{
  "api_version": "v1",
  "publisher": {
    "name": "Manazoo",
    "url": "https://manazoo.com/"
  },
  "license": "All rights reserved（AI の参照・要約と出典リンクつきの引用は可。転載・再配布・販売は不可）",
  "terms_url": "https://manazoo.com/terms/",
  "disclaimer": "内容の正確さには気をつけていますが、すべてが正しいとは限りません。テストや受験の前には、教科書や先生の説明でも確かめてください。間違いを見つけたら意見箱から知らせてください。",
  "attribution": "事実データの一部は Wikidata（CC0）による。単元の構成は文部科学省「中学校学習指導要領」「高等学校学習指導要領」（政府標準利用規約2.0）。",
  "kind": "quiz",
  "unit": {
    "slug": "complex-plane",
    "title": "複素数平面",
    "url": "https://manazoo.com/high/math-c/complex-plane/"
  },
  "canonical_url": "https://manazoo.com/high/math-c/complex-plane/quiz/",
  "questions": [
    {
      "id": "q-cnf-circle-equation-choice",
      "type": "choice",
      "prompt": "方程式|z－(1＋i)|＝3はどのような図形を表すか。",
      "choices": [
        "直線",
        "中心3、半径1＋iの円",
        "中心0、半径3の円",
        "中心1＋i、半径3の円"
      ],
      "answer": 3,
      "accept": [],
      "explanation": "|z－α|＝rは中心α、半径rの円を表す。α＝1＋i、r＝3なので中心1＋i、半径3の円になる。",
      "entities": [
        "complex-numbers-and-figures"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-cnf-distance-two-points",
      "type": "input",
      "prompt": "z1＝1＋2i、z2＝4＋6iのとき、2点間の距離を求めよ。",
      "choices": null,
      "answer": "5",
      "accept": [],
      "explanation": "距離は|z2－z1|で求まる。z2－z1＝3＋4iなので、|3＋4i|＝√(3²＋4²)＝5。",
      "entities": [
        "complex-numbers-and-figures"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-cnf-internal-division-real-part",
      "type": "input",
      "prompt": "z1＝2＋i、z2＝8＋7iを結ぶ線分を2:1に内分する点の実部を求めよ。",
      "choices": null,
      "answer": "6",
      "accept": [],
      "explanation": "内分点はw＝(n z₁＋m z₂)／(m＋n)。実部は(1×2＋2×8)／3＝18／3＝6。",
      "entities": [
        "complex-numbers-and-figures"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-cnf-rotation-90-choice",
      "type": "choice",
      "prompt": "点z＝2を原点のまわりに90°回転した点はどれか。",
      "choices": [
        "2i",
        "－2",
        "－2i",
        "2"
      ],
      "answer": 0,
      "accept": [],
      "explanation": "90°回転はzに(cos90°＋i sin90°)＝iを掛ける。2×i＝2iになる。",
      "entities": [
        "complex-numbers-and-figures"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-cpc-abs-value-3-4i",
      "type": "input",
      "prompt": "複素数z＝3＋4iの絶対値を求めよ。",
      "choices": null,
      "answer": "5",
      "accept": [],
      "explanation": "絶対値は|z|＝√(a²＋b²)で求まる。|z|＝√(3²＋4²)＝√25＝5。",
      "entities": [
        "complex-plane-and-conjugate"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-cpc-conjugate-choice",
      "type": "choice",
      "prompt": "2－5iの共役複素数はどれか。",
      "choices": [
        "2＋5i",
        "－2－5i",
        "2－5i",
        "－2＋5i"
      ],
      "answer": 0,
      "accept": [],
      "explanation": "共役複素数は実部をそのまま、虚部の符号を変える。2－5iの共役複素数は2＋5iだ。",
      "entities": [
        "complex-plane-and-conjugate"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-cpc-real-multiple-abs",
      "type": "input",
      "prompt": "|z|＝6のとき、|－3z|を求めよ。",
      "choices": null,
      "answer": "18",
      "accept": [],
      "explanation": "実数k倍の絶対値は|k|倍になる。|－3z|＝|－3|×|z|＝3×6＝18。",
      "entities": [
        "complex-plane-and-conjugate"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-cpc-sum-real-part",
      "type": "input",
      "prompt": "z1＝2＋3i、z2＝1－4iのとき、z1＋z2の実部を求めよ。",
      "choices": null,
      "answer": "3",
      "accept": [],
      "explanation": "和の実部は実部どうしを足す。2＋1＝3。",
      "entities": [
        "complex-plane-and-conjugate"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-cpf-abs-value-sqrt3-i",
      "type": "input",
      "prompt": "z＝√3＋iの絶対値を求めよ。",
      "choices": null,
      "answer": "2",
      "accept": [],
      "explanation": "絶対値はr＝√(a²＋b²)で求まる。r＝√((√3)²＋1²)＝√4＝2。",
      "entities": [
        "complex-polar-form"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-cpf-argument-sqrt3-i",
      "type": "input",
      "prompt": "z＝√3＋iの偏角を求めよ（0°以上360°未満）。",
      "choices": null,
      "answer": "30°",
      "accept": [],
      "explanation": "a＝√3、b＝1なのでtanθ＝1／√3。実軸から反時計回りに30°の位置にある。",
      "entities": [
        "complex-polar-form"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-cpf-multiply-by-i-choice",
      "type": "choice",
      "prompt": "複素数zにiをかけることは、何をすることと同じか。",
      "choices": [
        "原点のまわりに90°回転させる",
        "絶対値を2倍にする",
        "共役複素数にする",
        "実部と虚部を入れ替える"
      ],
      "answer": 0,
      "accept": [],
      "explanation": "iは絶対値1・偏角90°の複素数なので、iをかけると偏角が90°増える。つまり原点のまわりに90°回転する。",
      "entities": [
        "complex-polar-form"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-cpf-product-argument",
      "type": "input",
      "prompt": "絶対値2・偏角40°のz1と、絶対値3・偏角20°のz2がある。積z1z2の偏角を求めよ。",
      "choices": null,
      "answer": "60°",
      "accept": [],
      "explanation": "積の偏角はθ1＋θ2になる。40°＋20°＝60°。",
      "entities": [
        "complex-polar-form"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-dmt-hexagon-choice",
      "type": "choice",
      "prompt": "1の6乗根が単位円周上に作る図形はどれか。",
      "choices": [
        "正五角形",
        "正方形",
        "正三角形",
        "正六角形"
      ],
      "answer": 3,
      "accept": [],
      "explanation": "1のn乗根は単位円周上に等間隔で並ぶ正n角形の頂点になる。n＝6なので正六角形だ。",
      "entities": [
        "de-moivres-theorem"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-dmt-nth-root-argument",
      "type": "input",
      "prompt": "1の5乗根のうち、k＝2に対応する解の偏角を求めよ（0°以上360°未満）。",
      "choices": null,
      "answer": "144°",
      "accept": [],
      "explanation": "θ＝360°／n×kに当てはめる。360°／5×2＝144°。",
      "entities": [
        "de-moivres-theorem"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-dmt-power-abs-value",
      "type": "input",
      "prompt": "絶対値2、偏角20°の複素数を5乗したときの絶対値を求めよ。",
      "choices": null,
      "answer": "32",
      "accept": [],
      "explanation": "n乗の絶対値は元の絶対値のn乗になる。2⁵＝32。",
      "entities": [
        "de-moivres-theorem"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-dmt-power-argument",
      "type": "input",
      "prompt": "(cos30°＋i sin30°)⁴の偏角を求めよ（0°以上360°未満）。",
      "choices": null,
      "answer": "120°",
      "accept": [],
      "explanation": "ド・モアブルの定理より偏角はn倍になる。30°×4＝120°。",
      "entities": [
        "de-moivres-theorem"
      ],
      "verified": true,
      "placeholder": false
    }
  ]
}