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S1) A = +1. SUFF. S2) A(x^2)+1 = POS ==> A(x^2) = POS-1, thus A(x^2) could equal a positive number or zero. A could be any positive number or zero, which is not positive. INSUFF.

S1) A = +1. SUFF. S2) A(x^2)+1 = POS ==> A(x^2) = POS-1, thus A(x^2) could equal a positive number or zero. A could be any positive number or zero, which is not positive. INSUFF.

lets say X=5, and A=10 then 1) 25-10+10>0 but if x=5, and A=-10 then 25-10-10>10 same is true for A=0 so A is Insufficient

2) A can be either 0, or A>0, also insufficient IMO both are also insufficient.

I solved by undisguising the quadratic, but I may be wrong: \((x-1)(x-1) = x^2-2x+1,\) thus A = 1 = SUFF.

What's the answer?

I hate picking random numbers in problems like this because they don't always work: \(x^2 - 2x + A\) is positive for all \(x\)

If A = -10, then result must be positive for all X. So what if X is -2? \((-2)^2 - (-4) + (-10) =\) POSITIVE \(4 + 4 - 10 =\)NEGATIVE At least that's what I think.

1. \(x^2 - 2x + A\) is positive for all \(x\) 2. \(Ax^2 + 1\) is positive for all \(x\)

1) x^2 - 2x + A >0

A> -x^2+2x -- true for all values of x.

please note that here A is constant and x is variable..

Find out the possible values of "-x^2+2x"

"-x^2+2x" +ve when X=1 --(1) "-x^2+2x" -ve when X=3 --(2)

from (1) it is clear that A>+ve so it is positive. from (2) it is clear that A>-ve so it can positive or negative.

It has to satisfy both equations 1 and 2 A must be positive.

Suffcieint

2) Ax^2 + 1>0

A> -1/x^2 --> Tells that for all values X A> -ve number.

So A can be +ve or -ve Insuffcient

A is the Answer.

i dont think A can be negative because as soon as A becomes the negative, the parabola will be inverted, and you cannot gurantee that for all values of x, \(Ax^2 + 1\) is positive for all \(x\)[/quote]

but the equation sez other wise. i am confused !!!

Stmt 1: x^2 - 2x + A is positive for all x. That means this expression should be positive even for x = 0. With x = 0, the expression changes to A. And, if the expression is positive, A must be positive. Hence, sufficient.

Stmt2: Not sufficient. For a positive value as well as negative value of x, the expression can be positive.

1. \(x^2 - 2x + A\) is positive for all \(x\) 2. \(Ax^2 + 1\) is positive for all \(x\)

1) x^2 - 2x + A >0

A> -x^2+2x -- true for all values of x.

please note that here A is constant and x is variable..

Find out the possible values of "-x^2+2x"

"-x^2+2x" +ve when X=1 --(1) "-x^2+2x" -ve when X=3 --(2)

from (1) it is clear that A>+ve so it is positive. from (2) it is clear that A>-ve so it can positive or negative.

It has to satisfy both equations 1 and 2 A must be positive.

Suffcieint

2) Ax^2 + 1>0

A> -1/x^2 --> Tells that for all values X A> -ve number.

So A can be +ve or -ve Insuffcient

A is the Answer.

i dont think A can be negative because as soon as A becomes the negative, the parabola will be inverted, and you cannot gurantee that for all values of x, \(Ax^2 + 1\) is positive for all \(x\)

but the equation sez other wise. i am confused !!! [/quote]

Ax^2 + 1>0

A> -1/x^2

say x=1 A>-1 means A can be -1/2 or zero or Any postive value.
_________________

Your attitude determines your altitude Smiling wins more friends than frowning

here imagine that the value of x is +5 , then the equation becomes 25-10 +A ; now if A is -1/+1 OR -2/+2 etc doesnt make any diff. as the answer will always be +ve ; same is the case if you take x=-5 .

2 is obv not sufficient as many people have agreed