| {"doc": {"question": "Let k be the number of real solutions of the equation e^x + x - 2 = 0 in the interval [0, 1], and let n be the number of real solutions that are not in [0, 1]. Which of the following is true?", "subject": "college_mathematics", "choices": ["A", "B", "C", "D"], "answer": 1, "index": 0, "query": "Question: Let k be the number of real solutions of the equation e^x + x - 2 = 0 in the interval [0, 1], and let n be the number of real solutions that are not in [0, 1]. Which of the following is true?\n A. k = 0 and n = 1\n B. k = 1 and n = 0\n C. k = n = 1\n D. k > 1\nAnswer:", "gold": 1}, "task_name": "mmlu_college_mathematics:mc", "doc_id": 0, "native_id": 0, "label": 1, "requests": [{"request_type": "loglikelihood", "request": {"context": "The following are multiple choice questions (with answers) about college mathematics.\n\nQuestion: Let V be the set of all real polynomials p(x). Let transformations T, S be defined on V by T:p(x) -> xp(x) and S:p(x) -> p'(x) = d/dx p(x), and interpret (ST)(p(x)) as S(T(p(x))). Which of the following is true?\n A. ST = 0\n B. ST = T\n C. ST = TS\n D. ST - TS is the identity map of V onto itself.\nAnswer: D\n\nQuestion: A tank initially contains a salt solution of 3 grams of salt dissolved in 100 liters of water. A salt solution containing 0.02 grams of salt per liter of water is sprayed into the tank at a rate of 4 liters per minute. The sprayed solution is continually mixed with the salt solution in the tank, and the mixture flows out of the tank at a rate of 4 liters per minute. If the mixing is instantaneous, how many grams of salt are in the tank after 100 minutes have elapsed?\n A. 2\n B. 2 - e^-2\n C. 2 + e^-2\n D. 2 + e^-4\nAnswer: D\n\nQuestion: Let A be a real 2x2 matrix. Which of the following statements must be true?\r\nI. All of the entries of A^2 are nonnegative.\r\nII. The determinant of A^2 is nonnegative.\r\nIII. If A has two distinct eigenvalues, then A^2 has two distinct eigenvalues.\n A. I only\n B. II only\n C. III only\n D. II and III only\nAnswer: B\n\nQuestion: Suppose that f(1 + x) = f(x) for all real x. If f is a polynomial and f(5) = 11, then f(15/2)\n A. -11\n B. 0\n C. 11\n D. 33/2\nAnswer: C\n\nQuestion: Let A be the set of all ordered pairs of integers (m, n) such that 7m + 12n = 22. What is the greatest negative number in the set B = {m + n : (m, n) \\in A}?\n A. -5\n B. -4\n C. -3\n D. -2\nAnswer: B\n\nQuestion: Let k be the number of real solutions of the equation e^x + x - 2 = 0 in the interval [0, 1], and let n be the number of real solutions that are not in [0, 1]. Which of the following is true?\n A. k = 0 and n = 1\n B. k = 1 and n = 0\n C. k = n = 1\n D. k > 1\nAnswer:", "continuation": " A"}, "idx": 0}, {"request_type": "loglikelihood", "request": {"context": "The following are multiple choice questions (with answers) about college mathematics.\n\nQuestion: Let V be the set of all real polynomials p(x). Let transformations T, S be defined on V by T:p(x) -> xp(x) and S:p(x) -> p'(x) = d/dx p(x), and interpret (ST)(p(x)) as S(T(p(x))). Which of the following is true?\n A. ST = 0\n B. ST = T\n C. ST = TS\n D. ST - TS is the identity map of V onto itself.\nAnswer: D\n\nQuestion: A tank initially contains a salt solution of 3 grams of salt dissolved in 100 liters of water. A salt solution containing 0.02 grams of salt per liter of water is sprayed into the tank at a rate of 4 liters per minute. The sprayed solution is continually mixed with the salt solution in the tank, and the mixture flows out of the tank at a rate of 4 liters per minute. If the mixing is instantaneous, how many grams of salt are in the tank after 100 minutes have elapsed?\n A. 2\n B. 2 - e^-2\n C. 2 + e^-2\n D. 2 + e^-4\nAnswer: D\n\nQuestion: Let A be a real 2x2 matrix. Which of the following statements must be true?\r\nI. All of the entries of A^2 are nonnegative.\r\nII. The determinant of A^2 is nonnegative.\r\nIII. If A has two distinct eigenvalues, then A^2 has two distinct eigenvalues.\n A. I only\n B. II only\n C. III only\n D. II and III only\nAnswer: B\n\nQuestion: Suppose that f(1 + x) = f(x) for all real x. If f is a polynomial and f(5) = 11, then f(15/2)\n A. -11\n B. 0\n C. 11\n D. 33/2\nAnswer: C\n\nQuestion: Let A be the set of all ordered pairs of integers (m, n) such that 7m + 12n = 22. What is the greatest negative number in the set B = {m + n : (m, n) \\in A}?\n A. -5\n B. -4\n C. -3\n D. -2\nAnswer: B\n\nQuestion: Let k be the number of real solutions of the equation e^x + x - 2 = 0 in the interval [0, 1], and let n be the number of real solutions that are not in [0, 1]. Which of the following is true?\n A. k = 0 and n = 1\n B. k = 1 and n = 0\n C. k = n = 1\n D. k > 1\nAnswer:", "continuation": " B"}, "idx": 1}, {"request_type": "loglikelihood", "request": {"context": "The following are multiple choice questions (with answers) about college mathematics.\n\nQuestion: Let V be the set of all real polynomials p(x). Let transformations T, S be defined on V by T:p(x) -> xp(x) and S:p(x) -> p'(x) = d/dx p(x), and interpret (ST)(p(x)) as S(T(p(x))). Which of the following is true?\n A. ST = 0\n B. ST = T\n C. ST = TS\n D. ST - TS is the identity map of V onto itself.\nAnswer: D\n\nQuestion: A tank initially contains a salt solution of 3 grams of salt dissolved in 100 liters of water. A salt solution containing 0.02 grams of salt per liter of water is sprayed into the tank at a rate of 4 liters per minute. The sprayed solution is continually mixed with the salt solution in the tank, and the mixture flows out of the tank at a rate of 4 liters per minute. If the mixing is instantaneous, how many grams of salt are in the tank after 100 minutes have elapsed?\n A. 2\n B. 2 - e^-2\n C. 2 + e^-2\n D. 2 + e^-4\nAnswer: D\n\nQuestion: Let A be a real 2x2 matrix. Which of the following statements must be true?\r\nI. All of the entries of A^2 are nonnegative.\r\nII. The determinant of A^2 is nonnegative.\r\nIII. If A has two distinct eigenvalues, then A^2 has two distinct eigenvalues.\n A. I only\n B. II only\n C. III only\n D. II and III only\nAnswer: B\n\nQuestion: Suppose that f(1 + x) = f(x) for all real x. If f is a polynomial and f(5) = 11, then f(15/2)\n A. -11\n B. 0\n C. 11\n D. 33/2\nAnswer: C\n\nQuestion: Let A be the set of all ordered pairs of integers (m, n) such that 7m + 12n = 22. What is the greatest negative number in the set B = {m + n : (m, n) \\in A}?\n A. -5\n B. -4\n C. -3\n D. -2\nAnswer: B\n\nQuestion: Let k be the number of real solutions of the equation e^x + x - 2 = 0 in the interval [0, 1], and let n be the number of real solutions that are not in [0, 1]. Which of the following is true?\n A. k = 0 and n = 1\n B. k = 1 and n = 0\n C. k = n = 1\n D. k > 1\nAnswer:", "continuation": " C"}, "idx": 2}, {"request_type": "loglikelihood", "request": {"context": "The following are multiple choice questions (with answers) about college mathematics.\n\nQuestion: Let V be the set of all real polynomials p(x). Let transformations T, S be defined on V by T:p(x) -> xp(x) and S:p(x) -> p'(x) = d/dx p(x), and interpret (ST)(p(x)) as S(T(p(x))). Which of the following is true?\n A. ST = 0\n B. ST = T\n C. ST = TS\n D. ST - TS is the identity map of V onto itself.\nAnswer: D\n\nQuestion: A tank initially contains a salt solution of 3 grams of salt dissolved in 100 liters of water. A salt solution containing 0.02 grams of salt per liter of water is sprayed into the tank at a rate of 4 liters per minute. The sprayed solution is continually mixed with the salt solution in the tank, and the mixture flows out of the tank at a rate of 4 liters per minute. If the mixing is instantaneous, how many grams of salt are in the tank after 100 minutes have elapsed?\n A. 2\n B. 2 - e^-2\n C. 2 + e^-2\n D. 2 + e^-4\nAnswer: D\n\nQuestion: Let A be a real 2x2 matrix. Which of the following statements must be true?\r\nI. All of the entries of A^2 are nonnegative.\r\nII. The determinant of A^2 is nonnegative.\r\nIII. If A has two distinct eigenvalues, then A^2 has two distinct eigenvalues.\n A. I only\n B. II only\n C. III only\n D. II and III only\nAnswer: B\n\nQuestion: Suppose that f(1 + x) = f(x) for all real x. If f is a polynomial and f(5) = 11, then f(15/2)\n A. -11\n B. 0\n C. 11\n D. 33/2\nAnswer: C\n\nQuestion: Let A be the set of all ordered pairs of integers (m, n) such that 7m + 12n = 22. What is the greatest negative number in the set B = {m + n : (m, n) \\in A}?\n A. -5\n B. -4\n C. -3\n D. -2\nAnswer: B\n\nQuestion: Let k be the number of real solutions of the equation e^x + x - 2 = 0 in the interval [0, 1], and let n be the number of real solutions that are not in [0, 1]. Which of the following is true?\n A. k = 0 and n = 1\n B. k = 1 and n = 0\n C. k = n = 1\n D. k > 1\nAnswer:", "continuation": " D"}, "idx": 3}]} | |
| {"doc": {"question": "Up to isomorphism, how many additive abelian groups G of order 16 have the property that x + x + x + x = 0 for each x in G ?", "subject": "college_mathematics", "choices": ["A", "B", "C", "D"], "answer": 3, "index": 1, "query": "Question: Up to isomorphism, how many additive abelian groups G of order 16 have the property that x + x + x + x = 0 for each x in G ?\n A. 0\n B. 1\n C. 2\n D. 3\nAnswer:", "gold": 3}, "task_name": "mmlu_college_mathematics:mc", "doc_id": 1, "native_id": 1, "label": 3, "requests": [{"request_type": "loglikelihood", "request": {"context": "The following are multiple choice questions (with answers) about college mathematics.\n\nQuestion: Let V be the set of all real polynomials p(x). Let transformations T, S be defined on V by T:p(x) -> xp(x) and S:p(x) -> p'(x) = d/dx p(x), and interpret (ST)(p(x)) as S(T(p(x))). Which of the following is true?\n A. ST = 0\n B. ST = T\n C. ST = TS\n D. ST - TS is the identity map of V onto itself.\nAnswer: D\n\nQuestion: A tank initially contains a salt solution of 3 grams of salt dissolved in 100 liters of water. A salt solution containing 0.02 grams of salt per liter of water is sprayed into the tank at a rate of 4 liters per minute. The sprayed solution is continually mixed with the salt solution in the tank, and the mixture flows out of the tank at a rate of 4 liters per minute. If the mixing is instantaneous, how many grams of salt are in the tank after 100 minutes have elapsed?\n A. 2\n B. 2 - e^-2\n C. 2 + e^-2\n D. 2 + e^-4\nAnswer: D\n\nQuestion: Let A be a real 2x2 matrix. Which of the following statements must be true?\r\nI. All of the entries of A^2 are nonnegative.\r\nII. The determinant of A^2 is nonnegative.\r\nIII. If A has two distinct eigenvalues, then A^2 has two distinct eigenvalues.\n A. I only\n B. II only\n C. III only\n D. II and III only\nAnswer: B\n\nQuestion: Suppose that f(1 + x) = f(x) for all real x. If f is a polynomial and f(5) = 11, then f(15/2)\n A. -11\n B. 0\n C. 11\n D. 33/2\nAnswer: C\n\nQuestion: Let A be the set of all ordered pairs of integers (m, n) such that 7m + 12n = 22. What is the greatest negative number in the set B = {m + n : (m, n) \\in A}?\n A. -5\n B. -4\n C. -3\n D. -2\nAnswer: B\n\nQuestion: Up to isomorphism, how many additive abelian groups G of order 16 have the property that x + x + x + x = 0 for each x in G ?\n A. 0\n B. 1\n C. 2\n D. 3\nAnswer:", "continuation": " A"}, "idx": 0}, {"request_type": "loglikelihood", "request": {"context": "The following are multiple choice questions (with answers) about college mathematics.\n\nQuestion: Let V be the set of all real polynomials p(x). Let transformations T, S be defined on V by T:p(x) -> xp(x) and S:p(x) -> p'(x) = d/dx p(x), and interpret (ST)(p(x)) as S(T(p(x))). Which of the following is true?\n A. ST = 0\n B. ST = T\n C. ST = TS\n D. ST - TS is the identity map of V onto itself.\nAnswer: D\n\nQuestion: A tank initially contains a salt solution of 3 grams of salt dissolved in 100 liters of water. A salt solution containing 0.02 grams of salt per liter of water is sprayed into the tank at a rate of 4 liters per minute. The sprayed solution is continually mixed with the salt solution in the tank, and the mixture flows out of the tank at a rate of 4 liters per minute. If the mixing is instantaneous, how many grams of salt are in the tank after 100 minutes have elapsed?\n A. 2\n B. 2 - e^-2\n C. 2 + e^-2\n D. 2 + e^-4\nAnswer: D\n\nQuestion: Let A be a real 2x2 matrix. Which of the following statements must be true?\r\nI. All of the entries of A^2 are nonnegative.\r\nII. The determinant of A^2 is nonnegative.\r\nIII. If A has two distinct eigenvalues, then A^2 has two distinct eigenvalues.\n A. I only\n B. II only\n C. III only\n D. II and III only\nAnswer: B\n\nQuestion: Suppose that f(1 + x) = f(x) for all real x. If f is a polynomial and f(5) = 11, then f(15/2)\n A. -11\n B. 0\n C. 11\n D. 33/2\nAnswer: C\n\nQuestion: Let A be the set of all ordered pairs of integers (m, n) such that 7m + 12n = 22. What is the greatest negative number in the set B = {m + n : (m, n) \\in A}?\n A. -5\n B. -4\n C. -3\n D. -2\nAnswer: B\n\nQuestion: Up to isomorphism, how many additive abelian groups G of order 16 have the property that x + x + x + x = 0 for each x in G ?\n A. 0\n B. 1\n C. 2\n D. 3\nAnswer:", "continuation": " B"}, "idx": 1}, {"request_type": "loglikelihood", "request": {"context": "The following are multiple choice questions (with answers) about college mathematics.\n\nQuestion: Let V be the set of all real polynomials p(x). Let transformations T, S be defined on V by T:p(x) -> xp(x) and S:p(x) -> p'(x) = d/dx p(x), and interpret (ST)(p(x)) as S(T(p(x))). Which of the following is true?\n A. ST = 0\n B. ST = T\n C. ST = TS\n D. ST - TS is the identity map of V onto itself.\nAnswer: D\n\nQuestion: A tank initially contains a salt solution of 3 grams of salt dissolved in 100 liters of water. A salt solution containing 0.02 grams of salt per liter of water is sprayed into the tank at a rate of 4 liters per minute. The sprayed solution is continually mixed with the salt solution in the tank, and the mixture flows out of the tank at a rate of 4 liters per minute. If the mixing is instantaneous, how many grams of salt are in the tank after 100 minutes have elapsed?\n A. 2\n B. 2 - e^-2\n C. 2 + e^-2\n D. 2 + e^-4\nAnswer: D\n\nQuestion: Let A be a real 2x2 matrix. Which of the following statements must be true?\r\nI. All of the entries of A^2 are nonnegative.\r\nII. The determinant of A^2 is nonnegative.\r\nIII. If A has two distinct eigenvalues, then A^2 has two distinct eigenvalues.\n A. I only\n B. II only\n C. III only\n D. II and III only\nAnswer: B\n\nQuestion: Suppose that f(1 + x) = f(x) for all real x. If f is a polynomial and f(5) = 11, then f(15/2)\n A. -11\n B. 0\n C. 11\n D. 33/2\nAnswer: C\n\nQuestion: Let A be the set of all ordered pairs of integers (m, n) such that 7m + 12n = 22. What is the greatest negative number in the set B = {m + n : (m, n) \\in A}?\n A. -5\n B. -4\n C. -3\n D. -2\nAnswer: B\n\nQuestion: Up to isomorphism, how many additive abelian groups G of order 16 have the property that x + x + x + x = 0 for each x in G ?\n A. 0\n B. 1\n C. 2\n D. 3\nAnswer:", "continuation": " C"}, "idx": 2}, {"request_type": "loglikelihood", "request": {"context": "The following are multiple choice questions (with answers) about college mathematics.\n\nQuestion: Let V be the set of all real polynomials p(x). Let transformations T, S be defined on V by T:p(x) -> xp(x) and S:p(x) -> p'(x) = d/dx p(x), and interpret (ST)(p(x)) as S(T(p(x))). Which of the following is true?\n A. ST = 0\n B. ST = T\n C. ST = TS\n D. ST - TS is the identity map of V onto itself.\nAnswer: D\n\nQuestion: A tank initially contains a salt solution of 3 grams of salt dissolved in 100 liters of water. A salt solution containing 0.02 grams of salt per liter of water is sprayed into the tank at a rate of 4 liters per minute. The sprayed solution is continually mixed with the salt solution in the tank, and the mixture flows out of the tank at a rate of 4 liters per minute. If the mixing is instantaneous, how many grams of salt are in the tank after 100 minutes have elapsed?\n A. 2\n B. 2 - e^-2\n C. 2 + e^-2\n D. 2 + e^-4\nAnswer: D\n\nQuestion: Let A be a real 2x2 matrix. Which of the following statements must be true?\r\nI. All of the entries of A^2 are nonnegative.\r\nII. The determinant of A^2 is nonnegative.\r\nIII. If A has two distinct eigenvalues, then A^2 has two distinct eigenvalues.\n A. I only\n B. II only\n C. III only\n D. II and III only\nAnswer: B\n\nQuestion: Suppose that f(1 + x) = f(x) for all real x. If f is a polynomial and f(5) = 11, then f(15/2)\n A. -11\n B. 0\n C. 11\n D. 33/2\nAnswer: C\n\nQuestion: Let A be the set of all ordered pairs of integers (m, n) such that 7m + 12n = 22. What is the greatest negative number in the set B = {m + n : (m, n) \\in A}?\n A. -5\n B. -4\n C. -3\n D. -2\nAnswer: B\n\nQuestion: Up to isomorphism, how many additive abelian groups G of order 16 have the property that x + x + x + x = 0 for each x in G ?\n A. 0\n B. 1\n C. 2\n D. 3\nAnswer:", "continuation": " D"}, "idx": 3}]} | |
| {"doc": {"question": "Suppose P is the set of polynomials with coefficients in Z_5 and degree less than or equal to 7. If the operator D sends p(x) in P to its derivative p\u2032(x), what are the dimensions of the null space n and range r of D?", "subject": "college_mathematics", "choices": ["A", "B", "C", "D"], "answer": 3, "index": 2, "query": "Question: Suppose P is the set of polynomials with coefficients in Z_5 and degree less than or equal to 7. If the operator D sends p(x) in P to its derivative p\u2032(x), what are the dimensions of the null space n and range r of D?\n A. n = 1 and r = 6\n B. n = 1 and r = 7\n C. n = 2 and r = 5\n D. n = 2 and r = 6\nAnswer:", "gold": 3}, "task_name": "mmlu_college_mathematics:mc", "doc_id": 2, "native_id": 2, "label": 3, "requests": [{"request_type": "loglikelihood", "request": {"context": "The following are multiple choice questions (with answers) about college mathematics.\n\nQuestion: Let V be the set of all real polynomials p(x). Let transformations T, S be defined on V by T:p(x) -> xp(x) and S:p(x) -> p'(x) = d/dx p(x), and interpret (ST)(p(x)) as S(T(p(x))). Which of the following is true?\n A. ST = 0\n B. ST = T\n C. ST = TS\n D. ST - TS is the identity map of V onto itself.\nAnswer: D\n\nQuestion: A tank initially contains a salt solution of 3 grams of salt dissolved in 100 liters of water. A salt solution containing 0.02 grams of salt per liter of water is sprayed into the tank at a rate of 4 liters per minute. The sprayed solution is continually mixed with the salt solution in the tank, and the mixture flows out of the tank at a rate of 4 liters per minute. If the mixing is instantaneous, how many grams of salt are in the tank after 100 minutes have elapsed?\n A. 2\n B. 2 - e^-2\n C. 2 + e^-2\n D. 2 + e^-4\nAnswer: D\n\nQuestion: Let A be a real 2x2 matrix. Which of the following statements must be true?\r\nI. All of the entries of A^2 are nonnegative.\r\nII. The determinant of A^2 is nonnegative.\r\nIII. If A has two distinct eigenvalues, then A^2 has two distinct eigenvalues.\n A. I only\n B. II only\n C. III only\n D. II and III only\nAnswer: B\n\nQuestion: Suppose that f(1 + x) = f(x) for all real x. If f is a polynomial and f(5) = 11, then f(15/2)\n A. -11\n B. 0\n C. 11\n D. 33/2\nAnswer: C\n\nQuestion: Let A be the set of all ordered pairs of integers (m, n) such that 7m + 12n = 22. What is the greatest negative number in the set B = {m + n : (m, n) \\in A}?\n A. -5\n B. -4\n C. -3\n D. -2\nAnswer: B\n\nQuestion: Suppose P is the set of polynomials with coefficients in Z_5 and degree less than or equal to 7. If the operator D sends p(x) in P to its derivative p\u2032(x), what are the dimensions of the null space n and range r of D?\n A. n = 1 and r = 6\n B. n = 1 and r = 7\n C. n = 2 and r = 5\n D. n = 2 and r = 6\nAnswer:", "continuation": " A"}, "idx": 0}, {"request_type": "loglikelihood", "request": {"context": "The following are multiple choice questions (with answers) about college mathematics.\n\nQuestion: Let V be the set of all real polynomials p(x). Let transformations T, S be defined on V by T:p(x) -> xp(x) and S:p(x) -> p'(x) = d/dx p(x), and interpret (ST)(p(x)) as S(T(p(x))). Which of the following is true?\n A. ST = 0\n B. ST = T\n C. ST = TS\n D. ST - TS is the identity map of V onto itself.\nAnswer: D\n\nQuestion: A tank initially contains a salt solution of 3 grams of salt dissolved in 100 liters of water. A salt solution containing 0.02 grams of salt per liter of water is sprayed into the tank at a rate of 4 liters per minute. The sprayed solution is continually mixed with the salt solution in the tank, and the mixture flows out of the tank at a rate of 4 liters per minute. If the mixing is instantaneous, how many grams of salt are in the tank after 100 minutes have elapsed?\n A. 2\n B. 2 - e^-2\n C. 2 + e^-2\n D. 2 + e^-4\nAnswer: D\n\nQuestion: Let A be a real 2x2 matrix. Which of the following statements must be true?\r\nI. All of the entries of A^2 are nonnegative.\r\nII. The determinant of A^2 is nonnegative.\r\nIII. If A has two distinct eigenvalues, then A^2 has two distinct eigenvalues.\n A. I only\n B. II only\n C. III only\n D. II and III only\nAnswer: B\n\nQuestion: Suppose that f(1 + x) = f(x) for all real x. If f is a polynomial and f(5) = 11, then f(15/2)\n A. -11\n B. 0\n C. 11\n D. 33/2\nAnswer: C\n\nQuestion: Let A be the set of all ordered pairs of integers (m, n) such that 7m + 12n = 22. What is the greatest negative number in the set B = {m + n : (m, n) \\in A}?\n A. -5\n B. -4\n C. -3\n D. -2\nAnswer: B\n\nQuestion: Suppose P is the set of polynomials with coefficients in Z_5 and degree less than or equal to 7. If the operator D sends p(x) in P to its derivative p\u2032(x), what are the dimensions of the null space n and range r of D?\n A. n = 1 and r = 6\n B. n = 1 and r = 7\n C. n = 2 and r = 5\n D. n = 2 and r = 6\nAnswer:", "continuation": " B"}, "idx": 1}, {"request_type": "loglikelihood", "request": {"context": "The following are multiple choice questions (with answers) about college mathematics.\n\nQuestion: Let V be the set of all real polynomials p(x). Let transformations T, S be defined on V by T:p(x) -> xp(x) and S:p(x) -> p'(x) = d/dx p(x), and interpret (ST)(p(x)) as S(T(p(x))). Which of the following is true?\n A. ST = 0\n B. ST = T\n C. ST = TS\n D. ST - TS is the identity map of V onto itself.\nAnswer: D\n\nQuestion: A tank initially contains a salt solution of 3 grams of salt dissolved in 100 liters of water. A salt solution containing 0.02 grams of salt per liter of water is sprayed into the tank at a rate of 4 liters per minute. The sprayed solution is continually mixed with the salt solution in the tank, and the mixture flows out of the tank at a rate of 4 liters per minute. If the mixing is instantaneous, how many grams of salt are in the tank after 100 minutes have elapsed?\n A. 2\n B. 2 - e^-2\n C. 2 + e^-2\n D. 2 + e^-4\nAnswer: D\n\nQuestion: Let A be a real 2x2 matrix. Which of the following statements must be true?\r\nI. All of the entries of A^2 are nonnegative.\r\nII. The determinant of A^2 is nonnegative.\r\nIII. If A has two distinct eigenvalues, then A^2 has two distinct eigenvalues.\n A. I only\n B. II only\n C. III only\n D. II and III only\nAnswer: B\n\nQuestion: Suppose that f(1 + x) = f(x) for all real x. If f is a polynomial and f(5) = 11, then f(15/2)\n A. -11\n B. 0\n C. 11\n D. 33/2\nAnswer: C\n\nQuestion: Let A be the set of all ordered pairs of integers (m, n) such that 7m + 12n = 22. What is the greatest negative number in the set B = {m + n : (m, n) \\in A}?\n A. -5\n B. -4\n C. -3\n D. -2\nAnswer: B\n\nQuestion: Suppose P is the set of polynomials with coefficients in Z_5 and degree less than or equal to 7. If the operator D sends p(x) in P to its derivative p\u2032(x), what are the dimensions of the null space n and range r of D?\n A. n = 1 and r = 6\n B. n = 1 and r = 7\n C. n = 2 and r = 5\n D. n = 2 and r = 6\nAnswer:", "continuation": " C"}, "idx": 2}, {"request_type": "loglikelihood", "request": {"context": "The following are multiple choice questions (with answers) about college mathematics.\n\nQuestion: Let V be the set of all real polynomials p(x). Let transformations T, S be defined on V by T:p(x) -> xp(x) and S:p(x) -> p'(x) = d/dx p(x), and interpret (ST)(p(x)) as S(T(p(x))). Which of the following is true?\n A. ST = 0\n B. ST = T\n C. ST = TS\n D. ST - TS is the identity map of V onto itself.\nAnswer: D\n\nQuestion: A tank initially contains a salt solution of 3 grams of salt dissolved in 100 liters of water. A salt solution containing 0.02 grams of salt per liter of water is sprayed into the tank at a rate of 4 liters per minute. The sprayed solution is continually mixed with the salt solution in the tank, and the mixture flows out of the tank at a rate of 4 liters per minute. If the mixing is instantaneous, how many grams of salt are in the tank after 100 minutes have elapsed?\n A. 2\n B. 2 - e^-2\n C. 2 + e^-2\n D. 2 + e^-4\nAnswer: D\n\nQuestion: Let A be a real 2x2 matrix. Which of the following statements must be true?\r\nI. All of the entries of A^2 are nonnegative.\r\nII. The determinant of A^2 is nonnegative.\r\nIII. If A has two distinct eigenvalues, then A^2 has two distinct eigenvalues.\n A. I only\n B. II only\n C. III only\n D. II and III only\nAnswer: B\n\nQuestion: Suppose that f(1 + x) = f(x) for all real x. If f is a polynomial and f(5) = 11, then f(15/2)\n A. -11\n B. 0\n C. 11\n D. 33/2\nAnswer: C\n\nQuestion: Let A be the set of all ordered pairs of integers (m, n) such that 7m + 12n = 22. What is the greatest negative number in the set B = {m + n : (m, n) \\in A}?\n A. -5\n B. -4\n C. -3\n D. -2\nAnswer: B\n\nQuestion: Suppose P is the set of polynomials with coefficients in Z_5 and degree less than or equal to 7. If the operator D sends p(x) in P to its derivative p\u2032(x), what are the dimensions of the null space n and range r of D?\n A. n = 1 and r = 6\n B. n = 1 and r = 7\n C. n = 2 and r = 5\n D. n = 2 and r = 6\nAnswer:", "continuation": " D"}, "idx": 3}]} | |