Q50 / 160 · TS EAMCET 2024 (09 May Shift 1) · Mathematics

$x$ and $y$ are two complex numbers such that $|x|=|y|=1$. If $\operatorname{Arg}(x)=2 \alpha, \operatorname{Arg}(y)=3 \beta$ and $\alpha+\beta=\frac{\pi}{36}$, then $x^6 y^4+\frac{1}{x^6 y^4}=$

  • A
    0
  • B
    -1
  • Correct answer
    C
    1
  • D
    $\frac{1}{2}$

Solution

$x=e^{i 2 \alpha}, y=e^{i 3 \beta}$ $x^6 y^4=e^{12 i(\alpha+\beta)} ; \frac{1}{x^6 y^4}=e^{-12 i(\alpha+\beta)}$ $\begin{aligned} & x^6 y^4+\frac{1}{x^6 y^4}=e^{12 i(\alpha+\beta)}+e^{-12 i(\alpha+\beta)} \\ & =e^{i \frac{\pi}{3}}+e^{-i \frac{\pi}{3}}=2 \cos \frac{\pi}{3}=1\end{aligned}$

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

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