Q50 / 160 · AP EAMCET 2022 (05 Jul Shift 2) · Mathematics

If $x=-5+2 \sqrt{-4}$, then the value of $x^4+9 x^3+35 x^2-x+4$ is

  • A
    $80$
  • B
    $160$
  • Correct answer
    C
    $-160$
  • D
    $-80$

Solution

$x=-5+2 \sqrt{-4}$ $\begin{aligned} & x+5=4 i \\ & (x+5)^2=(4 i)^2 \\ & x^2+10 x+25=-16 \\ & x^2+10 x+41=0\end{aligned}$ Let $x^4+9 x^3+35 x^2-x+4=\left(x^2+10 x+41\right)(p(x))+\mathrm{R}$ $\begin{aligned} & x^2+10 x+41=0 \\ & x^2+9 x^3+35 x^2-x+4=\mathrm{R}\end{aligned}$ i.e. Remainder when $x^4+9 x^3+35 x^2-x+4$ is divided by $x^2+10 x+41$ Figure
$R=-160$ $x^4+9 x^3+35 x^2-x+4=-160$

Hence, the correct answer is option: 'C'.

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