Q51 / 160 · TS EAMCET 2024 (09 May Shift 1) · Mathematics

One of the roots of the equation $x^{14}+x^9-x^5-1=0$ is

  • A
    $\frac{1+\sqrt{3} i}{2}$
  • B
    $\frac{\sqrt{5}-1}{4}+i \frac{\sqrt{10-2 \sqrt{5}}}{4}$
  • C
    $\frac{1-\sqrt{3} i}{2}$
  • Correct answer
    D
    $\frac{\sqrt{5}+1}{4}+i \frac{\sqrt{10-2 \sqrt{5}}}{4}$

Solution

$x^{14}+x^9-x^5-1=0$ $\begin{aligned} & \Rightarrow\left(x^9-1\right)\left(x^5+1\right)=0 \\ & \Rightarrow x^5=-1 \text { or } x^9=1 \\ & \Rightarrow x^5=\cos 180^{\circ}+i \sin 180^{\circ} \\ & \Rightarrow x=\cos \left(\frac{180}{5}\right)+i \sin \left(\frac{180}{5}\right) \\ & \Rightarrow x=\cos 36^{\circ}+i \sin 36^{\circ} \\ & \Rightarrow x=\frac{\sqrt{5}+1}{4}+\frac{i \sqrt{10-2 \sqrt{5}}}{4} .\end{aligned}$

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

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