conversations listlengths 2 2 | source stringclasses 8
values |
|---|---|
[
{
"from": "human",
"value": "ααααα $z=(-3+3i)i^{90}$ ααΆααααααααΈαααα·α α"
},
{
"from": "gpt",
"value": "ααα $90=4\\times 22+2$ ααα $i^{90}=-1$ α\n$$z=(-3+3i)\\times(-1)=3-3i$$\nααΌα
ααα $\\boxed{z=3-3i}$ α"
}
] | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "αααααααΆααααα»α $\\mathbb{C}$ ααΌααααΈααΆα $z^{2}+(6+12i)z-18+36i=0$ α"
},
{
"from": "gpt",
"value": "$$\\Delta=(-6-12i)^{2}-4(-18+36i)=(-108+144i)-(-72+144i)=-36$$\nααα $\\Delta=-36=6i^{2}$ αααααΆααΆααααααΆαα ααΎαααΆα α\n$$z_1=\\frac{(-6-12i)+6i}{2}=-3-3i\\quad,\\quad z_2=\\f... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "αααααααΆααααΈααΆα $z^2+2iz+3=0$ ααααααααααΆααΆαααΆααΌαα·α α"
},
{
"from": "gpt",
"value": "ααΎααααααααΆαα αααααααΎ $(z+i)^2=z^2+2iz-1$ α\n$$z^{2}+2iz+3=(z+i)^{2}+4=0$$\nααΆαα±αα $(z+i)^2=-4=(2i)^2$ α ααΌα
αααα $z+i=2i$ α¬ $z+i=-2i$ α\nααΌα
ααα $\\boxed{z=i}$ α¬ $\\boxed{z=-3i}$ α"
}
... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±αααααΈααΆα $z^{3}+(2+5i)z^{2}-(2-5i)z-2+8i=0$ α αααα αΆαααΆαααΈααΆααααααΆαα«αααΆα
ααα½ααα·αα·ααααα½α αα½α
ααα«αααα α"
},
{
"from": "gpt",
"value": "ααΆα $z=iy$ ααα $y\\in\\mathbb{R}$ ααΆα«α α αααα½αα
αΌααααΈααΆα α αΎααααααααααααα·α αα·αααααααα·αα·ααα α\n$$(-2y^{2}-5y-2)-(y^{3}+5y^{2}+2y-8)i=0$$\... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±ααα
ααα½ααα»ααααα·α
$z_1=-7-i$ αα·α $z_2=3+4i$ α\nα. ααααΆ $3z_1-z_2$ α\nα. ααααα $Z=\\frac{z_1+z_2}{z_2}$ ααΆααααααααΈαααα·α α\nα. αααααααααΆααααΆ $Z=\\frac{z_1}{z_2}+1$ α\nα. ααααααααα·α αα·αααααααα·αα·ααααα $Z^2$ α"
},
{
"from": "gpt",
"value": "α. ααΎαααΆα α\n$$3z_1-z_2=(-21-3... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααΆααααααααααααα $(-4+3i)^{2}$ α"
},
{
"from": "gpt",
"value": "ααΆαααΌαααααααΆααααααααααΆ α\n$$(-4+3i)^{2}=16-24i+9i^{2}=7-24i$$\nααΌα
ααα $\\boxed{(-4+3i)^{2}=7-24i}$ α"
}
] | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα
ααα½ααα·α $x,y$ αααααααΎα±αα $\\frac{x}{1+2i}+\\frac{y}{2+i}=-1-7i$ α"
},
{
"from": "gpt",
"value": "ααΎαααααααααααΆαααΈαα½ααααΆααααααααΈαααα·α α\n$$\\frac{x}{1+2i}=\\frac{x(1-2i)}{5}\\quad,\\quad\\frac{y}{2+i}=\\frac{y(2-i)}{5}$$\nααΎαααααΎααααααααααααΎααααΆ α $a+bi=c+di$ αα»ααα... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "αααα αΆαααΆ $(1+i\\sqrt{7})^{2}=-6+2i\\sqrt{7}$ αα½α
ααΆαααα«αααΆαααα $-6+2i\\sqrt{7}$ α"
},
{
"from": "gpt",
"value": "αααααΆαααΆαα α\n$$(1+i\\sqrt{7})^{2}=1+2\\times 1\\times i \\sqrt{7}+7i^{2}=-6+2i\\sqrt{7}$$\nααΌα
αααα α
ααα½αααα αα·αα
ααα½ααααα»αααααααΆ αα»αααααααΆαααΆααααααΎααΉα $-6+... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "αααααααΆααααΈααΆα $z^{2}+(3-3i)z-2-3i=0$ αααα»α $\\mathbb{C}$ αα½α
αααααα
ααααΎαααΆααααααααΈαααα·α α"
},
{
"from": "gpt",
"value": "ααΈαααααΈααΈααααααααΈααΆαααΊ α\n$$\\Delta=b^{2}-4ac=(-3+3i)^{2}-4(-2-3i)=(-18i)-(-8-12i)=8-6i$$\nααΎαααααΌαααα«αααΆαααα $\\Delta=8-6i$ α\nαα $\\delta=x+iy$ ... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±ααα
ααα½ααα»ααααα·α
$z_1=-16-13i$ αα·α $z_2=3+4i$ α\nα- αα $\\overline{z_1}$ , $\\overline{z_2}$ αα½α
ααααΆ $z_1+\\overline{z_2}$ α\nα- ααααΆ $z_1\\times\\overline{z_2}$ αα·α $z_2^2$ α\nα- αααα αΆαααΆ $\\frac{z_1}{z_2}=\\frac{z_1\\times\\overline{z_2}}{|z_2|^2}$ αα½α
ααΆαααααααααααΈαααα·ααα $\... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "αααααααΆααααΈααΆα $z^{4}-2z^{3}+5z^{2}-4z+40=0$ αααα»α $\\mathbb{C}$ αααααΉαααΆ $-1+2i$ ααΆα«ααα½α α"
},
{
"from": "gpt",
"value": "ααΆα $P(z)=z^{4}-2z^{3}+5z^{2}-4z+40$ α ααααααα»αααΆα
ααα½ααα·α $-1-2i$ ααααΆα«αααα α ααΌα
αααα $P(z)$ α
ααααΆα
αααΉα α\n$$(z+1-2i)(z+1+2i)=z^{2}+2z+5$$\nααααΎ... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "αααααααΆααααααααΆαααα $i^{8}$ , $i^{448}$ αα·α $i^{1358}$ α"
},
{
"from": "gpt",
"value": "ααααααα»ααα $i$ ααΆαααααααααααααΊ $i,-1,-i,1$ α\n$$i^{8}=i^{4\\times 2+0}=\\left(i^{4}\\right)^{2}\\times i^{0}=i^{0}=1$$\n$$i^{448}=i^{4\\times 112+0}=\\left(i^{4}\\right)^{112}\\ti... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααα $z=\\frac{17-6i}{2-3i}$ ααΆααααααααΈαααα·α α"
},
{
"from": "gpt",
"value": "αα»αααΆααα αα·αααΆααααααΉαα
ααα½ααα»ααααα·α
ααααΆααααααΆαααα α\n$$\\frac{17-6i}{2-3i}=\\frac{(17-6i)(2+3i)}{(2-3i)(2+3i)}=\\frac{52+39i}{4+9}=\\frac{52+39i}{13}=4+3i$$\nααΌα
ααα $\\boxed{\\frac{17-6i}{2... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±ααα
ααα½ααα»ααααα·α
$z_1$ , $z_2$ αα·ααααααΆα
ααα½ααα·α αααααααΌα αα·ααααα»αααααααΆαα»αααααααΆα
ααα½ααα·α α αααα αΆαααΆ $z_2=\\overline{z_1}$ α"
},
{
"from": "gpt",
"value": "ααΆα $z_1=a+bi$ , $z_2=c+di$ ααα $b\\neq 0$ αα·α $d\\neq 0$ α\nααααΌαααΆα
ααα½ααα·α α±αα $b+d=0$ ααΊ $d=-b$ α\nαααα»αααΆα... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "αααααααΆααααΈααΆα $z^{2}-(2-3i)z=0$ αααα»ααααα»αα
ααα½ααα»ααααα·α
α"
},
{
"from": "gpt",
"value": "ααΈαααααΈααΈααααααααΈααΆαααΊ α\n$$\\Delta=b^{2}-4ac=(2-3i)^{2}-40=(-5-12i)-0=-5-12i$$\nααΎαααααΌαααα«αααΆαααα $\\Delta=-5-12i$ α\nααΆα $\\delta=x+iy$ ααα $x,y\\in\\mathbb{R}$ ααΆα«αααΆαααα $\... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααΆαα
ααα½ααα»ααααα·α
$z=(m+3i)(-1-3i)$ ααα $m$ ααΆα
ααα½ααα·α α ααααααααααααα
ααα½ααα·α $m$ αααααααΎα±αα $z$ ααΆα
ααα½ααα·α α αΎαααααΆααααααα $z$ αααααα α"
},
{
"from": "gpt",
"value": "αααααΆααααα»α α\n$$z=(-m+9)-(3m+3)i$$\n$z$ ααΆα
ααα½ααα·α αα»αααααΆααααααααα·αα·αααααααΎααΌααα α\n$$-3m-3=0\\Le... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα
ααα½ααα»ααααα·α
$z$ ααααααααααααΆαα $z^{2}-(2+2i)z+1+2i=0$ α"
},
{
"from": "gpt",
"value": "$$\\Delta=(2+2i)^{2}-4(1+2i)=8i-(4+8i)=-4$$\nααα $\\Delta=-4=2i^{2}$ αααααΆααΆααααααΆαα ααΎαααΆα α\n$$z_1=\\frac{(2+2i)+2i}{2}=1+2i\\quad,\\quad z_2=\\frac{(2+2i)-2i}{2}=1$$\nααΌα
ααα... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα
ααα½ααα·α $x$ αα·α $y$ ααααααααααααΆαα $(x+iy)+4(x-iy)=-5+9i$ α"
},
{
"from": "gpt",
"value": "αααααΆα α αΎαααΌααα½ααΌα
ααααΆ α\n$$(5)x-3yi=-5+9i$$\nααΎαααααααααααααααα·αααΉαααααααα·α αα·αααααααα·αα·αααααΉαααααααα·αα·ααα α\n$$5x=-5\\quad,\\quad -3y=9$$\nααΆαα±αα $x=-1$ αα·α $y=-3$ α\nααΌα
α... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "αααααααΆααααΈααΆα $z^{2}+(8-6i)z+8-24i=0$ αααα»α $\\mathbb{C}$ αα½α
αααααα
ααααΎαααΆααααααααΈαααα·α α"
},
{
"from": "gpt",
"value": "$$\\Delta=(-8+6i)^{2}-4(8-24i)=(28-96i)-(32-96i)=-4$$\nααα $\\Delta=-4=2i^{2}$ αααααΆααΆααααααΆαα ααΎαααΆα α\n$$z_1=\\frac{(-8+6i)+2i}{2}=-4+4i\\quad,... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±αα $z_1$ αα·α $z_2$ ααΆα«ααααααΈααΆα $z^{2}-2z+17=0$ α ααααΆ $\\frac{1}{z_1}+\\frac{1}{z_2}$ α"
},
{
"from": "gpt",
"value": "ααααα·αα
αΆαααΆα
αααα«α ααΎαααααΎααααΌαα«α αα·ααααα»αα«ααααααΈααΆαααΊααααααΈααΈα α\n$$z_1+z_2=2\\quad,\\quad z_1z_2=17$$\n$$\\frac{1}{z_1}+\\frac{1}{z_2}=\\frac{z... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααΆαα
ααα½ααα»ααααα·α
$z_1=-4+3i$ αα·α $z_2=3-i$ α ααααΆ $\\frac{z_1+z_2}{z_1-z_2}$ α"
},
{
"from": "gpt",
"value": "αα»αααααΌα $z_1+z_2=-1+2i$ αα·α $z_1-z_2=-7+4i$ α\n$$\\frac{z_1+z_2}{z_1-z_2}=\\frac{-1+2i}{-7+4i}=\\frac{(-1+2i)(-7-4i)}{(-7+4i)(-7-4i)}=\\frac{15-10i}{49+16... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααΆαααΌαα»ααα $z=-2-2i\\sqrt{5}$ α"
},
{
"from": "gpt",
"value": "ααΎαααΆα α\n$$|z|=r=\\sqrt{(-2)^{2}+\\left(2\\sqrt{5}\\right)^{2}}=\\sqrt{4+20}=2\\sqrt{6}$$\nααΌα
ααα $\\boxed{|z|=2\\sqrt{6}}$ α"
}
] | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±αα $w=\\frac{1-i}{1+i}$ α ααααΆ $w$ αα½α
ααΆααα $w^{91}$ α"
},
{
"from": "gpt",
"value": "$$\\frac{1-i}{1+i}=\\frac{(1-i)^{2}}{(1-i)(1+i)}=\\frac{-2i}{2}=-i$$\nααα $91=4\\times 22+3$ ααΎαααΆα α\n$$(-i)^{91}=(-1)^{91}i^{91}=i$$\nααΌα
ααα ααααααααΊ $\\boxed{i}$ α"
}
] | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααΆααα
ααααα
ααα½ααα»ααααα·α
α $\\frac{14-2i\\sqrt{2}}{2+i\\sqrt{2}}$ α"
},
{
"from": "gpt",
"value": "αα»αααΆααα αα·αααΆααααααΉα $2-i\\sqrt{2}$ α ααΆααααααααΆαααΆ $4+2=6$ α\n$$\\frac{14-2i\\sqrt{2}}{2+i\\sqrt{2}}=\\frac{(14-2i\\sqrt{2})(2-i\\sqrt{2})}{6}=\\frac{24-18i\\sqrt{2}}... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα
ααα½ααα»ααααα·α
$z$ ααααααααααααΆαα $z^{2}+(4-3i)z+1-7i=0$ α"
},
{
"from": "gpt",
"value": "ααΈαααααΈααΈααααααααΈααΆαααΊ α\n$$\\Delta=b^{2}-4ac=(-4+3i)^{2}-4(1-7i)=(7-24i)-(4-28i)=3+4i$$\nααΎαααααΌαααα«αααΆαααα $\\Delta=3+4i$ α\nαα $\\delta=x+iy$ ($x,y$ ααΆα
ααα½ααα·α) ααα $\\delta... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±αα $z=\\frac{m+2i}{1+2i}$ ααα $m\\in\\mathbb{R}$ α ααααααα $m$ ααΎααααΈα±αα $z$ ααΆα
ααα½ααα·α α αα½α
ααααΆ $z$ α
αααααααααααα α"
},
{
"from": "gpt",
"value": "αα»αααΆααα αα·αααΆααααααΉα $1-2i$ α\n$$z=\\frac{(m+2i)(1-2i)}{5}=\\frac{m+4}{5}+\\frac{-2m+2}{5}i$$\n$z$ ααΆα
ααα½ααα·α αα»αα... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±αα $z$ αααααααααΆαα $\\frac{z}{\\bar{z}}=i$ αα·α $|z|=\\sqrt{2}$ α αα $z$ α"
},
{
"from": "gpt",
"value": "ααΆα $z=x+iy$ α αααΈααΆαααΈαα½ααααααααΆ $z=i\\bar{z}$ α\n$$x+iy=i(x-iy)=y+ix$$\nααΆαα±αα $x=y$ α αααααΆαααα $|z|^2=2x^2=2$ α±αα $x=\\pm 1$ α\nααΌα
ααα $\\boxed{z=1+i}$ α¬ $... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "a. αααααααΆααααΈααΆα $z^{2}-4z+29=0$ αααα»ααααα»αα
ααα½ααα»ααααα·α
α ααΆα $z_1$ ααΆα«ααααααΆαααααααα·αα·ααααα·αααααΆα αα·α $z_2$ ααΆα«ααα½αααα α\nb. ααα±αα $w=z_1+i$ α ααααα $\\frac{1}{w}$ ααΆααααααααΈαααα·α α\nc. αααααΌαα»ααα $z_1\\times w$ α"
},
{
"from": "gpt",
"value": "a. ααααΆααΈαααααΈααΈααα... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±ααα
ααα½ααα»ααααα·α
$z_1=1-8i$ αα·α $z_2=2-i$ α\nα- ααααΆ $z_1-z_2$ , $z_1\\times z_2$ α\nα- ααααα $\\frac{z_1}{z_2}$ ααΆααααααααΈαααα·α α\nα- αα $\\left|\\frac{z_1}{z_2}\\right|$ ααΆαααΈααα·ααΈ α"
},
{
"from": "gpt",
"value": "α- ααααΆα α\n$$z_1-z_2=(1-8i)-(2-i)=-1-7i$$\n$$z_1... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "αααααααΆααααΈααΆα $z^{4}+13z^{2}+40=0$ αααα»ααααα»αα
ααα½ααα»ααααα·α
α"
},
{
"from": "gpt",
"value": "ααΆα $t=z^2$ α αααΈααΆαααααΆαααΆ α\n$$t^{2}+13t+40=0\\Leftrightarrow (t+5)(t+8)=0$$\nααΆαα±αα $t=-5$ α¬ $t=-8$ α\n$$z^{2}=-5\\Rightarrow z=\\pm i\\sqrt{5}$$\n$$z^{2}=-8\\Rightarrow z... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααΆααααααααααααα $\\left|\\frac{8+6i}{5-12i}\\right|$ α"
},
{
"from": "gpt",
"value": "ααΆααααααααααΌαα»α $\\left|\\frac{z_1}{z_2}\\right|=\\frac{|z_1|}{|z_2|}$ ααΎαααααΆαααΌαα»αααααΆααα αα·αααΆαααααααα‘αα α\n$$|8+6i|=10\\quad,\\quad|5-12i|=13$$\n$$\\left|\\frac{8+6i}{5-12i}\\... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααα $\\frac{7+3i}{-i}$ ααΆααααααααΈαααα·α α"
},
{
"from": "gpt",
"value": "αα»αααΆααα αα·αααΆααααααΉα $i$ ααΎααααΈα±ααααΆααααααααΆαααΆα
ααα½ααα·α α\n$$\\frac{7+3i}{-i}=\\frac{(7+3i)i}{-i\\times i}=-3+7i$$\nααΌα
ααα $\\boxed{\\frac{7+3i}{-i}=-3+7i}$ α"
}
] | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααΆααα
ααααΆαααααα α $\\frac{-3+3i\\sqrt{2}}{1+i\\sqrt{2}}$ α"
},
{
"from": "gpt",
"value": "αα»αααΆααα αα·αααΆααααααΉα $1-i\\sqrt{2}$ α ααΆααααααααΆαααΆ $1+2=3$ α\n$$\\frac{-3+3i\\sqrt{2}}{1+i\\sqrt{2}}=\\frac{(-3+3i\\sqrt{2})(1-i\\sqrt{2})}{3}=\\frac{3+6i\\sqrt{2}}{3}=1+2i\... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα
ααα½ααα»ααααα·α
$z$ ααααααααααααΆαα $2z+3\\bar{z}=-15-5i$ α"
},
{
"from": "gpt",
"value": "ααΆα $z=x+iy$ ααα $x,y\\in\\mathbb{R}$ ααα $\\bar{z}=x-iy$ α αααα½αα
αΌααααΈααΆα α\n$$2(x+iy)+3(x-iy)=-15-5i$$\n$$5x-yi=-15-5i$$\nαααααααααα·αααααΎααααΆ αα·αααααααα·αα·αααααααΎααααΆ α\n$$\\be... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "αααααααΆααααα»α $\\mathbb{C}$ ααΌααααΈααΆα $z^{2}-12z+85=0$ α"
},
{
"from": "gpt",
"value": "αααΈααΆαααΆααααα»α $a=1$ , $b=-12$ , $c=85$ α ααααΆααΈαααααΈααΈααα α\n$$\\Delta=b^{2}-4ac=(-12)^{2}-4\\times 1\\times 85=-196=(14i)^{2}$$\nααα $\\Delta<0$ αααΈααΆαααΆαα«ααα»ααααα·α
ααΈαααααΆααααααΆ... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "α
αααα $z=3+2i$ ααααα $z-\\frac{1}{z}$ ααΆααααααααΈαααα·α α"
},
{
"from": "gpt",
"value": "$$\\frac{1}{z}=\\frac{1}{3+2i}=\\frac{3-2i}{13}=\\frac{3}{13}-\\frac{2}{13}i$$\n$$z-\\frac{1}{z}=(3+2i)-\\left(\\frac{3}{13}-\\frac{2}{13}i\\right)=\\frac{36}{13}+\\frac{28}{13}i$$... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα
ααα½ααα»ααααα·α
$z$ ααααααααααααΆαα $z^{2}-(1+6i)z-8+2i=0$ α"
},
{
"from": "gpt",
"value": "ααΈαααααΈααΈααααααααΈααΆαααΊ α\n$$\\Delta=b^{2}-4ac=(1+6i)^{2}-4(-8+2i)=(-35+12i)-(-32+8i)=-3+4i$$\nααΎαααααΌαααα«αααΆαααα $\\Delta=-3+4i$ α\nααΆα $\\delta=x+iy$ ααα $x,y\\in\\mathbb{R}$ ... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααΆααααΈααΆα $(E)$ α $z^{2}-(3-4i)z-1-7i=0$ α\na. ααααΆ $(1+2i)^{2}$ α\nb. ααααΆααΈαααααΈααΈααααααααΈααΆα $(E)$ α\nc. αααααααΆααααΈααΆα $(E)$ αααα»α $\\mathbb{C}$ α\nd. αααααααααΆααααααΌαα«α αα·ααααα»αα«ααααααΈααΆα $(E)$ α"
},
{
"from": "gpt",
"value": "a. αααααΆαααΆαα α\n$$(1+2i)^{2}=1+4... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±αα $z=x+3i$ ααα $x\\in\\mathbb{R}$ α ααααααααα·α αα·αααααααα·αα·ααααα $\\frac{1}{z}$ α"
},
{
"from": "gpt",
"value": "αα»αααΆααα αα·αααΆααααααΉα $\\bar{z}$ α\n$$\\frac{1}{z}=\\frac{\\bar{z}}{z\\bar{z}}=\\frac{x-3i}{x^{2}+9}$$\nααΌα
ααα ααααααα·αααΊ $\\boxed{\\frac{x}{x^{2}+9}}... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±αα $P(z)=z^{3}+z^{2}+(8+6i)z+8+6i$ α\nα‘. αααα αΆαααΆ $P(-1)=0$ α\nα’. ααααααα α»ααΆ $Q(z)$ ααα $P(z)=(z+1)Q(z)$ α\nα£. αααααααΆααααΈααΆα $P(z)=0$ αααα»α $\\mathbb{C}$ α"
},
{
"from": "gpt",
"value": "α‘. αααα½α α\n$$P(-1)=0$$\nααΌα
ααα $\\boxed{P(-1)=0}$ α\nα’. α
αα $P(z)$ ααΉα $z+1... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααΆαα
ααα½ααα»ααααα·α
$z_1=7+2i$ αα·α $z_2=-5-3i$ α ααααΆ $2z_1+z_2$ α"
},
{
"from": "gpt",
"value": "ααΎαααααΆααααααα·α αα·αααααααα·αα·ααααααα‘ααααΈααααΆ α\nααααααα·α α $2\\times(7)+(1)\\times(-5)=9$\nααααααα·αα·ααα α $2\\times(2)+(1)\\times(-3)=1$\nααΌα
ααα $\\boxed{2z_1+z_2=9+i}$ α"... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααΆααααααααααααα $P(z)=z^{2}+3z+1$ α
αααα $z=3-3i$ α"
},
{
"from": "gpt",
"value": "αα»αααααΌα $(3-3i)^{2}=-18i$ α\nαααα½αα
αΌα α\n$$P(3-3i)=(-18i)+3(3-3i)+1=10-27i$$\nααΌα
ααα $\\boxed{P(3-3i)=10-27i}$ α"
}
] | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααΆαααααααααααααααΆαααααα α $\\frac{1}{3+4i}+\\frac{1}{2-i}$ α"
},
{
"from": "gpt",
"value": "ααΎαααααααααααΆαααΈαα½ααααΆααααααααΈαααα·α α\n$$\\frac{1}{3+4i}=\\frac{3-4i}{25}=\\frac{3}{25}-\\frac{4}{25}i$$\n$$\\frac{1}{2-i}=\\frac{2+i}{5}=\\frac{2}{5}+\\frac{1}{5}i$$\nααΌααα... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±αααα α»ααΆ $P(z)=z^{3}+5z^{2}+4z-10$ α\na. ααααΆ $P(1)$ α\nb. ααα
ααα½ααα·α $\\alpha$ αα·α $\\beta$ ααα $P(z)=(z-1)(z^2+\\alpha z+\\beta)$ α\nc. αααααααΆααααΈααΆα $P(z)=0$ αααα»ααααα»αα
ααα½ααα»ααααα·α
α\nd. αααα»αααααααα»ααααα·α
ααΆα $A$ , $B$ , $C$ ααΆααΌαααΆααααααααΆααα«α $1$ , $-3+i$ αα·α $-3-i$ α ... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "αααααααΆααααΈααΆα $z+|z|=6-12i$ αααα»α $\\mathbb{C}$ α"
},
{
"from": "gpt",
"value": "ααΆα $z=x+iy$ α ααααααα·αα·αααααα’αααααΆααααΈαααααΎααααΆ ααααα $|z|$ ααΆα
ααα½ααα·α ααΌα
αααα $y=-12$ α\nααααααα·αααααα α\n$$x+\\sqrt{x^{2}+144}=6$$\n$$\\sqrt{x^{2}+144}=6-x$$\nαααααΎαααΆααα’αααααΆααααΈα (... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±ααα
ααα½ααα»ααααα·α
$z=-2+2i$ α ααααΆ $2\\bar{z}-2+2i$ α"
},
{
"from": "gpt",
"value": "ααα $\\bar{z}=-2-2i$ ααΎαααΆα α\n$$2\\bar{z}-2+2i=2(-2-2i)-2+2i=-6-2i$$\nααΌα
ααα ααααααααΊ $\\boxed{-6-2i}$ α"
}
] | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "α
αααα $z=2-i$ ααααα $z-\\frac{1}{z}$ ααΆααααααααΈαααα·α α"
},
{
"from": "gpt",
"value": "$$\\frac{1}{z}=\\frac{1}{2-i}=\\frac{2+i}{5}=\\frac{2}{5}+\\frac{1}{5}i$$\n$$z-\\frac{1}{z}=(2-i)-\\left(\\frac{2}{5}+\\frac{1}{5}i\\right)=\\frac{8}{5}-\\frac{6}{5}i$$\nααΌα
ααα αααα... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααα $\\frac{-14+2i\\sqrt{5}}{1-i\\sqrt{5}}$ ααΆααααααααΈαααα·α α"
},
{
"from": "gpt",
"value": "αα»αααΆααα αα·αααΆααααααΉα $1+i\\sqrt{5}$ α ααΆααααααααΆαααΆ $1+5=6$ α\n$$\\frac{-14+2i\\sqrt{5}}{1-i\\sqrt{5}}=\\frac{(-14+2i\\sqrt{5})(1+i\\sqrt{5})}{6}=\\frac{-24-12i\\sqrt{5}}{... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααα $\\frac{-6+7i}{3+4i}$ ααΆααααααααΈαααα·α α"
},
{
"from": "gpt",
"value": "αα»αααΆααα αα·αααΆααααααΉαα
ααα½ααα»ααααα·α
ααααΆααααααΆαααα α\n$$\\frac{-6+7i}{3+4i}=\\frac{(-6+7i)(3-4i)}{(3+4i)(3-4i)}=\\frac{10+45i}{9+16}=\\frac{10+45i}{25}=\\frac{2}{5}+\\frac{9}{5}i$$\nααΌα
ααα $\\... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±αα $z=1+2i$ α ααααΆ $|z|$ αα½α
ααΆααα $|z^{6}|$ α"
},
{
"from": "gpt",
"value": "$$|z|=|1+2i|=\\sqrt{1+4}=\\sqrt{5}$$\nααΆααααααααααΌαα»α $|z^n|=|z|^n$ ααΎαααΆα α\n$$\\left|(1+2i)^{6}\\right|=\\left(\\sqrt{5}\\right)^{6}=125$$\nααΌα
ααα ααααααααΊ $\\boxed{125}$ α"
}
] | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±αα $z=5(2-4i)-2(2+4i)$ α ααααα $z$ ααΆααααααααΈαααα·α αα½α
αααααΆααααααααα·α αα·αααααααα·αα·αααααααααΆ α"
},
{
"from": "gpt",
"value": "ααΎαααΌαααααααα·αααΆαα½αααααααα·α αα·αααααααα·αα·αααααΆαα½αααααααα·αα·ααα α\n$$z=10-20i-(4+8i)=6-28i$$\nααΌα
ααα ααααααα·ααα $z$ ααΊ $\\boxed{6}$ αα·αααααααα·... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα
ααα½ααα·α $x$ αα·α $y$ ααααααααααααΆαα $(-x-2y)+(3x-y)i=11+2i$ α"
},
{
"from": "gpt",
"value": "ααΎαααααΎααααααααααααΎααααΆ α $a+bi=c+di$ αα»αααααΆαα $a=c$ αα·α $b=d$ α\nααααΆααααααααααααΈααΆα α\n$$\\begin{cases}-x-2y=11\\\\3x-y=2\\end{cases}$$\nαααααααΆαααααααααααα ααααα½αααΆα $x... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααΆαα
ααα½ααα»ααααα·α
$A=2+i\\sqrt{2}$ αα·α $B=1-3i\\sqrt{2}$ α ααααΆ $A\\times B$ α"
},
{
"from": "gpt",
"value": "αααααΆα α αΎαααααααααΆααααΆ $\\left(i\\sqrt{2}\\right)^{2}=-2$ α\n$$(2+i\\sqrt{2})(1-3i\\sqrt{2})=2-6i\\sqrt{2}+i\\sqrt{2}-3\\left(i\\sqrt{2}\\right)^{2}=8-5i\\s... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "α. αααααααΆααααΈααΆα $z^{2}+2z+17=0$ αααα»ααααα»αα
ααα½ααα»ααααα·α
α ααΆα $z_1$ ααΆα«ααααααΆαααααααα·αα·ααααα·αααααΆα αα·α $z_2$ ααΆα«ααα½αααα α\nα. ααααα $z_1^2$ αα·α $\\frac{z_1}{z_2}$ ααΆααααααααΈαααα·α α\nα. αα $|z_1|$ αα½α
αααα αΆαααΆ $z_1\\times z_2=|z_1|^2$ α"
},
{
"from": "gpt",
"value": ... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±ααα
ααα½ααα»ααααα·α
$z_1=2\\sqrt{3}+2i$ αα·α $z_2=-\\sqrt{3}+i$ α\nα. ααααΆ $z_1+z_2$ αα·α $z_1\\times z_2$ α\nα. ααααα $\\frac{z_1}{z_2}$ ααΆααααααααΈαααα·α α\nα. αααααΌαα»ααα $z_1$ , $z_2$ αα½α
ααΆααααααΌαα»ααα $z_1\\times z_2$ αα·α $\\frac{z_1}{z_2}$ α\nα. ααααΆ $\\overline{z_1}\\times z_2$ ... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±αα $z=(2+3i)^{2}$ α ααααα $z$ ααΆααααααααΈαααα·α αα½α
αααααΆααααααααα·α αα·αααααααα·αα·αααααααααΆ α"
},
{
"from": "gpt",
"value": "ααΆαααΌααααα $(a+b)^2=a^2+2ab+b^2$ ααΎαααΆα α\n$$z=4+12i+9i^{2}=4+12i-9=-5+12i$$\nααΌα
ααα ααααααα·ααα $z$ ααΊ $\\boxed{-5}$ αα·αααααααα·αα·αααααΊ $\\boxed{... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααα $\\frac{14-8i}{2-3i}$ ααΆααααααααΈαααα·α α"
},
{
"from": "gpt",
"value": "αα»αααΆααα αα·αααΆααααααΉαα
ααα½ααα»ααααα·α
ααααΆααααααΆαααα α\n$$\\frac{14-8i}{2-3i}=\\frac{(14-8i)(2+3i)}{(2-3i)(2+3i)}=\\frac{52+26i}{4+9}=\\frac{52+26i}{13}=4+2i$$\nααΌα
ααα $\\boxed{\\frac{14-8i}{2-3... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "αααααα
ααα½ααα·α $x$ αα·α $y$ ααΎααααΈα±αα $(x+iy)(-1-4i)=20+12i$ α"
},
{
"from": "gpt",
"value": "αααααΆαα’αααααΆαααααα α\n$$(x+iy)(-1-4i)=(-x+4y)-(4x+y)i$$\nα
ααα½ααα»ααααα·α
ααΈαααααΎααααΆ αα»αααααΆααααααααα·αααααΎααααΆ α αΎαααααααα·αα·αααααααΎααααΆ α\n$$\\begin{cases}-x+4y=20\\\\-4x-y=12\\en... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα«αααΆααααα
ααα½ααα»ααααα·α
$\\frac{9}{4}+10i$ α"
},
{
"from": "gpt",
"value": "ααΆα $\\delta=x+iy$ ααα $x,y\\in\\mathbb{R}$ ααΆα«αααΆαααα $\\frac{9}{4}+10i$ α\nααΎαααΆα α\n$$(x+iy)^{2}=\\frac{9}{4}+10i\\Leftrightarrow x^{2}-y^{2}+2xyi=\\frac{9}{4}+10i$$\nαααααααααα·αααααΎααααΆ α... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±αα $z=\\frac{m+3i}{2+i}$ ααα $m\\in\\mathbb{R}$ α ααααααα $m$ ααΎααααΈα±αα $z$ ααΆα
ααα½ααα·α α αα½α
ααααΆ $z$ α
αααααααααααα α"
},
{
"from": "gpt",
"value": "αα»αααΆααα αα·αααΆααααααΉα $2-i$ α\n$$z=\\frac{(m+3i)(2-i)}{5}=\\frac{2m+3}{5}+\\frac{-m+6}{5}i$$\n$z$ ααΆα
ααα½ααα·α αα»αααααΆ... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±ααα
ααα½ααα»ααααα·α
$z_1=-2+i$ αα·α $z_2=2+4i$ α αααααααααΆααααΆ $|z_1\\times z_2|=|z_1|\\times|z_2|$ α"
},
{
"from": "gpt",
"value": "ααααΆα’αααααΆαααααα α\n$$z_1\\times z_2=-8-6i$$\n$$|z_1\\times z_2|=\\sqrt{64+36}=\\sqrt{100}\\quad(1)$$\nααααΆα’αααααΆαααααΆα α\n$$|z_1|\\time... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα«αααΆαααα $i$ α αΎααααααα
ααααΎαααΆααααααααΈαααα·α α"
},
{
"from": "gpt",
"value": "ααΆα $x+iy$ ααα $x,y\\in\\mathbb{R}$ ααΆα«αααΆαααα $i$ α ααΎαααΆα α\n$$\\begin{cases}x^{2}-y^{2}=0\\\\2xy=1\\end{cases}$$\nααα $xy>0$ αααΈααΆαααΈαα½αα±αα $x=y$ α αααα½αα
αΌααααΈααΆαααΈααΈα α $2x^2=1$ ααΆαα±αα ... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "αααααα
ααα½ααα·α $m$ ααΎααααΈα±ααα
ααα½ααα»ααααα·α
$z=(m-1)+7i$ ααΆααααΌαα»αααααΎααΉα $25$ α"
},
{
"from": "gpt",
"value": "ααΎαααΆα α\n$$|z|=\\sqrt{(m-1)^{2}+49}=25$$\nααΎαααΆααα’αααααΆααααΈα α\n$$(m-1)^{2}+49=625\\Leftrightarrow (m-1)^{2}=576$$\n$m-1=24$ α¬ $ m-1=-24$\nααΌα
ααα $\\boxed{m=... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα
ααα½ααα·α $x$ αα·α $y$ ααααααααααααΆαα $(x+iy)+2(x-iy)=9+4i$ α"
},
{
"from": "gpt",
"value": "αααααΆα α αΎαααΌααα½ααΌα
ααααΆ α\n$$(3)x-1yi=9+4i$$\nααΎαααααΎααααααααααααΎααααΆ α $a+bi=c+di$ αα»αααααΆαα $a=c$ αα·α $b=d$ α\n$$3x=9\\quad,\\quad -y=4$$\nααΆαα±αα $x=3$ αα·α $y=-4$ α\nααΌα
ααα ... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "α. αααααααΆααααΈααΆα $z^{2}+6z+10=0$ αααα»ααααα»αα
ααα½ααα»ααααα·α
α ααΆα $z_1$ ααΆα«ααααααΆαααααααα·αα·ααααα·αααααΆα αα·α $z_2$ ααΆα«ααα½αααα α\nα. ααα±αα $w=z_1+3i$ α ααααα $\\frac{1}{w}$ ααΆααααααααΈαααα·α α\nα. αααααΌαα»ααα $z_1\\times w$ α"
},
{
"from": "gpt",
"value": "α. ααΎαααΆα α\n$$\\D... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααα $(2-6i)(-6-3i)$ ααΆαααααα $a+bi$ ααα $a,b\\in\\mathbb{R}$ α"
},
{
"from": "gpt",
"value": "αααααΆααααα»α α\n$$(2-6i)(-6-3i)=-12-6i+36i+18i^{2}=-12+30i-18=-30+30i$$\nααΌα
ααα αααα»αααααΎααΉα $\\boxed{-30+30i}$ α"
}
] | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "αααααααΆααααΈααΆα $(1-3i)z=(2+i)z+(6+7i)$ αααα»α $\\mathbb{C}$ αα½α
αααααα
ααααΎαααΆααααααααΈαααα·α α"
},
{
"from": "gpt",
"value": "αααααΌααα½ααΆα $z$ α\n$$\\left[(1-3i)-(2+i)\\right]z=6+7i$$\n$$(-1-4i)z=6+7i$$\n$$z=\\frac{6+7i}{-1-4i}=\\frac{(6+7i)(-1+4i)}{(-1-4i)(-1+4i)}=\\frac... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα
ααα½ααα·α $x$ ααα $|x-(4-4i)|=|x-(3-3i)|$ α"
},
{
"from": "gpt",
"value": "ααΎααααααααΆαααααΎααααΆααααΆαααααΌαα»α α\n$$(x-4)^{2}+16=(x-3)^{2}+9$$\nαααααΆα α αΎααα»ααα½ $x^2$ α\n$$-8x+32=-6x+18$$\n$$-2x=-14$$\nααΌα
ααα $\\boxed{x=7}$ α"
}
] | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±ααααααΌα $S=i+i^{2}+i^{3}+\\cdots+i^{765}$ α ααααΆ $S$ α"
},
{
"from": "gpt",
"value": "ααααααα»ααα½ααα½ααΆααααααΆααΆαααααΌαααΌααα ααααα $i+i^2+i^3+i^4=i-1-i+1=0$ α\nααααΌααααααΆααα½α
ααα½α $765$ α ααα $765=4\\times 191+1$ αααα»ααα½ααα½ααΆαα $191$ ααΆαααααΌαααΌααα α αΎααα
ααα $1$ αα½α
α»αααα... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "αααααααΆααααααααααααΈααΆα α $$\\begin{cases}z+w=6-4i\\\\2iz-w=7+5i\\end{cases}$$"
},
{
"from": "gpt",
"value": "ααΌααααΈααΆαααΆααααΈα ααΎααααΈααααΆαα $w$ α\n$$(1+2i)z=13+i$$\n$$z=\\frac{13+i}{1+2i}=\\frac{(13+i)(1-2i)}{(1+2i)(1-2i)}=\\frac{15-25i}{1+4}=\\frac{15-25i}{5}=3-5i$$\... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα
ααα½ααα»ααααα·α
$z$ ααα $|z|=10$ αα·αααααααα·ααα $z$ ααααΎααΉα $6$ α"
},
{
"from": "gpt",
"value": "ααΆα $z=6+iy$ ααα $y\\in\\mathbb{R}$ α\n$$|z|=\\sqrt{36+y^{2}}=10\\Rightarrow y^{2}=64\\Rightarrow y=\\pm 8$$\nααΌα
ααα $\\boxed{z=6+8i}$ α¬ $\\boxed{z=6-8i}$ α"
}
] | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα
ααα½ααα»ααααα·α
$z$ ααα $z+|z|=25-5i$ α"
},
{
"from": "gpt",
"value": "ααΆα $z=x+iy$ α ααααααα·αα·αααααα’αααααΆααααΈαααααΎααααΆ ααααα $|z|$ ααΆα
ααα½ααα·α ααΌα
αααα $y=-5$ α\nααααααα·αααααα α\n$$x+\\sqrt{x^{2}+25}=25$$\n$$\\sqrt{x^{2}+25}=25-x$$\nαααααΎαααΆααα’αααααΆααααΈα (αααααααα $x\... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±αα $z=-1+i$ α ααααΆ $z+\\bar{z}$ , $z-\\bar{z}$ αα·α $z\\times\\bar{z}$ α"
},
{
"from": "gpt",
"value": "ααΎαααΆα $\\bar{z}=-1-i$ α\n$$z+\\bar{z}=(-1+i)-1-i=-2$$\n$$z-\\bar{z}=(-1+i)-(-1-i)=2i$$\n$$z\\times\\bar{z}=(-1)^{2}+1^{2}=2$$\nααΌα
ααα $\\boxed{z+\\bar{z}=-2}$ α... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααα $z=(-1-4i)i^{184}$ ααΆααααααααΈαααα·α α"
},
{
"from": "gpt",
"value": "ααα $184=4\\times 46+0$ ααα $i^{184}=1$ α\n$$z=-1-4i$$\nααΌα
ααα $\\boxed{z=-1-4i}$ α"
}
] | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "αααααααΆααααΈααΆα $z^{2}-8z+20=0$ αααα»ααααα»αα
ααα½ααα»ααααα·α
α"
},
{
"from": "gpt",
"value": "αααα»α $b=-8$ ααΆα
ααα½αααΌ ααΌα
ααααααΎαααααΎ $b'=-4$ α\n$$\\Delta'=b'^{2}-ac=(-4)^{2}-20=-4=(2i)^{2}$$\nα«αααΆααααΈαααΊ α\n$$z_1=4+2i\\quad,\\quad z_2=4-2i$$\nααΌα
ααα αααα»αα«αααΊ $\\boxed{\\lef... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααΆααααΈααΆα $(E)$ α $z^{2}-(5-5i)z-2-14i=0$ α\nα‘. ααααΆ $(3+i)^{2}$ α\nα’. ααααΆααΈαααααΈααΈααααααααΈααΆα $(E)$ α\nα£. αααααααΆααααΈααΆα $(E)$ αααα»α $\\mathbb{C}$ α\nα€. αααααααααΆααααααΌαα«α αα·ααααα»αα«ααααααΈααΆα $(E)$ α"
},
{
"from": "gpt",
"value": "α‘. αααααΆαααΆαα α\n$$(3+i)^{2}=9+6i... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±αα $z=4i(-2+2i)-5+3i$ α ααααα $z$ ααΆααααααααΈαααα·α αα½α
αααααΆααααααααα·α αα·αααααααα·αα·αααααααααΆ α"
},
{
"from": "gpt",
"value": "ααΆα $i^2=-1$ ααΎαααΆα α\n$$z=-8i+8i^{2}-5+3i=-8i-8-5+3i=-13-5i$$\nααΌα
ααα $z=-13-5i$ ααΆαααααααα·α $\\boxed{-13}$ αα·αααααααα·αα·ααα $\\boxed{-5}$ α... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±ααα
ααα½ααα»ααααα·α
$z_1=8-i$ αα·α $z_2=2+i$ α\na. αα $\\overline{z_1}$ , $\\overline{z_2}$ αα½α
ααααΆ $z_1+\\overline{z_2}$ α\nb. ααααΆ $z_1\\times\\overline{z_2}$ αα·α $z_2^2$ α\nc. αααα αΆαααΆ $\\frac{z_1}{z_2}=\\frac{z_1\\times\\overline{z_2}}{|z_2|^2}$ αα½α
ααΆαααααααααααΈαααα·ααα $\\frac... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±ααα
ααα½ααα»ααααα·α
$z_1=-2+5i$ αα·α $z_2=1-4i$ α ααααΆ $\\frac{z_1+z_2}{z_1-z_2}$ α"
},
{
"from": "gpt",
"value": "αα»αααααΌα $z_1+z_2=-1+i$ αα·α $z_1-z_2=-3+9i$ α\n$$\\frac{z_1+z_2}{z_1-z_2}=\\frac{-1+i}{-3+9i}=\\frac{(-1+i)(-3-9i)}{(-3+9i)(-3-9i)}=\\frac{12+6i}{9+81}=\\... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα
ααα½ααα·α $x,y$ αααααααΎα±αα $\\frac{x}{3+i}+\\frac{y}{1+i}=4$ α"
},
{
"from": "gpt",
"value": "ααΎαααααααααααΆαααΈαα½ααααΆααααααααΈαααα·α α\n$$\\frac{x}{3+i}=\\frac{x(3-i)}{10}\\quad,\\quad\\frac{y}{1+i}=\\frac{y(1-i)}{2}$$\nα
ααα½ααα»ααααα·α
ααΈαααααΎααααΆ αα»αααααΆααααααααα·αααααΎααα... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "a. αααααααΆααααΈααΆα $z^{2}+6z+10=0$ αααα»ααααα»αα
ααα½ααα»ααααα·α
α ααΆα $z_1$ ααΆα«ααααααΆαααααααα·αα·ααααα·αααααΆα αα·α $z_2$ ααΆα«ααα½αααα α\nb. ααααα $z_1^2$ αα·α $\\frac{z_1}{z_2}$ ααΆααααααααΈαααα·α α\nc. αα $|z_1|$ αα½α
αααα αΆαααΆ $z_1\\times z_2=|z_1|^2$ α"
},
{
"from": "gpt",
"value": ... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααΆαααα»αααα
ααα½ααα»ααααα·α
α $(-1-i)(6-5i)$ α"
},
{
"from": "gpt",
"value": "ααΎααα»ααα½ααΈαα½ααααααααααα
αααΈαα½αααΉαααααααα
αααΈααΈα α\n$$(-1-i)(6-5i)=-6+5i-6i+5i^{2}=-6-i-5=-11-i$$\nααΌα
ααα αααα»αααααΎααΉα $\\boxed{-11-i}$ α"
}
] | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "αααα αΆαααΆ $(1+i)^{2}+(1-i)^{2}$ ααΆα
ααα½ααα·α αα½α
ααααΆαααααααααααΆ α"
},
{
"from": "gpt",
"value": "ααα $(1+i)^2=2i$ αα·α $(1-i)^2=-2i$ ααΎαααΆα α\n$$(1+i)^{2}=(2i)^{1}=2i$$\n$$(1-i)^{2}=(-2i)^{1}=-2i$$\n$$S=0$$\nααΌα
ααα $\\boxed{S=0}$ ααΆα
ααα½ααα·α α"
}
] | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±αααα α»ααΆ $P(z)=z^{3}-5z^{2}+8z-6$ α αααααΉαααΆ $z_0=1+i$ ααΆα«ααα $P(z)=0$ αααααααΆααααΈααΆα $P(z)=0$ αααα»α $\\mathbb{C}$ α"
},
{
"from": "gpt",
"value": "ααααααα»αααΆα
ααα½ααα·α α αΎα $1+i$ ααΆα«α ααα $1-i$ ααααΆα«αααα α\nαα α»ααΆα
ααααΆα
αααΉα α\n$$(z-(1+i))(z-(1-i))=z^{2}-2z+2$$\nα
αααα α»... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±αα $z=-2+2i$ α ααααα $\\frac{1}{z}$ ααΆααααααααΈαααα·α α"
},
{
"from": "gpt",
"value": "ααα $z\\bar{z}=|z|^2=8$ ααΎαααΆα $\\frac{1}{z}=\\frac{\\bar{z}}{|z|^2}$ α\n$$\\frac{1}{z}=\\frac{-2-2i}{8}=-\\frac{1}{4}-\\frac{1}{4}i$$\nααΌα
ααα $\\boxed{\\frac{1}{z}=-\\frac{1}{4}-... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααΆαα
ααα½ααα»ααααα·α
$z_1=1+3i$ αα·α $z_2=-4-i$ α αααααααααΆααααΆ $|z_1\\times z_2|=|z_1|\\times|z_2|$ α"
},
{
"from": "gpt",
"value": "ααααΆα’αααααΆαααααα α\n$$z_1\\times z_2=-1-13i$$\n$$|z_1\\times z_2|=\\sqrt{1+169}=\\sqrt{170}\\quad(1)$$\nααααΆα’αααααΆαααααΆα α\n$$|z_1|\\tim... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααΎ $-1+2i$ ααΆα«ααααααΈααΆα $z^{2}+(2-6i)z-4-6i=0$ α¬αα ? αααααα α"
},
{
"from": "gpt",
"value": "αααα½α $z=-1+2i$ αααα»αα’αααααΆαααααα ααΎαααΆαααααα α\n$$(-1+2i)^{2}+(2-6i)(-1+2i)-4-6i=3$$\nαααααααααα»αααΈααΌααα α\nααΌα
ααα $-1+2i$ αα·ααααααΆα«ααααααΈααΆααα α"
}
] | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααΎ $-2-i$ ααΆα«ααααααΈααΆα $z^{2}+(4-2i)z+7-4i=0$ α¬αα ?"
},
{
"from": "gpt",
"value": "αααα½α $z=-2-i$ α\n$$(-2-i)^{2}+(4-2i)(-2-i)+(7-4i)=(3+4i)-10+(7-4i)=0$$\nααΌα
ααα $-2-i$ ααΆα«ααααααΈααΆα α"
}
] | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±αα $P(z)=z^{3}-4z^{2}+6z-4$ α ααααΆ $P(2)$ αα½α
ααΆαα $P(z)$ ααΆαααααΆ α αΎααααααααΆα $P(z)=0$ α"
},
{
"from": "gpt",
"value": "αααα½α $z=2$ α $8-16+12-4=0$ α ααΌα
αααα $z=2$ ααΆα«α α αΎα $P(z)$ α
ααααΆα
αααΉα $z-2$ α\nα
αααα α»ααΆ ααΎαααΆα α\n$$P(z)=(z-2)(z^{2}-2z+2)$$\nαααααααΆα $z^{2}-2z... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±αααα α»ααΆ $P(z)=z^{3}+8z^{2}+37z+50$ α αααααΉαααΆ $z_0=-3+4i$ ααΆα«ααα $P(z)=0$ αααααααΆααααΈααΆα $P(z)=0$ αααα»α $\\mathbb{C}$ α"
},
{
"from": "gpt",
"value": "ααααααα»αααΆα
ααα½ααα·α α αΎα $-3+4i$ ααΆα«α ααα $-3-4i$ ααααΆα«αααα α\nαα α»ααΆα
ααααΆα
αααΉα α\n$$(z+3-4i)(z+3+4i)=z^{2}+6z+25$$\... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "α. αααααααΆααααααααααααΈααΆαααΆαααααααααα»ααααα»αα
ααα½ααα»ααααα·α
α\n$$\\begin{cases}z_1+iz_2=-1\\\\2z_1-z_2=3-5i\\end{cases}$$\nα. ααααΆ $|z_1|$ αα·α $|z_2|$ α\nα. ααααα $z_1\\times\\overline{z_2}$ ααΆααααααααΈαααα·α α\nα. ααΆααα $\\frac{z_1}{z_2}$ αααααααΎ $\\frac{z_1}{z_2}=\\frac{z_1\\times\... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααΆαα
ααα½ααα»ααααα·α
$z_1=-16-2i$ αα·α $z_2=2-3i$ α\nα- ααααΆ $2z_1-z_2$ α\nα- ααααα $Z=\\frac{z_1+z_2}{z_2}$ ααΆααααααααΈαααα·α α\nα- αααααααααΆααααΆ $Z=\\frac{z_1}{z_2}+1$ α\nα- ααααααααα·α αα·αααααααα·αα·ααααα $Z^2$ α"
},
{
"from": "gpt",
"value": "α- ααΎαααΆα α\n$$2z_1-z_2=(-32... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααΆαα
ααα½ααα»ααααα·α
$z_1=3+4i$ αα·α $z_2=3+i$ α ααααΆ $z_1\\times z_2$ α"
},
{
"from": "gpt",
"value": "ααΎααα»ααα½ααΈαα½ααααααααααα
αααΈαα½αααΉαααααααα
αααΈααΈα α\n$$(3+4i)(3+i)=9+3i+12i+4i^{2}=9+15i-4=5+15i$$\nααΌα
ααα αααα»αααααΎααΉα $\\boxed{5+15i}$ α"
}
] | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "αααα αΆαααΆ $2$ ααΆα«ααααααΈααΆα $z^{3}-6z^{2}+21z-26=0$ αα½α
ααα«αααΆααα’αααααα»α $\\mathbb{C}$ α"
},
{
"from": "gpt",
"value": "ααΆα $P(z)=z^{3}-6z^{2}+21z-26$ α αααα½α $z=2$ α $8-24+42-26=0$ α ααΌα
αααα $z=2$ ααΆα«α α αΎα $P(z)$ α
ααααΆα
αααΉα $z-2$ α\nα
αααα α»ααΆ ααΎαααΆα α\n$$P(z)=(z-2)(z^{2... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααα±αααα α»ααΆ $P(z)=z^{3}-z^{2}-4z-6$ α αααααΉαααΆ $z_0=-1+i$ ααΆα«ααα $P(z)=0$ αααααααΆααααΈααΆα $P(z)=0$ αααα»α $\\mathbb{C}$ α"
},
{
"from": "gpt",
"value": "ααααααα»αααΆα
ααα½ααα·α α αΎα $-1+i$ ααΆα«α ααα $-1-i$ ααααΆα«αααα α\nαα α»ααΆα
ααααΆα
αααΉα α\n$$(z+1-i)(z+1+i)=z^{2}+2z+2$$\nα
αααα α»ααΆ... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααΆα $z_1$ , $z_2$ ααΆα«ααααααααΈααΆα $z^{2}+8z+20=0$ αααα»α $\\mathbb{C}$ α ααααΆ $\\frac{z_1}{z_2}+\\frac{z_2}{z_1}$ α"
},
{
"from": "gpt",
"value": "ααααα·αα
αΆαααΆα
αααα«α ααΎαααααΎααααΌαα«α αα·ααααα»αα«ααααααΈααΆαααΊααααααΈααΈα α\n$$z_1+z_2=-8\\quad,\\quad z_1z_2=20$$\n$$\\frac{z_1}{z_2... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααα $\\frac{-5+6i\\sqrt{3}}{2-i\\sqrt{3}}$ ααΆααααααααΈαααα·α α"
},
{
"from": "gpt",
"value": "αα»αααΆααα αα·αααΆααααααΉα $2+i\\sqrt{3}$ α ααΆααααααααΆαααΆ $4+3=7$ α\n$$\\frac{-5+6i\\sqrt{3}}{2-i\\sqrt{3}}=\\frac{(-5+6i\\sqrt{3})(2+i\\sqrt{3})}{7}=\\frac{-28+7i\\sqrt{3}}{7}=-... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααΎ $3-2i$ ααΆα«ααααααΈααΆα $z^{2}-(6-5i)z+4-15i=0$ α¬αα ? αααααα α"
},
{
"from": "gpt",
"value": "αααα½α $z=3-2i$ αααα»αα’αααααΆαααααα ααΎαααΆαααααα α\n$$(3-2i)^{2}-(6-5i)(3-2i)+(4-15i)=1$$\nαααααααααα»αααΈααΌααα α\nααΌα
ααα $3-2i$ αα·ααααααΆα«ααααααΈααΆααα α"
}
] | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "ααααα½ααααααα $A=-2i^{260}-i^{73}-2i^{93}$ α"
},
{
"from": "gpt",
"value": "α
αααα·αααααααααΈαα½ααααΉα $4$ α\n$260=4\\times 65+0$ ααΆαα±αα $i^{260}=1$\n$73=4\\times 18+1$ ααΆαα±αα $i^{73}=i$\n$93=4\\times 23+1$ ααΆαα±αα $i^{93}=i$\nααΎαααΆα α\n$$A=-2(1)-i-2(i)=-2-3i$$\nααΌα
ααα $\\b... | BacII-Math-AI/complex |
[
{
"from": "human",
"value": "αααααα
ααα½ααα»ααααα·α
ααααΆααααα
ααα½ααα»ααααα·α
ααΈαα½αα α $z_1=1+i\\sqrt{3}$ , $z_2=6i$ , $z_3=-9+i$ , $z_4=-5$ α"
},
{
"from": "gpt",
"value": "α
ααα½ααα»ααααα·α
ααααΆαααα $a+bi$ ααΊ $a-bi$ ααΊααΎαααααΌααααααΆααααααα·αα·αααααααα»ααααα α\n$$\\overline{z_1}=1-i\\sqrt{3}$$\n$$\\overline{z_2}=... | BacII-Math-AI/complex |
End of preview. Expand in Data Studio
BacII Math 108K (FineTome Format)
This dataset is a conversational reformation of the Chandaro/BacII-Math-AI open Khmer math dataset into the ShareGPT / FineTome conversational format.
Every answer was verified and computed via SymPy, containing step-by-step worked solutions for Cambodia's Grade 12 National Exam (BacII) in Khmer with LaTeX math.
Dataset Structure
The dataset adheres to the FineTome structure (without score):
conversations: A list of turn dictionaries with:from:"human"(question / prompt) or"gpt"(solution / answer)value: message text in Khmer with LaTeX notation
source: topic identifier (e.g.BacII-Math-AI/complex,BacII-Math-AI/derivative, etc.)
Topics Breakdown (108,332 Total Pairs)
| Chapter | Source Tag | Number of Examples |
|---|---|---|
| Derivatives | BacII-Math-AI/derivative |
15,937 |
| Complex Numbers | BacII-Math-AI/complex |
9,914 |
| Conic Sections | BacII-Math-AI/conic |
7,843 |
| Differential Equations | BacII-Math-AI/diffeq |
13,201 |
| Integrals | BacII-Math-AI/integral |
17,984 |
| Limits | BacII-Math-AI/limit |
8,969 |
| Probability | BacII-Math-AI/probability |
19,684 |
| Solid Geometry (Space) | BacII-Math-AI/space |
14,800 |
| Total | 108,332 |
Original Source & Attribution
- Original project: Chandaro/BacII-Math-AI
- License: MIT License (Copyright 2026 Romduol)
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