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Question

# If α,β are real and distinct roots of ax2+bx−c=0 and p,q are real and distinct roots of ax2+bx−|c|=0, where (a>0), then

A
α,β(p,q)
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B
α,β[p,q]
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C
p,q(α,β)
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D
None of these
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Solution

## The correct option is A α,β∈[p,q]α,β are real and distinct roots of ax2+bx−c=0.Assume α<βTherefore,α,β=−b±√b2+4ac2ap, q be real and distinct roots of ax2+bx−|c|=0.Assume p<qTherefore,p,q=−b±√b2+4a|c|2aNow, c≤|c|⇒4ac≤4a|c|⇒√b2+4ac≤√b2+4a|c|⇒−b+√b2+4ac≤−b+√b2+4a|c|⇒−b+√b2+4ac2a≤−b+√b2+4a|c|2a⇒β≤qAlso, −√b2+4ac≥−√b2+4a|c|⇒−b−√b2+4ac≥−b−√b2+4a|c|⇒−b−√b2+4ac2a≥−b−√b2+4a|c|2a⇒α≥pHence, α,β∈[p,q].

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