all x
(x)(x) - 2(x) + A ---> Is positive
Assume "x" as 10 & A = -1
If x=0 and A=-1, then x²-2x+A < 0, which does not satisfy the constraint that x²-2x+A is positive for all values of x (including x=0).
To guarantee that x²-2x+A is positive FOR ALL VALUES OF X, A must be greater than 1, as shown in my solution above.
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Hi All,
The accumulated explanations seem to offer more complex ways of interpreting the prompt. I'm going to focus instead on pattern matching and TESTing Values, which will likely be easier to understand.
We're asked: Is A positive? This is a YES/NO question.
Fact 1: X^2 - 2X + A = positive FOR ALL VALUES OF X
This means that we need a value for A that will make the above result positive NO MATTER WHAT X IS.....
In this type of situation, I go looking at the possibilities:
If X = 0, then would have to be something POSITIVE
If X = 1, then A would have to be >1
If X = 2, then A would have to be something POSITIVE
If X = anything negative, then the A could be positive OR negative
Etc.
With this information, the ONLY way to account for all POSSIBLE values of X is for A > 1
This means that A MUST BE positive and the answer to the question is ALWAYS YES.
Fact 1 is SUFFICIENT
Fact 2: A(X^2) + 1 = positive FOR ALL VALUES OF X
Again, I'd look at the possibilities...
Since (X^2) = 0 or a POSITIVE, A would only need to be greater than OR EQUAL TO 0.
If A = 0, then the answer is NO.
If A > 0, then the answer is YES.
Fact 2 is INSUFFICIENT.
Final Answer: A
GMAT assassins aren't born, they're made,
Rich
The accumulated explanations seem to offer more complex ways of interpreting the prompt. I'm going to focus instead on pattern matching and TESTing Values, which will likely be easier to understand.
We're asked: Is A positive? This is a YES/NO question.
Fact 1: X^2 - 2X + A = positive FOR ALL VALUES OF X
This means that we need a value for A that will make the above result positive NO MATTER WHAT X IS.....
In this type of situation, I go looking at the possibilities:
If X = 0, then would have to be something POSITIVE
If X = 1, then A would have to be >1
If X = 2, then A would have to be something POSITIVE
If X = anything negative, then the A could be positive OR negative
Etc.
With this information, the ONLY way to account for all POSSIBLE values of X is for A > 1
This means that A MUST BE positive and the answer to the question is ALWAYS YES.
Fact 1 is SUFFICIENT
Fact 2: A(X^2) + 1 = positive FOR ALL VALUES OF X
Again, I'd look at the possibilities...
Since (X^2) = 0 or a POSITIVE, A would only need to be greater than OR EQUAL TO 0.
If A = 0, then the answer is NO.
If A > 0, then the answer is YES.
Fact 2 is INSUFFICIENT.
Final Answer: A
GMAT assassins aren't born, they're made,
Rich
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Uva@90
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Rich,[email protected] wrote:Hi All,
The accumulated explanations seem to offer more complex ways of interpreting the prompt. I'm going to focus instead on pattern matching and TESTing Values, which will likely be easier to understand.
We're asked: Is A positive? This is a YES/NO question.
Fact 1: X^2 - 2X + A = positive FOR ALL VALUES OF X
This means that we need a value for A that will make the above result positive NO MATTER WHAT X IS.....
In this type of situation, I go looking at the possibilities:
If X = 0, then would have to be something POSITIVE
If X = 1, then A would have to be >1
If X = 2, then A would have to be something POSITIVE
If X = anything negative, then the A could be positive OR negative
Etc.
With this information, the ONLY way to account for all POSSIBLE values of X is for A > 1
This means that A MUST BE positive and the answer to the question is ALWAYS YES.
Fact 1 is SUFFICIENT
Fact 2: A(X^2) + 1 = positive FOR ALL VALUES OF X
Again, I'd look at the possibilities...
Since (X^2) = 0 or a POSITIVE, A would only need to be greater than OR EQUAL TO 0.
If A = 0, then the answer is NO.
If A > 0, then the answer is YES.
Fact 2 is INSUFFICIENT.
Final Answer: A
GMAT assassins aren't born, they're made,
Rich
If A can have both POSITIVE as well as NEGATIVE how Statement1 is sufficient.If X = anything negative, then the A could be positive OR negative
Could you please explain ? I didn't get that part.
Thanks in advance.
Regards,
Uva.
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In Statement 1, the value of x² - 2x + A must be positive for ANY VALUE OF X.Uva@90 wrote: Could you please explain ? I didn't get that part.
It's possible that x² - 2x is negative.
For example:
If x=1/2, then x² - 2x = (1/2)² - 2(1/2) = -3/4.
Here, the value of A must be greater than 3/4 to yield a positive value for x² - 2x + A.
As the case above shows -- since x² - 2x can be negative -- there is only one way to GUARANTEE that the value of x² - 2x + A will ALWAYS be positive:
The value of A must be positive.
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Uva@90
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Mitch,GMATGuruNY wrote:In Statement 1, the value of x² - 2x + A must be positive for ANY VALUE OF X.Uva@90 wrote: Could you please explain ? I didn't get that part.
It's possible that x² - 2x is negative.
For example:
If x=1/2, then x² - 2x = (1/2)² - 2(1/2) = -3/4.
Here, the value of A must be greater than 3/4 to yield a positive value for x² - 2x + A.
As the case above shows -- since x² - 2x can be negative -- there is only one way to GUARANTEE that the value of x² - 2x + A will ALWAYS be positive:
The value of A must be positive.
Thanks for your reply.
Lets see different case:
When X = -1/2
Then X^2-2x = 5/4
Then 5/4 +A = Positive
Let A = 1
Then it is True
Let A = -1
5/4 -1 = 1/4 , still it is positive only right?
So we cant say surely A can take positive right ?
Do I getting anything wrong here . Please help me.
Thanks in advance.
Regards,
Uva.
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GMATGuruNY
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A=-1 is not a valid option in statement 1.Uva@90 wrote: Let A = -1
5/4 -1 = 1/4 , still it is positive only right?
So we cant say surely A can take positive right ?
Do I getting anything wrong here . Please help me.
Thanks in advance.
Regards,
Uva.
If A=-1 and x=1/2, then x² - 2x + A = -7/4, which is not positive.
If A=-1 and x=1/4, then x² - 2x + A = -23/16, which is not positive.
If A=-1 and x=1, then x² - 2x + A = -2, which is not positive.
Statement 1 indicates that x² - 2x + A must be positive for ANY VALUE OF X -- including x=1/2, x=1/4 and x=1.
Since A=-1 does not satisfy this constraint, A=-1 is NOT a valid option.
There is only one way to guarantee that x² - 2x + A will be positive for ANY value of x (including x=1/2, x=1/4 and x=1):
The value of A must be positive.
In my initial post, I proved that x² - 2x + A will be positive for ANY value of x only if A>1.
To verify this conclusion, test a few cases where A>1:
x² - 2x + 2
x² - 2x + 10
x² - 2x + 1.1
If you plug ANY value into the expressions above, the result in every case will be positive.
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Uva@90
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Mitch,GMATGuruNY wrote:A=-1 is not a valid option in statement 1.Uva@90 wrote: Let A = -1
5/4 -1 = 1/4 , still it is positive only right?
So we cant say surely A can take positive right ?
Do I getting anything wrong here . Please help me.
Thanks in advance.
Regards,
Uva.
If A=-1 and x=1/2, then x² - 2x + A = -7/4, which is not positive.
If A=-1 and x=1/4, then x² - 2x + A = -23/16, which is not positive.
If A=-1 and x=1, then x² - 2x + A = -2, which is not positive.
Statement 1 indicates that x² - 2x + A must be positive for ANY VALUE OF X -- including x=1/2, x=1/4 and x=1.
Since A=-1 does not satisfy this constraint, A=-1 is NOT a valid option.
There is only one way to guarantee that x² - 2x + A will be positive for ANY value of x (including x=1/2, x=1/4 and x=1):
The value of A must be positive.
In my initial post, I proved that x² - 2x + A will be positive for ANY value of x only if A>1.
To verify this conclusion, test a few cases where A>1:
x² - 2x + 2
x² - 2x + 10
x² - 2x + 1.1
If you plug ANY value into the expressions above, the result in every case will be positive.
Thanks a ton.
My bad, I understood question in different(WRONG) way.
Thanks for explaining so clearly.
Regards,
Uva.
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