dataset_name: college_mathematics
description: "The following are multiple choice questions (with answers) about college\
  \ mathematics.\n\nQ: 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 (B) ST =\
  \ T (C) ST = TS (D) ST - TS is the identity map of V onto itself.\nA: Let's think\
  \ step by step. For a given polynomial $p$ we have\n\\[ST(p) = (xp(x))\u2019 = p(x)\
  \ + xp\u2019(x)\\]\nand\n\\[TS(p) = xp\u2019(x).\\]\nHence \\[ST(p) - TS(p) = p(x)\
  \ + xp\u2019(x) - xp\u2019(x).\\] The answer is (D).\n\nQ: 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\
  \ (B) 0 (C) 11 (D) 33/2\nA: Let's think step by step. The only polynomial so that\
  \ $f(1 + x) = f(x)$ is a constant polynomial. Hence $f(5) = 11 = f(15/2)$. The answer\
  \ is (C).\n\nQ: Let A be a real 2x2 matrix. Which of the following statements must\
  \ be true?\nI. All of the entries of A^2 are nonnegative.\nII. The determinant of\
  \ A^2 is nonnegative.\nIII. If A has two distinct eigenvalues, then A^2 has two\
  \ distinct eigenvalues.\n(A) I only (B) II only (C) III only (D) II and III only\n\
  A: Let's think step by step. We have \\[ det(A^2) = (det(A))^2 \\geq 0,\\] hence\
  \ II holds.\nIII is false: as a counterexample take a diagonal matrix with -1 and\
  \ 1 on the diagonal. Then $A^2$ is the identity matrix. The answer is (B).\n\nQ:\
  \ 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 (B) -4 (C) -3 (D) -2\nA: 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 (B).\n\nQ: 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 (B) 2 - e^-2 (C) 2 + e^-2 (D) 2 + e^-4\nA: 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 $s(0) = 3$.\nWe 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 $[s(t)=2] \\Rightarrow\\left[s^{\\prime}(t)=0\right]$. For all $t \\in\
  \ \\mathbb{R}$,\n$$\n\frac{d}{d t}[\\ln (s-2)]=\frac{s^{\\prime}}{s-2}=\frac{-1}{25}=\f\
  rac{d}{d t}\\left[-\frac{t}{25}\right] .\n$$\nChoose $C \\in \\mathbb{R}$ 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 (D)."
include: _mmlu_flan_cot_fewshot_template_yaml
task: mmlu_flan_cot_fewshot_college_mathematics
