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Browse files- .gitattributes +1 -0
- models/SFT/added_tokens.json +5 -0
- models/SFT/chat_template.jinja +6 -0
- models/SFT/config.json +59 -0
- models/SFT/generation_config.json +6 -0
- models/SFT/math_eval_selfconsistent_aligned/choice/SFT_20250917_125406_analysis_report.md +0 -0
- models/SFT/math_eval_selfconsistent_aligned/console_log_choice_pure_regex_self_consistent_20250917_125611.txt +140 -0
- models/SFT/math_eval_selfconsistent_aligned/detailed_comparison_choice_pure_regex_self_consistent_20250917_125611.jsonl +140 -0
- models/SFT/math_eval_selfconsistent_aligned/summary_choice_pure_regex_self_consistent_20250917_125611.json +17 -0
- models/SFT/math_eval_selfconsistent_aligned/voting_details_choice_pure_regex_self_consistent_20250917_125611.jsonl +0 -0
- models/SFT/merges.txt +0 -0
- models/SFT/model-00001-of-00004.safetensors +3 -0
- models/SFT/model-00002-of-00004.safetensors +3 -0
- models/SFT/model-00003-of-00004.safetensors +3 -0
- models/SFT/model-00004-of-00004.safetensors +3 -0
- models/SFT/model.safetensors.index.json +347 -0
- models/SFT/special_tokens_map.json +20 -0
- models/SFT/tokenizer.json +3 -0
- models/SFT/tokenizer_config.json +43 -0
- models/SFT/vocab.json +0 -0
.gitattributes
CHANGED
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@@ -110,3 +110,4 @@ models/actor/tokenizer.json filter=lfs diff=lfs merge=lfs -text
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models/merged-sft-conceptandquiz-100/tokenizer.json filter=lfs diff=lfs merge=lfs -text
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models/qwen2-math-1.5b-grpo-concept-enhanced-20250923_164417/global_step_24/actor/huggingface/tokenizer.json filter=lfs diff=lfs merge=lfs -text
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models/qwen2-math-1.5b-grpo-concept-enhanced-20250923_164417/global_step_24/actor/tokenizer.json filter=lfs diff=lfs merge=lfs -text
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models/merged-sft-conceptandquiz-100/tokenizer.json filter=lfs diff=lfs merge=lfs -text
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models/qwen2-math-1.5b-grpo-concept-enhanced-20250923_164417/global_step_24/actor/huggingface/tokenizer.json filter=lfs diff=lfs merge=lfs -text
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models/qwen2-math-1.5b-grpo-concept-enhanced-20250923_164417/global_step_24/actor/tokenizer.json filter=lfs diff=lfs merge=lfs -text
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models/SFT/tokenizer.json filter=lfs diff=lfs merge=lfs -text
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models/SFT/added_tokens.json
ADDED
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{
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"<|endoftext|>": 151643,
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"<|im_end|>": 151645,
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"<|im_start|>": 151644
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}
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models/SFT/chat_template.jinja
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{% for message in messages %}{% if loop.first and messages[0]['role'] != 'system' %}{{ '<|im_start|>system
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You are a helpful assistant<|im_end|>
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' }}{% endif %}{{'<|im_start|>' + message['role'] + '
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' + message['content'] + '<|im_end|>' + '
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'}}{% endfor %}{% if add_generation_prompt %}{{ '<|im_start|>assistant
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' }}{% endif %}
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models/SFT/config.json
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{
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"architectures": [
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"Qwen2ForCausalLM"
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],
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"attention_dropout": 0.0,
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"bos_token_id": 151643,
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"eos_token_id": 151643,
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"hidden_act": "silu",
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"hidden_size": 3584,
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"initializer_range": 0.02,
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"intermediate_size": 18944,
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"layer_types": [
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention",
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"full_attention"
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],
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"max_position_embeddings": 4096,
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"max_window_layers": 28,
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"model_type": "qwen2",
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"num_attention_heads": 28,
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"num_hidden_layers": 28,
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"num_key_value_heads": 4,
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"rms_norm_eps": 1e-06,
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"rope_scaling": null,
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"rope_theta": 10000,
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"sliding_window": null,
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"tie_word_embeddings": false,
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"torch_dtype": "float16",
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"transformers_version": "4.55.4",
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"use_cache": true,
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"use_mrope": false,
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"use_sliding_window": false,
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"vocab_size": 152064
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}
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models/SFT/generation_config.json
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{
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"bos_token_id": 151643,
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"eos_token_id": 151643,
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"max_new_tokens": 2048,
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"transformers_version": "4.55.4"
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}
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models/SFT/math_eval_selfconsistent_aligned/choice/SFT_20250917_125406_analysis_report.md
ADDED
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The diff for this file is too large to render.
See raw diff
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models/SFT/math_eval_selfconsistent_aligned/console_log_choice_pure_regex_self_consistent_20250917_125611.txt
ADDED
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| 1 |
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[Question IDX: 0] GT: C | Voted Choice: C | Votes: {'A': 1, 'B': 0, 'C': 12, 'D': 2} | Correct: True
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| 2 |
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[Question IDX: 1] GT: C | Voted Choice: C | Votes: {'A': 3, 'B': 4, 'C': 7, 'D': 3} | Correct: True
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| 3 |
+
[Question IDX: 2] GT: B | Voted Choice: B | Votes: {'A': 1, 'B': 12, 'C': 3, 'D': 1} | Correct: True
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| 4 |
+
[Question IDX: 3] GT: B | Voted Choice: C | Votes: {'A': 3, 'B': 3, 'C': 8, 'D': 1} | Correct: False
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| 5 |
+
[Question IDX: 4] GT: D | Voted Choice: D | Votes: {'A': 4, 'B': 0, 'C': 2, 'D': 11} | Correct: True
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| 6 |
+
[Question IDX: 5] GT: D | Voted Choice: D | Votes: {'A': 0, 'B': 1, 'C': 8, 'D': 8} | Correct: True
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| 7 |
+
[Question IDX: 6] GT: A | Voted Choice: B | Votes: {'A': 3, 'B': 6, 'C': 2, 'D': 4} | Correct: False
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| 8 |
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[Question IDX: 7] GT: D | Voted Choice: A | Votes: {'A': 7, 'B': 3, 'C': 0, 'D': 7} | Correct: False
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| 9 |
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[Question IDX: 8] GT: B | Voted Choice: A | Votes: {'A': 15, 'B': 2, 'C': 0, 'D': 1} | Correct: False
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| 10 |
+
[Question IDX: 9] GT: B | Voted Choice: B | Votes: {'A': 3, 'B': 13, 'C': 2, 'D': 3} | Correct: True
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| 11 |
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[Question IDX: 10] GT: C | Voted Choice: B | Votes: {'A': 2, 'B': 15, 'C': 3, 'D': 0} | Correct: False
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| 12 |
+
[Question IDX: 11] GT: C | Voted Choice: C | Votes: {'A': 2, 'B': 0, 'C': 13, 'D': 3} | Correct: True
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| 13 |
+
[Question IDX: 12] GT: C | Voted Choice: C | Votes: {'A': 3, 'B': 3, 'C': 4, 'D': 3} | Correct: True
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| 14 |
+
[Question IDX: 13] GT: D | Voted Choice: C | Votes: {'A': 4, 'B': 3, 'C': 9, 'D': 2} | Correct: False
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| 15 |
+
[Question IDX: 14] GT: B | Voted Choice: B | Votes: {'A': 5, 'B': 11, 'C': 1, 'D': 1} | Correct: True
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| 16 |
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[Question IDX: 15] GT: B | Voted Choice: B | Votes: {'A': 3, 'B': 11, 'C': 0, 'D': 0} | Correct: True
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| 17 |
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[Question IDX: 16] GT: A | Voted Choice: A | Votes: {'A': 9, 'B': 7, 'C': 1, 'D': 2} | Correct: True
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| 18 |
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[Question IDX: 17] GT: A | Voted Choice: A | Votes: {'A': 6, 'B': 1, 'C': 4, 'D': 3} | Correct: True
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| 19 |
+
[Question IDX: 18] GT: C | Voted Choice: B | Votes: {'A': 6, 'B': 6, 'C': 3, 'D': 3} | Correct: False
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| 20 |
+
[Question IDX: 19] GT: C | Voted Choice: C | Votes: {'A': 0, 'B': 4, 'C': 9, 'D': 4} | Correct: True
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| 21 |
+
[Question IDX: 20] GT: B | Voted Choice: B | Votes: {'A': 2, 'B': 9, 'C': 4, 'D': 0} | Correct: True
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| 22 |
+
[Question IDX: 21] GT: B | Voted Choice: B | Votes: {'A': 3, 'B': 10, 'C': 2, 'D': 4} | Correct: True
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| 23 |
+
[Question IDX: 22] GT: B | Voted Choice: B | Votes: {'A': 4, 'B': 10, 'C': 2, 'D': 1} | Correct: True
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| 24 |
+
[Question IDX: 23] GT: D | Voted Choice: D | Votes: {'A': 2, 'B': 2, 'C': 3, 'D': 11} | Correct: True
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| 25 |
+
[Question IDX: 24] GT: B | Voted Choice: B | Votes: {'A': 2, 'B': 6, 'C': 5, 'D': 4} | Correct: True
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| 26 |
+
[Question IDX: 25] GT: A | Voted Choice: A | Votes: {'A': 16, 'B': 0, 'C': 2, 'D': 2} | Correct: True
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| 27 |
+
[Question IDX: 26] GT: C | Voted Choice: B | Votes: {'A': 1, 'B': 14, 'C': 3, 'D': 1} | Correct: False
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| 28 |
+
[Question IDX: 27] GT: C | Voted Choice: A | Votes: {'A': 14, 'B': 2, 'C': 2, 'D': 0} | Correct: False
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| 29 |
+
[Question IDX: 28] GT: B | Voted Choice: B | Votes: {'A': 3, 'B': 7, 'C': 3, 'D': 2} | Correct: True
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| 30 |
+
[Question IDX: 29] GT: D | Voted Choice: D | Votes: {'A': 2, 'B': 0, 'C': 3, 'D': 13} | Correct: True
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| 31 |
+
[Question IDX: 30] GT: B | Voted Choice: B | Votes: {'A': 2, 'B': 15, 'C': 0, 'D': 0} | Correct: True
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| 32 |
+
[Question IDX: 31] GT: C | Voted Choice: C | Votes: {'A': 1, 'B': 3, 'C': 12, 'D': 1} | Correct: True
|
| 33 |
+
[Question IDX: 32] GT: C | Voted Choice: C | Votes: {'A': 4, 'B': 3, 'C': 11, 'D': 1} | Correct: True
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| 34 |
+
[Question IDX: 33] GT: C | Voted Choice: C | Votes: {'A': 7, 'B': 0, 'C': 10, 'D': 1} | Correct: True
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| 35 |
+
[Question IDX: 34] GT: D | Voted Choice: D | Votes: {'A': 0, 'B': 1, 'C': 1, 'D': 19} | Correct: True
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| 36 |
+
[Question IDX: 35] GT: C | Voted Choice: C | Votes: {'A': 1, 'B': 5, 'C': 9, 'D': 2} | Correct: True
|
| 37 |
+
[Question IDX: 36] GT: C | Voted Choice: C | Votes: {'A': 3, 'B': 7, 'C': 7, 'D': 0} | Correct: True
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| 38 |
+
[Question IDX: 37] GT: D | Voted Choice: D | Votes: {'A': 0, 'B': 1, 'C': 0, 'D': 16} | Correct: True
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| 39 |
+
[Question IDX: 38] GT: A | Voted Choice: A | Votes: {'A': 8, 'B': 6, 'C': 3, 'D': 1} | Correct: True
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| 40 |
+
[Question IDX: 39] GT: D | Voted Choice: D | Votes: {'A': 1, 'B': 0, 'C': 2, 'D': 14} | Correct: True
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| 41 |
+
[Question IDX: 40] GT: D | Voted Choice: A | Votes: {'A': 12, 'B': 0, 'C': 1, 'D': 7} | Correct: False
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| 42 |
+
[Question IDX: 41] GT: B | Voted Choice: B | Votes: {'A': 2, 'B': 9, 'C': 1, 'D': 4} | Correct: True
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| 43 |
+
[Question IDX: 42] GT: A | Voted Choice: A | Votes: {'A': 7, 'B': 4, 'C': 2, 'D': 5} | Correct: True
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| 44 |
+
[Question IDX: 43] GT: C | Voted Choice: C | Votes: {'A': 2, 'B': 3, 'C': 11, 'D': 0} | Correct: True
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| 45 |
+
[Question IDX: 44] GT: D | Voted Choice: C | Votes: {'A': 1, 'B': 1, 'C': 8, 'D': 5} | Correct: False
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| 46 |
+
[Question IDX: 45] GT: B | Voted Choice: B | Votes: {'A': 1, 'B': 15, 'C': 2, 'D': 0} | Correct: True
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| 47 |
+
[Question IDX: 46] GT: D | Voted Choice: D | Votes: {'A': 1, 'B': 0, 'C': 3, 'D': 12} | Correct: True
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| 48 |
+
[Question IDX: 47] GT: C | Voted Choice: B | Votes: {'A': 4, 'B': 5, 'C': 0, 'D': 4} | Correct: False
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| 49 |
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[Question IDX: 48] GT: D | Voted Choice: D | Votes: {'A': 1, 'B': 2, 'C': 3, 'D': 14} | Correct: True
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| 50 |
+
[Question IDX: 49] GT: C | Voted Choice: C | Votes: {'A': 0, 'B': 1, 'C': 12, 'D': 2} | Correct: True
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| 51 |
+
[Question IDX: 50] GT: D | Voted Choice: B | Votes: {'A': 3, 'B': 8, 'C': 0, 'D': 5} | Correct: False
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| 52 |
+
[Question IDX: 51] GT: B | Voted Choice: B | Votes: {'A': 1, 'B': 12, 'C': 0, 'D': 1} | Correct: True
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| 53 |
+
[Question IDX: 52] GT: B | Voted Choice: C | Votes: {'A': 3, 'B': 1, 'C': 9, 'D': 0} | Correct: False
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| 54 |
+
[Question IDX: 53] GT: B | Voted Choice: B | Votes: {'A': 1, 'B': 13, 'C': 1, 'D': 4} | Correct: True
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| 55 |
+
[Question IDX: 54] GT: C | Voted Choice: C | Votes: {'A': 4, 'B': 1, 'C': 12, 'D': 2} | Correct: True
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| 56 |
+
[Question IDX: 55] GT: C | Voted Choice: D | Votes: {'A': 2, 'B': 3, 'C': 4, 'D': 4} | Correct: False
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| 57 |
+
[Question IDX: 56] GT: D | Voted Choice: D | Votes: {'A': 2, 'B': 3, 'C': 3, 'D': 11} | Correct: True
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| 58 |
+
[Question IDX: 57] GT: B | Voted Choice: B | Votes: {'A': 5, 'B': 6, 'C': 4, 'D': 4} | Correct: True
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| 59 |
+
[Question IDX: 58] GT: B | Voted Choice: D | Votes: {'A': 1, 'B': 3, 'C': 6, 'D': 10} | Correct: False
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| 60 |
+
[Question IDX: 59] GT: D | Voted Choice: B | Votes: {'A': 4, 'B': 6, 'C': 1, 'D': 1} | Correct: False
|
| 61 |
+
[Question IDX: 60] GT: B | Voted Choice: B | Votes: {'A': 1, 'B': 13, 'C': 2, 'D': 2} | Correct: True
|
| 62 |
+
[Question IDX: 61] GT: A | Voted Choice: A | Votes: {'A': 13, 'B': 4, 'C': 1, 'D': 2} | Correct: True
|
| 63 |
+
[Question IDX: 62] GT: D | Voted Choice: C | Votes: {'A': 3, 'B': 6, 'C': 10, 'D': 1} | Correct: False
|
| 64 |
+
[Question IDX: 63] GT: C | Voted Choice: A | Votes: {'A': 7, 'B': 2, 'C': 6, 'D': 2} | Correct: False
|
| 65 |
+
[Question IDX: 64] GT: A | Voted Choice: A | Votes: {'A': 10, 'B': 0, 'C': 3, 'D': 2} | Correct: True
|
| 66 |
+
[Question IDX: 65] GT: C | Voted Choice: C | Votes: {'A': 1, 'B': 2, 'C': 6, 'D': 2} | Correct: True
|
| 67 |
+
[Question IDX: 66] GT: B | Voted Choice: D | Votes: {'A': 3, 'B': 2, 'C': 4, 'D': 4} | Correct: False
|
| 68 |
+
[Question IDX: 67] GT: C | Voted Choice: A | Votes: {'A': 10, 'B': 0, 'C': 5, 'D': 0} | Correct: False
|
| 69 |
+
[Question IDX: 68] GT: C | Voted Choice: C | Votes: {'A': 2, 'B': 0, 'C': 18, 'D': 0} | Correct: True
|
| 70 |
+
[Question IDX: 69] GT: B | Voted Choice: B | Votes: {'A': 4, 'B': 6, 'C': 2, 'D': 3} | Correct: True
|
| 71 |
+
[Question IDX: 70] GT: D | Voted Choice: C | Votes: {'A': 4, 'B': 2, 'C': 8, 'D': 5} | Correct: False
|
| 72 |
+
[Question IDX: 71] GT: D | Voted Choice: D | Votes: {'A': 2, 'B': 1, 'C': 2, 'D': 13} | Correct: True
|
| 73 |
+
[Question IDX: 72] GT: A | Voted Choice: A | Votes: {'A': 12, 'B': 3, 'C': 2, 'D': 0} | Correct: True
|
| 74 |
+
[Question IDX: 73] GT: C | Voted Choice: C | Votes: {'A': 5, 'B': 1, 'C': 10, 'D': 1} | Correct: True
|
| 75 |
+
[Question IDX: 74] GT: B | Voted Choice: B | Votes: {'A': 5, 'B': 10, 'C': 2, 'D': 0} | Correct: True
|
| 76 |
+
[Question IDX: 75] GT: C | Voted Choice: B | Votes: {'A': 5, 'B': 5, 'C': 3, 'D': 5} | Correct: False
|
| 77 |
+
[Question IDX: 76] GT: B | Voted Choice: B | Votes: {'A': 4, 'B': 11, 'C': 1, 'D': 1} | Correct: True
|
| 78 |
+
[Question IDX: 77] GT: A | Voted Choice: A | Votes: {'A': 13, 'B': 5, 'C': 0, 'D': 0} | Correct: True
|
| 79 |
+
[Question IDX: 78] GT: D | Voted Choice: C | Votes: {'A': 2, 'B': 4, 'C': 6, 'D': 6} | Correct: False
|
| 80 |
+
[Question IDX: 79] GT: C | Voted Choice: A | Votes: {'A': 10, 'B': 0, 'C': 2, 'D': 8} | Correct: False
|
| 81 |
+
[Question IDX: 80] GT: A | Voted Choice: A | Votes: {'A': 11, 'B': 1, 'C': 2, 'D': 7} | Correct: True
|
| 82 |
+
[Question IDX: 81] GT: C | Voted Choice: B | Votes: {'A': 1, 'B': 14, 'C': 3, 'D': 1} | Correct: False
|
| 83 |
+
[Question IDX: 82] GT: B | Voted Choice: D | Votes: {'A': 4, 'B': 3, 'C': 2, 'D': 7} | Correct: False
|
| 84 |
+
[Question IDX: 83] GT: A | Voted Choice: A | Votes: {'A': 9, 'B': 6, 'C': 1, 'D': 0} | Correct: True
|
| 85 |
+
[Question IDX: 84] GT: B | Voted Choice: B | Votes: {'A': 8, 'B': 9, 'C': 1, 'D': 1} | Correct: True
|
| 86 |
+
[Question IDX: 85] GT: B | Voted Choice: C | Votes: {'A': 3, 'B': 3, 'C': 8, 'D': 4} | Correct: False
|
| 87 |
+
[Question IDX: 86] GT: C | Voted Choice: D | Votes: {'A': 4, 'B': 0, 'C': 6, 'D': 9} | Correct: False
|
| 88 |
+
[Question IDX: 87] GT: C | Voted Choice: A | Votes: {'A': 9, 'B': 2, 'C': 3, 'D': 1} | Correct: False
|
| 89 |
+
[Question IDX: 88] GT: D | Voted Choice: C | Votes: {'A': 3, 'B': 2, 'C': 7, 'D': 5} | Correct: False
|
| 90 |
+
[Question IDX: 89] GT: C | Voted Choice: A | Votes: {'A': 10, 'B': 1, 'C': 6, 'D': 2} | Correct: False
|
| 91 |
+
[Question IDX: 90] GT: B | Voted Choice: B | Votes: {'A': 2, 'B': 10, 'C': 1, 'D': 0} | Correct: True
|
| 92 |
+
[Question IDX: 91] GT: C | Voted Choice: C | Votes: {'A': 9, 'B': 1, 'C': 9, 'D': 1} | Correct: True
|
| 93 |
+
[Question IDX: 92] GT: A | Voted Choice: A | Votes: {'A': 11, 'B': 3, 'C': 2, 'D': 3} | Correct: True
|
| 94 |
+
[Question IDX: 93] GT: A | Voted Choice: A | Votes: {'A': 13, 'B': 1, 'C': 2, 'D': 3} | Correct: True
|
| 95 |
+
[Question IDX: 94] GT: B | Voted Choice: D | Votes: {'A': 0, 'B': 2, 'C': 4, 'D': 9} | Correct: False
|
| 96 |
+
[Question IDX: 95] GT: C | Voted Choice: A | Votes: {'A': 7, 'B': 5, 'C': 1, 'D': 4} | Correct: False
|
| 97 |
+
[Question IDX: 96] GT: D | Voted Choice: D | Votes: {'A': 2, 'B': 6, 'C': 1, 'D': 8} | Correct: True
|
| 98 |
+
[Question IDX: 97] GT: B | Voted Choice: C | Votes: {'A': 6, 'B': 2, 'C': 7, 'D': 1} | Correct: False
|
| 99 |
+
[Question IDX: 98] GT: C | Voted Choice: C | Votes: {'A': 1, 'B': 3, 'C': 7, 'D': 7} | Correct: True
|
| 100 |
+
[Question IDX: 99] GT: C | Voted Choice: C | Votes: {'A': 0, 'B': 3, 'C': 8, 'D': 3} | Correct: True
|
| 101 |
+
[Question IDX: 100] GT: C | Voted Choice: C | Votes: {'A': 3, 'B': 1, 'C': 12, 'D': 2} | Correct: True
|
| 102 |
+
[Question IDX: 101] GT: B | Voted Choice: C | Votes: {'A': 3, 'B': 2, 'C': 11, 'D': 1} | Correct: False
|
| 103 |
+
[Question IDX: 102] GT: B | Voted Choice: B | Votes: {'A': 2, 'B': 14, 'C': 0, 'D': 0} | Correct: True
|
| 104 |
+
[Question IDX: 103] GT: C | Voted Choice: A | Votes: {'A': 15, 'B': 0, 'C': 4, 'D': 1} | Correct: False
|
| 105 |
+
[Question IDX: 104] GT: D | Voted Choice: D | Votes: {'A': 2, 'B': 3, 'C': 1, 'D': 11} | Correct: True
|
| 106 |
+
[Question IDX: 105] GT: B | Voted Choice: D | Votes: {'A': 5, 'B': 6, 'C': 1, 'D': 8} | Correct: False
|
| 107 |
+
[Question IDX: 106] GT: C | Voted Choice: C | Votes: {'A': 1, 'B': 0, 'C': 19, 'D': 1} | Correct: True
|
| 108 |
+
[Question IDX: 107] GT: D | Voted Choice: A | Votes: {'A': 9, 'B': 4, 'C': 1, 'D': 7} | Correct: False
|
| 109 |
+
[Question IDX: 108] GT: C | Voted Choice: D | Votes: {'A': 2, 'B': 1, 'C': 5, 'D': 9} | Correct: False
|
| 110 |
+
[Question IDX: 109] GT: C | Voted Choice: C | Votes: {'A': 0, 'B': 0, 'C': 13, 'D': 7} | Correct: True
|
| 111 |
+
[Question IDX: 110] GT: C | Voted Choice: C | Votes: {'A': 0, 'B': 4, 'C': 10, 'D': 3} | Correct: True
|
| 112 |
+
[Question IDX: 111] GT: A | Voted Choice: A | Votes: {'A': 8, 'B': 1, 'C': 1, 'D': 8} | Correct: True
|
| 113 |
+
[Question IDX: 112] GT: A | Voted Choice: D | Votes: {'A': 5, 'B': 0, 'C': 5, 'D': 10} | Correct: False
|
| 114 |
+
[Question IDX: 113] GT: D | Voted Choice: C | Votes: {'A': 5, 'B': 1, 'C': 7, 'D': 4} | Correct: False
|
| 115 |
+
[Question IDX: 114] GT: A | Voted Choice: A | Votes: {'A': 12, 'B': 6, 'C': 1, 'D': 0} | Correct: True
|
| 116 |
+
[Question IDX: 115] GT: B | Voted Choice: B | Votes: {'A': 0, 'B': 16, 'C': 0, 'D': 1} | Correct: True
|
| 117 |
+
[Question IDX: 116] GT: C | Voted Choice: C | Votes: {'A': 3, 'B': 5, 'C': 8, 'D': 1} | Correct: True
|
| 118 |
+
[Question IDX: 117] GT: B | Voted Choice: B | Votes: {'A': 2, 'B': 10, 'C': 3, 'D': 0} | Correct: True
|
| 119 |
+
[Question IDX: 118] GT: C | Voted Choice: C | Votes: {'A': 1, 'B': 0, 'C': 15, 'D': 1} | Correct: True
|
| 120 |
+
[Question IDX: 119] GT: C | Voted Choice: D | Votes: {'A': 1, 'B': 0, 'C': 0, 'D': 16} | Correct: False
|
| 121 |
+
[Question IDX: 120] GT: C | Voted Choice: C | Votes: {'A': 0, 'B': 0, 'C': 14, 'D': 1} | Correct: True
|
| 122 |
+
[Question IDX: 121] GT: B | Voted Choice: B | Votes: {'A': 3, 'B': 10, 'C': 1, 'D': 1} | Correct: True
|
| 123 |
+
[Question IDX: 122] GT: C | Voted Choice: C | Votes: {'A': 0, 'B': 0, 'C': 19, 'D': 1} | Correct: True
|
| 124 |
+
[Question IDX: 123] GT: B | Voted Choice: B | Votes: {'A': 4, 'B': 5, 'C': 4, 'D': 1} | Correct: True
|
| 125 |
+
[Question IDX: 124] GT: B | Voted Choice: D | Votes: {'A': 2, 'B': 5, 'C': 5, 'D': 5} | Correct: False
|
| 126 |
+
[Question IDX: 125] GT: D | Voted Choice: B | Votes: {'A': 3, 'B': 7, 'C': 3, 'D': 4} | Correct: False
|
| 127 |
+
[Question IDX: 126] GT: D | Voted Choice: D | Votes: {'A': 2, 'B': 2, 'C': 6, 'D': 7} | Correct: True
|
| 128 |
+
[Question IDX: 127] GT: A | Voted Choice: A | Votes: {'A': 9, 'B': 9, 'C': 0, 'D': 2} | Correct: True
|
| 129 |
+
[Question IDX: 128] GT: B | Voted Choice: C | Votes: {'A': 1, 'B': 8, 'C': 9, 'D': 2} | Correct: False
|
| 130 |
+
[Question IDX: 129] GT: C | Voted Choice: C | Votes: {'A': 2, 'B': 0, 'C': 13, 'D': 0} | Correct: True
|
| 131 |
+
[Question IDX: 130] GT: C | Voted Choice: C | Votes: {'A': 6, 'B': 2, 'C': 7, 'D': 3} | Correct: True
|
| 132 |
+
[Question IDX: 131] GT: B | Voted Choice: C | Votes: {'A': 1, 'B': 6, 'C': 8, 'D': 1} | Correct: False
|
| 133 |
+
[Question IDX: 132] GT: B | Voted Choice: B | Votes: {'A': 3, 'B': 7, 'C': 1, 'D': 1} | Correct: True
|
| 134 |
+
[Question IDX: 133] GT: D | Voted Choice: D | Votes: {'A': 4, 'B': 0, 'C': 4, 'D': 11} | Correct: True
|
| 135 |
+
[Question IDX: 134] GT: B | Voted Choice: A | Votes: {'A': 10, 'B': 6, 'C': 0, 'D': 2} | Correct: False
|
| 136 |
+
[Question IDX: 135] GT: A | Voted Choice: A | Votes: {'A': 11, 'B': 6, 'C': 2, 'D': 1} | Correct: True
|
| 137 |
+
[Question IDX: 136] GT: D | Voted Choice: B | Votes: {'A': 2, 'B': 8, 'C': 3, 'D': 4} | Correct: False
|
| 138 |
+
[Question IDX: 137] GT: D | Voted Choice: B | Votes: {'A': 4, 'B': 10, 'C': 1, 'D': 2} | Correct: False
|
| 139 |
+
[Question IDX: 138] GT: A | Voted Choice: A | Votes: {'A': 11, 'B': 0, 'C': 7, 'D': 1} | Correct: True
|
| 140 |
+
[Question IDX: 139] GT: D | Voted Choice: D | Votes: {'A': 1, 'B': 0, 'C': 0, 'D': 18} | Correct: True
|
models/SFT/math_eval_selfconsistent_aligned/detailed_comparison_choice_pure_regex_self_consistent_20250917_125611.jsonl
ADDED
|
@@ -0,0 +1,140 @@
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| 1 |
+
{"idx": 0, "question": "If the determinant $\\left|\\begin{array}{rrr}1 & 2 & 5 \\\\ 1 & 3 & -2 \\\\ 2 & 5 & x\\end{array}\\right| = 0$, then $x = (\\ )$ A) 2 B) -2 C) 3 D) -3", "gt": "C", "final_choice": "C", "vote_counts": {"A": 1, "B": 0, "C": 12, "D": 2}, "valid_extractions": 15, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 2 |
+
{"idx": 1, "question": "In the polynomial $f(x) = \\left|\\begin{array}{rrrr}2 & x & -5 & 3 \\\\ 1 & 2 & 3 & 4 \\\\ -1 & 0 & -2 & -3 \\\\ -1 & 7 & -2 & -2\\end{array}\\right|$ with respect to $x$, the coefficient of the linear term is ( ) A) 1 B) 2 C) -1 D) -2", "gt": "C", "final_choice": "C", "vote_counts": {"A": 3, "B": 4, "C": 7, "D": 3}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 3 |
+
{"idx": 2, "question": "The $n$-order determinant $\\left|\\begin{array}{ccccc}0 & 0 & \\cdots & 0 & 1 \\\\ 0 & 0 & \\cdots & 1 & 0 \\\\ \\vdots & \\vdots & & \\vdots & \\vdots \\\\ 0 & 1 & \\cdots & 0 & 0 \\\\ 1 & 0 & \\cdots & 0 & 0\\end{array}\\right|$ equals ( ) A) $(-1)^{n^2}$ B) $(-1)^{\\frac{1}{2}n(n-1)}$ C) $(-1)^{\\frac{1}{2}n(n+1)}$ D) 1", "gt": "B", "final_choice": "B", "vote_counts": {"A": 1, "B": 12, "C": 3, "D": 1}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 4 |
+
{"idx": 3, "question": "The value of the determinant $\\left|\\begin{array}{cccc}0 & a & 0 & 0 \\\\ b & c & 0 & 0 \\\\ 0 & 0 & d & e \\\\ 0 & 0 & 0 & f\\end{array}\\right|$ is ( ) A) $abcdef$ B) $-abdf$ C) $abdf$ D) $cdf$", "gt": "B", "final_choice": "C", "vote_counts": {"A": 3, "B": 3, "C": 8, "D": 1}, "valid_extractions": 15, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 5 |
+
{"idx": 4, "question": "Let $|A|$ be an $n$-order determinant, $|B|$ an $m$-order determinant, both non-zero. Let $|C| = \\left|\\begin{array}{cc}A & O \\\\ O & B\\end{array}\\right|$, $|D| = \\left|\\begin{array}{cc}O & A \\\\ B & O\\end{array}\\right|$. Then the value of $|C| : |D|$ is ( ) A) 1 B) -1 C) $(-1)^n$ D) $(-1)^{mn}$", "gt": "D", "final_choice": "D", "vote_counts": {"A": 4, "B": 0, "C": 2, "D": 11}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 6 |
+
{"idx": 5, "question": "If all elements of an $n$-order determinant $(n>1)$ are either $+1$ or $-1$, then its value must be ( ) A) 1 B) -1 C) an odd number D) an even number", "gt": "D", "final_choice": "D", "vote_counts": {"A": 0, "B": 1, "C": 8, "D": 8}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 7 |
+
{"idx": 6, "question": "The value of the determinant $\\left|\\begin{array}{cccc}8 & 27 & 64 & 125 \\\\ 4 & 9 & 16 & 25 \\\\ 2 & 3 & 4 & 5 \\\\ 1 & 1 & 1 & 1\\end{array}\\right|$ is ( ) A) 12 B) -12 C) 16 D) -16", "gt": "A", "final_choice": "B", "vote_counts": {"A": 3, "B": 6, "C": 2, "D": 4}, "valid_extractions": 15, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 8 |
+
{"idx": 7, "question": "The value of the determinant $\\left|\\begin{array}{cccc}a_1 & 0 & b_1 & 0 \\\\ 0 & c_1 & 0 & d_1 \\\\ a_2 & 0 & b_2 & 0 \\\\ 0 & c_2 & 0 & d_2\\end{array}\\right|$ is ( ) A) $a_1c_1b_2d_2 - a_2b_1c_2d_1$ B) $(a_2b_2 - a_1b_1)(c_2d_2 - c_1d_1)$ C) $a_1a_2b_1b_2c_1c_2d_1d_2$ D) $(a_1b_2 - a_2b_1)(c_1d_2 - c_2d_1)$", "gt": "D", "final_choice": "A", "vote_counts": {"A": 7, "B": 3, "C": 0, "D": 7}, "valid_extractions": 17, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 9 |
+
{"idx": 8, "question": "Given $\\left|\\begin{array}{ccc}a_{11} & a_{12} & a_{13} \\\\ a_{21} & a_{22} & a_{23} \\\\ a_{31} & a_{32} & a_{33}\\end{array}\\right| = d$, then $\\left|\\begin{array}{ccc}3a_{31} & 3a_{32} & 3a_{33} \\\\ 2a_{21} & 2a_{22} & 2a_{23} \\\\ -a_{11} & -a_{12} & -a_{13}\\end{array}\\right|$ equals ( ) A) $-6d$ B) $6d$ C) $4d$ D) $-4d$", "gt": "B", "final_choice": "A", "vote_counts": {"A": 15, "B": 2, "C": 0, "D": 1}, "valid_extractions": 18, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 10 |
+
{"idx": 9, "question": "When does the following system have a unique solution? $\\begin{cases} b x_1 + x_2 + 2x_3 = 1 \\\\ 2x_1 - x_2 + 2x_3 = -4 \\\\ 4x_1 + x_2 + 4x_3 = -2 \\end{cases}$ A) Always B) $b \\ne 2$ C) $b = 2$ D) $b = 1$", "gt": "B", "final_choice": "B", "vote_counts": {"A": 3, "B": 13, "C": 2, "D": 3}, "valid_extractions": 21, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 11 |
+
{"idx": 10, "question": "Which of the following statements is incorrect? A) The cofactor of element $(i,j)$ in $|A|$ equals the minor times $(-1)^{i+j}$ B) Multiplying all elements in the first row by 2 and all in the second row leaves $|A|$ unchanged C) The determinant of the transpose is the negative of the original determinant D) Swapping the first and second row and then swapping the first and second column leaves $|A|$ unchanged", "gt": "C", "final_choice": "B", "vote_counts": {"A": 2, "B": 15, "C": 3, "D": 0}, "valid_extractions": 20, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 12 |
+
{"idx": 11, "question": "Which of the following statements is correct? A) Multiplying every element of an $n$-order determinant $(n>1)$ by 2 multiplies its value by 2 B) If the coefficient matrix determinant $|\\mathcal{A}| = 0$, then all solutions are zero C) If an upper triangular determinant has value 0, then at least one diagonal element is zero D) If all elements above the main diagonal are zero, the determinant is zero", "gt": "C", "final_choice": "C", "vote_counts": {"A": 2, "B": 0, "C": 13, "D": 3}, "valid_extractions": 18, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 13 |
+
{"idx": 12, "question": "Let $f(x) = \\left|\\begin{array}{ccc}1 & 1 & 2 \\\\ 1 & 1 & x^2 - 2 \\\\ 2 & x^2 + 1 & 1\\end{array}\\right|$, find the root(s) of $f(x) = 0$. A) 1, 1, 2, 2 B) -1, -1, 2, 2 C) 1, -1, 2, -2 D) -1, -1, -2, -2", "gt": "C", "final_choice": "C", "vote_counts": {"A": 3, "B": 3, "C": 4, "D": 3}, "valid_extractions": 13, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 14 |
+
{"idx": 13, "question": "Let $f(x) = \\left|\\begin{array}{rrrr}x - 2 & x - 1 & x - 2 & x - 3 \\\\ 2x - 2 & 2x - 1 & 2x - 2 & 2x - 3 \\\\ 3x - 3 & 3x - 2 & 4x - 5 & 3x - 5 \\\\ 4x & 4x - 3 & 5x - 7 & 4x - 3\\end{array}\\right|$, then $f(x)$ equals ( ) A) 0, 1, 2, 3 B) 0, 1, 2 C) 0, 1, 1 D) 0, 1", "gt": "D", "final_choice": "C", "vote_counts": {"A": 4, "B": 3, "C": 9, "D": 2}, "valid_extractions": 18, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 15 |
+
{"idx": 14, "question": "Let the determinant of the following $n \\times n$ tridiagonal matrix be $\\left|\\begin{array}{cccccc}2 & 1 & 0 & \\cdots & 0 & 0 \\\\ 1 & 2 & 1 & \\cdots & 0 & 0 \\\\ 0 & 1 & 2 & \\cdots & 0 & 0 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ 0 & 0 & 0 & \\cdots & 2 & 1 \\\\ 0 & 0 & 0 & \\cdots & 1 & 2\\end{array}\\right|$, what is its value? A) $n$ B) $n + 1$ C) $n - 1$ D) $n(n + 1)$", "gt": "B", "final_choice": "B", "vote_counts": {"A": 5, "B": 11, "C": 1, "D": 1}, "valid_extractions": 18, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 16 |
+
{"idx": 15, "question": "Let $A$ be an $m \\times k$ matrix and $B$ a $k \\times t$ matrix. If the $j$-th column of $B$ consists entirely of zeros, then which of the following is correct? A) The $j$-th row of $AB$ is all zero B) The $j$-th column of $AB$ is all zero C) The $j$-th row of $BA$ is all zero D) The $j$-th column of $BA$ is all zero ", "gt": "B", "final_choice": "B", "vote_counts": {"A": 3, "B": 11, "C": 0, "D": 0}, "valid_extractions": 14, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 17 |
+
{"idx": 16, "question": "Let $A$ be an $n$-order matrix. If $A$ satisfies which of the following conditions, then $\\mathcal{I}_n - A$ is invertible? A) $A^n = O$ B) $A$ is invertible C) $|A| = 0$ D) All diagonal entries of $A$ are zero", "gt": "A", "final_choice": "A", "vote_counts": {"A": 9, "B": 7, "C": 1, "D": 2}, "valid_extractions": 19, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 18 |
+
{"idx": 17, "question": "Let $A, B, C$ be $n \\times n$ matrices. Which of the following statements is correct? A) If $A$ is invertible, then $AB = AC$ implies $BA = CA$ B) If $A$ is invertible, then $AB = BA$ C) If $A \\ne O$, then $AB = AC$ implies $B = C$ D) If $B \\ne C$, then $AB \\ne AC$", "gt": "A", "final_choice": "A", "vote_counts": {"A": 6, "B": 1, "C": 4, "D": 3}, "valid_extractions": 14, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 19 |
+
{"idx": 18, "question": "Which of the following statements is incorrect? A) A product of several elementary matrices is invertible B) The sum of invertible matrices may not be invertible C) The product of two elementary matrices is still an elementary matrix D) Any invertible matrix can be written as a product of finitely many elementary matrices", "gt": "C", "final_choice": "B", "vote_counts": {"A": 6, "B": 6, "C": 3, "D": 3}, "valid_extractions": 18, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 20 |
+
{"idx": 19, "question": "Which of the following statements about invertible and non-invertible matrices of the same size is correct? A) The sum of two non-invertible matrices is always non-invertible B) The sum of two invertible matrices is always invertible C) The product of two non-invertible matrices is always non-invertible D) The product of a non-invertible matrix and an invertible matrix is always invertible", "gt": "C", "final_choice": "C", "vote_counts": {"A": 0, "B": 4, "C": 9, "D": 4}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 21 |
+
{"idx": 20, "question": "Which of the following statements about commutativity of matrix multiplication is incorrect? A) If $A$ is invertible, then $A$ and $A^{-1}$ commute B) Invertible matrices always commute with elementary matrices C) Any $n$-order matrix commutes with $cI_n$, where $c$ is a scalar Elementary matrices do not necessarily commute with each other", "gt": "B", "final_choice": "B", "vote_counts": {"A": 2, "B": 9, "C": 4, "D": 0}, "valid_extractions": 15, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 22 |
+
{"idx": 21, "question": "If $A$ is invertible, then which of the following is true? A) The product of $A$ with any same-size matrix is invertible B) If $B$ is a same-size elementary matrix, then $\\det(AB) \\ne 0$ C) If $B$ is a same-size invertible matrix, then $\\det(A + B) \\ne 0$ D) The product of $A$ with any scalar is still invertible", "gt": "B", "final_choice": "B", "vote_counts": {"A": 3, "B": 10, "C": 2, "D": 4}, "valid_extractions": 19, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 23 |
+
{"idx": 22, "question": "Let matrix $A$ be transformed into matrix $B$ by a finite sequence of elementary row operations. Which of the following is correct? A) If both $A$ and $B$ are $n$-order square matrices, then $|A| = |B|$ B) If both $A$ and $B$ are $n$-order square matrices, then $|A|$ and $|B|$ are both zero or both non-zero C) If $A$ is invertible, $B$ may not be invertible D) $A = B$", "gt": "B", "final_choice": "B", "vote_counts": {"A": 4, "B": 10, "C": 2, "D": 1}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 24 |
+
{"idx": 23, "question": "Let $A$ be an $n$-order square matrix and $A^*$ its adjugate matrix. Which of the following is incorrect? A) If $A$ is invertible, then $A^*$ is also invertible B) If $A$ is not invertible, then $A^*$ is also not invertible C) If $|A^*| \\ne 0$, then $A$ is invertible D) $|AA^*| = |A|$", "gt": "D", "final_choice": "D", "vote_counts": {"A": 2, "B": 2, "C": 3, "D": 11}, "valid_extractions": 18, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 25 |
+
{"idx": 24, "question": "Which of the following matrices can be expressed as a product of a finite number of elementary matrices? A) $\\begin{pmatrix} 1 & 2 & 3 \\\\ 0 & 4 & 2 \\end{pmatrix}$ B) $\\begin{pmatrix} 1 & 2 & 0 \\\\ 0 & -1 & 3 \\\\ 0 & 0 & 2 \\end{pmatrix}$ C) $\\begin{pmatrix} 0 & 1 & 0 \\\\ 1 & 0 & 1 \\\\ 1 & 0 & 1 \\end{pmatrix}$ D) $\\begin{pmatrix} -1 & 2 & 3 \\\\ 0 & -2 & -1 \\\\ -3 & 2 & 7 \\end{pmatrix}$", "gt": "B", "final_choice": "B", "vote_counts": {"A": 2, "B": 6, "C": 5, "D": 4}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 26 |
+
{"idx": 25, "question": "Which of the following statements about elementary matrices is correct? A) All elementary matrices are invertible B) The sum of elementary matrices is still an elementary matrix C) The determinant of every elementary matrix is 1 D) The product of elementary matrices is still an elementary matrix", "gt": "A", "final_choice": "A", "vote_counts": {"A": 16, "B": 0, "C": 2, "D": 2}, "valid_extractions": 20, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 27 |
+
{"idx": 26, "question": "Let\n$\nA = \\begin{pmatrix} a_{11} & \\cdots & a_{1n} \\\\ \\vdots & & \\vdots \\\\ a_{n1} & \\cdots & a_{nn} \\end{pmatrix}, \\quad B = \\begin{pmatrix} A_{11} & \\cdots & A_{1n} \\\\ \\vdots & & \\vdots \\\\ A_{n1} & \\cdots & A_{nn} \\end{pmatrix},\n$\nwhere $A_{ij}$ is the cofactor of $a_{ij}$. Then: A) $A$ is the adjugate of $B$ B) $B$ is the adjugate of $A$ C) $B$ is the adjugate of $A'$ (transpose of $A$) D) None of the above", "gt": "C", "final_choice": "B", "vote_counts": {"A": 1, "B": 14, "C": 3, "D": 1}, "valid_extractions": 19, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 28 |
+
{"idx": 27, "question": "Let $A, B$ be square matrices. Given the block diagonal matrix $C = \\begin{pmatrix} A & O \\\\ O & B \\end{pmatrix}$, what is $C^* = \\; ?$ A) $\\begin{pmatrix} A^* & O \\\\ O & B^* \\end{pmatrix}$ B) $\\begin{pmatrix} |A|A^* & O \\\\ O & |B|B^* \\end{pmatrix}$ C) $\\begin{pmatrix} |B|A^* & O \\\\ O & |A||B^* \\end{pmatrix}$ D) $\\begin{pmatrix} |A||B|A^* & O \\\\ O & |A||B|B^* \\end{pmatrix}$", "gt": "C", "final_choice": "A", "vote_counts": {"A": 14, "B": 2, "C": 2, "D": 0}, "valid_extractions": 18, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 29 |
+
{"idx": 28, "question": "Let $A$ be an $n$-order square matrix. Matrix $B$ is obtained by swapping two columns of $A$. If $|A| \\ne |B|$, then: A) $|A|$ may be zero B) $|A| \\ne 0$ C) $|A + B| \\ne 0$ D) $|A - B| \\ne 0$", "gt": "B", "final_choice": "B", "vote_counts": {"A": 3, "B": 7, "C": 3, "D": 2}, "valid_extractions": 15, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 30 |
+
{"idx": 29, "question": "Let $A, B, A + B$ be all invertible $n$-order matrices. Then $(A^{-1} + B^{-1})^{-1}$ equals: A) $A + B$ B) $A - B$ C) $(A + B)^{-1}$ D) $A (A + B)^{-1} B$", "gt": "D", "final_choice": "D", "vote_counts": {"A": 2, "B": 0, "C": 3, "D": 13}, "valid_extractions": 18, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 31 |
+
{"idx": 30, "question": "Suppose any $n$-dimensional vector can be expressed as a linear combination of $\\alpha_1, \\alpha_2, \\cdots, \\alpha_n$. Then $\\alpha_1, \\alpha_2, \\cdots, \\alpha_n$ (\\small $\\alpha$) are: A) Linearly dependent B) Have rank equal to $n$ C) Have rank less than $n$ D) None of the above", "gt": "B", "final_choice": "B", "vote_counts": {"A": 2, "B": 15, "C": 0, "D": 0}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 32 |
+
{"idx": 31, "question": "Let $A$ be an $n \\times n$ matrix with $|A| = 0$. Then: A) Some two rows (or columns) of $A$ are proportional B) At least one row (or column) of $A$ is entirely zero C) At least one row is a linear combination of the others D) Every row is a linear combination of the others", "gt": "C", "final_choice": "C", "vote_counts": {"A": 1, "B": 3, "C": 12, "D": 1}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 33 |
+
{"idx": 32, "question": "The maximal linearly independent subset of a vector group: A) Has a unique number of subsets B) May have non-unique subsets C) Has a unique number of vectors D) May have a non-unique number of vectors", "gt": "C", "final_choice": "C", "vote_counts": {"A": 4, "B": 3, "C": 11, "D": 1}, "valid_extractions": 19, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 34 |
+
{"idx": 33, "question": "Let $\\alpha_1, \\alpha_2, \\cdots, \\alpha_s$ $(s > 1, \\alpha_1 \\ne 0)$ be linearly dependent. Then: A) Every $\\alpha_i$ ($i > 1$) can be expressed by $\\alpha_1, \\cdots, \\alpha_{i-1}$ B) None of the $\\alpha_i$ ($i > 1$) can be expressed by the previous ones C) There exists at least one $\\alpha_i$ ($i > 1$) that can be expressed by the previous ones D) There exists one $\\alpha_i$ ($i > 1$) that cannot be expressed by the previous ones", "gt": "C", "final_choice": "C", "vote_counts": {"A": 7, "B": 0, "C": 10, "D": 1}, "valid_extractions": 18, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 35 |
+
{"idx": 34, "question": "Let $A$ be an $m \\times n$ matrix with its $n$ column vectors linearly independent. Then the rank of $A$ is: A) Greater than $m$ B) Greater than $n$ C) Equal to $m$ D) Equal to $n$", "gt": "D", "final_choice": "D", "vote_counts": {"A": 0, "B": 1, "C": 1, "D": 19}, "valid_extractions": 21, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 36 |
+
{"idx": 35, "question": "Let matrices $A$ and $B$ be equivalent, and suppose $A$ has a nonzero $k$-order minor. Then the rank of $B$ is ( ) $k$. A) $<$ B) $=$ C) $\\geq$ D) $\\leq$", "gt": "C", "final_choice": "C", "vote_counts": {"A": 1, "B": 5, "C": 9, "D": 2}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 37 |
+
{"idx": 36, "question": "Given $A = \\begin{pmatrix} 1 & 2 & 3 \\\\ 2 & 4 & t \\\\ 3 & 6 & 9 \\end{pmatrix}$, and $B$ is a $3 \\times 3$ nonzero matrix such that $BA = O$, then ( ). A) When $t = 6$, the rank of $B$ must be 1. B) When $t = 6$, the rank of $B$ must be 2. C) When $t \\ne 6$, the rank of $B$ must be 1. D) When $t \\ne 6$, the rank of $B$ must be 2.", "gt": "C", "final_choice": "C", "vote_counts": {"A": 3, "B": 7, "C": 7, "D": 0}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 38 |
+
{"idx": 37, "question": "The necessary and sufficient condition for the vector group $\\alpha_1, \\alpha_2, \\cdots, \\alpha_r$ to have rank $r$ is: A) No zero vector in the group B) No two vectors have proportional components C) At least one vector cannot be expressed as a combination of the others D) The vectors are linearly independent", "gt": "D", "final_choice": "D", "vote_counts": {"A": 0, "B": 1, "C": 0, "D": 16}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 39 |
+
{"idx": 38, "question": "Let $A$ be a $3 \\times 3$ matrix, and $A^*$ its adjugate. If all $2 \\times 2$ minors of $A$ are zero, then: A) $\\mathrm{r}(A) \\le 1$, $\\mathrm{r}(A^*) = 0$ B) $\\mathrm{r}(A) = 1$, $\\mathrm{r}(A^*) = 0$ C) $\\mathrm{r}(A) \\le 1$, $\\mathrm{r}(A^*) = 1$ D) $\\mathrm{r}(A) = 2$, $\\mathrm{r}(A^*) = 1$", "gt": "A", "final_choice": "A", "vote_counts": {"A": 8, "B": 6, "C": 3, "D": 1}, "valid_extractions": 18, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 40 |
+
{"idx": 39, "question": "For the following homogeneous linear system to have a nonzero solution, the necessary condition is: A) $m \\le n$ B) $m = n$ C) $m > n$ D) The rank of the coefficient matrix is less than $n$", "gt": "D", "final_choice": "D", "vote_counts": {"A": 1, "B": 0, "C": 2, "D": 14}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 41 |
+
{"idx": 40, "question": "Let $A$ be an $m \\times n$ matrix. If the non-homogeneous system $Ax = \\beta$ has more than one solution, then: A) The rank of $A$ is less than $m$ B) $m < n$ C) $A$ is a zero matrix D) The homogeneous system $Ax = 0$ has non-unique solutions", "gt": "D", "final_choice": "A", "vote_counts": {"A": 12, "B": 0, "C": 1, "D": 7}, "valid_extractions": 20, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 42 |
+
{"idx": 41, "question": "Let $A$ be an $m \\times n$ matrix. If the system $Ax = 0$ has a nonzero solution, then: A) $m < n$ B) $\\operatorname{r}(A) < n$ C) Some columns of $A$ are proportional D) The row vectors of $A$ are linearly dependent", "gt": "B", "final_choice": "B", "vote_counts": {"A": 2, "B": 9, "C": 1, "D": 4}, "valid_extractions": 16, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 43 |
+
{"idx": 42, "question": "Let $a_1, a_2, a_3$ be a basis for the solution space of the system $Ax = 0$. Then ( ) is also a basis for the solution space of this system. A) $a_1 + a_2 - a_3, a_1 + a_2 + 5a_3, 4a_1 + a_2 - 2a_3$ B) $a_1 + 2a_2 + a_3, 2a_1 + a_2 + 2a_3, a_1 + a_2 + a_3$ C) $a_1 + a_2, a_1 + a_2 + a_3$ D) $a_1 - a_2, a_2 - a_3, a_3 - a_1$", "gt": "A", "final_choice": "A", "vote_counts": {"A": 7, "B": 4, "C": 2, "D": 5}, "valid_extractions": 18, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 44 |
+
{"idx": 43, "question": "Find $t$ such that the vector group $\\alpha_1 = (2,1,0)$, $\\alpha_2 = (3,2,5)$, $\\alpha_3 = (5,4,t)$ is linearly dependent. A) 5 B) 10 C) 15 D) 20", "gt": "C", "final_choice": "C", "vote_counts": {"A": 2, "B": 3, "C": 11, "D": 0}, "valid_extractions": 16, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 45 |
+
{"idx": 44, "question": "Let $\\alpha_1 = (1,4,1)$, $\\alpha_2 = (2,1,-5)$, $\\alpha_3 = (6,2,-16)$, and $\\beta = (2,t,3)$. When $t = \\underline{\\qquad}$, $\\beta$ can be expressed as a linear combination of $\\alpha_1, \\alpha_2, \\alpha_3$. A) 1 B) 3 C) 6 D) 9", "gt": "D", "final_choice": "C", "vote_counts": {"A": 1, "B": 1, "C": 8, "D": 5}, "valid_extractions": 15, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 46 |
+
{"idx": 45, "question": "Let $\\varphi$ be a transformation on the 3D row vector space. Which of the following $\\varphi$ is not a linear transformation? A) $\\varphi(a_1,a_2,a_3) = (2a_1 - a_2 + a_3, a_2 + 5a_3, a_1 - a_3)$ B) $\\varphi(a_1,a_2,a_3) = (a_1^2, a_2^2, a_3^2)$ C) $\\varphi(a_1,a_2,a_3) = (0, a_1, 0)$ D) $\\varphi(a_1,a_2,a_3) = (3a_3, 3a_2, 3a_1)$ ", "gt": "B", "final_choice": "B", "vote_counts": {"A": 1, "B": 15, "C": 2, "D": 0}, "valid_extractions": 18, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 47 |
+
{"idx": 46, "question": "Let $\\varphi$ be a linear transformation on an $n$-dimensional vector space $V$. Which of the following $\\varphi$ is not an isomorphism? A) $\\varphi$ is injective B) $\\dim \\text{Im}\\, \\varphi = n$ C) $\\varphi$ is bijective D) $\\varphi^n = \\mathbf{0}$", "gt": "D", "final_choice": "D", "vote_counts": {"A": 1, "B": 0, "C": 3, "D": 12}, "valid_extractions": 16, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 48 |
+
{"idx": 47, "question": "Let $\\varphi$ be the linear transformation on the three-dimensional vector space $V$ over $F$ given by the matrix $\\begin{pmatrix} 1 & -1 & 2 \\\\ 2 & 0 & 1 \\\\ 1 & 2 & -1 \\end{pmatrix}$ in the basis $\\{e_1, e_2, e_3\\}$. What is the matrix representation of $\\varphi$ in the basis $\\{e_3, e_2, e_1\\}$? A) $\\begin{pmatrix} 1 & -1 & 2 \\\\ 2 & 0 & 1 \\\\ 1 & 2 & -1 \\end{pmatrix}$ B) $\\begin{pmatrix} 1 & 2 & 1 \\\\ -1 & 0 & 2 \\\\ 2 & 1 & -1 \\end{pmatrix}$ C) $\\begin{pmatrix} -1 & 2 & 1 \\\\ 1 & 0 & 2 \\\\ 2 & -1 & 1 \\end{pmatrix}$ D) $\\begin{pmatrix} 2 & -1 & 1 \\\\ 1 & 0 & 2 \\\\ -1 & 2 & 1 \\end{pmatrix}$", "gt": "C", "final_choice": "B", "vote_counts": {"A": 4, "B": 5, "C": 0, "D": 4}, "valid_extractions": 13, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 49 |
+
{"idx": 48, "question": "Let $V$ be an $n$-dimensional vector space. Let $\\varphi$ and $\\psi$ be linear transformations on $V$. A necessary and sufficient condition for $\\varphi$ and $\\psi$ to have the same image space dimension is: A) Both are invertible B) They have the same null space C) They have the same image space D) Their matrix representations have the same rank under any basi", "gt": "D", "final_choice": "D", "vote_counts": {"A": 1, "B": 2, "C": 3, "D": 14}, "valid_extractions": 20, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 50 |
+
{"idx": 49, "question": "Let $V$ be an $n$-dimensional vector space. The dimension of the vector space formed by all linear transformations on $V$ is: A) $n$ B) $\\frac{1}{2}n(n+1)$ C) $n^2$ D) Infinite", "gt": "C", "final_choice": "C", "vote_counts": {"A": 0, "B": 1, "C": 12, "D": 2}, "valid_extractions": 15, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 51 |
+
{"idx": 50, "question": "Let $V$ be an $n$-dimensional vector space with a basis. The subspaces generated by each basis vector are invariant under the linear transformation $\\varphi$ on $V$. What is the matrix of $\\varphi$ under this basis? A) It must be invertible. B) It must be an upper triangular matrix, but not necessarily diagonal. C) It must be a lower triangular matrix, but not necessarily diagonal. D) It must be a diagonal matrix.", "gt": "D", "final_choice": "B", "vote_counts": {"A": 3, "B": 8, "C": 0, "D": 5}, "valid_extractions": 16, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 52 |
+
{"idx": 51, "question": "Let $\\varphi, \\psi$ be linear transformations on an $n$-dimensional vector space $V$. Under which condition must $\\varphi = \\psi$? A) Same image and kernel space B) $\\varphi(\\alpha_i) = \\psi(\\alpha_i)$ for $n$ linearly independent vectors $\\alpha_i$ C) Both are invertible D) Same rank", "gt": "B", "final_choice": "B", "vote_counts": {"A": 1, "B": 12, "C": 0, "D": 1}, "valid_extractions": 14, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 53 |
+
{"idx": 52, "question": "Let $\\varphi$ be a nonlinear transformation on an $n$-dimensional vector space $V$, and we know that $\\varphi$ is not invertible. The condition below can guarantee that the kernel of $\\varphi$ is the intersection of the kernel space and its image space is ( ) A) The matrix representation of $\\varphi$ in the basis of $V$ satisfies $A^n = O$. B) The matrix representation of $\\varphi$ in some basis of $V$ satisfies $A^2 = A$. C) The dimension of the kernel space of $\\varphi$ is equal to the dimension of its image space. D) The dimension of the kernel space of $\\varphi$ is equal to the dimension of its image space plus $n$.", "gt": "B", "final_choice": "C", "vote_counts": {"A": 3, "B": 1, "C": 9, "D": 0}, "valid_extractions": 13, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 54 |
+
{"idx": 53, "question": "Which condition does NOT ensure a linear transformation $\\varphi$ on $n$-dimensional space $V$ is invertible? A) The determinant of its matrix representation is nonzero B) The matrix representation is symmetric C) $\\varphi$ maps $n$ linearly independent vectors to $n$ linearly independent vectors D) $\\varphi$ has no nontrivial invariant subspace", "gt": "B", "final_choice": "B", "vote_counts": {"A": 1, "B": 13, "C": 1, "D": 4}, "valid_extractions": 19, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 55 |
+
{"idx": 54, "question": "Let $V$ be a 2D real vector space. Which of the following matrices defines a linear transformation on $V$ that has no nontrivial invariant subspace? A) $\\begin{pmatrix} 1 & -1 \\\\ 0 & 1 \\end{pmatrix}$ B) $\\begin{pmatrix} 1 & 1 \\\\ 1 & 1 \\end{pmatrix}$ C) $\\begin{pmatrix} 0 & -1 \\\\ 1 & 0 \\end{pmatrix}$ D) $\\begin{pmatrix} 2 & 0 \\\\ 0 & 3 \\end{pmatrix}$", "gt": "C", "final_choice": "C", "vote_counts": {"A": 4, "B": 1, "C": 12, "D": 2}, "valid_extractions": 19, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 56 |
+
{"idx": 55, "question": "If the functions $f_1(x)$, $f_2(x)$, $f_3(x)$ are mutually independent, then ( ) A) $f_1(x)$, $f_2(x)$ must be independent B) $f_1(x)$, $f_2(x)$ are independent, or $f_1(x)$, $f_3(x)$ are independent, or $f_2(x)$, $f_3(x)$ are independent C) If $(f_1(x), f_2(x)) = d_1(x)$, $(f_2(x), f_3(x)) = d_2(x)$, then $(d_1(x), d_2(x)) = 1$ D) There exist $u(x), v(x)$ such that $f_3(x) = f_1(x) u(x) + f_2(x) v(x)$", "gt": "C", "final_choice": "D", "vote_counts": {"A": 2, "B": 3, "C": 4, "D": 4}, "valid_extractions": 13, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 57 |
+
{"idx": 56, "question": "Let $m$ and $n$ be integers greater than 1. Then the remainder of the division of $x^{3m} + x^{3n}$ by $x^2 + x + 1$ is ( ) A) $x + 1$ B) 0 C) 1 D) 2", "gt": "D", "final_choice": "D", "vote_counts": {"A": 2, "B": 3, "C": 3, "D": 11}, "valid_extractions": 19, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 58 |
+
{"idx": 57, "question": "If $(f(x), g(x)) = d(x)$, then the equation formed is ( ) A) $\\frac{f(x)}{d(x)}, g(x) = 1$ B) $\\frac{f(x)}{d(x)} \\cdot \\frac{g(x)}{d(x)} = 1$ C) $(f(x), d(x)) = 1$ D) $(f(x), d(x), g(x)) = 1$", "gt": "B", "final_choice": "B", "vote_counts": {"A": 5, "B": 6, "C": 4, "D": 4}, "valid_extractions": 19, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 59 |
+
{"idx": 58, "question": "If $(f(x), g(x)) = 1$, $(f(x), h(x)) = 1$, then the expression that is not necessarily mutually independent is ( ) A) $f(x), f(x) + g(x)$ B) $f(x), h(x) + g(x)$ C) $f(x), h(x) g(x)$ D) $f(x)g(x), f(x) + g(x)$", "gt": "B", "final_choice": "D", "vote_counts": {"A": 1, "B": 3, "C": 6, "D": 10}, "valid_extractions": 20, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 60 |
+
{"idx": 59, "question": "Let $f(x)$ be a polynomial over the field $F$, and let $K$ be a subfield of $F$. Then ( ) A) If $f(x)$ is not reducible over $F$, then $f(x)$ is also not reducible over $K$ B) If $f(x)$ is reducible over $F$, then $f(x)$ is also reducible over $K$ C) If $f(x)$ has a root in $F$, then $f(x)$ must have a root in $K$ D) If $f(x)$ is reducible over $K$, then $f(x)$ is also reducible over $F$", "gt": "D", "final_choice": "B", "vote_counts": {"A": 4, "B": 6, "C": 1, "D": 1}, "valid_extractions": 12, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 61 |
+
{"idx": 60, "question": "Let $f(x)$ be a polynomial with integer coefficients. Which of the following statements is true? A) $f(x)$ has a rational root if and only if it is reducible over the rationals B) If a reduced fraction $\\frac{q}{p}$ is a root of $f(x)$, then $q$ divides the constant term of $f(x)$ C) If a prime $p$ divides all coefficients of $f(x)$ except the leading term, then $f(x)$ is irreducible over the rationals D) If $f(x)$ has a repeated factor, then it must have a repeated root over the rationals", "gt": "B", "final_choice": "B", "vote_counts": {"A": 1, "B": 13, "C": 2, "D": 2}, "valid_extractions": 18, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 62 |
+
{"idx": 61, "question": "Let $f(x) = c p_1(x) p_2(x) \\cdots p_k(x)$ be a rational-coefficient polynomial where $p_i(x)$ are distinct irreducible monic polynomials of degree greater than 1. Then over the complex field: A) No repeated roots B) Possibly repeated roots C) No real roots D) Exactly $k$ real roots", "gt": "A", "final_choice": "A", "vote_counts": {"A": 13, "B": 4, "C": 1, "D": 2}, "valid_extractions": 20, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 63 |
+
{"idx": 62, "question": "Which of the following polynomials has repeated roots over the complex field? A) $x^n + 1$ B) $1 + x + \\frac{x^2}{2!} + \\cdots + \\frac{x^n}{n!}$ C) $x^n + x^{n-1} + \\cdots + 1$ D) Not clearly specified", "gt": "D", "final_choice": "C", "vote_counts": {"A": 3, "B": 6, "C": 10, "D": 1}, "valid_extractions": 20, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 64 |
+
{"idx": 63, "question": "Which of the following polynomials is not reducible over the field of rational numbers ( )? A) $x^{2n+1} + 1$ B) $x^4 + 4$ C) $x^6 + x^3 + 1 D) $x^4 - 3x^3 + 5x^2 + 2x - 5$", "gt": "C", "final_choice": "A", "vote_counts": {"A": 7, "B": 2, "C": 6, "D": 2}, "valid_extractions": 17, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 65 |
+
{"idx": 64, "question": "Which of the following statements is correct? A) If a complex number $c$ is a root of multiplicity $k$ of the polynomial $f(x)$, then $c$ is a root of multiplicity $k-1$ of $f'(x)$. B) If a complex number $c$ is a root of multiplicity $k$ of the polynomial $f(x)$, then $c$ is a root of multiplicity $k+1$ of $f'(x)$. C) If a complex number $c$ is a root of multiplicity $k$ of the polynomial $f(x)$, then $c$ is also a root of multiplicity $k$ of $f'(x)$. D) If the greatest common divisor of $f(x)$ and $f'(x)$ is a polynomial of degree $k$, then $f(x)$ has $k$ distinct roots.", "gt": "A", "final_choice": "A", "vote_counts": {"A": 10, "B": 0, "C": 3, "D": 2}, "valid_extractions": 15, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 66 |
+
{"idx": 65, "question": "Let $f(x)$ be a reducible polynomial over the rational numbers. Which conclusion is necessarily true? A) If $f(x) \\mid g(x) h(x)$, then $f(x) \\mid g(x)$ or $f(x) \\mid h(x)$ B) $f(x)$ must have a rational root C) If $p_1(x)$ and $p_2(x)$ are irreducible factors of $f(x)$ that are coprime over $\\mathbb{Q}$, then they share no common root over $\\mathbb{C}$ D) The degrees of all irreducible factors of $f(x)$ are at most 2", "gt": "C", "final_choice": "C", "vote_counts": {"A": 1, "B": 2, "C": 6, "D": 2}, "valid_extractions": 11, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 67 |
+
{"idx": 66, "question": "Let $f(x)$ be a real-coefficient polynomial. Which of the following statements is incorrect? A) If $\\deg f(x)$ is odd, then $f(x)$ must have a real root B) If $f(x)$ is reducible, then it must have a real root C) If all coefficients of $f(x)$ are positive real numbers, then it has no positive real root D) If $f(x)$ is reducible, then each irreducible factor has degree at most 2", "gt": "B", "final_choice": "D", "vote_counts": {"A": 3, "B": 2, "C": 4, "D": 4}, "valid_extractions": 13, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 68 |
+
{"idx": 67, "question": "The minimal degree of a rational-coefficient polynomial that has $\\sqrt{2} - 1 + i$ as a root is: A) 2 B) 3 C) 4 D) 6", "gt": "C", "final_choice": "A", "vote_counts": {"A": 10, "B": 0, "C": 5, "D": 0}, "valid_extractions": 15, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 69 |
+
{"idx": 68, "question": "Which of the following polynomials is a symmetric polynomial? A) $x_1^3 + x_2^3 - x_3^3$ B) $2x_1^3 + x_2^3 + x_3^3$ C) $x_1^3 + x_2^3 + x_3^3 + 2x_1x_2 + 2x_1x_3 + 2x_2x_3$ D) $x_1^2 + x_2^2 + x_3^2 + x_1x_2$", "gt": "C", "final_choice": "C", "vote_counts": {"A": 2, "B": 0, "C": 18, "D": 0}, "valid_extractions": 20, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 70 |
+
{"idx": 69, "question": "Which of the following pairs of polynomials share a common root? A) $x^3 - 4x^2 + 5x - 2$, $x^2 + x + 1$ B) $x^3 - 4x^2 + 5x - 2$, $x^2 + x - 2$ C) $x^4 - x^2 + 3x - 8$, $x^3 + x^2 + x + 1$ D) $x^4 - x^2 + 3x - 8$, $x^3 - 1$", "gt": "B", "final_choice": "B", "vote_counts": {"A": 4, "B": 6, "C": 2, "D": 3}, "valid_extractions": 15, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 71 |
+
{"idx": 70, "question": "Which of the following is *not* a necessary and sufficient condition for an $n$-order matrix $A$ to be invertible? A) All eigenvalues of $A$ are nonzero B) The determinant of $A$ is nonzero C) The constant term of the characteristic polynomial of $A$ is nonzero D) $A$ has $n$ linearly independent eigenvectors", "gt": "D", "final_choice": "C", "vote_counts": {"A": 4, "B": 2, "C": 8, "D": 5}, "valid_extractions": 19, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 72 |
+
{"idx": 71, "question": "If a matrix $A$ satisfies $A^2 = I$, then its eigenvalues can be: A) $0, 1$ B) $0, -1$ C) $0, 1, -1$ D) $1, -1$", "gt": "D", "final_choice": "D", "vote_counts": {"A": 2, "B": 1, "C": 2, "D": 13}, "valid_extractions": 18, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 73 |
+
{"idx": 72, "question": "Let $A$ be a $3 \\times 3$ matrix with eigenvalues $1, 0, -1$. Let $f(x) = x^2 - 2x - 1$. Then the eigenvalues of $f(A)$ are: A) $-2, -1, 2$ B) $-2, -1, -2$ C) $2, 1, -2$ D) $2, 0, -2$", "gt": "A", "final_choice": "A", "vote_counts": {"A": 12, "B": 3, "C": 2, "D": 0}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 74 |
+
{"idx": 73, "question": "Under which condition is an $n$-order matrix $A$ guaranteed to be similar to a diagonal matrix? A) $A$ has $n$ distinct eigenvectors B) $A$ is an upper triangular matrix C) $A$ has $n$ distinct eigenvalues D) $A$ is invertible", "gt": "C", "final_choice": "C", "vote_counts": {"A": 5, "B": 1, "C": 10, "D": 1}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 75 |
+
{"idx": 74, "question": "What is the necessary and sufficient condition for an $n$-order matrix $A$ to have *every* nonzero $n$-dimensional column vector as an eigenvector? A) $A$ is a diagonal matrix B) $A$ is a scalar matrix C) $A$ is the identity matrix D) $A$ is the zero matrix", "gt": "B", "final_choice": "B", "vote_counts": {"A": 5, "B": 10, "C": 2, "D": 0}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 76 |
+
{"idx": 75, "question": "Which of the following matrices is similar to a diagonal matrix? A) Not specified B) \\( \\begin{pmatrix}3 & 1\\end{pmatrix} \\) C) \\( \\begin{pmatrix}1 \\\\ -2\\end{pmatrix}, 0 \\) D) 5, 3", "gt": "C", "final_choice": "B", "vote_counts": {"A": 5, "B": 5, "C": 3, "D": 5}, "valid_extractions": 18, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 77 |
+
{"idx": 76, "question": "Which of the following statements is *false*? A) Eigenvectors corresponding to distinct eigenvalues are linearly independent B) Eigenvectors corresponding to the same eigenvalue are always linearly dependent C) Similar matrices must have the same eigenvalues D) Matrices with the same eigenvalues are not necessarily similar", "gt": "B", "final_choice": "B", "vote_counts": {"A": 4, "B": 11, "C": 1, "D": 1}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 78 |
+
{"idx": 77, "question": "Let $A$ be an $n$-order matrix. Swap the first and second rows, then swap the first and second columns to obtain matrix $B$. The relation between the eigenvalues of $A$ and $B$ is: A) Exactly the same B) Eigenvalues of $B$ are the negatives of those of $A$ C) Eigenvalues of $B$ are the squares of those of $A$ D) No certain relation", "gt": "A", "final_choice": "A", "vote_counts": {"A": 13, "B": 5, "C": 0, "D": 0}, "valid_extractions": 18, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 79 |
+
{"idx": 78, "question": "Let\n$\nA = \\begin{pmatrix}2 & 1 & 0 \\\\ 1 & 2 & 0 \\\\ 0 & 0 & t\\end{pmatrix}, \\quad B = \\begin{pmatrix}5 & 5 & 5 \\\\ 0 & 3 & 3 \\\\ 0 & 0 & 1\\end{pmatrix}.\n$\nTo make $A$ and $B$ similar, $t$ must be: A) 0 B) 1 C) 3 D) 5", "gt": "D", "final_choice": "C", "vote_counts": {"A": 2, "B": 4, "C": 6, "D": 6}, "valid_extractions": 18, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 80 |
+
{"idx": 79, "question": "If the matrix $A$ is similar to itself, then ( ) A) $A$ must be a scalar matrix B) $A$ must be a zero matrix C) $A$ must be a diagonal matrix D) $A$ can be any matrix", "gt": "C", "final_choice": "A", "vote_counts": {"A": 10, "B": 0, "C": 2, "D": 8}, "valid_extractions": 20, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 81 |
+
{"idx": 80, "question": "Which of the following is *not* an invertible transformation on a $\\lambda$-matrix? A) Multiply a row of the $\\lambda$-matrix by a polynomial $f(\\lambda)$ of degree > 0 B) A finite product of elementary transformations of $\\lambda$-matrices C) Swap two columns of the $\\lambda$-matrix D) Add $\\lambda$ times the first row to the second row", "gt": "A", "final_choice": "A", "vote_counts": {"A": 11, "B": 1, "C": 2, "D": 7}, "valid_extractions": 21, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 82 |
+
{"idx": 81, "question": "The necessary and sufficient condition for an $n \\times n$ $\\lambda$-matrix $A(\\lambda)$ to be invertible is: A) $A(\\lambda) \\neq 0$ B) $\\left|A(\\lambda)\\right| \\neq 0$ C) $\\left|A(\\lambda)\\right|$ is a nonzero constant D) All diagonal elements in the Smith normal form of $A(\\lambda)$ are nonzero", "gt": "C", "final_choice": "B", "vote_counts": {"A": 1, "B": 14, "C": 3, "D": 1}, "valid_extractions": 19, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 83 |
+
{"idx": 82, "question": "Which of the following $n \\times n$ matrices are necessarily similar? A) $A$ and $kA$, where $k$ is a nonzero real number B) $A$ and its transpose $A^{\\prime}$ C) $A$ and its adjugate $A^*$ D) $A$ and $AP$, where $P$ is an elementary matrix", "gt": "B", "final_choice": "D", "vote_counts": {"A": 4, "B": 3, "C": 2, "D": 7}, "valid_extractions": 16, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 84 |
+
{"idx": 83, "question": "Which of the following statements is correct? A) If matrices $A$ and $B$ have the same invariant factors, then they share the same minimal polynomial B) Matrices with identical characteristic and minimal polynomials must be similar C) Matrices with the same eigenvalues are similar D) Matrix equivalence implies matrix similarity", "gt": "A", "final_choice": "A", "vote_counts": {"A": 9, "B": 6, "C": 1, "D": 0}, "valid_extractions": 16, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 85 |
+
{"idx": 84, "question": "Which of the following statements is false? A) If matrix $A$ has $k$ elementary divisors, then its Jordan canonical form has $k$ Jordan blocks B) If matrix $A$ has $k$ non-constant invariant factors, then it has $k$ Jordan blocks C) If $A$ has an elementary divisor of degree $k$, then the corresponding Jordan block is of size $k \\times k$ D) The sum of the degrees of all elementary divisors equals the order of matrix $A$", "gt": "B", "final_choice": "B", "vote_counts": {"A": 8, "B": 9, "C": 1, "D": 1}, "valid_extractions": 19, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 86 |
+
{"idx": 85, "question": "If matrices $A$ and $B$ have the same elementary divisors, which of the following statements is *not necessarily* true? A) $A$ and $B$ have the same invariant factors B) There exists an invertible matrix $P$ such that $P^{-1}AP$ and $P^{-1}BP$ are both in Jordan form C) $A$ and $B$ are equivalent D) $|A| = |B|$", "gt": "B", "final_choice": "C", "vote_counts": {"A": 3, "B": 3, "C": 8, "D": 4}, "valid_extractions": 18, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 87 |
+
{"idx": 86, "question": "Which of the following real matrices must be similar over $\\mathbb{R}$ to a diagonal matrix? A) Elementary matrix B) Nonzero nilpotent matrix (i.e., $A^k = O$) C) Idempotent matrix (i.e., $A^2 = A$) D) Upper triangular matrix", "gt": "C", "final_choice": "D", "vote_counts": {"A": 4, "B": 0, "C": 6, "D": 9}, "valid_extractions": 19, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 88 |
+
{"idx": 87, "question": "Which of the following is a necessary and sufficient condition for matrix $A$ to be invertible? A) All invariant factors of $A$ are nonzero B) All determinantal divisors of $A$ are nonzero C) The last invariant factor of $A$ has a nonzero constant term D) At least one invariant factor of $A$ has a nonzero constant term", "gt": "C", "final_choice": "A", "vote_counts": {"A": 9, "B": 2, "C": 3, "D": 1}, "valid_extractions": 15, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 89 |
+
{"idx": 88, "question": "Let $A$ be an $8 \\times 8$ matrix with elementary divisors $(\\lambda - 1)^2, (\\lambda - 1)^2, (\\lambda + 1)^3, \\lambda$. What is the set of its invariant factors? A) $1, 1, 1, 1, \\lambda, (\\lambda - 1)^2, (\\lambda - 1)^2, (\\lambda + 1)^3$ B) $1, 1, 1, 1, 1, (\\lambda - 1)^2, (\\lambda - 1)^2, \\lambda (\\lambda + 1)^3$ C) $1, 1, 1, 1, 1, (\\lambda - 1)^2, \\lambda (\\lambda - 1)^2, (\\lambda + 1)^3$ D) $1, 1, 1, 1, 1, 1, (\\lambda - 1)^2, \\lambda (\\lambda - 1)^2 (\\lambda + 1)^3$", "gt": "D", "final_choice": "C", "vote_counts": {"A": 3, "B": 2, "C": 7, "D": 5}, "valid_extractions": 17, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 90 |
+
{"idx": 89, "question": "Given $A = \\begin{pmatrix}1 & 0 & 0 \\\\ 0 & 2 & 1 \\\\ 0 & 0 & 2 \\end{pmatrix}$, which of the following matrices is similar to $A$? A) $\\begin{pmatrix}1 & 0 & 0 \\\\ 0 & 2 & 0 \\\\ 0 & 0 & 2 \\end{pmatrix}$ B) $\\begin{pmatrix}1 & 1 & 0 \\\\ 0 & 2 & 0 \\\\ 0 & 0 & 2 \\end{pmatrix}$ C) $\\begin{pmatrix}1 & 0 & 0 \\\\ 0 & 2 & 0 \\\\ 0 & 1 & 2 \\end{pmatrix}$ D) $\\begin{pmatrix}1 & 1 & 0 \\\\ 0 & 1 & 1 \\\\ 0 & 0 & 2 \\end{pmatrix}$", "gt": "C", "final_choice": "A", "vote_counts": {"A": 10, "B": 1, "C": 6, "D": 2}, "valid_extractions": 19, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 91 |
+
{"idx": 90, "question": "Let $A$ be a $3 \\times 3$ matrix with minimal polynomial $(\\lambda - 1)^2$. Then the Jordan canonical form of $A$ is: A) $\\begin{pmatrix}1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1\\end{pmatrix}$ B) $\\begin{pmatrix}1 & 1 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1\\end{pmatrix}$ C) $\\begin{pmatrix}1 & 1 & 0 \\\\ 0 & 1 & 1 \\\\ 0 & 0 & 1\\end{pmatrix}$ D) $\\begin{pmatrix}1 & 1 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 0\\end{pmatrix}$", "gt": "B", "final_choice": "B", "vote_counts": {"A": 2, "B": 10, "C": 1, "D": 0}, "valid_extractions": 13, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 92 |
+
{"idx": 91, "question": "Let the characteristic polynomial of $A$ be $(\\lambda - 2)(\\lambda - 1)^3$ and its minimal polynomial be $(\\lambda - 2)(\\lambda - 1)^2$. Then: A) $A$ is a $4 \\times 4$ matrix and has 2 Jordan blocks B) $A$ is a $3 \\times 3$ matrix and has 2 Jordan blocks C) $A$ is a $4 \\times 4$ matrix and has 3 Jordan blocks D) $A$ is a $4 \\times 4$ matrix and is diagonalizable", "gt": "C", "final_choice": "C", "vote_counts": {"A": 9, "B": 1, "C": 9, "D": 1}, "valid_extractions": 20, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 93 |
+
{"idx": 92, "question": "Which of the following matrices necessarily has a Jordan canonical form that is not diagonal? A) Elementary matrix $E_{ij}$ with $i \\neq j$ B) Lower triangular matrix with distinct diagonal elements C) Elementary matrix D) Invertible matrix", "gt": "A", "final_choice": "A", "vote_counts": {"A": 11, "B": 3, "C": 2, "D": 3}, "valid_extractions": 19, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 94 |
+
{"idx": 93, "question": "If $A$ is a nonzero $10 \\times 10$ matrix satisfying $A^2 = O$, what is the maximum size of a Jordan block in the Jordan form of $A$? A) 2 B) 3 C) 4 D) 5", "gt": "A", "final_choice": "A", "vote_counts": {"A": 13, "B": 1, "C": 2, "D": 3}, "valid_extractions": 19, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 95 |
+
{"idx": 94, "question": "Given that matrix $A$ has an invariant factor $\\lambda^2 + \\lambda$, which of the following statements is correct? A) $A$ is similar to a diagonal matrix B) $A$ is a singular matrix C) All elementary divisors of $A$ are powers of $\\lambda$ or $\\lambda + 1$ D) The eigenvalues of $A$ are $0$ and $-1$ (possibly with multiplicity)", "gt": "B", "final_choice": "D", "vote_counts": {"A": 0, "B": 2, "C": 4, "D": 9}, "valid_extractions": 15, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 96 |
+
{"idx": 95, "question": "Given the matrix $A = \\begin{pmatrix}0 & \\frac{1}{\\sqrt{2}} & 1 \\\\ \\frac{1}{\\sqrt{2}} & 3 & -\\frac{3}{2} \\\\ 1 & -\\frac{3}{2} & 0 \\end{pmatrix}$, which of the following is its associated quadratic form? A) $x_1^2 + \\frac{1}{2}x_1x_2 + 2x_1x_3 - 3x_2x_3$ B) $2\\sqrt{2}x_1x_2 - 3x_2^2 + x_1x_3 - \\frac{3}{2}x_2x_3$ C) $\\sqrt{2}x_1x_2 + 3x_2^2 + 2x_1x_3 - 3x_2x_3$ D) $x_1x_2 - 3x_2^2 + x_1x_3 - 3x_2x_3$", "gt": "C", "final_choice": "A", "vote_counts": {"A": 7, "B": 5, "C": 1, "D": 4}, "valid_extractions": 17, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 97 |
+
{"idx": 96, "question": "Let $f(x_1, x_2, x_3) = x_1x_2 + x_1x_3 - x_2x_3$. What is its associated real symmetric matrix? A) $\\begin{pmatrix}1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & -1\\end{pmatrix}$ B) $\\begin{pmatrix}0 & 1 & 1 \\\\ 1 & 0 & -1 \\\\ 1 & -1 & 0\\end{pmatrix}$ C) $\\begin{pmatrix}0 & 1 & -1 \\\\ 0 & 0 & 1 \\\\ 0 & 0 & 0\\end{pmatrix}$ D) $\\begin{pmatrix}0 & \\frac{1}{2} & \\frac{1}{2} \\\\ \\frac{1}{2} & 0 & -\\frac{1}{2} \\\\ \\frac{1}{2} & -\\frac{1}{2} & 0\\end{pmatrix}$", "gt": "D", "final_choice": "D", "vote_counts": {"A": 2, "B": 6, "C": 1, "D": 8}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 98 |
+
{"idx": 97, "question": "Which of the following quadratic forms has a positive inertia index of 2? A) $f(x) = (x_1 + x_2 + x_3)^2 - 2x_2^2$ B) $f(x) = x_1^2 + x_2^2 + 5x_3^2 - 6x_1x_2 - 2x_1x_3 + 2x_2x_3$ C) $f(x) = x_1^2 + x_2^2 + x_3^2 - x_1x_2$ D) $f(x) = x_1^2 + x_2^2 + x_3^2 - 2x_1x_2 + 2x_1x_3 - 2x_2x_3$", "gt": "B", "final_choice": "C", "vote_counts": {"A": 6, "B": 2, "C": 7, "D": 1}, "valid_extractions": 16, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 99 |
+
{"idx": 98, "question": "Given $A = \\begin{pmatrix}-1 & 0 & 0 \\\\ 0 & \\frac{1}{3} & 0 \\\\ 0 & 0 & -2\\end{pmatrix}$, which matrix is congruent to $A$? A) $\\begin{pmatrix}-1 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 0 & 0\\end{pmatrix}$ B) $\\begin{pmatrix}1 & 0 & 0 \\\\ 0 & -2 & 0 \\\\ 0 & 0 & 1\\end{pmatrix}$ C) $\\begin{pmatrix}2 & 0 & 0 \\\\ 0 & -1 & 0 \\\\ 0 & 0 & -5\\end{pmatrix}$ D) $\\begin{pmatrix}2 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 3\\end{pmatrix}$", "gt": "C", "final_choice": "C", "vote_counts": {"A": 1, "B": 3, "C": 7, "D": 7}, "valid_extractions": 18, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 100 |
+
{"idx": 99, "question": "If a real symmetric matrix $A$ has rank $r$ and positive inertia index $m$, what is its signature (sign count difference)? A) $r$ B) $m - r$ C) $2m - r$ D) $r - m$", "gt": "C", "final_choice": "C", "vote_counts": {"A": 0, "B": 3, "C": 8, "D": 3}, "valid_extractions": 14, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 101 |
+
{"idx": 100, "question": "Which of the following quadratic forms is positive definite? A) $f = x_1^2 + x_2^2$ B) $f = x_1^2 + x_2^2 + 2x_1x_2 + x_3^2$ C) $f = 4x_1^2 + 3x_2^2 + 6x_3^2 - x_1x_2 - x_1x_3$ D) $f = x_1^2 + x_2^2 + x_3^2 + 2x_1x_2 + 2x_1x_3 + 2x_2x_3$", "gt": "C", "final_choice": "C", "vote_counts": {"A": 3, "B": 1, "C": 12, "D": 2}, "valid_extractions": 18, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 102 |
+
{"idx": 101, "question": "A real symmetric matrix $A$ is positive definite if and only if: A) $|A| > 0$ B) There exists an invertible matrix $C$ such that $A = C'C$ C) For all non-zero vectors $x$, $x'Ax > 0$ D) There exists a vector $\\alpha \\ne 0$ such that $\\alpha'A\\alpha > 0$", "gt": "B", "final_choice": "C", "vote_counts": {"A": 3, "B": 2, "C": 11, "D": 1}, "valid_extractions": 17, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 103 |
+
{"idx": 102, "question": "The quadratic form $f(x_1, \\dots, x_n)$ is positive definite if its coefficient matrix is: A) Real symmetric with all diagonal elements positive B) Real symmetric and all leading principal minors are positive C) Real symmetric with all entries positive D) Real symmetric and determinant is positive", "gt": "B", "final_choice": "B", "vote_counts": {"A": 2, "B": 14, "C": 0, "D": 0}, "valid_extractions": 16, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 104 |
+
{"idx": 103, "question": "If $A, B$ are both positive definite matrices, then: A) $AB$ and $A+B$ are both positive definite B) $AB$ is positive definite, $A+B$ is not C) $A+B$ is positive definite, $AB$ is not necessarily D) $AB$ is not positive definite, $A+B$ is positive definite", "gt": "C", "final_choice": "A", "vote_counts": {"A": 15, "B": 0, "C": 4, "D": 1}, "valid_extractions": 20, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 105 |
+
{"idx": 104, "question": "For the quadratic form $f = (k+1)x_1^2 + (k-1)x_2^2 + (k-2)x_3^2$, which condition on $k$ ensures $f$ is positive definite? A) $k > 0$ B) $k > 1$ C) $k = 1$ D) $k > 2$", "gt": "D", "final_choice": "D", "vote_counts": {"A": 2, "B": 3, "C": 1, "D": 11}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 106 |
+
{"idx": 105, "question": "Which of the following conditions does *not* guarantee that a real symmetric matrix $A$ is positive definite? A) $A^{-1}$ is positive definite B) The negative inertia index of $A$ is 0 C) The positive inertia index of $A$ equals $n$ D) $A$ is congruent to the identity matrix", "gt": "B", "final_choice": "D", "vote_counts": {"A": 5, "B": 6, "C": 1, "D": 8}, "valid_extractions": 20, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 107 |
+
{"idx": 106, "question": "If $A$ is an $n \\times n$ positive definite matrix, which of the following is false? A) $|A| > 0$ B) $A$ is non-singular C) All entries of $A$ are positive D) All diagonal elements of $A$ are positive", "gt": "C", "final_choice": "C", "vote_counts": {"A": 1, "B": 0, "C": 19, "D": 1}, "valid_extractions": 21, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 108 |
+
{"idx": 107, "question": "Let $A = \\begin{pmatrix}k & k & 1 \\\\ k & k & 0 \\\\ 1 & 0 & k^2\\end{pmatrix}$. For which values of $k$ is $A$ positive definite? A) $k > 1$ B) $k^2 > 1$ C) $k < 0$ D) No such $k$ exists", "gt": "D", "final_choice": "A", "vote_counts": {"A": 9, "B": 4, "C": 1, "D": 7}, "valid_extractions": 21, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 109 |
+
{"idx": 108, "question": "Which of the following matrices is congruent to the identity matrix? A) $\\begin{pmatrix}1 & 1 & 1 \\\\ 1 & 1 & 1 \\\\ 1 & 1 & 1\\end{pmatrix}$, $\\begin{pmatrix}1 & 2 & 1 \\\\ 2 & 7 & 1 \\\\ 1 & 1 & 1\\end{pmatrix}$ B) $\\begin{pmatrix}1 & 0 & 1 \\\\ 0 & 1 & 0 \\\\ 1 & 0 & 1\\end{pmatrix}$, $\\begin{pmatrix}2 & -1 & 2 \\\\ -1 & 3 & -\\frac{3}{2} \\\\ 2 & -\\frac{3}{2} & -4\\end{pmatrix}$ C) $\\begin{pmatrix}2 & 0 & 0 \\\\ 0 & 2 & 0 \\\\ 0 & 0 & 2\\end{pmatrix}$ D) $\\begin{pmatrix}1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1\\end{pmatrix}$", "gt": "C", "final_choice": "D", "vote_counts": {"A": 2, "B": 1, "C": 5, "D": 9}, "valid_extractions": 17, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 110 |
+
{"idx": 109, "question": "If $A$ is an $n \\times n$ real skew-symmetric matrix, then $A'A$ is: A) Positive definite B) Negative definite C) Positive semi-definite D) Negative semi-definite", "gt": "C", "final_choice": "C", "vote_counts": {"A": 0, "B": 0, "C": 13, "D": 7}, "valid_extractions": 20, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 111 |
+
{"idx": 110, "question": "Let $A$, $B$ be orthogonal matrices, $k$ be a nonzero real number, and $P$ an invertible matrix. Which of the following is true? A) $A + B$ is also an orthogonal matrix B) $kA$ is also an orthogonal matrix C) $AB$ is also an orthogonal matrix D) $P^{-1} A P$ is also an orthogonal matrix", "gt": "C", "final_choice": "C", "vote_counts": {"A": 0, "B": 4, "C": 10, "D": 3}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 112 |
+
{"idx": 111, "question": "Which of the following statements is correct? A) If nonzero vectors $u$ and $v$ are orthogonal, then $u$ and $v$ are linearly independent B) If vectors $v_1$ and $v_2$ are orthogonal, and $v_2$ and $v_3$ are orthogonal, then $v_1$ and $v_3$ are orthogonal C) If $U, W$ are subspaces of Euclidean space $V$ and $U \\cap W = \\{0\\}$, then $U$ and $W$ are orthogonal D) If $U, W$ are subspaces of Euclidean space $V$ such that $U \\cap W = \\{0\\}$ and $\\dim V = \\dim U + \\dim W$, then $U$ is the orthogonal complement of $W$", "gt": "A", "final_choice": "A", "vote_counts": {"A": 8, "B": 1, "C": 1, "D": 8}, "valid_extractions": 18, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 113 |
+
{"idx": 112, "question": "Which of the following matrices is orthogonally similar to $M = \\begin{pmatrix}1 \\\\ 0\\end{pmatrix} \\begin{pmatrix}0 & -1\\end{pmatrix}$? A) $\\begin{pmatrix}0 \\\\ 1\\end{pmatrix}$ B) $\\begin{pmatrix}1 & 2 \\\\ 0 & -1\\end{pmatrix}$ C) $\\begin{pmatrix}1 & 1 \\\\ & -1\\end{pmatrix}$ D) $\\begin{pmatrix}0 & 1 \\\\ -1 & 0\\end{pmatrix}$", "gt": "A", "final_choice": "D", "vote_counts": {"A": 5, "B": 0, "C": 5, "D": 10}, "valid_extractions": 20, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 114 |
+
{"idx": 113, "question": "Let $A$ be a real symmetric matrix of order $n$. Which of the following is true? A) The absolute values of all eigenvalues of $A$ are 1 B) $A$ has $n$ distinct eigenvalues C) Any $n$ linearly independent eigenvectors of $A$ are pairwise orthogonal D) There exists an orthogonal matrix $P$ such that $P'AP$ is diagonal", "gt": "D", "final_choice": "C", "vote_counts": {"A": 5, "B": 1, "C": 7, "D": 4}, "valid_extractions": 17, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 115 |
+
{"idx": 114, "question": "Which of the following statements is correct? A) Two similar real symmetric matrices must be orthogonally similar B) Positive definite matrices of the same size must be similar C) Matrices with the same eigenvalues must be similar D) Two congruent matrices must be similar", "gt": "A", "final_choice": "A", "vote_counts": {"A": 12, "B": 6, "C": 1, "D": 0}, "valid_extractions": 19, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 116 |
+
{"idx": 115, "question": "Let $A$ be an $n$-by-$n$ orthogonal matrix. Which of the following is true? A) All eigenvalues of $A$ are real B) All eigenvalues of $A$ have modulus 1 C) $A$ has $n$ distinct eigenvalues D) The linearly independent eigenvectors of $A$ are pairwise orthogonal", "gt": "B", "final_choice": "B", "vote_counts": {"A": 0, "B": 16, "C": 0, "D": 1}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 117 |
+
{"idx": 116, "question": "Let $\\varphi$ be a symmetric transformation on $n$-dimensional Euclidean space $V$. Which of the following is true? A) The matrix of $\\varphi$ in any basis is a real symmetric matrix B) The matrix of $\\varphi$ in any orthogonal basis is a real symmetric matrix C) The matrix of $\\varphi$ in any orthonormal basis is a real symmetric matrix D) The matrix of $\\varphi$ in any basis is orthogonally similar", "gt": "C", "final_choice": "C", "vote_counts": {"A": 3, "B": 5, "C": 8, "D": 1}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 118 |
+
{"idx": 117, "question": "Which of the following conditions guarantees that an $n$-by-$n$ matrix $A$ is orthogonal? A) $A$ transforms an orthogonal set of $n$-dimensional vectors into another orthogonal set B) For any $n$-dimensional vector $\\alpha$, $(A\\alpha, A\\alpha) = \\|\\alpha\\|^2$ C) $A$ preserves the angle between any two vectors D) All eigenvalues of $A$ are 1 or $-1$", "gt": "B", "final_choice": "B", "vote_counts": {"A": 2, "B": 10, "C": 3, "D": 0}, "valid_extractions": 15, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 119 |
+
{"idx": 118, "question": "A linear transformation $\\varphi$ on $n$-dimensional Euclidean space is orthogonal if and only if: A) The matrix of $\\varphi$ in any basis is orthogonal B) The matrix of $\\varphi$ in any orthogonal basis is orthogonal C) The matrix of $\\varphi$ in any orthonormal basis is orthogonal D) The matrix of $\\varphi$ in any orthonormal basis is symmetric", "gt": "C", "final_choice": "C", "vote_counts": {"A": 1, "B": 0, "C": 15, "D": 1}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 120 |
+
{"idx": 119, "question": "Let $u, v$ be vectors in $n$-dimensional Euclidean space $V$. Which of the following is incorrect? A) If the inner product of $u$ with each basis vector of $V$ is zero, then $u = 0$ B) If $e_1, \\dots, e_n$ is a basis of $V$, and $(u, e_i) = (v, e_i)$ for all $i$, then $u = v$ C) If $e_1, \\dots, e_n$ is a basis of $V$, and $(u, e_1)^2 + \\dots + (u, e_n)^2 = 1$, then $\\|u\\| = 1$ D) If $u$ and $v$ are unit vectors and distinct, then $u$ and $v$ are linearly dependent if and only if $u = -v$", "gt": "C", "final_choice": "D", "vote_counts": {"A": 1, "B": 0, "C": 0, "D": 16}, "valid_extractions": 17, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 121 |
+
{"idx": 120, "question": "The necessary and sufficient condition for an upper triangular matrix $A$ to be orthogonal is: A) $A$ is a diagonal matrix B) $A$ is the identity matrix C) $A$ is a diagonal matrix with entries $1$ or $-1$ on the diagonal D) $A$ is a diagonal matrix with entries $1$, $-1$, or $0$ on the diagonal", "gt": "C", "final_choice": "C", "vote_counts": {"A": 0, "B": 0, "C": 14, "D": 1}, "valid_extractions": 15, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 122 |
+
{"idx": 121, "question": "Let $A$ be an $n$-by-$n$ matrix. Which of the following is the necessary and sufficient condition for $A$ to be orthogonal? A) All eigenvalues of $A$ are $1$ or $-1$ B) The column vectors of $A$ form a standard orthonormal basis for $\\mathbb{R}^n$ C) The column vectors of $A$ are pairwise orthogonal D) $A$ is orthogonally similar to the identity matrix", "gt": "B", "final_choice": "B", "vote_counts": {"A": 3, "B": 10, "C": 1, "D": 1}, "valid_extractions": 15, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 123 |
+
{"idx": 122, "question": "Which of the following real matrices does not have real eigenvalues? A) Real symmetric matrix B) Real matrix of odd order C) Non-zero $2 \\times 2$ real skew-symmetric matrix D) Real upper triangular matrix", "gt": "C", "final_choice": "C", "vote_counts": {"A": 0, "B": 0, "C": 19, "D": 1}, "valid_extractions": 20, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 124 |
+
{"idx": 123, "question": "Two real symmetric matrices of order $n$ are similar if and only if: A) They are congruent B) They have real eigenvalues $\\lambda_1, \\lambda_2, \\dots, \\lambda_n$ C) They have real and distinct eigenvalues $\\lambda_1, \\lambda_2, \\dots, \\lambda_n$ D) They are both orthogonal matrices", "gt": "B", "final_choice": "B", "vote_counts": {"A": 4, "B": 5, "C": 4, "D": 1}, "valid_extractions": 14, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 125 |
+
{"idx": 124, "question": "Which of the following statements is correct? A) The linear combination of two orthogonal transformations is still an orthogonal transformation B) The linear combination of two symmetric transformations is still a symmetric transformation C) A symmetric transformation maps orthogonal vector sets to orthogonal vector sets D) A symmetric transformation is necessarily invertible", "gt": "B", "final_choice": "D", "vote_counts": {"A": 2, "B": 5, "C": 5, "D": 5}, "valid_extractions": 17, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 126 |
+
{"idx": 125, "question": "Which of the following transformations preserves the orthogonality of an orthogonal matrix $A$? A) Applying an elementary row operation to $A$ B) Similar transformation: $P^{-1} A P$, where $P$ is invertible C) Congruent transformation: $C' A C$, where $C$ is invertible D) Swapping the $i$-th and $j$-th rows and then swapping the $i$-th and $j$-th columns", "gt": "D", "final_choice": "B", "vote_counts": {"A": 3, "B": 7, "C": 3, "D": 4}, "valid_extractions": 17, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 127 |
+
{"idx": 126, "question": "Let $A$ be an orthogonal matrix. Which of the following is not necessarily an orthogonal matrix? A) $A'$ B) $A^*$ C) $-A$ D) $P^{-1} A P$, where $P$ is invertible", "gt": "D", "final_choice": "D", "vote_counts": {"A": 2, "B": 2, "C": 6, "D": 7}, "valid_extractions": 17, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 128 |
+
{"idx": 127, "question": "Let $A$ be a $2 \\times 2$ real normal matrix that is not symmetric. Then $A$ is orthogonal if and only if: A) $\\det(A) = 1$ B) $\\det(A) = -1$ C) $A$ is invertible D) $A$ is singular", "gt": "A", "final_choice": "A", "vote_counts": {"A": 9, "B": 9, "C": 0, "D": 2}, "valid_extractions": 20, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 129 |
+
{"idx": 128, "question": "Let $\\varphi, \\psi$ be self-adjoint operators on an $n$-dimensional unitary space. Which of the following linear transformations is still self-adjoint? A) $\\psi \\varphi$ B) $i(\\varphi \\psi - \\psi \\varphi)$ C) $i(\\varphi \\psi + \\psi \\varphi)$ D) $\\varphi \\psi - \\psi \\varphi$", "gt": "B", "final_choice": "C", "vote_counts": {"A": 1, "B": 8, "C": 9, "D": 2}, "valid_extractions": 20, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 130 |
+
{"idx": 129, "question": "Let $V$ be an $n$-dimensional Euclidean space, and $\\alpha_1, \\alpha_2, \\dots, \\alpha_n$ be a basis of $V$. Let $G$ be the metric matrix with respect to this basis. Suppose the matrix of linear transformation $\\varphi$ under this basis is $A$. Then the necessary and sufficient condition for $\\varphi$ to be self-adjoint is: A) $A$ is symmetric B) $A$ is orthogonal C) $A'G = GA$ D) $AG = GA$", "gt": "C", "final_choice": "C", "vote_counts": {"A": 2, "B": 0, "C": 13, "D": 0}, "valid_extractions": 15, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 131 |
+
{"idx": 130, "question": "Let $\\varphi: V \\to U$ be a linear mapping between finite-dimensional spaces, and let $\\varphi^*$ be its dual map. Then, ( ). A) If $\\varphi$ is injective, then $\\varphi^*$ is also injective B) If $\\varphi$ is surjective, then $\\varphi^*$ is also surjective C) If $\\varphi$ is injective, then $\\varphi^*$ is surjective D) If $\\varphi^*$ is injective, then $\\varphi$ is injective", "gt": "C", "final_choice": "C", "vote_counts": {"A": 6, "B": 2, "C": 7, "D": 3}, "valid_extractions": 18, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 132 |
+
{"idx": 131, "question": "Let $V$ be the three-dimensional space over field $F$. $V^*$ is its dual space, and $v_1, v_2, v_3$ is a basis of $V$. The dual basis of $v_1, v_1 + v_2, v_1 + v_2 + v_3$ is ( ). A) $v_1^*, v_1^* + v_2^*, v_1^* + v_2^* + v_3^*$ B) $v_1^* - v_2^*, v_2^* - v_3^*, v_3^*$ C) $v_1^*, v_1^* - v_2^*, v_2^* - v_3^*$ D) $v_1^* - v_2^*, v_1^* - v_3^*, v_2^* + v_3^*$", "gt": "B", "final_choice": "C", "vote_counts": {"A": 1, "B": 6, "C": 8, "D": 1}, "valid_extractions": 16, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 133 |
+
{"idx": 132, "question": "Let $\\varphi$ be a linear operator on an $n$-dimensional space $V$, and $\\varphi^*$ be its dual. If $V_1$ is a subspace of $V$, define $V_1^{\\perp} = \\{f \\in V^* \\mid f(V_1) = 0\\}$. Which of the following is correct? A) $\\dim(\\text{Ker}\\, \\varphi^*)^{\\perp} = \\dim \\text{Ker}\\, \\varphi$ B) $\\dim(\\text{Ker}\\, \\varphi)^{\\perp} = \\dim \\text{Im}\\, \\varphi$ C) $\\dim(\\text{Im}\\, \\varphi^*)^{\\perp} = \\dim \\text{Im}\\, \\varphi$ D) $\\dim \\text{Im}\\, \\varphi^* = \\dim(\\text{Im}\\, \\varphi)^{\\perp}$", "gt": "B", "final_choice": "B", "vote_counts": {"A": 3, "B": 7, "C": 1, "D": 1}, "valid_extractions": 12, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 134 |
+
{"idx": 133, "question": "Let $V$ be the three-dimensional row vector space over field $F$. Which of the following defines a bilinear map $f: V \\times V \\to F$? (Assume $x = (x_1, x_2, x_3), y = (y_1, y_2, y_3)$) A) $f(x,y) = x_1^2 + x_1 y_1 + 2 x_1 y_2$ B) $f(x,y) = x_1 y_1 + x_2 y_3 + 2 x_1 - 3 y_3$ C) $f(x,y) = x_1 x_2 + x_2 y_3 + 2 x_3 y_3$ D) $f(x,y) = 2 x_1 y_2 + x_2 y_1 - 5 x_3 y_2 + 2 x_3 y_3$", "gt": "D", "final_choice": "D", "vote_counts": {"A": 4, "B": 0, "C": 4, "D": 11}, "valid_extractions": 19, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 135 |
+
{"idx": 134, "question": "The representation matrices of a bilinear form $g: V \\times V \\to \\mathbb{F}$ in different bases of $V$ are ( ). A) necessarily similar B) necessarily congruent C) necessarily orthogonally similar D) necessarily equal", "gt": "B", "final_choice": "A", "vote_counts": {"A": 10, "B": 6, "C": 0, "D": 2}, "valid_extractions": 18, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 136 |
+
{"idx": 135, "question": "Let $g: U \\times V \\to \\mathbb{F}$ be a bilinear form, and let $\\varphi: V \\to U^*$ be defined by $\\varphi(v) = g(-, v)$. Then, ( ). A) $\\text{Ker}\\, \\varphi$ is the right radical of $g$ B) $\\text{Ker}\\, \\varphi$ is the left radical of $g$ C) $\\text{Im}\\, \\varphi$ is the right radical of $g$ D) $\\text{Im}\\, \\varphi$ is the left radical of $g$", "gt": "A", "final_choice": "A", "vote_counts": {"A": 11, "B": 6, "C": 2, "D": 1}, "valid_extractions": 20, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 137 |
+
{"idx": 136, "question": "Which of the following scalar products is non-degenerate? A) $g(x,y) = x_1 y_2 + 2 x_1 y_3 - 3 x_3 y_3$ for $x,y \\in F^4$ B) $g(x,y) = x_1 y_1 + 2 x_1 y_3 - 2 x_2 y_1 + x_2 y_2 + x_3 y_2 + 4 x_3 y_3$ for $x,y \\in F^4$ C) $g(x,y) = x' A y$ for $x,y \\in \\mathbb{R}^n$ where $A$ is an idempotent matrix not equal to $I_n$ D) $g(A,B) = \\text{tr}(AB)$ for $A,B$ in the space of $n \\times n$ matrices", "gt": "D", "final_choice": "B", "vote_counts": {"A": 2, "B": 8, "C": 3, "D": 4}, "valid_extractions": 17, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 138 |
+
{"idx": 137, "question": "Let $g,h$ be two non-degenerate scalar products on $V$, and let $v_1, \\dots, v_n$ be a basis of $V$. Suppose $A$ is the matrix of $g$ and $B$ the matrix of $h$ in this basis. If $h(x, y) = g(\\varphi(x), y)$, then the matrix of $\\varphi$ is ( ). A) $AB$ B) $A^{-1}B$ C) $A'B$ D) $(A^{-1})' B'$", "gt": "D", "final_choice": "B", "vote_counts": {"A": 4, "B": 10, "C": 1, "D": 2}, "valid_extractions": 17, "total_samples": 21, "is_correct": false, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 139 |
+
{"idx": 138, "question": "Two $n \\times n$ skew-symmetric matrices over a field $F$ are congruent if and only if ( ). A) They are equivalent B) They are similar C) They are unitarily similar (as complex matrices) D) They have the same eigenvalues", "gt": "A", "final_choice": "A", "vote_counts": {"A": 11, "B": 0, "C": 7, "D": 1}, "valid_extractions": 19, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
| 140 |
+
{"idx": 139, "question": "Let $U,V$ be $m$- and $n$-dimensional vector spaces over a field $F$. The dimension of the space of all bilinear forms from $U \\times V \\to \\mathbb{F}$ is ( ). A) $m$ B) $n$ C) $m + n$ D) $mn$", "gt": "D", "final_choice": "D", "vote_counts": {"A": 1, "B": 0, "C": 0, "D": 18}, "valid_extractions": 19, "total_samples": 21, "is_correct": true, "method_used": "pure_regex_self_consistent_voting", "confidence": "high", "needs_manual_review": false}
|
models/SFT/math_eval_selfconsistent_aligned/summary_choice_pure_regex_self_consistent_20250917_125611.json
ADDED
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{
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| 2 |
+
"acc": 64.28571428571429,
|
| 3 |
+
"pure_regex_acc": 64.28571428571429,
|
| 4 |
+
"total_questions": 140,
|
| 5 |
+
"correct_count": 90,
|
| 6 |
+
"total_samples_processed": 140,
|
| 7 |
+
"voting_enabled": true,
|
| 8 |
+
"manual_review_count": 0,
|
| 9 |
+
"api_calls_made": 0,
|
| 10 |
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"total_cost_usd": 0.0,
|
| 11 |
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"output_files": {
|
| 12 |
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"detailed_comparison": "/mnt/arc/zijunga1/mathevo/math-evaluation-harness/SFT/math_eval_selfconsistent_aligned/detailed_comparison_choice_pure_regex_self_consistent_20250917_125611.jsonl",
|
| 13 |
+
"summary": "/mnt/arc/zijunga1/mathevo/math-evaluation-harness/SFT/math_eval_selfconsistent_aligned/summary_choice_pure_regex_self_consistent_20250917_125611.json",
|
| 14 |
+
"voting_details": "/mnt/arc/zijunga1/mathevo/math-evaluation-harness/SFT/math_eval_selfconsistent_aligned/voting_details_choice_pure_regex_self_consistent_20250917_125611.jsonl",
|
| 15 |
+
"console_log": "/mnt/arc/zijunga1/mathevo/math-evaluation-harness/SFT/math_eval_selfconsistent_aligned/console_log_choice_pure_regex_self_consistent_20250917_125611.txt"
|
| 16 |
+
}
|
| 17 |
+
}
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models/SFT/math_eval_selfconsistent_aligned/voting_details_choice_pure_regex_self_consistent_20250917_125611.jsonl
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models/SFT/merges.txt
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models/SFT/model-00001-of-00004.safetensors
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models/SFT/model-00002-of-00004.safetensors
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version https://git-lfs.github.com/spec/v1
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models/SFT/model-00003-of-00004.safetensors
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version https://git-lfs.github.com/spec/v1
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models/SFT/model-00004-of-00004.safetensors
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version https://git-lfs.github.com/spec/v1
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models/SFT/model.safetensors.index.json
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| 331 |
+
"model.layers.8.self_attn.v_proj.bias": "model-00001-of-00004.safetensors",
|
| 332 |
+
"model.layers.8.self_attn.v_proj.weight": "model-00001-of-00004.safetensors",
|
| 333 |
+
"model.layers.9.input_layernorm.weight": "model-00002-of-00004.safetensors",
|
| 334 |
+
"model.layers.9.mlp.down_proj.weight": "model-00002-of-00004.safetensors",
|
| 335 |
+
"model.layers.9.mlp.gate_proj.weight": "model-00002-of-00004.safetensors",
|
| 336 |
+
"model.layers.9.mlp.up_proj.weight": "model-00002-of-00004.safetensors",
|
| 337 |
+
"model.layers.9.post_attention_layernorm.weight": "model-00002-of-00004.safetensors",
|
| 338 |
+
"model.layers.9.self_attn.k_proj.bias": "model-00002-of-00004.safetensors",
|
| 339 |
+
"model.layers.9.self_attn.k_proj.weight": "model-00002-of-00004.safetensors",
|
| 340 |
+
"model.layers.9.self_attn.o_proj.weight": "model-00002-of-00004.safetensors",
|
| 341 |
+
"model.layers.9.self_attn.q_proj.bias": "model-00002-of-00004.safetensors",
|
| 342 |
+
"model.layers.9.self_attn.q_proj.weight": "model-00002-of-00004.safetensors",
|
| 343 |
+
"model.layers.9.self_attn.v_proj.bias": "model-00002-of-00004.safetensors",
|
| 344 |
+
"model.layers.9.self_attn.v_proj.weight": "model-00002-of-00004.safetensors",
|
| 345 |
+
"model.norm.weight": "model-00003-of-00004.safetensors"
|
| 346 |
+
}
|
| 347 |
+
}
|
models/SFT/special_tokens_map.json
ADDED
|
@@ -0,0 +1,20 @@
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|
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|
| 1 |
+
{
|
| 2 |
+
"additional_special_tokens": [
|
| 3 |
+
"<|im_start|>",
|
| 4 |
+
"<|im_end|>"
|
| 5 |
+
],
|
| 6 |
+
"eos_token": {
|
| 7 |
+
"content": "<|endoftext|>",
|
| 8 |
+
"lstrip": false,
|
| 9 |
+
"normalized": false,
|
| 10 |
+
"rstrip": false,
|
| 11 |
+
"single_word": false
|
| 12 |
+
},
|
| 13 |
+
"pad_token": {
|
| 14 |
+
"content": "<|endoftext|>",
|
| 15 |
+
"lstrip": false,
|
| 16 |
+
"normalized": false,
|
| 17 |
+
"rstrip": false,
|
| 18 |
+
"single_word": false
|
| 19 |
+
}
|
| 20 |
+
}
|
models/SFT/tokenizer.json
ADDED
|
@@ -0,0 +1,3 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
version https://git-lfs.github.com/spec/v1
|
| 2 |
+
oid sha256:bcfe42da0a4497e8b2b172c1f9f4ec423a46dc12907f4349c55025f670422ba9
|
| 3 |
+
size 11418266
|
models/SFT/tokenizer_config.json
ADDED
|
@@ -0,0 +1,43 @@
|
|
|
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|
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|
|
|
|
|
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|
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|
|
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|
|
|
|
|
|
| 1 |
+
{
|
| 2 |
+
"add_prefix_space": false,
|
| 3 |
+
"added_tokens_decoder": {
|
| 4 |
+
"151643": {
|
| 5 |
+
"content": "<|endoftext|>",
|
| 6 |
+
"lstrip": false,
|
| 7 |
+
"normalized": false,
|
| 8 |
+
"rstrip": false,
|
| 9 |
+
"single_word": false,
|
| 10 |
+
"special": true
|
| 11 |
+
},
|
| 12 |
+
"151644": {
|
| 13 |
+
"content": "<|im_start|>",
|
| 14 |
+
"lstrip": false,
|
| 15 |
+
"normalized": false,
|
| 16 |
+
"rstrip": false,
|
| 17 |
+
"single_word": false,
|
| 18 |
+
"special": true
|
| 19 |
+
},
|
| 20 |
+
"151645": {
|
| 21 |
+
"content": "<|im_end|>",
|
| 22 |
+
"lstrip": false,
|
| 23 |
+
"normalized": false,
|
| 24 |
+
"rstrip": false,
|
| 25 |
+
"single_word": false,
|
| 26 |
+
"special": true
|
| 27 |
+
}
|
| 28 |
+
},
|
| 29 |
+
"additional_special_tokens": [
|
| 30 |
+
"<|im_start|>",
|
| 31 |
+
"<|im_end|>"
|
| 32 |
+
],
|
| 33 |
+
"bos_token": null,
|
| 34 |
+
"clean_up_tokenization_spaces": false,
|
| 35 |
+
"eos_token": "<|endoftext|>",
|
| 36 |
+
"errors": "replace",
|
| 37 |
+
"extra_special_tokens": {},
|
| 38 |
+
"model_max_length": 32768,
|
| 39 |
+
"pad_token": "<|endoftext|>",
|
| 40 |
+
"split_special_tokens": false,
|
| 41 |
+
"tokenizer_class": "Qwen2Tokenizer",
|
| 42 |
+
"unk_token": null
|
| 43 |
+
}
|
models/SFT/vocab.json
ADDED
|
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|
|
|