Datasets:

TIGER-Lab

/

TheoremQA

like 13

Follow

TIGER-Lab 109

How many ways are there to divide a set of 8 elements into 5 non-empty ordered subsets?

11760

integer

what is the value of $\int_{-infty}^{+infty} sin(3*t)*sin(t/\pi)/t^2 dt$?

1.0

float

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?

5

integer

Please solve the equation 2*x^3 + e^x = 10 using newton-raphson method.

1.42

float

How many ways are there to divide a set of 7 elements into 4 non-empty ordered subsets?

4200

integer

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.

[0, 5]

list of integer

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

[4, 1, 2, 0]

list of integer

Compute $\int_{|z| = 1} z^2 sin(1/z) dz$. The answer is Ai with i denoting the imaginary unit, what is A?

-1.047

float

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.

0.955

float

Let M be the set of bounded functions (i.e. \sup_{x\in[a,b]}|f(x)|<\infty) in C[0,1]. Is the set ${F(x)=\int_0^x f(t) dt | f \in M }$ a sequentially compact set? Answer 1 for yes and 0 for no. Furthermore, it can be proved using 1. Arzelà-Ascoli theorem, 2. Riesz representation theorem, 3. Banach fixed point theorem, 4. None of the above. Return the answers of the two questions in a list. For example, if you think the answer is no and Riesz representation theorem, then return [0,2].

[1, 1]

list of integer

Find the x value of the solutions to the linear system: 7x - y = 15x, -6x + 8y = 15y.

0

integer

In Jules Verne's 1865 story with this title, three men went to the moon in a shell fired from a giant cannon sunk in the earth in Florida. Find the minimum muzzle speed that would allow a shell to escape from the earth completely (the escape speed). Neglect air resistance, the earth's rotation, and the gravitational pull of the moon. The earth's radius and mass are $R_E}=$ $6.38 \times 10^6 m$ and $m_E=5.97 \times 10^{24} kg$. (Unit: 10 ^ 4 m/s)

1.12

float

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

False

bool

compute the line integral of \int_K xy dx, \int_L xy dx, where K is a straight line from (0,0) to (1,1) and L is the Parabola y=x^2 from (0,0) to (1,1). return the answer as a list

[0.333, 0.25]

list of float

True of false: one can draw a simple connected planar graph with 200 vertices and 400 faces

False

bool

Consider the basis B of R^2 consisting of vectors v_1 = [3,1] and v_2 = [-1, 3]. If x = [10, 10], find the B-coordinate vector of x

[4, 2]

list of integer

What is the number of labelled forests on 10 vertices with 5 connected components, such that vertices 1, 2, 3, 4, 5 all belong to different connected components?

50000

integer

Let g(x) be the inverse of f(x) = x + cos(x). What is g'(1)?

1

integer

Let V be the space of all infinite sequences of real numbers. Consider the transformation T(x_0, x_1, x_2, ...) = (x_1, x_2, x_3, ...) from V to V. Is the sequence (1,2,3,...) in the image of T?

True

bool

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)).

1.3733

float

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.

1.094

float

If u is the real part of a function, and v is the imaginary part, then the Cauchy-Riemann equations for u and v take the following form in polar coordinates: r\frac{\partial u}{\partial r} = \frac{\partial v}{\partial \theta} and r\frac{\partial v}{\partial r} = -\frac{\partial u}{\partial \theta}. Is this argument True or False?

True

bool

The shock absorbers in an old car with mass 1000 kg are completely worn out. When a 980-N person climbs slowly into the car at its center of gravity, the car sinks 2.8 cm. The car (with the person aboard) hits a bump, and the car starts oscillating up and down in SHM. Model the car and person as a single body on a single spring, and find the frequency of the oscillation. (Unit: Hz)

0.9

float

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.

59

integer

the matrix in ./mingyin/mc.png represents a markov chain. What is the period of state 0? What is the period of state 1? Return the two answers as a list.

[2, 2]

list of integer

$\lim_{x \to c}((x^2 - 5x - 6) / (x - c))$ exists. What is the value of c?

[-1, 6]

list of integer

Aisha graduates college and starts a job. She saves $1000 each quarter, depositing it into a retirement savings account. Suppose that Aisha saves for 30 years and then retires. At retirement she wants to withdraw money as an annuity that pays a constant amount every month for 25 years. During the savings phase, the retirement account earns 6% interest compounded quarterly. During the annuity payout phase, the retirement account earns 4.8% interest compounded monthly. Calculate Aisha’s monthly retirement annuity payout.

1898.27

float

Let $g_\theta(x_1,x_2)=f_\theta(x_1)f_\theta(x_2)$. Let $J_f(\theta)$ be the Fisher information of $f_\theta$. What is the relationship between $J_f(\theta)$ and $J_g(\theta)$? (a) $J_g(\theta) = 0.5J_f(\theta)$. (b) $J_g(\theta) = J_f(\theta)$. (c) $J_g(\theta) = 2J_f(\theta)$. (d) $J_g(\theta) = 4J_f(\theta)$. Which option is correct?

(c)

option

An auto magazine reports that a certain sports car has 53% of its weight on the front wheels and 47% on its rear wheels. (That is, the total normal forces on the front and rear wheels are 0.53w and 0.47w, respectively, where w is the car’s weight.) The distance between the axles is 2.46 m. How far in front of the rear axle is the car’s center of gravity?

1.3

float

A parachutist with mass m=80 kg is undergoing free fall. The drag force applied on him is $F_D = kv^2$, where v is the velocity measured relative to the air. The constant k=0.27 [Ns^2/m^2] is given. Find the distance traveled h in meters, until v=0.95$v_t$ is achieved, where $v_t$ is the terminal velocity. Return the numeric value.

345.0

float

ABCD is a parallelogram. E is the midpoint, F is also a midpoint. Area of AFG = 10, Area of EGH = 3. What is Area CDH?

7

integer

In how many ways can a set of 6 distinct letters be partitioned into 3 non-empty groups if each group must contain at least 2 letters?

15

integer

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

True

bool

For a\geq 0, we define $S_a={x | dist(x, S) \leq a}$, where $dist(x,S)=inf_{y\in S}||x-y||$. Suppose S is convex. Is S_a convex? Return 1 for yes and 0 for no.

1.0

float

suppose the 10-by-10 matrix A has the form: if i \neq j, A_{i,j}=a_i*b_j; if i=j, A_{i,j}=1+a_i*b_j for all 1<=i,j<=10. Here a_i = 1/i, b_i=1/(i+1). Find the determinant of A. return the numeric.

1.9

float

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.

5.333

float

Does the following transformation have an eigenvector: Counterclockwise rotation through an angle of 45 degrees followed by a scaling by 2 in R^2.

False

bool

How many ways are there to arrange 6 pairs of parentheses such that they are balanced?

132

integer

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.

4e-05

float

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

3025

integer

Suppose there is a 50-50 chance that an individual with logarithmic utility from wealth and with a current wealth of $20,000 will suffer a loss of $10,000 from a car accident. Insurance is competitively provided at actuarially fair rates. Compute the utility if the individual buys full insurance.

9.616

float

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.

-0.023669

float

Does $p(x) = x^5 + x − 1$ have any real roots?

True

bool

Find integer $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square.

3

integer

Does cos(x) = x^k have a solution for k = 2023?

True

bool

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

1.0472

float

A box contains 4 red, 3 green, and 2 blue balls. Balls are distinct even with the same color. In how many ways can we choose 4 balls, if at least 2 are red?

81

integer

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].

[0, 1]

list of integer

Sum the series $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty}\frac{m^2 n}{3^m(n3^m+m3^n)}$

0.28125

float

You want to move a 500-N crate across a level floor. To start thecrate moving, you have to pull with a 230-N horizontal force.Once the crate breaks loose and starts to move, you can keep itmoving at constant velocity with only 200 N. What are the coefficients of static and kinetic friction?

0.4

float

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%. The standard deviation is 15% over a 10-year period. Whats the market Sharpe Ratio?

0.33

float

What is the value of the integral $\int_2^4 \frac{\sqrt{log(9-x)}}{\sqrt{log(9-x)}+\sqrt{log(x+3)}} dx$?

1.0

float

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?

25

integer

A state issues a 15 year $1000 bond that pays $25 every six months. If the current market interest rate is 4%, what is the fair market value of the bond?

1111.97

float

In how many ways can we color a loop of 5 vertices with 3 colors such that no two adjacent vertices have the same color?

30

integer

The dependence between adjacent n-blocks of a stationary process grows linearly with n. True or False?

False

bool

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?

False

bool

What is the order of group Z_{18}?

18

integer

Is 80 dB twice as loud as 40 dB?

False

bool

Is x-1 a factor of 2*x^4+3*x^2-5x+7?

False

bool

A load dissipates 1.5kW of power in an ac series RC circuit. Given that the power factor is 0.75, what is its reactive power $(P_r)$? What is its apparent power $(P_a)$? Represent the answer in a list [$P_r, P_a$] with unit kVA and kVAR respectively.

[2.0, 1.32]

list of float

dy/dt = \sqrt{t}, y(1) = 1. What is y(4)?

5.667

float

For the two linear equations $2 * x + 3 * y = 10$ and $4 * x + 4 * y = 12$ iwth variables x and y. Use cramer's rule to solve these two variables.

[-1, 4]

list of integer

In how many ways can 10 distinct balls be placed into 4 identical boxes if each box must have at least 1 balls?

26335

integer

If x(n) and X(k) are an N-point DFT pair, then x(n+N)=x(n). Is it true?

True

bool

what is the limit of $2/\sqrt{\pi}*\sqrt{n}\int_0^1(1-x^2)^n dx$ as n goes to infinity?

1.0

float

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 rainy 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) = 0.55. The person has 60\% chance to bring an umbrella when the weather is rainy, 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 are the most likely weather from day 1 to day 3? Return the answer as a list of binary values, where 1 represents rain and 0 represents sunny.

[1, 0, 1]

list of integer

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?

True

bool

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?

25

integer

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?

1.372

float

Determine the multiplicity of the root ξ = 1, of the polynomial P(x) = x^5 - 2x^4 + 4x^3 - x^2 - 7x + 5 = 0 using synthetic division. What is P'(2) + P''(2)? Please return the decimal number.

163

integer

For the function $f(x,y)$ defined by $f(x,y)=1$ if $x=y$, $f(x,y)=0$ otherwise. Can we measure its integraion over the rectangle $[0,1]\times[0,1]$ using the Tonelli's Theorem? Answer true or false.

False

bool

The cross section for neutrons of energy 10 eV being captured by silver is 17 barns. What is the probability of a neutron being captured as it passes through a layer of silver 2 mm thick?

0.2

float

What is the determinant of matrix [[0, 1, 2], [7, 8, 3], [6, 5, 4]]?

-36

integer

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.

[1, 2]

list of integer

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?

1

integer

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?

1

integer

A glider with mass m = 0.200 kg sits on a frictionless horizontalair track, connected to a spring with force constant k = 5.00 N/m.You pull on the glider, stretching the spring 0.100 m, and release itfrom rest. The glider moves back toward its equilibrium position (x = 0).What is its x-velocity when x = 0.080 m? (Unit: m/s))

-0.3

float

Toss a coin repeatedly until two consecutive heads appear. Assume that the probability of the coin landing on heads is 3/7. Calculate the average number of times the coin needs to be tossed before the experiment can end.

7.77778

float

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.

1.667

float

Calculate the momentum uncertainty of a tennis ball constrained to be in a fence enclosure of length 35 m surrounding the court in kg m/s.

3e-36

float

Let {X_n: n \geq 1} be independent, identically distributed random variables taking integer values {1,-1}. Let S_0=0, S_n=\sum_{i=1}^n X_i. Let P(X_i=1)=0.8 and P(X_i=-1)=0.2. The range R_n of S_0,S_1,...,S_n is the number of distinct values taken by the sequence. Then what is the limit of n^{-1}E[R_n] as n goes to infinity? Here E[R_n] is the expectation over the random variable R_n.

0.6

float

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

1.132

float

Is the Fourier transform of the signal x(t)=(1-e^{-|t|})[u(t+1)-u(t-1)] real?

True

bool

compute the integral $\iint_V \frac{d x d y d z}{(1+x+y+z)^3}$, where V={(x, y, z): x, y, z \geq 0, x+y+z\leq 1}.

0.034

float

For matrix A = [[5, 4], [1, 2]], what are its eigen values?

[1, 6]

list of integer

What is the minimum number of people needed in a room to guarantee that there are 4 mutual friends or 4 mutual strangers?

18

integer

If polygon ABCDE ~ polygon PQRST, AB = BC = 8, AE = CD = 4, ED = 6, QR = QP, and RS = PT = 3, find the perimeter of polygon PQRST.

22.5

float

What is the effective rates (between 0 and 1) for 18% compounded quarterly? Return the numeric value.

0.1925

float

What is the effective rates for 3% compounded monthly?

0.0304

float

Let f be a real function on [0,1]. If the bounded variation of f on [0,1] equals f(1)-f(0), then: (a) f is increasing on [0,1]; (b) f is decreasing on [0,1]; (c) None of the above. Which one is correct?

(a)

option

If a stock pays a $5 dividend this year, and the dividend has been growing 6% annually, what will be the stock’s intrinsic value, assuming a required rate of return of 12%?

88.33

float

Malus' law: $I=I_0*cos^2($\theta$)$. Where I is the intensity of polarized light that has passed through the polarizer, I_0 is the intensity of polarized light before the polarizer, and $\theta$ is the angle between the polarized light and the polarizer. Unpolarized light passes through a polarizer. It then passes through another polarizer at angle 40 degree to the first, and then another at angle 15 degree to the second. What percentage of the original intensity was the light coming out of the second polarizer?

54.8

float

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

False

bool

Determine the number of positive real zero of the given function: $f(x)=x^5+4*x^4-3x^2+x-6$.

[3, 1]

list of integer

Consider a source X with a distortion measure $d(x, \hat{x})$ that satisfies the following property: all columns of the distortion matrix are permutations of the set $\{d_1, d_2, \ldots, d_m\}$. The function $\phi(D) = \max_{b:\sum_{i=1}^m p_i d_i \leq D} H(p)$ is concave. True or False?

True

bool

Consider a file with a size of 350 Kbytes storing in a web server. Client A sends a request to the server to retrieve the file from a remote location. It is known that the link capacity between client A and the server is 10 Mbps and the round trip time (RTT) between the server and client is fixed at 20ms. Assume that the segment size is 20 Kbytes and the client has a receiver buffer of 200Kbytes. Assume that the window size (W) is adjusted according to the congestion control procedures of TCP-Reno. How long (in ms) does client A take to receive the whole file from the server after sending a request? Given that the initial slow-start threshold is 32.

344

integer

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.

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

list of integer

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?

(b)

option

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

25.0

float