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TheoremQA_standardized

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The distortion rate function $D(R)=\min_{p(\hat{x}|x):I(X;\hat{X})\leq R} E(d(X,\hat{X}))$ is convex. True or False?

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True

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What is \lim_{x \to 1} ((x - 1) sin((\pi)/(x - 1))?

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0

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Suppose a European call option on a barrel of crude oil with a strike price of $50 and a maturity of one-month, trades for $5. What is the price of the put premium with identical strike price and time until expiration, if the one-month risk-free rate is 2% and the spot price of the underlying asset is $52?

instruction

2.92

output

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

instruction

[1, 2]

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Is W = {[x, y] in R^2: x >= 0 and y >= 0} a subspace of R^2?

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False

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Compute the integral $\iint_D xy^2 dA$, where $D$ is the rectangle defined by 0 <= x <= 2 and 0 <= y <= 1.

instruction

0.66667

output

Sally is driving along a straight highway in her 1965 Mustang. At when she is moving at in the positive x-direction, she passes a signpost at Her x-acceleration as a function of time is a_x = 2.0 m/s^2 - (0.10 m / s^3) t At X meter's, the car reaches maximum x-velocity? What is X?

instruction

517

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In an IPv4 datagram, the value of the total-length field is $(00 \mathrm{~A} 0)_{16}$ and the value of the headerlength (HLEN) is (5) $1_{16}$. How many bytes of payload are being carried by the datagram?

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140

output

In 1985 the space shuttle Challenger flew a cesium clock and compared its time with a fixed clock left on Earth. The shuttle orbited at approximately 330 km above Earth with a speed of 7712 m/s. Calculate the expected time lost per second (in picoseconds) for the moving clock and compare with the measured result of $-295.02 \pm 0.29 ps/s$, which includes a predicted effect due to general Relativity of $35.0 \pm 0.06 ps/s$

instruction

330.76

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Let A be an invertible n * n matrix and v and eigenvector of both A and B, is v necesarily an eigenvector of A + B?

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True

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Assume that the Black-Scholes framework holds. The price of a nondividened-paying stock is $30. The price of a put option on this stock is $4.00. You are given $(i) $\Delta=-0.28$. (ii) $\Gamma=0.10$ Using the delta-gamma approximation, determine the price of the put option if the stock price changes to $31.50.

instruction

3.7

output

Under some circumstances, a star can collapse into an extremely dense object made mostly of neutrons and called a neutron star. The density of a neutron star is roughly $10^14$ times as great as that of ordinary solid matter. Suppose we represent the star as a uniform, solid, rigid sphere, both before and after the collapse. The star's initial radius was $7 \tims 10^5$ km (comparable to our sun); its final radius is 16 km. If the original star rotated once in 30 days, find the angular speed (in rad/s) of the neutron star.

instruction

4600.0

output

In Image processing, closing is a process in which first dilation operation is performed and then erosion operation is performed. Is it true?

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True

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In how many ways can a group of 9 people be divided into 3 non-empty subsets?

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3025

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Coloring the edges of a complete graph with n vertices in 2 colors (red and blue), what is the smallest n that guarantees there is either a 4-clique in red or a 5-clique in blue?

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25

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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 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

(b)

output

for the matrix $A=(\begin{array}{rrrrr} 1 & 2 & 3 & 4 & -3 \1 & 2 & 0 & -5 & 1 \2 & 4 & -3 & -19 & 6 \3 & 6 & -3 & -24 & 7\end{array})$, what is its row rank and column rank? return the two numbers as a list.

instruction

[2, 2]

output

Let $P_5(x)$ be the fifth-degree Taylor polynomial approximation for f(x)=sin(x), centered at x=0. What is the Lagrange error of the polynomial approximation to sin(1)?.

instruction

0.000198

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If $u(x, y) = 4x^3y - 4xy^3$, is there a function v(x, y) such that u(x,y) + iv(x,y) is an analytical function?

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True

output

The Relativistic Heavy Ion Collider (RHIC) at the Brookhaven National Laboratory collides gold ions onto other gold ions head on. The energy of the gold ions is 100 GeV per nucleon. What is the center-of-mass energy of the collision in TeV?

instruction

39.4

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The asteroid Pallas has an orbital period of 4.62 years and an orbital eccentricity of 0.233. Find the semi-major axis of its orbit. (Unit: 10^11 m)

instruction

4.15

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Is there a y bewteen x and x+h such that $sin(x+h) - sinx = h * cos(y)$?

instruction

True

output

Let a undirected graph G with edges E = {<0,2>,<1,4>,<9,6>,<8,12>,<2,4>,<1,3>,<1,5>,<12,1>,<8,1>,<5,9>,<0,10>,<5,2>,<0,8>,<3,4>,<3,11>,<7,1>,<2,1>,<0,12>,<1,0>,<7,8>}, 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.

instruction

[0, 1, 2, 3, 8, 9]

output

Find the arc length of y = x^{-1} over the interval [1,2] using the Simpson's Rule S_8.

instruction

1.132

output

For an American perpetual option within the Black-Scholes framework, you are given: (i) $h_1 + h_2$ = 7/9 (ii) The continuously compounded risk-free interest rate is 5%. (iii) σ = 0.30. What is the value of $h_1$?

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1.51

output

Find the volume of a solid bounded by the elliptical paraboloid $z=2x^2 + y^2 + 1$, the plane x+y=1, and the coordinate planes.

instruction

0.75

output

Suppose there are 100 identical firms in a perfectly competitive industry. Each firm has a short-run total cost function of the form C(q) = rac{1}{300}q^3 + 0.2q^2 + 4q + 10. Suppose market demand is given by Q = -200P + 8,000. What will be the short-run equilibrium price?

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25

output

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

instruction

1.094

output

Please solve the equation sin(4*x) + x = 54 and provide all the roots using newton-raphson method.

instruction

[53.52, 54.25, 54.76]

output

What is the RC time constant of the circuit in seconds?

instruction

3800.0

output

In how many ways can a group of 7 people be divided into 2 non-empty subsets?

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63

output

In how many ways can a convex polygon with 8 sides be divided into triangles by connecting its vertices, with no intersecting lines?

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132

output

The electric flux through a spherical surface is $4.0\times 10^4 N \cdot m^2/C$. What is the net charge enclosed by the surface?

instruction

3.54e-07

output

How many ways are there to partition a set of 5 elements into 3 non-empty cycles?

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35

output

Calculate the Fermi temperature for copper in eV.

instruction

81600.0

output

What is the value of the inflection point of f(x) =(10 ln(x))/(x^2)?

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2.301

output

How many different 6-letter arrangements can be made from the letters in the word BANANA?

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60

output

Find the sum of $\sum_{n=1}^{\infty} (1/e^n + 1/(n*(n+1)))$

instruction

1.581

output

A monopolist can produce at constant average and marginal costs of AC = MC = 5. The firm faces a market demand curve given by Q = 53 - P. Calculate the consumer surplus obtained by consumers under perfect competition (where price = marginal cost)?

instruction

1152

output

Square ABCD. Rectangle AEFG. The degree of ∠AFG=20. Please find ∠AEB in terms of degree. Return the numeric value.

instruction

25.0

output

Coloring the edges of a complete graph with n vertices in 2 colors (red and blue), what is the smallest n that guarantees there is either a triangle in red or a 6-clique in blue?

instruction

18

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Does the following series $\sum_{i=0}^{\infty} \frac{n-1}{n^3+1}$ converge?

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1.0

output

Find the solutions y of the differential equation y'=(t^2+3y^2)/2ty with y(1) = 1. What is y(2)?

instruction

3.464

output

For an integer a > 0 and an integer b > 0, is there any other number c > 0 such that a^10 + b^10 = c^10?

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False

output

For $p(x)=f(x)g(x)$, if $f(2)=3$, $f'(2)=-4$, $g(2)=1$, and $g'(2)=6$, what is $p'(2)$?

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14

output

Adding a row to a channel transition matrix does not decrease capacity. True or False?

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True

output

Let $C$ be a variable length code that satisfies the Kraft inequality with equality but does not satisfy the prefix condition. Then $C$ has finite decoding delay. True or False?

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False

output

Let $X$ be uniformly distributed over $\{1, 2, \ldots, 256\}$. We ask random questions: Is $X\in S_1$? Is $X\in S_2$? ... until only one integer remains. All $2^256$ subsets of $\{1, 2, \ldots, 256\}$ are equally likely. How many deterministic questions are needed to determine $X$?

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8

output

Each day Paul, who is in third grade, eats lunch at school. He likes only Twinkies (t) and soda (s), and these provide him a utility of utility = U(t,s) = \sqrt{ts}. If Twinkies cost $0.10 each and soda costs $0.25 per cup, Paul's mom gives him $1, how many Twinkies should Paul buy to maximize utility?

instruction

5

output

Which of the following codeword lengths can be the word lengths of a 3-ary Huffman code? (a) (1, 2, 2, 2, 2). (b) (2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3).

instruction

(b)

output