72 lines
1.4 KiB
Markdown
72 lines
1.4 KiB
Markdown
#учеба #семестр_1 #высшая_математика
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1)
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Пример:
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$\mathrm{Re}\left( \frac{3-i}{1+3i} \right)\times \mathrm{Im}\left( \frac{2-i}{1+3i} \right)$
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Решение:
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$\mathrm{Re}:$
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$\frac{(3-i)(1-3i)}{1^{2}+3^{2}}=\frac{0-10i}{10}=-i$
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$\mathrm{Im}:$
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$\frac{(2-i)(1-3i)}{1^{2}+3^{2}}=\frac{-1-7i}{10}=-0.1-0.7i$
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$\mathrm{Re}\times \mathrm{Im}=0\times(-0.7)=0$
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3)
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Пример:
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$z^{2}+(2+4i)z+6+4i=0$
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Решение:
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$D=(2+4i)^{2}-24-16i=4-16+16i-24-16i=-36$
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$z_{1,2}=\frac{-2-4i\pm\sqrt{ -36 }}{2}=-1-5i;-1+i$
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4)
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Пример:
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$|z|+z=8+4i$
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Решение:
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$(z=a+ib)$
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$$
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\sqrt{ a^{2}+b^{2} }+a-8+ib-4i=0 \implies
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\begin{cases}
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\sqrt{ a^{2}+b^{2} }+a-8=0 \\
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ib-4i=0
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\end{cases}
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$$
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$b=4$
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$\sqrt{ a^{2}+16 }=8-a\implies a\leq 8$
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$a^{2}+16=a^{2}-16a+64$
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$a=3$
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$z=3+4i$
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5)
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Пример:
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$$
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\begin{cases}
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2z_{1}+3z_{2}=7-i \\
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iz_{1}-2z_{2}=-3+4i
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\end{cases}
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$$
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Решение:
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$z_{1}=3.5-0.5i-1.5z_{2}$
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$3.5i+0.5-1.5iz_{2}-2z_{2}=-3+4i$
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$z_{2}(2+1.5i)=3-4i+3.5i+0.5$
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$z_{2}=\frac{3.5-0.5i}{2+1.5i}=\frac{(3.5-0.5i)(2-1.5i)}{4+2.25}=\frac{7-0.75-i-5.25i}{6.25}=1-i$
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$2z_{1}+3-3i=7-i$
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$z_{1}=2+i$
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$z_{2}=1-i$
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7)
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Пример:
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$\frac{(1-i)(-3-i\sqrt{ 3 })}{(2+2i)(i+\sqrt{ 3 })}$
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Решение:
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$\frac{-\sqrt{ 3 }(1-i){(i+\sqrt{ 3 })}}{2(1+i){(i+\sqrt{ 3 })}}=$
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$-\frac{\sqrt{ 3 }}{2}\cdot\frac{(1-i)^{2}}{1^{2}+1^{2}}=$
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$-\frac{\sqrt{ 3 }}{2}\cdot\frac{-2i}{2}=\frac{i\sqrt{ 3 }}{2}$
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