First slide

Algebra of complex numbers

Question

If $3^{49} (x + i y) = {(\frac{3}{2} + \frac{i}{2} \sqrt{3})}^{100}, y \in N$ and $x = k y,$ then value of $k$ is

Moderate

A

$\pm 1 / 3$
B

$\pm 2 \sqrt{2}$
C

$\pm 1 / \sqrt{3}$
D

$\pm 1 \sqrt{3}$

Solution

We have
$\begin{array}{l} 3^{49} | x + i y | = | 3 / 2 + \sqrt{3} i / 2 |^{100} \\ \Rightarrow 3^{49} \sqrt{x^{2} + y^{2}} = (9 / 4 + 3 / 4)^{50} \\ \Rightarrow \sqrt{x^{2} + y^{2}} = 3 \Rightarrow (y) \sqrt{1 + k^{2}} = 3 \\ \Rightarrow \sqrt{1 + k^{2}} = 3, 3 / 2, 1, as y \in N \\ \Rightarrow k = \pm 2 \sqrt{2}, \pm \sqrt{5} / 2, 0 \end{array}$
Out of the given values, we have $k = \pm 2 \sqrt{2 .}$

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