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Câu hỏi: Cho số phức $z=\dfrac{-m+i}{1-m\left( m-2i \right)}, m\in \mathbb{R}$. Tìm số phức $\text{w}=\left( 3-2i \right)z$ khi $z$ có môđun lớn nhất.
A. $\text{w}=2+3i$.
B. $\text{w}=\dfrac{5}{2}+\dfrac{1}{2}i$.
C. $\text{w}=17+6i$.
D. $\text{w}=10-11i$.
A. $\text{w}=2+3i$.
B. $\text{w}=\dfrac{5}{2}+\dfrac{1}{2}i$.
C. $\text{w}=17+6i$.
D. $\text{w}=10-11i$.
Ta có: $z=\dfrac{-m+i}{1-m\left( m-2i \right)}=\dfrac{\left( -m+i \right)\left( 1-{{m}^{2}}-2mi \right)}{{{\left( 1-{{m}^{2}} \right)}^{2}}+4{{m}^{2}}}=\dfrac{m}{{{m}^{2}}+1}+\dfrac{i}{{{m}^{2}}+1}$
$\Rightarrow \left| z \right|=\sqrt{\dfrac{1}{{{m}^{2}}+1}}\le 1\Rightarrow {{\left| z \right|}_{\max }}=1\Leftrightarrow z=i$ khi $m=0$
$\Rightarrow \text{w}=\left( 3-2i \right)z=\left( 3-2i \right)i=2+3i$.
$\Rightarrow \left| z \right|=\sqrt{\dfrac{1}{{{m}^{2}}+1}}\le 1\Rightarrow {{\left| z \right|}_{\max }}=1\Leftrightarrow z=i$ khi $m=0$
$\Rightarrow \text{w}=\left( 3-2i \right)z=\left( 3-2i \right)i=2+3i$.
Đáp án A.