Question

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If two of the roots of equation x4 - 2x3 + ax2 + 8x + b = 0 are equal in magnitude but opposite in sig
then value of 4a + b is equal to :
(A) 16
(B) 8
(C) -16
theme values are also
28 ** ** *5 = o are equal in magnitude but opposite in
(D) -8​

Answer 1

$\textbf{Given:}$

$\text{Two of the roots of x^4 - 2x^3 +ax^2+8x+b=0 are equal but opposite in magnitude}$

$\textbf{To find:}$

$\text{The value of 4a+b }$

$\textbf{Solution:}$

$\text{Let the roots of x^4 - 2x^3 +ax^2+8x+b=0 be m,-m,n and p }$

$\text{Then,}$

$S_1=m+(-m)+n+p=2$

$\implies\bf\,n+p=2$.......(1)

$S_3=-m^2n-mnp+mnp-pm^2=-8$

$\implies\,-m^2n-pm^2=-8$

$\implies\,-m^2(n+p)=-8$

$\text{Using (1), we get}$

$\implies\,m^2(2)=8$

$\implies\,m^2=4$

$\implies\,m=\pm\,2$

$\text{Thus, we have 2 and -2 are two roots of the given equation}$

$\text{Since 2 is a root of x^4 - 2x^3 +ax^2+8x+b=0 , 2 satisfies the equation}$

$2^4-2(2)^3+a(2)^2+8(2)+b=0$

$16-16+4\,a+16+b=0$

$4\,a+16+b=0$

$\implies\boxed{\bf\,4\,a+b=-16}$

$\therefore\textbf{The value of 4a+b is -16}$

$\implies\textbf{Option (C) is correct}$

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Answer 2

Answer:

(c) -16 is correct answer

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