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Question

# If the roots of the quadratic equation x2âˆ’px+q=0 are real and differ by a quantity less than 1, then

A
None of these
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B
q>p24
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C
p214<q<1+p24
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D
p214<q<p24
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Solution

## The correct option is C p2−14<q<1+p24Given equation x2−px+q=0 Comparing it with standard equation ax2+bx+c=0 a=1,b=−p,c=q Given condition is roots of the given quadratic equation differs by a quantity less than 1. Mathematical representstion of this condition is |α−β|<1 We know that (α−β)2=Da2 Now, √D|a|<1 ⇒√(−p)2−4×1×q1<1 ⇒√p2−4q<1 Squaring the inequality ⇒|p2−4q|<1 ⇒−1<p2−4q<1 ⇒−1−p2<−4q<1−p2 ⇒−1−p24<−q<1−p24 ⇒p2−14<q<1+p24

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