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Question

If α,β,γ,δ are the roots of equation 2x4+4x33x2+3x+1=0, then value of 12α1+12β1+12γ1+12δ1 is

A
1619
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B
1619
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C
1916
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D
1916
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Solution

The correct option is B 1619
α,β,γ,δ are the roots of equation 2x4+4x33x2+3x+1=0
Let y=12α1
α=y+12y
Putting it in the equation, we get
2(1+y2y)4+4(1+y2y)33(1+y2y)+3(1+y2y)+1=02(1+y)4+8y(1+y)312y2(1+y)2+24y3(1+y)+16y4=02(y4+4y3+)+8y(y3+3y2+) 12y2(y2+2y+1)+24y3(1+y)+16y4=0

This equation has roots as 12α1,12β1,12γ1,12δ1
So,
12α1+12β1+12γ1+12δ1=Coefficient of y3Coefficient of y4=(8+2424+24)2+812+24+16=3238=1619

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