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TIGER-Lab 109

A company has 8 employees, including 3 managers and 5 engineers. How many different ways are there to form a team of 4 employees that includes at least 1 manager and at least 2 engineers?

60

integer

The mass of one of the small spheres of a Cavendish balance is 0.0100 kg, the mass of the nearest large sphere is 0.500 kg, and the center-to-center distance between them is 0.0500 m. Assuming the gravitational force on each sphere due to the other is $X * 10^{-10}$ N, what is X?

1.33

float

The image produced by a concave mirror is at -16.2m, and the magnification is 1.79. What is the object distance in terms of meter?

9.05

float

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

35

integer

Calculate the required memory size in Mebibytes (MiB) (in 3 sig.fig.) for storing a frame in 720p if the sampling scheme Y'CbCr 4:2:0 is used. Note that There are 1280 × 720 pixels in one 720p 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.

1.32

float

Find all positive integers $n<2^{250}$ for which simultaneously $n$ divides $2^n$, $n-1$ divides $2^n-1$, and $n-2$ divides $2^n - 2$. Return all positive integers as an ascending list.

[4, 16, 65536]

list of integer

What is \lim_{x o 9} ((x - 9)/(\sqrt{x} - 3))?

6

integer

Using n=8 approximate the value of $\int_{0}^4 cos(1 + \sqrt{x}) dx$ using the Simpson's rule.

-2.47160136

float

Compute $\int_{|z| = 2} (5z - 2) / (z * (z - 1)) dz$. The answer is Ai with i denoting the imaginary unit, what is A?

31.4

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. is the distance that B advertise to C infinity?

True

bool

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

2.301

float

suppose $x=\sqrt{17}/17$. what is the value of $\frac{2}{\pi} \int_0^{+\infty} \frac{\sin ^2 t}{t^2} cos(2xt) dt$? Rounding it to the hundredths place and return the value.

0.757

float

Let $X_1, X_2, \ldots$ be a sequence of independent indetically distributed random variables drawn according to the probability mass function $p(x) = N(0,1)$. Let $q(x)=N(1,1)$ be another probability mass function. Use natural logarithm to evaluate $\lim -\frac{1}{n}\log{q(X_1,X_2,\ldots,X_n)}$ as $n \to \infty$.

1.4

float

If a,b,c,d > 0 and c^2 + d^2 = (a^2 + b^2)^3, is a^3/c + b^3/d < 1?

False

bool

Phased Solutions Inc. has paid the following dividends per share from 2011 to 2020: 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 $0.70 | $0.80 | $0.925 | $1.095 | $1.275 | $1.455 | $1.590 | $1.795 | $1.930 | $2.110 If you plan to hold this stock for 10 years, believe Phased Solutions will continue this dividend pattern forever, and you want to earn 17% on your investment, what would you be willing to pay per share of Phased Solutions stock as of January 1, 2021?

60.23

float

Let $P(r,t,T)$ denote the price at time $t$ of $1 to be paid with certainty at time $T, t\leT$, if the short rate at time $t$ is equal to $r$. For a Vasicek model you are given: $P(0.04, 0, 2)=0.9445$, $P(0.05, 1, 3)=0.9321$, $P(r^*, 2, 4)=0.8960$. What is $r^*$?

0.08

float

Is 10 a quadratic residue modulo 19? Use Gauss's Lemma to answer it.

False

bool

Solve the following linear programming problems: maximize 3x + y subject to (1) -x + y <= 1, (2) 2x + y <= 4, (3) x>= 0 and y >= 0. What's [x, y] for the optimal solution?

[2, 0]

list of integer

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.

[1, 2]

list of integer

Incompressible oil of density 850 kg/m^3 is pumped through a cylindrical pipe at a rate of 9.5 liters per second. The second section of the pipe has a diameter of 4.0 cm. What are the flow speed in that section? (Unit: m/s)

7.6

float

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

12

integer

If r(t) = (6t+2)i + 5t^2j - 8tk, find the Binormal vector as [xi, yj, zk]. What are x, y, z? Return them as a list.

[0.8, 0.0, 0.6]

list of float

Consider the matrix of A=[[1, -1], [-1, 4]], is this a positive definite matrix?

True

bool

What is 3^(3^(3^3)) mod 100?

87

integer

which n <= 20 can be constructed a regular n-gonwith compass and straightedge? return all the possible numbers in a list

[3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20]

list of integer

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

[2, 4]

list of integer

What is the number of labelled rooted forests on 6 vertices

16807

integer

A group of 7 people is to be divided into 3 committees. Within each committee, people are ranked in a certain order. In how many ways can this be done?

12600

integer

Are the circuits shown in Fig. Qla and Fig. Q1b are identical? (Hint: Compare the Tranfer functions)

True

bool

Suppose f is analytic on the closed unit disk, f(0) = 0, and |f(z)| $\leq$ |e^z| whenever |z| = 1. How big can f((1 + i)/2) be? Return a numerical number.

1.9221

float

In triangle ABC, AB = 9x-1, CB = 5x-0.5, AC = 4x+1, and AC = CB. Find the measure of AB.

12.5

float

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

132

integer

For a $1,000 investment, what is the future value of the investment if the interest rate is 8% compounded annually for 3 years?

1259.71

float

What's phi(29791) where phi is Euler's Totient Function?

28830

integer

Let m and n be the roots of the equation 2x^2 + 15x + 16 = 0. What is the value of 1/m + 1/n?

-0.9375

float

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.

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

list of integer

Define f(x)=(4x+5)/(9-3x), is the function continuous at x=-1?

True

bool

assume you are Indonesian. In 2010, the rupiah exchange rate was around IDR15,000/USD, and the consumer price index in Indonesia and the United States was at 100. In 2019, the exchange rate changed to IDR14,000/USD. Simultaneously, Indonesia’s inflation rose 5% due to the consumer price index rising to 105. Meanwhile, the United States’ inflation rate rose 10% due to the consumer price index rising to 110. Whats the real exchange rate?

14666.67

float

Calculate the minimum kinetic energy of an electron that is localized within a typical nuclear radius of 6 x 10^-15 m in MeV.

15.9

float

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?

-3

integer

If $X(k)$ is the N-point DFT of a sequence $x(n)$, then circular time shift property is that N-point DFT of $x((n-I))_N$ is $X(k) e^{-j 2 \pi k \mid / N}$. Is it true?

True

bool

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?

1.73

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 (80,40,0,0,-1,0,0,1). 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

Consider a group of 10 people {A,B,C,D,E,F,G,H,I,J} and we are to choose a committee of 4 people from them. Given that (1) A and B should not be chosen together, and that (2) A, C, F should not be chosen together, then how many ways are there to choose such a committee?

176

integer

what is the limit of (2n)!!/(2n+1)!! as n goes to infinity?

0.0

float

You are given: (i) The current exchange rate is 0.011$/¥. (ii) A four-year dollar-denominated European put option on yen with a strike price of $0.008 sells for $0.0005. (iii) The continuously compounded risk-free interest rate on dollars is 3%. (iv) The continuously compounded risk-free interest rate on yen is 1.5%. Calculate the price of a four-year yen-denominated European put option on dollars with a strike price of ¥125.

42.77325

float

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 $\int_{|z|=r} f(z)/(z-z_1)(z-z_2) dz$?

0.0

float

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?

1.81

float

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. What is the P-value?

0.023

float

An ordinary deck of cards containing 26 red cards and 26 black cards is shuffled and dealt out one card at a time without replacement. Let $X_i$ be the color of the $i$th card. Compute $H(X_1,X_2,\ldots,X_{52})$ in bits.

48.8

float

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?

5

integer

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 (-20,60,-20,20,-20,21,-20,20). 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.

False

bool

Find acceleration in m/(min^2) at time t = 5 min of a helicopter whose height is s(t) = 300t - 4t^3 m.

-120

integer

A network with one primary and four secondary stations uses polling. The size of a data frame is 1000 bytes. The size of the poll, ACK, and NAK frames are 32 bytes each. Each station has 5 frames to send. How many total bytes are exchanged if each station can send only one frame in response to a poll?

21536

integer

What are the real eigenvalues of the matrix [[3, -2, 5], [1, 0, 7], [0, 0, 2]]?

[1, 2, 2]

list of integer

Suppose there are 10 independent random variables $X_1, X_2, \cdots, X_10$. Each of the $X_i$ lies within the range of [10, 11] with a mean value of 10.5. If we take the mean of the 10 random variables as $\hat{X_n}$. What is the upper bound of the probability that $\hat{X_n}$ is either smaller than 10.2 or larger than 10.8?

0.3305

float

What is \lim_{x \to 0} (csc(x) - cot(x))?

0

integer

Let’s assume Mutual Fund A has an annualized return of 15% and a downside deviation of 8%. Mutual Fund B has an annualized return of 12% and a downside deviation of 5%. The risk-free rate is 2.5%. What is the Sortino ratio for Fund A?

1.56

float

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

True

bool

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?

3.54e-07

float

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

0.66667

float

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.

0.363

float

A uniform plank of length L = 6.0 m and mass M = 90 kg rests on sawhorses separated by D = 1.5 m and equidistant from the center of the plank. Cousin Throckmorton wants to stand on the right-hand end of the plank. If the plank is to remain at rest, how massive can Throckmorton be? (Unit: kg)

30

integer

Let L^1[0,2] be the space of all the Lebesgue integrable functions on the interval [0,2], and C[0,2] be the space of all the continuous functions on the interval [0,2]. Suppose H=L^1[0,2], and X=C[0,2]. For any f\in L^1[0,2], define operator T as $(Tf)(x)=\int_0^x f(t)dt$. For the linear operator T from H to X, what is the norm of T? For the linear operator T from H to H, what is the norm of T? Return the answers of two questions as a list. For example, if the norm for the first question is 2, the second is 3, then return [2,3].

[1, 2]

list of integer

For matrix A = [[2, 4, 3], [3, 0, 1], [1, 2, 5]], what is its determinant?

-42

integer

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?

102.95

float

Passing to polar coordinates, calculate the double integral $\iint_S ydxdy$ with $y$ > 0, where S is a semicircle of a diameter 1 with center at point C(1/2, 0) above the X axis.

0.0833

float

In how many ways can 8 people be seated at 5 identical round tables? Each table must have at least 1 person seated.

1960

integer

Given a network in the figure, From Interface m1 of router R2 it can reach how many destinations?

4

integer

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

[53.52, 54.25, 54.76]

list of float

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

1.8

float

The product of two of the four roots of the quartic equation x^4 - 18x^3 +kx2 + 200x - 1984 = 0 is -32. Determine the value of k.

86

integer

Is the set of 3 * 3 matrices in reduced row-echelon form a subspace of R^{3 * 3}?

False

bool

A linear learning machine based on the kernel $k(x,x')=f(x)f(x')$ will always find a solution proportional to $f(x)$. True or false?

True

bool

Suppose 100 cars will be offered on the used-car market. Let 50 of them be good cars, each worth $10,000 to a buyer, and let 50 be lemons, each worth only $2,000. Suppose that there are enough buyers relative to sellers that competition among them leads cars to be sold at their maximum willingness to pay. What would the market equilibrium price for good cars be if sellers value good cars at $6,000?

6000

integer

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

True

bool

For any poitie integer $n$, let $\langle n\rangle$ denote the closest integer to $\sqrt{n}$. Evaluate $\sum_{n=1}^{\infty} \frac{2^{\langle n \rangle}+2^{-\langle n \rangle}}{2^n}$.

3.0

float

What is the total number of colors in RGB color space?

16777216

integer

Can we use the method of compass and straightedge construction to construct the edge length of a cube, such that the volume of the cube is equal to X time the volume of a given cube, where X belongs to the set {3,17,8,27,343,1331}? Return the answer list for the respective values of X with 1 for yes and 0 for no.

[0, 0, 1, 1, 1, 1]

list of integer

Calculate the Hamming pairwise distances and determine the minimum Hamming distance among the following codewords: 00000,10101,01010

2

integer

Find the smallest positive integer that leaves a remainder of 2 when divided by 3, a remainder of 3 when divided by 5, and a remainder of 1 when divided by 7.

8

integer

The mass of Earth is 5.97x10^24 kg, the mass of the Moon is 7.35x10^22 kg, and the mean distance of the Moon from the center of Earth is 3.84x105 km. The magnitude of the gravitational force exerted by Earth on the Moon is X * 10^20 N. What is X? Return a numeric value.

1.99

float

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.

0.0

float

ABC is a right triangle. AM is perpendicular to BC. The size of angle ABC is equal to 55 degrees. Find the size of angle MAC.

55

integer

Consider the 7.0-TeV protons that are produced in the LHC collider at CERN. Find the available center-of-mass energy if these protons collide with other protons in a fixed-target experiment in GeV.

114.5

float

suppose F(x,y,z)=0. What is $\frac{\partial x}{\partial y} \frac{\partial y}{\partial z} \frac{\partial z}{\partial x}$?

-1.0

float

In a sinusoidal sound wave of moderate loudness, the maximum pressure variations are about $3.0 \times 10 ^ {-2}$ Pa above and below atmospheric pressure. Find the corresponding maximum displacement if the frequency is 1000 Hz. In air at normal atmospheric pressure and density, the speed of sound is 344 m/s and the bulk modulus is $1.42 \times 10^5$ Pa. (Unit: $10 ^ {-8}$)

1.2

float

For the equation x^4 + 2*x^3 + x = 10, there are four roots. What is the sum of the roots using newton-raphson method.

-2.0

float

Portfolio | Portfolio 1 | Portfolio 2 | Portfolio 3 Expected Portfolio Return | 5.3% | 6.5% | 7.2% Portfolio Standard Deviation | 8.2% | 9.1% | 10.1% If we use Roy's safety-first criterion to decide with portfolio is optimal, with a threshold return of 5%. Is portfolio 2 the optimal one? Answer True or False.

False

bool

Suppose there are 8,000 hours in a year (actually there are 8,760) and that an individual has a potential market wage of $5 per hour. Suppose a rich uncle dies and leaves the individual an annual income of $4,000 per year. If he or she devotes 75 percent of full income to leisure, how many hours will be worked?

1400

integer

Suppose we are given the following information. Use this information to calculate abnormal return. Rf: 4% Rm: 12% Beta of the Portfolio: 1.8 Beginning Value of Portfolio: $50,000 Ending Value of Portfolio: $60,000 What is the abnormal return?

0.016

float

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?

False

bool

Finding all the real roots of the equation $\sqrt{x^2+x+1}+\sqrt{2 x^2+x+5}=\sqrt{x^2-3 x+13}$. Return the answer as a list with ascending order.

[-1.7807764064, 0.2807764064]

list of float

Suppose a student who was farsighted wears glasses that allows him to read at a distance of 20cm from his eyes to the book. His near-point distance is 63cm. If his glasses are 1.5cm from his eyes, what is the refractive power of his glasses lenses?

3.846

float

Traders in major financial institutions use the Black-Scholes formula in a backward fashion to infer other traders' estimation of $\sigma$ from option prices. In fact, traders frequently quote sigmas to each other, rather than prices, to arrange trades. Suppose a call option on a stock that pays no dividend for 6 months has a strike price of $35, a premium of $2.15, and time to maturity of 7 weeks. The current short-term T-bill rate is 7%, and the price of the underlying stock is $36.12. What is the implied volatility of the underlying security?

0.251

float

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 $35.0 \pm 0.06 ps/s$

330.76

float

An Aston Martin V8 Vantage sports car has a lateral acceleration of $0.96g = (0.96)(9.8 m / s^2) = 9.4 m / s^2$. This is the maximum centripetal acceleration the car can sustain without skidding out of a curved path. If the car is traveling at a constant 40m/s on level ground, what is the radius R of the tightest unbanked curve it can negotiate? (Unit: m))

170

integer

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 $\alpha = 0.05$?

True

bool

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

9

integer

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)

108000

integer