prompt stringlengths 155 3.83k | question stringlengths 48 3.72k | answer stringclasses 4
values | generations listlengths 10 10 | ave_token_logprob listlengths 10 10 | seq_len listlengths 10 10 | seq_logprob listlengths 10 10 | generations_gt listlengths 1 1 | ave_token_logprob_gt listlengths 1 1 | seq_len_gt listlengths 1 1 | seq_logprob_gt listlengths 1 1 |
|---|---|---|---|---|---|---|---|---|---|---|
The cyclic subgroup of Z_24 generated by 18 has order
A. 4
B. 8
C. 12
D. 6
Please think step by step and provide your final choice letter in the exact format:
The final answer is X.
| The cyclic subgroup of Z_24 generated by 18 has order
A. 4
B. 8
C. 12
D. 6
| The answer is A. | [
"To determine the order of the cyclic subgroup of \\(\\mathbb{Z}_{24}\\) generated by 18, we need to find the smallest positive integer \\(k\\) such that \\(18k\\) is congruent to 0 modulo 24. This is because the order of a cyclic subgroup generated by an element \\(a\\) in \\(\\mathbb{Z}_n\\) is the smallest posit... | [
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"The answer is A.<|im_end|>"
] | [
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Find the order of the factor group Z_6/<3>.
A. 2
B. 3
C. 6
D. 12
Please think step by step and provide your final choice letter in the exact format:
The final answer is X.
| Find the order of the factor group Z_6/<3>.
A. 2
B. 3
C. 6
D. 12
| The answer is B. | [
"To determine the order of the factor group \\( \\mathbb{Z}_6 / \\langle 3 \\rangle \\), we need to follow these steps:\n\n1. **Define the groups involved:**\n - \\( \\mathbb{Z}_6 \\) is the group of integers modulo 6. This group has 6 elements: \\( \\{0, 1, 2, 3, 4, 5\\} \\).\n - \\( \\langle 3 \\rangle \\) is... | [
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"The answer is B.<|im_end|>"
] | [
-8.5625
] | [
6
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-51.5
] |
Statement 1 | A permutation that is a product of m even permutations and n odd permutations is an even permutation if and only if n is even. Statement 2 | Every group is isomorphic to a group of permutations.
A. True, True
B. False, False
C. True, False
D. False, True
Please think step by step and provide your final ch... | Statement 1 | A permutation that is a product of m even permutations and n odd permutations is an even permutation if and only if n is even. Statement 2 | Every group is isomorphic to a group of permutations.
A. True, True
B. False, False
C. True, False
D. False, True
| The answer is A. | [
"To determine the truth of these statements, let's analyze each one step by step.\n\n**Statement 1: A permutation that is a product of m even permutations and n odd permutations is an even permutation if and only if n is even.**\n\nThis statement is true. According to the properties of permutations, a permutation i... | [
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"The answer is A.<|im_end|>"
] | [
-7.375
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6
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] |
Find the order of the factor group (Z_4 x Z_12)/(<2> x <2>)
A. 2
B. 3
C. 4
D. 12
Please think step by step and provide your final choice letter in the exact format:
The final answer is X.
| Find the order of the factor group (Z_4 x Z_12)/(<2> x <2>)
A. 2
B. 3
C. 4
D. 12
| The answer is C. | [
"To determine the order of the factor group \\((\\mathbb{Z}_4 \\times \\mathbb{Z}_{12})/(\\langle 2 \\rangle \\times \\langle 2 \\rangle)\\), we need to follow these steps:\n\n1. **Understand the groups involved:**\n - \\(\\mathbb{Z}_4\\) is the group of integers modulo 4.\n - \\(\\mathbb{Z}_{12}\\) is the grou... | [
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"The answer is C.<|im_end|>"
] | [
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] | [
6
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-45.5
] |
Find the maximum possible order for some element of Z_4 x Z_6.
A. 4
B. 6
C. 12
D. 24
Please think step by step and provide your final choice letter in the exact format:
The final answer is X.
| Find the maximum possible order for some element of Z_4 x Z_6.
A. 4
B. 6
C. 12
D. 24
| The answer is C. | [
"To determine the maximum possible order for some element of \\( \\mathbb{Z}_4 \\times \\mathbb{Z}_6 \\), we need to understand the order of an element in a direct product of two groups. The order of an element \\((a, b)\\) in \\(\\mathbb{Z}_4 \\times \\mathbb{Z}_6\\) is the least common multiple (LCM) of the order... | [
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"The answer is C.<|im_end|>"
] | [
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Statement 1 | The symmetric group S_3 is cyclic. Statement 2 | Every group is isomorphic to some group of permutations.
A. True, True
B. False, False
C. True, False
D. False, True
Please think step by step and provide your final choice letter in the exact format:
The final answer is X.
| Statement 1 | The symmetric group S_3 is cyclic. Statement 2 | Every group is isomorphic to some group of permutations.
A. True, True
B. False, False
C. True, False
D. False, True
| The answer is D. | [
"To determine the truth of the given statements, let's analyze each one step by step.\n\n**Statement 1: The symmetric group \\( S_3 \\) is cyclic.**\n\nThe symmetric group \\( S_3 \\) consists of all permutations of 3 elements. It has 6 elements and is denoted as \\( S_3 \\).\n\nA cyclic group is one in which there... | [
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"The answer is D.<|im_end|>"
] | [
-8.0625
] | [
6
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-48.5
] |
Statement 1 | If a and b are elements of finite order in an Abelian group, then |ab| is the lcm (|a|,|b|). Statement 2 | If g is a group element and g^n = e, then |g| = n.
A. True, True
B. False, False
C. True, False
D. False, True
Please think step by step and provide your final choice letter in the exact format:
The ... | Statement 1 | If a and b are elements of finite order in an Abelian group, then |ab| is the lcm (|a|,|b|). Statement 2 | If g is a group element and g^n = e, then |g| = n.
A. True, True
B. False, False
C. True, False
D. False, True
| The answer is B. | [
"To determine the truth of the given statements, let's analyze each one step by step.\n\n**Statement 1: If a and b are elements of finite order in an Abelian group, then |ab| is the lcm (|a|, |b|).**\n\nIn an Abelian group, the order of the product of two elements \\(a\\) and \\(b\\) is the least common multiple of... | [
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"The answer is B.<|im_end|>"
] | [
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] | [
6
] | [
-42.75
] |
Statement 1 | If f is a homomorphism from G to K and H is normal in G then f(H) is normal in K. Statement 2 | If f is a homomorphism from G to a group and H is finite subgroup of G, then |f(H)| divides |H|.
A. True, True
B. False, False
C. True, False
D. False, True
Please think step by step and provide your final choi... | Statement 1 | If f is a homomorphism from G to K and H is normal in G then f(H) is normal in K. Statement 2 | If f is a homomorphism from G to a group and H is finite subgroup of G, then |f(H)| divides |H|.
A. True, True
B. False, False
C. True, False
D. False, True
| The answer is D. | [
"To determine the correctness of the given statements, let's analyze each one step by step.\n\n### Statement 1: If \\( f \\) is a homomorphism from \\( G \\) to \\( K \\) and \\( H \\) is normal in \\( G \\), then \\( f(H) \\) is normal in \\( K \\).\n\n1. **Definition of Homomorphism**: A homomorphism \\( f: G \\t... | [
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"The answer is D.<|im_end|>"
] | [
-6.875
] | [
6
] | [
-41.25
] |
Find the maximum possible order for an element of S_n for n = 7.
A. 6
B. 12
C. 30
D. 105
Please think step by step and provide your final choice letter in the exact format:
The final answer is X.
| Find the maximum possible order for an element of S_n for n = 7.
A. 6
B. 12
C. 30
D. 105
| The answer is B. | [
"To determine the maximum possible order of an element in \\( S_7 \\), we need to understand the structure of the symmetric group \\( S_7 \\). The symmetric group \\( S_n \\) consists of all permutations of \\( n \\) elements. The order of an element in \\( S_n \\) is the least common multiple (LCM) of the lengths ... | [
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"The answer is B.<|im_end|>"
] | [
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] | [
6
] | [
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] |
Statement 1 | Every integral domain has a field of quotients. Statement 2 | A polynomial of degree n over a ring can have at most n zeros counting multiplicity.
A. True, True
B. False, False
C. True, False
D. False, True
Please think step by step and provide your final choice letter in the exact format:
The final answe... | Statement 1 | Every integral domain has a field of quotients. Statement 2 | A polynomial of degree n over a ring can have at most n zeros counting multiplicity.
A. True, True
B. False, False
C. True, False
D. False, True
| The answer is C. | [
"To determine the correct answer, let's analyze each statement separately.\n\n**Statement 1:** \"Every integral domain has a field of quotients.\"\n\nAn integral domain is a non-empty set that is closed under addition, subtraction, and multiplication, contains no zero divisors, and has no additive identity. A field... | [
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"The answer is C.<|im_end|>"
] | [
-9
] | [
6
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] |
Statement 1 | If a group has an element of order 10, then the number of elements of order 10 is divisible by 4. Statement 2 | If m and n are positive integers and phi is the Euler phi function, then phi(mn) = phi(m)phi(n).
A. True, True
B. False, False
C. True, False
D. False, True
Please think step by step and provide... | Statement 1 | If a group has an element of order 10, then the number of elements of order 10 is divisible by 4. Statement 2 | If m and n are positive integers and phi is the Euler phi function, then phi(mn) = phi(m)phi(n).
A. True, True
B. False, False
C. True, False
D. False, True
| The answer is B. | [
"To determine the truth of the given statements, we need to analyze them one by one.\n\n**Statement 1: If a group has an element of order 10, then the number of elements of order 10 is divisible by 4.**\n\nThis statement is false. Consider the group of integers under addition modulo 10. The element 10 has order 10,... | [
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"The answer is B.<|im_end|>"
] | [
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6
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Which of the following terms describes the body's ability to maintain its normal state?
A. Anabolism
B. Catabolism
C. Tolerance
D. Homeostasis
Please think step by step and provide your final choice letter in the exact format:
The final answer is X.
| Which of the following terms describes the body's ability to maintain its normal state?
A. Anabolism
B. Catabolism
C. Tolerance
D. Homeostasis
| The answer is D. | [
"The final answer is D.<|im_end|>",
"The final answer is D.<|im_end|>",
"The final answer is D.<|im_end|>",
"The final answer is D.<|im_end|>",
"To determine which term describes the body's ability to maintain its normal state, let's analyze each option:\n\nA. Anabolism: This term refers to the process of b... | [
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] | [
"The answer is D.<|im_end|>"
] | [
-9.6875
] | [
6
] | [
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Which of the following structures travel through the substance of the parotid gland?
A. The maxillary artery
B. The maxillary artery and retromandibular vein
C. The maxillary artery, retromandibular vein and facial artery
D. The maxillary artery, retromandibular vein, facial artery and buccal branch of the mandibular n... | Which of the following structures travel through the substance of the parotid gland?
A. The maxillary artery
B. The maxillary artery and retromandibular vein
C. The maxillary artery, retromandibular vein and facial artery
D. The maxillary artery, retromandibular vein, facial artery and buccal branch of the mandibular n... | The answer is B. | [
"The final answer is D.<|im_end|>",
"The final answer is D.<|im_end|>",
"The final answer is D.<|im_end|>",
"The final answer is D.<|im_end|>",
"To determine which structure travel through the substance of the parotid gland, we need to identify the vessels and nerves that pass through this gland.\n\n1. **Ma... | [
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"The answer is B.<|im_end|>"
] | [
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A medical practitioner places a stethoscope over the patient's seventh right intercostal space in the mid-axillary line. The stethoscope overlies the
A. upper lobe of the lung.
B. middle lobe of the lung.
C. lower lobe of the lung.
D. costo-diaphragmatic recess.
Please think step by step and provide your final choice l... | A medical practitioner places a stethoscope over the patient's seventh right intercostal space in the mid-axillary line. The stethoscope overlies the
A. upper lobe of the lung.
B. middle lobe of the lung.
C. lower lobe of the lung.
D. costo-diaphragmatic recess.
| The answer is C. | [
"The final answer is B.<|im_end|>",
"The final answer is C.<|im_end|>",
"The final answer is B.<|im_end|>",
"The final answer is B.<|im_end|>",
"To determine where the stethoscope over the patient's seventh right intercostal space in the mid-axillary line would be placed, we need to understand the anatomica... | [
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] | [
"The answer is C.<|im_end|>"
] | [
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6
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The maxillary sinus
A. is lined by stratified squamous epithelium.
B. drains into the superior meatus of the nasal cavities.
C. is innervated by branches of the maxillary division of the trigeminal nerve.
D. Receives its blood supply from the first part of the maxillary artery.
Please think step by step and provide you... | The maxillary sinus
A. is lined by stratified squamous epithelium.
B. drains into the superior meatus of the nasal cavities.
C. is innervated by branches of the maxillary division of the trigeminal nerve.
D. Receives its blood supply from the first part of the maxillary artery.
| The answer is C. | [
"The correct answer is C. The maxillary sinus is innervated by branches of the maxillary division of the trigeminal nerve.<|im_end|>",
"The final answer is C.<|im_end|>",
"The final answer is C.<|im_end|>",
"The final answer is C.<|im_end|>",
"To determine the correct answer, let's analyze each option step ... | [
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"The answer is C.<|im_end|>"
] | [
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Which of the following is flexible connective tissue that is attached to bones at the joints?
A. Adipose
B. Cartilage
C. Epithelial
D. Muscle
Please think step by step and provide your final choice letter in the exact format:
The final answer is X.
| Which of the following is flexible connective tissue that is attached to bones at the joints?
A. Adipose
B. Cartilage
C. Epithelial
D. Muscle
| The answer is B. | [
"The final answer is B.<|im_end|>",
"The final answer is B. Cartilage.<|im_end|>",
"The final answer is B.<|im_end|>",
"The final answer is B.<|im_end|>",
"To determine which option is the correct answer, let's analyze each choice step by step:\n\nA. Adipose - This refers to fat tissue. While fat tissue can... | [
-0.08740234375,
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7,
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-1.8515625,
-0.61328125
] | [
"The answer is B.<|im_end|>"
] | [
-9
] | [
6
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Tonsillar tissue is commonly found
A. on the posterior wall of the oropharynx.
B. under the mucosa of the ventral surface of the tongue.
C. between the palatoglossal and palatopharyngeal folds.
D. at all three sites.
Please think step by step and provide your final choice letter in the exact format:
The final answer is... | Tonsillar tissue is commonly found
A. on the posterior wall of the oropharynx.
B. under the mucosa of the ventral surface of the tongue.
C. between the palatoglossal and palatopharyngeal folds.
D. at all three sites.
| The answer is C. | [
"The final answer is A.<|im_end|>",
"The final answer is C.<|im_end|>",
"The final answer is A.<|im_end|>",
"The final answer is C.<|im_end|>",
"To determine the correct answer, let's analyze each option:\n\nA. on the posterior wall of the oropharynx.\n- The posterior wall of the oropharynx is the area behi... | [
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] | [
"The answer is C.<|im_end|>"
] | [
-7.5
] | [
6
] | [
-45
] |
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