Cách giải bài tập SGK Toán giải tích lớp 12, Bài 3. Phép chia số phức

Bài 1 trang 138

Bài 2 trang 138

Tìm nghịch đảo $\frac{1}{z}$$\frac{\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">1</span></span>}}{\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">z</span></span>}}$ của số phức $z$, biết:

a) $z=1+2i$$\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">z</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">=</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">1</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">+</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">2</span></span>}\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">i</span></span>}$;                         b) $z=\sqrt{2}-3i$$\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">z</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">=</span></span>\sqrt{\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">2</span></span>}}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">-</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">3</span></span>}\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">i</span></span>}$;

c) $z=i$$\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">z</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">=</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">i</span></span>}$;                                 d) $z=5+i\sqrt{3}$$\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">z</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">=</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">5</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">+</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">i</span></span>}\sqrt{\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">3</span></span>}}$.

Bài 3 trang 138

bài 4 trang 138

Lời giải chi tiết

a) Ta có $(3-2i)z+(4+5i)=7+3i\iff (3-2i)z=7+3i-4-5i$$<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">(</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">3</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">-</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">2</span></span>}\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">i</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">)</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">z</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">+</span></span><span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">(</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">4</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">+</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">5</span></span>}\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">i</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">)</span></span><span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">=</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">7</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">+</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">3</span></span>}\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">i</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">\iff </span></span><span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">(</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">3</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">-</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">2</span></span>}\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">i</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">)</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">z</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">=</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">7</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">+</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">3</span></span>}\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">i</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">-</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">4</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">-</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">5</span></span>}\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">i</span></span>}$

$\iff (3-2i)z=3-2i\iff z=\frac{3-2i}{3-2i}\iff z=1$$<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">\iff </span></span><span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">(</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">3</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">-</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">2</span></span>}\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">i</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">)</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">z</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">=</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">3</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">-</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">2</span></span>}\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">i</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">\iff </span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">z</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">=</span></span>\frac{\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">3</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">-</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">2</span></span>}\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">i</span></span>}}{\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">3</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">-</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">2</span></span>}\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">i</span></span>}}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">\iff </span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">z</span></span>}<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">=</span></span>\mathrm{<span\; style="font-family:arial,helvetica,sans-serif;"><span\; style="font-size:16px;">1</span></span>}$.

Vậy $z=1$

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