message stringlengths 1 1.07k | message_type stringclasses 2
values | message_id int64 0 1 | conversation_id int64 0 799 |
|---|---|---|---|
what is the value of $2/\pi*\prod_{k=1}^{\infty} \frac{(2*k)^2}{(2*k-1)(2*k+1)}$? | instruction | 0 | 200 |
1.0 | output | 1 | 200 |
What is the value of the series $\sum_{k=1}^{\infty} \frac{(-1)^{k-1}}{k} \sum_{n=0}^{\infty} \frac{1}{k 2^n+1}$? | instruction | 0 | 201 |
1.0 | output | 1 | 201 |
Compute the mean molecular speed v in the light gas hydrogen (H2) in m/s | instruction | 0 | 202 |
1750.0 | output | 1 | 202 |
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) | instruction | 0 | 203 |
108000 | output | 1 | 203 |
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. | instruction | 0 | 204 |
1.0 | output | 1 | 204 |
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. | instruction | 0 | 205 |
2.58 | output | 1 | 205 |
Approximate the area under the curve y=2^{x} between x=-1 and x=3 using the Trapezoidal rule with n=4 subintervals. | instruction | 0 | 206 |
11.25 | output | 1 | 206 |
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 $E(N)$. | instruction | 0 | 207 |
100.0 | output | 1 | 207 |
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 | instruction | 0 | 208 |
1.8 | output | 1 | 208 |
Tangent Circle at C. AB: common tangent. ∠OQB=112. What is ∠BAC? Return the numeric value. | instruction | 0 | 209 |
34.0 | output | 1 | 209 |
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? | instruction | 0 | 210 |
0.0758 | output | 1 | 210 |
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? | instruction | 0 | 211 |
-3 | output | 1 | 211 |
What is the number of labelled rooted forests on 6 vertices | instruction | 0 | 212 |
16807 | output | 1 | 212 |
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)? | instruction | 0 | 213 |
3.46621207 | output | 1 | 213 |
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? | instruction | 0 | 214 |
4.1e-08 | output | 1 | 214 |
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... | instruction | 0 | 215 |
False | output | 1 | 215 |
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. | instruction | 0 | 216 |
1.667 | output | 1 | 216 |
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? | instruction | 0 | 217 |
True | output | 1 | 217 |
Using n=8 approximate the value of $\int_{0}^4 cos(1 + \sqrt{x}) dx$ using the Simpson's rule. | instruction | 0 | 218 |
-2.47160136 | output | 1 | 218 |
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. | instruction | 0 | 219 |
True | output | 1 | 219 |
If $x=4*cost(t)$ and $y=8*sin(x)$, what is $y{''}_{xx}$ at t=pi/3? | instruction | 0 | 220 |
-4.0 | output | 1 | 220 |
Consider the matrix of A=[[1, -1], [-1, 4]], is this a positive definite matrix? | instruction | 0 | 221 |
True | output | 1 | 221 |
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.). | instruction | 0 | 222 |
3.59 | output | 1 | 222 |
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... | instruction | 0 | 223 |
0.5167 | output | 1 | 223 |
Are groups Z_4 * Z_2 and D_4 isomorphic? | instruction | 0 | 224 |
False | output | 1 | 224 |
Does 2^x +1/x = -4 have a solution? | instruction | 0 | 225 |
True | output | 1 | 225 |
When 30! is computed, it ends in 7 zeros. Find the digit that immediately precedes these zeros. | instruction | 0 | 226 |
8 | output | 1 | 226 |
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) | instruction | 0 | 227 |
0.4 | output | 1 | 227 |
Let x \in R with 0 < x < 1 and n \in N. Is (1 - x)^n >= 1/(1+nx)? | instruction | 0 | 228 |
False | output | 1 | 228 |
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) $\theta^{-2}$. | instruction | 0 | 229 |
(b) | output | 1 | 229 |
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? | instruction | 0 | 230 |
4.32 | output | 1 | 230 |
In how many ways can a committee of 2 men and 3 women be selected from a group of 6 men and 8 women? | instruction | 0 | 231 |
840 | output | 1 | 231 |
Find the orthogonal projection of 9e_1 onto the subspace of R^4 spanned by [2, 2, 1, 0] and [-2, 2, 0, 1]. | instruction | 0 | 232 |
[8, 0, 2, -2] | output | 1 | 232 |
Using Taylor's Approximation Theorem to show: What is $\lim_{x \to 0} \frac{e^\frac{x^4}{2}-\cos(x^2)}{x^4}$ | instruction | 0 | 233 |
1.0 | output | 1 | 233 |
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) | instruction | 0 | 234 |
8 | output | 1 | 234 |
What is 3^(3^(3^(...))) mod 100? There are 2012 3's in the expression. | instruction | 0 | 235 |
87 | output | 1 | 235 |
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? | instruction | 0 | 236 |
9.184 | output | 1 | 236 |
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 hypoth... | instruction | 0 | 237 |
0.023 | output | 1 | 237 |
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? | instruction | 0 | 238 |
102.95 | output | 1 | 238 |
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 $\Omega \cdot m$? | instruction | 0 | 239 |
2.5 | output | 1 | 239 |
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 capit... | instruction | 0 | 240 |
10.75 | output | 1 | 240 |
The diagonals of kite WXYZ intersect at P. If XP = 8, PZ = 8, WP = 6, and PY = 24, find ZY. | instruction | 0 | 241 |
25.3 | output | 1 | 241 |
Use the linear approximation to estimate (3.99)^3 (1.01)^4 (1.98)^{-1}. | instruction | 0 | 242 |
33.36 | output | 1 | 242 |
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) | instruction | 0 | 243 |
70 | output | 1 | 243 |
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. | instruction | 0 | 244 |
0.0 | output | 1 | 244 |
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 $\int_{|z|=r} f(z)/(z-z_1)(z-z_2) dz$? | instruction | 0 | 245 |
0.0 | output | 1 | 245 |
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. | instruction | 0 | 246 |
0.05 | output | 1 | 246 |
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. | instruction | 0 | 247 |
[0, 2, 0, 2, 1, 7, 1, 2, 0, 3, 0, 6] | output | 1 | 247 |
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 | 248 |
0.63662 | output | 1 | 248 |
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: $(0.5 \pi, 0.8,0.3)$ | instruction | 0 | 249 |
(a) | output | 1 | 249 |
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