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TheoremQA_standardized

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Is x-1 a factor of 2*x^4+3*x^2-5x+7?

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False

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A company has 8 employees, including 3 managers and 5 engineers. How many different ways are there to form a team of 4 employees that includes at least 1 manager and at least 2 engineers?

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60

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The open mapping theorem can be proved by (a) Baire category theorem; (b) Cauchy integral theorem; (c) random graph theorem; (d) None of the above. Which option is correct?

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(a)

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A bungee cord is 30.0 m long and, when stretched a distance x, it exerts a restoring force of magnitude kx. Your father-in-law (mass 95.0 kg) stands on a platform 45.0 m above the ground, and one end of the cord is tied securely to his ankle and the other end to the platform. You have promised him that when he steps off the platform he will fall a maximum distance of only 41.0 m before the cord stops him. You had several bungee cords to select from, and you tested them by stretching them out, tying one end to a tree, and pulling on the other end with a force of 380.0 N. When you do this, what distance (in m) will the bungee cord that you should select have stretched?

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0.602

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In how many ways can a set of 6 distinct letters be partitioned into 3 non-empty groups if each group must contain at least 2 letters?

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15

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The position of a point for any time t (t>0) s defined by the equations: x=2t, y=ln(t), z = t^2. Find the mean velocity of motion between times t=1 and t=10.

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11.25

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A scuba diver is wearing a head lamp and looking up at the surface of the water. If the minimum angle to the vertical resulting in total internal reflection is 25∘, what is the index of refraction of the water? $\theta_{air} = 1.00$.

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2.37

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Let $X_1, X_2, \ldots$ be a sequence of independent indetically distributed random variables drawn according to the probability mass function $p(x) = N(0,1)$. Let $q(x)=N(1,1)$ be another probability mass function. Use natural logarithm to evaluate $\lim -\frac{1}{n}\log{q(X_1,X_2,\ldots,X_n)}$ as $n \to \infty$.

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1.4

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What is the value of the integral $\int_0^{\pi/2} 1/(1+(tan(x))^{\sqrt{2}}) dx$?

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0.78539815

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How many trees are there on n (n > 1) labeled vertices with no vertices of degree 1 or 2?

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0

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Given the following spot rates: 1-year spot rate: 5%; 2-year spot rate: 6%. Determine the one-year forward rate (between 0 and 1) one year from today.

instruction

0.070095

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Let f be a real function on [0,1]. If the bounded variation of f on [0,1] equals f(1)-f(0), then: (a) f is increasing on [0,1]; (b) f is decreasing on [0,1]; (c) None of the above. Which one is correct?

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(a)

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Let X_2,X_3,... be independent random variables such that $P(X_n=n)=P(X_n=-n)=1/(2n\log (n)), P(X_n=0)=1-1/(n*\log(n))$. Does $n^{-1}\sum_{i=2}^n X_i$ converges in probability? Does $n^{-1}\sum_{i=2}^n X_i$ converges in almost surely? Return the answers of the two questions as a list.

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[1, 0]

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Calculate the total capacitive reactance in the figure. Answer in unit of Ohm (3 sig.fig.).

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3.18

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Are the vectors [1, 2], [2, 3], and [3, 4] linearly independent?

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False

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How many pairs of (a, b) can we substitute for a and b in 30a0b03 so that the resulting integer is divisible by 13?

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3

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A pizza parlor offers 8 different toppings. In how many ways can a customer order a pizza with 3 toppings?

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56

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Please solve x^3 + 2*x = 10 using newton-raphson method.

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1.8474

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Both A, B are n-by-n matrices with rank(A)=n, rank(A*B)=0. What is rank(B)?

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0.0

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Suppose a monopoly market has a demand function in which quantity demanded depends not only on market price (P) but also on the amount of advertising the firm does (A, measured in dollars). The specific form of this function is Q = (20 - P)(1 + 0.1A - 0.01A^2). The monopolistic firm's cost function is given by C = 10Q + 15 + A. Suppose there is no advertising (A = 0). What output will the profit-maximizing firm choose?

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5

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Calculate the interest rate (between 0 and 1) for an account that started with $5,000 and now has $13,000 and has been compounded annually for the past 12 years. Answer with the numeric value.

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0.0828

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Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were 71,76,80,82,and 91. What was the last score Mrs. Walter entered?

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80

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A hospital has a 3.0 x 10^14 Bq Co-60 source for cancer therapy. The rate of gamma rays incident on a patient of area 0.30 m^2 located 4.0 m from the source is $X*10^11$ Bq, what is X? Co-60 emits a 1.1- and a 1.3-MeV gamma ray for each disintegration.

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8.95

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Let V be the space of all infinite sequences of real numbers. Consider the transformation T(x_0, x_1, x_2, ...) = (x_1, x_2, x_3, ...) from V to V. Is the sequence (1,2,3,...) in the image of T?

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True

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Estimate the PEG ratio for a firm that has the following characteristics: Length of high growth = five years Growth rate in first five years = 25% Payout ratio in first five years = 20% Growth rate after five years = 8% Payout ratio after five years = 50% Beta = 1.0 Risk-free rate = T-bond rate = 6% Cost of equity = 6% + 1(5.5%) = 11.5% Risk premium = 5.5% What is the estimated PEG ratio for this firm?

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1.15

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In triangle ABC, AB = 9x-1, CB = 5x-0.5, AC = 4x+1, and AC = CB. Find the measure of AB.

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12.5

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Is the transformation T(M) = [[1, 2], [3, 4]]M from R^{2*2} to R^{2*2} an isomorphism?

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True

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Universal Fur is located in Clyde, Baffin Island, and sells high-quality fur bow ties throughout the world at a price of $5 each. The production function for fur bow ties (q) is given by q = 240x - 2x^2, where x is the quantity of pelts used each week. Pelts are supplied only by Dan's Trading Post, which obtains them by hiring Eskimo trappers at a rate of $10 per day. Dan's weekly production function for pelts is given by x = \sqrt{l}, where l represents the number of days of Eskimo time used each week. For a quasi-competitive case in which both Universal Fur and Dan's Trading Post act as price-takers for pelts, what will be the equilibrium price (p_x) for pelt?

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600

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Consider a periodic signal $x(t)$ with period $(T)$ equals to ten. Over one period (i.e., $-5 \leq t<5)$, it is defined as $$ x(t)=\left\{\begin{array}{cc} 2 & -5 \leq t<0 \\ -2 & 0 \leq t<5 \end{array}\right. $$ In Fourier series, the signal $x(t)$ is written in the form of $$ x(t)=\sum_{k=-\infty}^{\infty} c_k e^{\frac{j 2 \pi k t}{T}} $$ where the Fourier series coefficient $c_k$ is obtained as, $$ c_k=\frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} x(t) e^{-\frac{j 2 \pi k t}{T}} d t $$ Determine the value of $c_0$ (i.e., $\left.k=0\right)$

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0

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For every positive real number $x$, let $g(x)=\lim _{r \rightarrow 0}((x+1)^{r+1}-x^{r+1})^{1/r}$. What is the limit of $g(x)/x$ as $x$ goes to infinity?

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2.7182818

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Suppose there is a 50-50 chance that an individual with logarithmic utility from wealth and with a current wealth of $20,000 will suffer a loss of $10,000 from a car accident. Insurance is competitively provided at actuarially fair rates. Compute the utility if the individual buys full insurance.

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9.616

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Compute the double integrals over indicated rectangles $\iint\limits_{R}{{2x - 4{y^3}\,dA}}$, $R = [-5,4] \times [0, 3]

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-756

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In complex analysis, define U^n={(z_1, \cdots, z_n): |z_j|<1, j=1, \cdots, n} and B_n={(z_1, \cdots, z_n): \sum_{j=1}^n |z_j|^2<1 }. Are they conformally equivalent in C^n? Here C^n is the d-dimensional complex space. Return 1 for yes and 0 for no.

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0.0

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The diagonals of rhombus QRST intersect at P. If m∠QTS = 76, find m∠TSP.

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52

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In the figure, at what rate is thermal energy being generated in the 2R-resistor when $V_s = 12V$ and $R = 3.0\Omega$? Answer in unit of W.

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6

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Find integer $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square.

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3

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The current price of gold is $412 per ounce. The storage cost is $2 per ounce per year, payable quaterly in advance. Assuming a constant intrest rate of 9% compounded quarterly, what is the theoretial forward price of gold for delivery in 9 months?

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442.02

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Suppose we have the following differential equation with the initial condition: $\frac{\partial p}{\partial x} = 0.5 * x * (1-x)$ and $p(0)=2$. Use Euler's method to approximate p(2), using step of 1.

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2.0

output

what is the value of \sum_{n=0}^{\infty}(-1)^n \frac{1}{3 n+1}? Round the answer to the thousands decimal.

instruction

0.8356488482647211

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Maximize the entropy $H(X)$ of a non-negative integer-valued random variable $X$, taking values from 0 to infinity, subject to the constraint $E(X)=1$. Use base 2 logarithm to evaluate $H(X)$.

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2.0

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Does $p(x) = x^5 + x − 1$ have any real roots?

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True

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What is \lim_{x o 9} ((x - 9)/(\sqrt{x} - 3))?

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6

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Let S be the set of integers between 1 and 2^40 that contain two 1’s when written in base 2. What is the probability that a random integer from S is divisible by 9?

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0.1705

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Suppose 100 cars will be offered on the used-car market. Let 50 of them be good cars, each worth $10,000 to a buyer, and let 50 be lemons, each worth only $2,000. Suppose that there are enough buyers relative to sellers that competition among them leads cars to be sold at their maximum willingness to pay. What would the market equilibrium price for good cars be if sellers value good cars at $6,000?

instruction

6000

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Is the Fourier transform of the signal $x_1(t)=\left\{\begin{array}{cc}\sin \omega_0 t, & -\frac{2 \pi}{\omega_0} \leq t \leq \frac{2 \pi}{\omega_0} \\ 0, & \text { otherwise }\end{array}\right.$ imaginary?

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True

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Suppose H is a Banach space. Let A be a linear functional on the space H that maps H to H. Suppose operator A satisfies: for all $x\in H$, $||Ax||\geq a ||x||$ for some a>0. If A is not a compact operator on H, Is the dimension of H finite or infinite? Return 1 for finite dimension and 0 for infinite dimension

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0.0

output

what is the limit of $(n!)^{1/n}/n$ as n goes to infinity? Round the answer to the thousands decimal.

instruction

0.367879441

output

For (10236, 244), use the Euclidean algorithm to find their gcd.

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4

output

What is the order of the group S_3 * Z_2?

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12

output

What's the maximum number of edges in a simple triangle free planar graph with 30 vertices?

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56

output