TS – complexes 1 – Ex 1

Exercice 1

Déterminer le conjugué de chaque nombre complexe et donner sa forme algébrique.

  1. $z = (3+\ic)(-13 – 2\ic)$
    $\quad$
  2. $z=\ic(1-\ic)^3$
    $\quad$
  3. $z = \dfrac{2 – 3\ic}{8 + 5\ic}$
    $\quad$
  4. $z=\dfrac{2}{\ic + 1}-\dfrac{3}{1-\ic}$
    $\quad$

Correction

  1. $\overline{z} = \overline{(3+\ic)(-13 – 2\ic)}$ $= (3 – \ic)(-13 + 2\ic)$ $=-39 +6\ic + 13\ic + 2 $ $=-37 + 19\ic$
    $\quad$
  2. $\quad$
    $\begin{align*} \overline{z}& = \overline{\ic(1-\ic)^3}\\\\
    &= -\ic(1 + \ic)^3 \\\\
    &=-\ic(1+ \ic)(1 + \ic)^2 \\\\
    &= -\ic(1 + \ic)(1 + 2\ic – 1) \\\\
    &= (-\ic + 1)(2\ic) \\\\
    &=2 + 2\ic
    \end{align*}$
    $\quad$
  3. $\quad$
    $\begin{align*} \overline{z} &= \overline{\left(\dfrac{2 – 3\ic}{8 + 5\ic}\right)} \\\\
    & = \dfrac{2 + 3\ic}{8 – 5\ic} \\\\
    & = \dfrac{2 + 3\ic}{8 – 5\ic} \times \dfrac{8 + 5\ic}{8 + 5\ic}\\\\
    &= \dfrac{16 + 10\ic + 24\ic – 15}{8^2 + 5^2} \\\\
    &=\dfrac{1 + 34\ic}{89}
    \end{align*}$
    $\quad$
  4. $\quad$
    $\begin{align*} \overline{z} &= \overline{\dfrac{2}{\ic + 1}-\dfrac{3}{1-\ic}} \\\\
    &=\overline{\left(\dfrac{2(1 – \ic) – 3(\ic + 1)}{(\ic + 1)(1 – \ic)}\right)} \\\\
    &=\overline{\left(\dfrac{2 – 2\ic – 3\ic – 3}{1^2 + 1^2}\right)} \\\\
    &=\overline{\dfrac{-1 – 5\ic}{2}}\\\\
    &=\dfrac{-1 + 5\ic}{2}
    \end{align*}$