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TheoremQA_standardized

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A function f(x) is given by f(0)=3, f(2)=7, f(4)=11, f(6)=9, f(8)=3. Approximate the area under the curve y=f(x) between x=0 and x=8 using Trapezoidal rule with n=4 subintervals.

instruction

60.0

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Suppose H=L^2[0,1]. Operator $A: u(t) \mapsto t\times u(t)$ is a map from H to H. Then A is a bounded linear operator. Then the spectrum of A is: (a) [0,1], (b) [0,1/2], (c) [1/2, 1], (d) none of the above. Which one is correct?

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(a)

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Two sets of points are linearly separable if and only if their convex hulls are disjoint. True or False?

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True

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Suppose there are 8,000 hours in a year (actually there are 8,760) and that an individual has a potential market wage of $5 per hour. Suppose a rich uncle dies and leaves the individual an annual income of $4,000 per year. If he or she devotes 75 percent of full income to leisure, how many hours will be worked?

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1400

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Define f(x)=(4x+5)/(9-3x), is the function continuous at x=-1?

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True

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A radioactive sample contains two different isotopes, A and B. A has a half-life of 3 days, and B has a half-life of 6 days. Initially in the sample there are twice as many atoms of A as of B. In how many days will the ratio of the number of atoms of A to B be reversed?

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12.0

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Let $F_0(x)=log(x)$. For $n\geq 0$ and $x>0$, let $F_{n+1}(x)=\int_0^x F_n(t)dt$. Evaluate $\lim _{n \rightarrow \infty} (n! F_n(1))/(log(n))$.

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-1.0

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Find the absolute minimum value of the function $f(x,y)=x^2+y^2$ subject to the constraint $x^2+2*y^2=1$.

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0.5

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Which of these codes cannot be Huffman codes for any probability assignment? (a) {0, 10, 11}. (b) {00, 01, 10, 110}. (c) {0, 1}.

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(b)

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Suppose that $(X, Y, Z)$ are jointly Gaussian and that $X \rightarrow Y \rightarrow Z$ forms a Markov chain. Let $X$ and $Y$ have correlation coefficient 0.1 and let $Y$ and $Z$ have correlation coefficient 0.9. Find $I(X;Z)$ in bits.

instruction

0.00587

output

You throw a ball from your window $8.0 \mathrm{~m}$ above the ground. When the ball leaves your hand, it is moving at $10.0 \mathrm{~m} / \athrm{s}$ at an angle of $20^{\circ}$ below the horizontal. How far horizontally from your window will the ball hit the ground? Ignore air resistance. (Unit: m)

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9.2

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Compute the mean translational kinetic energy of a single ideal gas molecule in eV.

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0.038

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Does the function $y=xe^{-x^2/2}$, does it satisfy the equation $xy' = (1 - x^2)y$

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True

output

The planet Pluto (radius 1180 km) is populated by three species of purple caterpillar. Studies have established the following facts: 1. A line of 5 mauve caterpillars is as long as a line of 7 violet caterpillars. 2. A line of 3 lavender caterpillars and 1 mauve caterpillar is as long as a line of 8 violet caterpillars. 3. A line of 5 lavender caterpillars, 5 mauve caterpillars and 2 violet caterpillars is 1 m long in total. 4. A lavender caterpillar takes 10 s to crawl the length of a violet caterpillar. 5. Violet and mauve caterpillars both crawl twice as fast as lavender caterpillars. How many years would it take a mauve caterpillar to crawl around the equator of Pluto?

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23.0

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Consider an arbitrage-free securities market model, in which the risk-free interest rate is constant. There are two nondividend-paying stocks whose price processes are: $S_1(t)=S_1(0)e^{0.1t+0.2Z(t)}$ $S_2(t)=S_2(0)e^{0.125t+0.3Z(t)}$ where $Z(t)$ is a standard Brownian motion ant $t\ge0$. What is the continuously compounded risk-free interest rate?

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0.02

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Find the solutions to the second order boundary-value problem. y''-2y'+2y=0, y(0)=0, y(\pi/2) = 1. What is y(\pi/4)?

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0.322

output

At a waterpark, sleds with riders are sent along a slippery, horizontal surface by the release of a large compressed spring. The spring with force constant k = 40.0 N/cm and negligible mass rests on the frictionless horizontal surface. One end is in contact with a stationary wall. A sled and rider with total mass 70.0 kg are pushed against the other end, compressing the spring 0.375 m. The sled is then released with zero initial velocity. What is the sled's speed (in m/s) when the spring returns to its uncompressed length?

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2.83

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If x(n) and X(k) are an N-point DFT pair, then x(n+N)=x(n). Is it true?

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True

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Malus' law: $I=I_0*cos^2($\theta$)$. Where I is the intensity of polarized light that has passed through the polarizer, I_0 is the intensity of polarized light before the polarizer, and $\theta$ is the angle between the polarized light and the polarizer. Unpolarized light passes through a polarizer. It then passes through another polarizer at angle 40 degree to the first, and then another at angle 15 degree to the second. What percentage of the original intensity was the light coming out of the second polarizer?

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54.8

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The mass of Earth is 5.97x10^24 kg, the mass of the Moon is 7.35x10^22 kg, and the mean distance of the Moon from the center of Earth is 3.84x105 km. The magnitude of the gravitational force exerted by Earth on the Moon is X * 10^20 N. What is X? Return a numeric value.

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1.99

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In triangle ACD, B is located on the side AC, and E is located on the side AD. If AB = 3, AC = 5, CD = 3.5, ED = 3, and EB ∥ DC, what is the length of AD?

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7.5

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Consider a two-layer fully-connected neural network in which the hidden-unit nonlinear activation functions are given by logistic sigmoid functions. Does there exist an equivalent network in which the hidden unit nonlinear activation functions are given by hyperbolic tangent functions?

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True

output

There are only three active stations in a slotted Aloha network: A, B and C. Each station generates a frame in a time slot with the corresponding probabilities p_A=0.2, p_B=0.3 and p_C=0.4 respectively. What is the normalized throughput of the system?

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0.452

output

A linear learning machine based on the kernel $k(x,x')=f(x)f(x')$ will always find a solution proportional to $f(x)$. True or false?

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True

output

Does the utility function U(x,y) = xy/(x+y) has a convex indifference curve?

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True

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Assuming we are underground, and the only thing we can observe is whether a person brings an umbrella or not. The weather could be either rainy or sunny. Assuming the P(rain)=0.6 and P(sunny)=0.4. Assuming the weather on day $k$ is dependent on the weather on day $k-1$. We can write the transition probability as P(sunny $\mid$ sunny) = P(rain $\mid$ rain) = 0.7. The person has 60\% chance to bring an umbrella when the weather is rainy, and 40\% chance to bring an umbrella when the weather is sunny, i.e. P(umbrella $\mid$ rain) = 0.6 and P(umbrella $\mid$ sunny) = 0.4. If we observe that the person (1) brought an umbrella on day 1, (2) did not bring an umbrella on day 2, (3) brought an umbrella on day 3, (4) did not bring an umbrella on day 4. What are the most likely weather from day 1 to day 4? Return the answer as a list of binary values, where 1 represents rain and 0 represents sunny.

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[1, 1, 1, 1]

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Suppose that $X_1,X_2,...$ are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let $S=\sum_{i=1}^k X_i/2^i$, where $k$ is the least positive integer such that $X_k<X_{k+1}$, or $k=\infty$ if there is no such integer. Find the expected value of S.

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0.29744254

output

A network with one primary and four secondary stations uses polling. The size of a data frame is 1000 bytes. The size of the poll, ACK, and NAK frames are 32 bytes each. Each station has 5 frames to send. How many total bytes are exchanged if each station can send only one frame in response to a poll?

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21536

output

An auto magazine reports that a certain sports car has 53% of its weight on the front wheels and 47% on its rear wheels. (That is, the total normal forces on the front and rear wheels are 0.53w and 0.47w, respectively, where w is the car’s weight.) The distance between the axles is 2.46 m. How far in front of the rear axle is the car’s center of gravity?

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1.3

output

Assuming we are underground, and the only thing we can observe is whether a person brings an umbrella or not. The weather could be either rainy or sunny. Assuming the P(rain)=0.6 and P(sunny)=0.4. Assuming the weather on day $k$ is dependent on the weather on day $k-1$. We can write the transition probability as P(sunny $\mid$ sunny) = P(rain $\mid$ rain) = 0.55. The person has 60\% chance to bring an umbrella when the weather is rainy, and 40\% chance to bring an umbrella when the weather is sunny, i.e. P(umbrella $\mid$ rain) = 0.6 and P(umbrella $\mid$ sunny) = 0.4. If we observe that the person (1) brought an umbrella on day 1, (2) did not bring an umbrella on day 2, (3) brought an umbrella on day 3. What are the most likely weather from day 1 to day 3? Return the answer as a list of binary values, where 1 represents rain and 0 represents sunny.

instruction

[1, 0, 1]

output

Let a undirected graph G with edges E = {<0,4>,<4,1>,<0,3>,<3,4>,<3,2>,<1,3>}, which <A,B> represent Node A is connected to Node B. What is the minimum vertex cover of G? Represent the vertex cover in a list of ascending order.

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[3, 4]

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What are the real eigenvalues of the matrix [[3, -2, 5], [1, 0, 7], [0, 0, 2]]?

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[1, 2, 2]

output

The planet Mercury travels around the Sun with a mean orbital radius of 5.8x10^10 m. The mass of the Sun is 1.99x10^30 kg. Use Newton's version of Kepler's third law to determine how long it takes Mercury to orbit the Sun. Give your answer in Earth days.

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88.3

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In year N, the 300th day of the year is a Tuesday. In year N + 1, the 200th day is also a Tuesday. Suppose Monday is the 1-th day of the week, on which day of the week did the 100th day of the year N - 1 occur? Return a numeric between 1 and 7.

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4

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CheckMate forecasts that its dividend will grow at 20% per year for the next four years before settling down at a constant 8% forever. Dividend (current year,2016) = $12; expected rate of return = 15%. What is the fair value of the stock now?

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273.0

output

Consider a horizontal strip of N+2 squares in which the first and the last square are black and the remaining N squares are all white. Choose a white square uniformly at random, choose one of its two neighbors with equal probability, and color this neighboring square black if it is not already black. Repeat this process until all the remaining white squares have only black neighbors. Let $w(N)$ be the expected number of white squares remaining. What is the limit of $w(N)/N$ as $N$ goes to infinity?

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0.36787944

output

Find the arc length of the curve, where x=t, y=t^2 and z=2*t^3/3.

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7.333

output

Is the function of f(x) = sin(x) / |x| continuous everywhere?

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False

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What is \lim_{x \to 0} (csc(x) - cot(x))?

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0

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In a CSMA/CD network with a data rate of 10 Mbps, the minimum frame size is found to be 512 bits for the correct operation of the collision detection process. What should be the minimum frame size (in bits) if we increase the data rate to 1 Gbps?

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51200

output

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

instruction

1.42

output

For a $1,000 investment, what is the future value of the investment if the interest rate is 8% compounded annually for 3 years?

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1259.71

output

Water stands 12.0 m deep in a storage tank whose top is open to the atmosphere. What are the gauge pressures at the bottom of the tank? (Unit: 10 ^ 5 Pa)

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1.18

output

A group of 7 people is to be divided into 3 committees. Within each committee, people are ranked in a certain order. In how many ways can this be done?

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12600

output

How many people at least shall we include in one group, such that there must exist two different people in this group whose birthdays are in the same month?

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13

output

Light travel from water n=1.33 to diamond n=2.42. If the angle of incidence was 13 degree, determine the angle of refraction.

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7.1

output

Based on field experiments, a new variety green gram is expected to given an yield of 12.0 quintals per hectare. The variety was tested on 10 randomly selected farmers fields. The yield ( quintals/hectare) were recorded as 14.3,12.6,13.7,10.9,13.7,12.0,11.4,12.0,12.6,13.1. Do the results conform the expectation with Level of significance being 5%?

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True

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Let W(t) be the standard Brownian motion, and 0 < s < t. Find the conditional PDF of W(s = 1/2) given that W(t = 1) = 2. What are the mean and variance? Return the list of [mean, variance].

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[1.0, 0.25]

output

How many trees are there on 5 unlabeled vertices?

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3

output

The difference equation of a causal system is $y[n]+0.5 y[n-1]=x[n]-x[n-2]$, where $y[n]$ is its output and $x[n]$ is its input. Is the system a FIR filter?

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False

output