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TheoremQA_standardized

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Determine the number of positive real zero of the given function: $f(x)=x^5+4*x^4-3x^2+x-6$.

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[3, 1]

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A camper pours 0.300 kg of coffee, initially in a pot at 70.0°C into a 0.120-kg aluminum cup initially at 20.0°C. What is the equilibrium temperature? Assume that coffee has the same specific heat as water and that no heat is exchanged with the surroundings. (Unit: °C)

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66.0

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Suppose we are given the following information. Use this information to calculate abnormal return. Rf: 4% Rm: 12% Beta of the Portfolio: 1.8 Beginning Value of Portfolio: $50,000 Ending Value of Portfolio: $60,000 What is the abnormal return?

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0.016

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If polygon ABCDE ~ polygon PQRST, AB = BC = 8, AE = CD = 4, ED = 6, QR = QP, and RS = PT = 3, find the perimeter of polygon PQRST.

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22.5

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Are the vectors v_1 = [1,2,3], v_2 = [4,5,6], v_3 = [7,8,9] linearly independent?

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False

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The diagonals of rhombus FGHJ intersect at K. If m∠FJH = 82, find m∠KHJ.

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49

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Let’s assume that the 10-year annual return for the S&P 500 (market portfolio) is 10%, while the average annual return on Treasury bills (a good proxy for the risk-free rate) is 5%. The standard deviation is 15% over a 10-year period. Whats the market Sharpe Ratio?

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0.33

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Find the smallest positive integer that leaves a remainder of 5 when divided by 8, a remainder of 1 when divided by 3, and a remainder of 7 when divided by 11.

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205

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We know that $y'=(x+y) / 2$, we also know that $y(x=0) = 2, y(x=0.5) = 2.636, y(x=1) = 3.595, y(x=1.5) = 4.9868$, what is the value of y(2) using Adams bashforth predictor method.

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6.8731

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The equation of a digital filter is given by $y(n)=1 / 3(x(n)+x(n-1)+x(n-2))$, where $y(n)$ and $x(n)$ are, respectively, the nth samples of the output and input signals. Determine the pole(s) of the filter.

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0

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Obtain the number of real roots between 0 and 3 of the equation P(x) = x^4 -4x^3 + 3x^2 + 4x - 4 = 0 using Sturm's sequence.

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2

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A curve with a 120 m radius on a level road is banked at the correct angle for a speed of 20 m/s. If an automobile rounds this curve at 30 m/s, what is the minimum coefficient of static friction needed between tires and road to prevent skidding?

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0.34

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A court is investigating the possible occurrence of an unlikely event T. The reliability of two independent witnesses called Alf and Bob is known to the court: Alf tells the truth with probability \alpha and Bob with probability \beta, and there is no collusion between the two of them. Let A and B be the events that Alf and Bob assert (respectively) that T occurred, and let \tau=P(T). What is the probability that T occurred given that both Alf and Bob declare that T occurred? Suppose \alpha=\beta=9/10 and \tau=1/1000. Return the answer up to the thousands decimal.

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0.075

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In how many ways can we form a 7-digit number using the digits 1, 2, 2, 3, 3, 3, 4?

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420

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A glider with mass m = 0.200 kg sits on a frictionless horizontalair track, connected to a spring with force constant k = 5.00 N/m.You pull on the glider, stretching the spring 0.100 m, and release itfrom rest. The glider moves back toward its equilibrium position (x = 0).What is its x-velocity when x = 0.080 m? (Unit: m/s))

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-0.3

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What is the vector that spans the kernel of A = [[1, 0, 2, 4], [0, 1, -3, -1], [3, 4, -6, 8], [0, -1, 3, 4]]?

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[-2, 3, 1, 0]

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Every published book has a ten-digit ISBN-10 number that is usually of the form x_1 - x_2 x_3 x_4 - x_5 x_6 x_7 x_8 x_9 - x_{10} (where each x_i is a single digit). The first 9 digits identify the book. The last digit x_{10} is a check digit, it is chosen so that 10 x_1 + 9 x_2 + 8 x_3 + 7 x_4 + 6 x_5 + 5 x_6 + 4 x_7 + 3 x_8 + 2 x_9 + x_{10} = 0 (mod 11). Is 3-540-90518-9 a valid ISBN number?

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True

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Find the sum of $\sum_{n=1}^{\infty} (cost(1/n^2) - cost(1/(n+1)^2))$

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-0.459

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In Jules Verne's 1865 story with this title, three men went to the moon in a shell fired from a giant cannon sunk in the earth in Florida. Find the minimum muzzle speed that would allow a shell to escape from the earth completely (the escape speed). Neglect air resistance, the earth's rotation, and the gravitational pull of the moon. The earth's radius and mass are $R_E}=$ $6.38 \times 10^6 m$ and $m_E=5.97 \times 10^{24} kg$. (Unit: 10 ^ 4 m/s)

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1.12

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A model rocket follows the trajectory c(t) = (80t, 200t - 4.9t^2) until it hits the ground, with t in seconds and distance in meters. Find the rocket's maximum height in meters.

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2041

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Let a undirected graph G with edges E = {<0,1>,<0,2>,<0,3>,<3,5>,<2,3>,<2,4>,<4,5>}, which <A,B> represent Node A is connected to Node B. What is the shortest path from node 0 to node 5? Represent the path as a list.

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[0, 3, 5]

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The stock of the CCC Corporation is currently valued at $12 and is assumed to possess all the properties of geometric Brownian motion. It has an expected annual return of 15%, an annual volatility of 20%, and the annual risk-free is 10%. Using a binomial lattice, determine the price of a call option on CCC stock maturing in 10 monthes time with a strike price of $14 (Let the distance between nodes on your tree be 1 month in length).

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53.0

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Given the following circuit (with all current and voltage values in rms), find the value of $V_C$ in the unit of V.

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14.5

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Suppose the Markov Chain satisfies the diagram ./mingyin/diagram.png What is the period of state 0? What is the period of state 1? Return the two answers as a list.

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[2, 2]

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Is (t-y)y' - 2y +3t + y^2/t = 0 an Euler homogeneous equation?

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True

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Sum the series $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty}\frac{m^2 n}{3^m(n3^m+m3^n)}$

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0.28125

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Given that the Hue-Saturation subspace shown in Fig. Q2 is a perfect circle and that colors A, B and C can be represented as the 3 points shown in the subspace. Which color has the smallest saturation coefficient?

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(b)

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Assume a temperature of 300 K and find the wavelength of the photon necessary to cause an electron to jump from the valence to the conduction band in germanium in nm.

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1850.0

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How many ways are there to color the faces of a cube with three colors, up to rotation?

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57

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Your company has just written one million units of a one-year European asset-or-nothing put option on an equity index fund. The equity index fund is currently trading at 1000. It pays dividends continuously at a rate proportional to its price; the dividend yield is 2%. It has a volatility of 20%. The option’s payoff will be made only if the equity index fund is down by more than 40% at the end of one year. The continuously compounded risk-free interest rate is 2.5% Using the Black-Scholes model, determine the price of the asset-or-nothing put options. Give the answer in millions.

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3.6

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Astrophysical theory suggests that a burned-out star whose mass is at least three solar masses will collapse under its own gravity to form a black hole. If it does, the radius of its event horizon is X * 10^3 m, what is X?

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8.9

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For the 3 payments of $1000 each end-of-year, with 7% rate of return, what is the present value if the first payment is made at the end of fifth year?

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2002.0781

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Let {N(t), t=[0, \infty]} be a Poisson process with rate $\lambda = 5$. Find the probability of no arrivals in [3, 5)

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0.37

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Perform 2 iterations with the Müller method for the following equation: log_{10}(x) - x + 3 = 0, x_0 = 1/4, x_1 = 1/2, x_2 = 1. What's the decimal value of x_3?

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3.2

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Let $I(R)=\iint_{x^2+y^2 \leq R^2}(\frac{1+2 x^2}{1+x^4+6x^2y^2+y^4}-\frac{1+y^2}{2+x^4+y^4}) dx dy$. What is the limit of $I(R)$ as $R$ goes to infinity?

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1.53978589

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What is $(\frac{1 + cos(2x) + i*sin(2x)}{1 + cos(2x) - i*sin(2x)})^30$ with $x = \pi / 60$?

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-1.0

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George is seen to place an even-money $100,000 bet on the Bulls to win the NBA Finals. If George has a logarithmic utility-of-wealth function and if his current wealth is $1,000,000, what must he believe is the minimum probability that the Bulls will win?

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0.525

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The Chi-square statistic $\chi^2=\sum_c\frac{(P(x)-Q(x))^2}{Q(x)}$ is (twice) the first term in the Taylor series expansion of $D(P||Q)$ about $Q$. True or False?

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True

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Let a undirected graph G with edges E = {<0,1>,<1,3>,<0,3>,<3,4>,<0,4>,<1,2>,<2,5>,<2,7>,<2,6>,<6,7>,<6,10>,<5,8>,<10,9>,<5,10>,<6,8>,<7,8>,<6,9>,<7,10>,<8,10>,<9,11>,<9,12>,<9,13>,<13,12>,<13,11>,<11,14>}, which <A,B> represent Node A is connected to Node B. What is the shortest path from node 1 to node 14? Represent the path as a list.

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[1, 2, 6, 9, 11, 14]

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A distribution represented by a directed tree can be written as an equivalent distribution over the corresponding undirected tree. True or false?

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True

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compute the line integral of \int_K xy dx, \int_L xy dx, where K is a straight line from (0,0) to (1,1) and L is the Parabola y=x^2 from (0,0) to (1,1). return the answer as a list

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[0.333, 0.25]

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Consider the 7.0-TeV protons that are produced in the LHC collider at CERN. Find the available center-of-mass energy if these protons collide with other protons in a fixed-target experiment in GeV.

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114.5

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Solve the following linear programming problems: maximize 3x + y subject to (1) -x + y <= 1, (2) 2x + y <= 4, (3) x>= 0 and y >= 0. What's [x, y] for the optimal solution?

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[2, 0]

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ABCD is a Quadrilateral. E is the midpoint of BC. F is the midpoint of AD. Area of ABG=9 and Area of GEHF=21. What is the Area of CHD?

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12

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A train pulls out of the station at constant velocity. The received signal energy thus falls off with time as $1/i^2$. The total received signal at time $i$ is $Y_i = \frac{1}{i}X_i + Z_i$ where $Z_1, Z_2, \ldots$ are i.i.d. drawn from $N(0,1)$. The transmitter constraint for block length $n$ is $\frac{1}{n}\sum_{i=1}^n x_i^2(w) \leq 2 $ for $w \in \{1,2,\ldots, 2^{nR}\}$. Use Fano's inequality to find the capacity for this channel.

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0.0

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You have a coin and you would like to check whether it is fair or biased. More specifically, let $\theta$ be the probability of heads, $\theta = P(H)$. Suppose that you need to choose between the following hypotheses: H_0 (null hypothesis): The coin is fair, i.e. $\theta = \theta_0 = 1 / 2$. H_1 (the alternative hypothesis): The coin is not fair, i.e. $\theta > 1 / 2$. We toss 100 times and observe 60 heads. Can we reject H_0 at significance level $\alpha = 0.01$?

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False

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The marginal distribution for the variables $x_s$ in a factor $f_s(x_s)$ in a tree-structured factor graph, after running the sum-product message passing algorithm, can be written as the product of the message arriving at the factor node along all its links, times the local factor $f_s(x_s)$. True or false?

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True

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In a group of 10 people, each of whom has one of 3 different eye colors, at least how many people must have the same eye color?

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4

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Suppose there are 10 independent random variables $X_1, X_2, \cdots, X_10$. Each of the $X_i$ lies within the range of [10, 11] with a mean value of 10.5. If we take the mean of the 10 random variables as $\hat{X_n}$. What is the upper bound of the probability that $\hat{X_n}$ is either smaller than 10.2 or larger than 10.8?

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0.3305

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Assuming $x$ and $y$ are both 2-d random variable. The covariance matrix of $x=((1,2),(2,3),(3,5),(4,4))$, $y=((3,4),(1,5),(5,3),(3,3))$ is $Cov$. What is the trace of $Cov$?

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-0.166

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