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TheoremQA_standardized

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Find $\int_{0}^{\sqrt{3}} \frac{dx}{1+x^2}$.

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1.0472

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

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2.767

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The function f: U_5 o U_5 given by f(x) = x^2 is a homomorphism. What is K_f?

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[4, 1]

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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 silicon in nm.

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1130.0

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Lore Ltd. estimates that its dividend growth will be 13% per year for the next five years. It will then settle to a sustainable, constant, and continuing rate of 5%. Let’s say that the current year’s dividend is $14 and the required rate of return (or discount rate) is 12%. What is the current fair value of Lore Ltd. stock?

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291.45

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Use the Birge-Vieta method to find a real root correct to three decimals of the following equation: x^3 - 11x^2 + 32x - 22 = 0, p = 0.5

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1

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Let M be the inverse of the group element ((3, 5), (4, 6)) in Z_7. What is M[0][1]?

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6

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An airplane is flying at Mach 1.75 at an altitude of 8000 m, where the speed of sound is How long after the plane passes directly overhead will you hear the sonic boom? (Unit: m/s)

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560

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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 number of fissions.

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320000000.0

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A hydraulic press contains $0.25 m^3$ (250 L) of oil. Find the decrease in the volume of the oil when it is subjected to a pressure increase $\Delta p=1.6 \times 10^7 Pa$ (about 160 atm or 2300 psi). The bulk modulus of the oil is $B=5.0 \times 10^9 Pa$ (about $5.0 \times 10^4 atm$) and its compressibility is $k=1 / B=20 \times 10^{-6} atm^{-1}$. (Unit: 10^{-4} m^3)

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-0.8

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If z = arctan(e^{1 + (1 + x)^2}), what's the derivative of $\frac{\partial z}{\partial x}$ at x = 0.

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0.3017

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Suppose $f(x, y)= \begin{cases}1-x-y, & x+y \leqslant 1 \ 0, & x+y>1\end{cases}$. What is the integral of f(x,y) over the region I=[0,1]\times[0,1]?

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0.16667

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Compute the mean translational kinetic energy of a mole of ideal gas in J, both at room temperature 293 K.

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3650.0

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Let $R(D)$ be the rate distortion function for an i.i.d. process with probability mass function $p(x)$ and distortion function $d(x, \hat{x})$ , $x \in \mathcal{X}$ , $\hat{x} \in \hat{\mathcal{X}}$. If we add a new reproduction symbol $\hat{x}_0$ to $\hat{\mathcal{X}}$ with associated distortion $d(x, \hat{x}_0)$, $x \in \mathcal{X}$, $R(D)$ will decrease. True or False?

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True

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A single firm monopolizes the entire market for widgets and can produce at constant average and marginal costs of AC = MC = 10. Originally, the firm faces a market demand curve given by Q = 60 - P. Calculate the profit-maximizing price for the firm.

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35

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How many ways are there to arrange 9 people in a line such that no one is standing in their correct position?

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133496

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Two bicycle tires are set rolling with the same initial speed of 3.5 m/s on a long, straight road, and the distance each travels before its speed is reduced by half is measured. One tire is inflated to a pressure of 40 psi and goes 18.1 m; the other is at 105 psi and goes 92.9 m. What is the coefficient of rolling friction for each? Assume that the net horizontal force is due to rolling friction only.

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[0.0259, 0.00505]

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You are asked to determine the price of a European put option on a stock. Assuming the Black-Scholes framework holds, you are given: (i) The stock price is $100. (ii) The put option will expire in 6 months. (iii) The strike price is $98. (iv) The continuously compounded risk-free interest rate is r = 0.055. (v) δ = 0.01 (vi) σ = 0.50. What is the price of the put option?

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11.9

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determine the ratio of the radius of a uranium-238 nucleus to the radius of a helium-4 nucleus.

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3.9

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Consider $x(t)$ to be given as, $$ x(t)=10 \cos (20 \pi-\pi / 4)-5 \cos (50 \pi t) $$ What is minimum sampling rate (/Hz) such that $y(t)=x(t)$ ?

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50

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Find acceleration in m/(min^2) at time t = 5 min of a helicopter whose height is s(t) = 300t - 4t^3 m.

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-120

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For any triangle ABC, we have sin(A) + sin(B) + sin(C) $\le$ 3\sqrt(3)/2, is this true or false?

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True

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Find the ratio of forward-bias to reverse-bias currents when the same voltage 1.5 V is applied in both forward and reverse. Assume room temperature 293 K.

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-6e+25

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Consider the discrete memoryless channel $Y=XZ$ where $X$ and $Z$ are independent binary random variables that take on values 0 and 1. Let $P(Z=1)=0.5$. Find the capacity of this channel in bits.

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0.322

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The difference equation of a digital system is given by $$ y[n]-y[n-1]=2 x[n-1]-x[n-2], $$ where $x[n]$ and $y[n]$ are, respectively the current samples of the input and the output signals of the system. Determine if the system is a stable system.

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False

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Fig.Q3 shows an excerpt of the transmission phase of a TCP connection. Assume the length of the IP header is 20 bytes. What is the ACK number at message 6?

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839

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Consider the following graph, with links costs listed, and assume we are using shortest-path (or lowest-cost) routing, and that routing has equilibrated to a constant set of routing tables. The routing algorithm uses poisoned reverse, advertising an infinite weight for the poisoned paths. What distance does C advertise to B?

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5

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Compute the mean molecular speed v in the heavy gas radon (Rn) in m/s

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167.0

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The two-digit integers from 19 to 92 are written consecutively to form the large integer N = 192021 · · · 909192. Suppose that 3^k is the highest power of 3 that is a factor of N. What is k?

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1

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On a day when the speed of sound is the fundamental frequency of a particular stopped organ pipe is 220 Hz. The second overtone of this pipe has the same wavelength as the third harmonic of an open pipe. How long is the open pipe? (Unit: m)

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0.47

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Let (x_n) be a sequence defined by x_1 = 2 and x_{n+1} = 1 + 1/(1 + x_n). If (x_n) converges, what must its limit be in decimals?

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1.414

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For equation x^2*y^2-3y+2x^3=0, and suppose y=f(x). Then what is the derivate f'(1) near the point (1,1) and the point (1,2)? return the answer in a list.

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[8, -14]

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Square ABCD. CT: tangent to semicircle. Find the angle ∠CTD. Return the numeric value.

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63.4

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Passing to polar coordinates, calculate the double integral $\iint_S ydxdy$ with $y$ > 0, where S is a semicircle of a diameter 1 with center at point C(1/2, 0) above the X axis.

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0.0833

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Determine the time constant (i.e. τ ) of the circuit in the figure. Answer in unit of seconds (3 sig.fig.).

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3.93

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What is the minimum kinetic energy in MeV of a proton in a medium-sized nucleus having a diameter of 8.0 x 10^-15 m?

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0.08

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Is 10 a quadratic residue modulo 19? Use Gauss's Lemma to answer it.

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False

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A remote database contains 30 seconds of color motion-video. The video sequence is of the format (352 ́288 pixels) with RGB digitization at 30 frames per second. Find the the data rate for this motion-video in Mbits/s (3 sig. fig.).

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69.6

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What is the limit of the sequence a_n = n/(\sqrt{n^2 + 1})?

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1

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given a finite group A, and a collection of permutations B. Then (a) there exists B such that A is isomorphic to B; (b) for any B, A is isomorphic to B; (c) A can never be isomorphic to B; (d) none of the above. Which option is correct?

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(a)

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The product of two of the four roots of the quartic equation x^4 - 18x^3 +kx2 + 200x - 1984 = 0 is -32. Determine the value of k.

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86

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Evaluate $\lim _{x \rightarrow 1^{-}} \prod_{n=0}^{\infty}(\frac{1+x^{n+1}}{1+x^n})^{x^n}$?

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0.73575888

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Given the following equation: x - e^{-x} = 0. determine the initial approximations for finding the smallest positive root. Use these to find the root correct to three decimal places with Regula-Falsi method.

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0.567

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You are given: (i) The current exchange rate is 0.011$/¥. (ii) A four-year dollar-denominated European put option on yen with a strike price of $0.008 sells for $0.0005. (iii) The continuously compounded risk-free interest rate on dollars is 3%. (iv) The continuously compounded risk-free interest rate on yen is 1.5%. Calculate the price of a four-year yen-denominated European put option on dollars with a strike price of ¥125.

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42.77325

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Find the area of the region between the graphs of the functions f(x) = x^2 - 4x + 10, g(x) = 4x - x^2, 1 <= x <= 3.

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5.333

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A symmetric random walk on the three-dimensional cubic lattice Z^3 is transient or persistent? Return 1 for persistent and 0 for transient.

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0.0

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What is the Fisher information for the distribution family $f_\theta(x)=\theta e^{-\theta x}$, $x \geq 0$? (a) $\theta$. (b) $\theta^2$. (c) $\theta^{-1}$. (d) $\theta^{-2}$. Which option is correct?

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(d)

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Does \lim_{x \to 0} (cos(mx - 1)/(x^2) = -(m^2)/2 for m = 2?

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True

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For a parametric family $\{p_\theta(x)\}$ we know that $\lim_{\theta'\to\theta}\frac{1}{(\theta-\theta')^2}D(p_\theta||p_{\theta'}) = \alpha J(\theta)$, where $J(\theta)$ is the Fisher information. Use natural logarithm for KL divergence to compute $\alpha$.

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0.5

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A container weighs 3.22 lb force when empty. Filled with water at 60°F the mass of the container and its contents is 1.95 slugs. Find its volume in cubic feet. Assume density of water = 62.4 lb force/ft3.

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0.955

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