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kranthigv

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TheoremQA_standardized

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Does the following transformation have an eigenvector: Counterclockwise rotation through an angle of 45 degrees followed by a scaling by 2 in R^2.

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False

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Let h(x) = 1/(\sqrt{x} + 1). What is h''(x) when x = 1?

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0.125

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In how many ways can 6 people be seated at 2 identical round tables? Each table must have at least 1 person seated.

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225

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Determine the AC power gain for the emitter-follower in the figure. Assume that $\beta_{ac} = 175$

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24.1

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Let G_n(s) be the probability generating function of the size Z_n of the n-th generation of a branching process, where Z_0=1 and var(Z_1)>0. Let H_n be the inverse function of the function G_n, viewed as a function on the interval [0, 1]. Is M_n= {H_n(s)}^{Z_n} defines a martingale with respect to the sequence Z? Return 1 for yes and 0 for no.

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1.0

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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 30 degree to the first, and then another at angle 50 degree to the second. What percentage of the original intensity was the light coming out of the third polarizer?

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31.0

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What is the number of equivalent parameter settings due to interchange symmetries in a mixture model with 10 components?

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3628800

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A survey shows that a mayoral candidate is gaining votes at a rate of 2000t + 1000 votes per day, where t is the number of days since she announced her candidacy. How many supporters will the candidate have after 60 days, assuming that she had no supporters at t = 0?

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3660000

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Let g(x) be the inverse of f(x) = x + cos(x). What is g'(1)?

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1

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What is the minimum number of people needed in a room to guarantee that there are 4 mutual friends or 4 mutual strangers?

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18

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Let P[0,1] denotes all the polynomials on the interval [0,1]. Define the distance \rho(p, q)=\int_0^1|p(x)-q(x)| dx. Is (P[0,1],\rho) a complete space? Return 1 for yes and 0 for no.

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0.0

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In a sinusoidal sound wave of moderate loudness, the maximum pressure variations are about $3.0 \times 10 ^ {-2}$ Pa above and below atmospheric pressure. Find the corresponding maximum displacement if the frequency is 1000 Hz. In air at normal atmospheric pressure and density, the speed of sound is 344 m/s and the bulk modulus is $1.42 \times 10^5$ Pa. (Unit: $10 ^ {-8}$)

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1.2

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Find the fraction of the standard solar flux reaching the Earth (about 1000 W/m^2) available to a solar collector lying flat on the Earth’s surface at Miami (latitude 26°N) at noon on the winter solstice.

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0.656

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Given that each cone can contain two ice cream balls, how many different ice cream cones can you make if you have 6 flavors of ice cream and 5 types of cones?

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180

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Let L^1[0,2] be the space of all the Lebesgue integrable functions on the interval [0,2], and C[0,2] be the space of all the continuous functions on the interval [0,2]. Suppose H=L^1[0,2], and X=C[0,2]. For any f\in L^1[0,2], define operator T as $(Tf)(x)=\int_0^x f(t)dt$. For the linear operator T from H to X, what is the norm of T? For the linear operator T from H to H, what is the norm of T? Return the answers of two questions as a list. For example, if the norm for the first question is 2, the second is 3, then return [2,3].

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[1, 2]

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Let {N(t), t \in [0, \infty)} be a Poisson process with rate of $\lambda = 4$. Find it covariance function $C_N(t1, t2) for t1, t2 \in [0, \infy)$. What is C_N(2, 4)?

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8

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Suppose a stock has the following information. It is listed on the London stock exchange and operates throughout Europe. The yield on a UK 10 year treasury is 2.8%. The stock in question will earn 8.6% as per historical data. The Beta for the stock is 1.4, i.e., it is 140% volatile to the changes in the general stock market. What is the expected rate of return?

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10.92

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The spontaneous fission activity rate of U-238 is 6.7 fissions/kg s. A sample of shale contains 0.055% U-238 by weight. Calculate the number of spontaneous fissions in one day in a 106-kg pile of the shale by determining the mass of U-238 present in kg.

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550.0

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Estimate the PE 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 PE ratio for this firm?

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28.75

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Let a undirected graph G with edges E = {<2,1>,<2,0>,<2,3>,<1,4>,<4,3>}, 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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[2, 4]

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Is the Fourier transform of the signal x(t)=(1-e^{-|t|})[u(t+1)-u(t-1)] even?

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True

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Calculate the Hamming pairwise distances and determine the minimum Hamming distance among the following codewords: 000000,010101,101010,110110

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3

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For any triangle ABC, we have cos(A)cost(B)cos(C) $\leq$ 1/8, is this true or false?

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True

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Find the number of integers n, 1 ≤ n ≤ 25 such that n^2 + 3n + 2 is divisible by 6.

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13

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For a one-period binomial model for the price of a stock, you are given: (i) The period is one year. (ii) The stock pays no dividends. (iii) u =1.433, where u is one plus the rate of capital gain on the stock if the price goes up. (iv) d = 0.756 , where d is one plus the rate of capital loss on the stock if the price goes down. (v) The continuously compounded annual expected return on the stock is 10%. What is the true probability of the stock price going up.

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0.52

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Incompressible oil of density 850 kg/m^3 is pumped through a cylindrical pipe at a rate of 9.5 liters per second. The second section of the pipe has a diameter of 4.0 cm. What are the flow speed in that section? (Unit: m/s)

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7.6

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For the equation x^4 + 2*x^3 + x = 10, there are four roots. What is the sum of the roots using newton-raphson method.

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-2.0

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Let M be the set of bounded functions (i.e. \sup_{x\in[a,b]}|f(x)|<\infty) in C[0,1]. Is the set ${F(x)=\int_0^x f(t) dt | f \in M }$ a sequentially compact set? Answer 1 for yes and 0 for no. Furthermore, it can be proved using 1. Arzelà-Ascoli theorem, 2. Riesz representation theorem, 3. Banach fixed point theorem, 4. None of the above. Return the answers of the two questions in a list. For example, if you think the answer is no and Riesz representation theorem, then return [0,2].

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[1, 1]

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Are the circuits shown in Fig. Qla and Fig. Q1b are identical? (Hint: Compare the Tranfer functions)

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True

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Evaluate $\int_c 1 / (z^ + 4)^2 dz$ over the contour. This contour is a circle centered at (0, i) with a diameter of 3 on the (Re, Im) plane, the contour goes counter-clockwise.

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0.19634

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John's Lawn Mowing Service is a small business that acts as a price-taker (i.e., MR = P). The prevailing market price of lawn mowing is $20 per acre. John's costs are given by total cost = 0.1q^2 + 10q + 50, where q = the number of acres John chooses to cut a day. Calculate John's maximum daily profit.

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200

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Toss a coin repeatedly until two consecutive heads appear. Assume that the probability of the coin landing on heads is 3/7. Calculate the average number of times the coin needs to be tossed before the experiment can end.

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7.77778

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How many ways are there to divide a set of 6 elements into 3 non-empty ordered subsets?

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1200

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A Chord based distributed hash table (DHT) with 26 address space is used in a peer- to-peer file sharing network. There are currently 10 active peers in the network with node ID N1, N11, N15, N23, N31, N40, N45, N51, N60, and N63. Show all the target key (in ascending order, ignore the node's identifier itself) for N1.

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[2, 3, 5, 9, 17, 33]

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Find the mass and weight of the air at $20^{\circ} C$ in a living room with a $4.0 m \times 5.0 m$ floor and a ceiling 3.0 m high, and the mass and weight of an equal volume of water. (Unit: 10 ^ 5 N)

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5.9

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How many ways are there to arrange 6 pairs of parentheses such that they are balanced?

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132

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An investor has utility function $U(x) = x^{1/4}$ for salary. He has a new job offer which pays $80,000 with a bonus. The bonus will be $0, $10000, $20000, $30000, $40000, $50000, or $60000, each with equal probability. What is the certainty equivalent value of this job offer?

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108610

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A Chord based distributed hash table (DHT) with 25 address space is used in a peer- to-peer file sharing network. There are currently 5 active peers in the network with node ID N3, N8, N15, N19 and N30. Show all the target key (in ascending order, ignore the node's identifier itself) for N3.

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[4, 5, 7, 11, 19]

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A basketball team has 12 players, including 5 guards and 7 forwards. How many different starting lineups can be formed that include 3 guards and 2 forwards?

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210

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Consider a probability density $p_x(x)$ defined over a continuous variable x, and suppose that we make a nonlinear change of variable using $x = g(y)$. In the case of a linear transformation, the location of the maximum density transforms in the same way as the variable itself.

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True

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which n <= 20 can be constructed a regular n-gonwith compass and straightedge? return all the possible numbers in a list

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[3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20]

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A firm in a perfectly competitive industry has patented a new process for making widgets. The new process lowers the firm's average cost, meaning that this firm alone (although still a price taker) can earn real economic profits in the long run. Suppose a government study has found that the firm's new process is polluting the air and estimates the social marginal cost of widget production by this firm to be SMC = 0.5q. If the market price is $20, what should be the rate of a government-imposed excise tax to bring about optimal level of production?

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4

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What is the minimum number of people needed in a room to guarantee that there are 3 mutual friends or 3 mutual strangers?

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6

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Does r(t) = [8 - 4t^3, 2 + 5t^2, 9t^3] parametrize a line?

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False

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what is the value of $\int_{-infty}^{+infty} sin(3*t)*sin(t/\pi)/t^2 dt$?

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1.0

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What is the smallest number of vertices in a graph that guarantees the existence of a clique of size 3 or an independent set of size 2?

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3

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Let N be a spatial Poisson process with constant intensity $11$ in R^d, where d\geq2. Let S be the ball of radius $r$ centered at zero. Denote |S| to be the volume of the ball. What is N(S)/|S| as $r\rightarrow\infty$?

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11.0

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Clare manages a piano store. Her utility function is given by Utility = w - 100, where w is the total of all monetary payments to her and 100 represents the monetary equivalent of the disutility of exerting effort to run the store. Her next best alternative to managing the store gives her zero utility. The store's revenue depends on random factors, with an equal chance of being $1,000 or $400. If shareholders offered to share half of the store's revenue with her, what would her expected utility be?

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250

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How many ways are there to arrange the letters in the word *BANANA* up to the symmetries of the word?

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30

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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. 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.

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[1, 1, 1]

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