question
stringclasses 5
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class label 3
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dict | neg_question
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stringclasses 1
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stringclasses 5
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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? | [
"ST = 0",
"ST = T",
"ST = TS",
"ST - TS is the identity map of V onto itself."
] | 3D
| {
"content": "Given a text in the form of a short question, a question with long context, a fill in the blank query, or multiple statements. Negate the given text by making minimal changes. Only negate the main question or intent, don't negate the provided context of the question and don't complete the cloze query. Reply with the full question and context. Be concise and don't provide any explanation.\n\nText: 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\nNegated text:",
"role": "user"
} | 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 false? | 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?
A. ST = 0
B. ST = T
C. ST = TS
D. ST - TS is the identity map of V onto itself.
Answer: D | 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 false?
A. ST = 0
B. ST = T
C. ST = TS
D. ST - TS is the identity map of V onto itself.
Answer: A |
|||
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? | [
"2",
"2 - e^-2",
"2 + e^-2",
"2 + e^-4"
] | 3D
| {
"content": "Given a text in the form of a short question, a question with long context, a fill in the blank query, or multiple statements. Negate the given text by making minimal changes. Only negate the main question or intent, don't negate the provided context of the question and don't complete the cloze query. Reply with the full question and context. Be concise and don't provide any explanation.\n\nText: 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\nNegated text:",
"role": "user"
} | 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 NOT sprayed into the tank at a rate of 4 liters per minute. The sprayed solution is NOT continually mixed with the salt solution in the tank, and the mixture does NOT flow out of the tank at a rate of 4 liters per minute. After 100 minutes have elapsed, how many grams of salt are in the tank? | 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?
A. 2
B. 2 - e^-2
C. 2 + e^-2
D. 2 + e^-4
Answer: D | 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 NOT sprayed into the tank at a rate of 4 liters per minute. The sprayed solution is NOT continually mixed with the salt solution in the tank, and the mixture does NOT flow out of the tank at a rate of 4 liters per minute. After 100 minutes have elapsed, how many grams of salt are in the tank?
A. 2
B. 2 - e^-2
C. 2 + e^-2
D. 2 + e^-4
Answer: A |
|||
Let A be a real 2x2 matrix. Which of the following statements must be true?
I. All of the entries of A^2 are nonnegative.
II. The determinant of A^2 is nonnegative.
III. If A has two distinct eigenvalues, then A^2 has two distinct eigenvalues. | [
"I only",
"II only",
"III only",
"II and III only"
] | 1B
| {
"content": "Given a text in the form of a short question, a question with long context, a fill in the blank query, or multiple statements. Negate the given text by making minimal changes. Only negate the main question or intent, don't negate the provided context of the question and don't complete the cloze query. Reply with the full question and context. Be concise and don't provide any explanation.\n\nText: 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\nNegated text:",
"role": "user"
} | Let A be a real 2x2 matrix. Which of the following statements must NOT be true?
I. Some of the entries of A^2 are negative.
II. The determinant of A^2 is negative.
III. If A has two distinct eigenvalues, then A^2 does not have two distinct eigenvalues. | Let A be a real 2x2 matrix. Which of the following statements must be true?
I. All of the entries of A^2 are nonnegative.
II. The determinant of A^2 is nonnegative.
III. If A has two distinct eigenvalues, then A^2 has two distinct eigenvalues.
A. I only
B. II only
C. III only
D. II and III only
Answer: B | Let A be a real 2x2 matrix. Which of the following statements must NOT be true?
I. Some of the entries of A^2 are negative.
II. The determinant of A^2 is negative.
III. If A has two distinct eigenvalues, then A^2 does not have two distinct eigenvalues.
A. I only
B. II only
C. III only
D. II and III only
Answer: C |
|||
Suppose that f(1 + x) = f(x) for all real x. If f is a polynomial and f(5) = 11, then f(15/2) | [
"-11",
"0",
"11",
"33/2"
] | 2C
| {
"content": "Given a text in the form of a short question, a question with long context, a fill in the blank query, or multiple statements. Negate the given text by making minimal changes. Only negate the main question or intent, don't negate the provided context of the question and don't complete the cloze query. Reply with the full question and context. Be concise and don't provide any explanation.\n\nText: 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\nNegated text:",
"role": "user"
} | Suppose that f(1 + x) = f(x) for all real x. If f is a polynomial and f(5) = 11, then f(15/2).
Question: Is it not true that f(1 + x) is equal to f(x) for all real x? If f is a polynomial and f(5) = 11, is it not true that f(15/2) is equal to 11? | Suppose that f(1 + x) = f(x) for all real x. If f is a polynomial and f(5) = 11, then f(15/2)
A. -11
B. 0
C. 11
D. 33/2
Answer: C | Suppose that f(1 + x) = f(x) for all real x. If f is a polynomial and f(5) = 11, then f(15/2).
Question: Is it not true that f(1 + x) is equal to f(x) for all real x? If f is a polynomial and f(5) = 11, is it not true that f(15/2) is equal to 11?
A. -11
B. 0
C. 11
D. 33/2
Answer: D |
|||
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}? | [
"-5",
"-4",
"-3",
"-2"
] | 1B
| {
"content": "Given a text in the form of a short question, a question with long context, a fill in the blank query, or multiple statements. Negate the given text by making minimal changes. Only negate the main question or intent, don't negate the provided context of the question and don't complete the cloze query. Reply with the full question and context. Be concise and don't provide any explanation.\n\nText: 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\nNegated text:",
"role": "user"
} | Let A be the set of all ordered pairs of integers (m, n) such that 7m + 12n = 22. What is the greatest positive number in the set B = {m + n : (m, n) \in A}? | 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}?
A. -5
B. -4
C. -3
D. -2
Answer: B | Let A be the set of all ordered pairs of integers (m, n) such that 7m + 12n = 22. What is the greatest positive number in the set B = {m + n : (m, n) \in A}?
A. -5
B. -4
C. -3
D. -2
Answer: D |
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