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12 May 2023, 03:30
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
45%
(medium)
Question Stats:
54% (01:00) correct
46%
(01:10)
wrong
based on 91
sessions
History
Date
Time
Result
Not Attempted Yet
12 May 2023, 04:33
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Bunuel
A quadratic of the form \(ax^2 + bx + c\) has at least one real root if the discriminant \(b^2-4ac \geq 0\).
Conversely, if the discriminant \(b^2-4ac < 0\) we can conclude that the quadratic equation will not have any real roots.
Discriminant of \(x^2+px+q = 0\) ⇒ \(p^2 - 4q\)
Therefore for equation \(x^2+px+q = 0\) to have at least one real root \(p^2 - 4q\) must be greater than or equal to zero.
\(p^2\) is always non-negative, hence if p is less than zero we conclude that the equation has at least one real root.
Target Question: Is \(q < 0\)
Statement 1
(1) \(p<0\)
To answer the target question, we need to know the positive-negative nature of 'q'. Hence the information in Statement 1 is not sufficient to answer the target question. We can eliminate Options A and D.
Statement 2
(2) \(q<0\)
This is the information that we've been looking for. As q is negative, we can be sure that \(x^2+px+q = 0\) will have a root.
The statement alone is sufficient to answer the question.
Option B
13 May 2023, 06:26
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Does x^2+px+q=0 have a root?
(1) p<0
(2) q<0
For an equation to have roots, its discriminant must be greater than or equal to 0
p^2 - 4q >=0
(1) p<0
Which means p^2 is still positive but we dont know what is q and we cant say if p^2 is actually less than 4q (in which case it will not have any roots) or more than/equal to 4q (in which case it will have roots)
(1) alone is not enough
(2) q<0
p^2 is always positive which means that if q<0 then the discriminant p^2 - 4q will become p^2 + 4q. This will always be greater than or equal to zero but never less than 0 as we are adding two positive terms
Statement 2 alone is enough
(1) p<0
(2) q<0
For an equation to have roots, its discriminant must be greater than or equal to 0
p^2 - 4q >=0
(1) p<0
Which means p^2 is still positive but we dont know what is q and we cant say if p^2 is actually less than 4q (in which case it will not have any roots) or more than/equal to 4q (in which case it will have roots)
(1) alone is not enough
(2) q<0
p^2 is always positive which means that if q<0 then the discriminant p^2 - 4q will become p^2 + 4q. This will always be greater than or equal to zero but never less than 0 as we are adding two positive terms
Statement 2 alone is enough
