{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r04-eigenvalue-0001","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[2, 1], [0, 3]]。候選 λ=99、v=[0, 0] 是否為特徵對？","answer":{"kind":"short_text","value":"否","explanation_zh":"Av=[0, 0]；λv=[0, 0]；非零向量=否。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074001,"parameters":{"M":[[2,1],[0,3]],"lambda":99,"v":[0,0],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[\"zero_vector\"],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0002","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[-11, 0], [0, 3]]。候選 λ=3、v=[0, 1] 是否為特徵對？","answer":{"kind":"short_text","value":"是","explanation_zh":"Av=[0, 3]；λv=[0, 3]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074002,"parameters":{"M":[[-11,0],[0,3]],"lambda":3,"v":[0,1],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"-11\",\"0\"],[\"0\",\"3\"]],\"3\",[\"0\",\"1\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0003","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[-10, 0], [0, 4]]。候選 λ=-10、v=[1, 1] 是否為特徵對？","answer":{"kind":"short_text","value":"否","explanation_zh":"Av=[-10, 4]；λv=[-10, -10]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074003,"parameters":{"M":[[-10,0],[0,4]],"lambda":-10,"v":[1,1],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"-10\",\"0\"],[\"0\",\"4\"]],\"-10\",[\"1\",\"1\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0004","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[-9, 0], [0, 5]]。候選 λ=-8、v=[1, -1] 是否為特徵對？","answer":{"kind":"short_text","value":"否","explanation_zh":"Av=[-9, -5]；λv=[-8, 8]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074004,"parameters":{"M":[[-9,0],[0,5]],"lambda":-8,"v":[1,-1],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"-9\",\"0\"],[\"0\",\"5\"]],\"-8\",[\"1\",\"-1\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0005","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[4, 1], [1, 4]]。候選 λ=5、v=[1, 1] 是否為特徵對？","answer":{"kind":"short_text","value":"是","explanation_zh":"Av=[5, 5]；λv=[5, 5]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074005,"parameters":{"M":[[4,1],[1,4]],"lambda":5,"v":[1,1],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"4\",\"1\"],[\"1\",\"4\"]],\"5\",[\"1\",\"1\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0006","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[5, 0], [1, 6]]。候選 λ=5、v=[1, -1] 是否為特徵對？","answer":{"kind":"short_text","value":"是","explanation_zh":"Av=[5, -5]；λv=[5, -5]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074006,"parameters":{"M":[[5,0],[1,6]],"lambda":5,"v":[1,-1],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"5\",\"0\"],[\"1\",\"6\"]],\"5\",[\"1\",\"-1\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0007","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[-6, 0], [0, 8]]。候選 λ=-6、v=[1, 1] 是否為特徵對？","answer":{"kind":"short_text","value":"否","explanation_zh":"Av=[-6, 8]；λv=[-6, -6]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074007,"parameters":{"M":[[-6,0],[0,8]],"lambda":-6,"v":[1,1],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"-6\",\"0\"],[\"0\",\"8\"]],\"-6\",[\"1\",\"1\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0008","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[-5, 0], [0, 9]]。候選 λ=-4、v=[1, -1] 是否為特徵對？","answer":{"kind":"short_text","value":"否","explanation_zh":"Av=[-5, -9]；λv=[-4, 4]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074008,"parameters":{"M":[[-5,0],[0,9]],"lambda":-4,"v":[1,-1],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"-5\",\"0\"],[\"0\",\"9\"]],\"-4\",[\"1\",\"-1\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0009","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[-4, 0], [0, 10]]。候選 λ=-4、v=[1, 0] 是否為特徵對？","answer":{"kind":"short_text","value":"是","explanation_zh":"Av=[-4, 0]；λv=[-4, 0]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074009,"parameters":{"M":[[-4,0],[0,10]],"lambda":-4,"v":[1,0],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"-4\",\"0\"],[\"0\",\"10\"]],\"-4\",[\"1\",\"0\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0010","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[-3, 0], [0, 11]]。候選 λ=11、v=[0, 1] 是否為特徵對？","answer":{"kind":"short_text","value":"是","explanation_zh":"Av=[0, 11]；λv=[0, 11]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074010,"parameters":{"M":[[-3,0],[0,11]],"lambda":11,"v":[0,1],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"-3\",\"0\"],[\"0\",\"11\"]],\"11\",[\"0\",\"1\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0011","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[-2, 0], [0, 12]]。候選 λ=-2、v=[1, 1] 是否為特徵對？","answer":{"kind":"short_text","value":"否","explanation_zh":"Av=[-2, 12]；λv=[-2, -2]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074011,"parameters":{"M":[[-2,0],[0,12]],"lambda":-2,"v":[1,1],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"-2\",\"0\"],[\"0\",\"12\"]],\"-2\",[\"1\",\"1\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0012","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[-1, 0], [0, 13]]。候選 λ=0、v=[1, -1] 是否為特徵對？","answer":{"kind":"short_text","value":"否","explanation_zh":"Av=[-1, -13]；λv=[0, 0]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074012,"parameters":{"M":[[-1,0],[0,13]],"lambda":0,"v":[1,-1],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"-1\",\"0\"],[\"0\",\"13\"]],\"0\",[\"1\",\"-1\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0013","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[0, 0], [0, 14]]。候選 λ=0、v=[1, 0] 是否為特徵對？","answer":{"kind":"short_text","value":"是","explanation_zh":"Av=[0, 0]；λv=[0, 0]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074013,"parameters":{"M":[[0,0],[0,14]],"lambda":0,"v":[1,0],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"0\",\"0\"],[\"0\",\"14\"]],\"0\",[\"1\",\"0\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0014","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[1, 0], [0, 15]]。候選 λ=15、v=[0, 1] 是否為特徵對？","answer":{"kind":"short_text","value":"是","explanation_zh":"Av=[0, 15]；λv=[0, 15]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074014,"parameters":{"M":[[1,0],[0,15]],"lambda":15,"v":[0,1],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"1\",\"0\"],[\"0\",\"15\"]],\"15\",[\"0\",\"1\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0015","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[2, 0], [0, 16]]。候選 λ=2、v=[1, 1] 是否為特徵對？","answer":{"kind":"short_text","value":"否","explanation_zh":"Av=[2, 16]；λv=[2, 2]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074015,"parameters":{"M":[[2,0],[0,16]],"lambda":2,"v":[1,1],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"2\",\"0\"],[\"0\",\"16\"]],\"2\",[\"1\",\"1\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0016","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[3, 0], [0, 17]]。候選 λ=4、v=[1, -1] 是否為特徵對？","answer":{"kind":"short_text","value":"否","explanation_zh":"Av=[3, -17]；λv=[4, -4]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074016,"parameters":{"M":[[3,0],[0,17]],"lambda":4,"v":[1,-1],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"3\",\"0\"],[\"0\",\"17\"]],\"4\",[\"1\",\"-1\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0017","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[4, 0], [0, 18]]。候選 λ=4、v=[1, 0] 是否為特徵對？","answer":{"kind":"short_text","value":"是","explanation_zh":"Av=[4, 0]；λv=[4, 0]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074017,"parameters":{"M":[[4,0],[0,18]],"lambda":4,"v":[1,0],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"4\",\"0\"],[\"0\",\"18\"]],\"4\",[\"1\",\"0\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0018","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[5, 0], [0, 19]]。候選 λ=19、v=[0, 1] 是否為特徵對？","answer":{"kind":"short_text","value":"是","explanation_zh":"Av=[0, 19]；λv=[0, 19]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074018,"parameters":{"M":[[5,0],[0,19]],"lambda":19,"v":[0,1],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"5\",\"0\"],[\"0\",\"19\"]],\"19\",[\"0\",\"1\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0019","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[6, 0], [0, 20]]。候選 λ=6、v=[1, 1] 是否為特徵對？","answer":{"kind":"short_text","value":"否","explanation_zh":"Av=[6, 20]；λv=[6, 6]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074019,"parameters":{"M":[[6,0],[0,20]],"lambda":6,"v":[1,1],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"6\",\"0\"],[\"0\",\"20\"]],\"6\",[\"1\",\"1\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0020","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[7, 0], [0, 21]]。候選 λ=8、v=[1, -1] 是否為特徵對？","answer":{"kind":"short_text","value":"否","explanation_zh":"Av=[7, -21]；λv=[8, -8]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074020,"parameters":{"M":[[7,0],[0,21]],"lambda":8,"v":[1,-1],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"7\",\"0\"],[\"0\",\"21\"]],\"8\",[\"1\",\"-1\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0021","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[8, 0], [0, 22]]。候選 λ=8、v=[1, 0] 是否為特徵對？","answer":{"kind":"short_text","value":"是","explanation_zh":"Av=[8, 0]；λv=[8, 0]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074021,"parameters":{"M":[[8,0],[0,22]],"lambda":8,"v":[1,0],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"8\",\"0\"],[\"0\",\"22\"]],\"8\",[\"1\",\"0\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0022","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[9, 0], [0, 23]]。候選 λ=23、v=[0, 1] 是否為特徵對？","answer":{"kind":"short_text","value":"是","explanation_zh":"Av=[0, 23]；λv=[0, 23]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074022,"parameters":{"M":[[9,0],[0,23]],"lambda":23,"v":[0,1],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"9\",\"0\"],[\"0\",\"23\"]],\"23\",[\"0\",\"1\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0023","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[10, 0], [0, 24]]。候選 λ=10、v=[1, 1] 是否為特徵對？","answer":{"kind":"short_text","value":"否","explanation_zh":"Av=[10, 24]；λv=[10, 10]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074023,"parameters":{"M":[[10,0],[0,24]],"lambda":10,"v":[1,1],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"10\",\"0\"],[\"0\",\"24\"]],\"10\",[\"1\",\"1\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0024","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[11, 0], [0, 25]]。候選 λ=12、v=[1, -1] 是否為特徵對？","answer":{"kind":"short_text","value":"否","explanation_zh":"Av=[11, -25]；λv=[12, -12]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074024,"parameters":{"M":[[11,0],[0,25]],"lambda":12,"v":[1,-1],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"11\",\"0\"],[\"0\",\"25\"]],\"12\",[\"1\",\"-1\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0025","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"非零特徵向量候選","stem_zh":"實二階方陣，論域 R，M=[[12, 0], [0, 26]]。候選 λ=12、v=[1, 0] 是否為特徵對？","answer":{"kind":"short_text","value":"是","explanation_zh":"Av=[12, 0]；λv=[12, 0]；非零向量=是。必須同時有 Av=λv 且 v≠0；零向量不是特徵向量，λ=0 或負 λ 仍可合法。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零特徵向量候選"]},"variant":{"family_id":"qf-eigenpair-candidate","seed":202610074025,"parameters":{"M":[[12,0],[0,26]],"lambda":12,"v":[1,0],"operation":"eigenpair","domain":"reals","semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"12\",\"0\"],[\"0\",\"26\"]],\"12\",[\"1\",\"0\"]],\"operation\":\"eigenpair\"}"}},"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-r04-eigenvalue-0026","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(0,0)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{0}","explanation_zh":"互異特徵值={0}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074026,"parameters":{"a":0,"d":0,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"0\",\"0\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0027","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(-2,-2)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{-2}","explanation_zh":"互異特徵值={-2}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074027,"parameters":{"a":-2,"d":-2,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"-2\",\"-2\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0028","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(-10,3)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{-10, 3}","explanation_zh":"互異特徵值={-10, 3}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074028,"parameters":{"a":-10,"d":3,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"-10\",\"3\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0029","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(-9,4)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{-9, 4}","explanation_zh":"互異特徵值={-9, 4}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074029,"parameters":{"a":-9,"d":4,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"-9\",\"4\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0030","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(-8,5)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{-8, 5}","explanation_zh":"互異特徵值={-8, 5}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074030,"parameters":{"a":-8,"d":5,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"-8\",\"5\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0031","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(-7,6)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{-7, 6}","explanation_zh":"互異特徵值={-7, 6}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074031,"parameters":{"a":-7,"d":6,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"-7\",\"6\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0032","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(-6,7)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{-6, 7}","explanation_zh":"互異特徵值={-6, 7}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074032,"parameters":{"a":-6,"d":7,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"-6\",\"7\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0033","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(-5,8)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{-5, 8}","explanation_zh":"互異特徵值={-5, 8}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074033,"parameters":{"a":-5,"d":8,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"-5\",\"8\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0034","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(-4,9)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{-4, 9}","explanation_zh":"互異特徵值={-4, 9}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074034,"parameters":{"a":-4,"d":9,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"-4\",\"9\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0035","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(-3,10)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{-3, 10}","explanation_zh":"互異特徵值={-3, 10}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074035,"parameters":{"a":-3,"d":10,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"-3\",\"10\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0036","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(-2,11)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{-2, 11}","explanation_zh":"互異特徵值={-2, 11}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074036,"parameters":{"a":-2,"d":11,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"-2\",\"11\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0037","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(-1,12)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{-1, 12}","explanation_zh":"互異特徵值={-1, 12}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074037,"parameters":{"a":-1,"d":12,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"-1\",\"12\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0038","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(0,13)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{0, 13}","explanation_zh":"互異特徵值={0, 13}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074038,"parameters":{"a":0,"d":13,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"0\",\"13\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0039","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(1,14)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{1, 14}","explanation_zh":"互異特徵值={1, 14}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074039,"parameters":{"a":1,"d":14,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"1\",\"14\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0040","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(2,15)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{2, 15}","explanation_zh":"互異特徵值={2, 15}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074040,"parameters":{"a":2,"d":15,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"2\",\"15\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0041","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(3,16)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{3, 16}","explanation_zh":"互異特徵值={3, 16}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074041,"parameters":{"a":3,"d":16,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"3\",\"16\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0042","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(4,17)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{4, 17}","explanation_zh":"互異特徵值={4, 17}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074042,"parameters":{"a":4,"d":17,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"4\",\"17\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0043","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(5,18)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{5, 18}","explanation_zh":"互異特徵值={5, 18}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074043,"parameters":{"a":5,"d":18,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"5\",\"18\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0044","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(6,19)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{6, 19}","explanation_zh":"互異特徵值={6, 19}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074044,"parameters":{"a":6,"d":19,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"6\",\"19\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0045","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(7,20)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{7, 20}","explanation_zh":"互異特徵值={7, 20}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074045,"parameters":{"a":7,"d":20,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"7\",\"20\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0046","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(8,21)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{8, 21}","explanation_zh":"互異特徵值={8, 21}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074046,"parameters":{"a":8,"d":21,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"8\",\"21\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0047","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(9,22)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{9, 22}","explanation_zh":"互異特徵值={9, 22}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074047,"parameters":{"a":9,"d":22,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"9\",\"22\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0048","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(10,23)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{10, 23}","explanation_zh":"互異特徵值={10, 23}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074048,"parameters":{"a":10,"d":23,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"10\",\"23\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0049","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(11,24)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{11, 24}","explanation_zh":"互異特徵值={11, 24}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074049,"parameters":{"a":11,"d":24,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"11\",\"24\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0050","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"對角矩陣互異實特徵值集合","stem_zh":"實二階方陣，論域 R，diag(12,25)。求互異實特徵值集合，遞增列出且重複值只列一次。","answer":{"kind":"expression","value":"{12, 25}","explanation_zh":"互異特徵值={12, 25}。對角兩值都列入，若相同只留下單元素；不混用代數重數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["對角矩陣互異實特徵值集合"]},"variant":{"family_id":"qf-eigenvalue-diagonal-set","seed":202610074050,"parameters":{"a":12,"d":25,"operation":"diagonal_eigenvalues","domain":"reals","semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"12\",\"25\"],\"operation\":\"diagonal_eigenvalues\"}"}},"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-r04-eigenvalue-0051","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[2, 1], [0, 3]]。λ=99、v=[0, 0]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"是","explanation_zh":"核殘差=[0, 0]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074051,"parameters":{"M":[[2,1],[0,3]],"lambda":99,"v":[0,0],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[\"zero_vector\"],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0052","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[-11, 0], [0, 3]]。λ=3、v=[0, 1]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"是","explanation_zh":"核殘差=[0, 0]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074052,"parameters":{"M":[[-11,0],[0,3]],"lambda":3,"v":[0,1],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"-11\",\"0\"],[\"0\",\"3\"]],\"3\",[\"0\",\"1\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0053","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[-10, 0], [0, 4]]。λ=-10、v=[1, 1]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"否","explanation_zh":"核殘差=[0, 14]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074053,"parameters":{"M":[[-10,0],[0,4]],"lambda":-10,"v":[1,1],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"-10\",\"0\"],[\"0\",\"4\"]],\"-10\",[\"1\",\"1\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0054","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[-9, 0], [0, 5]]。λ=-8、v=[1, -1]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"否","explanation_zh":"核殘差=[-1, -13]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074054,"parameters":{"M":[[-9,0],[0,5]],"lambda":-8,"v":[1,-1],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"-9\",\"0\"],[\"0\",\"5\"]],\"-8\",[\"1\",\"-1\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0055","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[4, 1], [1, 4]]。λ=5、v=[1, 1]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"是","explanation_zh":"核殘差=[0, 0]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074055,"parameters":{"M":[[4,1],[1,4]],"lambda":5,"v":[1,1],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"4\",\"1\"],[\"1\",\"4\"]],\"5\",[\"1\",\"1\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0056","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[5, 0], [1, 6]]。λ=5、v=[1, -1]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"是","explanation_zh":"核殘差=[0, 0]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074056,"parameters":{"M":[[5,0],[1,6]],"lambda":5,"v":[1,-1],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"5\",\"0\"],[\"1\",\"6\"]],\"5\",[\"1\",\"-1\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0057","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[-6, 0], [0, 8]]。λ=-6、v=[1, 1]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"否","explanation_zh":"核殘差=[0, 14]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074057,"parameters":{"M":[[-6,0],[0,8]],"lambda":-6,"v":[1,1],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"-6\",\"0\"],[\"0\",\"8\"]],\"-6\",[\"1\",\"1\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0058","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[-5, 0], [0, 9]]。λ=-4、v=[1, -1]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"否","explanation_zh":"核殘差=[-1, -13]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074058,"parameters":{"M":[[-5,0],[0,9]],"lambda":-4,"v":[1,-1],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"-5\",\"0\"],[\"0\",\"9\"]],\"-4\",[\"1\",\"-1\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0059","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[-4, 0], [0, 10]]。λ=-4、v=[1, 0]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"是","explanation_zh":"核殘差=[0, 0]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074059,"parameters":{"M":[[-4,0],[0,10]],"lambda":-4,"v":[1,0],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"-4\",\"0\"],[\"0\",\"10\"]],\"-4\",[\"1\",\"0\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0060","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[-3, 0], [0, 11]]。λ=11、v=[0, 1]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"是","explanation_zh":"核殘差=[0, 0]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074060,"parameters":{"M":[[-3,0],[0,11]],"lambda":11,"v":[0,1],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"-3\",\"0\"],[\"0\",\"11\"]],\"11\",[\"0\",\"1\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0061","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[-2, 0], [0, 12]]。λ=-2、v=[1, 1]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"否","explanation_zh":"核殘差=[0, 14]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074061,"parameters":{"M":[[-2,0],[0,12]],"lambda":-2,"v":[1,1],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"-2\",\"0\"],[\"0\",\"12\"]],\"-2\",[\"1\",\"1\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0062","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[-1, 0], [0, 13]]。λ=0、v=[1, -1]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"否","explanation_zh":"核殘差=[-1, -13]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074062,"parameters":{"M":[[-1,0],[0,13]],"lambda":0,"v":[1,-1],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"-1\",\"0\"],[\"0\",\"13\"]],\"0\",[\"1\",\"-1\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0063","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[0, 0], [0, 14]]。λ=0、v=[1, 0]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"是","explanation_zh":"核殘差=[0, 0]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074063,"parameters":{"M":[[0,0],[0,14]],"lambda":0,"v":[1,0],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"0\",\"0\"],[\"0\",\"14\"]],\"0\",[\"1\",\"0\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0064","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[1, 0], [0, 15]]。λ=15、v=[0, 1]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"是","explanation_zh":"核殘差=[0, 0]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074064,"parameters":{"M":[[1,0],[0,15]],"lambda":15,"v":[0,1],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"1\",\"0\"],[\"0\",\"15\"]],\"15\",[\"0\",\"1\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0065","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[2, 0], [0, 16]]。λ=2、v=[1, 1]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"否","explanation_zh":"核殘差=[0, 14]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074065,"parameters":{"M":[[2,0],[0,16]],"lambda":2,"v":[1,1],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"2\",\"0\"],[\"0\",\"16\"]],\"2\",[\"1\",\"1\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0066","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[3, 0], [0, 17]]。λ=4、v=[1, -1]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"否","explanation_zh":"核殘差=[-1, -13]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074066,"parameters":{"M":[[3,0],[0,17]],"lambda":4,"v":[1,-1],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"3\",\"0\"],[\"0\",\"17\"]],\"4\",[\"1\",\"-1\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0067","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[4, 0], [0, 18]]。λ=4、v=[1, 0]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"是","explanation_zh":"核殘差=[0, 0]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074067,"parameters":{"M":[[4,0],[0,18]],"lambda":4,"v":[1,0],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"4\",\"0\"],[\"0\",\"18\"]],\"4\",[\"1\",\"0\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0068","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[5, 0], [0, 19]]。λ=19、v=[0, 1]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"是","explanation_zh":"核殘差=[0, 0]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074068,"parameters":{"M":[[5,0],[0,19]],"lambda":19,"v":[0,1],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"5\",\"0\"],[\"0\",\"19\"]],\"19\",[\"0\",\"1\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0069","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[6, 0], [0, 20]]。λ=6、v=[1, 1]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"否","explanation_zh":"核殘差=[0, 14]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074069,"parameters":{"M":[[6,0],[0,20]],"lambda":6,"v":[1,1],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"6\",\"0\"],[\"0\",\"20\"]],\"6\",[\"1\",\"1\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0070","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[7, 0], [0, 21]]。λ=8、v=[1, -1]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"否","explanation_zh":"核殘差=[-1, -13]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074070,"parameters":{"M":[[7,0],[0,21]],"lambda":8,"v":[1,-1],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"7\",\"0\"],[\"0\",\"21\"]],\"8\",[\"1\",\"-1\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0071","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[8, 0], [0, 22]]。λ=8、v=[1, 0]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"是","explanation_zh":"核殘差=[0, 0]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074071,"parameters":{"M":[[8,0],[0,22]],"lambda":8,"v":[1,0],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"8\",\"0\"],[\"0\",\"22\"]],\"8\",[\"1\",\"0\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0072","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[9, 0], [0, 23]]。λ=23、v=[0, 1]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"是","explanation_zh":"核殘差=[0, 0]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074072,"parameters":{"M":[[9,0],[0,23]],"lambda":23,"v":[0,1],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"9\",\"0\"],[\"0\",\"23\"]],\"23\",[\"0\",\"1\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0073","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[10, 0], [0, 24]]。λ=10、v=[1, 1]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"否","explanation_zh":"核殘差=[0, 14]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074073,"parameters":{"M":[[10,0],[0,24]],"lambda":10,"v":[1,1],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"10\",\"0\"],[\"0\",\"24\"]],\"10\",[\"1\",\"1\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0074","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[11, 0], [0, 25]]。λ=12、v=[1, -1]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"否","explanation_zh":"核殘差=[-1, -13]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074074,"parameters":{"M":[[11,0],[0,25]],"lambda":12,"v":[1,-1],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"11\",\"0\"],[\"0\",\"25\"]],\"12\",[\"1\",\"-1\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0075","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"候選純量的 kernel 成員","stem_zh":"實二階方陣，論域 R，M=[[12, 0], [0, 26]]。λ=12、v=[1, 0]。v 是否屬於 ker(M−λI)？λ 不預設為特徵值。","answer":{"kind":"short_text","value":"是","explanation_zh":"核殘差=[0, 0]。核含零；零向量總符合核條件，不能因此推出 λ 是特徵值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選純量的 kernel 成員"]},"variant":{"family_id":"qf-eigenvalue-kernel-membership","seed":202610074075,"parameters":{"M":[[12,0],[0,26]],"lambda":12,"v":[1,0],"operation":"kernel_membership","domain":"reals","semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"12\",\"0\"],[\"0\",\"26\"]],\"12\",[\"1\",\"0\"]],\"operation\":\"kernel_membership\"}"}},"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-r04-eigenvalue-0076","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[0, 0], [0, 0]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"1","explanation_zh":"特徵多項式係數=(1, 0, 0)；判別式=0。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074076,"parameters":{"M":[[0,0],[0,0]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"0\",\"0\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0077","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[1, -1], [2, 1]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"0","explanation_zh":"特徵多項式係數=(1, -2, 3)；判別式=-8。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074077,"parameters":{"M":[[1,-1],[2,1]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"2\",\"3\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0078","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[1, 1], [0, 1]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"1","explanation_zh":"特徵多項式係數=(1, -2, 1)；判別式=0。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074078,"parameters":{"M":[[1,1],[0,1]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"2\",\"1\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0079","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[-1, 1], [1, 1]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"2","explanation_zh":"特徵多項式係數=(1, 0, -2)；判別式=8。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074079,"parameters":{"M":[[-1,1],[1,1]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"0\",\"-2\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0080","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[2, -1], [3, 2]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"0","explanation_zh":"特徵多項式係數=(1, -4, 7)；判別式=-12。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074080,"parameters":{"M":[[2,-1],[3,2]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"4\",\"7\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0081","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[2, 1], [0, 2]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"1","explanation_zh":"特徵多項式係數=(1, -4, 4)；判別式=0。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074081,"parameters":{"M":[[2,1],[0,2]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"4\",\"4\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0082","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[-2, 1], [1, 2]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"2","explanation_zh":"特徵多項式係數=(1, 0, -5)；判別式=20。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074082,"parameters":{"M":[[-2,1],[1,2]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"0\",\"-5\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0083","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[3, -1], [4, 3]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"0","explanation_zh":"特徵多項式係數=(1, -6, 13)；判別式=-16。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074083,"parameters":{"M":[[3,-1],[4,3]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"6\",\"13\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0084","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[3, 1], [0, 3]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"1","explanation_zh":"特徵多項式係數=(1, -6, 9)；判別式=0。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074084,"parameters":{"M":[[3,1],[0,3]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"6\",\"9\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0085","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[-3, 1], [1, 3]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"2","explanation_zh":"特徵多項式係數=(1, 0, -10)；判別式=40。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074085,"parameters":{"M":[[-3,1],[1,3]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"0\",\"-10\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0086","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[4, -1], [5, 4]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"0","explanation_zh":"特徵多項式係數=(1, -8, 21)；判別式=-20。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074086,"parameters":{"M":[[4,-1],[5,4]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"8\",\"21\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0087","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[4, 1], [0, 4]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"1","explanation_zh":"特徵多項式係數=(1, -8, 16)；判別式=0。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074087,"parameters":{"M":[[4,1],[0,4]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"8\",\"16\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0088","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[-4, 1], [1, 4]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"2","explanation_zh":"特徵多項式係數=(1, 0, -17)；判別式=68。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074088,"parameters":{"M":[[-4,1],[1,4]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"0\",\"-17\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0089","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[5, -1], [6, 5]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"0","explanation_zh":"特徵多項式係數=(1, -10, 31)；判別式=-24。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074089,"parameters":{"M":[[5,-1],[6,5]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"10\",\"31\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0090","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[5, 1], [0, 5]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"1","explanation_zh":"特徵多項式係數=(1, -10, 25)；判別式=0。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074090,"parameters":{"M":[[5,1],[0,5]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"10\",\"25\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0091","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[-5, 1], [1, 5]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"2","explanation_zh":"特徵多項式係數=(1, 0, -26)；判別式=104。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074091,"parameters":{"M":[[-5,1],[1,5]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"0\",\"-26\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0092","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[6, -1], [7, 6]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"0","explanation_zh":"特徵多項式係數=(1, -12, 43)；判別式=-28。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074092,"parameters":{"M":[[6,-1],[7,6]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"12\",\"43\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0093","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[6, 1], [0, 6]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"1","explanation_zh":"特徵多項式係數=(1, -12, 36)；判別式=0。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074093,"parameters":{"M":[[6,1],[0,6]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"12\",\"36\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0094","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[-6, 1], [1, 6]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"2","explanation_zh":"特徵多項式係數=(1, 0, -37)；判別式=148。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074094,"parameters":{"M":[[-6,1],[1,6]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"0\",\"-37\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0095","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[7, -1], [8, 7]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"0","explanation_zh":"特徵多項式係數=(1, -14, 57)；判別式=-32。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074095,"parameters":{"M":[[7,-1],[8,7]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"14\",\"57\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0096","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[7, 1], [0, 7]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"1","explanation_zh":"特徵多項式係數=(1, -14, 49)；判別式=0。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074096,"parameters":{"M":[[7,1],[0,7]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"14\",\"49\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0097","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[-7, 1], [1, 7]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"2","explanation_zh":"特徵多項式係數=(1, 0, -50)；判別式=200。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074097,"parameters":{"M":[[-7,1],[1,7]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"0\",\"-50\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0098","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[8, -1], [9, 8]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"0","explanation_zh":"特徵多項式係數=(1, -16, 73)；判別式=-36。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074098,"parameters":{"M":[[8,-1],[9,8]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"16\",\"73\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0099","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[8, 1], [0, 8]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"1","explanation_zh":"特徵多項式係數=(1, -16, 64)；判別式=0。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074099,"parameters":{"M":[[8,1],[0,8]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"16\",\"64\"],\"operation\":\"real_eigenvalue_count\"}"}},"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-r04-eigenvalue-0100","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","language":"zh-Hant","source_mko_ids":["mko-eigenvalue"],"title_zh":"二階方陣的互異實特徵值數","stem_zh":"實二階方陣，論域 R，M=[[-8, 1], [1, 8]]。只求互異實特徵值的個數 0、1 或2，不求小數根。","answer":{"kind":"short_text","value":"2","explanation_zh":"特徵多項式係數=(1, 0, -65)；判別式=260。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階方陣的互異實特徵值數"]},"variant":{"family_id":"qf-eigenvalue-real-count","seed":202610074100,"parameters":{"M":[[-8,1],[1,8]],"operation":"real_eigenvalue_count","domain":"reals","semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"0\",\"-65\"],\"operation\":\"real_eigenvalue_count\"}"}},"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"}}