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Complex Numbers And Quadratic Equations

Question
CBSEENMA11015679
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If 2a + 3b + 6c =0, then at least one root of the equation ax2 + bx+ c = 0  lies in the interval

  • (0,1)

  • (1,2)

  • (2,3)

  • (1,3)

Solution

A.

(0,1)

Let space straight f apostrophe left parenthesis straight x right parenthesis space equals space ax squared space plus bx space plus straight c rightwards double arrow space straight f left parenthesis straight x right parenthesis space equals space ax cubed over 3 plus bx squared over 2 space plus cx space plus straight d straight f left parenthesis straight x right parenthesis space equals space fraction numerator 2 ax cubed space plus 3 bx squared space plus 6 cx space plus 6 straight d over denominator 6 end fraction straight f left parenthesis 1 right parenthesis space equals space fraction numerator 2 straight a space plus 3 straight b space plus 6 straight c space plus 6 straight d over denominator 6 end fraction space equals fraction numerator 6 straight d over denominator 6 end fraction space equals space straight d straight f left parenthesis 1 right parenthesis space space equals fraction numerator 6 straight d over denominator 6 end fraction space equals straight d straight f left parenthesis 0 right parenthesis space equals space straight f left parenthesis 1 right parenthesis straight f apostrophe left parenthesis straight x right parenthesis space equals space 0
∴ One of the roots of ax2 + bx + c = 0 lies between 0 and 1.

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