{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0001","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(0, 0)、平移向量 t=[0, 0]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"像點=(0, 0)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077001,"parameters":{"P":[0,0],"t":[0,0],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"0\",\"0\"],[\"0\",\"0\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0002","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(-11, 0)、平移向量 t=[0, 0]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(-11, 0)","explanation_zh":"像點=(-11, 0)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077002,"parameters":{"P":[-11,0],"t":[0,0],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"-11\",\"0\"],[\"0\",\"0\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0003","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(-10, -1)、平移向量 t=[1, 2]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(-9, 1)","explanation_zh":"像點=(-9, 1)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077003,"parameters":{"P":[-10,-1],"t":[1,2],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"-10\",\"-1\"],[\"1\",\"2\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0004","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(-9, -2)、平移向量 t=[-1, -1]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(-10, -3)","explanation_zh":"像點=(-10, -3)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077004,"parameters":{"P":[-9,-2],"t":[-1,-1],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"-9\",\"-2\"],[\"-1\",\"-1\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0005","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(-8, -3)、平移向量 t=[0, 1]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(-8, -2)","explanation_zh":"像點=(-8, -2)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077005,"parameters":{"P":[-8,-3],"t":[0,1],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"-8\",\"-3\"],[\"0\",\"1\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0006","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(-7, -4)、平移向量 t=[1, -2]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(-6, -6)","explanation_zh":"像點=(-6, -6)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077006,"parameters":{"P":[-7,-4],"t":[1,-2],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"-7\",\"-4\"],[\"1\",\"-2\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0007","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(-6, -5)、平移向量 t=[-1, 0]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(-7, -5)","explanation_zh":"像點=(-7, -5)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077007,"parameters":{"P":[-6,-5],"t":[-1,0],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"-6\",\"-5\"],[\"-1\",\"0\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0008","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(-5, -6)、平移向量 t=[0, 2]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(-5, -4)","explanation_zh":"像點=(-5, -4)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077008,"parameters":{"P":[-5,-6],"t":[0,2],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"-5\",\"-6\"],[\"0\",\"2\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0009","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(-4, -7)、平移向量 t=[1, -1]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(-3, -8)","explanation_zh":"像點=(-3, -8)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077009,"parameters":{"P":[-4,-7],"t":[1,-1],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"-4\",\"-7\"],[\"1\",\"-1\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0010","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(-3, -8)、平移向量 t=[-1, 1]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(-4, -7)","explanation_zh":"像點=(-4, -7)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077010,"parameters":{"P":[-3,-8],"t":[-1,1],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"-3\",\"-8\"],[\"-1\",\"1\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0011","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(-2, -9)、平移向量 t=[0, -2]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(-2, -11)","explanation_zh":"像點=(-2, -11)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077011,"parameters":{"P":[-2,-9],"t":[0,-2],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"-2\",\"-9\"],[\"0\",\"-2\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0012","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(-1, -10)、平移向量 t=[1, 0]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(0, -10)","explanation_zh":"像點=(0, -10)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077012,"parameters":{"P":[-1,-10],"t":[1,0],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"-1\",\"-10\"],[\"1\",\"0\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0013","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(0, -11)、平移向量 t=[-1, 2]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(-1, -9)","explanation_zh":"像點=(-1, -9)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077013,"parameters":{"P":[0,-11],"t":[-1,2],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"0\",\"-11\"],[\"-1\",\"2\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0014","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(1, -12)、平移向量 t=[0, -1]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(1, -13)","explanation_zh":"像點=(1, -13)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077014,"parameters":{"P":[1,-12],"t":[0,-1],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"1\",\"-12\"],[\"0\",\"-1\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0015","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(2, -13)、平移向量 t=[1, 1]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(3, -12)","explanation_zh":"像點=(3, -12)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077015,"parameters":{"P":[2,-13],"t":[1,1],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"2\",\"-13\"],[\"1\",\"1\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0016","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(3, -14)、平移向量 t=[-1, -2]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(2, -16)","explanation_zh":"像點=(2, -16)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077016,"parameters":{"P":[3,-14],"t":[-1,-2],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"3\",\"-14\"],[\"-1\",\"-2\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0017","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(4, -15)、平移向量 t=[0, 0]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(4, -15)","explanation_zh":"像點=(4, -15)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077017,"parameters":{"P":[4,-15],"t":[0,0],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"4\",\"-15\"],[\"0\",\"0\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0018","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(5, -16)、平移向量 t=[1, 2]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(6, -14)","explanation_zh":"像點=(6, -14)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077018,"parameters":{"P":[5,-16],"t":[1,2],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"5\",\"-16\"],[\"1\",\"2\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0019","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(6, -17)、平移向量 t=[-1, -1]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(5, -18)","explanation_zh":"像點=(5, -18)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077019,"parameters":{"P":[6,-17],"t":[-1,-1],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"6\",\"-17\"],[\"-1\",\"-1\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0020","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(7, -18)、平移向量 t=[0, 1]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(7, -17)","explanation_zh":"像點=(7, -17)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077020,"parameters":{"P":[7,-18],"t":[0,1],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"7\",\"-18\"],[\"0\",\"1\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0021","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(8, -19)、平移向量 t=[1, -2]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(9, -21)","explanation_zh":"像點=(9, -21)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077021,"parameters":{"P":[8,-19],"t":[1,-2],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"8\",\"-19\"],[\"1\",\"-2\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0022","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(9, -20)、平移向量 t=[-1, 0]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(8, -20)","explanation_zh":"像點=(8, -20)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077022,"parameters":{"P":[9,-20],"t":[-1,0],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"9\",\"-20\"],[\"-1\",\"0\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0023","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(10, -21)、平移向量 t=[0, 2]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(10, -19)","explanation_zh":"像點=(10, -19)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077023,"parameters":{"P":[10,-21],"t":[0,2],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"10\",\"-21\"],[\"0\",\"2\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0024","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(11, -22)、平移向量 t=[1, -1]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(12, -23)","explanation_zh":"像點=(12, -23)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077024,"parameters":{"P":[11,-22],"t":[1,-1],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"11\",\"-22\"],[\"1\",\"-1\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0025","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"平移後的精確像點","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(12, -23)、平移向量 t=[-1, 1]。求 P+t 的點座標。","answer":{"kind":"expression","value":"(11, -22)","explanation_zh":"像點=(11, -22)。兩分量按明示向量相加，像點位置須保留，不能只算兩點間差。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["平移後的精確像點"]},"variant":{"family_id":"qf-transform-translation-point","seed":202610077025,"parameters":{"P":[12,-23],"t":[-1,1],"operation":"translation_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-translation-point/{\"inputs\":[[\"12\",\"-23\"],[\"-1\",\"1\"]],\"operation\":\"translation_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0026","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(0, 0) 逆時針轉 0 個90°，求像點座標。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"像點=(0, 0)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077026,"parameters":{"P":[0,0],"turns":0,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[\"origin\"],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0027","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(-11, 2) 逆時針轉 1 個90°，求像點座標。","answer":{"kind":"expression","value":"(-2, -11)","explanation_zh":"像點=(-2, -11)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077027,"parameters":{"P":[-11,2],"turns":1,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"-11\",\"2\"],1],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0028","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(-10, 3) 逆時針轉 2 個90°，求像點座標。","answer":{"kind":"expression","value":"(10, -3)","explanation_zh":"像點=(10, -3)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077028,"parameters":{"P":[-10,3],"turns":2,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"-10\",\"3\"],2],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0029","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(-9, 4) 逆時針轉 3 個90°，求像點座標。","answer":{"kind":"expression","value":"(4, 9)","explanation_zh":"像點=(4, 9)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077029,"parameters":{"P":[-9,4],"turns":3,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"-9\",\"4\"],3],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0030","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(-8, 5) 逆時針轉 0 個90°，求像點座標。","answer":{"kind":"expression","value":"(-8, 5)","explanation_zh":"像點=(-8, 5)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077030,"parameters":{"P":[-8,5],"turns":0,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"-8\",\"5\"],0],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0031","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(-7, 6) 逆時針轉 1 個90°，求像點座標。","answer":{"kind":"expression","value":"(-6, -7)","explanation_zh":"像點=(-6, -7)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077031,"parameters":{"P":[-7,6],"turns":1,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"-7\",\"6\"],1],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0032","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(-6, 7) 逆時針轉 2 個90°，求像點座標。","answer":{"kind":"expression","value":"(6, -7)","explanation_zh":"像點=(6, -7)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077032,"parameters":{"P":[-6,7],"turns":2,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"-6\",\"7\"],2],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0033","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(-5, 8) 逆時針轉 3 個90°，求像點座標。","answer":{"kind":"expression","value":"(8, 5)","explanation_zh":"像點=(8, 5)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077033,"parameters":{"P":[-5,8],"turns":3,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"-5\",\"8\"],3],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0034","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(-4, 9) 逆時針轉 0 個90°，求像點座標。","answer":{"kind":"expression","value":"(-4, 9)","explanation_zh":"像點=(-4, 9)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077034,"parameters":{"P":[-4,9],"turns":0,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"-4\",\"9\"],0],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0035","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(-3, 10) 逆時針轉 1 個90°，求像點座標。","answer":{"kind":"expression","value":"(-10, -3)","explanation_zh":"像點=(-10, -3)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077035,"parameters":{"P":[-3,10],"turns":1,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"-3\",\"10\"],1],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0036","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(-2, 11) 逆時針轉 2 個90°，求像點座標。","answer":{"kind":"expression","value":"(2, -11)","explanation_zh":"像點=(2, -11)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077036,"parameters":{"P":[-2,11],"turns":2,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"-2\",\"11\"],2],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0037","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(-1, 12) 逆時針轉 3 個90°，求像點座標。","answer":{"kind":"expression","value":"(12, 1)","explanation_zh":"像點=(12, 1)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077037,"parameters":{"P":[-1,12],"turns":3,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"-1\",\"12\"],3],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0038","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(0, 13) 逆時針轉 0 個90°，求像點座標。","answer":{"kind":"expression","value":"(0, 13)","explanation_zh":"像點=(0, 13)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077038,"parameters":{"P":[0,13],"turns":0,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"0\",\"13\"],0],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0039","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(1, 14) 逆時針轉 1 個90°，求像點座標。","answer":{"kind":"expression","value":"(-14, 1)","explanation_zh":"像點=(-14, 1)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077039,"parameters":{"P":[1,14],"turns":1,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"1\",\"14\"],1],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0040","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(2, 15) 逆時針轉 2 個90°，求像點座標。","answer":{"kind":"expression","value":"(-2, -15)","explanation_zh":"像點=(-2, -15)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077040,"parameters":{"P":[2,15],"turns":2,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"2\",\"15\"],2],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0041","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(3, 16) 逆時針轉 3 個90°，求像點座標。","answer":{"kind":"expression","value":"(16, -3)","explanation_zh":"像點=(16, -3)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077041,"parameters":{"P":[3,16],"turns":3,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"3\",\"16\"],3],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0042","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(4, 17) 逆時針轉 0 個90°，求像點座標。","answer":{"kind":"expression","value":"(4, 17)","explanation_zh":"像點=(4, 17)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077042,"parameters":{"P":[4,17],"turns":0,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"4\",\"17\"],0],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0043","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(5, 18) 逆時針轉 1 個90°，求像點座標。","answer":{"kind":"expression","value":"(-18, 5)","explanation_zh":"像點=(-18, 5)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077043,"parameters":{"P":[5,18],"turns":1,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"5\",\"18\"],1],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0044","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(6, 19) 逆時針轉 2 個90°，求像點座標。","answer":{"kind":"expression","value":"(-6, -19)","explanation_zh":"像點=(-6, -19)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077044,"parameters":{"P":[6,19],"turns":2,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"6\",\"19\"],2],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0045","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(7, 20) 逆時針轉 3 個90°，求像點座標。","answer":{"kind":"expression","value":"(20, -7)","explanation_zh":"像點=(20, -7)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077045,"parameters":{"P":[7,20],"turns":3,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"7\",\"20\"],3],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0046","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(8, 21) 逆時針轉 0 個90°，求像點座標。","answer":{"kind":"expression","value":"(8, 21)","explanation_zh":"像點=(8, 21)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077046,"parameters":{"P":[8,21],"turns":0,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"8\",\"21\"],0],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0047","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(9, 22) 逆時針轉 1 個90°，求像點座標。","answer":{"kind":"expression","value":"(-22, 9)","explanation_zh":"像點=(-22, 9)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077047,"parameters":{"P":[9,22],"turns":1,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"9\",\"22\"],1],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0048","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(10, 23) 逆時針轉 2 個90°，求像點座標。","answer":{"kind":"expression","value":"(-10, -23)","explanation_zh":"像點=(-10, -23)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077048,"parameters":{"P":[10,23],"turns":2,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"10\",\"23\"],2],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0049","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(11, 24) 逆時針轉 3 個90°，求像點座標。","answer":{"kind":"expression","value":"(24, -11)","explanation_zh":"像點=(24, -11)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077049,"parameters":{"P":[11,24],"turns":3,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"11\",\"24\"],3],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0050","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"原點逆時針四分之一轉","stem_zh":"等單位正交座標系，兩軸採同一長度單位。以原點為中心，P=(12, 25) 逆時針轉 0 個90°，求像點座標。","answer":{"kind":"expression","value":"(12, 25)","explanation_zh":"像點=(12, 25)。每一轉將 (x,y) 換成 (−y,x)，順序及中心明示；此數值不單獨證明全平面性質。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原點逆時針四分之一轉"]},"variant":{"family_id":"qf-transform-quarter-turn-point","seed":202610077050,"parameters":{"P":[12,25],"turns":0,"operation":"quarter_turn_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-quarter-turn-point/{\"inputs\":[[\"12\",\"25\"],0],\"operation\":\"quarter_turn_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0051","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(0, 0) 關於 x軸 反射，求像點座標。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"像點=(0, 0)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077051,"parameters":{"P":[0,0],"axis":"x","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[\"origin\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0052","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(-11, 3) 關於 y軸 反射，求像點座標。","answer":{"kind":"expression","value":"(11, 3)","explanation_zh":"像點=(11, 3)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077052,"parameters":{"P":[-11,3],"axis":"y","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"-11\",\"3\"],\"y\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0053","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(-10, 4) 關於 直線 y=x 反射，求像點座標。","answer":{"kind":"expression","value":"(4, -10)","explanation_zh":"像點=(4, -10)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077053,"parameters":{"P":[-10,4],"axis":"diagonal","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"-10\",\"4\"],\"diagonal\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0054","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(-9, 5) 關於 直線 y=−x 反射，求像點座標。","answer":{"kind":"expression","value":"(-5, 9)","explanation_zh":"像點=(-5, 9)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077054,"parameters":{"P":[-9,5],"axis":"anti_diagonal","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"-9\",\"5\"],\"anti_diagonal\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0055","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(-8, 6) 關於 x軸 反射，求像點座標。","answer":{"kind":"expression","value":"(-8, -6)","explanation_zh":"像點=(-8, -6)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077055,"parameters":{"P":[-8,6],"axis":"x","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"-8\",\"6\"],\"x\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0056","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(-7, 7) 關於 y軸 反射，求像點座標。","answer":{"kind":"expression","value":"(7, 7)","explanation_zh":"像點=(7, 7)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077056,"parameters":{"P":[-7,7],"axis":"y","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"-7\",\"7\"],\"y\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0057","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(-6, 8) 關於 直線 y=x 反射，求像點座標。","answer":{"kind":"expression","value":"(8, -6)","explanation_zh":"像點=(8, -6)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077057,"parameters":{"P":[-6,8],"axis":"diagonal","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"-6\",\"8\"],\"diagonal\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0058","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(-5, 9) 關於 直線 y=−x 反射，求像點座標。","answer":{"kind":"expression","value":"(-9, 5)","explanation_zh":"像點=(-9, 5)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077058,"parameters":{"P":[-5,9],"axis":"anti_diagonal","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"-5\",\"9\"],\"anti_diagonal\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0059","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(-4, 10) 關於 x軸 反射，求像點座標。","answer":{"kind":"expression","value":"(-4, -10)","explanation_zh":"像點=(-4, -10)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077059,"parameters":{"P":[-4,10],"axis":"x","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"-4\",\"10\"],\"x\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0060","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(-3, 11) 關於 y軸 反射，求像點座標。","answer":{"kind":"expression","value":"(3, 11)","explanation_zh":"像點=(3, 11)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077060,"parameters":{"P":[-3,11],"axis":"y","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"-3\",\"11\"],\"y\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0061","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(-2, 12) 關於 直線 y=x 反射，求像點座標。","answer":{"kind":"expression","value":"(12, -2)","explanation_zh":"像點=(12, -2)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077061,"parameters":{"P":[-2,12],"axis":"diagonal","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"-2\",\"12\"],\"diagonal\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0062","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(-1, 13) 關於 直線 y=−x 反射，求像點座標。","answer":{"kind":"expression","value":"(-13, 1)","explanation_zh":"像點=(-13, 1)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077062,"parameters":{"P":[-1,13],"axis":"anti_diagonal","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"-1\",\"13\"],\"anti_diagonal\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0063","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(0, 14) 關於 x軸 反射，求像點座標。","answer":{"kind":"expression","value":"(0, -14)","explanation_zh":"像點=(0, -14)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077063,"parameters":{"P":[0,14],"axis":"x","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"0\",\"14\"],\"x\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0064","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(1, 15) 關於 y軸 反射，求像點座標。","answer":{"kind":"expression","value":"(-1, 15)","explanation_zh":"像點=(-1, 15)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077064,"parameters":{"P":[1,15],"axis":"y","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"1\",\"15\"],\"y\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0065","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(2, 16) 關於 直線 y=x 反射，求像點座標。","answer":{"kind":"expression","value":"(16, 2)","explanation_zh":"像點=(16, 2)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077065,"parameters":{"P":[2,16],"axis":"diagonal","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"2\",\"16\"],\"diagonal\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0066","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(3, 17) 關於 直線 y=−x 反射，求像點座標。","answer":{"kind":"expression","value":"(-17, -3)","explanation_zh":"像點=(-17, -3)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077066,"parameters":{"P":[3,17],"axis":"anti_diagonal","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"3\",\"17\"],\"anti_diagonal\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0067","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(4, 18) 關於 x軸 反射，求像點座標。","answer":{"kind":"expression","value":"(4, -18)","explanation_zh":"像點=(4, -18)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077067,"parameters":{"P":[4,18],"axis":"x","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"4\",\"18\"],\"x\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0068","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(5, 19) 關於 y軸 反射，求像點座標。","answer":{"kind":"expression","value":"(-5, 19)","explanation_zh":"像點=(-5, 19)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077068,"parameters":{"P":[5,19],"axis":"y","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"5\",\"19\"],\"y\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0069","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(6, 20) 關於 直線 y=x 反射，求像點座標。","answer":{"kind":"expression","value":"(20, 6)","explanation_zh":"像點=(20, 6)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077069,"parameters":{"P":[6,20],"axis":"diagonal","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"6\",\"20\"],\"diagonal\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0070","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(7, 21) 關於 直線 y=−x 反射，求像點座標。","answer":{"kind":"expression","value":"(-21, -7)","explanation_zh":"像點=(-21, -7)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077070,"parameters":{"P":[7,21],"axis":"anti_diagonal","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"7\",\"21\"],\"anti_diagonal\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0071","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(8, 22) 關於 x軸 反射，求像點座標。","answer":{"kind":"expression","value":"(8, -22)","explanation_zh":"像點=(8, -22)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077071,"parameters":{"P":[8,22],"axis":"x","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"8\",\"22\"],\"x\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0072","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(9, 23) 關於 y軸 反射，求像點座標。","answer":{"kind":"expression","value":"(-9, 23)","explanation_zh":"像點=(-9, 23)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077072,"parameters":{"P":[9,23],"axis":"y","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"9\",\"23\"],\"y\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0073","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(10, 24) 關於 直線 y=x 反射，求像點座標。","answer":{"kind":"expression","value":"(24, 10)","explanation_zh":"像點=(24, 10)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077073,"parameters":{"P":[10,24],"axis":"diagonal","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"10\",\"24\"],\"diagonal\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0074","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(11, 25) 關於 直線 y=−x 反射，求像點座標。","answer":{"kind":"expression","value":"(-25, -11)","explanation_zh":"像點=(-25, -11)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077074,"parameters":{"P":[11,25],"axis":"anti_diagonal","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"11\",\"25\"],\"anti_diagonal\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0075","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"指定原點直線的反射像","stem_zh":"等單位正交座標系，兩軸採同一長度單位。P=(12, 26) 關於 x軸 反射，求像點座標。","answer":{"kind":"expression","value":"(12, -26)","explanation_zh":"像點=(12, -26)。按指定直線翻轉或交換座標；單軸反射不混同原點對稱。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["指定原點直線的反射像"]},"variant":{"family_id":"qf-transform-axis-reflection-point","seed":202610077075,"parameters":{"P":[12,26],"axis":"x","operation":"axis_reflection_point","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-axis-reflection-point/{\"inputs\":[[\"12\",\"26\"],\"x\"],\"operation\":\"axis_reflection_point\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0076","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=0P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"退化常值映射","explanation_zh":"全域係數=0；長度倍率=0。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077076,"parameters":{"c":0,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"0\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0077","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=-1P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"全平面等距映射","explanation_zh":"全域係數=-1；長度倍率=1。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077077,"parameters":{"c":-1,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"1\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0078","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=2P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"非等距相似映射；倍率=2","explanation_zh":"全域係數=2；長度倍率=2。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077078,"parameters":{"c":2,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"2\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0079","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=-3P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"非等距相似映射；倍率=3","explanation_zh":"全域係數=-3；長度倍率=3。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077079,"parameters":{"c":-3,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"3\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0080","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=4P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"非等距相似映射；倍率=4","explanation_zh":"全域係數=4；長度倍率=4。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077080,"parameters":{"c":4,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"4\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0081","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=-5P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"非等距相似映射；倍率=5","explanation_zh":"全域係數=-5；長度倍率=5。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077081,"parameters":{"c":-5,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"5\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0082","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=6P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"非等距相似映射；倍率=6","explanation_zh":"全域係數=6；長度倍率=6。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077082,"parameters":{"c":6,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"6\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0083","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=-7P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"非等距相似映射；倍率=7","explanation_zh":"全域係數=-7；長度倍率=7。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077083,"parameters":{"c":-7,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"7\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0084","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=8P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"非等距相似映射；倍率=8","explanation_zh":"全域係數=8；長度倍率=8。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077084,"parameters":{"c":8,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"8\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0085","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=-9P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"非等距相似映射；倍率=9","explanation_zh":"全域係數=-9；長度倍率=9。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077085,"parameters":{"c":-9,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"9\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0086","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=10P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"非等距相似映射；倍率=10","explanation_zh":"全域係數=10；長度倍率=10。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077086,"parameters":{"c":10,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"10\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0087","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=-11P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"非等距相似映射；倍率=11","explanation_zh":"全域係數=-11；長度倍率=11。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077087,"parameters":{"c":-11,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"11\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0088","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=12P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"非等距相似映射；倍率=12","explanation_zh":"全域係數=12；長度倍率=12。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077088,"parameters":{"c":12,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"12\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0089","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=-13P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"非等距相似映射；倍率=13","explanation_zh":"全域係數=-13；長度倍率=13。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077089,"parameters":{"c":-13,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"13\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0090","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=14P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"非等距相似映射；倍率=14","explanation_zh":"全域係數=14；長度倍率=14。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077090,"parameters":{"c":14,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"14\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0091","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=-15P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"非等距相似映射；倍率=15","explanation_zh":"全域係數=-15；長度倍率=15。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077091,"parameters":{"c":-15,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"15\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0092","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=16P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"非等距相似映射；倍率=16","explanation_zh":"全域係數=16；長度倍率=16。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077092,"parameters":{"c":16,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"16\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0093","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=-17P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"非等距相似映射；倍率=17","explanation_zh":"全域係數=-17；長度倍率=17。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077093,"parameters":{"c":-17,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"17\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0094","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=18P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"非等距相似映射；倍率=18","explanation_zh":"全域係數=18；長度倍率=18。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077094,"parameters":{"c":18,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"18\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0095","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=-19P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"非等距相似映射；倍率=19","explanation_zh":"全域係數=-19；長度倍率=19。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077095,"parameters":{"c":-19,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"19\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0096","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=20P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"非等距相似映射；倍率=20","explanation_zh":"全域係數=20；長度倍率=20。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077096,"parameters":{"c":20,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"20\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0097","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=-21P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"非等距相似映射；倍率=21","explanation_zh":"全域係數=-21；長度倍率=21。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077097,"parameters":{"c":-21,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"21\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0098","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=22P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"非等距相似映射；倍率=22","explanation_zh":"全域係數=22；長度倍率=22。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077098,"parameters":{"c":22,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"22\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0099","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=-23P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"非等距相似映射；倍率=23","explanation_zh":"全域係數=-23；長度倍率=23。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077099,"parameters":{"c":-23,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"23\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r07-geometric-transformations-0100","batch_id":"batch-20261006-advance-20261007-r07-geometric-transformations-foundations","language":"zh-Hant","source_mko_ids":["mko-geometric-transformations"],"title_zh":"全平面均勻縮放分類","stem_zh":"等單位正交座標系，兩軸採同一長度單位。全平面縮放分類：原點中心映射 H(P)=24P，對整個 R² 判為退化常值、等距或非等距相似，並保留非等距長度倍率。","answer":{"kind":"short_text","value":"非等距相似映射；倍率=24","explanation_zh":"全域係數=24；長度倍率=24。依 |c| 分類全域公式，不從一對點推出性質；c=−1 是180°原點旋轉而非單軸反射。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全平面均勻縮放分類"]},"variant":{"family_id":"qf-transform-uniform-scale-class","seed":202610077100,"parameters":{"c":24,"operation":"uniform_scale_class","frame":"euclidean_orthonormal_equal_units","semantic_template":"qf-transform-uniform-scale-class/{\"inputs\":[\"24\"],\"operation\":\"uniform_scale_class\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}