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A young couple has made a non-refundable deposit of the first month's rent (equal to $1, 000) on a 6-month apartment lease. The next day they find a different apartment that they like just as well, but its monthly rent is only $900. They plan to be in the apartment only 6 months. Should they switch to the new apartment?

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0.0

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Company A is currently trading at $150 per share, and earnings per share are calculated as $10. What is the P/E ratio?

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15.0

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

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25

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

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11760

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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?

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1.56

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Is 7 a quadratic residue modulo 19? Use Gauss's Lemma to answer it.

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True

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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?

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60.23

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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.

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[4, 1, 2, 0]

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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.

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0.757

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what is the value of $\prod_{n=0}^{\infty}(1+(\frac{1}{2})^{2^n})$?

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2.0

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

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3

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Does f (x) = x2 + cx + 1 have a real root when c=0?

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False

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How many distinct directed trees can be constructed from a undirected tree with 100 nodes?

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100

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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.

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[-1.7807764064, 0.2807764064]

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A QCIF (176x144) image sequence is encoded using the MPEG video coding algorithm with the following Group Of Pictures (GOP). When a single bit error occurs in the 5th picture of a GOP, which pictures could possibly be affected by this error? Represent the answer in a list sorted in ascending order.

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[4, 6, 7, 8, 9, 10, 11, 12]

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Light of wavelength 400 nm is incident upon lithium (phi = 2.93 eV). Calculate the stopping potential in V.

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0.17

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

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-1.0

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Find the entropy rate of the Markov chain associated with a random walk of a king on the 3 by 3 chessboard. Use base 2 logarithm and return the entropy rate in bits.

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2.24

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How many labeled trees are there on 6 vertices?

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1296

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Find the fraction of the standard solar flux reaching the Earth (about 1000 W/m^22) available to a solar collector lying flat on the Earth’s surface at Regina, Saskatchewan (latitude 50°N) at noon on the summer solstice.

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0.891

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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?

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True

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Figure Q8 shows the contour of an object. Represent it with an 8-directional chain code. Represent the answer as a list with each digit as a element.

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[6, 7, 0, 6, 6, 4, 3, 4, 3, 1, 1]

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

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26

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In how many ways can 10 people be seated at 1 identical round tables? Each table must have at least 1 person seated.

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362880

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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.

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1.0

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If T_1 and T_2 are stopping times with respect to a filtration F. Is T_1+T_2 stopping time? Is max(T_1, T_2} stopping time? Is min(T_1, T_2} stopping time? Answer 1 for yes and 0 for no. Return the answers of the three questions as a list.

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[1, 1, 1]

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Let $N_1(t)$ and $N_2(t)$ be two independent Posson processes with rate $\lambda_1 = 1$ and $\lambda_2 = 2$, respectively. Let N(t) be the merged process N(t) = N_1(t) + N_2(t). Given that N(1) = 2, Find the probability that N_1(1) = 1.

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0.4444

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As scotch whiskey ages, its value increases. One dollar of scotch at year 0 is worth $V(t) = exp{2\sqrt{t} - 0.15t}$ dollars at time t. If the interest rate is 5 percent, after how many years should a person sell scotch in order to maximize the PDV of this sale?

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25

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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.

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18.25

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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 $r^*$?

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0.08

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If four points are picked independently at random inside the triangle ABC, what is the probability that no one of them lies inside the triangle formed by the other three?

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0.6667

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For the two linear equations $2 * x + 3 * y + z = 8$ and $4 * x + 4 * y + 4z = 12$ and $x + y + 8z = 10$ with variables x, y and z. Use cramer's rule to solve these three variables.

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[-1, 3, 1]

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Find the maximum entropy density $f$, defined for $x\geq 0$, satisfying $E(X)=\alpha_1$, $E(\ln{X})=\alpha_2$. Which family of densities is this? (a) Exponential. (b) Gamma. (c) Beta. (d) Uniform.

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(b)

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Let $X$ be uniformly distributed over $\{1, 2, \ldots, m\}$. Assume $m=2^n$ . We ask random questions: Is $X\in S_1$? Is $X\in S_2$? ... until only one integer remains. All $2^m$ subsets of $\{1, 2, \ldots, m\}$ are equally likely. Suppose we ask $n+\sqrt{n}$ random questions. Use Markov's inequality to find the probability of error (one or more wrong objects remaining) when $n$ goes to infinity?

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0.0

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If a cash flow of $100 has a discount rate of 5% and to be received in 5 years, what is the present value of the cash flow?

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78.3526

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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.

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[-1, 4]

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You wish to put a 1000-kg satellite into a circular orbit 300 km above the earth's surface. How much work must be done to the satellite to put it in orbit? 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: 10^10 J)

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3.26

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Given $V_s = 5V$, $R_1 = 480 \Omega$, $R_2 = 320 \Omega$, and $R_3 = 200 \Omega$, find the power dissipated by the 3 resistors $P_1, P_2, P_3$ in the figure. Represent your answer as a list [$P_1, P_2, P_3$] in the unit of mW.

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[12, 8, 5]

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Consider $x(t)$ to be given as, $$ x(t)=\cos (1000 \pi t) $$ . Let the sampling frequency be $2000 \mathrm{~Hz}$. Does aliasing occur?

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False

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Find the interval in which the smallest positive root of the following equations lies: tan x + tanh x = 0. Determine the roots correct to two decimal places using the bisection method

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2.37

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

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[3]

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Let $f(x) = 1/x$ on $(0, 1]$ and $f(x) = 3$ if $x = 0$. Is there a global maximum on interval $[0, 1]$?

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False

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Given 2 colors whose HSI representations are given as follows: (a) $(pi, 0.3,0.5)$, (b) $(0.5 pi, 0.8,0.3)$, which color is brighter?

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(a)

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A TCP entity sends 6 segments across the Internet. The measured round-trip times (RTTM) for the 6 segments are 68ms, 42ms, 65ms, 80ms, 38ms, and 75ms, respectively. Assume that the smooth averaged RTT (RTTs) and Deviation (RTTD) was respectively 70ms and 10ms just before the first of these six samples. According to the Jacobson's algorithm, the retransmission timeout (RTO) is given by one RTTs plus 4 times the value of RTTD. Determine the value of RTO (in ms) after the six segments using the Jacobson's algorithm if the exponential smoothing parameters (a and B) are 0.15 and 0.2 for calculating RTTs and RTTD respectively.

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114.28

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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.

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4.2

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Fig. Q7a shows the amplitude spectrum of a real-value discrete time signal x[n]. Determine the period of signal x[n] (in samples).

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8

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Passengers on a carnival ride move at constant speed in a horizontal circle of radius 5.0 m, making a complete circle in 4.0 s. What is their acceleration? (Unit: m/s^2))

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12

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Is differential equation $sin(t)y' + t^2e^yy' - y' = -ycos(t) - 2te^y$ exact or not?

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True

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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.

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True

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In how many ways can 8 people be seated at 2 identical round tables? Each table must have at least 1 person seated.

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13068

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