dataset_name: math
description: The following are multiple choice questions (with answers) about math.
fewshot_config:
  sampler: first_n
  samples:
  - question: "The symmetric group $S_n$ has $\\factorial{n}$ elements, hence it is not true that $S_{10}$ has 10 elements.
    Find the characteristic of the ring 2Z.
      (A) 0 (B) 30 (C) 3 (D) 10 (E) 12 (F) 50 (G) 2 (H) 100 (I) 20 (J) 5"
    target: "Let's think step by step. A characteristic of a ring is R is $n$ if the statement $ka = 0$ for all $a \\in 2Z$ implies that $k$ is a multiple of $n$. Assume that $ka = 0$ for all $a \\in 2Z$ for some $k$. In particular $2k = 0$. Hence $k=0$ and $n=0$. The answer is (A)."
  - question: "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 + TS is the identity map of V onto itself. (B) TS = 0 (C) ST = 1 (D) ST - TS = 0 (E) ST = T (F) ST = 0 (G) ST = TS (H) ST - TS is the identity map of V onto itself. (I) TS = T (J) ST = S"
    target: "Let's think step by step. For a given polynomial $p$ we have
    \\[ST(p) = (xp(x))’ = p(x) + xp’(x)\\]
    and
    \\[TS(p) = xp’(x).\\]
    Hence \\[ST(p) - TS(p) = p(x) + xp’(x) - xp’(x).\\] The answer is (H)."
  - question: "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) 0 (C) -3 (D) -7 (E) -4 (F) -6 (G) -1 (H) -2 (I) -9 (J) N/A"
    target: "Let's think step by step. We have 12n = 22 - 7m and one of the solutions is $m = -2$, $n = 3$. Then $m + n = 1$, hence we need to look for smaller $m$ in order to make $m + n$ negative. The next solution is $m = -14$ and $n = 10$. For smaller $m$ we have $m + n$ smaller than $-4$. The answer is (E)."
  - question: "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) 3 + e^-2 (B) 2 - e^-4 (C) 2 - e^-2 (D) 3 + e^-4 (E) 2 + e^-3 (F) 2 - e^-3 (G) 3 - e^-2 (H) 2 + e^-2 (I) 2 + e^-4 (J) 2"
    target: "Let's think step by step. For all $t \\in \\mathbb{R}$, let $s(t)$ denote the number grams of salt in the tank at the $t$ minute mark. Then $s(0) = 3$.
    We use $s$ and $s(t)$ interchangeably. We also use $s^{\\prime}$ and $s^{\\prime}(t)$ interchangeably. The solution sprayed into the tank adds $(0.02) 4=2 / 25$ grams of salt per minute. There are always 100 liters of liquid in the tank, containing $s$ grams of salt. So the density of salt in the tank is $s / 100$ grams per liter. The flow of water out of the tank therefore subtracts $4(s / 100)=s / 25$ grams of salt per minute. Then, for all $t \\in \\mathbb{R}$, we have $s^{\\prime}(t)=(2 / 25)-(s / 25)=(2-s) / 25$, and so $[s(t)=2] \\Rightarrow\\left[s^{\\prime}(t)=0ight]$. For all $t \\in \\mathbb{R}$,
    $$
    \\frac{d}{d t}[\\ln (s-2)]=\\frac{s^{\\prime}}{s-2}=\\frac{-1}{25}=\\frac{d}{d t}\\left[-\\frac{t}{25}ight] .
    $$
    Choose $C \\in \\mathbb{R}$ such that, for all $t \\in \\mathbb{R}, \\ln ((s(t)-2))=-[t / 25]+C$. Let $K:=e^{C}$. Then, for all $t \\in \\mathbb{R}$, we have $(s(t))-2=K e^{-t / 25}$, and so $s(t)=2+K e^{-t / 25}$. Then $3=s(0)=2+K e^{0}=2+K$, so $K=1$. Then $s(100)=2+K e^{-100 / 25}=2+1 \\cdot e^{-4}=2+e^{-4}$. The answer is (I)."
  - question: "A total of 30 players will play basketball at a park. There will be exactly 5 players on each team. Which statement correctly explains how to find the number of teams needed?
      (A) Multiply 5 by 5 to find 25 teams. (B) Divide 30 by 5 to find 6 teams. (C) Add 5 to 30 to find 35 teams. (D) Subtract 30 from 5 to find -25 teams. (E) Divide 5 by 30 to find 0.1667 teams. (F) Add 5 to 30 then divide by 2 to find 17.5 teams. (G) N/A (H) N/A (I) N/A (J) N/A"
    target: "Let's think step by step. We want to find the number of teams. We know that there are 5 players/team, and 30 players. Thus to get the number of teams we divide players by players/team, so 30 players / 5 players/team = 6 teams. The answer is (B)."
group: mmlu_pro_flan_cot_fewshot_stem
include: _mmlu_pro_flan_cot_fewshot_template_yaml
task: mmlu_pro_flan_cot_fewshot_math
