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TheoremQA_standardized

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what is the value of $2/\pi*\prod_{k=1}^{\infty} \frac{(2*k)^2}{(2*k-1)(2*k+1)}$?

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1.0

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What is the value of the series $\sum_{k=1}^{\infty} \frac{(-1)^{k-1}}{k} \sum_{n=0}^{\infty} \frac{1}{k 2^n+1}$?

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1.0

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Compute the mean molecular speed v in the light gas hydrogen (H2) in m/s

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1750.0

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Each of the four jet engines on an Airbus A380 airliner develops athrust (a forward force on the airliner) of 322,000 N (72,000 lb).When the airplane is flying at 250 m/s, what horsepower does each engine develop? (Unit: hp)

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108000

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Suppose H is a Banach space, and {x_n}\in H, x\in H. Then x_n weakly converges to x is equivalent to: ||x_n|| is bounded; for a dense set M* in H*, it holds \lim_{n\rightarrow\infty} f(x_n)=f(x) for all f\in M*. Is this correct? Answer 1 for yes and 0 for no.

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1.0

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Use Euler's Method to calculate the approximation of y(0.2) where y(x) is the solution of the initial-value problem that is as follows. y''+xy'+y=0 and y(0)=2 and y'(0) = 3.

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2.58

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Approximate the area under the curve y=2^{x} between x=-1 and x=3 using the Trapezoidal rule with n=4 subintervals.

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11.25

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Let $X_0, X_1, X_2, \ldots$ be drawn i.i.d. from $p(x)$, and $x\in\{1,2,3,\ldots,100\}. Let $N$ be the waiting time to the next occurrence of $X_0$. Compute $E(N)$.

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100.0

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Find the interval in which the smallest positive root of the following equations lies: x^3 - x - 4 = 0. Determine the roots correct to two decimal places using the bisection method

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1.8

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Tangent Circle at C. AB: common tangent. ∠OQB=112. What is ∠BAC? Return the numeric value.

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34.0

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Your firm is trying to decide whether to buy an e-commerce software company. The company has $100,000 in total capital assets: $60,000 in equity and $40,000 in debt. The cost of the company’s equity is 10%, while the cost of the company's debt is 5%. The corporate tax rate is 21%. What is the WACC of the company?

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0.0758

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Apply the Graeffe's root squaring method to find the roots of the following equation x^3 + 3x^2 - 4 = 0 correct to two decimals. What's the sum of these roots?

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-3

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What is the number of labelled rooted forests on 6 vertices

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16807

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Use the Runge-Kutta method with $h=0.1$ to find approximate values of the solution of $(y-1)^2 * y' = 2x + 3$ with y(1) = 4. What is y(0)?

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3.46621207

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In a particular semiconductor device, electrons that are accelerated through a potential of 5 V attempt to tunnel through a barrier of width 0.8 nm and height 10 V. What fraction of the electrons are able to tunnel through the barrier if the potential is zero outside the barrier?

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4.1e-08

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An investor who is bullish about a stock may wish to construct a bull spread for that stock. One way to construct such a spread is to buy a call with strke price $K_1$ and sell a call with the same expiration date but with a strike price of $K_2 > K_1$. If we draw the payoff curve for that a spread, the initial cost of the spread would be negative is this True? Answer True or False.

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False

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Julian is jogging around a circular track of radius 50 m. In a coordinate system with its origin at the center of the track, Julian's x-coordinate is changing at a rate of -1.25 m/s when his coordinates are (40, 30). Find dy/dt at this moment.

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1.667

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Let f = u(z) + iv(z) be an entire function in complex plane C. If |u(z)| < M for every z in C, where M is a positive constant, is f is a constant function?

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True

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Using n=8 approximate the value of $\int_{0}^4 cos(1 + \sqrt{x}) dx$ using the Simpson's rule.

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-2.47160136

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If $|x|$ is less than 0.7, then if we use fifth Maclaurin polynomial approximate $sin(x)$ the error is less than 0.0001. Is this correct? Answer True or False.

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True

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If $x=4*cost(t)$ and $y=8*sin(x)$, what is $y{''}_{xx}$ at t=pi/3?

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-4.0

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Consider the matrix of A=[[1, -1], [-1, 4]], is this a positive definite matrix?

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True

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Determine the value of R in order to get a phase angle of 35 degree between the source voltage and the total current in the figure. Give the answer in unit of $k\Omega$ (3 sig.fig.).

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3.59

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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 rain 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 rain, 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 is the probability that day 2 is raining?

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0.5167

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Are groups Z_4 * Z_2 and D_4 isomorphic?

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False

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Does 2^x +1/x = -4 have a solution?

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True

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When 30! is computed, it ends in 7 zeros. Find the digit that immediately precedes these zeros.

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8

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A door 1.00 m wide, of mass 15 kg, can rotate freely about a vertical axis through its hinges. A bullet with a mass of 10 g and a speed of 400 m/s strikes the center of the door, in a direction perpendicular to the plane of the door, and embeds itself there. Find the door's angular speed. (Unit: rad/s)

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0.4

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Let x \in R with 0 < x < 1 and n \in N. Is (1 - x)^n >= 1/(1+nx)?

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False

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What is the Cramer-Rao lower bound on $E_\theta(\hat{\theta}(X)-\theta)^2$, where $\hat{\theta}(X)$ is an unbaised estimator of $\theta$ for the distribution family $f_\theta(x)=\theta e^{-\theta x}$, $x \geq 0$? (a) $\theta$. (b) $\theta^2$. (c) $\theta^{-1}$. (d) $\theta^{-2}$.

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(b)

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An observer S who lives on the x-axis sees a flash of red light at x = 1210 m, then after 4.96 µs, a flash of blue at x = 480 m. Use subscripts R and B to label the coordinates of the events. What is the measured time interval (in µs) between these flashes?

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4.32

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In how many ways can a committee of 2 men and 3 women be selected from a group of 6 men and 8 women?

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840

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Find the orthogonal projection of 9e_1 onto the subspace of R^4 spanned by [2, 2, 1, 0] and [-2, 2, 0, 1].

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[8, 0, 2, -2]

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Using Taylor's Approximation Theorem to show: What is $\lim_{x \to 0} \frac{e^\frac{x^4}{2}-\cos(x^2)}{x^4}$

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1.0

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The bandwidth of an analog signal is 4kHz. An A/D converter is used to convert the signal from analog to digital. What is the minimum sampling rate for eliminating the aliasing problem? (in kHz)

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8

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What is 3^(3^(3^(...))) mod 100? There are 2012 3's in the expression.

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87

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Ms. Fogg is planning an around-the-world trip on which she plans to spend $10,000. The utility from the trip is a function of how much she actually spends on it (Y), given by U(Y) = ln Y. If there is a 25 percent probability that Ms. Fogg will lose $1,000 of her cash on the trip, what is the trip’s expected utility?

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9.184

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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. What is the P-value?

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0.023

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Suppose that: The 1-year spot rate is 3%; The 2-year spot rate is 4%; and The 3-year spot rate is 5%. What is the price of a 100-par value 3-year bond paying 6% annual coupon payment?

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102.95

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Consider a resistor made from a hollow cylinder of carbon as shown below. The inner radius of the cylinder is $R_i=0.2$mm and the outer radius is $R_o=0.3$mm. The length of the resistor is $L=0.9$mm. The resistivity of the carbon is $\rho=3.5 * 10^{-5} \Omega \cdot m$. What is the resistance in $\Omega \cdot m$?

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2.5

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For a two-period binomial model for stock prices, you are given: (i) Each period is 6 months. (ii) The current price for a nondividend-paying stock is $70.00. (iii) u =1.181, where u is one plus the rate of capital gain on the stock per period if the price goes up. (iv) d = 0.890 , where d is one plus the rate of capital loss on the stock per period if the price goes down. (v) The continuously compounded risk-free interest rate is 5%. What is the current price of a one-year American put option on the stock with a strike price of $80.00.

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10.75

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The diagonals of kite WXYZ intersect at P. If XP = 8, PZ = 8, WP = 6, and PY = 24, find ZY.

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25.3

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Use the linear approximation to estimate (3.99)^3 (1.01)^4 (1.98)^{-1}.

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33.36

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A glass contains 0.25 kg of Omni-Cola (mostly water) initially at 25°C. How much ice, initially at -20°C must you add to obtain a final temperature of 0°C with all the ice melted? Neglect the heat capacity of the glass. (Unit: g)

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70

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Suppose that feedback is used on a binary symmetric channel with parameter $p=0.5$. Each time a $Y$ is received, it becomes the next transmission. Thus $X_1$ is Bern(1/2), $X_2=Y_1$, $X_3=Y_2$, \ldots, X_n=Y_{n-1}. Find $\lim_{n\to\infty} \frac{1}{n} I(X_n;Y_n)$ in bits.

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0.0

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Let f be a bounded entire function, z_1,z_2 be two points in the ball B(0,r). What is the value of the integral $\int_{|z|=r} f(z)/(z-z_1)(z-z_2) dz$?

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0.0

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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%. Whats the market Treynor Ratio? Return the numeric value between 0 and 1.

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0.05

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Fig. Q4 shows the contour of an object. Represent it with an 8-directional chain code. The resultant chain code should be normalized with respect to the starting point of the chain code. Represent the answer as a list with each digit as a element.

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[0, 2, 0, 2, 1, 7, 1, 2, 0, 3, 0, 6]

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What is $\lim _{r \rightarrow \infty} (\int_0^{\pi/2} x^r sin(x) dx)/(r\int_0^{\pi/2} x^r cos(x) dx)$?

instruction

0.63662

output

Given 2 colors whose HSI representations are given as follows: which color looks closer to blue? (a) Color 1: $(\pi, 0.3,0.5)$, (b) Color 2: $(0.5 \pi, 0.8,0.3)$

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(a)

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