Question

Mathon
Question 17 ♡
Instructions:Select the ONE correct answer from the given options.
The sum of the complex roots of the equations (x - 1)' + 64 = 0 is
(1) 31
(2) 3
(3) 6i
(4) 6
0
0
0
0​

Answer 1

$\textbf{Given:}$

$(x-1)^3+64=0$

$\textbf{To find:}$

$\text{Sum of the roots of the equation}$

$\text{First we find the roots of the equation by using demovire's theorem}$

$(x-1)^3+64=0$

$(x-1)^3=-64$

$(x-1)^3=64(-1)$

$(x-1)^3=64[cos\,\pi+i\,sin\,\pi]$

$x-1=(64)^{\frac{1}{3}}[cos(\pi+2k\pi)+i\,sin(\pi+2k\pi)]^{\frac{1}{3}}$

$x-1=(4^3)^{\frac{1}{3}}[cos\,\frac{\pi+2k\pi}{3}+i\,sin\,\frac{\pi+2k\pi}{3}]$ $\;\text{where k=0,1,2}$

$x-1=4[cos\,\frac{\pi+2k\pi}{3}+i\,sin\,\frac{\pi+2k\pi}{3}]$ $\;\text{where k=0,1,2}$

$\text{For k=0, x-1=4[cos\frac{\pi}{3}+i\,sin\frac{\pi}{3}] }$

$\implies\,x-1=4[\frac{1}{2}+i\,\frac{\sqrt3}{2}]$

$\implies\,x-1=2+i\,2\sqrt{3}$

$\implies\boxed{\bf\,x=3+i\,2\sqrt{3}}$

$\text{For k=1, x-1=4[cos\,\pi+i\,sin\,\pi] }$

$\implies\,x-1=4[-1+i(0)]$

$\implies\,x-1=-4$

$\implies\boxed{\bf\,x=-3}$

$\text{For k=2, x-1=4[cos\frac{5\pi}{3}+i\,sin\frac{5\pi}{3}] }$

$\implies\,x-1=4[\frac{1}{2}-i\,\frac{\sqrt3}{2}]$

$\implies\,x-1=2-i\,2\sqrt{3}$

$\implies\boxed{\bf\,x=3-i\,2\sqrt{3}}$

$\textbf{The roots of the equation are}$

$\bf\,3+i\,2\sqrt{3},-3\;\text{and}\;3-i\,2\sqrt{3}$

$\therefore\textbf{The sum of the roots is 3}$

$\implies\textbf{option (2) is correct}$

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