Datasets:

kranthigv

/

TheoremQA_standardized

like 0

Let a undirected graph G with edges E = {<0,1>,<0,2>,<0,3>,<0,5>,<2,3>,<2,4>,<4,5>}, which <A,B> represent Node A is connected to Node B. What is the shortest path from node 0 to node 5? Represent the path as a list.

instruction

[0, 5]

output

Suppose a convex 3d-object has k pentagonal faces and m hexagonal faces. All faces are regular. What is k?

instruction

12

output

Three years ago, Fred invested $10,000 in the shares of ABC Corp. Each year, the company distributed dividends to its shareholders. Each year, Fred received $100 in dividends. Note that since Fred received $100 in dividends each year, his total income is $300. Today, Fred sold his shares for $12,000. What is the holding period return of his investment?

instruction

0.23

output

\lim_{x \to c} |f(x)| = 0. What is \lim_{x \to c} f(x)?

instruction

0

output

Use divergence therem to evaluate $\iint_S \vec{F} \cdot d \vec{S}$ where $\vec{F} = sin(\pi x) \vec{i} + (z y^3)\vec{j} + (z^2 + 4x)\vec{k}$ and $S$ is the suface of the box with $-1 \le x \le 2, 0 \le y \le 1$ and $1 \le z \le 4$. Note that all six sides of the box are included in $S$.

instruction

67.5

output

Consider $x(t)$ to be given as, $$ x(t)=\cos (1000 \pi t) $$ . Let the sampling frequency be $700 \mathrm{~Hz}$. Does aliasing occur?

instruction

True

output

Use Euler's method to find the solution to the differential equation dy/dx=y^2e^x at x=6 with the initial condition y(0)=0.01 and step size h=0.5. What is y(6)?

instruction

5.113

output

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. Can we reject H_0 at significance level $\alpha = 0.05$?

instruction

True

output

Let rectangle R = [1, 2.5] * [1, 2]. Calculate the Riemann Sum S_{3,2} for \int \int_{R} xy dA for the integral, using the lower-left vertex of rectangles as sample points.

instruction

2.812

output

Consider an m * n matrix A and an n * m matrix B (with n != m) such that AB = I_m. Are the columns of A linearly independent?

instruction

False

output

Consider two 5 year bonds: one has a 9% coupon and sells for 101.00; the other has a 7% coupon and sells for 93.20. What is the price of a 5-year zero-coupon bond.

instruction

65.9

output

A ship uses a sonar system to locate underwater objects. Find the wavelength of a 262-Hz wave in water. (Unit: m)

instruction

5.65

output

Let {N(t), t \in [0, \infty)} be a Poisson process with rate of $\lambda = 4$ and $X_1$ be the first arrival time. Given N(t) = 1, then what is $P(X_1 <= t / 2)$?

instruction

0.5

output

Let X_1, X_2,... be independent variables each taking values +1 or -1 with probabilities 1/2 and 1/2. It is know that $\sqrt{3/n^3}*\sum_{k=1}^n k*X_k$ converges in distribution normal distribution N(a,b) as n goes to infinity. Here a is the expectation and b is the variance. What are the values of a and b? Return the answers as a list. For example, if a=2, b=100, return [2,100].

instruction

[0, 1]

output

Consider an additive white Gaussian noise channel with an expected output power constraint $P=2$. Thus $Y = X + Z$, $Z \sim N(0, 1)$, $Z$ is independent of $X$, and $E(Y)^2 \leq 2$. Find the channel capacity in bits.

instruction

0.5

output

Calculate the minimum kinetic energy of a proton to be scattered from a fixed proton target to produce an antiproton in MeV.

instruction

5630.0

output

What is the Fisher information for the Gaussian distribution family $f_\theta(x)=N(0,\theta)$? (a) $2\theta$. (b) $2\theta^2$. (c) $0.5\theta^{-1}$. (d) $0.5\theta^{-2}$. Which option is correct?

instruction

(d)

output

Use euler's method to find the solution to the differential equation $\frac{\partial y}{\partial x} = 3x + 4y$ at $x=1$ with the initial condition y(0) = 0 and step size $h=0.25$. What is y(1)?

instruction

2.0625

output

Find the fraction of 7.7-MeV alpha particles that is deflected at an angle of 90° or more from a gold foil of 10^-6 m thickness.

instruction

4e-05

output

A box contains 4 red, 3 green, and 2 blue balls. Balls are identical besides of their colors. In how many ways can we choose 4 balls, if at least 2 are red?

instruction

6

output

If there exists an ordered numbering of the nodes such that for each node there are no links going to a lower-numbered node, then there are no directed cycles in a directed graph. True or false?

instruction

True

output

Assume the Black-Scholes framework. For $t \ge 0$, let $S(t)$ be the time-$t$ price of a nondividend-paying stock. You are given: (i) $S(0)=0.5 (ii) The stock price process is $\frac{dS(t)}{S(t)} = 0.05dt+0.2dZ(t)$ where $Z(t)$ is a standart Brownian motion. (iii) $E[S(1)^\alpha]=1.4$, where $\alpha$ is a negative constant. (iv) The continuously compounded risk-free interest rate is $3%$. Consider a contingent claim that pays $S(1)^\alpha$ at time 1. What is the time-0 price of the contigent claim?

instruction

1.372

output

A neutron at rest decays (breaks up) to a proton and an electron. Energy is released in the decay and appears as kinetic energy of the proton and electron. The mass of a proton is 1836 times the mass of an electron. What fraction of the total energy released goes into the kinetic energy of the proton?

instruction

0.000544

output

A debt of $25,000 is to be amortized over 7 years at 7% interest. What value of monthly payments will achieve this?

instruction

4638.83

output

If the quartic x^4 + 3x^3 + 11x^2 + 9x + A has roots k, l, m, and n such that kl = mn, find A.

instruction

9

output

Consider a source $X$ uniformly distributed on the set $\{1, 2, \dots, m\}$. The rate distortion function for this source with Hamming distortion is $R(D) = \log{m}-H(D)-D\log{(m-1)}$ for $0\leq D\leq 1-\frac{1}{m}$, and $R(D) = 0$ otherwise. True or False?

instruction

True

output

The returns on a stock are 2.45% at 2018, 5.42% at 2019, -13.83% at 2020. What is the compound annual rate (between -1 and 1) of return over the three years.

instruction

-0.023669

output

What are the generators of the additive cyclic group Z?

instruction

[1, -1]

output

True or false: there exists a graph with score (1, 2, 3, 4, 5).

instruction

False

output

The perfectly competitive videotape-copying industry is composed of many firms that can copy five tapes per day at an average cost of $10 per tape. Each firm must also pay a royalty to film studios, and the per-film royalty rate (r) is an increasing function of total industry output (Q): r = 0.002Q. Demand is given by Q = D(P) = 1,050 - 50P. Assuming the industry is in long-run equilibrium, what will be the equilibrium price of copied tapes?

instruction

11

output

For all $n>1$, define $a_n=\sum_{k=1}^{n-1} \frac{\sin (\frac{(2 k-1) \pi}{2 n})}{\cos ^2(\frac{(k-1) \pi}{2n}) \cos ^2 (\frac{k \pi}{2n})}$. What is the limit of $a_n/n^3$ as $n$ goes to infinity?

instruction

0.258

output

How many paths are there from the origin (0,0) to the point (10,10) on a grid such that the path only moves up or right and does not cross the diagonal line y = x?

instruction

16796

output

Let C[0,1] be all the continuous function on in the interval [0,1]. For the integral equation $x(t)-\lambda \int_0^1 e^{t-s} x(s) ds=y(t)$, where $y(t)\in C[0,1]$ is a given function. \lambda is a constant and |\lambda|<1. Then there exists a unique solution x(t)\in C[0,1]. This conclusion can be proved by: 1. Implicit function theorem, 2. Riesz representation theorem, 3. Banach fixed point theorem, 4. None of the above. Return the number as the answer.

instruction

3.0

output

What is the effective rates for 3% compounded monthly?

instruction

0.0304

output

what is the value of $\int_{0}^\pi (sin(123*x/2)/sin(x/2))^2dx$? Round the answer to the thousands decimal.

instruction

386.4158898

output

For the following functions, which are bounded entire functions? 1. f(x)=0; 2. f(x)= 1+i; 3. f(x)=sin(x); 4. f(x)=min{|cos(x)|,1}. Here i=\sqrt{-1} and $|\cdot|$ is the norm of a complex number. Return the numbers of the answers as a list.

instruction

[1, 2]

output

The polynomial $x^3 - Ax + 15$ has three real roots. Two of these roots sum to 5. What is |A|?

instruction

22.0

output

What's the value of a > 0, such that the tangent line to the graph of f(x) = (x^2) (e^(-x)) at x = a passes through the origin?

instruction

1

output

Let W(t) be the standard Brownian motion. Define X(t) = exp{W(t)}, for all t \in [0, \infty). Let 0 < s < t. Find Cov(X(s=1/2), X(t=1)).

instruction

1.3733

output

Consider a random walk on a connected graph with 4 edges. What is the lowest possible entropy rate? Use base 2 logarithm and return the entropy rate in bits.

instruction

0.75

output

For a simple random walk S_n with S_0=0 and P(S_n-S_{n-1}=1)=1/4, P(S_n-S_{n-1}=-1)=3/4. Let M=\max{S_n:n\geq 0}. What is the probability of the event {M\geq 5}? Round the answer to the thousands decimal.

instruction

0.01234567

output

Assume the half-life of the proton is 10^33 years. How many decays per year would you expect in a tank of water containing 350,000 liters of water?

instruction

0.08

output

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

instruction

1

output

While a person is walking, his arms swing through approximately a 45° angle in 0.5s.As a reasonable approximation, we can assume that the arm moves with constant speed during each swing. A typical arm is 70.0 cm long, measured from the shoulder joint. What is the acceleration (in metre per second squared) of a 1.0 g drop of blood in the fingertips at the bottom of the swing?

instruction

1.73

output

A teacher wants to invest $30,000 into an account that compounds annually. The interest rate at this bank is 1.8%. How much money will be in the account after 6 years?

instruction

33389.35

output

Let $X \sim N(0,1)$ and let the distortion measure be squared error. Here we do not allow block descriptions. Compute the minimum expected distortion for one bit quantization of $X$ using a squared error distortion measure.

instruction

0.363

output

A steel rod 2.0 m long has a cross-sectional area of $0.30 cm ^ 2$. It is hung by one end from a support, and a 550-kg milling machine is hung from its other end. Determine the stress on the rod and the resulting strain and elongation. (Unit: mm)

instruction

1.8

output

Find the smallest positive integer that leaves a remainder of 1 when divided by 2, a remainder of 2 when divided by 3, a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and a remainder of 5 when divided by 6.

instruction

59

output

Is cos(\pi/8) equal to (\sqrt{2+\sqrt{2}})/2?

instruction

True

output

One end of a 2.00-kg rope is tied to a support at the top of a mine shaft 80.0 m deep. The rope is stretched taut by a 20.0-kg box of rocks attached at the bottom. If a point on the rope is in transverse SHM with f = 2.00 Hz, how many cycles of the wave are there in the rope’s length?

instruction

1.81

output