74 lines
4.2 KiB
YAML
74 lines
4.2 KiB
YAML
dataset_name: college_mathematics
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description: The following are multiple choice questions (with answers) about college
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mathematics.
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fewshot_config:
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sampler: first_n
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samples:
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- question: 'Let V be the set of all real polynomials p(x). Let transformations T,
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S be defined on V by T:p(x) -> xp(x) and S:p(x) -> p''(x) = d/dx p(x), and interpret
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(ST)(p(x)) as S(T(p(x))). Which of the following is true?
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(A) ST = 0 (B) ST = T (C) ST = TS (D) ST - TS is the identity map of V onto
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itself.'
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target: "Let's think step by step. For a given polynomial $p$ we have\n\\[ST(p)\
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\ = (xp(x))\u2019 = p(x) + xp\u2019(x)\\]\nand\n\\[TS(p) = xp\u2019(x).\\]\n\
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Hence \\[ST(p) - TS(p) = p(x) + xp\u2019(x) - xp\u2019(x).\\] The answer is\
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\ (D)."
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- question: 'Suppose that f(1 + x) = f(x) for all real x. If f is a polynomial and
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f(5) = 11, then f(15/2)
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(A) -11 (B) 0 (C) 11 (D) 33/2'
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target: Let's think step by step. The only polynomial so that $f(1 + x) = f(x)$
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is a constant polynomial. Hence $f(5) = 11 = f(15/2)$. The answer is (C).
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- question: 'Let A be a real 2x2 matrix. Which of the following statements must be
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true?
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I. All of the entries of A^2 are nonnegative.
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II. The determinant of A^2 is nonnegative.
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III. If A has two distinct eigenvalues, then A^2 has two distinct eigenvalues.
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(A) I only (B) II only (C) III only (D) II and III only'
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target: 'Let''s think step by step. We have \[ det(A^2) = (det(A))^2 \geq 0,\]
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hence II holds.
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III is false: as a counterexample take a diagonal matrix with -1 and 1 on the
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diagonal. Then $A^2$ is the identity matrix. The answer is (B).'
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- question: 'Let A be the set of all ordered pairs of integers (m, n) such that 7m
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+ 12n = 22. What is the greatest negative number in the set B = {m + n : (m,
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n) \in A}?
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(A) -5 (B) -4 (C) -3 (D) -2'
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target: Let's think step by step. We have 12n = 22 - 7m and one of the solutions
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is $m = -2$, $n = 3$. Then $m + n = 1$, hence we need to look for smaller $m$
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in order to make $m + n$ negative. The next solution is $m = -14$ and $n = 10$.
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For smaller $m$ we have $m + n$ smaller than $-4$. The answer is (B).
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- question: 'A tank initially contains a salt solution of 3 grams of salt dissolved
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in 100 liters of water. A salt solution containing 0.02 grams of salt per liter
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of water is sprayed into the tank at a rate of 4 liters per minute. The sprayed
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solution is continually mixed with the salt solution in the tank, and the mixture
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flows out of the tank at a rate of 4 liters per minute. If the mixing is instantaneous,
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how many grams of salt are in the tank after 100 minutes have elapsed?
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(A) 2 (B) 2 - e^-2 (C) 2 + e^-2 (D) 2 + e^-4'
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target: "Let's think step by step. For all $t \\in \\mathbb{R}$, let $s(t)$ denote\
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\ the number grams of salt in the tank at the $t$ minute mark. Then $s(0) =\
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\ 3$.\nWe use $s$ and $s(t)$ interchangeably. We also use $s^{\\prime}$ and\
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\ $s^{\\prime}(t)$ interchangeably. The solution sprayed into the tank adds\
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\ $(0.02) 4=2 / 25$ grams of salt per minute. There are always 100 liters of\
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\ liquid in the tank, containing $s$ grams of salt. So the density of salt in\
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\ the tank is $s / 100$ grams per liter. The flow of water out of the tank therefore\
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\ subtracts $4(s / 100)=s / 25$ grams of salt per minute. Then, for all $t \\\
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in \\mathbb{R}$, we have $s^{\\prime}(t)=(2 / 25)-(s / 25)=(2-s) / 25$, and\
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\ so $[s(t)=2] \\Rightarrow\\left[s^{\\prime}(t)=0\right]$. For all $t \\in\
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\ \\mathbb{R}$,\n$$\n\frac{d}{d t}[\\ln (s-2)]=\frac{s^{\\prime}}{s-2}=\frac{-1}{25}=\f\
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rac{d}{d t}\\left[-\frac{t}{25}\right] .\n$$\nChoose $C \\in \\mathbb{R}$ such\
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\ that, for all $t \\in \\mathbb{R}, \\ln ((s(t)-2))=-[t / 25]+C$. Let $K:=e^{C}$.\
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\ Then, for all $t \\in \\mathbb{R}$, we have $(s(t))-2=K e^{-t / 25}$, and\
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\ so $s(t)=2+K e^{-t / 25}$. Then $3=s(0)=2+K e^{0}=2+K$, so $K=1$. Then $s(100)=2+K\
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\ e^{-100 / 25}=2+1 \\cdot e^{-4}=2+e^{-4}$. The answer is (D).\n\n"
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tag: mmlu_flan_cot_fewshot_stem
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include: _mmlu_flan_cot_fewshot_template_yaml
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task: mmlu_flan_cot_fewshot_college_mathematics
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