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TIGER-Lab 112

Suppose Host A wants to send a large file to Host B. The path from Host A to Host B has three links, of rates R1 = 500 kbps, R2 = 2 Mbps, and R3 = Mbps. Assuming no other traffic in the network, what is the throughput for the file transfer? (in kbps)

500

integer

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)

4.15

float

A cylindrical tank of height 4 m and radius 1 m is filled with water. Water drains through a square hole of side 2 cm in the bottom. How long does it take for the tank to go from full to empty?

7142

integer

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.

0.01234567

float

In year N, the 300th day of the year is a Tuesday. In year N + 1, the 200th day is also a Tuesday. Suppose Monday is the 1-th day of the week, on which day of the week did the 100th day of the year N - 1 occur? Return a numeric between 1 and 7.

4

integer

Use the Runge-Kutta method with $h=0.1$ to find approximate values for the solution of the initial value problem $y' + 2y = x^3e^{-2x}$ with y(0)=1 at $x=0.2$.

0.6705

float

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.

True

bool

Consider a periodic signal $x(t)$ with period $(T)$ equals to ten. Over one period (i.e., $-5 \leq t<5)$, it is defined as $$ x(t)=\left\{\begin{array}{cc} 2 & -5 \leq t<0 \\ -2 & 0 \leq t<5 \end{array}\right. $$ In Fourier series, the signal $x(t)$ is written in the form of $$ x(t)=\sum_{k=-\infty}^{\infty} c_k e^{\frac{j 2 \pi k t}{T}} $$ where the Fourier series coefficient $c_k$ is obtained as, $$ c_k=\frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} x(t) e^{-\frac{j 2 \pi k t}{T}} d t $$ Determine the value of $c_0$ (i.e., $\left.k=0\right)$

0

integer

Does the following series $\sum_{i=0}^{\infty} \frac{n^2 ln(n)}{n!}$ converge?

1.0

float

The Chi-square statistic $\chi^2=\sum_c\frac{(P(x)-Q(x))^2}{Q(x)}$ is (twice) the first term in the Taylor series expansion of $D(P||Q)$ about $Q$. True or False?

True

bool

Every group of order $5\cdot7\cdot47=1645 is abelian, and cyclic. Is this true? Answer true or false.

True

bool

What is \int_{-3}^1 (7x^2 + x +1)dx?

65.333

float

Find the mass and weight of the air at $20^{\circ} C$ in a living room with a $4.0 m \times 5.0 m$ floor and a ceiling 3.0 m high, and the mass and weight of an equal volume of water. (Unit: 10 ^ 5 N)

5.9

float

Compute the mean translational kinetic energy of a mole of ideal gas in J, both at room temperature 293 K.

3650.0

float

Let $F_0(x)=log(x)$. For $n\geq 0$ and $x>0$, let $F_{n+1}(x)=\int_0^x F_n(t)dt$. Evaluate $\lim _{n \rightarrow \infty} (n! F_n(1))/(log(n))$.

-1.0

float

How many trees are there on n (n > 1) labeled vertices with no vertices of degree 1 or 2?

0

integer

What is the order of the element 5 in U_8?

2

integer

Roughly how many bits are required on the average to describe to 3 digit accuracy the decay time (in years) of a radium atom if the half-life of radium is 80 years? Note that half-life is the median of the distribution.

19

integer

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

True

bool

Determine the AC power gain for the emitter-follower in the figure. Assume that $\beta_{ac} = 175$

24.1

float

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?

True

bool

The spontaneous fission activity rate of U-238 is 6.7 fissions/kg s. A sample of shale contains 0.055% U-238 by weight. Calculate the number of spontaneous fissions in one day in a 106-kg pile of the shale by determining the mass of U-238 present in kg.

550.0

float

Calculate the Fermi temperature for copper in eV.

81600.0

float

Use Stoke's Theorem to evaluate $\int_C \vec{F} \cdot d \vec{r}$ where $\vec{F} = z^2 \vec{i} + y^2 \vec{j} + x \vec{k}$ and $C$ is the triangle with vertices (1,0,0), (0,1,0) and (0,0,1) with counter-clockwise rotation.

-0.166

float

The function f: U_5 o U_5 given by f(x) = x^2 is a homomorphism. What is K_f?

[4, 1]

list of integer

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 30 degree to the first, and then another at angle 50 degree to the second. What percentage of the original intensity was the light coming out of the third polarizer?

31.0

float

An airplane is flying at Mach 1.75 at an altitude of 8000 m, where the speed of sound is How long after the plane passes directly overhead will you hear the sonic boom? (Unit: m/s)

560

integer

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

False

bool

If the spot rates for 1 and 2 years are $s_1=6.3%$ and $s_2=6.9%, what is the forward rate $f_{1,2}$?

0.075

float

For a parametric family $\{p_\theta(x)\}$ we know that $\lim_{\theta'\to\theta}\frac{1}{(\theta-\theta')^2}D(p_\theta||p_{\theta'}) = \alpha J(\theta)$, where $J(\theta)$ is the Fisher information. Use natural logarithm for KL divergence to compute $\alpha$.

0.5

float

A muon has a lifetime of 2 x 10^{-6} s in its rest frame. It is created 100 km above the earth and moves towards it at a speed of 2.97 x 10^8 m/s. At what altitude in km does it decay? Return a numeric number.

4.2

float

Suppose C[0,1] denotes the space of all the continuous functions on the interval [0,1]. Is (C[0,1],\|\cdot\|_1 ) a Banach space? Here $\|f(x)\|_1=\int_0^1 |f(t)|dt$ with $f\in C[0,1]$. Answer 1 for yes and 0 for no.

0.0

float

An athlete whirls a discus in a circle of radius 80.0 cm. At a certain instant, the athlete is rotating at 10.0 rad / s and the angular speed is increasing at 50.0 rad / s^2. At this instant, find the magnitude (Unit: m / s^2) of the acceleration. Return the numeric value.

89.4

float

The positive integers N and N^2 both end in the same sequence of four digits abcd when written in base 10, where digit a is nonzero. Find the three-digit number abc.

937

integer

Calculate the required memory size in Mebibytes (MiB) (in 3 sig.fig.) for storing a frame in 1080p if the sampling scheme R'G'B' 4:4:4 is used. Note that there are 1920 × 1080 pixels in one 1080p frame. Each pixel contains three primary-colour components. Each primary-colour component requires 1 byte of memory for storage. 1 Mebibyte has 1024^2 bytes.

5.93

float

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

1.0

float

The equation of a digital filter is given by $y(n)=1 / 3(x(n)+x(n-1)+x(n-2))$, where $y(n)$ and $x(n)$ are, respectively, the nth samples of the output and input signals. Determine the pole(s) of the filter.

0

integer

Given the following spot rates: 1-year spot rate: 5%; 2-year spot rate: 6%. Determine the one-year forward rate (between 0 and 1) one year from today.

0.070095

float

An object of height 5cm is placed 10 cm in front of a convex mirror that has a radius of curvature of 45.0 cm. Determine the magnification of the image.

1.8

float

For the 3 payments of $1000 each end-of-year, with 7% rate of return, what is the present value if the first payment is made at the end of fifth year?

2002.0781

float

Does the following series $\sum_{i=0}^{\infty} \frac{n!}{n^2 cos(n)}$ converge?

0.0

float

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

34.0

float

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.).

3.59

float

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

0

integer

In how many ways can we form a 7-digit number using the digits 1, 2, 2, 3, 3, 3, 4?

420

integer

Let {N(t), t=[0, \infty]} be a Poisson process with rate $\lambda = 5$. Find the probability of no arrivals in [3, 5)

0.37

float

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 speed of the gold ions as a fraction of the speed of light?

0.99996

float

The following data related the rubber percentage of two types of rubber plants, where the sample have been drawn independently. Test for their mean difference. Type 1: 6.21 5.70 6.04 4.47 5.22 4.45 4.84 5.84 5.88 5.82 6.09 5.59 6.06 5.59 6.74 5.55, Type 2: 4.28 7.71 6.48 7.71 7.37 7.20 7.06 6.40 8.93 5.91 5.51 6.36. Are there difference between these two rubber plants?

True

bool

In how many ways can 7 people be seated at 5 identical round tables? Each table must have at least 1 person seated.

175

integer

What is the minimum kinetic energy in MeV of a proton in a medium-sized nucleus having a diameter of 8.0 x 10^-15 m?

0.08

float

Given that each cone can contain two ice cream balls, how many different ice cream cones can you make if you have 6 flavors of ice cream and 5 types of cones?

180

integer

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

True

bool

Calculate the de Broglie Wavelength of a tennis ball of mass 57 g traveling 25 m/s in meters.

4.7e-34

float

If the peak voltage value of a signal is 20 times the peak voltage value of the noise, what is the SNR? What is the $\mathrm{SNR}_{\mathrm{dB}}$ (in 3 sig.fig.)?

26.0

float

Using n=6 approximate the value of $\int_{-1}^2 \sqrt{e^{-x^2} + 1} dx$ using the Simpson's rule.

3.70358145

float

What is the smallest number of vertices in a graph that guarantees the existence of a clique of size 3 or an independent set of size 2?

3

integer

Find the number of integers n, 1 ≤ n ≤ 25 such that n^2 + 3n + 2 is divisible by 6.

13

integer

Suppose Host A wants to send a large file to Host B. The path from Host A to Host B has three links, of rates R1 = 500 kbps, R2 = 2 Mbps, and R3 = Mbps. Suppose the file is 4 million bytes. Dividing the file size by the throughput, roughly how many seconds will it take to transfer the file to Host B?

64

integer

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 4-clique in blue?

18

integer

All walking animals, including humans, have a natural walking pace—a number of steps per minute that is more comfortable than a faster or slower pace. Suppose that this pace corresponds to the oscillation of the leg as a physical pendulum. Fossil evidence shows that T. rex, a two-legged dinosaur that lived about 65 million years ago, had a leg length L = 3.1 m and a stride length S = 4.0 m (the distance from one footprint to the next print of the same foot). Estimate the walking speed of T. rex. (Unit: m/s)

1.4

float

What is (6^83 + 8^83) mod 49?

35

integer

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)?

5.113

float

Suppose $f(x, y)= \begin{cases}1-x-y, & x+y \leqslant 1 \ 0, & x+y>1\end{cases}$. What is the integral of f(x,y) over the region I=[0,1]\times[0,1]?

0.16667

float

Let $I(R)=\iint_{x^2+y^2 \leq R^2}(\frac{1+2 x^2}{1+x^4+6x^2y^2+y^4}-\frac{1+y^2}{2+x^4+y^4}) dx dy$. What is the limit of $I(R)$ as $R$ goes to infinity?

1.53978589

float

If polygon ACDF is similar to polygon VWYZ, AF = 12, CD = 9, YZ = 10, YW = 6, and ZV = 3y-1, find the length of FD.

15

integer

What are the generators of the additive cyclic group Z?

[1, -1]

list of integer

Suppose $E \subset(0,2 \pi) is a measurable set. \left\{\xi_n ight\}$ is an arbitrary sequence of real numbers. If the Lebesgue measure of E is 2, what is $\lim _{n ightarrow \infty} \int_E \cos ^2 (n x+\xi_n ) dx$? Return the numeric.

1.0

float

Let $W(t)$ be a Bownian motion, Let $E[exp(i*W(t))]:= E[cos(W(t))+i*sin(W(t))]$, where $i=\sqrt{-1}$. Is $M(t):=exp(i*W(t))/E[exp(i*W(t))]$ a matingale? Return 1 for yes and 0 for no.

1.0

float

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).

(b)

option

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.

0.75

float

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

22.0

float

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?

1

integer

what is the limit of $(n!)^{1/n}/n$ as n goes to infinity? Round the answer to the thousands decimal.

0.367879441

float

Suppose ${X_n:n\geq 1}$ be independent and exponentially distributed with parameter 1. what is the probability $P(\limsup _{n \rightarrow infty} X_n/\log(n)=1)? Return a numeric value.

1.0

float

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

5630.0

float

The following signal $x_1(t)=\cos (3 \pi t)-4 \cos (5 \pi t-0.5 \pi)$ can be expressed as $x_1(t)=\operatorname{Real}\left(A e^{j \pi B t}\right)+\operatorname{Real}\left(D e^{j \pi E t}\right)$. What are B,E?

[3, 5]

list of integer

On a day when the speed of sound is the fundamental frequency of a particular stopped organ pipe is 220 Hz. The second overtone of this pipe has the same wavelength as the third harmonic of an open pipe. How long is the open pipe? (Unit: m)

0.47

float

A group of 5 patients treated with medicine. A is of weight 42,39,38,60 &41 kgs. Second group of 7 patients from the same hospital treated with medicine B is of weight 38, 42, 56, 64, 68, 69, & 62 kgs. Is there any difference between medicines under significance level of 5%?

False

bool

Coloring the edges of a complete graph with 6 vertices in 2 colors, how many triangles of the same color are there at least?

2

integer

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?

517

integer

John's Lawn Mowing Service is a small business that acts as a price-taker (i.e., MR = P). The prevailing market price of lawn mowing is $20 per acre. John's costs are given by total cost = 0.1q^2 + 10q + 50, where q = the number of acres John chooses to cut a day. Calculate John's maximum daily profit.

200

integer

suppose a,b,c,\alpha,\beta,\gamma are six real numbers with a^2+b^2+c^2>0. In addition, $a=b*cos(\gamma)+c*cos(\beta), b=c*cos(\alpha)+a*cos(\gamma), c=a*cos(\beta)+b*cos(\alpha)$. What is the value of $cos^2(\alpha)+cos^2(\beta)+cos^2(\gamma)+2*cos(\alpha)*cos(\beta)*cos(\gamma)? return the numeric.

1.0

float

Consider a source X uniform on $\{1,2,\ldots,m\}$ with a distortion measure $d(x, \hat{x})$ that satisfies the following property: all rows and columns of the distortion matrix are permutations of the set $\{d_1, d_2, \ldots, d_m\}$. Then the Shannon lower bound is tight. i.e. $R(D)=H(X)-\phi(D)$. True or False?

True

bool

Square ABCD center O. Right AEB. ∠ABE = 53. Find the numeric value of ∠OFC.

82.0

float

In the process of searching circles in an image, object O is detected. The contour of the object O is represented with the Fourier Descriptors (35,129,0,1,0,0,-1,0). Given that the Fourier Descriptors of a circle are (0,40,0,0,0,0,0,0). Is the object O a circle-like polygon in the image? Bear in mind that there is some high frequency noise in the image. You should take this into account when you make your judgment.

True

bool

Suppose C is a compact convex set in a linear normed space, and let T: C → C be a continuous mapping. Then, there exists a fixed point of T in C. Is this correct? Answer 1 for yes and 0 for no.

1.0

float

Suppose a convex 3d-object has 15 vertices and 39 edges. How many faces does it have?

26

integer

Find which digit is at 1001th place after the decimal point in the decimal expansion of the fraction 9/28.

2

integer

A one-hour color video in YUV format has a frame resolution of 1920x1080 with a 4:2:2 color sub-sampling format, 8 bits for each component, and a frame rate of 30 frames/s. Determine the storage requirement for the video in Gbytes (3 sig. fig.).

417

integer

Find the sum of all positive integers less than 196 and relatively prime to 98.

8232

integer

A 'fishbowl' of height 4r/3 is formed by removing the top third of a sphere of radius r=6. The fishbowl is fixed in sand so that its rim is parallel with the ground. A small marble of mass m rests at the bottom of the fishbowl. Assuming all surfaces are frictionless and ignoring air resistance, find the maximum initial velocity that could be given to the marble for it to land back in the fishbowl with g=9.8.

18.25

float

What is the minimum number of people needed in a room to guarantee that there are 3 mutual friends or 3 mutual strangers?

6

integer

The equation of a digital filter is given by $y(n)=1 / 3(x(n)+x(n-1)+x(n-2))$, where $y(n)$ and $x(n)$ are, respectively, the nth samples of the output and input signals. Is it a FIR?

True

bool

Suppose we have the following differential equation with the initial condition: $\frac{\partial p}{\partial x} = 0.5 * x * (1-x)$ and $p(0)=2$. Use Euler's method to approximate p(2), using step of 1.

2.0

float

At a waterpark, sleds with riders are sent along a slippery, horizontal surface by the release of a large compressed spring. The spring with force constant k = 40.0 N/cm and negligible mass rests on the frictionless horizontal surface. One end is in contact with a stationary wall. A sled and rider with total mass 70.0 kg are pushed against the other end, compressing the spring 0.375 m. The sled is then released with zero initial velocity. What is the sled's speed (in m/s) when the spring returns to its uncompressed length?

2.83

float

If a preferred share of stock pays dividends of $1.90 per year, and the required rate of return for the stock is 9%, then what is its intrinsic value?

22.11

float

Suppose that $(X, Y, Z)$ are jointly Gaussian and that $X \rightarrow Y \rightarrow Z$ forms a Markov chain. Let $X$ and $Y$ have correlation coefficient 0.1 and let $Y$ and $Z$ have correlation coefficient 0.9. Find $I(X;Z)$ in bits.

0.00587

float

A positive-definite kernel function satisfies the Cauchy-Schwartz inequality. True or false?

True

bool

Calculate the momentum uncertainty of an electron within the smallest diameter of a hydrogen atom in kg m/s.

1e-24

float

Clare manages a piano store. Her utility function is given by Utility = w - 100, where w is the total of all monetary payments to her and 100 represents the monetary equivalent of the disutility of exerting effort to run the store. Her next best alternative to managing the store gives her zero utility. The store's revenue depends on random factors, with an equal chance of being $1,000 or $400. If shareholders offered to share half of the store's revenue with her, what would her expected utility be?

250

integer