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If a1 and a2 are the real roots of x^2-px+12=0, then which of the

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If a1 and a2 are the real roots of x^2-px+12=0, then which of the  [#permalink]

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New post Updated on: 05 Oct 2019, 01:09
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If \(a1\) and \(a2\) are the real roots of \(x^2-px+12=0\), then which of the following statements is definitely true?

a) |\(a1+a2\)| \(\leq\) 2\(\sqrt{3}\)
b) |\(a1-a2\)| \(\leq\) 2\(\sqrt{3}\)
c) |\(a1+a2\)| \(\geq\) 4\(\sqrt{3}\)
d) |\(a1-a2\)| \(\geq\) 4\(\sqrt{3}\)
e) None of the above

Source: CAT 2019 India

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Originally posted by darshshah981 on 05 Oct 2019, 00:02.
Last edited by darshshah981 on 05 Oct 2019, 01:09, edited 3 times in total.
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If a1 and a2 are the real roots of x^2-px+12=0, then which of the  [#permalink]

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New post 05 Oct 2019, 00:28
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darshshah981 wrote:
If \(a1\) and \(a2\) are the real roots of \(x^2-px+12=0\), then which of the following statements is definitely true?

a) |\(a1+a2\)| \(\leq\) 2\(\sqrt{3}\)
b) |\(a1-a2\)| \(\leq\) 2\(\sqrt{3}\)
c) |\(a1+a2\)| \(\geq\) 4\(\sqrt{3}\)
d) |\(a1-a2\)| \(\geq\) 4\(\sqrt{3}\)
e) None of the above



\(x^2-px+12=0\)

Lets look at this quadratic equation is the form of \(ax^2+bx+c=0\)
We know that sum of the roots of a quadratic is \(-b/a\) and the product of the roots of a quadratic is \(c/a\)

Now, if the roots are real then the discriminant \(\geq\) 0, that is \(b^2 - 4ac\) \(\geq\) 0

>> \(p^2 - 4(1)(12)\) \(\geq\) 0
>> \(p^2\) \(\geq\) 48
>> |\(p\)| \(\geq\) \(\sqrt{48}\)
>> |\(p\)| \(\geq\) \(4\sqrt{3}\)

But p is the sum of the roots (a1+a2)...(Since sum of the roots of a quadratic equation is \(-b/a\))

Therefore, |\(a1+a2\)| \(\geq\) 4\(\sqrt{3}\) or C is the correct answer.


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Re: If a1 and a2 are the real roots of x^2-px+12=0, then which of the  [#permalink]

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New post 05 Oct 2019, 08:20
sum of roots should be greater than 0
hence \(p^2\)>48
or \(p>4\sqrt{3}\)
now p = sum of roots
hence C is the correct answer
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If a1 and a2 are the real roots of x^2-px+12=0, then which of the  [#permalink]

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New post 05 Oct 2019, 11:04
darshshah981 wrote:
If \(a1\) and \(a2\) are the real roots of \(x^2-px+12=0\), then which of the following statements is definitely true?

a) |\(a1+a2\)| \(\leq\) 2\(\sqrt{3}\)
b) |\(a1-a2\)| \(\leq\) 2\(\sqrt{3}\)
c) |\(a1+a2\)| \(\geq\) 4\(\sqrt{3}\)
d) |\(a1-a2\)| \(\geq\) 4\(\sqrt{3}\)
e) None of the above

Source: CAT 2019 India

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In the GMAT, there is almost always a way to solve questions regarding quadratics without using the quadratic formula.

For all quadratics in this form \(x^2 - ax + b = 0\), \(a\) is the sum of roots and \(b\) is the product of roots. Therefore we know \(a1*a2 = 12\) and \(a1 + a2 = p\).
If both roots are positive, we can minimize \(p\) when \(a1 = a2 = \sqrt{12}\), so that \(p = a1 + a2 = 2\sqrt{12} = 4\sqrt{3}\), which is a minimum so we have \(a1 + a2 >= 4\sqrt{3}\). We can do the opposite with negative roots to get a similar result, regardless all signs point to C.
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If a1 and a2 are the real roots of x^2-px+12=0, then which of the   [#permalink] 05 Oct 2019, 11:04
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